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Abstract: Abstract We study the existence and uniqueness of renormalized solutions for initial boundary value problems of the type $$\begin{aligned} \left( {\mathcal {P}}_{b}^{1}\right) \quad \left\{ \begin{aligned} u_{t}-\text {div}(a(t,x,\nabla u))=H(u)\mu \text { in }Q:=(0,T)\times \Omega ,\\ u(0,x)=u_{0}(x)\text { in }\Omega ,\ u(t,x)=0\text { on }(0,T)\times \partial \Omega , \end{aligned}\right. \end{aligned}$$ where \(u_{0}\in L^{1}(\Omega )\) , \(\mu \in {\mathcal {M}}_{b}(Q)\) is a general Radon measure on Q and \(H\in C_{b}^{0}({\mathbb {R}})\) is a continuous positive bounded function on \({\mathbb {R}}\) . The difficulties in the study of such problems concern the possibly very singular right-hand side that forces the choice of a suitable formulation that ensures both existence and uniqueness of solution. Using similar techniques, we will prove existence/nonexistence results of the auxiliary problem $$\begin{aligned} \left( {\mathcal {P}}_{b}^{2}\right) \quad \left\{ \begin{aligned}&u_{t}-\text {div}(a(t,x,\nabla u))+g(x,u) \nabla u ^{2}=\mu \text { in }Q:=(0,T)\times \Omega ,\\&u(0,x)=u_{0}(x)\text { in }\Omega ,\ u(t,x)=0\text { on }(0,T)\times \partial \Omega , \end{aligned}\right. \end{aligned}$$ under the assumption that g satisfies a sign condition and the nonlinear term depends on both x, u and its gradient. Thus, our results improve and complete the previous known existence results for problems \(\left( {\mathcal {P}}_{b}^{1,2}\right) \) . PubDate: 2022-12-13
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Abstract: Abstract Carmesin has extended Robertson and Seymour’s tree-of-tangles theorem to the infinite tangles of locally finite infinite graphs. We extend it further to the infinite tangles of all infinite graphs. Our result has a number of applications for the topology of infinite graphs, such as their end spaces and their compactifications. PubDate: 2022-11-29
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Abstract: Abstract In this article we study the relation between flat solvmanifolds and \(G_2\) -geometry. First, we give a classification of 7-dimensional flat splittable solvmanifolds using the classification of finite subgroups of \(\mathsf{GL}(n,\mathbb {Z})\) for \(n=5\) and \(n=6\) . Then, we look for closed, coclosed and divergence-free \(G_2\) -structures compatible with the flat metric on them. In particular, we provide explicit examples of compact flat manifolds with a torsion-free \(G_2\) -structure whose finite holonomy is cyclic and contained in \(G_2\) , and examples of compact flat manifolds admitting a divergence-free \(G_2\) -structure. PubDate: 2022-11-11 DOI: 10.1007/s12188-022-00261-7
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Abstract: Abstract In this paper we present a new approach to prove effective results in Diophantine approximation. This approach involves measures of local positivity of divisors combined with Faltings’s version of Siegel’s lemma instead of a zero estimate such as Dyson’s lemma. We then use it to prove an effective theorem on the simultaneous approximation of two algebraic numbers satisfying an algebraic equation with complex coefficients. PubDate: 2022-11-08 DOI: 10.1007/s12188-022-00260-8
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Abstract: Abstract In previous works, the authors investigated the relationships between linear stability of a generated linear series V on a curve C, and slope stability of the syzygy vector bundle \(M_{V,L} := \ker (V \otimes \mathcal {O}_C \rightarrow L)\) . In particular, the second named author and L. Stoppino conjecture that, for a complete linear system L , linear (semi)stability is equivalent to slope (semi)stability of \(M_{L}\) . The first and third named authors proved that this conjecture holds in the two opposite cases: hyperelliptic and generic curves. In this work we provide a counterexample to this conjecture on any smooth plane curve of degree 7. PubDate: 2022-05-30 DOI: 10.1007/s12188-022-00258-2
