We start with the Lorentz algebra $\text{\large$\mathfrak{L}$}=\text{\large$\mathfrak{o}$}_{\mathbb{R}}(1,3)$ over the reals and find a suitable basis $B$ such that the structure constants relative to it are integers. Thus we consider the $\mathbb{Z}$-algebra $\text{\large$\mathfrak{L}$}_{\mathbb{Z}}$ which is free as a $\mathbb{Z}$-module of which $B$ is $\mathbb{Z}$-basis. This allows us to define the Lorentz type algebra $\text{\large$\mathfrak{L}$}_K:=\text{\large$\mathfrak{L}$}_{\mathbb{Z}}\otimes_{\mathbb{Z}} K$ over any field $K$. In a similar way, we consider Poincaré type algebras over any field $K$. In this paper we study the ideal structure of Lorentz and of Poincaré type algebras over different fields. It turns out that Lorentz type algebras are simple if and only if the ground field has no square root of $-1$. Thus, they are simple over the reals but not over the complex. Also, if the ground field is of characteristic $2$ then Lorentz and Poincaré type algebras are neither simple nor semisimple. We extend the study of simplicity of the Lorentz algebra to the case of a ring of scalars where we have to use the notion of $\text{\large$\mathfrak{m}$}$-simplicity (relative to a maximal ideal $\text{\large$\mathfrak{m}$}$ of the ground ring of scalars). The Lorentz type algebras over a finite field $\mathbb{F}_q$ where $q=p^n$ and $p$ is odd are simple if and only if $n$ is odd and $p$ of the form $p=4k+3$. In case $p=2$ then the Lorentz type algebras are not simple. Once we know the ideal structure of the algebras, we get some information of their automorphism groups. For the Lorentz type algebras (except in the case of characteristic 2) we describe the affine group scheme of automorphisms and the derivation algebras. For the Poincaré algebras we restrict this program to the case of an algebraically closed field of characteristic other than 2.

In the present paper, we shall give necessary and sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator and its commutators on generalized weighted Orlicz-Morrey spaces $M^{\Phi,\varphi}_{w}({\mathbb R}^n)$. The main advance in comparison with the existing results is that we manage to obtain conditions for the boundedness not in integral terms but in less restrictive terms of supremal operators and we do not need $\Delta_2$-condition for the boundedness of the maximal operator. We also consider the vector-valued boundedness of the Hardy-Littlewood maximal operator.

For diophantine equations of the form $ax^2+bxy+cy^2+g=0$ over $\mathbb{Z}$ whose coefficients satisfy some assumptions, we show that a condition with respect to the Artin reciprocity map, which we call the Artin condition, is the only obstruction to the local-global principle for integral solutions of the equation. Some concrete examples are presented.

It is known that wave operators for three dimensional Schrödinger operators $-\Delta + V$ with threshold singularities are bounded in $L^p(\mathbb{R}^3)$ for $1<p<3$ in general and, for $1<p<\infty$ if and only if zero energy resonances are absent and all zero energy eigenfunctions $\phi$ of $-\Delta + V$ satisfy $\int V(x)x^\alpha \phi(x) dx=0$ for $|\alpha|\leq 1$. We prove here that they are bounded in $L^1(\mathbb{R}^3)$ if and only if zero energy resonances are absent. We also show that they are bounded in $L^\infty(\mathbb{R}^3)$ if no resonances are present and all zero energy eigenfunctions $\phi(x)$ satisfy $\int_{\mathbb{R}^3} x^\alpha V(x)\phi(x)dx=0$ for $0\leq |\alpha|\leq 2$. This fills the unknown parts of the $L^p$-boundedness problem for wave operators of three dimensional Schrödinger operators.

We will study the CR Yamabe equation for a partially integrable CR structure on the Heisenberg group which is deformed from the standard CR structure. By using the Lyapunov-Schmidt reduction, it is shown that a perturbed standard solution of the CR Yamabe equation is a solution of the deformed CR Yamabe equation, under certain conditions of the deformation. Especially, the deformed CR structure is only partially integrable, in general.

Kaimakamis and Panagiotidou in [11] introduced the notion of $^*$-Ricci soliton and studied the real hypersurfaces of a non-flat complex space form admitting a $^*$-Ricci soliton whose potential vector field is the structure vector field. In this article, we consider a real hypersurface of a non-flat complex space form which admits a $^*$-Ricci soliton whose potential vector field belongs to the principal curvature space and the holomorphic distribution.

