A construction of nonabelian simple {\acute{\mathrm{e}}}tale fundamental groups

Under the assumption of the generalized Riemann hypothesis (GRH), we show that there is a real quadratic field \(K\) such that the \({\acute{\mathrm{e}}}\) tale fundamental group \(\pi ^{\acute{\mathrm{et}}}_1(\mathrm {Spec}\;\mathcal {O}_K)\) of the spectrum of the ring of integers \(\mathcal {O}_K\) of \(K\) is isomorphic to \(A_5\) . The proof uses standard methods involving Odlyzko bounds, as well as the proof of Serre’s modularity conjecture. To the best of the author’s knowledge, this is the first example of a number field \(K\) for which \(\pi ^{\acute{\mathrm{et}}}_1(\mathrm {Spec}\;\mathcal {O}_K)\) is finite, nonabelian and simple under the assumption of the GRH.     

The theme of a vanishing period

Consider a multivalued formal function of the type 1 $$\begin{aligned} \varphi (s) : = \sum _{j=0}^k\,c_j(s).s^{\lambda + m_j}.(\mathrm{Log}\,s)^j, \end{aligned}$$ where \(\lambda \) is a positive rational number, \(c_j\) is in \({{\mathrm{\mathbb {C}}}}[[s]]\) and \(m_j \in \mathbb {N}\) for \(j \in [0,k-1]\) . The theme associated with such a \(\varphi \) is the “minimal filtered integral equation” satisfied by \(\varphi \) , in a sense which is made precise in this article. We study such objects and show that their isomorphism classes may be characterized by a finite set of complex numbers, when we assume the Bernstein polynomial of \(\varphi \) to be fixed. For a given \(\lambda \) , to fix the Bernstein polynomial is equivalent to fix a finite set of integers associated with the logarithm of the monodromy in the geometric situation described below. Our aim is to construct some analytic invariants, for instance in the following situation, let \(f : X \rightarrow D\) be a proper holomorphic function defined on a complex manifold \(X\) with values in a disc \(D\) . We assume that the only critical value is \(0 \in D\) and we consider this situation as a degenerating family of compact complex manifolds to a singular compact complex space \(f^{-1}(0)\) . To a smooth \((p+1)\) -form \(\omega \) on \(X\) such that \(\mathrm{d}\omega = 0 = \mathrm{d}f \wedge \omega \) and to a vanishing \(p\) -cycle \(\gamma \) chosen in the generic fiber \(f^{-1}(s_0), s_0 \in D \setminus \{0\}\) , we associated a “vanishing period” \(F_{\gamma }(s) : = \int _{\gamma _s} \omega \big /\mathrm{d}f \) which has an asymptotic expansion at \(0\) of the form \((1)\) above, when \(\gamma \) is chosen in the spectral subspace of \(H_p(f^{-1}(s_0), {{\mathrm{\mathbb {C}}}})\) for the eigenvalue \(\mathrm{e}^{2i\pi .\lambda }\) of the monodromy of \(f\) . Here \((\gamma _s)_{s \in D^*}\) is the horizontal multivalued family of \(p\) -cycles in the fibers of \(f\) obtained from the choice of \(\gamma \) . The aim of this article was to study the module generated by such a \(\varphi \) over the algebra \(\tilde{\mathcal {A}}\) , which is the \(b\) -completion of the algebra \(\mathcal {A}\) generated by the operators \(\mathrm{a} : = \times s\) and \(\mathrm{b} : = \int _{0}^{s}\) .     

Cohomology classes of conormal bundles of Schubert varieties and Yangian weight functions

