Köthe’s upper nil radical for modules

Let M be a left R-module. In this paper a generalization of the notion of an s-system of rings to modules is given. Let N be a submodule of M. Define $\mathcal{S}(N):=\{ {m\in M}:\, \mbox{every } s\mbox{-system containing } m \mbox{ meets}~N \}$ . It is shown that $\mathcal{S}(N)$ is equal to the intersection of all s-prime submodules of M containing N. We define $\mathcal{N}({}_{R}M) = \mathcal{S}(0)$ . This is called (Köthe’s) upper nil radical of M. We show that if R is a commutative ring, then $\mathcal{N}({}_{R}M) = {\mathop{\mathrm{rad}}\nolimits}_{R}(M)$ where ${\mathop{\mathrm{rad}}\nolimits}_{R}(M)$ denotes the prime radical of M. We also show that if R is a left Artinian ring, then ${\mathop{\mathrm{rad}}\nolimits}_{R}(M)=\mathcal{N}({}_{R}M)= {\mathop{\mathrm{Rad}}\nolimits}\, (M)= {\mathop{\mathrm{Jac}}\nolimits}\, (R)M$ where ${\mathop{\mathrm{Rad}}\nolimits}\, (M)$ denotes the Jacobson radical of M and ${\mathop{\mathrm{Jac}}\nolimits}\, (R)$ the Jacobson radical of the ring R. Furthermore, we show that the class of all s-prime modules forms a special class of modules.     

Zaks’ Lemma for Coherent Rings

Let A be a left and right coherent ring and C _A (resp., $C_{A^{\mathrm{op}}}$ ) a minimal cogenerator for right (resp., left) A-modules. We show that $\mathrm{flat \ dim \ }C_{A} = \mathrm{flat \ dim \ }C_{A^{\mathrm{op}}}$ whenever flat dim C _A ?<?∞ and $\mathrm{flat \ dim \ }C_{A^{\mathrm{op}}} < \infty$ , and that $\mathrm{flat \ dim \ }C_{A} = \mathrm{flat \ dim \ }C_{A^{\mathrm{op}}} < \infty$ if and only if the finitely presented right A-modules have bounded Gorenstein dimension.     

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.     

Strengthening the cohomological crepant resolution conjecture for Hilbert–Chow morphisms

Given any smooth toric surface $S$ , we prove a SYM-HILB correspondence which relates the 3-point, degree zero, extended Gromov–Witten invariants of the $n$ -fold symmetric product stack $[\mathrm{Sym}^n(S)]$ of $S$ to the 3-point extremal Gromov–Witten invariants of the Hilbert scheme $\mathrm{Hilb}^n(S)$ of $n$ points on $S$ . As we do not specialize the values of the quantum parameters involved, this result proves a strengthening of Ruan’s Cohomological Crepant Resolution Conjecture for the Hilbert–Chow morphism $\mathrm{Hilb}^n(S) \rightarrow \mathrm{Sym}^n(S)$ and yields a method of reconstructing the cup product for $\mathrm{Hilb}^n(S)$ from the orbifold invariants of $[\mathrm{Sym}^n(S)]$ .     

On the bicanonical map of primitive varieties with q(X) = \mathrm{dim }X: the degree and the Euler number

Let $X$ be a variety of maximal Albanese dimension and of general type. Assume that $q(X) = \mathrm{dim }X$ , the Albanese variety $\mathrm {Alb} (X)$ is a simple abelian variety, and the bicanonical map is not birational. We prove that the Euler number $\chi (X, \omega _X)$ is equal to 1, and $|2K_X|$ separates two distinct points over the same general point on $\mathrm {Alb} (X)$ via $\mathrm {alb}_X$ (Theorem 1.1).     

