In this paper we study the derivatives of Frobenius and the derivatives of Hodge-Tate weights for families of Galois representations with triangulations. We generalize the Fontaine-Mazur L-invariant and use it to build a formula which is a generalization of the Colmez-Greenberg-Stevens formula. For the purpose of proving this formula we show two auxiliary results called projection vanishing property and "projection vanishing implying L-invariants" property.

In this article, we investigate the shift of Abbes and Saito's ramification filtrations of the absolute Galois group of a complete discrete valuation field of positive characteristic under a purely inseparable extension. We also study a functoriality property for characteristic forms.

In the p-adic local Langlands correspondence for GL₂(Q_p), the following theorem of Berger and Breuil has played an important role:the locally algebraic representations of GL₂(Q_p) associated to crystabelline Galois representations admit a unique unitary completion. In this note, we give a new proof of the weaker statement that the locally algebraic representations admit at most one unitary completion and such a completion is automatically admissible. Our proof is purely representation theoretic, involving neither (φ, Γ)-module techniques nor global methods. When F is a finite extension of Q_p, we also get a simpler proof of a theorem of Vignéras for the existence of integral structures for (locally algebraic) special series and for (smooth) tamely ramified principal series.

We propose a conjecture on the relative twist formula of l-adic sheaves, which can be viewed as a generalization of Kato—Saito's conjecture. We verify this conjecture under some transversal assumptions. We also define a relative cohomological characteristic class and prove that its formation is compatible with proper push-forward. A conjectural relation is also given between the relative twist formula and the relative cohomological characteristic class.

Let X be a toric variety over a number field k with k[X]^×=k^×. Let W ⊂ X be a closed subset of codimension at least 2. We prove that X \ W satisfies strong approximation with algebraic Brauer-Manin obstruction.

We prove an anticyclotomic Iwasawa main conjecture proposed by Perrin-Riou for Heegner points for semi-stable elliptic curves E over a quadratic imaginary field K satisfying a certain generalized Heegner hypothesis, at an ordinary prime p. It states that the square of the index of the anticyclotomic family of Heegner points in E equals the characteristic ideal of the torsion part of its Bloch—Kato Selmer group (see Theorem 1.3 for precise statement). As a byproduct we also prove the equality in the Greenberg-Iwasawa main conjecture for certain Rankin-Selberg product (Theorem 1.7) under some local conditions, and an improvement of Skinner's result on a converse of Gross—Zagier and Kolyvagin theorem (Corollary 1.11).

We study the adjunction property of the Jacquet-Emerton functor in certain neighborhoods of critical points in the eigencurve. As an application, we construct two-variable p-adic L-functions around critical points via Emerton's representation theoretic approach.

Let F be a p-adic field of characteristic 0. We study a twisted local descent construction for the metaplectic groups Sp_2n(F), and also its relation to the corresponding local descent construction for odd special orthogonal groups via local theta correspondence. In consequence, we show that this descent construction gives irreducible supercuspidal genuine representations of Sp_2n(F) parametrized by a simple local L-parameter φτ corresponding to an irreducible supercuspidal representation τ of GL_2n(F) of symplectic type, and the genericity of the representations constructed can be indicated by a local epsilon factor condition. In particular, this local descent construction recovers the local Shimura correspondence for supercuspidal representations.

This article is the second article on the generalization of Kato's Euler system. The main subject of this article is to construct a family of Kato's Euler systems over the cuspidal eigencurve, which interpolate the Kato's Euler systems associated to the modular forms parametrized by the cuspidal eigencurve. We also explain how to use this family of Kato's Euler system to construct a family of distributions on Z_p over the cuspidal eigencurve; this distribution gives us a two-variable p-adic L function which interpolate the p-adic L function of modular forms.

This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers[Doc. Math., 21, 1607-1643 (2016)],[Taiwanese J. Math., 20(4), 723-741 (2016)], etc., the current authors and T. C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields. In this paper, we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number q. This establishes a key step that extends our previous explicit calculation of superspecial abelian surfaces to those of supersingular abelian surfaces. The second part is to introduce the notion of genera and idealcomplexes of abelian varieties with additional structures in a general setting. The purpose is to generalize the previous work by the second named author[Forum Math., 22(3), 565-582 (2010)] on abelian varieties with additional structures to similitude classes, which establishes more results on the connection between geometrically defined and arithmetically defined masses for further investigations.

