Constructive Axiomatic Approach to the Determination of the Orbit Spaces of Coregular Compact Linear Groups

We review the proposal of a constructive axiomatic approach to the determination of the orbit spaces of all the real compact linear groups, obtained through the computation of a metric matrix , which is defined only in terms of the scalar products between the gradients p₁(x),...,p_q(x) of the elements of a minimal integrity basis (MIB) for the ring [ⁿ]^G of G-invariant polynomials. The domain of semi-positivity of is known to realize the orbit space ⁿ/G of G as a semi-algebraic variety in the space ^q spanned by the variables p₁,...,p_q. The matrices can be obtained from the solutions of a universal differential equation (master equation), which satisfy convenient initial conditions. The master equation and the initial conditions involve as free parameters only the degrees d_a of the p_a(x)s. This approach tries to bypass the actual impossibility of explicitly determining a set of basic polynomial invariants for each group. Our results may be relevant in physical contexts where the study of covariant or invariant functions is important, like in the determination of patterns of spontaneous symmetry breaking in quantum field theory, in the analysis of phase spaces and structural phase transitions (Landaus theory), in covariant bifurcation theory, in crystal field theory and so on. Mathematics Subject Classifications (2000) 14L24, 13A50, 14L30.This paper is partially supported by INFN and MURST 40% and 60%.     

Geometric methods in the study of irregularities of distribution

Let be a signed measure on E ^d with E ^d =0 and ¦¦E^d<. DefineD _s() as sup ¦H¦ whereH is an open halfspace. Using integral and metric geometric techniques results are proved which imply theorems such as the following.Theorem A. Let be supported by a finite pointsetp _i. ThenD _s()>c _d(₁/ ₂)^1/2{ _{i(p i})²}^1/2 where ₁ is the minimum distance between two distinctp _i, and ₂ is the maximum distance. The numberc _d is an absolute dimensional constant. (The number .05 can be chosen forc ₂ in Theorem A.)Theorem B. LetD be a disk of unit area in the planeE ², andp ₁,p ₂,...,p _n be a set of points lying inD. If m if the usual area measure restricted toD, while nP _i=1/n defines an atomic measure _n, then independently of _n,nD _s(m – _n) .0335n ^1/4. Theorem B gives an improved solution to the Roth disk segment problem as described by Beck and Chen. Recent work by Beck shows thatnD _s(m – _n)cn ^1/4(logn)^–7/2.     

Daugavet type inequalities for operators on L p–spaces

Let T be a regular operator from L ^p L ^p. Then , where T_r denotes the regular norm of T, i.e., T_r=|T| where |T| denotes the modulus operator of a regular operator T. For p=1 every bounded linear operator is regular and T=T_r, so that the above inequality generalizes the Daugavet equation for operators on L ¹–spaces. The main result of this paper (Theorem 9) is a converse of the above result. Let T be a regular linear operator on L _p and denote by T _A the operator T_A. Then for all A with (A)>0 if and only if .     

Über die mittelfläche einer isotropen geradenkongruenz

Let be the middle surface of an isotropic rectilinear congruence of class C³ in the real Euclidean space E³. When the spherical image of is parametrized by special isothermal coordinates (u,v) G ², can be described by a generating harmonic function A(u,v). Using such a C-representation of , the basic properties of regularity and curvature of are discussed. Moreover, the cases that be a minimal (regular) surface ₁, or a plane surface ₂ are solved explicitly. In connection with the latter results (which are already well-known from Ribaucour) several new characterizations for being a regular surface ₁ resp. ₂ are given: they are based on special properties (like: being asymptotic lines resp. lines of curvature of ) of those curves c (-Spurlinien) in the tangents of which form in each point Xc a minimal angle with the straight line of passing through X.

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Herrn Prof. Dr. Karl Strubecker zum 80. Geburtstag gewidmet     

Volume of Hyperbolic 3-Manifolds with Iterated Pseudo-Anosov Amalgamations (Dedicated to Professor Mitsuyoshi Kato on his sixtieth birthday)

Let : be a pseudo-Anosov homeomorphism. We will study an asymptotic behaviour of the volume of closed hyperbolic 3-manifolds N _n obtained from certain 3-manifolds M, M by attaching their boundaries by the n-th iteration ⁿ of .     

