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We are studying topological properties of the Julia set of the map F(z, p) = *** of the complex projective plane PC² to itself. We show a relation between this rational function and an uncountable family of "paper folding" plane filling curves. [ABSTRACT FROM AUTHOR]

Important connections in representation theory arise from resolving a finite-dimensional algebra by an endomorphism algebra of a generator-cogenerator with finite global dimension; for instance, Auslander's correspondence, classical Schur–Weyl duality and Soergel's Struktursatz. Here, the module category of the resolution and the module category of the algebra being resolved are linked via an exact functor known as the Schur functor. In this paper, we investigate how to measure the quality of the connection between module categories of (projective) Noetherian algebras, B, and module categories of endomorphism algebras of generator-relative cogenerators over B which are split quasi-hereditary Noetherian algebras. In particular, we are interested in finding, if it exists, the highest degree n so that the endomorphism algebra of a generator-cogenerator provides an n-faithful cover, in the sense of Rouquier, of B. The degree n is known as the Hemmer–Nakano dimension of the standard modules. We prove that, in general, the Hemmer–Nakano dimension of standard modules with respect to a Schur functor from a split highest weight category over a field to the module category of a finite-dimensional algebra B is bounded above by the number of non-isomorphic simple modules of B. We establish methods for reducing computations of Hemmer–Nakano dimensions in the integral setup to computations of Hemmer–Nakano dimensions over finite-dimensional algebras, and vice-versa. In addition, we extend the framework to study Hemmer–Nakano dimensions of arbitrary resolving subcategories. In this setup, we find that the relative dominant dimension over (projective) Noetherian algebras is an important tool in the computation of these degrees, extending the previous work of Fang and Koenig. In particular, this theory allows us to derive results for Schur algebras and the BGG category \mathcal {O} in the integral setup from the finite-dimensional case. More precisely, we use the relative dominant dimension of Schur algebras to completely determine the Hemmer–Nakano dimension of standard modules with respect to Schur functors between module categories of Schur algebras over regular Noetherian rings and module categories of group algebras of symmetric groups over regular Noetherian rings. We exhibit several structural properties of deformations of the blocks of the Bernstein-Gelfand-Gelfand category \mathcal {O} establishing an integral version of Soergel's Struktursatz. We show that deformations of the combinatorial Soergel's functor have better homological properties than the classical one. [ABSTRACT FROM AUTHOR]

This paper studies topological definitions of chain recurrence and shadowing for continuous endomorphisms of topological groups generalizing the relevant concepts for metric spaces. It is proved that in this case the sets of chain recurrent points and chain transitive component of the identity are topological subgroups. Furthermore, we show that some dynamical properties are induced by the original system on quotient spaces. These results link an algebraic property to a dynamical property. [ABSTRACT FROM AUTHOR]

In this paper, we provide the basic properties of (semi)simple hypermodules. We show that if a hypermodule M is simple, then (E n d (M) , ·) is a group, where E n d (M) is the set of all normal endomorphisms of M. We prove that every simple hypermodule is normal projective with a zero singular subhypermodule. We also show that the class of semisimple hypermodules is closed under internal direct sums, factor hypermodules, and subhypermodules. In particular, we give a characterization of internal direct sums of subhypermodules of a hypermodule. [ABSTRACT FROM AUTHOR]

In this paper, we properly generalize the notion of co-Hopficity for modules to the concept of closed co-Hopficity. A module M is said to be closed co-Hopfian if any injective endomorphism of M has a closed submodule image. The aim of this paper is to study and investigate this class of modules. In addition, some relations for this class with other types of modules are provided. [ABSTRACT FROM AUTHOR]

In the past few decades, researchers have found that studying modules using endomorphisms is a powerful and useful tool. This has led to valuable works in this field. Recently, the study of (dual) Rickart modules has become an important approach as they are deeply connected to endomorphisms. Building on this work, the authors introduce a new perspective on (dual) Rickart modules using isomorphism. We also define virtually (dual) Rickart modules. It is shown that rings with all modules virtually Rickart are semisimple rings. The paper includes various examples to illustrate the concepts presented. [ABSTRACT FROM AUTHOR]

