On the number of reducible polynomials of bounded naive height

We prove an asymptotical formula for the number of reducible integer polynomials of degree d and of naive height at most T when \({T \to \infty}\) . The main term turns out to be of the form \({\kappa_d T^d}\) for each \({d \geq 3}\) , where the constant \({\kappa_d}\) is given in terms of some infinite Dirichlet series involving the volumes of symmetric convex bodies in \({\mathbb{R}^d}\) . For d = 2, we prove that there are asymptotically \({\kappa_2 T^2 \,\text{log} T}\) of such polynomials, where \({\kappa_2:=6(3\sqrt{5}+2\,\text{log} (1+\sqrt{5}) -2 \,\text{log}\, 2)/\pi^2}\) . Earlier results in this direction were given by van der Waerden, Pólya and Szegö, Dörge, Chela, and Kuba.     

Structure of \mathrm {Gal}(\mathbb {k}_2^{(2)}/\mathbb {k}) for some fields \mathbb {k}=\mathbb {Q}(\sqrt{2p_1p_2},i) with \mathbf {C}l_2(\mathbb {k})\simeq (2, 2, 2)

Let \(p_1 \equiv p_2 \equiv 5\pmod 8\) be different primes. Put \(i=\sqrt{-1}\) and \(d=2p_1p_2\) , then the bicyclic biquadratic field \(\mathbb {k}=\mathbb {Q}(\sqrt{d},i)\) has an elementary abelian 2-class group of rank \(3\) . In this paper we determine the nilpotency class, the coclass, the generators and the structure of the non-abelian Galois group \(\mathrm {Gal}(\mathbb {k}_2^{(2)}/\mathbb {k})\) of the second Hilbert 2-class field \(\mathbb {k}_2^{(2)}\) of \(\mathbb {k}\) . We study the capitulation problem of the 2-classes of \(\mathbb {k}\) in its seven unramified quadratic extensions \(\mathbb {K}_i\) and in its seven unramified bicyclic biquadratic extensions \(\mathbb {L}_i\) .     

On the Density of the Winding Number of Planar Brownian Motion

We obtain a formula for the density \(f(\theta , t)\) of the winding number of a planar Brownian motion \(Z_t\) around the origin. From this formula, we deduce an expansion for \(f(\log (\sqrt{t})\,\theta ,\,t)\) in inverse powers of \(\log \sqrt{t}\) and \((1+\theta ^2)^{1/2}\) which in particular yields the corrections of any order to Spitzer’s asymptotic law (in Spitzer, Trans. Am. Math. Soc. 87:187–197, 1958). We also obtain an expansion for \(f(\theta ,t)\) in inverse powers of \(\log \sqrt{t}\) , which yields precise asymptotics as \(t \rightarrow \infty \) for a local limit theorem for the windings.     

Blow-up solutions concentrated along minimal submanifolds for some supercritical elliptic problems on Riemannian manifolds

Let (M, g) and \({(K, \kappa)}\) be two Riemannian manifolds of dimensions m and k, respectively. Let \({\omega \in C^{2} (N), \omega > 0}\) . The warped product \({M \times_\omega K}\) is the (m +  k)-dimensional product manifold \({M \times K}\) furnished with metric \({g + \omega^{2} \kappa}\) . We prove that the supercritical problem $$- \Delta_{g + \omega^{2} \kappa} u + hu = u^{\frac{m+2}{m-2} \pm \varepsilon} ,\quad u > 0,\quad {\rm in}\,\, (M \times_{\omega} K, g + \omega^{2} \kappa)$$ has a solution concentrated along a k-dimensional minimal submanifold \({\Gamma}\) of \({M \times_{\omega } N}\) as the real parameter \({\varepsilon}\) goes to zero, provided the function h and the sectional curvatures along \({\Gamma}\) satisfy a suitable condition.     

