In the present paper, we prove an identity for the generating function of the quadruple zeta values with the action of the matrix ring $\mathrm{M}_{4}(\mathbb{Z})$. Taking homogeneous parts on both sides of the identity and substituting appropriate values for the variables, we obtain the sum formula for quadruple zeta values. We also obtain its weighted analogues, which include the formulas for this case proved by Guo and Xie (2009, J. Number Theory 129, 2747--2765) and by Ong, Eie, and Liaw (2013, Int. J. Number Theory 9, 1185--1198).

In this paper we introduce the notion of an unknotting index for virtual knots. We give some examples of computation by using writhe invariants, and discuss a relationship between the unknotting index and the virtual knot module. In particular, we show that for any non-negative integer $n$ there exists a virtual knot whose unknotting index is $(1,n)$.

In this paper, we study conjugacy invariants for 2-dimensional diffeomorphisms with homoclinic cubic tangencies (two-sided tangencies of the lowest order) under certain open conditions. Ordinary arguments used in past studies of conjugacy invariants associated with one-sided tangencies do not work in the two-sided case. We present a new method which is applicable to the two-sided case.

Ito, Miura, Okawa and Ueda have constructed a pair of Calabi-Yau threefolds $X$ and $Y$ that are L-equivalent and derived equivalent, but not stably birational. We complete the picture by showing that $X$ and $Y$ have isomorphic Chow motives.

We generalize the notion of biquandles to psyquandles and use these to define invariants of oriented singular links and pseudolinks. In addition to psyquandle counting invariants, we introduce Alexander psyquandles and corresponding invariants such as Alexander psyquandle polynomials and Alexander-Gröbner psyquandle invariants of oriented singular knots and links. We consider the relationship between Alexander psyquandle colorings of pseudolinks and $p$-colorings of pseudolinks. As a special case we define a generalization of the Alexander polynomial for oriented singular links and pseudolinks we call the Jablan polynomial and compute the invariant for all pseudoknots with up to five crossings and all 2-bouquet graphs with up to 6 classical crossings.

In this paper, we extend Roe's cyclic $1$-cocycle to relative settings. We also prove two relative index theorems for partitioned manifolds by using its cyclic cocycle, which are generalizations of index theorems on partitioned manifolds. One of these theorems is a variant of Theorem 3.3 in “Relative-partitioned index theorem” by M. Karami, A.H.S. Sadegh and M.E. Zadeh.

We consider a $C^r$ diffeomorphism having a Hopf point with $r\ge 5$. If there exists a homoclinic orbit associated with the Hopf point, we say that the diffeomorphism has a Hopf-homoclinic cycle. In this paper we prove that every $C^r$ diffeomorphism having a Hopf-homoclinic cycle can be $C^r$ approximated by diffeomorphisms with heterodimensional cycles. Moreover, we study stabilizations of such heterodimensional cycles.

We consider the commutators $[b,T]$ and $[b,I_{\rho}]$, where $T$ is a Calderón-Zygmund operator, $I_{\rho}$ is a generalized fractional integral operator and $b$ is a function in Campanato spaces with variable growth condition. It is known that these commutators are bounded on generalized Morrey spaces with variable growth condition. In this paper we discuss the compactness of these commutators.

We develop a general theory of quiver representations $\mathcal{F}$ which is motivated by investigating group representations in different characteristics simultaneously. A feature of $\mathcal{F}$ is to assign to each vertex $a$ of the quiver a finite set $\mathcal{F}_{a}$. This result provides a uniform explanation of many objects in group character theory. Furthermore, we give a characterization of when the prime graph $\Gamma(G)$ of a finite group $G$ is disconnected by using group characters. And then, this result on $\Gamma(G)$ can be rephrased in terms of certain quiver representations.

On a Shimura curve, the reduction modulo a prime $p$ of the Hecke correspondence $T(p)$ yields the congruence relation $\Pi\cup\Pi'$ with $\Pi$ being the graph of the Frobenius mapping from the Shimura curve modulo $p$ to itself, and $\Pi'$ its transpose. Starting with a curve $C$ of genus $g \geq 2$ over $\mathbb{F}_p$ together with a subset $\mathfrak{S}\subset C(\mathbb{F}_{p^2})$, Ihara studied the liftability to characteristic $0$ of $\Pi\cup\Pi'$ so that $\Pi$ and $\Pi'$ are separated outside $\mathfrak{S}$ in the lifting. In some case, Ihara obtained the uniqueness of the liftability to characteristic $0$ and gave some necessary and sufficient condition, described by some differential form on $C$, for $(C,\mathfrak{S})$ to be liftable to modulo $p^2$. In this paper, in case when $C$ is defined over $\mathbb{F}_{p^2}$, we compute complete tables of such $(C,{\mathfrak S})$ liftable to modulo $p^2$ for $g=2$ and $3\leq p \leq 13$ using computer, and as an application of this uniqueness, we identify some particular Shimura curve by its equation.

In this paper, we introduce the higher order hypergeometric Euler numbers and show several interesting expressions. In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli, Cauchy and Euler numbers. One advantage of hypergeometric numbers, including Bernoulli, Cauchy and Euler hypergeometric numbers, is the natural extension of determinant expressions of the numbers. As applications, we can get the inversion relations such that Euler numbers are elements in the determinant.

We compute the principal curvatures of homogeneous hypersurfaces in a Grassmann manifold $\widetilde{\text{Gr}}_{ 3}(\text{Im}\mathbb{O})$ by the $G_2$-action. As applications, we show that there is a unique orbit which is an austere submanifold, and that there are just two orbits which are proper biharmonic homogeneous hypersurfaces. We also show that the austere orbit is a weakly reflective submanifold.

In recent work Burns, Kurihara and Sano introduced a natural notion of (generalised) Stark elements of arbitrary rank and weight and conjectured a precise congruence relation between Stark elements of a fixed rank and different weights. This conjecture was shown to simultaneously generalise both the classical congruences of Kummer and the explicit reciprocity law of Artin-Hasse and Iwasawa. In this article, we show that the congruence conjecture also implies that the Iwasawa theoretical `zeta element’ that is conjecturally associated to the multiplicative group has good interpolation properties at arbitrary even integers.

A set of positive integers is called a Diophantine tuple if the product of any two elements in the set increased by 1 is a perfect square. A conjecture in this field asserts that any Diophantine triple can be uniquely extended to a Diophantine quadruple in some sense. This problem is reduced to study the coincidence between certain two binary recurrent sequences of integers. As an analogy of this, we consider a similar coincidence on the polynomial ring in one variable over the integers, and determine it completely. Our result is regarded as a generalization of a result in the paper “Complete solution of the polynomial version of a problem of Diophantus” by A. Dujella, C. Fuchs in Journal of Number Theory 106 (2004) on the polynomial variant of Diophantine tuples.