Asymptotic Density and the Asymptotics of Partition Functions

Let A be a set of positive integers with gcd (A) = 1, and let p _A (n) be the partition function of A. Let c ₀ = 2/3. If A has lower asymptotic density and upper asymptotic density , then lim inf log p _A (n)/c ₀ n and lim sup log p _A (n)/c ₀ n . In particular, if A has asymptotic density > 0, then log p _A (n) c₀n. Conversely, if > 0 and log p _A (n) c ₀ n, then the set A has asymptotic density .     

Nonclassical analogs of solitons in quantum field theory

The possible existence of quantum copies of classical soliton solutions that cannot exist if the effective Planck constant of the theory tends to zero is discussed. In the framework of the standard expansion in , such nonclassical solitons have energy of orderO() and thus occupy an intermediate position between purely classical configurations and second-quantized degrees of freedom. Analytic calculations are made for the profile functions, masses, and stability of such configurations in the ⁴ kink model.Physics Faculty, Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 94, No. 1, pp. 52–65, January, 1993.     

An additive function on a ring of integers in the imaginary quadratic field Q(√d) with class-number one

LetB (a) be an additive function on a ring of integers in the quadratic number fieldQ(d) given byB (a) = _p|a *N (p) with a fixed > 0, where the asterisk means that the summation is over the non-associate prime divisorsp of an integera inQ(d), N(a) is the norm ofa. In this paper we obtain the asymptotic formula of _N(a)x *B (a) in the case where the class-number ofQ(d) is one.Project supported by the National Natural Science Foundation of China.     

Elastic instabilities in cone- and -plate flow: Small gap theory

Consider the axisymmetric, inertialess cone- and -plate flow of a viscoelastic fluid. A perturbation method is used to obtain more tractable equations that describe the flow when the gap angle is small. A linear stability analysis of the base viscometric flow shows that there is a loss of stability when an elasticity parameter =DeWe, increases past a critical value. This purely elastic instability is of the oscillatory type. We obtain expressions for the critical elasticity number, frequency and wave number. The critical Deborah number varies as , the wave length as . and the wave speed as , where . is the gap angle. The most unstable mode exhibits infinitely many logarithmically-spaced roll cells which propagate inward towards the apex of the cone. These results are in agreement with experimental and numerical results.     

A proof for Theodorus' theorem by drawing diagrams

Summary In Plato's dialogue Theaetetus, the mathematician Theodorus (ca. 470-400 BC) is said to have proved the irrationality of 3, 5, ..., 17 by drawing diagrams. In this paper such a proof is presented and compared with another, more arithmetical, reconstructed proof by W. Knorr. Moreover a hypothetical early theory of similar rectangles is extracted from Euclid's Elements, which could have served as a basis for the geometrical proof of Theodorus' theorem.Herrn N. Stephanidis in Freundschaft gewidmet     

Relaxation of a Quadratic Functional Defined by a Nonnegative Unbounded Matrix

We study the lower semicontinuous envelope in L^p(), F, of a functional F of the form F(u)=A uudx where A=A(x) is not strictly elliptic and not bounded. We prove that F; may also be written as F;(u)= Buudx with B=AP A for a matrix P which is the matrix of an orthogonal projection. In the one-dimensional case, we characterize the domain of F and we explicit the matrix P.     

Some functionals over a compact Minkovskii space

In this paper we obtain estimates which are order-exact for the projection and Macphail constants of an arbitrary n-dimensional Banach space: 1(X)n, 1/n₁(X)1/n.Translated from Matematicheskii Zametki, Vol. 10, No. 4, pp. 453–457, 1971.     

Time decay for nonlinear wave equations in two space dimensions

The Cauchy Problem for the equation u_tt–u+|u|^p–1u=0 (x², t>0, >1) is studied. Smooth Cauchy data is prescribed, and no smallness condition is imposed. For >5, it is shown that the maximum amplitude of such a wave decays at the expected rate t^–1/2 as t. For 1+8<5, the maximum amplitude still decays, but at a slower rate. These results are then used to demonstrate the existence of the scattering operator when >_o, where _o is the root of the cubic equation ³-2²-7-8=0; thus _o4.15.Alfred P. Sloan Research Fellow     

An efficient heuristic algorithm for minimum matching

We give a new heuristic algorithm for minimum matching problems and apply it to the Euclidean problem with random vertices in 2 dimensions. The algorithm is based on simulated annealing and performs in practice faster than previous heuristic algorithms yielding suboptimal solutions of the same good quality. From configurations with up toN=20.000 vertices in the unit square we estimate that the length of a minimum matching scales asymptotically asLN with (=0.3123±0.0016.

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Zusammenfassung Wir stellen einen neuen heuristischen Algorithmus für minimale Matching-Probleme vor und wenden diesen auf das euklidische Problem mit zufÄlliger Punkteverteilung in 2 Dimensionen an. Auf Simulated Annealing basierend lÄuft der Algorithmus schneller als frühere heuristische Algorithmen und erreicht dabei suboptimale Lösungen gleich guter QualitÄt. Aus Konfigurationen mit bis zuN=20.000 Punkten im Einheitsquadrat schÄtzen wir, da\ für die LÄnge des minimalen Matchings asymptotischLN mit=0.3123±0.0016 gilt.

     

Gaussian Version of a Theorem of Milman and Schechtman

Using Gordon's inequalities, we give a short proof of the existence of an embedding T:l2k to ln such that ||T||||T-1|| c k/ln(1+n/k). In the same way, we give a new proof of a theorem of Milman and Schechtman (1995).     

