A superlinear space decomposition algorithm for constrained nonsmooth convex program

A class of constrained nonsmooth convex optimization problems, that is, piecewise C² convex objectives with smooth convex inequality constraints are transformed into unconstrained nonsmooth convex programs with the help of exact penalty function. The objective functions of these unconstrained programs are particular cases of functions with primal-dual gradient structure which has connection with VU space decomposition. Then a VU space decomposition method for solving this unconstrained program is presented. This method is proved to converge with local superlinear rate under certain assumptions. An illustrative example is given to show how this method works.     

Twisted first homology groups of the automorphism group of a free group

The automorphism group and outer automorphism group of a free group F_n of rank n act on the abelianized group H of F_n and the dual group H^* of H. The twisted first homology groups of and with coefficients in H and H^* are calculated.     

Boundary feedback stabilization of the undamped Timoshenko beam with both ends free

In this paper, we study a boundary feedback system of a class of nonuniform undamped Timoshenko beam with both ends free. We give some sufficient conditions and some necessary conditions for the system to have exponential stability. Our method is based on the operator semigroup technique, the multiplier technique, and the contradiction argument of the frequency domain method.     

Colorings of k-balanced matrices and integer decomposition property of related polyhedra

We show that a class of polyhedra, arising from certain 0,1 matrices introduced by Truemper and Chandrasekaran, has the integer decomposition property. This is accomplished by proving certain coloring properties of these matrices.     

A degree sum condition for graphs to be prism hamiltonian

Win, in 1975, and Jackson and Wormald, in 1990, found the best sufficient conditions on the degree sum of a graph to guarantee the properties of “having a k-tree” and “having a k-walk”, respectively. The property of “being prism hamiltonian” is an intermediate property between “having a 2-tree” and “having a 2-walk”. Thus, it is natural to ask what is the best degree sum condition for graphs to be prism hamiltonian. As an answer to this problem, in this paper, we show that a connected graph G of order n with σ₃(G)≥n is prism hamiltonian. The degree sum condition “σ₃(G)≥n” is best possible.     

Some decomposition method for analytic solving of certain nonlinear partial differential equations in physics with applications

Certain nonlinear partial differential equations (NPDEs) can be decomposed into several more simple equations, which can possess enough general analytic solutions. This approach and some interesting kinds of solutions (obtained by using this method) of some NPDEs in physics will be presented. The presented approach is somewhat similar to the homogeneous balance method, however they are different.     

A new transvectant algorithm for nilpotent normal forms

We offer an algorithm to determine the form of the normal form for a vector field with a nilpotent linear part, when the form of the normal form is known for each Jordan block of the linear part taken separately. The algorithm is based on the notion of transvectant, from classical invariant theory.     

On chromatic and flow polynomial unique graphs

It is known that the chromatic polynomial and flow polynomial of a graph are two important evaluations of its Tutte polynomial, both of which contain much information of the graph. Much research is done on graphs determined entirely by their chromatic polynomials and Tutte polynomials, respectively. Oxley asked which classes of graphs or matroids are determined by their chromatic and flow polynomials together. In this paper, we found several classes of graphs with this property. We first study which graphic parameters are determined by the flow polynomials. Then we study flow-unique graphs. Finally, we show that several classes of graphs, ladders, Möbius ladders and squares of n-cycle are determined by their chromatic polynomials and flow polynomials together. A direct consequence of our theorem is a result of de Mier and Noy [A. de Mier, M. Noy, On graphs determined by their Tutte polynomial, Graphs Comb. 20 (2004) 105-119] that these classes of graphs are Tutte polynomial unique.     

Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales

We prove that atomic decomposition for the Hardy spaces h₁ and H₁ is valid for noncommutative martingales. We also establish that the conditioned Hardy spaces of noncommutative martingales hp and bmo form interpolation scales with respect to both complex and real interpolations.     

A new construction of vertex algebras and quasi-modules for vertex algebras

In this paper, a new construction of vertex algebras from more general vertex operators is given and a notion of quasimodule for vertex algebras is introduced and studied. More specifically, a notion of quasilocal subset(space) of for any vector space W is introduced and studied, generalizing the notion of usual locality in the most possible way, and it is proved that on any maximal quasilocal subspace there exists a natural vertex algebra structure and that any quasilocal subset of generates a vertex algebra. Furthermore, it is proved that W is a quasimodule for each of the vertex algebras generated by quasilocal subsets of . A notion of Γ-vertex algebra is also introduced and studied, where Γ is a subgroup of the multiplicative group C^× of nonzero complex numbers. It is proved that any maximal quasilocal subspace of is naturally a Γ-vertex algebra and that any quasilocal subset of generates a Γ-vertex algebra. It is also proved that a Γ-vertex algebra exactly amounts to a vertex algebra equipped with a Γ-module structure which satisfies a certain compatibility condition. Finally, two families of examples are given, involving twisted affine Lie algebras and certain quantum torus Lie algebras.     