Let L be a second order positive, elliptic differential operator that is self-adjoint with respect to some C^∞ density dx on a compact connected manifold M. We proved that if 0 < α < 1, α/2 < s < α and f ∈ H^s(M) then the fractional Schrödinger propagator e^itLα/2 on M satisfies e^itLα/2 f(x) -f(x)=o(t^s/α-ε) almost everywhere as t → 0⁺, for any ε > 0.

We characterize the complex differential equations of the form

where a_j(x) are meromorphic functions in the variable x for j=0,..., n that admit either a Weierstrass first integral or a Weierstrass inverse integrating factor.

Let X be a C¹⁺ vector field on a compact Riemannian manifold M with dimension d ≥ 3. Let Λ be a transitive singular hypebolic set with positive volume. We show that Λ=M and Λ is a uniformly hyperbolic set without singularities.

In the high-dimensional setting, this article considers a canonical testing problem in multivariate analysis, namely testing coefficients in linear regression models. Several tests for high-dimensional regression coefficients have been proposed in the recent literature. However, these tests are based on the sum of squares type statistics, that perform well under the dense alternatives and suffer from low power under the sparse alternatives. In order to attack this issue, we introduce a new test statistic which is based on the maximum type statistic and magnifies the sparse signals. The limiting null distribution of the test statistic is shown to be the extreme value distribution of type I and the power of the test is analysed. In particular, it is shown theoretically and numerically that the test is powerful against sparse alternatives. Numerical studies are carried out to examine the numerical performance of the test and to compare it with other tests available in the literature.

Let D_r:={z=x + iy ∈ C:|z|< r}, r ≤ 1. For a normalized analytic function f in the unit disk D:=D₁, estimating the Dirichlet integral Δ(r, f)=∫ ∫_Dr|f'(z)|² dxdy, z=x + iy, is an important classical problem in complex analysis. Geometrically, Δ(r, f) represents the area of the image of D_r under f counting multiplicities. In this paper, our main objective is to estimate areas of images of D_r under non-vanishing analytic functions of the form (z/f)^μ, μ > 0, in principal powers, when f ranges over certain classes of analytic and univalent functions in D.

In this paper, we study superlinear elliptic equations with mixed boundary value conditions in annular domains. It is assumed that the nonlinearities depend on the derivative terms. Some results about existence of solutions are established by using the Nehari manifold technique, as well as iterative technique.

In this paper, we give an equitable presentation for the multiparameter quantum group associated to a symmetrizable Kac-Moody Lie algebra, which can be regarded as a natural generalization of the Terwilliger's equitable presentation for the one-parameter quantum group.

It is a common issue to compare treatment-specific survival and the weighted log-rank test is the most popular method for group comparison. However, in observational studies, treatments and censoring times are usually not independent, which invalidates the weighted log-rank tests. In this paper, we propose adjusted weighted log-rank tests in the presence of non-random treatment assignment and dependent censoring. A double-inverse weighted technique is developed to adjust the weighted log-rank tests. Specifically, inverse probabilities of treatment and censoring weighting are involved to balance the baseline treatment assignment and to overcome dependent censoring, respectively. We derive the asymptotic distribution of the proposed adjusted tests under the null hypothesis, and propose a method to obtain the critical values. Simulation studies show that the adjusted log-rank tests have correct sizes whereas the traditional weighted log-rank tests may fail in the presence of non-random treatment assignment and dependent censoring. An application to oropharyngeal carcinoma data from the Radiation Therapy Oncology Group is provided for illustration.

The main purpose of this paper is to define prime and introduce non-commutative arithmetics based on Thompson's group F. Defining primes in a non-abelian monoid is highly non-trivial, which relies on a concept called "castling". Three types of castlings are essential to grasp the arithmetics. The divisor function τ on Thompson's monoid S satisfies τ(uv) ≤ τ(u)τ(v) for any u, v ∈ S. Then the limit τ₀(u)=lim_n→∞(τ(u_n))^1/n exists. The quantity Ç(S)=sup_1≠u∈S τ₀(u)/τ(u) describes the complexity for castlings in S. We show that Ç(S)=1. Moreover, the Möbius function on S is calculated. And the Liouville function Ω on S is studied.

