Homogenization of an Elliptic System as the Cells of Periodicity are Refined in One Direction

We homogenize a second-order elliptic system with anisotropic fractal structure characteristic of many real objects: the cells of periodicity are refined in one direction. This problem is considered in the rectangle with Dirichlet conditions given on two sides and periodicity conditions on two other sides. An explicit formula for the homogenized operator is established, and an asymptotic estimate of the remainder is obtained. The accuracy of approximation depends on the exponent $\kappa$ ∈ (0, 1/2] of smoothness of the right-hand side with respect to slow variables (the Sobolev-Slobodetskii space) and is estimated by $O(h^\kappa )$ for $\kappa$ ∈ (0, 1/2) and by O(h ^1/2(1 + |log h|)) for $\kappa$ = 1/2.     

Asymptotic analysis of positive solutions of generalized Emden-Fowler differential equations in the framework of regular variation

Positive solutions of the nonlinear second-order differential equation $(p(t)|x'|^{\alpha - 1} x')' + q(t)|x|^{\beta - 1} x = 0,\alpha > \beta > 0,$ are studied under the assumption that p, q are generalized regularly varying functions. An application of the theory of regular variation gives the possibility of obtaining necessary and sufficient conditions for existence of three possible types of intermediate solutions, together with the precise information about asymptotic behavior at infinity of all solutions belonging to each type of solution classes.     

On C 1+α regularity of solutions of Isaacs parabolic equations with VMO coefficients

We prove that boundary value problems for fully nonlinear second-order parabolic equations admit L _p -viscosity solutions, which are in C ^1+α for an ${\alpha \in (0, 1)}$ . The equations have a special structure that the “main” part containing only second-order derivatives is given by a positive homogeneous function of second-order derivatives and as a function of independent variables it is measurable in the time variable and, so to speak, VMO in spatial variables.     

Distance between toroidal surgeries on hyperbolic knots in the -sphere

For a hyperbolic knot in the -sphere, at most finitely many Dehn surgeries yield non-hyperbolic -manifolds. As a typical case of such an exceptional surgery, a toroidal surgery is one that yields a closed -manifold containing an incompressible torus. The slope corresponding to a toroidal surgery, called a toroidal slope, is known to be integral or half-integral. We show that the distance between two integral toroidal slopes for a hyperbolic knot, except the figure-eight knot, is at most four.

     

Special values of symmetric hypergeometric functions

We discuss the -adic formula (0.3) of P. Th. Young, in the framework of Dwork's theory of the hypergeometric equation. We show that it gives the value at 0 of the Frobenius automorphism of the unit root subcrystal of the hypergeometric crystal. The unit disk at 0 is in fact singular for the differential equation under consideration, so that it's not a priori clear that the Frobenius structure should extend to that disk. But the singularity is logarithmic, and it extends to a divisor with normal crossings relative to in . We show that whenever the unit root subcrystal of the hypergeometric system has generically rank 1, it actually extends as a logarithmic -subcrystal to the unit disk at 0. So, in these optics, ``singular classes are not supersingular'. If, in particular, the holomorphic solution at 0 is bounded, the extended logarithmic -crystal has no singularity in the residue class of 0, so that it is an -crystal in the usual sense and the Frobenius operation is holomorphic. We examine in detail its analytic form.

     

Computing the tight closure in dimension two

We study computational aspects of the tight closure of a homogeneous primary ideal in a two-dimensional normal standard-graded domain. We show how to use slope criteria for the sheaf of relations for generators of the ideal to compute its tight closure. In particular, our method gives an algorithm to compute the tight closure of three elements under the condition that we are able to compute the Harder-Narasimhan filtration. We apply this to the computation of in , where is a homogeneous polynomial.

     

Static Electron–Positron Pair Creation in Strong Fields for a Non-linear Dirac Model

We consider the Hartree–Fock approximation of Quantum Electrodynamics, with the exchange term neglected. We prove that the probability of static electron–positron pair creation for the Dirac vacuum polarized by an external field of strength Z behaves as ${1-\exp(-\kappa Z^{2/3})}$ for Z large enough. Our method involves two steps. First, we estimate the vacuum expectation of general quasi-free states in terms of their total number of particles, which can be of general interest. Then, we study the asymptotics of the Hartree–Fock energy when ${Z \to+\infty}$ which gives the expected bounds.     

