Difference Inequalities for the Groups and Semigroups of Operators on Banach Spaces

Let ${\mathbf{T}=\{T(t)\} _{t\in\mathbb{R}}}$ be a ??(X, F)-continuous group of isometries on a Banach space X with generator A, where ??(X, F) is an appropriate local convex topology on X induced by functionals from ${ F\subset X^{\ast}}$ . Let ?? _A (x) be the local spectrum of A at ${x\in X}$ and ${r_{A}(x):=\sup\{\vert\lambda\vert :\lambda \in \sigma_{A}(x)\},}$ the local spectral radius of A at x. It is shown that for every ${x\in X}$ and ${\tau\in\mathbb{R},}$ $$\left\Vert T(\tau) x-x\right\Vert \leq \left\vert \tau \right\vert r_{A}(x)\left\Vert x\right\Vert.$$ Moreover if ${0\leq \tau r_{A}(x)\leq \frac{\pi}{2},}$ then it holds that $$\left\Vert T(\tau) x-T(-\tau)x\right\Vert \leq 2\sin \left(\tau r_{A}(x)\right)\left\Vert x\right\Vert.$$ Asymptotic versions of these results for C ₀-semigroup of contractions are also obtained. If ${\mathbf{T}=\{T(t)\}_{t\geq 0}}$ is a C ₀-semigroup of contractions, then for every ${x\in X}$ and ????? 0, $$\underset{t\rightarrow \infty }{\lim } \left\Vert T( t+\tau) x-T(t) x\right\Vert\leq\tau\sup\left\{ \left\vert \lambda \right\vert :\lambda \in\sigma_{A}(x)\cap i \mathbb{R} \right\} \left\Vert x\right\Vert. $$ Several applications are given.     

Mixing via the extended family

In this paper,the relationship between the extended family and several mixing properties in measuretheoretical dynamical systems is investigated.The extended family eF related to a given family F can be regarded as the collection of all sets obtained as"piecewise shifted"members of F.For a measure preserving transformation T on a Lebesgue space(X,B,μ),the sets of"accurate intersections of order k"defined below are studied,Nε(A0,A1,...,Ak)=n∈Z+:μk i=0T inAiμ(A0)μ(A1)μ(Ak)ε,for k∈N,A0,A1,...,Ak∈B and ε0.It is shown that if T is weakly mixing(mildly mixing)then for any k∈N,all the sets Nε(A0,A1,...,Ak)have Banach density 1(are in(eFip),i.e.,the dual of the extended family related to IP-sets).     

Homogenization of the Neumann problem in perforated domains: an alternative approach

The main result of this paper is a compactness theorem for families of functions in the space SBV (Special functions of Bounded Variation) defined on periodically perforated domains. Given an open and bounded set ${\Omega\subseteq\mathbb{R}^n}$ , and an open, connected, and (?1/2, 1/2) ⁿ -periodic set ${P\subseteq\mathbb{R}^n}$ , consider for any ???>?0 the perforated domain ?? _?? :=????????? P. Let ${(u_\varepsilon)\subset SBV^p(\Omega_{\varepsilon})}$ , p?>?1, be such that ${\int_{\Omega_{\varepsilon}}\left|{\nabla{u}_\varepsilon}\right|^pdx+\mathcal{H}^{n-1}(S_{u_\varepsilon}\,\cap\,\Omega_{\varepsilon}) +\left\Vert{u_\varepsilon}\right\Vert_{L^p(\Omega_{\varepsilon})}}$ is bounded. Then, we prove that, up to a subsequence, there exists ${u\in GSBV^p\,\cap\, L^p(\Omega)}$ satisfying ${\lim_\varepsilon\left\Vert{u-u_\varepsilon}\right\Vert_{L^1(\Omega_{\varepsilon})}=0}$ . Our analysis avoids the use of any extension procedure in SBV, weakens the hypotheses on P to the minimal ones and simplifies the proof of the results recently obtained in Focardi et?al. (Math Models Methods Appl Sci 19:2065?C2100, 2009) and Cagnetti and Scardia (J Math Pures Appl (9), to appear). Among the arguments we introduce, we provide a localized version of the Poincaré-Wirtinger inequality in SBV. As a possible application we study the asymptotic behavior of a brittle porous material represented by the perforated domain ?? _?? . Finally, we slightly extend the well-known homogenization theorem for Sobolev energies on perforated domains.     

