Infinite families of manifolds of positive $$k\mathrm{th}$$-intermediate Ricci curvature with k small
Abstract

Positive$$k\mathrm{th}$$$k\mathrm{th}$-intermediate Ricci curvature on a Riemanniann-manifold, to be denoted by$${{\,\mathrm{Ric}\,}}_k>0$$${\phantom{\rule{0ex}{0ex}}\mathrm{Ric}\phantom{\rule{0ex}{0ex}}}_{k}>0$, is a condition that interpolates between positive sectional and positive Ricci curvature (when$$k =1$$$k=1$and$$k=n-1$$$k=n-1$respectively). In this work, we produce many examples of manifolds of$${{\,\mathrm{Ric}\,}}_k>0$$${\phantom{\rule{0ex}{0ex}}\mathrm{Ric}\phantom{\rule{0ex}{0ex}}}_{k}>0$withksmall by examining symmetric and normal homogeneous spaces, along with certain metric deformations of fat homogeneous bundles. As a consequence, we show that every dimension$$n\ge 7$$$n\ge 7$congruent to$$3\,{{\,\mathrm{mod}\,}}4$$$3\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{mod}\phantom{\rule{0ex}{0ex}}4$supports infinitely many closed simply connected manifolds of pairwise distinct homotopy type, all of which admit homogeneous metrics of$${{\,\mathrm{Ric}\,}}_k>0$$${\phantom{\rule{0ex}{0ex}}\mathrm{Ric}\phantom{\rule{0ex}{0ex}}}_{k}>0$for some$$k$k. We also prove that each dimension$$n\ge 4$$$n\ge 4$congruent to 0 or$$1\,{{\,\mathrm{mod}\,}}4$$$1\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{mod}\phantom{\rule{0ex}{0ex}}4$supports closed manifolds which carry metrics of$${{\,\mathrm{Ric}\,}}_k>0$$${\phantom{\rule{0ex}{0ex}}\mathrm{Ric}\phantom{\rule{0ex}{0ex}}}_{k}>0$with$$k\le n/2$$$k\le n/2$, but do not admit metrics of positive sectional curvature. Authors: ; ; Publication Date: NSF-PAR ID: 10369318 Journal Name: Mathematische Annalen ISSN: 0025-5831 Publisher: Springer Science + Business Media Sponsoring Org: National Science Foundation ##### More Like this 1. Abstract Negative correlations in the sequential evolution of interspike intervals (ISIs) are a signature of memory in neuronal spike-trains. They provide coding benefits including firing-rate stabilization, improved detectability of weak sensory signals, and enhanced transmission of information by improving signal-to-noise ratio. Primary electrosensory afferent spike-trains in weakly electric fish fall into two categories based on the pattern of ISI correlations: non-bursting units have negative correlations which remain negative but decay to zero with increasing lags (Type I ISI correlations), and bursting units have oscillatory (alternating sign) correlation which damp to zero with increasing lags (Type II ISI correlations). Here, we predict and match observed ISI correlations in these afferents using a stochastic dynamic threshold model. We determine the ISI correlation function as a function of an arbitrary discrete noise correlation function$${{\,\mathrm{\mathbf {R}}\,}}_k$$${\phantom{\rule{0ex}{0ex}}R\phantom{\rule{0ex}{0ex}}}_{k}$, wherekis a multiple of the mean ISI. The function permits forward and inverse calculations of the correlation function. Both types of correlation functions can be generated by adding colored noise to the spike threshold with Type I correlations generated with slow noise and Type II correlations generated with fast noise. A first-order autoregressive (AR) process with a single parameter is sufficient to predict and accurately match both types of afferent ISImore » 2. Abstract Let us fix a primepand a homogeneous system ofmlinear equations$$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$${a}_{j,1}{x}_{1}+\cdots +{a}_{j,k}{x}_{k}=0$for$$j=1,\dots ,m$$$j=1,\cdots ,m$with coefficients$$a_{j,i}\in \mathbb {F}_p$$${a}_{j,i}\in {F}_{p}$. Suppose that$$k\ge 3m$$$k\ge 3m$, that$$a_{j,1}+\dots +a_{j,k}=0$$${a}_{j,1}+\cdots +{a}_{j,k}=0$for$$j=1,\dots ,m$$$j=1,\cdots ,m$and that every$$m\times m$$$m×m$minor of the$$m\times k$$$m×k$matrix$$(a_{j,i})_{j,i}$$${\left({a}_{j,i}\right)}_{j,i}$is non-singular. Then we prove that for any (large)n, any subset$$A\subseteq \mathbb {F}_p^n$$$A\subseteq {F}_{p}^{n}$of size$$|A|> C\cdot \Gamma ^n$$$|A|>C·{\Gamma }^{n}$contains a solution$$(x_1,\dots ,x_k)\in A^k$$$\left({x}_{1},\cdots ,{x}_{k}\right)\in {A}^{k}$to the given system of equations such that the vectors$$x_1,\dots ,x_k\in A$$${x}_{1},\cdots ,{x}_{k}\in A$are all distinct. Here,Cand$$\Gamma $$$\Gamma$are constants only depending onp,mandksuch that$$\Gamma $\Gamma . The crucial point here is the condition for the vectors$$x_1,\dots ,x_k$$${x}_{1},\cdots ,{x}_{k}$in the solution$$(x_1,\dots ,x_k)\in A^k$$$\left({x}_{1},\cdots ,{x}_{k}\right)\in {A}^{k}$to be distinct. If we relax this condition and only demand that$$x_1,\dots ,x_k$$${x}_{1},\cdots ,{x}_{k}$are not all equal, then the statement would follow easily from Tao’s slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments.

