On the projective differential geometry of a non-holonome surface in ordinary space

Sunto. The paper deals with many projective properties of the neighbourhoods of the third,4th, 5th order of a point on a non-holonome surfaceV ₃ ² inS ₃. Chiefly two remarkable projective correspondences are studied, between the tangent plane and the bundle of directions, with his contact point as centre. Many generalizations are obtained of geometrical loci (so as lines, planes, quadrics) projectively associated with the neighbourhoods of a point on an ordinary (holonome) surface. The last sections are concerned with the extension to theV ₃ ² of the quadric ofMoutard and theSegre's correspondence.      

Gitterpunkte unter algebraischen Kurven

Lattice points below Algebraic Curves. A generalization of the classical circle problem is treated. An asymptotic formula for the number of lattice points in a region whose boundary is an algebraic curve is obtained. This gives a mean value formula for the number of representations of the positive integers in the formn=g(x,y), whereg is a polynomial with coefficients 0 and leading terma _d0x^d*a_0dy^d. The caseg(x,y)=p₁(x)+p₂(y) was considered inKuba andNowak [4], andKuba [5]. The discrete Hardy-Littlewood method is used along with Rouché's theorem.

     

Integral transforms related to heat conduction

Summary Relations between such transcendental differential operators asexp tD _x ² andexp xD _t ^1/2 are obtained. A new characterization of temperature functions represented by the classicalPoisson integral is obtained, as well as one for positive temperature functions.     

On the Spectral Function of Some Higher Order Elliptic or Degenerate-elliptic Operators

^m σ/|α|=|β|=m D^α(a _αβD^β) is a higher order degenerate-elliptic operator on L²(R^N) and V ∈ L¹(R^N) we show in particular that s(λ) > 0 for all λ > 0 provided that ∫_RN V(x)dx≥ 0, V not equivalent to 0 and N ≤ 2m.     

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.

     

A model in which there are Jech-Kunen trees but there are no Kurepa trees

By an ω₁-tree we mean a tree of power ω₁ and height ω₁. We call an ω₁-tree a Jech-Kunen tree if it hask-many branches for somek strictly between ω₁ and 2^ω1. In this paper we construct the models of CH plus 2^ω1 > ω₂, in which there are Jech-Kunen trees and there are no Kurepa trees. The research of the first author was partially supported by the Basic Research Fund, Israeli Academy of Science, Publ. No. 466.     

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.     

Eine Ω-Abschätzung zu einem zweidimensionalen Gitterpunktproblem

LetG ₂ (R) denote the number of lattice points (x, y) (i. e. points of the plane with integer coordinates) in the domainx ^a +y ^a R,x0,y0 (with weight 1/2 ifxy=0) and letV ₂ (a)R ^2/a be the area of this region. Then for 0<a<1/3 it is known [2] thatG ₂ (R)–V ₂ (a)R ^2/a K(a)R ^(1/a)–1 . Combining a method due toBleicher andKnopp [1] with a result proved elsewhere [4], [5] by the author it is shown here thatG ₂ (R)=V ₂ (a)R ^2/a +(R ^(1/a)–1 ) for any fixed rational numbera with 1/3<a<1/2.     

Bestimmung der Eigenwerte und Eigenvektoren einer Matrix mit Hilfe des Quotienten-Differenzen-Algorithmus

Summary Two previous papers (in Vol. V) describe theory and some applications of the quotient-difference (=QD-) algorithm. Here we give an extension which allows the determination of the eigenvectors of a matrix. Letx ₍₀₎ ¹ , ...,x ₍₀₎ ⁿ be a coordinate system in whichA has Jacobi form (such a system may be constructed with methods ofC. Lanczos orW. Givens). Then the QD-algorithm allows the construction of a sequence of coordinate systemsx ₍₂₎ ¹ , ...,x ₍₂₎ ⁿ , (=0, 1, 2, ...) which converge for to the system of the eigenvectors ofA.     

Sopra un principio di trasformazione integrale dei problemi differenziali ed alcune sue applicazioni

Sunto Sviluppando ampiamente un'idea diWhittaker, si studiano trasformazioni del tipo che mutano le soluzioni di una equazione differenziale lineareL _tx+λM _tx=0, appartenenti ad una classe lineare (G), opportunamente precisata, in soluzioni di una seconda equazione differenzialeP _uy+λQ _uy=0, appartenenti ad un'altra classe lineare (Γ). Il primo problema si suppone autoaggiunto. Il nucleo K è una soluzione dell'equazione (L _t Q _u −M _t P _u)K=0, di classe (G) come funzione di t, di classe (Γ) come funzione di u. Il procedimento permette di stabilire proprietà integrali in base a sole verifiche qualitative; ne vengono fatte applicazioni alle funzioni diMathieue diBessele ai polinomi diLegendree diHermite. A Mauro Picone nel suo 70^mo compleanno.     

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.     

