Sharp Results Research Articles

Financial Market Ethics

Abstract We develop a model of psychological-games-played-on-a-network to demonstrate a role for endogenously determined, rationally chosen ethics. Our analysis produces sharp results about contagion of nonethical or ethical behavior and the possible equilibrium configurations of each type of behavior. We find, and quantify, critical densities for clusters of each type of behavior that determine everything about contagion. We introduce society as a third player to investigate ethical failures as externalities. We use these results to show how regulations and network structure can affect whether clusters of ethical behavior can survive and how large they can be in a financial market setting.

Rough solutions of the 3-D compressible Euler equations

We prove the local-in-time well-posedness for the solution of the compressible Euler equations in 3-D for the Cauchy data of the velocity, density and vorticity $(v,\varrho\mathfrak{w} \in H^s \times H^s \times H^{s'}$, $2\lt s'\lt s$. The result extends the sharp results of Smith-Tataru and of Wang, established in the irrotational case, i.e., $\mathfrak{w}=0$, which is known to be optimal for $s>2$. At the opposite extreme, in the incompressible case, i.e., with a constant density, the result is known to hold for $\mathfrak{w}\in H^s$, $s>3/2$ and fails for $s\le 3/2$. We therefore, conjecture that the optimal result should be $(v,\varrho,\mathfrak{w} \in H^s\times H^s\times H^{s'}$, $s>2$, $s' > \frac{3}{2}$. We view our work here as an important step in proving the conjecture. The main difficulty in establishing sharp well-posedness results for general compressible Euler flow is due to the highly nontrivial interaction between the sound waves, governed by quasilinear wave equations, and vorticity which is transported by the flow. To overcome this difficulty, we separate the dispersive part of sound wave from the transported part, and gain regularity significantly by exploiting the nonlinear structure of the system and the geometric structures of the acoustical spacetime.

Ramsey and Gallai-Ramsey Number for Wheels

Given a graph G and a positive integer k, define the Gallai-Ramsey number to be the minimum number of vertices n such that any k-edge coloring of \(K_n\) contains either a rainbow (all different colored) triangle or a monochromatic copy of G. Much like graph Ramsey numbers, Gallai-Ramsey numbers have gained a reputation as being very difficult to compute in general. As yet, still only precious few sharp results are known. In this paper, we obtain bounds on the Gallai-Ramsey number for wheels and the exact value for the wheel on 5 vertices.

Reflection algebras and conservation results for theories of iterated truth

We consider extensions of the language of Peano arithmetic by transfinitely iterated truth definitions satisfying uniform Tarskian biconditionals. Without further axioms, such theories are known to be conservative extensions of the original system of arithmetic. Much stronger systems, however, are obtained by adding either induction axioms or reflection axioms on top of them. Theories of this kind can interpret some well-known predicatively reducible fragments of second-order arithmetic such as iterated arithmetical comprehension.We obtain sharp results on the proof-theoretic strength of these systems using methods of provability logic. Reflection principles naturally define unary operators acting on the semilattice of axiomatizable extensions of our basic theory of iterated truth. The substructure generated by the top element of this algebra provides a canonical ordinal notation system for the class of theories under investigation.Using these notations we obtain conservativity relationships for iterated reflection principles of different logical complexity levels corresponding to the levels of the hyperarithmetical hierarchy, i.e., the analogs of Schmerl's formulas. These relationships, in turn, provide proof-theoretic analysis of our systems and of some related predicatively reducible theories. In particular, we uniformly calculate the ordinals characterizing the standard measures of their proof-theoretic strength, such as provable well-orderings, classes of provably recursive functions, and Π10-ordinals.

Sharp lifespan estimates for the weakly coupled system of semilinear damped wave equations in the critical case

The open question, which seems to be also the final part, in terms of studying the Cauchy problem for the weakly coupled system of damped wave equations or reaction-diffusion equations, is so far known as the sharp lifespan estimates in the critical case. In this paper, we mainly investigate lifespan estimates for solutions to the weakly coupled system of semilinear damped wave equations in the critical case. By using a suitable test function method associated with nonlinear differential inequalities, we catch upper bound estimates for the lifespan. Moreover, we establish polynomial-logarithmic type time-weighted Sobolev spaces to obtain lower bound estimates for the lifespan in low spatial dimensions. Then, together with the derived lifespan estimates, new and sharp results on estimates for the lifespan in the critical case are claimed. Finally, we give an application of our results to the semilinear reaction-diffusion system in the critical case.

