arXiv subjects

Alexander Segal

Publications and source records attributed to Alexander Segal.

6 recordsLinked to original sources

arXiv2025

The Scaled Polarity transform and related inequalities

In this paper we deal with generalizations of the Mahler volume product for log-concave functions. We show that the polarity transform $\mathcal A$ can be rescaled so that the Mahler product it induces has upper and lower bounds of the same asymptotics. We discuss a similar result for the $\mathcal J$ transform. As an application, we extend the K\"onig-Milman duality of entropy result to the class of geometric log-concave functions.

math.FA↗

arXiv2025

High-End Space Electronics: Active Shielding to Mitigate Catastrophic Single-Event Effects

Operating electronic systems in space environments presents significant challenges due to continuous exposure to cosmic, solar, and trapped radiation, which can induce catastrophic single-event effects. This paper introduces a novel nonintrusive mitigation apparatus designed to protect high-end commercial off-the-shelf electronics in space. The apparatus incorporates an array of real-time particle detectors coupled with a mitigation algorithm. Upon identifying potentially harmful particles, the system power cycles affected electronics, preempting permanent damage. The apparatus was evaluated using GEANT4 simulations, which were compared with empirical data from the "COTS-Capsule" experiment aboard the International Space Station, demonstrating strong agreement. Key results indicate that the system achieves a 95% detection accuracy with a power cycle rate of once every seven hours per square centimeter of sensitive electronics. The COTS-Capsule represents a cost-effective, flexible solution for integrating modern, high-end, non-space-qualified electronics into a variety of space missions, addressing critical challenges in the new-space era.

physics.ins-det↗

arXiv2019

Functional Brunn-Minkowski Inequalities Induced by Polarity

We prove a new family of inequalities, which compare the integral of a geometric convolution of non-negative functions with the integrals of the original functions. For classical inf-convolution, this type of inequality is called the Prékopa-Leindler inequality, which, restricted to indicators of convex bodies, gives the classical Brunn-Minkowski inequality. The convolution we consider is a different one, which arises from the study of the polarity transform for functions. While inf-convolution arises as the pull back of usual addition of convex functions under the Legendre transform, our geometric inf-convolution arises as the pull back of the second order reversing transform on geometric convex functions (called either polarity transform or $A$-transform). These are, up to linear terms, the only order reversing isomorphisms on the class of geometric convex functions. We prove that the integral of this new geometric convolution of two functions is bounded from below by the harmonic average of the individual integrals. Our inequality implies the Brunn-Minkowski inequality, as well as some other, new, inequalities for volumes of bodies. Our inequalities are intimately connected with Busemann's convexity theorem, a new variant of which we prove for 1-convex hulls and log-concave densities.

math.FA↗

arXiv2019

A Santaló-type Inequality for the ${\cal J}$ Transform

This paper deals with an analog of the Mahler volume product related to the ${\cal J}$ transform acting in the class of geometric convex functions ${\rm{Cvx}}_0({\mathbb R}^n)$. We provide asymptotically sharp bounds for the quantity $s^{\cal J}(f) = \frac{\int e^{-{\cal J} f}}{\int e^{-f}}$ and characterize all the extremal functions.

math.FA↗

arXiv2019

On the Linear Structures Induced by the Four Order Isomorphisms Acting on ${\rm{Cvx}}_0({\mathbb R}^n)$

It is known that the volume functional $\,ϕ\mapsto\int e^{-ϕ}\,$ satisfies certain concavity or convexity inequalities with respect to three of the four linear structures induced by the order isomorphisms acting on ${\rm{Cvx}}_0({\mathbb R}^n)$. In this note we define the fourth linear structure on ${\rm{Cvx}}_0({\mathbb R}^n)$ as the pullback of the standard linear structure under the ${\cal J}$ transform. We show that, interpolating with respect to this linear structure, no concavity or convexity inequalities hold, and prove that a quasi-convexity inequality is violated only by up to a factor of $2$. We also establish all the order relations which the four different interpolations satisfy.

math.FA↗

arXiv2015

Minkowski Symmetrizations of Star Shaped Sets

We provide sharp upper bounds for the number of symmetrizations required to transform a star shaped set in ${\mathbb R}^n$ arbitrarily close (in the Hausdorff metric) to the Euclidean ball.

math.MG↗