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Multiplication operators with deficiency indices (p,p) and sampling formulas in reproducing kernel Hilbert spaces of entire vector valued functions

2017-09-20
Harry Dym , Santanu Sarkar

Hypothesis testing and decision theoretic approach for fault detection in wireless sensor networks

2014-04-24
Mrinal Nandi , Amiya Nayak , Bimal Roy +1 more
Sensor networks aim at monitoring their surroundings for event detection and object tracking. In practice, due to failure or death of sensors, false signal can be transmitted. In this article, … Sensor networks aim at monitoring their surroundings for event detection and object tracking. In practice, due to failure or death of sensors, false signal can be transmitted. In this article, we consider the problem of fault detection in wireless sensor networks, in particular, addressing both the noise-related measurement error and sensor fault simultaneously in fault detection. We assume that the sensors are placed at the centre of a square (or hexagonal) cell in the region of interest (ROI) and, if the event occurs, it occurs at a particular cell of the ROI. We propose fault detection schemes that consider error probabilities in the optimal event detection process. We develop the schemes under the consideration of Neyman–Pearson hypothesis test and Bayes test.
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Model Selection Approach for Distributed Fault Detection in Wireless Sensor Networks

2014-01-01
Mrinal Nandi , Anup Dewanji , Bimal Roy +1 more
Sensor networks aim at monitoring their surroundings for event detection and object tracking. But, due to failure or death of sensors, false signal can be transmitted. In this paper, we … Sensor networks aim at monitoring their surroundings for event detection and object tracking. But, due to failure or death of sensors, false signal can be transmitted. In this paper, we consider the problems of distributed fault detection in wireless sensor network (WSN). In particular, we consider how to take decision regarding fault detection in a noisy environment as a result of false detection or false response of event by some sensors, where the sensors are placed at the center of regular hexagons and the event can occur at only one hexagon. We propose fault detection schemes that explicitly introduce the error probabilities into the optimal event detection process. We introduce two types of detection probabilities, one for the center node, where the event occurs, and the other one for the adjacent nodes. This second type of detection probability is new in sensor network literature. We develop schemes under the model selection procedure and multiple model selection procedure and use the concept of Bayesian model averaging to identify a set of likely fault sensors and obtain an average predictive error.
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On acyclic edge-coloring of the complete bipartite graphs <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> for odd prime <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg…

2015-08-13
Ayineedi Venkateswarlu , Santanu Sarkar

Counting Heron triangles with constraints

2013-01-25
Pantelimon Stănică , Santanu Sarkar , Sourav Sen Gupta +2 more
Heron triangles have the property that all three of their sides as well as their area are positive integers. In this paper, we give some estimates for the number of … Heron triangles have the property that all three of their sides as well as their area are positive integers. In this paper, we give some estimates for the number of Heron triangles with two of their sides fixed. We provide a general bound on this count H(a,b), where the sides a,b are fixed positive integers, and the estimate here is better than the one of Ionascu, Luca and St˘anic˘a for the general situation of fixed sides a,b .I n the case of primes sidesp,q, there is an additional hypothesis which helps us to drop the upper bounds on H(p,q). In particular, we prove that H(p,q) is less than or equal to 1 when p ! q 2 (mod 4). We also provide a count for the number of Heron triangles with a fixed height (there exists only one such when the height is prime). Moreover, we study the decomposability property of a Heron triangle into two similar ones, and provide some cases when a Heron triangle is not decomposable.

A Hybrid Inversive Congruential Pseudorandom Number Generator with High Period

2021-01-31
Constanza Riera , Tapabrata Roy , Santanu Sarkar +1 more
Though generating a sequence of pseudorandom numbers by linear methods (Lehmer generator) displays acceptable behavior under some conditions of the parameters, it also has undesirable features, which makes the sequence … Though generating a sequence of pseudorandom numbers by linear methods (Lehmer generator) displays acceptable behavior under some conditions of the parameters, it also has undesirable features, which makes the sequence unusable for various stochastic simulations. An extension which showed promise for such applications is a generator obtained by using a first-order recurrence based upon the inversive modulo a prime or a prime power, called inversive congruential generator (ICG). A lot of work has been dedicated to investigate the periods (under some conditions of the parameters), the lattice test passing, discrepancy and other statistical properties of such a generator. Here, we propose a new method, which we call hybrid inversive congruential generator (HICG), based upon a second order recurrence using the inversive modulo M, a power of 2. We investigate the period of this pseudorandom numbers generator (PRNG) and give necessary and sufficient conditions for our PRNG to have periods M (thereby doubling the period of the classical ICG) and M/2 (matching the one of the ICG). Moreover, we show that the lattice test complexity for a binary sequence associated to (a full period) HICG is precisely M/2.

On acyclic edge-coloring of complete bipartite graphs

2016-11-16
Ayineedi Venkateswarlu , Santanu Sarkar , Sai Mali Ananthanarayanan
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The defect sequence for contractive tuples

2012-09-15
Tirthankar Bhattacharyya , Bata Krishna Das , Santanu Sarkar
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Robust estimation in the errors variables model via weighted likelihood estimating equations

1997-06-01
Ayanendranath Basu , Santanu Sarkar

Progress in Cryptology – INDOCRYPT 2022

2022-01-01
Takanori Isobe , Santanu Sarkar

Vector valued de Branges spaces of entire functions based on pairs of Fredholm operator valued functions and functional model

2023-12-05
Subhankar Mahapatra , Santanu Sarkar
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On Universality of Quantum Fourier Transform

2012-03-01
Arpita Maitra , Santanu Sarkar
A methodology is presented to obtain the basis of qudits which are admissible to quantum Fourier transform (QFT) in the sense that the set of such kets are related by … A methodology is presented to obtain the basis of qudits which are admissible to quantum Fourier transform (QFT) in the sense that the set of such kets are related by the QFT in the same way as the kets of the computational basis. We first study this method for qubits to characterize the ensemble that works for the Hadamard transformation (QFT for two dimension). In this regard we identify certain incompleteness in the result of Maitra and Parashar (Int. J. Quantum Inform. 4 (2006) 653). Next we characterize the ensemble of qutrits for which QFT is possible. Further, some theoretical results related to higher dimensions are also discussed. Considering the unitary matrix Un related to QFT, the issue boils down to the problem of characterizing matrices that commute with Un.
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Hypothesis Testing and Decision Theoretic Approach for Fault Detection in Wireless Sensor Networks

2012-01-01
Mrinal Nandi , Amiya Nayak , Bimal Roy +1 more
Sensor networks aim at monitoring their surroundings for event detection and object tracking. But due to failure or death of sensors, false signal can be transmitted. In this paper, we … Sensor networks aim at monitoring their surroundings for event detection and object tracking. But due to failure or death of sensors, false signal can be transmitted. In this paper, we consider the problem of fault detection in wireless sensor network (WSN), in particular, addressing both the noise-related measurement error and sensor fault simultaneously in fault detection. We assume that the sensors are placed at the center of a square (or hexagonal) cell in region of interest (ROI) and, if the event occurs, it occurs at a particular cell of the ROI. We propose fault detection schemes that take into account error probabilities into the optimal event detection process. We develop the schemes under the consideration of Neyman-Pearson test and Bayes test.
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Spread and asymmetry of typical quantum coherence and their inhibition in response to glassy disorder

2023-08-01
George Biswas , Santanu Sarkar , Anindya Biswas +1 more
Abstract We consider the average quantum coherences of typical redits and qudits—vectors of real and complex Hilbert spaces—with the analytical forms stemming from the symmetry of Haar-uniformly distributed random pure … Abstract We consider the average quantum coherences of typical redits and qudits—vectors of real and complex Hilbert spaces—with the analytical forms stemming from the symmetry of Haar-uniformly distributed random pure states. We subsequently study the response to disorder in spread of the typical quantum coherence in response to glassy disorder. The disorder is inserted in the state parameters. Even in the absence of disorder, the quantum coherence distributions of redits and qudits are not uniform over the range of quantum coherence, and the spreads are relatively lower for higher dimensions. On insertion of disorder, the spreads decrease. This decrease in the spread of quantum coherence distribution in response to disorder is seen to be a generic feature of typical pure states: we observe the feature for different strengths of disorder and for various types of disorder distributions, viz. Gaussian, uniform, and Cauchy–Lorentz. We also find that the quantum coherence distributions become less asymmetric with increase in dimension and with infusion of glassy disorder.
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Analytic Kramer Sampling and Quasi Lagrange-Type Interpolation in Vector Valued RKHS

2024-08-23
Subhankar Mahapatra , Santanu Sarkar

Maximal Contractive Tuples

2013-10-06
B. Krishna Das , Jaydeb Sarkar , Santanu Sarkar
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Properties of singular integral operators $S_{\alpha,\beta}$

2015-05-20
Amit K. Samanta , Santanu Sarkar
For $\alpha, \beta \in L^{\infty} (S^1),$ the singular integral operator $S_{\alpha,\beta}$ on $L^2 (S^1)$ is defined by $S_{\alpha,\beta}f:= \alpha Pf+\beta Qf$, where $P$ denotes the orthogonal projection of $L^2(S^1)$ onto … For $\alpha, \beta \in L^{\infty} (S^1),$ the singular integral operator $S_{\alpha,\beta}$ on $L^2 (S^1)$ is defined by $S_{\alpha,\beta}f:= \alpha Pf+\beta Qf$, where $P$ denotes the orthogonal projection of $L^2(S^1)$ onto the Hardy space $H^2(S^1),$ and $Q$ denotes the orthogonal projection onto $H^2(S^1)^{\perp}.$ In a recent paper Nakazi and Yamamoto have studied the normality and self-adjointness of $S_{\alpha,\beta}.$ This work has shown that $S_{\alpha,\beta}$ may have analogous properties to that of the Toeplitz operator. In this paper we study several other properties of $S_{\alpha,\beta}.$

$\prod\limits_{i=1}^{n} \mathbb{Z}_{2^i}$-Additive Cyclic Codes

2017-08-23
Tapabrata Roy , Santanu Sarkar
In this paper we study $\prod\limits_{i=1}^{n} \mathbb{Z}_{2^i}$-Additive Cyclic Codes. These codes are identified as $\mathbb{Z}_{2^n}[x]$-submodules of $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}[x]/ \langle x^{\alpha_i}-1\rangle$; $\alpha_i$ and $\rm{i}$ being relatively prime for each $i=1,2,\ldots,n.$ We first … In this paper we study $\prod\limits_{i=1}^{n} \mathbb{Z}_{2^i}$-Additive Cyclic Codes. These codes are identified as $\mathbb{Z}_{2^n}[x]$-submodules of $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}[x]/ \langle x^{\alpha_i}-1\rangle$; $\alpha_i$ and $\rm{i}$ being relatively prime for each $i=1,2,\ldots,n.$ We first define a $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}$-additive cyclic code of a certain length. We then define the distance between two codewords and the minimum distance of such a code. Moreover we relate these to binary codes using the generalized Gray maps. We define the duals of such codes and show that the dual of a $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}$-additive cyclic code is also cyclic. We then give the polynomial definition of a $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}$-additive cyclic code of a certain length. We then determine the structure of such codes and derive a minimal spanning set for that. We also determine the total number of codewords in this code. We finally give an illustrative example of a $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}$-additive cyclic code.

Properties of singular integral operators $$\varvec{S}_{\varvec{\alpha ,\beta }}$$ S α , β

2018-02-01
Amit K. Samanta , Santanu Sarkar

Maximal contractive tuples

2013-06-04
B. Krishna Das , Jaydeb Sarkar , Santanu Sarkar
Maximality of a contractive tuple of operators is considered. Characterization of a contractive tuple to be maximal is obtained. Notion of maximality of a submodule of Drury-Arveson module on the … Maximality of a contractive tuple of operators is considered. Characterization of a contractive tuple to be maximal is obtained. Notion of maximality of a submodule of Drury-Arveson module on the $d$-dimensional unit ball $\mathbb{B}_d$ is defined. For $d=1$, it is shown that every submodule of the Hardy module over the unit disc is maximal. But for $d\ge 2$ we prove that any homogeneous submodule or submodule generated by polynomials is not maximal. A characterization of a submodule to be maximal is obtained.

Hypothesis Testing and Decision Theoretic Approach for Fault Detection in Wireless Sensor Networks

2012-03-02
Mrinal Nandi , Amiya Nayak , Bimal Roy +1 more
Sensor networks aim at monitoring their surroundings for event detection and object tracking. But due to failure or death of sensors, false signal can be transmitted. In this paper, we … Sensor networks aim at monitoring their surroundings for event detection and object tracking. But due to failure or death of sensors, false signal can be transmitted. In this paper, we consider the problem of fault detection in wireless sensor network (WSN), in particular, addressing both the noise-related measurement error and sensor fault simultaneously in fault detection. We assume that the sensors are placed at the center of a square (or hexagonal) cell in region of interest (ROI) and, if the event occurs, it occurs at a particular cell of the ROI. We propose fault detection schemes that take into account error probabilities into the optimal event detection process. We develop the schemes under the consideration of Neyman-Pearson test and Bayes test.

