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Abstract: Abstract The well-known Berwald square metric is a positively complete and projectively flat Finsler metric with vanishing flag curvature. In this paper, we study a positively complete square metric on a manifold. We show a rigidity result that if the Ricci curvature is constant, then it must be isometric to the Berwald square metric. This is not true without assumption on the completeness of the metric. PubDate: 2025-01-24 DOI: 10.1007/s10114-025-2578-y
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Abstract: Abstract The classical Mumford stability condition of vector bundles on a complex elliptic curve X, can be viewed as a Bridgeland stability condition on Db (Coh X), the bounded derived category of coherent sheaves on X. This point of view gives us infinitely many t-structures and hearts on Db (Coh X). In this paper, we answer the question which of these hearts are Noetherian or Artinian. PubDate: 2025-01-24 DOI: 10.1007/s10114-025-3286-3
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Abstract: Abstract In this paper we study a heat type equation associated to the curve shortening flow in the plane. We show the solutions become infinitely many times differentiable for a short time. The method of proof is to use the maximum principle following the Bernstein technique. PubDate: 2025-01-24 DOI: 10.1007/s10114-025-3057-1
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Abstract: Abstract In this paper, we study the relationship of balanced pairs in a recollement. As an application of balanced pairs, we introduce the notion of the relative tilting objects, and give a characterization of relative tilting objects, which is similar to Bazzoni characterization of n-tilting modules. Finally, we investigate the relationship of relative tilting objects in a recollement. PubDate: 2025-01-24 DOI: 10.1007/s10114-025-3331-2
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Abstract: Abstract In this article, we discuss the approach to solving a nonlinear PDE equation, specifically the p-Laplacian equation, with a general (nonlinear) boundary condition. We establish the existence and uniqueness of the solution, subject to certain assumptions outlined in this paper. To solve our nonlinear problem using the Finite Element Method (FEM), we derive an appropriate variational formulation. Additionally, we introduce a study of the residual a posteriori-error indicator, establishing both upper and lower bounds to control the error. The upper bound is determined using averaging interpolators in some quasi-norms defined by Barrett and Liu. Furthermore, we prove the equivalence between the residual error and the true error e = u − uh. Lastly, we perform a simulation of the p-Laplacian problem in the L-shape domain using a Matlab program in two-dimensional space. PubDate: 2025-01-24 DOI: 10.1007/s10114-025-3561-3
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Abstract: Abstract Let \({\cal C}\) be a triangulated category. We define m-term subcategories on \({\cal C}\) induced by n-rigid subcategories, which are extriangulated subcategories of \({\cal C}\) . Then we give a one-to-one correspondence between cotorsion pairs on 2-term subcategories \({\cal G}\) and support τ-tilting subcategories on an abelian quotient of \({\cal G}\) . If an m-term subcategory is induced by a co-t-structure, then we have a one-to-one correspondence between cotorsion pairs on it and cotorsion pairs on \({\cal C}\) under certain conditions. PubDate: 2025-01-24 DOI: 10.1007/s10114-025-2286-7
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Abstract: Abstract In this paper, we classify the finite non-abelian p-groups all of whose non-abelian proper subgroups have centers of the same order. PubDate: 2025-01-24 DOI: 10.1007/s10114-025-2325-4
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Abstract: Abstract We establish the Strassen’s law of the iterated logarithm (LIL for short) for independent and identically distributed random variables with \({\hat {\mathbb E}}[X_{1}]={\hat {\cal E}}[X_{1}]=0\) and \(C_{\mathbb V}[X_{1}^{2}]<\infty\) under a sub-linear expectation space with a countably sub-additive capacity \({\mathbb V}\) . We also show the LIL for upper capacity with \(\sigma={\overline \sigma}\) under some certain conditions. PubDate: 2025-01-24 DOI: 10.1007/s10114-025-2759-8
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Abstract: Abstract In this paper, we obtain some sufficient and necessary conditions for indecomposable positive definite integral lattices with discriminants 2, 3, 4 and 5 over \({\mathbb Z}\) being additively indecomposable lattices. Using these results, we prove that there exist additively indecomposable positive integral quadratic lattices with discriminants 2, 3, 4 and 5 and rank greater than or equal to 2 but for 35 exceptions. In the exceptions there are no lattices with the desired properties. We also give a lifting theorem of additively indecomposable positive definite integral lattices. PubDate: 2025-01-24 DOI: 10.1007/s10114-025-2562-6
