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Abstract: We generalise the surface-complexity of closed 3-manifolds to the compact case. This generalisation preserves the (natural generalisation of the) properties holding in the closed case: the surface-complexity on compact 3-manifolds is a natural number measuring how much the manifolds are complicated, it is subadditive under connected sum and it is finite-to-one on \(\mathbb {P}^2\) -irreducible and boundary-irreducible manifolds without essential annuli and Möbius strips. Moreover, for these manifolds, it equals the minimal number of cubes in an ideal cubulation of the manifold, except for a finite number of cases. We will also give estimations of the surface-complexity by means of ideal triangulations and Matveev complexity. PubDate: 2021-09-27
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Abstract: An analogue of the Mukai map \(m_g: {\mathcal {P}}_g \rightarrow {\mathcal {M}}_g\) is studied for the moduli \({\mathcal {R}}_{g, \ell }\) of genus g curves C with a level \(\ell \) structure. Let \({\mathcal {P}}^{\perp }_{g, \ell }\) be the moduli space of 4-tuples \((S, {\mathcal {L}}, {\mathcal {E}}, C)\) so that \((S, {\mathcal {L}})\) is a polarized K3 surface of genus g, \({\mathcal {E}}\) is orthogonal to \({\mathcal {L}}\) in \({{\,\mathrm{Pic}\,}}S\) and defines a standard degree \(\ell \) K3 cyclic cover of S, \(C \in \vert {\mathcal {L}} \vert \) . We say that \((S, {\mathcal {L}}, {\mathcal {E}})\) is a level \(\ell \) K3 surface. These exist for \(\ell \le 8\) and their families are known. We define a level \(\ell \) Mukai map \(r_{g, \ell }: {\mathcal {P}}^{\perp }_{g, \ell } \rightarrow {\mathcal {R}}_{g, \ell }\) , induced by the assignment of \((S, {\mathcal {L}}, {\mathcal {E}}, C)\) to \( (C, {\mathcal {E}} \otimes {\mathcal {O}}_C)\) . We investigate a curious possible analogy between \(m_g\) and \(r_{g, \ell }\) , that is, the failure of the maximal rank of \(r_{g, \ell }\) for \(g = g_{\ell } \pm 1\) , where \(g_{\ell }\) is the value of g such that \(\dim {\mathcal {P}}^{\perp }_{g, \ell } = \dim {\mathcal {R}}_{g,\ell }\) . This is proven here for \(\ell = 3\) . As a related open problem we discuss Fano threefolds whose hyperplane sections are level \(\ell \) K3 surfaces and their classification. PubDate: 2021-09-21
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Abstract: Let A be a positive bounded linear operator on a Hilbert space \(\left( { {\mathscr {H}}},\left\langle .,.\right\rangle \right) \) . The semi-inner product \(\left\langle x,y\right\rangle _{A}:=\left\langle Ax,y\right\rangle \) , x, y \(\in \) \({{\mathscr {H}}}\) , induces a semi-norm \(\left\ .\right\ _{A}\) on \({{\mathscr {H}}}\) . Let \(\omega _{A}\) \(\left( T\right) \) denote the A -numerical radius of an operator T in semi-Hilbertian space \(\left( { {\mathscr {H}}},\left\ .\right\ _{A}\right) \) . Our aim in this paper is to give new inequalities of A-numerical radius of operators in semi-Hilbertian spaces. In particular, we show that $$\begin{aligned} \omega _{A}^{n}\left( T\right) \le \frac{1}{2^{n-1}}\omega _{A}\left( T^{n}\right) +\left\ T\right\ _{A}\displaystyle \sum _{p=1}^{n-1}\frac{ 1}{2^{p}}\omega _{A}^{n-p-1}\left( T\right) \left\ T^{p}\right\ _{A}, \end{aligned}$$ for all \(n=2,3,\ldots \) Further, an extension of some inequalities of bounded linear operators on a Hilbert space due to Dragomir (Inequalities for the numerical radius of linear operators in Hilbert spaces. Springer briefs in mathematics, Springer, Berlin, 2013; Tamkang J Math 39(1):1–7, 2008) and Kittaneh et al. (Linear Algebra Appl 471:46–53, 2015) are proved on a semi-Hilbert space and some more related results are also obtained. PubDate: 2021-09-20
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Abstract: In this paper, we prove a strengthening of the generic vanishing result in characteristic \(p>0\) given in Hacon and Patakfalvi (Am J Math 138(4):963–998, 2016). As a consequence of this result, we show that irreducible \(\Theta \) divisors are strongly F-regular and we prove a related result for pluri-theta divisors. PubDate: 2021-09-06
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Abstract: The aim of this note is to find those commutative rings over which an exact analogue of the structure theory of injective modules over commutative noetherian rings holds for weak-injective modules, i.e. for modules M satisfying \(\mathop {\mathrm{Ext}}\nolimits _R^1(A,M)=0\) for all modules A of weak dimension \(\le 1\) . We will show that, surprisingly, but a very few commutative rings R possess the property that their weak-injective modules admit (up to isomorphism) unique decompositions into direct sums of indecomposable modules each of which is the injective or the weak-injective envelope of a cyclic module of the form \(R/{\mathsf {p}} \) with a prime ideal \({\mathsf {p}} \) . PubDate: 2021-09-01
