Angular Regularity Research Articles

Strichartz estimates for the 2D and 3D massless Dirac-Coulomb equations and applications

In this paper we continue the analysis of the dispersive properties of the 2D and 3D massless Dirac-Coulomb equations that has been started in [7] and [8]. We prove a priori estimates of the solution of the mentioned systems, in particular Strichartz estimates with an additional angular regularity, exploiting the tools developed in the previous works. As an application, we show local well-posedness results for a Dirac-Coulomb equation perturbed with Hartree-type nonlinearities.

Randomized final-state problem for the Zakharov system in dimension three

We consider the final-state problem for the Zakharov system in the energy space in three space dimensions. For without any size restriction, symmetry assumption or additional angular regularity, we perform a physical-space randomization on and an angular randomization on yielding random final states We obtain that for almost every ω, there is a unique solution of the Zakharov system scattering to the final state The key ingredient in the proof is the use of time-weighted norms and generalized Strichartz estimates which are accessible due to the randomization.

Angular regularization for unsupervised domain adaption on person re-identification

Angular regularization for unsupervised domain adaption on person re-identification

Small Data Scattering of Hartree Type Fractional Schrödinger Equations in a Scaling Critical Space

In this paper we study the small-data scattering of Hartree type fractional Schrödinger equations in space dimension 2, 3. It has Lévy index α between 1 and 2, and Hartree type nonlinearity F(u) = μ(|x|-γ * |u|2)u with 2d/(2d-1) < γ < 2, γ ≥ α > 1. This equation is scaling-critical in Ḣsc with sc = (γ-α)/2. We show that the solution scatters in Ḣsc,1, where Ḣsc,1 is also a scaling critical space of Sobolev type taking in angular regularity with norm defined by ||ϕ||Ḣsc,1 = ||ϕ||Ḣsc + ||∇Sϕ||Ḣsc. For this purpose we use the recently developed Strichartz estimate which is Lθ2-averaged on the unit sphere Sd-1.

M58. EVALUATING WEIGHTINGS OF PREDICTIVE AND POSTDICTIVE PROCESSES IN ABERRANT SENSE OF AGENCY IN SCHIZOPHRENIA

BackgroundSense of agency (SoA) refers to the subjective feeling that one is the agent who causes and controls one’s own actions and those effects on the external world. It’s widely known that patients with schizophrenia show aberrant SoA, but its cognitive and neural mechanisms remain unclear. Previous studies suggest that abnormal SoA is probably due to impaired predictive processes, such as predicting sensory feedbacks according to one’s own motor intention or motor commands. However, dysfunction of the prediction process alone cannot explain various symptoms of schizophrenia. The impairment in postdictive processes in patients with schizophrenia is also considered to be important to explain their symptoms. However, how the predictive and postdictive processes actually contribute to the judgment of agency in patients with schizophrenia remains unknown. In the present study, we focus on the functional contribution of the two processes at sensorimotor level to the judgment of agency at the cognitive level, rather than the performance of these functions. Our approach allows us to evaluate the weightings of each contribution quantitatively, and to associate such weightings to symptoms for each individual. The results of our study point to potential useful diagnosis, and can provide important knowledge for treatment in future research.MethodsWe use two sensorimotor tasks (i.e. the reaching task and the control detection task) and one explicit agency judgment task in this study. In all tasks, participants move one or several dots on a screen by moving a mouse. The speed of the dots’ motion always corresponds to participants’ mouse movement, but the direction of the dots’ motion is combined with both participants’ motion and prerecorded motion, and is applied with a certain angular bias. In the reaching task, participants move the dot to touch a destination on screen as quickly as possible. This task is designed to measure the extent of prediction errors at the sensorimotor level, which is considered as an important signal for the predictive process underlying agency judgment. In the control detection task, participants move three dots simultaneously, and identified one dot that can be partially controlled by their mouse movement, while the other two dots move in pre-recorded directions. This task does not require explicit judgment of agency. It measures the postdictive process that to detect the regularity between the visual feedbacks and one’s own movements at the sensorimotor level. At last, in the explicit agency judgment task, participants made binary judgment (i.e. yes or no) regarding whether they feel that they controlled the moving direction of the dot or not in each condition. In summary, the two sensorimotor tasks measure the output of predictive and postdictive processes at the sensorimotor level, respectively, and the explicit agency judgment task measures agency judgment at the cognitive level. We designed six experimental conditions, in which the actual angular error and regularity were systematically manipulate. Such manipulation creates variability in the performance of each task, and allows us to use structure equation modelling to analyze the contributions of the performance in the two sensorimotor tasks to the explicit agency judgment task, and to compare the results with healthy controls. We focus on the functional structure of the sensorimotor sub-component and the cognitive judgment of SoA.ResultsThe data collection is currently in progress. We plan to finish it by February.DiscussionWe predict weakened contributions from the two sub-components to SoA in patients, compared to healthy control. We plan to link the quantitatively estimated weightings to individual symptoms including both positive and negative symptoms.

