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SOME INEQUALITIES BETWEEN THE BEST SIMULTANEOUS APPROXIMATION AND MODULUS OF CONTINUITY IN THE WEIGHTED BERGMAN SPACE

Author: Saidusainov Muqim S.
Publication venue: Institute of Mathematics and Mechanics
Publication date: 27/12/2023
Field of study

Some inequalities between the best simultaneous approximation of functions and their intermediate derivatives, and the modulus of continuity in a weighted Bergman space are obtained. When the weight function is

\gamma(\rho)=\rho^\alpha,

\alpha>0

, some sharp inequalities between the best simultaneous approximation and an

m

th order modulus of continuity averaged with the given weight are proved. For a specific class of functions, the upper bound of the best simultaneous approximation in the space

B_{2,\gamma_{1}},

\gamma_{1}(\rho)=\rho^{\alpha},

\alpha>0

, is found. Exact values of several

n

-widths are calculated for the classes of functions

W_{p}^{(r)}(\omega_{m},q)

A NEW CHARACTERIZATION OF SYMMETRIC DUNKL AND $q$ -DUNKL-CLASSICAL ORTHOGONAL POLYNOMIALS

Author: Habbachi Yahia
Publication venue: Institute of Mathematics and Mechanics
Publication date: 27/12/2023
Field of study

In this paper, we consider the following

\mathcal{L}

-difference equation

\Phi(x) \mathcal{L}P_{n+1}(x)=(\xi_nx+\vartheta_n)P_{n+1}(x)+\lambda_nP_{n}(x),\quad n\geq0,

where

\Phi

is a monic polynomial (even),

\deg\Phi\leq2

\xi_n,\,\vartheta_n,\,\lambda_n,\,n\geq0

, are complex numbers and

\mathcal{L}

is either the Dunkl operator

T_\mu

or the the

q

-Dunkl operator

T_{(\theta,q)}

. We show that if

\mathcal{L}=T_\mu

, then the only symmetric orthogonal polynomials satisfying the previous equation are, up a dilation, the generalized Hermite polynomials and the generalized Gegenbauer polynomials and if

\mathcal{L}=T_{(\theta,q)}

, then the

q^2

-analogue of generalized Hermite and the

q^2

-analogue of generalized Gegenbauer polynomials are, up a dilation, the only orthogonal polynomials sequences satisfying the

\mathcal{L}

-difference equation

GRACEFUL CHROMATIC NUMBER OF SOME CARTESIAN PRODUCT GRAPHS

Author: Suparta I Nengah
Venkatachalam Mathiyazhagan
Gunadi I Gede Aris
Pratama Putu Andi Cipta
Publication venue: Institute of Mathematics and Mechanics
Publication date: 27/12/2023
Field of study

A graph

G(V,E)

is a system consisting of a finite non empty set of vertices

V(G)

and a set of edges

E(G)

. A (proper) vertex colouring of

G

is a function

f:V(G)\rightarrow \{1,2,\ldots,k\},

for some positive integer

k

such that

f(u)\neq f(v)

for every edge

uv\in E(G)

. Moreover, if

|f(u)-f(v)|\neq |f(v)-f(w)|

for every adjacent edges

uv,vw\in E(G)

, then the function

f

is called graceful colouring for

G

. The minimum number

k

such that

f

is a graceful colouring for

G

is called the graceful chromatic number of

G

. The purpose of this research is to determine graceful chromatic number of Cartesian product graphs

C_m \times P_n

for integers

m\geq 3

and

n\geq 2

, and

C_m \times C_n

for integers

m,n\geq 3

. Here,

C_m

and

P_m

are cycle and path with

m

vertices, respectively. We found some exact values and bounds for graceful chromatic number of these mentioned Cartesian product graphs

INEQUALITIES FOR A CLASS OF MEROMORPHIC FUNCTIONS WHOSE ZEROS ARE WITHIN OR OUTSIDE A GIVEN DISK

Author: Mir Mohd Yousf
Wali Shah Lubna
Shah Wali Mohammad
Publication venue: Institute of Mathematics and Mechanics
Publication date: 27/07/2023
Field of study

In this paper, we consider a class of meromorphic functions

r(z)

having an

s

-fold zero at the origin and establish some inequalities of Bernstein and Turán type for the modulus of the derivative of rational functions in the sup-norm on the disk in the complex plane. These results produce some sharper inequalities while taking into account the placement of zeros of the underlying rational function. Moreover, many inequalities for polynomials and polar derivatives follow as special cases. In particular, our results generalize as well as refine a result due Dewan et al. [6].

A PRESENTATION FOR A SUBMONOID OF THE SYMMETRIC INVERSE MONOID

Author: Sareeto Apatsara
Koppitz Jörg
Publication venue: Institute of Mathematics and Mechanics
Publication date: 27/12/2023
Field of study

In the present paper, we study a submonoid of the symmetric inverse semigroup

I_n

. Specifically, we consider the monoid of all order-, fence-, and parity-preserving transformations of

I_n

. While the rank and a set of generators of minimal size for this monoid are already known, we will provide a presentation for this monoid

ON ONE ZALCMAN PROBLEM FOR THE MEAN VALUE OPERATOR

Author: Volchkova Natalia P.
Volchkov Vitaliy V.
Publication venue: Institute of Mathematics and Mechanics
Publication date: 27/07/2023
Field of study

