Dual graph homomorphism functions

For any two graphs F and G, let hom(F,G) denote the number of homomorphisms F→G, that is, adjacency preserving maps V(F)→V(G) (graphs may have loops but no multiple edges). We characterize graph parameters f for which there exists a graph F such that f(G)=hom(F,G) for each graph G.The result may be considered as a certain dual of a characterization of graph parameters of the form hom(.,H), given by Freedman, Lovász and Schrijver [M. Freedman, L. Lovász, A. Schrijver, Reflection positivity, rank connectivity, and homomorphisms of graphs, J. Amer. Math. Soc. 20 (2007) 37-51]. The conditions amount to the multiplicativity of f and to the positive semidefiniteness of certain matrices N(f,k).     

Homomorphisms between the chevalley groups over any fields of characteristic zero

Given an irreducible root system ∑, let G(F,L) denote the Cheval- ley group over a field F corresponding to a lattice L between the root lattice and the weight lattice of ∑,. We will determine all nontnvial homomorphisms from G(k,L ₁) to G(K,L ₂when k and K are any fields of characteristic zero, and we will verify that any nontrivial homomorphism from G(k,L ₁) to G(K,L ₂are induced by a field homomorphism from k to K by multiplying an automorphism of G(K,L ₂.     

On the extension of vertex maps to graph homomorphisms

A reflexive graph is a simple undirected graph where a loop has been added at each vertex. If G and H are reflexive graphs and U⊆V(H), then a vertex map f:U→V(G) is called nonexpansive if for every two vertices x,y∈U, the distance between f(x) and f(y) in G is at most that between x and y in H. A reflexive graph G is said to have the extension property (EP) if for every reflexive graph H, every U⊆V(H) and every nonexpansive vertex map f:U→V(G), there is a graph homomorphism φ_f:H→G that agrees with f on U. Characterizations of EP-graphs are well known in the mathematics and computer science literature. In this article we determine when exactly, for a given “sink”-vertex s∈V(G), we can obtain such an extension φ_f;s that maps each vertex of H closest to the vertex s among all such existing homomorphisms φ_f. A reflexive graph G satisfying this is then said to have the sink extension property (SEP). We then characterize the reflexive graphs with the unique sink extension property (USEP), where each such sink extensions φ_f;s is unique.     

A New Class of Generalized Thompson's Groups and Their Normal Subgroups

For a given group G and a homomorphism ?: G → G × G, we construct groups ?_?(G), 𝒯_?(G), and 𝒱_?(G) that blend Thompson's groups F, T, and V with G, respectively. Furthermore, we describe the lattice of normal subgroups of the groups ?_Δ(G), where Δ: G → G × G is the diagonal homomorphism, Δ(g) = (g, g).     

The Exact Value of Normal Structure Coefficients and WCS coefficients in a Class of Orlicz Function Spaces

Let φ be an N-function. Then the normal structure coefficients N and the weakly convergent sequence coefficients WCS of the Orlicz function spaces L ^φ[0, 1] generated by φ and equipped with the Luxemburg and Orlicz norms have the following exact values. (i) If F _φ(t) = t _?(t)/φ(t) is decreasing and 1 < C _φ < 2 (where \(C_\Phi = \lim _{t \to + \infty } t\varphi (t)/\Phi (t)\)), then N(L ^(φ)[0, 1]) = N(L ^φ[0, 1]) = WCS(L ^(φ)[0, 1]) = WCS(L ^φ[0, 1]) = 2^1?1/Cφ. (ii) If F _φ(t) is increasing and C _φ > 2, then N(L ^(φ)[0, 1]) = N(L ^φ[0, 1]) = WCS(L ^(φ)[0, 1]) = WCS(L ^φ[0, 1]) = 2^1/Cφ.     

