Hermitian decomposition of continuous functions on a fractal surface

In this paper the Théodoresco transform is used to show that, under additional assumptions, each Hölder continuous function f defined on the boundary Γ of a fractal domain Ω ? ?²ⁿ can be expressed as f = Ψ⁺ ? Ψ^?, where Ψ^± are Hölder continuous functions on Γ and Hermitian monogenically extendable to Ω and to ?²ⁿ ? (Ω ∪ Γ) respectively.     

Tensor products and relation quantales

A classical tensor product \({A \otimes B}\) of complete lattices A and B, consisting of all down-sets in \({A \times B}\) that are join-closed in either coordinate, is isomorphic to the complete lattice Gal(A,B) of Galois maps from A to B, turning arbitrary joins into meets. We introduce more general kinds of tensor products for closure spaces and for posets. They have the expected universal property for bimorphisms (separately continuous maps or maps preserving restricted joins in the two components) into complete lattices. The appropriate ingredient for quantale constructions is here distributivity at the bottom, a generalization of pseudocomplementedness. We show that the truncated tensor product of a complete lattice B with itself becomes a quantale with the closure of the relation product as multiplication iff B is pseudocomplemented, and that the tensor product has a unit element iff B is atomistic. The pseudocomplemented complete lattices form a semicategory in which the hom-set between two objects is their tensor product. The largest subcategory of that semicategory has as objects the atomic boolean complete lattices, which is equivalent to the category of sets and relations. More general results are obtained for closure spaces and posets.     

Boundary criterion for integral operators

Integral operators of the form \(L_K^{ - 1} f(x) = \int\limits_\Omega {K(x,t)f(t)dt}\) for the case of a finite domain Ω ? Rⁿ with smooth boundary ?Ω are considered. Conditions on the real kernel K(x, t) of an integral operator under which this operator satisfies a well-defined boundary condition for the corresponding differential equation are found. The application of the results is demonstrated on the example of a Sturm–Liouville equation, for which the derivation of the general form of well-posed boundary value problems is presented.     

On the Monge–Ampère equation with boundary blow-up: existence,uniqueness and asymptotics

We consider the Monge–Ampère equation det D ² u = b(x)f(u) > 0 in Ω, subject to the singular boundary condition u = ∞ on ?Ω. We assume that \(b\in C^\infty(\overline{\Omega})\) is positive in Ω and non-negative on ?Ω. Under suitable conditions on f, we establish the existence of positive strictly convex solutions if Ω is a smooth strictly convex, bounded domain in \({\mathbb R}^N\) with N ≥ 2. We give asymptotic estimates of the behaviour of such solutions near ?Ω and a uniqueness result when the variation of f at ∞ is regular of index q greater than N (that is, \(\lim_{u\to \infty} f(\lambda u)/f(u)=\lambda^q\) , for every λ > 0). Using regular variation theory, we treat both cases: b > 0 on ?Ω and \(b\equiv 0\) on ?Ω.     

A spectral method for nonlinear elliptic equations

Let Ω be an open, simply connected, and bounded region in \(\mathbb {R}^{d}\), d ≥ 2, and assume its boundary ?Ω is smooth and homeomorphic to \(\mathbb {S}^{d-1}\). Consider solving an elliptic partial differential equation L u = f(?, u) over Ω with zero Dirichlet boundary value. The function f is a nonlinear function of the solution u. The problem is converted to an equivalent elliptic problem over the open unit ball \(\mathbb {B}^{d}\) in \(\mathbb {R}^{d}\), say \(\widetilde {L}\widetilde {u} =\widetilde {f}(\cdot ,\widetilde {u})\). Then a spectral Galerkin method is used to create a convergent sequence of multivariate polynomials \(\widetilde {u} _{n}\) of degree ≤ n that is convergent to \(\widetilde {u}\). The transformation from Ω to \(\mathbb {B}^{d}\) requires a special analytical calculation for its implementation. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For \(u\in C^{\infty } \left (\overline {\Omega }\right ) \) and assuming ?Ω is a C ^∞ boundary, the convergence of \(\left \Vert \widetilde {u} -\widetilde {u}_{n}\right \Vert _{H^{1}}\) to zero is faster than any power of 1/n. The error analysis uses a reformulation of the boundary value problem as an integral equation, and then it uses tools from nonlinear integral equations to analyze the numerical method. Numerical examples illustrate experimentally an exponential rate of convergence. A generalization to ?Δu + γ u = f(u) with a zero Neumann boundary condition is also presented.     

