A natural generalization of graph Ramsey theory is the study of unavoidable subgraphs in large colored graphs. In this paper, we find a minimal family of unavoidable graphs in two-edgecolored graphs. Namely, for a positive even integer k, let Sk be the family of two-edge-colored graphs on k vertices such that one of the colors forms either two disjoint Kk/2s or simply one Kk/2. Bollobás conject...

We give algorithms for approximation by low-rank positive semidefinite (PSD) matrices. For symmetric input matrix A ∈ Rn×n, target rank k, and error parameter ε > 0, one algorithm finds with constant probability a PSD matrix Ỹ of rank k such that ‖A− Ỹ ‖2F ≤ (1+ε)‖A−Ak,+‖ 2 F , where Ak,+ denotes the best rank-k PSD approximation to A, and the norm is Frobenius. The algorithm takes time O(nnz(A...

and Applied Analysis 3 = [− 0.5k 1 x 1.5 1 + 0.5k 1 x 1 + 0.5 (r 1 − ε 12 − 0.25σ 2 1 ) x 0.5 1 − 0.5 (r 1 − ε 12 − 0.5σ 2 1 )] dt + [− 0.5k 2 x 1.5 2 + 0.5k 2 x 2 + 0.5 (r 2 − ε 21 − 0.25σ 2 2 ) x 0.5 2 − 0.5 (r 2 − ε 21 − 0.5σ 2 2 )] dt + [0.5 (x −0.5 1 − x −1 1 ) ε 12 x 2 + 0.5 (x −0.5 2 − x −1 2 ) ε 21 x 1 ] dt + 0.5 (x 0.5 1 − 1) σ 1 dB 1 (t) + 0.5 (x 0.5 2 − 1) σ 2 dB 2 (t) . (11) There e...

For uniform systems of spin-less fermions in d spatial dimensions with d > 1, interacting through the isotropic two-body potential v(r − r), a celebrated theorem due to Luttinger (1961) states that under the assumption of validity of the many-body perturbation theory the self-energy Σ(k; ε), with 0 ≤ k∼3kF (where kF stands for the Fermi wavenumber), satisfies the following universal asymptotic ...

Let (M, g̃) be an N -dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen-Cahn equation ε∆g̃u + (1− u)u = 0 in M, where ε is a small parameter. Let K ⊂M be an (N − 1)-dimensional smooth minimal submanifold that separatesM into two disjoint components. Assume that K is non-degenerate in the sense that it does not support non-trivial Jacobi fields, and that |AK...

This paper presents a filter-based artificial fish swarm algorithm for solving nonconvex constrained global optimization problems. Convergence to an ε-global minimizer is guaranteed. At each iteration k, the algorithm requires a (ρ(k),ε(k))-global minimizer of a bound constrained bi-objective subproblem, where as k→ ∞, ρ(k)→ 0 gives the constraint violation tolerance and ε(k)→ ε is the error bo...

and Applied Analysis 3 2. Preliminaries In this section, we complete notations and recall some known definitions. Let K X be the class of all nonempty and compact subset of X. If A ∈ K X we define the ε-neighbourhood of A as the set N A, ε {x ∈ X | d x,A < ε}, 2.1 where d x,A infa∈A‖x − a‖. The Hausdorff separation ρ A,B of A,B ∈ K X is defined by ρ A,B inf{ε > 0 | A ⊆ N B, ε }. 2.2 The Hausdor...

Let {Rt, 0 ≤ t ≤ 1} be a symmetric α-stable Riemann-Liouville process with Hurst parameterH > 0. Consider a translation invariant, β-self-similar, and p-pseudo-additive functional semi-norm ||.||. We show that if H > β+1/p and γ = (H − β− 1/p), then lim ε↓0 ε logP [||R|| ≤ ε] = −K ∈ [−∞, 0), with K finite in the Gaussian case α = 2. If α < 2, we prove that K is finite when R is continuous and H...

In this paper we investigate linear error correcting codes and projective caps related to the Grassmann embedding ε k of an orthogonal Grassmannian ∆k. In particular, we determine some of the parameters of the codes arising from the projective system determined by ε k (∆k). We also study special sets of points of ∆k which are met by any line of ∆k in at most 2 points and we show that their imag...