1D Schrödinger Operators with Short Range Interactions: Two-Scale Regularization of Distributional Potentials

For real ${L_\infty(\mathbb{R})}$ -functions ${\Phi}$ and ${\Psi}$ of compact support, we prove the norm resolvent convergence, as ${\varepsilon}$ and ${\nu}$ tend to 0, of a family ${S_{\varepsilon \nu}}$ of one-dimensional Schrödinger operators on the line of the form $$S_{\varepsilon \nu} = -\frac{d^2}{dx^2} + \frac{\alpha}{\varepsilon^2} \Phi \left( \frac{x}{\varepsilon} \right) + \frac{\beta}{\nu} \Psi \left(\frac{x}{\nu} \right),$$ provided the ratio ${\nu/\varepsilon}$ has a finite or infinite limit. The limit operator S ₀ depends on the shape of ${\Phi}$ and ${\Psi}$ as well as on the limit of ratio ${\nu/\varepsilon}$ . If the potential ${\alpha\Phi}$ possesses a zero-energy resonance, then S ₀ describes a non trivial point interaction at the origin. Otherwise S ₀ is the direct sum of the Dirichlet half-line Schrödinger operators.     

Asymptotic analysis of a second-order singular perturbation model for phase transitions

We study the asymptotic behavior, as ${\varepsilon}$ tends to zero, of the functionals ${F^k_\varepsilon}$ introduced by Coleman and Mizel in the theory of nonlinear second-order materials; i.e., $$F^k_\varepsilon(u):=\int\limits_{I} \left(\frac{W(u)}{\varepsilon}-k\,\varepsilon\,(u')^2+\varepsilon^3(u'')^2\right)\,dx,\quad u\in W^{2,2}(I),$$ where k?>?0 and ${W:\mathbb{R}\to[0,+\infty)}$ is a double-well potential with two potential wells of level zero at ${a,b\in\mathbb{R}}$ . By proving a new nonlinear interpolation inequality, we show that there exists a positive constant k ₀ such that, for k?<?k ₀, and for a class of potentials W, ${(F^k_\varepsilon)}$ ??(L ¹)-converges to $$F^k(u):={\bf m}_k \, \#(S(u)),\quad u\in BV(I;\{a,b\}),$$ where m _k is a constant depending on W and k. Moreover, in the special case of the classical potential ${W(s)=\frac{(s^2-1)^2}{2}}$ , we provide an upper bound on the values of k such that the minimizers of ${F_\varepsilon^k}$ cannot develop oscillations on some fine scale and a lower bound on the values for which oscillations occur, the latter improving a previous estimate by Mizel, Peletier and Troy.     

Perturbed asymptotically linear problems

The aim of this paper is to investigate the existence of solutions of the semilinear elliptic problem $$\left\{\begin{array}{ll} -\Delta u\ =\ p(x, u) + \varepsilon g(x, u)\quad {\rm in}\,\, \Omega, \\ u=0 \quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\,\,\,\,{\rm on}\,\, \partial\Omega, \end{array} \right. \quad\quad\quad(0.1) $$ where Ω is an open bounded domain of ${\mathbb{R}^N}$ , ${\varepsilon\in\mathbb{R}, p}$ is subcritical and asymptotically linear at infinity, and g is just a continuous function. Even when this problem has not a variational structure on ${H^1_0(\Omega)}$ , suitable procedures and estimates allow us to prove that the number of distinct critical levels of the functional associated to the unperturbed problem is “stable” under small perturbations, in particular obtaining multiplicity results if p is odd, both in the non-resonant and in the resonant case.     

