On definability in some lattices of semigroup varieties

An identity of the form x ₁?x _n ??x _1?? x _2?? ?x _n?? where ?? is a non-trivial permutation on the set {1,??,n} is called a permutation identity. If u??v is a permutation identity, then ?(u??v) [respectively r(u??v)] is the maximal length of the common prefix [suffix] of the words u and v. A variety that satisfies a permutation identity is called permutative. If $\mathcal{V}$ is a permutative variety, then $\ell=\ell(\mathcal{V})$ [respectively $r=r(\mathcal{V})$ ] is the least ? [respectively r] such that $\mathcal{V}$ satisfies a permutation identity ?? with ?(??)=? [respectively r(??)=r]. A?variety that consists of nil-semigroups is called a nil-variety. If ?? is a set of identities, then $\operatorname {var}\varSigma$ denotes the variety of semigroups defined by ??. If $\mathcal{V}$ is a variety, then $L (\mathcal{V})$ denotes the lattice of all subvarieties of $\mathcal{V}$ . For ?,r??0 and n>1 let $\mathfrak{B}_{\ell,r,n}$ denote the set that consists of n! identities of the form $$t_1\cdots t_\ell x_1x_2 \cdots x_n z_{1}\cdots z_{r}\approx t_1\cdots t_\ell x_{1\pi}x_{2\pi} \cdots x_{n\pi}z_{1}\cdots z_{r}, $$ where ?? is a permutation on the set {1,??,n}. We prove that for each permutative nil-variety $\mathcal{V}$ and each $\ell\ge\ell(\mathcal{V})$ and $r\ge r(\mathcal{V})$ there exists n>1 such that $\mathcal{V}$ is definable by a first-order formula in $L(\operatorname{var}{\mathfrak{B}}_{l,r,n})$ if ???r or $\mathcal{V}$ is definable up to duality in $L(\operatorname{var}{\mathfrak{B}}_{\ell,r,n})$ if ?=r.     

Diophantine approximation with sign constraints

Let $\alpha $ and $\beta $ be real numbers such that $1$ , $\alpha $ and $\beta $ are linearly independent over $\mathbb {Q}$ . A classical result of Dirichlet asserts that there are infinitely many triples of integers $(x_0,x_1,x_2)$ such that $|x_0+\alpha x_1+\beta x_2| < \max \{|x_1|,|x_2|\}^{-2}$ . In 1976, Schmidt asked what can be said under the restriction that $x_1$ and $x_2$ be positive. Upon denoting by $\gamma \cong 1.618$ the golden ratio, he proved that there are triples $(x_0,x_1,x_2) \in \mathbb {Z}^3$ with $x_1,x_2>0$ for which the product $|x_0 + \alpha x_1 + \beta x_2| \max \{|x_1|,|x_2|\}^\gamma $ is arbitrarily small. Although Schmidt later conjectured that $\gamma $ can be replaced by any number smaller than $2$ , Moshchevitin proved very recently that it cannot be replaced by a number larger than $1.947$ . In this paper, we present a construction of points $(1,\alpha ,\beta )$ showing that the result of Schmidt is in fact optimal. These points also possess strong additional Diophantine properties that are described in the paper.     

Pfaffian representations of cubic surfaces

Let $\mathbb{K }$ be a field of characteristic zero. We describe an algorithm which requires a homogeneous polynomial $F$ of degree three in $\mathbb{K }[x_{0},x_1,x_{2},x_{3}]$ and a zero ${\mathbf{a }}$ of $F$ in $\mathbb{P }^{3}_{\mathbb{K }}$ and ensures a linear Pfaffian representation of $\text{ V}(F)$ with entries in $\mathbb{K }[x_{0},x_{1},x_{2},x_{3}]$ , under mild assumptions on $F$ and ${\mathbf{a }}$ . We use this result to give an explicit construction of (and to prove the existence of) a linear Pfaffian representation of $\text{ V}(F)$ , with entries in $\mathbb{K }^{\prime }[x_{0},x_{1},x_{2},x_{3}]$ , being $\mathbb{K }^{\prime }$ an algebraic extension of $\mathbb{K }$ of degree at most six. An explicit example of such a construction is given.     