Uniform boundedness of a family of set functions

Let be a ring of sets, X a normed space, : X ( ) a bounded family of triangular functions. The following generalized Nikodym theorem is established: the family {} is uniformly bounded on if and only if it is bounded on every sequence of pairwise disjoint sets of which the union is a part of some set in . An analogous criterion is established also for semiadditive functions. In addition, it is shown that uniform boundedness of a family of triangular functions is preserved in passing from a ring to the -ring it generates.Translated from Matematicheskie Zametki, Vol. 23, No. 6, pp. 855–861, June, 1978.     

On a continuous time stochastic approximation problem

This paper is concerned with a continuous time stochastic approximation/optimization problem. The algorithm is given by a pair of differential-integral equations. Our main effort is to derive the asymptotic properties of the algorithm. It is shown that ast , a suitably normalized sequence of the estimation error,t(¯x _tr–) is equivalent to a scaled sequence of the random noise process, namely, (1/t) ₀ ^tr _sds. Consequently, the asymptotic normality is obtained via a functional invariance theorem, and the asymptotic covariance matrix is shown to be the optimal one. As a result, the algorithm is asymptotically efficient.Supported in part by the National Science Foundation, and in part by Wayne State University.Supported in part by Wayne State University through a research assistantship.     

Embeddings of Lorentz—Marcinkiewicz spaces with mixed norms

X(Y) f -:X(Y)={fM(×): f_{X(Y)=f(x,.)YX<} . =(0, ), M (×) — , ×, X, Y, Z— . X(Y) Z(×).     

A note on unit and class number of real quadratic fields

Letp be an odd prime withp1 (mod 4) andQ(p) be the real quadratic field. Also let andh denote the fundamental unit and the class number of Q(p), respectively. The main purpose of this paper is to study the explicit expressions of ^h and ^2h , and to discuss the problems related to the conjecture of Ankeny-Artin-Chowla.     

Calculation of the temperature of polymer or composite surfaces placed in an infrared laser field

We consider methods for calculating the surface temperature T_s of polymers upon exposure to infrared laser radiation during modification, pyrolysis, ablation, ignition, etc. We have shown that we need to calculate the values of T_s for polymers using different analytical expressions after estimating the value of the dimensionless parameter u = at_e, where is the absorption coefficient for the laser radiation, a is the thermal diffusivity, and t_e is the radiation exposure time. Under these conditions, we achieve the smallest calculation error and we can take into account volume absorption of radiation.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 4, pp. 540–546, July–August, 1995.     

Blow-up of solutions of nonlinear wave equations in three space dimensions

Let u(x,t) be a solution, uA|u|^p for xIR₃, t0 where is the d'Alembertian, and A, p are constants with A>0, 10–|x–x⁰|, if the initial data u(x,0), u_t(x,0) have their support in the ball |x–x⁰|t₀. In particular global solutions of u=A|u|^p with initial data of compact support vanish identically. On the other hand for A>0, p>1+2 global solutions of u=A|u|^p exist, if the initial data are of compact support and u is sufficiently small in a suitable norm. For p=2 the time at which u becomes infinite is of order u^–2.Dedicated to Hans Lewy and Charles B. Morrey, Jr.The research for this paper was performed at the Courant Institute and supported by the Office of Naval Research under Contract No. N00014-76-C-0301. Reproduction in whole or part is permitted for any purpose of the United States Government.     

On equivalence of rearrangements of the Haar system in dyadic Hardy and BMO spaces

H={h ₁,I } — , . : , I ¦(I)¦=¦I¦, ¦I¦ — I. H H ={h _(I),I} . , , . L ^p .

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Dedicated to Professor B. Szökefalvi-Nagy on his 75th birthday

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This research was supported in part by MTA-NSF Grants INT-8400708 and 8620153.     

Cutting disjoint disks by straight lines

Fork>0 letf(k) denote the minimum integerf such that, for any family ofk pairwise disjoint congruent disks in the plane, there is a direction such that any line having direction intersects at mostf of the disks. We determine the exact asymptotic behavior off(k) by proving that there are two positive constantsd ₁,d ₂ such thatd ₁k logkf(k)d ₂k logk. This result has been motivated by problems dealing with the separation of convex sets by straight lines.The work of the first author was supported in part by the Allon Fellowship, by the Bat Sheva de Rothschild Foundation, by the Fund for Basic Research administered by the Israel Academy of Sciences, and by the Center for Absorbtion in Science. Work by the second author was supported by the Technion V. P.R. Fund, Grant No. 100-0679. The third author's work was supported by the Natural Sciences and Engineering Research Council, Canada, and the joint project Combinatorial Optimization of the Natural Science and Engineering Research Council (NSERC), Canada, and the German Research Association (Deutsche Forschungsgemeinschaft, SFB 303).     

Unramified Kummer extensions of prime power degree

We determine, for which xR^* the Kummer extension K(Pⁿx) is unramified over K, where K is a local field of char. 0 and res.char. p, _pnK, and R is the valuation ring of K. This builds on techniques of Hasse. Actually, a much more general result concerning cyclic Galois extensions of commutative rings is proved.     

On a Test Statistic for Linear Trend

Let {W(s)} _s 0 be a standard Wiener process. The supremum of the squared Euclidian norm Y (t)², of the R²-valued process Y(t)=(1/t W(t), {12/t ³ int_{0 t} s dW (s)– {3/t} W(t)), t [, 1], is the asymptotic, large sample distribution, of a test statistic for a change point detection problem, of appearance of linear trend. We determine the asymptotic behavior P {sup _t [, 1] Y(t)² > u as u , of this statistic, for a fixed (0,1), and for a moving = (u) 0 at a suitable rate as u . The statistical interest of our results lie in their use as approximate test levels.