Calixto P. Calderón

Modeling tumor growth

The meaning and limitations of certain mathematical models of tumor growth are discussed, and some new derivations of the existing models are given. A theoretical justification for Gompertzs law of growth for tumors is presented. An age-dependent Von Bertalanffys equation and diffusion models are introduced, and existence and uniqueness problems are addressed.

Initial values of solutions of the Navier-Stokes equations

This paper considers weak solutions to the Navier-Stokes equations in the sense considered in previous papers of the author and of Fabes, Jones, and Riviere. Results concerning pointwise a.e. convergence of the solutions to the initial values are established. The initial values that are considered here are divergence free vector functions belonging to L p (R n ), for p greater than or equal to the dimension n

On the classical trapping problem

We consider the problem of finding the probability density for the location of an untrapped pest satisfying a diffusion equation with the scaled Laplacian. Then by taking the initial data to the diffusion process to be in some Lp spaces, we establish the existence and uniqueness of local solutions in these spaces and the existence and uniqueness of weak global solutions in Lp,q; p,q greater than 3. The interest of this method relies on the fact that it is by successive approximations and hence amenable to numerical treatment.

Coarse-Graining the Cyclic Lotka-Volterra Model: SSA and Local Maximum Likelihood Estimation

When the output of an atomistic simulation (such as the Gillespie stochastic simulation algorithm, SSA) can be approximated as a diffusion process, we may be interested in the dynamic features of the deterministic (drift) component of this diffusion. We perform traditional scientific computing tasks (integration, steady state and closed orbit computation, and stability analysis) on such a drift component using a SSA simulation of the Cyclic Lotka-Volterra system as our illustrative example. The results of short bursts of appropriately initialized SSA simulations are used to fit local diffusion models using Ait-Sahalia’s transition density expansions [1], [2], [3] in a maximum likelihood framework. These estimates are then coupled with standard numerical algorithms (such as Newton-Raphson or numerical integration routines) to help design subsequent SSA experiments. A brief discussion of the validity of the local diffusion approximation of the SSA simulation (a jump process) is included.

Lacunary differentiability of functions in Rn

Abstract The existence of directional derivatives of functions in the Sobolev spaces Lpk(Rn) is studied. The novelty consists in calculating them through lacunary incremental quotients. Under these conditions no restrictions on p are necessary; the condition p > ( n k ) can be dropped.

Some new results on H summability of Fourier series

Abstract In this paper we shall be concerned with Hα summability, for 0 < α ≤ 2 of the Fourier series of arbitrary L1([−π, π]) functions. The methods employed here are a modification of the real variable ones introduced by J. Marcinkiewicz. The needed modifications give direct proofs of maximal theorems with respect to A1 weights. We also give a counter-example of a measure such that there is no convergence a.e. to the density of the measure. Finally, we present a Kakutani type of theorem, proving the ω*-density, in the space of of probability measures defined on [−π, π] of Borel measures for which there is no H2 summability a.e.

Potential Operators with Mixed Homogeneity

In 1966 Cora Sadosky introduced a number of results in a remarkable paper “A note on Parabolic Fractional and Singular Integrals”, see Sadosky (Studia Math 26:295–302, 1966), in particular, a quasi homogeneous version of Sobolev’s immersion theorem was discussed in the paper. Later, C. P. Calderon and T. Kwembe, following those ideas and incorporating the context of Fabes-Riviere homogeneity (Fabes and Riviere, Studia Math 27:19–38, 1966), proved a similar results for potential operators with kernels having mixed homogeneity. Calderon-Kwembe’s (Dispersal models. X Latin American School of Mathematics (Tanti, 1991). Rev Un Mat Argent 37(3–4):212–229, 1991/1992) basic theorem was very much in the spirit of Sadosky’s result. The natural extension of Sadosky’s paper is nevertheless the joint paper by C. Sadosky and M. Cotlar (On quasi-homogeneous Bessel potential operators. In: Singular integrals. Proceedings of symposia in pure mathematics, Chicago, 1966. American Mathematical Society, Providence, 1967, pp 275–287) which constitutes a true tour de force through, what is now considered, local properties of solutions of parabolic partial differential equations. The tools are the introduction of “Parabolic Bessel Potentials” combined with mixed homogeneity local smoothness estimates.

Remembrances and Silhouettes

My life in Mathematics began when I transferred from The University of Cuyo, San Juan, to the University of Buenos Aires in 1961. My brother Alberto helped me economically and morally for the jump. Upon my arrival to Buenos Aires, Dr. Alberto Gonzalez Dominguez (1904–1982) helped and oriented me with the change. The first subject I took in the Math Department, School of Exact Sciences, was Funciones Reales I, first course on Lebesgue Integration. Prof. Evelio Oklander was the instructor.

Some Non Standard Applications of the Laplace Method

In this paper we consider two nonstandard applications of the Laplace method. The first one is referred to the Inversion Formula of D.V. Widder and E.L. Post, from which a maximal theorem is proved. The second one is a special Calderon–Zygmund partition that gives us a genuine generalization of Natanson’s lemma in this context.

Approximation units and summation of independent random variables

Abstract Let {Xn} be a sequence of independent, identically distributed random variables. Assume that X1 has a density function with E(X1) = 0 and σn2(X1) ƒ n (x) the density of ( 1 n ) ∑ i − 1 n X i . The Weak Law of Large Numbers gives ∝ |x|>ϵ ƒ n (x) dx → 0 for every e > 0 whereas ∝ ƒ n dx = 1 . This tells us that ƒ n ∗ g → g in the L1 metric whenever g ϵ L1(−∝, ∝). This can be readily seen in the case when g is continuous with compact support. The general case follows by a density argument as a consequence of Youngs inequality. Throughout this paper we show that if in addition the characteristic function of X1 belongs to some class Lx, large α, then ƒ n ∗ g converges a.e. to g. Similar results are discussed for the case when σ = ∞. It is shown that these results can be phrased in terms of a more general theorem concerning approximation units.