Signsolvability revisited

For a real matrix A, Q(A) denotes the set of all matrices with the same sign pattern as A. A linear system Ax=b is signsolvable if solvability and Q(x) depend only on Q(A) and Q(b). The study of signsolvability can be decomposed into the study of L-matrices and of S-matrices, where A is an L-matrix [S-matrix] if the nullspace of each member of Q(A) is {0} [is a line intersecting the open positive orthant]. The problem of recognizing L-matrices is shown to be NP-complete, even in the [almost square] case. Recognition of square L-matrices was transformed into a graph-theoretic problem by Bassett, Maybee, and Quirk in 1968. The complexity of this problem remains open, but that of some related graph-theoretic problems is determined. The relation between S-matrices and L-matrices is studied, and it is shown that a certain recursive construction yields all S-matrices, thus proving a 1964 conjecture of Gorman.     

Error bounds for linear complementarity problems for SB-matrices

S-strictly dominant B-matrices (SB-matrices) are introduced by Li et al. (Numer Linear Algebra Appl 14:391?C405, 2007). In this paper, we give error bounds for the linear complementarity problem when the matrix involved is an SB-matrix, which generalize those of DB-matrix linear complementarity problem and show advantages with respect to the computational cost. Then the perturbation bounds of SB-matrices linear complementarity problems are also provided. The preliminary numerical results show the sharpness of the bounds.     

Bidimensional allocation of seats via zero-one matrices with given line sums

In some proportional electoral systems with more than one constituency the number of seats allotted to each constituency is pre-specified, as well as, the number of seats that each party has to receive at a national level. “Bidimensional allocation” of seats to parties within constituencies consists of converting the vote matrix V into an integer matrix of seats “as proportional as possible” to V, satisfying constituency and party totals and an additional “zero-vote zero-seat” condition. In the current Italian electoral law this Bidimensional Allocation Problem (or Biproportional Apportionment Problem—BAP) is ruled by an erroneous procedure that may produce an infeasible allocation, actually one that is not able to satisfy all the above conditions simultaneously. In this paper we focus on the feasibility aspect of BAP and, basing on the theory of (0,1)-matrices with given line sums, we formulate it for the first time as a “Matrix Feasibility Problem”. Starting from some previous results provided by Gale and Ryser in the 60’s, we consider the additional constraint that some cells of the output matrix must be equal to zero and extend the results by Gale and Ryser to this case. For specific configurations of zeros in the vote matrix we show that a modified version of the Ryser procedure works well, and we also state necessary and sufficient conditions for the existence of a feasible solution. Since our analysis concerns only special cases, its application to the electoral problem is still limited. In spite of this, in the paper we provide new results in the area of combinatorial matrix theory for (0,1)-matrices with fixed zeros which have also a practical application in some problems related to graphs.     

Integral modular data and congruences

We compute all fusion algebras with symmetric rational S-matrix up to dimension 12. Only two of them may be used as S-matrices in a modular datum: the S-matrices of the quantum doubles of ℤ/2ℤ and S ₃. Almost all of them satisfy a certain congruence which has some interesting implications, for example for their degrees. We also give explicitly an infinite sequence of modular data with rational S- and T-matrices which are neither tensor products of smaller modular data nor S-matrices of quantum doubles of finite groups. For some sequences of finite groups (certain subdirect products of S ₃,D ₄,Q ₈,S ₄), we prove the rationality of the S-matrices of their quantum doubles.     

Indecomposable (1,3)-Groups and a matrix problem

Almost completely decomposable groups with a critical typeset of type (1, 3) and a p-primary regulator quotient are studied. It is shown that there are, depending on the exponent of the regulator quotient p ^k , either no indecomposables if k ? 2; only six near isomorphism types of indecomposables if k = 3; and indecomposables of arbitrary large rank if k ? 4.     

A remark on a theorem by Claire Amiot

Claire Amiot has classified the connected triangulated k-categories with finitely many isoclasses of indecomposables satisfying suitable hypotheses. We remark that her proof shows that these triangulated categories are determined by their underlying k-linear categories. We observe that, if the connectedness assumption is dropped, the triangulated categories are still determined by their underlying k-categories together with the action of the suspension functor on the set of isoclasses of indecomposables.     

