Maximal Monotone Multifunctions of Br?ndsted–Rockafellar Type

We consider whether the inequality-splitting property established in the Brøndsted–Rockafellar theorem for the subdifferential of a proper convex lower semicontinuous function on a Banach space has an analog for arbitrary maximal monotone multifunctions. We introduce the maximal monotone multifunctions of type (ED), for which an inequality-splitting property does hold. These multifunctions form a subclass of Gossez"s maximal monotone multifunctions of type (D); however, in every case where it has been proved that a multifunction is maximal monotone of type (D) then it is also of type (ED). Specifically, the following maximal monotone multifunctions are of type (ED): ultramaximal monotone multifunctions, which occur in the study of certain nonlinear elliptic functional equations; single-valued linear operators that are maximal monotone of type (D); subdifferentials of proper convex lower semicontinuous functions; subdifferentials of certain saddle-functions. We discuss the negative alignment set of a maximal monotone multifunction of type (ED) with respect to a point not in its graph – a mysterious continuous curve without end-points lying in the interior of the first quadrant of the plane. We deduce new inequality-splitting properties of subdifferentials, almost giving a substantial generalization of the original Brøndsted–Rockafellar theorem. We develop some mathematical infrastructure, some specific to multifunctions, some with possible applications to other areas of nonlinear analysis: the formula for the biconjugate of the pointwise maximum of a finite set of convex functions – in a situation where the obvious formula for the conjugate fails; a new topology on the bidual of a Banach space – in some respects, quite well behaved, but in other respects, quite pathological; an existence theorem for bounded linear functionals – unusual in that it does not assume the existence of any a priori bound; the 'big convexification" of a multifunction.     

On Upper and Lower Almost β-continuous Multifunctions

In this paper, the authors define a multifunction F : X Y to be upper (lower) almost -continuous if F+(V) (F- (V)) is -open in X for every regular open set V of Y. They obtain some characterizations and several properties concerning upper (lower) almost -continuous multifunctions.     

Discrete Newton methods and iterated defect corrections

Summary In this paper we present a general theory for discrete Newton methods, iterated defect corrections via neighbouring problems and deferred corrections based on asymptotic expansions of the discretization error.Dedicated to Professor Dr. J. Weisinger on the occasion of his sixty-fifth birthday     

Die Nuklearität der Ultradistributionsräume und der Satz vom Kern II

Continuing the research of part I conditions equivalent to ()- or ()-nuclearity of spaces of ultradifferential functions and their duals as well as some applications are given. To get these results it is shown that tensor products of smooth sequence spaces, power series spaces, and spaces S(M_q) introduced in part I are isomorphic to suitable sequence spaces of the same class, which are stable provided the factors are stable power series spaces. Hence it is possible to establish isomorphisms between different functions spaces, to calculate the nuclearity types of tensor products by the nuclearity types of the factors, and to prove that the class of ()- or ()-nuclear spaces is closed under forming tensor products iff is multiplicatively stable.     

Maximal inequalities,convexity inequality,and their duality. II

, , . . . [1], , . , , ., , L logL. , , . . . . [5]. , .     

On Riesz means with respect to a cylindric distance function

(1–) ₊ , — R ⁿ =R ^j ×R ^k , ()=max{¦ ₁¦, ¦ ₁¦},=( ₁, ₂), ₁R ^J , ₂R ^k ,j,k1,n=j+k. n=3 , (1–) ₊ [L ¹(R ⁿ )]¹, >1/2; j=4, (1–) ₊ R L ^p (R ⁿ ). .

---

---

The author would like to thank Professor W. Trebels for encouragement and valuable advice.     

Quasianapole Interaction of Charged Fermions

We obtain the analytic expression for the total cross section of the reaction e ^– e ⁺l ^– l ⁺ (l=,) taking possible quasianapole interaction effects into account. We find numerical restrictions on the interaction parameter value from data for the reaction e ^– e ^+–+ in the energy domain below the Z ⁰ peak.     

An extension of Edelstein's theorem to a uniform space

Summary The paper contains fixed point theorems of Edelstein's type for contractive mappings in (,)-chainable uniform spaces.     

Нормаляный поридок аддитявных арифметичыских функций на множестве дсдвинутых” пностых чясел

.      

Local invariants of differential equations

We consider an analytic system X=(X) in the neighborhood of the fixed point X=0. Depending on the characteristic numbers of the matrix (/x)₀, we define the integer d 0 as the dimension of the normal form or as the multiplicity of the resonance. We show that a system with d=1, subject to certain additional assumptions, has a finite number of invariants relative to reversible formal changes of variablesx = (Y). All these invariants are the coefficients of some normal form. We touch upon questions concerning invariants of relatively smooth and continuous substitutions.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 499–507, October, 1973.     

