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Abstract: Abstract We give an example of a regular 1-dimensional foliation along a genus 3 curve whose Ueda type is one and normal bundle is of order two. PubDate: 2024-04-12

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Abstract: Abstract We study the singularities of commuting vector fields of a real submanifold of a Kähler manifold Z. PubDate: 2024-04-04

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Abstract: Abstract As is well-known, numerical experiments show that Napoleon’s Theorem for planar triangles does not extend to a similar statement for triangles on the unit sphere \(S^2\) . Spherical triangles for which an extension of Napoleon’s Theorem holds are called Napoleonic, and until now the only known examples have been equilateral. In this paper we determine all Napoleonic spherical triangles, including a class corresponding to points on a 2-dimensional ellipsoid, whose Napoleonisations are all congruent. Other new classes of examples are also found, according to different versions of Napoleon’s Theorem for the sphere. The classification follows from successive simplifications of a complicated original algebraic condition, exploiting geometric symmetries and algebraic factorisations. PubDate: 2024-04-03

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Abstract: Abstract We consider some local entropy properties of dynamical systems under the assumption of shadowing. In the first part, we give necessary and sufficient conditions for shadowable points to be certain entropy points. In the second part, we give some necessary and sufficient conditions for (non) h-expansiveness under the assumption of shadowing and chain transitivity; and use the result to present a counter-example for a question raised by Artigue et al. (Proc Am Math Soc 150:3369–3378, 2022). PubDate: 2024-04-02

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Abstract: Abstract We prove the existence of positive solutions for a system of quasilinear equations driven by \(p_i\) -Laplacian operators, with Dirichlet boundary conditions, variable exponents, convection terms, and depending on two parameters. No upper bound on the variable exponents, neither in u nor in \(\nabla u\) , is imposed. It is for the first time when such systems are investigated. The range of solvability for the involved parameters is explicitly determined. Our approach relies on a version of sub-supersolution method for systems that is adapted to the specific character of our problem. PubDate: 2024-03-31

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Abstract: Abstract The generalized three dimensional Navier–Stokes equations with damping are considered. Firstly, existence and uniqueness of strong solutions in the periodic domain \({\mathbb {T}}^{3}\) are proved for \(\frac{1}{2}<\alpha <1,~~ \beta +1\ge \frac{6\alpha }{2\alpha -1}\in (6,+\infty )\) . Then, in the whole space \(R^3,\) if the critical situation \(\beta +1= \frac{6\alpha }{2\alpha -1}\) and if \(u_{0}\in H^{1}(R^{3}) \bigcap {\dot{H}}^{-s}(R^{3})\) with \(s\in [0,1/2]\) , the decay rate of solution has been established. We give proofs of these two results, based on energy estimates and a series of interpolation inequalities, the key of this paper is to give an explanation for that on the premise of increasing damping term, the well-posedness and decay can still preserve at low dissipation \(\alpha <1,\) and the relationship between dissipation and damping is given. PubDate: 2024-03-27

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Abstract: Abstract In this work we study evaluation codes defined on the points of a subset \(\mathcal {X}\) of an affine space over a finite field, whose vanishing ideal admits a Gröbner basis of a certain type, which occurs for subsets considered in several well-known examples of evaluation codes, like Reed-Solomon codes, Reed-Muller codes and affine cartesian codes. We determine properties of the polynomials in this basis which allow the determination of the footprint of the vanishing ideal and the explicit construction of indicator functions for the points of \(\mathcal {X}\) . We then consider generalized monomial evaluation codes and find information on their duals, and the dimension of their hulls. We present several examples of applications of the results we found. PubDate: 2024-03-20

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Abstract: Abstract This paper is concerned with the existence and concentration of ground state solutions for the following class of elliptic Kirchhoff–Boussinesq type problems given by $$\begin{aligned} \Delta ^{2} u \pm \Delta _{p} u +(1+\lambda V(x))u= f(u)\quad \text {in}\ {\mathbb {R}}^{4}, \end{aligned}$$ where \(2< p< 4,\) \(f\in C( {\mathbb {R}}, {\mathbb {R}})\) is a nonlinearity which has subcritical or critical exponential growth at infinity and \(V\in C({\mathbb {R}}^4,{\mathbb {R}})\) is a potential that vanishes on a bounded domain \(\Omega \subset {\mathbb {R}}^4.\) Using variational methods, we show the existence of ground state solutions, which concentrates on a ground state solution of a Kirchhoff–Boussinesq type equation in \(\Omega .\) PubDate: 2024-03-15

