An official website of the United States government
Abstract We compute the 2‐adic effective slice spectral sequence (ESSS) for the motivic stable homotopy groups of , a motivic analogue of the connective ‐local sphere over prime fields of characteristic not two. Together with the analogous computation over algebraically closed fields, this yields information about the motivic ‐local sphere over arbitrary base fields of characteristic not two. To compute the spectral sequence, we prove several results that may be of independent interest. We describe the ‐differentials in the slice spectral sequence in terms of the motivic Steenrod operations over general base fields, building on analogous results of Ananyevskiy, Röndigs, and Østvær for the very effective cover of Hermitian K‐theory. We also explicitly describe the coefficients of certain motivic Eilenberg–MacLane spectra and compute the ESSS for the very effective cover of Hermitian K‐theory over prime fields.
more »
« less
Riesz Energy with a Radial External Field: When is the Equilibrium Support a Sphere?
Abstract We consider Riesz energy problems with radial external fields. We study the question of whether or not the equilibrium measure is the uniform distribution on a sphere. We develop general necessary and general sufficient conditions on the external field that apply to powers of the Euclidean norm as well as certain Lennard – Jones type fields. Additionally, in the former case, we completely characterize the values of the power for which a certain dimension reduction phenomenon occurs: the support of the equilibrium measure becomes a sphere. We also briefly discuss the relationship between these problems and certain constrained optimization problems. Our approach involves the Frostman characterization, the Funk–Hecke formula, and the calculus of hypergeometric functions.
more »
« less
- Award ID(s):
- 2202877
- PAR ID:
- 10562678
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Potential Analysis
- Volume:
- 63
- Issue:
- 2
- ISSN:
- 0926-2601
- Format(s):
- Medium: X Size: p. 705-738
- Size(s):
- p. 705-738
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract When compressed, certain lattices undergo phase transitions that may allow nuclei to gain sig- nificant kinetic energy. To explore the dynamics of this phenomenon, we develop a methodology to study Coulomb coupled N-body systems constrained to a sphere, as in the Thomson problem. We initialize N total Boron nuclei as point particles on the surface of the sphere, allowing them to equilibrate via Coulomb scattering with a viscous damping term. To simulate a phase transition, we remove Nrm particles, forcing the system to rearrange into a new equilibrium. With this model, we consider the Thomson problem as a dynamical system, providing a framework to explore how non-zero temperature affects structural imperfections in Thomson minima. We develop a scaling relation for the average peak kinetic energy attained by a single particle as a function of N and Nrm. For certain values of N , we find an order of magnitude energy gain when increasing Nrm from 1 to 6. The model may help to design a lattice that maximizes the energy output.more » « less
-
Abstract The primary motivation behind quantitative work in international trade and many other fields is to shed light on the economic consequences of policy changes and other shocks. To help assess and potentially strengthen the credibility of such quantitative predictions, we introduce an IV-based goodness-of-fit measure that provides the basis for testing causal predictions in arbitrary general equilibrium environments as well as for estimating the average misspecification in these predictions. As an illustration of how to use the measure in practice, we revisit the welfare consequences of the U.S.-China trade war predicted by Fajgelbaum et al. (2020).more » « less
-
Interaction of electric fields with biological cells is indispensable for many physiological processes. Thermal electrical noise in the cellular environment has long been considered as the minimum threshold for detection of electrical signals by cells. However, there is compelling experimental evidence that the minimum electric field sensed by certain cells and organisms is many orders of magnitude weaker than the thermal electrical noise limit estimated purely under equilibrium considerations. We resolve this discrepancy by proposing a nonequilibrium statistical mechanics model for active electromechanical membranes and hypothesize the role of activity in modulating the minimum electrical field that can be detected by a biological membrane. Active membranes contain proteins that use external energy sources to carry out specific functions and drive the membrane away from equilibrium. The central idea behind our model is that active mechanisms, attributed to different sources, endow the membrane with the ability to sense and respond to electric fields that are deemed undetectable based on equilibrium statistical mechanics. Our model for active membranes is capable of reproducing different experimental data available in the literature by varying the activity. Elucidating how active matter can modulate the sensitivity of cells to electric signals can open avenues for a deeper understanding of physiological and pathological processes.more » « less
-
Given a Zariski-dense, discrete group, Γ, of isometries acting on (n + 1)- dimensional hyperbolic space, we use spectral methods to obtain a sharp asymptotic formula for the growth rate of certain Γ-orbits. In particular, this allows us to obtain a best-known effective error rate for the Apollonian and (more generally) Kleinian sphere packing counting problems, that is, counting the number of spheres in such with radius bounded by a growing parameter. Our method extends the method of Kontorovich [Kon09], which was itself an extension of the orbit counting method of Lax-Phillips [LP82], in two ways. First, we remove a compactness condition on the discrete subgroups considered via a technical cut- off and smoothing operation. Second, we develop a coordinate system which naturally corresponds to the inversive geometry underlying the sphere counting problem, and give structure theorems on the arising Casimir operator and Haar measure in these coordinates.more » « less