The class A(R,S) of (0, 1)-matrices

Let R and S be two vectors whose components are m and n non-negative integers, respectively. Let $\text{A}$(R, S) be the class consisting of all (0, 1)-matrices of size m by n with row sum vector R and column sum vector S. In this paper we derive a lower bound to the cardinality of the class $\text{A}$(R, S), which can be computed readily.     

Completely mixed games and M-matrices

Any non-singular M-matrix is a completely mixed matrix game with positive value. We exploit this property to give game-theoretic proofs of several well-known characterizations of such matrices. The same methods yield also many theorems on S₀-irreducible matrices that are closely related to M-matrices.     

Linear transformations that preserve certain positivity classes of matrices

For each of several S ? R^n,n, those linear transformations $\text{L}\text{: R}{}^{\mathrm{n,n}}\text{→ R}{}^{\mathrm{n,n}}$ which map S onto S are characterized. Each class is a familiar one which generalizes the notion of positivity to matrices. The classes include: the matrices with nonnegative principal minors, the M-matrices, the totally nonnegative matrices, the D-stable matrices, the matrices with positive diagonal Lyapunov solutions, and the H-matrices, as well as other related classes. The set of transformations is somewhat different from case to case, but the strategy of proof, while differing in detail, is similar.     

Generalized inverse-positivity and splittings of M-matrices

The concepts of matrix monotonicity, generalized inverse-positivity and splittings are investigated and are used to characterize the class of all M-matrices A, extending the well-known property that A^?1?0 whenever A is nonsingular. These conditions are grouped into classes in order to identify those that are equivalent for arbitrary real matrices A. It is shown how the nonnegativity of a generalized left inverse of A plays a fundamental role in such characterizations, thereby extending recent work by one of the authors, by Meyer and Stadelmaier and by Rothblum. In addition, new characterizations are provided for the class of M-matrices with “property c”; that is, matrices A having a representation A=sI?B, s>0, B?0, where the powers of $\text{(}\text{1}\text{s}\text{)B}$ converge. Applications of these results to the study of iterative methods for solving arbitrary systems of linear equations are given elsewhere.     

Multiplicities of discriminants

We compute the multiplicity of the discriminant of a line bundle ￡ over a nonsingular varietyS at a given sectionX, in terms of the Chern classes of ￡ and of the cotangent bundle ofS, and the Segre classes of the jacobian scheme ofX inS. ForS a surface, we obtain a precise formula that expresses the multiplicity as a sum of a term due to the non-reduced components of the section, and a term that depends on the Milnor numbers of the singularities ofX _red. Also, under certain hypotheses, we provide formulas for the “higher discriminants” that parametrize sections with a singular point of prescribed multiplicity. As an application, we obtain criteria for the various discriminants to be “small”. Supported in part by the Max-Planck-Institut für Mathematik     

On Supersets of Wavelet Sets

Considering a single dyadic orthonormal wavelet ψ in L ²(?), it is still an open problem whether the support of $\widehat{\psi}$ always contains a wavelet set. As far as we know, the only result in this direction is that if the Fourier support of a wavelet function is “small” then it is either a wavelet set or a union of two wavelet sets. Without assuming that a set S is the Fourier support of a wavelet, we obtain some necessary conditions and some sufficient conditions for a “small” set S to contain a wavelet set. The main results, which are in terms of the relationship between two explicitly constructed subsets A and B of S and two subsets T ₂ and D ₂ of S intersecting itself exactly twice translationally and dilationally respectively, are (1) if $A\cup B\not\subseteq T_{2}\cap D_{2}$ then S does not contain a wavelet set; and (2) if A∪B?T ₂∩D ₂ then every wavelet subset of S must be in S?(A∪B) and if S?(A∪B) satisfies a “weak” condition then there exists a wavelet subset of S?(A∪B). In particular, if the set S?(A∪B) is of the right size then it must be a wavelet set.     