On the Topology of Kac–Moody groups

We study the topology of spaces related to Kac–Moody groups. Given a Kac–Moody group over $\mathbb C $ , let $\text {K}$ denote the unitary form with maximal torus ${{\mathrm{T}}}$ having normalizer ${{\mathrm{N}}}({{\mathrm{T}}})$ . In this article we study the cohomology of the flag manifold $\text {K}/{{{\mathrm{T}}}}$ as a module over the Nil-Hecke algebra, as well as the (co)homology of $\text {K}$ as a Hopf algebra. In particular, if $\mathbb F $ has positive characteristic, we show that $\text {H}_*(\text {K},\mathbb F )$ is a finitely generated algebra, and that $\text {H}^*(\text {K},\mathbb F )$ is finitely generated only if $\text {K}$ is a compact Lie group . We also study the stable homotopy type of the classifying space $\text {BK}$ and show that it is a retract of the classifying space $\text {BN(T)}$ of ${{\mathrm{N}}}({{\mathrm{T}}})$ . We illustrate our results with the example of rank two Kac–Moody groups.     

Tilting Theory and Functor Categories II. Generalized Tilting

In this paper we continue the project of generalizing tilting theory to the category of contravariant functors $\mathrm{Mod}(\mathcal{C})$ , from a skeletally small preadditive category $\mathcal{C}$ to the category of abelian groups, initiated in [15]. We introduce the notion of a generalized tilting category $\mathcal{T}$ , and we concentrate here on extending Happel’s theorem to $\mathrm{Mod}(\mathcal{C})$ ; more specifically, we prove that there is an equivalence of triangulated categories $\mathcal{D}^{b}( \mathrm{Mod}(\mathcal{C}))\cong \mathcal{D}^{b}(\mathrm{Mod}(\mathcal{T}))$ . We then add some restrictions on our category $\mathcal{C}$ , in order to obtain a version of Happel’s theorem for the categories of finitely presented functors. We end the paper proving that some of the theorems for artin algebras, relating tilting with contravariantly finite categories proved in Auslander and Reiten (Adv Math 12(3):306–366, 1974; Adv Math 86(1):111–151, 1991), can be extended to the category of finitely presented functors $\mathrm{mod}(\mathcal{C})$ , with $\mathcal{C}$ a dualizing variety.     

Period polynomial relations between double zeta values

The even weight period polynomial relations in the double shuffle Lie algebra  $\mathfrak{ds}$ were discovered by Ihara, and completely classified in Schneps (J. Lie Theory 16(1): 19–37, 2006) by relating them to restricted even period polynomials associated to cusp forms on  $\mathrm{SL} _{2}(\mathbb{Z})$ . In an article published in the same year, Gangl et al. (Double zeta values and modular forms. Automorphic forms and zeta functions, pp. 71–106, 2006) displayed certain linear combinations of odd-component double zeta values which are equal to scalar multiples of simple zeta values in even weight, and also related them to restricted even period polynomials. In this paper, we relate the two sets of relations, showing how they can be deduced from each other by duality.     

A class of semi-supervised support vector machines by DC programming

This paper investigate a class of semi-supervised support vector machines ( $\text{ S }^3\mathrm{VMs}$ ) with arbitrary norm. A general framework for the $\text{ S }^3\mathrm{VMs}$ was first constructed based on a robust DC (Difference of Convex functions) program. With different DC decompositions, DC optimization formulations for the linear and nonlinear $\text{ S }^3\mathrm{VMs}$ are investigated. The resulting DC optimization algorithms (DCA) only require solving simple linear program or convex quadratic program at each iteration, and converge to a critical point after a finite number of iterations. The effectiveness of proposed algorithms are demonstrated on some UCI databases and licorice seed near-infrared spectroscopy data. Moreover, numerical results show that the proposed algorithms offer competitive performances to the existing $\text{ S }^3\mathrm{VM}$ methods.     