We consider the conormal bundle of a Schubert variety \(S_I\) in the cotangent bundle \(T^*\!{{\mathrm{\mathrm {Gr}}}}\) of the Grassmannian \({{\mathrm{\mathrm {Gr}}}}\) of \(k\) -planes in \({{\mathrm{\mathbb {C}}}}^n\) . This conormal bundle has a fundamental class \({\kappa _I}\) in the equivariant cohomology \(H^*_{{{\mathrm{\mathbb T}}}}(T^*\!\!{{\mathrm{\mathrm {Gr}}}})\) . Here \({{\mathrm{\mathbb T}}}=({{\mathrm{\mathbb {C}}}}^*)^n\times {{\mathrm{\mathbb {C}}}}^*\) . The torus \(({{\mathrm{\mathbb {C}}}}^*)^n\) acts on \(T^*\!{{\mathrm{\mathrm {Gr}}}}\) in the standard way and the last factor \({{\mathrm{\mathbb {C}}}}^*\) acts by multiplication on fibers of the bundle. We express this fundamental class as a sum \(Y_I\) of the Yangian \(Y(\mathfrak {gl}_2)\) weight functions \((W_J)_J\) . We describe a relation of \(Y_I\) with the double Schur polynomial \([S_I]\) . A modified version of the \(\kappa _I\) classes, named \(\kappa '_I\) , satisfy an orthogonality relation with respect to an inner product induced by integration on the non-compact manifold \(T^*\!{{\mathrm{\mathrm {Gr}}}}\) . This orthogonality is analogous to the well known orthogonality satisfied by the classes of Schubert varieties with respect to integration on \({{\mathrm{\mathrm {Gr}}}}\) . The classes \((\kappa '_I)_I\) form a basis in the suitably localized equivariant cohomology \(H^*_{{{\mathrm{\mathbb T}}}}(T^*\!\!{{\mathrm{\mathrm {Gr}}}})\) . This basis depends on the choice of the coordinate flag in \({{\mathrm{\mathbb {C}}}}^n\) . We show that the bases corresponding to different coordinate flags are related by the Yangian R-matrix.     

The Hardy–Rellich Inequality and Uncertainty Principle on the Sphere

Let \(\Delta _0\) be the Laplace–Beltrami operator on the unit sphere \(\mathbb {S}^{d-1}\) of \({\mathbb R}^d\) . We show that the Hardy–Rellich inequality of the form $$\begin{aligned} \mathop \int \limits _{\mathbb {S}^{d-1}} \left| f (x)\right| ^2 \mathrm{d}{\sigma }(x) \le c_d \min _{e\in \mathbb {S}^{d-1}} \mathop \int \limits _{\mathbb {S}^{d-1}} (1- {\langle }x, e {\rangle }) \left| (-\Delta _0)^{\frac{1}{2}}f(x) \right| ^2 \mathrm{d}{\sigma }(x) \end{aligned}$$ holds for \(d =2\) and \(d \ge 4\) but does not hold for \(d=3\) with any finite constant, and the optimal constant for the inequality is \(c_d = 8/(d-3)^2\) for \(d =2, 4, 5,\) and, under additional restrictions on the function space, for \(d\ge 6\) . This inequality yields an uncertainty principle of the form $$\begin{aligned} \min _{e\in \mathbb {S}^{d-1}} \mathop \int \limits _{\mathbb {S}^{d-1}} (1- {\langle }x, e {\rangle }) |f(x)|^2 \mathrm{d}{\sigma }(x) \mathop \int \limits _{\mathbb {S}^{d-1}}\left| \nabla _0 f(x)\right| ^2 \mathrm{d}{\sigma }(x) \ge c'_d \end{aligned}$$ on the sphere for functions with zero mean and unit norm, which can be used to establish another uncertainty principle without zero mean assumption, both of which appear to be new.     

Divide and Conquer Roadmap for Algebraic Sets

Let \(\mathrm{R}\) be a real closed field and \(\hbox {D}\subset \mathrm{R}\) an ordered domain. We describe an algorithm that given as input a polynomial \(P \in \hbox {D}[ X_{1} , \ldots ,X_{{ k}} ]\) and a finite set, \(\mathcal {A}= \{ p_{1} , \ldots ,p_{m} \}\) , of points contained in \(V= {\mathrm{{Zer}}} ( P, \mathrm{R}^{{ k}})\) described by real univariate representations, computes a roadmap of \(V\) containing \(\mathcal {A}\) . The complexity of the algorithm, measured by the number of arithmetic operations in \(\hbox {D}\) , is bounded by \(\big ( \sum _{i=1}^{m} D^{O ( \log ^{2} ( k ) )}_{i} +1 \big ) ( k^{\log ( k )} d )^{O ( k\log ^{2} ( k ))}\) , where \(d= \deg ( P )\) and \(D_{i}\) is the degree of the real univariate representation describing the point \(p_{i}\) . The best previous algorithm for this problem had complexity card \(( \mathcal {A} )^{O ( 1 )} d^{O ( k^{3/2} )}\) (Basu et al., ArXiv, 2012), where it is assumed that the degrees of the polynomials appearing in the representations of the points in \(\mathcal {A}\) are bounded by \(d^{O ( k )}\) . As an application of our result we prove that for any real algebraic subset \(V\) of \(\mathbb {R}^{k}\) defined by a polynomial of degree \(d\) , any connected component \(C\) of \(V\) contained in the unit ball, and any two points of \(C\) , there exists a semi-algebraic path connecting them in \(C\) , of length at most \(( k ^{\log (k )} d )^{O ( k\log ( k ) )}\) , consisting of at most \(( k ^{\log (k )} d )^{O ( k\log ( k ) )}\) curve segments of degrees bounded by \(( k ^{\log ( k )} d )^{O ( k \log ( k) )}\) . While it was known previously, by a result of D’Acunto and Kurdyka (Bull Lond Math Soc 38(6):951–965, 2006), that there always exists a path of length \(( O ( d ) )^{k-1}\) connecting two such points, there was no upper bound on the complexity of such a path.     

