<abstract xmlns="http://www.w3.org/1999/xhtml">

A <italic>submetric space</italic> is a topological space with continuous metrics, generating a metric topology weaker than the original one (e.g. a separable Hilbert space with the weak topology).
We demonstrate that on submetric spaces there exists a theory of convergence in probability, in law etc. equally effective as the Probability Theory on metric spaces. In the theory on submetric spaces the central role is played by a version of the Skorokhod almost sure representation, proved by the author some 25 years ago and in 2010 rediscovered by specialists in stochastic partial differential equations in the form of “stochastic compactness method”.
</abstract>

Probability on Submetric Spaces

<abstract xmlns="http://www.w3.org/1999/xhtml">

We approach the problem of integration for rough integrands and integrators, typically representing trajectories of stochastic processes possessing only some Hölder regularity of possibly low order, in the framework of para-control calculus. For this purpose, we first decompose integrand and integrator into Paley–Littlewood packages along the Haar–Schauder system. By careful estimation of the components of products of packages of the integrand and derivatives of the integrator we obtain a characterization of Young’s integral. For the most interesting case of functions with Hölder regularities that sum up to an order below 1 we have to employ the concept of para-control of integrand and integrator with respect to a reference function for which a version of antisymmetric Lévy area is known to exist. This way we obtain an interpretation of the rough path integral. Lévy areas being known for most frequently used stochastic processes such as (fractional) Brownian motion, this integral serves as a basis for pathwise stochastic calculus, as the integral in classical rough path analysis.
</abstract>

A Fourier Analysis Based New Look at Integration

<abstract xmlns="http://www.w3.org/1999/xhtml">

Let (<italic>S, </italic>+) be an abelian semigroup, let (<italic>H, </italic>+) be an abelian group which is uniquely 2-divisible, and let <italic>ϕ </italic>be an endomorphism of <italic>S</italic>. We find the solutions <italic>f, h </italic>: <italic>S </italic>→ <italic>H </italic>of each of the functional equations
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2023-0015_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>ϕ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>=</mo><mi>h</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mo>∘</mo><mi>ϕ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>,</mo><mi> </mi><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>S</mi><mo>,</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>ϕ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>=</mo><mi>h</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mn>2</mn><mi>f</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>,</mo><mi> </mi><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>S</mi><mo>,</mo></mrow></mtd></mtr></mtable></mrow>
</math>
<tex-math>\matrix{ {f\left( {x + y} \right) + f\left( {x + \varphi \left( y \right)} \right) = h\left( x \right) + f\left( y \right) + f \circ \varphi \left( y \right),\,x,y \in S,} \hfill \cr {f\left( {x + y} \right) + f\left( {x + \varphi \left( y \right)} \right) = h\left( x \right) + 2f\left( y \right),\,x,y \in S,} \hfill \cr }</tex-math>
</alternatives>
</disp-formula>
in terms of additive and bi-additive maps. Moreover, as applications, we determine the solutions of some related functional equations.
</abstract>

New Pexiderizations of Drygas’ Functional Equation on Abelian Semigroups

<abstract xmlns="http://www.w3.org/1999/xhtml">

The (fuzzy) filter of the Sheffer stroke Hilbert algebra was addressed by Oner, Katican and Borumand Saeid. The weak version of the filter was discussed by Jun and Oner. In this manuscipt, with the fuzzy version of the weak filter in mind, the notions of fuzzy weak filters and (<italic>∈, ∈ ∨q</italic>)-fuzzy weak filters are introduced, and their properties are explored. Conditions under which <italic>t</italic>-level set, <italic>Q</italic>t-set and <italic>t</italic>-<italic>∈ ∨q</italic>-set become weak filters are explored in relation to fuzzy weak filters and (<italic>∈, ∈ ∨q</italic>)-fuzzy weak filters. The relationship and characterization of the fuzzy weak filter and the (<italic>∈, ∈ ∨q</italic>)-fuzzy weak filter are investigated.
</abstract>