Model Selection Approach for Distributed Fault Detection in Wireless Sensor Networks

2013-01-21
Mrinal Nandi , Anup Dewanji , Bimal Roy +1 more
Sensor networks aim at monitoring their surroundings for event detection and object tracking. But, due to failure, or death of sensors, false signal can be transmitted. In this paper, we … Sensor networks aim at monitoring their surroundings for event detection and object tracking. But, due to failure, or death of sensors, false signal can be transmitted. In this paper, we consider the problems of distributed fault detection in wireless sensor network (WSN). In particular, we consider how to take decision regarding fault detection in a noisy environment as a result of false detection or false response of event by some sensors, where the sensors are placed at the center of regular hexagons and the event can occur at only one hexagon. We propose fault detection schemes that explicitly introduce the error probabilities into the optimal event detection process. We introduce two types of detection probabilities, one for the center node, where the event occurs and the other one for the adjacent nodes. This second type of detection probability is new in sensor network literature. We develop schemes under the model selection procedure, multiple model selection procedure and use the concept of Bayesian model averaging to identify a set of likely fault sensors and obtain an average predictive error.

On Acyclic Edge-Coloring of Complete Bipartite Graphs

2015-03-11
Ayineedi Venkateswarlu , Santanu Sarkar , Ajit S Mali
An acyclic edge-coloring of a graph is a proper edge-coloring without bichromatic ($2$-colored) cycles. The acyclic chromatic index of a graph $G$, denoted by $a'(G)$, is the least integer $k$ … An acyclic edge-coloring of a graph is a proper edge-coloring without bichromatic ($2$-colored) cycles. The acyclic chromatic index of a graph $G$, denoted by $a'(G)$, is the least integer $k$ such that $G$ admits an acyclic edge-coloring using $k$ colors. Let $\Delta = \Delta(G)$ denote the maximum degree of a vertex in a graph $G$. A complete bipartite graph with $n$ vertices on each side is denoted by $K_{n,n}$. Basavaraju, Chandran and Kummini proved that $a'(K_{n,n}) \ge n+2 = \Delta + 2$ when $n$ is odd. Basavaraju and Chandran provided an acyclic edge-coloring of $K_{p,p}$ using $p+2$ colors and thus establishing $a'(K_{p,p}) = p+2 = \Delta + 2$ when $p$ is an odd prime. The main tool in their approach is perfect $1$-factorization of $K_{p,p}$. Recently, following their approach, Venkateswarlu and Sarkar have shown that $K_{2p-1,2p-1}$ admits an acyclic edge-coloring using $2p+1$ colors which implies that $a'(K_{2p-1,2p-1}) = 2p+1 = \Delta + 2$, where $p$ is an odd prime. In this paper, we generalize this approach and present a general framework to possibly get an acyclic edge-coloring of $K_{n,n}$ which possess a perfect $1$-factorization using $n+2 = \Delta+2$ colors. In this general framework, we show that $K_{p^2,p^2}$ admits an acyclic edge-coloring using $p^2+2$ colors and thus establishing $a'(K_{p^2,p^2}) = p^2+2 = \Delta + 2$ when $p\ge 5$ is an odd prime.

The Defect Sequence for Contractive Tuples

2012-11-26
Tirthankar Bhattacharyya , Bata Krishna Das , Santanu Sarkar
We introduce the defect sequence for a contractive tuple of Hilbert space operators and investigate its properties. The defect sequence is a sequence of numbers, called defect dimensions associated with … We introduce the defect sequence for a contractive tuple of Hilbert space operators and investigate its properties. The defect sequence is a sequence of numbers, called defect dimensions associated with a contractive tuple. We show that there are upper bounds for the defect dimensions. The tuples for which these upper bounds are obtained, are called maximal contractive tuples. The upper bounds are different in the non-commutative and in the com- mutative case. We show that the creation operators on the full Fock space and the co ordinate multipliers on the Drury-Arveson space are maximal. We also study pure tuples and see how the defect dimensions play a role in their irreducibility.

Settling the mystery of Z r=r in RC4.

2017-01-01
Sabyasachi Dey , Santanu Sarkar
In this paper, using a matrix, at first we revisit the work of Mantin on finding the probability distribution of the RC4 permutation after the completion of the KSA. After … In this paper, using a matrix, at first we revisit the work of Mantin on finding the probability distribution of the RC4 permutation after the completion of the KSA. After that, we extend the same idea to analyse the probabilities during any iteration of the Pseudo Random Generation Algorithm. Next, we study the bias of Zr = r (where Zr is the r-th output keystream byte), which is one of the significant biases observed in the RC4 output keystream. This bias has played an important role in the plaintext recovery attack proposed by Isobe et al. in FSE 2013. However, the accurate theoretical explanation of the bias of Zr = r is still a mystery. Though several attempts have been made to prove this bias, none of those provides an accurate justification. Here, using the results found with the help of the probability transition matrix we justify this bias of Zr = r accurately and settle this issue. The bias obtained from our proof matches the experimental observations perfectly.

New Results on Modular Inversion Hidden Number Problem and Inversive Congruential Generator.

2019-01-01
Jun Xu , Santanu Sarkar , Lei Hu +2 more
The Modular Inversion Hidden Number Problem (MIHNP), introduced by Boneh, Halevi and Howgrave-Graham in Asiacrypt 2001, is briefly described as follows: Let \({\mathrm {MSB}}_{\delta }(z)\) refer to the \(\delta \) … The Modular Inversion Hidden Number Problem (MIHNP), introduced by Boneh, Halevi and Howgrave-Graham in Asiacrypt 2001, is briefly described as follows: Let \({\mathrm {MSB}}_{\delta }(z)\) refer to the \(\delta \) most significant bits of z. Given many samples \(\left( t_{i}, {\mathrm {MSB}}_{\delta }((\alpha + t_{i})^{-1} \bmod {p})\right) \) for random \(t_i \in \mathbb {Z}_p\), the goal is to recover the hidden number \(\alpha \in \mathbb {Z}_p\). MIHNP is an important class of Hidden Number Problem.

Spread and asymmetry of typical quantum coherence and their inhibition in response to glassy disorder

2022-01-01
George Biswas , Santanu Sarkar , Anindya Biswas +1 more
We consider the average quantum coherences of typical redits and qudits - vectors of real and complex Hilbert spaces - with the analytical forms stemming from the symmetry of Haar-uniformly … We consider the average quantum coherences of typical redits and qudits - vectors of real and complex Hilbert spaces - with the analytical forms stemming from the symmetry of Haar-uniformly distributed random pure states. We subsequently study the response to disorder in spread of the typical quantum coherence in response to glassy disorder. The disorder is inserted in the state parameters. Even in the absence of disorder, the quantum coherence distributions of redits and qudits are not uniform over the range of quantum coherence, and the spreads are lower for higher dimensions. On insertion of disorder, the spreads decrease. This decrease in the spread of quantum coherence distribution in response to disorder is seen to be a generic feature of typical pure states: we observe the feature for different strengths of disorder and for various types of disorder distributions, viz. Gaussian, uniform, and Cauchy-Lorentz. We also find that the quantum coherence distributions become less asymmetric with increase in dimension and with infusion of glassy disorder.
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Model Selection Approach for Distributed Fault Detection in Wireless Sensor Networks

2013-01-01
Mrinal Nandi , Anup Dewanji , Bimal Roy +1 more
Sensor networks aim at monitoring their surroundings for event detection and object tracking. But, due to failure, or death of sensors, false signal can be transmitted. In this paper, we … Sensor networks aim at monitoring their surroundings for event detection and object tracking. But, due to failure, or death of sensors, false signal can be transmitted. In this paper, we consider the problems of distributed fault detection in wireless sensor network (WSN). In particular, we consider how to take decision regarding fault detection in a noisy environment as a result of false detection or false response of event by some sensors, where the sensors are placed at the center of regular hexagons and the event can occur at only one hexagon. We propose fault detection schemes that explicitly introduce the error probabilities into the optimal event detection process. We introduce two types of detection probabilities, one for the center node, where the event occurs and the other one for the adjacent nodes. This second type of detection probability is new in sensor network literature. We develop schemes under the model selection procedure, multiple model selection procedure and use the concept of Bayesian model averaging to identify a set of likely fault sensors and obtain an average predictive error.
View PDF Chat

Maximal contractive tuples

2013-01-01
B. Krishna Das , Jaydeb Sarkar , Santanu Sarkar
Maximality of a contractive tuple of operators is considered. Characterization of a contractive tuple to be maximal is obtained. Notion of maximality of a submodule of Drury-Arveson module on the … Maximality of a contractive tuple of operators is considered. Characterization of a contractive tuple to be maximal is obtained. Notion of maximality of a submodule of Drury-Arveson module on the $d$-dimensional unit ball $\mathbb{B}_d$ is defined. For $d=1$, it is shown that every submodule of the Hardy module over the unit disc is maximal. But for $d\ge 2$ we prove that any homogeneous submodule or submodule generated by polynomials is not maximal. A characterization of a submodule to be maximal is obtained.
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$\prod\limits_{i=1}^{n} \mathbb{Z}_{2^i}$-Additive Cyclic Codes

2017-01-01
Tapabrata Roy , Santanu Sarkar
In this paper we study $\prod\limits_{i=1}^{n} \mathbb{Z}_{2^i}$-Additive Cyclic Codes. These codes are identified as $\mathbb{Z}_{2^n}[x]$-submodules of $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}[x]/ \langle x^{\alpha_i}-1\rangle$; $\alpha_i$ and $\rm{i}$ being relatively prime for each $i=1,2,\ldots,n.$ We first … In this paper we study $\prod\limits_{i=1}^{n} \mathbb{Z}_{2^i}$-Additive Cyclic Codes. These codes are identified as $\mathbb{Z}_{2^n}[x]$-submodules of $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}[x]/ \langle x^{\alpha_i}-1\rangle$; $\alpha_i$ and $\rm{i}$ being relatively prime for each $i=1,2,\ldots,n.$ We first define a $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}$-additive cyclic code of a certain length. We then define the distance between two codewords and the minimum distance of such a code. Moreover we relate these to binary codes using the generalized Gray maps. We define the duals of such codes and show that the dual of a $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}$-additive cyclic code is also cyclic. We then give the polynomial definition of a $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}$-additive cyclic code of a certain length. We then determine the structure of such codes and derive a minimal spanning set for that. We also determine the total number of codewords in this code. We finally give an illustrative example of a $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}$-additive cyclic code.
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On Acyclic Edge-Coloring of Complete Bipartite Graphs

2015-01-01
Ayineedi Venkateswarlu , Santanu Sarkar , A. Sai Mali
An acyclic edge-coloring of a graph is a proper edge-coloring without bichromatic ($2$-colored) cycles. The acyclic chromatic index of a graph $G$, denoted by $a'(G)$, is the least integer $k$ … An acyclic edge-coloring of a graph is a proper edge-coloring without bichromatic ($2$-colored) cycles. The acyclic chromatic index of a graph $G$, denoted by $a'(G)$, is the least integer $k$ such that $G$ admits an acyclic edge-coloring using $k$ colors. Let $\Delta = \Delta(G)$ denote the maximum degree of a vertex in a graph $G$. A complete bipartite graph with $n$ vertices on each side is denoted by $K_{n,n}$. Basavaraju, Chandran and Kummini proved that $a'(K_{n,n}) \ge n+2 = \Delta + 2$ when $n$ is odd. Basavaraju and Chandran provided an acyclic edge-coloring of $K_{p,p}$ using $p+2$ colors and thus establishing $a'(K_{p,p}) = p+2 = \Delta + 2$ when $p$ is an odd prime. The main tool in their approach is perfect $1$-factorization of $K_{p,p}$. Recently, following their approach, Venkateswarlu and Sarkar have shown that $K_{2p-1,2p-1}$ admits an acyclic edge-coloring using $2p+1$ colors which implies that $a'(K_{2p-1,2p-1}) = 2p+1 = \Delta + 2$, where $p$ is an odd prime. In this paper, we generalize this approach and present a general framework to possibly get an acyclic edge-coloring of $K_{n,n}$ which possess a perfect $1$-factorization using $n+2 = \Delta+2$ colors. In this general framework, we show that $K_{p^2,p^2}$ admits an acyclic edge-coloring using $p^2+2$ colors and thus establishing $a'(K_{p^2,p^2}) = p^2+2 = \Delta + 2$ when $p\ge 5$ is an odd prime.
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The Defect Sequence for Contractive Tuples

