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In this paper we continue our investigation concerning the concept of a liken. 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 (ln(n+1))∞ n=0. The set of all likens can be parameterized by the points of some infinite dimensional, complete metric space. In this space of likens we consider elements up to isomorphism and define properties of likens 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.

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.

L. Losonczi [4] determined local solutions of the generalized Cauchy equation f(F(x, y))= f(x) + f(y) on components of the denition of a given associative rational function F. The class of the associative rational function was described by A. Chéritat [1] and his work was followed by paper [3] of the author. The aim of the present paper is to describe local solutions of the equation considered for some singular associative rational functions.

Student knowledge assessment is a key element of the pedagogical process, as it provides students, parents, and educators with important feedback information on students’ knowledge and skills. Assessment of students’ mathematical knowledge is complex, as several factors are normally included in students’ final grade, and simply calculating the average of students’ achievements may not provide a complete picture of their knowledge. Therefore, in this paper, we aimed to investigate the possibility of using fuzzy logic to assess students’ knowledge and competencies by considering (1) students’ overall academic performance, (2) the quality of students’ project work on a topic from the history of mathematics and (3) the regularity of handing in homework. The study was conducted considering 22 Italian high school students. The results show that students’ academic performance is similar to student grades obtained with fuzzy logic.

In this paper, we discuss the no-arbitrage condition in a discrete financial market model which does not hold the same interest rate assumptions. Our research was based on, essentially, one of the most important results in mathematical finance, called the Fundamental Theorem of Asset Pricing. For the standard approach a risk-free bank account process is used as numeraire. In those models it is assumed that the interest rates for borrowing and saving money are the same. In our paper we consider the model of a market (with d risky assets), which does not hold the same interest rate assumptions. We introduce two predictable processes for modelling deposits and loans. We propose a new concept of a martingale pair for the market and prove that if there exists a martingale pair for the considered market, then there is no arbitrage opportunity. We also consider special cases in which the existence of a martingale pair is necessary and the sufficient conditions for these markets to be arbitrage free

This paper is devoted to the study of a multi-step method with divided differences for solving nonlinear equations in Banach spaces. In earlier studies, hypotheses on the Fréchet derivative up to the sixth order of the operator under consideration is used to prove the convergence of the method. That restricts the applicability of the method. In this paper we extended the applicability of the sixth-order multi-step method by using only hypotheses on the first derivative of the operator involved. Our convergence conditions are weaker than the conditions used in earlier studies. Numerical examples where earlier results cannot be applied to solve equations but our results can be applied are also given in this study.

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/D of a pseudo-M algebra A via a translation deductive system D of A is given.

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.

It is widely believed that math classes in primary school belong into the category of least favorite classes. Entire teams of psychologists, educators and mathematicians have studied the student aversion (and, to a certain extent, parent antipathy) to math classes, but they have also studied methods and ways to overcome this problem. They have tried to find a way to improve students’ activities, to make them more engaged in classes, and to increase their motivation. One of the opinions is that the student success requires their strong will, persistence, diligence and perseverance. Certainly, the competence, motivation and consistency of teachers, wholead them to the world of knowledge, have a great influence. One of the methods of achieving a positive influence of teachers, in the domain of increasing interest in mathematics and reducing aversion to it, is the use of interesting mathematical problems, in which the numerical value of the current calendar year or even the current date appears. The aim of this paper is to point out how the motivation of primary school students to adopt mathematical content can be increased. With concrete examples of interesting algebra problems, which tickle students’ fancy and make them think, we show how some lessons can be learned.

The topic of our paper is the hat problem in which each of n players is randomly fitted with a blue or red hat. Then everybody can try to guess simultaneously his own hat color by looking at the hat colors of the other players. The team wins if at least one player guesses his hat color correctly, and no one guesses his hat color wrong; otherwise the team loses. The aim is to maximize the probability of a win. There are known many variations of the hat problem. In this paper we give a comprehensive list of variations considered in the literature. We describe the applications of the hat problem and its variations, and their connections to different areas of science. We give the full bibliography of any papers, books, and electronic publications about the hat problem.

