Superficie cubiche di\mathbb{A}_\mathbb{C}^3 a curve sottoinsieme intersezione completa

Summary In this paper we classify the algebraic cubic surfaces of the affine space is the complex field, whose algebraic curves are set-theoretic complete intersections of ; in other words surfaces such that every prime ideal of height 1in the coordinate ring [] of is the radical of a principal ideal; if is non singular in codimension 1this means that [] is semifactorial. We give the equations of such surfaces within linear isomorphisms of providing also methods by which one can construct the equations of the surfaces cutting on its curves as set-theoretic complete intersections. Moreover for each of these surfaces we determine the minimum positive number such that every algebraic curve of with multiplicity of intersection , is complete intersection of itself with another surface § 8where the results are summarized). We tackle also the problem of such a classification over algebraically closed fields k different from .

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Lavoro eseguito nell'ambito del G.N.S.A.G.A. del C.N.R.     

Quadratically constrained quadratic programming: Some applications and a method for solution

The problem (QPQR) considered here is: minimizeQ ₁ (x) subject toQ _i (x) 0,i M ₁ {2,...,m},x P R ⁿ, whereQ _i (x), i M {1} M ₁ are quadratic forms with positive semi-definite matrices, andP a compact nonempty polyhedron of Rⁿ. Applications of (QPQR) and a new method to solve it are presented.Letu S={u R ^m;u 0, u _i= l}be fixed;then the problem:iM minimize u _iQ_i (x (u)) overP, always has an optimal solutionx (u), which is either feasible, iM i.e. u C₁ {u S;Q _i (x (u)) 0,i M ₁} or unfeasible, i.e. there exists ani M ₁ withu C {u S; Q_i(x(u)) 0}.Let us defineC _i C_i S _i withS _i {u S; u _i=0}, i M. A constructive method is used to prove that C _i is not empty and thatx (û) withiM û C _i characterizes an optimal solution to (QPQR). Quite attractive numerical results have been reached with this method.

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Zusammenfassung Die vorliegende Arbeit befaßt sich mit Anwendungen und einer neuen Lösungsmethode der folgenden Aufgabe (QPQR): man minimiere eine konvexe quadratische ZielfunktionQ _i (x) unter Berücksichtigung konvexer quadratischer RestriktionenQ _i (x) 0, iM ₁ {2,...,m}, und/oder linearer Restriktionen.·Für ein festesu S {u R ^m;u 0, u _i=1},M {1} M₁ besitzt das Problem:iM minimiere die konvexe quadratische Zielfunktion u _i Q_i (x (u)) über dem durch die lineareniM Restriktionen von (QPQR) erzeugten, kompakten und nicht leeren PolyederP R ⁿ, immer eine Optimallösungx (u), die entweder zulässig ist: u C₁ {u S;Q ₁ (x (u)) 0,i M ₁} oder unzulässig ist, d.h. es existiert eini M ₁ mitu C_i {u S;Q _i (x(u))0}.Es seien folgende MengenC _i C_i S _i definiert, mitS _i {u S;u _i=0}, i M. Es wird konstruktiv bewiesen, daß C _i 0 undx (û) mitû C _i eine Optimallösung voniM iM (QPQR) ist; damit ergibt sich eine Methode zur Lösung von (QPQR), die sich als sehr effizient erwiesen hat. Ein einfaches Beispiel ist angegeben, mit dem alle Schritte des Algorithmus und dessen Arbeitsweise graphisch dargestellt werden können.

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An earlier version of this paper was written during the author's stay at the Institute for Operations Research, Swiss Federal Institute of Technology, Zürich.     

Ergodic properties of semi-Markovian operators on the Z1-part

Summary In this note we consider a semi-Markovian operator, that is a positive linear mapping T: L ¹ L ¹ such that sup T ⁿ <. We study the behavior of T ⁿ on the Z ¹-part of the space (the disappearing part in Sucheston's terminology). We show in particular, that if the operator T has a non-trivial conservative part in Z ¹, then the ratio theorem must fail.Research supported by the U.S.Army Research Office (Durham) under contract DA-31-124-ARO(D)-288.     

Fourier series of additive vector measures and their term-by-term differentiation

On a measurable space (T, , ) we choose an additive measure: Z (Z is a Banach space) with the following property: for alle , we have ; this measure defines an indefinite integral over the measure onL ² (T, ,). We prove that if { _n (t)} _n ^=1/ is an orthonormal basis inL ² and _n (e)=e _n (t) d, then any additive measure: Z whose Radon-Nikodým derivatived/d belongs toL ² is uniquely expandable in a series(e)= _n ^=1/ _n n(e) that converges to(e) uniformly with respect toe can be differentiated term-by-term, and satisfies _n ^=1/ _n ^/2 <. In the caseL ²[0,2],Z=, the Fourier series of a 2-periodic absolutely continuous functionF(t) such thatF'(t) L ²[0, 2] is superuniformly convergent toF(t).Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 180–184, August, 1998.     

