<abstract xmlns="http://www.w3.org/1999/xhtml">We announce a new result on determining the Conley index of the Poincaré map for a time-periodic non-autonomous ordinary differential equation. The index is computed using some singular cycles related to an index pair of a small-step discretization of the equation. We indicate how the result can be applied to computer-assisted proofs of the existence of bounded and periodic solutions. We provide also some comments on computer-assisted proving in dynamics.</abstract>

On Computer-Assisted Proving The Existence Of Periodic And Bounded Orbits

<abstract xmlns="http://www.w3.org/1999/xhtml">We show that the solution to the orthogonal additivity problem in real inner product spaces depends continuously on the given function and provide an application of this fact.</abstract>

On The Continuous Dependence Of Solutions To Orthogonal Additivity Problem On Given Functions

<abstract xmlns="http://www.w3.org/1999/xhtml">A set <italic>S</italic> ⊂ <italic>V</italic> is a <italic>dominating set</italic> of a graph <italic>G</italic> = (<italic>V</italic>, <italic>E</italic>) if every vertex <italic>υ</italic> ∈ <italic>V</italic> which does not belong to <italic>S</italic> has a neighbour in <italic>S</italic>. The <italic>domination number γ</italic>(<italic>G</italic>) of the graph <italic>G</italic> is the minimum cardinality of a dominating set in <italic>G</italic>. A dominating set <italic>S</italic> is a <italic>γ-set</italic> in <italic>G</italic> if <italic>|S|</italic> = γ(<italic>G</italic>).Some graphs have exponentially many <italic>γ</italic>-sets, hence it is worth to ask a question if a <italic>γ</italic>-set can be obtained by some transformations from another <italic>γ</italic>-set. The study of gamma graphs is an answer to this reconfiguration problem. We give a partial answer to the question which graphs are gamma graphs of trees. In the second section gamma graphs <italic>γ.T</italic> of trees with diameter not greater than five will be presented. It will be shown that hypercubes <italic>Qk</italic> are among <italic>γ.T</italic> graphs. In the third section <italic>γ.T</italic> graphs of certain trees with three pendant vertices will be analysed. Additionally, some observations on the diameter of gamma graphs will be presented, in response to an open question, published by Fricke et al., if diam(<italic>T</italic> (<italic>γ</italic>)) = <italic>O</italic>(<italic>n</italic>)?</abstract>

Gamma Graphs Of Some Special Classes Of Trees

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we investigate the general solution and Hyers–Ulam–Rassias stability of a new mixed type of additive and quintic functional equation of the form
<disp-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/amsil-2015-0004_i1.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>f</mi><mrow><mo stretchy="false">(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo stretchy="false">)</mo></mrow><mo>−</mo><mn>5</mn><mi>f</mi><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo stretchy="false">)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mo stretchy="false">)</mo></mrow><mo>+</mo><mn>10</mn><mi>f</mi><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo stretchy="false">)</mo></mrow><mo>−</mo><mn>5</mn><mi>f</mi><mrow><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mo stretchy="false">)</mo></mrow><mo>=</mo><mn>10</mn><mi>f</mi><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mo>+</mo><mn>4</mn><mi>f</mi><mrow><mo stretchy="false">(</mo><mrow><mn>2</mn><mi>x</mi></mrow><mo stretchy="false">)</mo></mrow><mo>−</mo><mn>8</mn><mi>f</mi><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math><tex-math>$$f\left( {3x + y} \right) - 5f\left( {2x + y} \right) + f\left( {2x - y} \right) + 10f\left( {x + y} \right) - 5f\left( {x - y} \right) = 10f\left( y \right) + 4f\left( {2x} \right) - 8f\left( x \right)$$</tex-math></alternatives></disp-formula>
in the set of real numbers.</abstract>

Department of Mathematics, Garmsar Branch, Islamic Azad University, Garmsar, Iran

Department of Mathematics, Thiruvalluvar University College of Arts and Science, Gazhalnayaganpatti, Tirupattur-635 901 Tamil Nadu, India

Department of Mathematics, Sacred Heart College, Tirupattur-635 601, TamilNadu, India

Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

Mixed Type Of Additive And Quintic Functional Equations

<abstract xmlns="http://www.w3.org/1999/xhtml">The abstract Cauchy problem on scales of Banach space was considered by many authors. The goal of this paper is to show that the choice of the space on scale is significant. We prove a theorem that the selection of the spaces in which the Cauchy problem <italic>ut</italic> − Δ<italic>u</italic> = <italic>u|u|s</italic> with initial–boundary conditions is considered has an influence on the selection of index <italic>s</italic>. For the Cauchy problem connected with the heat equation we will study how the change of the base space influents the regularity of the solutions.</abstract>

An Application Of The Theory Of Scale Of Banach Spaces

<abstract xmlns="http://www.w3.org/1999/xhtml">Let 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/amsil-2015-0006_i1.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mi>∞</mi></msubsup><mrow><msub><mi>α</mi><mi>n</mi></msub><msup><mi>z</mi><mi>n</mi></msup></mrow></mstyle></math><tex-math>$f(z) = \sum\nolimits_{n = 0}^\infty {\alpha _n z^n }$</tex-math></alternatives></inline-formula>
 be a function defined by power series with complex coefficients and convergent on the open disk <italic>D</italic> (0, <italic>R</italic>) ⊂ ℂ, <italic>R</italic> > 0. For any <italic>x</italic>, <italic>y</italic> ∈ ℬ, a Banach algebra, with ‖<italic>x</italic>‖, ‖<italic>y</italic>‖ &lt; <italic>R</italic> we show among others that
<disp-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/amsil-2015-0006_i2.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>‖</mo><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>‖</mo></mrow><mo>≤</mo><mrow><mo>‖</mo><mrow><mi>y</mi><mo>−</mo><mi>x</mi></mrow><mo>‖</mo></mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mrow><msubsup><mi>f</mi><mi>a</mi><mo>′</mo></msubsup></mrow></mrow></mstyle><mo stretchy="false">(</mo><mrow><mo>‖</mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>t</mi><mo stretchy="false">)</mo><mi>x</mi><mo>+</mo><mi>t</mi><mi>y</mi></mrow><mo>‖</mo></mrow><mo stretchy="false">)</mo><mi>d</mi><mi>t</mi></math><tex-math>$$\left\| {f(y) - f(x)} \right\| \le \left\| {y - x} \right\|\int_0^1 {f_a^\prime } (\left\| {(1 - t)x + ty} \right\|)dt$$</tex-math></alternatives></disp-formula>
where
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/amsil-2015-0006_i3.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>f</mi><mi>a</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mi>∞</mi></msubsup><mrow><mo>|</mo><msub><mi>α</mi><mi>n</mi></msub><mo>|</mo></mrow></mstyle><mtext> </mtext><msup><mi>z</mi><mi>n</mi></msup></math><tex-math>$f_a (z) = \sum\nolimits_{n = 0}^\infty {|\alpha _n |} \;z^n$</tex-math></alternatives></inline-formula>
. Inequalities for the commutator such as
<disp-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/amsil-2015-0006_i4.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>‖</mo><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>‖</mo></mrow><mo>≤</mo><mn>2</mn><msub><mi>f</mi><mi>a</mi></msub><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><msubsup><mi>f</mi><mi>a</mi><mo>′</mo></msubsup><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mrow><mo>‖</mo><mrow><mi>y</mi><mo>−</mo><mi>x</mi></mrow><mo>‖</mo></mrow><mo>,</mo></math><tex-math>$$\left\| {f(x)f(y) - f(y)f(x)} \right\| \le 2f_a (M)f_a^\prime (M)\left\| {y - x} \right\|,$$</tex-math></alternatives></disp-formula>
if ‖<italic>x</italic>‖, ‖<italic>y</italic>‖ ≤ <italic>M</italic> &lt; <italic>R</italic>, as well as some inequalities of Hermite–Hadamard type are also provided.</abstract>

Mathematics, School of Engineering & Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia

School of Computer Science & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa, URL:

Inequalities Of Lipschitz Type For Power Series In Banach Algebras

<abstract xmlns="http://www.w3.org/1999/xhtml">We examine the class of spaces in which the second player has a winning strategy in the open-open game. We show that this spaces are not universally Kuratowski–Ulam. We also show that the games <italic>G</italic> and <italic>G</italic>7 introduced by P. Daniels, K. Kunen, H. Zhou [Fund. Math. <bold>145</bold> (1994), no. 3, 205–220] are not equivalent.</abstract>