3. Abstract

Let$$\phi$$$\varphi$be a positive map from the$$n\times n$$$n×n$matrices$$\mathcal {M}_n$$${M}_{n}$to the$$m\times m$$$m×m$matrices$$\mathcal {M}_m$$${M}_{m}$. It is known that$$\phi$$$\varphi$is 2-positive if and only if for all$$K\in \mathcal {M}_n$$$K\in {M}_{n}$and all strictly positive$$X\in \mathcal {M}_n$$$X\in {M}_{n}$,$$\phi (K^*X^{-1}K) \geqslant \phi (K)^*\phi (X)^{-1}\phi (K)$$$\varphi \left({K}^{\ast }{X}^{-1}K\right)⩾\varphi {\left(K\right)}^{\ast }\varphi {\left(X\right)}^{-1}\varphi \left(K\right)$. This inequality is not generally true if$$\phi$$$\varphi$is merely a Schwarz map. We show that the corresponding tracial inequality$${{\,\textrm{Tr}\,}}[\phi (K^*X^{-1}K)] \geqslant {{\,\textrm{Tr}\,}}[\phi (K)^*\phi (X)^{-1}\phi (K)]$$$\phantom{\rule{0ex}{0ex}}\text{Tr}\phantom{\rule{0ex}{0ex}}\left[\varphi \left({K}^{\ast }{X}^{-1}K\right)\right]⩾\phantom{\rule{0ex}{0ex}}\text{Tr}\phantom{\rule{0ex}{0ex}}\left[\varphi {\left(K\right)}^{\ast }\varphi {\left(X\right)}^{-1}\varphi \left(K\right)\right]$holds for a wider class of positive maps that is specified here. We also comment on the connections of this inequality with various monotonicity statements that have found wide use in mathematical physics, and apply it, and a close relative, to obtain some new, definitive results.

4. Abstract

Fix a positive integernand a finite field$${\mathbb {F}}_q$$${F}_{q}$. We study the joint distribution of the rank$${{\,\mathrm{rk}\,}}(E)$$$\phantom{\rule{0ex}{0ex}}\mathrm{rk}\phantom{\rule{0ex}{0ex}}\left(E\right)$, then-Selmer group$$\text {Sel}_n(E)$$${\text{Sel}}_{n}\left(E\right)$, and then-torsion in the Tate–Shafarevich group Equation missing<#comment/>asEvaries over elliptic curves of fixed height$$d \ge 2$$$d\ge 2$over$${\mathbb {F}}_q(t)$$${F}_{q}\left(t\right)$. We compute this joint distribution in the largeqlimit. We also show that the “largeq, then large height” limit of this distribution agrees with the one predicted by Bhargava–Kane–Lenstra–Poonen–Rains.

5. Abstract

We develop a new heavy quark transport model, QLBT, to simulate the dynamical propagation of heavy quarks inside the quark-gluon plasma (QGP) created in relativistic heavy-ion collisions. Our QLBT model is based on the linear Boltzmann transport (LBT) model with the ideal QGP replaced by a collection of quasi-particles to account for the non-perturbative interactions among quarks and gluons of the hot QGP. The thermal masses of quasi-particles are fitted to the equation of state from lattice QCD simulations using the Bayesian statistical analysis method. Combining QLBT with our advanced hybrid fragmentation-coalescence hadronization approach, we calculate the nuclear modification factor$$R_\mathrm {AA}$$${R}_{\mathrm{AA}}$and the elliptic flow$$v_2$$${v}_{2}$ofDmesons at the Relativistic Heavy-Ion Collider and the Large Hadron Collider. By comparing our QLBT calculation to the experimental data on theDmeson$$R_\mathrm {AA}$$${R}_{\mathrm{AA}}$and$$v_2$$${v}_{2}$, we extract the heavy quark transport parameter$$\hat{q}$$$\stackrel{^}{q}$and diffusion coefficient$$D_\mathrm {s}$$${D}_{s}$in the temperature range of$$1-4~T_\mathrm {c}$$$1-4\phantom{\rule{0ex}{0ex}}{T}_{c}$, and compare them with the lattice QCD results and other phenomenological studies.