Nash equilibria on topological semilattices

A function ${u : X \to \mathbb{R}}$ defined on a partially ordered set is quasi-Leontief if, for all ${x \in X}$ , the upper level set ${\{x\prime \in X : u(x\prime) \geq u(x)\}}$ has a smallest element; such an element is an efficient point of u. An abstract game ${u_{i} : \prod^{n}_{j=1} X_j \to \mathbb{R}, i \in \{1, \ldots , n\}}$ , is a quasi-Leontief game if, for all i and all ${(x_{j})_{j \neq i} \in \prod_{j \neq i} X_{j}, u_{i}((x_{j})_{j \neq i};-) : X_{i} \to \mathbb{R}}$ is quasi-Leontief; a Nash equilibrium x* of an abstract game ${u_{i} :\prod^{n}_{j=1} X_{j} \to \mathbb{R}}$ is efficient if, for all ${i, x^{*}_{i}}$ is an efficient point of the partial function ${u_{i}((x^{*}_{j})_{j \neq i};-) : X_{i} \to \mathbb{R}}$ . We establish the existence of efficient Nash equilibria when the strategy spaces X _i are topological semilattices which are Peano continua and Lawson semilattices.     

The reflexive least squares solutions of the matrix equation A_1X_1B_1+A_2X_2B_2+\cdots +A_lX_lB_l=Cwith a submatrix constraint

In this paper, an efficient algorithm is presented for minimizing $\|A_1X_1B_1 + A_2X_2B_2+\cdots +A_lX_lB_l-C\|$ where $\|\cdot \|$ is the Frobenius norm, $X_i\in R^{n_i \times n_i}(i=1,2,\cdots ,l)$ is a reflexive matrix with a specified central principal submatrix $[x_{ij}]_{r\leq i,j\leq n_i-r}$ . The algorithm produces suitable $[X_1,X_2,\cdots ,X_l]$ such that $\|A_1X_1B_1+A_2X_2B_2+\cdots +A_lX_lB_l-C\|=\min $ within finite iteration steps in the absence of roundoff errors. We show that the algorithm is stable any case. The algorithm requires little storage capacity. Given numerical examples show that the algorithm is efficient.     

Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps

We consider the Markov chain ${\{X_n^x\}_{n=0}^\infty}$ on ${\mathbb{R}^d}$ defined by the stochastic recursion ${X_{n}^{x}= \psi_{\theta_{n}} (X_{n-1}^{x})}$ , starting at ${x\in\mathbb{R}^d}$ , where ?? ₁, ?? ₂, . . . are i.i.d. random variables taking their values in a metric space ${(\Theta, \mathfrak{r})}$ , and ${\psi_{\theta_{n}} :\mathbb{R}^d\mapsto\mathbb{R}^d}$ are Lipschitz maps. Assume that the Markov chain has a unique stationary measure ??. Under appropriate assumptions on ${\psi_{\theta_n}}$ , we will show that the measure ?? has a heavy tail with the exponent ???>?0 i.e. ${\nu(\{x\in\mathbb{R}^d: |x| > t\})\asymp t^{-\alpha}}$ . Using this result we show that properly normalized Birkhoff sums ${S_n^x=\sum_{k=1}^n X_k^x}$ , converge in law to an ??-stable law for ${\alpha\in(0, 2]}$ .     

On the boundary of the attainable set of the dirichlet spectrum

Denoting by ${\varepsilon\subseteq\mathbb{R}^2}$ the set of the pairs ${(\lambda_1(\Omega),\,\lambda_2(\Omega))}$ for all the open sets ${\Omega\subseteq\mathbb{R}^N}$ with unit measure, and by ${\Theta\subseteq\mathbb{R}^N}$ the union of two disjoint balls of half measure, we give an elementary proof of the fact that ${\partial\varepsilon}$ has horizontal tangent at its lowest point ${(\lambda_1(\Theta),\,\lambda_2(\Theta))}$ .     

A hybrid inequality of Erdös–Turán–Koksma for digital sequences

For bases $\mathbf{b}=(b_1, \ldots , b_s)$ of $s$ not necessarily distinct integers $b_i\ge 2$ , we prove a version of the inequality of Erdös–Turán–Koksma for the hybrid function system composed of the Walsh functions in base $\mathbf{b}^{(1)}=(b_1, \ldots , b_{s_1})$ and, as second component, the $\mathbf{b}^{(2)}$ -adic functions, $\mathbf{b}^{(2)}=(b_{s_1+1}, \ldots , b_s)$ , with $s=s_1+s_2$ , $s_1$ and $s_2$ not both equal to 0. Further, we point out why this choice of a hybrid function system covers all possible cases of sequences that employ addition of digit vectors as their main construction principle.     

