On Generalized Derivations and Centralizers of Operator Algebras with Involution

Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H and A(H) ? B(H) be a standard operator algebra which is closed under the adjoint operation. Let F: A(H)→ B(H) be a linear mapping satisfying F(AA*A) = F(A)A*A + Ad(A*)A + AA*d(A) for all A ∈ A(H), where the associated linear mapping d: A(H) → B(H) satisfies the relation d(AA*A) = d(A)A*A + Ad(A*)A + AA*d(A) for all A ∈ A(H). Then F is of the form F(A) = SA ? AT for all A ∈ A(H) and some S, T ∈ B(H), that is, F is a generalized derivation. We also prove some results concerning centralizers on A(H) and semisimple H^*-algebras.     

Annihilating and power-commuting generalized skew derivations on lie ideals in prime rings

Let R be a prime ring of characteristic different from 2 and 3, Q_r its right Martindale quotient ring, C its extended centroid, L a non-central Lie ideal of R and n ≥ 1 a fixed positive integer. Let α be an automorphism of the ring R. An additive map D: R → R is called an α-derivation (or a skew derivation) on R if D(xy) = D(x)y + α(x)D(y) for all x, y ∈ R. An additive mapping F: R → R is called a generalized α-derivation (or a generalized skew derivation) on R if there exists a skew derivation D on R such that F(xy) = F(x)y + α(x)D(y) for all x, y ∈ R.     

Every graph is (2,3)-choosable

A total weighting of a graph G is a mapping ? that assigns to each element z ∈ V (G)∪E(G) a weight ?(z). A total weighting ? is proper if for any two adjacent vertices u and v, ∑ _e∈E(u) ?(e)+?(u)≠∑ _e∈E(v) ?(e)+?(v). This paper proves that if each edge e is given a set L(e) of 3 permissible weights, and each vertex v is given a set L(v) of 2 permissible weights, then G has a proper total weighting ? with ?(z) ∈ L(z) for each element z ∈ V (G)∪E(G).     

Distance domination of generalized de Bruijn and Kautz digraphs

Let G = (V,A) be a digraph and k ≥ 1 an integer. For u, v ∈ V, we say that the vertex u distance k-dominate v if the distance from u to v at most k. A set D of vertices in G is a distance k-dominating set if each vertex of V D is distance k-dominated by some vertex of D. The distance k-domination number of G, denoted by γ _k (G), is the minimum cardinality of a distance k-dominating set of G. Generalized de Bruijn digraphs G _B (n, d) and generalized Kautz digraphs G _K (n, d) are good candidates for interconnection networks. Denote Δ _k := (∑ _j=0 ^k d ^j )^?1. F. Tian and J. Xu showed that ?nΔ _k ? γ _k (G _B (n, d)) ≤?n/d ^k? and ?nΔ _k ? ≤ γ _k (G _K (n, d)) ≤ ?n/d ^k ?. In this paper, we prove that every generalized de Bruijn digraph G _B (n, d) has the distance k-domination number ?nΔ _k ? or ?nΔ _k ?+1, and the distance k-domination number of every generalized Kautz digraph G _K (n, d) bounded above by ?n/(d ^k?1+d ^k )?. Additionally, we present various sufficient conditions for γ _k (G _B (n, d)) = ?nΔ _k ? and γ _k (G _K (n, d)) = ?nΔ _k ?.     

Fast Algorithm of Square Rooting in Some Finite Fields of Odd Characteristic

It was proved that the complexity of square root computation in the Galois field GF(3^s), s = 2^kr, is equal to O(M(2^k)M(r)k + M(r) log₂r) + 2^kkr^1+o(1), where M (n) is the complexity of multiplication of polynomials of degree n over fields of characteristics 3. The complexity of multiplication and division in the field GF(3^s) is equal to O(M(2^k)M(r)) and O(M(2^k)M(r)) + r^1+o(1), respectively. If the basis in the field GF(3^r) is determined by an irreducible binomial over GF(3) or is an optimal normal basis, then the summands 2^kkr^1+o(1) and r^1+o(1) can be omitted. For M(n) one may take n log₂nψ(n) where ψ(n) grows slower than any iteration of the logarithm. If k grow and r is fixed, than all the estimates presented here have the form O_r (M (s) log ₂s) = s (log ₂s)²ψ(s).     

