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.

In this paper we analyze the convergence of the following type of series

\begin{eqnarray}T_{N}^{{\mathcal{L}}}f(x)=\mathop{\sum }_{j=N_{1}}^{N_{2}}v_{j}\big(e^{-a_{j+1}{\mathcal{L}}}f(x)-e^{-a_{j}{\mathcal{L}}}f(x)\big),\quad x\in \mathbb{R}^{n},\end{eqnarray} $$\begin{eqnarray}T_{N}^{{\mathcal{L}}}f(x)=\mathop{\sum }_{j=N_{1}}^{N_{2}}v_{j}\big(e^{-a_{j+1}{\mathcal{L}}}f(x)-e^{-a_{j}{\mathcal{L}}}f(x)\big),\quad x\in \mathbb{R}^{n},\end{eqnarray}$$

{\{e^{-t{\mathcal{L}}}\}}_{t>0}${\{e^{-t{\mathcal{L}}}\}}_{t>0}$

{\mathcal{L}}=-\unicode[STIX]{x1D6E5}+V${\mathcal{L}}=-\unicode[STIX]{x1D6E5}+V$

\unicode[STIX]{x1D6E5}$\unicode[STIX]{x1D6E5}$

V$V$

RH_{q}$RH_{q}$

q>n/2$q>n/2$

n\geqslant 3$n\geqslant 3$

N=(N_{1},N_{2})\in \mathbb{Z}^{2}$N=(N_{1},N_{2})\in \mathbb{Z}^{2}$

N_{1}<N_{2}$N_{1}<N_{2}$

{\{v_{j}\}}_{j\in \mathbb{Z}}${\{v_{j}\}}_{j\in \mathbb{Z}}$

{\{a_{j}\}}_{j\in \mathbb{Z}}${\{a_{j}\}}_{j\in \mathbb{Z}}$

Our analysis will consist in the boundedness, in L^{p}(\mathbb{R}^{n})$L^{p}(\mathbb{R}^{n})$ and in BMO(\mathbb{R}^{n})$BMO(\mathbb{R}^{n})$, of the operators T_{N}^{{\mathcal{L}}}$T_{N}^{{\mathcal{L}}}$ and its maximal operator T^{\ast }f(x)=\sup _{N}T_{N}^{{\mathcal{L}}}f(x)$T^{\ast }f(x)=\sup _{N}T_{N}^{{\mathcal{L}}}f(x)$.

It is also shown that the local size of the maximal differential transform operators (with V=0$V=0$) is the same with the order of a singular integral for functions f$f$ having local support. Moreover, if {\{v_{j}\}}_{j\in \mathbb{Z}}\in \ell ^{p}(\mathbb{Z})${\{v_{j}\}}_{j\in \mathbb{Z}}\in \ell ^{p}(\mathbb{Z})$, we get an intermediate size between the local size of singular integrals and Hardy–Littlewood maximal operator.

A ring \unicode[STIX]{x1D6EC}$\unicode[STIX]{x1D6EC}$ is called right Köthe if every right \unicode[STIX]{x1D6EC}$\unicode[STIX]{x1D6EC}$-module is a direct sum of cyclic modules. In this paper, we give a characterization of basic hereditary right Köthe rings in terms of their Coxeter valued quivers. We also give a characterization of basic right Köthe rings with radical square zero. Therefore, we give a solution to Köthe’s problem in these two cases.

This work is concerned with Bielawski’s hyperkähler slices in the cotangent bundles of homogeneous affine varieties. One can associate such a slice with the data of a complex semisimple Lie group G$G$, a reductive subgroup H\subseteq G$H\subseteq G$, and a Slodowy slice S\subseteq \mathfrak{g}:=\text{Lie}(G)$S\subseteq \mathfrak{g}:=\text{Lie}(G)$, defining it to be the hyperkähler quotient of T^{\ast }(G/H)\times (G\times S)$T^{\ast }(G/H)\times (G\times S)$ by a maximal compact subgroup of G$G$. This hyperkähler slice is empty in some of the most elementary cases (e.g., when S$S$ is regular and (G,H)=(\text{SL}_{n+1},\text{GL}_{n})$(G,H)=(\text{SL}_{n+1},\text{GL}_{n})$, n\geqslant 3$n\geqslant 3$), prompting us to seek necessary and sufficient conditions for non-emptiness.