Bounded factorization property for Fréchet spaces

An operator T ∈ L(E, F) factors over G if T = RS for some S ∈ L(E, G) and R ∈ L(G, F); the set of such operators is denoted by L^G(E, F). A triple (E, G, F) satisfies bounded factorization property (shortly, (E, G, F) ∈ ???) if L^G(E, F) ? LB(E, F), where LB(E, F) is the set of all bounded linear operators from E to F. The relationship (E, G, F) ∈ ??? is characterized in the spirit of Vogt's characterisation of the relationship L(E, F) = LB(E, F) [23]. For triples of K?othe spaces the property ??? is characterized in terms of their K?othe matrices. As an application we prove that in certain cases the relations L(E, G₁) = LB(E, G₁) and L(G₂, F) = LB(G₂, F) imply (E, G, F) ∈ ??? where G is a tensor product of G₁ and G₂.     

Module homomorphisms and multipliers on locally compact quantum groups

For a Banach algebra A with a bounded approximate identity, we investigate the A-module homomorphisms of certain introverted subspaces of A^∗, and show that all A-module homomorphisms of A^∗ are normal if and only if A is an ideal of A∗∗. We obtain some characterizations of compactness and discreteness for a locally compact quantum group G. Furthermore, in the co-amenable case we prove that the multiplier algebra of L¹(G) can be identified with M(G). As a consequence, we prove that G is compact if and only if LUC(G)=WAP(G) and M(G)≅Z(LUC^∗(G)); which partially answer a problem raised by Volker Runde.     

An “almost” full embedding of the category of graphs into the category of groups

We construct a functor F:Graphs→Groups which is faithful and “almost” full, in the sense that every nontrivial group homomorphism FX→FY is a composition of an inner automorphism of FY and a homomorphism of the form Ff, for a unique map of graphs f:X→Y. When F is composed with the Eilenberg-Mac Lane space construction K(FX,1) we obtain an embedding of the category of graphs into the unpointed homotopy category which is full up to null-homotopic maps.We provide several applications of this construction to localizations (i.e. idempotent functors); we show that the questions:

(1)

Is every orthogonality class reflective?

(2)

Is every orthogonality class a small-orthogonality class?

have the same answers in the category of groups as in the category of graphs. In other words they depend on set theory: (1) is equivalent to weak Vopěnka's principle and (2) to Vopěnka's principle. Additionally, the second question, considered in the homotopy category, is also equivalent to Vopěnka's principle.     

Extensions of regular mappings and the ?ojasiewicz exponent at infinity

Let F:V→Cm be a regular mapping, where V⊂Cn is an algebraic set of positive dimension and m?n?2, and let L_∞(F) be the ?ojasiewicz exponent at infinity of F. We prove that F has a polynomial extension G:Cn→Cm such L_∞(G)=L_∞(F). Moreover, we give an estimate of the degree of the extension G. Additionally, we prove that if then for any β∈Q, β?L_∞(F), the mapping F has a polynomial extension G with L_∞(G)=β. We also give an estimate of the degree of this extension.     

Polynomial Identities of RA and RA2 Loop Algebras

Let F be an algebraically closed field of characteristic zero and L an RA loop. We prove that the loop algebra FL is in the variety generated by the split Cayley–Dickson algebra Z _F over F. For RA2 loops of type M(Dih(A), ^?1,g ₀), we prove that the loop algebra is in the variety generated by the algebra  ₃ which is a noncommutative simple component of the loop algebra of a certain RA2 loop of order 16. The same does not hold for the RA2 loops of type M(G, ^?1,g ₀), where G is a non-Abelian group of exponent 4 having exactly 2 squares.     

Fundamental groups of asymptotic cones

We show that for any metric space M satisfying certain natural conditions, there is a finitely generated group G, an ultrafilter ω, and an isometric embedding ι of M to the asymptotic cone Cone_ω(G) such that the induced homomorphism ι^*:π₁(M)→π₁(Cone_ω(G)) is injective. In particular, we prove that any countable group can be embedded into a fundamental group of an asymptotic cone of a finitely generated group.     

Multiplicative mappings at some points on matrix algebras

Let M_n be the algebra of all n×n matrices, and let φ:M_n→M_n be a linear mapping. We say that φ is a multiplicative mapping at G if φ(ST)=φ(S)φ(T) for any S,T∈M_n with ST=G. Fix G∈M_n, we say that G is an all-multiplicative point if every multiplicative linear bijection φ at G with φ(I_n)=I_n is a multiplicative mapping in M_n, where I_n is the unit matrix in M_n. We mainly show in this paper the following two results: (1) If G∈M_n with detG=0, then G is an all-multiplicative point in M_n; (2) If φ is an multiplicative mapping at I_n, then there exists an invertible matrix P∈M_n such that either φ(S)=PSP^-1 for any S∈M_n or φ(T)=PT^trP^-1 for any T∈M_n.     

