Mixing and some integro-functional equations

Summary It is proved that the operatorP: L ¹ (0, ) L ¹(0, ), given byPg(z) = _z/c [g(x)/cx]dx, is completely mixing, i.e.,P ⁿ g ₁ 0 forg L ¹(0, ) with g dx = 0. This implies that, forc (0, 1), each continuous and bounded solution of the equationf(x)= ₀ ^cx f(t)dt/(cx) (x (0, 1]) is constant.     

Monotonicity properties of zeros of generalized Airy functions

Summary We show, among other things, that the positive zeros of a solution ofy +x y=0,y(0)=0 decrease to 1 as increases, 0.

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Sommario Si dimostra, tra l'altro, che gli zeri positivi d'una soiuzione diy +x y=0,y(0)=0 decrescono al limite 1, quando cresce, 0.

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To the memory of Milo Háik

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This research was supported by grants from the Natural Sciences and Engineering Research Council (Canada) and Consiglio Nazionale delle Ricerche (Italy). Some of the work was done while the second-named author was visiting the Department of Mathematics, University of Torino.     

Localization of the spectrum of certain non-self-adjoint operators

Let the self-adjoint operator A and the bounded operator B be specified in Hilbert space We let denote the spectral family of the operator A. If (E – E _N ) B ₂+E–_NB ² 0 npnN , then in the complex plane z=+ there will exist the curve ¦ ¦ =f (), limf () = 0 for ± such that the entire spectrum of the operator A+B lies within the region ¦ ¦ f(). In particular, the condition of the theorem will be satisfied when B is a completely continuous operator.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 415–420, April, 1968.The author expresses his appreciation to R. S. Ismagilov for his discussion of the results.     

Surjectivity of quotient maps for algebraic (ℂ,+)-actions and polynomial maps with contractible fibers

In this paper we establish two results concerning algebraic (,+)-actions on ⁿ . First, let be an algebraic (,+)-action on ³. By a result of Miyanishi, its ring of invariants is isomorphic to [t ₁,t ₂]. Iff ₁,f ₂ generate this ring, the quotient map of is the mapF:³²,x(f ₁(x), f₂(x)). By using some topological arguments we prove thatF is always surjective. Secon, we are interested in dominant polynomial mapsF: ^{n n-1} whose connected components of their generic fibers are contractible. For such maps, we prove the existence of an algebraic (,+)-action on ⁿ for whichF is invariant. Moreover we give some conditions so thatF*([t ₁,...,t _n-1 ]) is the ring of invariants of .Dedicated to all my friends and my family     

Measure ratio of equidecomposable sets

Let n 2. There are Lebesgue measurable sets A and B in ³ such that (B)/(A)=r and A _n B if and only if 2/n r n/2.     

Generalized Variation of Mappings with Applications to Composition Operators and Multifunctions

We study (set-valued) mappings of bounded -variation defined on the compact interval I and taking values in metric or normed linear spaces X. We prove a new structural theorem for these mappings and extend Medvedev's criterion from real valued functions onto mappings with values in a reflexive Banach space, which permits us to establish an explicit integral formula for the -variation of a metric space valued mapping. We show that the linear span GV (I;X) of the set of all mappings of bounded -variation is automatically a Banach algebra provided X is a Banach algebra. If h:I× X Y is a given mapping and the composition operator is defined by (f)(t)=h(t,f(t)), where tI and f:I X, we show that :GV (I;X) GV (I;Y) is Lipschitzian if and only if h(t,x)=h₀(t)+h₁(t)x, tI, xX. This result is further extended to multivalued composition operators with values compact convex sets. We prove that any (not necessarily convex valued) multifunction of bounded -variation with respect to the Hausdorff metric, whose graph is compact, admits regular selections of bounded -variation.     

On unfoldable cardinals, ω-closed cardinals,and the beginning of the inner model hierarchy

Let be a cardinal, and let H be the class of sets of hereditary cardinality less than ; let () > be the height of the smallest transitive admissible set containing every element of {}H. We show that a ZFC-definable notion of long unfoldability, a generalisation of weak compactness, implies in the core model K, that the mouse order restricted to H is as long as . (It is known that some weak large cardinal property is necessary for the latter to hold.) In other terms we delimit its strength as follows: TheoremCon(ZFC+²- ¹₁-Determinacy) Con(ZFC+V=K+ a long unfoldable cardinal Con(ZFC+X(X^# exists) + is universally Baire rR(DL(r))), and this is set-generically absolute). We isolate a notion of -closed cardinal which is weaker than an ₁-Erd\ os cardinal, and show that this bounds the first long unfoldable: Theorem Let be -closed. Then there is a long unfoldable <.Mathematics Subject Classification (2000): 03E45, 03E15, 03E55, 03E60The author wishes to gratefully acknowledge support from Nato Grant PST.CLG 975324.     

The measure of a translated ball in uniformly convex spaces

Let (E, ¦·¦) be a uniformly convex Banach space with the modulus of uniform convexity of power type. Let be the convolution of the distribution of a random series inE with independent one-dimensional components and an arbitrary probability measure onE. Under some assumptions about the components and the smoothness of the norm we show that there exists a constant such that |{·<t}–{·+r<t}|r ^q , whereq depends on the properties of the norm. We specify it in the case ofL spaces, >1.     

Blow-up solutions of nonlinear elliptic equations in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> with critical exponent <Emphasis Type="Italic">Dedicated to Philippe Dufour,whose exquisite horology artwork ‘‘Simplicity’’ inspires so much.</Emphasis>

For an integer n3 and any positive number , we establish the existence of smooth functions K on ⁿ {0} with |K–1|, such that the equation has a smooth positive solution which blows up at the origin (i.e., u does not have slow decay near the origin). Furthermore, we show that in some situations K can be extended as a Lipschitz function on ⁿ . These provide counter-examples to a conjecture of C.-S. Lin when n>4, and a question of Taliaferro. Mathematics Subject Classification (2000)Primary 35J60; Secondary 53C21     

Thermorelaxation resonance in the quantum theory of Brownian motion

The Brownian motion of a quantum particle in a thermal reservoir possessing a finite correlation time _c is considered. Non-Markov Langevin equations for a stationary nonequilibrium state are obtained. At low temperaturesT of the thermal reservoir, the correlation time _c =/2T is fairly long. It is shown that allowance for the damping of the particle momentum over the correlation times _c : _c 1, leads to an oscillating temperature dependence of the relaxation coefficient (1/T) in the region of low temperatures of the thermal reservoir.N. I. Lobachevski Nizhegorod State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 1, pp. 119–126, July, 1992.