Some variations on Kac's formula for Brownian motion

We show that under suitable conditions $$\begin{gathered} E_x f\left\{ {a + \int_0^t \beta \left[ {b + \int_0^s {\alpha \left( {X_r } \right)dr, c + s, X_s } } \right]ds, b + \int_0^t {\alpha \left( {X_s } \right)ds, c + t, X_t } } \right\} \hfill \\ = e^{tG} f\left[ {a, b, c, x} \right] \hfill \\ \end{gathered} $$ whereX _t is a Brownian motion andG is the generator of a (C ₀) contraction semigroupe ^tG.     

Presentations of the amalgamated free product of two infinite cycles

LetH=〈a,b;a ^k =b ^l 〉, wherek,l≧2 andk+l>4. McCool and Pietrowski have proved that any pair of generators forH is Nielsen equivalent to a pairx=a ^r andy=b ^s where $$(a){\text{ }}gcd(r, s) = gcd(r, k) = gcd(s, l) = 1,$$ $$(b){\text{ }}0< 2r \leqq ks{\text{ }}and{\text{ }}0< 2s \leqq lr.$$ In terms ofx andy,H can be presented as $$G = \left\langle {x,{\text{ }}y;{\text{ }}x^{ks} = y^{lr} ,\left[ {x,{\text{ }}y^l } \right] = \left[ {x^k ,{\text{ }}y} \right] = 1} \right\rangle$$ and Zieschang has shown that ifr=1 ors=1, thenH can be defined by a single relation inx andy. We establish the exact converse of Zieschang's result, namely thatH is not defined by a single relation inx andy unlessr=1 ors=1. The proof is based on an observation of Magnus which associates polynomials with relators and some elementary facts about cyclotomic polynomials.     

Lower semicontinuous perturbations of maximal monotone differential inclusions

We prove existence of solutions to 1 $$\dot x \in - Ax + F\left( {t,x} \right),x\left( a \right) = x^0 ,$$ whereA is a maximal monotone operator inR ⁿ andF is a multifunction measurable in (t, x) and l.s.c. inx, satisfying a sublinear growth condition.     

Existence of positive solutions to nonlinear elliptic equations involving critical Sobolev exponents

In this paper we extend the results of Brezis and Nirenberg in [4] to the problem $$\left\{ \begin{gathered} Lu = - D_i (a_{ij} (x)D_j u) = b(x)u^p + f(x,u) in\Omega , \hfill \\ p = (n + 2)/(n - 2) \hfill \\ u > 0 in\Omega , u = 0 \partial \Omega , \hfill \\ \end{gathered} \right.$$ whereL is a uniformly elliptic operator,b(x)≥0,f(x,u) is a lower order perturbation ofu ^p at infinity. The existence of solutions to (A) is strongly dependent on the behaviour ofa _ij (x), b(x) andf(x, u). For example, for any bounded smooth domain Ω, we have \(a_{ij} \left( x \right) \in C\left( {\bar \Omega } \right)\) such thatLu=u ^p possesses a positive solution inH ₀ ¹ (Ω). We also prove the existence of nonradial solutions to the problem ?Δu=f(|x|, u) in Ω,u>0 in Ωu=0 on ?Ω, Ω=B(0,1). for a class off(r, u).     

О РАцИОНАльНых сОстА ВльУЩИх МЕРОМОРФНых ФУНкцИИ И Их пРОИжВОД Ных

LetG be an arbitrary domain in \(\bar C\) ,f a function meromorphic inG, $$M_f \mathop = \limits^{def} \mathop {\lim \sup }\limits_{G \mathrel\backepsilon z \to \partial G} \left| {f(z)} \right|< \infty ,$$ andR the sum of the principal parts in the Laurent expansions off with respect to all its poles inG. We set $$f_G (z) = R(z) - \alpha ,{\mathbf{ }}where{\mathbf{ }}\alpha = \mathop {\lim }\limits_{z \to \infty } (f(z) - R(z))$$ in case ∞?G, andα=0 in case ∞?G. It is proved that $$\left\| {f_G } \right\|_{C(\partial G)} \leqq 50(\deg f_G )M_f ,{\mathbf{ }}\left\| {f'_G } \right\|_{L_1 (\partial G)} \leqq 50(\deg f_G )V(\partial G)M_f ,$$ where $$V(\partial G) = \sup \left\{ {\left\| {r'} \right\|_{L_1 (\partial G)} :r(z) = a/(z - b),{\mathbf{ }}\left\| r \right\|_{G(\partial G)} \leqq 1} \right\}.$$      

