We propose the complex dipole simulation method (CDSM) which approximates a holomorphic function by linear combination of $1 / (z - \zeta)$ with the use of its boundary values. In this paper, we treat a function $f$ which is holomorphic in $\Omega$ and continuous on $\overline{\Omega}$ in the case where $\Omega$ is a disk or the exterior domain of a disk. Then we establish the following fact: if $f$ is holomorphic in some neighborhood of $\overline{\Omega}$, the error of an approximate function $f^{(N)}$ decays exponentially with respect to $N$, where $N$ is the number of the charge points.

In this note, we give a cohomological characterization of all rank 2 split vector bundles on Hirzebruch surfaces.

We consider the $\mathbf{R}$-valued Vassiliev invariants of degrees 2 and 3 normalized by the conditions that they take values 0 on the unknot and 1 on the trefoil. We give certain answers for a problem due to N. Okuda about these two invariants. Moreover, we prove a conjecture due to Simon Willerton concerning the degree-3 Vassiliev invariant in the case of torus knots.

In this paper we study general conditions to prove the infiniteness of the genus of certain towers of function fields over a perfect field. We show that many known examples of towers with infinite genus are particular cases of these conditions. In the case of tame towers we show that the infiniteness of their genus is actually equivalent to these conditions.

In this paper we study the John-Nirenberg inequalities with variable exponents on a probability space. Let $Y$ be a rearrangement invariant Banach function space defined on $(\Omega,\mathcal{F},P)$ and a measurable function $p(\cdot): \Omega\rightarrow \mathbf{R}^+$ be a variable exponent. We prove that if the stochastic basis is regular, then $$BMO_{\phi,Y}=BMO_{\phi,p(\cdot)}\,,\quad \forall 1\leq p(\cdot)<\infty\,,$$ where $\phi(r)=1/r\Phi^{-1}(1/r)$ and $\Phi$ is a concave function with proper condition.

In this paper, we consider a classification problem for continuous fractional binary operations on $\mathbf K$, where $\mathbf K$ denotes the real field $\mathbf R$ or the complex field $\mathbf C$. We first show that there exist exactly two continuous fractional binary operations on $\mathbf R$ up to isomorphism. In the complex case, we describe completely all continuous fractional binary operations on $\mathbf C$ in terms of ordinary fraction. Applying this description, we give a partial solution to the classification problem in the complex case. Moreover we show that there exist exactly two homogeneous cancellative binary operations on $\mathbf K$ up to isomorphism.

We deal with the existence and non-existence of positive solutions for the problem \begin{eqnarray*} \displaystyle \left \{ \begin{array}{@{\,}c} -\Delta_p u +m(x)u^{p-1} = a(x)f(u) + \lambda b(x)g(u)~~ \text{in}~~ \mathbf{R}^N\,,\\ \\ \displaystyle u > 0~~ \text{in}~~\mathbf{R}^N\,,~~~ u(x)\rightarrow 0 \ \text{when} \ |x|\rightarrow\infty\,, \end{array} \right. \end{eqnarray*} where $\Delta_p$ is the $p$-Laplacian operator, $1<p<N$, $\lambda>0$ is a real parameter, $ f, g: (0, \infty)\rightarrow (0,\infty)$ and $m, a, b: \mathbf{R}^N\rightarrow[0,\infty )$; $ a, b\neq 0$ are continuous functions. In this work we consider, for example, nonlinearities with combined effects of concave and convex terms, besides allowing the presence of singularities. For existence of solutions, we exploit the lower and upper solutions method, combined with a technique of monotone-regularization on the nonlinearities $f$ and $g$ and for non-existence we use a consequence of Picone identity.

We refer generalized degenerate zigzag nanotubes as periodic metric graphs which consist of segments of length 1 and rings of length 2 throughout this paper. In this paper, we consider the case where there are one segment and three rings in the basic period cell and analyze the spectrum of periodic Schrödinger operators on the generalized degenerate zigzag nanotube. We obtain the relationship between the structure of the metric graph and the nondegenerate spectral gaps of the Schrödinger operators.

Let $S$ be an $\mathbf{N}^d$-graded algebra over a noetherian ring and a finitely generated $\mathbf{N}^d$-graded $S$-module $M$. This paper will study the relationship of filter-regular sequences of $M$ to joint reductions and homogeneous parameter systems. As an application, we show that any maximal filter-regular sequence is a joint reduction of $(S_1,\ldots,S_d)$ with respect to $M,$ and any maximal strong-filter-regular sequence is a reduction of $S_+$ with respect to $M$. And we characterize the existence of parts of homogeneous parameter systems for $M$ consisting of elements of total degree 1 via strong-filter-regular sequences.

