Bases in semilinear spaces over zerosumfree semirings

This paper investigates the cardinality of a basis and the characterizations of a basis in semilinear space of n-dimensional vectors over zerosumfree semirings. First, it discusses the cardinality of a basis and gives some necessary and sufficient conditions that each basis has the same number of elements. Then it presents some conditions that a set of vectors is a basis and that a set of linearly independent vectors can be extended to a basis. In the end, it shows a necessary and sufficient condition that two semilinear spaces are isomorphic.     

A generalization of steinhagen’s theorem

The theorem of Steinhagen establishes a relation between inradius and width of a convex set. The half of the width can be interpreted as the minimum of inradii of all 1-dimensional orthogonal projections of a convex set. By considering i-dimensional projections we obtain series ofi-dimensional inradii. In this paper we study some relations between these inradii and by this we find a natural generalization of Steinhagen’s theorem. Further we show in the course of our investigation that the minimal error of the triangle inequality for a set of vectors cannot be too large.     

Matrix multiplication is determined by orthogonality and trace

Any associative bilinear multiplication on the set of n-by-n matrices over some field of characteristic not two, that makes the same vectors orthogonal and has the same trace as ordinary matrix multiplication, must be ordinary matrix multiplication or its opposite.     

Gevrey vectors in involutive tube structures and Gevrey regularity for the solutions of certain classes of semilinear systems

In this work, we introduce the notion of s-Gevrey vectors in locally integrable structures of tube type. Under the hypothesis of analytic hypoellipticity, we study the Gevrey regularity of such vectors and also show how this notion can be applied to the study of the Gevrey regularity of solutions of certain classes of semilinear equations.     

Bases in semilinear spaces over join-semirings

This paper investigates the cardinality of a basis in semilinear spaces of n-dimensional vectors over join-semirings. First, it introduces the notion of an irredundant decomposition of an element in a join-semiring, then discusses the cardinality of a basis and proves that the cardinality of each basis is n if and only if the multiplicative identity element 1 is join-irreducible. If 1 is not a join-irreducible element then each basis need not have the same number of elements in semilinear spaces of n-dimensional vectors over join-semirings. This gives an answer to an open problem raised by Di Nola et al. in their work [Algebraic analysis of fuzzy systems, Fuzzy Sets and Systems 158 (2007) 1-22].     

On open questions in the geometric approach to structural learning Bayesian nets

The basic idea of an algebraic approach to learning Bayesian network (BN) structures is to represent every BN structure by a certain uniquely determined vector, called the standard imset. In a recent paper [18], it was shown that the set S of standard imsets is the set of vertices (=extreme points) of a certain polytope P and natural geometric neighborhood for standard imsets, and, consequently, for BN structures, was introduced.The new geometric view led to a series of open mathematical questions. In this paper, we try to answer some of them. First, we introduce a class of necessary linear constraints on standard imsets and formulate a conjecture that these constraints characterize the polytope P. The conjecture has been confirmed in the case of (at most) 4 variables. Second, we confirm a former hypothesis by Raymond Hemmecke that the only lattice points (=vectors having integers as components) within P are standard imsets. Third, we give a partial analysis of the geometric neighborhood in the case of 4 variables.     

Boundary controllability for the semilinear Schrödinger equations on Riemannian manifolds

We study the boundary exact controllability for the semilinear Schrödinger equation defined on an open, bounded, connected set Ω of a complete, n-dimensional, Riemannian manifold M with metric g. We prove the locally exact controllability around the equilibria under some checkable geometrical conditions. Our results show that exact controllability is geometrical characters of a Riemannian metric, given by the coefficients and equilibria of the semilinear Schrödinger equation. We then establish the globally exact controllability in such a way that the state of the semilinear Schrödinger equation moves from an equilibrium in one location to an equilibrium in another location.     

Note on cardinality of bases in semilinear spaces over zerosumfree semirings

A necessary and sufficient condition under which all bases in a semilinear space of n-dimensional vectors over zerosumfree semirings have exactly n elements is established.     