Quasi-implication algebras,Part I: Elementary theory

Quasi-implication algebras (QIA's) are intended to generalize orthomodular lattices (OML's) in the same way that implication algebras (J. C. Abbott) generalize Boolean lattices. A QIA is defined to be a setQ together with a binary operation → satisfying the following conditions (a→b is denotedab). (Q1) $$\left( {ab} \right)a = a$$ (Q2) $$\left( {ab} \right)\left( {ac} \right) = \left( {ba} \right)\left( {bc} \right)$$ (Q3) $$\left( {\left( {ab} \right)\left( {ba} \right)} \right)a = \left( {\left( {ba} \right)\left( {ab} \right)} \right)b$$ Every OML induces a QIA, wherea → b=a ^⊥?(a?b). On the other hand, every QIA induces a join semi-lattice with a greatest element 1, where 1=aa,a≤b iffab=1, anda?b=((ab)(ba))a. A bounded QIA is defined to be a QIA with a least element 0 (w.r.t.≤). The QIA associated with any OML is bounded, the zero elements being the same. Conversely, every bounded QIA induces an OML, wherea ^⊥=a0, anda?b=((ab)(a0))0. The relationC of compatibility is defined so thataCb iffa≤ba, and it is shown that every compatible sub-QIA of a QIA is an implication algebra.     

Regularity results for a class of obstacle problems in Heisenberg groups

We study regularity results for solutions u ∈ HW ^1,p (Ω) to the obstacle problem $$\int_\Omega \mathcal{A} \left( {x,\nabla _{\mathbb{H}^u } } \right)\nabla _\mathbb{H} \left( {v - u} \right)dx \geqslant 0 \forall v \in \mathcal{K}_{\psi ,u} \left( \Omega \right)$$ such that u ? ψ a.e. in Ω, where $xxx$ , in Heisenberg groups ? ⁿ . In particular, we obtain weak differentiability in the T-direction and horizontal estimates of Calderon-Zygmund type, i.e. $$\begin{gathered} T\psi \in HW_{loc}^{1,p} \left( \Omega \right) \Rightarrow Tu \in L_{loc}^p \left( \Omega \right), \hfill \\ \left| {\nabla _{\mathbb{H}\psi } } \right|^p \in L_{loc}^q \left( \Omega \right) \Rightarrow \left| {\nabla _{\mathbb{H}^u } } \right|^p \in L_{loc}^q \left( \Omega \right), \hfill \\ \end{gathered}$$ where 2 < p < 4, q > 1.     

Asymptotics for the minimization of a Ginzburg-Landau functional

LetΩ ? ?² be a smooth bounded simply connected domain. Consider the functional $$E_\varepsilon (u) = \frac{1}{2}\int\limits_\Omega {\left| {\nabla u} \right|^2 + \frac{1}{{4\varepsilon ^2 }}} \int\limits_\Omega {(|u|^2 - 1)^2 } $$ on the classH _g ¹ ={u εH ¹(Ω; ?);u=g on ?Ω} whereg:?Ω? → ? is a prescribed smooth map with ¦g¦=1 on ?Ω? and deg(g, ?Ω)=0. Let uu _ε be a minimizer for E_ε onH _g ¹ . We prove that u_ε → u₀ in \(C^{1,\alpha } (\bar \Omega )\) as ε → 0, where u₀ is identified. Moreover \(\left\| {u_\varepsilon - u_0 } \right\|_{L^\infty } \leqslant C\varepsilon ^2 \) .     

Some new estimates of the Fourier-Bessel transform in the space \mathbb{L}_2 \left( {\mathbb{R}_ + } \right)

The Fourier-Bessel integral transform $$g\left( x \right) = F\left[ f \right]\left( x \right) = \frac{1} {{2^p \Gamma \left( {p + 1} \right)}}\int\limits_0^{ + \infty } {t^{2p + 1} f\left( x \right)j_p \left( {xt} \right)dt}$$ is considered in the space $\mathbb{L}_2 \left( {\mathbb{R}_ + } \right)$ . Here, j _p (u) = ((2 ^p Γ(p+1))/(u ^p ))J _p (u) and J _p (u) is a Bessel function of the first kind. New estimates are proved for the integral $$\delta _N^2 \left( f \right) = \int\limits_N^{ + \infty } {x^{2p + 1} g^2 \left( x \right)dx, N > 0,}$$ in $\mathbb{L}_2 \left( {\mathbb{R}_ + } \right)$ for some classes of functions characterized by a generalized modulus of continuity.     