A strict minimax inequality criterion and some of its consequences

In this paper, we point out a very flexible scheme within which a strict minimax inequality occurs. We then show the fruitfulness of this approach presenting a series of various consequences. Here is one of them: Let Y be a finite-dimensional real Hilbert space, J : Y ?? R a C ¹ function with locally Lipschitzian derivative, and ${\varphi : Y \to [0, + \infty[}$ a C ¹ convex function with locally Lipschitzian derivative at 0 and ${\varphi^{-1}(0) = \{0\}}$ . Then, for each ${x_0 \in Y}$ for which J??(x ₀)??? 0, there exists ???> 0 such that, for each ${r \in ]0, \delta[}$ , the restriction of J to B(x ₀, r) has a unique global minimum u _r which satisfies $$J(u_r)\leq J(x)-\varphi(x-u_r)$$ for all ${x \in B(x_0, r)}$ , where ${B(x_0, r) = \{x \in Y : \|x-x_0\|\leq{r}\}.}$      

Lower and upper bounds of Dirichlet eigenvalues for totally characteristic degenerate elliptic operators

Let λkbe the k-th Dirichlet eigenvalue of totally characteristic degenerate elliptic operator-ΔB defined on a stretched cone B0 ■ [0,1) × X with boundary on {x1 = 0}. More precisely,ΔB=(x1αx1)2+ α2x2+ + α2xnis also called the cone Laplacian. In this paper,by using Mellin-Fourier transform,we prove thatλk Cnk2 n for any k 1,where Cn=(nn+2)(2π)2(|B0|Bn)-2n,which gives the lower bounds of the Dirchlet eigenvalues of-ΔB. On the other hand,by using the Rayleigh-Ritz inequality,we deduce the upper bounds ofλk,i.e.,λk+1 1 +4n k2/nλ1. Combining the lower and upper bounds of λk,we can easily obtain the lower bound for the first Dirichlet eigenvalue λ1 Cn(1 +4n)-12n2.     

Computable representations for convex hulls of low-dimensional quadratic forms

Let ${\mathcal{C}}$ be the convex hull of points ${{\{{1 \choose x}{1 \choose x}^T \,|\, x\in \mathcal{F}\subset \Re^n\}}}$ . Representing or approximating ${\mathcal{C}}$ is a fundamental problem for global optimization algorithms based on convex relaxations of products of variables. We show that if n ≤ 4 and ${\mathcal{F}}$ is a simplex, then ${\mathcal{C}}$ has a computable representation in terms of matrices X that are doubly nonnegative (positive semidefinite and componentwise nonnegative). We also prove that if n = 2 and ${\mathcal{F}}$ is a box, then ${\mathcal{C}}$ has a representation that combines semidefiniteness with constraints on product terms obtained from the reformulation-linearization technique (RLT). The simplex result generalizes known representations for the convex hull of ${{\{(x_1, x_2, x_1x_2)\,|\, x\in\mathcal{F}\}}}$ when ${\mathcal{F}\subset\Re^2}$ is a triangle, while the result for box constraints generalizes the well-known fact that in this case the RLT constraints generate the convex hull of ${{\{(x_1, x_2, x_1x_2)\,|\, x\in\mathcal{F}\}}}$ . When n = 3 and ${\mathcal{F}}$ is a box, we show that a representation for ${\mathcal{C}}$ can be obtained by utilizing the simplex result for n = 4 in conjunction with a triangulation of the 3-cube.     

Composition Operators on Spaces of Fractional Cauchy Transforms

For ?? > 0, the Banach space ${\mathcal{F}_{\alpha}}$ is defined as the collection of functions f which can be represented as integral transforms of an appropriate kernel against a Borel measure defined on the unit circle T. Let ?? be an analytic self-map of the unit disc D. The map ?? induces a composition operator on ${\mathcal{F}_{\alpha}}$ if ${C_{\Phi}(f) = f \circ \Phi \in \mathcal{F}_{\alpha}}$ for any function ${f \in \mathcal{F}_{\alpha}}$ . Various conditions on ?? are given, sufficient to imply that C _?? is bounded on ${\mathcal{F}_{\alpha}}$ , in the case 0 < ?? < 1. Several of the conditions involve ???? and the theory of multipliers of the space ${\mathcal{F}_{\alpha}}$ . Relations are found between the behavior of C _?? and the membership of ?? in the Dirichlet spaces. Conditions given in terms of the generalized Nevanlinna counting function are shown to imply that ?? induces a bounded composition operator on ${\mathcal{F}_{\alpha}}$ , in the case 1/2 ?? ?? < 1. For such ??, examples are constructed such that ${\| \Phi \|_{\infty} = 1}$ and ${C_{\Phi}: \mathcal{F}_{\alpha} \rightarrow \mathcal{F}_{\alpha}}$ is bounded.     

