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Abstract: We compute the cones of effective divisors on blowups of \(\mathbb P^1 \times \mathbb P^2\) and \(\mathbb P^1 \times \mathbb P^3\) in up to 6 points. We also show that all these varieties are log Fano, giving a conceptual explanation for the fact that all the cones we compute are rational polyhedral. PubDate: 2022-05-18

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Abstract: We study linear and algebraic structures in sets of Dirichlet series with maximal Bohr’s strip. More precisely, we consider a set \({\mathscr {M}}\) of Dirichlet series which are uniformly continuous on the right half plane and whose strip of uniform but not absolute convergence has maximal width, i.e., \(\nicefrac {1}{2}\) . Considering the uniform norm, we show that \({\mathscr {M}}\) contains an isometric copy of \(\ell _1\) (except zero) and is strongly \(\aleph _0\) -algebrable. Also, there is a dense \(G_\delta \) set such that any of its elements generates a free algebra contained in \({\mathscr {M}}\cup \{0\}\) . Furthermore, we investigate \(\mathscr {M}\) as a subset of the Hilbert space of Dirichlet series whose coefficients are square-summable. In this case, we prove that \({\mathscr {M}}\) contains an isometric copy of \(\ell _2\) (except zero). PubDate: 2022-05-17

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Abstract: In the infinite dimensional Heisenberg group, we construct a left invariant weak Riemannian metric that gives a degenerate geodesic distance. The same construction yields a degenerate sub-Riemannian distance. We show how the standard notion of sectional curvature adapts to our framework, but it cannot be defined everywhere and it is unbounded on suitable sequences of planes. The vanishing of the distance precisely occurs along this sequence of planes, so that the degenerate Riemannian distance appears in connection with an unbounded sectional curvature. In the 2005 paper by Michor and Mumford, this phenomenon was first observed in some specific Fréchet manifolds. PubDate: 2022-05-16

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Abstract: In this paper we consider nonnegative functions f on \(\mathbb {R}^n\) which are defined either by \(f(x)=\min \,(f_1(x_1),\ldots ,f_n(x_n))\) or by \(f(x)=\min \,(f_1(\hat{x}_1),\ldots ,f_n(\hat{x}_n)).\) Such minimum-functions are useful, in particular, in embedding theorems. We prove sharp estimates of rearrangements and Lorentz type norms for these functions, and we find the link between their Lorentz norms and geometric properties of their level sets. PubDate: 2022-05-01

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Abstract: We study integrability conditions for existence and nonexistence of a local-in-time integral solution of fractional semilinear heat equations with rather general growing nonlinearities in uniformly local \(L^p\) spaces. Our main results about this matter consist of Theorems 1.4, 1.6, 5.1 and 5.3. We introduce a supersolution of an integral equation which can be applied to a nonlocal parabolic equation. When the nonlinear term is \(u^p\) or \(e^u\) , a local-in-time solution can be constructed in the critical case, and integrability conditions for the existence and nonexistence are completely classified. Our analysis is based on the comparison principle, Jensen’s inequality and \(L^p\) - \(L^q\) type estimates. PubDate: 2022-05-01

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Abstract: In this paper we establish a natural framework for the stability of mean curvature flow solitons in warped product spaces. These solitons are regarded as stationary immersions for a weighted volume functional. Under this point of view, we are able to find geometric conditions for finiteness of the index and some characterizations of stable solitons. We also prove some non-existence results for solitons as applications of a comparison principle which suits well the structure of the diffusion elliptic operator associated to the weighted measures we are considering. PubDate: 2022-05-01

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Abstract: We improve the discretization technique for weighted Lorentz norms by eliminating all “non-degeneracy” restrictions on the involved weights. We use the new method to provide equivalent estimates on the optimal constant C such that the inequality $$\begin{aligned} \left( \int _0^L (f^*(t))^{q} w(t)\,\mathrm {d} t\right) ^\frac{1}{q} \le C \left( \int _0^L \left( \int _0^t u(s)\,\mathrm {d} s\right) ^{-p} \left( \int _0^t f^*(s) u(s) \,\mathrm {d} s\right) ^p v(t) \,\mathrm {d} t\right) ^\frac{1}{p} \end{aligned}$$ holds for all relevant measurable functions, where \(L\in (0,\infty ]\) , \(p, q \in (0,\infty )\) and u, v, w are locally integrable weights, u being strictly positive. In the case of weights that would be otherwise excluded by the restrictions, it is shown that additional limit terms naturally appear in the characterizations of the optimal C. A weak analogue for \(p=\infty \) is also presented. PubDate: 2022-05-01

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Abstract: We consider left-invariant (purely) coclosed G \(_2\) -structures on 7-dimensional 2-step nilpotent Lie groups. According to the dimension of the commutator subgroup, we obtain various criteria characterizing the Riemannian metrics induced by left-invariant purely coclosed G \(_2\) -structures. Then, we use them to determine the isomorphism classes of 2-step nilpotent Lie algebras admitting such type of structures. As an intermediate step, we show that every metric on a 2-step nilpotent Lie algebra admitting coclosed G \(_2\) -structures is induced by one of them. Finally, we use our results to give the explicit description of the metrics induced by purely coclosed G \(_2\) -structures on 2-step nilpotent Lie algebras with derived algebra of dimension at most two, up to automorphism. PubDate: 2022-05-01