There are six classes of endomorphisms for a graph. The sets of these endomorphisms form a chain under the inclusion of sets. In order to systematically study these endomorphisms, Böttcher and Knauer defined the concepts of the endomorphism spectrum and endomorphism type of a graph in 1992. In this paper, based on the property and structure of the endomorphism monoids of graphs, six classes of endomorphisms of double-edge fan graphs are described. In particular, we give the endomorphism spectra and endomorphism types of double-edge fan graphs. [ABSTRACT FROM AUTHOR]

In one of our former papers Endomorphisms of the measure algebra of commutative hypergroups we considered exponential monomials on hypergroups and higher order derivations of the corresponding measure algebra. Continuing with this, we are now looking for the connection between the generalized exponential polynomials of a commutative hypergroup and the higher order derivations of the corresponding measure algebra. [ABSTRACT FROM AUTHOR]

There are six different classes of endomorphisms for a graph. The sets of these endomorphisms always form a chain under the inclusion of sets. In order to study these different endomorphisms more systematically, Böttcher and Knauer proposed the concept of the endomorphism type of a graph in 1992. In this paper, we explore the six different classes of endomorphisms of graph P (3 m + 1 , 3) . In particular, the endomorphism type of P (3 m + 1 , 3) is given. [ABSTRACT FROM AUTHOR]

In the paper, we study the semiring of skew polynomials over a Rickart Bezout semiring. Namely, let every left annihilator ideal of a semiring be an ideal. Then the semiring of skew polynomials is a semiring without nilpotent elements, and every its finitely generated left monic ideal is principal if and only if is a left Rickart left Bezout semiring, is a rigid endomorphism, and is invertible for any nonzerodivisor . We also obtain a characterization of the semiring in terms of Pierce stalks of the semiring . The structure of left monic ideals of the semiring of skew polynomials over a left Rickart left Bezout semiring is clarified. [ABSTRACT FROM AUTHOR]

Let S be a semigroup, and let φ, ψ: S → S 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 f (x ϕ (y)) + f (ψ (y) x) = 2 f (x) f (y) , x , y ∈ S , where f : S → ℂ is the unknown function by expressing its solutions in terms of multiplicative functions. Some consequences of this result are presented. [ABSTRACT FROM AUTHOR]

Let be a class of Abelian groups. A group is said to be determined by its endomorphism semigroup in the class if every isomorphism , where , implies the isomorphism . The paper describes those Abelian groups in the class of completely decomposable quotient divisible Abelian groups which are determined by their endomorphism semigroups in the class . [ABSTRACT FROM AUTHOR]

In this paper, we describe ss-supplement submodules in terms of a special class of endomorphisms. Let R be a ring with semisimple radical and P be a projective R-module. We show that there is a bijection between ss-supplement submodules of P and ss-supplement submodules of EndR(P) . Moreover, we define radical-s-projective modules as a generalization of projective modules. We prove that every ss-supplement submodule of a projective R-module is radical-s-projective over the ring R with semisimple radical. We show that over SSI -ring R, every radical-s-projective R-module is projective. We provide that over a ring R with semisimple radical, every ss-supplement submodule of a projective R-module is a direct summand if and only if every radical-s-projective R-module is projective. [ABSTRACT FROM AUTHOR]

In this paper we study properties of endomorphisms of \mathbb {P}^k using a symmetric product construction (\mathbb {P}^1)^k/\mathfrak {S}_k \cong \mathbb {P}^k. Symmetric products have been used to produce examples of endomorphisms of \mathbb {P}^k with certain characteristics, k\geq 2. In the present note, we discuss the use of these maps to enlighten stability phenomena in parameter spaces. In particular, we study k-deep post-critically finite maps and characterize families of Lattès maps. [ABSTRACT FROM AUTHOR]

In this paper we study grouplike monoids, defined as being monoids that contain a group to which we add an ordered set of idempotents. We classify finite categories with two objects having grouplike endomorphism monoids, by presenting a construction theorem for such categories, and proving that every grouplike category comes from this construction. Studying the algebraic properties of the endomorphism monoids allows us to gather extra information on the category itself, which in particular helps the counting problem because the nature of the monoids affects greatly the structure of the category. At the end of the paper, we give a count of certain categories with grouplike monoids, concluded from the properties of grouplike monoids that are studied in the paper. [ABSTRACT FROM AUTHOR]

