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We construct a two discrete parameter family of compact minimal surfaces embedded in the Berger sphere which may be considered as the analogue of the helicoidal Karcher-Scherk surfaces.

This appendix gives a lower bound of the Billingsley-Hausdorff dimension of a level set related to Birkhoff average in the “non-compact” symbolic space $\mathbb{N}^{\mathbb{N}}$, defined by Gibbs measure.

It is a singular fact that the first and pithy de Finetti’s essay on exchangeability has not earned the same reputation as that of others of his papers about the same subject. In fact, this paper contains, on the one hand, all the main results on sequences of exchangeable events, together with the right subjectivistic interpretation of the role they play in the study of the connections between probability and frequencies. On the other hand, the paper makes use of mathematical methods abandoned, immediately after its publication, by de Finetti himself. The center of this methods is the so–called characteristic function of a random phenomenon. Independently of the destiny of the paper, we think that, apart from its undoubted historical value, it contains ideas susceptible of interesting new developments. Therefore, we have deemed it suitable to give here a detailed and faithful account of its content, for the benefit of the colleagues who are not in a position to understand Italian. Moreover, to emphasize the value of the paper at issue, we develop de Finetti’s brief hint to the extendibility of exchangeable sequences of events, to obtain a new explicit necessary and sufficient condition of an algebraic nature.

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For an involution near ring 𝒩 and its ideal ℐ, the text introduces an involution ideal-based zero- divisor graph Γℐ*(𝒩) which is an undirected graph with vertex set { x ∈ 𝒩 - ℐ: x𝒩y ⊂ ℐ (or y𝒩x ⊂ ℑ ) for some y ∈ 𝒩-ℐ}, where two distinct vertices x and y are adjacent if and only if y𝒩x* ⊂ ℐ or x𝒩y* ⊂ ℐ. The paper provides characterizations of Γℐ*(𝒩) when it forms a complete graph or a star graph. It also explores the structure of Γℐ*(𝒩), investigates its properties like connectedness with diam(Γℐ*(𝒩)) ≤ 3 and analyzes the connection of Γℐ*(𝒩) with Γℐ*(𝒩/ℐ). Furthermore, the paper discusses the chromatic number and clique number of the graph. It also characterizes all right permutable *-near-rings 𝒩 for which the graph Γℐ*(𝒩) can be with a finite chromatic number.

n the present paper, the authors introduce a new subclass namely; RL(φ4L, ν) of Kamali-type starlike functions defined in the open unit disk D connected with four-leaf shaped domain. We investigate the bounds of some initial coefficients, Fekete-Szegö inequality and other results of logarithmic coefficients for the function belonging to above class. Relevant connections of the results derived in this paper with those of earlier works are indicated.

Copyright of Enfance en Difficulté is the property of UQTR for the purpose & interests in Revue Enfance en difficulte & Education Inclusive and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Strong convergence theorems that approximate common fixed points of two nonlinear mappings are presented. Our method is based on the Martinez-Yanes–Xu iteration, which extends Nakajo and Takahashi’s CQ method. In this paper, by exploiting the mean-valued iteration procedure, we further develop Nakajo and Takahashi’s CQ method and Takahashi, Takeuchi, and Kubota’s shrinking projection method. The approach of this paper does not require that the two mappings be continuous or commutative. The types of mappings considered in this paper include nonexpansive mappings and other well-known classes of mappings as special cases.

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.

In 1985 Crapo introduced in [3] a new mathematical object that he called geometry of circuits. Four years later, in 1989, Manin and Schechtman defined in [13] the same object and called it discriminantal arrangement, the name by which it is known now a days. Those discriminantal arrangements $\mathcal{B}(n,k,\mathcal{A}^0)$ are builded from an arrangement $\mathcal{A}^0$ of $n$ hyperplanes in general position in a $k$-dimensional space and their combinatorics depends on the arrangement $\mathcal{A}^0$. On this basis, in 1997 Bayer and Brandt (see [2]) distinguished two different type of arrangements $\mathcal{A}^0$ calling very generic the ones for which the intersection lattice of $\mathcal{B}(n,k,\mathcal{A}^0)$ has maximum cardinality and non-very generic the others. Results on the combinatorics of $\mathcal{B}(n,k,\mathcal{A}^0)$ in the very generic case already appear in Crapo [3] and in 1997 in Athanasiadis [1] while the first known result on non-very generic case is due to Libgober and the first author in 2018. In their paper [12] they provided a necessary and sufficient condition on $\mathcal{A}^0$ for which the cardinality of rank 2 intersections in $\mathcal{B}(n,k,\mathcal{A}^0)$ is not maximal anymore. In this paper we further develop their result providing a sufficient condition on $\mathcal{A}^0$ for which the cardinality of rank r, $r \ge 2$, intersections in $\mathcal{B}(n,k,\mathcal{A}^0)$ decreases.