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Abstract: A locally convex space (lcs) E is said to have an \(\omega ^{\omega }\) -base if E has a neighborhood base \(\{U_{\alpha }:\alpha \in \omega ^\omega \}\) at zero such that \(U_{\beta }\subseteq U_{\alpha }\) for all \(\alpha \le \beta \) . The class of lcs with an \(\omega ^{\omega }\) -base is large, among others contains all (LM)-spaces (hence (LF)-spaces), strong duals of distinguished Fréchet lcs (hence spaces of distributions \(D^{\prime }(\Omega )\) ). A remarkable result of Cascales-Orihuela states that every compact set in an lcs with an \(\omega ^{\omega }\) -base is metrizable. Our main result shows that every uncountable-dimensional lcs with an \(\omega ^{\omega }\) -base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional vector space \(\varphi \) endowed with the finest locally convex topology has an \(\omega ^\omega \) -base but contains no infinite-dimensional compact subsets. It turns out that \(\varphi \) is a unique infinite-dimensional locally convex space which is a \(k_{\mathbb {R}}\) -space containing no infinite-dimensional compact subsets. Applications to spaces \(C_{p}(X)\) are provided. PubDate: 2022-05-01

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Abstract: This paper concerns the study of the asymptotic behavior of the solutions to a family of fractional type problems on a bounded domain, satisfying homogeneous Dirichlet boundary conditions. The family of differential operators includes the fractional \(p_n\) -Laplacian when \(p_n\rightarrow \infty \) as a particular case, tough it could be extended to a function of the Hölder quotient of order s, whose primitive is an Orlicz function satisfying appropriated growth conditions. The limit equation involves the Hölder infinity Laplacian. PubDate: 2022-05-01

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Abstract: We consider a Dirichlet problem driven by the anisotropic (p, q)-Laplacian and with a reaction that has the competing effects of a singular term and of a parametric superlinear perturbation. Based on variational tools along with truncation and comparison techniques, we prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter varies. PubDate: 2022-05-01

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Abstract: In the setting of the semigroup generated by the Schrödinger operator \(L= -\Delta +V\) with the potential V satisfying an appropriate reverse Hölder condition, we consider some non-local fractional differentiation operators. We study their behaviour on suitable weighted smoothness spaces. Actually, we obtain such continuity results for positive powers of L as well as for the mixed operators \(L^{\alpha /2}V^{\sigma /2}\) and \(L^{-\alpha /2}V^{\sigma /2}\) with \(\sigma >\alpha \) , together with their adjoints. PubDate: 2022-05-01

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Abstract: We consider the problem of finding the limit at infinity (corresponding to the downward Morse flow) of a Higgs bundle in the nilpotent cone under the natural \(\mathbb {C}^*\) -action on the moduli space. For general rank we provide an answer for Higgs bundles with regular nilpotent Higgs field, while in rank three we give the complete answer. Our results show that the limit can be described in terms of data defined by the Higgs field, via a filtration of the underlying vector bundle. PubDate: 2022-05-01

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Abstract: Consider the class QS of all non-degenerate planar quadratic systems and its subclass QSE of all its systems possessing an invariant ellipse. This is an interesting family because on one side it is defined by an algebraic geometric property and on the other, it is a family where limit cycles occur. Note that each quadratic differential system can be identified with a point of \({{\mathbb {R}}}^{12}\) through its coefficients. In this paper we provide necessary and sufficient conditions for a system in QS to have at least one invariant ellipse. We give the global “bifurcation” diagram of the family QS which indicates where an ellipse is present or absent and in case it is present, the diagram indicates if the ellipse is or it is not a limit cycle. The diagram is expressed in terms of affine invariant polynomials and it is done in the 12-dimensional space of parameters. This diagram is also an algorithm for determining for any quadratic system if it possesses an invariant ellipse and whether or not this ellipse is a limit cycle. PubDate: 2022-05-01

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Abstract: In this manuscript we deal with existence/uniqueness and regularity issues of suitable weak solutions to nonlocal problems driven by fractional Laplace type operators. Different from previous researches, in our approach we consider gradient non-linearity sources with subcritical growth, as well as appropriated measures as sources and boundary datum. We provide an in-depth discussion on the notions of solutions involved together with existence/uniqueness results in different regimes and for different boundary value problems. Finally, this work extends previous ones by dealing with more general nonlocal operators, source terms and boundary data. PubDate: 2022-05-01

