Institute of Mathematics AS CR, v. v. i.

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Partitioning planar graph of girth 5 into two forests with maximum degree 4

Author: Chen Min
Raspaud André
Wang Weifan
Yu Weiqiang
Publication venue: Matematický ústav AV ČR
Publication date: 01/01/2024
Field of study

summary:Given a graph

G=(V, E)

, if we can partition the vertex set

V

into two nonempty subsets

V_1

and

V_2

which satisfy

\Delta (G[V_1])\le d_1

and

\Delta (G[V_2])\le d_2

, then we say

G

has a

(\Delta _{d_{1}},\Delta _{d_{2}})

-partition. And we say

G

admits an

(F_{d_{1}}, F_{d_{2}})

-partition if

G[V_1]

and

G[V_2]

are both forests whose maximum degree is at most

d_{1}

and

d_{2}

, respectively. We show that every planar graph with girth at least 5 has an

(F_4, F_4)

-partition

Bounds for the derivative of certain meromorphic functions and on meromorphic Bloch-type functions

Author: Bhowmik Bappaditya
Sen Sambhunath
Publication venue: Matematický ústav AV ČR
Publication date: 01/01/2024
Field of study

summary:It is known that if

f

is holomorphic in the open unit disc

{\mathbb D}

of the complex plane and if, for some

c>0

|f(z)|\leq 1/(1-|z|^2)^c

z\in {\mathbb D}

, then

|f'(z)|\leq 2(c+1)/(1-|z|^2)^{c+1}

. We consider a meromorphic analogue of this result. Furthermore, we introduce and study the class of meromorphic Bloch-type functions that possess a nonzero simple pole in

{\mathbb D}

. In particular, we obtain bounds for the modulus of the Taylor coefficients of functions in this class

On certain $GL(6)$ form and its Rankin-Selberg convolution

Author: Kaur Amrinder
Sankaranarayanan Ayyadurai
Publication venue: Matematický ústav AV ČR
Publication date: 01/01/2024
Field of study

summary:We consider

L_G(s)

to be the

L

-function attached to a particular automorphic form

G

GL(6)

. We establish an upper bound for the mean square estimate on the critical line of Rankin-Selberg

L

-function

L_{G \times G}(s)

. As an application of this result, we give an asymptotic formula for the discrete sum of coefficients of

L_{G \times G}(s)

Generalized absolute convergence of single and double Vilenkin-Fourier series and related results

Author: Kalsariya Nayna Govindbhai
Ghodadra Bhikha Lila
Publication venue: Matematický ústav AV ČR
Publication date: 01/01/2024
Field of study

summary:We consider the Vilenkin orthonormal system on a Vilenkin group

G

and the Vilenkin-Fourier coefficients

\hat {f}(n)

n\in \mathbb {N}

, of functions

f\in L^p(G)

for some

1<p\le 2

. We obtain certain sufficient conditions for the finiteness of the series

\sum _{n=1}^{\infty }a_n|\hat {f}(n)|^r

, where

\{a_n\}

is a given sequence of positive real numbers satisfying a mild assumption and

0<r<2

. We also find analogous conditions for the double Vilenkin-Fourier series. These sufficient conditions are in terms of (either global or local) moduli of continuity of

f

and give multiplicative analogue of some results due to Móricz (2010), Móricz and Veres (2011), Golubov and Volosivets (2012), and Volosivets and Kuznetsova (2020)

A generalization of reflexive rings

Author: Çalcı Mete Burak
Chen Huanyin
Halıcıoğlu Sait
Publication venue: Matematický ústav AV ČR
Publication date: 01/01/2024
Field of study

summary:We introduce a class of rings which is a generalization of reflexive rings and

J

-reversible rings. Let

R

be a ring with identity and

J(R)

denote the Jacobson radical of

R

. A ring

R

is called

J

-reflexive if for any

a, b \in R

aRb = 0

implies

bRa \subseteq J(R)

. We give some characterizations of a

J

-reflexive ring. We prove that some results of reflexive rings can be extended to

J

-reflexive rings for this general setting. We conclude some relations between

J

-reflexive rings and some related rings. We investigate some extensions of a ring which satisfies the

J

-reflexive property and we show that the

J

-reflexive property is Morita invariant

Geometric visualisation – how to develop it in the form of a non-interactive game Triangular Domino