The purpose of this note is to correct statements and proofs of results in the original paper. [ABSTRACT FROM AUTHOR]

The renowned 2D invertible Hénon map turns into 1D noninvertible quadratic map when its leading parameter b becomes zero. This well-known link was studied by Mira who demonstrated that the bifurcation set of Hénon diffeomorphism is similar to his "box-within-a-box" bifurcation structure of 1D endomorphism. In general, such similarity has not been strictly established, especially in multidimensional cases. In this paper, we proved that the Mira bifurcation structure of a quadratic noninvertible map persists when the parameter increases from zero and the map turns into an invertible multidimensional generalized Hénon map. The changes of periodic and homoclinic orbits and chaotic attractors at this transition are described. We proved the existence of Newhouse regions is different from those Mira boxes that accumulate to the homoclinic bifurcations. [ABSTRACT FROM AUTHOR]

We show that all finite Brouwerian semilattices have strong endomorphism kernel property (SEKP), give a new proof that all finite relative Stone algebras have SEKP and also fully characterize dual generalized Boolean algebras which possess SEKP. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with multiplicative identity, C a coassociative and counital Rcoalgebra, B an R-bialgebra. A clean comodule is a generalization and dualization of a clean module. An R-module M is called a clean module if the endomorphism ring of M over R (denoted by EndR(M)) is clean. Thus, any element of EndR(M) can be expressed as a sum of a unit and an idempotent element of EndR(M). Moreover, for a right C-comodule M, the endomorphism set of C-comodule M denoted by EndC(M) is a subring of EndR(M). A C-comodule M is a clean comodule if the EndC(M) is a clean ring. A Hopf module M over B is a B-module and a B-comodule that satisfies the compatible conditions. This paper considers the notions of a clean ring, clean module, clean coalgebra, and clean comodule in relation to the Hopf Module. We divide our discussion into two parts, i.e., clean and bi-clean Hopf modules. A B-Hopf module M is said to be clean if the endomorphism ring of M is clean, and M is a bi-clean Hopf module if M is clean as a module over B and also clean as a comodule over B. Moreover, we give sufficient conditions of (bi)-clean bialgebras and Hopf modules related to the cleanness concept of modules and comodules. [ABSTRACT FROM AUTHOR]

Let R be a ring with involution and let α and β be endomorphisms of R. In this paper we characterise generalised Jordan (α, β)-higher derivations and related maps on (semi)-prime rings with involution. As consequences of our main theorems, many known results can be either generalised or deduced. [ABSTRACT FROM AUTHOR]

A finitely generated group Γ is called strongly scale-invariant if there exists an injective endomorphism φ : Γ → Γ with the image φ(Γ) of finite index in r and the subgroup ∩n>0 φn (Γ) finite. The only known examples of such groups are virtually nilpotent, or equivalently, all examples have polynomial growth. A question by Nekrashevych and Pete asks whether these groups are the only possibilities for such endomorphisms, motivated by the positive answer due to Gromov in the special case of expanding group morphisms. In this paper, we study this question for the class of virtually polycyclic groups, i.e. the virtually solvable groups for which every subgroup is finitely generated. Using the ℚ-algebraic hull, which allows us to extend the injective endomorphisms of certain virtually polycyclic groups to a linear algebraic group, we show that the existence of such an endomorphism implies that the group is virtually nilpotent. Moreover, we fully characterize which virtually nilpotent groups have a morphism satisfying the condition above, related to the existence of a positive grading on the corresponding radicable nilpotent group. As another application of the methods, we generalize a result of Fel'shtyn and Lee about which maps on infra-solvmanifolds can have finite Reidemeister number for all iterates. [ABSTRACT FROM AUTHOR]