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Abstract: Abstract This paper concerns two-player zero-sum stochastic differential games with nonanticipative strategies against closed-loop controls in the case where the coefficients of mean-field stochastic differential equations and cost functional depend on the joint distribution of the state and the control. In our game, both the (lower and upper) value functions and the (lower and upper) second-order Bellman–Isaacs equations are defined on the Wasserstein space \({\cal P}_{2}({\mathbb R}^{n})\) which is an infinite dimensional space. The dynamic programming principle for the value functions is proved. If the (upper and lower) value functions are smooth enough, we show that they are the classical solutions to the second-order Bellman–Isaacs equations. On the other hand, the classical solutions to the (upper and lower) Bellman–Isaacs equations are unique and coincide with the (upper and lower) value functions. As an illustrative application, the linear quadratic case is considered. Under the Isaacs condition, the explicit expressions of optimal closed-loop controls for both players are given. Finally, we introduce the intrinsic notion of viscosity solution of our second-order Bellman–Isaacs equations, and characterize the (upper and lower) value functions as their viscosity solutions. PubDate: 2025-01-24 DOI: 10.1007/s10114-025-2666-z
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Abstract: Abstract Let (Ωn+1, g) be an (n + 1)-dimensional smooth compact connected Riemannian manifold with connected boundary Σ = ∂Ω, whose principal curvatures are bounded from below by a positive constant. In this paper, we provide some sharp lower bounds of the first eigenvalue of the Laplacian on the boundary Σ. In the two dimensional case, we establish our estimates for manifolds whose Gaussian curvature K satisfies K ≥ ±1; in the higher dimensional case, we give our estimates for manifolds with the so-called Ric k Σ condition (see Section 2). PubDate: 2025-01-24 DOI: 10.1007/s10114-025-3403-3
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Abstract: Abstract Extriangulated categories were introduced by Nakaoka and Palu, which unify exact categories and extension-closed subcategories of triangulated categories. In this paper, we develop the relative homology aspect of Auslander bijection in extriangulated categories. Namely, we introduce the notion of generalized ARS-duality relative to an additive subfunctor, and prove that there is a bijective triangle which involves the generalized ARS-duality and the restricted Auslander bijection relative to the subfunctor. We give the Auslander’s defect formula in terms of the generalized ARS-duality, show the interplay of morphisms being determined by objects and a kind of extriangles, and characterize a class of objects which are somehow controlled by this kind of extriangles. We also give a realization for a functor relating the Auslander bijection and the generalized ARS-duality. PubDate: 2025-01-24 DOI: 10.1007/s10114-025-2074-4
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Abstract: Abstract A graph G is edge-k-choosable if, for any assignment of lists L(e)of at least k colors to all edges e ∈ E(G), there exists a proper edge coloring such that the color of e belongs to L(e) for all e ∈ E(G). One of Vizing’s classic conjectures asserts that every graph is edge-(Δ + 1)-choosable. It is known since 1999 that this conjecture is true for general graphs with Δ ≤ 4. More recently, in 2015, Bonamy confirmed the conjecture for planar graph with Δ ≥ 8, but the conjecture is still open for planar graphs with 5 ≤ Δ ≤ 7. We confirm the conjecture for planar graphs with Δ ≥ 6 in which every 7-cycle (if any) induces a C7 (so, without chords), thereby extending a result due to Dong, Liu and Li. PubDate: 2025-01-24 DOI: 10.1007/s10114-025-2761-1
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Abstract: Abstract This paper investigates tangent measures in the sense of Preiss for homogeneous Cantor sets on ℝd generated by specific iterated function systems that satisfy the strong separation condition. Through the dynamics of “zooming in” on typical point, we derive an explicit and uniform formula for the tangent measures associated with this category of homogeneous Cantor sets on ℝd. PubDate: 2025-01-24 DOI: 10.1007/s10114-025-3326-z
