Related papers: An exactly solved model of biological evolution
The transformation of the free-energy landscape from smooth to hierarchical is one of the richest features of mean-field disordered systems. A well-studied example is the de Almeida-Thouless transition for spin glasses in a magnetic field,…
We study the adaptation dynamics of an initially maladapted asexual population with genotypes represented by binary sequences of length $L$. The population evolves in a maximally rugged fitness landscape with a large number of local optima.…
We derive an expression for the quasinormal modes of scalar perturbations in near extreme d-dimensional Schwarzschild-de Sitter and Reissner-Nordstrom-de Sitter black holes. We show that, in the near extreme limit, the dynamics of the…
Deep, overparameterized regression models are notorious for their tendency to overfit. This problem is exacerbated in heteroskedastic models, which predict both mean and residual noise for each data point. At one extreme, these models fit…
Motivated by the recent empirical successes of deep generative models, we study the computational complexity of the following unsupervised learning problem. For an unknown neural network $F:\mathbb{R}^d\to\mathbb{R}^{d'}$, let $D$ be the…
We introduce new finite-dimensional spaces specifically designed to approximate the solutions to high-frequency Helmholtz problems with smooth variable coefficients in dimension $d$. These discretization spaces are spanned by Gaussian…
By introducing the generalized master function of order up to four together with corresponding weight function, we have obtained all quasi-exactly solvable second order differential equations. It is shown that these differntial equations…
We study the sample complexity of differentially private optimization of quasi-concave functions. For a fixed input domain $\mathcal{X}$, Cohen et al. (STOC 2023) proved that any generic private optimizer for low sensitive quasi-concave…
We present a comprehensive study of the Kerr-Newman quasinormal mode spectrum in the Dudley-Finley approximation, where the linear gravitoelectromagnetic perturbations are decoupled by "freezing" either one of the fields to its background…
In these proceedings, we first summarize some general properties of phase transitions in the presence of quenched disorder, with emphasis on the following points: the need to distinguish typical and averaged correlations, the possible…
We consider the quantum evolution of a fermion-hole pair in a d-dimensional gas of non-interacting fermions in the presence of random phase scattering. This system is mapped onto an effective Ising model, which enables us to show rigorously…
Phenotypically structured equations arise in population biology to describe the interaction of species with their environment that brings the nutrients. This interaction usually leads to selection of the fittest individuals. Models used in…
Einstein's General Relativity theory simplifies dramatically in the limit that the spacetime dimension D is very large. This could still be true in the gravity theory with higher derivative terms. In this paper, as the first step to study…
This paper develops a quasispecies model that incorporates the SOS response. We consider a unicellular, asexually replicating population of organisms, whose genomes consist of a single, double-stranded DNA molecule, i.e. one chromosome. We…
The nonequilibrium stationary state of an exclusive genetic switch is considered. The model comprises two competing species and a single binding site which, when bound to by a protein of one species, causes the other species to be…
Deep neural networks (DNNs) have recently emerged as effective tools for approximating solution operators of partial differential equations (PDEs) including evolutionary problems. Classical numerical solvers for such PDEs often face…
We consider the parabolic Anderson model $\frac{\partial}{\partial t} v_n=\kappa\Delta_n v_n + \xi_n v_n$ on the $n$-dimensional hypercube $\{-1,+1\}^n$ with random i.i.d. potential $\xi_n$. We parametrize time by volume and study $v_n$ at…
Given a large, high-dimensional sample from a spiked population, the top sample covariance eigenvalue is known to exhibit a phase transition. We show that the largest eigenvalues have asymptotic distributions near the phase transition in…
Results are presented for the time evolution of fermions initially in a non-zero temperature normal phase, following the switch on of an attractive interaction. The dynamics are studied in the disordered phase close to the critical point,…
In three spatial dimensions, we study the Cauchy problem for the model wave equation $- \partial_t^2 \Psi + (1 + \Psi)^P \Delta \Psi = 0$ for $P \in \lbrace 1,2 \rbrace$. We exhibit a stable form of finite-time Tricomi-type degeneracy…