Viewpoint-Aware Loss with Angular Regularization for Person Re-Identification

Although great progress in supervised person re-identification (Re-ID) has been made recently, due to the viewpoint variation of a person, Re-ID remains a massive visual challenge. Most existing viewpoint-based person Re-ID methods project images from each viewpoint into separated and unrelated sub-feature spaces. They only model the identity-level distribution inside an individual viewpoint but ignore the underlying relationship between different viewpoints. To address this problem, we propose a novel approach, called Viewpoint-Aware Loss with Angular Regularization (VA-reID). Instead of one subspace for each viewpoint, our method projects the feature from different viewpoints into a unified hypersphere and effectively models the feature distribution on both the identity-level and the viewpoint-level. In addition, rather than modeling different viewpoints as hard labels used for conventional viewpoint classification, we introduce viewpoint-aware adaptive label smoothing regularization (VALSR) that assigns the adaptive soft label to feature representation. VALSR can effectively solve the ambiguity of the viewpoint cluster label assignment. Extensive experiments on the Market1501 and DukeMTMC-reID datasets demonstrated that our method outperforms the state-of-the-art supervised Re-ID methods.

Small data scattering of the inhomogeneous cubic–quintic NLS in 2 dimensions

The aim of this paper is to show the small data scattering for 2D ICQNLS: iut=−Δu+K1(x)|u|2u+K2(x)|u|4u.Under the assumption that |∂jKl|≲|x|bl−j for j=0,1,2,l=1,2 and 0≤bl≤l−23, we prove the small data scattering in an angularly regular Sobolev space Hθ1,1. We use the decaying property of angularly regular functions, which are defined as functions in Sobolev space Hθ1,1⊂H1 with angular regularity such that ‖∂θf‖H1<∞, and also use the recently developed angularly averaged Strichartz estimates (Tao, 2000; Cho and Lee, 2013; Guo et al., 2018). In addition, we suggest a sufficient condition for non-existence of scattering.

Linear adjoint restriction estimates for paraboloid

We prove a class of modified paraboloid restriction estimates with a loss of angular derivatives for the full set of paraboloid restriction conjecture indices. This result generalizes the paraboloid restriction estimate in radial case from [Shao, Rev. Mat. Iberoam. 25(2009), 1127–1168], as well as the result from [Miao et al. Proc. AMS 140(2012), 2091–2102]. As an application, we show a local smoothing estimate for a solution of the linear Schrödinger equation under the assumption that the initial datum has additional angular regularity.

Small data scattering of semirelativistic Hartree equation

In this paper we study the small data scattering of Hartree type semirelativistic equation in space dimension 3. The Hartree type nonlinearity is [V∗|u|2]u and the potential V which generalizes the Yukawa has some growth condition. We show that the solution scatters to linear solution if an initial data given in Hs,1 is sufficiently small and s>14. Here, Hs,1 is Sobolev type space taking in angular regularity with norm defined by ‖φ‖Hs,1=‖φ‖Hs+‖∇Sφ‖Hs. To establish the results we employ the recently developed Strichartz estimate which is Lθ2-averaged on the unit sphere S2 and construct the resolution space based on Up-Vp space.

Transference of bilinear restriction estimates to quadratic variation norms and the Dirac–Klein–Gordon system

Firstly, bilinear Fourier Restriction estimates --which are well-known for free waves-- are extended to adapted spaces of functions of bounded quadratic variation, under quantitative assumptions on the phase functions. This has applications to nonlinear dispersive equations, in particular in the presence of resonances. Secondly, critical global well-posedness and scattering results for massive Dirac-Klein-Gordon systems in dimension three are obtained, in resonant as well as in non-resonant regimes. The results apply to small initial data in scale-invariant Sobolev spaces exhibiting a small amount of angular regularity.

Scattering for the 3D Gross–Pitaevskii Equation

We study the Cauchy problem for the 3D Gross-Pitaevskii equation. The global well-posedness in the natural energy space was proved by G\'erard \cite{Gerard}. In this paper we prove scattering for small data in the same space with some additional angular regularity, and in particular in the radial case we obtain small energy scattering.

On the cubic Dirac equation with potential and the Lochak–Majorana condition

We study a cubic Dirac equation on R×R3i∂tu+Du+V(x)u=〈βu,u〉βu perturbed by a large potential with almost critical regularity. We prove global existence and scattering for small initial data in H1 with additional angular regularity. The main tool is an endpoint Strichartz estimate for the perturbed Dirac flow. In particular, the result covers the case of spherically symmetric data with small H1 norm. When the potential V has a suitable structure, we prove global existence and scattering for large initial data having a small chiral component, related to the Lochak–Majorana condition.

