Classification of 7-dimensional einstein nilradicals

The problem of classifying Einstein solvmanifolds, or equivalently, Ricci soliton nilmanifolds, is known to be equivalent to a question on the variety $ {\mathfrak{N}_n}\left( \mathbb{C} \right) $ of n-dimensional complex nilpotent Lie algebra laws. Namely, one has to determine which GL _n (?)-orbits in $ {\mathfrak{N}_n}\left( \mathbb{C} \right) $ have a critical point of the squared norm of the moment map. In this paper, we give a classification result of such distinguished orbits for n?=?7. The set $ {{{{\mathfrak{N}_n}\left( \mathbb{C} \right)}} \left/ {{{\text{G}}{{\text{L}}_7}\left( \mathbb{C} \right)}} \right.} $ is formed by 148 nilpotent Lie algebras and 6 one-parameter families of pairwise non-isomorphic nilpotent Lie algebras. We have applied to each Lie algebra one of three main techniques to decide whether it has a distinguished orbit or not.     

Height of a Superposition

We observe that, given a poset ${\left( {E,{\user1{\mathcal{R}}}} \right)}$ and a finite covering ${\user1{\mathcal{R}}} = {\user1{\mathcal{R}}}_{1} \cup \cdots \cup {\user1{\mathcal{R}}}_{n} $ of its ordering, the height of the poset does not exceed the natural product of the heights of the corresponding sub-relations: $$\mathfrak{h}{\left( {E,{\user1{\mathcal{R}}}} \right)} \leqslant \mathfrak{h}{\left( {E,{\user1{\mathcal{R}}}_{1} } \right)} \otimes \cdots \otimes \mathfrak{h}{\left( {E,{\user1{\mathcal{R}}}_{n} } \right)}.$$ Conversely for every finite sequence $(\xi_1,\cdots,\xi_n)$ of ordinals, every poset ${\left( {E,{\user1{\mathcal{R}}}} \right)}$ of height at most $\xi_1\otimes\cdots\otimes\xi_n$ admits a partition ${\left( {{\user1{\mathcal{R}}}_{1} , \cdots ,{\user1{\mathcal{R}}}_{n} } \right)}$ of its ordering ${\user1{\mathcal{R}}}$ such that each ${\left( {E,{\user1{\mathcal{R}}}_{k} } \right)}$ has height at most $\xi_k$ . In particular for every finite sequence $(\xi_1,\cdots,\xi_n)$ of ordinals, the ordinal $$\xi _{1} \underline{ \otimes } \cdots \underline{ \otimes } \xi _{n} : = \sup {\left\{ {{\left( {\xi ^{\prime }_{1} \otimes \cdots \otimes \xi ^{\prime }_{n} } \right)} + 1:\xi ^{\prime }_{1} < \xi _{1} , \cdots ,\xi ^{\prime }_{n} < \xi _{n} } \right\}}$$ is the least $\xi$ for which the following partition relation holds $$\mathfrak{H}_{\xi } \to {\left( {\mathfrak{H}_{{\xi _{1} }} , \cdots ,\mathfrak{H}_{{\xi _{n} }} } \right)}^{2} $$ meaning: for every poset ${\left( {A,{\user1{\mathcal{R}}}} \right)}$ of height at least $\xi$ and every finite covering ${\left( {{\user1{\mathcal{R}}}_{1} , \cdots ,{\user1{\mathcal{R}}}_{n} } \right)}$ of its ordering ${\user1{\mathcal{R}}}$ , there is a $k$ for which the relation ${\left( {A,{\user1{\mathcal{R}}}_{k} } \right)}$ has height at least $\xi_k$ . The proof will rely on analogue properties of vertex coverings w.r.t. the natural sum.     

Good gradings of basic Lie superalgebras

We classify good ?-gradings of basic Lie superalgebras over an algebraically closed field $\mathbb{F}$ of characteristic zero. Good ?-gradings are used in quantum Hamiltonian reduction for affine Lie superalgebras, where they play a role in the construction of super W-algebras. We also describe the centralizer of a nilpotent even element and of an $\mathfrak{s}\mathfrak{l}_2$ -triple in $\mathfrak{g}\mathfrak{l}\left( {\left. m \right|n} \right)$ and $\mathfrak{o}\mathfrak{s}\mathfrak{p}\left( {\left. m \right|2n} \right)$ .     

