Beneficial to both beginning students and researchers, Asymptotic Analysis and Perturbation Theory immediately introduces asymptotic notation and then applies this tool to familiar problems, including limits, inverse functions, and integrals. …
Beneficial to both beginning students and researchers, Asymptotic Analysis and Perturbation Theory immediately introduces asymptotic notation and then applies this tool to familiar problems, including limits, inverse functions, and integrals. Suitable for those who have completed the standard calculus sequence, the book assumes no prior knowledge o
This note discusses the analytic behavior of the general polylogarithm Li s (z) at the singularity z = 1. We consider both integer and complex values for s. Using these …
This note discusses the analytic behavior of the general polylogarithm Li s (z) at the singularity z = 1. We consider both integer and complex values for s. Using these results, we derive yet another way of extending the Riemann zeta function to the entire plane.
In this paper, we study the fractional iterates of the exponential function. This is an unresolved problem, not due to a lack of a known solution, but because there are …
In this paper, we study the fractional iterates of the exponential function. This is an unresolved problem, not due to a lack of a known solution, but because there are an innite number of solutions, and there is no agreement as to which solution is "best." We will approach the problem by rst solving Abel's functional equation <TEX>${\alpha}(e^x)={\alpha}(x)+1$</TEX> by perturbing the exponential function so as to produce a real xed point, allowing a unique holomorphic solution. We then use this solution to nd a solution to the unperturbed problem. However, this solution will depend on the way we rst perturbed the exponential function. Thus, we then strive to remove the dependence of the perturbed function. Finally, we produce a solution that is in a sense more natural than other solutions.
We will study the best way to reveal a hidden perfect matching in a balanced bipartite graph by eliminating edges, one by one, in the hope that the eliminated edge …
We will study the best way to reveal a hidden perfect matching in a balanced bipartite graph by eliminating edges, one by one, in the hope that the eliminated edge is not part of the mystery perfect matching. We will look for the strategy that maximizes the odds of finding the perfect matching without revealing a fixed number of the edges in that perfect matching. For a complete bipartite graph, this is equivalent to finding a mystery permutation via negative guesses with only a fixed number of incorrect negative guesses.
In 1848, chess master Max Bezzel proposed a problem of placing 8 queens on a standard 8 × 8 chessboard so that no two queens can attack each other [3]. …
In 1848, chess master Max Bezzel proposed a problem of placing 8 queens on a standard 8 × 8 chessboard so that no two queens can attack each other [3]. Soon after this, Carl Gauss extended the prob...
When matching n pairs of socks, drawn randomly one at a time, there is a question of how big the pile of unmatched socks is expected to get. Each permutation …
When matching n pairs of socks, drawn randomly one at a time, there is a question of how big the pile of unmatched socks is expected to get. Each permutation of socks drawn has a corresponding Dyck path, with the total number of Dyck paths equaling the n-th Catalan number. However, some discussions failed to take into account that not every Dyck path is equally likely in the process of sorting socks. In this paper we will take the probabilities of the Dyck paths into account, and find a method of finding the expected maximum size of the unmatched sock pile. We also find the first two terms of the asymptotic series for this maximum, and give a conjecture on the third term.
The new edition of Abstract Algebra: An Interactive Approach presents a hands-on and traditional approach to learning groups, rings, and fields. It then goes further to offer optional technology use …
The new edition of Abstract Algebra: An Interactive Approach presents a hands-on and traditional approach to learning groups, rings, and fields. It then goes further to offer optional technology use to create opportunities for interactive learning and computer use. This new edition offers a more traditional approach offering additional topics to the primary syllabus placed after primary topics are covered. This creates a more natural flow to the order of the subjects presented. This edition is transformed by historical notes and better explanations of why topics are covered. This innovative textbook shows how students can better grasp difficult algebraic concepts through the use of computer programs. It encourages students to experiment with various applications of abstract algebra, thereby obtaining a real-world perspective of this area. Each chapter includes, corresponding Sage notebooks, traditional exercises, and several interactive computer problems that utilize Sage and Mathematica® to explore groups, rings, fields and additional topics. This text does not sacrifice mathematical rigor. It covers classical proofs, such as Abel's theorem, as well as many topics not found in most standard introductory texts. The author explores semi-direct products, polycyclic groups, Rubik's Cube®-like puzzles, and Wedderburn's theorem. The author also incorporates problem sequences that allow students to delve into interesting topics, including Fermat's two square theorem.