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.

Copyright of Filozofski Vestnik is the property of Scientific Research Centre of Slovenian Academy of Sciences & Arts and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Mesure et Évaluation en Éducation is the property of ADMEE-Canada and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The purpose of the present paper is to investigate the Sălăgean - type probability distribution of order α. In this paper, we obtain moments, factorial moments and moment generating function for this distribution. Finally, we obtain a entropy of this distribution. Our results improve and generalize the results of Porwal [Starlike and convex type probability distribution, Afr. Mat. 30 (2019)].

Anantharaman and Le Masson proved that any family of eigenbases of the adjacency operators of a family of graphs is quantum ergodic (a form of delocalization) assuming the graphs satisfy conditions of expansion and high girth. In this paper, we show that neither of these two conditions is sufficient by itself to necessitate quantum ergodicity. We also show that having conditions of expansion and a specific relaxation of the high girth constraint present in later papers on quantum ergodicity is not sufficient. We do so by proving new properties of the Cartesian product of two graphs where one is infinite.

The paper summarizes the main core of the last results that we obtained in [8, 4, 17] on the regularity of the value function for a Bolza problem of a one-dimensional, vectorial problem of the calculus of variations. We are concerned with a nonautonomous Lagrangian that is possibly highly discontinuous in the state and velocity variables, nonconvex in the velocity variable and non coercive. The main results are achieved under the assumption that the Lagrangian is convex on the one-dimensional lines of the velocity variable and satisfies a local Lipschitz continuity condition w.r.t. the time variable, known in the literature as Property (S), and strictly related to the validity of the Erdmann–Du-Bois Reymond equation.Under our assumptions, there exists a minimizing sequence of Lipschitz functions. A first consequence is that we can exclude the presence of the Lavrentiev phenomenon. Moreover, under a further mild growth assumption satisfied by the minimal length functional, fully described in the paper, the above sequence may be taken with the same Lipschitz rank, even when the initial datum and initial value vary on a compact set. The Lipschitz regularity of the value function follows.

Problems in mathematics are a frequent major complaint in neuropediatric departments, for which there are two explanatory theoretical models: the hypothesis of a genetic and modular origin (with a number sense deficit) and a multidetermined origin. The purpose of this paper is to review the mathematical difficulties described in Turner syndrome and Fragile X syndrome, because a specific mathematical disorder is usually reported in these populations, supporting the existence of a number sense. Analysis of the literature reveals highly variable cognitive phenotypes in these populations, especially regarding mathematical abilities. Performance heterogeneity might be related to different factors such as the abilities needed to perform the task, the variability of definitions, the different tests used in the studies and the heterogeneity of the syndromes themselves. A number sense deficit is usually described in these syndromes, but variable cognitive impairments are also observed. The idea of a modular functioning is then debated and we argue for the necessity of a global cognitive evaluation approach., (Copyright © 2018. Published by Elsevier SAS.)

The last three decades two distinct trends have emerged within Early Childhood Education, namely Early Childhood Science Education and Early Childhood Special Education. The current paper tries to move along these areas and illuminate the common ground that lie between them. In particular, drawing from the formulated empiricist, Piagetian and socio-cognitive trends within Early Childhood Science Education, it tries to explore to what extend they could expand in special needs context. In addition, it proposes diverse, specific strategies that emerge from the above-mentioned trends and could be implemented in teaching science within special needs settings. By so doing, the present paper aspires to set the frame of a newly born and well-promising trend, that of Early Childhood Special Science Education.

In this paper we present a short overview of results that provide relationships among scalar, matrix-valued and certain operator-valued positive definite kernels. We refine and extend some of them in order that they may be applied for strict positive definiteness as well. This is a topic not well explored in the literature but that has potential usefulness in the characterization of several classes of positive definite and strictly positive definite kernels. This is ratified in the paper with the inclusion of a number of applications and examples.

