Quantitative Version Research Articles

Quantitative unique continuation of solutions to the bi‐Laplace equations

In this paper, we prove a three‐ball inequality for y satisfying an equation of the form in some open, connected set Ω of with and . The derivation of such estimate relies on a delicate Carleman estimate for the bi‐Laplace equation and some Caccioppoli inequalities to estimate the lower‐terms. Based on three‐ball inequality, we then derive the vanishing order of y is less than , where | · |∞ means the L∞ norm, which is a quantitative version of the strong unique continuation property for y. Furthermore, under some priori assumptions on Vj and y, we prove that the nontrivial solution y satisfies the decay property around the point at infinity. In particular, if , this decaying rate can be improved to .

Quantitative combinatorial geometry for concave functions

We prove several exact quantitative versions of Helly's and Tverberg's theorems, which guarantee that a finite family of convex sets in Rd has a large intersection. Our results characterize conditions that are sufficient for the intersection of a family of convex sets to contain a “witness set” which is large under some concave or log-concave measure. The possible witness sets include ellipsoids, zonotopes, and H-convex sets. Our results show that several new optimization problems can be solved with algorithms for LP-type problems. We obtain colorful and fractional variants of all our Helly-type theorems.

Nanda-Hamner Curves Show Huge Latitudinal Variation but No Circadian Components in Drosophila Montana Photoperiodism.

Insect species with a wide distribution offer a great opportunity to trace latitudinal variation in the photoperiodic regulation of traits important in reproduction and stress tolerances. We measured this variation in the photoperiodic time-measuring system underlying reproductive diapause in Drosophila montana, using a Nanda-Hamner (NH) protocol. None of the study strains showed diel rhythmicity in female diapause proportions under a constant day length (12 h) and varying night lengths in photoperiods ranging from 16 to 84 h at 16°C. In the northernmost strains (above 55°N), nearly all females entered diapause under all photoperiods and about half of them even in continuous darkness, while the females of the southern strains showed high diapause proportions only in the circadian 24 h photoperiod. Significant correlation between the strains’ mean diapause proportions in ≥ 24 h photoperiods and critical day length (CDL; half of the females enter diapause) suggests at least partial causal connection between the traits. Interestingly, females of the northern strains entered diapause even in ≤ 24 h photoperiods, where the night length was shorter than their critical night length (24 h - CDL), but where the females experienced a higher number of Light:Dark cycles than in 24 h photoperiods. NH experiments, performed on the control and selection lines in our previous selection experiment, and completed here, gave similar results and confirmed that selection for shorter, southern-type CDL decreases female diapausing rate in non-circadian photoperiods. Overall, our study shows that D. montana females measure night length quantitatively, that the photoperiodic counter may play a prominent but slightly different role in extra short and extra long photoperiods and that northern strains show high stability against perturbations in the photoperiod length and in the presence of LD cycles. These features are best explained by the quantitative versions of the damped external coincidence model.

A quantitative stability theorem for convolution on the Heisenberg group

Although the convolution operators on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this paper, we use the expansion method to prove a quantitative version of this characterization.

Quantitative homogenization of the parabolic and elliptic Green’s functions on percolation clusters

We study the heat kernel and the Green’s function on the infinite supercritical percolation cluster in dimension d≥2 and prove a quantitative homogenization theorem for these functions with an almost optimal rate of convergence. These results are a quantitative version of the local central limit theorem proved by Barlow and Hambly in (Electron. J. Probab. 14 (2009) 1–27). The proof relies on a structure of renormalization for the infinite percolation cluster introduced in (Comm. Pure Appl. Math. 71 (2018) 1717–1849), Gaussian bounds on the heat kernel established by Barlow in (Ann. Probab. 32 (2004) 3024–3084) and tools of the theory of quantitative stochastic homogenization. An important step in the proof is to establish a C0,1-large-scale regularity theory for caloric functions on the infinite cluster and is of independent interest.

Weak Harnack inequalities for eigenvalues and constant rank theorems

We consider convex solutions of nonlinear elliptic equations which satisfy the structure condition of Bian and Guan. We prove a weak Harnack inequality for the eigenvalues of the Hessian of these solutions. This can be viewed as a quantitative version of the constant rank theorem.