In this paper, we consider composite quantile regression for partial functional linear regression model with polynomial spline approximation. Under some mild conditions, the convergence rates of the estimators and mean squared prediction error, and asymptotic normality of parameter vector are obtained. Simulation studies demonstrate that the proposed new estimation method is robust and works much better than the least-squares based method when there are outliers in the dataset or the random error follows heavy-tailed distributions. Finally, we apply the proposed methodology to a spectroscopic data sets to illustrate its usefulness in practice.

This paper is mainly concerned with the following nonlinear p-Laplacian equation
-Δ_pu(x) + (λa(x) + 1)|u|^p-2(x)u(x)=f(x, u(x)), in V
on a locally finite graph G=(V, E) with more general nonlinear term, where Δ_p is the discrete p-Laplacian on graphs, p ≥ 2. Under some suitable conditions on f and a(x), we can prove that the equation admits a positive solution by the Mountain Pass theorem and a ground state solution u_λ via the method of Nehari manifold, for any λ > 1. In addition, as λ → +∞, we prove that the solution u_λ converge to a solution of the following Dirichlet problem

where Ω={x ∈ V:a(x)=0} is the potential well and ∂Ω denotes the the boundary of Ω.

Lin-Lu-Yau introduced a notion of Ricci curvature for graphs and obtained a complete classification for all Ricci-flat graphs with girth at least five. In this paper, we characterize all Ricci-flat graphs of girth four with vertex-disjoint 4-cycles.

We consider the inverse curvature flows of smooth, closed and strictly convex rotation hypersurfaces in space forms M_κⁿ⁺¹ with speed function given by F^-α, where α ∈ (0, 1] for κ=0, -1, α=1 for κ=1 and F is a smooth, symmetric, strictly increasing and 1-homogeneous function of the principal curvatures of the evolving hypersurfaces. We show that the curvature pinching ratio of the evolving hypersurface is controlled by its initial value, and prove the long time existence and convergence of the flows. No second derivatives conditions are required on F.

In this paper, we study the regularity of the complex Hessian equation when the right hand has pole singularity. We show the H¨ older continuity of the solution to the Dirichlet problem. In particular, for the complex Monge-Amp` ere equation, we improve a result of[7].

In order to study the Yamabe changing-sign problem, Bahri and Xu proposed a conjecture which is a universal inequality for p points in R^m. They have verified the conjecture for p ≤ 3. In this paper, we first simplify this conjecture by giving two sufficient and necessary conditions inductively. Then we prove the conjecture for the basic case m=1 with arbitrary p. In addition, for the cases when p=4, 5 and m ≥ 2, we manage to reduce them to the basic case m=1 and thus prove them as well.

In this paper we will show how the Bohr-Sommerfeld levels of a quantum completely integrable system can be computed modulo O(?^∞) by an inductive procedure starting at stage zero with the Bohr-Sommerfeld levels of the corresponding classical completely integrable system.

In this paper, we consider Ricci flow on four dimensional closed manifold with bounded scalar curvature, noncollasping volume and bounded diameter. Under such conditions, we can show that the manifold has finitely many diffeomorphism types, which generalizes Cheeger-Naber's result to bounded scalar curvature along Ricci flow. In particular, this implies the manifold has uniform L² Riemann curvature bound. As an application, we point out that four dimensional Ricci flow would not have uniform scalar curvature upper bound if the initial metric only satisfying lower Ricci curvature bound, lower volume bound and upper diameter bound.

In this paper, we prove gradient estimates for the positive solutions of Lu=0 and Lu=∂u/∂t on conformal solitons, where L(·)=Δ(·) +

In this paper, the curve shortening flow in a general Riemannian manifold is studied, Altschuler's results about the flow for space curves are generalized. For any n-dimensional (n ≥ 2) Riemannian manifold (M, g) with some natural assumptions, we prove the planar phenomenon when the curve shortening flow blows up.

This short note is concerned with a measure version criterion for hypersurfaces to be minimal. Certain natural flows and associated reflections for many minimal hypercones, including minimal isoparametric hypercones and area-minimizing hypercones, are studied.