A cohomological description of property (T) for quantum groups

We prove a Delorme-Guichardet type theorem for discrete quantum groups expressing property (T) of the quantum group in question in terms of its first cohomology groups. As an application, we show that the first L²-Betti number of a discrete property (T) quantum group vanishes.     

Parseval frames for ICC groups

We analyze Parseval frames generated by the action of an ICC group on a Hilbert space. We parametrize the set of all such Parseval frames by operators in the commutant of the corresponding representation. We characterize when two such frames are strongly disjoint. We prove an undersampling result showing that if the representation has a Parseval frame of equal norm vectors of norm , the Hilbert space is spanned by an orthonormal basis generated by a subgroup. As applications we obtain some sufficient conditions under which a unitary representation admits a Parseval frame which is spanned by a Riesz sequences generated by a subgroup. In particular, every subrepresentation of the left-regular representation of a free group has this property.     

Improving the high order nonlinearity lower bound for Boolean functions with given algebraic immunity

Algebraic immunity is a recently introduced cryptographic parameter for Boolean functions used in stream ciphers. If pAI(f) and pAI(f⊕1) are the minimum degree of all annihilators of f and f⊕1 respectively, the algebraic immunity AI(f) is defined as the minimum of the two values. Several relations between the new parameter and old ones, like the degree, the r-th order nonlinearity and the weight of the Boolean function, have been proposed over the last few years.In this paper, we improve the existing lower bounds of the r-th order nonlinearity of a Boolean function f with given algebraic immunity. More precisely, we introduce the notion of complementary algebraic immunity defined as the maximum of pAI(f) and pAI(f⊕1). The value of can be computed as part of the calculation of AI(f), with no extra computational cost. We show that by taking advantage of all the available information from the computation of AI(f), that is both AI(f) and , the bound is tighter than all known lower bounds, where only the algebraic immunity AI(f) is used.     

Local regularity theorems for the stationary thermistor problem with oscillating degeneracy

In this paper we present some regularity results for solutions to the system −Δu=σ(u)²|∇φ|, div(σ(u)∇φ)=0 in the case where σ(u) is allowed to oscillate between 0 and a positive number as u→∞. In particular, we show that u is locally bounded if σ(u) is bounded below by a suitable exponential function.     

Letter to the Editor: A note on the preconditioned Gauss-Seidel (GS) method for linear systems

Wen Li (J. Comput. Appl. Math., 182 (2005) 81-90) asserted that there are some errors in article by Hiroshi Niki, Kyouji Harada, Munenori Morimoto and Michio Sakakihara (J. Comput. Appl. Math., 164-165 (2004) 587-600). And Li presented a new proof for the corresponding results in H. Niki et al. In this paper, we point out some errors in Li’s assertion. Moreover, we show that a new proof presented by Li is imperfect.     

A sufficient condition for pancyclability of graphs

Let G be a graph of order n and S be a vertex set of q vertices. We call G,S-pancyclable, if for every integer i with 3≤i≤q there exists a cycle C in G such that |V(C)∩S|=i. For any two nonadjacent vertices u,v of S, we say that u,v are of distance two in S, denoted by d_S(u,v)=2, if there is a path P in G connecting u and v such that |V(P)∩S|≤3. In this paper, we will prove that if G is 2-connected and for all pairs of vertices u,v of S with d_S(u,v)=2, , then there is a cycle in G containing all the vertices of S. Furthermore, if for all pairs of vertices u,v of S with d_S(u,v)=2, , then G is S-pancyclable unless the subgraph induced by S is in a class of special graphs. This generalizes a result of Fan [G. Fan, New sufficient conditions for cycles in graphs, J. Combin. Theory B 37 (1984) 221-227] for the case when S=V(G).     

A first-order differential double subordination with applications

Let q₁ and q₂ belong to a certain class of normalized analytic univalent functions in the open unit disk of the complex plane. Sufficient conditions are obtained for normalized analytic functions p to satisfy the double subordination chain q₁(z)?p(z)?q₂(z). The differential sandwich-type result obtained is applied to normalized univalent functions and to Φ-like functions.     

Some applications of free Fisher information on frame theory

We introduce the notion of operator-valued free Fisher information with respect to a positive map of a random variable in an operator-valued noncommutative probability space and point out its close relations to the modular frames arising from conditional expectations. Then we can apply this notion on the study of frame theory, especially on the disjointness problem of modular frames arising from conditional expectations.     

A nonlinear functional for general scalar hyperbolic conservation laws

A generalized entropy functional was introduced in [T.-P. Liu, T. Yang, A new entropy functional for scalar conservation laws, Comm. Pure Appl. Math. 52 (1999) 1427-1442] for the scalar hyperbolic conservation laws with convex flux function. This functional was crucially used in the functional approach to the L¹ stability study on the system of hyperbolic conservation laws when each characteristic field is either genuinely nonlinear or linearly degenerate. However, how to construct the generalized entropy functional for scalar conservation laws with general flux, and then how to apply the functional approach to the L¹ study on general systems are still open. In this paper, we construct a new nonlinear functional which gives some partial answer to this question and we expect the analysis will shed some light on the future investigation in this direction.