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Abstract: Abstract We prove the positive energy conjecture for a class of asymptotically Horowitz-Myers (AHM) metrics on \({\mathbb{R}^2} \times {\mathbb{T}^{n - 2}}\) . This generalizes the previous results of Barzegar et al. (2020) as well as Liang and Zhang (2020). PubDate: 2022-09-19

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Abstract: Abstract When solving differential equations by a spectral method, it is often convenient to shift from Chebyshev polynomials Tn(x) with coefficients an to modified basis functions that incorporate the boundary conditions. For homogeneous Dirichlet boundary conditions, u(±1) = 0, popular choices include the “Chebyshev difference basis” ςn(x) ≡ Tn+2(x) − Tn(x) with coefficients here denoted by bn and the “quadratic factor basis functions” ϱn(x) ≡ (1 − x2)Tn(x) with coefficients cn. If u(x) is weakly singular at the boundaries, then the coefficients an decrease proportionally to \({\cal O}\left( {A\left( n \right)/{n^\kappa }} \right)\) for some positive constant κ, where A(n) is a logarithm or a constant. We prove that the Chebyshev difference coefficients bn decrease more slowly by a factor of 1/n while the quadratic factor coefficients cn decrease more slowly still as \({\cal O}\left( {A\left( n \right)/{n^{\kappa - 2}}} \right)\) . The error for the unconstrained Chebyshev series, truncated at degree n = N, is \({\cal O}\left( {\left {A\left( N \right)} \right /{N^\kappa }} \right)\) in the interior, but is worse by one power of N in narrow boundary layers near each of the endpoints. Despite having nearly identical error norms, the error in the Chebyshev basis is concentrated in boundary layers near both endpoints, whereas the error in the quadratic factor and difference basis sets is nearly uniform oscillations over the entire interval in x. Meanwhile, for Chebyshev polynomials and the quadratic factor basis, the value of the derivatives at the endpoints is \({\cal O}\left( {{N^2}} \right)\) , but only \({\cal O}\left( N \right)\) for the difference basis. Furthermore, we give the asymptotic coefficients and rigorous error estimates of the approximations in these three bases, solved by the least squares method. We also find an interesting fact that on the face of it, the aliasing error is regarded as a bad thing, actually, the error norm associated with the downward curving spectral coefficients decreases even faster than the error norm of infinite truncation. But the premise is contrasted under the same basis, and when involving different bases, it may not yet establish. PubDate: 2022-09-08

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Abstract: Abstract For a compact operator tuple A, if its projective spectrum P(A*) is smooth, there exists a natural Hermitian holomorphic line bundle EA over P(A*) which is a unitary invariant for A. This paper shows that under some additional spectral conditions, EA is a complete unitary invariant, i.e., EA can determine the compact operator tuple up to unitary equivalence. PubDate: 2022-09-07

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Abstract: Abstract This paper deals with regularity properties for minimizing sequences of some integral functionals related to the nonlinear elasticity theory. Under some structural conditions, we derive that the minimizing sequence and the derivatives of the sequences have some regularity properties by using the Ekeland variational principle. PubDate: 2022-09-02

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Abstract: Abstract We give an elementary proof to the asymptotic expansion formula of Rochon and Zhang (2012) for the unique complete Kähler-Einstein metric of Cheng and Yau (1980), Kobayashi (1984), Tian and Yau (1987) and Bando (1990) on quasi-projective manifolds. The main tools are the solution formula for second-order ordinary differential equations (ODEs) with constant coefficients and spectral theory for the Laplacian operator on a closed manifold. PubDate: 2022-09-01

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Abstract: Abstract In this paper, I introduce a new generalization of the concept of an operad, further generalizing the concept of an opetope introduced by Baez and Dolan (1998), who used this for the definition of their version of non-strict n-categories. Opetopes arise from iterating a certain construction on operads called the +-construction, starting with monoids. The first step gives rise to plain operads, i.e., operads without symmetries. The permutation axiom in a symmetric operad, however, is an additional structure resulting from permutations of variables, independent of the structure of a monoid. Even though we can apply the +-construction to symmetric operads, there is the possibility of introducing a completely different kind of permutations on the higher levels by again permuting variables without regard to the structures on the previous levels. Defining and investigating these structures is the main purpose of this paper. The structures obtained in this way are what I call n-actads. In n-actads with n > 1, the permutations on the different levels give rise to a certain special kind of n-fold category. I also explore the concept of iterated algebras over an n-actad (generalizing an algebra and a module over an operad), and various types of iterated units. I give some examples of algebras over 2-actads, and show how they can be used to construct certain new interesting homotopy types of operads. I also discuss a connection between actads and ordinal notation. PubDate: 2022-09-01

