Generalized Hausdorff dimension distortion in Euclidean spaces under Sobolev mappings

We investigate how the integrability of the derivatives of Orlicz-Sobolev mappings defined on open subsets of Rn affect the sizes of the images of sets of Hausdorff dimension less than n. We measure the sizes of the image sets in terms of generalized Hausdorff measures.     

Bi-Sobolev mappings and elliptic equations in the plane

Suppose that f=(u,v) is a homeomorphism in the plane of the Sobolev class such that its inverse is of the same Sobolev class. We prove that u and v have the same set of critical points. As an application we show that u and v are distributional solutions to the same non-trivial degenerate elliptic equation in divergence form. We study similar properties also in higher dimensions.     

Regularity of the inverse of a homeomorphism with finite inner distortion

Let f:Ω→ f(Ω) R~n be a W~(1,1)-homeomorphism with L~1-integrable inner distortion.We show that finiteness of min{lip_f(x),k_f/(x)},for every x ∈Ω\E,implies that f~(-1) ∈ W~(1,n)and has finite distortion,provided that the exceptional set E has cr-finite H~1-measure.Moreover,/ has finite distortion,differentiable a.e.and the Jacobian J_f 0 a.e.     

Mappings of finite distortion: Formation of cusps III

We give sharp integrability conditions on the distortion of a planar homeomorphism that maps a standard cusp onto the unit disk.     

Refined weight of edges in normal plane maps

The weight $w\left(e\right)$ of an edge $e$ in a normal plane map (NPM) is the degree-sum of its end-vertices. An edge $e=uv$ is of type $(i,j)$ if $d\left(u\right)\le i$ and $d\left(v\right)\le j$. In 1940, Lebesgue proved that every NPM has an edge of one of the types $(3,11)$, $(4,7)$, or $(5,6)$, where 7 and 6 are best possible. In 1955, Kotzig proved that every 3-connected planar graph has an edge $e$ with $w\left(e\right)\le 13$, which bound is sharp. Borodin (1989), answering Erd?s’ question, proved that every NPM has either a $(3,10)$-edge, or $(4,7)$-edge, or $(5,6)$-edge.A vertex is simplicial if it is completely surrounded by 3-faces. In 2010, Ferencová and Madaras conjectured (in different terms) that every 3-polytope without simplicial 3-vertices has an edge $e$ with $w\left(e\right)\le 12$. Recently, we confirmed this conjecture by proving that every NPM has either a simplicial 3-vertex adjacent to a vertex of degree at most 10, or an edge of types $(3,9)$, $(4,7)$, or $(5,6)$.By a $k^{\left(?\right)}$-vertex we mean a $k$-vertex incident with precisely $?$ triangular faces. The purpose of our paper is to prove that every NPM has an edge of one of the following types: $(3^{\left(3\right)},10)$, $(3^{\left(2\right)},9)$, $(3^{\left(1\right)},7)$, $(4^{\left(4\right)},7)$, $(4^{\left(3\right)},6)$, $(5^{\left(5\right)},6)$, or $(5,5)$, where all bounds are best possible. In particular, this implies that the bounds in $(3,10)$, $(4,7)$, and $(5,6)$ can be attained only at NPMs having a simplicial 3-, 4-, or 5-vertex, respectively.     

Additive maps preserving M-P inverses of matrices over Fields

Suppose F is a field of characteristic not 2 or 3. A characterization is given for all additive maps, on the algebra of all n × n matrices over F. which preserve Moore -Penrose(M-P) Inverses of matrices.     

Piecewise constructions of inverses of cyclotomic mapping permutation polynomials

Given a permutation polynomial of a large finite field, finding its inverse is usually a hard problem. Based on a piecewise interpolation formula, we construct the inverses of cyclotomic mapping permutation polynomials of arbitrary finite fields.     

Weight of faces in plane maps

Precise upper bounds are obtained for the minimum weight of minor faces in normal plane maps and 3-polytopes with specified maximum vertex degree. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 648–657, November, 1998. The research of the first named author was supported in part by the Visiting Fellowship Research Grant GR/K00561 from the Engineering and Physical Sciences Research Council and by the Russian Foundation for Basic Research under grant No. 96-01-01614 and No. 97-01-01075.     

有限偏差函数与调和函数

分别借助解析函数与调和函数两类函数的Dirichlet积分,利用相关文献给定边界值的拟共形映射极值伸缩商的估计方法,通过有限偏差函数和拟共形映射的关系估计了具有给定边界值的有限偏差函数的极值伸缩商.得到了解析函数的Dirichlet积分在有限偏差函数下具有拟不变性,同时给出有限偏差函数极值伸缩商的下界估计.     