Isolated boundary singularities of semilinear elliptic equations

Given a smooth domain ${\Omega\subset\mathbb{R}^N}$ such that ${0 \in \partial\Omega}$ and given a nonnegative smooth function ?? on ???, we study the behavior near 0 of positive solutions of ???u?=?u ^q in ?? such that u =? ?? on ???\{0}. We prove that if ${\frac{N+1}{N-1} < q < \frac{N+2}{N-2}}$ , then ${u(x)\leq C |x|^{-\frac{2}{q-1}}}$ and we compute the limit of ${|x|^{\frac{2}{q-1}} u(x)}$ as x ?? 0. We also investigate the case ${q= \frac{N+1}{N-1}}$ . The proofs rely on the existence and uniqueness of solutions of related equations on spherical domains.     

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.     

On restricting Cauchy–Pexider functional equations to submanifolds

Sufficient geometric conditions are given which determine when the Cauchy–Pexider functional equation f(x)g(y) = h(x + y) restricted to x, y lying on a hypersurface in ${\mathbb{R}^d}$ has only solutions which extend uniquely to exponential affine functions ${\mathbb{R}^d \to \mathbb{C}}$ (when f, g, h are assumed to be measurable and non-trivial). The Cauchy–Pexider-type functional equations ${\prod_{j=0}^df_j(x_j)=F(\sum_{j=0}^dx_j)}$ for ${x_0, \ldots,x_d}$ lying on a curve and ${f_1(x_1)f_2(x_2)f_3(x_3)=F(x_1+x_2+x_3)}$ for x ₁, x ₂, x ₃ lying on a hypersurface are also considered.     

Traces of Permuting n-Additive Maps and Permuting n-Derivations of Rings

Let R be a prime ring with center Z(R). For a fixed positive integer n, a permuting n-additive map ${\Delta : R^n \to R}$ is known to be permuting n-derivation if ${\Delta(x_1, x_2, \ldots, x_i x'_{i},\ldots, x_n) = \Delta(x_1, x_2, \ldots, x_i, \ldots, x_n)x'_i + x_i \Delta(x_1, x_2, \ldots, x'_i, \ldots, x_n)}$ holds for all ${x_i, x'_i \in R}$ . A mapping ${\delta : R \to R}$ defined by δ(x) = Δ(x, x, . . . ,x) for all ${x \in R}$ is said to be the trace of Δ. In the present paper, we have proved that a ring R is commutative if there exists a permuting n-additive map ${\Delta : R^n \to R}$ such that ${xy + \delta(xy) = yx + \delta(yx), xy- \delta(xy) = yx - \delta(yx), xy - yx = \delta(x) \pm \delta(y)}$ and ${xy + yx = \delta(x) \pm \delta(y)}$ holds for all ${x, y \in R}$ . Further, we have proved that if R is a prime ring with suitable torsion restriction then R is commutative if there exist non-zero permuting n-derivations Δ₁ and Δ₂ from ${R^n \to R}$ such that Δ₁(δ ₂(x), x, . . . ,x) =  0 for all ${x \in R,}$ where δ ₂ is the trace of Δ₂. Finally, it is shown that in a prime ring R of suitable torsion restriction, if ${\Delta_1, \Delta_2 : R^n \longrightarrow R}$ are non-zero permuting n-derivations with traces δ ₁, δ ₂, respectively, and ${B : R^n \longrightarrow R}$ is a permuting n-additive map with trace f such that δ ₁ δ ₂(x) =  f(x) holds for all ${x \in R}$ , then R is commutative.     

Continuity of solutions of a problem in the calculus of variations

We study the problem of minimizing ${\int_{\Omega} L(x,u(x),Du(x))\,{\rm d}x}$ over the functions ${u\in W^{1,p}(\Omega)}$ that assume given boundary values ${\phi}$ on ???. We assume that L(x, u, Du)?=?F(Du)?+?G(x, u) and that F is convex. We prove that if ${\phi}$ is continuous and ?? is convex, then any minimum u is continuous on the closure of ??. When ?? is not convex, the result holds true if F(Du)?=?f(|Du|). Moreover, if ${\phi}$ is Lipschitz continuous, then u is H?lder continuous.     

Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones

We prove that if ${U\subset \mathbb {R}^n}$ is an open domain whose closure ${\overline U}$ is compact in the path metric, and F is a Lipschitz function on ?U, then for each ${\beta \in \mathbb {R}}$ there exists a unique viscosity solution to the β-biased infinity Laplacian equation $$\beta |\nabla u| + \Delta_\infty u=0$$ on U that extends F, where ${\Delta_\infty u= |\nabla u|^{-2} \sum_{i,j} u_{x_i}u_{x_ix_j} u_{x_j}}$ . In the proof, we extend the tug-of-war ideas of Peres, Schramm, Sheffield and Wilson, and define the β-biased ${\epsilon}$ -game as follows. The starting position is ${x_0 \in U}$ . At the kth step the two players toss a suitably biased coin (in our key example, player I wins with odds of ${\exp(\beta\epsilon)}$ to 1), and the winner chooses x _k with ${d(x_k,x_{k-1}) < \epsilon}$ . The game ends when ${x_k \in \partial U}$ , and player II pays the amount F(x _k ) to player I. We prove that the value ${u^{\epsilon}(x_0)}$ of this game exists, and that ${\|u^\epsilon - u\|_\infty \to 0}$ as ${\epsilon \to 0}$ , where u is the unique extension of F to ${\overline{U}}$ that satisfies comparison with β-exponential cones. Comparison with exponential cones is a notion that we introduce here, and generalizing a theorem of Crandall, Evans and Gariepy regarding comparison with linear cones, we show that a continuous function satisfies comparison with β-exponential cones if and only if it is a viscosity solution to the β-biased infinity Laplacian equation.     

On Families of Graphs of Large Cycle Indicator, Matrices of Large Order and Key Exchange Protocols With Nonlinear Polynomial Maps of Small Degree

This paper studies the group theoretical protocol of Diffie?CHellman key exchange in the case of symmetrical group ${S_{p^n}}$ and more general Cremona group ${C(\mathbb K^n)}$ of polynomial automorphisms of free module ${\mathbb K^n}$ over arbitrary commutative ring ${\mathbb K}$ . This algorithm depends very much on the choice of the base ${g_n \in C( \mathbb K^n)}$ . It is important to work with the base ${g_n \in C( \mathbb K^n)}$ , which is a polynomial map of a small degree and a large order such that the degrees of all powers ${g_n^k}$ are also bounded by a small constant. We suggest fast algorithms for generation of a map ${g_n={f_n} \xi_nf_n^{-1}}$ , where ?? _n is an affine transformation (degree is 1) of a large order and f _n is a fixed nonlinear polynomial map in n variables such that ${f_n^{-1}}$ is also a polynomial map and both maps f _n and ${f_n^{-1}}$ are of small degrees. The method is based on properties of infinite families of graphs with a large cycle indicator and families of graphs of a large girth in particular. It guaranties that the order of g _n is tending to infinity as the dimension n tends to infinity. We propose methods of fast generation of special families of cubical maps f _n such that ${f_n^{-1}}$ is also of degree 3 based on properties of families of graphs of a large girth and graphs with a large cycle indicator. At the end we discuss cryptographical applications of maps of the kind ?? f _n ??^?1 and some graph theoretical problems motivated by such applications.     

On All Fractional （a,b, k）-Critical Graphs

Let a,b,k,r be nonnegative integers with 1≤a≤b and r≥2.LetG be a graph of order n with n(a+b)(r(a+b)-2)+ak/a.In this paper,we first show a characterization for all fractional(a,b,k)-critical graphs.Then using the result,we prove that G is all fractional(a,b,k)-critical if δ(G)≥(r-1)b2/a+k and |NG(x1)∪NG(x2)∪···∪NG(xr)|≥bn+ak/a+b for any independent subset {x1,x2,...,xr} in G.Furthermore,it is shown that the lower bound on the condition|NG(x1)∪NG(x2)∪···∪NG(xr)|≥bn+ak/a+b is best possible in some sense,and it is an extension of Lu's previous result.     