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Abstract: Abstract Let {T(t)}t⩾0 be a C0-semigroup on an infinite-dimensional separable Hilbert space; a suitable definition of near {T(t)*}t⩾0 invariance of a subspace is presented in this paper. A series of prototypical examples for minimal nearly {S(t)*}t⩾0 invariant subspaces for the shift semigroup {S(t)}t⩾0 on L2(0, ∞) are demonstrated, which have close links with near \(T_\theta^*\) invariance on Hardy spaces of the unit disk for an inner function θ. Especially, the corresponding subspaces on Hardy spaces of the right half-plane and the unit disk are related to model spaces. This work further includes a discussion on the structure of the closure of certain subspaces related to model spaces in Hardy spaces. PubDate: 2022-09-01

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Abstract: Abstract This work is concerned with the time-fractional doubly parabolic Keller-Segel system in ℝN (N ≽ 1), and we derive some refined results on the large time behavior of solutions which are presupposed to enjoy some uniform boundedness properties. Moreover, the well-posedness and the asymptotic stability of solutions in Marcinkiewicz spaces are studied. The results are achieved by means of an appropriate estimation of the system nonlinearities in the course of an analysis based on Duhamel-type representation formulae and the Kato-Fujita framework which consists in constructing a fixed-point argument by using a suitable time-dependent space. PubDate: 2022-09-01

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Abstract: Abstract Let X be a projective manifold and let {θ}∈ H1,1(X, ℝ) be a nonzero pseudo-effective (transcendental) class, where θ is a smooth closed real (1, 1)-form. We prove that if for any one-dimensional complex submanifold C ⊂ X and ϕ ∈ SPsh(C, θ∣C) with a single analytic singularity at some point p ∈ C, there exists a function \(\tilde \varphi \in {\rm{Psh}}(X,\theta )\) such that \(\tilde \varphi \left {_C = \varphi } \right.\) and \({\tilde \varphi }\) is continuous at points of C {p}, then {θ} is a Kähler class. PubDate: 2022-09-01

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Abstract: Abstract The first aim of this article is to study the sharp singular (two-weight) Trudinger-Moser inequalities with Finsler norms on ℝ2. The second goal is to propose a different approach to study a vanishing-concentration-compactness principle for the Trudinger-Moser inequalities and use this to investigate the existence and the nonexistence of the maximizers for the Trudinger-Moser inequalities in the subcritical regions. Finally, by applying our Finsler Trudinger-Moser inequalities to suitable Finsler norms, we derive the sharp affine Trudinger-Moser inequalities which are essentially stronger than the Trudinger-Moser inequalities with standard energy of the gradient. PubDate: 2022-09-01

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Abstract: Abstract In this paper, we consider the asymptotic dynamics of the skew-product semiflow generated by the following time almost periodically-forced scalar reaction-diffusion equation: \({u_t} = {u_{xx}} + f(t,u,{u_x}),\,\,\,\,\,t > 0,\,\,\,\,\,0 < x < L\) with the periodic boundary condition \(u(t,0) = u(t,L),\,\,\,\,\,\,\,{u_x}(t,0) = {u_x}(t,L),\) where f is uniformly almost periodic in t. In particular, we study the topological structure of the limit sets of the skew-product semiflow. It is proved that any compact minimal invariant set (throughout this paper, we refer to it as a minimal set) can be residually embedded into an invariant set of some almost automorphically-forced flow on a circle S1 = ℝ/Lℤ (see Definition 2.4 for “residually embedded”). Particularly, if f (t,u,p) = f(t,u, −p), then the flow on a minimal set can be embedded into an almost periodically-forced minimal flow on ℝ (see Definition 2.4 for “embedded”). Moreover, it is proved that the ω-limit set of any bounded orbit contains at most two minimal sets that cannot be obtained from each other by phase translation. In addition, we further consider the asymptotic dynamics of the skew-product semiflow generated by (0.1) with the Neumann boundary condition ux(t, 0) = ux(t, L) = 0 or the Dirichlet boundary condition u(t, 0) = u(t, L) = 0. For such a system, it has been known that the ω-limit set of any bounded orbit contains at most two minimal sets. By applying the new results for (0.1) + (0.2), under certain direct assumptions on f, we prove in this paper that the flow on any minimal set of (0.1) with the Neumann boundary condition or the Dirichlet boundary condition can be embedded into an almost periodically-forced minimal flow on ℝ. Finally, a counterexample is given to show that even for quasi-periodically-forced equations, the results we obtain here cannot be further improved in general. PubDate: 2022-09-01

