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Abstract: For a number field F and a prime number p, the ℤp-torsion module of the Galois group of the maximal abelian pro-p extension of F unramified outside p over F, denoted by \({{\cal T}_p}(F)\) , is an important subject in abelian p-ramification theory. In this paper, we study the group \({{\cal T}_2}(F) = {{\cal T}_2}(m)\) of the quadratic field \(F = \mathbb{Q}(\sqrt m )\) . Firstly, assuming m > 0, we prove an explicit 4-rank formula for quadratic fields that \({\rm{r}}{{\rm{k}}_4}({{\cal T}_2}( - m)) = {\rm{r}}{{\rm{k}}_2}({{\cal T}_2}( - m)) - {\rm{rank}}(R)\) , where R is a certain explicitly described Rédei matrix over \({\mathbb{F}_2}\) . Furthermore, using this formula, we obtain the 4-rank density formula of \({{\cal T}_2}\) -groups of imaginary quadratic fields. Secondly, for l an odd prime, we obtain results about the 2-power divisibility of orders of \({{\cal T}_2}( \pm l)\) and \({{\cal T}_2}( \pm 2l)\) , both of which are cyclic 2-groups. In particular, we find that \(\# {{\cal T}_2}(l) \equiv 2\# {{\cal T}_2}(2l) \equiv {h_2}( - 2l)\) (mod 16) if l ≡ 7 (mod 8), where h2(−2l) is the 2-class number of \(\mathbb{Q}(\sqrt { - 2l} )\) . We then obtain density results for \({{\cal T}_2}( \pm l)\) and \({{\cal T}_2}( \pm 2l)\) when the orders are small. Finally, based on our density results and numerical data, we propose distribution conjectures about \({{\cal T}_p}(F)\) when F varies over real or imaginary quadratic fields for any prime p, and about \({{\cal T}_2}( \pm l)\) and \({{\cal T}_2}( \pm 2l)\) when l varies, in the spirit of Cohen-Lenstra heuristics. Our conjecture in the \({{\cal T}_2}(l)\) case is closely connected to Shanks-Sime-Washington’s speculation on the distributions of the zeros of 2-adic L-functions and to the distributions of the fundamental units. PubDate: 2022-05-11

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Abstract: On an almost Hermitian manifold, there are two Hermitian scalar curvatures associated with a canonical Hermitian connection. In this paper, two explicit formulas on these two scalar curvatures are obtained in terms of the Riemannian scalar curvature, norms of the components of the covariant derivative of the fundamental 2-form with respect to the Levi-Civita connection, and the codifferential of the Lee form. Then we use them to get characterization results of the Kähler metric, the balanced metric, the locally conformal Kähler metric or the k-Gauduchon metric. As corollaries, we show partial results related to a problem given by Lejmi and Upmeier (2020) and a conjecture by Angella et al. (2018). PubDate: 2022-05-09

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Abstract: In this paper, we investigate the large-time behavior of strong solutions to the Cauchy problem for one-dimensional compressible isentropic magnetohydrodynamic equations near a stable equilibrium. The difference between the one-dimensional and multi-dimensional cases is a feature for compressible flows and also brings new difficulties. In contrast to the multi-dimensional case, the decay rates of nonlinear terms may not be faster than linear terms in dimension one. To handle this, we shall present a new energy estimate in terms of a combination of the solutions with small initial data. We aim to establish the sharp upper and lower bounds on the L2-decay rates of the solutions and all their spatial derivatives when the initial perturbation is small in L1(ℝ) ∩ H2(ℝ). It is worth noticing that there is no decay loss for the highest-order spatial derivatives of the solutions so that the large-time behavior for the hyperbolic-parabolic system is exactly sharp. As a byproduct, the above result is also valid for compressible Navier-Stokes equations. Our approach is based on various interpolation inequalities, energy estimates, spectral analysis, and Fourier time-splitting and high-low frequency decomposition methods. PubDate: 2022-05-07

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Abstract: In the study of the number of limit cycles of near-Hamiltonian systems, the first order Melnikov function plays an important role. This paper aims to generalize Horozov-Iliev’s method to estimate the upper bound of the number of zeros of the function. PubDate: 2022-05-06

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Abstract: Abstract In this paper, we establish the local existence of weak solutions with higher regularity of the three-dimensional half-space compressible isentropic Navier-Stokes equations with the adiabatic exponent γ > 1 in the presence of vacuum. Here we do not need any smallness of the initial data. PubDate: 2022-05-01

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Abstract: Abstract We investigate convergence properties of random Taylor series whose coefficients are ψ-mixing random variables. In particular, we give sufficient conditions such that the circle of the convergence of the series forms almost surely a natural boundary. PubDate: 2022-05-01