2012-01-01
Tirthankar Bhattacharyya , Bata Krishna Das , Santanu Sarkar
We introduce the defect sequence for a contractive tuple of Hilbert space operators and investigate its properties. The defect sequence is a sequence of numbers, called defect dimensions associated with … We introduce the defect sequence for a contractive tuple of Hilbert space operators and investigate its properties. The defect sequence is a sequence of numbers, called defect dimensions associated with a contractive tuple. We show that there are upper bounds for the defect dimensions. The tuples for which these upper bounds are obtained, are called maximal contractive tuples. The upper bounds are different in the non-commutative and in the com- mutative case. We show that the creation operators on the full Fock space and the co ordinate multipliers on the Drury-Arveson space are maximal. We also study pure tuples and see how the defect dimensions play a role in their irreducibility.
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Boundedness of composition operator on several variable Paley-Wiener space

2022-12-09
Amit K. Samanta , Santanu Sarkar

Vector valued de Branges spaces of entire functions based on pairs of Fredholm operator valued functions and functional model

2023-01-01
Subhankar Mahapatra , Santanu Sarkar
In this paper, we have considered vector valued reproducing kernel Hilbert spaces (RKHS) $\mathcal{H}$ of entire functions associated with operator valued kernel functions. de Branges operators $\mathfrak{E}=(E_- , E_+)$ analogous … In this paper, we have considered vector valued reproducing kernel Hilbert spaces (RKHS) $\mathcal{H}$ of entire functions associated with operator valued kernel functions. de Branges operators $\mathfrak{E}=(E_- , E_+)$ analogous to de Branges matrices have been constructed with the help of pairs of Fredholm operator valued entire functions on $\mathfrak{X}$, where $\mathfrak{X}$ is a complex seperable Hilbert space. A few explicit examples of these de Branges operators are also discussed. The newly defined RKHS $\mathcal{B}(\mathfrak{E})$ based on the de Branges operator $\mathfrak{E}=(E_-,E_+)$ has been characterized under some special restrictions. The complete parametrizations and canonical descriptions of all selfadjoint extensions of the closed, symmetric multiplication operator by the independent variable have been given in terms of unitary operators between ranges of reproducing kernels. A sampling formula for the de Branges spaces $\mathcal{B}(\mathfrak{E})$ has been discussed. A particular class of entire operators with infinite deficiency indices has been dealt with and shown that they can be considered as the multiplication operator for a specific class of these de Branges spaces. Finally, a brief discussion on the connection between the characteristic function of a completely nonunitary contraction operator and the de Branges spaces $\mathcal{B}(\mathfrak{E})$ has been given.

On vector valued de Branges spaces: Factorization, Isometric Embedding and Associated Functions

2023-01-01
Subhankar Mahapatra , Santanu Sarkar
This paper deals with certain aspects of the vector valued de Branges spaces of entire functions that are associated with operator valued kernel functions. Some factorization and isometric embedding results … This paper deals with certain aspects of the vector valued de Branges spaces of entire functions that are associated with operator valued kernel functions. Some factorization and isometric embedding results have been extended from the scalar valued theory of de Branges spaces. In particular, global factorization and analytic equivalence of Fredholm operator valued entire functions are discussed. Additionally, the operator valued entire functions associated with these de Branges spaces are studied.

Representing the inverse map as a composition of quadratics in a finite field of characteristic $2$

2023-01-01
Florian Luca , Santanu Sarkar , Pantelimon Stănică
In 1953, Carlitz~\cite{Car53} showed that all permutation polynomials over $\F_q$, where $q>2$ is a power of a prime, are generated by the special permutation polynomials $x^{q-2}$ (the inversion) and $ … In 1953, Carlitz~\cite{Car53} showed that all permutation polynomials over $\F_q$, where $q>2$ is a power of a prime, are generated by the special permutation polynomials $x^{q-2}$ (the inversion) and $ ax+b$ (affine functions, where $0\neq a, b\in \F_q$). Recently, Nikova, Nikov and Rijmen~\cite{NNR19} proposed an algorithm (NNR) to find a decomposition of the inverse function in quadratics, and computationally covered all dimensions $n\leq 16$. Petrides~\cite{P23} found a class of integers for which it is easy to decompose the inverse into quadratics, and improved the NNR algorithm, thereby extending the computation up to $n\leq 32$. Here, we extend Petrides' result, as well as we propose a number theoretical approach, which allows us to cover easily all (surely, odd) exponents up to~$250$, at least.

Uncertainty principle for Weyl transform and Fourier--Wigner transform

2023-01-01
Amit K. Samanta , Santanu Sarkar
We prove an uncertainty type principle for finite linear combinations of Fourier--Wigner transforms. This is equivalent to proving a finite rank theorem for the Weyl transform. We prove an uncertainty type principle for finite linear combinations of Fourier--Wigner transforms. This is equivalent to proving a finite rank theorem for the Weyl transform.
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Properties of singular integral operators $S_{α,β}$

2015-01-01
Amit K. Samanta , Santanu Sarkar
For $α, β\in L^{\infty} (S^1),$ the singular integral operator $S_{α,β}$ on $L^2 (S^1)$ is defined by $S_{α,β}f:= αPf+βQf$, where $P$ denotes the orthogonal projection of $L^2(S^1)$ onto the Hardy space … For $α, β\in L^{\infty} (S^1),$ the singular integral operator $S_{α,β}$ on $L^2 (S^1)$ is defined by $S_{α,β}f:= αPf+βQf$, where $P$ denotes the orthogonal projection of $L^2(S^1)$ onto the Hardy space $H^2(S^1),$ and $Q$ denotes the orthogonal projection onto $H^2(S^1)^{\perp}.$ In a recent paper Nakazi and Yamamoto have studied the normality and self-adjointness of $S_{α,β}.$ This work has shown that $S_{α,β}$ may have analogous properties to that of the Toeplitz operator. In this paper we study several other properties of $S_{α,β}.$

Analytic Kramer sampling and quasi Lagrange-type interpolation in vector valued RKHS

2024-05-30
Subhankar Mahapatra , Santanu Sarkar
This paper discusses an abstract Kramer sampling theorem for functions within a reproducing kernel Hilbert space (RKHS) of vector valued holomorphic functions. Additionally, we extend the concept of quasi Lagrange-type … This paper discusses an abstract Kramer sampling theorem for functions within a reproducing kernel Hilbert space (RKHS) of vector valued holomorphic functions. Additionally, we extend the concept of quasi Lagrange-type interpolation for functions within a RKHS of vector valued entire functions. The dependence of having quasi Lagrange-type interpolation on an invariance condition under the generalized backward shift operator has also been discussed. Furthermore, the paper establishes the connection between quasi Lagrange-type interpolation, operator of multiplication by the independent variable, and de Branges spaces of vector valued entire functions.
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de Branges matrices and associated de Branges spaces of vector valued entire functions

2024-06-21
Subhankar Mahapatra , Santanu Sarkar
This paper extends the concept of de Branges matrices to any finite $m\times m$ order where $m=2n$. We shall discuss these matrices along with the theory of de Branges spaces … This paper extends the concept of de Branges matrices to any finite $m\times m$ order where $m=2n$. We shall discuss these matrices along with the theory of de Branges spaces of $\mathbb{C}^n$-valued entire functions and their associated functions. A parametrization of these matrices is obtained using the Smirnov maximum principle for matrix valued functions. Additionally, a factorization of matrix valued meromorphic functions is discussed.
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J-contractive operator valued functions, vector valued de Branges spaces and functional models

2024-11-04
B. P. Garg , Santanu Sarkar
In this paper, we study the reproducing kernel Hilbert spaces (RKHS) which are constructed from \( J \)-contractive operator valued analytic functions on the upper half plane. First, we discuss … In this paper, we study the reproducing kernel Hilbert spaces (RKHS) which are constructed from \( J \)-contractive operator valued analytic functions on the upper half plane. First, we discuss the Potapov-Ginzburg transform of a specific class of Fredholm operators, investigate some of its properties, and outline sufficient conditions for a \( J \)-contractive operator to be \( J \)-bicontractive. Subsequently, we construct certain classes of vector valued de Branges spaces and aim to explore the significance of these classes. Using these spaces, we present a functional model for simple Volterra operator nodes. Additionally, we analyze their connections with another classes of de Branges spaces which are based on a pair of Fredholm operator valued analytic functions. Finally, we discuss a functional model for simple, closed, densely defined, symmetric operators with infinite deficiency indices within this framework.
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J-contractive operator valued functions, vector valued de Branges spaces and functional models

2025-04-01
B. P. Garg , Santanu Sarkar

Characterizations for minimal codes: graph theory approach and algebraic approach over finite chain rings

2025-05-10
Makhan Maji , Sihem Mesnager , Santanu Sarkar +1 more

Characterizations for minimal codes: graph theory approach and algebraic approach over finite chain rings

2025-05-10
Makhan Maji , Sihem Mesnager , Santanu Sarkar +1 more

J-contractive operator valued functions, vector valued de Branges spaces and functional models

2025-04-01
B. P. Garg , Santanu Sarkar

J-contractive operator valued functions, vector valued de Branges spaces and functional models

2024-11-04
B. P. Garg , Santanu Sarkar
In this paper, we study the reproducing kernel Hilbert spaces (RKHS) which are constructed from \( J \)-contractive operator valued analytic functions on the upper half plane. First, we discuss … In this paper, we study the reproducing kernel Hilbert spaces (RKHS) which are constructed from \( J \)-contractive operator valued analytic functions on the upper half plane. First, we discuss the Potapov-Ginzburg transform of a specific class of Fredholm operators, investigate some of its properties, and outline sufficient conditions for a \( J \)-contractive operator to be \( J \)-bicontractive. Subsequently, we construct certain classes of vector valued de Branges spaces and aim to explore the significance of these classes. Using these spaces, we present a functional model for simple Volterra operator nodes. Additionally, we analyze their connections with another classes of de Branges spaces which are based on a pair of Fredholm operator valued analytic functions. Finally, we discuss a functional model for simple, closed, densely defined, symmetric operators with infinite deficiency indices within this framework.
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Analytic Kramer Sampling and Quasi Lagrange-Type Interpolation in Vector Valued RKHS

2024-08-23
Subhankar Mahapatra , Santanu Sarkar

de Branges matrices and associated de Branges spaces of vector valued entire functions

2024-06-21
Subhankar Mahapatra , Santanu Sarkar
This paper extends the concept of de Branges matrices to any finite $m\times m$ order where $m=2n$. We shall discuss these matrices along with the theory of de Branges spaces … This paper extends the concept of de Branges matrices to any finite $m\times m$ order where $m=2n$. We shall discuss these matrices along with the theory of de Branges spaces of $\mathbb{C}^n$-valued entire functions and their associated functions. A parametrization of these matrices is obtained using the Smirnov maximum principle for matrix valued functions. Additionally, a factorization of matrix valued meromorphic functions is discussed.
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Analytic Kramer sampling and quasi Lagrange-type interpolation in vector valued RKHS

2024-05-30
Subhankar Mahapatra , Santanu Sarkar
This paper discusses an abstract Kramer sampling theorem for functions within a reproducing kernel Hilbert space (RKHS) of vector valued holomorphic functions. Additionally, we extend the concept of quasi Lagrange-type … This paper discusses an abstract Kramer sampling theorem for functions within a reproducing kernel Hilbert space (RKHS) of vector valued holomorphic functions. Additionally, we extend the concept of quasi Lagrange-type interpolation for functions within a RKHS of vector valued entire functions. The dependence of having quasi Lagrange-type interpolation on an invariance condition under the generalized backward shift operator has also been discussed. Furthermore, the paper establishes the connection between quasi Lagrange-type interpolation, operator of multiplication by the independent variable, and de Branges spaces of vector valued entire functions.
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Vector valued de Branges spaces of entire functions based on pairs of Fredholm operator valued functions and functional model

2023-12-05
Subhankar Mahapatra , Santanu Sarkar
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Spread and asymmetry of typical quantum coherence and their inhibition in response to glassy disorder

2023-08-01
George Biswas , Santanu Sarkar , Anindya Biswas +1 more
Abstract We consider the average quantum coherences of typical redits and qudits—vectors of real and complex Hilbert spaces—with the analytical forms stemming from the symmetry of Haar-uniformly distributed random pure … Abstract We consider the average quantum coherences of typical redits and qudits—vectors of real and complex Hilbert spaces—with the analytical forms stemming from the symmetry of Haar-uniformly distributed random pure states. We subsequently study the response to disorder in spread of the typical quantum coherence in response to glassy disorder. The disorder is inserted in the state parameters. Even in the absence of disorder, the quantum coherence distributions of redits and qudits are not uniform over the range of quantum coherence, and the spreads are relatively lower for higher dimensions. On insertion of disorder, the spreads decrease. This decrease in the spread of quantum coherence distribution in response to disorder is seen to be a generic feature of typical pure states: we observe the feature for different strengths of disorder and for various types of disorder distributions, viz. Gaussian, uniform, and Cauchy–Lorentz. We also find that the quantum coherence distributions become less asymmetric with increase in dimension and with infusion of glassy disorder.
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Vector valued de Branges spaces of entire functions based on pairs of Fredholm operator valued functions and functional model

2023-01-01
Subhankar Mahapatra , Santanu Sarkar
In this paper, we have considered vector valued reproducing kernel Hilbert spaces (RKHS) $\mathcal{H}$ of entire functions associated with operator valued kernel functions. de Branges operators $\mathfrak{E}=(E_- , E_+)$ analogous … In this paper, we have considered vector valued reproducing kernel Hilbert spaces (RKHS) $\mathcal{H}$ of entire functions associated with operator valued kernel functions. de Branges operators $\mathfrak{E}=(E_- , E_+)$ analogous to de Branges matrices have been constructed with the help of pairs of Fredholm operator valued entire functions on $\mathfrak{X}$, where $\mathfrak{X}$ is a complex seperable Hilbert space. A few explicit examples of these de Branges operators are also discussed. The newly defined RKHS $\mathcal{B}(\mathfrak{E})$ based on the de Branges operator $\mathfrak{E}=(E_-,E_+)$ has been characterized under some special restrictions. The complete parametrizations and canonical descriptions of all selfadjoint extensions of the closed, symmetric multiplication operator by the independent variable have been given in terms of unitary operators between ranges of reproducing kernels. A sampling formula for the de Branges spaces $\mathcal{B}(\mathfrak{E})$ has been discussed. A particular class of entire operators with infinite deficiency indices has been dealt with and shown that they can be considered as the multiplication operator for a specific class of these de Branges spaces. Finally, a brief discussion on the connection between the characteristic function of a completely nonunitary contraction operator and the de Branges spaces $\mathcal{B}(\mathfrak{E})$ has been given.