Fuzzy Weak Filters of Sheffer Stroke Hilbert Algebras

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this paper, we use the finite element method to solve the fractional space-time diffusion equation over finite fields. This equation is obtained from the standard diffusion equation by replacing the first temporal derivative with the new fractional derivative recently introduced by Caputo and Fabrizion and the second spatial derivative with the Riemann–Liouville fractional derivative. The existence and uniqueness of the numerical solution and the result of error estimation are given. Numerical examples are used to support the theoretical results.
</abstract>

Djillali Liabes University of Sidi Bel-Abbes

Department of Mathematical Analysis and Numerical Mathematics

Numeric Fem’s Solution for Space-Time Diffusion Partial Differential Equations with Caputo–Fabrizion and Riemann–Liouville Fractional Order’s Derivatives

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this paper, we introduce one-parameter generalization of dual-hyperbolic Jacobsthal numbers – dual-hyperbolic <italic>r</italic>-Jacobsthal numbers. We present some properties of them, among others the Binet formula, Catalan, Cassini, and d’Ocagne identities. Moreover, we give the generating function and summation formula for these numbers. The presented results are a generalization of the results for the dual-hyperbolic Jacobsthal numbers.
</abstract>

One-Parameter Generalization of Dual-Hyperbolic Jacobsthal Numbers

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this short paper, I recall the history of dealing with the lack of compactness of a sequence in the case of an unbounded domain and prove the vanishing Lions-type result for a sequence of Lebesgue-measurable functions. This lemma generalizes some results for a class of Orlicz–Sobolev spaces. What matters here is the behavior of the integral, not the space.
</abstract>

A Generalized Version of the Lions-Type Lemma

<abstract xmlns="http://www.w3.org/1999/xhtml">

For a continuous and positive function <italic>w</italic> (<italic>λ</italic>) , λ <italic>></italic> 0 and <italic>µ</italic> a positive measure on (0, ∞) we consider the following <italic>integral transform</italic> 
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2023-0008_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>𝒟</mi><mrow><mo>(</mo><mrow><mi>w</mi><mo>,</mo><mi>μ</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mrow><msubsup><mo>∫</mo><mn>0</mn><mo>∞</mo></msubsup><mrow><mi>w</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow><msup><mrow><mrow><mrow><mo>(</mo><mrow><mi>λ</mi><mo>+</mo><mi>T</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>d</mi><mi>μ</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow><mo>,</mo></mrow></mrow></mrow></math>
<tex-math>\mathcal{D}\left( {w,\mu } \right)\left( T \right): = \int_0^\infty {w\left( \lambda \right){{\left( {\lambda + T} \right)}^{ - 1}}d\mu \left( \lambda \right),}</tex-math>
</alternatives>
</disp-formula>
where the integral is assumed to exist for <italic>T</italic> a positive operator on a complex Hilbert space <italic>H</italic>.
Assume that <italic>A ≥ α ></italic> 0, <italic>δ ≥ B ></italic> 0 and 0 <italic>&lt; m ≤ B − A ≤ M</italic> for some constants <italic>α, δ, m, M.</italic> Then 
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2023-0008_eq_002.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mn>0</mn><mo>≤</mo><mo>-</mo><mi>m</mi><msup><mi>𝒟</mi><mo>′</mo></msup><mrow><mo>(</mo><mrow><mi>w</mi><mo>,</mo><mi>μ</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>δ</mi><mo>)</mo></mrow><mo>≤</mo><mi>𝒟</mi><mrow><mo>(</mo><mrow><mi>w</mi><mo>,</mo><mi>μ</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>-</mo><mi>𝒟</mi><mrow><mo>(</mo><mrow><mi>w</mi><mo>,</mo><mi>μ</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow><mo>≤</mo><mo>-</mo><mi>M</mi><msup><mi>𝒟</mi><mo>′</mo></msup><mrow><mo>(</mo><mrow><mi>w</mi><mo>,</mo><mi>μ</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>,</mo></mrow></math>
<tex-math>0 \le - m\mathcal{D}'\left( {w,\mu } \right)\left( \delta \right) \le \mathcal{D}\left( {w,\mu } \right)\left( A \right) - \mathcal{D}\left( {w,\mu } \right)\left( B \right) \le - M\mathcal{D}'\left( {w,\mu } \right)\left( \alpha \right),</tex-math>
</alternatives>
</disp-formula>
where <italic>D</italic>′(<italic>w, µ</italic>) (<italic>t</italic>) is the derivative of <italic>D</italic>(<italic>w, µ</italic>) (<italic>t</italic>) as a function of <italic>t ></italic> 0.
If <italic>f</italic> : [0, <italic>∞</italic>) → ℝ is operator monotone on [0, <italic>∞</italic>) with <italic>f</italic> (0) = 0, then 
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2023-0008_eq_003.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mtable><mtr><mtd><mrow><mn>0</mn><mo>≤</mo><mfrac><mi>m</mi><mrow><msup><mrow><mi>δ</mi></mrow><mn>2</mn></msup></mrow></mfrac><mrow><mo>[</mo> <mrow><mi>f</mi><mrow><mo>(</mo><mi>δ</mi><mo>)</mo></mrow><mo>-</mo><msup><mi>f</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>δ</mi><mo>)</mo></mrow><mi>δ</mi><mo>≤</mo><mi>f</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><msup><mrow><mi>A</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>-</mo><mi>f</mi><msup><mrow><mrow><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow></mrow></mrow><mrow><mi>B</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow> <mo>]</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mo>≤</mo><mfrac><mi>M</mi><mrow><msup><mrow><mi>α</mi></mrow><mn>2</mn></msup></mrow></mfrac><mrow><mo>[</mo> <mrow><mi>f</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>-</mo><msup><mi>f</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mi>α</mi></mrow> <mo>]</mo></mrow><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math>
<tex-math>\matrix{ {0 \le {m \over {{\delta ^2}}}\left[ {f\left( \delta \right) - f'\left( \delta \right)\delta \le f\left( A \right){A^{ - 1}} - f{{\left( B \right)}^{B - 1}}} \right]} \cr { \le {M \over {{\alpha ^2}}}\left[ {f\left( \alpha \right) - f'\left( \alpha \right)\alpha } \right].} \cr }</tex-math>
</alternatives>
</disp-formula>

