The purpose of this paper is to construct infinite families of real abelian number fields $K$ of degree four with $\lambda_{2}(K)= \mu_{2}(K) =0$ and $\nu_{2}(K) >0$.

In this paper we discuss functional relations for the multi-variable version of Witten zeta-functions associated with the Lie algebra $\mathfrak{sl}(r)$ ($r=3, 4, 5$).

Let $\A_j$ be a uniform algebra with a Choquet boundary $Ch\A_j$, $j = 1, 2$. In this paper we prove that if $\phi : \exp\A_1 \to \exp\A_2$ is a surjection and satisfies the equality \begin{equation*} \max \left\{ \left\| \frac{\phi (f)}{\phi (g)} -1 \right\|_\infty, \left\| \frac{\phi (g)}{\phi (f)} -1 \right\|_\infty \right \} =\max \left\{ \left\| \frac{f}{g} -1 \right\|_\infty, \left\| \frac{g}{f} -1 \right\|_\infty \right \} \end{equation*} for any $f, g \in \exp\A_1$, then $\phi$ is of the form \begin{equation*} \phi(f)(y) = \left\{ \begin{array}{ll} \phi(1)(y) f(\varphi(y))^{\kappa(y)} & \text{for}~ y \in K, \\ \phi(1)(y) \overline{f(\varphi(y))}^{\kappa(y)} & \text{for}~ y \in Ch\A_2 \setminus K \\ \end{array} \right. \end{equation*} for any $f \in \exp\A_1$, where $\kappa$ is a continuous function from $Ch\A_2$ into $\{ 1, -1 \}$, $\varphi$ is a homeomorphism from $Ch\A_2$ onto $Ch\A_1$ and $K$ is a clopen subset of $Ch\A_2$.

We derive an analytic formula for the dual Jacobian matrix of a generalised hyperbolic tetrahedron. Two cases are considered: a mildly truncated and a prism truncated tetrahedron. The Jacobian for the latter arises as an analytic continuation of the former, that falls in line with a similar behaviour of the corresponding volume formulae. Also, we obtain a volume formula for a hyperbolic $n$-gonal prism: the proof requires the above mentioned Jacobian, employed in the analysis of the edge lengths behaviour of such a prism, needed later for the Schläfli formula.

Let $K_n$ be the $n$-th layer of the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}(\sqrt{5})$ and $h_n$ the class number of $K_n$. We prove that, if $\ell$ is a prime number less than $6\cdot10^4$, then $\ell$ does not divide $h_n$ for any non-negative integer $n$.

In this paper we study the interface regularity of the solutions to the differential systems defined by differential forms (for example, stationary Maxwell systems) on $N(\geq 3)$-dimensional Riemannian manifolds. Our results are natural extensions of the results of \textit{Interface regularity of the solutions for the rotation free and the divergence free systems} and \textit{Interface vanishing for solutions to Maxwell and Stokes systems}.

We classify the isomorphism classes of finitely generated torsion $\mathcal{O}_E[[T]]$-modules which are free over $\mathcal{O}_E$ of rank $4$, where $\mathcal{O}_E$ is the ring of the integers of a local field $E$. We apply this classification to the Iwasawa module associated to the cyclotomic $\mathbb{Z}_p$-extension of an imaginary quadratic field.

The symmetries of complex molecular structures can be modeled by the {\em topological symmetry group} of the underlying embedded graph. It is therefore important to understand which topological symmetry groups can be realized by particular abstract graphs. This question has been answered for complete graphs [7]; it is natural next to consider complete bipartite graphs. In previous work we classified the complete bipartite graphs that can realize topological symmetry groups isomorphic to $A_4$, $S_4$ or $A_5$ [12]; in this paper we determine which complete bipartite graphs have an embedding in $S^3$ whose topological symmetry group is isomorphic to $\Z_m$, $D_m$, $\Z_r \x \Z_s$ or $(\Z_r \x \Z_s) \ltimes \Z_2$.

