We provide a complete description of ergodic perturbed monomials on 2-adic spheres around the unity.
We provide a complete description of ergodic perturbed monomials on 2-adic spheres around the unity.
By observing the equivalence of assertions on determining the jump of a function by its differentiated or integrated Fourier series, we generalize a previous result of Kvernadze, Hagstrom and Shapiro …
By observing the equivalence of assertions on determining the jump of a function by its differentiated or integrated Fourier series, we generalize a previous result of Kvernadze, Hagstrom and Shapiro to the whole class of functions of harmonic bounded variation and without finiteness assumption on the number of discontinuities. New results on determination of jump discontinuities by means of the tails of integrated generalized Fourier-Jacobi series are derived.
By observing the equivalence of assertions on determining the jump of a function by its differentiated or integrated Fourier series, we generalize a previous result of Kvernadze, Hagstrom and Shapiro …
By observing the equivalence of assertions on determining the jump of a function by its differentiated or integrated Fourier series, we generalize a previous result of Kvernadze, Hagstrom and Shapiro to the whole class of functions of harmonic bounded variation and without finiteness assumption on the number of discontinuities. Two results on determination of jump discontinuities by means of the tails of integrated Fourier-Chebyshev series are derived.
We derive closed formula for the heat kernel $K_\mathbb{H}$ associated to Maass-Laplacian operator $D_k$ for any real $k$ and prove that heat kernel $K_\mathbb{H}$ is strictly monotone decreasing function. We …
We derive closed formula for the heat kernel $K_\mathbb{H}$ associated to Maass-Laplacian operator $D_k$ for any real $k$ and prove that heat kernel $K_\mathbb{H}$ is strictly monotone decreasing function. We also derive some important asymptotic formulae for the heat kernel $K_\mathbb{H}$.
Geometry is a very interesting, applicable and beautiful part of mathematics. However, geometry is often difficult for students to understand and demanding for teachers to teach [1]. Constructing proofs in …
Geometry is a very interesting, applicable and beautiful part of mathematics. However, geometry is often difficult for students to understand and demanding for teachers to teach [1]. Constructing proofs in geometric problems turns out to be particularly difficult, even for high attaining students [2]. Sometimes, students do not even know where to start when trying to solve these [3].
We derive closed formula for the heat kernel $K_{\mathbb H, k}$ associated to the Maass-Laplacian operator $D_k$ for any real weight $k$ and prove that the heat kernel $K_{\mathbb H, …
We derive closed formula for the heat kernel $K_{\mathbb H, k}$ associated to the Maass-Laplacian operator $D_k$ for any real weight $k$ and prove that the heat kernel $K_{\mathbb H, k}$ is strictly monotone decreasing function of the hyperbolic distance. We derive small time and large time asymptotic formulae for the heat kernel $K_{\mathbb H, k}$ and describe its behavior as a function of the real weight $k$.
We derive closed formula for the heat kernel $K_\mathbb{H}$ associated to Maass-Laplacian operator $D_k$ for any real $k$ and prove that heat kernel $K_\mathbb{H}$ is strictly monotone decreasing function. We …
We derive closed formula for the heat kernel $K_\mathbb{H}$ associated to Maass-Laplacian operator $D_k$ for any real $k$ and prove that heat kernel $K_\mathbb{H}$ is strictly monotone decreasing function. We also derive some important asymptotic formulae for the heat kernel $K_\mathbb{H}$.
By observing the equivalence of assertions on determining the jump of a function by its differentiated or integrated Fourier series, we generalize a previous result of Kvernadze, Hagstrom and Shapiro …
By observing the equivalence of assertions on determining the jump of a function by its differentiated or integrated Fourier series, we generalize a previous result of Kvernadze, Hagstrom and Shapiro to the whole class of functions of harmonic bounded variation and without finiteness assumption on the number of discontinuities. Two results on determination of jump discontinuities by means of the tails of integrated Fourier-Chebyshev series are derived.
We study the theta function and the Hurwitz-type zeta function associated to the Lucas sequence $U=\{U_n(P,Q)\}_{n\geq 0}$ of the first kind determined by the real numbers $P,Q$ under certain natural …
We study the theta function and the Hurwitz-type zeta function associated to the Lucas sequence $U=\{U_n(P,Q)\}_{n\geq 0}$ of the first kind determined by the real numbers $P,Q$ under certain natural assumptions on $P$ and $Q$. We deduce an asymptotic expansion of the theta function $\theta_U(t)$ as $t\downarrow 0$ and use it to obtain a meromorphic continuation of the Hurwitz-type zeta function $\zeta_{U}\left( s,z\right) =\sum\limits_{n=0}^{\infty }\left(z+U_{n}\right) ^{-s}$ to the whole complex $s-$plane. Moreover, we identify the residues of $\zeta_{U}\left( s,z\right)$ at all poles in the half-plane $\Re(s)\leq 0$.