Elliptic function of level 4

The article is devoted to the theory of elliptic functions of level n. An elliptic function of level n determines a Hirzebruch genus called an elliptic genus of level n. Elliptic functions of level n are also of interest because they are solutions of the Hirzebruch functional equations. The elliptic function of level 2 is the Jacobi elliptic sine function, which determines the famous Ochanine–Witten genus. It is the exponential of the universal formal group of the form F(u, v) = (u² ? v²)/(uB(v) ? vB(u)), B(0) = 1. The elliptic function of level 3 is the exponential of the universal formal group of the form F(u, v) = (u²A(v) ? v²A(u))/(uA(v)² ? vA(u)²), A(0) = 1, A″(0) = 0. In the present study we show that the elliptic function of level 4 is the exponential of the universal formal group of the form F(u, v) = (u²A(v) ? v²A(u))/(uB(v) ? vB(u)), where A(0) = B(0) = 1 and for B′(0) = A″(0) = 0, A′(0) = A₁, and B″(0) = 2B₂ the following relation holds: (2B(u) + 3A₁u)² = 4A(u)³ ? (3A₁² ? 8B₂)u²A(u)². To prove this result, we express the elliptic function of level 4 in terms of the Weierstrass elliptic functions.     

Finite groups with given σ-embedded and σ-n-embedded subgroups

Let G be a finite group and σ = {σ _i |i∈I} be a partition of the set of all primes P. A set H of subgroups of G is said to be a complete Hall σ-set of G if every non-identity member of H is a Hall σ _i -subgroup of G and H contains exactly one Hall σ _i -subgroup of G for every σ _i ∈ σ(G). A subgroup H is said to be σ-permutable if G possesses a complete Hall σ-set H such that HA ^x = A ^x H for all A ∈ H and all x ∈ G. Let H be a subgroup of G. Then we say that: (1) H is σ-embedded in G if there exists a σ-permutable subgroup T of G such that HT = H ^σG and H ∩ T ≤ H _σG , where H _σG is the subgroup of H generated by all those subgroups of H which are σ-permutable in G, and H ^σG is the σ-permutable closure of H, that is, the intersection of all σ-permutable subgroups of G containing H. (2) H is σ-n-embedded in G if there exists a normal subgroup T of G such that HT = H ^G and H ∩ T ≤ H _σG . In this paper, we study the properties of the new embedding subgroups and use them to determine the structure of finite groups.     

Minimum implicit degree condition restricted to claws for hamiltonian cycles

Let id(v) denote the implicit degree of a vertex v in a graph G. We define G to be implicit 1-heavy (implicit 2-heavy) if at least one (two) of the end vertices of each induced claw has (have) implicit degree at least n/2. In this paper, we prove that: (a) Let G be a 2-connected graph of order n ≥ 3. If G is implicit 2-heavy and |N(u) ∩ N(v)| ≥ 2 for every pair of vertices u and v with d(u, v) = 2 and max{id(u), id(v)} < n/2, then G is hamiltonian. (b) Let G be a 3-connected graph of order n ≥ 3. If G is implicit 1-heavy and |N(u) ∩ N(v)| ≥ 2 for each pair of vertices u and v with d(u, v) = 2 and max{id(u), id(v)} < n/2, then G is hamiltonian.     

Polyboxes,Cube Tilings and Rigidity

A non-empty subset A of X=X ₁×???×X _d is a (proper) box if A=A ₁×???×A _d and A _i ?X _i for each i. Suppose that for each pair of boxes A, B and each i, one can only know which of the three states takes place: A _i =B _i , A _i =X _i ?B _i , A _i ?{B _i ,X _i ?B _i }. Let F and G be two systems of disjoint boxes. Can one decide whether ∪F=∪G? In general, the answer is ‘no’, but as is shown in the paper, it is ‘yes’ if both systems consist of pairwise dichotomous boxes. (Boxes A, B are dichotomous if there is i such that A _i =X _i ?B _i .) Several criteria that enable to compare such systems are collected. The paper includes also rigidity results, which say what assumptions have to be imposed on F to ensure that ∪F=∪G implies F=G. As an application, the rigidity conjecture for 2-extremal cube tilings of Lagarias and Shor is verified.     

An algorithm to compute <Emphasis Type="Italic">ω</Emphasis>-primality in a numerical monoid

Let M be a commutative, cancellative, atomic monoid and x a nonunit in M. We define ω(x)=n if n is the smallest positive integer with the property that whenever x∣a ₁???a _t , where each a _i is an atom, there is a T?{1,2,…,t} with |T|≤n such that x∣∏_k∈T a _k . The ω-function measures how far x is from being prime in M. In this paper, we give an algorithm for computing ω(x) in any numerical monoid. Simple formulas for ω(x) are given for numerical monoids of the form 〈n,n+1,…,2n?1〉, where n≥3, and 〈n,n+1,…,2n?2〉, where n≥4. The paper then focuses on the special case of 2-generator numerical monoids. We give a formula for computing ω(x) in this case and also necessary and sufficient conditions for determining when x is an atom. Finally, we analyze the asymptotic behavior of ω(x) by computing \(\lim_{x\rightarrow \infty}\frac{\omega(x)}{x}\).     

