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The purpose of this paper is to provide visualization tools for the fuzzy discrete dynamical systems introduced in [7]. [ABSTRACT FROM AUTHOR]

This paper aims to introduce the concept of a fuzzy discrete dynamical system as a fuzzy subgroupoid of X×Z×X, with Z representing the group of integer numbers. Additionally, we provide Maple procedures for the random generation of a fuzzy discrete dynamical system. [ABSTRACT FROM AUTHOR]

The purpose of this note is to highlight the relationship between T-fuzzy groupoids and left (or right) invariant T-fuzzy equivalence relations. We will also show that there is a 1 - 1 correspondence between the T-fuzzy reductions of a T-fuzzy groupoid and the T-fuzzy reductions of the associated T-fuzzy equivalence relation, as well as a similar correspondence between disjoint unions. [ABSTRACT FROM AUTHOR]

By a T-fuzzy action of a groupoid G on a set X we mean a T-fuzzy subgroupoid of the blow-up groupoid G[X, σ] of G corresponding to a map σ: X → G(0) (the momentum map of the action). For the fuzzification of a crisp action of a groupoid G on a set X we can use a T-fuzzy equivalence relation on X satisfying an invariance condition. Conversely, using a T-fuzzy action of a groupoid G on a set X, we can define an equivalence relation on X, a subgroupoid H of G and a crisp action of H on the quotient of X. If the T-fuzzy action is obtained by fuzzification of a crisp action, then H=G and the initial action can be recovered on a quotient space of X. In general, we can only partially recover the crip action. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a new way to think about groupoid actions through the intermediate of a generalization. We reformulate and extend various notion of groupoid morphisms using this notion of generalized action. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to provide visualization tools for analysis of the family of groupoids associated in [5] to a discrete nonautonomous system. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a way to associate "relevant" reductions to subgroupoids G of X×Z×X with G(0) finite. In order to obtain such a reduction we use singular value decomposition of a matrix associated with the groupoid. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to associate a family of groupoids and morphisms to a discrete nonautonomous system that does not necessarily satisfy the semigroup property. This approach provides a unifying framework to study discrete nonautonomous systems and to uncover similarities between seemingly unrelated systems. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a novel concept of groupoid action and to obtain a characterization of a groupoid homomorphism as an action that commutes with the multiplication on the range. We also define the corresponding fuzzy concepts. This approach allows the study within a unified framework of various fuzzy structures viewed as fuzzy subgroupoids of appropriate groupoids. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a notion of morphism between two groupoid defined in terms of groupoid actions and we prove that groupoids as objects and equivalence classes of certain type of such morhisms form category which is equivalent to the category introduced in [2]. [ABSTRACT FROM AUTHOR]

Dana P. Williams raised in [Proc. Amer. Math. Soc., Ser. B 3 (2016), pp. 1-8] the following question: Must a second countable, locally compact, transitive groupoid have open range map? This paper gives a negative answer to that question. Although a second countable, locally compact transitive groupoid G may fail to have open range map, we prove that we can replace its topology with a topology which is also second countable, locally compact, and with respect to which G is a topological groupoid whose range map is open. Moreover, the two topologies generate the same Borel structure and coincide on the fibres of G. [ABSTRACT FROM AUTHOR]

The purpose of this paper is provide Maple procedures to quantify how close are the original 3D-object and the 3D-object formed using an approximate development obtained by applying the well-known gore method to a surface of revolution. [ABSTRACT FROM AUTHOR]

The purpose of this short note is to introduce a notion of T-fuzzy groupoid fuzzifying not only the set but also the groupoid operations (the partially defined multiplication as well as the inversion) and to associate to a fuzzy action in the sense [D. Boixader and J. Recasens, Fuzzy Sets and Systems, 2018] such a fuzzy groupoid. [ABSTRACT FROM AUTHOR]

