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.     

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.     

Characterization of derivations on B(X) by local actions

Let A be a unital algebra and M be a unital A-bimodule. A linear map δ : A →M is said to be Jordan derivable at a nontrivial idempotent P ∈ A if δ(A) ? B + A ? δ(B) =δ(A ? B) for any A, B ∈ A with A ? B = P, here A ? B = AB + BA is the usual Jordan product. In this article, we show that if A = Alg N is a Hilbert space nest algebra and M = B(H), or A = M = B(X), then, a linear map δ : A → M is Jordan derivable at a nontrivial projection P ∈ N or an arbitrary but fixed nontrivial idempotent P ∈ B(X) if and only if it is a derivation. New equivalent characterization of derivations on these operator algebras was obtained.     

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 $\mathbf{e}\in {\mathbb{L}}^2(\Omega )$.     

A negative answer to a question about leading terms ideals of polynomial ideals

We construct an example of a finitely generated ideal $I$ of $R\left[X\right]$, where $R$ is a one-dimensional domain, whose leading terms ideal is not finitely generated. This gives a negative answer to the open question of whether if $R$ is a domain with Krull dimension ≤1, then for any finitely generated ideal $I$ of $R\left[X\right]$, the leading terms ideal of $I$ is also finitely generated. Moreover, as a positive part of our answer, we prove that for any one-dimensional domain $R$ and any $a,b\in R$, the ideal of $R\left[X\right]$ generated by the leading terms of $〈1+aX,b〉$ is finitely generated.     

A remark on the orbit structure of the complexification of a semisimple symmetric space

We consider the action of a real semisimple Lie group G on the complexification $G_{\mathbf{C}}/H_{\mathbf{C}}$ of a semisimple symmetric space $G/H$ and we present a refinement of Matsuki?s results (Matsuki, 1997 [1]) in this case. We exhibit a finite set of points in $G_{\mathbf{C}}/H_{\mathbf{C}}$, sitting on closed G-orbits of locally minimal dimension, whose slice representation determines the G-orbit structure of $G_{\mathbf{C}}/H_{\mathbf{C}}$. Every such point $\overline{p}$ lies on a compact torus and occurs at specific values of the restricted roots of the symmetric pair $(\mathfrak{g},\mathfrak{h})$. The slice representation at $\overline{p}$ is equivalent to the isotropy representation of a real reductive symmetric space, namely $Z_G(p^4)/G_{\overline{p}}$. In principle, this gives the possibility to explicitly parametrize all G-orbits in $G_{\mathbf{C}}/H_{\mathbf{C}}$.     

Triangular C1-bialgebra defined as the direct sum of matrix algebras

Let $M_{*}(\mathbf{C})$ denote the ${\mathrm{C}}^1$-algebra defined as the direct sum of all matrix algebras $\{M_n(\mathbf{C}):n?1\}$. It is known that $M_{*}(\mathbf{C})$ has a non-cocommutative comultiplication ${\mathrm{\Delta }}_{\phi }$. From a certain set of transformations of integers, we construct a universal R-matrix R of the ${\mathrm{C}}^1$-bialgebra $(M_{*}(\mathbf{C})\text{,}{\mathrm{\Delta }}_{\phi })$ such that the quasi-cocommutative ${\mathrm{C}}^1$-bialgebra $(M_{*}(\mathbf{C})\text{,}{\mathrm{\Delta }}_{\phi }\text{,}R)$ is triangular. Furthermore, it is shown that certain linear Diophantine equations are corresponded to the Yang–Baxter equations of R.     

Positive effects of repulsion on boundedness in a fully parabolic attraction–repulsion chemotaxis system with logistic source

In this paper we study the global boundedness of solutions to the fully parabolic attraction–repulsion chemotaxis system with logistic source: $u_t=\mathrm{\Delta }u?\chi \mathrm{?}?(u\mathrm{?}v)+\xi \mathrm{?}?(u\mathrm{?}w)+f(u)$, $v_t=\mathrm{\Delta }v?\beta v+\alpha u$, $w_t=\mathrm{\Delta }w?\delta w+\gamma u$, subject to homogeneous Neumann boundary conditions in a bounded and smooth domain $\mathrm{\Omega }?{\mathbb{R}}^n$ ($n\ge 1$), where χ, α, ξ, γ, β and δ are positive constants, and $f:\mathbb{R}\to \mathbb{R}$ is a smooth function generalizing the logistic source $f(s)=a?b{s}^{\theta }$ for all $s\ge 0$ with $a\ge 0$, $b>0$ and $\theta \ge 1$. It is shown that when the repulsion cancels the attraction (i.e. $\chi \alpha =\xi \gamma$), the solution is globally bounded if $n\le 3$, or $\theta >{\theta }_n:=\min ?\{\frac{n+2}{4},\frac{n\sqrt{n^2+6n+17}?n^2?3n+4}{4}\}$ with $n\ge 2$. Therefore, due to the inhibition of repulsion to the attraction, in any spatial dimension, the exponent θ is allowed to take values less than 2 such that the solution is uniformly bounded in time.