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Abstract: We study the regularity properties of Green’s function \(G_T({\mathbf {x}},t {\mathbf {x}}_0,t_0)\) associated to Robin’s problem for a class of second order operators on \(\Omega \times ]-T,T[,\) \(\Omega \subset {\mathbb {R}}^n\) a bounded open set with regular boundary, including the hyperbolic heat equation for anisotropic non homogeneous bodies with constant thermal properties as a particular case. We show that for every source point \(({\mathbf {x}}_0,t_0)\) and every \(k\in {\mathbb {N}}\) there is an open set \({\mathcal {O}}_k\subset \Omega \times ]-T,T[\) such that \(G_T({\mathbf {x}},t {\mathbf {x}}_0,t_0)\) is k times differentiable with continuity on \({\mathcal {O}}_k\) and \(\mu \bigl ((\Omega \times ]-T,T[)\backslash {\mathcal {O}}_k\bigr )=0.\) PubDate: 2021-09-01
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Abstract: The singular set of a viscosity solution to a Hamilton–Jacobi equation is known to propagate, from any noncritical singular point, along singular characteristics which are curves satisfying certain differential inclusions. In the literature, different notions of singular characteristics were introduced. However, a general uniqueness criterion for singular characteristics, not restricted to mechanical systems or problems in one space dimension, is missing at the moment. In this paper, we prove that, for a Tonelli Hamiltonian on \(\mathbb {R}^2\) , two different notions of singular characteristics coincide up to a bi-Lipschitz reparameterization. As a significant consequence, we obtain a uniqueness result for the class of singular characteristics that was introduced by Khanin and Sobolevski in the paper [On dynamics of Lagrangian trajectories for Hamilton-Jacobi equations. Arch. Ration. Mech. Anal., 219(2):861–885, 2016]. PubDate: 2021-09-01
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Abstract: Let D be an integral domain with quotient field K and X an indeterminate over K. The set \(\mathrm {Int}(D):=\{f\in K[X]: f(D)\subseteq D\}\) , of integer-valued polynomials over D, is known to be a ring. The purpose of this note is to calculate the Krull dimension of rings between D[X] and \(\mathrm {Int}(D)\) when D is either locally essential or t-locally Noetherian. By the way, we find a problem in the proof of the first main result of Fontana and Kabbaj (Proc Am Math Soc 132:2529–2535) and then we will recover it. Also, we give some examples to illustrate our main result. PubDate: 2021-09-01
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Abstract: We show that in a noetherian commutative unital topological algebra, the prime ideals associated with a closed ideal as well as its isolated primary components are closed. We obtain a version of the closed graph theorem. An example of a noetherian (even principal) commutative unital semi-simple and incomplete normed algebra whose each ideal is closed is also given. PubDate: 2021-09-01
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Abstract: In this paper, our purpose is to establish new Nomizu–Smyth type results concerning complete Riemannian immersions in the hyperbolic and de Sitter spaces. Assuming that the image of the Gauss map of such an immersion lies in a totally umbilical hypersurface and supposing the existence of a point of the immersion where the Gauss–Kronecker curvature does not vanish, we prove that the immersion must be a totally umbilical hypersurface of the ambient space. PubDate: 2021-09-01
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Abstract: An estimate of the Dunkl kernels associated with the root systems of type \(B_2 \) is derived, providing an improvement for the well-known exponential-type estimate. PubDate: 2021-09-01
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Abstract: We prove a regularity result for minimal configurations of variational problems involving both bulk and surface energies in some bounded open region \(\varOmega \subseteq {\mathbb {R}}^n\) . We will deal with the energy functional \({\mathscr {F}}(v,E):=\int _\varOmega [F(\nabla v)+1_E G(\nabla v)+f_E(x,v)]\,dx+P(E,\varOmega )\) . The bulk energy depends on a function v and its gradient \(\nabla v\) . It consists in two strongly quasi-convex functions F and G, which have polinomial p-growth and are linked with their p-recession functions by a proximity condition, and a function \(f_E\) , whose absolute valuesatisfies a q-growth condition from above. The surface penalization term is proportional to the perimeter of a subset E in \(\varOmega \) . The term \(f_E\) is allowed to be negative, but an additional condition on the growth from below is needed to prove the existence of a minimal configuration of the problem associated with \({\mathscr {F}}\) . The same condition turns out to be crucial in the proof of the regularity result as well. If (u, A) is a minimal configuration, we prove that u is locally Hölder continuous and A is equivalent to an open set \({\tilde{A}}\) . We finally get \(P(A,\varOmega )={\mathscr {H}}^{n-1}(\partial {\tilde{A}}\cap \varOmega \) ). PubDate: 2021-09-01
