Recollections of Fred Almgren

While visiting Princeton for the 1997–98 year I have had the privilege of using Fred’s office and helping Jean to go through his papers. It has been moving to see the magnitude of Fred’s helpful correspondence with students and colleagues, some of which Jean and I returned to them with an invitation to contribute short recollections. This article gives some selections from their responses. For more recollections see Mackenzie’s article on “Fred Almgren: Lover of Mathematics, Family, and Life’s Adventures” (Notices AMS 44, 1997, 1102–1106) and the memorial issue ofExperimental Mathematics (Vol. 6, 1997, 1–12).     

Periodic solutions of a weakly nonlinear system of partial differential equations with pulse influence

We establish conditions of the existence of solutions periodic in t with period T for a weakly nonlinear system of partial differential equations with pulse influence. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 4, pp. 601–605, April, 1997.     

On the maximal conjugate of a totally real algebraic integer

Let α be an algebraic integer of degreed whosed conjugates are all real. We give a lower bound for the absolute value of conjugates of α in terms ofd and of the number of conjugates outside the interval [−2; 2]. Combining this with a lower bound for Mahler's measure of a polynomial, we obtain a lower bound for the maximal conjugate of a totally real algebraic integer. Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 37, No. 1, pp. 18–25, January–March, 1997.     

Orders of intersections of sylowp-subgroups in finite groups

Using the classification of finite simple groups, we tackle problem 5.12 from the “Kourovka Notebook,” for G a finite group satisfying O_p=1 for any prime p. Supported by RFFR. Translated fromAlgebra i Logika, Vol. 36, No. 2, pp. 156–165, March–April, 1997.     

A quantum analog of the Poincare-Birkhoff-Witt theorem

We reduce the basis construction problem for character Hopf algebras to a study of special elements, called “super-letters,” which are defined by Shirshov standard words. It is shown that character Hopf algebras having not more than finitely many “hard” super-letters share some of the properties of universal envelopings of finite-dimensional lie algebras. The background for our proofs is the construction of a filtration such that the associated graded algebra is obtained by iterating the skew polynomials construction, possibly followed with factorization. Supported by the National Society of Researchers, México (SNI, exp. 18740, 1997–2000) Translated fromAlgebra i Logika, Vol. 38, No. 4, pp. 476–507, July–August, 1999.     

On the zero-velocity-limit solutions to the Navier–Stokes equations of compressible flow

We consider the zero-velocity stationary problem of the Navier–Stokes equations of compressible isentropic flow describing the distribution of the density ϱ of a fluid in a spatial domain Ω⊂ℝ ^N driven by a time-independent potential external force b=∇F. A sharp condition in terms of F is given for the problem to possess a unique nonnegative solution ϱ having a prescribed mass m > 0. Received: 20 October 1997     

A simple derivation of the MacWilliams identity for linear ordered codes and orthogonal arrays

In (Can J Math 51(2):326–346, 1999), Martin and Stinson provide a generalized MacWilliams identity for linear ordered orthogonal arrays and linear ordered codes (introduced by Rosenbloom and Tsfasman (Prob Inform Transm 33(1):45–52, 1997) as “codes for the m-metric”) using association schemes. We give an elementary proof of this generalized MacWilliams identity using group characters and use it to derive an explicit formula for the dual type distribution of a linear ordered code or orthogonal array.      

On the Multiplicity of a C^∞-differentiable Function-germ

Risler ＆ Trotman in 1997 proved that the multiplicity of an analytic function germ is left-right lipschitz invariant, which provided a partial answer to Zariski conjecture. In this note, based on the recent work of Comte, Milman ＆ Trotman, we generalize the work of them to prove that the multiplicity of a C^∞ differentiable function germ is also left-right lipschitz invariant.     

On the long time behavior of the stochastic heat equation

We consider the stochastic heat equation in one space dimension and compute – for a particular choice of the initial datum – the exact long time asymptotic. In the Carmona-Molchanov approach to intermittence in non stationary random media this corresponds to the identification of the sample Lyapunov exponent. Equivalently, by interpreting the solution as the partition function of a directed polymer in a random environment, we obtain a weak law of large numbers for the quenched free energy. The result agrees with the one obtained in the physical literature via the replica method. The proof is based on a representation of the solution in terms of the weakly asymmetric exclusion process. Received: 11 November 1997 / Revised version: 31 July 1998     

A solution of the problem of generating the second harmonic

We propose an algorithm for a numerical solution of the problem of generating the second harmonic. The algorithm is based on the finite-difference method. Bibliography: 6 titles. Translated fromObchyslyuval'na ta Prykladna Matermatyka, No. 81, 1997, pp. 9–14.     