The tail behaviour of a random sum of subexponential random variables and vectors

Let $\left\{ X,X_{i},i=1,2,...\right\} $ denote independent positive random variables having common distribution function (d.f.) F(x) and, independent of X, let ν denote an integer valued random variable. Using X ₀=0, the random sum Z=∑ _i=0 ^ν X _i has d.f. $G(x)=\sum_{n=0}^{\infty }\Pr\{\nu =n\}F^{n\ast }(x)$ where F ^n?(x) denotes the n-fold convolution of F with itself. If F is subexponential, Kesten’s bound states that for each ε>0 we can find a constant K such that the inequality $$ 1-F^{n\ast }(x)\leq K(1+\varepsilon )^{n}(1-F(x))\, , \qquad n\geq 1,x\geq 0 \, , $$ holds. When F is subexponential and E(1 +ε) ^ν <∞, it is a standard result in risk theory that G(x) satisfies $$ 1 - G{\left( x \right)} \sim E{\left( \nu \right)}{\left( {1 - F{\left( x \right)}} \right)},\,\,x \to \infty \,\,{\left( * \right)} $$ In this paper, we show that (*) holds under weaker assumptions on ν and under stronger conditions on F. Stam (Adv. Appl. Prob. 5:308–327, 1973) considered the case where $ \overline{F}(x)=1-F(x)$ is regularly varying with index –α. He proved that if α>1 and $E{\left( {\nu ^{{\alpha + \varepsilon }} } \right)} < \infty $ , then relation (*) holds. For 0<α<1, it is sufficient that Eν<∞. In this paper we consider the case where $\overline{F}(x)$ is an O-regularly varying subexponential function. If the lower Matuszewska index $\beta (\overline{F})<-1$ , then the condition ${\text{E}}{\left( {\nu ^{{{\left| {\beta {\left( {\overline{F} } \right)}} \right|} + 1 + \varepsilon }} } \right)} < \infty$ is sufficient for (*). If $\beta (\overline{F} )>-1$ , then again Eν<∞ is sufficient. The proofs of the results rely on deriving bounds for the ratio $\overline{F^{n\ast }}(x)/\overline{F} (x)$ . In the paper, we also consider (*) in the special case where X is a positive stable random variable or has a compound Poisson distribution derived from such a random variable and, in this case, we show that for n≥2, the ratio $\overline{F^{n\ast }}(x)/\overline{F}(x)\uparrow n$ as x↑∞. In Section 3 of the paper, we briefly discuss an extension of Kesten’s inequality. In the final section of the paper, we discuss a multivariate analogue of (*).     

A note on da Costa-Doria “exotic formalizations”

We analyze N. C. A. da Costa and F. A. Doria’s “exotic formalization” of the conjecture P = NP [3–7]. For any standard axiomatic PA extension T and any number-theoretic sentence ${\varphi }$ , we let ${\varphi ^{\star} := \varphi \vee \lnot \mathsf{Con}\left( \mathsf{T}\right)}$ and prove the following “exotic” inferences 1–3. 1. ${\mathsf{T}+\varphi ^{\star}}$ is consistent, if so is T, 2. ${\mathsf{T}+\varphi}$ is consistent, provided that ${\mathsf{T}+\varphi ^{\star}}$ is ω-consistent, 3. ${\mathsf{T}+\varphi}$ is consistent, provided that T is consistent and has the same provably total recursive functions as ${\mathsf{T}+\left( \varphi \leftrightarrow \varphi ^{\star }\right) }$ . Furthermore we show that 1–3 continue to hold for ${\varphi ^{\star} := \varphi _{S} :=\varphi \vee \lnot S}$ , where ${S=\forall x\exists yR\left( x,y\right)}$ is any ${\Pi _{2}^{0}}$ sentence satisfying: 4. ${\left( \forall n\in \omega \right) \left( \mathsf{T}\vdash S_{x}\left[ \underline{n}\right] \right) }$ , 5. ${\mathsf{Con}\left( \mathsf{T}\right) \Rightarrow \mathsf{T}\nvdash S}$ . We observe that if ${\varphi :=\left[ \mathsf{P}=\mathsf{NP}\right] }$ and ${S:= \left[\digamma total\right] }$ , where ${\digamma=\digamma _{\mathsf{T}}}$ is da Costa-Doria “exotic” function with respect to T, then 4, 5 are satisfied for most familiar (presumably) consistent T in question, while ${\varphi _{S}}$ becomes equivalent to da Costa-Doria “exotic formalization” ${\left[ \mathsf{P}=\mathsf{NP}\right]^{\digamma}}$ . Moreover, the corresponding “exotic” inferences 1–3 generalize analogous da Costa-Doria results. Hence these “exotic” inferences are universal for all number-theoretic sentences and not characteristic to the conjecture P = NP. Nor do they infer relative consistency of P = NP (see Conclusion 15 in the text).     

