An open question akin to the slice-ribbon conjecture asks whether every ribbon knot can be represented as a symmetric union. Next to this basic existence question sits the question of uniqueness of such representations. Eisermann and Lamm investigated the latter question by introducing a notion of symmetric equivalence among symmetric union diagrams and showing that non-equivalent diagrams can be detected using a refined version of the Jones polynomial. We prove that every topological spin model gives rise to many effective invariants of symmetric equivalence, which can be used to distinguish infinitely many Reidemeister equivalent but symmetrically non-equivalent symmetric union diagrams. We also show that such invariants are not equivalent to the refined Jones polynomial and we use them to provide a partial answer to a question left open by Eisermann and Lamm.

Let k$k$ be an arbitrary positive integer and let \unicode[STIX]{x1D6FE}(n)$\unicode[STIX]{x1D6FE}(n)$ stand for the product of the distinct prime factors of n$n$. For each integer n\geqslant 2$n\geqslant 2$, let a_{n}$a_{n}$ and b_{n}$b_{n}$ stand respectively for the maximum and the minimum of the k$k$ integers \unicode[STIX]{x1D6FE}(n+1),\unicode[STIX]{x1D6FE}(n+2),\ldots ,\unicode[STIX]{x1D6FE}(n+k)$\unicode[STIX]{x1D6FE}(n+1),\unicode[STIX]{x1D6FE}(n+2),\ldots ,\unicode[STIX]{x1D6FE}(n+k)$. We show that \liminf _{n\rightarrow \infty }a_{n}/b_{n}=1$\liminf _{n\rightarrow \infty }a_{n}/b_{n}=1$. We also prove that the same result holds in the case of the Euler function and the sum of the divisors function, as well as the functions \unicode[STIX]{x1D714}(n)$\unicode[STIX]{x1D714}(n)$ and \unicode[STIX]{x1D6FA}(n)$\unicode[STIX]{x1D6FA}(n)$, which stand respectively for the number of distinct prime factors of n$n$ and the total number of prime factors of n$n$ counting their multiplicity.

We approximate the Riemann Zeta-Function by polynomials and Dirichlet polynomials with restricted zeros.

Let \{\mathbf{F}(n)\}_{n\in \mathbb{N}}$\{\mathbf{F}(n)\}_{n\in \mathbb{N}}$ and \{\mathbf{G}(n)\}_{n\in \mathbb{N}}$\{\mathbf{G}(n)\}_{n\in \mathbb{N}}$ be linear recurrence sequences. It is a well-known Diophantine problem to determine the finiteness of the set {\mathcal{N}}${\mathcal{N}}$ of natural numbers such that their ratio \mathbf{F}(n)/\mathbf{G}(n)$\mathbf{F}(n)/\mathbf{G}(n)$ is an integer. In this paper we study an analogue of such a divisibility problem in the complex situation. Namely, we are concerned with the divisibility problem (in the sense of complex entire functions) for two sequences F(n)=a_{0}+a_{1}f_{1}^{n}+\cdots +a_{l}f_{l}^{n}$F(n)=a_{0}+a_{1}f_{1}^{n}+\cdots +a_{l}f_{l}^{n}$ and G(n)=b_{0}+b_{1}g_{1}^{n}+\cdots +b_{m}g_{m}^{n}$G(n)=b_{0}+b_{1}g_{1}^{n}+\cdots +b_{m}g_{m}^{n}$, where the f_{i}$f_{i}$ and g_{j}$g_{j}$ are nonconstant entire functions and the a_{i}$a_{i}$ and b_{j}$b_{j}$ are non-zero constants except that a_{0}$a_{0}$ can be zero. We will show that the set {\mathcal{N}}${\mathcal{N}}$ of natural numbers such that F(n)/G(n)$F(n)/G(n)$ is an entire function is finite under the assumption that f_{1}^{i_{1}}\cdots f_{l}^{i_{l}}g_{1}^{j_{1}}\cdots g_{m}^{j_{m}}$f_{1}^{i_{1}}\cdots f_{l}^{i_{l}}g_{1}^{j_{1}}\cdots g_{m}^{j_{m}}$ is not constant for any non-trivial index set (i_{1},\ldots ,i_{l},j_{1},\ldots ,j_{m})\in \mathbb{Z}^{l+m}$(i_{1},\ldots ,i_{l},j_{1},\ldots ,j_{m})\in \mathbb{Z}^{l+m}$.