The representation of numbers is assumed to interact with two visuo-motor functions. On the one hand, following the observation that children often use their fingers to learn the counting sequence and basic arithmetic operations, numbers were assumed to interact with finger movements. On the other hand, following the recurrent observation that small numbers are preferentially associated with the left side of space while large numbers are preferentially associated with the right side of space, numbers were assumed to interact with space. In this paper, we will examine the role vision plays in shaping these interactions between fingers and numbers and between numbers and space. To this aim, different experiments with blind and sighted people will be detailed. The role of developmental vision on the development of the finger-numeral representation will first be examined. Then we will investigate the influence of developmental vision on the visuo-spatial representation of numbers., (© 2016 médecine/sciences – Inserm.)

The purpose of this paper is to establish necessary and sufficient optimality conditions and some duality results for weakly efficient solutions of a constrained bilevel multiobjective fractional programming problem (P) with an extremal-value function. Using parametric approach, the problem (P) is first equivalently transformed into a parametric problem (P µ) with µ ∈ Rp, for which we construct then a dual problem. This is achieved in terms of conjugate duality theory. Under appropriate assumptions, the weak and strong duality results for (Pµ) are presented. These results permit us to give dual characterizations for the weakly efficient solutions of the problem (P).

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Copyright of RAIRO -- Operations Research is the property of EDP Sciences and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Canadian Journal of Education / Revue Canadienne de l'Éducation is the property of Canadian Society for the Study of Education and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Apart from his great creative work in mathematical science, Mihailo Petrović Alas, one of the greatest Serbian mathematicians, managed to write several works of methodological character which were intended for teachers at elementary and secondary schools. In these works Mihailo Petrović’s aim was to popularize mathematics and show how teachers can also be creative and inventors within the domain of teaching. We give an overview of these works in our paper and acquaint the readers in detail with his interesting application of the Pythagorean theorem. A short retrospect of the complete scientific and pedagogical opus of Mihailo Petrović Alas will also be given in the paper.

In the paper [M. O. Olatinwo, Stability of coupled fixed point iterations and the continuous dependence of coupled fixed points, Communications on Applied Nonlinear Analysis 19 (2012), 71-83], the author has extended the notion of stability of fixed point iterative procedures contained in the paper [A. M. Harder and T. L. Hicks, Stability results for fixed point iteration procedures, Math. Japonica 33 (1988), 693-706], as well as the continuous dependence of fixed points to the coupled fixed point settings by employing the contractive conditions and the coupled fixed point iteration in the article [F. Sabetghadam, H. P. Masiha and A. H. Sanatpour, Some coupled fixed point theorems in cone metric spaces, Fixed Point Theory and Applications, Article ID 125426 (2009)]. In the present paper, we obtain some results on stability of coupled fixed point iterative procedures by using rational type contractive conditions.

In the papers Iterates of Stancu Operators, via Contraction Principle (2002), respectively Iterates of Bernstein Operators, via Contraction Principle (2004), author I. A. Rus studied the existence of fixed points for Stancu operators Pn,α,β and Bernstein operators Bn. The aim of this paper is to find conditions for which the Stancu operators Pn,α,β are contractions on the graph, in order to demonstrate that the contraction principle can be applied for the study of the existence of fixed points for iterates of Stancu operators. The method used for this paper is the spectral method, which was also used in the paper Over-iterates of Bernstein-Stancu operators (2007), authors Gonska, Piţul and Raşa. The study began with finding constant C∈[0,1[ that would satisfy the inequality ||Pn,α,β2 (f)-Pn,α,β (f)|| ≤ C ||Pn,α,β (f)-f||, for any f∈ C[0,1]. The conclusion is that there are conditions for which the Stancu operators are contractions on the graph, and the methods used for the study of the existence of fixed points of their iterates can also be extended to the study of the existence of fixed points of other linear operators.

This paper attempts to revisit the geometry of the demand and supply functions in economic analysis, taking mathematical exigencies into account. Leaning on the fundamental principles that guide the theory of demand and supply, the paper uses the analytical approach of data analysis to address the objectives of the investigation. Results of the study indicate that whether expressed in the standard mathematical form (Quantity-price relation) or casual mathematical form (Price-quantity relation), the demand and supply functions obey the laws of demand and supply. Also, the results reveal that the first component in the computation of the price-elasticity remains unchanged and takes into account the quantity-price relationship, irrespective of the functional form (Quantity-price function or price-quantity function) used in defining the demand and supply functions. Furthermore, the results show that the consumer and producer surpluses can still be evaluated without violating mathematical requirements with the quantity-price relation model of describing the demand and supply functions to accommodate welfare issues associated with the demand and supply concepts. As a result, it is suggested that mathematical requirements of logic, rigor, consistency, and objectivity be strictly respected when subjecting the analysis of real world economic phenomena to mathematical treatments.