Sharp quantitative stability for isoperimetric inequalities with homogeneous weights

We prove the sharp quantitative stability for a wide class of weighted isoperimetric inequalities. More precisely, we consider isoperimetric inequalities in convex cones with homogeneous weights.Inspired by the proof of such isoperimetric inequalities through the ABP method (see X. Cabré, X. Ros-Oton, and J. Serra [J. Eur. Math. Soc. (JEMS) 18 (2016), pp. 2971–2998]), we construct a new convex coupling (i.e., a map that is the gradient of a convex function) between a generic setEEand the minimizer of the inequality (as in Gromov’s proof of the isoperimetric inequality). Even if this map does not come from optimal transport, and even if there is a weight in the inequality, we adapt the methods of [Figalli, Maggi, and Pratelli [Invent. Math. 182 (2010), pp. 167–211] and prove that ifEEis almost optimal for the inequality then it is quantitatively close to a minimizerup to translations. Then, a delicate analysis is necessary to rule out the possibility of translations.As a step of our proof, we establish a sharp regularity result forrestrictedconvex envelopes of a function that might be of independent interest.

Irregularities of distribution and geometry of planar convex sets

We consider a planar convex body C and we prove several analogs of Roth's theorem on irregularities of distribution. When ∂C is C2 regardless of curvature, we prove that for every set PN of N points in T2 we have the sharp bound∫01∫T2|card(PN∩(τC+t))−τ2N|C||2dtdτ⩾cN1/2. When ∂C is only piecewise C2 and is not a polygon we prove the sharp bound∫01∫T2|card(PN∩(τC+t))−τ2N|C||2dtdτ⩾cN2/5. We also give a whole range of intermediate sharp results between N2/5 and N1/2. Our proofs depend on a lemma of Cassels-Montgomery, on ad hoc constructions of finite point sets, and on a geometric type estimate for the average decay of the Fourier transform of the characteristic function of C.

Oscillation result for half-linear delay difference equations of second-order.

In this paper, we obtain the new single-condition criteria for the oscillation of second-order half-linear delay difference equation. Even in the linear case, the sharp result is new and, to our knowledge, improves all previous results. Furthermore, our method has the advantage of being simple to prove, as it relies just on sequentially improved monotonicities of a positive solution. Examples are provided to illustrate our results.

BSNet: Dynamic Hybrid Gradient Convolution Based Boundary-Sensitive Network for Remote Sensing Image Segmentation

Boundary information is essential for the semantic segmentation of remote sensing images. However, most existing methods were designed to establish strong contextual information while losing detailed information, making it challenging to extract and recover boundaries accurately. In this paper, a boundary-sensitive network (BSNet) is proposed to address this problem via dynamic hybrid gradient convolution (DHGC) and coordinate sensitive attention (CSA). Specifically, in the feature extraction stage, we propose dynamic hybrid gradient convolution (DHGC) to replace vanilla convolution, which adaptively aggregates one vanilla convolution kernel and two gradient convolution kernels (GCKs) into a new operator to enhance boundary information extraction. The GCKs are proposed to explicitly encode boundary information, which are inspired by traditional Sobel operators. In the feature recovery stage, the coordinate sensitive attention (CSA) is introduced. This module is used to reconstruct the sharp and detailed segmentation results by adaptively modeling the boundary information and long-range dependencies in the low-level features as the assistance of high-level features. Note that DHGC and CSA are plug-and-play modules. We evaluate the proposed BSNet on three public data sets: the ISPRS 2-D semantic labeling Vaihingen, Potsdam benchmark and iSAID data set. The experimental results indicate that BSNet is a highly effective architecture that produces sharper predictions around object boundaries and significantly improves the segmentation accuracy. Our method demonstrates superior performance on the Vaihingen, Potsdam benchmark and iSAID data set, in terms of the mean F1, with improvements of 4.6%, 2.3% and 2.4% over strong baselines, respectively. The code and models will be made publicly available.

