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

The aim of this paper is to present background information in relation with some fractional-order type operators in the complex plane, which is designed by the fractional-order derivative operator(s). Next we state various implications of that operator and then we show some interesting-special results of those applications.
</abstract>

An extensive note on various fractional-order type operators and some of their effects to certain holomorphic functions

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

In this work, we characterize the class of compact matrix operators from <italic>c</italic>0(<italic>Q</italic>), <italic>c</italic>(<italic>Q</italic>) and <italic>ℓ</italic>∞ (<italic>Q</italic>) into <italic>c</italic>0, <italic>c</italic> and <italic>ℓ</italic>∞, respectively, with the notion of the Hausdorff measure of noncompactness. Moreover, we determine some geometric properties of the sequence space <italic>ℓp</italic>(<italic>Q</italic>).
</abstract>

Recep Tayyip Erdogan University, Mathematics

Compactness of quadruple band matrix operator and geometric properties

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

Using the notion of operations on a generalized topological space (<italic>X, µ</italic>) and a hereditary class we have introduced the notion of <italic>γµ</italic>-compactness modulo a hereditary class ℋ termed as <italic>γµ</italic>ℋ -compactness. We have studied <italic>γµ</italic>ℋ-compact spaces and <italic>γµ</italic>ℋ-compact sets relative to <italic>µ</italic>.
</abstract>

<italic>γµ</italic>ℋ-compactness in GTS

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

BCK-sequences and <italic>n</italic>-commutative BCK-algebras were introduced by T. Traczyk, together with two related problems. The first one, whether BCK-sequences are always prolongable. The second one, if the class of all <italic>n</italic>-commutative BCK-algebras is characterised by one identity. W. A. Dudek proved that the answer to the former question is positive in some special cases, e.g. when BCK-algebra is linearly ordered. T. Traczyk showed that the answer to the latter is a˚rmative for <italic>n</italic> = 1, 2. Nonetheless, by providing counterexamples, we proved that the answers to both those open problems are negative.
</abstract>

On Traczyk’s BCK-sequences

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

The classic method of solving the cubic and the quartic equations using Tschirnhaus transformation yields true as well as false solutions. Recently some papers on this topic are published, in which methods are given to get only the true solutions of cubic and quartic equations. However these methods have some limitations. In this paper the author presents a method of solving cubic and quartic equations using Tschirnhaus transformation, which yields only the true solutions. The proposed method is much simpler than the methods published earlier.
</abstract>

A simpler method to get only the true solutions of cubic and quartic equations using Tschirnhaus transformation

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

Since <italic>Geronimo Cardano</italic>, algebraic equations of degree 4 have been solved analytically. Frequently, the solution algorithm is given in its entirety. We discovered two algorithms that lead to the same <italic>resolvente</italic>, each with two solutions; therefore, six formal solutions appear to solve an algebraic equation of degree four. Given that a square was utilized to derive the solution in both instances, it is critical to verify each solution. This check reveals that the four Cardanic solutions are the only four solutions to an algebraic equation of degree four. This demonstrates that <italic>Carl Friedrich Gauss</italic>’ (1799) <italic>fundamental theorem of algebra</italic> is not simple, despite the fact that it is a fundamental theorem. This seems to be a novel insight.
</abstract>

Solution to algebraic equations of degree 4 and the fundamental theorem of algebra by Carl Friedrich Gauss

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

In this paper we continue our investigation concerning the concept of a <italic>liken</italic>. This notion has been defined as a sequence of non-negative real numbers, tending to infinity and closed with respect to addition in ℝ. The most important examples of likens are clearly the set of natural numbers ℕ with addition and the set of positive natural numbers ℕ* with multiplication, represented by the sequence <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_aupcsm-2022-0007_eq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mrow><mrow><mo>(</mo><mrow><mo>ln</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mo>∞</mo></msubsup></mrow></math>
<tex-math>\left( {\ln \left( {n + 1} \right)} \right)_{n = 0}^\infty</tex-math>
</alternatives>
</inline-formula>. The set of all likens can be parameterized by the points of some infinite dimensional, complete metric space. In this <italic>space of likens</italic> we consider elements up to isomorphism and define <italic>properties of likens</italic> as such that are isomorphism invariant. The main result of this paper is a theorem characterizing the liken ℕ* of natural numbers with multiplication in the space of all likens.
</abstract>

On a certain characterisation of the semigroup of positive natural numbers with multiplication

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

The class of pseudo-M algebras contains pseudo-BCK, pseudo-BCI, pseudo-BCH, pseudo-BE, pseudo-CI algebras and many other algebras of logic. In this paper, the notion of deductive system in a pseudo-M algebra is introduced and its elementary properties are investigated. Closed deductive systems are defined and studied. The homomorphic properties of (closed) deductive systems are provided. The concepts of translation deductive systems and R-congruences in pseudo-M algebras are introduced and investigated. It is shown that there is a bijection between closed translation deductive systems and R-congruences. Finally, the construction of quotient algebra 𝒜/<italic>D</italic> of a pseudo-M algebra 𝒜 via a translation deductive system <italic>D</italic> of 𝒜 is given.
</abstract>

Deductive systems of pseudo-M algebras

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Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

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