A spectral method for elliptic equations: the Dirichlet problem

Let Ω be an open, simply connected, and bounded region in ? ^d , d?≥?2, and assume its boundary \(\partial\Omega\) is smooth. Consider solving an elliptic partial differential equation Lu?=?f over Ω with zero Dirichlet boundary values. The problem is converted to an equivalent elliptic problem over the unit ball B; and then a spectral Galerkin method is used to create a convergent sequence of multivariate polynomials u _n of degree ≤?n that is convergent to u. The transformation from Ω to B requires a special analytical calculation for its implementation. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For \(u\in C^{\infty}( \overline{\Omega})\) and assuming \(\partial\Omega\) is a C ^?∞? boundary, the convergence of \(\left\Vert u-u_{n}\right\Vert _{H^{1}}\) to zero is faster than any power of 1/n. Numerical examples in ?² and ?³ show experimentally an exponential rate of convergence.     

Sturm–Liouville problems in weighted spaces in domains with non-smooth edges. II

We consider a (generally, noncoercive) mixed boundary value problem in a bounded domain D of Rⁿ for a second order elliptic differential operator A(x, ?). The differential operator is assumed to be of divergent form in D and the boundary operator B(x, ?) is of Robin type on ?D. The boundary of D is assumed to be a Lipschitz surface. Besides, we distinguish a closed subset Y ? ?D and control the growth of solutions near Y. We prove that the pair (A, B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, the weight function being a power of the distance to the singular set Y. Moreover, we prove the completeness of root functions related to L.     

Symmetrization of Cuntz’ Picture for the Kasparov <Emphasis Type="Italic">KK</Emphasis>-Bifunctor

Given C^*-algebras A and B, we generalize the notion of a quasi-homomorphism from A to B in the sense of Cuntz by considering quasi-homomorphisms from some C^*-algebra C to B such that C surjects onto A and the two maps forming the quasi-homomorphism agree on the kernel of this surjection. Under an additional assumption, the group of homotopy classes of such generalized quasi-homomorphisms coincides with KK(A, B). This makes the definition of the Kasparov bifunctor slightly more symmetric and provides more flexibility in constructing elements of KK-groups. These generalized quasi-homomorphisms can be viewed as pairs of maps directly from A (instead of various C’s), but these maps need not be *-homomorphisms.     

Catenoidal Layers for the Allen-Cahn Equation in Bounded Domains

This paper presents a new family of solutions to the singularly perturbed Allen-Cahn equation α~2Δu + u(1- u~2) = 0 in a smooth bounded domain Ω R~3, with Neumann boundary condition and α 0 a small parameter. These solutions have the property that as α→ 0, their level sets collapse onto a bounded portion of a complete embedded minimal surface with finite total curvature intersecting ?Ω orthogonally and that is non-degenerate respect to ?Ω. The authors provide explicit examples of surfaces to which the result applies.     

Squares and their centers

We study the relationship between the size of two sets B, S ? R², when B contains either the whole boundary or the four vertices of a square with axes-parallel sides and center in every point of S. Size refers to cardinality, Hausdorff dimension, packing dimension, or upper or lower box dimension. Perhaps surprisingly, the results vary depending on the notion of size under consideration. For example, we construct a compact set B of Hausdorff dimension 1 which contains the boundary of an axes-parallel square with center in every point in [0, 1]², prove that such a B must have packing and lower box dimension at least 7/4, and show by example that this is sharp. For more general sets of centers, the answers for packing and box counting dimensions also differ. These problems are inspired by the analogous problems for circles that were investigated by Bourgain, Marstrand and Wolff, among others.     