Strongly amenable involutive representations of involutive Banach algebras

Let \(\mathfrak{A }\) be a Banach \(*\) -algebra and let \(\varphi \) be a nonzero self-adjoint character on \(\mathfrak{A }\) . For a   \(*\) -representation \(\pi \) of \(\mathfrak{A }\) on a Hilbert space \(\mathcal{H }\) , we introduce and study strong \(\varphi \) -amenability of \(\pi \) in terms of certain states on the von Neumann algebra of bounded operators on \(\mathcal{H }\) . We then give some characterizations of this notion in terms of certain positive functionals on \(\mathfrak{A }\) . We finally investigate some hereditary properties of strong \(\varphi \) -amenability of Banach algebras.     

Improved online algorithms for the batch scheduling of equal-length jobs with incompatible families to maximize the weighted number of early jobs

We consider the online scheduling of equal-length jobs with incompatible families on \(m\) identical batch machines. Each job has a release time, a deadline and a weight. Each batch machine can process up to \(b\) jobs (which come from the same family) simultaneously as a batch, where \(b\) is called the capacity of the machines. Our goal is to determine a preemption-restart schedule which maximizes the weighted number of early jobs. For this problem, Li et al. (Inf Process Lett 112:503–508, 2012) provided an online algorithm of competitive ratio \(3+2\sqrt{2}\) for both \(b=\infty \) and \(b<\infty \) . In this paper, we study two special cases of this problem. For the case that \(m=2\) , we first present a lower bound 2, and then provide an online algorithm with a competitive ratio of 3 for both \(b=\infty \) and \(b<\infty \) . For the case in which \(m=3\) , \(b=\infty \) and all jobs come from a common family, we present an online algorithm with a competitive ratio of \((8+2\sqrt{15})/3\approx 5.249\) .     

\mathcal {A}_{p, {\mathbb {E}}} Weights,Maximal Operators,and Hardy Spaces Associated with a Family of General Sets

Suppose that \({\mathbb {E}}:=\{E_r(x)\}_{r\in {\mathcal {I}}, x\in X}\) is a family of open subsets of a topological space \(X\) endowed with a nonnegative Borel measure \(\mu \) satisfying certain basic conditions. We establish an \(\mathcal {A}_{{\mathbb {E}}, p}\) weights theory with respect to \({\mathbb {E}}\) and get the characterization of weighted weak type (1,1) and strong type \((p,p)\) , \(1<p\le \infty \) , for the maximal operator \({\mathcal {M}}_{{\mathbb {E}}}\) associated with \({\mathbb {E}}\) . As applications, we introduce the weighted atomic Hardy space \(H^1_{{\mathbb {E}}, w}\) and its dual \(BMO_{{\mathbb {E}},w}\) , and give a maximal function characterization of \(H^1_{{\mathbb {E}},w}\) . Our results generalize several well-known results.     

Supercyclic abelian semigroups of matrices on \mathbb {C}^{n}

We give a complete characterization of a supercyclic abelian semigroup of matrices on \(\mathbb {C}^{n}\) . For finitely generated semigroups, this characterization is explicit and it is used to determine the minimal number of matrices in normal form over \(\mathbb {C}\) that form a supercyclic abelian semigroup on \({\mathbb {C}}^{n}\) . In particular, no abelian semigroup generated by \(n-1\) matrices on \(\mathbb {C}^{n}\) can be supercyclic.     

Special values of anticyclotomic L-functions modulo λ

The purpose of this article is to generalize some results of Vatsal on the special values of Rankin–Selberg L-functions in an anticyclotomic \({\mathbb{Z}_{p}}\) -extension. Let g be a cuspidal Hilbert modular newform of parallel weight \({(2,\ldots,2)}\) and level \({\mathcal{N}}\) over a totally real field F, and let K/F be a totally imaginary quadratic extension of relative discriminant \({\mathcal{D}}\) . We study the l-adic valuation of the special values \({L(g,\chi,\frac{1}{2})}\) as \({\chi}\) varies over the ring class characters of K of \({\mathcal{P}}\) -power conductor, for some fixed prime ideal \({\mathcal{P}}\) . We prove our results under the only assumption that the prime to \({\mathcal{P}}\) part of \({\mathcal{N}}\) is relatively prime to \({\mathcal{D}}\) .     