Convergence rate of the Allen-Cahn equation to generalized motion by mean curvature

We investigate the singular limit, as ${\varepsilon \to 0}$ , of the Allen-Cahn equation ${u^\varepsilon_t=\Delta u^\varepsilon+\varepsilon^{-2}f(u^\varepsilon)}$ , with f a balanced bistable nonlinearity. We consider rather general initial data u ₀ that is independent of ${{\varepsilon}}$ . It is known that this equation converges to the generalized motion by mean curvature ?? in the sense of viscosity solutions??defined by Evans, Spruck and Chen, Giga, Goto. However, the convergence rate has not been known. We prove that the transition layers of the solutions ${u^{\varepsilon}}$ are sandwiched between two sharp ??interfaces?? moving by mean curvature, provided that these ??interfaces?? sandwich at t?=?0 an ${\mathcal O({\varepsilon}|\,{\rm ln}\,{\varepsilon}|)}$ neighborhood of the initial layer. In some special cases, which allow both extinction and pinches off phenomenon, this enables to obtain an ${\mathcal O({\varepsilon}|\,{\rm ln}\,{\varepsilon}|)}$ estimate of the location and the thickness measured in space-time of the transition layers. A result on the regularity of the generalized motion by mean curvature is also provided in the Appendix.     

Axiomatizability by {{\forall}{\exists}!}-sentences

A ${\forall\exists!}$ -sentence is a sentence of the form ${\forall x_{1}\cdots x_{n}\exists!y_{1}\cdots y_{m}O(\overline{x},\overline{y})}$ , where O is a quantifier-free formula, and ${\exists!}$ stands for ??there exist unique??. We prove that if ${\mathcal{C}}$ is (up to isomorphism) a finite class of finite models then ${\mathcal{C}}$ is axiomatizable by a set of ${\forall\exists!}$ -sentences if and only if ${\mathcal{C}}$ is closed under isomorphic images, ${\mathcal{C}}$ has the intersection property, and ${\mathcal{C}}$ is closed under fixed-point submodels. This result is employed to characterize the subclasses of finitely generated discriminator varieties axiomatizable by sentences of the form ${\forall\exists!\bigwedge p=q}$ .     

Full regularity of a free boundary problem with two phases

Let ?? be a bounded domain in ${\mathbb{R}^{n}, n\geq2}$ . We use ${\mathcal{M}_{\Omega}}$ to denote the collection of all pairs of (A, u) such that ${A\subset\Omega}$ is a set of finite perimeter and ${u\in H^{1}\left( \Omega\right)}$ satisfies $$u\left( x\right) =0\quad\text{a.e.}x\in A.$$ We consider the energy functional $$E_{\Omega}\left( A,u\right) =\int\limits_{\Omega}\left\vert\triangledown u\right\vert ^{2}+P_{\Omega}\left( A\right)$$ defined on ${\mathcal{M}_{\Omega}}$ , where P _??(A) denotes the perimeter of A inside ??. Let ${\left( A,u\right)\in\mathcal{M}_{\Omega}}$ be a minimizer with volume constraint. Our main result is that when n????7, u is locally Lipschitz and the free boundary ?A is analytic in ??.     

Strong convergence of a proximal point algorithm with bounded error sequence

Given any maximal monotone operator ${A: D(A)\subset H \rightarrow 2^H}$ in a real Hilbert space H with ${A^{-1}(0) \ne \emptyset}$ , it is shown that the sequence of proximal iterates ${x_{n+1}=(I+\gamma_n A)^{-1}(\lambda_n u+(1-\lambda_n)(x_n+e_n))}$ converges strongly to the metric projection of u on A ^?1(0) for (e _n ) bounded, ${\lambda_n \in (0,1)}$ with ${\lambda_n \to 1}$ and γ _n  > 0 with ${\gamma_n \to\infty}$ as ${n \to \infty}$ . In comparison with our previous paper (Boikanyo and Moro?anu in Optim Lett 4(4):635–641, 2010), where the error sequence was supposed to converge to zero, here we consider the classical condition that errors be bounded. In the case when A is the subdifferential of a proper convex lower semicontinuous function ${\varphi :H \to (-\infty,+ \infty]}$ , the algorithm can be used to approximate the minimizer of φ which is nearest to u.