On vector valued de Branges spaces: Factorization, Isometric Embedding and Associated Functions

2023-01-01
Subhankar Mahapatra , Santanu Sarkar
This paper deals with certain aspects of the vector valued de Branges spaces of entire functions that are associated with operator valued kernel functions. Some factorization and isometric embedding results … This paper deals with certain aspects of the vector valued de Branges spaces of entire functions that are associated with operator valued kernel functions. Some factorization and isometric embedding results have been extended from the scalar valued theory of de Branges spaces. In particular, global factorization and analytic equivalence of Fredholm operator valued entire functions are discussed. Additionally, the operator valued entire functions associated with these de Branges spaces are studied.

Representing the inverse map as a composition of quadratics in a finite field of characteristic $2$

2023-01-01
Florian Luca , Santanu Sarkar , Pantelimon Stănică
In 1953, Carlitz~\cite{Car53} showed that all permutation polynomials over $\F_q$, where $q>2$ is a power of a prime, are generated by the special permutation polynomials $x^{q-2}$ (the inversion) and $ … In 1953, Carlitz~\cite{Car53} showed that all permutation polynomials over $\F_q$, where $q>2$ is a power of a prime, are generated by the special permutation polynomials $x^{q-2}$ (the inversion) and $ ax+b$ (affine functions, where $0\neq a, b\in \F_q$). Recently, Nikova, Nikov and Rijmen~\cite{NNR19} proposed an algorithm (NNR) to find a decomposition of the inverse function in quadratics, and computationally covered all dimensions $n\leq 16$. Petrides~\cite{P23} found a class of integers for which it is easy to decompose the inverse into quadratics, and improved the NNR algorithm, thereby extending the computation up to $n\leq 32$. Here, we extend Petrides' result, as well as we propose a number theoretical approach, which allows us to cover easily all (surely, odd) exponents up to~$250$, at least.

Uncertainty principle for Weyl transform and Fourier--Wigner transform

2023-01-01
Amit K. Samanta , Santanu Sarkar
We prove an uncertainty type principle for finite linear combinations of Fourier--Wigner transforms. This is equivalent to proving a finite rank theorem for the Weyl transform. We prove an uncertainty type principle for finite linear combinations of Fourier--Wigner transforms. This is equivalent to proving a finite rank theorem for the Weyl transform.
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Boundedness of composition operator on several variable Paley-Wiener space

2022-12-09
Amit K. Samanta , Santanu Sarkar

Spread and asymmetry of typical quantum coherence and their inhibition in response to glassy disorder

2022-01-01
George Biswas , Santanu Sarkar , Anindya Biswas +1 more
We consider the average quantum coherences of typical redits and qudits - vectors of real and complex Hilbert spaces - with the analytical forms stemming from the symmetry of Haar-uniformly … We consider the average quantum coherences of typical redits and qudits - vectors of real and complex Hilbert spaces - with the analytical forms stemming from the symmetry of Haar-uniformly distributed random pure states. We subsequently study the response to disorder in spread of the typical quantum coherence in response to glassy disorder. The disorder is inserted in the state parameters. Even in the absence of disorder, the quantum coherence distributions of redits and qudits are not uniform over the range of quantum coherence, and the spreads are lower for higher dimensions. On insertion of disorder, the spreads decrease. This decrease in the spread of quantum coherence distribution in response to disorder is seen to be a generic feature of typical pure states: we observe the feature for different strengths of disorder and for various types of disorder distributions, viz. Gaussian, uniform, and Cauchy-Lorentz. We also find that the quantum coherence distributions become less asymmetric with increase in dimension and with infusion of glassy disorder.
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Progress in Cryptology – INDOCRYPT 2022

2022-01-01
Takanori Isobe , Santanu Sarkar

A Hybrid Inversive Congruential Pseudorandom Number Generator with High Period

2021-01-31
Constanza Riera , Tapabrata Roy , Santanu Sarkar +1 more
Though generating a sequence of pseudorandom numbers by linear methods (Lehmer generator) displays acceptable behavior under some conditions of the parameters, it also has undesirable features, which makes the sequence … Though generating a sequence of pseudorandom numbers by linear methods (Lehmer generator) displays acceptable behavior under some conditions of the parameters, it also has undesirable features, which makes the sequence unusable for various stochastic simulations. An extension which showed promise for such applications is a generator obtained by using a first-order recurrence based upon the inversive modulo a prime or a prime power, called inversive congruential generator (ICG). A lot of work has been dedicated to investigate the periods (under some conditions of the parameters), the lattice test passing, discrepancy and other statistical properties of such a generator. Here, we propose a new method, which we call hybrid inversive congruential generator (HICG), based upon a second order recurrence using the inversive modulo M, a power of 2. We investigate the period of this pseudorandom numbers generator (PRNG) and give necessary and sufficient conditions for our PRNG to have periods M (thereby doubling the period of the classical ICG) and M/2 (matching the one of the ICG). Moreover, we show that the lattice test complexity for a binary sequence associated to (a full period) HICG is precisely M/2.

New Results on Modular Inversion Hidden Number Problem and Inversive Congruential Generator.

2019-01-01
Jun Xu , Santanu Sarkar , Lei Hu +2 more
The Modular Inversion Hidden Number Problem (MIHNP), introduced by Boneh, Halevi and Howgrave-Graham in Asiacrypt 2001, is briefly described as follows: Let \({\mathrm {MSB}}_{\delta }(z)\) refer to the \(\delta \) … The Modular Inversion Hidden Number Problem (MIHNP), introduced by Boneh, Halevi and Howgrave-Graham in Asiacrypt 2001, is briefly described as follows: Let \({\mathrm {MSB}}_{\delta }(z)\) refer to the \(\delta \) most significant bits of z. Given many samples \(\left( t_{i}, {\mathrm {MSB}}_{\delta }((\alpha + t_{i})^{-1} \bmod {p})\right) \) for random \(t_i \in \mathbb {Z}_p\), the goal is to recover the hidden number \(\alpha \in \mathbb {Z}_p\). MIHNP is an important class of Hidden Number Problem.

Properties of singular integral operators $$\varvec{S}_{\varvec{\alpha ,\beta }}$$ S α , β

2018-02-01
Amit K. Samanta , Santanu Sarkar

Multiplication operators with deficiency indices (p,p) and sampling formulas in reproducing kernel Hilbert spaces of entire vector valued functions

2017-09-20
Harry Dym , Santanu Sarkar

$\prod\limits_{i=1}^{n} \mathbb{Z}_{2^i}$-Additive Cyclic Codes

2017-08-23
Tapabrata Roy , Santanu Sarkar
In this paper we study $\prod\limits_{i=1}^{n} \mathbb{Z}_{2^i}$-Additive Cyclic Codes. These codes are identified as $\mathbb{Z}_{2^n}[x]$-submodules of $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}[x]/ \langle x^{\alpha_i}-1\rangle$; $\alpha_i$ and $\rm{i}$ being relatively prime for each $i=1,2,\ldots,n.$ We first … In this paper we study $\prod\limits_{i=1}^{n} \mathbb{Z}_{2^i}$-Additive Cyclic Codes. These codes are identified as $\mathbb{Z}_{2^n}[x]$-submodules of $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}[x]/ \langle x^{\alpha_i}-1\rangle$; $\alpha_i$ and $\rm{i}$ being relatively prime for each $i=1,2,\ldots,n.$ We first define a $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}$-additive cyclic code of a certain length. We then define the distance between two codewords and the minimum distance of such a code. Moreover we relate these to binary codes using the generalized Gray maps. We define the duals of such codes and show that the dual of a $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}$-additive cyclic code is also cyclic. We then give the polynomial definition of a $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}$-additive cyclic code of a certain length. We then determine the structure of such codes and derive a minimal spanning set for that. We also determine the total number of codewords in this code. We finally give an illustrative example of a $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}$-additive cyclic code.

Settling the mystery of Z r=r in RC4.

2017-01-01
Sabyasachi Dey , Santanu Sarkar
In this paper, using a matrix, at first we revisit the work of Mantin on finding the probability distribution of the RC4 permutation after the completion of the KSA. After … In this paper, using a matrix, at first we revisit the work of Mantin on finding the probability distribution of the RC4 permutation after the completion of the KSA. After that, we extend the same idea to analyse the probabilities during any iteration of the Pseudo Random Generation Algorithm. Next, we study the bias of Zr = r (where Zr is the r-th output keystream byte), which is one of the significant biases observed in the RC4 output keystream. This bias has played an important role in the plaintext recovery attack proposed by Isobe et al. in FSE 2013. However, the accurate theoretical explanation of the bias of Zr = r is still a mystery. Though several attempts have been made to prove this bias, none of those provides an accurate justification. Here, using the results found with the help of the probability transition matrix we justify this bias of Zr = r accurately and settle this issue. The bias obtained from our proof matches the experimental observations perfectly.

$\prod\limits_{i=1}^{n} \mathbb{Z}_{2^i}$-Additive Cyclic Codes

2017-01-01
Tapabrata Roy , Santanu Sarkar
In this paper we study $\prod\limits_{i=1}^{n} \mathbb{Z}_{2^i}$-Additive Cyclic Codes. These codes are identified as $\mathbb{Z}_{2^n}[x]$-submodules of $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}[x]/ \langle x^{\alpha_i}-1\rangle$; $\alpha_i$ and $\rm{i}$ being relatively prime for each $i=1,2,\ldots,n.$ We first … In this paper we study $\prod\limits_{i=1}^{n} \mathbb{Z}_{2^i}$-Additive Cyclic Codes. These codes are identified as $\mathbb{Z}_{2^n}[x]$-submodules of $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}[x]/ \langle x^{\alpha_i}-1\rangle$; $\alpha_i$ and $\rm{i}$ being relatively prime for each $i=1,2,\ldots,n.$ We first define a $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}$-additive cyclic code of a certain length. We then define the distance between two codewords and the minimum distance of such a code. Moreover we relate these to binary codes using the generalized Gray maps. We define the duals of such codes and show that the dual of a $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}$-additive cyclic code is also cyclic. We then give the polynomial definition of a $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}$-additive cyclic code of a certain length. We then determine the structure of such codes and derive a minimal spanning set for that. We also determine the total number of codewords in this code. We finally give an illustrative example of a $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}$-additive cyclic code.
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On acyclic edge-coloring of complete bipartite graphs

2016-11-16
Ayineedi Venkateswarlu , Santanu Sarkar , Sai Mali Ananthanarayanan
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On acyclic edge-coloring of the complete bipartite graphs <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> for odd prime <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg…