In the paper we examined the some effects of derivation, trace of permuting tri-derivation and endomorphism on each other in prime and semiprime ring. Let R be a 2, 3-torsion free prime ring and F : R x R x R → R be a permuting tri-derivation with trace f, d : R → R be a derivation. In particular, the following assertions have been proved: • if [d(r), r] = f(r) for all r • R, then R is commutative or d = 0 (Theorem 1); • if g : R → R is an endomorphism such that F(d(r), r, r) = g(r) for all r • R, then F = 0 or d = 0 (Theorem 2); • if F(d(r), r, r) = f(r) for all r • R, then (i) F = 0 or d = 0, (ii) d(r) • f(r) = 0 for all r • R (Theorem 3). In the other hand, if there exist permuting tri-derivations F1, F2 : R x R x R → R such that F1(f2(r), r, r) = f1(r) for all r • R, where f1 and f2 are traces of F1 and F2, respectively, then (i) F1 = 0 or F2 = 0, (ii) f1(r) • f2(r) = 0 for all r • R (Theorem 4). [ABSTRACT FROM AUTHOR]

Let be an Abelian group. A class of Abelian groups is called a -class (a -class) if, for any groups and in the class , the isomorphism of the groups and (the isomorphism of the endomorphism rings and and of the groups and ) implies the isomorphism of the groups and . In the paper, we study conditions that must be satisfied by a vector group for some class of homogeneously decomposable torsion-free Abelian groups to be a class (Theorem 1), and also, for some in the class of vector groups, for some class of homogeneously decomposable torsion-free Abelian groups to be a -class (Theorem 2). [ABSTRACT FROM AUTHOR]

In this paper we give a classification of endomorphisms and additive mappings of a prime ring satisfying certain algebraic identities. Moreover, we provide an example proving that the primeness hypothesis imposed in our theorems is not superfluous. [ABSTRACT FROM AUTHOR]

Let H be an acylindrically hyperbolic group without nontrivial finite normal subgroups. We show that any finite system S of equations with constants from H is equivalent to a single equation. We also show that the algebraic set associated with S is, up to conjugacy, a projection of the algebraic set associated with a single splitted equation (such an equation has the form w.x1;: :: ; xn/D h, where w 2 F.X/, h 2 H). From this we deduce the following statement: Let G be an arbitrary overgroup of the above group H. Then H is verbally closed in G if and only if it is algebraically closed in G. These statements have interesting implications; here we give only two of them: If H is a noncyclic torsion-free hyperbolic group, then every (possibly infinite) system of equations with finitely many variables and with constants from H is equivalent to a single equation. We give a positive solution to Problem 5.2 from the paper [J. Group Theory 17 (2014), 29-40] of Myasnikov and Roman'kov: Verbally closed subgroups of torsion-free hyperbolic groups are retracts. Moreover, we describe solutions of the equation xnym D anbm in acylindrically hyperbolic groups (AH-groups), where a, b are non-commensurable jointly special loxodromic elements and n;m are integers with sufficiently large common divisor. We also prove the existence of special test words in AH-groups and give an application to endomorphisms of AH-groups. [ABSTRACT FROM AUTHOR]

This paper initiates a discrete Hodge theory for cellular sheaves taking values in a category of lattices and Galois connections. The key development is the Tarski Laplacian, an endomorphism on the cochain complex whose fixed points yield a cohomology that agrees with the global section functor in degree zero. This has immediate applications in consensus and distributed optimization problems over networks and broader potential applications. [ABSTRACT FROM AUTHOR]

In this paper we study the dynamics of continuous endomorphisms on Polish groups. We offer necessary and sufficient conditions for a continuous endomorphism on a Polish group to be weakly mixing. We prove that any continuous endomorphism of an abelian Polish group can be extended in a natural way to a topologically mixing endomorphism on the countable infinite product of said group. [ABSTRACT FROM AUTHOR]

In this paper we study p-adic principal series representation of a p-adic group G as a module over the maximal compact subgroup G0. We show that there are no non-trivial G0-intertwining maps between principal series representations attached to characters whose restrictions to the torus of G0 are distinct, and there are no non-scalar endomorphisms of a fixed principal series representation. This is surprising when compared with another result which we prove: that a principal series representation may contain infinitely many closed G0-invariant subspaces. As for the proof, we work mainly in the setting of Iwasawa modules, and deduce results about G0-representations by duality. [ABSTRACT FROM AUTHOR]