Antibasis theorems for {\Pi^0_1} classes and the jump hierarchy

We prove two antibasis theorems for ${\Pi^0_1}$ classes. The first is a jump inversion theorem for ${\Pi^0_1}$ classes with respect to the global structure of the Turing degrees. For any ${P\subseteq 2^\omega}$ , define S(P), the degree spectrum of P, to be the set of all Turing degrees a such that there exists ${A \in P}$ of degree a. For any degree ${{\bf a \geq 0'}}$ , let ${\textrm{Jump}^{-1}({\bf a) = \{b : b' = a \}}}$ . We prove that, for any ${{\bf a \geq 0'}}$ and any ${\Pi^0_1}$ class P, if ${\textrm{Jump}^{-1} ({\bf a}) \subseteq S(P)}$ then P contains a member of every degree. For any degree ${{\bf a \geq 0'}}$ such that a is recursively enumerable (r.e.) in 0', let ${Jump_{\bf \leq 0'} ^{-1}({\bf a)=\{b : b \leq 0' \textrm{and} b' = a \}}}$ . The second theorem concerns the degrees below 0'. We prove that for any ${{\bf a\geq 0'}}$ which is recursively enumerable in 0' and any ${\Pi^0_1}$ class P, if ${\textrm{Jump}_{\bf \leq 0'} ^{-1}({\bf a)} \subseteq S(P)}$ then P contains a member of every degree.     

A Note on Classical and p-adic Fréchet Functional Equations with Restrictions

Given X,Y two ${\mathbb{Q}}$ -vector spaces, and f : X → Y, we study under which conditions on the sets ${B_{k} \subseteq X, k=1,\ldots,s}$ , if ${\Delta_{h_1h_2 \cdots h_s}f(x) = 0}$ for all ${x \in X}$ and ${h_k \in B_k, k = 1,2,\ldots,s}$ , then ${\Delta_{h_1h_2\cdots h_{s}}f(x) = 0}$ for all ${(x,h_{1},\ldots,h_{s}) \in X^{s+1}}$ .     

Some 0/1 polytopes need exponential size extended formulations

We prove that there are 0/1 polytopes ${P \subseteq \mathbb{R}^{n}}$ that do not admit a compact LP formulation. More precisely we show that for every n there is a set ${X \subseteq \{ 0,1\}^n}$ such that conv(X) must have extension complexity at least ${2^{n/2\cdot(1-o(1))}}$ . In other words, every polyhedron Q that can be linearly projected on conv(X) must have exponentially many facets. In fact, the same result also applies if conv(X) is restricted to be a matroid polytope. Conditioning on ${\mathbf{NP}\not\subseteq \mathbf{P_{/poly}}}$ , our result rules out the existence of a compact formulation for any ${\mathbf{NP}}$ -hard optimization problem even if the formulation may contain arbitrary real numbers.     

Spectral self-affine measures with prime determinant

The self-affine measure $\mu _{M,D}$ relating to an expanding matrix $M\in M_{n}(\mathbb Z )$ and a finite digit set $D\subset \mathbb Z ^n$ is a unique probability measure satisfying the self-affine identity with equal weight. In the present paper, we shall study the spectrality of $\mu _{M,D}$ in the case when $|\det (M)|=p$ is a prime. The main result shows that under certain mild conditions, if there are two points $s_{1}, s_{2}\in \mathbb R ^{n}, s_{1}-s_{2}\in \mathbb Z ^{n}$ such that the exponential functions $e_{s_{1}}(x), e_{s_{2}}(x)$ are orthogonal in $L^{2}(\mu _{M,D})$ , then the self-affine measure $\mu _{M,D}$ is a spectral measure with lattice spectrum. This gives some sufficient conditions for a self-affine measure to be a lattice spectral measure.     

A covering lemma for {K(\mathbb {R})}

The Dodd–Jensen Covering Lemma states that “if there is no inner model with a measurable cardinal, then for any uncountable set of ordinals X, there is a ${Y\in K}$ such that ${X\subseteq Y}$ and |X| = |Y|”. Assuming ZF+AD alone, we establish the following analog: If there is no inner model with an ${\mathbb {R}}$ –complete measurable cardinal, then the real core model ${K(\mathbb {R})}$ is a “very good approximation” to the universe of sets V; that is, ${K(\mathbb {R})}$ and V have exactly the same sets of reals and for any set of ordinals X with ${|{X}|\ge\Theta}$ , there is a ${Y\in K(\mathbb {R})}$ such that ${X\subseteq Y}$ and |X| = |Y|. Here ${\mathbb {R}}$ is the set of reals and ${\Theta}$ is the supremum of the ordinals which are the surjective image of ${\mathbb {R}}$ .     