The second pinching theorem for hypersurfaces with constant mean curvature in a sphere

We generalize the second pinching theorem for minimal hypersurfaces in a sphere due to Peng–Terng, Wei–Xu, Zhang, and Ding–Xin to the case of hypersurfaces with small constant mean curvature. Let $M^n$ be a compact hypersurface with constant mean curvature $H$ in $S^{n+1}$ . Denote by $S$ the squared norm of the second fundamental form of $M$ . We prove that there exist two positive constants $\gamma (n)$ and $\delta (n)$ depending only on $n$ such that if $|H|\le \gamma (n)$ and $\beta (n,H)\le S\le \beta (n,H)+\delta (n)$ , then $S\equiv \beta (n,H)$ and $M$ is one of the following cases: (i) $S^{k}\Big (\sqrt{\frac{k}{n}}\Big )\times S^{n-k}\Big (\sqrt{\frac{n-k}{n}}\Big )$ , $\,1\le k\le n-1$ ; (ii) $S^{1}\Big (\frac{1}{\sqrt{1+\mu ^2}}\Big )\times S^{n-1}\Big (\frac{\mu }{\sqrt{1+\mu ^2}}\Big )$ . Here $\beta (n,H)=n+\frac{n^3}{2(n-1)}H^2+\frac{n(n-2)}{2(n-1)} \sqrt{n^2H^4+4(n-1)H^2}$ and $\mu =\frac{n|H|+\sqrt{n^2H^2+ 4(n-1)}}{2}$ .     

Some totally geodesic submanifolds of the nonlinear Grassmannian of a compact symmetric space

Let \(M\) and \(N\) be two connected smooth manifolds, where \(M\) is compact and oriented and \(N\) is Riemannian. Let \(\mathcal {E}\) be the Fréchet manifold of all embeddings of \(M\) in \(N\) , endowed with the canonical weak Riemannian metric. Let \(\sim \) be the equivalence relation on \(\mathcal {E}\) defined by \(f\sim g\) if and only if \(f=g\circ \phi \) for some orientation preserving diffeomorphism \(\phi \) of \(M\) . The Fréchet manifold \(\mathcal {S}= \mathcal {E}/_{\sim }\) of equivalence classes, which may be thought of as the set of submanifolds of \(N\) diffeomorphic to \(M\) and is called the nonlinear Grassmannian (or Chow manifold) of \(N\) of type \(M\) , inherits from \( \mathcal {E}\) a weak Riemannian structure. We consider the following particular case: \(N\) is a compact irreducible symmetric space and \(M\) is a reflective submanifold of \(N\) (that is, a connected component of the set of fixed points of an involutive isometry of \( N\) ). Let \(\mathcal {C}\) be the set of submanifolds of \(N\) which are congruent to \(M\) . We prove that the natural inclusion of \(\mathcal {C}\) in \(\mathcal {S}\) is totally geodesic.     

Anderson’s Orthogonality Catastrophe for One-Dimensional Systems

The overlap, \({\mathcal{D}_N}\) , between the ground state of N free fermions and the ground state of N fermions in an external potential in one spatial dimension is given by a generalized Gram determinant. An upper bound is \({\mathcal{D}_N\leq\exp(-\mathcal{I}_N)}\) with the so-called Anderson integral \({\mathcal{I}_N}\) . We prove, provided the external potential satisfies some conditions, that in the thermodynamic limit \({\mathcal{I}_N = \gamma\ln N + O(1)}\) as \({N\to\infty}\) . The coefficient γ > 0 is given in terms of the transmission coefficient of the one-particle scattering matrix. We obtain a similar lower bound on \({\mathcal{D}_N}\) concluding that \({\tilde{C} N^{-\tilde{\gamma}} \leq \mathcal{D}_N \leq CN^{-\gamma}}\) with constants C, \({\tilde{C}}\) , and \({\tilde{\gamma}}\) . In particular, \({\mathcal{D}_N\to 0}\) as \({N\to\infty}\) which is known as Anderson’s orthogonality catastrophe.     