The notion of metric compactification was introduced by Gromov and later rediscovered by Rieffel. It has been mainly studied on proper geodesic metric spaces. We present here a generalization of the metric compactification that can be applied to infinite-dimensional Banach spaces. Thereafter we give a complete description of the metric compactification of infinite-dimensional \ell _{p}$\ell _{p}$ spaces for all 1\leqslant p<\infty$1\leqslant p<\infty$. We also give a full characterization of the metric compactification of infinite-dimensional Hilbert spaces.

In this paper, we study a class of homogeneous Finsler metrics of vanishing S$S$-curvature on a (4n+3)$(4n+3)$-dimensional sphere. We find a second order ordinary differential equation that characterizes Einstein metrics with constant Ricci curvature 1$1$ in this class. Using this equation we show that there are infinitely many homogeneous Einstein metrics on S^{4n+3}$S^{4n+3}$ of constant Ricci curvature 1$1$ and vanishing S$S$-curvature. They contain the canonical metric on S^{4n+3}$S^{4n+3}$ of constant sectional curvature 1$1$ and the Einstein metric of non-constant sectional curvature given by Jensen in 1973.

Infinitely many new Einstein Finsler metrics are constructed on several homogeneous spaces. By imposing certain conditions on the homogeneous spaces, it is shown that the Ricci constant condition becomes an ordinary differential equation. The regular solutions of this equation lead to a two parameter family of Einstein Finsler metrics with vanishing S$S$ curvature.

We describe all degenerations of three-dimensional anticommutative algebras \mathfrak{A}\mathfrak{c}\mathfrak{o}\mathfrak{m}_{3}$\mathfrak{A}\mathfrak{c}\mathfrak{o}\mathfrak{m}_{3}$ and of three-dimensional Leibniz algebras \mathfrak{L}\mathfrak{e}\mathfrak{i}\mathfrak{b}_{3}$\mathfrak{L}\mathfrak{e}\mathfrak{i}\mathfrak{b}_{3}$ over \mathbb{C}$\mathbb{C}$. In particular, we describe all irreducible components and rigid algebras in the corresponding varieties.

Given systems of two (inhomogeneous) quadratic equations in four variables, it is known that the Hasse principle for integral points may fail. Sometimes this failure can be explained by some integral Brauer–Manin obstruction. We study the existence of a non-trivial algebraic part of the Brauer group for a family of such systems and show that the failure of the integral Hasse principle due to an algebraic Brauer–Manin obstruction is rare, as for a generic choice of a system the algebraic part of the Brauer-group is trivial. We use resolvent constructions to give quantitative upper bounds on the number of exceptions.

We produce an isomorphism E_{\infty }^{m,-m-1}\cong \text{Nrd}_{1}(A^{\otimes m})$E_{\infty }^{m,-m-1}\cong \text{Nrd}_{1}(A^{\otimes m})$ between terms of the \text{K}$\text{K}$-theory coniveau spectral sequence of a Severi–Brauer variety X$X$ associated with a central simple algebra A$A$ and a reduced norm group, assuming A$A$ has equal index and exponent over all finite extensions of its center and that \text{SK}_{1}(A^{\otimes i})=1$\text{SK}_{1}(A^{\otimes i})=1$ for all i>0$i>0$.

Free binary systems are shown not to admit idempotent means. This refutes a conjecture of the author. It is also shown that the extension of Hindman’s theorem to nonassociative binary systems formulated and conjectured by the author is false.

