Growth rates of algebras,III: finite solvable algebras

We investigate how the behavior of the function d_A(n), which gives the size of a least size generating set for Aⁿ, influences the structure of a finite solvable algebra A.     

Non-low<Subscript>2</Subscript>-ness and computable Lipschitz reducibility

In this paper, we prove that if a c.e. Turing degree d is non-low₂, then there are two left-c.e. reals β ₀, β ₁ in d, such that, if β ₀ is wtt-reducible to a left-c.e. real α, then β ₁ is not computable Lipschitz (cl-) reducible to α. As a corollary, d contains a left-c.e. real which is not cl-reducible to any complex (wtt-complete) left-c.e. real.     

Searching for a Best Least Absolute Deviations Solution of an Overdetermined System of Linear Equations Motivated by Searching for a Best Least Absolute Deviations Hyperplane on the Basis of Given Data

We consider the problem of searching for a best LAD-solution of an overdetermined system of linear equations Xa=z, X∈?^m×n, m≥n, \(\mathbf{a}\in \mathbb{R}^{n}, \mathbf {z}\in\mathbb{R}^{m}\). This problem is equivalent to the problem of determining a best LAD-hyperplane x?a ^T x, x∈? ⁿ on the basis of given data \((\mathbf{x}_{i},z_{i}), \mathbf{x}_{i}= (x_{1}^{(i)},\ldots,x_{n}^{(i)})^{T}\in \mathbb{R}^{n}, z_{i}\in\mathbb{R}, i=1,\ldots,m\), whereby the minimizing functional is of the form

$F(\mathbf{a})=\|\mathbf{z}-\mathbf{Xa}\|_1=\sum_{i=1}^m|z_i-\mathbf {a}^T\mathbf{x}_i|.$

An iterative procedure is constructed as a sequence of weighted median problems, which gives the solution in finitely many steps. A criterion of optimality follows from the fact that the minimizing functional F is convex, and therefore the point a ^?∈? ⁿ is the point of a global minimum of the functional F if and only if 0∈?F(a ^?).

Motivation for the construction of the algorithm was found in a geometrically visible algorithm for determining a best LAD-plane (x,y)?αx+βy, passing through the origin of the coordinate system, on the basis of the data (x _i ,y _i ,z _i ),i=1,…,m.     

Geometric Realizations of Lusztig’s Symmetries of Symmetrizable Quantum Groups

Let U be the quantum group and f be the Lusztig’s algebra associated with a symmetrizable generalized Cartan matrix. The algebra f can be viewed as the positive part of U. Lusztig introduced some symmetries T _i on U for all i ∈ I. Since T _i (f) is not contained in f, Lusztig considered two subalgebras _i f and ⁱ f of f for any i ∈ I, where _i f={x ∈ f | T _i (x) ∈ f} and \({^{i}\mathbf {f}}=\{x\in \mathbf {f}\,\,|\,\,T^{-1}_{i}(x)\in \mathbf {f}\}\). The restriction of T _i on _i f is also denoted by \(T_{i}:{_{i}\mathbf {f}}\rightarrow {^{i}\mathbf {f}}\). The geometric realization of f and its canonical basis are introduced by Lusztig via some semisimple complexes on the variety consisting of representations of the corresponding quiver. When the generalized Cartan matrix is symmetric, Xiao and Zhao gave geometric realizations of Lusztig’s symmetries in the sense of Lusztig. In this paper, we shall generalize this result and give geometric realizations of _i f, ⁱ f and \(T_{i}:{_{i}\mathbf {f}}\rightarrow {^{i}\mathbf {f}}\) by using the language ’quiver with automorphism’ introduced by Lusztig.     

Curvature inequalities for operators of the Cowen–Douglas class

Let T be an operator tuple in the Cowen–Douglas class B _n (Ω) for Ω ? C ^m . The kernels Ker(T ? w) ^l , for w ∈ Ω, l = 1, 2, ···, define Hermitian vector bundles E _T ^l over Ω. We prove certain negativity of the curvature of E _T ^l . We also study the relation between certain curvature inequality and the contractive property of T when Ω is a planar domain.     

On Ramanujan sums over the Gaussian integers

This note deals with Ramanujan sums c _m (n) over the ring ?[i], in particular with asymptotics for sums of c _m (n) taken over both variables m, n.     

Homogenization of hyperbolic equations

We consider a self-adjoint matrix elliptic operator A _{ε, ε} > 0, on L ₂(R ^d ;C ⁿ ) given by the differential expression b(D)*g(x/ε)b(D). The matrix-valued function g(x) is bounded, positive definite, and periodic with respect to some lattice; b(D) is an (m × n)-matrix first order differential operator such that m ≥ n and the symbol b(ξ) has maximal rank. We study the operator cosine cos(τA _ε ^1/2 ), where τ ∈ R. It is shown that, as ε → 0, the operator cos(τA _ε ^1/2 ) converges to cos(τ(A ⁰)^1/2) in the norm of operators acting from the Sobolev space H ^s (R ^d ;C ⁿ ) (with a suitable s) to L ₂(R ^d ;C ⁿ ). Here A ₀ is the effective operator with constant coefficients. Sharp-order error estimates are obtained. The question about the sharpness of the result with respect to the type of the operator norm is studied. Similar results are obtained for more general operators. The results are applied to study the behavior of the solution of the Cauchy problem for the hyperbolic equation ? _τ ² u _ε (x, τ) = ?A _ε u _ε (x, τ).     

