Finite-element based sparse approximate inverses for block-factorized preconditioners

In this work we analyse a method to construct numerically efficient and computationally cheap sparse approximations of some of the matrix blocks arising in the block-factorized preconditioners for matrices with a two-by-two block structure. The matrices arise from finite element discretizations of partial differential equations. We consider scalar elliptic problems, however the approach is appropriate also for other types of problems such as parabolic problems or systems of equations. The technique is applicable for both selfadjoint and non-selfadjoint problems, in two as well as in three space dimensions. We analyse in detail the two-dimensional case and provide extensive numerical evidence for the efficiency of the proposed matrix approximations, both serial and parallel. Two- and three-dimensional tests are included.     

Minimizing and maximizing the Euclidean norm of the product of two polynomials

We consider the problem of minimizing or maximizing the quotient $$f_{m,n}(p,q):=\frac{\|{pq}\|}{\|{p}\|\|{q}\|} \ ,$$ where $p=p_0+p_1x+\dots+p_mx^m$ , $q=q_0+q_1x+\dots+q_nx^n\in{\mathbb K}[x]$ , ${\mathbb K}\in\{{\mathbb R},{\mathbb C}\}$ , are non-zero real or complex polynomials of maximum degree $m,n\in{\mathbb N}$ respectively and $\|{p}\|:=(|p_0|^2+\dots+|p_m|^2)^{\frac{1}{2}}$ is simply the Euclidean norm of the polynomial coefficients. Clearly f _m,n is bounded and assumes its maximum and minimum values min f _m,n ?=?f _m,n (p _min, q _min) and max f _m,n ?=?f(p _max, q _max). We prove that minimizers p _min, q _min for ${\mathbb K}={\mathbb C}$ and maximizers p _max, q _max for arbitrary ${\mathbb K}$ fulfill $\deg(p_{\min})=m=\deg(p_{\max})$ , $\deg(q_{\min})=n=\deg(q_{\max})$ and all roots of p _min, q _min, p _max, q _max have modulus one and are simple. For ${\mathbb K}={\mathbb R}$ we can only prove the existence of minimizers p _min, q _min of full degree m and n respectively having roots of modulus one. These results are obtained by transferring the optimization problem to that of determining extremal eigenvalues and corresponding eigenvectors of autocorrelation Toeplitz matrices. By the way we give lower bounds for min f _m,n for real polynomials which are slightly better than the known ones and inclusions for max f _m,n .     

Optimal Embeddings of Spaces of Generalized Smoothness in the Critical Case

We study necessary and sufficient conditions for embeddings of Besov and Triebel-Lizorkin spaces of generalized smoothness B^(n/p,Y)_p,q(\mathbbRⁿ)B^{(n/p,\Psi)}_{p,q}(\mathbb{R}^{n}) and F^(n/p,Y)_p,q(\mathbbRⁿ)F^{(n/p,\Psi)}_{p,q}(\mathbb{R}^{n}), respectively, into generalized H?lder spaces L_￥,r^m(·)( \mathbb Rⁿ)\Lambda_{\infty,r}^{\mu(\cdot)}(\ensuremath {\ensuremath {\mathbb {R}}^{n}}). In particular, we are able to characterize optimal embeddings for this class of spaces provided q>1. These results improve the embedding assertions given by the continuity envelopes of B^(n/p,Y)_p,q(\mathbbRⁿ)B^{(n/p,\Psi)}_{p,q}(\mathbb{R}^{n}) and F^(n/p,Y)_p,q(\mathbbRⁿ)F^{(n/p,\Psi)}_{p,q}(\mathbb{R}^{n}), which were obtained recently solving an open problem of D.D. Haroske in the classical setting.     

Representation of natural numbers by sums of four squares of integers having a special form

This paper obtains an asymptotic formula for the number of solutions to the equation $ l_1^2 + { }l_2^2 + l_3^2 + l_4^2 = N $ in integers l ₁, l ₂, l ₃, l ₄ such that a < {??l _j } < b, where ?? is a quadratic irrational number, 0 ?? a < b ?? 1, j = 1, 2, 3, 4.     

The asymptotic expansion for n! and the Lagrange inversion formula

We obtain an explicit simple formula for the coefficients of the asymptotic expansion for the factorial of a natural number, $$n!=n^n\sqrt{2\pi n}\mbox{e}^{-n}\biggl\{1+\frac{a_1}{n}+\frac{a_2}{n^2}+\frac{a_3}{n^3}+\cdots\biggr\},$$ in terms of derivatives of powers of an elementary function that we call normalized left truncated exponential function. The unique explicit expression for the a _k that appears to be known is that of Comtet in (Advanced Combinatorics, Reidel, 1974), which is given in terms of sums of associated Stirling numbers of the first kind. By considering the bivariate generating function of the associated Stirling numbers of the second kind, another expression for the coefficients in terms of them follows also from our analysis. Comparison with Comtet??s expression yields an identity which is somehow unexpected if considering the combinatorial meaning of the terms. It suggests by analogy another possible formula for the coefficients, in terms of a normalized left truncated logarithm, that in fact proves to be true. The resulting coefficients, as well as the first ones are identified via the Lagrange inversion formula as the odd coefficients of the inverse of a pair of formal series. This in particular leads to the identification of a couple of simple implicit equations, which permits us to obtain also some recurrences related to the a _k ??s.     