Uniqueness of diffusion operators and capacity estimates

Let Ω be a connected open subset of R ^d . We analyse L ₁-uniqueness of real second-order partial differential operators ${H = - \sum^d_{k,l=1} \partial_k c_{kl} \partial_l}$ and ${K = H + \sum^d_{k=1}c_k \partial_k + c_0}$ on Ω where ${c_{kl} = c_{lk} \in W^{1,\infty}_{\rm loc}(\Omega), c_k \in L_{\infty,{\rm loc}}(\Omega), c_0 \in L_{2,{\rm loc}}(\Omega)}$ and C(x) = (c _kl (x)) > 0 for all ${x \in \Omega}$ . Boundedness properties of the coefficients are expressed indirectly in terms of the balls B(r) associated with the Riemannian metric C ^?1 and their Lebesgue measure |B(r)|. First, we establish that if the balls B(r) are bounded, the Täcklind condition ${\int^\infty_R dr r({\rm log}|B(r)|)^{-1} = \infty}$ is satisfied for all large R and H is Markov unique then H is L ₁-unique. If, in addition, ${C(x) \geq \kappa (c^{T} \otimes c)(x)}$ for some ${\kappa > 0}$ and almost all ${x \in \Omega}$ , ${{\rm div} c \in L_{\infty,{\rm loc}}(\Omega)}$ is upper semi-bounded and c ₀ is lower semi-bounded, then K is also L ₁-unique. Secondly, if the c _kl extend continuously to functions which are locally bounded on ?Ω and if the balls B(r) are bounded, we characterize Markov uniqueness of H in terms of local capacity estimates and boundary capacity estimates. For example, H is Markov unique if and only if for each bounded subset A of ${\overline\Omega}$ there exist ${\eta_n \in C_c^\infty(\Omega)}$ satisfying , where ${\Gamma(\eta_n) = \sum^d_{k,l=1}c_{kl} (\partial_k \eta_n) (\partial_l \eta_n)}$ , and for each ${\varphi \in L_2(\Omega)}$ or if and only if cap(?Ω) = 0.     

Slopes of vector bundles on projective curves and applications to tight closure problems

We study different notions of slope of a vector bundle over a smooth projective curve with respect to ampleness and affineness in order to apply this to tight closure problems. This method gives new degree estimates from above and from below for the tight closure of a homogeneous -primary ideal in a two-dimensional normal standard-graded algebra in terms of the minimal and the maximal slope of the sheaf of relations for some ideal generators. If moreover this sheaf of relations is semistable, then both degree estimates coincide and we get a vanishing type theorem.

     

Estimates for the Poisson kernel and the evolution kernel on the Heisenberg group

We obtain an upper estimate for the Poisson kernel for the class of second-order left invariant differential operators on the semi-direct product of the 2n?+?1-dimensional Heisenberg group ${\mathcal H_n}$ and an Abelian group ${A = \mathbb {R}^k.}$ We also give an upper estimate for the transition probabilities of the evolution on ${\mathcal H_{n}}$ driven by the Brownian motion (with drift) in ${\mathbb {R}^k.}$      

On the gonality sequence of an algebraic curve

For any smooth irreducible projective curve X, the gonality sequence ${\{d_r \;| \; r \in \mathbb N\}}$ is a strictly increasing sequence of positive integer invariants of X. In most known cases d _r+1 is not much bigger than d _r . In our terminology this means the numbers d _r satisfy the slope inequality. It is the aim of this paper to study cases when this is not true. We give examples for this of extremal curves in ${{\mathbb P}^r}$ , for curves on a general K3-surface in ${{\mathbb P}^r}$ and for complete intersections in ${{\mathbb P}^3}$ .     

The rank and generating set for inverse limits of wreath products of Abelian groups

We derive an exact formula for the topological rank d(W) of the inverse limit ${W = \ldots \wr A_2 \wr A_1}$ of iterated wreath products of arbitrary nontrivial finite Abelian groups. By using the language of automorphisms of a spherically homogeneous rooted tree, we construct and study a topological generating set for W with cardinality ${d(A_1) + \rho'}$ , where ${\rho'}$ is the topological rank of the profinite Abelian group ${A_2 \times A_3 \times \cdots}$ . In particular, if the group A ₁ is cyclic, this approach gives a minimal generating set for W.     

The homological degree of a module

A homological degree of a graded module is an extension of the usual notion of multiplicity tailored to provide a numerical signature for the module even when is not Cohen-Macaulay. We construct a degree, , that behaves well under hyperplane sections and the modding out of elements of finite support. When carried out in a local algebra this degree gives a simulacrum of complexity à la Castelnuovo-Mumford's regularity. Several applications for estimating reduction numbers of ideals and predictions on the outcome of Noether normalizations are given.