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Abstract: We establish Picone’s identity for p-Laplace operator and biharmonic operator on hyperbolic space. We consider both, the ball model \({\mathbb {B}}^N\) and the half space model \(\mathbb {H}^N.\) As an application of Picone’s identity, we show that the principal eigenvalue \(\lambda _1\) of \(- \Delta _{p, \mathbb {H}}\) is simple. We also obtain a Hardy type inequality on hyperbolic space. PubDate: 2021-09-01
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Abstract: We study a family of surfaces of general type with \(p_g=q=2\) and \(K^2=7\) , originally constructed by C. Rito in [35]. We provide an alternative construction of these surfaces, that allows us to describe their Albanese map and the corresponding locus \(\mathcal {M}\) in the moduli space of surfaces of general type. In particular we prove that \(\mathcal {M}\) is an open subset, and it has three connected components, all of which are 2-dimensional, irreducible and generically smooth PubDate: 2021-08-11
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Abstract: The Kuramoto–Velarde equation describes slow space-time variations of disturbances at interfaces, diffusion–reaction fronts and plasma instability fronts. It also describes Benard–Marangoni cells that occur when there is large surface tension on the interface in a microgravity environment. Under appropriate assumption on the initial data, of the time T, and the coefficients of such equation, we prove the well-posedness of the classical solutions for the Cauchy problem, associated with this equation. PubDate: 2021-07-20
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Abstract: We prove a generalization of the Fulton–Hansen connectedness theorem, where \({\mathbb {P}}^n\) is replaced by a normal variety on which an algebraic group acts with a dense orbit. PubDate: 2021-07-15
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Abstract: In this paper, we present several numerical radius and norm inequalities for sum of Hilbert space operators. These inequalities improve some earlier related inequalities. For an operator T ∈ B(H), we prove that $$ {\omega}^{2} \left( T \right) \le \frac{1}{2}\omega \left( {T^{2} } \right) + \frac{1}{{2\sqrt 2 }}\omega \left( {\left T \right ^{2} + i\left {T^{\ast} } \right ^{2} } \right) .$$ PubDate: 2021-07-07
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Abstract: We give finite presentations for the fundamental group of moduli spaces due to Miranda of smooth Weierstrass curves over \({\mathbf {P}}^1\) which extend the classical result for elliptic curves to the relative situation over the projective line. We thus get natural generalisations of \(SL_2{{\mathbb {Z}}}\) presented in terms of \(\Bigg (\begin{array}{ll} 1&{}1\\ 0&{}1\end{array} \Bigg )\) , \(\Bigg (\begin{array}{ll} 1&{}0\\ {-1}&{}1\end{array} \Bigg )\) on one hand and the first examples of fundamental groups of moduli stacks of elliptic surfaces on the other. Our approach exploits the natural \({\mathbb {Z}}_2\) -action on Weierstrass curves and the identification of \({\mathbb {Z}}_2\) -fixed loci with smooth hypersurfaces in an appropriate linear system on a projective line bundle over \({{\mathbf {P}}}^1\) . The fundamental group of the corresponding discriminant complement can be presented in terms of finitely many generators and relations using methods in the Zariski tradition. PubDate: 2021-06-28
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Abstract: Every Krull monoid has a transfer homomorphism onto a monoid of zero-sum sequences over a subset of its class group. This transfer homomorphism is a crucial tool for studying the arithmetic of Krull monoids. In the present paper, we strengthen and refine this tool for Krull monoids with finitely generated class group. PubDate: 2021-06-28
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Abstract: An inequality proved firstly by Remak and then generalized by Friedman shows that there are only finitely many number fields with a fixed signature and whose regulator is less than a prescribed bound. Using this inequality, Astudillo, Diaz y Diaz, Friedman and Ramirez-Raposo succeeded to detect all fields with small regulators having degree less or equal than 7. In this paper we show that a certain upper bound for a suitable polynomial, if true, can improve Remak–Friedman’s inequality and allows a classification for some signatures in degree 8 and better results in degree 5 and 7. The validity of the conjectured upper bound is extensively discussed. PubDate: 2021-06-22
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