A Tits polygon is a bipartite graph in which the neighborhood of every vertex is endowed with an “opposition relation” satisfying certain properties. Moufang polygons are precisely the Tits polygons in which these opposition relations are all trivial. There is a standard construction that produces a Tits polygon whose opposition relations are not all trivial from an arbitrary pair (\unicode[STIX]{x1D6E5},T)$(\unicode[STIX]{x1D6E5},T)$, where \unicode[STIX]{x1D6E5}$\unicode[STIX]{x1D6E5}$ is a building of type \unicode[STIX]{x1D6F1}$\unicode[STIX]{x1D6F1}$, \unicode[STIX]{x1D6F1}$\unicode[STIX]{x1D6F1}$ is a spherical, irreducible Coxeter diagram of rank at least 3$3$, and T$T$ is a Tits index of absolute type \unicode[STIX]{x1D6F1}$\unicode[STIX]{x1D6F1}$ and relative rank 2$2$. A Tits polygon is called k$k$-plump if its opposition relations satisfy a mild condition that is satisfied by all Tits triangles coming from a pair (\unicode[STIX]{x1D6E5},T)$(\unicode[STIX]{x1D6E5},T)$ such that every panel of \unicode[STIX]{x1D6E5}$\unicode[STIX]{x1D6E5}$ has at least k+1$k+1$ chambers. We show that a 5$5$-plump Tits triangle is parametrized and uniquely determined by a ring R$R$ that is alternative and of stable rank 2$2$. We use the connection between Tits triangles and the theory of Veldkamp planes as developed by Veldkamp and Faulkner to show existence.

A theorem of Burgess and Stephenson asserts that in an exchange ring with central idempotents, every maximal left ideal is also a right ideal. The proof uses sheaf-theoretic techniques. In this paper, we give a short elementary proof of this important theorem.

We present a Hopf boundary point lemma for the difference between two Hölder continuously differentiable functions, each weak solutions to a divergence-form quasilinear equation, under mild boundedness assumptions on the coefficients of this equation.

In this paper, we investigate the holomorphic sections of holomorphic Finsler bundles over both compact and non-compact complete complex manifolds. We also inquire into the holomorphic vector fields on compact and non-compact complete complex Finsler manifolds. We get vanishing theorems in each case according to different certain curvature conditions. This work can be considered as generalizations of the classical results on Kähler manifolds and hermitian bundles.

The classical Alexandrov–Bakelman–Pucci estimate for the Laplacian states

\begin{eqnarray}\max _{x\in \unicode[STIX]{x03A9}}|u(x)|\leqslant \max _{x\in \unicode[STIX]{x2202}\unicode[STIX]{x03A9}}|u(x)|+c_{s,n}\text{diam}(\unicode[STIX]{x03A9})^{2-\frac{n}{s}}\Vert \unicode[STIX]{x0394}u\Vert _{L^{s}(\unicode[STIX]{x03A9})},\end{eqnarray} $$\begin{eqnarray}\max _{x\in \unicode[STIX]{x03A9}}|u(x)|\leqslant \max _{x\in \unicode[STIX]{x2202}\unicode[STIX]{x03A9}}|u(x)|+c_{s,n}\text{diam}(\unicode[STIX]{x03A9})^{2-\frac{n}{s}}\Vert \unicode[STIX]{x0394}u\Vert _{L^{s}(\unicode[STIX]{x03A9})},\end{eqnarray}$$

\unicode[STIX]{x03A9}\subset \mathbb{R}^{n}$\unicode[STIX]{x03A9}\subset \mathbb{R}^{n}$

u\in C^{2}(\unicode[STIX]{x03A9})\cap C(\overline{\unicode[STIX]{x03A9}})$u\in C^{2}(\unicode[STIX]{x03A9})\cap C(\overline{\unicode[STIX]{x03A9}})$

s>n/2$s>n/2$

s=n/2$s=n/2$

n\geqslant 3$n\geqslant 3$

\unicode[STIX]{x03A9}\subset \mathbb{R}^{n}$\unicode[STIX]{x03A9}\subset \mathbb{R}^{n}$

\begin{eqnarray}\max _{x\in \unicode[STIX]{x03A9}}|u(x)|\leqslant \max _{x\in \unicode[STIX]{x2202}\unicode[STIX]{x03A9}}|u(x)|+c_{n}\Vert \unicode[STIX]{x0394}u\Vert _{L^{\frac{n}{2},1}(\unicode[STIX]{x03A9})},\end{eqnarray} $$\begin{eqnarray}\max _{x\in \unicode[STIX]{x03A9}}|u(x)|\leqslant \max _{x\in \unicode[STIX]{x2202}\unicode[STIX]{x03A9}}|u(x)|+c_{n}\Vert \unicode[STIX]{x0394}u\Vert _{L^{\frac{n}{2},1}(\unicode[STIX]{x03A9})},\end{eqnarray}$$