We start from the question raised by Williams (Proc. Am. Math. Soc. 2016): Must a second countable, locally compact, transitive groupoid G have open range map? If the answer is positive, the topology of G is in fact locally transitive (in the sense of [Seda, 1976]). We prove that even if the answer is negative, we can replace the original topology of G with a local transitive topology so that the topologies of the r-fibres are not affected. The new topology is locally compact Hausdorff but not necessary second countable. However its full C*-algebra (introduced in [Renault, 1980]) is still isomorphic to C*(H)⊗K(L2(μ)), where H is the isotropy group at a unit u and μ is a positive Radon measure on the unit space. We also present a few remarks concerning the Haar systems on locally compact groupoids and for every locally compact groupoid having paracompact unit space and second countable r-fibres, we prove the existence of a pre-Haar system bounded on the compact sets. [ABSTRACT FROM AUTHOR]

The purpose of this note is to introduce a notion of T-fuzzy groupoid (in the sense of Brandt) that allows to study within a unified framework various fuzzy structures such as fuzzy sets, fuzzy groups, fuzzy equivalence relations and fuzzy transformation groups [ABSTRACT FROM AUTHOR]

The aim of this paper is provide a set of Maple procedures to evaluate of how close the 3D-object formed using an approximate development is from the original 3D-object. The approximate developments taking into consideration are obtained extending the well-known zone method from spheres. [ABSTRACT FROM AUTHOR]

Starting from the observation that through groupoids we can express in a unified way the notions of fundamental system of entourages of a uniform structure on a space X, respectively the system of neighborhoods of the unity of a topological group that determines its topology, we introduce in this paper a notion of G-uniformity for a groupoid G. The topology induced by a G-uniformity turns G into a topological locally transitive groupoid. We also prove a Urysohn type lemma for groupoids and obtain metrization theorems for groupoids unifying in two ways the Alexandroff--Urysohn Theorem and Birkhoff- Kakutani Theorem. [ABSTRACT FROM AUTHOR]

The purpose of this note is to introduce four different groupoids associated to a function f:X×N×N→X, where X is a uniform space. Thesegroupoids allow to study within a unified framework theasymptotic behavior of thediscrete systems of the form xn+1 = fn,n0 (xn,,xn-1,..., xn0 ) that do not necessarily satisfy the semigroup property of a process. For these systems the groupoidsare associated with the functionsf defined by f(x,n,n0) := fn,n0 (xn-1,,xn-2,..., xn0 ) with x=xn0. [ABSTRACT FROM AUTHOR]

The purpose of this paper is provide a set of Maple procedures to construct approximate developments of a general surface of revolution generalizing the well-known gore method for sphere. [ABSTRACT FROM AUTHOR]

The purpose of this short note is to prove that if the category of all subgroupoids of trivial groupoids with group ࡌ is seen as a model category M in the sense [M. Nielsen A. Joyal and G. Winskel, 1996] and [E. Haghverdi, P. Tabuada, G.J. Pappas, 2002], then we can highlight a path category P (a subcategory of M serving as an abstract notion of time) such that the P-open morphisms are exactly the groupoid homomorphisms. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the isomorphisms of the category of all subgroupoids of trivial groupoids on various sets X with group Z. We also provide Maple procedures for obtain simplified models of a discrete dynamical system seen as a subgroupoid a trivial groupoid with group Z. [ABSTRACT FROM AUTHOR]

The purpose of this paper is provide various Maple procedures to visualize the behavior of a general fractional order discrete systems perturbed by a sequence of real random variables. [ABSTRACT FROM AUTHOR]

The purpose of this paper is provide a set of Maple procedures to simulate a general fractional order discrete systems perturbed by a sequence of real random variables. [ABSTRACT FROM AUTHOR]