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Abstract: Abstract In this short note, we establish a sharp Morrey regularity theory for an even order elliptic system of Rivière type: $$\Delta^{m}u=\sum_{l=0}^{m-1}\Delta^{l}\langle{V}_{l}, du\rangle+\sum_{l=0}^{m-2}\Delta^{l}\delta(w_{l}du)+f \quad {\rm in} \; B^{2m}$$ under minimal regularity assumptions on the coefficients functions Vl, wl and that f belongs to certain Morrey space. This can be regarded as a further extension of the recent Lp-regularity theory obtained by Guo–Xiang–Zheng [J. Math. Pures Appl. (9), 165, 286–324 (2022)], and generalizes [Proc. Amer. Math. Soc., 152(10), 4261–4268 (2024)], [Acta Math. Sci. Ser. B (Engl. Ed.), 44(2), 420–430 (2024)] for second and fourth order elliptic systems. PubDate: 2025-01-24 DOI: 10.1007/s10114-025-3353-9
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Abstract: Abstract Let Ω be a domain of ℝn with n ≥ 2 and p(·) be a local Lipschitz funcion in Ω with 1 < p(x) < ∞ in Ω. We build up an interior quantitative second order Sobolev regularity for the normalized p(·)-Laplace equation −Δ p(·) N u = 0 in Ω as well as the corresponding inhomogeneous equation −Δ p(·) N u =f in Ω with f ∈ C0(Ω). In particular, given any viscosity solution u to −Δ p(·) N u = 0 in Ω, we prove the following: in dimension n = 2, for any subdomain U ⋐ Ω and any β ≥ 0, one has ∣Du∣βDu ∈ L loc 2+δ (U) with a quantitative upper bound, and moreover, the map \((x_{1},x_{2})\rightarrow\vert Du\vert^{\beta}(u_{x_{1}},-u_{x_{2}})\) is quasiregular in U in the sense that $$\vert D[\vert Du\vert^{\beta}\;Du]\vert^{2}\leq-C\;\text{det}\;D[\vert Du\vert^{\beta}\;Du]\;\;\;\;\;\text{a.e.}\;\text{in}\;U.$$ in dimension n ≥ 3, for any subdomain U ⋐ Ω with infU p(x) > 1 and \(\text{sup}_{U}\;p(x)<3+{2\over{n-2}}\) , one has D2u ∈ L loc 2+δ (U) with a quantitative upper bound, and also with a pointwise upper bound $$\vert D^{2}u\vert^{2}\leq-C\sum_{1\leq i<j\leq n}[u_{x_{i}x_{j}}u_{x_{j}x_{i}}-u_{x_{i}x_{i}}u_{x_{j}x_{j}}]\;\;\;\;\;\text{a.e}\;\text{in}\;U.$$ Here constants δ > 0 and C ≥ 1 are independent of u. These extend the related results obtaind by Adamowicz–Hästö [Mappings of finite distortion and PDE with nonstandard growth. Int. Math. Res. Not. IMRN, 10, 1940–1965 (2010)] when n = 2 and β = 0. PubDate: 2025-01-01 DOI: 10.1007/s10114-025-3356-6
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Abstract: Abstract In this paper, we establish a weighted maximal L2 estimate of operator-valued Bochner–Riesz means. The proof is based on noncommutative square function estimates and a sharp weighted noncommutative Hardy–Littlewood maximal inequality. PubDate: 2025-01-01 DOI: 10.1007/s10114-025-3315-2
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Abstract: Abstract In this paper, we consider the jump and variational inequalities of truncated singular integral operator with rough kernel $$T_{\Omega,\beta,\varepsilon}f(x)=\int_{\mid y\mid>\varepsilon}{\Omega(y)\over \mid y\mid ^{n-\beta}}f(x-y)dy,$$ where the kernel \(\Omega \in (L(\log^{+}L)^{2})^{n \over{n-\beta}}(\mathbb{S}^{n-1})\) satisfies the vanishing condition and the homogeneous condition of degree 0. This kind of singular integral appears in the approximation of the surface quasi-geostrophic (SQG) equation from the generalized SQG equation. We establish the (Lp, Lq) estimate of the jump and variational inequalities of the families {TΩ,β,ε}ε>0 for \({1\over q}={1\over p}-{\beta\over n}\) and 0 < β < 1. Moreover, one can get the Lp boundedness of the Calderón–Zygmund operator with the same kernel by letting β → 0+. PubDate: 2025-01-01 DOI: 10.1007/s10114-025-3462-5
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Abstract: Abstract For s ∈ [0, 1], b ∈ ℝ and p ∈ [1, ∞), let \(\dot{B}_{p,\infty}^{s,b}(\mathbb{R}^{n})\) be the logarithmic-Gagliardo–Lipschitz space, which arises as a limiting interpolation space and coincides to the classical Besov space when b = 0 and s ∈ (0, 1). In this paper, the authors study restricting principles of the Riesz potential space \(\cal{I}_{\alpha}(\dot{B}_{p,\infty}^{s,b}(\mathbb{R}^{n}))\) into certain Radon–Campanato space. PubDate: 2025-01-01 DOI: 10.1007/s10114-025-3458-1
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Abstract: Abstract We extend the (outer) measure \(\gamma_{\cal{I}}\) associated to an operator ideal \(\cal{I}\) to a measure \(\gamma_{\frak{J}}\) for bounded bilinear operators. If \(\cal{I}\) is surjective and closed, and \(\frak{J}\) is the class of those bilinear operators such that \(\gamma_{\frak{J}}(T)=0\) , we prove that \(\frak{J}\) coincides with the composition bideal \(\cal{I}\circ\frak{B}\) . If \(\cal{I}\) satisfies the Σr-condition, we establish a simple necessary and sufficient condition for an interpolated operator by the real method to belong to \(\frak{J}\) . Furthermore, if in addition \(\cal{I}\) is symmetric, we prove a formula for the measure \(\gamma_{\frak{J}}\) of an operator interpolated by the real method. In particular, results apply to weakly compact operators. PubDate: 2025-01-01 DOI: 10.1007/s10114-025-3506-x
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