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Abstract: Abstract We characterise the numerical semigroups with a monotone Apéry set (MANS-semigroups for short). Moreover, we describe the families of MANS-semigroups when we fix the multiplicity and the ratio. PubDate: 2024-03-14

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Abstract: Abstract The invariant subspace problem (ISP) for Hilbert spaces asks if every bounded linear operator has a non-trivial closed invariant subspace. Due to the existence of universal operators (in the sense of Rota) the ISP can be solved by proving that every minimal invariant subspace of a universal operator is one dimensional. In this work, we obtain conditions for \(T^{*}_{\varphi } _{M}\) to have a non-trivial subspace where \(M\subset H^{2}({\mathbb {D}}^{2})\) is an invariant subspace of the Toeplitz operator \(T_{\varphi }^{*}\) on the Hardy space over the bidisk \(H^{2}({\mathbb {D}}^{2})\) induced by the symbol \(\varphi \in H^{\infty }({\mathbb {D}})\) . We then use this fact to obtain sufficient conditions for the ISP to be true. PubDate: 2024-03-04

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Abstract: Abstract Lattice Lipschitz operators define a new class of nonlinear Banach-lattice-valued maps that can be written as diagonal functions with respect to a certain basis. In the n-dimensional case, such a map can be represented as a vector of size n of real-valued functions of one variable. In this paper we develop a method to approximate almost diagonal maps by means of lattice Lipschitz operators. The proposed technique is based on the approximation properties and error bounds obtained for these operators, together with a pointwise version of the interpolation of McShane and Whitney extension maps that can be applied to almost diagonal functions. In order to get the desired approximation, it is necessary to previously obtain an approximation to the set of eigenvectors of the original function. We focus on the explicit computation of error formulas and on illustrative examples to present our construction. PubDate: 2024-02-13 DOI: 10.1007/s00574-024-00385-9

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Abstract: Abstract In this paper, we study the universal inequalities for eigenvalues of a clamped plate problem of the drifting Laplacian in several cases, and establish some universal inequalities that are different from those obtained previously in (Du et al. in Z Angew Math Phys 66(3):703–726, 2015). PubDate: 2024-02-06 DOI: 10.1007/s00574-024-00384-w

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Abstract: Abstract In this work, we are concerned with the main mechanism for possible blow-up criteria of smooth solutions to the 3D incompressible Boussinesq equations. The main results state that the finite-time blowup/global existence of smooth solutions to the Boussinesq equation is controlled by either of the criteria $$\begin{aligned} u_{h}\in L^{2}\left( 0,T;\dot{B}_{\infty ,\infty }^{0}({\mathbb {R}} ^{3})\right) \quad \text {or}\quad \nabla _{h}u_{h}\in L^{1}\left( 0,T;\dot{B} _{\infty ,\infty }^{0}\left( {\mathbb {R}}^{3}\right) \right) , \end{aligned}$$ where \(u_{h}\) and \(\nabla _{h}\) denote the horizontal components of the velocity field and partial derivative with respect to the horizontal variables, respectively. We present a new simple proof for the regularity of this system without using the higher-order energy law and without any assumptions on the temperature \(\theta .\) Our results extend the Navier–Stokes equations results in Dong and Zhang (Nonlinear Anal Real World Appl 11:2415–2421, 2010), Dong and Chen (J Math Anal Appl 338:1–10, 2008) and Gala and Ragusa (Electron J Qual Theory Differ Equ, 2016a) to Boussinesq equations. PubDate: 2024-02-02 DOI: 10.1007/s00574-024-00383-x

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Abstract: Abstract In this paper, we investigate the analytical solutions to the cylindrically symmetric compressible Navier–Stokes equations with density-dependent viscosity and vacuum free boundary. The shear and bulk viscosity coefficients are assumed to be a power function of the density and a positive constant, respectively, and the free boundary is assumed to move in the radial direction with the radial velocity, which will affect the angular velocity but does not affect the axial velocity. We obtain a global analytical solution by using some ansatzs and reducing the original partial differential equations into a nonlinear ordinary differential equation about the free boundary. The free boundary is shown to grow at least sub-linearly in time and not more than linearly in time for the analytical solution by using a new averaged quantity. PubDate: 2024-01-23 DOI: 10.1007/s00574-023-00382-4