When matching n pairs of socks, drawn randomly one at a time, there is a question of how big the pile of unmatched socks is expected to get. Each permutation …
When matching n pairs of socks, drawn randomly one at a time, there is a question of how big the pile of unmatched socks is expected to get. Each permutation of socks drawn has a corresponding Dyck path, with the total number of Dyck paths equaling the n-th Catalan number. However, some discussions failed to take into account that not every Dyck path is equally likely in the process of sorting socks. In this paper we will take the probabilities of the Dyck paths into account, and find a method of finding the expected maximum size of the unmatched sock pile. We also find the first two terms of the asymptotic series for this maximum, and give a conjecture on the third term.
In 1848, chess master Max Bezzel proposed a problem of placing 8 queens on a standard 8 × 8 chessboard so that no two queens can attack each other [3]. …
In 1848, chess master Max Bezzel proposed a problem of placing 8 queens on a standard 8 × 8 chessboard so that no two queens can attack each other [3]. Soon after this, Carl Gauss extended the prob...
The new edition of Abstract Algebra: An Interactive Approach presents a hands-on and traditional approach to learning groups, rings, and fields. It then goes further to offer optional technology use …
The new edition of Abstract Algebra: An Interactive Approach presents a hands-on and traditional approach to learning groups, rings, and fields. It then goes further to offer optional technology use to create opportunities for interactive learning and computer use. This new edition offers a more traditional approach offering additional topics to the primary syllabus placed after primary topics are covered. This creates a more natural flow to the order of the subjects presented. This edition is transformed by historical notes and better explanations of why topics are covered. This innovative textbook shows how students can better grasp difficult algebraic concepts through the use of computer programs. It encourages students to experiment with various applications of abstract algebra, thereby obtaining a real-world perspective of this area. Each chapter includes, corresponding Sage notebooks, traditional exercises, and several interactive computer problems that utilize Sage and Mathematica® to explore groups, rings, fields and additional topics. This text does not sacrifice mathematical rigor. It covers classical proofs, such as Abel's theorem, as well as many topics not found in most standard introductory texts. The author explores semi-direct products, polycyclic groups, Rubik's Cube®-like puzzles, and Wedderburn's theorem. The author also incorporates problem sequences that allow students to delve into interesting topics, including Fermat's two square theorem.
In this paper, we study the fractional iterates of the exponential function. This is an unresolved problem, not due to a lack of a known solution, but because there are …
In this paper, we study the fractional iterates of the exponential function. This is an unresolved problem, not due to a lack of a known solution, but because there are an innite number of solutions, and there is no agreement as to which solution is "best." We will approach the problem by rst solving Abel's functional equation <TEX>${\alpha}(e^x)={\alpha}(x)+1$</TEX> by perturbing the exponential function so as to produce a real xed point, allowing a unique holomorphic solution. We then use this solution to nd a solution to the unperturbed problem. However, this solution will depend on the way we rst perturbed the exponential function. Thus, we then strive to remove the dependence of the perturbed function. Finally, we produce a solution that is in a sense more natural than other solutions.
Beneficial to both beginning students and researchers, Asymptotic Analysis and Perturbation Theory immediately introduces asymptotic notation and then applies this tool to familiar problems, including limits, inverse functions, and integrals. …
Beneficial to both beginning students and researchers, Asymptotic Analysis and Perturbation Theory immediately introduces asymptotic notation and then applies this tool to familiar problems, including limits, inverse functions, and integrals. Suitable for those who have completed the standard calculus sequence, the book assumes no prior knowledge o
We will study the best way to reveal a hidden perfect matching in a balanced bipartite graph by eliminating edges, one by one, in the hope that the eliminated edge …
We will study the best way to reveal a hidden perfect matching in a balanced bipartite graph by eliminating edges, one by one, in the hope that the eliminated edge is not part of the mystery perfect matching. We will look for the strategy that maximizes the odds of finding the perfect matching without revealing a fixed number of the edges in that perfect matching. For a complete bipartite graph, this is equivalent to finding a mystery permutation via negative guesses with only a fixed number of incorrect negative guesses.
This note discusses the analytic behavior of the general polylogarithm Li s (z) at the singularity z = 1. We consider both integer and complex values for s. Using these …
This note discusses the analytic behavior of the general polylogarithm Li s (z) at the singularity z = 1. We consider both integer and complex values for s. Using these results, we derive yet another way of extending the Riemann zeta function to the entire plane.
