On the number of matrices and a random matrix with prescribed row and column sums and 0-1 entries

We consider the set Sigma(R,C) of all mxn matrices having 0-1 entries and prescribed row sums R=(r_1, ..., r_m) and column sums C=(c_1, ..., c_n). We prove an asymptotic estimate for the cardinality |Sigma(R, C)| via the solution to a convex optimization problem. We show that if Sigma(R, C) is sufficiently large, then a random matrix D in Sigma(R, C) sampled from the uniform probability measure in Sigma(R,C) with high probability is close to a particular matrix Z=Z(R,C) that maximizes the sum of entropies of entries among all matrices with row sums R, column sums C and entries between 0 and 1. Similar results are obtained for 0-1 matrices with prescribed row and column sums and assigned zeros in some positions.Comment: 26 pages, proofs simplified, results strengthene

Four problems regarding representable functors

Let $R$, $S$ be two rings, $C$ an $R$-coring and ${}_{R}^C{\mathcal M}$ the category of left $C$-comodules. The category ${\bf Rep}\, ( {}_{R}^C{\mathcal M}, {}_{S}{\mathcal M} )$ of all representable functors ${}_{R}^C{\mathcal M} \to {}_{S}{\mathcal M}$ is shown to be equivalent to the opposite of the category ${}_{R}^C{\mathcal M}_S$. For $U$ an $(S,R)$-bimodule we give necessary and sufficient conditions for the induction functor $U\otimes_R - : {}_{R}^C\mathcal{M} \to {}_{S}\mathcal{M}$ to be: a representable functor, an equivalence of categories, a separable or a Frobenius functor. The latter results generalize and unify the classical theorems of Morita for categories of modules over rings and the more recent theorems obtained by Brezinski, Caenepeel et al. for categories of comodules over corings.Comment: 16 pages, the second versio

Brunn-Minkowski Inequalities for Contingency Tables and Integer Flows

Given a non-negative mxn matrix W=(w_ij) and positive integer vectors R=(r_1, >..., r_m) and C=(c_1, ..., c_n), we consider the total weight T(R, C; W) of mxn non-negative integer matrices (contingency tables) D with the row sums r_i, the column sums c_j, and the weight of D=(d_ij) equal to product of w_ij^d_ij. In particular, if W is a 0-1 matrix, T(R, C; W) is the number of integer feasible flows in a bipartite network. We prove a version of the Brunn-Minkowski inequality relating the numbers T(R, C; W) and T(R_k, C_k; W), where (R, C) is a convex combination of (R_k, C_k) for k=1, ..., p.Comment: 16 page

Shear critical R/C columns under increasing axial load

Structural elements in old reinforced concrete (R/C) frame buildings are often prone to shear or flexure-shear failure, which can eventually lead to loss of axial load capacity of vertical elements and initiate vertical progressive collapse of a building. An experimental investigation of shear and flexure-shear critical R/C elements subjected to increasing axial load is reported herein. The focus is on the effect of vertical load redistribution from axially failing columns on the non-linear (pre- and post-peak) response of neighboring shear-dominated members. The test results along with an analysis of the recorded deformation, strength, stiffness and energy dissipation characteristics shed light on the performance of sub-standard columns under constant and increasing axial load subsequent, or just prior, to failing in shear, thus providing useful insights into the assessment of existing R/C structures

Integrated optimization of nonlinear R/C frames with reliability constraints

A structural optimization algorithm was researched including global displacements as decision variables. The algorithm was applied to planar reinforced concrete frames with nonlinear material behavior submitted to static loading. The flexural performance of the elements was evaluated as a function of the actual stress-strain diagrams of the materials. Formation of rotational hinges with strain hardening were allowed and the equilibrium constraints were updated accordingly. The adequacy of the frames was guaranteed by imposing as constraints required reliability indices for the members, maximum global displacements for the structure and a maximum system probability of failure