arXiv subjects

Stephen Schecter

Publications and source records attributed to Stephen Schecter.

4 recordsLinked to original sources

arXiv2026

When Do Riemann Solutions Consist of Rarefactions, Jumps, and Constants?

A solution of a Riemann problem for a strictly hyperbolic system of conservation laws is traditionally expected to consist of rarefaction waves, jump discontinuities, and constant states. In this paper, we investigate whether a Riemann solution has this structure when the solution is only assumed to be measurable and essentially bounded. To discriminate continuous and discontinuous features in an $L^\infty$ solution, we introduce one-sided accumulation sets based on local essential images. Supposing that throughout a bounded open interval a solution is continuous in the essential image (ess-im) sense, we prove that it is a rarefaction wave if it is resonant (the characteristic speed equals $x/t$), and otherwise it is constant. Although an ess-im discontinuity might not be a jump discontinuity, we show that all ess-im accumulation states lie on a common Hugoniot locus and have the same speed. Anomalies are possible if there are limit points of ess-im discontinuities, but if the set of ess-im discontinuities is finite, then an $L^\infty$ Riemann solution has bounded variation and is composed of finitely many rarefaction waves, jump discontinuities, and constant states.

math.AP↗

arXiv2020

Geometric Singular Perturbation Theory Analysis of an Epidemic Model with Spontaneous Human Behavioral Change

We consider a model due to Piero Poletti and collaborators that adds spontaneous human behavioral change to the standard SIR epidemic model. In its simplest form, the Poletti model adds one differential equation, motivated by evolutionary game theory, to the SIR model. The new equation describes the evolution of a variable $x$ that represents the fraction of the population using normal behavior. The remaining fraction $1-x$ uses altered behavior such as staying home, social isolation, mask wearing, etc. Normal behavior offers a higher payoff when the number of infectives is low; altered behavior offers a higher payoff when the number is high. We show that the entry-exit function of geometric singular perturbation theory can be used to analyze the model in the limit in which behavior changes on a much faster time scale than that of the epidemic. In particular, behavior does not change as soon as a different behavior has a higher payoff; current behavior is sticky. The delay until behavior changes in predicted by the entry-exit function.

q-bio.PE↗

arXiv2015

The entry-exit function and geometric singular perturbation theory

For small $\epsilon>0$, the system $\dot x = \epsilon$, $\dot z = h(x,z,\epsilon)z$, with $h(x,0,0)<0$ for $x<0$ and $h(x,0,0)>0$ for $x>0$, admits solutions that approach the $x$-axis while $x<0$ and are repelled from it when $x>0$. The limiting attraction and repulsion points are given by the well-known entry-exit function. For $h(x,z,\epsilon)z$ replaced by $h(x,z,\epsilon)z^2$, we explain this phenomenon using geometric singular perturbation theory. We also show that the linear case can be reduced to the quadratic case, and we discuss the smoothness of the return map to the line $z=z_0$, $z_0>0$, in the limit $\epsilon\to0$.

math.DS↗

arXiv2012

Morse theory for Lagrange multipliers and adiabatic limits

Given two Morse functions $f, \mu$ on a compact manifold $M$, we study the Morse homology for the Lagrange multiplier function on $M \times {\mathbb R}$ which sends $(x, \eta)$ to $f(x) + \eta \mu(x)$. Take a product metric on $M \times {\mathbb R}$, and rescale its ${\mathbb R}$-component by a factor $\lambda^2$. We show that generically, for large $\lambda$, the Morse-Smale-Witten chain complex is isomorphic to the one for $f$ and the metric restricted to ${\mu^{-1}(0)}$, with grading shifted by one. On the other hand, let $\lambda\to 0$, we obtain another chain complex, which is geometrically quite different but has the same homology as the singular homology of $\mu^{-1}(0)$ and the isomorphism between them is provided by the homotopy by varying $\lambda$. Our proofs contain both the implicit function theorem on Banach manifolds and geometric singular perturbation theory.

math.GT↗