In this paper we show that if p is a polynomial which bifurcates then the product map (z,w)↦(p(z),q(w)) can be approximated by polynomial skew products possessing special dynamical objets called blenders. Moreover, these objets can be chosen to be of two types: repelling or saddle. As a consequence, such product map belongs to the closure of the interior of two different sets: the bifurcation locus of Hd(P²) and the set of endomorphisms having an attracting set of non-empty interior. In an independent part, we use perturbations of Hénon maps to obtain examples of attracting sets with repelling points and also of quasi-attractors which are not attracting sets. [ABSTRACT FROM AUTHOR]

The notion of a pre-period of an algebra A is defined by means of the notion of the pre-period λ(f) of a monounary algebra hA; fi: it is determined by sup{λ(f) | f is an endomorphism of A}. In this paper we focus on the pre-period of a finite modular lattice. The main result is that the pre-period of any finite modular lattice is less than or equal to the length of the lattice; also, necessary and sufficient conditions under which the pre-period of the glued sum is equal to the length of the lattice, are shown. Moreover, we show the triangle inequality of the pre-period of the glued sum. [ABSTRACT FROM AUTHOR]

The metallic-like (a, b) -manifold is a manifold endowed with a polynomial structure of second degree which unifies the almost product, complex structures and includes metallic structures. We introduce the metallic-like maps between metallic-like (a, b) -manifolds and we give a criterion for the nonconstancy of these maps. We prove that an almost contact structure on a Riemannian manifold induces a metallic-like (a, b) -structure and we give an example of a nonconstant metallic-like endomorphism of a particular almost contact manifold. [ABSTRACT FROM AUTHOR]

In the paper, Problem 18.3 of the book “Abelian groups” (2015) by L. Fuchs is solved in the case of Abelian groups with finite p-ranks. For an Abelian group A, a sequence of groups (An) is considered, where A0 = A and An+1 = End An. It is shown that, if all p-ranks of the group A are finite, then this sequence can stabilize either after A0 or after A1. [ABSTRACT FROM AUTHOR]

All rings are commutative. Let M be a module. We introduce the property (P): Every endomorphism of M has a non-trivial invariant submodule. We determine the structure of all vector spaces having (P) over any field and all semisimple modules satisfying (P) over any ring. Also, we provide a structure theorem for abelian groups having this property. We conclude the paper by characterizing the class of rings for which every module satisfies (P) as that of the rings R for which R/m is an algebraically closed field for every maximal ideal m of R. [ABSTRACT FROM AUTHOR]

Let X be a nonempty set and P (X) the power set of X. The aim of this paper is to provide an explicit description of the monoid End 1 P (X) (P (X)) of unital ring endomorphisms of the Boolean ring P (X) and the automorphism group Aut (P (X)) when X is finite. Among other facts, it is shown that if X has cardinality n ≥ 1 , then End 1 P (X) (P (X)) ≅ T n o p , where T n is the full transformation monoid on the set X and Aut (P (X)) ≅ S n . [ABSTRACT FROM AUTHOR]

A ring K is a unique addition ring (a UA-ring) if its multiplicative semigroup ( K, · ) can be equipped with a unique binary operation + transforming this semigroup to a ring ( K, ·, +). An Abelian group is called an End-UA-group if its endomorphism ring is a UA-ring. In the paper, we find End-UA-groups in the class of nonreduced Abelian groups. [ABSTRACT FROM AUTHOR]

We study invariants associated to Smale spaces obtained from an expanding endomorphism on a (closed connected Riemannian) flat manifold. Specifically, the relevant invariants are the K-theory of the associated C*-algebras and Putnam's homology theory for Smale spaces. The latter is isomorphic to the groupoid homology of the groupoids used to construct the C*-algebras. [ABSTRACT FROM AUTHOR]