On the rank of incidence matrices in projective Hjelmslev spaces

Let \(R\) be a finite chain ring with \(|R|=q^m\) , \(R/{{\mathrm{Rad}}}R\cong \mathbb {F}_q\) , and let \(\Omega ={{\mathrm{PHG}}}({}_RR^n)\) . Let \(\tau =(\tau _1,\ldots ,\tau _n)\) be an integer sequence satisfying \(m=\tau _1\ge \tau _2\ge \cdots \ge \tau _n\ge 0\) . We consider the incidence matrix of all shape \(\varvec{m}^s=(\underbrace{m,\ldots ,m}_s)\) versus all shape \(\tau \) subspaces of \(\Omega \) with \(\varvec{m}^s\preceq \tau \preceq \varvec{m}^{n-s}\) . We prove that the rank of \(M_{\varvec{m}^s,\tau }(\Omega )\) over \(\mathbb {Q}\) is equal to the number of shape \(\varvec{m}^s\) subspaces. This is a partial analog of Kantor’s result about the rank of the incidence matrix of all \(s\) dimensional versus all \(t\) dimensional subspaces of \({{\mathrm{PG}}}(n,q)\) . We construct an example for shapes \(\sigma \) and \(\tau \) for which the rank of \(M_{\sigma ,\tau }(\Omega )\) is not maximal.     

Fixed points and homology of superelliptic Jacobians

Let \(\eta : C_{f,N}\rightarrow \mathbb {P}^1\) be a cyclic cover of \(\mathbb {P}^1\) of degree \(N\) which is totally and tamely ramified for all the ramification points. We determine the group of fixed points of the cyclic covering group \({{\mathrm{Aut}}}(\eta )\simeq \mathbb {Z}/ N \mathbb {Z}\) acting on the Jacobian \(J_N:={{\mathrm{Jac}}}(C_{f,N})\) . For each prime \(\ell \) distinct from the characteristic of the base field, the Tate module \(T_\ell J_N\) is shown to be a free module over the ring \(\mathbb {Z}_\ell [T]/(\sum _{i=0}^{N-1}T^i)\) . We also study the subvarieties of \(J_N\) and calculate the degree of the induced polarization on the new part \(J_N^\mathrm {new}\) of the Jacobian.     

On some integrals over the product of three Legendre functions

The definite integrals \(\int_{-1}^{1}(1-x^{2})^{(\nu-1)/2}[P_{\nu}(x)]^{3}\, \mathrm{d}x\) , \(\int_{-1}^{1}(1-x^{2})^{(\nu-1)/2} [P_{\nu}(x)]^{2}P_{\nu}(-x)\, \mathrm{d}x\) , \(\int_{-1}^{1}x(1-x^{2})^{(\nu-1)/2}[P_{\nu+1}(x)]^{3}\,\mathrm{d}x\) , and \(\int_{-1}^{1}x(1-x^{2})^{(\nu-1)/2} [P_{\nu+1}(x)]^{2}P_{\nu +1}(-x)\,\mathrm{d}x \) are evaluated in closed form, where P _ν is the Legendre function of degree ν, and \(\operatorname{Re}\nu>-1\) . Special cases of these formulae are related to certain integrals over elliptic integrals that have arithmetic interest.     

