Non-degenerate Invariant (Super)Symmetric Bilinear Forms on Simple Lie (Super)Algebras

We review the list of non-degenerate invariant (super)symmetric bilinear forms (briefly: NIS) on the following simple (relatives of) Lie (super)algebras: (a) with symmetrizable Cartan matrix of any growth, (b) with non-symmetrizable Cartan matrix of polynomial growth, (c) Lie (super)algebras of vector fields with polynomial coefficients, (d) stringy a.k.a. superconformal superalgebras, (e) queerifications of simple restricted Lie algebras. Over algebraically closed fields of positive characteristic, we establish when the deform (i.e., the result of deformation) of the known finite-dimensional simple Lie (super)algebra has a NIS. Amazingly, in most of the cases considered, if the Lie (super)algebra has a NIS, its deform has a NIS with the same Gram matrix after an identification of bases of the initial and deformed algebras. We do not consider odd parameters of deformations. Closely related with simple Lie (super)algebras with NIS is the notion of doubly extended Lie (super)algebras of which affine Kac–Moody (super)algebras are the most known examples.     

The lattice of convex subgroupoids of an ordered groupoid

Let V(A) (V_o (A)) be the lattice consisting of ϕ and all convex subgroupoids (with zero) of an o-groupoid A (with zero) with respect to the union B∨C which is the convex subgroupoid generated by B∩C and the set-theoretical intersection. In this paper, we study partially ordered groupoids (o-groupoids with zero) without proper convex subgroupoids (with zero), o-groupoids (with zero) for which V(A) (V_o (A)) is a chain, is the Cartesian product of two chains, is modular, is distributive, is complemented, or has a finite length, and also o-groupoids with zero and with associative powers (o-semigroups) in which every subgroupoid with zero (sub-semigroup) is convex. Finally, we give some results about lattice-closed classes of o-groupoids.     

Radial DEA models without inputs or without outputs

In this paper we consider radial DEA models without inputs (or without outputs), and radial DEA models with a single constant input (or with a single constant output). We demonstrate that (i) a CCR model without inputs (or without outputs) is meaningless; (ii) a CCR model with a single constant input (or with a single constant output) coincides with the corresponding BCC model; (iii) a BCC model with a single constant input (or a single constant output) collapses to a BCC model without inputs (or without outputs); and (iv) all BCC models, including those without inputs (or without outputs), can be condensed to models having one less variable (the radial efficiency score) and one less constraint (the convexity constraint).     

A BLOW-UP RESULT FOR A CLASS DOUBLY NONLINEAR PARABOLIC EQUATIONS WITH VARIABLE-EXPONENT NONLINEARITIES

This paper deals with the following doubly nonlinear parabolic equations(u + |u|~(r(x)-2)u)t-div(|?u|~(m(x)-2)?u) = |u|~(p(x)-2)u, where the exponents of nonlinearity r(x), m(x) and p(x) are given functions. Under some appropriate assumptions on the exponents of nonlinearity, and with certain initial data, a blow-up result is established with positive initial energy.     

指数Diophantine方程(bn)~x+(2n)~y=((b+2)n)~z的例外解

设b是大于3的正奇数.运用初等方法讨论了方程(bn)^x+(2n)x+(2n)^y=((b+2)n)y=((b+2)n)^z适合(x,y,z)≠(1,1,1)的正整数解(x,y,z,n).证明了:i)对于任何给定的正整数N,存在无穷多个b可使该方程有满足min{x,y,z}≥N的正整数解(x,y,z,n);ii)对于任何给定的b,该方程仅有有限多组正整数解(x,y,z,n)满足y>z=x.     

Effect of the shape and size of the cell on the compressive strength of honeycomb cores

The effect of cell shape and size and panel thickness on the strength of honeycomb materials is studied. Three types (shapes) of core cells are investigated: 1) regular (symmetric) hexahedral; 2) hexahedral offset by one-half pitch; 3) reinforced hexahedral. It is found that the strength of the investigated types of core increases in the following order; 1) cores with hexahedral cells offset by one-half pitch; 2) cores with regular hexahedral cells, and 3) cores with reinforced hexahedral cells. It is shown that cores with reinforced cells obey all the laws characteristic of other cores. It is confirmed that the facing strengthens the core material.Mekhanika Polimerov, Vol. 1, No. 3, pp. 29–32, 1965     