Search Externalities in Firm-to-Firm Trade

I develop a model of firm-to-firm search and matching to show that the impact of falling trade costs on firm sourcing decisions and consumer welfare depends on the relative size of search externalities in domestic and international markets. These externalities can be positive if firms share information about potential matches, or negative if the market is congested. Using unique firm-to-firm transaction-level data from Uganda, I document empirical evidence consistent with positive externalities in international markets and negative externalities in domestic markets. I then build a dynamic quantitative version of the model and show that, in Uganda, a 25% reduction in trade costs led to a 3.7% increase in consumer welfare, 12% of which was due to search externalities.

On Eagleson's theorem in the non‐stationary setup

A classical result due to Eagleson states (in particular) that if appropriately normalized Birkhoff sums generated by a measurable function and an ergodic probability preserving transformation converge in distribution, then they also converge in distribution with respect to any probability measure which is absolutely continuous with respect to the invariant one. In this note, we prove several quantitative and infinite-dimensional versions of Eagleson's theorem for some classes of non-stationary stochastic processes which satisfy certain type of decay of correlations.

Quantitative results on a Halpern-type proximal point algorithm

We apply proof mining methods to analyse a result of Boikanyo and Moroşanu on the strong convergence of a Halpern-type proximal point algorithm. As a consequence, we obtain quantitative versions of this result, providing uniform effective rates of asymptotic regularity and metastability.

False-twin-free graphs with a fixed number of negative eigenvalues

We prove a quantitative version of a result of Torgašev concerning graphs with a fixed number of negative eigenvalues. We also establish a structural result stating that if for a hereditary family of graphs every graph of order N+1 and N+2 has false twins, then every graph from this family of order greater than N has false twins.

Sharp Caffarelli–Kohn–Nirenberg inequalities on Riemannian manifolds: the influence of curvature

We first establish a family of sharp Caffarelli–Kohn–Nirenberg type inequalities (shortly, sharp CKN inequalities) on the Euclidean spaces and then extend them to the setting of Cartan–Hadamard manifolds with the same best constant. The quantitative version of these inequalities also is proved by adding a non-negative remainder term in terms of the sectional curvature of manifolds. We next prove several rigidity results for complete Riemannian manifolds supporting the Caffarelli–Kohn–Nirenberg type inequalities with the same sharp constant as in the Euclidean space of the same dimension. Our results illustrate the influence of curvature to the sharp CKN inequalities on the Riemannian manifolds. They extend recent results of Kristály (J. Math. Pures Appl. 119 (2018), 326–346) to a larger class of the sharp CKN inequalities.

A quantitative stability inequality for fractional capacities

<abstract><p>The aim of this work is to show a non-sharp quantitative stability version of the fractional isocapacitary inequality. In particular, we provide a lower bound for the isocapacitary deficit in terms of the Fraenkel asymmetry. In addition, we provide the asymptotic behaviour of the $ s $-fractional capacity when $ s $ goes to $ 1 $ and the stability of our estimate with respect to the parameter $ s $.</p></abstract>

Quantitative ergodicity for the symmetric exclusion process with stationary initial data

We consider the symmetric exclusion process on the $d$-dimensional lattice with translational invariant and ergodic initial data. It is then known that as $t$ diverges the distribution of the process at time $t$ converges to a Bernoulli product measure. Assuming a summable decay of correlations of the initial data, we prove a quantitative version of this convergence by obtaining an explicit bound on the Ornstein $\bar d$-distance. The proof is based on the analysis of a two species exclusion process with annihilation.

A simple completion of Fisher's fundamental theorem of natural selection.

Fisher's fundamental theorem of natural selection shows that the part of the rate of change of mean fitness that is due to natural selection equals the additive genetic variance in fitness. Fisher embedded this result in a model of total fitness, adding terms for deterioration of the environment and density dependence. Here, a quantitative genetic version of this neglected model is derived that relaxes its assumptions that the additive genetic variance in fitness and the rate of deterioration of the environment do not change over time, allows population size to vary, and includes an input of mutational variance. The resulting formula for total rate of change in mean fitness contains two terms more than Fisher's original, representing the effects of stabilizing selection, on the one hand, and of mutational variance, on the other, making clear for the first time that the fundamental theorem deals only with natural selection that is directional (as opposed to stabilizing) on the underlying traits. In this model, the total (rather than just the additive) genetic variance increases mean fitness. The unstructured population allows an explanation of Fisher's concept of fitness as simply birth rate minus mortality rate, and building up to the definition in structured populations.