The aim of this paper is to understand the dynamics of the theory of order in the nineteenth century and to reveal a specific approach to mathematics, science, philosophy and decorative art in which order plays a prominent role. We will analyze the singular meaning that Poinsot assigns to the notion of order in the mathematical sciences, before describing the circulation of his writings on the order in the nineteenth century. Poinsot is one of the main sources of Cournot, who places the notions of order and form as the basis of his knowledge system. Then we will study the writings of Bourgoin who develops a combinatorics of ornaments based on the categories of order and form.

Copyright of Canadian Journal of Learning & Technology is the property of Canadian Network for Innovation in Education and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

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.

Over the past decade, the assessment of the general educational value of computer programming and computational thinking has been constantly increasing and, as a result, they are introduced to increasingly younger ages. In parallel, educational programming environments are significantly progressing, providing a variety of options for different ages. This paper presents an overview of the modern learning programming environments for primary education and proposes a classification system with categories corresponding to the technological and educational dimensions of the area. The paper aims to support teachers in learning design for the interdisciplinary approach of programming and the development of computational thinking.

In this paper we continue our study of the Frobenius instability locus in the coarse moduli space of semi-stable vector bundles of rank $r$ and degree $0$ over a smooth projective curve defined over an algebraically closed field of characteristic $p>0$. In a previous paper we identified the "maximal" Frobenius instability strata with opers (more precisely as opers of type $1$ in the terminology of the present paper) and related them to certain Quot-schemes of Frobenius direct images of line bundles. The main aim of this paper is to describe for any integer $q \geq 1$ a conjectural generalization of this correspondence between opers of type $q$ (which we introduce here) and Quot-schemes of Frobenius direct images of vector bundles of rank $q$. We also give a conjectural formula for the dimension of the Frobenius instability locus.

The bicategory of Landau-Ginzburg models denoted by ℒ𝒢K possesses adjoints and this helps in explaining a certain duality that exists in the setting of Landau-Ginzburg models in terms of some specified relations. The construction of ℒ𝒢K is reminiscent of, but more complex than, the construction of the bicategory of associative algebras and bimodules. In this paper, we review this complex but very inspiring construction in order to expose it more to pure mathematicians. In particular, we spend some time explaining the intricate construction of unit morphisms in this bicategory from a new vantage point. Besides, we briefly discuss how this bicategory could be constructed in more than one way using the variants of the Yoshino tensor product. Furthermore, without resorting to Atiyah classes, we prove that the left and right unitors in this bicategory have direct right inverses but do not have direct left inverses.

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 objective of this paper is to generalize and improve some results in fixed point theorems in both complete metric space and Menger space. These results are generalizations of the analogous ones recently proved by Khojasteh [5], Demma [1], Yildirim [13], where we establish a dynamic information about their other fixed points if there exist i.e the distance between two fixed point in case of metric space and their equivalent in probabilistic metric space. Some illustrative examples are furnished, which demonstrate the validity of the hypotheses. As an application to our main result, we derive a uniqueness fixed point theorem for a self-mapping under strong conditions.

Our aim in this paper is to show that the modulus of smoothness and the $K$-functionals constructed from the Sobolev-type space corresponding to the Dunkl operator are equivalent on the interval $(-1,1)$.

Let $R$ be a commutative noetherian local ring which is singular and has an isolated singularity. Let $\mathsf {D_{sg}}(R)$ be the singularity category of $R$ in the sense of Buchweitz and Orlov. In this paper, we find real numbers $t$ such that the complexity $\delta _t(G,X)$ in the sense of Dimitrov, Haiden, Katzarkov and Kontsevich vanishes for any split generator $G$ of $\mathsf {D_{sg}}(R)$ and any object $X$ of $\mathsf {D_{sg}}(R)$. In particular, the entropy $\mathrm{h}_t(F)$ of an exact endofunctor $F$ of $\mathsf {D_{sg}}(R)$ is not defined for such numbers $t$.

