Multiplizitäten “unendlich-ferner” Spitzen

LetX be the quotient of a bounded symmetric domainD by an arithmetically defined subgroup of all analytic automorphisms ofD and letX ^* be theSatake-compactification ofX. In the present note, the multiplicities of the local rings of the zero-dimensional boundary components ofX ^* will be computed in a completely elementary manner using reduction-theory in selfadjoint homogeneous cones.     

Sulla rappresentazione asintotica delle funzioni di Bessel di uguale ordine ed argomento

Sunto Partendo da una rappresentazione integrale delle funzioni diHankel H _v ⁽¹⁾ (v) e H _v ⁽²⁾ (v) e facendo uso della teoria delle serie inviluppanti diJ. G. van der Corput si perviene ad uno sviluppo asintotico delle funzioni diBessel J_v(v) e Y_v(v) e si valuta il termine complementare di tale sviluppo.

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Summary From an integral rapresentation of theHankel functions H _v ⁽¹⁾ (v) and H _v ⁽²⁾ (v) and by usingvan der Corput's theory of enveloping series we obtain an asymptotic expansion forBessel functions J_v(v) and Y_v(v). An upper bound for the error term is also obtained.

     

Sulle trasformazioni continue

Sunto Si dimostra che ogni trasformazione piana, rappresentata da funzioni assolutamente continue secondoTonelli con dervate parziali prime integrabiliL ²→ α con α>0, sono a variazione limitata e assolutamente continue secondoBanach eVitali. Si danno inoltre esempi effettivi di trasformazioni piane, rappresentate da funzioni assolutamente continue secondoTonelli con derivate parziali prime integrabiliL ², che non sono ne a variazione limitata ne assolutamente continue secondoBanacheVitali. Lavoro eseguito nel Seminario Matematico della R. Scuola Normale Superiore di Pisa.     

Instances of the Conjecture of Chang

We consider various forms of the Conjecture of Chang. Part A constitutes an introduction. Donder and Koepke have shown that if ρ is a cardinal such that ρ ≧ ω₁, and (ρ⁺⁺,ρ⁺↠(ρ⁺, ρ), then 0⁺ exists. We obtain the same conclusion in Part B starting from some other forms of the transfer hypothesis. As typical corollaries, we get: Theorem A.Assume that there exists cardinals λ, κ, such that λ ≧ K ⁺ ≧ω₂ and (λ⁺, λ)↠(K ⁺,K. Then 0⁺ exists. Theorem B.Assume that there exists a singularcardinal κ such that(K ⁺,K↠(ω₁, ω₀. Then 0⁺ exists. Theorem C.Assume that (λ ⁺⁺, λ). Then 0⁺ exists (also ifK=ω ₀. Remark. Here, as in the paper of Donder and Koepke, “O⁺ exists” is a matter of saying that the hypothesis is strictly stronger than “L(μ) exists”. Of course, the same proof could give a few more sharps overL(μ), but the interest is in expecting more cardinals, coming from a larger core model. Theorem D.Assume that (λ ⁺⁺, λ)↠(K ⁺, K) and thatK≧ω ₁. Then 0⁺ exists. Remark 2. Theorem B is, as is well-known, false if the hypothesis “κ is singular” is removed, even if we assume thatK≧ω ₂, or that κ is inaccessible. We shall recall this in due place. Comments. Theorem B and Remark 2 suggest we seek the consistency of the hypothesis of the form:K ⁺, K↠(ω_n +1, ω_n), for κ singular andn≧0. 0266 0152 V 3 The consistency of several statements of this sort—a prototype of which is (N _ω+1,N _ω)↠(ω₁, ω₀) —have been established, starting with an hypothesis slightly stronger than: “there exists a huge cardinal”, but much weaker than: “there exists a 2-huge cardinal”. These results will be published in a joint paper by M. Magidor, S. Shelah, and the author of the present paper.     

Some Geometric Characterizations of the Hopf Fibrations of the Three-Sphere

The Fréchet manifold of all embeddings (up to orientation preserving reparametrizations) of the circle in S ³ has a canonical weak Riemannian metric. We use the characterization obtained by H. Gluck and F. Warner of the oriented great circle fibrations of S ³ to prove that among all such fibrations π:S ³→B, the manifold B consisting of the oriented fibers is totally geodesic in , or has minimum volume or diameter with the induced metric, exactly when π is a Hopf fibration. Partially supported by foncyt, Antorchas, ciem (conicet) and secyt (unc).     

Metrica hermitiana ellittica in uno spazio proiettivo quaternionale

Riassunto Si definisce e si studia una nozione di distanza nello spazio proiettivo q aternionaleT _n ^Q , la quale si riduce nell'infinitesimo alla meirica riemanniana del modello metrico reale diT _n ^Q assegnato daMartinelli, e subordina le nozioni di distanza diCayley-Klein e diFubini-Study negli spazi proiettivi reale e complessoT _n ^R ,T _n ^C subordinati aT _n ^Q . A M. Enrico Bompiani nel suo giubileo scientifico Lavoro eseguito nell'ambito del Gruppo di ricerca n. 37 del C. N. R.     