Tables of good lattices in four and five dimensions

Summary The good lattice theory yields a powerful method of computing approximations for the integral of functions defined on [0,1]^s through averaged sums ofm evaluations. We present a continuation of the only existing table of best lattices fors=4 up tom=3298, and the first table fors=5 up tom=772.From the Ecole Polytechnique de Montréal, département de Mathématiques appliquées, C.P. 6079 Station A, Montréal QC, Canada H3C 3A7. This research was supported by Grant A3087 from the National Science and Engineering Research Council of Canada     

Regular homomorphisms of generalized projective planes

A generalized projective plane is an incidence structure together with a relation distant on the set of points and also on the set of lines, such that any two distant points A,B (lines a,b) have a unique common line (A,B) (common point (a,b)) and three further axioms hold. Every commutative ring with 1 supplies a model. A homomorphism of into an incidence structure is called regular if the following condition and its dual are valid: A distant B and c IA,B implies c=(A,B). We shall prove the following two theorems. Let be a generalized projective plane satisfying a richness condition called (U). Let M I m. If and are regular homomorphisms of such that X = M X = M for each point X of the line m then A = B A = B for any two points A,B. If is a projective plane over a commutative ring such that (U) holds then the surjective regular homomorphisms of are induced by the ideals of the ring; in particular, the image of under a regular homomorphism is again a projective plane over a ring, and preserves distant.     

Sur les égalités du miroir et certaines formes faibles de la conjecture de Greenberg

Let p be an odd prime. For any CM number field K containing a primitive p^th-root of unity, class field theory and Kummer theory put together yield the well known reflection inequality ^+– between the plus and minus parts of the -invariant of K. Greenbergs conjecture asserts that ⁺ is always trivial. We study here a weak form of this conjecture, namely ⁺=^– if and only if ⁺=^–=0.     

Criteria for the Boundedness and Compactness of Operators with Power-Logarithmic Kernels

In the present work, necessary and sufficient conditions are given in terms of a nonnegative Borel measure which ensure the boundedness and compactness of operators with power-logarithmic kernels from L ^p (0, a) to L ^p (0, a) (or to L ^q (0, a)), where 0 < a < , 1 < p, q < , > 1/p and 0.     

Minimality of root vectors of operator functions analytic in an angle

We study the minimality of elementsx _h,j,k of canonical systems of root vectors. These systems correspond to the characteristic numbers _k of operator functionsL() analytic in an angle; we assume that operators act in a Hilbert space . In particular, we consider the case whereL()=I+T()c, >0,I is an identity operator,C is a completely continuous operator, (I- C)^–1c for ¦arg¦, 0<<, the operator functionT() is analytic, and T()c for ¦arg¦<. It is proved that, in this case, there exists >0 such that the system of vectorsC ^v x _h,j,k is minimal in for arbitrary positive <1+, provided that ¦_k¦>.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 545–566, May, 1994.This research was partially supported by the Ukrainian State Committee of Science and Technology.     

Fine Analysis of the Quasi-Orderings on the Power Set

We pursue the fine analysis of the quasi-orderings and on the power set of a quasi-ordering (Q,). We set X Y if every xX is majorized in by some yY, and X Y if every yY is minorized in by some xX. We show that both these quasi-orderings are -wqo if and only if the original quasi-ordering is ( )-wqo. For this holds also restricted to finite subsets, thus providing an example of a finitary operation on quasi-orderings which does not preserve wqo but preserves bqo.     

Hardy spaces of predictable martingales

, P _p Q _p .

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This research was partially supported by the Hungarian Scientific Foundation.     

Uniqueness for isotropic diffusions with a linear drift

This paper proves that-valued solutions to the SDE are unique in distribution, when D^d is convex and open, D, c>0, is positive and locally Lipschitz on D and zero on D, and {xD:g(x)r} is convex for r sufficiently small. The proof (for =0) is based on the transformation X_te^ctX_t, which removes the drift, and a random time change. Although the set-up is rather specialized the result gives uniqueness for some SDEs that cannot be treated by any of the conventional techniques.Mathematics Subject Classification (2000):60J60, 60H10     

On Differences of Semicubical Powers

For X,Y,>0, let and define I ₈(X,Y,) to be the cardinality of the set. In this paper it is shown that, for >0, Y ²/X ³=O(), =O(Y ³/X ³) and X=O (Y ²), one has I ₈(X,Y,)=O(X ² Y ²+X min (X ^{3/2} Y ³, X ^{11/2} Y ^{–1})+X min (^{1/3} X ² Y ³, X ^{14/3} Y ^{1/3})), with the implicit constant depending only on . There is a brief report on an application of this that leads, by way of the Bombieri-Iwaniec method for exponential sums, to some improvement of results on the mean squared modulus of a Dirichlet L-function along a short interval of its critical line.     

Solution of two sequencing problems

The solution of the following problems is offered. Suppose a multiset J (¦J¦=p) is given. For each pair of elements and J, a number 1 P is given. Moreover, if 1 < x<p then x is undefined. If x=1, then x=p. Problem 1. Find the permutation ₁..._F of elements of the multiset J satisfying the following conditions. Let _i, _i=. If i,j < x, thenj <i. If i,j > x, then i<j. Such a permutation is called a PC-schedule. Problem 2. Find a PC-schedule in which the following property holds: if i < x < j, _i=, _j=, then. Such a PC-schedule is called an SC-schedule. The conditions under which these problems have solutions are studied. For their solution an algorithm of shifts is used with the complexity O(¦B(J)¦²¦J¦).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 124, pp. 44–72, 1983.     

Surfaces of general type with <Emphasis Type="Italic">p</Emphasis><Subscript><Emphasis Type="Italic">g</Emphasis></Subscript>=0, <Emphasis Type="Italic">K</Emphasis><Superscript>2</Superscript>=6 and non birational bicanonical map

Let S be a minimal surface of general type with p_g=0 and K²=6, such that its bicanonical map is not birational. The map is a morphism of degree 4 onto a surface. The case of deg = 4 is completely classified in [Topology, 40 (5) (2001), 977–991] and the present paper completes the characterization of these surfaces. It is proven that the degree of cannot be equal to 3, and the geometry of surfaces with deg = 2 is analysed in detail. The last section contains three examples of such surfaces, two of which appear to be new.Mathematics Subject Classification (2000): 14J29     

A commutativity criterion for certain algebras of invariant differential operators on nilpotent homogeneous spaces

Let G be a connected, simply connected real nilpotent Lie group with Lie algebra , H a connected closed subgroup of G with Lie algebra and f a linear form on satisfying f([, ]) = {0} Let _f be the unitary character of H with differential at the origin. Let _f be the unitary representation of G induced from the character _f of H. We consider the algebra (, , f) of differential operators invariant under the action of G on the bundle with basis G/H associated to these data. We show that (, , f) is commutative if and only if _f is of finite multiplicities. This proves a conjecture of Corwin-Greenleaf and Duflo. Mathematics Subject Classification (1991):43A80, 43A85, 22E25, 22E27, 22E30UMR n 7539 du CNRS, Analyse, Géométrie et Applications.UMR n 7586 du CNRS, Théorie des Groupes, Représentations, Applications.     

Stability and Attraction to Normality for Lévy Processes at Zero and at Infinity

We prove some limiting results for a Lévy process X _t as t0 or t, with a view to their ultimate application in boundary crossing problems for continuous time processes. In the present paper we are mostly concerned with ideas related to relative stability and attraction to the normal distribution on the one hand and divergence to large values of the Lévy process on the other. The aim is to find analytical conditions for these kinds of behaviour which are in terms of the characteristics of the process, rather than its distribution. Some surprising results occur, especially for the case t0; for example, we may have X _t /t ^P + (t0) (weak divergence to +), whereas X _t /t a.s. (t0) is impossible (both are possible when t), and the former can occur when the negative Lévy spectral component dominates the positive, in a certain sense. Almost sure stability of X _t , i.e., X _t tending to a nonzero constant a.s. as t or as t0, after normalisation by a non-stochastic measurable function, reduces to the same type of convergence but with normalisation by t, thus is equivalent to strong law behaviour. Boundary crossing problems which are amenable to the methods we develop arise in areas such as sequential analysis and option pricing problems in finance.     

Location of blow-up set for a semilinear parabolic equation with large diffusion

This paper is concerned with where is a bounded smooth domain in R ^N , T _D >0, D>0, and p>1 with (N–2)pN+2. Let P ₂ be the projection from L ²() onto the second Neumann eigenspace. We prove that, if P ₂0 in and D is sufficiently large, the solution u of (P) blows up only near the set , where . Mathematics Subject Classification (2000):35K20, 35K55, 58K57     

Wilson's functional equation on C

Summary We find the complete set of continuous solutionsf, g of Wilson's functional equation _{n = 0} ^{N – 1} f(x + wⁿy) = Nf(x)g(y), x, y C, given a primitiveN ^th rootw of unity.Disregarding the trivial solutionf = 0 andg any complex function, it is known thatg satisfies a version of d'Alembert's functional equation and so has the formg(z) = g (z) = N^–1 _{n = 0} ^{N – 1} E(wⁿz) for some C². HereE ₍₁, ₂)(x + iy) = exp( ₁x + ₂).For fixedg = g the space of solutionsf of Wilson's functional equation can be decomposed into theN isotypic subspaces for the action of Z _N on the continuous functions on C. We prove that ther ^th component, wherer {0, 1, ,N – 1}, of any solution satisfies the signed functional equation _{n = 0} ^{N – 1} f(x + wⁿy)w^nr = Ng(x)f(y), x, y C. We compute the solution spaces of each of these signed equations: They are 1-dimensional and spanned byz _{n = 0} ^{N – 1} w^nr E(wⁿz), except forg = 1 andr 0 where they are spanned by andz ^{N – r}. Adding the components we get the solution of Wilson's equation. Analogous results are obtained with the action ofZ _N on C replaced by that ofSO(2).The case ofg = 0 in the signed equations is special and solved separately both for Z _N andSO(2).