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Abstract: Abstract Let \(G\left( k,n\right) \) be the Grassmannian of oriented subspaces of dimension k of \(\mathbb {R}^{n}\) with its canonical Riemannian metric. We study the energy of maps assigning to each \(P\in G\left( k,n\right) \) a unit vector normal to P. They are sections of a sphere bundle \(E_{k,n}^{1}\) over \(G\left( k,n\right) \) . The octonionic double and triple cross products induce in a natural way such sections for \(k=2\) , \(n=7\) and \(k=3\) , \(n=8\) , respectively. We prove that they are harmonic maps into \(E_{k,n}^{1}\) endowed with the Sasaki metric. This, together with the well-known result that Hopf vector fields on odd dimensional spheres are harmonic maps into their unit tangent bundles, allows us to conclude that all unit normal sections of the Grassmannians associated with cross products are harmonic. In a second instance we analyze the energy of maps assigning an orthogonal complex structure \(J\left( P\right) \) on \(P^{\bot }\) to each \(P\in G\left( 2,8\right) \) . We prove that the one induced by the octonionic triple product is a harmonic map into a suitable sphere bundle over \(G\left( 2,8\right) \) . This generalizes the harmonicity of the canonical almost complex structure of \(S^{6}\) . PubDate: 2022-04-26

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Abstract: Abstract The \(\mathbb Z\) -genus of a link L in \(S^3\) is the minimal genus of a locally flat, embedded, connected surface in \(D^4\) whose boundary is L and with the fundamental group of the complement infinite cyclic. We characterise the \(\mathbb Z\) -genus of boundary links in terms of their single variable Blanchfield forms, and we present some applications. In particular, we show that a variant of the shake genus of a knot, the \(\mathbb Z\) -shake genus, equals the \(\mathbb Z\) -genus of the knot. PubDate: 2022-04-15

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Abstract: Abstract Let \(\{h_n\}\) be a sequence in \({\mathbb {R}}^d\) tending to infinity and let \(\{T_{h_n}\}\) be the corresponding sequence of shift operators given by \((T_{h_n}f)(x)=f(x-h_n)\) for \(x\in {\mathbb {R}}^d\) . We prove that \(\{T_{h_n}\}\) converges weakly to the zero operator as \(n\rightarrow \infty \) on a separable rearrangement-invariant Banach function space \(X({\mathbb {R}}^d)\) if and only if its fundamental function \(\varphi _X\) satisfies \(\varphi _X(t)/t\rightarrow 0\) as \(t\rightarrow \infty \) . On the other hand, we show that \(\{T_{h_n}\}\) does not converge weakly to the zero operator as \(n\rightarrow \infty \) on all Marcinkiewicz endpoint spaces \(M_\varphi ({\mathbb {R}}^d)\) and on all non-separable Orlicz spaces \(L^\varPhi ({\mathbb {R}}^d)\) . Finally, we prove that if \(\{h_n\}\) is an arithmetic progression: \(h_n = nh\) , \(n \in {\mathbb {N}}\) with an arbitrary \(h\in {\mathbb {R}}^d{\setminus }\{0\}\) , then \(\{T_{nh}\}\) does not converge weakly to the zero operator on any non-separable rearrangement-invariant Banach function space \(X({\mathbb {R}}^d)\) as \(n\rightarrow \infty \) . PubDate: 2022-03-22

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Abstract: Abstract A closed Riemann surface S is called a generalized Fermat curve of type (p, n), where \(n,p \ge 2\) are integers such that \((p-1)(n-1)>2\) , if it admits a group \(H \cong {\mathbb Z}_{p}^{n}\) of conformal automorphisms with quotient orbifold S/H of genus zero with exactly \(n+1\) cone points, each one of order p; in this case H is called a generalized Fermat group of type (p, n). In this case, it is known that S is non-hyperelliptic and that H is its unique generalized Fermat group of type (p, n). Also, explicit equations for them, as a fiber product of classical Fermat curves of degree p, are known. For p a prime integer, we describe those subgroups K of H acting freely on S, together with algebraic equations for S/K, and determine those K such that S/K is hyperelliptic. PubDate: 2022-02-24 DOI: 10.1007/s13163-022-00422-5

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Abstract: Abstract In this manuscript, we will study the asymptotic behavior for a class of nonlocal diffusion equations associated with the weighted fractional \(\wp (\cdot )\) -Laplacian operator involving constant/variable exponent, with \(\wp ^{-}:=\min _{(x,y) \in {\overline{\Omega }}\times {\overline{\Omega }}} \wp (x,y)\geqslant \max \left\{ 2N/(N+2s),1\right\} \) and \(s\in (0,1).\) In the case of constant exponents, under some appropriate conditions, we will study the existence of solutions and asymptotic behavior of solutions by employing the subdifferential approach and we will study the problem when \(\wp \) goes to \(\infty \) . Already, for case the weighted fractional \(\wp (\cdot )\) -Laplacian operator, we will also study the asymptotic behavior of the problem solution when \(\wp (\cdot )\) goes to \(\infty \) , in the whole or in a subset of the domain (the problem involving the fractional \(\wp (\cdot )\) -Laplacian presents a discontinuous exponent). To obtain the results of the asymptotic behavior in both problems it will be via Mosco convergence. PubDate: 2022-01-31 DOI: 10.1007/s13163-021-00419-6