2015-08-13
Ayineedi Venkateswarlu , Santanu Sarkar

Properties of singular integral operators $S_{\alpha,\beta}$

2015-05-20
Amit K. Samanta , Santanu Sarkar
For $\alpha, \beta \in L^{\infty} (S^1),$ the singular integral operator $S_{\alpha,\beta}$ on $L^2 (S^1)$ is defined by $S_{\alpha,\beta}f:= \alpha Pf+\beta Qf$, where $P$ denotes the orthogonal projection of $L^2(S^1)$ onto … For $\alpha, \beta \in L^{\infty} (S^1),$ the singular integral operator $S_{\alpha,\beta}$ on $L^2 (S^1)$ is defined by $S_{\alpha,\beta}f:= \alpha Pf+\beta Qf$, where $P$ denotes the orthogonal projection of $L^2(S^1)$ onto the Hardy space $H^2(S^1),$ and $Q$ denotes the orthogonal projection onto $H^2(S^1)^{\perp}.$ In a recent paper Nakazi and Yamamoto have studied the normality and self-adjointness of $S_{\alpha,\beta}.$ This work has shown that $S_{\alpha,\beta}$ may have analogous properties to that of the Toeplitz operator. In this paper we study several other properties of $S_{\alpha,\beta}.$

On Acyclic Edge-Coloring of Complete Bipartite Graphs

2015-03-11
Ayineedi Venkateswarlu , Santanu Sarkar , Ajit S Mali
An acyclic edge-coloring of a graph is a proper edge-coloring without bichromatic ($2$-colored) cycles. The acyclic chromatic index of a graph $G$, denoted by $a'(G)$, is the least integer $k$ … An acyclic edge-coloring of a graph is a proper edge-coloring without bichromatic ($2$-colored) cycles. The acyclic chromatic index of a graph $G$, denoted by $a'(G)$, is the least integer $k$ such that $G$ admits an acyclic edge-coloring using $k$ colors. Let $\Delta = \Delta(G)$ denote the maximum degree of a vertex in a graph $G$. A complete bipartite graph with $n$ vertices on each side is denoted by $K_{n,n}$. Basavaraju, Chandran and Kummini proved that $a'(K_{n,n}) \ge n+2 = \Delta + 2$ when $n$ is odd. Basavaraju and Chandran provided an acyclic edge-coloring of $K_{p,p}$ using $p+2$ colors and thus establishing $a'(K_{p,p}) = p+2 = \Delta + 2$ when $p$ is an odd prime. The main tool in their approach is perfect $1$-factorization of $K_{p,p}$. Recently, following their approach, Venkateswarlu and Sarkar have shown that $K_{2p-1,2p-1}$ admits an acyclic edge-coloring using $2p+1$ colors which implies that $a'(K_{2p-1,2p-1}) = 2p+1 = \Delta + 2$, where $p$ is an odd prime. In this paper, we generalize this approach and present a general framework to possibly get an acyclic edge-coloring of $K_{n,n}$ which possess a perfect $1$-factorization using $n+2 = \Delta+2$ colors. In this general framework, we show that $K_{p^2,p^2}$ admits an acyclic edge-coloring using $p^2+2$ colors and thus establishing $a'(K_{p^2,p^2}) = p^2+2 = \Delta + 2$ when $p\ge 5$ is an odd prime.

On Acyclic Edge-Coloring of Complete Bipartite Graphs

2015-01-01
Ayineedi Venkateswarlu , Santanu Sarkar , A. Sai Mali
An acyclic edge-coloring of a graph is a proper edge-coloring without bichromatic ($2$-colored) cycles. The acyclic chromatic index of a graph $G$, denoted by $a'(G)$, is the least integer $k$ … An acyclic edge-coloring of a graph is a proper edge-coloring without bichromatic ($2$-colored) cycles. The acyclic chromatic index of a graph $G$, denoted by $a'(G)$, is the least integer $k$ such that $G$ admits an acyclic edge-coloring using $k$ colors. Let $\Delta = \Delta(G)$ denote the maximum degree of a vertex in a graph $G$. A complete bipartite graph with $n$ vertices on each side is denoted by $K_{n,n}$. Basavaraju, Chandran and Kummini proved that $a'(K_{n,n}) \ge n+2 = \Delta + 2$ when $n$ is odd. Basavaraju and Chandran provided an acyclic edge-coloring of $K_{p,p}$ using $p+2$ colors and thus establishing $a'(K_{p,p}) = p+2 = \Delta + 2$ when $p$ is an odd prime. The main tool in their approach is perfect $1$-factorization of $K_{p,p}$. Recently, following their approach, Venkateswarlu and Sarkar have shown that $K_{2p-1,2p-1}$ admits an acyclic edge-coloring using $2p+1$ colors which implies that $a'(K_{2p-1,2p-1}) = 2p+1 = \Delta + 2$, where $p$ is an odd prime. In this paper, we generalize this approach and present a general framework to possibly get an acyclic edge-coloring of $K_{n,n}$ which possess a perfect $1$-factorization using $n+2 = \Delta+2$ colors. In this general framework, we show that $K_{p^2,p^2}$ admits an acyclic edge-coloring using $p^2+2$ colors and thus establishing $a'(K_{p^2,p^2}) = p^2+2 = \Delta + 2$ when $p\ge 5$ is an odd prime.
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Properties of singular integral operators $S_{α,β}$

2015-01-01
Amit K. Samanta , Santanu Sarkar
For $α, β\in L^{\infty} (S^1),$ the singular integral operator $S_{α,β}$ on $L^2 (S^1)$ is defined by $S_{α,β}f:= αPf+βQf$, where $P$ denotes the orthogonal projection of $L^2(S^1)$ onto the Hardy space … For $α, β\in L^{\infty} (S^1),$ the singular integral operator $S_{α,β}$ on $L^2 (S^1)$ is defined by $S_{α,β}f:= αPf+βQf$, where $P$ denotes the orthogonal projection of $L^2(S^1)$ onto the Hardy space $H^2(S^1),$ and $Q$ denotes the orthogonal projection onto $H^2(S^1)^{\perp}.$ In a recent paper Nakazi and Yamamoto have studied the normality and self-adjointness of $S_{α,β}.$ This work has shown that $S_{α,β}$ may have analogous properties to that of the Toeplitz operator. In this paper we study several other properties of $S_{α,β}.$

Hypothesis testing and decision theoretic approach for fault detection in wireless sensor networks

2014-04-24
Mrinal Nandi , Amiya Nayak , Bimal Roy +1 more
Sensor networks aim at monitoring their surroundings for event detection and object tracking. In practice, due to failure or death of sensors, false signal can be transmitted. In this article, … Sensor networks aim at monitoring their surroundings for event detection and object tracking. In practice, due to failure or death of sensors, false signal can be transmitted. In this article, we consider the problem of fault detection in wireless sensor networks, in particular, addressing both the noise-related measurement error and sensor fault simultaneously in fault detection. We assume that the sensors are placed at the centre of a square (or hexagonal) cell in the region of interest (ROI) and, if the event occurs, it occurs at a particular cell of the ROI. We propose fault detection schemes that consider error probabilities in the optimal event detection process. We develop the schemes under the consideration of Neyman–Pearson hypothesis test and Bayes test.
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Model Selection Approach for Distributed Fault Detection in Wireless Sensor Networks

2014-01-01
Mrinal Nandi , Anup Dewanji , Bimal Roy +1 more
Sensor networks aim at monitoring their surroundings for event detection and object tracking. But, due to failure or death of sensors, false signal can be transmitted. In this paper, we … Sensor networks aim at monitoring their surroundings for event detection and object tracking. But, due to failure or death of sensors, false signal can be transmitted. In this paper, we consider the problems of distributed fault detection in wireless sensor network (WSN). In particular, we consider how to take decision regarding fault detection in a noisy environment as a result of false detection or false response of event by some sensors, where the sensors are placed at the center of regular hexagons and the event can occur at only one hexagon. We propose fault detection schemes that explicitly introduce the error probabilities into the optimal event detection process. We introduce two types of detection probabilities, one for the center node, where the event occurs, and the other one for the adjacent nodes. This second type of detection probability is new in sensor network literature. We develop schemes under the model selection procedure and multiple model selection procedure and use the concept of Bayesian model averaging to identify a set of likely fault sensors and obtain an average predictive error.
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Maximal Contractive Tuples

2013-10-06
B. Krishna Das , Jaydeb Sarkar , Santanu Sarkar
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Maximal contractive tuples

2013-06-04
B. Krishna Das , Jaydeb Sarkar , Santanu Sarkar
Maximality of a contractive tuple of operators is considered. Characterization of a contractive tuple to be maximal is obtained. Notion of maximality of a submodule of Drury-Arveson module on the … Maximality of a contractive tuple of operators is considered. Characterization of a contractive tuple to be maximal is obtained. Notion of maximality of a submodule of Drury-Arveson module on the $d$-dimensional unit ball $\mathbb{B}_d$ is defined. For $d=1$, it is shown that every submodule of the Hardy module over the unit disc is maximal. But for $d\ge 2$ we prove that any homogeneous submodule or submodule generated by polynomials is not maximal. A characterization of a submodule to be maximal is obtained.

Counting Heron triangles with constraints

2013-01-25
Pantelimon Stănică , Santanu Sarkar , Sourav Sen Gupta +2 more
Heron triangles have the property that all three of their sides as well as their area are positive integers. In this paper, we give some estimates for the number of … Heron triangles have the property that all three of their sides as well as their area are positive integers. In this paper, we give some estimates for the number of Heron triangles with two of their sides fixed. We provide a general bound on this count H(a,b), where the sides a,b are fixed positive integers, and the estimate here is better than the one of Ionascu, Luca and St˘anic˘a for the general situation of fixed sides a,b .I n the case of primes sidesp,q, there is an additional hypothesis which helps us to drop the upper bounds on H(p,q). In particular, we prove that H(p,q) is less than or equal to 1 when p ! q 2 (mod 4). We also provide a count for the number of Heron triangles with a fixed height (there exists only one such when the height is prime). Moreover, we study the decomposability property of a Heron triangle into two similar ones, and provide some cases when a Heron triangle is not decomposable.

Model Selection Approach for Distributed Fault Detection in Wireless Sensor Networks

2013-01-21
Mrinal Nandi , Anup Dewanji , Bimal Roy +1 more
Sensor networks aim at monitoring their surroundings for event detection and object tracking. But, due to failure, or death of sensors, false signal can be transmitted. In this paper, we … Sensor networks aim at monitoring their surroundings for event detection and object tracking. But, due to failure, or death of sensors, false signal can be transmitted. In this paper, we consider the problems of distributed fault detection in wireless sensor network (WSN). In particular, we consider how to take decision regarding fault detection in a noisy environment as a result of false detection or false response of event by some sensors, where the sensors are placed at the center of regular hexagons and the event can occur at only one hexagon. We propose fault detection schemes that explicitly introduce the error probabilities into the optimal event detection process. We introduce two types of detection probabilities, one for the center node, where the event occurs and the other one for the adjacent nodes. This second type of detection probability is new in sensor network literature. We develop schemes under the model selection procedure, multiple model selection procedure and use the concept of Bayesian model averaging to identify a set of likely fault sensors and obtain an average predictive error.

Model Selection Approach for Distributed Fault Detection in Wireless Sensor Networks

2013-01-01
Mrinal Nandi , Anup Dewanji , Bimal Roy +1 more
Sensor networks aim at monitoring their surroundings for event detection and object tracking. But, due to failure, or death of sensors, false signal can be transmitted. In this paper, we … Sensor networks aim at monitoring their surroundings for event detection and object tracking. But, due to failure, or death of sensors, false signal can be transmitted. In this paper, we consider the problems of distributed fault detection in wireless sensor network (WSN). In particular, we consider how to take decision regarding fault detection in a noisy environment as a result of false detection or false response of event by some sensors, where the sensors are placed at the center of regular hexagons and the event can occur at only one hexagon. We propose fault detection schemes that explicitly introduce the error probabilities into the optimal event detection process. We introduce two types of detection probabilities, one for the center node, where the event occurs and the other one for the adjacent nodes. This second type of detection probability is new in sensor network literature. We develop schemes under the model selection procedure, multiple model selection procedure and use the concept of Bayesian model averaging to identify a set of likely fault sensors and obtain an average predictive error.
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Maximal contractive tuples

2013-01-01
B. Krishna Das , Jaydeb Sarkar , Santanu Sarkar
Maximality of a contractive tuple of operators is considered. Characterization of a contractive tuple to be maximal is obtained. Notion of maximality of a submodule of Drury-Arveson module on the … Maximality of a contractive tuple of operators is considered. Characterization of a contractive tuple to be maximal is obtained. Notion of maximality of a submodule of Drury-Arveson module on the $d$-dimensional unit ball $\mathbb{B}_d$ is defined. For $d=1$, it is shown that every submodule of the Hardy module over the unit disc is maximal. But for $d\ge 2$ we prove that any homogeneous submodule or submodule generated by polynomials is not maximal. A characterization of a submodule to be maximal is obtained.
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The Defect Sequence for Contractive Tuples

2012-11-26
Tirthankar Bhattacharyya , Bata Krishna Das , Santanu Sarkar
We introduce the defect sequence for a contractive tuple of Hilbert space operators and investigate its properties. The defect sequence is a sequence of numbers, called defect dimensions associated with … We introduce the defect sequence for a contractive tuple of Hilbert space operators and investigate its properties. The defect sequence is a sequence of numbers, called defect dimensions associated with a contractive tuple. We show that there are upper bounds for the defect dimensions. The tuples for which these upper bounds are obtained, are called maximal contractive tuples. The upper bounds are different in the non-commutative and in the com- mutative case. We show that the creation operators on the full Fock space and the co ordinate multipliers on the Drury-Arveson space are maximal. We also study pure tuples and see how the defect dimensions play a role in their irreducibility.