Gysin maps and cycle classes for Hodge cohomology

Iff:X→Y is a projective morphism between regular varieties over a field, we construct Gysin maps $$f_ * :H^i \left( {X,\Omega _{X/Z}^j } \right) \to H_{f(x)}^{i + d} \left( {X,\Omega _{Y/Z}^j } \right)$$ for the Hodge cohomology groups, whered-dimY-dimX. These Gysin maps have the expected properties, and in particular may be used to construct a cycle class map $$Cl_X :CH^i \left( {X,S} \right) \to H^i \left( {X,\Omega _{X/Z}^i } \right)$$ whereX is quasi-projective over a field,S is the singular locus, andCH ⁱ(X, S) is the relative Chow group of codimension-i cycles modulo rational equivalence. Simple properties of this cycle map easily imply the infinite dimensionality theorem for the Chow group of zero cycles of a normal projective varietyX overC with \(H^n \left( {X,\mathcal{O}_X } \right) \ne 0\) , wheren=dimX. One also recovers examples of Nori of affinen-dimensional varieties which support indecomposable vector bundles of rankn.     

Оценки отклонений оп тимальных сеток вL p -норме и теория квад ратурных формул

Let E^s=[0, 1]^s be then-dimensional unit cube, 1<p<∞, anda=(a ₁, ...,a _s ) some set of natural numbers. Denote byL _p ^(a) , (E ^s ) the class of functionsf: E ^s → C for which $$\left\| {\frac{{\partial ^{b_1 + \cdots + b_s } f}}{{\partial x_1^{b_1 } \cdots \partial x_s^{b_s } }}} \right\|_p \leqslant 1,$$ where $$0< b_1< a_1 , ..., 0< b_s< a_s .$$ Set $$R_p^{\left( a \right)} \left( N \right) = \mathop {\inf }\limits_{card \mathfrak{S} = N} R_\mathfrak{S} \left( {L_p^{\left( a \right)} \left( {E^s } \right)} \right),$$ where $R_\mathfrak{S} \left( {L_p^{\left( a \right)} \left( {E^s } \right)} \right)$ is the error of the quadrature formulas on the mesh $\mathfrak{S}$ (for the classL _p ^(a) (E ^s )), consisting of N nodes and weights, and the infimum is taken with respect to all possibleN nodes and weights. In this paper, the two-sided estimate $$\frac{{\left( {\log N} \right)^{{{\left( {l - 1} \right)} \mathord{\left/ {\vphantom {{\left( {l - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} }}{{N^d }} \ll _{p, a} R^{\left( a \right)} \left( N \right) \ll _{p, a} \frac{{\left( {\log N} \right)^{{{\left( {l - 1} \right)} \mathord{\left/ {\vphantom {{\left( {l - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} }}{{N^d }}$$ is proved for every natural numberN > 1, whered=min{a ₁, ...,a _s }, whilel is the number of those components of a which coincide withd. An analogous result is proved for theL _p -norm of the deviation of meshes.     

L p inequalities for polynomials with restricted zeros

LetP(z) be a polynomial of degreen which does not vanish in the disk |z|<k. It has been proved that for eachp>0 andk≥1, $$\begin{gathered} \left\{ {\frac{1}{{2\pi }}\int_0^{2\pi } {\left| {P^{(s)} (e^{i\theta } )} \right|^p d\theta } } \right\}^{1/p} \leqslant n(n - 1) \cdots (n - s + 1) B_p \hfill \\ \times \left\{ {\frac{1}{{2\pi }}\int_0^{2\pi } {\left| {P(e^{i\theta } )} \right|^p d\theta } } \right\}^{1/p} , \hfill \\ \end{gathered} $$ where $B_p = \left\{ {\frac{1}{{2\pi }}\int_0^{2\pi } {\left| {k^s + e^{i\alpha } } \right|^p d\alpha } } \right\}^{ - 1/p} $ andP ^(s)(z) is thesth derivative ofP(z). This result generalizes well-known inequality due to De Bruijn. Asp→∞, it gives an inequality due to Govil and Rahman which as a special case gives a result conjectured by Erdös and first proved by Lax.     

Behavior of solutions of quasilinear elliptic inequalities in an unbounded domain

We consider the solutions of the inequalityLu≤?(¦gradu¦), whereL is a uniformly elliptic homogeneous operator and ? is a function increasing faster than any linear function but not faster thanξ lnξ, in the unbounded domain $$\left\{ {x \in \mathbb{R}^n |\sum\limits_{i = 2}^n {x_i^2< (\psi (x_1 ))^2 ,} {\text{ }} - \infty< x_1< \infty } \right\},$$ , whereψ is a bounded function with bounded derivative. We estimate the growth of the solutions in terms of $\int_0^{x_1 } {(1/\psi (r))dr}$ . For the special case in which?(ξ)=aξ lnξ+C, the solutionsu(x ₁,x ₂,...,x _n ) grow as $\left( {\int_0^{x_1 } {(1/\psi (r))dr} } \right)^N$ , whereN is any given number anda=a(N).     