Tetration $F$ as the analytic solution of equations $F(z-1)=\ln (F(z))$, $F(0)=1$ is considered. The representation is suggested through the integral equation for values of $F$ at the imaginary axis. Numerical …
Tetration $F$ as the analytic solution of equations $F(z-1)=\ln (F(z))$, $F(0)=1$ is considered. The representation is suggested through the integral equation for values of $F$ at the imaginary axis. Numerical analysis of this equation is described. The straightforward iteration converges within tens of cycles; with double precision arithmetics, the residual of order of 1.e-14 is achieved. The numerical solution for $F$ remains finite at the imaginary axis, approaching fixed points $L$, $L^{*}$ of logarithm ($L=\ln L$). Robustness of the convergence and smallness of the residual indicate the existence of unique tetration $F(z)$, that grows along the real axis and approaches $L$ along the imaginary axis, being analytic in the whole complex $z$-plane except for singularities at integer the $z<-1$ and the cut at $z<-2$. Application of the same method for other cases of the Abel equation is discussed.
This paper contains an attempt to develop for discrete semigroups of infinite order matrices with nonnegative elements a simple theory analogous to the Perron-Frobenius theory of finite matrices.It is assumed …
This paper contains an attempt to develop for discrete semigroups of infinite order matrices with nonnegative elements a simple theory analogous to the Perron-Frobenius theory of finite matrices.It is assumed throughout that the matrix is irreducible, but some consideration is given to the periodic case.The main topics considered are (i) nonnegative solutions to the inequalities ^xj (r>0) (ii) nonnegative solutions to the inequalities r 2 χ kt k j ^ Xj (r > 0) (iii) the limiting behaviour of sums Pj(n; r) = âs n -> oo ?where {Uk} is arbitrary nonnegative vector.An extensive use is made of generating function techniques.
This article describes an efficient and robust algorithm and implementation for the evaluation of the Wright ω function in IEEE double precision arithmetic over the complex plane.
This article describes an efficient and robust algorithm and implementation for the evaluation of the Wright ω function in IEEE double precision arithmetic over the complex plane.
A slender beam has two spatially nonhomogeneous damping terms. The first one acts opposite to the bending moment time derivative and is sometimes called structural damping, while the second acts …
A slender beam has two spatially nonhomogeneous damping terms. The first one acts opposite to the bending moment time derivative and is sometimes called structural damping, while the second acts opposite to the velocity and is called viscous damping. When these damping coefficients are constant, it is known that structural damping causes a strong attenuation rate that is frequency-proportional, whereas viscous damping causes a constant attenuation rate for all frequencies. In this paper, using the method of Birkhof [Trans. Amer. Math. Soc., 9 (1908), pp. 219–231 ], [Trans. Amer. Math. Soc., 9 (1908), pp. 373–395], and Birkhofl and Langer [Proc. Amer. Acad. Arts Sci., (2) 58 (1923), pp. 51–128] explicit asymptotic expressions for the eigenfrequencies of the nonhomogeneous damping problem are derived. It is shown that the asymptotic patterns of the eigenspectrum remain similar to the constant coefficients case. The viscous damping effect is also shown to cause a constant shift to both the attenuation rates and the frequencies; thus it is overwhelmed by the structural damping effect. Because experimentally it has been observed that all eigenfrequencies of light beams essentially lie within the asymptotic regime, the asymptotic formulas derived herein should be useful in determining the pole assignment for feedback stabilization.
We introduce the concept of regular super-functions at a fixed point. It is derived from the concept of regular iteration. A super-function F of h is a solution of F(z+1)=h(F(z)). …
We introduce the concept of regular super-functions at a fixed point. It is derived from the concept of regular iteration. A super-function F of h is a solution of F(z+1)=h(F(z)). We provide a condition for F being entire, we also give two uniqueness criteria for regular super-functions. In the particular case h(x)=b$\hat {\phantom {x}}$x we call F super-exponential. h has two real fixed points for b between 1 and e$\hat {\phantom {x}}$(1/e). Exemplary we choose the base b=sqrt(2) and portray the four classes of real regular super-exponentials in the complex plane. There are two at fixed point 2 and two at fixed point 4. Each class is given by the translations along the x-axis of a suitable representative. Both super-exponentials at fixed point 4âone strictly increasing and one strictly decreasingâare entire. Both super-exponentials at fixed point 2âone strictly increasing and one strictly decreasingâare holomorphic on a right half-plane. All four super-exponentials are periodic along the imaginary axis. Only the strictly increasing super-exponential at 2 can satisfy F(0)=1 and can hence be called tetrational. We develop numerical algorithms for the precise evaluation of these functions and their inverses in the complex plane. We graph the two corresponding different half-iterates of h(z)=sqrt(2)$\hat {\phantom {x}}$z. An apparent symmetry of the tetrational to base sqrt(2) disproved.
I solved this problem originally for the case of p = P.Dr. Gronwall sugcested to me the possibility of generalization.
I solved this problem originally for the case of p = P.Dr. Gronwall sugcested to me the possibility of generalization.