Discreteness and openness of mappings with distortion in Orlicz spaces on ℍ-type Carnot groups

We prove discreteness and openness of the mappings with finite distortion defined on ℍ-type Carnot groups whose distortion functions belong to certain Orlicz spaces. Similar results in Euclidean spaces were established bymany authors under different assumptions about the mappings and their distortion functions. The text was submitted by the authors in English.     

Bounds on the inverses of nonnegative triangular matrices with monotonicity properties

Some bounds on the entries and on the norm of the inverse of triangular matrices with nonnegative and monotone entries are found. All the results are obtained by exploiting the properties of the fundamental matrix of the recurrence relation which generates the sequence of the entries of the inverse matrix. One of the results generalizes a theorem contained in a recent article of one of the authors about Toeplitz matrices.     

An example concerning the zero set of the Jacobian

Let f∈W1,1(Ω,Rn) be a homeomorphism of finite distortion K. It is known that if K1/(n−1)∈L¹(Ω), then the Jacobian Jf of f is positive almost everywhere in Ω. We will show that this integrability assumption on K is sharp in any Orlicz-scale: if α is increasing function (satisfying minor technical assumptions) such that limt→∞α(t)=∞, then there exists f such that K1/(n−1)/α(K)∈L¹(Ω) and Jf vanishes in a set of positive measure.     

Describing 3-faces in normal plane maps with minimum degree 4

In 1940, Lebesgue proved that every 3-polytope with minimum degree at least 4 contains a 3-face for which the set of degrees of its vertices is majorized by one of the following sequences:

(4,4,∞),(4,5,19),(4,6,11),(4,7,9),(5,5,9),(5,6,7).$(4,4,\infty ),(4,5,19),(4,6,11),(4,7,9),(5,5,9),(5,6,7)\text{.}$

     

Enumeration of maps on the projective plane

1. IntroductionA lnap is rooted if an edge is distinguished togetl1er with an end and a side of the edge.An edge belo11ging to only one face is called double (or 8ingular by some author), al1 othersbelonging to exactly two faces are called s1ngle. The enumeration of rooted p1anar maps wasfirst introduced by Tutte['], Techniques originated by Tutte [2,3l for enumerating variousclasses of rooted Inaps on tIle sphere are here applied to the c1asses of alI rooted maps onthe projective plane. Th…     

On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers

Let A_n=Circ(F₁,F₂,…,F_n) and B_n=Circ(L₁,L₂,…,L_n) be circulant matrices, where F_n is the Fibonacci number and L_n is the Lucas number. We prove that A_n is invertible for n > 2, and B_n is invertible for any positive integer n. Afterwards, the values of the determinants of matrices A_n and B_n can be expressed by utilizing only the Fibonacci and Lucas numbers. In addition, the inverses of matrices A_n and B_n are derived.     

Chaos in orientation reversing twist maps of the plane

We study forcing of periodic points in orientation reversing twist maps. First, we observe that the fourth iterate of an orientation reversing twist map can be expressed as the composition of four orientation preserving positive twist maps. We then reformulate the problem in terms of parabolic flows, which form the natural dynamics on a certain space of braid diagrams. Second, we focus our attention on period-4 points, which we classify in terms of their corresponding braid diagrams. They can be categorized in two types. If an orientation reversing twist map has a period-4 point of one type, then there is a semi-conjugacy to symbolic dynamics and the system is forced to be chaotic. We also show that this result is sharp in the sense that the remaining type does not necessarily lead to chaos.     

Fitting maps of the plane to experimental data

This paper is motivated by the problem of fitting a differential equation to experimental data; a first step is to fit the time 1 map of the corresponding flow, as described in Section 1, and these time 1 maps are the diffeomorphisms in the title of the present paper. The negative part of statement (ii) of Theorem 3.1 implies that there is no algorithm which fits diffeomorphisms of the plane to observed data in a way that depends continuously on the observed data and only on the observed data.     

The number of nonseparable maps on the projective plane

In this paper, we study the rooted nonseparable maps on the sphere and the projective plane with the valency of root-face and the number of edges as parameters. Explicit expression of enumerating functions are obtained for such maps on the sphere and the projective plane. A parametric expression of the generating function is obtained for such maps on the projective plane, from which asymptotic evaluations are derived. Moreover, if the number of edges is sufficiently large, then almost all nonseparable maps on the projective plane are not triangulation.