Quasirandom permutations are characterized by 4-point densities

For permutations ${\pi}$ and ${\tau}$ of lengths ${|\pi|\le|\tau|}$ , let ${t(\pi,\tau)}$ be the probability that the restriction of ${\tau}$ to a random ${|\pi|}$ -point set is (order) isomorphic to ${\pi}$ . We show that every sequence ${\{\tau_j\}}$ of permutations such that ${|\tau_j|\to\infty}$ and ${t(\pi,\tau_j)\to 1/4!}$ for every 4-point permutation ${\pi}$ is quasirandom (that is, ${t(\pi,\tau_j)\to 1/|\pi|!}$ for every ${\pi}$ ). This answers a question posed by Graham.     

On additive decompositions of the set of primitive roots modulo p

It is conjectured that the set ${\mathcal {G}}$ of the primitive roots modulo p has no decomposition (modulo p) of the form ${\mathcal {G}= \mathcal {A} +\mathcal {B}}$ with ${|\mathcal {A}|\ge 2}$ , ${|\mathcal {B} |\ge 2}$ . This conjecture seems to be beyond reach but it is shown that if such a decomposition of ${\mathcal {G}}$ exists at all, then ${|\mathcal {A} |}$ , ${|\mathcal {B} |}$ must be around p ^1/2, and then this result is applied to show that ${\mathcal {G}}$ has no decomposition of the form ${\mathcal {G} =\mathcal {A} + \mathcal {B} + \mathcal {C}}$ with ${|\mathcal {A} |\ge 2}$ , ${|\mathcal {B} |\ge 2}$ , ${|\mathcal {C} |\ge 2}$ .     

Symmetry and quasi-centrality of the operator space projective tensor product

For C*-algebras A and B, the operator space projective tensor product ${A\widehat{\otimes}B}$ and the Banach space projective tensor product ${A\otimes_{\gamma}B}$ are shown to be symmetric. We also show that ${A\widehat{\otimes}B}$ is a weakly Wiener algebra. Finally, quasi-centrality and the unitary group of ${A\widehat{\otimes}B}$ are discussed.     

A minimum principle for the problem of St-Venant in {\mathbb{R}^N, N \geq 2}

This paper deals mainly with the St-Venant problem in a convex domain ?? of ${\mathbb{R}^N, N \geq 2}$ . A minimum principle for a combination of the stress function ${\psi}$ and ${|\nabla \psi|}$ is derived. Some possible applications are indicated.     

Large free sets in powers of universal algebras

We prove that for each universal algebra ${(A, \mathcal{A})}$ of cardinality ${|A| \geq 2}$ and infinite set X of cardinality ${|X| \geq | \mathcal{A}|}$ , the X-th power ${(A^{X}, \mathcal{A}^{X})}$ of the algebra ${(A, \mathcal{A})}$ contains a free subset ${\mathcal{F} \subset A^{X}}$ of cardinality ${|\mathcal{F}| = 2^{|X|}}$ . This generalizes the classical Fichtenholtz–Kantorovitch–Hausdorff result on the existence of an independent family ${\mathcal{I} \subset \mathcal{P}(X)}$ of cardinality ${|\mathcal{I}| = |\mathcal{P}(X)|}$ in the Boolean algebra ${\mathcal{P}(X)}$ of subsets of an infinite set X.     

Strengthening Kazhdan’s property (T) by Bochner methods

In this paper, we propose a property which is a natural generalization of Kazhdan??s property (T) and prove that many, but not all, groups with property (T) also have this property. Let ?? be a finitely generated group. One definition of ?? having property (T) is that ${H^{1}(\Gamma, \pi, {\mathcal{H}}) = 0}$ where the coefficient module ${{\mathcal{H}}}$ is a Hilbert space and ?? is a unitary representation of ?? on ${{\mathcal{H}}}$ . Here we allow more general coefficients and say that ?? has property ${F \otimes {H}}$ if ${H^{1}(\Gamma, \pi_{1}{\otimes}\pi_{2}, F{\otimes} {\mathcal{H}}) = 0}$ if (F, ?? ₁) is any representation with dim(F) <??? and ${({\mathcal{H}}, \pi_{2})}$ is a unitary representation. The main result of this paper is that a uniform lattice in a semisimple Lie group has property ${F \otimes {H}}$ if and only if it has property (T). The proof hinges on an extension of a Bochner-type formula due to Matsushima?CMurakami and Raghunathan. We give a new and more transparent derivation of this formula as the difference of two classical Weitzenb?ck formula??s for two different structures on the same bundle. Our Bochner-type formula is also used in our work on harmonic maps into continuum products (Fisher and Hitchman in preparation; Fisher and Hitchman in Int Math Res Not 72405:1?C19, 2006). Some further applications of property ${F\otimes {H}}$ in the context of group actions will be given in Fisher and Hitchman (in preparation).     