Some examples for operator convex functions as well as for integral transforms <italic>D</italic> (<italic>·, ·</italic>) related to the exponential and logarithmic functions are also provided.
</abstract>

Gradient Inequalities for an Integral Transform of Positive Operators in Hilbert Spaces

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this article, we derive a great number of identities involving the <italic>ω</italic> function counting distinct prime divisors of a given number <italic>n</italic>. These identities also include Pochhammer symbols, Fibonacci and Lucas numbers and many more.
</abstract>

Identities Arising from Binomial-Like Formulas Involving Divisors of Numbers

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this paper, we present the Sugeno integral of Hermite–Hadamard inequality for the case of quasi-arithmetically convex (q-ac) functions which acts as a generator for all quasi-arithmetic means in the frame work of Sugeno integral.
</abstract>

Sugeno Integral for Hermite–Hadamard Inequality and Quasi-Arithmetic Means

<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/j_amsil-2023-0016_eq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>P</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow><mi>n</mi></msubsup><mrow><msub><mrow><mi>a</mi></mrow><mi>j</mi></msub><msup><mrow><mi>z</mi></mrow><mi>j</mi></msup></mrow></mrow></math>
<tex-math>P\left( z \right) = \sum\nolimits_{j = 0}^n {{a_j}{z^j}}</tex-math>
</alternatives>
</inline-formula> be a polynomial of degree <italic>n </italic>such that <italic>a</italic><italic>n </italic> ≥ <italic>a</italic><italic>n−</italic>1 ≥ . . . ≥ <italic>a</italic>1 ≥ <italic>a</italic>0 ≥ 0. Then according to Eneström-Kakeya theorem all the zeros of <italic>P </italic>(<italic>z</italic>) lie in <italic>|z| ≤ </italic>1. This result has been generalized in various ways (see [1, 3, 4, 6, 7]). In this paper we shall prove some generalizations of the results due to Aziz and Zargar [1, 2] and Nwaeze [7].
</abstract>

On the Zeros of Polynomials with Restricted Coefficients

Report of Meeting: The Twenty-second Debrecen–Katowice Winter Seminar on Functional Equations and Inequalities Hajdúszoboszló (Hungary), February 1–4, 2023

AHEAD OF PRINT

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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 