Simulation tools (in Matlab) for linear discrete-time fractional order system as well as for discrete-time linear fractional order stochastic systems (with additive noise) can be found in [3]. In this paper we intend to approach the nonlinear case. More precisely, the purpose of this paper is to provide a small collection of procedures for simulating nonlinear fractional order stochastic discrete-time control systems. Compared with the procedures offered in [1], these procedures allow the implementation of models of discrete-time fractional order stochastic control systems with multidimensional stochastic perturbations. [ABSTRACT FROM AUTHOR]

The purpose of this short note is to introduce a groupoid associated to a function f: Xx ▯ → X where X is a uniform space. The motivation is given by the discrete time systems of the form Xn+1 = fn(Xn,,Xn-1,-,X0) that do not satisfy the semigroup property of a process. For these systems the groupoid is induced by the function f defined by f(x,n) := fn(xn-1, xn-2, ...,x0) with X0=x (or ... = x for a fixed n0). The notions of stability and asymptotic stability are introduced in the groupoid framework. [ABSTRACT FROM AUTHOR]

The purpose of this short note is to introduce a groupoid associated to a function f: X×   →X where X is a uniform space. The motivation is given by the discrete time systems of the form xn+1 = fn(xn, xn-1,,...,x0) that do not satisfy the semigroup property of a process. For these systems the groupoid is induced by the function f defined by f(x,n) := fn(xn-1,,xxn-2,,...,x0) with x0=x (or Xn0 = X for a fixed n>0 ). The notions of stability and asymptotic stability are introduced in the groupoid framework. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to transfer the construction of the Haar measure of a locally compact group to the groupoid settings. The groupoid is endowed with a family of subsets containing the unit space. [ABSTRACT FROM AUTHOR]

The goal of this article is to go over certain problems concerning teaching of mathematics to engineering students,based on finds paradoxical situation in both Europe and the North America,that although there is a spectacular expansion of mathematics in all fields of knowledge,including engineering,mathematical education at all levels degrades. In these circumstances,the question is whether widespread use of computer technology might improve the mathematical training of students. We give some examples that highlight both the advantages and disadvantages of using specialized software packages. [ABSTRACT FROM AUTHOR]

We sτarτ wiτh a locally diagonally compacτ groupoid (G,τG) in τhe sense of [4], a pre- Haar sysτem n on G and a family A of lefτ and righτ uniformly conτinuous on fibres funcτions on G. The purpose of τhis shorτ noτe is τo inτroduce a new locally diagonally compacτ τopology τ(τG,A,v ) on G such τhaτ τG and τ(τG,A,v) induce τhe same subspace τopology on τhe groupoid fibres and τhe pre-Haar sysτem n becomes conτinuous for τhe funcτions in A. [ABSTRACT FROM AUTHOR]

We start with a groupoid G endowed with a family W of subsets mimicking the properties of a neighborhood basis of the unit space (of a topological groupoid with paracompact unit space). Using the family W we endow each G-space with a uniform structure. The uniformities of the G-spaces allow us to define various notions of amenability for the G-equivariant maps. As in [1], the amenability of the groupoid G is defined as the amenability of its range map. If the groupoid G is a group, all notions of amenability that we introduce coincide with the classical notion of amenability for topological (not necessarily locally-compact) groups. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove that if each point of a topological groupoid G has a neighborhood basis of diagonally compact sets, then the same is true for the principal groupoid R of G endowed with the transported topology from G, as well as for G endowed with the modified topology with respect to R introduced in [3]. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a way for studying subgroupoids G of X×Z×X (and especially, restrictions of the groupoids associated to discrete dynamical system) taking advantage of Maple symbolic and arbitrary precision computing capabilities. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the topology of the orbit space of an irreversible discrete dynamical system (X, φ) seen as a principal groupoid associated to the groupoid G(X,φ,E) introduced in [1] (where E is an equivalence relation on X) [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a groupoid associated to a dynamical system. We find this groupoid to be adequate for study of the dynamical system via computer visualization. The principal innovation in our approach to classical ([4] and [7]) is the use of a uniform structure on phase space which induce a non-Hausdorff topology. [ABSTRACT FROM AUTHOR]

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