In this paper, we study discrete Bessel functions which are solutions to the discretization of Bessel differential equations when the forward and the backward difference replace the time derivative. We …
In this paper, we study discrete Bessel functions which are solutions to the discretization of Bessel differential equations when the forward and the backward difference replace the time derivative. We focus on the discrete Bessel equations with the backward difference and derive their solutions. We then study the transformation properties of those functions, describe their asymptotic behaviour and compute Laplace transform. As an application, we study the discrete wave equation on the integers in timescale $T=\mathbb{Z}$ and express its fundamental and general solution in terms of the discrete $J$-Bessel function. Going further, we show that the first fundamental solution of this equation oscillates with the exponentially decaying amplitude as time tends to infinity.
In this paper we derive some new identities involving the Fibonacci and Lucas polynomials and the Chebyshev polynomials of the first and the second kind. Our starting point is a …
In this paper we derive some new identities involving the Fibonacci and Lucas polynomials and the Chebyshev polynomials of the first and the second kind. Our starting point is a finite trigonometric sum which equals the resolvent kernel on the discrete circle with $m$ vertices and which can be evaluated in two different ways. An expression for this sum in terms of the Chebyshev polynomials was deduced in \cite{JKS} and the expression in terms of the Fibonacci and Lucas polynomials is deduced in this paper. As a consequence, we establish some further identities involving trigonometric sums and Fibonacci, Lucas, Pell and Pell-Lucas polynomials and numbers, thus providing a "physical" interpretation for those identities. Moreover, the finite trigonometric sum of the type considered in this paper can be related to the effective resistance between any two vertices of the $N$-cycle graph with four nearest neighbors $C_{N}(1,2)$. This yields further identities involving Fibonacci numbers.
In this paper, we generalise an interesting geometry problem from the 1995 edition of the International Mathematical Olympiad (IMO) using analytic geometry tools.
In this paper, we generalise an interesting geometry problem from the 1995 edition of the International Mathematical Olympiad (IMO) using analytic geometry tools.
In this paper, we generalise an interesting geometry problem from the 1995 edition of the International Mathematical Olympiad (IMO) using analytic geometry tools.
In this paper, we generalise an interesting geometry problem from the 1995 edition of the International Mathematical Olympiad (IMO) using analytic geometry tools.
In this paper we derive some new identities involving the Fibonacci and Lucas polynomials and the Chebyshev polynomials of the first and the second kind. Our starting point is a …
In this paper we derive some new identities involving the Fibonacci and Lucas polynomials and the Chebyshev polynomials of the first and the second kind. Our starting point is a finite trigonometric sum which equals the resolvent kernel on the discrete circle with $m$ vertices and which can be evaluated in two different ways. An expression for this sum in terms of the Chebyshev polynomials was deduced in \cite{JKS} and the expression in terms of the Fibonacci and Lucas polynomials is deduced in this paper. As a consequence, we establish some further identities involving trigonometric sums and Fibonacci, Lucas, Pell and Pell-Lucas polynomials and numbers, thus providing a "physical" interpretation for those identities. Moreover, the finite trigonometric sum of the type considered in this paper can be related to the effective resistance between any two vertices of the $N$-cycle graph with four nearest neighbors $C_{N}(1,2)$. This yields further identities involving Fibonacci numbers.
In this paper, we study discrete Bessel functions which are solutions to the discretization of Bessel differential equations when the forward and the backward difference replace the time derivative. We …
In this paper, we study discrete Bessel functions which are solutions to the discretization of Bessel differential equations when the forward and the backward difference replace the time derivative. We focus on the discrete Bessel equations with the backward difference and derive their solutions. We then study the transformation properties of those functions, describe their asymptotic behaviour and compute Laplace transform. As an application, we study the discrete wave equation on the integers in timescale $T=\mathbb{Z}$ and express its fundamental and general solution in terms of the discrete $J$-Bessel function. Going further, we show that the first fundamental solution of this equation oscillates with the exponentially decaying amplitude as time tends to infinity.