On the Fourier--Haar Coefficients of Functions of Several Variables with Bounded Vitali Variation

In this paper, we study the behavior of the Fourier--Haar coefficients $a_{m_1 , \ldots ,m_n } \left( f \right)$ of functions $f$ Lebesgue integrable on the $n$ -dimensional cube $D_n = \left[ {0,1} \right]^n $ and having a bounded Vitali variation $V_{D_n } f$ on it. It is proved that $$\sum\limits_{m_1 = 2}^\infty \cdots \sum\limits_{m_n = 2}^\infty {\left| {a_{m_1 , \ldots ,m_n } \left( f \right)} \right|} \leqslant \left( {\frac{{2 + \sqrt 2 }}{3}} \right)^n {\text{ }}.{\text{ }}V_{D_n } f$$ and shown that this estimate holds for some function of bounded finite nonzero Vitali variation.     

A Construction of Generalized Harish-Chandra Modules for Locally Reductive Lie Algebras

We study cohomological induction for a pair $ {\left( {\mathfrak{g},\mathfrak{k}} \right)} $ , $ \mathfrak{g} $ being an infinitedimensional locally reductive Lie algebra and $ \mathfrak{k} \subset \mathfrak{g} $ being of the form $ \mathfrak{k}_{0} \subset C_{\mathfrak{g}} {\left( {\mathfrak{k}_{0} } \right)} $ , where $ \mathfrak{k}_{0} \subset \mathfrak{g} $ is a finite-dimensional reductive in $ \mathfrak{g} $ subalgebra and $ C_{\mathfrak{g}} {\left( {\mathfrak{k}_{0} } \right)} $ is the centralizer of $ \mathfrak{k}_{0} $ in $ \mathfrak{g} $ . We prove a general nonvanishing and $ \mathfrak{k} $ -finiteness theorem for the output. This yields, in particular, simple $ {\left( {\mathfrak{g},\mathfrak{k}} \right)} $ -modules of finite type over k which are analogs of the fundamental series of generalized Harish-Chandra modules constructed in [PZ1] and [PZ2]. We study explicit versions of the construction when $ \mathfrak{g} $ is a root-reductive or diagonal locally simple Lie algebra.     

On Certain Integral Operators of Fractional Type

In this paper we study integral operators of the form $$T\,f\left( x \right) = \int {k_1 \left( {x - a_1 y} \right)k_2 \left( {x - a_2 y} \right)...k_m \left( {x - a_m y} \right)f\left( y \right)dy} ,$$ $$k_i \left( y \right) = \sum\limits_{j \in Z} {2^{\frac{{jn}}{{q_i }}} } \varphi _{i,j} \left( {2^j y} \right),\,1 \leqq q_i < \infty ,\frac{1}{{q_1 }} + \frac{1}{{q_2 }} + ... + \frac{1}{{q_m }} = 1 - r,$$ $0 \leqq r < 1$ , and $\varphi _{i,j}$ satisfying suitable regularity conditions. We obtain the boundedness of $T:L^p \left( {R^n } \right) \to T:L^q \left( {R^n } \right)$ for $1 < p < \frac{1}{r}$ and $\frac{1}{q} = \frac{1}{p} - r$ .     

On the integrability of radial basis functions

In this paper we investigate the integrability of certain radial basis functions. From the following forms of function σ, $$\varphi \left( r \right) = \left\{ \begin{gathered} \sum\limits_{k = 0}^{d + [a]} {c_k r^{a - k} + g(r) } r > A, \hfill \\ \sum\limits_{k = 0}^{d + [a]} {c_k r^{a - k} \ln r + g(r), } r > A. \hfill \\ \end{gathered} \right.$$ where A≧0 and $g \circ || \circ || \in L^1 \left( {R^d } \right)$ , we construct the function $$\psi (t) = \sum\limits_{j \in J} {a_j \varphi \left( {||t - t_j ||} \right),} $$ where J is a finite index set, $\left\{ {a_j } \right\}_{j \in J} \subseteq R$ and $\left\{ {t_j } \right\}_{j \in J} \subseteq R^d $ . We show that if $\hat \psi $ is continuous at the origin, the ψ is integrable in R^d.     