Tensor products of tilting modules

We consider whether the tilting properties of a tilting A-module T and a tilting B-module T′ can convey to their tensor product T ? T′: The main result is that T ? T′ turns out to be an (n + m)-tilting A ? B-module, where T is an m-tilting A-module and T′ is an n-tilting B-module.     

Depth of Boolean functions over an arbitrary infinite basis

Realization of Boolean functions by circuits is considered over an arbitrary infinite complete basis. The depth of a circuit is defined as the greatest number of functional elements constituting a directed path from an input of the circuit to its output. The Shannon function of the depth is defined for a positive integer n as the minimal depth D _B (n) of the circuits sufficient to realize every Boolean function on n variables over a basis B. It is shown that, for each infinite basis B, either there exists a constant β ? 1 such that D _B (n) = β for all sufficiently large n or there exist an integer constant γ ? 2 and a constant δ such that log _γ n ? D _B (n) ? log _γ n + δ for all n.     

Polar functions of multiparameter bifractional Brownian sheets

Let B^H,K ： （B^H,K（t）, t ∈R＋^N} be an （N,d）-bifractional Brownian sheet with Hurst indices H = （H1,..., HN） ∈ （0, 1）^N and K = （K1,..., KN）∈ （0, 1]^N. The characteristics of the polar functions for B^H,K are investigated. The relationship between the class of continuous functions satisfying the Lipschitz condition and the class of polar-functions of B^H,K is presented. The Hausdorff dimension of the fixed points and an inequality concerning the Kolmogorov＇s entropy index for B^H,K are obtained. A question proposed by LeGall about the existence of no-polar, continuous functions statisfying the Holder condition is also solved.     

Chowla’s cosine problem

A continuous linear map T from a Banach algebra A into another B approximately preserves the zero products if ‖T(a)T(b)‖ ≤ α‖a‖‖b‖ (a,b ∈ A, ab = 0) for some small positive α. This paper is mainly concerned with the question of whether any continuous linear surjective map T: A → B that approximately preserves the zero products is close to a continuous homomorphism from A onto B with respect to the operator norm. We show that this is indeed the case for amenable group algebras.     

Sturm–Liouville problems in weighted spaces in domains with nonsmooth edges. I

We consider (in general noncoercive) mixed problems in a bounded domain D in ? ⁿ for a second-order elliptic partial differential operator A(x, ?). It is assumed that the operator is written in divergent form in D, the boundary operator B(x, ?) is the restriction of a linear combination of the function and its derivatives to ?D and the boundary of D is a Lipschitz surface. We separate a closed set Y ? ?D and control the growth of solutions near Y. We prove that the pair (A,B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, where the weight is a power of the distance to the singular set Y. Finally, we prove the completeness of the root functions associated with L.The article consists of two parts. The first part published in the present paper, is devoted to exposing the theory of the special weighted Sobolev–Slobodetskii? spaces in Lipschitz domains. We obtain theorems on the properties of these spaces; namely, theorems on the interpolation of these spaces, embedding theorems, and theorems about traces. We also study the properties of the weighted spaces defined by some (in general) noncoercive forms.     

Kirchberger-type theorems for separation by convex domains

We say that a convex set K in ? ^d strictly separates the set A from the set B if A ? int(K) and B ? cl K = ø. The well-known Theorem of Kirchberger states the following. If A and B are finite sets in ? ^d with the property that for every T ? A?B of cardinality at most d + 2, there is a half space strictly separating T ? A and T ? B, then there is a half space strictly separating A and B. In short, we say that the strict separation number of the family of half spaces in ? ^d is d + 2.In this note we investigate the problem of strict separation of two finite sets by the family of positive homothetic (resp., similar) copies of a closed, convex set. We prove Kirchberger-type theorems for the family of positive homothets of planar convex sets and for the family of homothets of certain polyhedral sets. Moreover, we provide examples that show that, for certain convex sets, the family of positive homothets (resp., the family of similar copies) has a large strict separation number, in some cases, infinity. Finally, we examine how our results translate to the setting of non-strict separation.     

