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...

Abstract Kernelized Gram matrix $W$ constructed from data points $\{x_i\}_{i=1}^N$ as $W_{ij}= k_0( \frac{ \| x_i - x_j \|^2} {\sigma ^2} ) $ is widely used in graph-based geometric analysis and unsupervised learning. An important question how to choose the kernel bandwidth $\sigma $, a common practice called self-tuned adaptively sets _i$ at each point $x_i$ by $k$-nearest neighbor (kNN) dista...

The maintenance scheduling of thermal generators is a large-scale combinatorial optimization problem with constraints. In this paper we introduce the Rank-Based Ant System algorithm based version of the Ant System. This algorithm reinforces local search in neighborhood of the best solution found in each iteration while implementing methods to slow convergence and facilitate exploration.Rank-Bas...