Spectral-free methods in the theory of hereditarily indecomposable Banach spaces

We give new and simple proofs of some classical properties of hereditarily indecomposable Banach spaces, including the result by W. T. Gowers and B. Maurey that a hereditarily indecomposable Banach space cannot be isomorphic to a proper subspace of itself. These proofs do not make use of spectral theory and therefore, they work in real spaces as well as in complex spaces. We use our method to prove some new results. For example, we give a quantitative version of the latter result by Gowers and Maurey and deduce that Banach spaces that are isometric to all of their subspaces should have an unconditional basis with unconditional constant arbitrarily close to $1$. We also study the homotopy relation between into isomorphisms from hereditarily indecomposable spaces.

BOUNDS FOR INDEXES OF NILPOTENCY IN COMMUTATIVE RING THEORY: A PROOF MINING APPROACH

AbstractIt is well-known that an element of a commutative ring with identity is nilpotent if, and only if , it lies in every prime ideal of the ring. A modification of this fact is amenable to a very simple proof mining analysis. We formulate a quantitative version of this modification and obtain an explicit bound. We present an application. This proof mining analysis is theleitmotiffor some comments and observations on the methodology of computational extraction. In particular, we emphasize that the formulation of quantitative versions of ordinary mathematical theorems is ofindependent interestfrom proof mining metatheorems.

A quantitative version of the Kolmogorov–Riesz theorem

In the present paper, we prove a generalization of the celebrated Kolmogorov–Riesz theorem in the $$L^{p}([0,1],X)$$ spaces, with $$1\le p <\infty$$ and X a Banach space. Specifically, our result is a quantitative version of such theorem. Our main tool is the so-called degree of nondensifiability, which measures (in the specified sense), the distance from a given convex subset of a Banach space to the class of its Peano Continua.

Strong limit multiplicity for arithmetic hyperbolic surfaces and 3-manifolds

We show that every sequence of torsion-free arithmetic congruence lattices in $\mathrm{PGL}(2,\mathbb R)$ or $\mathrm{PGL}(2,\mathbb C)$ satisfies a strong quantitative version of the Limit Multiplicity property. We deduce that for $R>0$ in certain range, growing linearly in the degree of the invariant trace field, the volume of the $R$-thin part of any congruence arithmetic hyperbolic surface or congruence arithmetic hyperbolic $3$-manifold $M$ is of order at most $\mathrm{Vol}(M)^{11/12}$. As an application we prove Gelander's conjecture on homotopy type of arithmetic hyperbolic $3$-manifolds: We show that there are constants $A,B$ such that every such manifold $M$ is homotopy equivalent to a simplicial complex with at most $A\mathrm{Vol}(M)$ vertices, all of degrees bounded by $B$.

Homogenization and hypocoercivity for Fokker–Planck equations driven by weakly compressible shear flows

Abstract We study the long-time dynamics of 2D linear Fokker–Planck equations driven by a drift that can be decomposed in the sum of a large shear component and the gradient of a regular potential depending on one spatial variable. The problem can be interpreted as that of a passive scalar advected by a slightly compressible shear flow, and undergoing small diffusion. For the corresponding stochastic differential equation, we give explicit homogenization rates in terms of a family of time-scales depending on the parameter measuring the strength of the incompressible perturbation. This is achieved by exploiting an auxiliary Poisson problem, and by computing the related effective diffusion coefficients. Regarding the long-time behavior of the solution of the Fokker–Planck equation, we provide explicit decay rates to the unique invariant measure by employing a quantitative version of the classical hypocoercivity scheme. From a fluid mechanics perspective, this turns out to be equivalent to quantifying the phenomenon of enhanced diffusion for slightly compressible shear flows.

Notes on the Szegő minimum problem. I. Measures with deep zeroes

The classical Szegő polynomial approximation theorem states that the polynomials are dense in the space L2(ρ), where ρ is a measure on the unit circle, if and only if the logarithmic integral of the measure ρ diverges. In this note we give a quantitative version of Szegő’s theorem in the special case when the divergence of the logarithmic integral is caused by deep zeroes of the measure ρ on a sufficiently rare subset of the circle.