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Abstract: Abstract We explicitly construct global attractors of fully nonlinear parabolic equations in one spatial dimension. These attractors are decomposed as equilibria (time independent solutions) and heteroclinic orbits (solutions that converge to distinct equilibria backwards and forwards in time). In particular, we state necessary and sufficient conditions for the occurrence of heteroclinics between hyperbolic equilibria, which is accompanied by a method that computes such conditions. PubDate: 2022-08-03

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Abstract: Abstract In this paper we study the impact of the zero order term in the study of Dirichlet problems with convection or drift terms. PubDate: 2022-07-20

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Abstract: Abstract In the present paper, we consider the k-Nearest Neighbors (k-NN) method in the single index regression model in the case of a functional predictor and a scalar response. The main result of this work is the establishment of the almost complete convergence rates for the quasi-associated data. The obtained results rely on the classical functional kernel estimate. Some simulation studies are carried out to show the finite sample performances of the k-NN estimator. Finally, we show how to use the k-NN estimator of the functional single index regression model in the analysis of air quality to illustrate the effectiveness of our methodology. PubDate: 2022-07-06

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Abstract: Abstract We continue the study of the space \(BV^\alpha ({\mathbb {R}}^n)\) of functions with bounded fractional variation in \({\mathbb {R}}^n\) of order \(\alpha \in (0,1)\) introduced in our previous work (Comi and Stefani in J Funct Anal 277(10):3373–3435, 2019). After some technical improvements of certain results of Comi and Stefani (2019) which may be of some separated insterest, we deal with the asymptotic behavior of the fractional operators involved as \(\alpha \rightarrow 1^-\) . We prove that the \(\alpha \) -gradient of a \(W^{1,p}\) -function converges in \(L^p\) to the gradient for all \(p\in [1,+\infty )\) as \(\alpha \rightarrow 1^-\) . Moreover, we prove that the fractional \(\alpha \) -variation converges to the standard De Giorgi’s variation both pointwise and in the \(\Gamma \) -limit sense as \(\alpha \rightarrow 1^-\) . Finally, we prove that the fractional \(\beta \) -variation converges to the fractional \(\alpha \) -variation both pointwise and in the \(\Gamma \) -limit sense as \(\beta \rightarrow \alpha ^-\) for any given \(\alpha \in (0,1)\) . PubDate: 2022-06-20

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Abstract: Abstract We consider a Dirichlet double phase problem with variable exponents and nonstandard growth. The reaction has the competing effects of a parametric concave term and a superlinear perturbation (“concave–convex problem”). We show that for all small values of the parameter the problem has at least two nontrivial bounded solutions. PubDate: 2022-06-18

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Abstract: Abstract In this paper, we develop the notion of c-almost periodicity for functions defined on vertical strips in the complex plane. As a generalization of Bohr’s concept of almost periodicity, we study the main properties of this class of functions which was recently introduced for the case of one real variable. In fact, we extend some important results of this theory which were already demonstrated for some particular cases. In particular, given a non-null complex number c, we prove that the family of vertical translates of a prefixed c-almost periodic function defined in a vertical strip U is relatively compact on any vertical substrip of U, which leads to proving that every c-almost periodic function is also almost periodic and, in fact, \(c^m\) -almost periodic for each integer number m. PubDate: 2022-06-16

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Abstract: Given a family \({\mathcal {Z}}=\{\Vert \cdot \Vert _{Z_Q}\}\) of norms or quasi-norms with uniformly bounded triangle inequality constants, where each Q is a cube in \({\mathbb {R}}^n\) , we provide an abstract estimate of the form $$\begin{aligned} \Vert f-f_{Q,\mu }\Vert _{Z_Q}\le c(\mu )\psi ({\mathcal {Z}})\Vert f\Vert _{\mathrm {BMO}(\mathrm {d}\mu )} \end{aligned}$$ for every function \(f\in \mathrm {BMO}(\mathrm {d}\mu )\) , where \(\mu \) is a doubling measure in \({\mathbb {R}}^n\) and \(c(\mu )\) and \(\psi ({\mathcal {Z}})\) are positive constants depending on \(\mu \) and \({\mathcal {Z}}\) , respectively. That abstract scheme allows us to recover the sharp estimate $$\begin{aligned} \Vert f-f_{Q,\mu }\Vert _{L^p \left( Q,\frac{\mathrm {d}\mu (x)}{\mu (Q)}\right) }\le c(\mu )p\Vert f\Vert _{\mathrm {BMO}(\mathrm {d}\mu )}, \qquad p\ge 1 \end{aligned}$$ for every cube Q and every \(f\in \mathrm {BMO}(\mathrm {d}\mu )\) , which is known to be equivalent to the John–Nirenberg inequality, and also enables us to obtain quantitative counterparts when \(L^p\) is replaced by suitable strong and weak Orlicz spaces and \(L^{p(\cdot )}\) spaces. Besides the aforementioned results we also generalize [(Ombrosi in Isr J Math 238:571-591, 2020), Theorem 1.2] to the setting of doubling measures and obtain a new characterization of Muckenhoupt’s \(A_\infty \) weights. PubDate: 2022-06-07

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Abstract: Abstract We introduce meromorphic nearby cycle functors and study their functorial properties. Moreover we apply them to monodromies of meromorphic functions in various situations. Combinatorial descriptions of their reduced Hodge spectra and Jordan normal forms will be obtained. PubDate: 2022-05-27 DOI: 10.1007/s13163-022-00431-4

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Abstract: 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 DOI: 10.1007/s13163-022-00425-2

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Abstract: 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 DOI: 10.1007/s13163-022-00426-1

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Abstract: 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 DOI: 10.1007/s13163-022-00430-5

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Abstract: 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 DOI: 10.1007/s13163-021-00396-w

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Abstract: 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 DOI: 10.1007/s13163-021-00389-9

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Abstract: 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 DOI: 10.1007/s13163-021-00394-y

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Abstract: 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 DOI: 10.1007/s13163-021-00399-7

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Abstract: 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 DOI: 10.1007/s13163-021-00392-0

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Abstract: 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 DOI: 10.1007/s13163-021-00397-9

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Abstract: 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 DOI: 10.1007/s13163-021-00390-2

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Abstract: 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 DOI: 10.1007/s13163-021-00395-x

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Abstract: 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 DOI: 10.1007/s13163-021-00393-z