Fictitious states,coupled laws and local time

Summary The work of Ray and Neveu has established that, for any transition function P on a countable set E, (i) there exists a best possible entrance boundary E ⁺ supporting a right continuous, strong Markov process X with transition function P and that (ii) the points y of E ⁺ are in one-one correspondence with the extremal entrance laws g ^y of P. Here, it is shown that, if a point y of E ⁺ is regular for itself, then the derived characteristic f ^y of the local time at y is a regular extremal entrance law coupled with g ^y in the sense of Neveu. Further, coupled laws arise only in this fashion. By using excursion theory, a simple explicit formula for f ^y in terms of g ^y may be obtained. The paper contains a conjecture about the intrinsic character of the Ray-Neveu topology and an example which shows emphatically that, in general, local time is not a derivative of occupation time.     

Prime ideals in the C*-algebra of a nilpotent group

We say that a locally compact groupG hasT ₁ primitive ideal space if the groupC ^*-algebra,C ^*(G), has the property that every primitive ideal (i.e. kernel of an irreducible representation) is closed in the hull-kernel topology on the space of primitive ideals ofC ^*(G), denoted by PrimG. This means of course that every primitive ideal inC ^*(G) is maximal. Long agoDixmier proved that every connected nilpotent Lie group hasT ₁ primitive ideal space. More recentlyPoguntke showed that discrete nilpotent groups haveT ₁ primitive ideal space and a few month agoCarey andMoran proved the same property for second countable locally compact groups having a compactly generated open normal subgroup. In this note we combine the methods used in [3] with some ideas in [9] and show that for nilpotent locally compact groupsG, having a compactly generated open normal subgroup, closed prime ideals inC ^*(G) are always maximal which implies of course that PrimG isT ₁.     

Operatorenkalkül über freien Monoiden I: Strukturen

In part I algebraic structures (esp. rings) on the sets of polynomials and formal power series on an at most countable alphabetA are considered. Given a partial order onA the words ofA ^* are mixed together in consistence with it. It is shown that the structures derived are associative iff the given partial order is of linear type. The coefficients appearing at these operations are identified as generalizations of the ordinary binomial coefficients and a number of relations involving them are listed up.(Part II will bring a generalization ofRota's theory of polynomial sequences of binomial type to the structures studied in I.In Part III the theory of special binomial systems will be continued until the analogue of Lagrange inversion and a short development of generalized Sheffer polynomials will be given).     

A class of quasilinear stochastic partial differential equations of McKean-Vlasov type with mass conservation

Summary A system ofN particles inR ^d with mean field interaction and diffusion is considered. Assuming adiabatic elimination of the momenta the positions satisfy a stochastic ordinary differential equation driven by Brownian sheets (microscopic equation), where all coefficients depend on the position of the particles and on the empirical mass distribution process. This empirical mass distribution process satisfies a quasilinear stochastic partial differential equation (SPDE). This SPDE (mezoscopic equation) is solved for general measure valued initial conditions by extending the empirical mass distribution process from point measure valued initial conditions with total mass conservation. Starting with measures with densities inL ₂(R ^d ,dr), wheredr is the Lebesgue measure, the solution will have densities inL ₂(R ^d ,dr) and strong uniqueness (in the Itô sense) is obtained. Finally, it is indicated how to obtain (macroscopic) partial differential equations as limits of the so constructed SPDE's.This research was supported by NSF grant DMS92-11438 and ONR grant N00014-91J-1386     

Notiz zu den quadratfreien Zahlen in arithmetischen Progressionen

Some lower estimates for the number of squarefree (or nearly squarefree) numbers in arthmetic progressions are proved; it follows that a sharpened version of a result ofPrachar on 3-free numbers holds true. For the least squarefree number of the serieskm ²+l the estimate <k ³⁺ is proved. A very simple proof of a theorem ofErdös andPrachar is also given.     

A generalization of Riemann's method for partial differential equations

Summary A bilinear divergence identity is obtained, which differs from the usualLagrange divergence identity employed byRiemann. In the case of two independent variables, this new identity is used to unify the treatment ofCauchy's problem for hyperbolic equations, the initial value problem for parabolic equations, and theDirichlet problem for elliptic equations. This research was supported in whole or in part by the United States Air Force under Contract N^o. AF18(600)-573 monitored by the Office of Scientific Research, Air Research and Development Command.     

On Weighted Hardy Spaces on Complex Semigroups

H^p (S,α) on a complex open Ol'shanskii semigroup S = G Exp (iW), where 1 ≤p≤∞ and α is an absolute value on the involutive semigroup X. For 1 < p < ∞ we prove the existence of an isometric boundary value map H ^p (S,α) → L ^p (G) generalizing the corresponding result of Ol'shanskii for p = 2 and α = 1. In the second part we use the fine structure of the space H ² (S,1) to prove the existence of a bounded holomorphic function on S whose absolute value has a unique maximum in the boudary point 1Β G and therefore complete the proof of the approximation property of the Poisson kernel and the uniqueness of G as a Shilov boundary of S whenever W does not contain affine line.     