The defect sequence for contractive tuples

2012-09-15
Tirthankar Bhattacharyya , Bata Krishna Das , Santanu Sarkar
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Hypothesis Testing and Decision Theoretic Approach for Fault Detection in Wireless Sensor Networks

2012-03-02
Mrinal Nandi , Amiya Nayak , Bimal Roy +1 more
Sensor networks aim at monitoring their surroundings for event detection and object tracking. But due to failure or death of sensors, false signal can be transmitted. In this paper, we … Sensor networks aim at monitoring their surroundings for event detection and object tracking. But due to failure or death of sensors, false signal can be transmitted. In this paper, we consider the problem of fault detection in wireless sensor network (WSN), in particular, addressing both the noise-related measurement error and sensor fault simultaneously in fault detection. We assume that the sensors are placed at the center of a square (or hexagonal) cell in region of interest (ROI) and, if the event occurs, it occurs at a particular cell of the ROI. We propose fault detection schemes that take into account error probabilities into the optimal event detection process. We develop the schemes under the consideration of Neyman-Pearson test and Bayes test.

On Universality of Quantum Fourier Transform

2012-03-01
Arpita Maitra , Santanu Sarkar
A methodology is presented to obtain the basis of qudits which are admissible to quantum Fourier transform (QFT) in the sense that the set of such kets are related by … A methodology is presented to obtain the basis of qudits which are admissible to quantum Fourier transform (QFT) in the sense that the set of such kets are related by the QFT in the same way as the kets of the computational basis. We first study this method for qubits to characterize the ensemble that works for the Hadamard transformation (QFT for two dimension). In this regard we identify certain incompleteness in the result of Maitra and Parashar (Int. J. Quantum Inform. 4 (2006) 653). Next we characterize the ensemble of qutrits for which QFT is possible. Further, some theoretical results related to higher dimensions are also discussed. Considering the unitary matrix Un related to QFT, the issue boils down to the problem of characterizing matrices that commute with Un.
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Hypothesis Testing and Decision Theoretic Approach for Fault Detection in Wireless Sensor Networks

2012-01-01
Mrinal Nandi , Amiya Nayak , Bimal Roy +1 more
Sensor networks aim at monitoring their surroundings for event detection and object tracking. But due to failure or death of sensors, false signal can be transmitted. In this paper, we … Sensor networks aim at monitoring their surroundings for event detection and object tracking. But due to failure or death of sensors, false signal can be transmitted. In this paper, we consider the problem of fault detection in wireless sensor network (WSN), in particular, addressing both the noise-related measurement error and sensor fault simultaneously in fault detection. We assume that the sensors are placed at the center of a square (or hexagonal) cell in region of interest (ROI) and, if the event occurs, it occurs at a particular cell of the ROI. We propose fault detection schemes that take into account error probabilities into the optimal event detection process. We develop the schemes under the consideration of Neyman-Pearson test and Bayes test.
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The Defect Sequence for Contractive Tuples

2012-01-01
Tirthankar Bhattacharyya , Bata Krishna Das , Santanu Sarkar
We introduce the defect sequence for a contractive tuple of Hilbert space operators and investigate its properties. The defect sequence is a sequence of numbers, called defect dimensions associated with … We introduce the defect sequence for a contractive tuple of Hilbert space operators and investigate its properties. The defect sequence is a sequence of numbers, called defect dimensions associated with a contractive tuple. We show that there are upper bounds for the defect dimensions. The tuples for which these upper bounds are obtained, are called maximal contractive tuples. The upper bounds are different in the non-commutative and in the com- mutative case. We show that the creation operators on the full Fock space and the co ordinate multipliers on the Drury-Arveson space are maximal. We also study pure tuples and see how the defect dimensions play a role in their irreducibility.
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Robust estimation in the errors variables model via weighted likelihood estimating equations

1997-06-01
Ayanendranath Basu , Santanu Sarkar
Coauthor Papers Together
Subhankar Mahapatra 6
Mrinal Nandi 6
Bimal Roy 6
Amit K. Samanta 5
Ayineedi Venkateswarlu 4
Jaydeb Sarkar 3
Tirthankar Bhattacharyya 3
Bata Krishna Das 3
Pantelimon Stănică 3
Tapabrata Roy 3
Anup Dewanji 3
Amiya Nayak 3
Anindya Biswas 2
B. P. Garg 2
Ujjwal Sen 2
George Biswas 2
B. Krishna Das 2
Huaxiong Wang 1
Jun Xu 1
Sabyasachi Dey 1
Florian Luca 1
Ajit S Mali 1
A. Sai Mali 1
Harry Dym 1
Arpita Maitra 1
Sihem Mesnager 1
Lei Hu 1
Kalyan Hansda 1
B. Krishna Das 1
Makhan Maji 1
Constanza Riera 1
Sai Mali Ananthanarayanan 1
Nirupam Kar 1
Takanori Isobe 1
Yanbin Pan 1
Sourav Sen Gupta 1
Ayanendranath Basu 1
Subhamoy Maitra 1

Some Hilbert spaces of analytic functions. I

1963-01-01
Louis de Branges
The spaces of analytic functions studied arise from the problem of finding invariant subspaces for bounded linear transformations in Hilbert space. Two fundamental problems are (1) to determine the invariant … The spaces of analytic functions studied arise from the problem of finding invariant subspaces for bounded linear transformations in Hilbert space. Two fundamental problems are (1) to determine the invariant subspaces of any bounded transformation, (2) to reconstruct a transformation from its invariant subspaces. Satisfactory answers are known for self-adjoint transformations, for unitary transformations, and more generally for normal transformations, but for all other kinds of transformations the known results are less complete. Beurling [2] illustrates how to find invariant subspaces for isometric transformations which are not unitary, and in so doing uncovers an important connection with analytic function theory. Aronszajn and Smith [1] are able to show the existence of invariant subspaces for completely continuous transformations, a result which they ascribe to von Neumann in the Hilbert space case with which we are now concerned. A general study of transformations T with TT* completely continuous is started by Livsic [11] and continued by Brodskil and Livsic [9]. The Livsic approach forms an interesting link between the methods of analytic function theory and those which depend on compactness in linear spaces. When the spectrum of the transformation is a point, Gohberg and Krein [10] give an integral representation of the transformation in terms of invariant subspaces. This construction makes an interesting contrast with the spectral representation of a self-adjoint operator. Relations between the existence of invariant subspaces and the factorization of related operator valued entire functions are obtained by Brodskil' [7; 8]. Our purpose now is to give an exposition of the function theoretic background to this interesting observation. Recall that we have previously [3-6] made a study of Hilbert spaces, whose elements are entire functions and which have these properties: (H1) Whenever F(z) is in the space and has a nonreal zero w, the function F(z)(z w)/(z w) is in the space and has the same norm as F(z). (H2) For every complex number w, the linear functional defined on the space by F(z) -+ F(w) is continuous. (H3) Whenever F(z) is in the space, the function F*(z)=F(i) is in the space and has the same norm as F(z). The axiom (H2) which appears here is conjectured to be a consequence of (Hi). Several apparently weaker conditions, of various degrees of subtlety, are known to imply (H2), and one of these is quoted
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Some Hilbert spaces of analytic functions. III

1965-10-01
Louis de Branges
This paper continues the study of Hilbert spaces of analytic functions which are involved in the structural analysis of nonself-adjoint transformations in Hilbert space. The theory of nonself-adjoint transformations originates … This paper continues the study of Hilbert spaces of analytic functions which are involved in the structural analysis of nonself-adjoint transformations in Hilbert space. The theory of nonself-adjoint transformations originates in the quantum mechanical problems of nuclear scattering theory. Although the ordinary use of the Schriidinger equation leads to self-adjoint transformations, it is customary for physicists to subdivide the nuclear reaction in such a way that nonself-adjoint transformations occur. This observation, made by Livgic [l], caused him to found a general theory of nonself-adjoint transformations. Although LivBic’s theory is extensive and powerful, it has received very little recognition because no one, apparently, can follow his arguments. The difficulties are so great that Dolph and Penzlin [2] have attempted an independent derivation of the main results. The trouble is due not so much to logical gaps as it is to insufficient motivation of the main tool, the characteristic operator function. This quantity arises in the description of fundamental solutions of formally self-adjoint differential equations under variations of the boundary conditions. Yet the characteristic operator function is applied to transformations which have no connection with differential equations. The trouble is that LivGc has missed the meaning of the characteristic operator function, which is to be found in the construction of certain Hilbert spaces of analytic functions [3]. To explain how these spaces originate, we must go back to the basic work of Stone [4] and to our previous work with entire functions [5-81. Stone’s book has two different objectives, apart from a general formulation of concepts. The first is the study of self-adjoint transformations. It is important to note how he goes about the study of these transformations. From his point of view the structure theorem is an abstract analogue of the integral representation of functions which are analytic and have a nonnegative real part in the upper half-plane. (See, for example, Nevanlinna and Nieminen [9] for a more detailed reconstruction of the same argument.) It has

HILBERT SPACES OF ENTIRE FUNCTIONS

1969-11-01
J. L. B. Cooper

Subalgebras of C*-algebras III: Multivariable operator theory

1998-01-01
Whilliam Arveson
A d-contraction is a d-tuple (T1, . . . , Td) of mutually commuting operators acting on a common Hilbert space H such that ‖T1ξ1 + T2ξ2 + · · … A d-contraction is a d-tuple (T1, . . . , Td) of mutually commuting operators acting on a common Hilbert space H such that ‖T1ξ1 + T2ξ2 + · · · + Tdξd‖ ≤ ‖ξ1‖ + ‖ξ2‖ + · · · + ‖ξd‖ for all ξ1, ξ2, . . . , ξd ∈ H. These are the higher dimensional counterparts of contractions. We show that many of the operator-theoretic aspects of function theory in the unit disk generalize to the unit ball Bd in complex d-space, including von Neumann’s inequality and the model theory of contractions. These results depend on properties of the d-shift, a distinguished d-contraction which acts on a new H2 space associated with Bd, and which is the higher dimensional counterpart of the unilateral shift. H 2 and the d-shift are highly unique. Indeed, by exploiting the noncommutative Choquet boundary of the d-shift relative to its generated C∗-algebra we find that there is more uniqueness in dimension d ≥ 2 than there is in dimension one.
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Some Hilbert spaces of analytic functions. II

1965-01-01
Louis de Branges

Isometric dilations for infinite sequences of noncommuting operators

1989-01-01
Gelu Popescu
This paper develops a dilation theory for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace upper T Subscript n Baseline right-brace Subscript n equals 1 Superscript normal infinity"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> … This paper develops a dilation theory for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace upper T Subscript n Baseline right-brace Subscript n equals 1 Superscript normal infinity"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:msubsup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">\{ {T_n}\} _{n = 1}^\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> an infinite sequence of noncommuting operators on a Hilbert space, when the matrix <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper T 1 comma upper T 2 comma ellipsis right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>T</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[{T_1},{T_2}, \ldots ]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a contraction. A Wold decomposition for an infinite sequence of isometries with orthogonal final spaces and a minimal isometric dilation for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace upper T Subscript n Baseline right-brace Subscript n equals 1 Superscript normal infinity"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:msubsup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">\{ {T_n}\} _{n = 1}^\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are obtained. Some theorems on the geometric structure of the space of the minimal isometric dilation and some consequences are given. This results are used to extend the Sz.-Nagy-Foiaş lifting theorem to this noncommutative setting.
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A note on acyclic edge coloring of complete bipartite graphs

2009-02-24
Manu Basavaraju , L. Sunil Chandran

On Coloring Problems

2013-01-01
Weifan Wang , Yuehua Bu

Improved bounds on coloring of graphs

2012-01-07
Sokol Ndreca , Aldo Procacci , Benedetto Scoppola
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The acyclic edge chromatic number of a random d-regular graph is d + 1

2005-05-01
Jaroslav Nešetřil , Nicholas Wormald
We prove the theorem from the title: the acyclic edge chromatic number of a random d-regular graph is asymptotically almost surely equal to d + 1. This improves a result … We prove the theorem from the title: the acyclic edge chromatic number of a random d-regular graph is asymptotically almost surely equal to d + 1. This improves a result of Alon, Sudakov, and Zaks and presents further support for a conjecture that Δ(G) + 2 is the bound for the acyclic edge chromatic number of any graph G. It also represents an analog of a result of Robinson and the second author on edge chromatic number. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 69–74, 2005 AMS classification: 05C15 (primary: graph coloring) 68R05 (secondary: combinatorics).