In this paper, we study quandles of cyclic type, which form a particular subclass of finite quandles. The main result of this paper describes the set of isomorphism classes of quandles of cyclic type in terms of certain cyclic permutations. By using our description, we give a direct classification of quandles of cyclic type with cardinality up to 12.

We prove an extension of a theorem of A. Ros on \textit{a characterization of seven compact Kaehler submanifolds by holomorphic pinching} [5] to certain submanifolds of the complex Grassmannian manifolds.

In this paper, we prove, for primes $l$ satisfying some conditions, the $l$-parts of the Galois Brumer-Stark conjecture, which is formulated by Dejou and Roblot for Galois CM-extensions with dihedral or generalized quaternion Galois group of specified degrees.

We present new blow-up results for nonlocal reaction-diffusion equations with nonlocal nonlinearities. The nonlocal source terms we consider are of several types, and are relevant to various models in physics and engineering. They may involve an integral of an unknown function, either in space, in time, or both in space and time, or they may depend on localized values of the solution. We first show the existence and uniqueness of the solution to problem relying on contraction mapping fixed point theorem. Then, the comparison principles for problem are established through a standard method. Finally, for the radially symmetric and non-increasing initial data, we give a complete classification in terms of global and single point blow-up according to the parameters. Moreover, the blow-up rates are also determined in each case.

We will study the Chern-Ricci flow on non-Kähler properly elliptic surfaces. These surfaces are compact complex surfaces whose first Betti number is odd, Kodaira dimension is equal to 1 and admit an elliptic fibration to a smooth compact curve. We will show that a solution of the Chern-Ricci flow converges in $C^\alpha$-topology on these elliptic surfaces by choosing a special initial metric.

Euler's famous formula written in symbolic notation as $(B_0+B_0)^n=-n B_{n-1}-(n-1)B_n$ was extended to $(B_{l_1}+\cdots+B_{l_m})^n$ for $m\ge 2$ and arbitrary fixed integers $l_1,\dots,l_m\ge 0$. In this paper, we consider the higher-order recurrences for Cauchy numbers $(c_{l_1}+\cdots+c_{l_m})^n$, where the $n$-th Cauchy number $c_n$ ($n\ge 0$) is defined by the generating function $x/\ln(1+x)=\sum_{n=0}^\infty c_n x^n/n!$. In special, we give an explicit expression in the case $l_1=\cdots=l_m=0$ for any integers $n\ge 1$ and $m\ge 2$. We also discuss the case for Cauchy numbers of the second kind $\widehat c_n$ in similar ways.

We are concerned with finitely nested square roots which are roots of iterations of a real quadratic polynomial $x^2-c$ with $c\geq 2$, and the limits of such nested square roots. We investigate how they are related to a Poincaré function $f(x)$ satisfying the functional equation $f(sx)=f(x)^2-c$, where $s=1+\sqrt{1+4c}$. Our main theorems can be viewed as a natural generalization of the work of Wiernsberger and Lebesgue for the case $c=2$. The key ingredients of the proof are some analytic properties of $F(x)$, which have been intensively studied by the second author using infinite compositions.

This paper is concerned with semilinear Volterra diffusion equations with spatial inhomogeneity and advection. We intend to study the effects of interaction among diffusion, advection and Volterra integral under spatially inhomogeneous environments. Since the existence and uniqueness result of global-in-time solutions can be proved in the standard manner, our main interest is to study their asymptotic behavior as $t\to \infty$. For this purpose, we study the related stationary problem by the monotone method and establish some sufficient conditions on the existence of a unique positive solution. Its global attractivity is also studied with use of a suitable Lyapunov functional.

Moura, Neves and Schneider proved the trace theorem for 2-microlocal Besov spaces with variable integrability and smoothness, where the summability parameter was constant (Math.\,Nachr.,286 (2013) 1240--1254). In this paper, we extend the trace theorem for the case that the summability parameter is also a variable exponent.