Efficient algorithms for multivariate and ∞-variate integration with exponential weight

Using the Multivariate Decomposition Method (MDM), we develop an efficient algorithm for approximating the ∞-variate integral $$\mathcal{I}_{\infty}(f) = \lim\limits_{d\rightarrow \infty} \int\limits_{\mathcal{R}_{+}^{d}}f(x_{1},\ldots,x_{d},0,0,\ldots)\cdot \exp\left(-\sum\limits_{j=1}^{d} x_{j}\right) \mathrm{d} \mathbf{x} $$ for a class of functions f that are once differentiable with respect to each variable. MDM requires efficient algorithms for d-variate versions of the problem. Such algorithms are provided by Smolyak’s construction which is based on efficient algorithms for the univariate integration $$ I \left(f\right) = \int_{0}^{\infty} f\left(x\right)^{-x} \mathrm{d} \mathbf{x}. $$ Detailed analysis and development of (nearly) optimal quadratures for I(f) is the main contribution of the current paper.     

Polynomials for \mathrm{GL}_p\times \mathrm{GL}_q orbit closures in the flag variety

The subgroup \(K=\mathrm{GL}_p \times \mathrm{GL}_q\) of \(\mathrm{GL}_{p+q}\) acts on the (complex) flag variety \(\mathrm{GL}_{p+q}/B\) with finitely many orbits. We introduce a family of polynomials specializing representatives for cohomology classes of the orbit closures in the Borel model. We define and study \(K\) -orbit determinantal ideals to support the geometric naturality of these representatives. Using a modification of these ideals, we describe an analogy between two local singularity measures: the \(H\) -polynomials and the Kazhdan–Lusztig–Vogan polynomials.     

Critical points for functionals with quasilinear singular Euler–Lagrange equations

For a bounded, open subset Ω of ${\mathbb{R}^{N}}$ with N > 2, and a measurable function a(x) satisfying 0 < α ≤ a(x) ≤ β, a.e. ${x \in \Omega}$ , we study the existence of positive solutions of the Euler–Lagrange equation associated to the non-differentiable functional $$\begin{array}{ll}J(v) = \frac{1}{2} \int \limits_{\Omega} [a(x)+|v|^{\gamma}]| \nabla v|^{2}- \frac{1}{p} \int \limits_{\Omega}(v_{+})^p,\end{array}$$ if γ > 0 and p > 1. Special emphasis is placed on the case ${2^{*} < p < \frac{2^{*}}{2} ( \gamma +2 )}$ .     

A note on quadratic convergence of a smoothing Newton algorithm for the LCP

The linear complementarity problem (LCP) is to find ${(x,s)\in\mathfrak{R}^n\times\mathfrak{R}^n}$ such that (x, s) ≥ 0, s = Mx + q, x ^T s = 0 with ${M\in\mathfrak{R}^{n\times n}}$ and ${q\in\mathfrak{R}^n}$ . The smoothing Newton algorithm is one of the most efficient methods for solving the LCP. To the best of our knowledge, the best local convergence results of the smoothing Newton algorithm for the LCP up to now were obtained by Huang et al. (Math Program 99:423–441, 2004). In this note, by using a revised Chen–Harker–Kanzow–Smale smoothing function, we propose a variation of Huang–Qi–Sun’s algorithm and show that the algorithm possesses better local convergence properties than those given in Huang et al. (Math Program 99:423–441, 2004).     