L^{p,q}$L^{p,q}$

L^{p}$L^{p}$

n=2$n=2$

c>0$c>0$

\unicode[STIX]{x03A9}\subset \mathbb{R}^{2}$\unicode[STIX]{x03A9}\subset \mathbb{R}^{2}$

\begin{eqnarray}\max _{x\in \unicode[STIX]{x03A9}}|u(x)|\leqslant \max _{x\in \unicode[STIX]{x2202}\unicode[STIX]{x03A9}}|u(x)|+c\max _{x\in \unicode[STIX]{x03A9}}\int _{y\in \unicode[STIX]{x03A9}}\max \left\{1,\log \left(\frac{|\unicode[STIX]{x03A9}|}{\Vert x-y\Vert ^{2}}\right)\right\}|\unicode[STIX]{x0394}u(y)|dy.\end{eqnarray} $$\begin{eqnarray}\max _{x\in \unicode[STIX]{x03A9}}|u(x)|\leqslant \max _{x\in \unicode[STIX]{x2202}\unicode[STIX]{x03A9}}|u(x)|+c\max _{x\in \unicode[STIX]{x03A9}}\int _{y\in \unicode[STIX]{x03A9}}\max \left\{1,\log \left(\frac{|\unicode[STIX]{x03A9}|}{\Vert x-y\Vert ^{2}}\right)\right\}|\unicode[STIX]{x0394}u(y)|dy.\end{eqnarray}$$

The object of this paper is to study Yamabe solitons on almost co-Kähler manifolds as well as on (k,\unicode[STIX]{x1D707})$(k,\unicode[STIX]{x1D707})$-almost co-Kähler manifolds. We also study Ricci solitons on (k,\unicode[STIX]{x1D707})$(k,\unicode[STIX]{x1D707})$-almost co-Kähler manifolds.

We prove that Krivine’s Function Calculus is compatible with integration. Let (\unicode[STIX]{x1D6FA},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D707})$(\unicode[STIX]{x1D6FA},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D707})$ be a finite measure space, X$X$ a Banach lattice, \mathbf{x}\in X^{n}$\mathbf{x}\in X^{n}$, and f:\mathbb{R}^{n}\times \unicode[STIX]{x1D6FA}\rightarrow \mathbb{R}$f:\mathbb{R}^{n}\times \unicode[STIX]{x1D6FA}\rightarrow \mathbb{R}$ a function such that f(\cdot ,\unicode[STIX]{x1D714})$f(\cdot ,\unicode[STIX]{x1D714})$ is continuous and positively homogeneous for every \unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}$\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}$, and f(\mathbf{s},\cdot )$f(\mathbf{s},\cdot )$ is integrable for every \mathbf{s}\in \mathbb{R}^{n}$\mathbf{s}\in \mathbb{R}^{n}$. Put F(\mathbf{s})=\int f(\mathbf{s},\unicode[STIX]{x1D714})\,d\unicode[STIX]{x1D707}(\unicode[STIX]{x1D714})$F(\mathbf{s})=\int f(\mathbf{s},\unicode[STIX]{x1D714})\,d\unicode[STIX]{x1D707}(\unicode[STIX]{x1D714})$ and define F(\mathbf{x})$F(\mathbf{x})$ and f(\mathbf{x},\unicode[STIX]{x1D714})$f(\mathbf{x},\unicode[STIX]{x1D714})$ via Krivine’s Function Calculus. We prove that under certain natural assumptions F(\mathbf{x})=\int f(\mathbf{x},\unicode[STIX]{x1D714})\,d\unicode[STIX]{x1D707}(\unicode[STIX]{x1D714})$F(\mathbf{x})=\int f(\mathbf{x},\unicode[STIX]{x1D714})\,d\unicode[STIX]{x1D707}(\unicode[STIX]{x1D714})$, where the right hand side is a Bochner integral.