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Abstract: Abstract In 2021 the authors D. Bravo, M. Lanzilotta, O. Mendoza and J. Vivero gave a generalization of the concept of Igusa-Todorov algebra and proved that those algebras, named Lat-Igusa-Todorov (LIT for short), satisfy the finitistic dimension conjecture. In this paper we explore the scope of that generalization and give conditions for a triangular matrix algebra to be LIT in terms of the algebras and the bimodule used in its definition. As an application we obtain that the tensor product of an LIT \(\mathbb {K}\) -algebra with a path algebra of a quiver whose underlying graph is a tree, is LIT. PubDate: 2022-05-24 DOI: 10.1007/s12188-022-00257-3
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Abstract: Abstract We give a formula for certain values and derivatives of Siegel series and use them to compute Fourier coefficients of derivatives of the Siegel Eisenstein series of weight \(\frac{g}{2}\) and genus g. When \(g=4\) , the Fourier coefficient is approximated by a certain Fourier coefficient of the central derivative of the Siegel Eisenstein series of weight 2 and genus 3, which is related to the intersection of 3 arithmetic modular correspondences. Applications include a relation between weighted averages of representation numbers of symmetric matrices. PubDate: 2022-04-27 DOI: 10.1007/s12188-022-00256-4
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Abstract: Abstract Let p be an odd prime. In this paper we compute the Fourier coefficients of the Siegel Eisenstein series of degree 2, level p with the trivial or the quadratic character, associated to a certain cusp. For that we need to define the p-factor of the special type of Siegel series with character. PubDate: 2022-01-18 DOI: 10.1007/s12188-021-00255-x
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Abstract: Abstract We already have the concept of isotropicity of a minimal surface in a Riemannian 4-manifold and a space-like or time-like surface in a neutral 4-manifold with zero mean curvature vector. In this paper, based on the understanding of it, we define and study isotropicity of a space-like or time-like surface in a Lorentzian 4-manifold N with zero mean curvature vector. If the surface is space-like, then the isotropicity means either the surface has light-like or zero second fundamental form or it is an analogue of complex curves in Kähler surfaces. In addition, if N is a space form, then the isotropicity means that the surface has both the properties. If the surface is time-like and if N is a space form, then the isotropicity means that the surface is totally geodesic. PubDate: 2022-01-03 DOI: 10.1007/s12188-021-00254-y
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Abstract: Abstract It is proved that decomposable graphs are set recognizable and that the index graph of the canonical decomposition as well as the graphs induced on the maximal autonomous sets of vertices are set reconstructible. From these results, we obtain set reconstructibility for many decomposable graphs as well as a concise description of the decomposable graphs for which set reconstruction remains an open problem. PubDate: 2022-01-03 DOI: 10.1007/s12188-021-00252-0
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Abstract: Abstract We study sign changes and non-vanishing of a certain double sequence of Fourier coefficients of cusp forms of half-integral weight. PubDate: 2021-12-14 DOI: 10.1007/s12188-021-00253-z
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Abstract: Abstract Combining theorems of Voisin and Marian, Shen, Yin and Zhao, we compute the dimensions of the orbits under rational equivalence in the Mukai system of rank two and genus two. We produce several examples of algebraically coisotropic and constant cycle subvarieties. PubDate: 2021-10-01 DOI: 10.1007/s12188-021-00251-1
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Abstract: Abstract In this paper, we consider a kind of area-preserving flow for closed convex planar curves which will decrease the length of the evolving curve and make the evolving curve more and more circular during the evolution process. And the final shape of the evolving curve will be a circle as time \(t\rightarrow +\infty \) . PubDate: 2021-10-01 DOI: 10.1007/s12188-021-00249-9
