Coincidences of projections and linear n-valued maps of tori

We prove that the Nielsen fixed point number N(φ) of an n-valued map φ:X?X of a compact connected triangulated orientable q-manifold without boundary is equal to the Nielsen coincidence number of the projections of the graph of φ, a subset of X×X, to the two factors. For certain q×q integer matrices A, there exist “linear” n-valued maps Φn,A,σ:Tq?Tq of q-tori that generalize the single-valued maps fA:Tq→Tq induced by the linear transformations TA:Rq→Rq defined by TA(v)=Av. By calculating the Nielsen coincidence number of the projections of its graph, we calculate N(Φn,A,σ) for a large class of linear n-valued maps.     

On the computation of the hull number of a graph

Let G be a graph. If u,v∈V(G), a u-vshortest path of G is a path linking u and v with minimum number of edges. The closed interval I[u,v] consists of all vertices lying in some u-v shortest path of G. For S⊆V(G), the set I[S] is the union of all sets I[u,v] for u,v∈S. We say that S is a convex set if I[S]=S. The convex hull of S, denoted I_h[S], is the smallest convex set containing S. A set S is a hull set of G if I_h[S]=V(G). The cardinality of a minimum hull set of G is the hull number of G, denoted by hn(G). In this work we prove that deciding whether hn(G)≤k is NP-complete.We also present polynomial-time algorithms for computing hn(G) when G is a unit interval graph, a cograph or a split graph.     

Higher rank numerical ranges and low rank perturbations of quantum channels

For a positive integer k, the rank-k numerical range Λk(A) of an operator A acting on a Hilbert space H of dimension at least k is the set of scalars λ such that PAP=λP for some rank k orthogonal projection P. In this paper, a close connection between low rank perturbation of an operator A and Λk(A) is established. In particular, for 1?r<k it is shown that Λk(A)⊆Λk−r(A+F) for any operator F with rank(F)?r. In quantum computing, this result implies that a quantum channel with a k-dimensional error correcting code under a perturbation of rank at most r will still have a (k−r)-dimensional error correcting code. Moreover, it is shown that if A is normal or if the dimension of A is finite, then Λk(A) can be obtained as the intersection of Λk−r(A+F) for a collection of rank r operators F. Examples are given to show that the result fails if A is a general operator. The closure and the interior of the convex set Λk(A) are completely determined. Analogous results are obtained for Λ_∞(A) defined as the set of scalars λ such that PAP=λP for an infinite rank orthogonal projection P. It is shown that Λ_∞(A) is the intersection of all Λk(A) for k=1,2,…. If A−μI is not compact for all μ∈C, then the closure and the interior of Λ_∞(A) coincide with those of the essential numerical range of A. The situation for the special case when A−μI is compact for some μ∈C is also studied.     

The fractional chromatic number, the Hall ratio, and the lexicographic product

Let χ_f denote the fractional chromatic number and ρ the Hall ratio, and let the lexicographic product of G and H be denoted GlexH. Main results: (i) ρ(GlexH)≤χ_f(G)ρ(H); (ii) if ρ(G)=χ_f(G) then ρ(GlexH)=ρ(G)ρ(H) for all H; (iii) χ_f−ρ is unbounded. In addition, the question of how big χ_f/ρ can be is discussed.     

Formality theorem for Lie bialgebras and quantization of twists and coboundary r-matrices

Let (g,δ?) be a Lie bialgebra. Let (U?(g),Δ?) a quantization of (g,δ?) through Etingof-Kazhdan functor. We prove the existence of a L_∞-morphism between the Lie algebra C(g)=Λ(g) and the tensor algebra (without unit) T₊U=T₊(U?(g)[−1]) with Lie algebra structure given by the Gerstenhaber bracket. When s is a twist for (g,δ), we deduce from the formality morphism the existence of a quantum twist F. When (g,δ,r) is a coboundary Lie bialgebra, we get the existence of a quantization R of r.     

Some paranormed sequence spaces of non-absolute type derived by weighted mean

The sequence spaces ?_∞(p), c(p) and c₀(p) were introduced and studied by Maddox [I.J. Maddox, Paranormed sequence spaces generated by infinite matrices, Proc. Cambridge Philos. Soc. 64 (1968) 335-340]. In the present paper, the sequence spaces λ(u,v;p) of non-absolute type which are derived by the generalized weighted mean are defined and proved that the spaces λ(u,v;p) and λ(p) are linearly isomorphic, where λ denotes the one of the sequence spaces ?_∞, c or c₀. Besides this, the β- and γ-duals of the spaces λ(u,v;p) are computed and the basis of the spaces c₀(u,v;p) and c(u,v;p) is constructed. Additionally, it is established that the sequence space c₀(u,v) has AD property and given the f-dual of the space c₀(u,v;p). Finally, the matrix mappings from the sequence spaces λ(u,v;p) to the sequence space μ and from the sequence space μ to the sequence spaces λ(u,v;p) are characterized.     