Acyclic chromatic index and linear arboricity of graphs

1991-01-01
Filip Guldan

ON THE EXTENSIONS AND THE CHARACTERISTIC FUNCTION OF A SYMMETRIC OPERATOR

1968-02-28
A V Štraus
In this paper we describe a class of dissipative and accumulative extensions of a symmetric operator A acting in a Hilbert space H. The concept of the characteristic function of … In this paper we describe a class of dissipative and accumulative extensions of a symmetric operator A acting in a Hilbert space H. The concept of the characteristic function of an operator A is defined in terms of such extensions, and its boundary values are studied. In the general case the domain of definition of the operator A is not assumed to be dense in H.

Defect indices of powers of a contraction

2010-01-13
Hwa-Long Gau , Pei Yuan Wu

Multiplication operators with deficiency indices (p,p) and sampling formulas in reproducing kernel Hilbert spaces of entire vector valued functions

2017-09-20
Harry Dym , Santanu Sarkar

A Family of Perfect Factorisations of Complete Bipartite Graphs

2002-05-01
Darryn Bryant , Barbara Maenhaut , Ian M. Wanless

Boundary Estimation in Sensor Networks: Theory and Methods

2003-01-01
Robert Nowak , Urbashi Mitra

M.G. Krein’s Lectures on Entire Operators

1997-01-01
М. Л. Горбачук , В. И. Горбачук
This book is devoted to the theory of entire Hermitian operators, an important branch of functional analysis harmoniously combining the methods of operator theory and the theory of analytic functions. This book is devoted to the theory of entire Hermitian operators, an important branch of functional analysis harmoniously combining the methods of operator theory and the theory of analytic functions.

Some Hilbert spaces of entire functions

1959-10-01
Louis de Branges
We study here the structure of spaces whose elements are entire functions and which have the following three properties. (H1) Wheneverf(z) is in the and w is a nonreal zero … We study here the structure of spaces whose elements are entire functions and which have the following three properties. (H1) Wheneverf(z) is in the and w is a nonreal zero of f(z), the function f(z) (z -wD)/(z w) is in the and has the same norm. (H2) Whenever w is any nonreal complex number, the linear functional defined on the by f(z) ->f(w), which gives each function in the its value at w, is continuous. (H3) Whenever f(z) is in the space, the function f*(z) -f(z) is in the and has the samne norm. We are able to give a fairly complete discussion of the structure of such spaces. We would like to pose the problem of structure for spaces whose elements are entire functions and which satisfy H1, H2, and H3 with the word Hilbert space replaced bv Banach The reader who is interested in linear transformations in may want to consider a linear transformation, multiplication by z, defined byf(z)-+zf(z) whenever f(z) and zf(z) are in the space. The hypothesis H2 implies that the transformation has a closed graph. The hypothesis Hi implies that the transformation is symmetric and has deficiency index (1, 1). The hypothesis H3 provides a conjugation with respect to which the transformation is real. We will not go into the interpretation now, but the reader who is interested in it should be able to fill in some of the details by reference to Stone [6]. We use the letter 3a to stand for a whose elements are entire functions satisfying HI and H2, and usually H3. If f(z) is in the space, we write the nlorm JIf(t)JI as if t were a dummy variable of integration. If f(z) and g(z) are in the space, the inner product is written (f(t), g(t)). For each nonreal complex member w, K(w, z) as a function of complex z is to be the unique element of the such that for each f(z) in the
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Fundamental aspects of the representation theory of Hermitian operators with deficiency index (𝑚,𝑚)

1971-01-01
Michael Krein

Acyclic edge colorings of graphs

2001-07-01
Noga Alon , Benny Sudakov , Ayal Zaks
A proper coloring of the edges of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G, denoted by … A proper coloring of the edges of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G, denoted by a′(G), is the least number of colors in an acyclic edge coloring of G. For certain graphs G, a′(G) ≥ Δ(G) + 2 where Δ(G) is the maximum degree in G. It is known that a′(G) ≤ 16 Δ(G) for any graph G. We prove that there exists a constant c such that a′(G) ≤ Δ(G) + 2 for any graph G whose girth is at least cΔ(G) log Δ(G), and conjecture that this upper bound for a′(G) holds for all graphs G. We also show that a′(G) ≤ Δ + 2 for almost all Δ-regular graphs. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 157–167, 2001

Intrinsic randomness as a measure of quantum coherence

2015-08-27
Xiao Yuan , Hongyi Zhou , Zhu Cao +1 more
Based on the theory of quantum mechanics, intrinsic randomness in measurement distinguishes quantum effects from classical ones. From the perspective of states, this quantum feature can be summarized as coherence … Based on the theory of quantum mechanics, intrinsic randomness in measurement distinguishes quantum effects from classical ones. From the perspective of states, this quantum feature can be summarized as coherence or superposition in a specific (classical) computational basis. Recently, by regarding coherence as a physical resource, Baumgratz et al.~present a comprehensive framework for coherence measures. Here, we propose a quantum coherence measure essentially using the intrinsic randomness of measurement. The proposed coherence measure provides an answer to the open question in completing the resource theory of coherence. Meanwhile, we show that the coherence distillation process can be treated as quantum extraction, which can be regarded as an equivalent process of classical random number extraction. From this viewpoint, the proposed coherence measure also clarifies the operational aspect of quantum coherence. Finally, our results indicate a strong similarity between two types of quantumness --- coherence and entanglement.
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Quantification of the differences between quenched and annealed averaging for RNA secondary structures

2005-12-12
Tsunglin Liu , Ralf Bundschuh
The analytical study of disordered system is usually difficult due to the necessity to perform a quenched average over the disorder. Thus, one may resort to the easier annealed ensemble … The analytical study of disordered system is usually difficult due to the necessity to perform a quenched average over the disorder. Thus, one may resort to the easier annealed ensemble as an approximation to the quenched system. In the study of RNA secondary structures, we explicitly quantify the deviation of this approximation from the quenched ensemble by looking at the correlations between neighboring bases. This quantified deviation then allows us to propose a constrained annealed ensemble which predicts physical quantities much closer to the results of the quenched ensemble without becoming technically intractable.
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Acyclic Edge Colouring of Outerplanar Graphs

2007-06-25
Rahul Muthu , N. Narayanan , C. R. Subramanian

Weighted Norm Inequalities for Some Singular Integral Operators

1997-08-01
Takahiko Nakazi , Takanori Yamamoto

A 2 rebit gate universal for quantum computing

2002-01-01
Terry Rudolph , Lov K. Grover
We show, within the circuit model, how any quantum computation can be efficiently performed using states with only real amplitudes (a result known within the Quantum Turing Machine model). This … We show, within the circuit model, how any quantum computation can be efficiently performed using states with only real amplitudes (a result known within the Quantum Turing Machine model). This allows us to identify a 2-qubit (in fact 2-rebit) gate which is universal for quantum computing, although it cannot be used to perform arbitrary unitary transformations.

Transversals of Latin squares and their generalizations

1975-08-01
Sherman K. Stein
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Full distribution of work done on a quantum system for arbitrary initial states

2015-10-23
Paolo Solinas , Simone Gasparinetti
We propose an approach to define and measure the statistics of work, internal energy and dissipated heat in a driven quantum system. In our framework the presence of a physical … We propose an approach to define and measure the statistics of work, internal energy and dissipated heat in a driven quantum system. In our framework the presence of a physical detector arises naturally and work and its statistics can be investigated in the most general case. In particular, we show that the quantum coherence of the initial state can lead to measurable effects on the moments of the work done on the system. At the same time, we recover the known results if the initial state is a statistical mixture of energy eigenstates. Our method can also be applied to measure the dissipated heat in an open quantum system. By sequentially coupling the system to a detector, we can track the energy dissipated in the environment while accessing only the system degrees of freedom.
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Norms and Essential Norms of the Singular Integral Operator with Cauchy Kernel on Weighted Lebesgue Spaces

2010-03-18
Takahiko Nakazi , Takanori Yamamoto

Operational Resource Theory of Coherence

2016-03-24
Andreas Winter , Dong Yang
We establish an operational theory of coherence (or of superposition) in quantum systems, by focusing on the optimal rate of performance of certain tasks. Namely, we introduce the two basic … We establish an operational theory of coherence (or of superposition) in quantum systems, by focusing on the optimal rate of performance of certain tasks. Namely, we introduce the two basic concepts-"coherence distillation" and "coherence cost"-in the processing quantum states under so-called incoherent operations [Baumgratz, Cramer, and Plenio, Phys. Rev. Lett. 113, 140401 (2014)]. We, then, show that, in the asymptotic limit of many copies of a state, both are given by simple single-letter formulas: the distillable coherence is given by the relative entropy of coherence (in other words, we give the relative entropy of coherence its operational interpretation), and the coherence cost by the coherence of formation, which is an optimization over convex decompositions of the state. An immediate corollary is that there exists no bound coherent state in the sense that one would need to consume coherence to create the state, but no coherence could be distilled from it. Further, we demonstrate that the coherence theory is generically an irreversible theory by a simple criterion that completely characterizes all reversible states.
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NORMS OF SOME SINGULAR INTEGRAL OPERATORS AND THEIR INVERSE OPERATORS

1997-09-01
Takahiko Nakazi , Takanori Yamamoto
Let and be bounded measurable functions on the unit circle T. Then the singular integral operator S, is defined by S, f = P +f + P f, (f 2 … Let and be bounded measurable functions on the unit circle T. Then the singular integral operator S, is defined by S, f = P +f + P f, (f 2 L 2 (T)) where P+ is an analytic projection and P is a co-analytic projection. In this paper, the norms of S, and its inverse operator on the Hilbert space L 2 (T) are calculated in general, using , and ¯ + H 1 . Moreover, the relations between these and the norms of Hankel operators are established. As an application, in some special case in which and are nonconstant functions, the norm of S, is calculated in a completely explicit form. If and are constant functions, then it is well known that the norm of S, on L 2 (T) is equal to max{| |,| |}. If and are nonzero constant functions, then it is also known that S, on L 2 (T) has an inverse operator S 1, 1 whose norm is equal to max ˘ | | 1 ,| | 1 ¯ .

Understanding quantum entanglement: Qubits, rebits and the quaternionic approach

2003-05-01
Josep Amengual i Batle , A. Plastino , M. Casas +1 more
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Measuring Quantum Coherence with Entanglement

2015-07-08
Alexander Streltsov , Uttam Singh , Himadri Shekhar Dhar +2 more
Quantum coherence is an essential ingredient in quantum information processing and plays a central role in emergent fields such as nanoscale thermodynamics and quantum biology. However, our understanding and quantitative … Quantum coherence is an essential ingredient in quantum information processing and plays a central role in emergent fields such as nanoscale thermodynamics and quantum biology. However, our understanding and quantitative characterization of coherence as an operational resource are still very limited. Here we show that any degree of coherence with respect to some reference basis can be converted to entanglement via incoherent operations. This finding allows us to define a novel general class of measures of coherence for a quantum system of arbitrary dimension, in terms of the maximum bipartite entanglement that can be generated via incoherent operations applied to the system and an incoherent ancilla. The resulting measures are proven to be valid coherence monotones satisfying all the requirements dictated by the resource theory of quantum coherence. We demonstrate the usefulness of our approach by proving that the fidelity-based geometric measure of coherence is a full convex coherence monotone, and deriving a closed formula for it on arbitrary single-qubit states. Our work provides a clear quantitative and operational connection between coherence and entanglement, two landmark manifestations of quantum theory and both key enablers for quantum technologies.
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Operator-valued inner functions analytic on the closed disc

1972-04-01
Stephen L. Campbell
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Invariant Subspaces and Nevanlinna–Pick Kernels

2000-12-01
Scott McCullough , Tavan T. Trent

Description of quantum coherence in thermodynamic processes requires constraints beyond free energy

2015-03-10
Matteo Lostaglio , David Jennings , Terry Rudolph
Recent studies have developed fundamental limitations on nanoscale thermodynamics, in terms of a set of independent free energy relations. Here we show that free energy relations cannot properly describe quantum … Recent studies have developed fundamental limitations on nanoscale thermodynamics, in terms of a set of independent free energy relations. Here we show that free energy relations cannot properly describe quantum coherence in thermodynamic processes. By casting time-asymmetry as a quantifiable, fundamental resource of a quantum state we arrive at an additional, independent set of thermodynamic constraints that naturally extend the existing ones. These asymmetry relations reveal that the traditional Szilard engine argument does not extend automatically to quantum coherences, but instead only relational coherences in a multipartite scenario can contribute to thermodynamic work. We find that coherence transformations are always irreversible. Our results also reveal additional structural parallels between thermodynamics and the theory of entanglement.
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On acyclic edge-coloring of the complete bipartite graphs <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> for odd prime <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg…

2015-08-13
Ayineedi Venkateswarlu , Santanu Sarkar

Quantum mechanics on a real Hilbert space

1999-01-01
Jan Myrheim
The complex Hilbert space of standard quantum mechanics may be treated as a real Hilbert space. The pure states of the complex theory become mixed states in the real formulation. … The complex Hilbert space of standard quantum mechanics may be treated as a real Hilbert space. The pure states of the complex theory become mixed states in the real formulation. It is then possible to generalize standard quantum mechanics, keeping the same set of physical states, but admitting more general observables. The standard time reversal operator involves complex conjugation, in this sense it goes beyond the complex theory and may serve as an example to motivate the generalization. Another example is unconventional canonical quantization such that the harmonic oscillator of angular frequency $ω$ has any given finite or infinite set of discrete energy eigenvalues, limited below by $\hbarω/2$.

