On Extensions of the Frank-Wolfe Theorems

In this paper we consider optimization problems defined by a quadratic objective function and a finite number of quadratic inequality constraints. Given that the objective function is bounded over the feasible set, we present a comprehensive study of the conditions under which the optimal solution set is nonempty, thus extending the so-called Frank-Wolfe theorem. In particular, we first prove a general continuity result for the solution set defined by a system of convex quadratic inequalities. This result implies immediately that the optimal solution set of the aforementioned problem is nonempty when all the quadratic functions involved are convex. In the absence of the convexity of the objective function, we give examples showing that the optimal solution set may be empty either when there are two or more convex quadratic constraints, or when the Hessian of the objective function has two or more negative eigenvalues. In the case when there exists only one convex quadratic inequality constraint (together with other linear constraints), or when the constraint functions are all convex quadratic and the objective function is quasi-convex (thus allowing one negative eigenvalue in its Hessian matrix), we prove that the optimal solution set is nonempty.     

On the J-Normal Extensions of Operators

A bounded linear operator T acting on a Hilbert space H is said to be J- subnormal with order n if on some \Pi _n-Pontrjagin space \Pi containing H, there exists a bounded J-normal operator \tilde T such that \tilde Tf=Tf for every f in H and that \Pi is spanned by the elements of the form $\tilde T^{*k}f$, where f \in H and k = 0, 1, 2,\cdots. Let H be a Hilbert space and let I7 be in B(H). The main purpose of this paper is to prove that the following statements are equivalent: (1) Tis J-subnormal with order n; (2) For each non-negative integer r and for each set {x_ik: i, k = 0, 1,\cdots, r} of elements of H, the Hermitian form $\sum\limits_{i,j,k,l=0}^r(T^jx_ik,T^ix_jl)\alpha_ik\bar \alpha_jl$ has at most n negative squares, and for at least one choice of r and {x_ik}, it has exactly n negative squares; (3) The operator function is quasi-positive befinite with order n in the complex plane. This result is an extension of the theorems of Halmos and Bram,     

On the Number of Precolouring Extensions

We investigate the number of proper λ -colourings of a hypergraph extending a given proper precolouring. We prove that this number agrees with a polynomial in λ for any sufficiently largeλ , and we establish a generalization of Whitney’s broken circuit theorem by applying a recent improvement of the inclusion–exclusion principle.     

On Extensions of AO-Groups

The notion of an extension is important in the study of partially ordered groups. In the present paper the notion of a lexicographic extension of a partially ordered group by an AO-group is studied. A result is obtained concerning an AO-group G which is a lexicographic extension of a directed subgroup of G.__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 277–281, 2003.     

可换序半群的分式扩张

In this paper we extend the construction of the field of rational numbers from the ring of integers to an arbitrary commutative ordered semigroup.We first construct a fractional ordered semigroup and a homomorphism ψs:R→S-1R.Secondly,we characterize the commutative ordered semigroup so constructed by a universal mapping property.     

On the Numerical Stability of Fourier Extensions

An effective means to approximate an analytic, nonperiodic function on a bounded interval is by using a Fourier series on a larger domain. When constructed appropriately, this so-called Fourier extension is known to converge geometrically fast in the truncation parameter. Unfortunately, computing a Fourier extension requires solving an ill-conditioned linear system, and hence one might expect such rapid convergence to be destroyed when carrying out computations in finite precision. The purpose of this paper is to show that this is not the case. Specifically, we show that Fourier extensions are actually numerically stable when implemented in finite arithmetic, and achieve a convergence rate that is at least superalgebraic. Thus, in this instance, ill-conditioning of the linear system does not prohibit a good approximation.In the second part of this paper we consider the issue of computing Fourier extensions from equispaced data. A result of Platte et al. (SIAM Rev. 53(2):308–318, 2011) states that no method for this problem can be both numerically stable and exponentially convergent. We explain how Fourier extensions relate to this theoretical barrier, and demonstrate that they are particularly well suited for this problem: namely, they obtain at least superalgebraic convergence in a numerically stable manner.     

关于Boutroux—Cartan定理的推广

推广了著名的Boutroux—Cartan定理。设aμ（μ=1，2，…，n）为复平面上任意的n个点，H为任意的一个正数，则在平面上同时使得n∏μ=1 ｜z-aμ｜≤（H/e）^n和n∑μ=1 1/｜z-aμ｜≥nlog（en）/H成立的点z可被含于总数不超过n，半径总和不超过2H的一组圈内。     

On Minimal Extensions of Rings

Given two rings R ? S, S is said to be a minimal ring extension of R, if R is a maximal subring of S. In this article, we study minimal extensions of an arbitrary ring R, with particular focus on those possessing nonzero ideals that intersect R trivially. We will also classify the minimal ring extensions of prime rings, generalizing results of Dobbs, Dobbs &; Shapiro, and Ferrand &; Olivier, on commutative minimal extensions.     

On Commutativity of Ideal Extensions

In this article, we examine commutativity of ideal extensions. We introduce methods of constructing such extensions. In particular, we construct a noncommutative ring T which contains a central and idempotent ideal I such that T/I is a field. This answers a question from [1 Andruszkiewicz, R. R., Puczy?owski, E. R. (1995). On commutative idempotent rings. Proc. Roy. Soc. Edinburgh Sect. A 125(2):341–349.[Crossref] , [Google Scholar]]. Moreover, we classify fields of characteristic zero which can be obtained as T/I for some T.     

On Extensions of Myers' Theorem

Let M be a compact Riemannian manifold, and let h be a smoothfunction on M. Let p^h(x) = inf||–₁(Ric_x(,)–2Hess(h_x(,)).Here Ric_x denotes the Ricci curvature at x and Hess(h) is theHessian of h. Then M has finite fundamental group if ^h–p^h<0. Here ^h =:+2L_h is the Bismut-Witten Laplacian. This leadsto a quick proof of recent results on extension of Myers' theoremto manifolds with mostly positive curvature. There is also asimilar result for noncompact manifolds.     

On the Density of Elliptic Curves

We show that 17.9% of all elliptic curves over Q, ordered by their exponential height, are semistable, and that there is a positive density subset of elliptic curves for which the root numbers are uniformly distributed. Moreover, for any > 1/6 (resp. > 1/12) the set of Frey curves (resp. all elliptic curves) for which the generalized Szpiro Conjecture |(E)| N _E ¹² is false has density zero. This implies that the ABC Conjecture holds for almost all Frey triples. These results remain true if we use the logarithmic or the Faltings height. The proofs make use of the fibering argument in the square-free sieve of Gouvêa and Mazur. We also obtain conditional as well as unconditional lower bounds for the number of curves with Mordell–Weil rank 0 and 2, respectively.     

On the Iwasawa Theory of p-Adic Lie Extensions

In this paper, the new techniques and results concerning the structure theory of modules over noncommutative Iwasawa algebras are applied to arithmetic: we study Iwasawa modules over p-adic Lie extensions k of number fields k 'up to pseudo-isomorphism'. In particular, a close relationship is revealed between the Selmer group of Abelian varieties, the Galois group of the maximal Abelian unramified p-extension of k as well as the Galois group of the maximal Abelian p-extension unramified outside S where S is a certain finite setof places of k. Moreover, we determine the Galois module structure of local units and other modules arising from Galois cohomology.     

On Elliptic Problems in Besov Spaces

Suppose that the distribution function of a standardized sum of independent identically distributed random variables tends to a stable law as n°. Some differences in moments and pseudomoments, inequalities of the BERRY-ESSEEN-Type and asymptotic expansions are characterized when the limit law is either normal or non-normal stable.