We, in this paper, investigate two-sample quantile difference by empirical likelihood method when the responses with high-dimensional covariates of the two populations are missing at random. In particular, based on sufficient dimension reduction technique, we construct three empirical loglikelihood ratios for the quantile difference between two samples by using inverse probability weighting imputation, regression imputation as well as augmented inverse probability weighting imputation, respectively, and prove their asymptotic distributions. At the same time, we give a test to check whether two populations have the same distribution. A simulation study is carried out to investigate finite sample behavior of the proposed methods too.

Let X={X(t) ∈ R^d, t ∈ R^N } be a centered space-time anisotropic Gaussian random field whose components are independent and satisfy some mild conditions. We study the packing dimension of range X(E) under the anisotropic (time variable) metric space (R^N, ρ) and (space variable) metric space (R^d, τ), where E ⊂ R^N is a Borel set. Our results generalize the corresponding results of Estrade, Wu and Xiao (Commun. Stoch. Anal., 5, 41-64 (2011)) for time-anisotropic Gaussian random fields to space-time anisotropic Gaussian fields.

Let K be a genus g alternating knot with Alexander polynomial Δ_K(T)=Σ_i=-g^g a_iTⁱ. We show that if|a_g|=|a_g-1|, then K is the torus knot T_2g+1,±2. This is a special case of the Fox Trapezoidal Conjecture. The proof uses Ozsváth and Szabó's work on alternating knots.

Let (X, T) be a weakly mixing minimal system, p₁,..., p_d be integer-valued generalized polynomials and (p₁, p₂,..., p_d) be non-degenerate. Then there exists a residual subset X₀ of X such that for all x ∈ X₀, {(T ^p₁(n)x,..., T^p_d(n)x):n ∈ Z} is dense in X^d.

This paper is concerned with the following Chern-Simons-Schrödinger equation -Δu + V (|x|)u +(∫_|x|^∞h(s)/su²(s)ds + h²(|x|)/|x|²)u=a(|x|)f(u) in R², where h(s)=∫₀^sl/2u²(l)dl, V, a:R⁺ → R are radially symmetric potentials and the nonlinearity f:R → R is of subcritical or critical exponential growth in the sense of Trudinger-Moser. We give some new sufficient conditions on f to obtain the existence of nontrivial solutions or ground state solutions. In particular, some new estimates and techniques are used to overcome the difficulty arising from the critical growth of Trudinger-Moser type.

The purpose of the present paper is to deliberate ∗-conformal Yamabe soliton, whose potential vector field is torse-forming on Kenmotsu manifold. Here, we have shown the nature of the soliton and find the scalar curvature when the manifold admitting ∗-conformal Yamabe soliton on Kenmotsu manifold. Next, we have evolved the characterization of the vector field when the manifold satisfies ∗-conformal Yamabe soliton. Also we have embellished some applications of vector field as torse-forming in terms of ∗-conformal Yamabe soliton on Kenmotsu manifold. We have developed an example of ∗-conformal Yamabe soliton on 3-dimensional Kenmotsu manifold to prove our findings.

Let T be a strongly singular Calderón-Zygmund operator and b ∈ L_loc(Rⁿ). This article finds out a class of non-trivial subspaces BMOω,p,u(Rⁿ) of BMO(Rⁿ) for certain ω ∈ A₁, 0 < p ≤ 1 and 1 < u ≤ ∞, such that the commutator[b, T] is bounded from weighted Hardy space H_ω^p(Rⁿ) to weighted Lebesgue space L_ω^p(Rⁿ) if b ∈ BMOω,p,∞(Rⁿ), and is bounded from weighted Hardy space H_ω^p(Rⁿ) to itself if T^∗1=0 and b ∈ BMOω,p,u(Rⁿ) for 1 < u < 2.

In this paper, we give some sufficient conditions for extended neighborhood coronas to have Laplacian perfect state transfer. We also give some conditions for extended neighborhood coronas to be periodic.

We study several stronger versions of sensitivity for minimal group actions, including n-sensitivity, thick n-sensitivity and blockily thick n-sensitivity, and characterize them by the regionally proximal relation.

In this article, we showed that positive relative entropy implies mean Li-Yorke chaos along some good sequences in the fiber. Our result extended Li and Qiao's result in[Monatsh. Math., 186 (2018), 153-173].