Jensen inequalities for functions with higher monotonicities

Summary We investigate generalizations of the classical Jensen and Chebyshev inequalities. On one hand, we restrict the class of functions and on the other we enlarge the class of measures which are allowed. As an example, consider the inequality (J)(f(x) d) A (f(x) d, d d = 1. Iff is an arbitrary nonnegativeL _x function, this holds if 0, is convex andA = 1. Iff is monotone the measure need not be positive for (J) to hold for all convex withA = 1. If has higher monotonicity, e.g., is also convex, then we get a version of (J) withA < 1 and measures that need not be positive.     

Computation of suboptimal randomized strategies for steering the random motion of a point under partial observation

Consider the random motion in the plane of a pointM, whose velocityv=(v ₁,v ₂) is perturbed by an ²-valued Gaussian white noise. Only noisy nonlinear observations taken on the point location (state) are available toM. The velocityv is of the formv(y)= _u (u ₁,u ₂) _y (du), wherey denotes the value of the observed signal,U is the range of the velocity, and, for eachy, _y is a probability measure on (U). Using the available observations, the pointM wishes to steer itself into a given target set by choosing a randomized strategy ={ _y :y ²}. Sufficient conditions on weak optimal randomized strategies are derived. An algorithm for computing weak suboptimal randomized strategies is suggested, and the strategies are computed for a variety of cases.This work was partially supported by a grant from Control Data.     

On dominant operators

This paper is devoted to the study of dominant operators with an emphasis on their spectral properties. In particular the equation (T–)f() x (T a dominant or hyponormal operator on the Hilbert space ,x andf a function from the open setU to ) is investigated in an effort to discover necessary and/or sufficient conditions for the analyticity off.Supported in part by the National Science Foundation.     

Endomorphism monoids of algebras whose subalgebra lattices are chains

A monoidM and a latticeL arealgebraic if there is an algebraA with endomorphism monoid EndA M and subalgebra lattice SuA L. For each chainC we characterize those monoidsM for whichM and C are algebraic. In particular we show that a finite monoidM is algebraic with the three-chain iff the equalizers ofM form a chainE 3. The same assertion however fails for infinite monoids. This generalizes the corresponding result for two-chains and solves a problem posed by B. Jónsson ([2], p. 147). We settle the same question for all longer chainsK. Presented by Ivo Rosenberg.     

Systèmes de sortie ℱ Dt prévisibles

Summary Given a random closed setM, adapted to a filtration ( _t ), we construct a local time ofM which is both ( _t ) adapted and ( _Dt ) predictable, whereD _t =inf{s>t: sM}. Similarly an exit system, both ( _t ) optional and ( _Dt ) predictable, is associated withM and with the process representing the future at each timet. The paper is motivated by the markovian case, where the general results are applied to Ray processes.     

Arithmetic distance on compact symmetric spaces

Let M be a compact Riemannian symmetric space. Then M=G/K, where G is the identity component of the isometry group of M and K is the isotropy subgroup of G at a point. In 1965 Nagano studied and classified the geometric transformation groups of compact symmetric spaces. Roughly speaking they are larger groups L that act on M, (i) G/L; (ii) L is a Lie transformation group acting effectively on M; (iii) L preserves the symmetric structure of M; and (iv) L is simple.Using Helgason spheres, S(), the minimal totally geodesic spheres in a compact irreducible symmetric space, we define an arithmetic distance for compact irreducible symmetric spaces and prove: THEOREM. Let M=G ^p(K ⁿ ), K=, H, or R, or M=AI(n), of rank greater that 1 and dimension greater that 3, let L be the geometric transformation group of M. Let L={: MM: is a diffeomorphism and preserves arithmetic distance}. Then L=L     

Weak Variation of Gaussian Processes

Let X(t) (tR) be a real-valued centered Gaussian process with stationary increments. We assume that there exist positive constants ₀, C ₁, and c ₂ such that for any tR and hR with |h|₀ and for any 0r<min{|t|, ₀} where is regularly varying at zero of order (0 < < 1). Let be an inverse function of near zero such that (s)=(s) log log(1/s) is increasing near zero. We obtain exact estimates for the weak -variation of X(t) on [0,a].     

On steiner ratio conjectures

LetM be a metric space andP a finite set of points inM. The Steiner ratio inM is defined to be(M)=inf{L _s(P)/L _m(P) |P M}, whereL _s(P) andL _m(P) are the lengths of the Steiner minimal tree and the minimal spanning tree onP, respectively. In this paper, we study various conjectures on(M). In particular, we show that forn-dimensional Euclidean space _n ,( _n )>0.615.Supported in part by the National Science Foundation of China.     

Mittelwert der Eulerschen φ-Funktion und des Quadrates der Dirichletschen Teilerfunktion in algebraischen Zahlkörpern

LetK be an algebraic number field, and for every integer K let () andd(), respectively, denote the number of relatively prime residue classes and the number of divisors of the principal ideal (). Asymptotic equalities are proved for the sums () and d ²(), where runs through certain finite sets of integers ofK.     