The effects of indefinite nonlinear boundary conditions on the structure of the positive solutions set of a logistic equation

We investigate a semilinear elliptic equation with a logistic nonlinearity and an indefinite nonlinear boundary condition, both depending on a parameter λ. Overall, we analyze the effect of the indefinite nonlinear boundary condition on the structure of the positive solutions set. Based on variational and bifurcation techniques, our main results establish the existence of three nontrivial non-negative solutions for some values of λ, as well as their asymptotic behavior. These results suggest that the positive solutions set contains an S-shaped component in some case, as well as a combination of a C-shaped and an S-shaped components in another case.     

Random reduction consistency of the Weber set,the core and the anti-core

In this paper we introduce a new consistency condition and provide characterizations for several solution concepts in TU cooperative game theory. Our new consistency condition, which we call the random reduction consistency, requires the consistency of payoff vectors assigned by a solution concept when one of the players is removed with some probability. We show that the random reduction consistency and other standard properties characterize the Weber set, the convex hull of the marginal contribution vectors. Another salient feature of random reduction consistency is that, by slightly changing its definition, we can characterize the core and the anti-core in a parallel manner. Our result enables us to compare the difference between the three solution concepts from the viewpoint of consistency.     

Least-squares orthogonalization using semidefinite programming

We consider the problem of constructing an optimal set of orthogonal vectors from a given set of vectors in a real Hilbert space. The vectors are chosen to minimize the sum of the squared norms of the errors between the constructed vectors and the given vectors. We show that the design of the optimal vectors, referred to as the least-squares (LS) orthogonal vectors, can be formulated as a semidefinite programming (SDP) problem. Using the many well-known algorithms for solving SDPs, which are guaranteed to converge to the global optimum, the LS vectors can be computed very efficiently in polynomial time within any desired accuracy.By exploiting the connection between our problem and a quantum detection problem we derive a closed form analytical expression for the LS orthogonal vectors, for vector sets with a broad class of symmetry properties. Specifically, we consider geometrically uniform (GU) sets with a possibly non-abelian generating group, and compound GU sets which consist of subsets that are GU.     

Profile polytopes of some classes of families

The profile vector of a family F of subsets of an n-element set is (f ₀,f ₁,…,f _n ) where f _i denotes the number of the i-element members of F. The extreme points of the set of profile vectors for some class of families has long been studied. In this paper we introduce the notion of k-antichainpair families and determine the extreme points of the set of profile vectors of these families, extending results of Engel and P.L. Erd?s regarding extreme points of the set of profile vectors of intersecting, co-intersecting Sperner families. Using this result we determine the extreme points of the set of profile vectors for some other classes of families, including complement-free k-Sperner families and self-complementary k-Sperner families. We determine the maximum cardinality of intersecting k-Sperner families, generalizing a classical result of Milner from k = 1.     

A representation of convex semilinear sets

If F is an ordered field, a subset of n-space over F is said to be semilinear just in case it is a finite Boolean combination of translates of closed halfspaces, where a closed halfspace is the set of all points obeying a homogeneous weak linear inequality with coefficients from F. Andradas, Rubio, and Vélez have shown that closed (open) convex semilinear sets are finite intersections of translates of closed (open) halfspaces (an open halfspace is defined as before, but with a strict inequality). This paper represents arbitrary convex semilinear sets in a manner analogous to that of Andradas, Rubio, and Vélez.     

Existence and multiplicity of solutions for Neumann problems

In this paper we examine semilinear and nonlinear Neumann problems with a nonsmooth locally Lipschitz potential function. Using variational methods based on the nonsmooth critical point theory, for the semilinear problem we prove a multiplicity result under conditions of double resonance at higher eigenvalues. Our proof involves a nonsmooth extension of the reduction method due to Castro-Lazer-Thews. The nonlinear problem is driven by the p-Laplacian. So first we make some observations about the beginning of the spectrum of (−Δp,W1,p(Z)). Then we prove an existence and multiplicity result. The existence result permits complete double resonance. The multiplicity result specialized in the semilinear case (i.e. p=2) corresponds to the super-sub quadratic situation.     