Generalized Lax-Milgram theorem in Banach spaces and its application to the elliptic system of boundary value problems

We generalize the well-known Lax-Milgram theorem on the Hilbert space to that on the Banach space. Suppose that ${a(\cdot, \cdot)}$ is a continuous bilinear form on the product ${X\times Y}$ of Banach spaces X and Y, where Y is reflexive. If null spaces N _X and N _Y associated with ${a(\cdot, \cdot)}$ have complements in X and in Y, respectively, and if ${a(\cdot, \cdot)}$ satisfies certain variational inequalities both in X and in Y, then for every ${F \in N_Y^{\perp}}$ , i.e., ${F \in Y^{\ast}}$ with ${F(\phi) = 0}$ for all ${\phi \in N_Y}$ , there exists at least one ${u \in X}$ such that ${a(u, \varphi) = F(\varphi)}$ holds for all ${\varphi \in Y}$ with ${\|u\|_X \le C\|F\|_{Y^{\ast}}}$ . We apply our result to several existence theorems of L ^r -solutions to the elliptic system of boundary value problems appearing in the fluid mechanics.     

The Lower Weyl Spectrum of a Positive Operator

For the lower Weyl spectrum $$\sigma_{\rm w}^-(T) = \bigcap_{0 \le K \in \mathcal{K}(E) \le T} \sigma(T - K),$$ where T is a positive operator on a Banach lattice E, the conditions for which the equality ${\sigma_{\rm w}^-(T) = \sigma_{\rm w}^-(T^*)}$ holds, are established. In particular, it is true if E has order continuous norm. An example of a weakly compact positive operator T on ? _∞ such that the spectral radius ${r(T) \in \sigma_{\rm w}^-(T) {\setminus} (\sigma_{\rm f}(T) \cup \sigma_{\rm w}^-(T^*))}$ , where σ _f(T) is the Fredholm spectrum, is given. The conditions which guarantee the order continuity of the residue T _?1 of the resolvent R(., T) of an order continuous operator T ≥ 0 at ${r(T) \notin \sigma_{\rm f}(T)}$ , are discussed. For example, it is true if T is o-weakly compact. It follows from the proven results that a Banach lattice E admitting an order continuous operator T ≥ 0, ${r(T) \notin \sigma_{\rm f}(T)}$ , can not have the trivial band ${E_n^\sim}$ of order continuous functionals in general. It is obtained that a non-zero order continuous operator T : E → F can not be approximated in the r-norm by the operators from ${E_\sigma^\sim \otimes F}$ , where F is a Banach lattice, ${E_\sigma^\sim}$ is a disjoint complement of the band ${E_n^\sim}$ of E*.     

Singular limits for Liouville-type equations on the flat two-torus

We consider the problem $$\begin{aligned} -\Delta u=\varepsilon ^{2}e^{u}- \frac{1}{|\Omega |}\int _\Omega \varepsilon ^{2} e^{u}+ {4\pi N\over |\Omega |} - 4 \pi N\delta _p, \quad \text{ in} {\Omega }, \quad \int _\Omega u=0 \end{aligned}$$ in a flat two-torus $\Omega $ with periodic boundary conditions, where $\varepsilon >0,\,|\Omega |$ is the area of the $\Omega $ , $N>0$ and $\delta _p$ is a Dirac mass at $p\in \Omega $ . We prove that if $1\le m<N+1$ then there exists a family of solutions $\{u_\varepsilon \}_{\varepsilon }$ such that $\varepsilon ^{2}e^{u_\varepsilon }\rightharpoonup 8\pi \sum _{i=1}^m\delta _{q_i}$ as $\varepsilon \rightarrow 0$ in measure sense for some different points $q_{1}, \ldots , q_{m}$ . Furthermore, points $q_i$ , $i=1,\dots ,m$ are different from $p$ .