Let

\mathcal{D}'(\mathbb{R}^n)

and

\mathcal{E}'(\mathbb{R}^n)

be the spaces of distributions and compactly supported distributions on

\mathbb{R}^n

n\geq 2

respectively, let

\mathcal{E}'_{\natural}(\mathbb{R}^n)

be the space of all radial (invariant under rotations of the space

mathbb{R}^n

) distributions in

\mathcal{E}'(\mathbb{R}^n)

, let

\widetilde{T}

be the spherical transform (Fourier–Bessel transform) of a distribution

T\in\mathcal{E}'_{\natural}(\mathbb{R}^n)

, and let

\mathcal{Z}_{+}(\widetilde{T})

be the set of all zeros of an even entire function

\widetilde{T}

lying in the half-plane

\mathrm{Re} \, z\geq 0

and not belonging to the negative part of the imaginary axis. Let

\sigma_{r}

be the surface delta function concentrated on the sphere

S_r=\{x\in\mathbb{R}^n: |x|=r\}

. The problem of L. Zalcman on reconstructing a distribution

f\in \mathcal{D}'(\mathbb{R}^n)

from known convolutions

f\ast \sigma_{r_1}

and

f\ast \sigma_{r_2}

is studied. This problem is correctly posed only under the condition

r_1/r_2\notin M_n

, where

M_n

is the set of all possible ratios of positive zeros of the Bessel function

J_{n/2-1}

. The paper shows that if

r_1/r_2\notin M_n

, then an arbitrary distribution

f\in \mathcal{D}'(\mathbb{R}^n)

can be expanded into an unconditionally convergent series

f=\sum\limits_{\lambda\in\mathcal{Z}_{+}(\widetilde{\Omega}_{r_1})}\,\,\, \sum\limits_{\mu\in\mathcal{Z}_+(\widetilde{\Omega}_{r_2})}\frac{4\lambda\mu}{(\lambda^2-\mu^2) \widetilde{\Omega}_{r_1}^{\,\,\,\displaystyle{'}}(\lambda)\widetilde{\Omega}_{r_2}^{\,\,\,\displaystyle{'}}(\mu)}\Big(P_{r_2} (\Delta) \big((f\ast\sigma_{r_2})\ast \Omega_{r_1}^{\lambda}\big)-P_{r_1} (\Delta) \big((f\ast\sigma_{r_1})\ast \Omega_{r_2}^{\mu}\big)\Big)

in the space

\mathcal{D}'(\mathbb{R}^n)

, where

\Delta

is the Laplace operator in

\mathbb{R}^n

P_r

is an explicitly given polynomial of degree

[(n+5)/4]

, and

\Omega_{r}

and

\Omega_{r}^{\lambda}

are explicitly constructed radial distributions supported in the ball

|x|\leq r

. The proof uses the methods of harmonic analysis, as well as the theory of entire and special functions. By a similar technique, it is possible to obtain inversion formulas for other convolution operators with radial distributions

BIHARMONIC GREEN FUNCTION AND BISUPERMEDIAN ON INFINITE NETWORKS

Author: Varadha Raj Manivannan
Madhu Venkataraman
Publication venue: Institute of Mathematics and Mechanics
Publication date: 29/12/2022
Field of study

In this article, we have discussed Biharmonic Green function on an infinite network and bimedian functions. We have proved some standard results in terms of supermedian and bimedian. Also, we have proved the Discrete Riquier problem in the setting of bimedian functions

ON ONE INEQUALITY OF DIFFERENT METRICS FOR TRIGONOMETRIC POLYNOMIALS

Author: Arestov Vitalii V.
Deikalova Marina V.
Publication venue: Institute of Mathematics and Mechanics
Publication date: 29/12/2022
Field of study

We study the sharp inequality between the uniform norm and

L^p(0,\pi/2)

-norm of polynomials in the system

\mathscr{C}=\{\cos (2k+1)x\}_{k=0}^\infty

of cosines with odd harmonics. We investigate the limit behavior of the best constant in this inequality with respect to the order

n

of polynomials as

n\to\infty

and provide a characterization of the extremal polynomial in the inequality for a fixed order of polynomials

A QUADRUPLE INTEGRAL INVOLVING THE EXPONENTIAL LOGARITHM OF QUOTIENT RADICALS IN TERMS OF THE HURWITZ-LERCH ZETA FUNCTION

Author: Reynolds Robert
Stauffer Allan
Publication venue: Institute of Mathematics and Mechanics
Publication date: 29/12/2022
Field of study

With a possible connection to integrals used in General Relativity, we used our contour integral method to write a closed form solution for a quadruple integral involving exponential functions and logarithm of quotient radicals. Almost all Hurwitz–Lerch Zeta functions have an asymmetrical zero distribution. All the results in this work are new

ANALYSIS OF THE GROWTH RATE OF FEMININE MOSQUITO THROUGH DIFFERENCE EQUATIONS

Author: Murugesan Regan
Kumaravel Sathish Kumar
Rasappan Suresh
Al Majrafi Wardah Abdullah
Publication venue: Institute of Mathematics and Mechanics
Publication date: 29/12/2022
Field of study

The mosquito life cycle is developed mathematically with the concept of difference equation. The qualitative properties of the life-cycle are analyzed. The Lyapunov function is defined for difference equation to stabilize the system of mosquito life cycle. A novel technique is applied for deriving stability criterion, especially the back-stepping control technique is applied for discrete time system. The bifurcation analysis is also furnished for the model of mosquito life cycle. The new technique is applied in the mosquito life cycle model and its results are examined through MATLAB

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