Bifurcation of Hyperbolic Planforms

Motivated by a model for the perception of textures by the visual cortex in primates, we analyze the bifurcation of periodic patterns for nonlinear equations describing the state of a system defined on the space of structure tensors, when these equations are further invariant with respect to the isometries of this space. We show that the problem reduces to a bifurcation problem in the hyperbolic plane D\mathcal {D} (Poincaré disc). We make use of the concept of a periodic lattice in D\mathcal {D} to further reduce the problem to one on a compact Riemann surface D/\varGamma\mathcal {D}/\varGamma, where Γ is a cocompact, torsion-free Fuchsian group. The knowledge of the symmetry group of this surface allows us to use the machinery of equivariant bifurcation theory. Solutions which generically bifurcate are called “H-planforms”, by analogy with the “planforms” introduced for pattern formation in Euclidean space. This concept is applied to the case of an octagonal periodic pattern, where we are able to classify all possible H-planforms satisfying the hypotheses of the Equivariant Branching Lemma. These patterns are, however, not straightforward to compute, even numerically, and in the last section we describe a method for computation illustrated with a selection of images of octagonal H-planforms.     

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 .     

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.     

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.     

Computing Syzygies by Faug��re��s F5 algorithm

In this paper, we introduce a new algorithm for computing a set of generators for the syzygies on a sequence of polynomials. For this, we extend a given sequence of polynomials to a Gr?bner basis using Faugère??s F₅ algorithm (A new efficient algorithm for computing Gr?bner bases without reduction to zero (F ₅). ISSAC, ACM Press, pp 75?C83, 2002). We show then that if we keep all the reductions to zero during this computation, then at termination (by adding principal syzygies) we obtain a basis for the module of syzygies on the input polynomials. We have implemented our algorithm in the computer algebra system Magma, and we evaluate its performance via some examples.     

Bounds for Zeros of Solutions of Second Order Differential Equations with Polynomial Coefficients

We consider the equation ${u''=P(z)u\;\;(z\in\mathbb{C})}$ where P(z) is a polynomial. Let z _k (u), k = 1, 2,... be the zeros of a solution u(z) to that equation. Bounds for the sums $$\sum_{k=1}^{j} \frac {1} {|z_k(u)|}\;(j=1, 2, \ldots)$$ are established. Some applications of these bounds are also considered.     

Diffusion-dominated asymptotics of solution to chemotaxis model

The paper contains results on the asymptotic behavior, as t → +∞, of small solutions to simplified Keller–Segel problem modeling chemotaxis in the whole space \mathbb R²{\mathbb R^2}. We prove that the multiple of the heat kernel is a surprisingly good approximation of solutions.     

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).     

Suspending the Cartan Embedding of {{\mathbb H}P^n} Through Spindles and Generators of Homotopy Groups

We provide an equivariant suspension of the Cartan embedding of the symmetric space ${S^{4n+3} \to \mathbb {H}P^n \hookrightarrow Sp(n+1)}$ ; this construction furnishes geometric generators of the homotopy group of ?? _4n+6 Sp(n?+?1). We study the topology and geometry of the image of this generator; in particular we show that it is a spindle, minimal with respect to the biinvariant metric from Sp(n?+?1). This spindle also admits a different metric of positive curvature away from the cone singular point.