Covering shadows with a smaller volume

For n?2 a construction is given for convex bodies K and L in Rn such that the orthogonal projection Lu onto the subspace u^⊥ contains a translate of Ku for every direction u, while the volumes of K and L satisfy Vn(K)>Vn(L).A more general construction is then given for n-dimensional convex bodies K and L such that each orthogonal projection Lξ onto a k-dimensional subspace ξ contains a translate of Kξ, while the mth intrinsic volumes of K and L satisfy Vm(K)>Vm(L) for all m>k.For each k=1,…,n, we then define the collection Cn,k to be the closure (under the Hausdorff topology) of all Blaschke combinations of suitably defined cylinder sets (prisms).It is subsequently shown that, if L∈Cn,k, and if the orthogonal projection Lξ contains a translate of Kξ for every k-dimensional subspace ξ of Rn, then Vn(K)?Vn(L).The families Cn,k, called k-cylinder bodies of Rn, form a strictly increasing chain

Cn,1⊂Cn,2⊂?⊂Cn,n−1⊂Cn,n,     

A method of finding automorphism groups of endomorphism monoids of relational systems

For any set X and any relation ρ on X, let T(X,ρ) be the semigroup of all maps a:X→X that preserve ρ. Let S(X) be the symmetric group on X. If ρ is reflexive, the group of automorphisms of T(X,ρ) is isomorphic to N_S(X)(T(X,ρ)), the normalizer of T(X,ρ) in S(X), that is, the group of permutations on X that preserve T(X,ρ) under conjugation. The elements of N_S(X)(T(X,ρ)) have been described for the class of so-called dense relations ρ. The paper is dedicated to applications of this result.     

Bijective linear maps on semimodules spanned by Boolean matrices of fixed rank

Let M_m,n(B) be the semimodule of all m×n Boolean matrices where B is the Boolean algebra with two elements. Let k be a positive integer such that 2?k?min(m,n). Let B(m,n,k) denote the subsemimodule of M_m,n(B) spanned by the set of all rank k matrices. We show that if T is a bijective linear mapping on B(m,n,k), then there exist permutation matrices P and Q such that T(A)=PAQ for all A∈B(m,n,k) or m=n and T(A)=PA^tQ for all A∈B(m,n,k). This result follows from a more general theorem we prove concerning the structure of linear mappings on B(m,n,k) that preserve both the weight of each matrix and rank one matrices of weight k². Here the weight of a Boolean matrix is the number of its nonzero entries.     

Bitableau bases for Garsia-Haiman modules of hollow type

Garsia-Haiman modules C[Xn,Yn]/Iγ are quotient rings in the variables Xn={x₁,x₂,…,xn} and Yn={y₁,y₂,…,yn} that generalize the quotient ring C[Xn]/I, where I is the ideal generated by the elementary symmetric polynomials ej(Xn) for 1?j?n. A bitableau basis for the Garsia-Haiman modules of hollow type is constructed. Applications of this basis to representation theory and other related polynomial spaces are considered.     

On Harary index of graphs

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. For a connected graph G=(V,E) and two nonadjacent vertices v_i and v_j in V(G) of G, recall that G+v_iv_j is the supergraph formed from G by adding an edge between vertices v_i and v_j. Denote the Harary index of G and G+v_iv_j by H(G) and H(G+v_iv_j), respectively. We obtain lower and upper bounds on H(G+v_iv_j)−H(G), and characterize the equality cases in those bounds. Finally, in this paper, we present some lower and upper bounds on the Harary index of graphs with different parameters, such as clique number and chromatic number, and characterize the extremal graphs at which the lower or upper bounds on the Harary index are attained.     

Some bounds on the p-domination number in trees

Let p be a positive integer and G=(V,E) a graph. A subset S of V is a p-dominating set if every vertex of V-S is dominated at least p times, and S is a p-dependent set of G if the subgraph induced by the vertices of S has maximum degree at most p-1. The minimum cardinality of a p-dominating set a of G is the p-domination number γ_p(G) and the maximum cardinality of a p-dependent set of G is the p-dependence number β_p(G). For every positive integer p?2, we show that for a bipartite graph G, γ_p(G) is bounded above by (|V|+|Y_p|)/2, where Y_p is the set of vertices of G of degree at most p-1, and for every tree T, γ_p(T) is bounded below by β_p-1(T). Moreover, we characterize the trees achieving equality in each bound.     

Maximum nullity of outerplanar graphs and the path cover number

Let G=(V,E) be a graph with V={1,2,…,n}. Define S(G) as the set of all n×n real-valued symmetric matrices A=[a_ij] with a_ij≠0,i≠j if and only if ij∈E. By M(G) we denote the largest possible nullity of any matrix A∈S(G). The path cover number of a graph G, denoted P(G), is the minimum number of vertex disjoint paths occurring as induced subgraphs of G which cover all the vertices of G.There has been some success with relating the path cover number of a graph to its maximum nullity. Johnson and Duarte [5], have shown that for a tree T,M(T)=P(T). Barioli et al. [2], show that for a unicyclic graph G,M(G)=P(G) or M(G)=P(G)-1. Notice that both families of graphs are outerplanar. We show that for any outerplanar graph G,M(G)?P(G). Further we show that for any partial 2-path G,M(G)=P(G).     