Some singular integral operators and Helson-Szegö measures

1990-02-01
Takahiko Nakazi , Takanori Yamamoto

Hilbert spaces of analytic functions, inverse scattering and operator models.II

1985-03-01
Daniel Alpay , Harry Dym

Some Hilbert spaces of entire functions

1961-01-01
Louis de Branges
We study here the structure of spaces whose elements are entire functions and which have the following three properties. (H1) Wheneverf(z) is in the and w is a nonreal zero … We study here the structure of spaces whose elements are entire functions and which have the following three properties. (H1) Wheneverf(z) is in the and w is a nonreal zero of f(z), the function f(z) (z -wD)/(z w) is in the and has the same norm. (H2) Whenever w is any nonreal complex number, the linear functional defined on the by f(z) ->f(w), which gives each function in the its value at w, is continuous. (H3) Whenever f(z) is in the space, the function f*(z) -f(z) is in the and has the samne norm. We are able to give a fairly complete discussion of the structure of such spaces. We would like to pose the problem of structure for spaces whose elements are entire functions and which satisfy H1, H2, and H3 with the word Hilbert space replaced bv Banach The reader who is interested in linear transformations in may want to consider a linear transformation, multiplication by z, defined byf(z)-+zf(z) whenever f(z) and zf(z) are in the space. The hypothesis H2 implies that the transformation has a closed graph. The hypothesis Hi implies that the transformation is symmetric and has deficiency index (1, 1). The hypothesis H3 provides a conjugation with respect to which the transformation is real. We will not go into the interpretation now, but the reader who is interested in it should be able to fill in some of the details by reference to Stone [6]. We use the letter 3a to stand for a whose elements are entire functions satisfying HI and H2, and usually H3. If f(z) is in the space, we write the nlorm JIf(t)JI as if t were a dummy variable of integration. If f(z) and g(z) are in the space, the inner product is written (f(t), g(t)). For each nonreal complex member w, K(w, z) as a function of complex z is to be the unique element of the such that for each f(z) in the
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Some Hilbert spaces of entire functions

1960-01-01
Louis de Branges
A Hilbert space, whose elements are entire functions, is of especial interest if it satisfies three axioms.(HI) Whenever F(z) is in the Hilbert space and w is a nonreal zero … A Hilbert space, whose elements are entire functions, is of especial interest if it satisfies three axioms.(HI) Whenever F(z) is in the Hilbert space and w is a nonreal zero of F(z), the function F(z)(z -w)/(z -w) is in the Hilbert space and has the same norm as F(z).(H2) Whenever wis a nonreal complex number, the linear functional defined on the Hilbert space by F(z)->F(w), which gives each function in the Hilbert space its value at w, is continuous.(H3) Whenever F(z) is in the Hilbert space, F*(z) = F(z) is in the Hilbert space and has the same norm as F(z).If E(z) is an entire function such that \E(z)\ <\E(z)\ for y>0, the set 3C(£) of entire functions F(z) such that Received by the editors October 9, 1959.for y>0, where Ec(z) = Cc(z)-iSc(z) and C,(z) = Ca(z)Ci(z) -Sa(z)So(z), Sc(z) = Ca(z)S1(z) + Sa(z)C0(z) and A (z) is defined and analytic for y>0 and \ A (z) | ^ 1.The discussion of Sturm-Liouville differential equations by Stone [ll] and of the Hamburger moment problem by Shohat and Tamarkin [10] and Stone [ll] leads to the situation of Theorem VII.Theorem X.A. Let Co, S0, G, Si be real numbers such that C0Ci+505i = 1.Let a(t), b(t), c(t) be locally integrable real-valued functions of t^O such that a(t) 2:0, c(t) StO, and b2(t) ^a(t)c(t).For each complex z, there exist unique absolutely continuous functions C0(t, z), So(t, z), Ci(t, z), Si(t, z) of t^O such that C0(0, z) = Co, S0(0, z) = So, Ci(0, z) = Ci, Si(0, z) = Si and Co' (t, z) = -za(t)S0(t, z) -zb(t)C0(t, z),a.e. for t^O.For each t^O, Co(t, z), So(t, z), Ci(t, z), Si(t, z) are entire functions which are real for real z and satisfy ( 5) and ( 6).If Eo(t, z) = Co(t, z)-iSo(t, z)and Ei(t, z) = Ci(t, z)-iSi(t, z), then \E0(t, z)\ ^\Eo(t, z)\ and \Ei(t, z)\ ^\Ei(t, z)\ for y>0.Theorem X.B.Let Co, So, Ci, Si be real numbers such that CoCi-\-SoSi = 1.Let an, bn, cn be real numbers defined for n = 0, 1, 2, • ■ • , such that a"S:0, Cns^O, and b2n = ancn.Let Co(n, z), So(n, z), Ci(n, z), Si(n, z) be the polynomials in z defined inductively by C0(n, z) = Co -z E»<, a"So(m, z) -z 22"<n bmC0(m, z), So(n, Z) = So -Z X)m<n 6»So(w, z) + Z Xlm<n CmCo(m, z), ( 14) Ci(n, z) = Ci -z ^m<" cJS^m, z) + z 2Zm<" bmCi(m, z), Sx(n, z) = Si -z 2Zm<" bmSi(m, z) + z ^m<" ad(m, z).Then for each n, Co(n, z), S0(n, z), Ci(n, z), Si(n, z) are entire functions which are real for real z and satisfy (5) and ( 6).If E0(n, z) = Co(n, z)-iS0(n, z) and Ei(n, z) = Ci(n, z)-iSi(n, z), then \Eo(n, z)\ ^|£o(w, z)\ and \Ei(n, z)\ ^ \Ei(n, z)\ for y>0.Our proofs use the Poisson representation of a function positive and harmonic in a half plane, discussed by Loomis and Widder [9].We use a lemma on linear fractional transformations.
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The extraction of work from quantum coherence

2016-02-18
Kamil Korzekwa , Matteo Lostaglio , Jonathan Oppenheim +1 more
The interplay between quantum-mechanical properties, such as coherence, and classical notions, such as energy, is a subtle topic at the forefront of quantum thermodynamics. The traditional Carnot argument limits the … The interplay between quantum-mechanical properties, such as coherence, and classical notions, such as energy, is a subtle topic at the forefront of quantum thermodynamics. The traditional Carnot argument limits the conversion of heat to work; here we critically assess the problem of converting coherence to work. Through a careful account of all resources involved in the thermodynamic transformations within a fully quantum-mechanical treatment, we show that there exist thermal machines extracting work from coherence arbitrarily well. Such machines only need to act on individual copies of a state and can be reused. On the other hand, we show that for any thermal machine with finite resources not all the coherence of a state can be extracted as work. However, even bounded thermal machines can be reused infinitely many times in the process of work extraction from coherence.
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Bayesian model averaging: a tutorial (with comments by M. Clyde, David Draper and E. I. George, and a rejoinder by the authors

1999-11-01
Jennifer A. Hoeting , David Madigan , Adrian E. Raftery +1 more
Standard statistical practice ignores model uncertainty. Data analysts typically select a model from some class of models and then proceed as if the selected model had generated the data. This … Standard statistical practice ignores model uncertainty. Data analysts typically select a model from some class of models and then proceed as if the selected model had generated the data. This approach ignores the uncertainty in model selection, leading to over-confident inferences and decisions that are more risky than one thinks they are. Bayesian model averaging (BMA)provides a coherent mechanism for accounting for this model uncertainty. Several methods for implementing BMA have recently emerged. We discuss these methods and present a number of examples.In these examples, BMA provides improved out-of-sample predictive performance. We also provide a catalogue of currently available BMA software.
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Hypothesis testing and decision theoretic approach for fault detection in wireless sensor networks

2014-04-24
Mrinal Nandi , Amiya Nayak , Bimal Roy +1 more
Sensor networks aim at monitoring their surroundings for event detection and object tracking. In practice, due to failure or death of sensors, false signal can be transmitted. In this article, … Sensor networks aim at monitoring their surroundings for event detection and object tracking. In practice, due to failure or death of sensors, false signal can be transmitted. In this article, we consider the problem of fault detection in wireless sensor networks, in particular, addressing both the noise-related measurement error and sensor fault simultaneously in fault detection. We assume that the sensors are placed at the centre of a square (or hexagonal) cell in the region of interest (ROI) and, if the event occurs, it occurs at a particular cell of the ROI. We propose fault detection schemes that consider error probabilities in the optimal event detection process. We develop the schemes under the consideration of Neyman–Pearson hypothesis test and Bayes test.
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Acyclic coloring of graphs

1991-09-01
Noga Alon , Colin McDiarmid , Bruce Reed
Abstract A vertex coloring of a graph G is called acyclic if no two adjacent vertices have the same color and there is no two‐colored cycle in G. The acyclic … Abstract A vertex coloring of a graph G is called acyclic if no two adjacent vertices have the same color and there is no two‐colored cycle in G. The acyclic chromatic number of G, denoted by A(G) , is the least number of colors in an acyclic coloring of G. We show that if G has maximum degree d , then A(G) = 0(d 4/3 ) as d → ∞. This settles a problem of Erdös who conjectured, in 1976, that A(G) = o(d 2 ) as d → ∞. We also show that there are graphs G with maximum degree d for which A(G) = Ω(d 4/3 /(log d) 1/3 ); and that the edges of any graph with maximum degree d can be colored by 0(d) colors so that no two adjacent edges have the same color and there is no two‐colored cycle. All the proofs rely heavily on probabilistic arguments.

Essential norms of some singular integral operators

1999-12-01
Takahiko Nakazi
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Work extraction and thermodynamics for individual quantum systems

2014-06-27
Paul Skrzypczyk , Anthony Short , Sandu Popescu
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Nanoscale Heat Engine Beyond the Carnot Limit

2014-01-22
J. Roßnagel , Obinna Abah , F. Schmidt‐Kaler +2 more
We consider a quantum Otto cycle for a time-dependent harmonic oscillator coupled to a squeezed thermal reservoir. We show that the efficiency at maximum power increases with the degree of … We consider a quantum Otto cycle for a time-dependent harmonic oscillator coupled to a squeezed thermal reservoir. We show that the efficiency at maximum power increases with the degree of squeezing, surpassing the standard Carnot limit and approaching unity exponentially for large squeezing parameters. We further propose an experimental scheme to implement such a model system by using a single trapped ion in a linear Paul trap with special geometry. Our analytical investigations are supported by Monte Carlo simulations that demonstrate the feasibility of our proposal. For realistic trap parameters, an increase of the efficiency at maximum power of up to a factor of 4 is reached, largely exceeding the Carnot bound.
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Generating random density matrices

2011-06-01
Karol Życzkowski , K. A. Penson , Ion Nechita +1 more
We study various methods to generate ensembles of random density matrices of a fixed size N, obtained by partial trace of pure states on composite systems. Structured ensembles of random … We study various methods to generate ensembles of random density matrices of a fixed size N, obtained by partial trace of pure states on composite systems. Structured ensembles of random pure states, invariant with respect to local unitary transformations are introduced. To analyze statistical properties of quantum entanglement in bi-partite systems we analyze the distribution of Schmidt coefficients of random pure states. Such a distribution is derived in the case of a superposition of k random maximally entangled states. For another ensemble, obtained by performing selective measurements in a maximally entangled basis on a multi--partite system, we show that this distribution is given by the Fuss-Catalan law and find the average entanglement entropy. A more general class of structured ensembles proposed, containing also the case of Bures, forms an extension of the standard ensemble of structureless random pure states, described asymptotically, as N \to \infty, by the Marchenko-Pastur distribution.
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On Factorizations of Analytic Operator-Valued Functions and Eigenvalue Multiplicity Questions

2014-12-12
Fritz Gesztesy , Helge Holden , Roger Nichols
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