Szemerédi’s regularity lemma is an important tool in graph theory which has applications throughout combinatorics. In this paper we prove an analogue of Szemerédi’s regularity lemma in the context of abelian groups and use it to derive some results in additive number theory. The simplest is a structure theorm for sets which are almost sum-free. If A ⊆ {1, . . . , N} has δN2 triples (a1, a2, a3)...

Let G be an infinite connected graph with minimum degree δ and maximum degree ∆. Let Gp be a random induced subgraph of G obtained by selecting each vertex of G independently with probability p, 0 < p < 1, and let G≤k p be the induced subgraph of Gp obtained by deleting all vertices of Gp with degree greater than k in Gp. We show that if δ ≥ 6 and ∆/δ is not too large then G≤3 p almost surely h...

A 3 − arc of a graph G is a 4-tuple (y, a, b, x) of vertices such that both (y, a, b) and (a, b, x) are paths of length two in G. Let ←→ G denote the symmetric digraph of a graph G. The 3-arc graph X(G) of a given graph G is defined to have vertices the arcs of ←→ G . Two vertices (ay), (bx) are adjacent in X(G) if and only if (y, a, b, x) is a 3-arc of G. The purpose of this work is to study t...

Labelling Neural Matrix and Collective Dynamics Mau-Hsiang Shih 施茂祥 National Normal University, Taiwan Email: [email protected] Abstract The brain is considered to be a complex, self-organizing system; it consists of enormous numbers of interacting neurons and perpetually weaves its intricate web. Scientists believe that collective dynamics of the brain is deeply entwined with the neural ...

1. Proof. (a) Let x ∈ [0, 1] and δ > 0, we want to show that there exist n such that nλ mod 1 ∈ (x−δ, x+δ), for which it suffices to show that there exists n0 and y ∈ (−δ, δ) such that n0λ ≡ y (mod 1). Then take n to be an appropriate multiple of n0. To prove the reduced proposition, choose m such that 10−m < δ. Note that there exist n1 < n2 such that (n1λ mod 1) and (n2λ mod 1) have the same f...

RATIONALE Differential effects of δ(B) and δ(C) subtypes of Ca²⁺/calmodulin-dependent protein kinase (CaMKII) on cardiomyocyte Ca²⁺ handling and survival have been suggested to result from their respective nuclear versus cytosolic localizations. CaMKIIδ subtype localization and its relationship to enzyme activation and target phosphorylation have not, however, been systematically evaluated. O...

In this paper, we derive some algebraic identities on right and left neighbors R(F ) and L(F ) of an indefinite binary quadratic form F = F (x, y) = ax + bxy + cy of discriminant Δ = b − 4ac. We prove that the proper cycle of F can be given by using its consecutive left neighbors. Also we construct a connection between right and left neighbors of F . Keywords—Quadratic form, indefinite form, cy...

and Applied Analysis 3 V = ±√−α − 5B 2 β + 20CAβ, δ = 4β (−8CAB 2 + B 4 + 16A 2 C 2 ) , (8b) where A, B, and C are arbitrary constants, but C cannot be zero. Case 3. One has a −1 = 0, a 2 = 0, a 1 = 0, a −2 = 2A 2 √ 30β γ , a 0 = 2AC√ 30β γ , B = 0, V = ±√−α − 60CAβ, δ = −16βA 2 C 2 . (8c) Case 4. One has a −1 = 0, a 2 = 0, a 1 = 0, a −2 = 2A 2 √ 30β γ , a 0 = − −15 ± √165 15 CA√ 30β γ , B = 0,...

Abstract We derive the asymptotic behavior of following ruin probability: $$ \mathrm{P}\left\{\exists t\in G\left(\delta \right):{B}_H(t)-{c}_1t&gt;{q}_1u,{B}_H(t)-{c}_2t&gt;{q}_2u\right\},\kern0.72em H\in \left(0,1\right),u\to \infty, P ∃ t ∈ <m...

Let Δ = ABC be a reference triangle and Δ′ = A′B′C′ an inscribed triangle of Δ. We define the cevian projection of Δ′ as the cevian triangle ΔP of a certain point P . Given a point P not on a sideline, all inscribed triangles with common cevian projection ΔP form a family DP = {Δ(t) = AtBtCt, t ∈ R}. The parallels of the lines AAt, BBt, CCt through any point of a certain circumconic CP intersec...