A weighted Hardy inequality and nonexistence of positive solutions

In this article, we prove that the following weighted Hardy inequality $$\begin{array}{ll}\big(\frac{|{d-p}|}{p}\big)^{p}\, \int\limits_{\Omega}\, \frac{|{u}|^{p}}{|{x}|^{p}}\;d\mu \\ \quad \quad \le \int\limits_{\Omega}\,|{\nabla u}|^{p}\;d\mu+ \big(\frac{|{d-p}|}{p}\big)^{p-1}\,\textrm{sgn}(d-p)\,\int\limits_{\Omega}|{u}|^{p}\,\frac{(x^{t}Ax)^{p/2}}{|{x}|^{p}}\; d\mu \quad \quad \quad (1) \end{array}$$ holds with optimal Hardy constant ${\big(\frac{|d-p|}{p}\big)^{p}}$ for all ${u \in W^{1,p}_{\mu,0}(\Omega)}$ if the dimension d ≥ 2, 1 < p < d, and for all ${u \in W^{1,p}_{\mu,0}(\Omega{\setminus}\{0\})}$ if p > d ≥ 1. Here we assume that Ω is an open subset of ${\mathbb{R}^{d}}$ with ${0 \in \Omega}$ , A is a real d × d-symmetric positive definite matrix, c > 0, and $$ d \mu: = \rho(x) \,dx \qquad \textrm{with} \quad \rho(x) = c \cdot \exp(-\frac{1}{p}(x^{t}Ax)^{p/2}), \quad x \in\Omega.\quad \quad (2) $$ If p > d ≥ 1, then we can deduce from (1) a weighted Poincaré inequality on ${W^{1,p}_{\mu,0}(\Omega \setminus\{0\})}$ . Due to the optimality of the Hardy constant in (1), we can establish the nonexistence (locally in time) of positive weak solutions of a p-Kolmogorov parabolic equation perturbed by a singular potential in dimension d = 1, for 1 < p <  + ∞, and when Ω =  ]0, + ∞[.     

A zoo of diffeomorphism groups on \mathbb{R }^{n}

We consider the groups ${\mathrm{Diff }}_\mathcal{B }(\mathbb{R }^n)$ , ${\mathrm{Diff }}_{H^\infty }(\mathbb{R }^n)$ , and ${\mathrm{Diff }}_{\mathcal{S }}(\mathbb{R }^n)$ of smooth diffeomorphisms on $\mathbb{R }^n$ which differ from the identity by a function which is in either $\mathcal{B }$ (bounded in all derivatives), $H^\infty = \bigcap _{k\ge 0}H^k$ , or $\mathcal{S }$ (rapidly decreasing). We show that all these groups are smooth regular Lie groups.     

Exact tail asymptotics of the supremum of strongly dependent Gaussian processes over a random interval

Let $ \mathcal{T} $ be a positive random variable independent of a real-valued stochastic process $ \left\{ {X(t),t\geqslant 0} \right\} $ . In this paper, we investigate the asymptotic behavior of $ \mathrm{P}\left( {{\sup_{{t\in \left[ {0,\mathcal{T}} \right]}}}X(t)>u} \right) $ as u→∞ assuming that X is a strongly dependent stationary Gaussian process and $ \mathcal{T} $ has a regularly varying survival function at infinity with index λ ∈ [0, 1). Under asymptotic restrictions on the correlation function of the process, we show that $ \mathrm{P}\left( {{\sup_{{t\in \left[ {0,\mathcal{T}} \right]}}}X(t)>u} \right)={c^{\lambda }}\mathrm{P}\left( {\mathcal{T}>m(u)} \right)\left( {1+o(1)} \right) $ with some positive finite constant c and function m(·) defined in terms of the local behavior of the correlation function and the standard Gaussian distribution.     