On the numerical integration of nonlinear initial value problems by linear multistep methods

We study the numerical solution of the nonlinear initial value problem $$\left\{ {\begin{array}{*{20}c} {{{du(t)} \mathord{\left/ {\vphantom {{du(t)} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} + Au(t) = f(t),t > 0} \\ {u(0) = c,} \\ \end{array} } \right.$$ whereA is a nonlinear operator in a real Hilbert space. The problem is discretized using linear multistep methods, and we assume that their stability regions have nonempty interiors. We give sharp bounds for the global error by relating the stability region of the method to the monotonicity properties ofA. In particular we study the case whereAu is the gradient of a convex functional φ(u).     

Error bounds for the finite element approximation of a degenerate quasilinear parabolic variational inequality

In this paper, we establish some error bounds for the continuous piecewise linear finite element approximation of the following problem: Let Ω be an open set in ? ^d , withd=1 or 2. GivenT>0,p ∈ (1, ∞),f andu ₀; findu ∈K, whereK is a closed convex subset of the Sobolev spaceW ₀ ^1,p (Ω), such that for anyv∈K $$\begin{gathered} \int\limits_\Omega {u_1 (\upsilon - u) dx + } \int\limits_\Omega {\left| {\nabla u} \right|^{p - 2} } \nabla u \cdot \nabla (\upsilon - u) dx \geqslant \int\limits_\Omega {f(\upsilon - u) dx for} a.e. t \in (0,T], \hfill \\ u = 0 on \partial \Omega \times (0,T] and u(0,x) = u_0 (x) for x \in \Omega . \hfill \\ \end{gathered} $$ We prove error bounds in energy type norms for the fully discrete approximation using the backward Euler time discretisation. In some notable cases, these error bounds converge at the optimal rate with respect to the space discretisation, provided the solutionu is sufficiently regular.     

Discontinuous superlinear elliptic equations of divergence form

We prove existence and global Hölder regularity of the weak solution to the Dirichlet problem $\left\{ {\begin{array}{lc} {{\rm div} \left( a^{ij} (x,u)D_{j} u \right) = b(x,u,Du) \quad {\rm in}\, \Omega \subset {\mathbb R}^{n}, \, n \ge 2,} \\ {u = 0 \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,{\rm on}\, \partial\Omega \in C^{1}. } \\ \end{array}} \right.$ The coefficients a ^ij (x, u) are supposed to be VMO functions with respect to x while the term b(x, u, Du) allows controlled growth with respect to the gradient Du and satisfies a sort of sign-condition with respect to u. Our results correct and generalize the announcements in Ragusa (Nonlinear Differ Equ Appl 13:605–617, 2007, Erratum in Nonlinear Differ Equ Appl 15:277–277, 2008).     

H convergence for quasi-linear elliptic equations with quadratic growth

We consider in this paper the limit behavior of the solutionsu ^? of the problem $$\begin{gathered} - div(a^\varepsilon Du^\varepsilon ) + \gamma u^\varepsilon = H^\varepsilon (x, u^\varepsilon , Du^\varepsilon ), \hfill \\ u^\varepsilon \in H_0^1 (\Omega ) \cap L^\infty (\Omega ), \hfill \\ \end{gathered}$$ whereH ^? has quadratic growth inDu ^? anda ^? (x) is a family of matrices satisfying the general assumptions of abstract homogenization. We also consider the problem $$\begin{gathered} - div(a^\varepsilon Du^\varepsilon ) + G^\varepsilon (x, u^\varepsilon , Du^\varepsilon ) = f \in H^{ - 1} (\Omega ), \hfill \\ u^\varepsilon \in H_0^1 (\Omega ), G^\varepsilon (x, u^\varepsilon , Du^\varepsilon ) \in L^1 (\Omega ), u^\varepsilon G^\varepsilon (x, u^\varepsilon , Du^\varepsilon ) \in L^1 (\Omega ) \hfill \\ \end{gathered}$$ whereG ^? has quadratic growth inDu ^? and satisfiesG ^? (x, s, ξ)s ≥ 0. Note that in this last modelu ^? is in general unbounded, which gives extra difficulties for the homogenization process. In both cases we pass to the limit and obtain an homogenized equation having the same structure.     

On spectra ofp-hyponormal operators

The purpose of this paper is to show the following: Let 0<p<1/2. IfT=U|T| is a p-hyponormal operator with a unitaryU on a Hilbert space, then $$\sigma (T) = \mathop \cup \limits_{0 \leqslant k \leqslant 1} \sigma (T_{\left[ k \right]} ),$$ where $$T_{\left[ k \right]} = U[(1 - k)S_U^ - (\left| T \right|^{2p} ) + kS_U^ + (\left| T \right|^{2p} ]^{\tfrac{1}{{2p}}} $$ andS _U ^± (T) denote the polar symbols ofT.