The Picard Theorem on S-metric spaces

Recently, the notion of an S-metric space is defined and extensively studied as a generalization of a metric space. In this paper, we define the notion of the S∞-space and prove its completeness. We obtain a new generalization of the classical "Picard Theorem".     

On spherically convex sets and Q-matrices

Let K be a closed spherically convex subset of S^n?1 that is contained in a hemisphere, and x?(K) the radial projection onto S^n?1 of the centroid of K. Then p^Tx?(K)>0 for all p ? K. A specialization of this result to spherical simplices is used to derive a necessary condition for Q-matrices, i.e., matrices for which every corresponding linear complementarity problem has at least one solution.     

Tree Modules and Counting Polynomials

We give a formula for counting tree modules for the quiver S _g with g loops and one vertex in terms of tree modules on its universal cover. This formula, along with work of Helleloid and Rodriguez-Villegas, is used to show that the number of d-dimensional tree modules for S _g is polynomial in g with the same degree and leading coefficient as the counting polynomial $A_{S_g}(d, q)$ for absolutely indecomposables over $\mathbb{F}_q$ , evaluated at q?=?1.     

Behaviorism and belief

The examination of now-abandoned behaviorist analysis of the concept of belief can bring to light defects in perspectives such as functionalism and physicalism that are still considered viable. Most theories have in common that they identify the holding of the belief that p by a subject S with some matter of fact in or about S that is distinct from and independent of p. In the case of behaviorism it is easy to show that this feature of the theory generates incoherence in the first-person point of view since it gives footing to the possibility that S could correctly assert “I believe that p,” (that is, “I have the complex disposition the behaviorist theory identifies with holding the belief that p”) and at the same time deny that p is the case. Parallel incoherence can be developed in the context of other philosophically popular accounts of the nature of belief.     

Quantum scattering at low energies

For a class of negative slowly decaying potentials, including V(x):=−γ|x|^−μ with 0<μ<2, we study the quantum mechanical scattering theory in the low-energy regime. Using appropriate modifiers of the Isozaki-Kitada type we show that scattering theory is well behaved on the whole continuous spectrum of the Hamiltonian, including the energy 0. We show that the modified scattering matrices S(λ) are well-defined and strongly continuous down to the zero energy threshold. Similarly, we prove that the modified wave matrices and generalized eigenfunctions are norm continuous down to the zero energy if we use appropriate weighted spaces. These results are used to derive (oscillatory) asymptotics of the standard short-range and Dollard type S-matrices for the subclasses of potentials where both kinds of S-matrices are defined. For potentials whose leading part is −γ|x|^−μ we show that the location of singularities of the kernel of S(λ) experiences an abrupt change from passing from positive energies λ to the limiting energy λ=0. This change corresponds to the behaviour of the classical orbits. Under stronger conditions one can extract the leading term of the asymptotics of the kernel of S(λ) at its singularities.     

Nonnegative rank-preserving operators

Analogues of characterizations of rank-preserving operators on field-valued matrices are determined for matrices witheentries in certain structures $\text{S}$ contained in the nonnegative reals. For example, if $\text{S}$ is the set of nonnegative members of a real unique factorization domain (e.g. the nonnegative reals or the nonnegative integers), M is the set of m×n matrices with entries in $\text{S}$, and min(m,n)?4, then a “linear” operator on M preserves the “rank” of each matrix in M if and only if it preserves the ranks of those matrices in M of ranks 1, 2, and 4. Notions of rank and linearity are defined analogously to the field-valued concepts. Other characterizations of rank-preserving operators for matrices over these and other structures $\text{S}$ are also given.     

Uniform Uniform Exponential Growth of Subgroups of the Mapping Class Group

Let Mod(S) denote the mapping class group of a compact, orientable surface S. We prove that finitely generated subgroups of Mod(S) which are not virtually abelian have uniform exponential growth with minimal growth rate bounded below by a constant depending only, and necessarily, on S. For the proof, we find in any such subgroup explicit free group generators which are “short” in any word metric. Besides bounding growth, this allows a bound on the return probability of simple random walks.