In this Note, we consider traveling waves in a reaction-diffusion equation dimension one. Motivated by the motion of dislocations crystals, introduce an additive parameter σ reaction term, which may be interpreted as exterior force applied on crystal. Under certain natural assumptions and for every value σ∈[σ - ,σ + ], show existence ϕ velocity c. The range σ∈(σ ) corresponds to bistable cases ...

In 2020, Oh and Thomas constructed a virtual cycle [ X stret...

Hint . Without loss of generality, assume that L = L(G), where G = (V,Σ, P, S) is in Chomsky normal form, and let R = L(D), for some DFA D = (Q,Σ, δ, q0, F ). Use a kind of cross-product construction as sketched below. Construct a CFG G2 whose set of nonterminals is Q×N ×Q ∪ {S0}, where S0 is a new nonterminal, and whose productions are of the form: S0 → (q0, S, f), for every f ∈ F ; (p,A, δ(p,...

 d1△u− u+ u p vq + σ = 0 in Ω, d2△v − v + u r vs = 0 in Ω, ∂u ∂ν = ∂v ∂ν = 0 on ∂Ω where △ is the Laplace operator, Ω is a bounded smooth domain in R, n ≥ 1 and ν is the unit outer normal to ∂Ω. Under suitable conditions on the exponents p, q, r and s, different types of a priori estimates are obtained, existence and non-existence results of nontrivial solutions are derived, for both σ > 0 and...

A classical approach to accurately estimating the covariance matrix Σ of a p-variate normal distribution is to draw a sample of size n > p and form a sample covariance matrix. However, many modern applications operate with much smaller sample sizes, thus calling for estimation guarantees in the regime n p. We show that a sample of size n = O(m log p) is sufficient to accurately estimate in oper...

Let Tn(x) = Pn j=1 ±e2πij 2x where ± stands for a random choice of sign with equal probability. It is shown here that with high probability minx∈[0,1) |Tn(x)| < n−σ provided n is large and σ < 1/12. Similar results are proved for other powers than squares. The problem of determining the optimal σ is open. For the case Tn(x) = Pn j=1 rje 2πijdx, where d = 2, 3, . . . is fixed and with standard n...

In this article, semi-ring and semi-algebra of sets are formalized so as to construct a measure of a given set in the next step. Although a semi-ring of sets has already been formalized in [12], that is, strictly speaking, a definition of a quasi semi-ring of sets suggested in the last few decades [14]. We adopt a classical definition of a semi-ring of sets here to avoid such a confusion. Ring ...

We prove several dichotomy theorems which extend some known results on σ -bounded and σ -compactpointsets. In particular we show that, given a finite number of Δ1 equivalence relations F1 , . . . , Fn , any Σ1 set A of the Baire space either is covered by compact Δ1 sets and lightface Δ1 equivalence classes of the relations Fi , or A contains a superperfect subset which is pairwise Fi -inequiva...

Abstract In this study, the solution of Neumann problem associated with CR Yamabe operator on a subset Ω \Omega manifold mathvariant="double-struck">S 3 {{\mathbb{S}}}^{3} bounded by Clifford torus mathvariant="normal">Σ \Sigma is dis...