The paper gives a general construction of all continuous solutions of inequality (1) fulfilling one of conditions (5) or (26). This paper is a continuation of [3].

In Plato’s later ethics, the key challenge to one’s personal progress is unorderly thoughts and motions of soul. This disorder should be corrected by recognising and repeating the orderly thoughts and motions of cosmic gods. This paper examines the theoretical assumptions of Plato’s later ethics. A closer analysis of the Timaeus reveals a peculiar conception of mathematics, according to which numbers and their relations not only reveal facts about the reality, but also express values and ideals. By learning mathematics human beings come to know the everyday ethical life of cosmic gods and become like them.

O objetivo deste artigo é analisar, a partir dos textos de Platão e de comentadores, a apresentação de argumentos a favor da utilização da matemática e da geometria como propedêutica à aprendizagem da filosofia, bem como investigar as reverberações da ontologia e da epistemologia platônicas nesse programa pedagógico. Pretende-se, ainda, apontar comparativamente similaridades entre crises nos fundamentos da matemática e seu impacto na concepção de racionalidade, tanto no universo grego antigo como na contemporaneidade.The aim of this paper is to analyze, based on Plato's and commentators' texts, the presentation of arguments for the use of mathematics and geometry as propaedeutics to philosophy learning, as well as to investigate the reverberations of platonic ontology and epistemology in this pedagogical program. The paper will also consider similarities between crises in the foundations of mathematics and the impact of these crises on the conception of rationality, both in ancient Greek universe and in contemporary world.

This paper investigates the problem of the existence and uniqueness of solutions under the generalized self-similar forms to the space-fractional diffusion equation. Therefore, through applying the properties of Schauder's and Banach's fixed point theorems; we establish several results on the global existence and blow-up of generalized self-similar solutions to this equation.

This paper investigates the problem of the existence and uniqueness of solutions under the generalized self-similar forms to the space-fractional diffusion equation. Therefore, through applying the properties of Schauder's and Banach's fixed point theorems; we establish several results on the global existence and blow-up of generalized self-similar solutions to this equation.

In this paper, we use a characterization of the mutual multifractal Hausdorff dimension in terms of auxiliary measures to investigate the projections of measures with small supports.

The aim of this paper is to prove the existence and stability of solutions of a system of quadratic integral equations in the Banach algebra of continuous and bounded functions on unbounded rectangle. The main tool used in our considerations is the multiple fixed point theorem which is a consequence of Darbo's fixed point theorem and the technique associated with measures of noncompactness. We also present an illustrative example.

The main purpose of the present paper is to demonstrate that as early as 1904 pre-eminent American mathematician Maxime Bôcher was an adherent to the presently relevant argument of reasonableness, or even necessity of parallel development of two philosophical methods of reflection on mathematics, so that its essence could be more fully comprehended. The goal of the research gives rise to the question: what two types of philosophical deliberation on mathematics were proposed by Bôcher?

In this paper, we introduce multi-Jensen-cubic mappings and unify the system of functional equations defining the multi-Jensen-cubic mapping to a single equation. Applying a fixed point theorem, we establish the generalized Hyers-Ulam stability of multi-Jensen-cubic mappings. As a known outcome, we show that every approximate multi-Jensen-cubic mapping can be multi-Jensen-cubic.

In the present paper, we study in the harmonic analysis associated to the Weinstein operator, the boundedness on Lp of the uncentered maximal function. First, we establish estimates for the Weinstein translation of characteristic function of a closed ball with radius ε centered at 0 on the upper half space Rd-1× ]0,+∞[. Second, we prove weak-type L1-estimates for the uncentered maximal function associated with the Weinstein operator and we obtain the Lp-boundedness of this operator for 1 < p ≤+∞. As application, we define a large class of operators such that each operator of this class satisfies these Lp-inequalities. In particular, the maximal operator associated respectively with the Weinstein heat semigroup and the Weinstein-Poisson semigroup belong to this class.

This paper presents a theorem dealing with absolute matrix summability of infinite series. This theorem has been proved taking quasi β-power increasing sequence instead of almost increasing sequence.

In this paper, we derive several subordination results and integral means result for certain class of analytic functions defined by means of q-differential operator. Some interesting corollaries and consequences of our results are also considered.