On the homotopy type of (<Emphasis Type="Italic">n</Emphasis>-1)-connected (3<Emphasis Type="Italic">n</Emphasis>+1)-dimensional free chain Lie algebra

Let R be a subring ring of Q. We reserve the symbol p for the least prime which is not a unit in R; if R ?Q, then p=∞. Denote by DGL _n ^np , n≥1, the category of (n-1)-connected np-dimensional differential graded free Lie algebras over R. In [1] D. Anick has shown that there is a reasonable concept of homotopy in the category DGL _n ^np . In this work we intend to answer the following two questions: Given an object (L(V), ?) in DGL _n ³ⁿ⁺² and denote by S(L(V), ?) the class of objects homotopy equivalent to (L(V), ?). How we can characterize a free dgl to belong to S(L(V), ?)? Fix an object (L(V), ?) in DGL _n ³ⁿ⁺² . How many homotopy equivalence classes of objects (L(W), δ) in DGL _n ³ⁿ⁺² such that H _* (W, d′)?H _* (V, d) are there? Note that DGL _n ³ⁿ⁺² is a subcategory of DGL _n ^np when p>3. Our tool to address this problem is the exact sequence of Whitehead associated with a free dgl.     

Homoclinic points and isomorphism rigidity of algebraic ℤ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>-actions on zero-dimensional compact abelian groups

Letd>1, and letα andβ be mixing ? ^d -actions by automorphisms of zero-dimensional compact abelian groupsX andY, respectively. By analyzing the homoclinic groups of certain sub-actions ofα andβ we prove that, if the restriction ofα to some subgroup Γ ? ? ^d of infinite index is expansive and has completely positive entropy, then every measurable factor mapφ: (X, α)→(Y, β) is almost everywhere equal to an affine map. The hypotheses of this result are automatically satisfied if the actionα contains an expansive automorphismα ⁿ ,n ∈ ? ^d , or ifα arises from a nonzero prime ideal in the ring of Laurent polynomials ind variables with coefficients in a finite prime field. Both these corollaries generalize the main theorem in [9]. In several examples we show that this kind of isomorphism rigidity breaks down if our hypotheses are weakened.     

Isotopes of the alternative monster and the Skosyrsky algebra

We prove that the isotopes of the alternative monster and the Skosyrsky algebra satisfy the identity П_i=1⁴ [x_i, y_i] = 0. Hence, the algebras themselves satisfy the identity П_i=1⁴ (c, x_i, y_i) = 0. We also show that none of the identities П_i=1ⁿ(c, x_i, y_i) = 0 holds in all commutative alternative nil-algebras of index 3. Thus, we refute the Grishkov–Shestakov hypothesis about the structure of the free finitely generated commutative alternative nil-algebras of index 3.     

Some Structural Aspects of the Katětov Order on Borel Ideals

We prove that the Katětov order on Borel ideals (1) contains a copy of \(\mathcal {P}(\omega )/\mathbf {Fin}\), consequently it has increasing and decreasing chains of lenght ??; (2) the sequence F i n ^α (α < ω ₁) is a strictly increasing chain; and (3) in the Cohen model, Katětov order does not contain any increasing nor decreasing chain of length ??, answering a question of Hru?ák (2011).     

Equations in finite fields with restricted solution sets. I (Character sums)

In earlier papers, for “large” (but otherwise unspecified) subsets A, B of Z _p and for h(x) ∈ Z _p [x], Gyarmati studied the solvability of the equations a + b = h(x), resp. ab = h(x) with a ∈ A, b ∈ B, x ∈ Z _p , and for large subsets A, B, C, D of Z _p Sárközy showed the solvability of the equations a + b = cd, resp. ab + 1 = cd with a ∈ A, b ∈ B, c ∈ C, d ∈ D. In this series of papers equations of this type will be studied in finite fields. In particular, in Part I of the series we will prove the necessary character sum estimates of independent interest some of which generalize earlier results.     