Understanding the behavior of a black-box model with probabilistic inputs can be based on the decomposition of a parameter of interest (e.g., its variance) into contributions attributed to each coalition of inputs (i.e., subsets of inputs). In this paper, we produce conditions for obtaining unambiguous and interpretable decompositions of very general parameters of interest. This allows recovering known decompositions, holding under weaker assumptions than the literature states.

In this paper, we study the relative homological dimension based on the class of projectively coresolved Gorenstein flat modules (PGF-modules), that were introduced by Saroch and Stovicek in [26]. The resulting PGF-dimension of modules has several properties in common with the Gorenstein projective dimension, the relative homological theory based on the class of Gorenstein projective modules. In particular, there is a hereditary Hovey triple in the category of modules of finite PGF-dimension, whose associated homotopy category is triangulated equivalent to the stable category of PGF-modules. Studying the finiteness of the PGF global dimension reveals a connection between classical homological invariants of left and right modules over the ring, that leads to generalizations of certain results by Jensen [24], Gedrich and Gruenberg [17] that were originally proved in the realm of commutative Noetherian rings.

The optimal transport map between the standard Gaussian measure and an $\alpha $-strongly log-concave probability measure is $\alpha ^{-1/2}$-Lipschitz, as first observed in a celebrated theorem of Caffarelli. In this paper, we apply two classical covariance inequalities (the Brascamp–Lieb and Cramér–Rao inequalities) to prove a sharp bound on the Lipschitz constant of the map that arises from entropically regularized optimal transport. In the limit as the regularization tends to zero, we obtain an elegant and short proof of Caffarelli’s original result. We also extend Caffarelli’s theorem to the setting in which the Hessians of the log-densities of the measures are bounded by arbitrary positive definite commuting matrices.

This paper studies the unique continuation problem for the heat equation. We prove a so-called conditional stability estimate for the solution. We are interested in local estimates that are Hölder stable with the weakest possible norms of data on the right-hand side. Such an estimate is useful for the convergence analysis of computational methods dealing with data assimilation. We focus on the case of a known solution at initial time and in some subdomain but that is unknown on the boundary. To the best of our knowledge, this situation has not yet been studied in the literature.

The last decade has seen an abundance of congruences for $b_\ell (n)$, the number of $\ell $-regular partitions of $n$. Notably absent are congruences modulo $4$ for $b_3(n)$. In this paper, we introduce Ramanujan type congruences modulo $4$ for $b_3(2n)$ involving some primes $p$ congruent to $ 11, 13, 17, 19, 23$ modulo $24$.

A conjecture by the second author, proven by Bonnafé–Rouquier, says that the multiplicity matrix for baby Verma modules over the restricted rational Cherednik algebra has rank one over $\mathbb{Q}$ when restricted to each block of the algebra.In this paper, we show that if $H$ is a prime algebra that is a free Frobenius extension over a regular central subalgebra $R$, and the centre of $H$ is normal Gorenstein, then each central quotient $A$ of $H$ by a maximal ideal $\mathfrak{m}$ of $R$ satisfies the rank-one property with respect to the Cartan matrix of $A$. Examples where the result is applicable include graded Hecke algebras, extended affine Hecke algebras, quantized enveloping algebras at roots of unity, non-commutative crepant resolutions of Gorenstein domains and 3 and 4 dimensional PI Sklyanin algebras.In particular, since the multiplicity matrix for restricted rational Cherednik algebras has the rank-one property if and only if its Cartan matrix does, our result provides a different proof of the original conjecture.

In this paper, we consider the class of normalized analytic functions of the form f(z)=z+∑n=2∞ anzn. Following this functions, we define the functions whose coefficients are probabilities of the geometric distribution series and other special modes of this series. Also, we consider different special classes of f(z). In the following we consider some lemmas that make connection between defined special classes with the function f(z). Follower of this topic we will consider the theorems that make connection between defined classes with the functions whose coefficients are probabilities of geometric distribution series. Also we define Alexander-type integral operator and find the necessary and sufficient conditions for being this operator to defined general classes.

In this paper, we will present the author's interpretation and embellishment of five lectures on cluster theory given by Kiyoshi Igusa during the Spring semester of 2022 at Brandeis University. They are meant to be used as an introduction to cluster theory from a representation-theoretic point of view.