Let f be a polynomial in two complex variables. We say that f is nearly irreducible if any two nonconstant polynomial factors of f have a common zero in C2. In the paper we give a criterion of nearly irreducibility for a given polynomial f in terms of its Newton diagram.

This paper presents some generalizations of BCI algebras (the RM, tRM, *RM, RM**, *RM**, aRM**, *aRM**, BCH**, BZ, pre-BZ and pre-BCI algebras). We investigate the p-semisimple property for algebras mentioned above; give some examples and display various conditions equivalent to p-semisimplicity. Finally, we present a model of mereology without antisymmetry (NAM) which could represent a tRM algebra.

The object of the present paper is to investigate the nature of Ricci solitons on D-homothetically deformed Kenmotsu manifold with generalized weakly symmetric and generalized weakly Ricci symmetric curvature restrictions.

In the present paper, two theorems of absolute summability have been proved by using the definition of almost increasing sequence.

In this paper we establish a new fractional identity involving a function oftwo independent variables, and then we derive some fractionalHermite-Hadamard type integral inequalities for functions whose modulus ofthe mixed derivatives are co-ordinated s-preinvex in the second sense.

In this paper, a known result dealing with |N, pn|k summability of infinite series has been generalized to the φ-|N, pn;δ|k infinite series by using an almost increasing sequence.

The convex hull of the subgraph of the prime counting function x → π(x) is a convex set, bounded from above by a graph of some piecewise affine function x → ε(x). The vertices of this function form an infinite sequence of points (ek,π(ek))1∞. The elements of the sequence (ek)1∞ shall be called the extremal prime numbers. In this paper we present some observations about the sequence (ek)1∞ and we formulate a number of questions inspired by the numerical data. We prove also two - it seems - interesting results. First states that if the Riemann Hypothesis is true, then lim(ek+1/ek)=1. The second, also depending on Riemann Hypothesis, describes the order of magnitude of the differences between consecutive extremal prime numbers.

In this paper we study existence and uniqueness of solutions for a coupled system consisting of fractional differential equations of Caputo type, subject to Riemann–Liouville fractional integral boundary conditions. The uniqueness of solutions is established by Banach contraction principle, while the existence of solutions is derived by Leray–Schauder’s alternative. We also study the Hyers–Ulam stability of mentioned system. At the end, examples are also presented which illustrate our results.

It has been a long-standing problem posed by the first author in a conference in Marseille in 1990 to characterize semitopological semigroups which have common fixed point property when acting on a nonempty weak* compact convex subset of a dual Banach space as weak* continuous and norm nonexpansive mappings. Our investigation in the paper centers around this problem. Our main results rely on the well-known Ky Fan's inequality for convex functions.

In this paper I argue that Aristotle's understanding of mathematical continuity constrains the mathematical ontology he can consistently hold. On my reading, Aristotle can only be a mathematical abstractionist of a certain sort. To show this, I first present an analysis of Aristotle's notion of continuity by bringing together texts from his Metaphysica and Physica, to show that continuity is, for Aristotle, a certain kind of per se unity, and that upon this rests his distinction between continuity and contiguity. Next I argue briefly that Aristotle intends for his discussion of continuity to apply to pure mathematical objects such as lines and figures, as well as to extended bodies. I show that this leads him to a difficulty, for it does not at first appear that the distinction between continuity and contiguity can be preserved for abstract mathematicals. Finally, I present a solution according to which Aristotle's understanding of continuity can only be saved if he holds a certain kind of mathematical ontology.

Each family M of means has a natural, partial order (point-wise order), that is M ≤ N iff M(x) ≤ N(x) for all admissible x. In this setting we can introduce the notion of interval-type set (a subset I ⊂ M such that whenever M ≤ P ≤ N for some M,N ∈ I and P ∈ M then P ∈ I). For example, in the case of power means there exists a natural isomorphism between interval-type sets and intervals contained in real numbers. Nevertheless there appear a number of interesting objects for a families which cannot be linearly ordered. In the present paper we consider this property for Gini means and Hardy means. Moreover some results concerning L∞ metric among (abstract) means will be obtained.