Howe correspondence and Springer correspondence for real reductive dual pairs

We consider a real reductive dual pair (G′, G) of type I, with rank ${({\rm G}^{\prime}) \leq {\rm rank(G)}}$ . Given a nilpotent coadjoint orbit ${\mathcal{O}^{\prime} \subseteq \mathfrak{g}^{{\prime}{*}}}$ , let ${\mathcal{O}^{\prime}_\mathbb{C} \subseteq \mathfrak{g}^{{\prime}{*}}_\mathbb{C}}$ denote the complex orbit containing ${\mathcal{O}^{\prime}}$ . Under some condition on the partition λ′ parametrizing ${\mathcal{O}^{\prime}}$ , we prove that, if λ is the partition obtained from λ by adding a column on the very left, and ${\mathcal{O}}$ is the nilpotent coadjoint orbit parametrized by λ, then ${\mathcal{O}_\mathbb{C}= \tau (\tau^{\prime -1}(\mathcal{O}_\mathbb{C}^{\prime}))}$ , where ${\tau, \tau^{\prime}}$ are the moment maps. Moreover, if ${chc(\hat\mu_{\mathcal{O}^{\prime}}) \neq 0}$ , where chc is the infinitesimal version of the Cauchy-Harish-Chandra integral, then the Weyl group representation attached by Wallach to ${\mu_{\mathcal{O}^{\prime}}}$ with corresponds to ${\mathcal{O}_\mathbb{C}}$ via the Springer correspondence.     

Tail homogeneity of invariant measures of multidimensional stochastic recursions in a critical case

We consider the stochastic recursion ${X_{n+1} = M_{n+1}X_{n} + Q_{n+1}, (n \in \mathbb{N})}$ , where ${Q_n, X_n \in \mathbb{R}^d }$ , M _n are similarities of the Euclidean space ${ \mathbb{R}^d }$ and (Q _n , M _n ) are i.i.d. We study asymptotic properties at infinity of the invariant measure for the Markov chain X _n under assumption ${\mathbb{E}{[\log|M|]}=0}$ i.e. in the so called critical case.     

A general LIL for trimmed sums of random fields in banach spaces

Given a field of independent identically distributed (i.i.d.) random variables $ \left\{ {X_{\bar n} ;\bar n \in \aleph ^d } \right\} $ indexed by d-tuples of positive integers and taking values in a separable Banach space B, let $ X_{\bar n}^{(r)} = X_{\bar m} $ is the r-th maximum of $ \left\{ {\left\| {X_{\bar k} } \right\|;\bar k \leqq \bar n} \right\} $ and let $ ^{(r)} S_{\bar n} = S_{\bar n} - \left( {X_{\bar n}^{(1)} + \cdots + X_{\bar n}^{(r)} } \right) $ be the trimmed sums, where $ S_{\bar n} = \sum\nolimits_{\bar k \leqq \bar n} {X_{\bar k} } $ . This paper aims to obtain a general law of the iterated logarithm (LIL) for the trimmed sums which improves previous works.     

Value-sets of polynomials at p-adic integers

In this paper we are concerned with the classification of the subsets A of ${\mathbb{Z}_p}$ which occur as images ${f(\mathbb{Z}_p^r)}$ of polynomial functions ${f:\mathbb{Z}_p^r\to \mathbb{Z}_p}$ , limiting ourselves to compact-open subsets (i.e. finite unions of open balls). We shall prove three main results: (i) Every compact-open ${A\subset \mathbb{Z}_p}$ is of the shape ${A=f(\mathbb{Z}_p^r)}$ for suitable r and ${f\in\mathbb{Z}_p[X_1,\ldots ,X_r]}$ . (ii) For each r ₀ there is a compact-open A such that in (i) we cannot take r < r ₀. (iii) For any compact-open set ${A\subset \mathbb{Z}_p}$ there exists a polynomial ${f\in\mathbb{Q}_p[X]}$ such that ${f(\mathbb{Z}_p)=A}$ . We shall also discuss in more detail which sets A can be represented as ${f(\mathbb{Z}_p)}$ for a polynomial ${f\in\mathbb{Z}_p[X]}$ in a single variable.     

Some results on metric n-Lie algebras

We study the structure of a metric n-Lie algebra G over the complex field C. Let G = SR be the Levi decomposition, where R is the radical of G and S is a strong semisimple subalgebra of G. Denote by m(G) the number of all minimal ideals of an indecomposable metric n-Lie algebra and R ⊥ the orthogonal complement of R. We obtain the following results. As S-modules, R ⊥ is isomorphic to the dual module of G/R. The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on G is equal to that of the vector space of certain linear transformations on G; this dimension is greater than or equal to m(G) + 1. The centralizer of R in G is equal to the sum of all minimal ideals; it is the direct sum of R ⊥ and the center of G. Finally, G has no strong semisimple ideals if and only if R⊥■R.     