Generalizations of some zero sum theorems

Given an abelian group G of order n, and a finite non-empty subset A of integers, the Davenport constant of G with weight A, denoted by D _A (G), is defined to be the least positive integer t such that, for every sequence (x ₁,..., x _t ) with x _i ?∈?G, there exists a non-empty subsequence \((x_{j_1},\ldots, x_{j_l})\) and a _i ?∈?A such that \(\sum_{i=1}^{l}a_ix_{j_i} = 0\). Similarly, for an abelian group G of order n, E _A (G) is defined to be the least positive integer t such that every sequence over G of length t contains a subsequence \((x_{j_1} ,\ldots, x_{j_n})\) such that \(\sum_{i=1}^{n}a_ix_{j_i} = 0\), for some a _i ?∈?A. When G is of order n, one considers A to be a non-empty subset of {1,..., n???1 }. If G is the cyclic group \({\Bbb Z}/n{\Bbb Z}\), we denote E _A (G) and D _A (G) by E _A (n) and D _A (n) respectively.In this note, we extend some results of Adhikari et al (Integers 8 (2008) Article A52) and determine bounds for \(D_{R_n}(n)\) and \(E_{R_n}(n)\), where \(R_n = \{x^2 : x \in (\mathbb{Z}/n\mathbb{ Z})^*\}\). We follow some lines of argument from Adhikari et al (Integers 8 (2008) Article A52) and use a recent result of Yuan and Zeng (European J. Combinatorics 31 (2010) 677–680), a theorem due to Chowla (Proc. Indian Acad. Sci. (Math. Sci.) 2 (1935) 242–243) and Kneser’s theorem (Math. Z. 58 (1953) 459–484; 66 (1956) 88–110; 61 (1955) 429–434).     

Simplex—Karyon Algorithm of Multidimensional Continued Fraction Expansion

A simplex–karyon algorithm for expanding real numbers α = (α₁,..., α _d ) in multidimensional continued fractions is considered. The algorithm is based on a (d + 1)-dimensional superspace S with embedded hyperplanes: a karyon hyperplane K and a Farey hyperplane F. The approximation of numbers α by continued fractions is performed on the hyperplane F, and the degree of approximation is controlled on the hyperplane K. A local ?(r)-strategy for constructing convergents is chosen, with a free objective function ?(r) on the hyperplane K.     

Norms on L of Periodic Interpolation Splines with Equidistant Nodes

We consider the set S _r,n of periodic (with period 1) splines of degree r with deficiency 1 whose nodes are at n equidistant points x_i=i / n. For n-tuples y = (y₀, ... , y_n-1), we take splines s _r,n (y, x) from S _r,n solving the interpolation problem

$$s_{r,n} (y,t_i ) = y_i,$$

where t _i = x _i if r is odd and t _i is the middle of the closed interval [x _i , x _i+1 ] if r is even. For the norms L _r,n ^* of the operator y → s _r,n (y, x) treated as an operator from l¹ to L¹ [0, 1] we establish the estimate

$$L_{r,n}^ * = \frac{4}{{\pi ^2 n}}log min(r,n) + O\left( {\frac{1}{n}} \right)$$

with an absolute constant in the remainder. We study the relationship between the norms L _r,n ^* and the norms of similar operators for nonperiodic splines.

     

Asymptotic Analysis for Random Walks with Nonidentically Distributed Jumps Having Finite Variance

Let ξ₁,ξ₂,... be independent random variables with distributions F₁F₂,... in a triangular array scheme (F _i may depend on some parameter). Assume that Eξ _i = 0, Eξ _i ² < ∞, and put \(S_n = \sum {_{i = 1}^n \;} \xi _i ,\;\overline S _n = \max _{k \leqslant n} S_k\). Assuming further that some regularly varying functions majorize or minorize the “averaged” distribution \(F = \frac{1}{n}\sum {_{i = 1}^n F_i }\), we find upper and lower bounds for the probabilities P(S _n > x) and \(P(\bar S_n > x)\). We also study the asymptotics of these probabilities and of the probabilities that a trajectory {S _k } crosses the remote boundary {g(k)}; that is, the asymptotics of P(max_k≤n(S _k ? g(k)) > 0). The case n = ∞ is not excluded. We also estimate the distribution of the first crossing time.     

Calculation of Geometric Probabilities Using Covariogram of Convex Bodies

In the paper, a formula to calculate the probability that a random segment L(ω, u) in R ⁿ with a fixed direction u and length l lies entirely in the bounded convex body D ? R ⁿ (n ≥ 2) is obtained in terms of covariogram of the body D. For any dimension n ≥ 2, a relationship between the probability P(L(ω, u) ? D) and the orientation-dependent chord length distribution is also obtained. Using this formula, we obtain the explicit form of the probability P(L(ω, u) ? D) in the cases where D is an n-dimensional ball (n ≥ 2), or a regular triangle on the plane.     

Hamiltonian cycles in Dirac graphs

We prove that for any n-vertex Dirac graph (graph with minimum degree at least n/2) G=(V,E), the number, Ψ(G), of Hamiltonian cycles in G is at least

$exp_2 [2h(G) - n\log e - o(n)],$

where h(G)=maxΣ _e x _e log(1/x _e ), the maximum over x: E → ?⁺ satisfying Σ _e?υ x _e = 1 for each υ ∈ V, and log =log₂. (A second paper will show that this bound is tight up to the o(n).)

We also show that for any (Dirac) G of minimum degree at least d, h(G) ≥ (n/2) logd, so that Ψ(G) > (d/(e + o(1))) ⁿ . In particular, this says that for any Dirac G we have Ψ(G) > n!/(2 + o(1)) ⁿ , confirming a conjecture of G. Sárközy, Selkow, and Szemerédi which was the original motivation for this work.