We construct an $L^2$-model of “very small” irreducible unitary representations of simple Lie groups $G$ which, up to finite covering, occur as conformal groups Co$(V)$ of simple Jordan algebras $V$. If $V$ is split and $G$ is not of type $A_n$, then the representations are minimal in the sense that the annihilators are the Joseph ideals. Our construction allows the case where $G$ does not admit minimal representations. In particular, applying to Jordan algebras of split rank one we obtain the entire complementary series representations of $SO(n,1)_0$. A distinguished feature of these representations in all cases is that they attain the minimum of the Gelfand-Kirillov dimensions among irreducible unitary representations. Our construction provides a unified way to realize the irreducible unitary representations of the Lie groups in question as Schrödinger models in $L^2$-spaces on Lagrangian submanifolds of the minimal real nilpotent coadjoint orbits. In this realization the Lie algebra representations are given explicitly by differential operators of order at most two, and the key new ingredient is a systematic use of specific second-order differential operators (Bessel operators) which are naturally defined in terms of the Jordan structure.

In this paper, we apply the Hopf's strong maximum principle in order to obtain a suitable characterization of the complete linear Weingarten hypersurfaces immersed in the hyperbolic space $\mathbb H^{n+1}$. Under the assumption that the mean curvature attains its maximum and supposing an appropriated restriction on the norm of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder of $\mathbb H^{n+1}$.

We prove that the strong polarized relation $\binom{\theta}{\omega} \rightarrow \binom{\theta}{\omega}^{1,1}_2$, applied simultaneously for every $\theta\in[\aleph_1,2^{\aleph_0}]$, is consistent with ZFC. Consequently, $\binom{inv}{\omega} \rightarrow \binom{inv}{\omega}^{1,1}_2$ is consistent for every cardinal invariant of the continuum. Some results in this direction are generalized to higher cardinals.

Nous prouvons que la relation polarisée forte $\binom{\theta}{\omega} \rightarrow \binom{\theta}{\omega}^{1,1}_2$, appliquée simultanément à chaque cardinal $\theta\in[\aleph_1,2^{\aleph_0}]$, est en accord avec ZFC. Par conséquent, la relation $\binom{inv}{\omega} \rightarrow \binom{inv}{\omega}^{1,1}_2$ est en accord avec ZFC pour chaque caractéristique sur le continu. Nous étudions plusieurs généralisations pour certains cardinaux élevés.

We introduce a numerical invariant, called trivializing number, of knots and investigate it. The trivializing number gives an upper bound of unknotting number and canonical genus for knots. We present a table of trivializing numbers for up to 10 crossings knots. We conjecture that twice of the unknotting number of any positive knot is equal to the trivializing number of it and give a partial answer.

A converse theorem for double Dirichlet series is established. As an application, we show that certain zeta functions introduced by Shintani are actually Weyl group multiple Dirichlet series associated to metaplectic Eisenstein series on $GL(2)$.

In this paper we classify the elliptic fibrations on $K3$ surfaces which are the double cover of a blow up of $\mathbb{P}^2$ branched along rational curves and we give equations for many of these elliptic fibrations. Thus we obtain a classification of the van Geemen-Sarti involutions (which are symplectic involutions induced by a translation by a 2-torsion section on an elliptic fibration) on such a surface. Each van Geemen-Sarti involution induces a 2-isogeny between two $K3$ surfaces, which is described in this paper.

It is shown that a space $X$ is strongly paracompact if and only if for every complete metric space $(Y,\rho)$, every l.s.c. mapping from $X$ into the nonempty closed subsets of $Y$ has a separable-valued Hausdorff continuous section. Several applications are demonstrated as well.

In this paper, by clarifying the concept of relative K-stability in [28], we shall solve the stability part of an extremal Kähler version of Donaldson-Tian-Yau's Conjecture. This extends the results in [15] and [27]. We then propose a program to solve the existence part of the conjecture.

The concept of behavior spaces introduced by Shiba plays an important role of systematic investigation of abelian differentials on an open Riemann surface. A Shiba behavior space consists of harmonic differentials which satisfy a certain period condition and boundary behavior. In this paper, for any open Riemann surface of infinite genus we construct Shiba behavior spaces with arbitrarily prescribed period condition and with specific boundary behavior.

The aim of this work is the description of the isomorphism classes of all Leavitt path algebras coming from graphs satisfying Condition (Sing) with up to three vertices. In particular, this classification recovers the one achieved by Abrams et al. [1] in the case of graphs whose Leavitt path algebras are purely infinite simple. The description of the isomorphism classes is given in terms of a series of invariants including the K$_0$ group, the socle, the number of loops with no exits and the number of hereditary and saturated subsets of the graph.

In this article, we make use of geometry of sections of elliptic surfaces and elementary arithmetic on the Mordell-Weil group in order to study existence problem of dihedral covers with given reduced curves as the branch loci. As an application, we give some examples of Zariski pairs $(B_1, B_2)$ for “conic-line arrangements” satisfying the following conditions:

(i) $\deg B_1 = \deg B_2 = 7$.

(ii) Irreducible components of $B_i$ $(i = 1, 2)$ are lines and conics.

(iii) Singularities of $B_i$ ($i = 1, 2$) are nodes, tacnodes and ordinary triple points.

In this article, we show that, $Q:_A\frak{m}^t\subseteq\frak{m}^t$ for all integers $t$ > 0, and for all parameter ideals $Q\subseteq\frak{m}^{2t-1}$ in a one-dimensional Cohen-Macaulay local ring $(A,\frak{m})$ provided that $A$ is not a regular local ring. The assertion obtained by Wang can be extended to one-dimensional (hence, arbitrary dimensional) local rings after some mild modifications. We refer to these quotient ideals $I = Q:_A\frak{m}^t$, $t$-th quasi-socle ideals of $Q$. Examples are explored.

Previous work by the authors (this journal, 60 (2008), 1009-1044) produced equations that hold on certain loci of the Jacobian of a cyclic $C_{rs}$ curve. A curve of this type generalizes elliptic curves, and the equations in question are given in terms of (Klein's) generalization of Weierstrass' $\sigma$-function. The key tool is a matrix with entries that are polynomial in the coordinates of the affine plane model of the curve, thus can be expressed in terms of $\sigma$ and its derivatives. The key geometric loci on the Jacobian of the curve give a stratification of Brill-Noether type. The results are of the type of Riemann-Kempf singularity theorem, the methods are germane to those used by J. D. Fay, who gave vanishing tables for Riemann's $\theta$-function and its derivatives. The main objects we use were developed by several contemporary authors, aside from the classical definitions: meromorphic differentials were expressed in terms of the coordinates mainly by V. M. Buchstaber, J. C. Eilbeck, V. Z. Enolski, D. V. Leykin, and Taylor expansions for $\sigma$ in terms of Schur polynomials also contributed by A. Nakayashiki, in terms of Sato's $\tau$-function. Within this framework, following specific results for $\sigma$-derivatives given by Y. Ônishi, we arrive at our main results, namely statements on the vanishing on given strata of the partial derivatives of $\sigma$ indexed by Young-diagrams subsets that can be worked out in terms of the Weierstrass semigroup of the curve at its point at infinity. The combinatorial statements hold not only for Jacobians but for the stratification of Sato's infinite-dimensional Grassmann manifold as well.