The Discrete Spectrum Asymptotics with Respect to a Large Coupling Constant under Strong Nonnegative Perturbations

Let A be a self-adjoint operator, let (, ) be an inner gap in the spectrum of the operator A, and let B(t) = A + tW ^* W, where the operator W(A – iI)^-1 is not necessarily bounded. Conditions are obtained under which the spectrum of B(t) in (, ) is discrete. Let N(, A, W, ), (, ), > 0, be the number of eigenvalues of the operator B(t) passing the point (, ) as t increases from 0 to . The asymptotics of N(, A, W, ) as + is obtained in terms of the spectral asymptotics of a certain self-adjoint compact operator. Bibliography: 5 titles.     

A chord-stretching map of a convex loop is an isometry

Let ⁿ be n-dimensional Euclidean space, and let : [0, L] ⁿ and : [0, L] ⁿ be closed rectifiable arcs in ⁿ of the same total length L which are parametrized via their arc length. is said to be a chord-stretched version of if for each 0s tL, |(t)–(s)| |(t)–(s)|. is said to be convex if is simple and if ([0, L]) is the frontier of some plane convex set. Individual work by Professors G. Choquet and G. T. Sallee demonstrated that if were simple then there existed a convex chord-stretched version of . This result led Professor Yang Lu to conjecture that if were convex and were a chord-stretched version of then and would be congruent, i.e. any chord-stretching map of a convex arc is an isometry. Professor Yang Lu has proved this conjecture in the case where and are C ² curves. In this paper we prove the conjecture in general.     

Positive Solutions for a Singular Nonlinear Problem on a Bounded Domain in R 2

For a bounded regular Jordan domain in R ², we introduce and study a new class of functions K() related on its Green function G. We exploit the properties of this class to prove the existence and the uniqueness of a positive solution for the singular nonlinear elliptic equation u+(x,u)=0, in D(), with u=0 on and uC(), where is a nonnegative Borel measurable function in ×(0,) that belongs to a convex cone which contains, in particular, all functions (x,t)=q(x)t ^–,>0 with nonnegative functions qK(). Some estimates on the solution are also given.     

β-Expansions and Frequency of Zero

We study how the frequency of zero in the -expansion of x varies with respect to for almost all x.     

The intersection of complete geodesics on <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript> with positive curvature

Let M be a Riemannian manifold. A complete geodesic on M means that :(-,+)M is a normalized geodesic. In this paper, we prove that on (S²,g) with positive curvature, any two complete geodesics must intersect an infinite number of times, and a complete geodesic must self-intersect an infinite number of times. Mathematics Subject Classification (2000) 53C40 (53C22)     

On Sets of Planes in Projective Spaces Intersecting Mutually in One Point

Let be a projective space. In this paper we consider sets of planes of such that any two planes of intersect in exactly one point. Our investigation will lead to a classification of these sets in most cases. There are the following two main results:- If is a set of planes of a projective space intersecting mutually in one point, then the set of intersection points spans a subspace of dimension 6. There are up to isomorphism only three sets where this dimension is 6. These sets are related to the Fano plane.- If is a set of planes of PG(d,q) intersecting mutually in one point, and if q3, 3(q²+q+1), then is either contained in a Klein quadric in PG(5,q), or is a dual partial spread in PG(4,q), or all elements of pass through a common point.     

Posterior Sensitivity to the Sampling Distribution and the Prior: More than One Observation

Sensitivity of a posterior quantity (f, P) to the choice of the sampling distribution f and prior P is considered. Sensitivity is measured by the range of (f, P) when f and P vary in nonparametric classes ^f and ^P respectively. Direct and iterative methods are described which obtain the range of (f, P) over f ^f when prior P is fixed, and also the overall range over f ^f and P ^P . When multiple i.i.d. observations X ₁,...,X _k are observed from f, the posterior quantity (f, P) is not a ratio-linear function of f. A method of steepest descent is proposed to obtain the range of (f, P). Several examples illustrate applications of these methods.     

Bergman-Vekua theory in differential form

We develop Bergman-Vekua integral operator theory on the basis of a new kind of series suggested by polynomial operators (classP _j operators); in a sense, this is an analog of the Weierstrass approach as opposed to the Cauchy-Riemann approach in classical complex analysis, which also has advantages over the original form in boundary value problems and other applications.Dedicated to the memory of my friend, Peter Henrici     

Stream vectors in three dimensional aerodynamics

Summary This work deals with the decomposition of a vector fieldu intou=×+. Non homogeneous boundary conditions on or are investigated; applications to the computation of inviscid flows are given; finally a conforming finite element implementation is studied and tested.     

Transferring estimates for zonal convolution operators

_p - , . Ad- .