Suppose that D\subset \mathbb{C}$D\subset \mathbb{C}$ is a simply connected subdomain containing the origin and f(z_{1})$f(z_{1})$ is a normalized convex (resp., starlike) function on D$D$. Let

\begin{eqnarray}\unicode[STIX]{x1D6FA}_{N}(D)=\bigg\{(z_{1},w_{1},\ldots ,w_{k})\in \mathbb{C}\times \mathbb{C}^{n_{1}}\times \cdots \times \mathbb{C}^{n_{k}}:\Vert w_{1}\Vert _{p_{1}}^{p_{1}}+\cdots +\Vert w_{k}\Vert _{p_{k}}^{p_{k}}<\frac{1}{\unicode[STIX]{x1D706}_{D}(z_{1})}\bigg\},\end{eqnarray} $$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{N}(D)=\bigg\{(z_{1},w_{1},\ldots ,w_{k})\in \mathbb{C}\times \mathbb{C}^{n_{1}}\times \cdots \times \mathbb{C}^{n_{k}}:\Vert w_{1}\Vert _{p_{1}}^{p_{1}}+\cdots +\Vert w_{k}\Vert _{p_{k}}^{p_{k}}<\frac{1}{\unicode[STIX]{x1D706}_{D}(z_{1})}\bigg\},\end{eqnarray}$$

p_{j}\geqslant 1$p_{j}\geqslant 1$

N=1+n_{1}+\cdots +n_{k}$N=1+n_{1}+\cdots +n_{k}$

w_{1}\in \mathbb{C}^{n_{1}},\ldots ,w_{k}\in \mathbb{C}^{n_{k}}$w_{1}\in \mathbb{C}^{n_{1}},\ldots ,w_{k}\in \mathbb{C}^{n_{k}}$

\unicode[STIX]{x1D706}_{D}$\unicode[STIX]{x1D706}_{D}$

D$D$

\begin{eqnarray}\unicode[STIX]{x1D6F7}_{N,1/p_{1},\ldots ,1/p_{k}}(f)(z_{1},w_{1},\ldots ,w_{k})=(f(z_{1}),(f^{\prime }(z_{1}))^{1/p_{1}}w_{1},\ldots ,(f^{\prime }(z_{1}))^{1/p_{k}}w_{k})\end{eqnarray} $$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{N,1/p_{1},\ldots ,1/p_{k}}(f)(z_{1},w_{1},\ldots ,w_{k})=(f(z_{1}),(f^{\prime }(z_{1}))^{1/p_{1}}w_{1},\ldots ,(f^{\prime }(z_{1}))^{1/p_{k}}w_{k})\end{eqnarray}$$

\unicode[STIX]{x1D6FA}_{N}(D)$\unicode[STIX]{x1D6FA}_{N}(D)$

D$D$

\mathbb{C}^{2}$\mathbb{C}^{2}$

\unicode[STIX]{x1D6F7}_{N,1/p_{1},\ldots ,1/p_{k}}(f)$\unicode[STIX]{x1D6F7}_{N,1/p_{1},\ldots ,1/p_{k}}(f)$

\unicode[STIX]{x1D6FC}$\unicode[STIX]{x1D6FC}$

f$f$

\unicode[STIX]{x1D6FC}$\unicode[STIX]{x1D6FC}$

Let \ell \in \mathbb{N}$\ell \in \mathbb{N}$ and p\in (1,\infty ]$p\in (1,\infty ]$. In this article, the authors establish several equivalent characterizations of Sobolev spaces W^{2\ell +2,p}(\mathbb{R}^{n})$W^{2\ell +2,p}(\mathbb{R}^{n})$ in terms of derivatives of ball averages. The novelty in the results of this article is that these equivalent characterizations reveal some new connections between the smoothness indices of Sobolev spaces and the derivatives on the radius of ball averages and also that, to obtain the corresponding results for higher order Sobolev spaces, the authors first establish the combinatorial equality: for any \ell \in \mathbb{N}$\ell \in \mathbb{N}$ and k\in \{0,\ldots ,\ell -1\}$k\in \{0,\ldots ,\ell -1\}$, \sum _{j=0}^{2\ell }(-1)^{j}\binom{2\ell }{j}|\ell -j|^{2k}=0$\sum _{j=0}^{2\ell }(-1)^{j}\binom{2\ell }{j}|\ell -j|^{2k}=0$.