Quadratic base change of θ10

In case ofGL _n overp-adic fields, it is known that Shintani base change is well behaved. However, things are not so simple for general reductive groups. In the first part of this paper, we present a counterexample to the existence of quadratic base change descent for some Galois invariant representations. These are representations of type θ₁₀. In the second part, we compute the localL-factor of θ₁₀. Unlike many other supercuspidal representations, we find that theL-factor of θ₁₀ has two poles. Finally, we discuss these two results in relation to the local Langlands correspondence. The authors are supported in part by NSF grants.     

RESEARCH ANNOUNCEMENTS Structure of Solvable Quadratic Lie Algebras

Killing form plays a key role in the theory of semisimple Lie algebras. It is natural to extend the study to Lie algebras with a nondegenerate symmetric invariant bilinear form. Such a Lie algebra is generally called a quadratic Lie algebra which occur naturally in physics. Besides semisimple Lie algebras, interesting quadratic Lie algebras include the Kac-Moody algebras and the Extended Affine Lie algebras.     

Congruences for Quadratic Units of Norm −1

Let D 1 (mod 4) be a positive integer. Let R be the ring {x + y(1 + )/2 : x, y }. Suppose that R contains a unit of norm –1 as well as an element of norm 2, and thus an element of norm –2. It is not hard to see that ±1(mod ²). In this paper we determine modulo ³ and modulo ³ using only elementary techniques. This determination extends a recent result of Mastropietro, which was proved using class field theory.     

A Note on 3-Divisibility of Class Number of Quadratic Field

In this paper, the authors show that there exists infinitely many family of pairs of quadratic fields Q(D^1/2) and Q((D+n)(1/2)) with D, n∈Z whose class numbers are both divisible by 3.     

Indivisibility of Class Numbers and Iwasawa λ-Invariants of Real Quadratic Fields

Let D>0 be the fundamental discriminant of a real quadratic field, and h(D) its class number. In this paper, by refining Ono's idea, we show that for any prime p>3, {0<D<X|h(D)0(mod p)}>> _p (X)/logX.     

An Important Property of the Weak Focus of a Quadratic Differential System

It was proved in [1] that the order of a weak focus of a quadratic differential system is at most3,i.e.,if v_1=v_3=v_5=v_7=0 in Bautin's symbol,then the critical point is a center.By using theDulac function method we prove in this paper that,if a quadratic differential system has two weakfoci,then each focus must be of order 1;and also a known result of L.A.Cherkas:under the aloveconditon no limit cycle can exist.     

2-Sylow Subgroups of K_2O_F for Real Quadratic Field and Class Group for Imaginary Quadratic Field

Inthispaper,westudytherelationbetweenthe2-Syl0w'subgroups0fK,OFf0rrealquadraticfie1dF=Q(N)andthec1assgroupC(E)forimaginaryquadraticfie1dE=Q(In).ForgeneralnumberfieldF,manyauth0rshavecharacterizedthe2-SylowsubgroupstructureofK2OFandobtained2'-rankK2OFformulas(n>1).Especially,forquadraticfieldF,QinHuorong['Jgottheformulas0f4-orderelementsinK2OFonthebasis0fBrowkin-Sch1nzel[2].Inthispaper,wediscussthemfurther.Lemma(Llj,Theorems2.2and3-3)LetF=Q(H),d>2beasquarefreeinteger,andmap0sitive…     

The Problem on Class Numbers of Quadratic Number Fields

It is a survey of the problem on class numbers of quadratic number fields.     

RESEARCH ANNOUNCEMENTS——Structure of Solvable Quadratic Lie Algebras

Killing form plays a key role in the theory of semisimple Lie algebras. It is natural to extend the study to Lie algebras with a nondegenerate symmetric invariant bilinear form. Such a Lie algebra is generally called a quadratic Lie algebra which occur naturally in physics^[10，12，13]. Besides semisimple Lie algebras, interesting quadratic Lie algebras include the Kac-Moody algebras and the Extended Affine Lie algebras. In this paper, we study solvable quadratic Lie algebras. In Section 1, we study quadratic solvable Lie algebras whose Cartan subalgebras consist of semi-simple elements. In Section 2,we present a procedure to construct a class of quadratic Lie algebras, and we can exhaust all solvable quadratic Lie algebras in such a way. All Lie algebras mentioned in this paper are finite dimensional Lie algebras over a field F of characteristic 0.     