Small quotients in Euclidean algorithms

Numbers whose continued fraction expansion contains only small digits have been extensively studied. In the real case, the Hausdorff dimension ?? _M of the reals with digits in their continued fraction expansion bounded by M was considered, and estimates of ?? _M for M???? were provided by Hensley (J. Number Theory 40:336?C358, 1992). In the rational case, first studies by Cusick (Mathematika 24:166?C172, 1997), Hensley (In: Proc. Int. Conference on Number Theory, Quebec, pp. 371?C385, 1987) and Vallée (J. Number Theory 72:183?C235, 1998) considered the case of a fixed bound M when the denominator N tends to ??. Later, Hensley (Pac. J. Math. 151(2):237?C255, 1991) dealt with the case of a bound M which may depend on the denominator N, and obtained a precise estimate on the cardinality of rational numbers of denominator less than N whose digits (in the continued fraction expansion) are less than M(N), provided the bound M(N) is large enough with respect to N. This paper improves this last result of Hensley towards four directions. First, it considers various continued fraction expansions; second, it deals with various probability settings (and not only the uniform probability); third, it studies the case of all possible sequences M(N), with the only restriction that M(N) is at least equal to a given constant M ₀; fourth, it refines the estimates due to Hensley, in the cases that are studied by Hensley. This paper also generalises previous estimates due to Hensley (J. Number Theory 40:336?C358, 1992) about the Hausdorff dimension ?? _M to the case of other continued fraction expansions. The method used in the paper combines techniques from analytic combinatorics and dynamical systems and it is an instance of the Dynamical Analysis paradigm introduced by Vallée (J. Théor. Nr. Bordx. 12:531?C570, 2000), and refined by Baladi and Vallée (J. Number Theory 110:331?C386, 2005).     

Varieties of interlaced bilattices

The paper contains some algebraic results on several varieties of algebras having an (interlaced) bilattice reduct. Some of these algebras have already been studied in the literature (for instance bilattices with conflation, introduced by M. Fitting), while others arose from the algebraic study of O. Arieli and A. Avron??s bilattice logics developed in the third author??s PhD dissertation. We extend the representation theorem for bounded interlaced bilattices (proved, among others, by A. Avron) to unbounded bilattices and prove analogous representation theorems for the other classes of bilattices considered. We use these results to establish categorical equivalences between these structures and well-known varieties of lattices.     

An application of the maximum principle to describe the layer behavior of large solutions and related problems

This work is devoted to the analysis of the asymptotic behavior of positive solutions to some problems of variable exponent reaction-diffusion equations, when the boundary condition goes to infinity (large solutions). Specifically, we deal with the equations ??u = u ^p(x), ??u = ?m(x)u?+?a(x)u ^p(x) where a(x)??? a ₀ >?0, p(x)??? 1 in ??, and ??u = e ^p(x) where p(x)??? 0 in ??. In the first two cases p is allowed to take the value 1 in a whole subdomain ${\Omega_c\subset \Omega}$ , while in the last case p can vanish in a whole subdomain ${\Omega_c\subset \Omega}$ . Special emphasis is put in the layer behavior of solutions on the interphase ?? _i :?= ??? _c ???. A similar study of the development of singularities in the solutions of several logistic equations is also performed. For example, we consider ???u = ?? m(x)u?a(x) u ^p(x) in ??, u = 0 on ???, being a(x) and p(x) as in the first problem. Positive solutions are shown to exist only when the parameter ?? lies in certain intervals: bifurcation from zero and from infinity arises when ?? approaches the boundary of those intervals. Such bifurcations together with the associated limit profiles are analyzed in detail. For the study of the layer behavior of solutions the introduction of a suitable variant of the well-known maximum principle is crucial.     

Explicit Green operators for quantum mechanical Hamiltonians. I. The hydrogen atom

We study a new approach to determine the asymptotic behaviour of quantum many-particle systems near coalescence points of particles which interact via singular Coulomb potentials. This problem is of fundamental interest in electronic structure theory in order to establish accurate and efficient models for numerical simulations. Within our approach, coalescence points of particles are treated as embedded geometric singularities in the configuration space of electrons. Based on a general singular pseudo-differential calculus, we provide a recursive scheme for the calculation of the parametrix and corresponding Green operator of a nonrelativistic Hamiltonian. In our singular calculus, the Green operator encodes all the asymptotic information of the eigenfunctions. Explicit calculations and an asymptotic representation for the Green operator of the hydrogen atom and isoelectronic ions are presented.