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Abstract: We prove the Mond conjecture for wave fronts which states that the number of parameters of a frontal versal unfolding is less than or equal to the number of spheres in the image of a stable frontal deformation with equality if the wave front is weighted homogeneous. We give two different proofs. The first one depends on the fact that wave fronts are related to discriminants of map germs and we then use the analogous result proved by Damon and Mond in this context. The second one is based on ideas by Fernández de Bobadilla, Nuño-Ballesteros and Peñafort Sanchis and by Nuño-Ballesteros and Fernández-Hernández. The advantage of the second approach is that most results are valid for any frontal, not only wave fronts, and thus give important tools which may be useful to prove the conjecture for frontals in general. PubDate: 2025-03-21
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Abstract: By deriving Reilly type inequality we prove a lower bound estimate of the first Dirichlet eigenvalue of the Finsler Laplacian operator on a compact Finsler measure space with smooth boundary $$(M,F,d\mu )$$ and weighted Ricci curvature bound from below $$Ric _N\ge (N-1)k>0$$, whose boundary has non-positive $$d\mu $$-mean curvature. Moreover, we show that the lower bound is achieved if and only if M is isometric to a forward (backward) geodesic ball of radius $$\frac{\pi }{2\sqrt{k}}$$ and F has constant radial flag curvature equal to k. PubDate: 2025-03-03
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Abstract: Given a smooth partial action $$\alpha $$ of a Lie groupoid G on a smooth manifold M, we provide necessary and sufficient conditions for $$\alpha $$ to be globalizable with smooth globalization. As an application, we provide results on the differentiable structure of orbit and stabilizer spaces induced by $$\alpha ,$$ which leads to other criteria for its globalization in terms of its orbit maps in the case that $$\alpha $$ is free and transitive. Further, under the assumption that $$\alpha $$ is free and proper, we prove that there exists exactly one differentiable structure on the quotient structure of the orbit equivalence space M/G such that the quotient map $$\pi :M\rightarrow M/G$$ is a submersion. PubDate: 2025-02-18
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Abstract: Following the concept of L-sets in the dual of a Banach space and using the class of almost Dunford-Pettis sets in a Banach lattice, the notion of L-almost Dunford-Pettis sets is introduced and studied. As an application, Banach lattices with the strong relatively compact Dunford-Pettis property, weak Dunford-Pettis property, and positive Schur property are characterized. Moreover, the behavior of almost Dunford-Pettis completely continuous operators is described and then the domination problem and the adjoint problem for this class of operators are addressed. PubDate: 2025-02-14
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Abstract: In this paper we prove that for topologically mixing metric Anosov flows their equilibrium states corresponding to Hölder potentials satisfy a strong rigidity property: they are determined only by their disintegrations on (strong) stable or unstable leaves. As a consequence we deduce: the corresponding horocyclic foliations of such systems are uniquely quasi-ergodic, provided that the corresponding Jacobian is Hölder, without any restriction on the dimension of the invariant distributions. This gives another proof of a result of Babillott-Ledrappier. PubDate: 2025-02-08
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Abstract: In this paper we consider the planar non-collinear central configurations with n bodies with power-law potentials like $$\sum m_im_jr^{-a}$$, $$a>0$$, in which is possible to remove one body and still have a central configuration. This kind of central configurations is called a (n, 1)-stacked central configuration. We prove that the unique planar (n, 1)-stacked central configuration is formed by a regular polygon with equal masses at the vertices and one arbitrary mass at the barycenter, for $$4\le n PubDate: 2025-02-06
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Abstract: We show that the solutions to the nonlocal obstacle problems for the nonlocal $$-\Delta _p^s$$ operator, when the fractional parameter $$s\rightarrow \sigma $$ for $$0 PubDate: 2025-01-15
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Abstract: In 1982, Wu-Yi Hsiang, in an article published in the Journal of Differential Geometry, classified constant mean curvature hypersurfaces in Euclidean space, invariant by the action of the group $$O(p)\times O(q).$$ In his work, he conjectured that there is only one of such hypersurfaces in the Euclidean space, invariant by the action of the group $$O(p)\times O(q),$$ whose profile curve has a singularity at the origin. In this paper, we prove this conjecture using blowing up techniques for degenerate singularities and invariant manifold theory for a tridimensional system of ordinary differential equations. We remark that the noninvariance of the constant mean curvature equation by homotheties prevents us from using the method developed by E. Bombieri, E. De Giorgi, and E. Giusti, in an article published in the Inventiones Mathematicae in 1969, to classify minimal hypersurfaces in the Euclidean space and transform the constant mean curvature equation in a bidimensional system of ordinary differential equations. This forces us to analyze a tridimensional system of ordinary differential equations with degenerate singularities. PubDate: 2025-01-15
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Abstract: Let $$n \ge 8$$ be even, and let $$G = \langle x, y \mid x^2 = y^{n/2}, y^n = 1, yx = xy^s \rangle $$, where $$s^2 \equiv 1 \pmod n$$ and $$s \not \equiv \pm 1 \pmod n$$. In this paper, we provide the precise values of some zero-sum constants over G, namely the small Davenport constant, $$\eta $$-constant, Gao constant, and Erdős-Ginzburg-Ziv constant. In particular, the Gao’s and Zhuang-Gao’s Conjectures hold for G. We also solve the associated inverse problems when $$n \equiv 0 \pmod 4$$. PubDate: 2025-01-10