Sopra gli scorrimenti proiettivi

Sunto. In questo lavoro diamo senza dimostrazioni (i) alcuni risultati di geometria proiettiva differenziale dovuti alla nozione di scorrimento proiettivo che si deve alBortolotti. Si giunge a questi, dando una nuova caratterizzazione di questa nozione, ciò che permette a determinare tutte le superficie che ammettono scorrimenti proiettivi in sè. Caso particolare di queste sono quelle le cui linee diDarboux di un sistema sono pangeodetiche diFubini e che perciò corrispondono ad un problema diE. Bompiani. Si pone poi sotto una nuova luce risultati recenti diLingenberg sulle superficie che ammettono ∞ ¹ collineazioni in sè. A Enrico Bompiani in occasione del suo Giubileo scientifico     

Amalgamation bases for commutative rings without nilpotent elements

SCR=the class of commutative rings without nilpotent elements.Theorem,R is an amalgamation base for SCR iff rad (I=Ann²(I) forI ⊆R finitely generated. Supplement. IfR ε SCR thenR is contained in an amalgamation basis for SCR having no new idempotent elements. CR=the class of commutative rings. Theorem.R is an amalgamation base for CR iffR is a pureR-submodule of any commutative ring extendingR.     

Über die Theorie der relativ-Abel'schen Zahlkörper

Ohne Zusammenfassung Mit geringen ?nderungen abgedruckt aus den Nachrichten der Kgl. Ges. der Wiss. zu G?ttingen 1898. Inzwischen sind folgende auf diesen Gegenstand bezügliche Inaugural-Dissertationen in G?ttingen erschienen:Das quadratische Reciprocit?tsgesetz im quadratischen Zahlk?rper mit der Classenzahl 1. vonH. D?rrie 1898,Tafel der Klassenanzahlen für kubische Zahlk?rper vonL. W. Reid 1899,Das allgemeine quadratische Reciprocit?tsgesetz in ausgew?hlten Kreisk?rpern der 2 ^{h ten} Einheitswurzeln vonK. S. Hilbert 1900,Quadratische Reciprocit?tsgesetze in algebraischen Zahlk?rpern vonG. Rückle 1901. Insbesondere die letzte Dissertation enth?lt zahlreiche und interessante Beispiele zu der hier entwickelten Theorie.     

An analytically oriented discretization technique for boundary value problems

Dedicated to ProfessorJ.W. Schmidt on the occasion of his 60 ^th birthday     

Sulla geometria proiettiva differenziale di una superficie auolonoma nello spazio ordinario

Sunto. Osservazioni, ragguagli, notizie su precedenti lavori concernenti le superficie anolonome e le quadriche diMoutard per esse; ulteriori sviluppi concernenti gli intorni del3 ^o e del4 ^o ordine di un punto su di una superficie anolonoma.     

Admissibility of Unbounded Operators and Wellposedness of Linear Systems in Banach Spaces

We study linear systems, described by operators A, B, C for which the state space X is a Banach space.We suppose that − A generates a bounded analytic semigroup and give conditions for admissibility of B and C corresponding to those in G. Weiss’ conjecture. The crucial assumptions on A are boundedness of an H^∞-calculus or suitable square function estimates, allowing to use techniques recently developed by N. Kalton and L. Weis. For observation spaces Y or control spaces U that are not Hilbert spaces we are led to a notion of admissibility extending previous considerations by C. Le Merdy. We also obtain a characterisation of wellposedness for the full system. We give several examples for admissible operators including point observation and point control. At the end we study a heat equation in X = L^p(Ω), 1 < p < ∞, with boundary observation and control and prove its wellposedness for several function spaces Y and U on the boundary ∂Ω.     

Teoremi di immersione su iperpiani coordinati per funzioni di un certo spazio funzionale

Riassunto In questa nota si considera uno spazio funzionale L _p ^P (E ^n,, introdotto daL. Cattabriga e che rientra nella classe considerata daL. R. Volevic—B. P. Panejah; si ottengono in particolare alcuni teoremi di immersione per le ?restrizioni? a iperpiani coordinati delle funzioni di tale spazio.

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Résumé En cette note l'on considére un espace fonctionnel, signé L _p ^P (E ^n,, qui a été introduit parM. L. Cattabriga et qui appartient à la classe considéré parM. M. L. R. Volevic—B. P. Panejah; en particulier l'on obtient quelques théorèmes d'immersion pour les ?restrictions? aux iperplans cohordonnés des foctions du dit espace.

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Lavoro eseguito nell'ambito dell'attività dei gruppi di ricerca matematici del C.N.R.