In this paper, we study *-g-frames in tensor products of Hilbert C*-modules. We show that a tensor product of two *-g-frames is a *-g-frame, and we get some result.

In this paper, we define a new subclass of bi-univalent functions involving q-difference operator in the open unit disk. For functions belonging to this class, we obtain estimates on the first two Taylor-Maclaurin coefficients |a2| and |a3|.

In this paper, we define a new subclass of bi-univalent functions involving a Hohlov operator in the open unit disk. For functions belonging to this class, we obtain estimates on the first two Taylor-Maclaurin coefficients |a2| and | a3|.

In the article some philosophical issues regarding the emergence of the modern cosmology are discussed. The mathematical field equations enabled for the first time in the history to set a cosmological problem as a physical one. The “power” of mathematics was not easy to be read by the pioneers of cosmology. Also the philosophical perspective on the Universe changed in front of this “power” expressed by Einstein and de Sitter in their models. In the paper we propose also some new ideas related to the mathematics of cosmology as a physical science.

In the present paper, we define a notion of an m2-topological space by introducing a count of openness of a multiset (mset in short) and study the properties of m2-subspaces, mgp-maps etc. Decomposition theorems involving m-topologies and m2-topologies are established. The behaviour of the functional image and functional preimage of an m2-topologies, the continuity of the identity mapping and a constant mapping in m2-topologies are also examined.

The first result in partition relations topic belongs to Ramsey (1930). Since that this topic has been still explored. Probably the most famous partition theorem is Erdős-Rado theorem (1956). On the other hand in 60's of the last century Efimov introduced strong sequences method, which was used for proving some famous theorems in dyadic spaces. The aim of this paper is to generalize theorem on strong sequences and to show that it is equivalent to generalized version of well-known Erdős-Rado theorem. It will be also shown that this equivalence holds for singulars. Some applications and conclusions will be presented too.

Without doubt, the principle of least action is a fundamental principle in classical mechanics. Contemporary physicists, however, consider the PLA as a purely mathematical principle – even an axiom which they cannot completely justify. Such an account stands in sharp contrast with the historical meaning of the PLA. When the principle was introduced in the 1740s, by Pierre-Louis Moreau de Maupertuis, its meaning was much more versatile. For Maupertuis the principle of least action signified that nature is thrifty or economical in all its actions, i.e., that nature avoids to do anything unnecessary. Maupertuis understood the principle in teleological terms and even considered the principle as an expression of God’s wisdom. It has been correctly pointed out by historians that Maupertuis in his later years moved towards a more speculative and metaphysical approach, whereas his contemporary Euler and later Lagrange, wanted to avoid such theological and metaphysical implications and frame the PLA (in line with contemporary standards) in purely mathematical terms. Such readings, however, have had the unintended side-effect that they lose out of sight the question how the mathematical and metaphysical aspect of the principle of least action fit together within Maupertuis’ own work. Investigating if and how the mathematical and metaphysical aspects of the PLA are compatible within Maupertuis’ thought will be the main goal of this paper.

Wolff and Kant agree that the «mathematical method» is, generally speaking, the axiomatic-deductive method of Euclid’s Elements. Each demonstration is carried out with recourse to an individual geometric figure in order to prove a general proposition about said figure with apodictic certainty. Although this figure is an individual, say a triangle, it can be used to prove propositions which hold for all triangles: it is an individual which represents the universal. While Wolff and Kant agree that these are the steps comprising the mathematical method, they disagree on the scope of its application. Wolff thinks that a thing in nature can also be treated like a geometric figure, i.e., that one can demonstrate general and certain propositions from it. This paper will discuss the metaphysical underpinnings of Wolff’s application of the mathematical method to natural philosophy. Specifically, it will take up Wolff’s modal metaphysics, as it pertains essences, which, on my interpretation, explains why Wolff thinks he can treat a thing in nature as an individual which represents the universal. I will then present Kant’s arguments against such a theory of essences and explain why this precludes the employment of the mathematical method in natural philosophy, for Kant.

