The zero-divisor graph Π(R) of a commutative ring R is the whose vertices are elements such that u and v adjacent if only uv = 0. If graphs G H have same chromatic polynomial, then we say they chromatically equivalent (or χ−equivalent), written as ∼ H. Suppose uniquely determined by its polynomial. Then it said to be unique χ-unique). In this paper, discuss question: For which numbers n Π(Zn) χ...

For an irreducible complex character χ of the finite group G, let π(χ) denote set prime divisors degree χ(1) χ. Denote then by ρ(G) union all sets and σ(G) largest value |π(χ)|, as runs in Irr(G). The ρ-σ conjecture, formulated Bertram Huppert 80's, predicts that |ρ(G)|≤3σ(G) always holds, whereas |ρ(G)|≤2σ(G) holds if G is solvable; moreover, O. Manz T.R. Wolf proposed a “strengthened” form co...

A new analysis of (63)Cu and (17)O NMR shift data on La(1.85)Sr(0.15)CuO(4) is reported that supports earlier work arguing for a two-component description of La(1.85)Sr(0.15)CuO(4), but conflicts with the widely held view that the cuprates are a one-component system. The data are analyzed in terms of two components A and B with susceptibilities χ(AA), χ(AB)(= χ(BA)) and χ(BB). We find that abov...

Definition 9.1.3. An additive character of F is a character of the additive group F, i.e. a map χ : (F,+) → T, with χ(a+ b) = χ(a)χ(b). Now, let Fq denote the field of order q = p . We can define it by Fq := Fp[x]/(f), where f is any irreducible polynomial of degree k. We know that Fq is a cyclic group of order q − 1, and is generated by some g ∈ Fq (in other word Fq = 〈g〉). That is, g = 1 and ...

We prove that on a punctured oriented surface with Euler characteristic χ < 0, the maximal cardinality of a set of essential simple arcs that are pairwise nonhomotopic and intersecting at most once is 2|χ|(|χ|+1). This gives a cubic estimate in |χ| for a set of curves pairwise intersecting at most once on a closed surface. We also give polynomial estimates in |χ| for sets of arcs and curves pai...

For a character χ of finite group G, the co-degree is c (1)=[G:kerχ] χ(1). Let p be prime and let e positive integer. In this paper, we first show that if G p-solvable such e+1 ∤χ (1), for every irreducible then p-length not greater than e. Next, study groups satisfying condition 2 does divide co-degrees their characters.

Let V be a finite-dimensional complex vector space, and let W ⊂ GL(V ) be a complex reflection group. Let q be an indeterminate. For an irreducible character χ ∈ Irr(W ), let R(χ) denote its fake degree, a polynomial in q. We define R(χ) for reducible characters as well, by extending linearly. Let N∗ be the number of reflections in W , and let Ω = (ωχ,χ′)χ,χ′∈Irr(W ) be the square matrix with e...

A hereditary class G of graphs is χ-bounded if there is a χ-binding function, say f such that χ(G) ≤ f(ω(G)), for every G ∈ G, where χ(G) (ω(G)) denote the chromatic (clique) number of G. It is known that for every 2K2-free graph G, χ(G) ≤ ( ω(G)+1 2 ) , and the class of (2K2, 3K1)-free graphs does not admit a linear χ-binding function. In this paper, we are interested in classes of 2K2-free gr...

We derive general conditions for 100% frequency conversion in any doubly resonant nonlinear cavity, for both secondand third-harmonic generation via χ (2) and χ (3) nonlinearities. We find that conversion efficiency is optimized for a certain “critical” power depending on the cavity parameters, and assuming reasonable parameters we predict 100% conversion using milliwatts of power or less. Thes...