Congruence and conjunctivity of matrices to their adjoints

Every square matrix over a field F is involutorily congruent over F to its transpose, and hence each such matrix is the product of a symmetric matrix and an involutory matrix over F. In the usual complex case every matrix which is conjunctive with its adjoint (=conjugate-transpose) is involutorily conjunctive with its adjoint and hence is the product of a hermitian matrix and an involutory matrix; furthermore every such matrix is conjunctive with a real matrix. These three conditions on a matrix, (1) being conjunctive with its adjoint, (2) being involutorily conjunctive with its adjoint, and (3) being conjunctive with a real matrix, are studied in the more general context of a field F with involution, and it is shown in general that (3) implies (2), that (2) implies (3) if char F≠2 (a 2×2 counterexample exists for each F with char F=2), and that (1) does not in general imply (2) (a 2×2 counterexample in the complexification of the rational field is presented). The problem of deciding which matrices satisfy (2) is equivalent (even in this general context) to the problem of deciding which pairs of self-adjoint (“hermitian”) matrices are involutorily conjunctive. For the general 2×2 case, the three conditions are characterized in terms of norms.     

平衡范畴与半群的双序

研究范畴与半群通过幂等元双序建立的一种自然联系.对每个有幂等元的半群S,其幂等元生成的左、右主理想之集通过双序ω~e,ω~r自然确定两个有子对象、有像且每个包含都右可裂的范畴L(S),R(S),其中态射的性质与S中元素的富足性、正则性有自然对应.利用这个联系,我们定义了"平衡(富足、正规)范畴"概念.对任一平衡(富足、正规)范畴■,我们构造其"锥半群"■,证明■左富足(富足、正则),且每个平衡(富足、正规)范畴■都与某左富足(富足、正则)半群S的左主理想范畴L(S)(作为有子对象的范畴)同构.     

Local tracking and stability for degenerate viscous shock waves

We study the pointwise behavior of perturbed degenerate (sonic) shock waves for scalar conservation laws with nonconstant diffusion. Building on the pointwise Green's function approach of Zumbrun and Howard (Indiana U. Math. J. 47(3) (1998) 741) we extend the linear analysis to an equation with nonintegrable coefficients. In lieu of working with the integrated equation, we employ a tracking mechanism that we expect will allow degenerate waves to be incorporated into the general framework for nondegenerate systems (Indiana U. Math. J. 47(3) (1998) 741).     

Some Theorems for Hypersurface of Randers Spaces

In this paper, we consider the hypersurfaces of Randers space with constant flag curvature. (1) Let (Mⁿ⁺¹, F) be a Randers-Minkowski space. If (Mⁿ, F) is a hypersurface of (Mⁿ⁺¹, F) with constant flag curvature K=1, then we can prove that M is Riemannian. (2) Let (Mⁿ⁺¹, F) be a Randers space with constant flag curvature. Assume (M, F) is a compact hypersurface of (Mⁿ⁺¹, F) with constant mean curvature|H|. Then a pinching theorem is established, which generalizes the result of[Proc. Amer. Math. Soc., 120, 1223-1229 (1994)] from the Riemannian case to the Randers space.     

On the spectral radius of unicyclic graphs with prescribed degree sequence

We consider the set of unicyclic graphs with prescribed degree sequence. In this set we determine the (unique) graph with the largest spectral radius (or index) with respect to the adjacency matrix. In addition, we give a conjecture about the (unique) graph with the largest index in the set of connected graphs with prescribed degree sequence.     

Properties of Complex Oscillation of Solutions of a Class of Higher Order Linear Differential Equations

Let A(z) be an entire function with μ(A) 1/2 such that the equation f~((k))+A(z)f = 0, where k ≥ 2, has a solution f with λ(f) μ(A), and suppose that A_1 = A+h,where h■0 is an entire function with ρ(h) μ(A). Then g~((k))+ A_1(z)g = 0 does not have a solution g with λ(g) ∞.     

Convergence Properties of Generalized Fourier Series on a Parallel Hexagon Domain

A new Rogosinski-type kernel function is constructed using kernel function of partial sums Sn（f; t） of generalized Fourier series on a parallel hexagon domain Ω associating with threedirection partition. We prove that an operator Wn（f; t） with the new kernel function converges uniformly to any continuous function f（t） ∈ Cn（Ω） （the space of all continuous functions with period Ω） on Ω. Moreover, the convergence order of the operator is presented for the smooth approached function.     

(Semi-)Nonrelativisitic Limit of the Nonlinear Dirac Equations

We consider the nonlinear Dirac equation (NLD) with time dependent external electro-magnetic potentials, involving a dimensionless parameter $ε\in(0,1]$ which is inversely proportional to the speed of light. In the nonrelativistic limit regime $ε\ll1$ (speed of light tends to infinity), we decompose the solution into the eigenspaces associated with the 'free Dirac operator' and construct an approximation to the NLD with $O(ε^2)$ error. The NLD converges (with a phase factor) to a coupled nonlinear Schrödinger system (NLS) with external electric potential in the nonrelativistic limit as $ε\to0^+$, and the error of the NLS approximation is first order $O(ε)$. The constructed $O(ε^2)$ approximation is well-suited for numerical purposes.     