We give a spherical-geometric characterization of the non-empty hyperkähler slices that arise when S=S_{\text{reg}}$S=S_{\text{reg}}$ is a regular Slodowy slice, proving that non-emptiness is equivalent to the so-called \mathfrak{a}$\mathfrak{a}$-regularity of (G,H)$(G,H)$. This \mathfrak{a}$\mathfrak{a}$-regularity condition is formulated in several equivalent ways, one being a concrete condition on the rank and complexity of G/H$G/H$. We also provide a classification of the \mathfrak{a}$\mathfrak{a}$-regular pairs (G,H)$(G,H)$ in which H$H$ is a reductive spherical subgroup. Our arguments make essential use of Knop’s results on moment map images and Losev’s algorithm for computing Cartan spaces.

Motivated by works on extension sets in standard domains, we introduce a notion of the Carathéodory set that seems better suited for the methods used in proofs of results on characterization of extension sets. A special stress is put on a class of two-dimensional submanifolds in the tridisc that not only turns out to be Carathéodory but also provides examples of two-dimensional domains for which the celebrated Lempert Theorem holds. Additionally, a recently introduced notion of universal sets for the Carathéodory extremal problem is studied and new results on domains admitting (no) finite universal sets are given.

We study coisotropic submanifolds of b-symplectic manifolds. We prove that b-coisotropic submanifolds (those transverse to the degeneracy locus) determine the b-symplectic structure in a neighborhood, and provide a normal form theorem. This extends Gotay’s theorem in symplectic geometry. Further, we introduce strong b-coisotropic submanifolds and show that their coisotropic quotient, which locally is always smooth, inherits a reduced b-symplectic structure.

In this paper, we completely solve the L^{2}\to L^{r}$L^{2}\to L^{r}$ extension conjecture for the zero radius sphere over finite fields. We also obtain the sharp L^{p}\to L^{4}$L^{p}\to L^{4}$ extension estimate for non-zero radii spheres over finite fields, which improves the previous result of the first and second authors significantly.

We study immersed surfaces in {\mathbb R}^3${\mathbb R}^3$ that are critical points of the Willmore functional under boundary constraints. The two cases considered are when the surface meets a plane orthogonally along the boundary and when the boundary is contained in a line. In both cases we derive weak forms of the resulting free boundary conditions and prove regularity by reflection.

Let f and g be two cuspidal modular forms and let {\mathcal {F}}${\mathcal {F}}$ be a Coleman family passing through f, defined over an open affinoid subdomain V of weight space \mathcal {W}$\mathcal {W}$ . Using ideas of Pottharst, under certain hypotheses on f and g,$g,$ we construct a coherent sheaf over V \times \mathcal {W}$V \times \mathcal {W}$ that interpolates the Bloch–Kato Selmer group of the Rankin–Selberg convolution of two modular forms in the critical range (i.e, the range where the p-adic L-function L_p$L_p$ interpolates critical values of the global L-function). We show that the support of this sheaf is contained in the vanishing locus of L_p$L_p$ .

We construct a scheme B(r; {\mathbb {A}}^n)$B(r; {\mathbb {A}}^n)$ such that a map X \to B(r; {\mathbb {A}}^n)$X \to B(r; {\mathbb {A}}^n)$ corresponds to a degree-n étale algebra on X equipped with r generating global sections. We then show that when n=2$n=2$ , i.e., in the quadratic étale case, the singular cohomology of B(r; {\mathbb {A}}^n)({\mathbb {R}})$B(r; {\mathbb {A}}^n)({\mathbb {R}})$ can be used to reconstruct a famous example of S. Chase and to extend its application to showing that there is a smooth affine r-1$r-1$ -dimensional {\mathbb {R}}${\mathbb {R}}$ -variety on which there are étale algebras {\mathcal {A}}_n${\mathcal {A}}_n$ of arbitrary degrees n that cannot be generated by fewer than r elements. This shows that in the étale algebra case, a bound established by U. First and Z. Reichstein in [6] is sharp.

We study exceptional Jordan algebras and related exceptional group schemes over commutative rings from a geometric point of view, using appropriate torsors to parametrize and explain classical and new constructions, and proving that over rings, they give rise to nonisomorphic structures.

We begin by showing that isotopes of Albert algebras are obtained as twists by a certain \mathrm F_4$\mathrm F_4$ -torsor with total space a group of type \mathrm E_6$\mathrm E_6$ and, using this, that Albert algebras over rings in general admit nonisomorphic isotopes even in the split case, as opposed to the situation over fields. We then consider certain \mathrm D_4$\mathrm D_4$ -torsors constructed from reduced Albert algebras, and show how these give rise to a class of generalised reduced Albert algebras constructed from compositions of quadratic forms. Showing that this torsor is nontrivial, we conclude that the Albert algebra does not uniquely determine the underlying composition, even in the split case. In a similar vein, we show that a given reduced Albert algebra can admit two coordinate algebras which are nonisomorphic and have nonisometric quadratic forms, contrary, in a strong sense, to the case over fields, established by Albert and Jacobson.