Best Proximity Pair Results for Relatively Nonexpansive Mappings in Geodesic Spaces

Given A and B two nonempty subsets in a metric space, a mapping T: A ∪ B → A ∪ B is relatively nonexpansive if d(Tx, Ty) ≤ d(x, y) for every x ∈ A, y ∈ B. A best proximity point for such a mapping is a point x ∈ A ∪ B such that d(x, Tx) = dist(A, B). In this work, we extend the results given in Eldred et al. (2005) [A.A. Eldred, W.A. Kirk, P. Veeramani, Proximal normal structure and relatively nonexpansive mappings, Studia Math. 171, 283–293] for relatively nonexpansive mappings in Banach spaces to more general metric spaces. Namely, we give existence results of best proximity points for cyclic and noncyclic relatively nonexpansive mappings in the context of Busemann convex reflexive metric spaces. Moreover, particular results are proved in the setting of CAT(0) and uniformly convex geodesic spaces. Finally, we show that proximal normal structure is a sufficient but not necessary condition for the existence in A × B of a pair of best proximity points.     

Viscosity Approximation Methods for Multivalued Mappings in Banach Spaces

Let K be a nonempty closed and convex subset of a real reflexive Banach space X that has weakly sequentially continuous duality mapping J. Let T: K → K be a multivalued non-expansive non-self-mapping satisfying the weakly inwardness condition as well as the condition T(y) = {y} for any y ∈ F(T) and such that for a contraction f: K → K and any t ∈ (0, 1), there exists x _t  ∈ K satisfying x _t  ∈ tf(x _t ) + (1 ? t)Tx _t . Then it is proved that {x _t } ? K converges strongly to a fixed point of T, which is also a solution of certain variational inequality. Moreover, the convergence of two explicit methods are also investigated.     

Extensions of Banach's Contraction Principle

Let A and B be nonempty subsets of a metric space. As a non-self mapping T: A → B does not necessarily have a fixed point, it is of considerable interest to find an element x that is as close to Tx as possible. In other words, if the fixed point equation Tx = x has no exact solution, then it is contemplated to find an approximate solution x such that the error d(x, Tx) is minimum. Indeed, best proximity point theorems investigate the existence of such optimal approximate solutions, called best proximity points, of the fixed point equation Tx = x when there is no exact solution. As d(x, Tx) is at least d(A, B), a best proximity point theorem achieves an absolute minimum of the error d(x, Tx) by stipulating an approximate solution x of the fixed point equation Tx = x to satisfy the condition that d(x, Tx) = d(A, B). This article furnishes extensions of Banach's contraction principle to the case of non-self mappings. On account of the preceding argument, the proposed generalizations are formulated as best proximity point theorems for non-self contractions.     

Common best proximity points: global minimal solutions

Let us suppose that A and B are nonempty subsets of a metric space. Let S:A?B and T:A?B be nonself-mappings. Considering the fact S and T are nonself-mappings, it is feasible that the equations Sx=x and Tx=x have no common solution, designated as a common fixed point of the mappings S and T. Eventually, when the equations have no common solution, one contemplates to figure out an element x that is in close proximity to Sx and Tx in the sense that d(x,Sx) and d(x,Tx) are minimum. In fact, common best proximity point theorems scrutinize the existence of such optimal approximate solutions, known as common best proximity points, to the equations Sx=x and Tx=x in the event that the equations have no common solution. Further, one can perceive that the real-valued functions x?d(x,Sx) and x?d(x,Tx) estimate the magnitude of the error involved for any common approximate solution of the equations Sx=x and Tx=x. In light of the fact that the distance between x and Sx, and the distance between x and Tx are at least the distance between A and B for all x in A, a common best proximity point theorem ascertains global minimum of both functions x?d(x,Sx) and x?d(x,Tx) by limiting a common approximate solution of the equations Sx=x and Tx=x to fulfil the requirement that d(x,Sx)=d(A,B) and d(x,Tx)=d(A,B). This article discusses a common best proximity point theorem for a pair of nonself-mappings, one of which dominates the other proximally, thereby yielding common optimal approximate solutions of some fixed point equations when there is no common solution.     