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Abstract: In this paper, we study results of well-posedness and regularity of higher order in time abstract non-autonomous semilinear Cauchy problems associated with Newton’s binomial theorem and the theory of sectorial operators. Our approach to parabolic problems of arbitrarily order n apparently has never been addressed earlier in the existing literature. Also, we present applications to evolutionary equations involving the fractional Laplacian in bounded smooth domains of \({\mathbb {R}}^N\) . PubDate: 2024-01-11 DOI: 10.1007/s00574-023-00381-5

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Abstract: Abstract Applying Geometric Invariant Theory (GIT), we study the stability of foliations of degree 3 on \(\mathbb {P}^{2}\) with a unique singular point of multiplicity 1, 2, or 3 and Milnor number 13. In particular, we characterize those foliations for multiplicity 2 in three cases: stable, strictly semistable, and unstable. PubDate: 2023-12-22 DOI: 10.1007/s00574-023-00379-z

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Abstract: Abstract We construct for every proper algebraic space over a ground field an Albanese map to a para-abelian variety, which is unique up to unique isomorphism. This holds in the absence of rational points or ample sheaves, and also for reducible or non-reduced spaces, under the mere assumption that the structure morphism is in Stein factorization. It also works under suitable assumptions in families. In fact the treatment of the relative setting is crucial, even to understand the situation over ground fields. This also ensures that Albanese maps are equivariant with respect to actions of group schemes. Our approach depends on the notion of families of para-abelian varieties, where each geometric fiber admits the structure of an abelian variety, and representability of tau-parts in relative Picard groups, together with structure results on algebraic groups. PubDate: 2023-12-20 DOI: 10.1007/s00574-023-00378-0

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Abstract: Abstract We consider a complete noncompact minimal hypersurface \(\Sigma ^n\) in a product manifold \({{\mathbb {S}}}^{n}(\sqrt{2(n-1)})\times {{\mathbb {R}}}\) \((n\ge 3)\) . We get that there admits no nontrivial \(L^2\) harmonic 1-forms on \(\Sigma \) if the square of \(L^n\) -norm of the second fundamental form is less than \(\frac{\alpha ^2n}{2C_0(n-1)}\) or the square of the length of the second fundamental form is less than \(\frac{n\alpha ^2}{2(n-1)}\) . Here \(\alpha \) is an angle function and \(C_0\) is the Sobolev constant depending only on n. PubDate: 2023-12-20 DOI: 10.1007/s00574-023-00380-6

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Abstract: Abstract Kwapień’s theorem asserts that every continuous linear operator from \(\ell _{1}\) to \(\ell _{p}\) is absolutely \(\left( r,1\right) \) -summing for \(1/r=1-\left 1/p-1/2\right .\) When \(p=2\) it recovers the famous Grothendieck’s theorem. In this paper we investigate multilinear variants of these theorems and related issues. Among other results we present a unified version of Kwapień’s and Grothendieck’s results that encompasses the cases of multiple summing and absolutely summing multilinear operators. PubDate: 2023-12-14 DOI: 10.1007/s00574-023-00377-1

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Abstract: Abstract Given a strictly increasing continuous function \(\phi :\mathbb {R}_{\ge 0} \longrightarrow \mathbb {R}\cup \{-\infty \}\) with \(\lim _{t\rightarrow \infty }\phi (t) = \infty \) , a function \(f:[a,b] \longrightarrow \mathbb {R}_{\ge 0}\) is said to be \(\phi \) -concave if \(\phi \circ f\) is concave. When \(\phi (t) = t^p\) , \(p>0\) , this notion is that of p-concavity whereas for \(\phi (t) = \log (t)\) it leads to the so-called log-concavity. In this work, we show that if the cross-sections volume function of a compact set \(K\subset \mathbb {R}^n\) (of positive volume) w.r.t. some hyperplane H passing through its centroid is \(\phi \) -concave, then one can find a sharp lower bound for the ratio \(\textrm{vol}(K^{-})/\textrm{vol}(K)\) , where \(K^{-}\) denotes the intersection of K with a halfspace bounded by H. When K is convex, this inequality recovers a classical result by Grünbaum. Moreover, other related results for \(\phi \) -concave functions (and involving the centroid) are shown. PubDate: 2023-11-29 DOI: 10.1007/s00574-023-00376-2