Square functions in the Hermite setting for functions with values in UMD spaces

In this paper, we characterize the Lebesgue Bochner spaces \(L^p({\mathbb{R }}^{n},B),\, 1 , by using Littlewood–Paley \(g\) -functions in the Hermite setting, provided that \(B\) is a UMD Banach space. We use \(\gamma \) -radonifying operators \(\gamma (H,B)\) where \(H=L^2((0,\infty ),\frac{\mathrm{d}t}{t})\) . We also characterize the UMD Banach spaces in terms of \(L^p({\mathbb{R }}^{n},B)-L^p({\mathbb{R }}^{n},\gamma (H,B))\) boundedness of Hermite Littlewood–Paley \(g\) -functions.     

On the Dynamics of WKB Wave Functions Whose Phase are Weak KAM Solutions of H–J Equation

In the framework of toroidal Pseudodifferential operators on the flat torus \({\mathbb {T}}^n := ({\mathbb {R}} / 2\pi {\mathbb {Z}})^n\) we begin by proving the closure under composition for the class of Weyl operators \(\mathrm {Op}^w_\hbar (b)\) with symbols \(b \in S^m (\mathbb {T}^n \times \mathbb {R}^n)\) . Subsequently, we consider \(\mathrm {Op}^w_\hbar (H)\) when \(H=\frac{1}{2} |\eta |^2 + V(x)\) where \(V \in C^\infty ({\mathbb {T}}^n)\) and we exhibit the toroidal version of the equation for the Wigner transform of the solution of the Schrödinger equation. Moreover, we prove the convergence (in a weak sense) of the Wigner transform of the solution of the Schrödinger equation to the solution of the Liouville equation on \(\mathbb {T}^n \times {\mathbb {R}}^n\) written in the measure sense. These results are applied to the study of some WKB type wave functions in the Sobolev space \(H^{1} (\mathbb {T}^n; {\mathbb {C}})\) with phase functions in the class of Lipschitz continuous weak KAM solutions (positive and negative type) of the Hamilton–Jacobi equation \(\frac{1}{2} |P+ \nabla _x v (P,x)|^2 + V(x) = \bar{H}(P)\) for \(P \in \ell {\mathbb {Z}}^n\) with \(\ell >0\) , and to the study of the backward and forward time propagation of the related Wigner measures supported on the graph of \(P+ \nabla _x v\) .     

Differentiability in the Sobolev space W^{1,n-1}

Let \(\Omega \subset {\mathbb {R}}^{n}\) be a domain, \(n \ge 2\) . We show that a continuous, open and discrete mapping \(f \in W_{\mathrm{loc }}^{1,n-1}(\Omega , {\mathbb {R}}^{n})\) with integrable inner distortion is differentiable almost everywhere on \(\Omega \) . As a corollary we get that the branch set of such a mapping has measure zero.     

The \bar{\partial }-Equation on an Annulus Between Two Strictly q-Convex Domains with Smooth Boundaries

In this paper, we study the global boundary regularity of the \(\bar{\partial }\) - equation on an annulus domain \(\Omega \) between two strictly \(q\) -convex domains with smooth boundaries in \(\mathbb{C }^n\) for some bidegree. To this finish, we first show that the \(\bar{\partial }\) -operator has closed range on \(L^{2}_{r, s}(\Omega )\) and the \(\bar{\partial }\) -Neumann operator exists and is compact on \(L^{2}_{r,s}(\Omega )\) for all \(r\ge 0\) , \(q\le s\le n-q- 1\) . We also prove that the \(\bar{\partial }\) -Neumann operator and the Bergman projection operator are continuous on the Sobolev space \(W^{k}_{r,s}(\Omega )\) , \(k\ge 0\) , \(r\ge 0\) , and \(q\le s\le n-q-1\) . Consequently, the \(L^{2}\) -existence theorem for the \(\bar{\partial }\) -equation on such domain is established. As an application, we obtain a global solution for the \(\bar{\partial }\) equation with Hölder and \(L^p\) -estimates on strictly \(q\) -concave domain with smooth \(\mathcal C ^2\) boundary in \(\mathbb{C }^n\) , by using the local solutions and applying the pushing out method of Kerzman (Commun Pure Appl Math 24:301–380, 1971).     