The exact formula for the Hausdorff dimension of the graph of a continuous self-affine function is obtained. The Hausdorff dimension of some class of Borel probability measures is computed. The …
The exact formula for the Hausdorff dimension of the graph of a continuous self-affine function is obtained. The Hausdorff dimension of some class of Borel probability measures is computed. The Hausdorff measures corresponding to the functions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi Subscript c Baseline left-parenthesis t right-parenthesis equals t Superscript upper H upper D left-parenthesis graph left-parenthesis f right-parenthesis right-parenthesis Baseline exp left-parenthesis c StartRoot log 1 slash t log log log 1 slash t EndRoot"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>φ<!-- φ --></mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>H</mml:mi> <mml:mi>D</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>graph</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:mi>exp</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>c</mml:mi> <mml:msqrt> <mml:mi>log</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>t</mml:mi> <mml:mi>log</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mi>log</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mi>log</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>t</mml:mi> </mml:msqrt> </mml:mrow> <mml:annotation encoding="application/x-tex">{\varphi _c}(t) = {t^{HD({\text {graph}}(f))}}\exp (c\sqrt {\log 1/t\log \log \log 1/t}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are studied.
The two regular super-exponentials to base exp(1/e) are constructed. An efficient algorithm for the evaluation of these super-exponentials and their inverse functions is suggested and compared to the already published …
The two regular super-exponentials to base exp(1/e) are constructed. An efficient algorithm for the evaluation of these super-exponentials and their inverse functions is suggested and compared to the already published results.
Abstract If T is a linear transformation on ℝ n with singular values α 1 ≥ α 2 ≥ … ≥ α n , the singular value function ø s …
Abstract If T is a linear transformation on ℝ n with singular values α 1 ≥ α 2 ≥ … ≥ α n , the singular value function ø s is defined by where m is the smallest integer greater than or equal to s . Let T 1 , …, T k be contractive linear transformations on ℝ n . Let where the sum is over all finite sequences (i 1 , …, i r ) with 1 ≤ i j ≤ k. Then for almost all (a 1 , …, a k ) ∈ ℝ nk , the unique non-empty compact set F satisfying has Hausdorff dimension min{ d, n }. Moreover the ‘box counting’ dimension of F is almost surely equal to this number.
Abstract The determination of the figure of the earth is considered as a local free boundary value problem of potential theory. In the C 1,ε ‐topology local existence is proved …
Abstract The determination of the figure of the earth is considered as a local free boundary value problem of potential theory. In the C 1,ε ‐topology local existence is proved by means of Legendre transforms. This method also provides an elementary argument for the regularity of the solutions.
We study the maximum of a Brownian motion with a parabolic drift; this is a random variable that often occurs as a limit of the maximum of discrete processes whose …
We study the maximum of a Brownian motion with a parabolic drift; this is a random variable that often occurs as a limit of the maximum of discrete processes whose expectations have a maximum at an interior point. We give new series expansions and integral formulas for the distribution and the first two moments, together with numerical values to high precision.
This paper develops some properties of matrices which have nonnegative elements and act as bounded operators on one of the sequence spaces l p or l p (μ), where μ …
This paper develops some properties of matrices which have nonnegative elements and act as bounded operators on one of the sequence spaces l p or l p (μ), where μ is a measure on the integers.Its chief aim is to relate the operator properties of such matrices to the matrix properties (the value of the convergence parameter R, UN-recurrence and i?-positivity) described in detail in Part I.The relationships between the convergence norm (equal to 1/22), the spectral radius, and the operator norm are discussed, and conditions are set up for the convergence norm to lie in the point spectrum.It is shown that the correspondence between matrix and operator properties depends very much on the choice of the underlying space.Other topics considered are the problem of characterizing the operator norm in matrix terms, and a general theorem on the structure of a positive contraction operator on l p .
Beneficial to both beginning students and researchers, Asymptotic Analysis and Perturbation Theory immediately introduces asymptotic notation and then applies this tool to familiar problems, including limits, inverse functions, and integrals. …
Beneficial to both beginning students and researchers, Asymptotic Analysis and Perturbation Theory immediately introduces asymptotic notation and then applies this tool to familiar problems, including limits, inverse functions, and integrals. Suitable for those who have completed the standard calculus sequence, the book assumes no prior knowledge o
We study the mapping properties of a nonconstant entire solution of the equation <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis z plus 1 right-parenthesis equals e Superscript f left-parenthesis z …
We study the mapping properties of a nonconstant entire solution of the equation <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis z plus 1 right-parenthesis equals e Superscript f left-parenthesis z right-parenthesis Baseline minus 1 period"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1.</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">f(z + 1) = {e^{f(z)}} - 1.</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula>