Dirichlet forms,quasiregular functions and Brownian motion

Summary Given a quasiregular function on an open setU in ⁿ it is shown that there exists a diffusionX _t inU such that mapsX _t inton-dimensional Brownian motion. The process is constructed from a Dirichlet form which can be described explicitly. This enables us to apply stochastic methods in the investigation of quasiregular mappings. Some examples of applications are given, including boundary behaviour and value distribution.     

Conjugation for polynomial mappings

We consider Keller's functions, namely polynomial functionsf:C ⁿ C ⁿ with detf(x)=1 at allx C ⁿ. Keller conjectured that they are all bijective and have polynomial inverses. The problem is still open.Without loss of generality assumef(0)=0 andf'(0)=I. We study the existence of certain mappingsh , > 1, defined by power series in a ball with center at the origin, such thath(0)=I andh (f(x))=h (x). So eachh conjugates f to its linear part I in a ball where it is injective.We conjecture that for Keller's functionsf of the homogeneous formf(x)=x +g(x),g(sx)=s ^dg(x),g(x)ⁿ=0,xC ⁿ,sC the conjugationh for f is anentire function.     

New lower semicontinuity results for polyconvex integrals

Summary We study integral functionals of the formF(u, )= f(u)dx, defined foru C¹(;R ^k), R ⁿ . The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) c₀¦(A)¦ for a suitable constant c₀ > 0, where (A) is then-vector whose components are the determinants of all minors of thek×n matrixA. We prove thatF is lower semicontinuous onC ¹(;R ^k) with respect to the strong topology ofL ¹(;R ^k). Then we consider the relaxed functional , defined as the greatest lower semicontinuous functional onL ¹(;R ^k ) which is less than or equal toF on C¹(;R ^k). For everyu BV(;R ^k) we prove that (u,) f(u)dx+c₀¦D^su¦(), whereDu=u dx+D^su is the Lebesgue decomposition of the Radon measureDu. Moreover, under suitable growth conditions onf, we show that (u,)= f(u)dx for everyu W^1,p(;R ^k), withp min{n,k}. We prove also that the functional (u, ) can not be represented by an inte- gral for an arbitrary functionu BV_loc(R ⁿ;R ^k). In fact, two examples show that, in general, the set function (u, ) is not subadditive whenu BV_loc(R ⁿ;R ^k), even ifu W _loc ^1,p (R ⁿ;R ^k) for everyp < min{n,k}. Finally, we examine in detail the properties of the functionsu BV(;R ^k) such that (u, )= f(u)dx, particularly in the model casef(A)=¦(A)¦.     

Finitely Valued f-Modules, An Addendum

In an -group M with an appropriate operator set it is shown that the -value set (M) can be embedded in the value set (M). This embedding is an isomorphism if and only if each convex -subgroup is an -subgroup. If (M) has a.c.c. and M is either representable or finitely valued, then the two value sets are identical. More generally, these results hold for two related operator sets ₁ and ₂ and the corresponding -value sets and . If R is a unital -ring, then each unital -module over R is an f-module and has exactly when R is an f-ring in which 1 is a strong order unit.     

L 2-approximation by the translates of a function and related attenuation factors

Summary Let be thek-dimensional subspace spanned by the translates (·–2j/k),j=0, 1, ...,k–1, of a continuous, piecewise smooth, complexvalued, 2-periodic function . For a given functionfL ²(–, ), its least squares approximantS _kf from can be expressed in terms of an orthonormal basis. Iff is continuous,S _kf can be computed via its discrete analogue by fast Fourier transform. The discrete least squares approximant is used to approximate Fourier coefficients, and this complements the works of Gautschi on attenuation factors. Examples of include the space of trigonometric polynomials where is the de la Valleé Poussin kernel, algebraic polynomial splines where is the periodic B-spline, and trigonometric polynomial splines where is the trigonometric B-spline.     

Some properties of the moduli of families of curves

Let A={a₁,...,a_n} and B={b₁,...,b_m} be systems of distinct points in , let be a family of homotopic classes H_i,i=1,..., j+m, of closed Jordan curves on, where the classes H_j+l, l=1,...,m, consist of curves that are homotopic to a point curve in b. Let =₁,..., _j+m be a system of positive numbers and letU be the modulus of the extremal-metric problem for the family and the system . In this paper we investigate the dependence of the modulusU=U(,A,B) on the parameters _i and on the disposition of the points a_k and b. One shows thatU is a smooth function of the indicated arguments and one obtains expressions for the derivatives U, U, and U. One gives some applications of these results.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 72–82, 1985.     

über straffe Wahrscheinlichkeitsverteilungen im Raum der Schwartzschen Distributionen

Zusammenfassung In den letzten Jahren erschien eine Reihe von Arbeiten, die sich systematisch mit Wahrscheinlichkeitsverteilungen auf topologischen Gruppen, Halbgruppen, topologischen RÄumen und topologischen linearen RÄumen beschÄftigten. Als besonders geeignet für eine topologische Wahrscheinlichkeitstheorie erwiesen sich hierbei die sogenannten straffen (tight) Wahrscheinlichkeitsverteilungen (vgl. Le Cam [3], Hildenbrand [11], Prochoeov [20], Varadarajan [25]).Die vorliegende Arbeit befa\t sich mit straffen Wahrscheinlichkeitsverteilungen im Raum D, dem topologischen Dualraum des Raumes D der auf der reellen Zahlengeraden definierten beliebig oft differenzierbaren Funktionen mit kompaktem TrÄger Tr .Der Ausgangspunkt für die Untersuchung von Zufallselementen mit Werten in linearen RÄumen, die nicht notwendig BanachrÄume sind, war wohl der von GELFAND [8] eingeführte Begriff des verallgemeinerten stochastischen Prozesses (VSP). Solange man bei einem solchen Proze\ Eigenschaften untersucht, die sich mit Hilfe seiner endlichdimensionalen Randverteilungen Q_{1,...,n}, _i D, beschreiben lassen, wird man sich wie im Fall eines gewöhnlichen stochastischen Prozesses natürlich die Frage stellen, ob ein geeigneter Standard-stichprobenraum existiert, etwa der Raum D, so da\ sich jeder VSP auffassen lÄ\t als Wahrscheinlichkeitsverteilung auf einem geeigneten hinreichend umfangreichen -Ring von Teilmengen des Raumes D. Die fundamentale Arbeit von MINLOS [18] gab hierzu die Lösung: Durch ein vertrÄgliches System endlichdimensionaler Wahrscheinlichkeitsverteilungen Q_{1,...,n}, _i D, mit gewissen Eigenschaften, die denen der Randverteilungen eines VSP entsprechen, lÄ\t sich auf dem SystemB der Zylindermengen des Raumes D eine sogenannte schwache Verteilung definieren, von der gezeigt wird, da\ sie -additiv ist. Durch EinschrÄnkung des Raumes der sogenannten Testfunktionen auf den metrisierbaren Teilraum D _K{ D:Tr K, K kompakt in } von D lÄ\t sich dieses Ergebnis wie folgt verschÄrfen: Die durch ein vertrÄgliches System endlichdimensionaler Randverteilungen Q_{1,...,n}, _i D, mit entsprechenden Eigenschaften, auf dem System B _K der Zylindermengen des Raumes D_K definierte schwache Verteilung K ist straff bezüglich der schwachen Topologie (D_K, D_K) in D_K.Die Frage nach der Gültigkeit einer entsprechenden VerschÄrfung für das Dualsystem >DD<, bzw. allgemeiner für ein Dualsystem E, F mit nicht notwendig metrisierbarem F, bildete den Gegenstand neuerer Untersuchungen, über deren Ergebnisse auf dem letzten Berkeley Symposium E. Mourier berichtete (vgl. [19]).Im ersten Kapitel der vorliegenden Arbeit des Verfassers wird demgegenüber eine Methode aufgezeigt, mit deren Hilfe, unter Verwendung des Minlosschen Satzes in seiner ursprünglichen Form, auf direktem Wege für das Dualsystem >D, D< der Nachweis gelingt, da\ eine schwache Verteilung auf B nicht nur -additiv, sondern automatisch straff ist (bzgl. der schwachen Topologie (D, D) in D) und sich somit eindeutig fortsetzen lÄ\t zu einer straffen Wahrscheinlichkeitsverteilung auf dem System 83 der Boreischen Mengen in D, welches den von den Zylindermengen erzeugten -Ring (B) umfa\t. Mit anderen Worten wird damit gezeigt, da\ man jeden VSP auffassen kann als straffe Wahrscheinlichkeitsverteilung auf den Boreischen Mengen in D. Wir sprechen dann auch von einer zufÄlligen Distribution.Im zweiten Kapitel betrachten wir spezielle zufÄllige Distributionen, nÄmlich Normal-verteilungen v, die aus Randverteilungen hervorgehen, welche n-dimensionale Normal-verteilungen sind, und beschÄftigen uns mit dem Problem der Äquivalenz und SingularitÄtzweier Normalverteilungen v₁ und v₂ in D. Für den Fall v₁ = v, v₂= v_{f 0}, wo v_{f 0}(Z) =v(Z – f₀), ZB fD, zeigte DUDLEY [6], da\ entweder Äquivalenz oder SingularitÄt vorliegt, wobei er ein notwendiges und hinreichendes Kriterium für den Fall der Äquivalenz angibt. Aus der Theorie der gewöhnlichen stochastischen Prozesse ist nun bekannt, da\ die beiden Wahrschein-lichkeitsma\e, die zwei beliebigen Gau\schen Prozessen auf dem Raum ihrer Realisierungen entsprechen, entweder Äquivalent oder singular sind. Es lag deshalb nahe, nach einem Kriterium zu suchen, welches es einerseits gestattet, im Fall zweier beliebiger Normalverteilungen v₁ und v₂ in D zu entscheiden, wann Äquivalenz vorliegt, und welches andererseits die naheliegende Vermutung bestÄtigt, da\ für zwei Normalverteilungen in D dieselbe Alternative wie im eben zitierten klassischen Fall vorliegt. Dieses Problem wird gelöst, indem wir zeigen, da\ sich ein von Kallianfur-Oodaira [13] aufgestelltes Kriterium für die Äquivalenz zweier Normalverteilungen auf den Boreischen Mengen eines separablen Hilbertraumes auf den Distributionsraum D übertragen lÄ\t.Im dritten Kapitel beschÄftigen wir uns mit der Frage der Äquivalenz zweier beliebiger (nicht notwendig normaler) Wahrscheinlichkeitsverteilungen in D.Abschlie\end möchte der Autor Herrn Professor Dr. K. Krickeberg (Heidelberg) für die Anregung zu dieser Arbeit sowie für die Unterstützung wÄhrend ihrer Durchführung herzlich danken.     

Finely holomorphic functions and quasi-analytic classes

We study the set of functions in quasi-analytic classes and the set of finely holomorphic functions. We show that no one of these two sets is contained in the other.LetI denote the set of complex functionsf: for which there exists a quasi-analytic classC{M _n} containingf. Let denote the set of complex functionsf: for which there exist a fine domainU containing the real line and a function finely holomorphic onU satisfyingf(x)= (x) for allx . The power of unique continuation is incomparable in these two cases (I\ is non-empty, \I is non-empty).Research supported by the grant No. 201/93/2174 of Czech Grant Agency and by the grant No. 354 of Charles University.     

k-regular factors and semi-k-regular factors in bipartite graphs

LetG be a graph, andk1 an integer. LetU be a subset ofV(G), and letF be a spanning subgraph ofG such that deg _F (x)=k for allx V(G)–U. If deg _F (x)k for allxU, thenF is called an upper semi-k-regular factor with defect setU, and if deg _F (x)k for allxU, thenF is called a lower semi-k-regular factor with defect setU. Now letG=(X, Y;E(G)) be a bipartite graph with bipartition (X,Y) such that X=Yk+2. We prove the following two results.(1) Suppose that for each subsetU ₁X such that U ₁=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.(2) Suppose that for each subsetU ₁X such that U ₁=X–1/k+1,G has a lower semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=X–1/k+1,G has a lower semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.     

Subordination des résolvantes

Résumé Soit (V ₎₀ une résolvante définie sur un espace mesurable telle que le noyau initial est borné; on trouve une condition nécéssaire et suffisante pour qu'un noyau borné U possède une résolvante (U ₎₀ telle que U V pour tout 0. On donne plusieurs applications de ce résultat.     

A Homotopy 2-Groupoid of a Hausdorff Space

If X is a Hausdorff space we construct a 2-groupoid G ₂ X with the following properties. The underlying category of G ₂ X is the `path groupoid" of X whose objects are the points of X and whose morphisms are equivalence classes f, g of paths f, g in X under a relation of thin relative homotopy. The groupoid of 2-morphisms of G ₂ X is a quotient groupoid X / N X, where X is the groupoid whose objects are paths and whose morphisms are relative homotopy classes of homotopies between paths. N X is a normal subgroupoid of X determined by the thin relative homotopies. There is an isomorphism G ₂ X(f,f) ₂(X, f(0)) between the 2-endomorphism group of f and the second homotopy group of X based at the initial point of the path f. The 2-groupoids of function spaces yield a 2-groupoid enrichment of a (convenient) category of pointed spaces.We show how the 2-morphisms may be regarded as 2-tracks. We make precise how cubical diagrams inhabited by 2-tracks can be pasted.     

Schwarz-Pick Inequalities for Hyperbolic Domains in the Extended Plane

Let and be two hyperbolic simply connected domains in the extended complex plane C¯ = C¯ {}. We derive sharp upper bounds for the modulus of the nth derivative of a holomorphic, resp. meromorphic function f: at a point z ₀ . The bounds depend on the densities (z ₀) and (f(z ₀)) of the Poincaré metrics and on the hyperbolic distances of the points z ₀ and f(z ₀) to the point .     

On monotone and convex approximation by algebraic polynomials

The following results are obtained: If >0, 2, [3, 4], andf is a nondecreasing (convex) function on [–1, 1] such thatE _n (f) n ^– for any n>, then E _n ⁽¹⁾ (f)Cn ^– (E _n ⁽²⁾ (f)Cn ^–) for n>, where C=C(), E_n(f) is the best uniform approximation of a continuous function by polynomials of degree (n–1), and E _n ⁽¹⁾ (f) (E _n ⁽²⁾ (f)) are the best monotone and convex approximations, respectively. For =2 ( [3, 4]), this result is not true.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1266–1270, September, 1994.     

Characterization theorem of generalized polynomial of best approximation having bounded coefficients

Let the set of generalized polynomials having bounded coefficients beK={p= _jg_j. _{j j j},j=1, 2, ...,n}, whereg ₁,g ₂, ...,g _n are linearly independent continuous functions defined on the interval [a, b], _j, _j are extended real numbers satisfying _j<+, _j>-, and _{j j}. Assume thatf is a continuous function defined on a compact setX [a, b]. This paper gives the characterization theorem forp being the best uniform approximation tof fromK, and points out that the characterization theorem can be applied in calculating the approximate solution of best approximation tof fromK.     

On the optimality of a characterization theorem for the gamma function using the multiplication formula

Summary The following Artin type characterization of : _{+ +} is proved: Assume thatf: _{+ +} satisfies the Gauss multiplication formula for some fixedp 2,f is absolutely continuous on [l/p, 1 + ] for some > 0 and lim _{x 0} xf(x) = 1. Thenf(x) = (x) forx > 0.The optimality of this result is checked by means of counterexamples. For instance, it is shown that the result is no longer true, if f is absolutely continuous is replaced by f is continuous and of finite variation.     

Über die relativistische Elektronenoptik elektrostatis cher Beschleuniger

Summary Relativistic and nonrelativistic data for the coaxial two-cylinder electrostatic electron-lens are compared. Values of the potential along the axis were measured on a resistance network analogue, whilst trajectories were calculated on a high speed digital computer.IndexR indicating relativistic values, andN non-relativistic ones, quotientsF _R/F_N of the focal point coordinates (image sided),f _R/f_N of the focal length (image sided) and differences of the principal plane coordinates (H _R–H _N)/R (object sided) and (H _R–H _N)/R (image sided) are represented graphically as functions of the imagespace potential andU. The gap between the cylinders was in all cases 0,8R (R radius of the cylinders).U being the potential on the object side, andU on the image side, =U/U was varied from 0,1 to 0,966 andU from 0 to 2,000 kV; in the relativistic case a maximal departure of about 20% from the nonrelativistic values was found for focal length. A ten-lens accelerator of 1,5 MeV overall tension has been calculated both ways.     

Multi-Faithful Spanning Trees of Infinite Graphs

For an end and a tree T of a graph G we denote respectively by m() and m _T () the maximum numbers of pairwise disjoint rays of G and T belonging to , and we define tm() := min{m _T(): T is a spanning tree of G}. In this paper we give partial answers — affirmative and negative ones — to the general problem of determining if, for a function f mapping every end of G to a cardinal f() such that tm() f() m(), there exists a spanning tree T of G such that m _T () = f() for every end of G.     

Stop times in fock space stochastic calculus

Summary A stop time S in the boson Fock space over L ₂()₊ is a spectral measure in [0,] such that {S([0,t])} is an adapted process. Following the ideas of Hudson [6], to each stop time S a canonical shift operator U ^Sis constructed in . When S({}) has the vacuum as a null vector U ^Sbecomes an isometry. When S({})=0 it is shown that admits a factorisation _S]{S where _{S is the range of U ^Sand _S] is a suitable subspace of called the Fock space upto time S. This, in particular, implies the strong Markov property of quantum Brownian motion in the boson as well as fermion sense and the Dynkin-Hunt property that the classical Brownian motion begins afresh at each stop time. The stopped Weyl and fermion processes are defined and their properties studied. A composition operation is introduced in the space of stop time to make it a semigroup. Stop time integrals are introduced and their properties constitute the basic tools for the subject.     

Limits of balayage measures

LetA be a subset of a balayage space (X,W) and a measure onX. It is shown that for every sequence _n of measures such that lim_nn and lim_{n n} ^A = the limit measure is of the formf+[(1-f)]^A for some (unique) Borel function 0f1_Cb(A). Furthermore, conditions are given such that any such functionf occurs.     

The embedding of linearly ordered sets

It is shown that if a linearly ordered set B does not contain as subsets sets of order type and ^* then B can be embedded in 2 . We construct an example of a set satisfying the above conditions which cannot be embedded in any 2 if < . Simultaneously we show that for any ordinal, 2 ⁺¹ cannot be embedded in 2 and that there exists at least ₊₁ distinct dense order types of cardinality 2 .Translated from Matematicheskie Zametki, Vol. 11, No. 1, pp. 83–88, January, 1972.In conclusion, I wish to take the opportunity to thank Yu. L. Ershov for kindness and assistance in this work.