Positive-Part Moments via the Fourier–Laplace Transform

Integral expressions for positive-part moments E X₊^p\mathsf{E}\,X_{+}^{p} (p>0) of random variables X are presented, in terms of the Fourier–Laplace or Fourier transforms of the distribution of X. A necessary and sufficient condition for the validity of such an expression is given. This study was motivated by extremal problems in probability and statistics, where one needs to evaluate such positive-part moments.     

An Extension of Hall's Theorem

Let (A_i) _{i ? I} (A_i) _{i \in I} and (B_i) _{i ? I} (B_i) _ {i \in I} be two (possibly infinite) families of finite sets. Let cl(P) denote the closure of the set P : = { (A_i, B_i ): i ? I } P := \{ ({A_i}, {B_i} ): i \in I \} of the pairs with respect to the componentwise union and intersection operations. Then there exists an injective map è_{i ? I} A_i ? è_{i ? I} B_i {\displaystyle \bigcup _ {i \in I}} A_i \rightarrow {\displaystyle \bigcup _ {i \in I }} B_i such that f (A_i) í B_i f (A_i) \subseteq B_i for every i if, and only if, card (A) ￡ (A) \leq card (B) for every pair (A, B) ? cl (P) (A, B) \in cl (P) .     

Model equations for the Eiffel Tower profile: historical perspective and new results

Model equations for the shape of the Eiffel Tower are investigated. One model purported to be based on Eiffel's writing does not give a tower with the correct curvature. A second popular model not connected with Eiffel's writings provides a fair approximation to the tower's skyline profile of 29 contiguous panels. Reported here is a third model derived from Eiffel's concern about wind loads on the tower, as documented in his communication to the French Civil Engineering Society on 30 March 1885. The result is a nonlinear, integro-differential equation which is solved to yield an exponential tower profile. It is further verified that, as Eiffel wrote, “in reality the curve exterior of the tower reproduces, at a determined scale, the same curve of the moments produced by the wind”. An analysis of the actual tower profile shows that it is composed of two piecewise continuous exponentials with different growth rates. This is explained by specific safety factors for wind loading that Eiffel & Company incorporated in the design of the free-standing tower. To cite this article: P. Weidman, I. Pinelis, C. R. Mecanique 332 (2004).     

Some estimates for large deviations and their application to strong law of large numbers

Siberian Mathematical Journal -     

A microsystem for sensing and patterning oxidative microgradients during cell culture

We present the design, modeling, fabrication and testing of a microsystem for the electrolytic patterning and sensing of oxidative microgradients within 1 x 1 mm2 area during cell culture. The system employs an array of microfabricated electrodes (3-40 microm in width) embedded in gas-permeable microchannels to generate precise doses of dissolved oxygen (ranging from 10 fmol O2 mm(-2) s(-1) to 100 nmol O2 mm(-2) s(-1)) via electrolysis. The microgradients generated by different microelectrodes in the array can be superimposed to pattern multi-dimensional oxygen profiles not possible with other methods. We demonstrate the patterning, sensing and quantification of dissolved oxygen microgradients in the 0 to 40% dO2 range using this microsystem. Reactive oxygen species generation and dosing is also quantified. Lastly, we demonstrate how the microtechnology enables new types of experiments in three different cell culture models: localized hyperoxia-induced apoptosis in C2C12 myoblasts, dynamic aerotaxis assays of Bacillus subtilis, and studies of calcium release in an ischemia/re-oxygenation myoblast model.     

On the Minimal Number of Even Submatrices of 0-1 Matrices

An asymptotic formula for the minimum possible number of even p x q submatrices of an m x n 0-1 matrix A is obtained. It is shown that if Ais considered random and pq is even, then the distributionof the number of the even p x q submatricesof A is highly skewed to the right, the left endpointof the distribution being very close to its mean, while its rightendpoint is twice the mean. A relation to Turán numbersis indicated.