From submodule categories to preprojective algebras

Let \(S(n)\) be the category of invariant subspaces of nilpotent operators with nilpotency index at most \(n\) . Such submodule categories have been studied already in 1934 by Birkhoff, they have attracted a lot of attention in recent years, for example in connection with some weighted projective lines (Kussin, Lenzing, Meltzer). On the other hand, we consider the preprojective algebra \(\Pi _n\) of type \(\mathbb {A}_n\) ; the preprojective algebras were introduced by Gelfand and Ponomarev, they are now of great interest, for example they form an important tool to study quantum groups (Lusztig) or cluster algebras (Geiss, Leclerc, Schröer). We are going to discuss the connection between the submodule category \(\mathcal {S}(n)\) and the module category \(\hbox {mod}\;\Pi _{n-1}\) of the preprojective algebra \(\Pi _{n-1}\) . Dense functors \(\mathcal {S}(n) \rightarrow \hbox {mod}\;\Pi _{n-1}\) are known to exist: one has been constructed quite a long time ago by Auslander and Reiten, recently another one by Li and Zhang. We will show that these two functors are full, dense, objective functors with index \(2n\) , thus \(\hbox {mod}\;\Pi _{n-1}\) is obtained from \(\mathcal {S}(n)\) by factoring out an ideal which is generated by \(2n\) indecomposable objects. As a byproduct we also obtain new examples of ideals in triangulated categories, namely ideals \(\mathcal {I}\) in a triangulated category \(\mathcal {T}\) which are generated by an idempotent such that the factor category \(\mathcal {T}/\mathcal {I}\) is an abelian category.     

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.     

Hall Polynomials and Composition Algebra of Representation Finite Algebras

Let \(\mathcal{A}\) be a representation finite algebra over finite field k such that the indecomposable \(\mathcal{A}\) -modules are determined by their dimension vectors and for each \(M, L \in ind(\mathcal{A})\) and \(N\in mod(\mathcal{A})\) , either \(F^{M}_{N L}=0\) or \(F^{M}_{L N}=0\) . We show that \(\mathcal{A}\) has Hall polynomials and the rational extension of its Ringel–Hall algebra equals the rational extension of its composition algebra. This result extend and unify some known results about Hall polynomials. As a consequence we show that if \(\mathcal{A}\) is a representation finite simply-connected algebra, or finite dimensional k-algebra such that there are no short cycles in \(mod(\mathcal{A})\) , or representation finite cluster tilted algebra, then \(\mathcal{A}\) has Hall polynomials and \(\mathcal{H}(\mathcal{A})\otimes_\mathbb{Z}Q=\mathcal{C}(\mathcal{A})\otimes_\mathbb{Z}Q\) .     

Simply transitive geodesic ball packings to \mathbf{S}^2\!\times \!\mathbf{R} space groups generated by glide reflections

The \(\mathbf{S}^2\!\times \!\mathbf{R}\) geometry can be derived by the direct product of the spherical plane \(\mathbf{S}^2\) and the real line \(\mathbf{R}\) . In (Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry) 42:235–250, 2001), Farkas has classified and given the complete list of the space groups of \(\mathbf{S}^2\!\times \!\mathbf{R}\) . The \(\mathbf{S}^2\!\times \!\mathbf{R}\) manifolds were classified by Molnár and Farkas in [2] by similarity and diffeomorphism. In Szirmai (Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry) 52(2):413–430, 2011), we have studied the geodesic balls and their volumes in \(\mathbf{S}^2\!\times \!\mathbf{R}\) space; moreover, we have introduced the notion of geodesic ball packing and its density and have determined the densest geodesic ball packing for generalized Coxeter space groups of \(\mathbf{S}^2\!\times \!\mathbf{R}\) . In this paper, we study the locally optimal ball packings to the \(\mathbf{S}^2\!\times \!\mathbf{R}\) space groups having Coxeter point groups, and at least one of the generators is a glide reflection. We determine the densest simply transitive geodesic ball arrangements for the above space groups; moreover, we compute their optimal densities and radii. The density of the densest packing is \(\approx 0.80407553\) , may be surprising enough in comparison with the Euclidean result \(\frac{\pi }{\sqrt{18}}\approx 0.74048\) . Molnár has shown in (Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry) 38(2):261–288, 1997) that the homogeneous 3-spaces have a unified interpretation in the real projective 3-sphere \(\mathcal PS ^3(\mathbf{V}^4,\varvec{V}_4,\mathbb R )\) . In our work, we shall use this projective model of \(\mathbf{S}^2\!\times \!\mathbf{R}\) geometry.     

