On critical normal sections for two-dimensional immersions in $${\mathbb R^{n+2}}$$

We study orthonormal normal sections of two-dimensional immersions in , which are critical for a functional of total torsion and which we call Coulomb sections. In particular, we establish upper bounds for the torsion coefficients in the case of non-flat normal bundles. With these notes we continue a foregoing paper on surfaces in      

Regular complete permutation polynomials over $${\mathbb {F}}_{2^{n}}$$ F 2 n

Designs, Codes and Cryptography - Applications of permutation polynomials in cryptography are closely related to their cycle structures. For example, many block ciphers use permutation polynomials...     

Lagrangian Klein Bottles in {\mathbb{R}^{2n}}

It is shown that the n-dimensional Klein bottle admits a Lagrangian embedding into \mathbbR²ⁿ{\mathbb{R}^{2n}} if and only if n is odd.     

Cyclic codes over $${{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}$$

In this work, we focus on cyclic codes over the ring \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} , which is not a finite chain ring. We use ideas from group rings and works of AbuAlrub et.al. in (Des Codes Crypt 42:273–287, 2007) to characterize the ring (\mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂)/(xⁿ-1){({{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2})/(x^n-1)} and cyclic codes of odd length. Some good binary codes are obtained as the images of cyclic codes over \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} under two Gray maps that are defined. We also characterize the binary images of cyclic codes over \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} in general.     

Maximal q-Plurisubharmonic Functions in {\mathbb{C}^{n}}

In this paper we will study maximal q-plurisubharmonic functions in ${\mathbb{C}^n}$ . At the same time, we define a notion above weakly q-plurisubharmonic functions and describe the relation between these functions and maximal q-plurisubharmonic functions.     

Ruled Weingarten Hypersurfaces in Hyperbolic Space {{\mathbb{H}}^{n+1}}

In this work we give a local classification of connected ruled Weingarten hypersurfaces M ⁿ , n ≥ 3 in the hyperbolic space ${{\mathbb{H}}^{n+1} \subset {\mathbb{L}}^{n+2}}$ .     

Permutation polynomials of the type $$x^rg(x^{s})$$ over $${\mathbb {F}}_{q^{2n}}$$Fq2n

We provide some new families of permutation polynomials of \({\mathbb {F}}_{q^{2n}}\) of the type \(x^rg(x^{s})\), where the integers r, s and the polynomial \(g \in {\mathbb {F}}_q[x]\) satisfy particular restrictions. Some generalizations of known permutation binomials and trinomials that involve a sort of symmetric polynomials are given. Other constructions are based on the study of algebraic curves associated to certain polynomials. In particular we generalize families of permutation polynomials constructed by Gupta–Sharma, Li–Helleseth, Li–Qu–Li–Fu.     

Difference Sets in {\mathbb{Z}}_4^m and {\mathbb{F}}_2^{2m}

Let R=GR(4,m) be the Galois ring of cardinality 4^m and let T be the Teichmüller system of R. For every map λ of T into { -1,+1} and for every permutation Π of T, we define a map φ _λ ^Π of Rinto { -1,+1} as follows: if x∈R and if x=a+2b is the 2-adic representation of x with x∈T and b∈T, then φ _λ ^Π (x)=λ(a)+2Tr(Π(a)b), where Tr is the trace function of R . For i=1 or i=-1, define D _i as the set of x in R such thatφ _λ ^Π =i. We prove the following results: 1) D _i is a Hadamard difference set of (R,+). 2) If φ is the Gray map of R into ${\mathbb{F}}_2^{2m}$ , then (D _i) is a difference set of ${\mathbb{F}}_2^{2m}$ . 3) The set of D _i and the set of φ(D _i) obtained for all maps λ and Π, both are one-to-one image of the set of binary Maiorana-McFarland difference sets in a simple way. We also prove that special multiplicative subgroups of R are difference sets of kind D _i in the additive group of R. Examples are given by means of morphisms and norm in R.     

Closed characteristics on non-compact hypersurfaces in $${\mathbb {R}^{2n}}$$

Viterbo demonstrated that any (2n − 1)-dimensional compact hypersurface of contact type has at least one closed characteristic. This result proved the Weinstein conjecture for the standard symplectic space (, ω). Various extensions of this theorem have been obtained since, all for compact hypersurfaces. In this paper we consider non-compact hypersurfaces coming from mechanical Hamiltonians, and prove an analogue of Viterbo’s result. The main result provides a strong connection between the top half homology groups H _i (M), i = n, . . . , 2n − 1, and the existence of closed characteristics in the non-compact case (including the compact case). J. B. van den Berg is supported by NWO VENI grant 639.031.204. R. C. Vandervorst and F. Pasquotto are supported by NWO VIDI grant 639.032.202. This research is also partially supported by the RTN project ‘Fronts-Singularities’.     

On lattices of convex sets in {\mathbb{R}}^{n}

Properties of several sorts of lattices of convex subsets of are examined. The lattice of convex sets containing the origin turns out, for n > 1, to satisfy a set of identities strictly between those of the lattice of all convex subsets of and the lattice of all convex subsets of The lattices of arbitrary, of open bounded, and of compact convex sets in all satisfy the same identities, but the last of these is join-semidistributive, while for n > 1 the first two are not. The lattice of relatively convex subsets of a fixed set satisfies some, but in general not all of the identities of the lattice of “genuine” convex subsets of To the memory of Ivan RivalReceived April 22, 2003; accepted in final form February 16, 2005.This revised version was published online in August 2005 with a corrected cover date.     

Trajectory of an Observer Tracking Object Motion around Convex Obstacles in $${{\mathbb{R}}^{2}}$$ and $${{\mathbb{R}}^{3}}$$

Doklady Mathematics - An autonomous object moves at a constant speed along the shortest path, while bypassing an ordered collection of pairwise disjoint convex sets. The object is tracked by an...     

An upper bound for the height for regular affine automorphisms of {{{\mathbb{A}}^ n}}

Kawaguchi (Math. Ann. 335(2):285–310, 359–374, 2006) proved a height inequality for ${h\bigl(f(P)\bigr)}$ when f is a regular affine automorphism of ${{\mathbb{A}}^2}$ , and he conjectured that a similar estimate is also true for regular affine automorphisms of ${{\mathbb{A}}^n}$ for n ≥ 3. In this paper we prove Kawaguchi’s conjecture. This implies that Kawaguchi’s theory of canonical heights for regular affine automorphisms of projective space is true in all dimensions.     

Existence of isoperimetric regions in {\mathbb{R}^{n}} with density

We prove the existence of isoperimetric regions in ${\mathbb{R}^{n}}$ with density under various hypotheses on the growth of the density. Along the way, we prove results on the boundedness of isoperimetric regions.     

Inequalities for the Volume of the Unit Ball in {\mathbb{R}^{n}}

The volume of the unit ball in ${\mathbb{R}^{n}}$ is defined by $$\Omega_{n} = \frac{\pi^{n/2}}{\Gamma(n/2+1)},\qquad n = 1,2,3,\ldots,$$ where Γ denotes the classical gamma function of Euler. In several recently published papers numerous authors studied properties of Ω _n . In particular, various inequalities involving Ω _n are given in the literature. In this paper, we continue the work on this subject and offer new inequalities. More precisely, we offer sharp upper and lower bounds for $$\frac{\Omega_{n}^{2}}{\Omega_{n-1} \Omega_{n+1}},\quad\frac{\Omega_{n}}{\Omega_{n-1}+\Omega_{n+1}} \quad {\rm and} \quad\Omega_{n}.$$