Blow-up rates for higher-order semilinear parabolic equations and systems and some Fujita-type theorems

In this paper, we derive blow-up rates for higher-order semilinear parabolic equations and systems. Our proof is by contradiction and uses a scaling argument. This procedure reduces the problems of blow-up rate to Fujita-type theorems. In addition, we also give some new Fujita-type theorems for higher-order semilinear parabolic equations and systems with the time variable on R. These results are not restricted to positive solutions.     

On the convexity of reachability sets of controlled initial-boundary value problems

For a functional-operator equation describing a broad class of controlled initial-boundary value problems, we introduce the notion of abstract reachability set. We obtain sufficient conditions for the convexity and precompactness of that set. The situation of a Nash ?-equilibrium is justified in the sense of program strategies in noncooperative functional-operator games with many players. As an example of reduction of a controlled initial-boundary value problem to the equation under study, we consider the Cauchy problem for a semilinear wave equation with two space variables.     

The topological Tverberg theorem and related topics

The classical Borsuk–Ulam theorem, established some eighty years ago, may now be seen as a consequence of the nonvanishing of the mod 2 cohomology Euler class of a certain vector bundle over a real projective space. A theorem of Kakutani from the 1940s that any continuous real-valued function on the 2–sphere must be constant on some set of three orthogonal vectors may be deduced similarly from the nontriviality of some mod 3 cohomology Euler class. The more recent topological Tverberg theorem of Bárány, Shlosman and Szücs, concerning a prime p, and the extensions of that theorem which have appeared in the last few years in the work of Blagojevi?, Karasev, Matschke, Ziegler and others, may be proved by showing that some mod p Euler class is nonzero. This paper presents a survey of these, and related, results from the viewpoint of topological fibrewise fixed–point theory.     

Model recovery for Hammerstein systems using the auxiliary model based orthogonal matching pursuit method

This article investigates parameter and order identification of a block-oriented Hammerstein system by using the orthogonal matching pursuit method in the compressive sensing theory which deals with how to recover a sparse signal in a known basis with a linear measurement model and a small set of linear measurements. The idea is to parameterize the Hammerstein system into the linear measurement model containing a measurement matrix with some unknown variables and a sparse parameter vector by using the key variable separation principle, then an auxiliary model based orthogonal matching pursuit algorithm is presented to recover the sparse vector.The standard orthogonal matching pursuit algorithm with a known measurement matrix is a popular recovery strategy by picking the supporting basis and the corresponding non-zero element of a sparse signal in a greedy fashion. In contrast to this, the auxiliary model based orthogonal matching pursuit algorithm has unknown variables in the measurement matrix. For a K-sparse signal, the standard orthogonal matching pursuit algorithm takes a fixed number of K stages to pick K columns (atoms) in the measurement matrix, while the auxiliary model based orthogonal matching pursuit algorithm takes steps larger than K to pick K atoms in the measurement matrix with the process of picking and deleting atoms, due to the gradually accurate estimates of the unknown variables step by step.The auxiliary model based orthogonal matching pursuit algorithm can simultaneously identify parameters and orders of the Hammerstein system, and has a high efficient identification performance.     

Comparison Method of Stability Analysis of Semilinear Time-Delay Stochastic Evolution Equation

Abstract

In this paper, we set up the comparison theorem between the mild solution of semilinear time-delay stochastic evolution equation with general time-delay variable and the solution of a class (1-dimension) deterministic functional differential equation, by using the Razumikhin–Lyapunov type functional and the theory of functional differential inequalities. By applying this comparison theorem, we give various types of the stability comparison criteria for the semilinear time-delay stochastic evolution equations. With the aid of these comparison criteria, one can reduce the stability analysis of semilinear time-delay stochastic evolution equations in Hilbert space to that of a class (1-dimension) deterministic functional differential equations. Furthermore, these comparison criteria in special case have been applied to derive sufficient conditions for various stability of the mild solution of semilinear time-delay stochastic evolution equations. Finally, the theories are illustrated with some examples.