On Galois Algebras Satisfying the Fundamental Theorem

Let B be a Galois algebra over a commutative ring R with Galois group G such that B ^H is a separable subalgebra of B for each subgroup H of G. Then it is shown that B satisfies the fundamental theorem if and only if B is one of the following three types: (1) B is an indecomposable commutative Galois algebra, (2) B = Re ⊕ R(1 ? e) where e and 1 ? e are minimal central idempotents in B, and (3) B is an indecomposable Galois algebra such that for each separable subalgebra A, V _B (A) = ?∑ _g∈G(A) J _g , and the centers of A and B ^G(A) are the same where V _B (A) is the commutator subring of A in B, J _g  = {b ∈ B | bx = g(x)b for each x ∈ B} for a g ∈ G, and G(A) = {g ∈ G | g(a) = a for all a ∈ A}.     

Remotal Sets in Orlicz Spaces

Let X be a Banach space, (I, μ) be a finite measure space. By L ^Φ(I, X), let us denote the space of all X-valued Bochner Orlicz integrable functions on the unit interval I equipped with the Luxemburg norm. A closed bounded subset G of X is called remotal if for any x ∈ X, there exists g ∈ G such that ‖x ? g‖ = ρ(x, G) = sup {‖x ? y‖: y ∈ G}. In this article, we show that for a separable remotal set G ? X, the set of Bochner integrable functions, L ^Φ(I, G) is remotal in L ^Φ(I, X). Some other results are presented.     

Varieties of strange plane curves

Abstract

Let A be a commutative ring with identity, let X, Y be indeterminates and let F(X,Y), G(X, Y) ∈ A[X, Y] be homogeneous. Then the pair F(X, Y), G(X, Y) is said to be radical preserving with respect to A if Rad((F(x, y), G(x, y))R) = Rad((x,y)R) for each A-algebra R and each pair of elements x, y in R. It is shown that infinite sequences of pairwise radical preserving polynomials can be obtained by homogenizing cyclotomic polynomials, and that under suitable conditions on a ?-graded ring A these can be used to produce an infinite set of homogeneous prime ideals between two given homogeneous prime ideals P ? Q of A such that ht(Q/P) = 2.     

Commuting Traces of Biadditive Maps on Invertible Elements

Let R be a simple unital ring. Under a mild technical restriction on R, we will characterize biadditive mappings G: R² → R satisfying G(u, u)u = uG(u, u), and G(1, r) = G(r, 1) = r for all unit u ∈ R and r ∈ R, respectively. As an application, we describe bijective linear maps θ: R → R satisfying θ(xyx^?1y^?1) = θ(x)θ(y)θ(x)^?1θ(y)^?1 for all invertible x, y ∈ R. This solves an open problem of Herstein on multiplicative commutators. More precisely, we will show that θ is an isomorphism. Furthermore, we shall see the existence of a unital simple ring R′ without nontrivial idempotents, that admits a bijective linear map f: R′ → R′, preserving multiplicative commutators, that is not an isomorphism.     

Approximation Methods for a Common Fixed Point of a Finite Family of Nonexpansive Mappings

Let K be a nonempty closed and convex subset of a real Banach space E. Let T: K → E be a continuous pseudocontractive mapping and f:K → E a contraction, both satisfying weakly inward condition. Then for t ? (0, 1), there exists a sequence {y _t } ? K satisfying the following condition: y _t  = (1 ? t)f(y _t ) + tT(y _t ). Suppose further that {y _t } is bounded or F(T) ≠  and E is a reflexive Banach space having weakly continuous duality mapping J _? for some gauge ?. Then it is proved that {y _t } converges strongly to a fixed point of T, which is also a solution of certain variational inequality. Moreover, an explicit iteration process that converges strongly to a common fixed point of a finite family of nonexpansive mappings and hence to a solution of a certain variational inequality is constructed.     

On the Invariant Fields and Rings of Some Groups of Polynomial Automorphisms

Abstract

We consider the group G of C-automorphisms of C(x, y) (resp. C[x, y]) generated by s, t such that t(x) = y, t(y) = x and s(x) = x, s(y) = ? y + u(x) where u ∈ C[x] is of degree k ≥ 2. Using Galois's theory, we show that the invariant field and the invariant algebra of G are equal to C.     

