On the equation −Δu + e u − 1 = 0 with measures as boundary data

If Ω is a bounded domain in ${\mathbb{R}^N}$ , we study conditions on a Radon measure μ on ?Ω for solving the equation ?Δu + e ^u ? 1 = 0 in Ω with u = μ on ?Ω. The conditions are expressed in terms of Orlicz capacities.     

On the Various Realizations of the Basic Representation of An−1(1) and the Combinatorics of Partitions

The infinite dimensional Lie algebra l _n = A _n–1 ⁽¹⁾ can be realized in several ways as an algebra of differential operators. The aim of this note is to show that the intertwining operators between the realizations of l _n corresponding to all partitions of n can be described very simply by using combinatorial constructions.     

Lattice-theoretic characterizations of the group classes {\mathfrak{N}^{k-1}\mathfrak{A}} and {\mathfrak{N}^k} for k?�� 2

Let ${2\leq k\in \mathbb{N}}$ . Recently, Costantini and Zacher obtained a lattice-theoretic characterization of the classes ${\mathfrak{N}^k}$ of finite soluble groups with nilpotent length at most k. It is the aim of this paper to give a lattice-theoretic characterization of the classes ${\mathfrak{N}^{k-1}\mathfrak{A}}$ of finite groups with commutator subgroup in ${\mathfrak{N}^{k-1}}$ ; in addition, our method also yields a new characterization of the classes ${\mathfrak{N}^k}$ . The main idea of our approach is to use two well-known theorems of Gaschütz on the Frattini and Fitting subgroups of finite groups.     

On the Multiplicative Orders of γ and γ+γ−1 over finite fields

Using bounds of character sums we show that one of the open questions about the possible relation between the multiplicative orders of γ and γ+γ⁻¹ has a negative answer. In fact we show that in some sense the multiplicative orders of these elements are independent.     

On the Independence Numbers of Distance Graphs with Vertices in {–1, 0, 1}n

Doklady Mathematics - In this work, we find new bounds for the independence numbers of distance graphs with vertices in {–1, 0, 1}.     

Classification of hypersurfaces with constant M?bius curvature in {\mathbb{S}^{m+1}}

Let ${x: M^{m} \rightarrow \mathbb{S}^{m+1}}$ be an m-dimensional umbilic-free hypersurface in an (m?+?1)-dimensional unit sphere ${\mathbb{S}^{m+1}}$ , with standard metric I?= dx · dx. Let II be the second fundamental form of isometric immersion x. Define the positive function ${\rho=\sqrt{\frac{m}{m-1}}\|II-\frac{1}{m}tr(II)I\|}$ . Then positive definite (0,2) tensor ${\mathbf{g}=\rho^{2}I}$ is invariant under conformal transformations of ${\mathbb{S}^{m+1}}$ and is called M?bius metric. The curvature induced by the metric g is called M?bius curvature. The purpose of this paper is to classify the hypersurfaces with constant M?bius curvature.     

The elliptic functions and integrals of the “{1 \mathord{\left/ {\vphantom {1 9}} \right. \kern-\nulldelimiterspace} 9}” problem

We describe the functions needed in the determination of the rate of convergence of best $L^\infty $ rational approximation to $\exp ( - x)$ on [0,∞) when the degree n of the approximation tends to ∞ (“1/9” problem).     

On the Periodic Structure of Delayed Difference Equations of the Form x n = f ( x n − k ) on I and S 1

The aim of this paper is to give an account of some results recently obtained in Combinatorial Dynamics and apply them to get for k S 2 the periodic structure of delayed difference equations of the form x n = f ( x n m k ) on I and S 1 .     

Characterization of {v μ+1 + 2v μ,v μ + 2v μ − 1;t,q}-min · hypers and its applications to error-correcting codes

Recently, Hamada [5] characterized all {v ₂ + 2v ₁,v ₁ + 2v ₀;t,q}-min · hypers for any integert 2 and any prime powerq 3 wherev _l = (q ^l – 1)/(q – 1) for any integerl 0. The purpose of this paper is to characterize all {v _{+ 1} + 2v ,v + 2v _{– 1};t,q}-min · hypers for any integerst, and any prime powerq such thatt 3, 2 t – 1 andq 5 and to characterize all (n, k, d; q)-codes meeting the Griesmer bound (1.1) for the casek 3, d = q ^k-1 – (2q ^-1 +q ) andq 5 using the results in Hamada [3, 4, 5].     