Approximation by M. Riesz Kernels in L p for p<1

Let α > 0. We consider the linear span $\mathfrak{X}_\alpha \left( {\mathbb{R}^n } \right)$ of scalar Riesz's kernels $\left\{ {\tfrac{1}{{\left| {x - a} \right|^\alpha }}} \right\}_{a \in \mathbb{R}^n }$ and the linear span $\mathfrak{Y}_\alpha \left( {\mathbb{R}^n } \right)$ of vector Riesz's kernels $\left\{ {\tfrac{1}{{\left| {x - a} \right|^{\alpha + 1} }}\left( {x - a} \right)} \right\}_{a \in \mathbb{R}^n }$ . We study the following problems. (1) When is the intersection $\mathfrak{X}_\alpha \left( {\mathbb{R}^n } \right) \cap L^p \left( {\mathbb{R}^n } \right)$ dense in L^p(?ⁿ)? (2) When is the intersection $\mathfrak{Y}_\alpha \left( {\mathbb{R}^n } \right) \cap L^p \left( {\mathbb{R}^n ,\mathbb{R}^n } \right)$ dense in L^p(?ⁿ, ?ⁿ)? Bibliography: 15 titles.     

Linear recurring sequences and subfield subcodes of cyclic codes

Linear recurring sequences over finite fields play an important role in coding theory and cryptography. It is known that subfield subcodes of linear codes yield some good codes. In this paper, we study linear recurring sequences and subfield subcodes. Let Mqm(f(x)) denote the set of all linear recurring sequences over Fqm with characteristic polynomial f(x) over Fqm . Denote the restriction of Mqm(f(x)) to sequences over Fq and the set after applying trace function to each sequence in Mqm(f(x)) by Mqm(f(x)) | Fq and Tr( Mqm(f(x))), respectively. It is shown that these two sets are both complete sets of linear recurring sequences over Fq with some characteristic polynomials over Fq. In this paper, we firstly determine the characteristic polynomials for these two sets. Then, using these results, we determine the generator polynomials of subfield subcodes and trace codes of cyclic codes over Fqm .     

On Total Incomparability of Mixed Tsirelson Spaces

We give criteria of total incomparability for certain classes of mixed Tsirelson spaces. We show that spaces of the form $T\left[ {\left( {\mathcal{M}_k ,\theta _k } \right)_{k = 1}^l } \right]$ with index $i\left( {\mathcal{M}_k } \right)$ finite are either c ₀ or $\ell _p $ saturated for some p and we characterize when any two spaces of such a form are totally incomparable in terms of the index $i\left( {\mathcal{M}_k } \right)$ and the parameter θ _k . Also, we give sufficient conditions of total incomparability for a particular class of spaces of the form $T\left[ {\left( {\mathcal{A}_k ,\theta _k } \right)_{k = 1}^\infty } \right]$ in terms of the asymptotic behaviour of the sequence $\left\| {\sum\limits_{j = 1}^n {e_i } } \right\|$ where (e _i is the canonical basis.     

Bethe subalgebras of the group algebra of the symmetric group

We introduce families $ \mathcal{B}_n^S\left( {{z_1},\ldots,{z_n}} \right) $ and $ \mathcal{B}_{{n,\hbar}}^S\left( {{z_1},\ldots,{z_n}} \right) $ of maximal commutative subalgebras, called Bethe subalgebras, of the group algebra $ \mathbb{C}\left[ {\mathfrak{S}n} \right] $ of the symmetric group. Bethe subalgebras are deformations of the Gelfand?Zetlin subalgebra of $ \mathbb{C}\left[ {\mathfrak{S}n} \right] $ . We describe various properties of Bethe subalgebras.     

Constant-Sign Periodic and Almost Periodic Solutions for a System of Integral Equations

We consider the following system of integral equations $${u_{i}(t)=\int\nolimits_{I} g_{i}(t, s)f(s, u_{1}(s), u_{2}(s), \cdots, u_{n}(s))ds, \quad t \in I, \ 1 \leq i\leq n}$$ where I is an interval of $\mathbb{R}$ . Our aim is to establish criteria such that the above system has a constant-sign periodic and almost periodic solution (u ₁, u ₂,…,u _n ) when I is an infinite interval of $\mathbb{R}$ , and a constant-sign periodic solution when I is a finite interval of $\mathbb{R}$ . The above problem is also extended to that on $\mathbb{R}$ $$u_{i} {\left( t \right)} = {\int_\mathbb{R} {g_{i} {\left( {t,s} \right)}f_{i} {\left( {s,u_{1} {\left( s \right)},u_{2} {\left( s \right)}, \cdots ,u_{n} {\left( s \right)}} \right)}ds\quad t \in \mathbb{R},\quad 1 \leqslant i \leqslant n.} }$$      

Orthogonality properties of linear combinations of orthogonal polynomials II

Let and be polynomials orthogonal on the unit circle with respect to the measures dσ and dμ, respectively. In this paper we consider the question how the orthogonality measures dσ and dμ are related to each other if the orthogonal polynomials are connected by a relation of the form , for , where . It turns out that the two measures are related by if , where and are known trigonometric polynomials of fixed degree and where the 's are the zeros of on . If the 's and 's are uniformly bounded then (under some additional conditions) much more can be said. Indeed, in this case the measures dσ and dμ have to be of the form and , respectively, where are nonnegative trigonometric polynomials. Finally, the question is considered to which weight functions polynomials of the form where denotes the reciprocal polynomial of , can be orthogonal. This revised version was published online in June 2006 with corrections to the Cover Date.     