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Abstract: Abstract We are concerned with partial dimension reduction for the conditional mean function in the presence of controlling variables. We suggest a profile least square approach to perform partial dimension reduction for a general class of semi-parametric models. The asymptotic properties of the resulting estimates for the central partial mean subspace and the mean function are provided. In addition, a Wald-type test is proposed to evaluate a linear hypothesis of the central partial mean subspace, and a generalized likelihood ratio test is constructed to check whether the nonparametric mean function has a specific parametric form. These tests can be used to evaluate whether there exist interactions between the covariates and the controlling variables, and if any, in what form. A Bayesian information criterion (BIC)-type criterion is applied to determine the structural dimension of the central partial mean subspace. Its consistency is also established. Numerical studies through simulations and real data examples are conducted to demonstrate the power and utility of the proposed semi-parametric approaches. PubDate: 2022-09-01

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Abstract: Abstract Ring epimorphisms often induce silting modules and cosilting modules, termed minimal silting or minimal cosilting. The aim of this paper is twofold. Firstly, we determine the minimal tilting and minimal cotilting modules over a tame hereditary algebra. In particular, we show that a large cotilting module is minimal if and only if it has an adic module as a direct summand. Secondly, we discuss the behavior of minimality under ring extensions. We show that minimal cosilting modules over a commutative noetherian ring extend to minimal cosilting modules along any flat ring epimorphism. Similar results are obtained for commutative rings of small homological dimension. PubDate: 2022-09-01

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Abstract: Abstract Principal component analysis (PCA) has been widely used in analyzing high-dimensional data. It converts a set of observed data points of possibly correlated variables into a set of linearly uncorrelated variables via an orthogonal transformation. To handle streaming data and reduce the complexities of PCA, (subspace) online PCA iterations were proposed to iteratively update the orthogonal transformation by taking one observed data point at a time. Existing works on the convergence of (subspace) online PCA iterations mostly focus on the case where the samples are almost surely uniformly bounded. In this paper, we analyze the convergence of a subspace online PCA iteration under more practical assumption and obtain a nearly optimal finite-sample error bound. Our convergence rate almost matches the minimax information lower bound. We prove that the convergence is nearly global in the sense that the subspace online PCA iteration is convergent with high probability for random initial guesses. This work also leads to a simpler proof of the recent work on analyzing online PCA for the first principal component only. PubDate: 2022-08-30

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Abstract: Abstract We consider the following fractional Schrödinger equation: (0.1) $${\left( { - \Delta } \right)^s}u + V\left( y \right)u = {u^p},\,\,\,\,\,\,u > 0\,\,\,\,{\rm{in}}\,\,{\mathbb{R}^N},$$ where s ∈ (0, 1), \(1 < p < {{N + 2s} \over {N - 2s}}\) , and V(y) is a positive potential function and satisfies some expansion condition at infinity. Under the Lyapunov-Schmidt reduction framework, we construct two kinds of multi-spike solutions for (0.1). The first k-spike solution uk is concentrated at the vertices of the regular k-polygon in the (y1, y2)-plane with k and the radius large enough. Then we show that uk is non-degenerate in our special symmetric workspace, and glue it with an n-spike solution, whose centers lie in another circle in the (y3, y4)-plane, to construct infinitely many multi-spike solutions of new type. The nonlocal property of (−Δ)s is in sharp contrast to the classical Schrödinger equations. A striking difference is that although the nonlinear exponent in (0.1) is Sobolev-subcritical, the algebraic (not exponential) decay at infinity of the ground states makes the estimates more subtle and difficult to control. Moreover, due to the non-locality of the fractional operator, we cannot establish the local Pohozaev identities for the solution u directly, but we address its corresponding harmonic extension at the same time. Finally, to construct new solutions we need pointwise estimates of new approximation solutions. To this end, we introduce a special weighted norm, and give the proof in quite a different way. PubDate: 2022-08-26