Complexity of approximate realizations of Lipschitz functions by schemes in continuous bases

We show that any function satisfying the Lipschitz condition on a given closed interval can be approximately computed by a scheme (nonbranching program) in the basis composed of functions $$x - y, \left| x \right|, x*y = \min (\max (x,0),1)\min (\max (y,0),1),$$ and all constants from the closed interval [0, 1]; here the complexity of the scheme is $O\left( {{1 \mathord{\left/ {\vphantom {1 {\sqrt \varepsilon }}} \right. \kern-0em} {\sqrt \varepsilon }}} \right)$ , where ? is the accuracy of the approximation. This estimate of complexity, is in general, order-sharp.     

Exponential sums over primes and the prime twin problem

The purpose of this paper is to obtain an effective estimate of the exponential sum $\sum_{n\le x}\Lambda(n)e\left(\left(\frac{a}{q}+\beta\right)n\right)$ (where e(??)=e ^{2?? i ??} , ??,?????, (a,q)=1 and ?? is the von Mangoldt function) in the range ${(\log x)}^{1/2+\varepsilon}\le q\le \frac{x}{{(\log\log\log x)}^{1+\varepsilon}}$ and $|\beta|<\frac{1}{q{(\log\log\log x)}^{1+\varepsilon}}$ . It improves Daboussi??s estimate [2, Theorem 1] in the range q??(log?x) ^D and x(log?x)^?D ??q, D>0 and is valid in a wider range for ??.     

Symmetric 2-Structures,a Classification

We classify symmetric 2-structures ${(P, \mathfrak{G}_1, \mathfrak{G}_2, \mathfrak{K})}$ , i.e. chain structures which correspond to sharply 2-transitive permutation sets (E, Σ) satisfying the condition: “ ${(*) \, \, \forall \sigma, \tau \in \Sigma : \sigma \circ \tau^{-1} \circ \sigma \in \Sigma}$ ”. To every chain ${K \in \mathfrak{K}}$ one can associate a reflection ${\widetilde{K}}$ in K. Then (*) is equivalent to “ ${(**) \, \, \forall K \in \mathfrak{K} : \widetilde{K}(\mathfrak{K}) = \mathfrak{K}}$ ” and one can define an orthogonality “ ${\perp}$ ” for chains ${K, L \in \mathfrak{K}}$ by “ ${K \perp L \Leftrightarrow K \neq L \wedge \widetilde{K}(L) = L}$ ”. The classification is based on the cardinality of the set of chains which are orthogonal to a chain K and passing through a point p of K. For one of these classes (called point symmetric 2-structures) we proof that in each point there is a reflection and that the set of point reflections forms a regular involutory permutation set.     

Properties of finite unrefinable chains of ring topologies

Let R(+, ·) be a nilpotent ring and $ \left( {\mathfrak{M}, < } \right) $ be the lattice of all ring topologies on R(+, ·) or the lattice of all such ring topologies on R(+, ·) in each of which the ring R possesses a basis of neighborhoods of zero consisting of subgroups. Let ?? and ??? be ring topologies from $ \mathfrak{M} $ such that $ \tau = {\tau_0}{ \prec_\mathfrak{M}}{\tau_1}{ \prec_\mathfrak{M}} \cdots { \prec_\mathfrak{M}}{\tau_n} = \tau ^{\prime} $ . Then k????n for every chain $ \tau = {\tau ^{\prime}_0} < {\tau ^{\prime}_1} < \cdots < {\tau ^{\prime}_k} = \tau ^{\prime} $ of topologies from $ \mathfrak{M} $ , and also n?=?k if and only if $ {\tau ^{\prime}_i}{ \prec_\mathfrak{M}}{\tau ^{\prime}_{i + 1}} $ for all 0????i?<?k.     