Universally Kuratowski–Ulam Spaces And Open-Open Games

<abstract xmlns="http://www.w3.org/1999/xhtml">The paper is devoted to the communication complexity of lattice operations in linearly ordered finite sets. All well known techniques ([4, Chapter 1]) to determine the communication complexity of the infimum function in linear lattices disappoint, because a gap between the lower and upper bound is equal to <italic>O</italic>(log2<italic>n</italic>), where <italic>n</italic> is the cardinality of the lattice. Therefore our aim will be to investigate the communication complexity of the function more carefully. We consider a family of so called interval protocols and we construct the interval protocols for the infimum. We prove that the constructed protocols are optimal in the family of interval protocols. It is still open problem to compute the communication complexity of constructed protocols but the numerical experiments show that their complexity is less than the complexity of known protocols for the infimum function.</abstract>

Communication Complexity And Linearly Ordered Sets

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, a general fixed point theorem for cyclic multi-valued mappings satisfying an implicit relation from [19] different from implicit relations used in [13] and [23], generalizing some results from [22], [15], [13], [14], [16], [10] and from other papers, is proved.</abstract>

A General Fixed Point Theorem For Implicit Cyclic Multi-Valued Contraction Mappings

<abstract xmlns="http://www.w3.org/1999/xhtml">The main goal of the paper is to examine the dimension of the vector space spanned by powers of linear forms. We also find a lower bound for the number of summands in the presentation of zero form as a sum of <italic>d</italic>-th powers of linear forms.</abstract>

Linear Dependence Of Powers Of Linear Forms

<abstract xmlns="http://www.w3.org/1999/xhtml">Markov operators arising from graph directed constructions of iterated function systems are considered. Exponential convergence to an invariant measure is proved.</abstract>

Exponential Convergence For Markov Systems

Report of Meeting

AHEAD OF PRINT

Volume 38 (2024): Issue 1 (March 2024)

Volume 37 (2023): Issue 2 (September 2023)

Volume 37 (2023): Issue 1 (March 2023)

Volume 36 (2022): Issue 2 (September 2022)

Volume 36 (2022): Issue 1 (March 2022)

Volume 35 (2021): Issue 2 (September 2021)

Volume 35 (2021): Issue 1 (February 2021)

Volume 34 (2020): Issue 2 (September 2020)

Volume 34 (2020): Issue 1 (July 2020)

Volume 33 (2019): Issue 1 (September 2019)

Volume 32 (2018): Issue 1 (September 2018)

Volume 31 (2017): Issue 1 (September 2017)

Volume 30 (2016): Issue 1 (September 2016)

Volume 29 (2015): Issue 1 (September 2015)

Annales Mathematicae Silesianae

Annales Mathematicae Silesianae publishes significant research and survey papers from all branches of pure and applied mathematics, and reports of meetings. It welcomes contributed papers that develop important, new mathematical ideas and results or solve outstanding problems.  Article processing chargesThe journal does not have article processing charges (APCs) nor article submission charges.  Archiving  Sciendo archives the contents of this journal in Portico - digital long-term preservation service of scholarly books, journals and collections.  Plagiarism Policy  The editorial board is participating in a growing community of Similarity Check System's users in order to ensure that the content published is original and trustworthy. Similarity Check is a medium that allows for comprehensive manuscripts screening, aimed to eliminate plagiarism and provide a high standard and quality peer-review process.  Peer Review process  Annales Mathematicae Silesianae is a single-blind peer reviewed journal.Normally the papers are peer reviewed by two independent referees who are experts in the appropriate branch of mathematics. According to their opinions the manuscripts are either accepted or rejected. In the case their opinions contradict each other we send the paper to the third reviewer.Reviewers are asked to provide detailed, constructive comments that will help both the editors to make a decision on the publication and the author(s) to improve their manuscript. Preparing the review the reviewers may use the following scheme: review_scheme.pdf. Reviewers are also asked to declare that there is no conflict of interests between them and the authors.  Target Group  - researchers in all branches of pure and applied mathematics 