Best Uniform Rational Approximations of Functions by Orthoprojections

Let C[-1,1] be the Banach space of continuous complex functions $f$ on the interval [-1,1] equipped with the standard maximum norm $\left\| f \right\|$ ; let $\omega \left( \cdot \right) = \omega \left( { \cdot ,f} \right)$ be the modulus of continuity of $f$ ; and let $R_n = R_n \left( f \right)$ be the best uniform approximation of $f$ by rational functions (r.f.) whose degrees do not exceed $n = 1, 2, \ldots $ . The space C[-1,1] is also regarded as a pre-Hilbert space with respect to the inner product given by $\left( {f,g} \right) = \left( {1/\pi } \right)\int_{ - 1}^1 {f\left( x \right)g\left( x \right)} \left( {1 - x^2 } \right)^{ - 1/2} dx$ . Let $z_n = \{ z_1 , z_2 , \ldots z_n \} $ be a set of points located outside the interval [-1,1]. By $F\left( { \cdot ,f,z_n } \right)$ we denote an orthoprojection operator acting from the pre-Hilbert space C[-1,1] onto its ( ${n + 1}$ )-dimensional subspace consisting of rational functions whose poles (with multiplicity taken into account) can only be points of the set $z_n $ . In this paper, we show that if $f$ is not a rational function of degree $ \leqslant n$ , then we can find a set of points $z_n = z_n \left( f \right)$ such that $\left\| {f\left( \cdot \right) - F\left( { \cdot ,f,z_n } \right)} \right\| \leqslant 12R_n ln\frac{3}{{\omega ^{ - 1} \left( {R_n /3} \right)}}.$      

Distribution of integral functionals of a Brownian motion process

In this paper one considers methods which enable one to determine the distribution of certain functionals of a Brownian motion process. Among such functionals we have: the positive continuous additive functional of a Brownian motion, defined by the formula $$A\left( t \right) = \int\limits_{ - \infty }^\infty {\hat t\left( {t, y} \right)dF\left( y \right),} $$ where \(\hat t\left( {t, y} \right)\) is the Brownian local time process while F(y) is a monotonically increasing right continuous function; the functional $$B\left( t \right) = \mathop {\mathop \smallint \limits_{ - \infty } }\nolimits^\infty f\left( {y,\hat t\left( {t, y} \right)} \right)dy,$$ where f(y, x) is a continuous function; and the functional $$C\left( t \right) = \mathop {\mathop \smallint \limits_0 }\nolimits^t f\left( {w\left( s \right),\hat t\left( {sr} \right)} \right)ds$$ As an application of these methods one considers some concrete functionals such that \(\hat t^{ - 1} \left( z \right) = \min \left\{ {s:\hat t\left( {s, o} \right) = z} \right\},\mathop {\mathop \smallint \limits_{ - \infty } }\nolimits^\infty \hat t^2 \left( {t, y} \right)dy,\mathop {\sup }\limits_{y \in R^1 } \hat t\left( {T, y} \right)\) , where T is an exponential random time, independent of \(\hat t\left( {t, y} \right)\) .     

Sharp weak-type inequalities for Hilbert transform and Riesz projection

The paper is devoted to the study of the weak norms of the classical operators in the vector-valued setting.

1. Let S, H denote the singular integral involution operator and the Hilbert transform on $L^p \left( {\mathbb{T}, \ell _\mathbb{C}^2 } \right)$ , respectively. Then for 1 ≤ p ≤ 2 and any f, $$\left\| {\mathcal{S}f} \right\|_{p,\infty } \leqslant \left( {\frac{1} {\pi }\int_{ - \infty }^\infty {\frac{{\left| {\tfrac{2} {\pi }\log \left| t \right|} \right|^p }} {{t^2 + 1}}dt} } \right)^{ - 1/p} \left\| f \right\|p,$$ $$\left\| {\mathcal{H}f} \right\|_{p,\infty } \leqslant \left( {\frac{1} {\pi }\int_{ - \infty }^\infty {\frac{{\left| {\tfrac{2} {\pi }\log \left| t \right|} \right|^p }} {{t^2 + 1}}dt} } \right)^{ - 1/p} \left\| f \right\|p.$$ Both inequalities are sharp.
2. Let P ₊ and P _? stand for the Riesz projection and the co-analytic projection on $L^p \left( {\mathbb{T}, \ell _\mathbb{C}^2 } \right)$ , respectively. Then for 1 ≤ p ≤ 2 and any f, $$\left\| {P + f} \right\|_{p,\infty } \leqslant \left\| f \right\|_p ,$$ $$\left\| {P - f} \right\|_{p,\infty } \leqslant \left\| f \right\|_p .$$ Both inequalities are sharp.
3. We establish the sharp versions of the estimates above in the nonperiodic case.