One-dimensional local rings with reduced associated graded ring and their Hilbert functions

Let A be a one-dimensional reduced local ring with finite normalization. Let G(A) be the associated graded ring of A. In this paper we analyse the two conditions: Proj (G(A)) reduced and G(A) reduced together with their relations with the equality H(n)=H_R (n), where H(n) and H_R (n) are respectively the Hilbert function of the ring A and of the local ring R of G(A)_red=G(A)/nil (G(A)) at its homogenous maximal ideal. As a consequence of our results we get a class of ordinary singularities with H(n) locally decreasing for any embedding dimension H(1) greater then 4.     

Sufficient conditions for total ill-posedness in linear semi-infinite optimization

This paper deals with the so-called total ill-posedness of linear optimization problems with an arbitrary (possibly infinite) number of constraints. We say that the nominal problem is totally ill-posed if it exhibits the highest unstability in the sense that arbitrarily small perturbations of the problem’s coefficients may provide both, consistent (with feasible solutions) and inconsistent problems, as well as bounded (with finite optimal value) and unbounded problems, and also solvable (with optimal solutions) and unsolvable problems. In this paper we provide sufficient conditions for the total ill-posedness property exclusively in terms of the coefficients of the nominal problem.     

The Construction of Basis Functions for Curved Elements in the Finite Element Method

Part basis (wedge) functions are constructed for plane triangularelements with two straight and one curved side, using ideasfrom three dimensional geometry. Such elements are desirablefor regions with curved boundaries or interfaces. Lagrange interpolatingfunctions which are in turn (i) linear, (ii) quadratic, and(iii) cubic along the straight sides of the triangle are considered.Curved elements which had previously been dealt with using isoparametricco-ordinates appear as special cases of this more general geometricaltreatment. The ideas can be extended in a straightforward mannerto deal with tetrahedral elements with three plane and one curvedsurface.     

Certain Partial Conservativeness Properties of Intuitionistic Set Theory with the Principle of Double Complement of Sets

The Zermelo–Fraenkel set theory with the underlying intuitionistic logic (for brevity, we refer to it as the intuitionistic Zermelo–Fraenkel set theory) in a two-sorted language (where the sort 0 is assigned to numbers and the sort 1, to sets) with the collection scheme used as the replacement scheme of axioms (the ZFI2C theory) is considered. Some partial conservativeness properties of the intuitionistic Zermelo–Fraenkel set theory with the principle of double complement of sets (DCS) with respect to a certain class of arithmetic formulas (the class all so-called AEN formulas) are proved. Namely, let T be one of the theories ZFI2C and ZFI2C + DCS. Then (1) the theory T+ECT is conservative over T with respect to the class of AEN formulas; (2) the theory T+ECT+M is conservative over T+M{su?} with respect to the class of AEN formulas. Here ECT stands for the extended Church’s thesis, Mis the strong Markov principle, and M{su?} is the weak Markov principle. The following partial conservativeness properties are also proved: (3) T+ECT+M is conservative over T with respect to the class of negative arithmetic formulas; (4) the classical theory ZF2 is conservative over ZFI2C with respect to the class of negative arithmetic formulas.     

一类高阶线性微分方程解的复振荡

In this paper,we shall use Nevanlinna theory of meromorphic functions to investigate the complex oscillation theory of solutions of some higher order linear differential equation.Suppose that A is a transcendental entire function with ρ(A)<1/2.Suppose that k≥2 and f(k)+A(z)f=0 has a solution f with λ(f)<ρ(A),and suppose that A1=A+h,where h≡0 is an entire function with ρ(h)<ρ(A).Then g(k)+A1(z)g=0 does not have a solution g with λ(g)<∞.     

Two classes of operators with irreducibility and the small and compact perturbations of them

This paper gives the concepts of finite dimensional irreducible operators((FDI) operators)and infinite dimensional irreducible operators((IDI) operators). Discusses the relationships of(FDI)operators,(IDI) operators and strongly irreducible operators((SI) operators) and illustrates some properties of the three classes of operators. Some sufficient conditions for the finite-dimensional irreducibility of operators which have the forms of upper triangular operator matrices are given. This paper proves that every operator with a singleton spectrum is a small compact perturbation of an(FDI) operator on separable Banach spaces and shows that every bounded linear operator T can be approximated by operators in(Σ FDI)(X) with respect to the strong-operator topology and every compact operator K can be approximated by operators in(Σ FDI)(X) with respect to the norm topology on a Banach space X with a Schauder basis, where(ΣFDI)(X) := {T∈B(X) : T=Σki=1Ti, Ti ∈(FDI), k ∈ N}.