Some homological properties of commutative semitrivial ring extensions

LetR be a commutative ring with 1 andM anR-module. If:M _R MR is anR-module homomorphism satisfying(mm)=(mm) and(mm)m=m(mm), the additive abelian groupRM becomes a commutative ring, if multiplication is defined by (r,m)(r,m)=(rr+(mm),rm+rm). This ring is called the semitrivial extension ofR byM and and it is denoted byR M. This generalizes the notion of a trivial extension and leads to a more interesting variety of examples. The purpose of this paper is to studyR M; in particular, we are interested in some homological properties ofR M as that of being Cohen-Macaulay, Gorenstein or regular. A sample result: Let (R,m) be a local Noetherian ring,M a finitely generatedR-module and Im() m. ThenR M is Gorenstein if and only if eitherRM is Gorenstein orR is Gorenstein,M is a maximal Cohen-Macaulay module andMM ^*, where the isomorphism is given by the adjoint of.     

Über den algorithmischen Nachweis von Ungleichungen I

We shall develop a method to prove inequalities in a unified manner. The idea is as follows: It is quite often possible to find a continuous functional : ⁿ , such that the left- and the right-hand side of a given inequality can be written in the form (u)(v) for suitable points,v=v(u). If one now constructs a map ^{n n} , which is functional increasing (i.e. for each x ⁿ (which is not a fixed point of ) the inequality (x)<((x)) should hold) one specially gets the chain (u)( u))( ²(u))... ⁿ (u)). Under quite general conditions one finds that the sequence { ⁿ (u)} _n converges tov=v(u). As a consequence one obtains the inequality (u)(v).     

Factorization of a selfadjoint nonanalytic operator function II. Single spectral zone

We consider a selfadjoint and smooth enough operator-valued functionL() on the segment [a, b]. LetL(a)0,L(b)0, and there exist two positive numbers and such that the inequality |(L()f, f)|< ([a, b] f=1) implies the inequality (L'()f, f)>. Then the functionL() admits a factorizationL()=M()(I-Z) whereM() is a continuous and invertible on [a, b] operator-valued function, and operatorZ is similar to a selfadjoint one. This result was obtained in the first part of the present paper [10] under a stronge conditionL()0 ( [a,b]). For analytic functionL() the result of this paper was obtained in [13].     

On the order of approximation to functions ofH p (R) (0<p≦1) by certain means of Fourier integrals

H ^P (R ₊ ² ) — R ₊ ² ={zC: Imz>0} ^p (R) — H ^p (R ₊ ² ). P ^k (f,x) — ë- — ,W ^k (f,x) — — R ^k, (f,x) — f H (R) (. §1,1)–3)); _k (, f) _p — - . , fH ^p (R) 0<p1,kN; (1+)^–1<p1, 0<<,kN.     

φ-Factors for Function Seminorms

Let (, A, ) be a measure space, a function seminorm on M, the space of measurable functions on , and M the space {f M : (f) < }. Every Borel measurable function : [0, ) [0, ) induces a function : M M by (f)(x) = (|f(x)|). We introduce the concepts of -factor and -invariant space. If is a -subadditive seminorm function, we give, under suitable conditions over , necessary and sufficient conditions in order that M be invariant and prove the existence of -factors for . We also give a characterization of the best -factor for a -subadditive function seminorm when is -finite. All these results generalize those about multiplicativity factors for function seminorms proved earlier.     

Taut Distance-Regular Graphs of Odd Diameter

Let denote a bipartite distance-regular graph with diameter D 4, valency k 3, and distinct eigenvalues ₀ > ₁ > ··· > _D. Let M denote the Bose-Mesner algebra of . For 0 i D, let E _i denote the primitive idempotent of M associated with _i . We refer to E ₀ and E _D as the trivial idempotents of M. Let E, F denote primitive idempotents of M. We say the pair E, F is taut whenever (i) E, F are nontrivial, and (ii) the entry-wise product E F is a linear combination of two distinct primitive idempotents of M. We show the pair E, F is taut if and only if there exist real scalars , such that _{i + 1 i + 1} – _{i – 1 i – 1} = _i ( _{i + 1} – _{i – 1}) + _i ( _{i + 1} – _{i – 1}) + (1 i D – 1)where ₀, ₁, ..., _D and ₀, ₁, ..., _D denote the cosine sequences of E, F, respectively. We define to be taut whenever has at least one taut pair of primitive idempotents but is not 2-homogeneous in the sense of Nomura and Curtin. Assume is taut and D is odd, and assume the pair E, F is taut. We show

About generalized ideals in a distributive lattice

Let L be a distributive lattice characterized by a ternary operation (, ,), where (a,b,c)=(ab)(bc)(ac)=(ab)(ac)(bc), a,b,cL. The note considers convex sublattices of L, called generalized ideals of L generated by the operation (, ,). Some remarks have been stated about the graph of a distributive lattice.