In this paper, we study the chord index of virtual knots, which can be thought of as an extension of the chord parity. We show how to use the chord index to enhance the quandle coloring invariants. The notion of indexed quandle is introduced, which generalizes the quandle idea. Some applications of this new invariant is discussed. We also study how to define a generalized chord index via a fixed finite biquandle. Finally, the chord index and its applications in twisted knot theory are discussed.

Let M be a von Neumann algebra and let {αt}t∊R be a σ-weakly continuous flow on M; i.e., suppose that {αt}t∊R is a one-parameter group of *-automorphisms of M such that for each ρ in the predual, M∗, of M and for each x ∊ M, the function of t, ρ(αt(x)), is continuous on R. In recent years, considerable attention has been focused on the subspace of M, H∞(α), which is defined to be

where H∞(R) is the classical Hardy space consisting of the boundary values of functions bounded analytic in the upper half-plane. In Theorem 3.15 of [8] it is proved that in fact H∞(α) is a σ-weakly closed subalgebra of M containing the identity operator such that

is σ-weakly dense in M, and such that

Let $V$ be an analytic variety in some open set in ${{\mathbb{C}}^{n}}$ which contains the origin and which is purely $k$-dimensional. For a curve $\gamma $ in ${{\mathbb{C}}^{n}}$, defined by a convergent Puiseux series and satisfying $\gamma (0)\,=\,0$, and $d\,\ge \,1$, define ${{V}_{t}}\,:=\,{{t}^{-d}}\,\left( V\,-\,\gamma \left( t \right) \right)$. Then the currents defined by ${{V}_{t}}$ converge to a limit current ${{T}_{\gamma ,d}}\left[ V \right]$ as $t$ tends to zero. ${{T}_{\gamma ,d}}\left[ V \right]$ is either zero or its support is an algebraic variety of pure dimension $k$ in ${{\mathbb{C}}^{n}}$. Properties of such limit currents and examples are presented. These results will be applied in a forthcoming paper to derive necessary conditions for varieties satisfying the local Phragmén-Lindelöf condition that was used by Hörmander to characterize the constant coefficient partial differential operators which act surjectively on the space of all real analytic functions on ${{\mathbb{R}}^{n}}$.

We are concerned in this paper with the study of homomorphisms between different algebras of continuous functions, especially the algebras of real functions which are either weakly continuous on bounded sets or weakly uniformly continuous on bounded sets on a Banach space (see definitions below).

These spaces of weakly [uniformly] continuous functions appeared in relation with some questions in Infinite-dimensional Approximation Theory (see [4], [6], [11], [12], [13] and [16]); and since the structure of these function spaces is closely related with properties of different weak topologies (the bounded-weak and bounded-weak* topologies, respectively) and with the structure of Banach spaces on which they are defined, their study also presents interest from the point of view of Banach space theory, as can be seen in [2], [12] or [17].

We enlarge the class of Rapoport–Zink spaces of Hodge type by modifying the centers of the associated $p$-adic reductive groups. Such obtained Rapoport–Zink spaces are said to be of abelian type. The class of Rapoport–Zink spaces of abelian type is strictly larger than the class of Rapoport–Zink spaces of Hodge type, but the two type spaces are closely related as having isomorphic connected components. The rigid analytic generic fibers of Rapoport–Zink spaces of abelian type can be viewed as moduli spaces of local $G$-shtukas in mixed characteristic in the sense of Scholze.

We prove that Shimura varieties of abelian type can be uniformized by the associated Rapoport–Zink spaces of abelian type. We construct and study the Ekedahl–Oort stratifications for the special fibers of Rapoport–Zink spaces of abelian type. As an application, we deduce a Rapoport–Zink type uniformization for the supersingular locus of the moduli space of polarized K3 surfaces in mixed characteristic. Moreover, we show that the Artin invariants of supersingular K3 surfaces are related to some purely local invariants.