The results are new even if the operators act on complex-valued functions. The proof rests on the construction of an appropriate plurisubharmonic function and probabilistic techniques.     

Mixing via the extended family

In this paper,the relationship between the extended family and several mixing properties in measuretheoretical dynamical systems is investigated.The extended family eF related to a given family F can be regarded as the collection of all sets obtained as"piecewise shifted"members of F.For a measure preserving transformation T on a Lebesgue space(X,B,μ),the sets of"accurate intersections of order k"defined below are studied,Nε(A0,A1,...,Ak)=n∈Z+:μk i=0T inAiμ(A0)μ(A1)μ(Ak)ε,for k∈N,A0,A1,...,Ak∈B and ε0.It is shown that if T is weakly mixing(mildly mixing)then for any k∈N,all the sets Nε(A0,A1,...,Ak)have Banach density 1(are in(eFip),i.e.,the dual of the extended family related to IP-sets).     

Compatible Adapted Pairs and a Common Slice Theorem for Some Centralizers

Let $ \mathfrak{g} $ be a simple Lie algebra and $ x \in \mathfrak{g} $ nilpotent. We derive a criterion for when the G-orbits in $ \mathfrak{g}* $ and the G ^x -orbits in $ {\left( {\mathfrak{g}^{x} } \right)}^{*} $ admit a common slice, applicable (in principle) when x is of Bala–Carter type. When $ \mathfrak{g} $ is of type A, or if x is the highest root vector with g not of type E₈, we show that this criterion is satisfied. In these cases we also show that the Mishchenko–Fomenko shift of argument produces a maximal Poisson commutative polynomial subalgebra of $ S{\left( {\mathfrak{g}^{x} } \right)} $ which maps isomorphically by restriction of functions to an affine translate of a subspace of $ {\left( {\mathfrak{g}^{x} } \right)}^{*} $ . It is conjectured that the above criterion is satisfied when the dimensions of certain weights spaces, which can be computed purely combinatorially, are decreasing.     

Representation of cyclotomic fields and their subfields

Let $\mathbb{K}$ be a finite extension of a characteristic zero field $\mathbb{F}$ . We say that a pair of n × n matrices (A,B) over $\mathbb{F}$ represents $\mathbb{K}$ if $\mathbb{K} \cong {{\mathbb{F}\left[ A \right]} \mathord{\left/ {\vphantom {{\mathbb{F}\left[ A \right]} {\left\langle B \right\rangle }}} \right. \kern-0em} {\left\langle B \right\rangle }}$ , where $\mathbb{F}\left[ A \right]$ denotes the subalgebra of $\mathbb{M}_n \left( \mathbb{F} \right)$ containing A and 〈B〉 is an ideal in $\mathbb{F}\left[ A \right]$ , generated by B. In particular, A is said to represent the field $\mathbb{K}$ if there exists an irreducible polynomial $q\left( x \right) \in \mathbb{F}\left[ x \right]$ which divides the minimal polynomial of A and $\mathbb{K} \cong {{\mathbb{F}\left[ A \right]} \mathord{\left/ {\vphantom {{\mathbb{F}\left[ A \right]} {\left\langle {q\left( A \right)} \right\rangle }}} \right. \kern-0em} {\left\langle {q\left( A \right)} \right\rangle }}$ . In this paper, we identify the smallest order circulant matrix representation for any subfield of a cyclotomic field. Furthermore, if p is a prime and $\mathbb{K}$ is a subfield of the p-th cyclotomic field, then we obtain a zero-one circulant matrix A of size p × p such that (A, J) represents $\mathbb{K}$ , where J is the matrix with all entries 1. In case, the integer n has at most two distinct prime factors, we find the smallest order 0, 1-companion matrix that represents the n-th cyclotomic field. We also find bounds on the size of such companion matrices when n has more than two prime factors.