in this paper, we first define spaces of single difference sequences defined by a sequence of orlicz functions without convexity and investigate their properties. then we extend this idea to spaces of double sequences and present a new matrix theoretic approach construction of such double sequence spaces.

Let Jα k be a real power of the integration operator Jk defined on Sobolev space W k p [0, 1]. We investigate the spectral properties of the operator Ak = ⊕n j=1 λjJ α k defined on ⊕n j=1W k p [0, 1]. Namely, we describe the commutant {Ak} ′, the double commutant {Ak} ′′ and the algebra AlgAk. Moreover, we describe the lattices LatAk and HypLatAk of invariant and hyperinvariant subspaces of Ak,...

We give three formulas expressing the Smale invariant of an immersion f of a (4k− 1)-sphere into (4k + 1)-space. The terms of the formulas are geometric characteristics of any generic smooth map g of any oriented 4k-dimensional manifold, where g restricted to the boundary is an immersion regularly homotopic to f in (6k − 1)-space. The formulas imply that if f and g are two non-regularly homotop...

Let X1, X2, . . . , Xn be independent random variables and Sk = Pk i=1 Xi. We show that for any constants ak, P( max 1≤k≤n ||Sk| − ak| > 11t) ≤ 30 max 1≤k≤n P(||Sk| − ak| > t). We also discuss similar inequalities for sums of Hilbert and Banach space valued random vectors.

In this section, we show the derivation of the posterior distribution over cluster-weights, π, conditioned on the cluster labels, z. We begin with the definition of a Dirichlet process from [1]. Definition A.1 (Dirichlet Process). Let H be a measure on a measureable space, Ω. If for any finite partition, (A1, A2, · · · , AK) of the space, the measure, G, on the partition follows the following D...

We describe a new algorithm called Frequent Directions for deterministic matrix sketching in the row-updates model. The algorithm is presented an arbitrary input matrix A ∈ Rn×d one row at a time. It performed O(d`) operations per row and maintains a sketch matrix B ∈ R`×d such that for any k < ` ‖AA−BB‖2 ≤ ‖A−Ak‖F /(`− k) and ‖A− πBk(A)‖F ≤ ( 1 + k `− k ) ‖A−Ak‖F . Here, Ak stands for the mini...

Note that μ0 is simply normalized Lebesque measure on B. The k-th Bergman space, Ak, is defined as the space of analytic functions on B which are square integrable with respect to the measure μk. Note that Ak = H (μk), where H (μk) be the L2(μk) -closure of the ball algebra A, and that Aj c Ak for j < k. The standard orthonormal base for Ak is given by eka = c(a, k, n)z a = c{a, ky n)r^e i0Clθl...

The statistics of the digits of a continued fraction, also known as partial quotients, have been studied at least since the time of Gauss. The usual measure m on the open interval (0, 1) gives a probability space U . Let ak, k ≥ 1 be integer-valued random variables which take α ∈ (0, 1) to the k partial quotient or digit in the continued fraction expansion α = 1/(a1 + 1/(a2 + · · · )). Let Sr =...