On a subsemigroup of the universal covering of Lie semigroups

Let \(G\) be a connected Lie group and \(S\) a generating Lie semigroup. An important fact is that generating Lie semigroups admit simply connected covering semigroups. Denote by \(\widetilde{S}\) the simply connected universal covering semigroup of \(S\) . In connection with the problem of identifying the semigroup \(\Gamma (S)\) of monotonic homotopy with a certain subsemigroup of the simply connected covering semigroup \(\widetilde{S}\) we consider in this paper the following subsemigroup $$\begin{aligned} \widetilde{S}_{L}=\overline{\left\langle \mathrm {Exp}(\mathbb {L} (S))\right\rangle } \subset \widetilde{S}, \end{aligned}$$ where \(\mathrm {Exp}:\mathbb {L}(S)\rightarrow S\) is the lifting to \( \widetilde{S}\) of the exponential mapping \(\exp :\mathbb {L}(S)\rightarrow S\) . We prove that \(\widetilde{S}_{L}\) is also simply connected under the assumption that the Lie semigroup \(S\) is right reversible. We further comment how this result should be related to the identification problem mentioned above.     

On Sharp Aperture-Weighted Estimates for Square Functions

Let \(S_{\alpha ,\psi }(f)\) be the square function defined by means of the cone in \({\mathbb R}^{n+1}_{+}\) of aperture \(\alpha \) , and a standard kernel \(\psi \) . Let \([w]_{A_p}\) denote the \(A_p\) characteristic of the weight \(w\) . We show that for any \(1<p<\infty \) and \(\alpha \ge 1\) , $$\begin{aligned} \Vert S_{\alpha ,\psi }\Vert _{L^p(w)}\lesssim \alpha ^n[w]_{A_p}^{\max \left( \frac{1}{2},\frac{1}{p-1}\right) }. \end{aligned}$$ For each fixed \(\alpha \) the dependence on \([w]_{A_p}\) is sharp. Also, on all class \(A_p\) the result is sharp in \(\alpha \) . Previously this estimate was proved in the case \(\alpha =1\) using the intrinsic square function. However, that approach does not allow to get the above estimate with sharp dependence on \(\alpha \) . Hence we give a different proof suitable for all \(\alpha \ge 1\) and avoiding the notion of the intrinsic square function.     

Biharmonic elliptic problems involving the 2nd Hessian operator

In this paper we will study the equation $$\begin{aligned} \Delta ^2 u=S_2(D^2u),\quad \Omega \subset \mathbb {R}^N, \end{aligned}$$ with \(N=3,\) where \( S_2(D^2u)(x)=\sum _{1\le i , being \(\lambda _i,\) the solutions to the equation $$\begin{aligned} \mathrm{det}\left( \lambda I-D^2u(x)\right) =0, \end{aligned}$$ \(i=1,\dots ,N,\) and \(\Omega \) is a bounded domain with smooth boundary. We deal with several boundary conditions looking for the appropriate framework to get existence and multiplicity of nontrivial solutions. This kind of equation is related to some models of growth, and for this reason it is natural to study the effect of zero order local reaction terms of the type \(F_{\lambda }(x,u)=\lambda |u|^{p-1}u\) , with \(\lambda \in \mathbb {R}\) , \(\lambda >0\) , and \(0 , and also the solvability of the boundary problems with a source term \(f\) satisfying some integrability hypotheses.     

Isomorphisms between Jacobson graphs

Let \(R\) be a commutative ring with a non-zero identity and \(\mathfrak {J}_R\) be its Jacobson graph. We show that if \(R\) and \(R'\) are finite commutative rings, then \(\mathfrak {J}_R\cong \mathfrak {J}_{R'}\) if and only if \(|J(R)|=|J(R')|\) and \(R/J(R)\cong R'/J(R')\) . Also, for a Jacobson graph \(\mathfrak {J}_R\) , we obtain the structure of group \(\mathrm {Aut}(\mathfrak {J}_R)\) of all automorphisms of \(\mathfrak {J}_R\) and prove that under some conditions two semi-simple rings \(R\) and \(R'\) are isomorphic if and only if \(\mathrm {Aut}(\mathfrak {J}_R)\cong \mathrm {Aut}(\mathfrak {J}_{R'})\) .     