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Abstract: The main focus of the paper is on a class of semi-infinite fractional interval-valued mathematical programming problems with vanishing constraints under data locally Lipschitzian ((NSIMPVC) for short). The definitions of Mordukhovich-convexity and a new version of the Abadie constraint qualifications are employed. Besides, two dual models of the Wolfe and Mond-Weir types for (NSIMPVC) are constructed through Mordukhovich subdifferentials. In addition, we derive the characterization of optimality and duality for (NSIMPVC) and its Wolfe and Mond-Weir types dual model in which three weak, strong, and converse duality relations for the same are examined. We also provide the VC-KKT-type necessary optimality conditions for (weakly) LU-optimal solution of (NSIMPVC) using the definition of the (ACQ) and (VC-ACQ) types Abadie constraint qualification along with the closedness of an aggregate set at the given vector. First-order necessary optimality condition becomes sufficient optimality condition under suitable assumptions on the generalized convexity of objective and constraint functions. PubDate: 2025-01-09
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Abstract: In this article, we investigate expansivity, Li–Yorke chaos and shadowing properties for composition operators $$C_{\phi }f = f \circ \phi $$ induced by affine self-maps $$\phi $$ of the right half-plane $${\mathbb {C}}_{+}$$ on the Hardy–Hilbert space $$H^{2}(\mathbb {C_{+}})$$. PubDate: 2025-01-08
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Abstract: We provide a systematic way of calculating the quiver region associated with a given exceptional collection. As an application, we prove that $$\mu $$-stable sheaves, obtained as the cokernel of morphisms of exceptional $$\mu $$-stable bundles, are Bridgeland stable, using the determinant. In the subsequent sections, we focus on the case of even rank 2 instantons over $$\mathbb {P}^3$$ and $$Q_3$$, where we prove that instanton sheaves, instanton bundles and perverse instantons are Bridgeland stable and offer a description of the moduli space near its only actual wall. PubDate: 2024-12-30
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Abstract: In this paper, we establish a connection between evolution algebras of dimension two and Hopf algebras, via the algebraic group of automorphisms of an evolution algebra. Initially, we describe the Hopf algebra associated with the automorphism group of a 2-dimensional evolution algebra. Subsequently, for a 2-dimensional evolution algebra A over a field K, we detail the relation between the algebra associated with the (tight) universal associative and commutative representation of A, referred to as the (tight) p-algebra, and the corresponding Hopf algebra, $$\mathscr {H}$$, representing the affine group scheme $${{\,\textrm{Aut}\,}}(A)$$. Our analysis involves the computation of the (tight) $$p-$$algebra associated with any 2-dimensional evolution algebra, whenever it exists. We find that $${{\,\textrm{Aut}\,}}(A)=1$$ if and only if there is no faithful associative and commutative representation for A. Moreover, there is a faithful associative and commutative representation for A if and only if $$\mathscr {H}\not \cong K$$ and $$\text {char} (K)\ne 2$$, or $$\mathscr {H}\not \cong K(\epsilon )$$ (the dual numbers algebra) and $$\mathscr {H}\not \cong K$$ in case of $$\text {char} (K)= 2$$. Furthermore, if A is perfect and has a faithful tight p-algebra, then this p-algebra is isomorphic to $$\mathscr {H}$$ (as algebras). Finally, we derive implications for arbitrary finite-dimensional evolution algebras. PubDate: 2024-12-26
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Abstract: In this paper, we prove that manifolds of finite volume with Anosov geodesic flows have dense periodic orbits. The same result works for conservative Anosov flows in non-compact cases. PubDate: 2024-12-10
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Abstract: An automorphism of a projective curve over an algebraically closed field is said to be non-classical, if the image of any smooth point under the automorphism lies on the tangent line at the point. This paper shows that in characteristic zero, there does not exist a smooth plane curve admitting a non-classical automorphism. This result is a generalization of results according to Levcovitz in 1991. This paper also considers the relation between non-classical automorphisms and Galois points. PubDate: 2024-12-10
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Abstract: Discussed here is a regularized version of the classical Gardner equation $$ u_t + u_x + uu_x + A u^2u_x - u_{xxt} \, = \, 0, $$ u t + u x + u u x + A u 2 u x - u xxt = 0 , that arises in hydrodynamics and plasma physics. This initial-value problem posed on all of $${\mathbb {R}}$$ will be considered with bore-like initial data. That is, the initial wave configuration will consist of a moderately smooth function that asymptotes to zero as the spatial variable $$x \rightarrow +\infty $$, but converges to $$r> 0$$ as $$x \rightarrow -\infty $$. Such initial profiles can arise in internal wave propagation, for example. In their idealized versions set on all of $${\mathbb {R}}$$, they possess an infinite amount of potential energy. This makes the analysis of the initial-value problem a slightly more subtle than the common situation where the initial profile is assumed to be localized, so being modelled by Sobolev-class initial data. PubDate: 2024-12-07
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Abstract: We consider in this paper various theoretical and numerical issues on classical one dimensional models of internal waves with surface tension. They concern the Cauchy problem, including the long time dynamic, localized solitons or multisolitons, the soliton resolution property. We survey known results, present a few new ones together with open questions and conjectures motivated by numerical simulations. A major issue is to emphasize the differences of the qualitative behavior of solutions with those of the same equations without the capillary term. PubDate: 2024-12-07
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Abstract: In the present paper, we show that when the mass of the Proca field is large enough, the Dirac-Proca equations are approximated by the cubic Dirac equation. This justifies the four-fermion interaction approximation of the intermediate vector boson model in a mathematically rigorous sense. The proof is based on the Strichartz estimate of the Proca sector with mass parameter. PubDate: 2024-12-07
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