The topic of this paper is the relationship between characters of irreducible supercuspidal representations of the p-adic unramified 3 x 3 unitary group and Fourier transforms of invariant measures on elliptic adjoint orbits in the Lie algebra. We prove that most supercuspidal representations have the property that, on some neighbourhood of zero, the character composed with the exponential map coincides with the formal degree of the representation times the Fourier transform of a measure on one elliptic orbit. For the remainder, a linear combination of the Fourier transforms of measures on two elliptic orbits must be taken. As a consequence of these relations between characters and Fourier transforms, the coefficients in the local character expansions are expressed in terms of values of Shalika germs. By calculating which of the values of the Shalika germs associated to regular nilpotent orbits are nonzero, we determine which irreducible supercuspidal representations have Whittaker models. Finally, the coefficients in the local character expansions of three families of supercuspidal representations are computed.

Let A be a separable C*-algebra. Two representations π and π1 of A on the Hilbert spaces H and H1, respectively are said to be quasi-equivalent (denoted by π ~ π1) if projections of H ⊕ H1 on the invariant subspaces H and H1 of (π ⊕ π1)(A) have the same central support in the commutant (π ⊕ π1) (A)′ of (π ⊕ π1) (A), or equivalently, if there is an isomorphism ϕ of π(A)″ onto π1(A)″ such that ϕ(π(x)) = π(x) for all x ∊ A (cf. [5, § 5]). A representation π of A is said to be a factor representation if the center of π(A)″ consists of scalar multiples of the identity.

In [1] Atiyah described how to use the complex representations of the symmetric group, Sn, to define and investigate operations in complex topological K-theory. In this paper operations for more general Grothendieck groups are described in terms of the integral representations of Sn using the representations directly without passing to the dual as Atiyah did. The principal tool, which is proved in the first section, is the theorem that the direct sum of the Grothendieck groups of finite integral representations of Sn form a bialgebra isomorphic to a polynomial ring with a sequence of divided powers. A consequence of this theorem is that the only operations that can be constructed from the symmetric groups will be polynomials in the symmetric powers.

We consider the usual topological vector space H(G) of all functions holomorphic in a region G ⊂ ℂ. If G satisfies certain conditions, it is possible to introduce the Hadamard product as multiplication in H(G), and then H(G) turns out to be a commutative topological algebra. In [5] we characterized the invertible elements in H(G), and the aim of this paper is to study the closed ideals and some further questions.

Postnikov constructed a cellular decomposition of the totally nonnegative Grassmannians. The poset of cells can be described (in particular) via Grassmann necklaces. We study certain quiver Grassmannians for the cyclic quiver admitting a cellular decomposition, whose cells are naturally labeled by Grassmann necklaces. We show that the posets of cells coincide with the reversed cell posets of the cellular decomposition of the totally nonnegative Grassmannians. We investigate algebro-geometric and combinatorial properties of these quiver Grassmannians. In particular, we describe the irreducible components, study the action of the automorphism groups of the underlying representations, and describe the moment graphs. We also construct a resolution of singularities for each irreducible component; the resolutions are defined as quiver Grassmannians for an extended cyclic quiver.

Let $E$ be an elliptic curve, and let ${{L}_{n}}$ be the Kummer extension generated by a primitive ${{p}^{n}}$-th root of unity and a ${{p}^{n}}$-th root of $a$ for a fixed $a\,\in \,{{\mathbb{Q}}^{\times }}\,-\,\left\{ \pm 1 \right\}$. A detailed case study by Coates, Fukaya, Kato and Sujatha and $V$. Dokchitser has led these authors to predict unbounded and strikingly regular growth for the rank of $E$ over ${{L}_{n}}$ in certain cases. The aim of this note is to explain how some of these predictions might be accounted for by Heegner points arising from a varying collection of Shimura curve parametrisations.

A new method for classifying naturally reductive spaces is presented. This method relies on a new construction and the structure theory of naturally reductive spaces recently developed by the author. This method is applied to obtain the classification of all naturally reductive spaces in dimension 7 and 8.

We study representations of o-orders, that is, of o-regular -algebras, in the case that o is a Dedekind domain. Our main concern is with those -modules, called -representation modules, which are regular as o-modules. For any -module M we denote by D(M) the ideal consisting of the elements x ∈ o such that x.Ext1(M, N) = 0 for all -modules N, where Ext = Ext(,0) is the relative functor of Hochschild (5). To compute D(M) we need the small amount of homological algebra presented in § 1. In § 2 we show that the -representation modules with rational hulls isomorphic to direct sums of right ideal components of the rational hull A of , called principal-modules, are characterized by the property that D(M) ≠ 0. The (, o)-projective -modules are those with D(M) = 0. We observe that D(M) divides the ideal I() of (2) for every M , and give another proof of the fact that I() ≠ 0 if and only if A is separable. Up to this point, o can be taken to be an arbitrary integral domain.

