Conditional p-adic probability logic

In this paper we present the proof-theoretical approach to p-adic valued conditional probabilistic logics. We introduce two such logics denoted by ${\mathit{CPL}}_{{\mathbf{Z}}_{\mathbf{p}}}$ and ${\mathit{CPL}}_{{\mathbf{Q}}_{\mathbf{p}}}^{\mathit{fin}}$. Each of these logics extends classical propositional logic with a list of binary (conditional probability) operators. Formulas are interpreted in Kripke-like models that are based on p-adic probability spaces. Axiomatic systems with infinitary rules of inference are given and proved to be sound and strongly complete. The decidability of the satisfiability problem for each logic is proved.     

String shuffle: Circuits and graphs

We show that shuffle, the problem of determining whether a string w can be composed from an order preserving shuffle of strings x and y, is not in ${\mathbf{AC}}^0$, but it is in ${\mathbf{AC}}^1$. The fact that shuffle is not in ${\mathbf{AC}}^0$ is shown by a reduction of parity to shuffle and invoking the seminal result of Furst et al., while the fact that it is in ${\mathbf{AC}}^1$ is implicit in the results of Mansfield. Together, the two results provide a lower and upper bound on the complexity of this combinatorial problem. We also explore an interesting relationship between graphs and the shuffle problem, namely what types of graphs can be represented with strings exhibiting the anti-Monge condition.     

The Carroll and Chang conjecture of equal Indscal components when Candecomp/Parafac gives perfect fit

The Candecomp/Parafac algorithm approximates a set of matrices ${\mathbf{X}}_1\text{,}\dots \text{,}{\mathbf{X}}_I$ by products of the form ${\mathbf{AC}}_i{\mathbf{B}}^{\prime }$, with ${\mathbf{C}}_i$ diagonal, $i=1\text{,}\dots \text{,}I$. Carroll and Chang have conjectured that, when the matrices are symmetric, the resulting $\mathbf{A}$ and $\mathbf{B}$ will be column wise proportional. For cases of perfect fit, Ten Berge et al. have shown that the conjecture holds true in a variety of cases, but may fail when there is no unique solution. In such cases, obtaining proportionality by changing (part of) the solution seems possible. The present paper extends and further clarifies their results. In particular, where Ten Berge et al. solved all $I\times 2\times 2$ cases, now all $I\times 3\times 3$ cases, and also the $I\times 4\times 4$ cases for $I=2\text{,}8$, and 9 are clarified. In a number of cases, $\mathbf{A}$ and $\mathbf{B}$ necessarily have column wise proportionality when Candecomp/Parafac is run to convergence. In other cases, proportionality can be obtained by using specific methods. No cases were found that seem to resist proportionality.     

The limiting distributions of large heavy Wigner and arbitrary random matrices

A heavy Wigner matrix $X_N$ is defined similarly to a classical Wigner one. It is Hermitian, with independent sub-diagonal entries. The diagonal entries and the non-diagonal entries are identically distributed. Nevertheless, the moments of the entries of $\sqrt{N}{X}_N$ tend to infinity with N, as for matrices with truncated heavy tailed entries or adjacency matrices of sparse Erdös–Rényi graphs. Consider a family ${\mathbf{X}}_N$ of independent heavy Wigner matrices and an independent family ${\mathbf{Y}}_N$ of arbitrary random matrices with a bound condition and converging in ^?-distribution in the sense of free probability. We characterize the possible limiting joint ^?-distributions of $({\mathbf{X}}_N,{\mathbf{Y}}_N)$, giving explicit formulas for joint ^?-moments. We find that they depend on more than the ^?-distribution of ${\mathbf{Y}}_N$ and that in general ${\mathbf{X}}_N$ and ${\mathbf{Y}}_N$ are not asymptotically ^?-free. We use the traffic distributions and the associated notion of independence [21] to encode the information on ${\mathbf{Y}}_N$ and describe the limiting ^?-distribution of $({\mathbf{X}}_N,{\mathbf{Y}}_N)$. We develop this approach for related models and give recurrence relations for the limiting ^?-distribution of heavy Wigner and independent diagonal matrices.     

Inhabitants of interesting subsets of the Bousfield lattice

In 1979, Bousfield defined an equivalence relation on the stable homotopy category. The set of Bousfield classes has some important subsets such as the distributive lattice DL of all classes $〈E〉$ which are smash idempotent and the complete Boolean algebra cBA of closed classes. We provide examples of spectra that are in DL, but not in cBA; in particular, for every prime p, the Bousfield class of the Eilenberg–MacLane spectrum $〈H{\mathbb{F}}_p〉$ is in $\mathbf{DL}?\mathbf{cBA}$.     

Maximal determinant over a certain class of matrices and its application to D-optimality of designs

It is known that some optimality criteria of experimental designs are functionals of the eigenvalues of their information matrices. In this context we study the problem of maximizing the determinant of $\alpha {\mathbf{I}}_t-(\mathbf{P}+{\mathbf{P}}^T)$, $\alpha >2$, over the class of t-by-t permutation matrices, and the determinant of $\alpha {\mathbf{I}}_t+\mathbf{P}+{\mathbf{P}}^T$, $\alpha \ge 2.5$, over the class of t-by-t permutation matrices with zero diagonal (derangement matrices). The results are used to characterize D-optimal complete block designs under an interference model.     

Linear preservers of term ranks of matrices over semirings

The term rank of a matrix $A$ over a semiring $\mathbf{S}$ is the least number of lines (rows or columns) needed to include all the nonzero entries in $A$. In this paper, we study linear operators that preserve term ranks of matrices over $\mathbf{S}$. In particular, we show that a linear operator $T$ on matrix space over $\mathbf{S}$ preserves term rank if and only if $T$ preserves term ranks $1$ and $\alpha (\ge 2)$ if and only if $T$ preserves two consecutive term ranks in a restricted condition. Other characterizations of term-rank preservers are also given.     

Expression of Dirichlet boundary conditions in terms of the strain tensor in linearized elasticity

In a previous work, it was shown how the linearized strain tensor field $\mathbf{e}:=\frac{1}{2}(\mathbf{?}{\mathit{u}}^T+\mathbf{?}\mathit{u})\in {\mathbb{L}}^2(\Omega )$ can be considered as the sole unknown in the Neumann problem of linearized elasticity posed over a domain $\Omega ?{\mathbb{R}}^3$, instead of the displacement vector field $\mathit{u}\in {\mathit{H}}^1(\Omega )$ in the usual approach. The purpose of this Note is to show that the same approach applies as well to the Dirichlet–Neumann problem. To this end, we show how the boundary condition $\mathit{u}=\mathbf{0}$ on a portion ${\Gamma }_0$ of the boundary of Ω can be recast, again as boundary conditions on ${\Gamma }_0$, but this time expressed only in terms of the new unknown $\mathbf{e}\in {\mathbb{L}}^2(\Omega )$.