Non-Lipschitz points and the {\textit{SBV}} regularity of the minimum time function

This paper is devoted to the study of the Hausdorff dimension of the singular set of the minimum time function \(T\) under controllability conditions which do not imply the Lipschitz continuity of \(T\) . We consider first the case of normal linear control systems with constant coefficients in \({\mathbb {R}}^N\) . We characterize points around which \(T\) is not Lipschitz as those which can be reached from the origin by an optimal trajectory (of the reversed dynamics) with vanishing minimized Hamiltonian. Linearity permits an explicit representation of such set, that we call \(\mathcal {S}\) . Furthermore, we show that \(\mathcal {S}\) is countably \(\mathcal {H}^{N-1}\) -rectifiable with positive \(\mathcal {H}^{N-1}\) -measure. Second, we consider a class of control-affine planar nonlinear systems satisfying a second order controllability condition: we characterize the set \(\mathcal {S}\) in a neighborhood of the origin in a similar way and prove the \(\mathcal {H}^1\) -rectifiability of \(\mathcal {S}\) and that \(\mathcal {H}^1(\mathcal {S})>0\) . In both cases, \(T\) is known to have epigraph with positive reach, hence to be a locally \(BV\) function (see Colombo et al.: SIAM J Control Optim 44:2285–2299, 2006; Colombo and Nguyen.: Math Control Relat 3: 51–82, 2013). Since the Cantor part of \(DT\) must be concentrated in \(\mathcal {S}\) , our analysis yields that \(T\) is locally \(SBV\) , i.e., the Cantor part of \(DT\) vanishes. Our results imply also that \(T\) is differentiable outside a \(\mathcal {H}^{N-1}\) -rectifiable set. With small changes, our results are valid also in the case of multiple control input.     

A Theory of Stationary Trees and the Balanced Baumgartner–Hajnal–Todorcevic Theorem for Trees

Building on early work by Stevo Todorcevic, we develop a theory of stationary subtrees of trees of successor-cardinal height. We define the diagonal union of subsets of a tree, as well as normal ideals on a tree, and we characterize arbitrary subsets of a non-special tree as being either stationary or non-stationary. We then use this theory to prove the following partition relation for trees: Main Theorem. Let \({\kappa}\) be any infinite regular cardinal, let ξ be any ordinal such that \({2^{|\xi|} < \kappa}\) , and let k be any natural number. Then $$non-(2^{<\kappa})-special\, tree \rightarrow (\kappa + \xi)^{2}_k.$$ This is a generalization to trees of the Balanced Baumgartner–Hajnal–Todorcevic Theorem, which we recover by applying the above to the cardinal \({(2^{< \kappa})^{+}}\) , the simplest example of a non- \({(2^{< \kappa})}\) -special tree. As a corollary, we obtain a general result for partially ordered sets: Theorem. Let \({\kappa}\) be any infinite regular cardinal, let ξ be any ordinal such that \({2^{|\xi|} < \kappa}\) , and let k be any natural number. Let P be a partially ordered set such that \({P \rightarrow (2^{< \kappa})^{1}_{2^{< \kappa}} }\) . Then $$P \rightarrow (\kappa + \xi)^{2}_{k}.$$