Matrices whose entries belong to certain rings of algebraic integers can be associated with discrete groups of transformations of inversive $n$-space or hyperbolic $(n+1)-\text{space}\,{{\text{H}}^{n+1}}$. For small $n$, these may be Coxeter groups, generated by reflections, or certain subgroups whose generators include direct isometries of ${{\text{H}}^{n+1}}$. We show how linear fractional transformations over rings of rational and (real or imaginary) quadratic integers are related to the symmetry groups of regular tilings of the hyperbolic plane or 3-space. New light is shed on the properties of the rational modular group $\text{PS}{{\text{L}}_{2}}(\mathbb{Z})$, the Gaussian modular (Picard) group $\text{PS}{{\text{L}}_{2}}(\mathbb{Z}[i])$, and the Eisenstein modular group $\text{PS}{{\text{L}}_{2}}(\mathbb{Z}\left[ \omega \right])$.

We are concerned here with the problem of describing the direct integral decomposition of a unitary representation obtained by restriction from a larger group. This is the dual problem to the more commonly investigated problem of decomposing induced representations. In this paper we work in the context of completely solvable Lie groups—more general than nilpotent, but less general than exponential solvable. Moreover, the groups involved are simply connected. The restriction problem was considered originally in [2] and in [6] for nilpotent groups.

Let C be an n-square Hermitian matrix, presented in partitioned form as

where A is a-square and B is b-square. Let denote the eigenvalues of C, A, B, respectively. In a recent paper [10] the following inequality was established:

1.1

if

1.2

This inequality is a simplification and a sharpening of an inequality established earlier in [6], and is a wide generalization of an inequality of Aronszajn [4].

Suppose that N is 2-coloured. Then there are infinitely many monochromatic solutions to $x+y=z^{2}$. On the other hand, there is a 3-colouring of N with only finitely many monochromatic solutions to this equation.

In this paper, we study the boundedness of є-families of operators on Triebel-Lizorkin with wide range of parameters. We also prove that є -families of operators are bounded from Triebel-Lizorkin spaces into (generalized) tent spaces, and obtain a characterization of certain Triebel-Lizorkin spaces in terms of tent spaces. In particular, the boundedness of fractional operators in Triebel-Lizorkin, and a sharp version of T\theorem for generalized Calderón-Zygmund operators on Triebel-Lizorkin spaces can be considered as applications of (proofs of) these results.

Let $A$ be a separable amenable purely infinite simple ${{C}^{*}}$-algebra which satisfies the Universal Coefficient Theorem. We prove that $A$ is weakly semiprojective if and only if ${{K}_{i}}(A\text{)}$ is a countable direct sum of finitely generated groups $\left( i\,=\,0,\,1 \right)$. Therefore, if $A$ is such a ${{C}^{*}}$-algebra, for any $\varepsilon \,>\,0$ and any finite subset $\mathcal{F}\,\subset \,A$ there exist $\delta \,>\,0$ and a finite subset $G\,\subset \,A$ satisfying the following: for any contractive positive linear map $L\,:\,A\,\to \,B$ (for any ${{C}^{*}}$-algebra $B$) with $||L\left( ab \right)\,-\,L\left( a \right)L\left( b \right)||\,<\,\delta$ for $a,\,b\,\in \,\mathcal{G}$ there exists a homomorphism $h:\,A\,\to \,B$ such that $||\,h\left( a \right)\,-\,L\left( a \right)||\,<\,\varepsilon$ for $a\,\in \,\mathcal{F}$.

In this paper we first generalize to the case of partial flags a result proved both by Spaltenstein and by Steinberg that relates the relative position of two complete flags and the irreducible components of the flag variety in which they lie, using the Robinson–Schensted–Knuth correspondence. Then we use this result to generalize the mirabolic Robinson–Schensted–Knuth correspondence defined by Travkin, to the case of two partial flags and a line.