On a lower bound of the rate of convergence in the central limit theorem form-dependent random variables

We derive a lower bound of the rate of convergence in the central limit theorem form-dependent random variables under the finiteness of the fifth absolute moments of summands. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 37, No. 3, pp. 388–398, July–September, 1997.     

Estimate for the second-order derivatives of solutions of curvature-type equations

A local estimate for the second-order solutions to the curvature-type equations is established. Bibliography: 9 titles. Translated fromProblemy Matematicheskogo Analiza. No. 16. 1997, pp. 134–144.     

Generalized killing vectors in the five-dimensional lorentz manifold

Solutions of the generalized Killing equation are obtained for the metric with an initialL =t ² ⊕so(3) symmetry in the five-dimensional manifold of signature −3. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 2, pp. 249–253, August, 1997.     

Two analogs of the Rudin-Carleson theorem

In this article, a new asymptotic estimate for embeddings of singular measures into H^1,∞ is proved and an application to multidimensional analogs of the theorem of Rudin-Carleson is given. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 200–209. Translated by S. V. Kislyakov.     

On Fefferman and Burkholder–Davis–Gundy inequalities for ℰ-martingales

In a previous paper we introduced a new concept, the notion of ℰ-martingales and we extended the well-known Doob inequality (for 1 < p < + ∞) and the Burkholder–Davis–Gundy inequalities (for p = 2) to ℰ-martingales. After showing new Fefferman-type inequalities that involve sharp brackets as well as the space bmo _q , we extend the Burkholder–Davis–Gundy inequalities (for 1 < p < + ∞) to ℰ-martingales. By means of these inequalities we give sufficient conditions for the closedness in L ^p of a space of stochastic integrals with respect to a fixed ℝ^d-valued semimartingale, a question which arises naturally in the applications to financial mathematics. Finally we investigate the relation between uniform convergence in probability and semimartingale topology. Received: 22 July 1997 / Revised version: 3 July 1998     

Approximation of the classes C β ψ H ω by generalized de la Valiée-Poussin sums

On the classes of periodic functions C _β^ψ H _β, we study approximating properties of trigonometric polynomials generated by methods for summation of Fourier series. Slavyansk Pedagogical Institute, Slavyansk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 4, pp. 606–610, April, 1997.     

Construction of the extremal function for a functional on the classH ω (n)

For a functional on the classH _ω ⁽ⁿ⁾ ,n≥3, we construct the extremal function on which the upper bound obtained by A. I. Stepanets is attained. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 519–529, April, 1997. Translated by N. K. Kulman     

Sharp inequality for deviation of Rogozinski sums and the second continuity modulus in the space of periodic continuous functions

The sharp constant (uniformly in n) is found in a Jackson-type inequality involving the Rogozinski sums of order n and the second modulus of continuity with step π/(n+1). Bibliography: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 26–45. Translated by S. Yu. Pilyugin.     

Maximum principle for fully nonlinear equations via the iterated comparison function method

We present various versions of generalized Aleksandrov–Bakelman–Pucci (ABP) maximum principle for L ^p -viscosity solutions of fully nonlinear second-order elliptic and parabolic equations with possibly superlinear-growth gradient terms and unbounded coefficients. We derive the results via the “iterated” comparison function method, which was introduced in our previous paper (Koike and Święch in Nonlin. Diff. Eq. Appl. 11, 491–509, 2004) for fully nonlinear elliptic equations. Our results extend those of (Koike and Święch in Nonlin. Diff. Eq. Appl. 11, 491–509, 2004) and (Fok in Comm. Partial Diff. Eq. 23(5–6), 967–983) in the elliptic case, and of (Crandall et al. in Indiana Univ. Math. J. 47(4), 1293–1326, 1998; Comm. Partial Diff. Eq. 25, 1997–2053, 2000; Wang in Comm. Pure Appl. Math. 45, 27–76, 1992) and (Crandall and Święch in Lecture Notes in Pure and Applied Mathematics, vol. 234. Dekker, New York, 2003) in the parabolic case. Dedicated to Hitoshi Ishii on the occasion of his 60th birthday.     

Existence of the Vejvoda-Shtedry spaces

We investigate the linear periodic problem u _tt −u _xx =F(x, t), u(x+2π, t)=u(x, t+T)=u(x, t), ∈ ℝ², and establish conditions for the existence of its classical solution in spaces that are subspaces of the Vejvoda-Shtedry spaces. Ternopol’ Pedagogical Institute, Ternopol’. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 302–308, February, 1997.