Investigation of Exponential Dichotomy of Itô Stochastic Systems by Using Quadratic Forms

For linear stochastic systems, we obtain sufficient conditions for mean-square exponential dichotomy in terms of Lyapunov functions that are quadratic forms.     

A Necessary and Sufficient Condition for Admissibility of Nonnegative Quadratic Estimator

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The First Power Mean of the Inversion of L-Functions Weighted by Quadratic Gauss Sums

The main purpose of this paper is to use estimates for character sums and analytic methods to study the first power mean of the inversion of Dirichlet L-functions with the weight of general quadratic Gauss sums, and three asymptotic formulae are obtained.     

Inexact Alternating Direction Methods of Multipliers with Logarithmic–Quadratic Proximal Regularization

In the literature, it was shown recently that the Douglas–Rachford alternating direction method of multipliers can be combined with the logarithmic-quadratic proximal regularization for solving a class of variational inequalities with separable structures. This paper studies the inexact version of this combination, where the resulting subproblems are allowed to be solved approximately subject to different inexactness criteria. We prove the global convergence and establish worst-case convergence rates for the derived inexact algorithms.     

Unknot Diagrams Requiring a Quadratic Number of Reidemeister Moves to Untangle

Given any knot diagram E, we present a sequence of knot diagrams of the same knot type for which the minimum number of Reidemeister moves required to pass to E is quadratic with respect to the number of crossings. These bounds apply both in S ² and in ?².     

Ergodic Type Bellman Equations of First Order with Quadratic Hamiltonian

We consider Bellman equations of ergodic type in first order. The Hamiltonian is quadratic on the first derivative of the solution. We study the structure of viscosity solutions and show that there exists a critical value among the solutions. It is proved that the critical value has the representation by the long time average of the kernel of the max-plus Schrödinger type semigroup. We also characterize the critical value in terms of an invariant density in max-plus sense, which can be understood as a counterpart of the characterization of the principal eigenvalue of the Schrödinger operator by an invariant measure.     

Maximal Quadratic Modules on ∗-rings

We generalize the notion of and results on maximal proper quadratic modules from commutative unital rings to ?-rings and discuss the relation of this generalization to recent developments in noncommutative real algebraic geometry. The simplest example of a maximal proper quadratic module is the cone of all positive semidefinite complex matrices of a fixed dimension. We show that the support of a maximal proper quadratic module is the symmetric part of a prime ?-ideal, that every maximal proper quadratic module in a Noetherian ?-ring comes from a maximal proper quadratic module in a simple artinian ring with involution and that maximal proper quadratic modules satisfy an intersection theorem. As an application we obtain the following extension of Schmüdgen’s Strict Positivstellensatz for the Weyl algebra: Let c be an element of the Weyl algebra \(\mathcal{W}(d)\) which is not negative semidefinite in the Schrödinger representation. It is shown that under some conditions there exists an integer k and elements \(r_1,\ldots,r_k \in \mathcal{W}(d)\) such that ∑ _j=1 ^k r _j c r _j ^? is a finite sum of hermitian squares. This result is not a proper generalization however because we don’t have the bound k ≤d.     

Positive Solution of a Quadratic Integral Equation Using Generalization of Darbo’s Fixed Point Theorem

The objective of this article is to establish a fixed point theorem using different class of control functions, which is a generalization of Darbo’s fixed point theorem (DFPT) and its various versions obtained by several authors. Using this result we obtain existence of at least one positive solution for a quadratic integral equation of Fredholm type.     

Comments on “On the Indefinite Quadratic Fractional Optimization with Two Quadratic Constraints”

In this note, with reference to a paper by the same authors, we add an extra assumption, correct the statement of Lemma 2.1 and subsequently correct the proof of this lemma.