On the Schröder equation and iterative sequences of C r diffeomorphisms in {\mathbb{R}^{N}} space

Let ${U \subset \mathbb{R}^{N}}$ be a neighbourhood of the origin and a function ${F:U\rightarrow U}$ be of class C ^r , r ≥ 2, F(0) = 0. Denote by F ⁿ the n-th iterate of F and let ${0<|s_1|\leq \cdots \leq|s_N| <1 }$ , where ${s_1, \ldots , s_N}$ are the eigenvalues of dF(0). Assume that the Schröder equation ${\varphi(F(x))=S\varphi(x)}$ , where S: = dF(0) has a C ² solution φ such that dφ(0) = id. If ${\frac{log|s_1|}{log|s_N|} <2 }$ then the sequence {S ^?n F ⁿ (x)} converges for every point x from the basin of attraction of F to a C ² solution φ of (1). If ${2\leq\frac{log|s_1|}{log|s_N|} }$ then this sequence can be diverging. In this case we give some sufficient conditions for the convergence and divergence of the sequence {S ^?n F ⁿ (x)}. Moreover, we show that if F is of class C ^r and ${r>\big[\frac{log|s_1|}{log|s_N|} \big ]:=p \geq 2}$ then every C ^r solution of the Schröder equation such that dφ(0) = id is given by the formula $$\begin{array}{ll}\varphi (x)={\lim\limits_{n \rightarrow \infty}} (S^{-n}F^n(x) + {\sum\limits _{k=2}^{p}} S^{-n}L_k (F^n(x))),\end{array}$$ where ${L_k:\mathbb{R}^{N} \rightarrow \mathbb{R}^{N}}$ are some homogeneous polynomials of degree k, which are determined by the differentials d ^(j) F(0) for 1 < j ≤  p.     

Notes on the Kazhdan–Lusztig Theorem on Equivalence of the Drinfeld Category and the Category of $\boldsymbol{U}_{\!\boldsymbol q}{\boldsymbol {\mathfrak g}}$-Modules

We discuss the proof of Kazhdan and Lusztig of the equivalence of the Drinfeld category \({\mathcal D}({\mathfrak g},\hbar)\) of \({\mathfrak g}\)-modules and the category of finite dimensional \(U_q{\mathfrak g}\)-modules, \(q=e^{\pi i\hbar}\), for \(\hbar\in{\mathbb C}\setminus{\mathbb Q}^*\). Aiming at operator algebraists the result is formulated as the existence for each \(\hbar\in i{\mathbb R}\) of a normalized unitary 2-cochain \({\mathcal F}\) on the dual \(\hat G\) of a compact simple Lie group G such that the convolution algebra of G with the coproduct twisted by \({\mathcal F}\) is *-isomorphic to the convolution algebra of the q-deformation G _q of G, while the coboundary of \({\mathcal F}^{-1}\) coincides with Drinfeld’s KZ-associator defined via monodromy of the Knizhnik–Zamolodchikov equations.     

A new optimal quaternary sequence family of length 2(2 n − 1) obtained from the orthogonal transformation of Families {\mathcal{B}} and {\mathcal{C}}

In this paper, a general orthogonal transformation on the optimal quaternary sequence Families ${\mathcal{B}}$ and ${\mathcal{C}}$ is presented. Consequently, the known optimal Family ${\mathcal{D}}$ and a new optimal Family ${\mathcal{E}}$ are produced in a uniform method. In contrast to the known optimal Family ${\mathcal{D}}$ , the new Family ${\mathcal{E}}$ has the same parameters such as the sequence length 2(2 ⁿ ? 1), the family size 2 ⁿ , and the maximal nontrivial correlation value ${2^{\frac{n+1}{2}}+2}$ , where n is a positive integer, but with a different correlation function.