Vertex transitive graphs from injective linear mappings

Based on a motivation coming from the study of the metric structure of the category of finite dimensional vector spaces over a finite field \(\mathbb {F}\) , we examine a family of graphs, defined for each pair of integers \(1 \le k \le n\) , with vertex set formed by all injective linear transformations \(\mathbb {F}^k \rightarrow \mathbb {F}^n\) and edges corresponding to pairs of mappings, \(f\) and \(g\) , with \(\lambda (f,g)= \dim \mathrm{Im }(f-g)=1 \) . For \(\mathbb {F}\cong \mathrm{GF }(q)\) , this graph will be denoted by \(\mathrm{INJ }_q(k,n)\) . We show that all such graphs are vertex transitive and Hamiltonian and describe the full automorphism group of each \(\mathrm{INJ }_q (k,n)\) for \(k . Using the properties of line-transitive groups, we completely determine which of the graphs \(\mathrm{INJ }_q (k,n)\) are Cayley and which are not. The Cayley ones consist of three infinite families, corresponding to pairs \((1,n),\,(n-1,n)\) , and \((n,n)\) , with \(n\) and \(q\) arbitrary, and of two sporadic examples \(\mathrm{INJ }_{2} (2,5)\) and \(\mathrm{INJ }_{2}(3,5)\) . Hence, the overwhelming majority of our graphs is not Cayley.     

Pseudo-effective cone of Grassmann bundles over a curve

Let \(E\) be a vector bundle over a smooth projective curve \(X\) defined over an algebraically closed field \(k\) . For any integer \(1\,\le \, r\, <\, \mathrm{rank}(E)\) , let \(\mathrm{Gr}_r(E)\,\longrightarrow \, X\) be a Grassmann bundle parametrizing all \(r\) dimensional quotients of the fibers of \(E\) . We compute the pseudo-effective cone in the real Néron–Severi group \(\mathrm{NS}(\mathrm{Gr}_r(E))_\mathbb{R }\) . We prove that this cone coincides with the nef cone in \(\mathrm{NS}(\mathrm{Gr}_r(E))_\mathbb{R }\) if and only if the vector bundle \(E\) is semistable (respectively, strongly semistable) when the characteristic of \(k\) is zero (respectively, positive). Examples are given to show that this characterization of (strong) semistability is not true for vector bundles on higher dimensional projective varieties.     

L^2-properties of the \overline{\partial } and the \overline{\partial }-Neumann operator on spaces with isolated singularities

Let \(X\) be a Hermitian complex space of pure dimension with only isolated singularities and \(\pi : M\rightarrow X\) a resolution of singularities. Let \(\Omega \subset \subset X\) be a domain with no singularities in the boundary, \(\Omega ^*=\Omega {\setminus }\!{{\mathrm{Sing}}}X\) and \(\Omega '=\pi ^{-1}(\Omega )\) . We relate \(L^2\) -properties of the \(\overline{\partial }\) and the \(\overline{\partial }\) -Neumann operator on \(\Omega ^*\) to properties of the corresponding operators on \(\Omega '\) (where the situation is classically well understood). Outside some middle degrees, there are compact solution operators for the \(\overline{\partial }\) -equation on \(\Omega ^*\) exactly if there are such operators on the resolution \(\Omega '\) , and the \(\overline{\partial }\) -Neumann operator is compact on \(\Omega ^*\) exactly if it is compact on \(\Omega '\) .     

Existence of the generalized Fučík spectrum for nonhomogeneous elliptic operators

By variational methods and Morse theory, we prove the existence of uncountably many \((\alpha ,\beta )\in \mathbb R ^2\) for which the equation \(-\mathrm{div}\, A(x, \nabla u)=\alpha u_+^{p-1} -\beta u_-^{p-1}\) in \(\Omega \) , has a sign changing solution under the Neumann boundary condition, where a map \(A\) from \(\overline{\Omega }\times \mathbb R ^N\) to \(\mathbb R ^N\) satisfying certain regularity conditions. As a special case, the above equation contains the \(p\) -Laplace equation. However, the operator \(A\) is not supposed to be \((p-1)\) -homogeneous in the second variable. In particular, it is shown that generally the Fu?ík spectrum of the operator \(-\mathrm{div}\, A(x, \nabla u)\) on \(W^{1,p}(\Omega )\) contains some open unbounded subset of \(\mathbb R ^2\) .