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Abstract: Abstract Ever since the famous Erdős-Ko-Rado theorem initiated the study of intersecting families of subsets, extremal problems regarding intersecting properties of families of various combinatorial objects have been extensively investigated. Among them, studies about families of subsets, vector spaces and permutations are of particular concerns. Recently, we proposed a new quantitative intersection problem for families of subsets: For \({\cal F} \subseteq \left({\matrix{{[n]} \cr k \cr}} \right)\) , define its total intersection number as \({\cal I}({\cal F}) = \sum\nolimits_{{F_1},{F_2} \in {\cal F}} {\left {{F_1} \cap {F_2}} \right } \) . Then, what is the structure of \({\cal F}\) when it has the maximal total intersection number among all the families in \(\left({\matrix{{[n]} \cr k \cr}} \right)\) with the same family size' In a recent paper, Kong and Ge (2020) studied this problem and characterized extremal structures of families maximizing the total intersection number of given sizes. In this paper, we consider the analogues of this problem for families of vector spaces and permutations. For certain ranges of family size, we provide structural characterizations for both families of subspaces and families of permutations having maximal total intersection numbers. To some extent, these results determine the unique structure of the optimal family for some certain values of \(\left {\cal F} \right \) and characterize the relationship between having maximal total intersection number and being intersecting. Besides, we also show several upper bounds on the total intersection numbers for both families of subspaces and families of permutations of given sizes. PubDate: 2022-05-01

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Abstract: Abstract The present paper is concerned with a class of quasi-linear elliptic degenerate equations. The degenerate operator arises from analysis of manifolds with singularities. The variational methods are applied here to verify the existence of infinitely many solutions for the problem. PubDate: 2022-05-01

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Abstract: Abstract Let T be a right exact functor from an abelian category ℬ into another abelian category \({\mathscr A}\) . Then there exists a functor p from the product category \({\mathscr A} \times {\mathscr B}\) to the comma category ( \(\left( {T \downarrow {\mathscr A}} \right)\) ). In this paper, we study the property of the extension closure of some classes of objects in ( \(\left( {T \downarrow {\mathscr A}} \right)\) ), the exactness of the functor p and the detailed description of orthogonal classes of a given class \({\rm{P}}\left( {{\cal X},{\cal Y}} \right)\) in ( \(\left( {T \downarrow {\mathscr A}} \right)\) ). Moreover, we characterize when special precovering classes in abelian categories \({\mathscr A}\) and ℬ can induce special precovering classes in ( \(\left( {T \downarrow {\mathscr A}} \right)\) ). As an application, we prove that under suitable cases, the class of Gorenstein projective left Λ-modules over a triangular matrix ring \({\rm{\Lambda }} = \left( {\matrix{ R & M \cr 0 & S \cr } } \right)\) is special precovering if and only if both the classes of Gorenstein projective left R-modules and left S-modules are special precovering. Consequently, we produce a large variety of examples of rings such that the class of Gorenstein projective modules is special precovering over them. PubDate: 2022-05-01

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Abstract: Abstract In this paper, we present the concavity of the minimal L2 integrals related to multiplier ideal sheaves on the weakly pseudoconvex Kähler manifolds which implies the sharp effectiveness results of the strong openness conjecture and a conjecture posed by Demailly and Kollár (2001) on weakly pseudoconvex Kähler manifolds. We obtain the relation between the concavity and the L2 extension theorem. PubDate: 2022-05-01

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Abstract: Abstract This paper is about an optimal pricing control under a Markov chain model. The objective is to dynamically adjust the product price over time to maximize a discounted reward function. It is shown that the optimal control policy is of threshold type. Closed-form solutions are obtained. A numerical example is also provided to illustrate our results. PubDate: 2022-05-01

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Abstract: Abstract This paper continues to derive the globally hyperbolic moment model of arbitrary order for the three-dimensional special relativistic Boltzmann equation with the Anderson-Witting collision. The method is the model reduction by the operator projection. Finding an orthogonal basis of the weighted polynomial space is crucial and built on infinite families of the complicate relativistic Grad type orthogonal polynomials depending on a parameter and the real spherical harmonics instead of the irreducible tensors. We study the properties of those functions carefully, including their recurrence relations, their derivatives with respect to the independent variable and parameter, and the zeros of the orthogonal polynomials. Our moment model is proved to be globally hyperbolic and linearly stable. Moreover, the Lorentz-covariance and the quasi-one-dimensional case, the non-relativistic and ultra-relativistic limits are also studied. PubDate: 2022-05-01