Este ensayo atiende al momento seguramente más ambicioso de la historia de la filosofía: la conversión, en el seno del Idealismo Alemán del modus exponendi de aquélla, primero en Crítica (Kant), luego en Doctrina (Fichte), en Sistema (Schelling), y finalmente en Ciencia (Hegel), al mismo tiempo que, en el modus componendi, la Metafísica se transfigura en Lógica. Ello supone, contra una interpretación manualística ya apenas seguida, no abandono o menosprecio por parte de Hegel de las ciencias experimentales, sino al contrario: una minuciosa utilización de las mismas como origen, condición y material de elaboración del propio desarrollo lógico, de manera que, mientras el plano de la exposición (el methodus) permanece relativamente fijo, el de la composición, mediado por la Realphilosophie, se modifica y altera al ritmo de las teorías y los acontecimientos del día de los hombres, como si su “peso” fuera dejando una huella en el reino lógico de las sombras. [This paper examines the probably most ambitious moment of the history of philosophy: the transformation, at the center of the German Idealism, of Philosophy into Critique (Kant), Doctrine (Fichte), System (Schelling) and finally Science (Hegel), as regards the philosophical modus exponendi, and simultaneously the transfiguration of Metaphysics into Logic, in its modus componendi. Rejecting the old-fashioned handbook interpretations, I claim these transformations do not make Hegel neglect or contempt the experimental sciences. On the contrary, they prove he meticulously used the experimental sciences as origin, condition and elaboration material of the logical development, so that while the exposition level (methodus) remains relatively firm, the composition level, mediated by the Realphilosophie, is modified and altered according to the human theories and the human daily events, as if its "weight" left its traces in the logical realm of shadows.]

In this paper, We study the superstablity problem of the cosine and sine type functional equations: f(xσ(y)a)+f(xya)=2f(x)f(y) and f(xσ(y)a)-f(xya)=2f(x)f(y), where f:S → C is a complex valued function; S is a semigroup; σ is an involution of S and a is a fixed element in the center of S.

In this paper, we propose and analyse an iterative algorithm for the approximation of a common solution for a finite family of k-strict pseudo-contractions and two finite families of generalized equilibrium problems in the setting of Hilbert spaces. Strong convergence results of the proposed iterative algorithm together with some applications to solve the variational inequality problems are established in such setting. Our results generalize and improve various existing results in the current literature.

For an arbitrary triangle ABC and an integer n we define points Dn, En, Fn on the sides BC, CA, AB respectively, in such a manner that |AC|n/|AB|n =|CDn|/|BDn|, |AB|n/|BC|n = |AEn|/|CEn|, |BC|n/|AC|n =|BFn|/|AFn|. Cevians ADn, BEn, CFn are said to be the Maneeals of order n. In this paper we discuss some properties of the Maneeals and related objects.

For two-dimensional manifold M with locally symmetric connection ∇ and with ∇-parallel volume element vol one can construct a flat connection on the vector bundle TM⊕E, where E is a trivial bundle. The metrizable case, when M is a Riemannian manifold of constant curvature, together with its higher dimension generalizations, was studied by A.V. Shchepetilov [J. Phys. A: 36 (2003), 3893-3898]. This paper deals with the case of non-metrizable locally symmetric connection. Two flat connections on TM⊕(R⨯M) and two on TM⊕(R2⨯M) are constructed. It is shown that two of those connections - one from each pair - may be identified with the standard flat connection in RN, after suitable local affine embedding of (M,∇) into RN.

In this paper we define strong ideals and horizontal ideals in pseudo-BCH-algebras and investigate the properties and characterizations of them.

The aim of this paper is to study the superstability problem of the d'Alembert type functional equation f(x+y+z)+f(x+y+σ(z))+f(x+σ(y)+z)+f(σ(x)+y+z)=4f(x)f(y)f(z) for all x,y,z ∈ G, where G is an abelian group and σ: G → G is an endomorphism such that σ(σ(x))=x for an unknown function f from G into C or into a commutative semisimple Banach algebra.

In this paper we study the comparative growth properties of a composition of entire and meromorphic functions on the basis of the relative order (relative lower order) of Wronskians generated by entire and meromorphic functions.