Designs associated with maximum independent sets of a graph

A (v, β _o , μ)-design over regular graph G = (V, E) of degree d is an ordered pair D = (V, B), where |V| = v and B is the set of maximum independent sets of G called blocks such that if i, j ∈ V, i ≠ j and if i and j are not adjacent in G then there are exactly μ blocks containing i and j. In this paper, we study (v, β _o , μ)-designs over the graphs K _n × K _n , T(n)-triangular graphs, L ₂(n)-square lattice graphs, Petersen graph, Shrikhande graph, Clebsch graph and the Schläfli graph and non-existence of (v, β _o , μ)-designs over the three Chang graphs T ₁(8), T ₂(8) and T ₃(8).     

Meromorphic Solutions for a Class of Differential Equations and Their Applications

In this note, we study the admissible meromorphic solutions for algebraic differential equation fⁿf' + P_n?1(f) = R(z)e^α(z), where P_n?1(f) is a differential polynomial in f of degree ≤ n ? 1 with small function coefficients, R is a non-vanishing small function of f, and α is an entire function. We show that this equation does not possess any meromorphic solution f(z) satisfying N(r, f) = S(r, f) unless P_n?1(f) ≡ 0. Using this result, we generalize a well-known result by Hayman.     

Generalized 3-edge-connectivity of Cartesian product graphs

The generalized k-connectivity κ _k (G) of a graph G was introduced by Chartrand et al. in 1984. As a natural counterpart of this concept, Li et al. in 2011 introduced the concept of generalized k-edge-connectivity which is defined as λ _k (G) = min{λ(S): S ? V (G) and |S| = k}, where λ(S) denotes the maximum number l of pairwise edge-disjoint trees T ₁, T ₂, …, T _l in G such that S ? V (T _i ) for 1 ? i ? l. In this paper we prove that for any two connected graphs G and H we have λ ₃(G □ H) ? λ ₃(G) + λ ₃(H), where G □ H is the Cartesian product of G and H. Moreover, the bound is sharp. We also obtain the precise values for the generalized 3-edge-connectivity of the Cartesian product of some special graph classes.     

On idempotent <Emphasis Type="Italic">τ</Emphasis>-measurable operators affiliated to a von Neumann algebra

Let τ be a faithful normal semifinite trace on a von Neumann algebra M, let p, 0 < p < ∞, be a number, and let L_p(M, τ) be the space of operators whose pth power is integrable (with respect to τ). Let P and Q be τ-measurable idempotents, and let A ≡ P ? Q. In this case, 1) if A ≥ 0, then A is a projection and QA = AQ = 0; 2) if P is quasinormal, then P is a projection; 3) if Q ∈ M and A ∈ L_p(M, τ), then A² ∈ L_p(M, τ). Let n be a positive integer, n > 2, and A = Aⁿ ∈ M. In this case, 1) if A ≠ 0, then the values of the nonincreasing rearrangement μ_t(A) belong to the set {0} ∪ [‖A^n?2‖^?1, ‖A‖] for all t > 0; 2) either μ_t(A) ≥ 1 for all t > 0 or there is a t₀ > 0 such that μ_t(A) = 0 for all t > t₀. For every τ-measurable idempotent Q, there is aunique rank projection P ∈ M with QP = P, PQ = Q, and PM = QM. There is a unique decomposition Q = P + Z, where Z² = 0, ZP = 0, and PZ = Z. Here, if Q ∈ L_p(M, τ), then P is integrable, and τ(Q) = τ(P) for p = 1. If A ∈ L₁(M, τ) and if A = A³ and A ? A² ∈ M, then τ(A) ∈ R.     

Non-associative Ore extensions

We introduce non-associative Ore extensions, S = R[X; σ, δ], for any nonassociative unital ring R and any additive maps σ, δ: R → R satisfying σ(1) = 1 and δ(1) = 0. In the special case when δ is either left or right R _δ -linear, where R _δ = ker(δ), and R is δ-simple, i.e. {0} and R are the only δ-invariant ideals of R, we determine the ideal structure of the nonassociative differential polynomial ring D = R[X; id _R , δ]. Namely, in that case, we show that all non-zero ideals of D are generated by monic polynomials in the center Z(D) of D. We also show that Z( _D ) = R _δ [p] for a monic p ∈ R _δ [X], unique up to addition of elements from Z(R) _δ . Thereby, we generalize classical results by Amitsur on differential polynomial rings defined by derivations on associative and simple rings. Furthermore, we use the ideal structure of D to show that D is simple if and only if R is δ-simple and Z(D) equals the field R _δ ∩ Z(R). This provides us with a non-associative generalization of a result by Öinert, Richter and Silvestrov. This result is in turn used to show a non-associative version of a classical result by Jordan concerning simplicity of D in the cases when the characteristic of the field R _δ ∩ Z(R) is either zero or a prime. We use our findings to show simplicity results for both non-associative versions of Weyl algebras and non-associative differential polynomial rings defined by monoid/group actions on compact Hausdorff spaces.     

