Let G be a nonabelian group. We define the noncommuting graph ?(G) of G as follows: its vertex set is G\Z(G), the set of non-central elements of G, and two different vertices x and y are joined by an edge if and only if x and y do not commute as elements of G, i.e., [x,y]≠1. We prove that if L ∈ {L₄(7), L₄(11), L₄(13), L₄(16), L₄(17)} and G is a finite group such that ?(G)??(L), then G?L.

The drift, the risk-free interest rate, and the volatility change over time horizon in realistic financial world. These frustrations break the necessary assumptions in the Black-Scholes model (BSM) in which all parameters are assumed to be constant. To better model the real markets, a modified BSM is proposed for numerically evaluating options price–changeable parameters are allowed through the backward Markov regime switching. The method of fundamental solutions (MFS) is applied to solve the modified model and price a given option. A series of numerical simulations are provided to illustrate the effect of the changing market on option pricing.

We shall derive two sufficient conditions for complete finitedimensional Alexandrov spaces of nonnegative curvature to be contractible. One of the new technical tools used in our proof is a quadrangle comparison theorem inspired by Perelman.

Let Q² = [0, 1]² be the unit square in two-dimensional Euclidean space ?2. We study the L^p boundedness of the oscillatory integral operator T_α,β defined on the set ?(?2+n) of Schwartz test functions byTα,βf(u,v,x)=∫Q2f(u-t,v-s,x-γ(t,s))t1+α1s1+α2eit-β1s-β2dtds,where x∈?n, (u,v)∈?2, (t,s,γ(t,s))=(t,s,tp1sq1,tp2sq2,?,tpnsqn) is a surface on ?n+2, and β₁>α₁, β₂>α₂. Our results extend some known results on ?3.

People studied the properties and structures of restricted Lie algebras all whose elements are semisimple. It is the main objective of this paper to continue the investigation in order to obtain deeper structure theorems. We obtain some sufficient conditions for the commutativity of restricted Lie algebras, generalize some results of R. Farnsteiner and characterize some properties of a finite-dimensional semisimple restricted Lie algebra all whose elements are semisimple. Moreover, we show that a centralsimple restricted Lie algebra all whose elements are semisimple over a field of characteristic p>7 is a form of a classical Lie algebra.

In this paper, we focus on the existence of symmetric λ-configurations with λ = 2, 3, and 4. Three new spatial configurations (v₈)₂ for v = 30, 31, and 32 are constructed. The existence of a spatial configuration (v_k)₂ are updated for k≤10. The existence tables for symmetric λ-configurations for λ = 3, 4, and small k are also given.

Alternating direction method (ADM) has been well studied in the context of linearly constrained convex programming problems. Recently, because of its significant efficiency and easy implementation in novel applications, ADM is extended to the case where the number of separable parts is a finite number. The algorithmic framework of the extended method consists of two phases. At each iteration, it first produces a trial point by using the usual alternating direction scheme, and then the next iterate is updated by using a distance-descent direction offered by the trial point. The generated sequence approaches the solution set monotonically in the Fejér sense, and the method is called alternating direction-based contraction (ADBC) method. In this paper, in order to simplify the subproblems in the first phase, we add a proximal term to the objective function of the minimization subproblems. The resulted algorithm is called proximal alternating direction-based contraction (PADBC) methods. In addition, we present different linearized versions of the PADBC methods which substantially broaden the applicable scope of the ADBC method. All the presented algorithms are guided by a general framework of the contraction methods for monotone variational inequalities, and thus, the convergence follows directly.

It is shown that there exists a quantum superdeterminant sdet_qT for the quantum super group OSP_q(1|2n). It is also shown that the quantum superdeterminant sdet_qT is a group-like element and central, and that the square of sdet_qT for OSP_q(1|2n) is equal to 1.

We extend Yamada-Watababe’s criterion [J. Math. Kyoto Univ., 1971, 11: 553-563] on the pathwise uniqueness of one-dimensional stochastic differential equations to a special class of multi-dimensional stochastic differential equations.

In this paper, we consider a triple of Gromov-Hausdorff convergence: Ai→dGHA, Bi→dGHB and maps f_i : A_i → B_i converge to a map f : A → B, where A_i are compact Alexandrov n-spaces and B_i are compact Riemannian m-manifolds such that the curvature, diameter and volume are suitably bounded (non-collapsing). When f is a submetry, we give a necessary and sufficient condition for the sequence to be stable, that is, for i large, there are homeomorphisms, Ψ_i : A_i → A, Φ_i : B_i → B such that f ? Ψ_i = Φ_i ? f_i. When f is an ?-submetry with ?>0, we obtain a sufficient condition for the stability in the case that A_i are Riemannian manifolds. Our results generalize the stability/finiteness results on fiber bundles by Riemannian submersions and by submetries.

This paper gives a Noether type inequality of a minimal Gorenstein 3-fold of general type whose canonical map is generically finite.

In this paper, we introduce the concept of weakly s-semipermutable subgroups. Let G be a finite group. Using the condition that the minimal subgroups or subgroups of order p² of a given Sylow p-subgroup of G are weakly s-semipermutable in G, we give a criterion for p-nilpotency of G and get some results about formation.

We introduce two residual type a posteriori error estimators for second-order elliptic partial differential equations with its right-hand side in L^p (1<p≤2) space. Both estimators are proved to yield global upper and local lower bounds for the W^1,p seminorm of the error. We adopt the estimators as the indicators in h-mesh adaptive method to solve two typical model problems. It is verified by the numerical results that the estimators lead to optimal orders of convergence.