are well known. When {Ij}j∈Z is the collection of dyadic intervals, i.e., I0 = {0} and Ij = sgn(j)[2 , 2) for |j| > 0, the estimate (1.2) is the classical Littlewood–Paley inequality which is valid (as well as the reverse estimate with ≥ in place of ≤) for all p ∈ (1,∞). If the Ij are disjoint intervals of equal length, then (1.2) holds if and only if p ∈ [2,∞); this was first proved by L. Carl...

1. Introduction. By a labeled graph we shall mean a finite (non-empty) graph Γ, without loops or multiple edges, each of whose edges is labeled by an integer greater than or equal to 2. Let the vertices of Γ be s 1 , s 2 ,. .. , s n , and let the label on an edge with endpoints s i and s j be m ij ≥ 2. Define ab m to be the word abab. .. of length m. Then the Artin group AΓ associated with the ...

For n ∈ , we consider the problem of partitioning the interval [0, n) into k subintervals of positive integer lengths `1, . . . , `k such that the lengths satisfy a set of simple constraints of the form `i ij `j where ij is one of <, >, or =. In the full information case, ij is given for all 1 ≤ i, j ≤ k. In the sequential information information case, ij is given for all 1 < i < k and j = i ± ...

The maximum cut problem this paper deals with can be formulated as follows. Given an undirected simple graph G = (V,E) where V and E stand for the node and edge sets respectively, and given weights assigned to the edges: (wij)ij∈E , a cut δ(S), with S ⊆ V is de ned as the set of edges in E with exactly one endnode in S, i.e. δ(S) = {ij ∈ E | |S ∩ {i, j}| = 1}. The weight w(S) of the cut δ(S) is...

We study the behavior of the Josephson current IJ flowing in a finite-size Josephson junction in a superconducting ring in the presence of an externally applied magnetic field H, taking into account the effect of the shielding currents. The set of self-consistent equations for the system can be solved explicitly for IJ in the small self-inductance coefficient limit for not negligible effective ...

Abstract The paper is devoted to the generalization of Vinberg theory homogeneous convex cones. Such a cone described as set “positive definite matrices” in commutative algebra ? n Hermitian T-matrices. These algebras are Euclidean Jordan and consist × matrices A = ( ij ), where ii ? ?, entry for i &lt; j belongs some vector space V ; ????) $$ {a}_{ji}={a}_{ij}^{\ast }=\mathfrak{g}\left({a}_{ij...

Theorem 2. Picard-Lindelof existence/uniqueness theorem Suppose f : (I × U) ⊆ (R × R) → R is continuous in (t, y) and uniformly Lipschitz in y. Then ∀(τ, ξ) ∈ (I × U), the ODE [ẏ = f(t, y) with y(τ) = ξ] has a unique solution on {t ∈ I : |t− τ | < α} for some α > 0. Depends on: Completeness of R, Gronwalls lemma (for uniqueness Proof idea: 5 steps: 1. Choose a, b s.t. {t : |t − τ | < a} ⊆ I and...

A signed k-partite graph (signed multipartite graph) is a k-partite graph in which each edge is assigned a positive or a negative sign. If G(V1, V2, · · · , Vk) is a signed k-partite graph with Vi = {vi1, vi2, · · · , vini}, 1 ≤ i ≤ k, the signed degree of vij is sdeg(vij) = dij = d + ij − d − ij , where 1 ≤ i ≤ k, 1 ≤ j ≤ ni and d + ij(d − ij) is the number of positive (negative) edges inciden...

Let M m;n (respectively, H n) denote the space of m n complex matrices (respectively, n n Hermitian matrices). Let S H n be a closed convex set. We obtain necessary and suucient conditions for X 0 2 S to attain the maximum in the following concave maximization problem: maxf min (A + X) : X 2 Sg where A 2 H n is a xed matrix. Let denote the Hadamard (entrywise) product, i.e., given matrices A = ...

A Latin square L = L(`ij) over the set S = {0, 1, . . . , n − 1} is called totally non-polynomial over Zn iff 1. there are no polynomials Ui(y) ∈ Zn[y] such that Ui(j) = `ij for all i, j ∈ Zn; 2. there are no polynomials Vj(x) ∈ Zn[x] such that Vj(i) = `ij for all i, j ∈ Zn. In the presented paper we describe four possible constructions of such Latin squares which might be of particular interes...