A Note on Generalized Lagrangians of Non-uniform Hypergraphs

Set \(A\subset {\mathbb N}\) is less than \(B\subset {\mathbb N}\) in the colex ordering if m a x(A△B)∈B. In 1980’s, Frankl and Füredi conjectured that the r-uniform graph with m edges consisting of the first m sets of \({\mathbb N}^{(r)}\) in the colex ordering has the largest Lagrangian among all r-uniform graphs with m edges. A result of Motzkin and Straus implies that this conjecture is true for r=2. This conjecture seems to be challenging even for r=3. For a hypergraph H=(V,E), the set T(H)={|e|:e∈E} is called the edge type of H. In this paper, we study non-uniform hypergraphs and define L(H) a generalized Lagrangian of a non-uniform hypergraph H in which edges of different types have different weights. We study the following two questions: 1. Let H be a hypergraph with m edges and edge type T. Let C _m,T denote the hypergraph with edge type T and m edges formed by taking the first m sets with cardinality in T in the colex ordering. Does L(H)≤L(C _m,T ) hold? If T={r}, then this question is the question by Frankl and Füredi. 2. Given a hypergraph H, find a minimum subhypergraph G of H such that L(G) = L(H). A result of Motzkin and Straus gave a complete answer to both questions if H is a graph. In this paper, we give a complete answer to both questions for {1,2}-hypergraphs. Regarding the first question, we give a result for {1,r ₁,r ₂,…,r _l }-hypergraph. We also show the connection between the generalized Lagrangian of {1,r ₁,r ₂,? ,r _l }-hypergraphs and {r ₁,r ₂,? ,r _l }-hypergraphs concerning the second question.     

On Ranks of Polynomials

Let V be a vector space over a field k, P : V → k, d ≥?3. We show the existence of a function C(r, d) such that rank(P) ≤ C(r, d) for any field k, char(k) > d, a finite-dimensional k-vector space V and a polynomial P : V → k of degree d such that rank(?P/?t) ≤ r for all t ∈ V ??0. Our proof of this theorem is based on the application of results on Gowers norms for finite fields k. We don’t know a direct proof even in the case when k = ?.     

Structure of Abelian rings

Let R be a ring with identity. We use J(R); G(R); and X(R) to denote the Jacobson radical, the group of all units, and the set of all nonzero nonunits in R; respectively. A ring is said to be Abelian if every idempotent is central. It is shown, for an Abelian ring R and an idempotent-lifting ideal N ? J(R) of R; that R has a complete set of primitive idempotents if and only if R/N has a complete set of primitive idempotents. The structure of an Abelian ring R is completely determined in relation with the local property when X(R) is a union of 2; 3; 4; and 5 orbits under the left regular action on X(R) by G(R): For a semiperfect ring R which is not local, it is shown that if G(R) is a cyclic group with 2 ∈ G(R); then R is finite. We lastly consider two sorts of conditions for G(R) to be an Abelian group.     

Paranormal Elements in Normed Algebra

For a normed algebra A and natural numbers k we introduce and investigate the ∥ · ∥ closed classes P _k (A). We show that P₁(A) is a subset of P _k (A) for all k. If T in P₁(A), then Tⁿ lies in P₁(A) for all natural n. If A is unital, U, V ∈ A are such that ∥U∥ = ∥V∥ = 1, VU = I and T lies in P _k (A), then UTV lies in P _k (A) for all natural k. Let A be unital, then 1) if an element T in P₁(A) is right invertible, then any right inverse element T^?1 lies in P₁(A); 2) for ßßIßß = 1 the class P₁(A) consists of normaloid elements; 3) if the spectrum of an element T, T ∈ P₁(A) lies on the unit circle, then ∥TX∥ = ∥X∥ for all X ∈ A. If A = B(H), then the class P₁(A) coincides with the set of all paranormal operators on a Hilbert space H.     

Valuation ideals and primary <Emphasis Type="Italic">w</Emphasis>-ideals

Let D be an integral domain, V (D) (resp., t-V (D)) be the set of all valuation (resp., t-valuation) ideals of D, and w-P(D) be the set of primary w-ideals of D. Let D[X] be the polynomial ring over D, c(f) be the ideal of D generated by the coefficients of f ∈ D[X], and N _v = {f ∈ D[X] | c(f) _v = D}. In this paper, we study integral domains D in which w-P(D) ? t-V (D), t-V (D) ? w-P(D), or t-V (D) = w-P(D). We also study the relationship between t-V (D) and \(V\left( {D{{\left[ X \right]}_{{N_v}}}} \right)\), and characterize when t-V (A + XB[X]) ? w-P(A + XB[X]) holds for a proper extension A ? B of integral domains.