Elliptic function of level 4

The article is devoted to the theory of elliptic functions of level n. An elliptic function of level n determines a Hirzebruch genus called an elliptic genus of level n. Elliptic functions of level n are also of interest because they are solutions of the Hirzebruch functional equations. The elliptic function of level 2 is the Jacobi elliptic sine function, which determines the famous Ochanine–Witten genus. It is the exponential of the universal formal group of the form F(u, v) = (u² ? v²)/(uB(v) ? vB(u)), B(0) = 1. The elliptic function of level 3 is the exponential of the universal formal group of the form F(u, v) = (u²A(v) ? v²A(u))/(uA(v)² ? vA(u)²), A(0) = 1, A″(0) = 0. In the present study we show that the elliptic function of level 4 is the exponential of the universal formal group of the form F(u, v) = (u²A(v) ? v²A(u))/(uB(v) ? vB(u)), where A(0) = B(0) = 1 and for B′(0) = A″(0) = 0, A′(0) = A₁, and B″(0) = 2B₂ the following relation holds: (2B(u) + 3A₁u)² = 4A(u)³ ? (3A₁² ? 8B₂)u²A(u)². To prove this result, we express the elliptic function of level 4 in terms of the Weierstrass elliptic functions.     

Inversion and subspaces of a finite field

Consider two F _q -subspaces A and B of a finite field, of the same size, and let A ^?1 denote the set of inverses of the nonzero elements of A. The author proved that A ^?1 can only be contained in A if either A is a subfield, or A is the set of trace zero elements in a quadratic extension of a field. Csajbók refined this to the following quantitative statement: if A ^?1 ? B, then the bound |A ^?1∩B| ≤ 2|B|/q ? 2 holds. He also gave examples showing that his bound is sharp for |B| ≤ q ³. Our main result is a proof of the stronger bound |A ^?1 ∩ B| ≤ |B|/q · (1 + O _d (q ^?1/2)), for |B| = q ^d with d > 3. We also classify all examples with |B| ≤ q ³ which attain equality or near-equality in Csajbók’s bound.     

Twisted Frobenius Extensions of Graded Superrings

We define twisted Frobenius extensions of graded superrings. We develop equivalent definitions in terms of bimodule isomorphisms, trace maps, bilinear forms, and dual sets of generators. The motivation for our study comes from categorification, where one is often interested in the adjointness properties of induction and restriction functors. We show that A is a twisted Frobenius extension of B if and only if induction of B-modules to A-modules is twisted shifted right adjoint to restriction of A-modules to B-modules. A large (non-exhaustive) class of examples is given by the fact that any time A is a Frobenius graded superalgebra, B is a graded subalgebra that is also a Frobenius graded superalgebra, and A is projective as a left B-module, then A is a twisted Frobenius extension of B.     

The Richardson scheme for the singularly perturbed parabolic reaction-diffusion equation in the case of a discontinuous initial condition

The Dirichlet problem for a singularly perturbed parabolic reaction-diffusion equation with a piecewise continuous initial condition in a rectangular domain is considered. The higher order derivative in the equation is multiplied by a parameter ?², where ? ∈ (0, 1]. When ? is small, a boundary and an interior layer (with the characteristic width ?) appear, respectively, in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristic of the reduced equation passing through the discontinuity point of the initial function; for fixed ?, these layers have limited smoothness. Using the method of additive splitting of singularities (induced by the discontinuities of the initial function and its low-order derivatives) and the condensing grid method (piecewise uniform grids that condense in a neighborhood of the boundary layers), a finite difference scheme is constructed that converges ?-uniformly at a rate of O(N ^?2ln² N + n ₀ ^?1 ), where N + 1 and N ₀ + 1 are the numbers of the mesh points in x and t, respectively. Based on the Richardson technique, a scheme that converges ?-uniformly at a rate of O(N ^?3 + N ₀ ^?2 ) is constructed. It is proved that the Richardson technique cannot construct a scheme that converges in ?-uniformly in x with an order greater than three.