Noncommutative Artin motives

In this article, we introduce the category of noncommutative Artin motives as well as the category of noncommutative mixed Artin motives. In the pure world, we start by proving that the classical category ${{\mathrm{AM}}}(k)_\mathbb Q $ of Artin motives (over a base field k) can be characterized as the largest category inside Chow motives which fully embeds into noncommutative Chow motives. Making use of a refined bridge between pure motives and noncommutative pure motives, we then show that the image of this full embedding, which we call the category ${{\mathrm{NAM}}}(k)_\mathbb Q $ of noncommutative Artin motives, is invariant under the different equivalence relations and modification of the symmetry isomorphism constraints. As an application, we recover the absolute Galois group $\mathrm{Gal}(\overline{k}/k)$ from the Tannakian formalism applied to ${{\mathrm{NAM}}}(k)_\mathbb Q $ . Then, we develop the base-change formalism in the world of noncommutative pure motives. As an application, we obtain new tools for the study of motivic decompositions and Schur/Kimura finiteness. Making use of this theory of base-change, we then construct a short exact sequence relating $\mathrm{Gal}(\overline{k}/k)$ with the noncommutative motivic Galois groups of k and $\overline{k}$ . Finally, we describe a precise relationship between this short exact sequence and the one constructed by Deligne–Milne. In the mixed world, we introduce the triangulated category ${{\mathrm{NMAM}}}(k)_\mathbb Q $ of noncommutative mixed Artin motives and construct a faithful functor from the classical category ${{\mathrm{MAM}}}(k)_\mathbb Q $ of mixed Artin motives to it. When k is a finite field, this functor is an equivalence. On the other hand, when k is of characteristic zero ${{\mathrm{NMAM}}}(k)_\mathbb Q $ is much richer than ${{\mathrm{MAM}}}(k)_\mathbb Q $ since its higher Ext-groups encode all the (rationalized) higher algebraic $K$ -theory of finite étale k-schemes. In the appendix, we establish a general result about short exact sequences of Galois groups which is of independent interest. As an application, we obtain a new proof of Deligne–Milne’s short exact sequence.     

Multiple solutions for nonhomogeneous Schrödinger–Maxwell and Klein– Gordon–Maxwell equations on R 3

In this paper we study the following nonhomogeneous Schrödinger–Maxwell equations $\left\{\begin{array}{ll} {-\triangle u+V(x)u+ \phi u=f(x,u)+h(x),} \quad {\rm in}\,\,\,{\mathbf{R}}^3,\\ {-\triangle \phi=u^2, \qquad\qquad\qquad\qquad\qquad\qquad\,\,\, {\rm in} \,\,{\mathbf{R}}^3,} \end{array} \right.$ where f satisfies the Ambrosetti–Rabinowitz type condition. Under appropriate assumptions on V, f and h, the existence of multiple solutions is proved by using the Ekeland’s variational principle and the Mountain Pass Theorem in critical point theory. Similar results for the nonhomogeneous Klein–Gordon–Maxwell equations $\left\{\begin{array}{ll} {-\triangle u+[m^2-(\omega+\phi)^2]u=|u|^{q-2}u+h(x), \quad {\rm in} \,\,\,{\mathbf{R}}^3,}\\ {-\triangle \phi+ \phi u^2=-\omega u^2, \qquad\qquad\qquad\qquad\qquad\,\,\, {\rm in} \,\,\,{\mathbf{R}}^3,} \end{array} \right.$ are also obtained when 2 < q < 6.     

Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities

We study the following nonlinear Schrödinger equations $$\begin{array}{lll}(-i\varepsilon\nabla+A(x))^2 w + V(x)w = W(x)g(|w|)w; \quad \quad \quad \quad \quad \quad \quad \quad \quad (0.1)\\(-i\varepsilon\nabla+A(x))^2 w + V(x)w = W(x)\left(g(|w|)+|w|^{2^*-2}\right)w,\quad \quad \quad\,\,(0.2)\end{array}$$ for ${w \in H^1\left( \mathbb{R}^N, \mathbb{C} \right)}$ , where g(|w|)w is super linear and subcritical, 2* = 2N/(N ? 2) if N > 2 and = ∞ if N = 2, min V > 0 and inf W > 0. Under proper assumptions we explore the existence and concentration phenomena of semiclassical solutions of (0.1). The most interesting result obtained here refers to the critical case. We establish the existence and describe the concentration of semiclassical ground states of (0.2) provided either min V <  τ ₀ for some τ₀ > 0, or ${\max W > \kappa_{0}}$ for some ${\kappa_0 > 0}$ .