Derived semidistributive lattices

For L a finite lattice, let ${\mathbb {C}(L) \subseteq L^2}$ denote the set of pairs γ = (γ ₀, γ ₁) such that ${\gamma_0 \prec \gamma_1}$ and order it as followsγ ≤ δ iff γ ₀ ≤ δ ₀, ${\gamma_{1} \nleq \delta_0,}$ and γ ₁ ≤ δ ₁. Let ${\mathbb {C}(L, \gamma)}$ denote the connected component of γ in this poset. Our main result states that, for any ${\gamma, \mathbb {C}(L, \gamma)}$ is a semidistributive lattice if L is semidistributive, and that ${\mathbb {C}(L, \gamma)}$ is a bounded lattice if L is bounded. Let ${\mathcal{S}_{n}}$ be the Permutohedron on n letters and let ${\mathcal{T}_{n}}$ be the Associahedron on n + 1 letters. Explicit computations show that ${\mathbb {C}(\mathcal{S}_{n}, \alpha) = \mathcal{S}_{n-1}}$ and ${\mathbb {C}(\mathcal {T}_n, \alpha) = \mathcal {T}_{n-1}}$ , up to isomorphism, whenever α1 is an atom of ${\mathcal{S}_{n}}$ or ${\mathcal{T}_{n}}$ . These results are consequences of new characterizations of finite join-semidistributive and of finite lower bounded lattices: (i) a finite lattice is join-semidistributive if and only if the projection sending ${\gamma \in \mathbb {C}(L)}$ to ${\gamma_0 \in L}$ creates pullbacks, (ii) a finite join-semidistributive lattice is lower bounded if and only if it has a strict facet labelling. Strict facet labellings, as defined here, are a generalization of the tools used by Caspard et al. to prove that lattices of finite Coxeter groups are bounded.     

Nikodym Maximal Functions Associated with Variable Planes in {\mathbb{R}^{3}}

Given a vector field ${\mathfrak{a}}$ on ${\mathbb{R}^3}$ , we consider a mapping ${x\mapsto \Pi_{\mathfrak{a}}(x)}$ that assigns to each ${x\in\mathbb{R}^3}$ , a plane ${\Pi_{\mathfrak{a}}(x)}$ containing x, whose normal vector is ${\mathfrak{a}(x)}$ . Associated with this mapping, we define a maximal operator ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^1_{loc}(\mathbb{R}^3)}$ for each ${N\gg 1}$ by $$\mathcal{M}^{\mathfrak{a}}_Nf(x)=\sup_{x\in\tau} \frac{1}{|\tau|} \int_{\tau}|f(y)|\,dy$$ where the supremum is taken over all 1/N ×? 1/N?× 1 tubes τ whose axis is embedded in the plane ${\Pi_\mathfrak{a}(x)}$ . We study the behavior of ${\mathcal{M}^{\mathfrak{a}}_N}$ according to various vector fields ${\mathfrak{a}}$ . In particular, we classify the operator norms of ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^2(\mathbb{R}^3)}$ when ${\mathfrak{a}(x)}$ is the linear function of the form (a ₁₁ x ₁?+?a ₂₁ x ₂, a ₁₂ x ₁?+?a ₂₂ x ₂, 1). The operator norm of ${\mathcal{M}^\mathfrak{a}_N}$ on ${L^2(\mathbb{R}^3)}$ is related with the number given by $$D=(a_{12}+a_{21})^2-4a_{11}a_{22}.$$      

A new class of bent and hyper-bent Boolean functions in polynomial forms

Bent functions are maximally nonlinear Boolean functions and exist only for functions with even number of inputs. This paper is a contribution to the construction of bent functions over ${\mathbb{F}_{2^{n}}}$ (n = 2m) having the form ${f(x) = tr_{o(s_1)} (a x^ {s_1}) + tr_{o(s_2)} (b x^{s_2})}$ where o(s _i ) denotes the cardinality of the cyclotomic class of 2 modulo 2 ⁿ ? 1 which contains s _i and whose coefficients a and b are, respectively in ${F_{2^{o(s_1)}}}$ and ${F_{2^{o(s_2)}}}$ . Many constructions of monomial bent functions are presented in the literature but very few are known even in the binomial case. We prove that the exponents s ₁ = 2 ^m ? 1 and ${s_2={\frac {2^n-1}3}}$ , where ${a\in\mathbb{F}_{2^{n}}}$ (a ?? 0) and ${b\in\mathbb{F}_{4}}$ provide a construction of bent functions over ${\mathbb{F}_{2^{n}}}$ with optimum algebraic degree. For m odd, we give an explicit characterization of the bentness of these functions, in terms of the Kloosterman sums. We generalize the result for functions whose exponent s ₁ is of the form r(2 ^m ? 1) where r is co-prime with 2 ^m  + 1. The corresponding bent functions are also hyper-bent. For m even, we give a necessary condition of bentness in terms of these Kloosterman sums.