Rate of escape of conditioned Brownian motion

We study the norm of the two-dimensional Brownian motion conditioned to stay outside the unit disk at all times. By conditioning the process is changed from barely recurrent to slightly transient. We obtain sharp results on the rate of escape to infinity of the process of future minima: we find an integral test on the function g so that the future minima process drops below the barrier exp{lnt×g(lnlnt)} at arbitrary large times; we show that the future minima process exceeds K t×lnlnlntat arbitrary large times with probability 0 [resp., 1] if K is larger [resp., smaller] than some positive constant. For this, we introduce a renewal structure attached to record times and values. Additional results are given for the long time behavior of the norm.

Coefficient functionals for a class of bounded turning functions related to modified sigmoid function

<abstract><p>The main objective of the present article is to define the class of bounded turning functions associated with modified sigmoid function. Also we investigate and determine sharp results for the estimates of four initial coefficients, Fekete-Szegö functional, the second-order Hankel determinant, Zalcman conjucture and Krushkal inequality. Furthermore, we evaluate bounds of the third and fourth-order Hankel determinants for the class and for the 2-fold and 3-fold symmetric functions.</p></abstract>

Local H1-norm error analysis of a mixed finite element method for a time-fractional biharmonic equation

In this work, a time-fractional biharmonic equation with a Caputo derivative of fractional order α∈(0,1) is considered, whose solutions exhibit a weak singularity at initial time t=0. For this problem, a system of two second-order differential equations is derived by introducing a intermediate variable p=−Δu, then discretised the system using the standard finite element method in space together with the L1 discretisation of Caputo derivative on graded mesh in time. The H1-norm stability result of the method is established, and then a sharp H1-norm local convergent result is presented. Finally, numerical experiments are provided to further verify our theoretical analysis for each fixed value of α.

The Bohr phenomenon for analytic functions on shifted disks

In this paper, we investigate the Bohr phenomenon for the class of analytic functions defined on the simply connected domain
 \(\Omega_{\gamma}=\bigg\{z\in\mathbb{C} \colon \bigg|z+\frac{\gamma}{1-\gamma}\bigg|<\frac{1}{1-\gamma}\bigg\}\) for \(0\leq \gamma<1.\)
 We study improved Bohr radius, Bohr-Rogosinski radius and refined Bohr radius for the class of analytic functions defined in \(\Omega_{\gamma}\), and obtain several sharp results.

A sharp relative-error bound for the Helmholtz h-FEM at high frequency

For the h-finite-element method (h-FEM) applied to the Helmholtz equation, the question of how quickly the meshwidth h must decrease with the frequency k to maintain accuracy as k increases has been studied since the mid 80’s. Nevertheless, there still do not exist in the literature any k-explicit bounds on the relative error of the FEM solution (the measure of the FEM error most often used in practical applications), apart from in one dimension. The main result of this paper is the sharp result that, for the lowest fixed-order conforming FEM (with polynomial degree, p, equal to one), the condition “h^2 k^3 sufficiently small" is sufficient for the relative error of the FEM solution in 2 or 3 dimensions to be controllably small (independent of k) for scattering of a plane wave by a nontrapping obstacle and/or a nontrapping inhomogeneous medium. We also prove relative-error bounds on the FEM solution for arbitrary fixed-order methods applied to scattering by a nontrapping obstacle, but these bounds are not sharp for pge 2. A key ingredient in our proofs is a result describing the oscillatory behaviour of the solution of the plane-wave scattering problem, which we prove using semiclassical defect measures.

Global bifurcation of solitary waves to the Boussinesq abcd system

The Boussinesq abcd system arises in the modeling of long wave small amplitude water waves in a channel, where the four parameters (a,b,c,d) satisfy one constraint. In this paper we focus on the solitary wave solutions to such a system. In particular we work in two parameter regimes where the system does not admit a Hamiltonian structure (corresponding to b≠d). We prove via analytic global bifurcation techniques the existence of solitary waves in such parameter regimes. Some qualitative properties of the solutions are also derived, from which sharp results can be obtained for the global solution curves.Specifically, we first construct solutions bifurcating from the stationary waves, and obtain a global continuous curve of solutions that exhibits a loss of ellipticity in the limit. The second family of solutions bifurcate from the classical Boussinesq supercritical waves. We show that the curve associated to the second class either undergoes a loss of ellipticity in the limit or becomes arbitrarily close to having a stagnation point.