On isotropic tensor-functions and the measure of deformation

Zusammenfassung Es wird gezeigt, dass eine isotrope Beziehung zwischen zwei symmetrischen Tensoren zweiten Ranges auf die Form gebracht werden kann, wobei sowie dieA _n,B _n,C _n ähnliche Potenzreihen in den Hauptinvarianten vonx _s ^r sind. Die Konstantenk müssen durch geeignete Versuche bestimmt werden, die über den Zusammenhang zwischen den physikalischen Grössenx _s ^r undy _s ^r Auskunft geben. Die können aber auch Konstanten sein. Dies trifft beispielsweise zu, wennx _s ^r undy _s ^r verschiedene Masse der Deformation sind. In diesem Falle kann man zum Beispiel das logarithmische Hencky-Mass durch das Almansi-Mass ausdrücken. Tut man dies, dann ergeben sich aus einer quasilinearen Spannungs-Fluss-Beziehung gewisse Effekte zweiter Ordnung in zähen Flüssigkeiten, auch dann, wenn das Medium unelastisch und die Zähigkeit gering ist. Damit liesse sich ein Zentripetalpump-Effekt in Luft, der neulich beobachtet worden ist, theoretisch erklären.

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de|The symbols used are A indicatingAlmansi - A _n,B _n,C _n coefficients - C indicatingCauchy - e strain - G indicatingGreen - g _rs metric tensor - H indicatingHencky - i, j, k principal directions - K _n constant coefficient - l length - l ₀ initial length - l ₁ final length - p _s ^r stress tensor - S amount of shear, also indicatingSwainger - V volume - v velocity - x _s ^r ,y _s ^r symmetrical tensors of rank two - coefficient - l elongation - Kronecker's delta - coefficient of viscosity - _c coefficient of cross viscosity - Lamé's constant - shear modulus - I, II, III principal invariants The research reported in this document has been sponsored in part by the Air Research and Development Command, United States Air Force, under Contract AF 61 (514)-871, through the European Office, ARDC.     

Degenerate Self-adjoint Evolution Equations on the Unit Interval

a∈ C ¹([0,1]) vanishing only at 0, 1 and b∈ C ([0,1]), we study existence, qualitative properties, asymptotic behaviour and approximation of the solutions of the degenerate evolution problem u _t = (au _z ) _z - bu in ]0,1[x]0, + ∞[, with u(x,0) = u ₀ (x) and boundary conditions a(x)u _x (x,t) → 0 as x → 0,1, t > 0.     

Spectral invariance,ellipticity, and the Fredholm property for pseudodifferential operators on weighted Sobolev spaces

The pseudodifferential operators with symbols in the Grushin classes \~S _inf0 ^sup, , 0 < 1, of slowly varying symbols are shown to form spectrally invariant unital Frécher-*-algebras (*-algebras) in L(L ²(R ⁿ )) and in L(H ^st ) for weighted Sobolev spaces H _inf ^supst defined via a weight d function . In all cases, the Fredholm property of an operator can be characterized by uniform ellipticity of the symbol. This gives a converse to theorems of Grushin and Kumano-Ta-Taniguchi. Both, the spectrum and the Fredholm spectrum of an operator turn out to be independent of the choices of s, t and .The characterization of the Fredholm property by uniform ellipticity leads to an index theorem for the Fredholm operators in these classes, extending results of Fedosov and Hörmander.     

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     

On cubic polynomials I. Hua's estimate of exponential sums

Hua andChen gave estimates of sums wheree(z)=e ^2iz and is a polynomial of the typef(x)/q wheref(x)=a _k x ^k +...+a ₁ x with integer coefficients having gcd (q, a _k ,...,a ₁)=1 But no good estimates hold for these sums whenq is small in comparison tok. We therefore consider here a related but different class of polynomials. Special emphasis is given to the cubic case.In subsequent papers of this series we shall deal with cubic exponential sums in many variables and withp-adic and rational zeros of systems of cubic forms.Partially supported by NSF contract NSF-MCS-8015356.     

Le equazioni diofanteeax 4-bx 2+1=□, 4x 4+x 2 y 2+4y 4=□.

Summary Si determinano tre classi di quartiche u²=ax⁴+1, a e b interi, b²−4a≠□ per le quali l'appartenenza di un punto razionale (x₀,y₀),x₀≠0 porta l'esistenza nella stessa quartica di infiniti punti razionali. Si dimostra pure che l'equazione 4x²+x²y²+4y⁴=□ non ammette soluzioni intere con |xy| > 1. A BeniaminoSegre nel suo 70^mo compleanno. Entrata in Redazione il 26 febbraio 1973.     

Moltiplicatori(L^{\vec p} - L^{\vec q} ) degli integrali di Fourierdegli integrali di Fourier

Riassunto In questa nota si considerano gli spazi funzionali con norma mista, studiati daA. Benedek eR. Panzone, e vengono date delle condizioni sufficienti perchè una funzione misurabile sia un moltiplicatore dell’integrale di Fourier da a . Tali condizioni sono l’estensione di quelle trovate daP. I. Lizorkin nel caso degli spazi di LebesgueL ^p (R _n ).

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Résumé Dans cette note on considère les espaces fonctionnels avec norme mixte, étudiés parA. Benedek etR. Panzone, et l’on obtient quelques conditions suffisantes afin que une fonction mesurable soit un multiplicateur de l’intégral de Fourier. Ces conditions sont la généralisation de ceux découvertes parP. I. Lizorkin dans le cas des espaces de LebesgueL ^p (R _n ).

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Lavoro esequito nell’ambito dell’attività dei gruppi di ricerca matematici del C.N.R.     

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.