Preserving Zeros of xy and xy*

We describe surjective additive maps θ: A → B which preserve zero products, where A is a ring with a nontrivial idempotent and B is a prime ring. We also characterize surjective additive maps θ: A → B such that for all x, y ∈ A we have θ(x)θ(y)* = 0 if and only if xy* = 0. Here A is a unital prime ring with involution that contains a nontrivial idempotent and B is a prime ring with involution.     

Common Best Proximity Points: Global Optimal Solutions

Let S:A→B and T:A→B be given non-self mappings, where A and B are non-empty subsets of a metric space. As S and T are non-self mappings, the equations Sx=x and Tx=x do not necessarily have a common solution, called a common fixed point of the mappings S and T. Therefore, in such cases of non-existence of a common solution, it is attempted to find an element x that is closest to both Sx and Tx in some sense. Indeed, common best proximity point theorems explore the existence of such optimal solutions, known as common best proximity points, to the equations Sx=x and Tx=x when there is no common solution. It is remarked that the functions x→d(x,Sx) and x→d(x,Tx) gauge the error involved for an approximate solution of the equations Sx=x and Tx=x. In view of the fact that, for any element x in A, the distance between x and Sx, and the distance between x and Tx are at least the distance between the sets A and B, a common best proximity point theorem achieves global minimum of both functions x→d(x,Sx) and x→d(x,Tx) by stipulating a common approximate solution of the equations Sx=x and Tx=x to fulfill the condition that d(x,Sx)=d(x,Tx)=d(A,B). The purpose of this article is to elicit common best proximity point theorems for pairs of contractive non-self mappings and for pairs of contraction non-self mappings, yielding common optimal approximate solutions of certain fixed point equations. Besides establishing the existence of common best proximity points, iterative algorithms are also furnished to determine such optimal approximate solutions.     

Inverse Problems for Time-Dependent Singular Heat Conductivities: Multi-Dimensional Case

We consider an inverse boundary value problem for the heat equation ? _t u = div (γ? _x u) in (0, T) × Ω, u = f on (0, T) × ?Ω, u| _t=0 = u ₀, in a bounded domain Ω ? ? ⁿ , n ≥ 2, where the heat conductivity γ(t, x) is piecewise constant and the surface of discontinuity depends on time: γ(t, x) = k ² (x ∈ D(t)), γ(t, x) = 1 (x ∈ Ω?D(t)). Fix a direction e* ∈ 𝕊 ^n?1 arbitrarily. Assuming that ?D(t) is strictly convex for 0 ≤ t ≤ T, we show that k and sup {e*·x; x ∈ D(t)} (0 ≤ t ≤ T), in particular D(t) itself, are determined from the Dirichlet-to-Neumann map : f → ?_ν u(t, x)|_{(0, T)×?Ω}. The knowledge of the initial data u ₀ is not used in the proof. If we know min_0≤t≤T (sup _x∈D(t) x·e*), we have the same conclusion from the local Dirichlet-to-Neumann map. Numerical examples of stationary and moving circles inside the unit disk are shown. The results have applications to nondestructive testing. Consider a physical body consisting of homogeneous material with constant heat conductivity except for a moving inclusion with different conductivity. Then the location and shape of the inclusion can be monitored from temperature and heat flux measurements performed at the boundary of the body. Such a situation appears for example in blast furnaces used in ironmaking.     

On the Genus of the Nilpotent Graphs of Finite Groups

The nilpotent graph of a group G is a simple graph whose vertex set is G?nil(G), where nil(G) = {y ∈ G | ? x, y ? is nilpotent ? x ∈ G}, and two distinct vertices x and y are adjacent if ? x, y ? is nilpotent. In this article, we show that the collection of finite non-nilpotent groups whose nilpotent graphs have the same genus is finite, derive explicit formulas for the genus of the nilpotent graphs of some well-known classes of finite non-nilpotent groups, and determine all finite non-nilpotent groups whose nilpotent graphs are planar or toroidal.     