Polynomial-Exponential Bounds for Some Trigonometric and Hyperbolic Functions

Recent advances in mathematical inequalities suggest that bounds of polynomial-exponential-type are appropriate for evaluating key trigonometric functions. In this paper, we innovate in this sense by establishing new and sharp bounds of the form (1−αx2)eβx2 for the trigonometric sinc and cosine functions. Our main result for the sinc function is a double inequality holding on the interval (0, π), while our main result for the cosine function is a double inequality holding on the interval (0, π/2). Comparable sharp results for hyperbolic functions are also obtained. The proofs are based on series expansions, inequalities on the Bernoulli numbers, and the monotone form of the l’Hospital rule. Some comparable bounds of the literature are improved. Examples of application via integral techniques are given.

Comparison Theorems for 3D Manifolds With Scalar Curvature Bound

Abstract Two sharp comparison results are derived for 3D complete noncompact manifolds with scalar curvature bounded from below. The 1st one concerns the Green’s function. When the scalar curvature is nonnegative, it states that the rate of decay of an energy quantity over the level set is strictly less than that of the Euclidean space unless the manifold itself is isometric to the Euclidean space. The result is in turn converted into a sharp area comparison for the level set of the Green’s function when in addition the Ricci curvature of the manifold is assumed to be asymptotically nonnegative at infinity. The 2nd result provides a sharp upper bound of the bottom spectrum in terms of the scalar curvature lower bound, in contrast to the classical result of Cheng, which involves a Ricci curvature lower bound.

Functional Inequalities: Nonlinear Flows and Entropy Methods as a Tool for Obtaining Sharp and Constructive Results

Interpolation inequalities play an essential role in analysis with fundamental consequences in mathematical physics, nonlinear partial differential equations (PDEs), Markov processes, etc., and have a wide range of applications in various other areas of Science. Research interests have evolved over the years: while mathematicians were originally focussed on abstract properties (for instance appropriate notions of functional spaces for the existence of weak solutions in PDEs), more qualitative questions (for instance, bifurcation diagrams, multiplicity of the solutions in PDEs and their qualitative behaviour) progressively emerged. The use of entropy methods in nonlinear PDEs is a typical example: in some cases, the optimal constant in the inequality can be interpreted as an optimal rate of decay of an entropy for an associated evolution equation. Much more has been learned by adopting this point of view. This paper aims at illustrating some of these recent aspect of entropy-entropy production inequalities, with applications to stability in Gagliardo–Nirenberg–Sobolev inequalities and symmetry results in Caffarelli–Kohn–Nirenberg inequalities. Entropy methods provide a framework which relates nonlinear regimes with their linearized counterparts. This framework allows to prove optimality results, symmetry results and stability estimates. Some emphasis will be put on the hidden structure which explain such properties. Related open problems will be listed.

Some sharp results about the global existence and blowup of solutions to a class of coupled pseudo-parabolic equations

In this paper, we investigate an initial boundary value problem to a class of system pseudo-parabolic equations with power type source terms in low dimensional. By introducing a family of potential wells we not only obtain the invariant sets and vacuum isolating of solutions but also give some threshold results of global existence, blow-up phenomenon of weak solutions for the subcritical, critical and supercritical initial energy cases respectively. Furthermore, we study the decay rates for the global solutions and lifespan, blow-up rates of the blow-up solutions on some special cases.

Global attractivity of delayed and nonlocal diffusive logistic models with variable coefficients

In this paper, we consider a reaction-diffusion equation with continuous delay and spatial variable coefficients which models the evolution of a single species. We establish a sharp threshold dynamic result: there exists a critical value λ ( a ) such that if λ ( a ) < 0 the positive steady state solution of the equation is globally attractive, while if λ ( a ) ≥ 0 the trivial steady state is globally attractive. To this end, we analyze the ω -limit set of the equation and prove that it is a singleton. Moreover, we apply our method to obtain global attractivity of the positive steady state of a spatially nonlocal diffusive logistic model.