We study the theta function and the Hurwitz-type zeta function associated to the Lucas sequence $U=\{U_n(P,Q)\}_{n\geq 0}$ of the first kind determined by the real numbers $P,Q$ under certain natural …
We study the theta function and the Hurwitz-type zeta function associated to the Lucas sequence $U=\{U_n(P,Q)\}_{n\geq 0}$ of the first kind determined by the real numbers $P,Q$ under certain natural assumptions on $P$ and $Q$. We deduce an asymptotic expansion of the theta function $\theta_U(t)$ as $t\downarrow 0$ and use it to obtain a meromorphic continuation of the Hurwitz-type zeta function $\zeta_{U}\left( s,z\right) =\sum\limits_{n=0}^{\infty }\left(z+U_{n}\right) ^{-s}$ to the whole complex $s-$plane. Moreover, we identify the residues of $\zeta_{U}\left( s,z\right)$ at all poles in the half-plane $\Re(s)\leq 0$.
Geometry is a very interesting, applicable and beautiful part of mathematics. However, geometry is often difficult for students to understand and demanding for teachers to teach [1]. Constructing proofs in …
Geometry is a very interesting, applicable and beautiful part of mathematics. However, geometry is often difficult for students to understand and demanding for teachers to teach [1]. Constructing proofs in geometric problems turns out to be particularly difficult, even for high attaining students [2]. Sometimes, students do not even know where to start when trying to solve these [3].
We derive closed formula for the heat kernel $K_\mathbb{H}$ associated to Maass-Laplacian operator $D_k$ for any real $k$ and prove that heat kernel $K_\mathbb{H}$ is strictly monotone decreasing function. We …
We derive closed formula for the heat kernel $K_\mathbb{H}$ associated to Maass-Laplacian operator $D_k$ for any real $k$ and prove that heat kernel $K_\mathbb{H}$ is strictly monotone decreasing function. We also derive some important asymptotic formulae for the heat kernel $K_\mathbb{H}$.
We derive closed formula for the heat kernel $K_{\mathbb H, k}$ associated to the Maass-Laplacian operator $D_k$ for any real weight $k$ and prove that the heat kernel $K_{\mathbb H, …
We derive closed formula for the heat kernel $K_{\mathbb H, k}$ associated to the Maass-Laplacian operator $D_k$ for any real weight $k$ and prove that the heat kernel $K_{\mathbb H, k}$ is strictly monotone decreasing function of the hyperbolic distance. We derive small time and large time asymptotic formulae for the heat kernel $K_{\mathbb H, k}$ and describe its behavior as a function of the real weight $k$.
We derive closed formula for the heat kernel $K_\mathbb{H}$ associated to Maass-Laplacian operator $D_k$ for any real $k$ and prove that heat kernel $K_\mathbb{H}$ is strictly monotone decreasing function. We …
We derive closed formula for the heat kernel $K_\mathbb{H}$ associated to Maass-Laplacian operator $D_k$ for any real $k$ and prove that heat kernel $K_\mathbb{H}$ is strictly monotone decreasing function. We also derive some important asymptotic formulae for the heat kernel $K_\mathbb{H}$.
We provide a complete description of ergodic perturbed monomials on 2-adic spheres around the unity.
We provide a complete description of ergodic perturbed monomials on 2-adic spheres around the unity.
By observing the equivalence of assertions on determining the jump of a function by its differentiated or integrated Fourier series, we generalize a previous result of Kvernadze, Hagstrom and Shapiro …
By observing the equivalence of assertions on determining the jump of a function by its differentiated or integrated Fourier series, we generalize a previous result of Kvernadze, Hagstrom and Shapiro to the whole class of functions of harmonic bounded variation and without finiteness assumption on the number of discontinuities. New results on determination of jump discontinuities by means of the tails of integrated generalized Fourier-Jacobi series are derived.
By observing the equivalence of assertions on determining the jump of a function by its differentiated or integrated Fourier series, we generalize a previous result of Kvernadze, Hagstrom and Shapiro …
By observing the equivalence of assertions on determining the jump of a function by its differentiated or integrated Fourier series, we generalize a previous result of Kvernadze, Hagstrom and Shapiro to the whole class of functions of harmonic bounded variation and without finiteness assumption on the number of discontinuities. Two results on determination of jump discontinuities by means of the tails of integrated Fourier-Chebyshev series are derived.
By observing the equivalence of assertions on determining the jump of a function by its differentiated or integrated Fourier series, we generalize a previous result of Kvernadze, Hagstrom and Shapiro …
By observing the equivalence of assertions on determining the jump of a function by its differentiated or integrated Fourier series, we generalize a previous result of Kvernadze, Hagstrom and Shapiro to the whole class of functions of harmonic bounded variation and without finiteness assumption on the number of discontinuities. Two results on determination of jump discontinuities by means of the tails of integrated Fourier-Chebyshev series are derived.