Mrówka maximal almost disjoint families for uncountable cardinals

We consider generalizations of a well-known class of spaces, called by S. Mrówka, N∪R, where R is an infinite maximal almost disjoint family (MADF) of countable subsets of the natural numbers N. We denote these generalizations by ψ=ψ(κ,R) for κ?ω. Mrówka proved the interesting theorem that there exists an R such that |βψ(ω,R)?ψ(ω,R)|=1. In other words there is a unique free z-ultrafilter p₀ on the space ψ. We extend this result of Mrówka to uncountable cardinals. We show that for κ?c, Mrówka's MADF R can be used to produce a MADF M⊂ω[κ] such that |βψ(κ,M)?ψ(κ,M)|=1. For κ>c, and every M⊂ω[κ], it is always the case that |βψ(κ,M)?ψ(κ,M)|≠1, yet there exists a special free z-ultrafilter p on ψ(κ,M) retaining some of the properties of p₀. In particular both p and p₀ have a clopen local base in βψ (although βψ(κ,M) need not be zero-dimensional). A result for κ>c, that does not apply to p₀, is that for certain κ>c, p is a P-point in βψ.     

Labeling bipartite permutation graphs with a condition at distance two

An L(p,q)-labeling of a graph G is an assignment f from vertices of G to the set of non-negative integers {0,1,…,λ} such that |f(u)−f(v)|≥p if u and v are adjacent, and |f(u)−f(v)|≥q if u and v are at distance 2 apart. The minimum value of λ for which G has L(p,q)-labeling is denoted by λ_p,q(G). The L(p,q)-labeling problem is related to the channel assignment problem for wireless networks.In this paper, we present a polynomial time algorithm for computing L(p,q)-labeling of a bipartite permutation graph G such that the largest label is at most (2p−1)+q(bc(G)−2), where bc(G) is the biclique number of G. Since λ_p,q(G)≥p+q(bc(G)−2) for any bipartite graph G, the upper bound is at most p−1 far from optimal.     

Greedy F-colorings of graphs

Let G=(V,E) be a connected graph. For a symmetric, integer-valued function δ on V×V, where K is an integer constant, N₀ is the set of nonnegative integers, and Z is the set of integers, we define a C-mapping by F(u,v,m)=δ(u,v)+m−K. A coloring c of G is an F-coloring if F(u,v,|c(u)−c(v)|)?0 for every two distinct vertices u and v of G. The maximum color assigned by c to a vertex of G is the value of c, and the F-chromatic number F(G) is the minimum value among all F-colorings of G. For an ordering of the vertices of G, a greedy F-coloring c of s is defined by (1) c(v₁)=1 and (2) for each i with 1?i<n, c(v_i+1) is the smallest positive integer p such that F(v_j,v_i+1,|c(v_j)−p|)?0, for each j with 1?j?i. The greedy F-chromatic number gF(s) of s is the maximum color assigned by c to a vertex of G. The greedy F-chromatic number of G is gF(G)=min{gF(s)} over all orderings s of V. The Grundy F-chromatic number is GF(G)=max{gF(s)} over all orderings s of V. It is shown that gF(G)=F(G) for every graph G and every F-coloring defined on G. The parameters gF(G) and GF(G) are studied and compared for a special case of the C-mapping F on a connected graph G, where δ(u,v) is the distance between u and v and .     

Some new spaces of double sequences

In this study, we define the double sequence spaces BS, BS(t), CSp, CSbp, CSr and BV, and also examine some properties of those sequence spaces. Furthermore, we show that these sequence spaces are complete paranormed or normed spaces under some certain conditions. We determine the α-duals of the spaces BS, BV, CSbp and the β(?)-duals of the spaces CSbp and CSr of double series. Finally, we give the conditions which characterize the class of four-dimensional matrix mappings defined on the spaces CSbp, CSr and CSp of double series.     

On a new notion of the solution to an ill-posed problem

A new understanding of the notion of the stable solution to ill-posed problems is proposed. The new notion is more realistic than the old one and better fits the practical computational needs. A method for constructing stable solutions in the new sense is proposed and justified. The basic point is: in the traditional definition of the stable solution to an ill-posed problem Au=f, where A is a linear or nonlinear operator in a Hilbert space H, it is assumed that the noisy data {f_δ,δ} are given, ‖f−f_δ‖≤δ, and a stable solution u_δ:=R_δf_δ is defined by the relation lim_δ→0‖R_δf_δ−y‖=0, where y solves the equation Au=f, i.e., Ay=f. In this definition y and f are unknown. Any f∈B(f_δ,δ) can be the exact data, where B(f_δ,δ):={f:‖f−f_δ‖≤δ}.The new notion of the stable solution excludes the unknown y and f from the definition of the solution. The solution is defined only in terms of the noisy data, noise level, and an a priori information about a compactum to which the solution belongs.