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Abstract: Abstract In this paper, we mainly study how to estimate the error density in the ultrahigh dimensional sparse additive model, where the number of variables is larger than the sample size. First, a smoothing method based on B-splines is applied to the estimation of regression functions. Second, an improved two-stage refitted cross-validation (RCV) procedure by random splitting technique is used to obtain the residuals of the model, and then the residual-based kernel method is applied to estimate the error density function. Under suitable sparse conditions, the large sample properties of the estimator including the weak and strong consistency, as well as normality and the law of the iterated logarithm are obtained. Especially, the relationship between the sparsity and the convergence rate of the kernel density estimator is given. The methodology is illustrated by simulations and a real data example, which suggests that the proposed method performs well. PubDate: 2022-05-01

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Abstract: Abstract In this paper, we study the uniformly strong convergence of the Kähler-Ricci flow on a Fano manifold with varied initial metrics and smoothly deformed complex structures. As an application, we prove the uniqueness of Kähler-Ricci solitons in the sense of diffeomorphism orbits. The result generalizes Tian-Zhu’s theorem for the uniqueness of of Kähler-Ricci solitons on a compact complex manifold, and it is also a generalization of Chen-Sun’s result of the uniqueness of Kähler-Einstein metric orbits. PubDate: 2022-04-29

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Abstract: In this paper, we accomplish the unified convergence analysis of a second-order method of multipliers (i.e., a second-order augmented Lagrangian method) for solving the conventional nonlinear conic optimization problems. Specifically, the algorithm that we investigate incorporates a specially designed nonsmooth (generalized) Newton step to furnish a second-order update rule for the multipliers. We first show in a unified fashion that under a few abstract assumptions, the proposed method is locally convergent and possesses a (nonasymptotic) superlinear convergence rate, even though the penalty parameter is fixed and/or the strict complementarity fails. Subsequently, we demonstrate that for the three typical scenarios, i.e., the classic nonlinear programming, the nonlinear second-order cone programming and the nonlinear semidefinite programming, these abstract assumptions are nothing but exactly the implications of the iconic sufficient conditions that are assumed for establishing the Q-linear convergence rates of the method of multipliers without assuming the strict complementarity. PubDate: 2022-04-25

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Abstract: Abstract We study the toric degeneration of Weyl group translated Schubert divisors of a partial flag variety \(F{\ell _{{n_1}, \ldots,{n_k};n}}\) via Gelfand-Cetlin polytopes. We propose a conjecture that Schubert varieties of appropriate dimensions intersect transversally up to translation by Weyl group elements, and verify it in various cases, including the complex Grassmannian Gr(2, n) and the complete flag variety Fℓ1,2,3,4. PubDate: 2022-04-15

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Abstract: Abstract In this paper, we study irreducible non-weight modules over the mirror Heisenberg-Virasoro algebra \({\cal D}\) , including Whittaker modules, \({\cal U}\left( {{\mathbb{C}d_0}} \right)\) -free modules and their tensor products. More precisely, we give the necessary and sufficient conditions for the Whittaker modules to be irreducible. We determine all the \({\cal D}\) -module structures on \({\cal U}\left( {{\mathbb{C}d_0}} \right)\) , and find the necessary and sufficient conditions for these modules to be irreducible. At last, we determine the necessary and sufficient conditions for the tensor products of Whittaker modules and \({\cal U}\left( {{\mathbb{C}d_0}} \right)\) -free modules to be irreducible, and obtain that any two such tensor products are isomorphic if and only if the corresponding Whittaker modules and \({\cal U}\left( {{\mathbb{C}d_0}} \right)\) -free modules are isomorphic. These lead to many new irreducible non-weight modules over \({\cal D}\) . PubDate: 2022-04-11

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Abstract: Abstract The initial value problem of the multi-dimensional drift-flux model for two-phase flow is investigated in this paper, and the global existence of weak solutions with finite energy is established for general pressure-density functions without the monotonicity assumption. PubDate: 2022-04-07

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Abstract: Abstract In this paper we prove the strong unique continuation property for a class of fourth order elliptic equations involving strongly singular potentials. Our argument is to establish some Hardy-Rellich type inequalities with boundary terms and introduce an Almgren’s type frequency function to show some doubling conditions for the solutions to the above-mentioned equations. PubDate: 2022-04-01

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Abstract: Abstract In this paper, the Harnack inequality for G-SDEs with degenerate noise is derived by the method of coupling by change of measure. Moreover, for any bounded and continuous function f, the gradient estimate $$\left {\nabla {{\bar P}_t}f} \right \leqslant c(p,t){(\bar P_t{\left f \right ^p})^{{1 \over p}}},\;\;\;\;\;\;p > 1,\;\;\;\;\;t> 0$$ is obtained for the associated nonlinear semigroup \({\bar P_t}\) . As an application of the Harnack inequality, we prove the existence of the weak solution to degenerate G-SDEs under some integrable condition. Finally, an example is presented. All of the above results extend the existing ones in the linear expectation setting. PubDate: 2022-04-01