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Abstract: Abstract We study p-adic L-functions \(L_p(s,\chi )\) for Dirichlet characters \(\chi \) . We show that \(L_p(s,\chi )\) has a Dirichlet series expansion for each regularization parameter c that is prime to p and the conductor of \(\chi \) . The expansion is proved by transforming a known formula for p-adic L-functions and by controlling the limiting behavior. A finite number of Euler factors can be factored off in a natural manner from the p-adic Dirichlet series. We also provide an alternative proof of the expansion using p-adic measures and give an explicit formula for the values of the regularized Bernoulli distribution. The result is particularly simple for \(c=2\) , where we obtain a Dirichlet series expansion that is similar to the complex case. PubDate: 2021-08-30 DOI: 10.1007/s12188-021-00244-0
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Abstract: Abstract The moduli spaces of flat \({\text{SL}}_2\) - and \({\text{PGL}}_2\) -connections are known to be singular SYZ-mirror partners. We establish the equality of Hodge numbers of their intersection (stringy) cohomology. In rank two, this answers a question raised by Tamás Hausel in Remark 3.30 of “Global topology of the Hitchin system”. PubDate: 2021-08-21 DOI: 10.1007/s12188-021-00246-y
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Abstract: Abstract We give generators and relations for the graded rings of Hermitian modular forms of degree two over the rings of integers in \({\mathbb {Q}}(\sqrt{-7})\) and \({\mathbb {Q}}(\sqrt{-11})\) . In both cases we prove that the subrings of symmetric modular forms are generated by Maass lifts. The computation uses a reduction process against Borcherds products which also leads to a dimension formula for the spaces of modular forms. PubDate: 2021-08-12 DOI: 10.1007/s12188-021-00245-z
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Abstract: Abstract Motivated by calculations of motivic homotopy groups, we give widely attained conditions under which operadic algebras and modules thereof are preserved under (co)localization functors. PubDate: 2021-07-22 DOI: 10.1007/s12188-021-00240-4
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Abstract: Abstract In this paper we investigate growth properties and the zero distribution of polynomials attached to arithmetic functions g and h, where g is normalized, of moderate growth, and \(0<h(n) \le h(n+1)\) . We put \(P_0^{g,h}(x)=1\) and $$\begin{aligned} P_n^{g,h}(x) := \frac{x}{h(n)} \sum _{k=1}^{n} g(k) \, P_{n-k}^{g,h}(x). \end{aligned}$$ As an application we obtain the best known result on the domain of the non-vanishing of the Fourier coefficients of powers of the Dedekind \(\eta \) -function. Here, g is the sum of divisors and h the identity function. Kostant’s result on the representation of simple complex Lie algebras and Han’s results on the Nekrasov–Okounkov hook length formula are extended. The polynomials are related to reciprocals of Eisenstein series, Klein’s j-invariant, and Chebyshev polynomials of the second kind. PubDate: 2021-06-14 DOI: 10.1007/s12188-021-00241-3
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Abstract: Abstract Let M be a compact connected smooth pseudo-Riemannian manifold that admits a topologically transitive G-action by isometries, where \(G = G_1 \ldots G_l\) is a connected semisimple Lie group without compact factors whose Lie algebra is \({\mathfrak {g}}= {\mathfrak {g}}_1 \oplus {\mathfrak {g}}_2 \oplus \cdots \oplus {\mathfrak {g}}_l\) . If \(m_0,n_0,n_0^i\) are the dimensions of the maximal lightlike subspaces tangent to M, G, \(G_i\) , respectively, then we study G-actions that satisfy the condition \(m_0=n_0^1 + \cdots + n_0^{l}\) . This condition implies that the orbits are non-degenerate for the pseudo Riemannian metric on M and this allows us to consider the normal bundle to the orbits. Using the properties of the normal bundle to the G-orbits we obtain an isometric splitting of M by considering natural metrics on each \(G_i\) . PubDate: 2021-06-07 DOI: 10.1007/s12188-021-00242-2
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