     

Turán Extremal Problem for Periodic Functions with Small Support and Its Applications

We study a Turán extremal problem on the largest mean value of a 1-periodic even function with nonnegative Fourier coefficients, fixed value at zero, and support on a closed interval $[ - h,h],0 < h \leqslant 1/2$ . We show how the solution of this extremal problem for rational numbers h=p/q is related to the solution of two finite-dimensional problems of linear programming. The solution of the Turán problem for rational numbers h of the form 2/q, 3/q, 4/q, $p/(2p + 1)$ is obtained. Applications of the Turán problem to analytic number theory are given.     

Nicely generated and chaotic ideals

We show that if the real line is the disjoint union of meager sets such that every meager set is contained in a countable union of them, then . This answers a question addressed by Jacek Cicho\'{n}. We also prove two theorems saying roughly that any attempt to produce the isomorphism type of the meager ideal in the Cohen real and the random real extensions must fail. All our results hold for meager replaced by null, as well.

     

Characterization of Tangent Cones of Noncollapsed Limits with Lower Ricci Bounds and Applications

Consider a limit space ${(M_\alpha,g_\alpha,p_\alpha)\stackrel{GH}{\rightarrow} (Y,d_Y,p)}$ , where the ${M_\alpha^n}$ have a lower Ricci curvature bound and are volume noncollapsed. The tangent cones of Y at a point ${p\in Y}$ are known to be metric cones C(X), however they need not be unique. Let ${\overline\Omega_{Y,p}\subseteq\mathcal{M}_{GH}}$ be the closed subset of compact metric spaces X which arise as cross sections for the tangents cones of Y at p. In this paper we study the properties of ${\overline\Omega_{Y,p}}$ . In particular, we give necessary and sufficient conditions for an open smooth family ${\Omega\equiv (X,g_s)}$ of closed manifolds to satisfy ${\overline\Omega =\overline\Omega_{Y,p}}$ for some limit Y and point ${p\in Y}$ as above, where ${\overline\Omega}$ is the closure of Ω in the set of metric spaces equipped with the Gromov–Hausdorff topology. We use this characterization to construct examples which exhibit fundamentally new behaviors. The first application is to construct limit spaces (Y ⁿ , d _Y , p) with n ≥ 3 such that at p there exists for every 0 ≤ k ≤ n?2 a tangent cone at p of the form , where X ^n-k-1 is a smooth manifold not isometric to the standard sphere. In particular, this is the first example which shows that a stratification of a limit space Y based on the Euclidean behavior of tangent cones is not possible or even well defined. It is also the first example of a three dimensional limit space with nonunique tangent cones. The second application is to construct a limit space (Y ⁵ , d _Y , p), such that at p the tangent cones are not only not unique, but not homeomorphic. Specifically, some tangent cones are homeomorphic to cones over while others are homeomorphic to cones over .     

Preconditioning in H and applications

We consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator I. The natural setting for such problems is in the Hilbert space H and the variational formulation is based on the inner product in H. We show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques. These preconditioners are shown to be spectrally equivalent to the inverse of the operator. As a consequence, they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations, with the number independent of the mesh discretization. We describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems.

     

Characterization and the pre-Schwarzian norm estimate for concave univalent functions

Let Co(α) denote the class of concave univalent functions in the unit disk ${\mathbb{D}}$ . Each function ${f\in Co(\alpha)}$ maps the unit disk ${\mathbb{D}}$ onto the complement of an unbounded convex set. In this paper we find the exact disk of variability for the functional ${(1-|z|^2)\left ( f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)}$ . In particular, this gives sharp upper and lower estimates for the pre-Schwarzian norm of concave univalent functions. Next we obtain the set of variability of the functional ${(1-|z|^2)\left(f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)}$ whenever f′′(0) is fixed. We also give a characterization for concave functions in terms of Hadamard convolution. In addition to sharp coefficient inequalities, we prove that functions in Co(α) belong to the H ^p space for p < 1/α.     

Derived Equivalences of Actions of a Category

Let $\mathbb{k}$ be a commutative ring and I a category. As a generalization of a $\mathbb{k}$ -category with a (pseudo) action of a group we consider a family of $\mathbb{k}$ -categories with a (pseudo, lax, or oplax) action of I, namely an oplax functor from I to the 2-category of small $\mathbb{k}$ -categories. We investigate derived equivalences of those oplax functors, and establish a Morita type theorem for them. This gives a base of investigations of derived equivalences of Grothendieck constructions of those oplax functors.