Universal commutative operator algebras and transfer function realizations of polynomials

To each finite-dimensional operator space E is associated a commutative operator algebra UC(E), so that E embeds completely isometrically in UC(E) and any completely contractive map from E to bounded operators on Hilbert space extends uniquely to a completely contractive homomorphism out of UC(E). The unit ball of UC(E) is characterized by a Nevanlinna factorization and transfer function realization. Examples related to multivariable von Neumann inequalities are discussed.     

Control problems governed by Sobolev partial differential equations

Let G be a bounded subset of Rⁿ with a smooth boundary and Q = G × (0, T]. We consider a control problem governed by the Sobolev initial-value problem My_t(u) + Ly(u) = u in L²(Q), y(·, 0; u) = 0 in L₂(G), where M = M(x) and L = L(x) are symmetric uniformly strongly elliptic operators of orders 2m and 2l, respectively. The problem is to find the control u₀ of L²(Q)-norm at most b that steers to within a prescribed tolerance ? of a given function Z in L²(G) and that minimizes a certain energy functional. Our main results establish regularity properties of u₀. We also give results concerning the existence and uniqueness of the optimal control, the controllability of Sobolev initial-value problems, and properties of the Lagrange multipliers associated with the problem constraints.     

Automatic continuity over Moore groups

LetG be a Moore group, letB be a Banach algebra, and let :L ¹(G)B be a homomorphism. We show that is continuous if and only if its restriction to the center ofL ¹(G) is continuous. As a consequence, we obtain that (i) every homomorphism fromL ¹(G) orC ^*(G) onto a dense subalgebra of a semisimple Banach algebra, and (ii) every epimorphism fromC ^*(G) onto a Banach algebra is automatically continuous.     

Computation of Nielsen numbers for maps of compact surfaces with boundary

In this paper we study the Nielsen number of a self-map f:M→M of a compact connected surface with boundary. Let G=π₁(M) be the fundamental group of M which is a finitely generated free group. We introduce a new algebraic condition called “bounded solution length” on the induced endomorphism φ:G→G of f and show that many maps which have no remnant satisfy this condition. For a map f that has bounded solution length, we describe an algorithm for computing the Nielsen number N(f).     

Verma Modules Over Generalized Block Algebras B(b (1), b (2))

ABSTRACT

Given a field F with characteristic zero, a free Abelian group G with rank two, and a total order ? on G which is compatible with the addition, we define Verma modules M([ddot], ?) over the generalized Block algebra B(b ⁽¹⁾, b ⁽²⁾) with b ⁽¹⁾, b ⁽²⁾ ∈ F. The irreducibility of the module M([ddot], ?) is completely determined in this article.     

On minimum rank and zero forcing sets of a graph

For a graph G on n vertices and a field F, the minimum rank of G over F, written as mr^F(G), is the smallest possible rank over all n×n symmetric matrices over F whose (i,j)th entry (for ) is nonzero whenever ij is an edge in G and is zero otherwise. The maximum nullity of G over F is M^F(G)=n-mr^F(G). The minimum rank problem of a graph G is to determine mr^F(G) (or equivalently, M^F(G)). This problem has received considerable attention over the years. In [F. Barioli, W. Barrett, S. Butler, S.M. Cioab?, D. Cvetkovi?, S.M. Fallat, C. Godsil, W. Haemers, L. Hogben, R. Mikkelson, S. Narayan, O. Pryporova, I. Sciriha, W. So, D. Stevanovi?, H. van der Holst, K.V. Meulen, A.W. Wehe, AIM Minimum Rank-Special Graphs Work Group, Zero forcing sets and the minimum rank of graphs, Linear Algebra Appl. 428 (2008) 1628-1648], a new graph parameter Z(G), the zero forcing number, was introduced to bound M^F(G) from above. The authors posted an attractive question: What is the class of graphs G for which Z(G)=M^F(G) for some field F? This paper focuses on exploring the above question.