The index ind(F) of a fixed point class F is a classical invariant in the Nielsen fixed point theory. In the recent paper [13], the authors introduced a new invariant ichr(F) called the improved characteristic, and proved that ind(F) ≤ ichr(F) for all fixed point classes of π1-injective selfmaps of connected finite graphs. In this note, we show that the two homotopy invariants mentioned above are exactly the same. Since the improved characteristic is totally determined by the endomorphism of the fundamental group, we give a group-theoretical approach to compute indices of fixed point classes of graph selfmaps. As a consequence, we give a new criterion of a fixed point, which extends the one due to Gaboriau, Jaeger, Levitt and Lustig. [ABSTRACT FROM AUTHOR]

We will show how to constructed an Skew-Field with trace-preserving endomorphisms of the affine plane. Earlier in my paper, we doing a detailed description of endomorphisms algebra and trace-preserving endomorphisms algebra in an affine plane, and we have constructed an associative unitary ring for which trace-preserving endomorphisms. In this paper we formulate and prove an important Lemma, which enables us to construct a particular trace-preserving endomorphism, with the help of which we can construct the inverse trace-preserving endomorphisms of every tracepreserving endomorphism. At the end of this paper we have proven that the set of tracepreserving endomorphisms together with the actions of'addition' and 'composition' (which is in the role of 'multiplication') forms a skew field. [ABSTRACT FROM AUTHOR]

In this paper we present some commutativity theorems for prime rings R with involution - of the second kind in which endomorphisms satisfy certain algebraic identities. Furthermore, we provide examples to show that various restrictions imposed in the hypotheses of our theorems are not super- uous. [ABSTRACT FROM AUTHOR]

In this paper, we employ automated deduction techniques to prove and generalize some well-known theorems in group theory that involve power maps x n . The difficulty lies in the fact that the term x n cannot be expressed in the syntax of first-order logic when n is an integer variable. Here we employ a new concept of "power-like functions" by extracting relevant equational properties valid for all power functions and implement these equational rules in Prover9, a first-order theorem prover. We recast the original theorems and prove them in this new context of power-like functions. Consequently these first-order proofs remain valid for all n but the length and complexity of the proofs remain constant independent of the value of n. To give an example, it is well-known (Baer in Proc Am Math Soc 4:15–26, 1953, Alperin in Can J Math 21:1238–1244 1969) that every torsion-free group in which the power map f (x) = x n is an endomorphism is abelian. Here we show that every torsion-free group in which a power-like map is an endomorphism is, indeed, abelian. Also, we generalize similar theorems from groups to a class of cancellative semigroups, and once again, Prover9 happily proves all these new generalizations as well. [ABSTRACT FROM AUTHOR]

A ring R is called reversible if for a , b ∈ R , a b = 0 implies b a = 0 . These rings play an important role in the study of noncommutative ring theory. Kafkas et al. (Algebra Discrete Math. 12 (2011) 72–84) generalized the notion of reversible rings to central reversible rings. In this paper, we extend the notion of central reversibility of rings to ring endomorphisms. We investigate various properties of these rings and answer relevant questions that arise naturally in the process of development of these rings, and as a consequence many new results related to central reversible rings are also obtained as corollaries to our results. [ABSTRACT FROM AUTHOR]

In this paper we propose an alternative approach for the determination of the Fibonacci numbers and some results of Foata, Ramanujan and other results on Chebychev polynomials of first and second kinds. This approach is based on the action of the symmetrizes endomorphism operators ... on the sequences ... Satisfactory results were obtained in almost all of the treated cases. Moreover, we give new results for the Stirling numbers. [ABSTRACT FROM AUTHOR]

An extended derivation (endomorphism) of a (restricted) Lie algebra L is an assignment of a derivation (respectively) of L' for any (restricted) Lie morphism f:L\to L', functorial in f in the obvious sense. We show that (a) the only extended endomorphisms of a restricted Lie algebra are the two obvious ones, assigning either the identity or the zero map of L' to every f; and (b) if L is a Lie algebra in characteristic zero or a restricted Lie algebra in positive characteristic, then L is in canonical bijection with its space of extended derivations (so the latter are all, in a sense, inner). These results answer a number of questions of G. Bergman. In a similar vein, we show that the individual components of an extended endomorphism of a compact connected group are either all trivial or all inner automorphisms. [ABSTRACT FROM AUTHOR]