Author: Slezáková Jana
Publication venue: Union of Czech Mathematicians and Physicists
Publication date: 01/01/2024
Field of study

summary:Existuje mnoho nástrojů, které žákům pomáhají lépe upevňovat osvojené vědomosti a dovednosti. Tradiční výuka mnohdy opomíjí trénování geometrické představivosti. To lze zajistit např. zařazením takových her do výuky, které v procesu učení propojují mozek s manipulativními činnostmi. Cílem předloženého příspěvku je poukázat na důležitost výuky geometrie a geometrické představivosti, a to formou neinterakční geometrické hry Trojúhelníkové domino. Uvedená geometrická hra je vhodná mimo jiné pro vzdělávání žáků intaktních a také žáků se speciálními vzdělávacími potřebami. Ve výuce geometrie nachází další využití, např. při určování velikostí vnitřních úhlů u vrcholů jednotlivých nekonvexních mnohoúhelníků. Trojúhelníkové domino je hra, která podporuje seberealizaci, kreativitu, samostatnost a svobodnou volbu při dosahování výukových cílů.summary:Many tools help students better manage the acquired knowledge and skills. Traditional teaching often neglects to practice geometric visualisation. This can be ensured by incorporating geometric games into teaching that connect the brain with manipulative activities in a learning process. The presented contribution highlights the importance of teaching geometry and geometric visualisation using a non-interactive geometric game, Triangular Domino. The mentioned geometric game is, among others, suitable for educating intact pupils and pupils with special educational needs. In teaching geometry, it finds other uses, e.g., in determining the sizes of interior angles at the vertices of individual non-convex polygons. Triangular Domino is a game that enables self-realisation, creativity, independence, and free choice in achieving learning goals

Summer School on Geometry 2023

Author: Moravcová Vlasta
Publication venue: Union of Czech Mathematicians and Physicists
Publication date: 01/01/2024
Field of study

summary:Krátké sdělení informuje o proběhlém vzdělávacím semináři, který je určen především stávajícím i budoucím učitelům matematiky základních a středních škol

The canonical constructions of connections on total spaces of fibred manifolds

Author: Mikulski Włodzimierz M.
Publication venue: Department of Mathematics, Faculty of Science of Masaryk University, Brno
Publication date: 01/01/2024
Field of study

summary:We classify classical linear connections

A(\Gamma ,\Lambda ,\Theta )

on the total space

Y

of a fibred manifold

Y\rightarrow M

induced in a natural way by the following three objects: a general connection

\Gamma

Y\rightarrow M

, a classical linear connection

\Lambda

M

and a linear connection

\Theta

in the vertical bundle

VY\rightarrow Y

. The main result says that if

\mathrm{dim}(M)\ge 3

and

\mathrm{dim}(Y)-\mathrm{dim}(M) \ge 3

then the natural operators

A

under consideration form the

17

dimensional affine space

Rings with divisibility on descending chains of ideals

Author: Es Safi Oussama Aymane
Mahdou Najib
Tekir Ünsal
Publication venue: Matematický ústav AV ČR
Publication date: 01/01/2024
Field of study

summary:This paper deals with the rings which satisfy

DCC_{d}

condition. This notion has been introduced recently by R. Dastanpour and A. Ghorbani (2017) as a generalization of Artnian rings. It is of interest to investigate more deeply this class of rings. This study focuses on commutative case. In this vein, we present this work in which we examine the transfer of these rings to the trivial, amalgamation and polynomial ring extensions. We also investigate the relationship between this class of rings and the well known ones. Furthermore, many new results are presented in the scope of this paper. For example, there is one which concerns the decomposition of ideals on prime ones and another which investigate the Krull dimension of the ring satisfying

DCC_{d}

condition. At the end of this work, we provide a result which concerns the modules over such rings

Homological dimensions for endomorphism algebras of Gorenstein projective modules

Author: Zhang Aiping
Lei Xueping
Publication venue: Matematický ústav AV ČR
Publication date: 01/01/2024
Field of study

summary:Let

A

be a CM-finite Artin algebra with a Gorenstein-Auslander generator

E

M

be a Gorenstein projective

A

-module and

B = {\rm End}_A M

. We give an upper bound for the finitistic dimension of

B

in terms of homological data of

M

. Furthermore, if

A

n

-Gorenstein for

2 \leq n < \infty

, then we show the global dimension of

B

is less than or equal to

n

plus the

B

-projective dimension of

{\rm Hom}_A(M, E).

As an application, the global dimension of

{\rm End}_A E

is less than or equal to

n

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