A conjecture on a relation between two different Rubin-Stark elements was recently proposed by the author, and also by Mazur and Rubin. For a tower of finite extensions of global fields $K/L/k$ such that $K/k$ is abelian, this conjecture gives a relation between Rubin-Stark elements $\varepsilon_{K,S,T,V}$ and $\varepsilon_{L,S,T,V'}$, where $S, T, V$ and $V'$ are suitable sets of places of $k$. In this paper, we prove this conjecture under the following three assumptions: (i) $V$ contains all infinite places of $k$; (ii) all $v\in S$ split completely in $L$; (iii) $\mathrm{Gal}(K/L)$ is the direct product of the inertia groups at $v\in S\setminus V$.

Let $X,Y$ be locally compact Hausdorff spaces, $A$ be a complex subspace of $C_0(X)$ and $T: A \longrightarrow C_0(Y)$ be a real-linear isometry, whose range is not assumed to be a complex subspace of $C_0(Y)$. In this paper, using the set $\Theta(A)$ and $\tau(A)$ consisting of all extremely strong boundary points and strong boundary points of $A$, respectively we introduce appropriate subsets $Y_0$ and $Y_1$ of $Y$ and give a description of $T$ on these sets. More precisely, we show that there exist continuous functions $\Phi:Y_0\longrightarrow \Theta(A)$, $\alpha:Y_0\longrightarrow [-1,1]$ and $w:Y_0\longrightarrow \mathbf{T}$, where $\mathbf{T}$ is the unit circle, such that \[Tf(y)=w(y) \cdot (\mathrm{Re}(f(\Phi(y)))+\alpha(y) i \, \mathrm{Im}(f(\Phi(y))) \] for all $f\in A$ and $y\in Y_0$. The result is improved in the case where either i) $T(A)$ is a complex subspace of $C_0(Y)$ and $\Theta(A)= \mathrm{ch}(A)$, where $\mathrm{ch}(A)$ is the Choquet boundary of $A$ or ii) $T(A)$ satisfies a certain separating property.

In the first case we show that there exists a clopen subset $K$ of $Y_0$ such that \[ (Tf)(y)= w(y)\left\lbrace \begin{array}{@{}cl} f( \Phi(y)) & \,\, y\in K,\\ \overline{f(\Phi(y))} & \,\, y\notin K, \end{array} \right. \] for each $f\in A$ and $y\in Y_0$. In the second case we obtain similar results for $\tau(A)\cap \mathrm{ch}(A)$ and $Y_1$ instead of $\Theta(A)$ and $Y_0$.

For a given oriented spherical closed curve with $n$ transversal double points, we assign a cyclic word of length $2n$ on two letters $L$ standing left and $R$ standing right by reading the crossing sign so that each crossing point is read once $L$ and once $R$. The LR number of the curve is the number of appearance of subwords $LR$ in the cyclic word. We completely determine oriented spherical closed curves whose LR numbers are less than or equal to three.

We prove a generalization of the Strichartz estimate for the inhomogeneous wave equation $\square u(t,x) =f(t,x)$ in the space-time $\boldsymbol{R}^{1+n}$. We estimate the solution in vector-valued homogeneous Besov spaces $\dot{B}^\theta_{q,2}(\boldsymbol{R}; \dot{B}^\sigma_{r,2}(\boldsymbol{R}^n))$. Such an estimate shows the time differentiability of the solution of fractional order.

In this paper, we introduce the countable chain condition for C*-algebras and study its fundamental properties. We show independence from $\mathsf{ZFC}$ of the statement that this condition is preserved under the tensor products of C*-algebras.

Torsion-free finitely generated nilpotent groups are called $\mathcal{T}$-groups. This article gives a class of $\mathcal{T}$-groups in which zeta functions of groups determine the isomorphism classes.

As was pointed out by Nikulin [8] and Vinberg [10], a right-angled polyhedron of finite volume in the hyperbolic $n$-space $\mathbf{H}^n$ has at least one cusp for $n\geq 5$. We obtain non-trivial lower bounds on the number of cusps of such polyhedra. For example, right-angled polyhedra of finite volume must have at least three cusps for $n=6$. Our theorem also says that the higher the dimension of a right-angled polyhedron becomes, the more cusps it must have.

For a closed 4-manifold $X$, any knot $K$ in the boundary of punctured $X$ bounds a non-orientable and null-homologous embedded surface in punctured $X$. Thus we can define an invariant $\gamma_X^0(K)$ to be the smallest first Betti number of such surfaces. Note that $\gamma^0_{S^4}$ is equal to the non-orientable 4-ball genus. While it is very likely that for a given $X$, $\gamma^0_X$ has no upper bound, it is difficult to show it. Recently, Batson showed that $\gamma^0_{S^4}$ has no upper bound. In this paper we show that for any positive integer $n$, $\gamma^0_{n\mathbf{C}P^2}$ has no upper bound.

In this paper, we prove the equivalence of kinetic solutions and entropy solutions for the initial-boundary value problem with a non-homogeneous boundary condition for a multi-dimensional scalar first-order conservation law with a multiplicative noise. We somewhat generalized the definitions of kinetic solutions and of entropy solutions given in Kobayasi and Noboriguchi [8] and Bauzet, Vallet and Wittobolt [1], respectively.