Pillai’s generalized function

The distribution of values of Pillai’s function over the ring of Gauss integers $ \mathbb{Z}\left[ i \right] $ is studied. The asymptotic formulas for the summators $ {\varSigma_{{N\left( \alpha \right)\leq x}}}\frac{{g\left( \alpha \right)}}{{{N^{\alpha }}\left( \alpha \right)}} $ , where $ a\in \mathbb{R} $ , are found.     

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 ??.     

A decomposition of the universal embedding space for the near polygon {{\mathbb H}_n}

Let ${{\mathbb H}_n, n \geq 1}$ , be the near 2n-gon defined on the 1-factors of the complete graph on 2n?+?2 vertices, and let e denote the absolutely universal embedding of ${{\mathbb H}_n}$ into PG(W), where W is a ${\frac{1}{n+2} \left(\begin{array}{c}2n+2 \\ n+1\end{array}\right)}$ -dimensional vector space over the field ${{\mathbb F}_2}$ with two elements. For every point z of ${{\mathbb H}_n}$ and every ${i \in {\mathbb N}}$ , let Δ _i (z) denote the set of points of ${{\mathbb H}_n}$ at distance i from z. We show that for every pair {x, y} of mutually opposite points of ${{\mathbb H}_n, W}$ can be written as a direct sum ${W_0 \oplus W_1 \oplus \cdots \oplus W_n}$ such that the following four properties hold for every ${i \in \{0,\ldots,n \}}$ : (1) ${\langle e(\Delta_i(x) \cap \Delta_{n-i}(y)) \rangle = {\rm PG}(W_i)}$ ; (2) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(x) \right) \right\rangle = {\rm PG}(W_0 \oplus W_1 \oplus \cdots \oplus W_i)}$ ; (3) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(y) \right) \right\rangle = {\rm PG}(W_{n-i}\oplus W_{n-i+1} \oplus \cdots \oplus W_n)}$ ; (4) ${\dim(W_i) = |\Delta_i(x) \cap \Delta_{n-i}(y)| = \left(\begin{array}{c}n \\ i\end{array}\right)^2 - \left(\begin{array}{c}n \\ i-1\end{array}\right) \cdot \left(\begin{array}{c}n \\ i+1\end{array}\right)}$ .     

Existence of semiclassical states for a quasilinear Schrödinger equation with critical exponent in {\mathbb{R}^N}

We consider the following perturbed version of quasilinear Schrödinger equation $$\begin{array}{lll}-\varepsilon^2\Delta u +V(x)u-\varepsilon^2\Delta (u^2)u=h(x,u)u+K(x)|u|^{22^*-2}u\end{array}$$ in ${\mathbb{R}^N}$ , where N ≥ 3, 22* = 4N/(N ? 2), V(x) is a nonnegative potential, and K(x) is a bounded positive function. Using minimax methods, we show that this equation has at least one positive solution provided that ${\varepsilon \leq \mathcal{E}}$ ; for any ${m\in\mathbb{N}}$ , it has m pairs of solutions if ${\varepsilon \leq \mathcal{E}_m}$ , where ${\mathcal{E}}$ and ${\mathcal{E}_m}$ are sufficiently small positive numbers. Moreover, these solutions ${u_\varepsilon \to 0}$ in ${H^1(\mathbb{R}^N)}$ as ${\varepsilon \to 0}$ .     