In this work we show that every quotient of a torus endomorphism has a parabolic orbifold, answering a question of Mario Bonk and Daniel Meyer posed in [ Expanding Thurston maps , American Mathematical Society, Providence, RI, 2017]. [ABSTRACT FROM AUTHOR]

We examine the heap of linear connections on anchored vector bundles and Lie algebroids. Naturally, this covers the example of affine connections on a manifold. We present some new interpretations of classical results via this ternary structure of connections. Endomorphisms of linear connections are studied, and their ternary structure, in particular the endomorphism truss, is explicitly presented. We remark that the use of ternary structures in differential geometry is novel and that the endomorphism truss of linear connections provides a concrete geometric example of a truss. [ABSTRACT FROM AUTHOR]

Let σ be a linear *-endomorphism on a C* -algebra A so that σ(A) acts on a Hilbert space H which including K(H) and let {α}teR be a σ-C*-dynamical system on A with the generator δ: In this paper, we demonstrate some conditions under which ftgt2R is implemented by a C0-groups of unitaries on H. In particular, we prove that for a rank- one projection p 2 A; which is invariant under t; there is a C0-group futgt2R of unitaries in B(H) such that t(a) = utσ(a)u t: Furthermore, introducing the concepts of σ-inner endomorphism and σ-bijective map, we prove that each σ-bijective linear endomorphism on A is a σ-inner endomorphism, where σ ia idempotent. Finally, as an application, we characterize each so-called σ-C -dynamical system on the concrete C - algebra A:= B(H) B(H); where H is a separable Hilbert space and σ is the linear-endomorphism σ(S; T) = (0; T) on A:. [ABSTRACT FROM AUTHOR]

The discrete logarithm problem (DLP) in a group is a fundamental assumption that underpins the security of many systems. Hence evaluating its hardness is important. For efficient implementation of cryptographic algorithms, sometimes groups with additional structures are preferred. However, these structures may be used to obtain faster attacks. By using equivalence classes, Galbraith and Ruprai proposed a faster algorithm to solve the DLP in an interval of size N, with expected running time of 1.361 N group operations. Liu generalized their algorithm to the 2-dimensional case, which required 1.450 N group operations. Further, for an elliptic curve with an efficiently computable endomorphism, Liu reduced the complexity to 1.026 N . In this paper, we propose a variant of the Galbraith–Ruprai algorithm. This variant has average-case asymptotic complexity close to 1.253 N for sufficiently large N. For certain practical parameters, the complexity is 1.275 N in the 1-dimensional case and 1.393 N in the 2-dimensional case. Then we extend the algorithm for the case of larger equivalence classes. In particular, for the 2-dimensional DLP in a rectangle on an elliptic curve with an efficiently computable endomorphism, we reduce the complexity to 0.985 N for certain fixed parameters. We also discuss some possible further improvements. [ABSTRACT FROM AUTHOR]

Let C be an Abelian group. A class X of Abelian groups is called a CE•H-class if, for every groups A, B ∈ X, the isomorphisms E•(A) ≅ E•(B) and Hom(C, A) ≅ Hom(C, B) imply the isomorphism A ≅ B. In the paper, necessary and sufficient conditions on a completely decomposable torsion-free Abelian group C are described under which a given class of torsion-free Abelian groupsisa CE•H-class. [ABSTRACT FROM AUTHOR]

This paper is an attempt to stress the usefulness of multi-variable special functions by expressing them in terms of the corresponding Lie algebra or Lie group. The problem of framing the 2-index 4-variable 1-parameter Hermite polynomials (2I4V1PHP) into the context of the irreducible representations ↑ωμ of G(0,1) and ωμ of K5 is considered. Certain relations involving 2I4V1PHP Hm,n(x, y, z, u, ρ) are obtained using the approach adopted by Miller. Certain examples involving other forms of Hermite polynomials are derived as special cases. Further, some properties of the 2I4V1PHP Hm,n(x, y, z, u, ρ) are obtained by using a quadratic combination of four operators defined on a Lie algebra of endomorphisms of a vector space. [ABSTRACT FROM AUTHOR]

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