On the Conjecture of Lehmer, Limit Mahler Measure of Trinomials and Asymptotic Expansions

AbstractLetn ≥2 be an integer and denote byθnthe real root in (0, 1) of the trinomialGn(X) = −1 +X+Xn. The sequence of Perron numbers(θn−1)n≥2$(\theta _n^{ - 1} )_{n \ge 2} $tends to 1. We prove that the Conjecture of Lehmer is true for{θn−1|n≥2}$\{ \theta _n^{ - 1} |n \ge 2\} $by the direct method of Poincaré asymptotic expansions (divergent formal series of functions) of the rootsθn,zj,n, ofGn(X) lying in|z|&lt; 1, as a function ofn, jonly. This method, not yet applied to Lehmer’s problem up to the knowledge of the author, is successfully introduced here. It first gives the asymptotic expansion of the Mahler measuresM(Gn)=M(θn)=M(θn−1)${\rm{M}}(G_n ) = {\rm{M}}(\theta _n ) = {\rm{M}}(\theta _n^{ - 1} )$of the trinomialsGnas a function ofnonly, without invoking Smyth’s Theorem, and their unique limit point above the smallest Pisot number. Comparison is made with Smyth’s, Boyd’s and Flammang’s previous results. By this method we obtain a direct proof that the conjecture of Schinzel-Zassenhaus is true for{θn−1|n≥2}$\{ \theta _n^{ - 1} |n \ge 2\} $, with a minoration of the house, and a minoration of the Mahler measure M(Gn) better than Dobrowolski’s one. The angular regularity of the roots ofGn, near the unit circle, and limit equidistribution of the conjugates, forntending to infinity (in the sense of Bilu, Petsche, Pritsker), towards the Haar measure on the unit circle, are described in the context of the Erdős-Turán-Amoroso-Mignotte theory, with uniformly bounded discrepancy functions.

Sharp spherically averaged Strichartz estimates for the Schrödinger equation

We prove generalized Strichartz estimates with weaker angular integrability for the Schrödinger equation. Our estimates are sharp except some endpoints. Then we apply these new estimates to prove scattering for the 3D Zakharov system with small data in the energy space with low angular regularity. Our results improve the results obtained recently in Guo Z et al (2014 Generalized Strichartz estimates and scattering for 3D Zakharov system Commun. Math. Phys. 331 239–59).

Spherically Averaged Maximal Function and Scattering for the Two-Dimensional Cubic Derivative Schrödinger Equation

We prove scattering for the 2D cubic derivative Schr\"odinger equation with small data in the critical Besov space with one degree angular regularity. The main new ingredient is that we prove a spherically averaged maximal function estimate for the 2D Schr\"odinger equation. We also prove a global well-posedness result for the 2D Schr\"odinger map in the critical Besov space with one degree angular regularity. The key ingredients for the latter results are the spherically averaged maximal function estimate, null form structure observed in \cite{Bej}, as well as the generalised spherically averaged Strichartz estimates obtained in \cite{Guo2} in order to exploit the null form structure.

Optimal Forward and Reverse Estimates of Morawetz and Kato–Yajima Type with Angular Smoothing Index

For the solution of the free Schrodinger equation, we obtain the optimal constants and characterise extremisers for forward and reverse smoothing estimates which are global in space and time, contain a homogeneous and radial weight in the space variable, and incorporate a certain angular regularity. This will follow from a more general result which permits analogous sharp forward and reverse smoothing estimates and a characterisation of extremisers for the solution of the free Klein–Gordon and wave equations. The nature of extremisers is shown to be sensitive to both the dimension and the size of the smoothing index relative to the dimension. Furthermore, in four spatial dimensions and certain special values of the smoothing index, we obtain an exact identity for each of these evolution equations.

Generalized Strichartz Estimates and Scattering for 3D Zakharov System

We obtain scattering for the 3D Zakharov system with non-radial small data in the energy space with angular regularity of degree one. The main ingredient is a generalized Strichartz estimate for the Schrödinger equation in the space of L 2 angular integrability.

Profile decompositions of fractional Schrödinger equations with angularly regular data

We study the fractional Schrödinger equations in R1+d, d⩾3, of order d/(d−1)<α<2. Under the angular regularity assumption we prove linear and nonlinear profile decompositions which extend the previous results [9] to data without radial assumption. As applications we show blowup phenomena of solutions to mass-critical fractional Hartree equations.

STRICHARTZ ESTIMATES FOR WAVE EQUATION WITH INVERSE SQUARE POTENTIAL

In this paper, we study the Strichartz-type estimates of the solution for the linear wave equation with inverse square potential. Assuming the initial data possesses additional angular regularity, especially the radial initial data, the range of admissible pairs is improved. As an application, we show the global well-posedness of the semi-linear wave equation with inverse-square potential [Formula: see text] for power p being in some regime when the initial data are radial. This result extends the well-posedness result in Planchon, Stalker, and Tahvildar-Zadeh.

Almost Global Existence for Some Semilinear Wave Equations with Almost Critical Regularity

For any subcritical index of regularity s > 3/2, we prove the almost global well posedness for the 2-dimensional semilinear wave equation with the cubic nonlinearity in the derivatives, when the initial data are small in the Sobolev space H s × H s−1 with certain angular regularity. The lifespan is known to be sharp in general. The main new ingredient in the proof is an endpoint version of the generalized Strichartz estimates in the space . In the last section, we also consider the general semilinear wave equations with the spatial dimension n ≥ 2 and the order of nonlinearity p ≥ 3.