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

Let <italic>S</italic> be a semigroup, and let <italic>φ, ψ</italic>: <italic>S</italic> → <italic>S</italic> be two endomorphisms (which are not necessarily involutive). Our main goal in this paper is to solve the following generalized variant of d’Alembert’s functional equation
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2022-0004_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mi>ϕ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mrow><mi>ψ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mi>x</mi></mrow><mo>)</mo></mrow><mo>=</mo><mn>2</mn><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>f</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>,</mo><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi>x</mi><mo>,</mo><mi>y</mi><mi> </mi><mo>∈</mo><mi> </mi><mi>S</mi><mo>,</mo></mrow></math>
<tex-math>f\left( {x\varphi \left( y \right)} \right) + f\left( {\psi \left( y \right)x} \right) = 2f\left( x \right)f\left( y \right),\,\,\,\,\,\,x,y\, \in \,S,</tex-math>
</alternatives>
</disp-formula>
where <italic>f</italic> : <italic>S</italic> → ℂ is the unknown function by expressing its solutions in terms of multiplicative functions. Some consequences of this result are presented.
</abstract>

Department of Mathematics, Faculty of Sciences

A Variant of D’Alembert’s Functional Equation on Semigroups with Endomorphisms

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

This paper presents a constructive proof of the existence of a regular non-atomic strictly-positive measure on any second-countable non-atomic locally compact Hausdorff space. This construction involves a sequence of finitely-additive set functions defined recursively on an ascending sequence of rings of subsets with a set function limit that is extendable to a measure with the desired properties. Non-atomicity of the space provides a meticulous way to ensure that the set function limit is <italic>σ</italic>-additive.
</abstract>

Construction of Regular Non-Atomic Strictly-Positive Measures in Second-Countable Non-Atomic Locally Compact Hausdorff Spaces

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

We construct a separately continuous function <italic>f</italic> : ℚ × ℚ → [0; 1] and a dense subset <italic>D</italic> ⊆ ℚ × ℚ such that <italic>f</italic>[D] is not dense in f[ℚ × ℚ], in other words, <italic>f</italic> is separately continuous and not somewhat (feebly) continuous.
</abstract>

A Separately Continuous Function Not Somewhat Continuous

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

The primary object of study is the “cosine-sine” functional equation <italic>f</italic>(<italic>xy</italic>) = <italic>f</italic>(<italic>x</italic>)<italic>g</italic>(<italic>y</italic>)+<italic>g</italic>(<italic>x</italic>)<italic>f</italic>(<italic>y</italic>)+<italic>h</italic>(<italic>x</italic>)<italic>h</italic>(<italic>y</italic>) for unknown functions <italic>f, g, h</italic> : <italic>S →</italic> ℂ, where <italic>S</italic> is a semigroup. The name refers to the fact that it contains both the sine and cosine addition laws. This equation has been solved on groups and on semigroups generated by their squares. Here we find the solutions on a larger class of semigroups and discuss the obstacles to finding a general solution for all semigroups. Examples are given to illustrate both the results and the obstacles.
We also discuss the special case <italic>f</italic>(<italic>xy</italic>) = <italic>f</italic>(<italic>x</italic>)<italic>g</italic>(<italic>y</italic>) + <italic>g</italic>(<italic>x</italic>)<italic>f</italic>(<italic>y</italic>) <italic>− g</italic>(<italic>x</italic>)<italic>g</italic>(<italic>y</italic>) separately, since it has an independent direct solution on a general semigroup.
We give the continuous solutions on topological semigroups for both equations.
</abstract>

The Cosine-Sine Functional Equation on Semigroups

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

In this miniature note we generalize the classical Gauss congruences for integers to rings of integers in algebraic number fields.
</abstract>

Gauss Congruences in Algebraic Number Fields

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

Jacobsthal numbers are a special case of numbers defined recursively by the second order linear relation and for these reasons they are also named as numbers of the Fibonacci type. They have many interpretations, representations and applications in distinct areas of mathematics. In this paper we present the Jacobsthal representation hybrinomials, i.e. polynomials, which are a generalization of Jacobsthal hybrid numbers.
</abstract>

Jacobsthal Representation Hybrinomials

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

Let <italic>S</italic> be a semigroup and α, β ∈ ℝ. The purpose of this paper is to determine the general solution <italic>f</italic> : ℝ2 → <italic>S</italic> of the following parametric functional equation <disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2022-0001_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>x</mi></mrow><mn>1</mn></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mn>2</mn></msub><mo>+</mo><mi>α</mi><msub><mrow><mi>y</mi></mrow><mn>1</mn></msub><msub><mrow><mi>y</mi></mrow><mn>2</mn></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mn>1</mn></msub><msub><mrow><mi>y</mi></mrow><mn>2</mn></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mn>2</mn></msub><msub><mrow><mi>y</mi></mrow><mn>1</mn></msub><mo>+</mo><mi>β</mi><msub><mrow><mi>y</mi></mrow><mn>1</mn></msub><msub><mrow><mi>y</mi></mrow><mn>2</mn></msub></mrow><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>x</mi></mrow><mn>1</mn></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mn>1</mn></msub></mrow><mo>)</mo></mrow><mi>f</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>x</mi></mrow><mn>2</mn></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mn>2</mn></msub></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math>
<tex-math>f\left( {{x_1} + {x_2} + \alpha {y_1}{y_2},{x_1}{y_2} + {x_2}{y_1} + \beta {y_1}{y_2}} \right) = f\left( {{x_1},{y_1}} \right)f\left( {{x_2},{y_2}} \right),</tex-math>
</alternatives>
</disp-formula> for all (<italic>x</italic>1, <italic>y</italic>1), (<italic>x</italic>2, <italic>y</italic>2) ∈ ℝ2, that generalizes some functional equations arising from number theory and is connected with the characterizations of the determinant of matrices.
</abstract>

A Parametric Functional Equation Originating from Number Theory

Report of Meeting: The Twenty-first Katowice–Debrecen Winter Seminar on Functional Equations and Inequalities Brenna (Poland), February 2–5, 2022

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