Group Laws [x,y −1] ≡ u(x,y) and Varietal Properties

Let F = ?x, y? be a free group. It is known that the commutator [x, y ^?1] cannot be expressed in terms of basic commutators, in particular in terms of Engel commutators. We show that the laws imposing such an expression define specific varietal properties. For a property 𝒫 we consider a subset U(𝒫) ? F such that every law of the form [x, y ^?1] ≡ u, u ∈ U(𝒫) provides the varietal property 𝒫. For example, we show that each subnormal subgroup is normal in every group of a variety 𝔙 if and only if 𝔙 satisfies a law of the form [x, y ^?1] ≡ u, where u ∈ [F′, ?x?].     

Best proximity points: global optimal approximate solutions

Let A and B be non-empty subsets of a metric space. As a non-self mapping \({T:A\longrightarrow B}\) does not necessarily have a fixed point, it is of considerable interest to find an element x in A that is as close to Tx in B as possible. In other words, if the fixed point equation Tx = x has no exact solution, then it is contemplated to find an approximate solution x in A such that the error d(x, Tx) is minimum, where d is the distance function. Indeed, best proximity point theorems investigate the existence of such optimal approximate solutions, called best proximity points, to the fixed point equation Tx = x when there is no exact solution. As the distance between any element x in A and its image Tx in B is at least the distance between the sets A and B, a best proximity pair theorem achieves global minimum of d(x, Tx) by stipulating an approximate solution x of the fixed point equation Tx = x to satisfy the condition that d(x, Tx) = d(A, B). The purpose of this article is to establish best proximity point theorems for contractive non-self mappings, yielding global optimal approximate solutions of certain fixed point equations. Besides establishing the existence of best proximity points, iterative algorithms are also furnished to determine such optimal approximate solutions.     

Separativity of Regular Rings in Which 2 Is Invertible

A regular ring R is separative provided that for all finitely generated projective right R-modules A and B, A⊕ A? A⊕ B? A⊕ B implies that A? B. We prove, in this article, that a regular ring R in which 2 is invertible is separative if and only if each a ∈ R satisfying R(1 ? a ²)R = Rr(a) = ?(a)R and i(End _R (aR)) = ∞ is unit-regular if and only if each a ∈ R satisfying R(1 ? a ²)R ∩ RaR = Rr(a) ∩ ?(a)R ∩ RaR and i(End _R (aR)) = ∞ is unit-regular. Further equivalent characterizations of such regular rings are also obtained.     

The group PSL(2,q) is not the multiplication group of a loop

ABSTRACT

We introduce the notion of weak transitivity for torsion-free abelian groups. A torsion-free abelian group G is called weakly transitive if for any pair of elements x, y ∈ G and endomorphisms ?, ψ ∈ End(G) such that x? = y, yψ = x, there exists an automorphism of G mapping x onto y. It is shown that every suitable ring can be realized as the endomorphism ring of a weakly transitive torsion-free abelian group, and we characterize up to a number-theoretical property the separable weakly transitive torsion-free abelian groups.     

A Principle for Critical Point Under Generalized Regular Constraint and Ill-Posed Lagrange Multipliers Under Nonregular Constraints

In this article, a kind of nonregular constraint and a principle for seeking critical point under the constraint are presented, where no Lagrange multiplier is involved. Let E, F be two Banach spaces, g: E → F a c ¹ map defined on an open set U in E, and the constraint S = the preimage g ^?1(y ₀), y ₀ ∈ F. A main deference between the nonregular constraint and regular constraint is that g′(x) at any x ∈ S is not surjective. Recently, the critical point theory under the nonregular constraint is a concerned focus in optimization theory. The principle also suits the case of regular constraint. Coordinately, the generalized regular constraint is introduced and the critical point principle on generalized regular constraint is established. Let f: U → ? be a nonlinear functional. While the Lagrange multiplier L in classical critical point principle is considered, its expression is given by using generalized inverse g′⁺(x) of g′(x) as follows: if x ∈ S is a critical point of f| _S , then L = f′(x) ○ g′⁺(x) ∈ F*. Moreover, it is proved that if S is a regular constraint, then the Lagrange multiplier L is unique; otherwise, L is ill-posed. Hence, in case of the nonregular constraint, it is very difficult to solve Euler equations; however, it is often the case in optimization theory. So the principle here seems to be new and applicable. By the way, the following theorem is proved: if A ∈ B(E, F) is double split, then the set of all generalized inverses of A, GI(A) is smooth diffeomorphic to certain Banach space. This is a new and interesting result in generalized inverse analysis.