Representation of cyclotomic fields and their subfields

Let $\mathbb{K}$ be a finite extension of a characteristic zero field $\mathbb{F}$ . We say that a pair of n × n matrices (A,B) over $\mathbb{F}$ represents $\mathbb{K}$ if $\mathbb{K} \cong {{\mathbb{F}\left[ A \right]} \mathord{\left/ {\vphantom {{\mathbb{F}\left[ A \right]} {\left\langle B \right\rangle }}} \right. \kern-0em} {\left\langle B \right\rangle }}$ , where $\mathbb{F}\left[ A \right]$ denotes the subalgebra of $\mathbb{M}_n \left( \mathbb{F} \right)$ containing A and 〈B〉 is an ideal in $\mathbb{F}\left[ A \right]$ , generated by B. In particular, A is said to represent the field $\mathbb{K}$ if there exists an irreducible polynomial $q\left( x \right) \in \mathbb{F}\left[ x \right]$ which divides the minimal polynomial of A and $\mathbb{K} \cong {{\mathbb{F}\left[ A \right]} \mathord{\left/ {\vphantom {{\mathbb{F}\left[ A \right]} {\left\langle {q\left( A \right)} \right\rangle }}} \right. \kern-0em} {\left\langle {q\left( A \right)} \right\rangle }}$ . In this paper, we identify the smallest order circulant matrix representation for any subfield of a cyclotomic field. Furthermore, if p is a prime and $\mathbb{K}$ is a subfield of the p-th cyclotomic field, then we obtain a zero-one circulant matrix A of size p × p such that (A, J) represents $\mathbb{K}$ , where J is the matrix with all entries 1. In case, the integer n has at most two distinct prime factors, we find the smallest order 0, 1-companion matrix that represents the n-th cyclotomic field. We also find bounds on the size of such companion matrices when n has more than two prime factors.     

A contribution to stability theory for nonlinear Neumann problem

In this paper we study the stability of the solutions of some nonlinear Neumann problems, under perturbations of the domains in the Hausdorff complementary topology. We consider the problem $${{\left\{\begin{array}{c}-\text{ div}\;\left(a\left( x,\nabla u_{\Omega}\right)\right)=0 \;\text{in}\; \Omega \\ {a\left( x, \nabla u_{\Omega}\right) \cdot \nu=0\; \text{on}\; \partial\Omega}\end{array}\right.}}$$ where ${{\mathbf{R}^n \times \mathbf{R}^n \rightarrow \mathbf{R}^n}}$ is a Caratheodory function satisfying the standard monotonicity and growth conditions of order p, 1?<?p?<???. If ?? _h is a uniformly bounded sequence of connected open sets in R ⁿ , n ??? 2, we prove that if ${{\Omega_{h}^{c} \rightarrow \Omega^{c}}}$ in the Hausdorff metric, ${|\Omega_{h}| \rightarrow |\Omega|}$ and the geodetic distances satisfy the inequality ${d_{\Omega}\left( x,y\right) \leq \liminf_{h} d_{\Omega_{h}} \left( x,y\right)}$ for every ${x, y \in \Omega,}$ then ${\nabla u_{\Omega_h} \rightarrow\nabla u_{\Omega}}$ strongly in L ^p , provided that W ^{1, ??}(??) is dense in the space L ^{1, p} (??) of all functions whose gradient belongs to L ^p (??, R ⁿ ).     

Compatible Adapted Pairs and a Common Slice Theorem for Some Centralizers

Let $ \mathfrak{g} $ be a simple Lie algebra and $ x \in \mathfrak{g} $ nilpotent. We derive a criterion for when the G-orbits in $ \mathfrak{g}* $ and the G ^x -orbits in $ {\left( {\mathfrak{g}^{x} } \right)}^{*} $ admit a common slice, applicable (in principle) when x is of Bala–Carter type. When $ \mathfrak{g} $ is of type A, or if x is the highest root vector with g not of type E₈, we show that this criterion is satisfied. In these cases we also show that the Mishchenko–Fomenko shift of argument produces a maximal Poisson commutative polynomial subalgebra of $ S{\left( {\mathfrak{g}^{x} } \right)} $ which maps isomorphically by restriction of functions to an affine translate of a subspace of $ {\left( {\mathfrak{g}^{x} } \right)}^{*} $ . It is conjectured that the above criterion is satisfied when the dimensions of certain weights spaces, which can be computed purely combinatorially, are decreasing.