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Abstract: Abstract In this paper, we first establish the existence of blow-up solutions with two antipodal points to the fourth order mean field equations on \(\mathbb{S}^{4}\) . Moreover, we construct non-axially symmetric solutions with blow-up points at the vertices of regular configurations, i.e., equilateral triangles on a great circle, regular tetrahedrons, cubes, octahedrons, icosahedrons and dodecahedrons. The bubbling rates of these blow-up solutions rely on various bubbling configurations. PubDate: 2022-08-23

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Abstract: Abstract Tang and Zhang (2020) and Choe and Hoppe (2018) showed independently that one can produce minimal submanifolds in spheres via the Clifford type minimal product of minimal submanifolds. In this paper, we show that the minimal product is immersed by its first eigenfunctions (of its Laplacian) if and only if the two beginning minimal submanifolds are immersed by their first eigenfunctions. Moreover, we give the estimates of the Morse index and the nullity of the minimal product. In particular, we show that the Clifford minimal submanifold \((\sqrt {{{{n_1}} \over n}} {S^{{n_1}}}, \ldots ,\sqrt {{{{n_k}} \over n}} {S^{{n_k}}}) \subset {S^{n + k - 1}}\) has the index (k − 1)(n + k + 1) and the nullity (k − 1) ∑1⩽i<j⩽k(ni + 1)(nj + 1) (with n = ∑nj). PubDate: 2022-08-18

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Abstract: Abstract In this paper, we study the following critical elliptic problem with a variable exponent: $$\left\{ {\matrix{{ - \Delta u = {u^{p + \epsilon a\left( x \right)}}} \hfill & {{\rm{in}}\,\,\Omega ,} \hfill \cr {u > 0} \hfill & {{\rm{in}}\,\,\Omega ,} \hfill \cr {u = 0} \hfill & {{\rm{on}}\,\partial \Omega ,} \hfill \cr } } \right.$$ where \(a\left( x \right) \in {C^2}\left( {\overline \Omega } \right),\,p = {{N + 2} \over {N - 2}},\,\,\epsilon > 0\) , and Ω is a smooth bounded domain in ℝN (N ≽ 4). We show that for ∊ small enough, there exists a family of bubble solutions concentrating at the negative stable critical point of the function a(x). This is a new perturbation to the critical elliptic equation in contrast to the usual subcritical or supercritical perturbation, and gives the first existence result for the critical elliptic problem with a variable exponent. PubDate: 2022-08-17

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Abstract: Abstract For a closed hypersurface Mn ⊂ Sn+1(1) with constant mean curvature and constant non-negative scalar curvature, we show that if \({\rm{tr}}\left({{{\cal A}^k}} \right)\) are constants for k = 3, …, n − 1 and the shape operator \({\cal A}\) then M is isoparametric. The result generalizes the theorem of de Almeida and Brito (1990) for n = 3 to any dimension n, strongly supporting the Chern conjecture. PubDate: 2022-08-05

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Abstract: Abstract Fisher matrix is one of the most important statistics in multivariate statistical analysis. Its eigenvalues are of primary importance for many applications, such as testing the equality of mean vectors, testing the equality of covariance matrices and signal detection problems. In this paper, we establish the limiting spectral distribution of high-dimensional noncentral Fisher matrices and investigate its analytic behavior. In particular, we show the determination criterion for the support of the limiting spectral distribution of the noncentral Fisher matrices, which is the base of investigating the high-dimensional problems concerned with noncentral Fisher matrices. PubDate: 2022-08-04