In this paper, we express the logarithmic invariant of knots in terms of derivatives of the colored Jones invariants. Logarithmic invariant is defined by using the Jacobson radicals of the restricted quantum group $\overline{\mathcal U}_\xi(sl_2)$ where $\xi$ is a root of unity. We also propose a version of the volume conjecture stating a relation between the logarithmic invariants and the hyperbolic volumes of the cone manifolds along a knot, which is proved for the figure-eight knot.

The purpose of this paper is to formulate problem about existence or non-existence of functional relations for the Euler-Zagier multiple zeta-functions and solve this problem. This problem is a functional analogue of the problem about existence or non-existence of relations among the multiple zeta values. By our results, we can solve a functional analogue of the problem about the dimension of the $\mathbb{Q}$-vector space spanned by the multiple zeta values.

In this paper, we give a description about the first and second complex interpolation between $L^\infty$ and the generalized Morrey spaces. Our result can be viewed as a supplement of the complex interpolation of generalized Morrey spaces, discussed in [9]. We also give an explicit description of some closed subspaces of generalized Morrey spaces and their complex interpolation spaces.

A quasi contact metric manifold is a natural generalization of a contact metric manifold based on the geometry of the corresponding quasi Kahler cones. In this paper we prove that if a quasi contact metric manifold has constant sectional curvature $c$, then $c=1$; additionally, if the characteristic vector field is Killing, then the manifold is Sasakian. These facts are some generalizations of Olszak's theorem to quasi contact metric manifolds.

In this paper we prove, under some assumptions, that every polynomial $D(-1)$-triple in $\mathbb{Z}[X]$ can only be extended to a polynomial $D(-1;1)$-quadruple in $\mathbb{Z}[X]$ by polynomials $d^{\pm}$. More precisely, if $\{a,b,c;d\}$ is a polynomial $D(-1;1)$-quadruple, then $$d=d^{\pm}=-(a+b+c)+2(abc\pm rst)\,,$$ where $r$, $s$ and $t$ are polynomials from $\mathbb{Z}[X]$ with positive leading coefficients that satisfy $ab-1=r^2$, $ac-1=s^2$ and $bc-1=t^2$.

In this paper, we define the truncated Bernoulli-Carlitz numbers and the truncated Cauchy-Carlitz numbers as analogues of hypergeometric Bernoulli numbers and hypergeometric Cauchy numbers, and as extensions of Bernoulli-Carlitz numbers and the Cauchy-Carlitz numbers. These numbers can be expressed explicitly in terms of incomplete Stirling-Carlitz numbers.

In the study of order estimation of the Riemann zeta-function $\zeta(s) = \sum_{n=1}^\infty n^{-s}$, solving Lindelöf hypothesis is an important theme. As one of the relationships, asymptotic behavior of mean values has been studied. Furthermore, the theory of the mean values is also noted in the double zeta-functions, and the mean values of the Euler-Zagier type of double zeta-function and Mordell-Tornheim type of double zeta-function were studied. In this paper, we prove asymptotic formulas for mean square values of the Barnes double zeta-function $\zeta_2 (s, \alpha ; v, w ) = \sum_{m=0}^\infty \sum_{n=0}^\infty (\alpha+vm+wn)^{-s}$ with respect to $\text{Im}(s)$ as $\text{Im}(s) \rightarrow + \infty$.

We define two parabolic flows on almost complex manifolds, which coincide with the pluriclosed flow and the Hermitian curvature flow respectively on complex manifolds. We study the relationship between these parabolic evolution equations on a compact almost Hermitian manifold.

In this article, we introduce the notion of star-Ricci tensors in the real hypersurfaces of complex quadric $Q^m$. It is proved that there exist no Hopf hypersurfaces in $Q^m,m\geq3$, with commuting star-Ricci tensor or parallel star-Ricci tensor. As a generalization of star-Einstein metric, star-Ricci solitons on $M$ are considered. In this case we show that $M$ is an open part of a tube around a totally geodesic $\mathbb{C}P^\frac{m}{2}\subset Q^{m},m\geq4$.

We study the first hitting time to $b$ of a Bessel process with index $\nu$ starting from $a$, which is denoted by $\tau_{a,b}^{(\nu)}$, in the case when $0<b<a$. When $\nu>1$ and $\nu-1/2$ is not an integer, we obtain that $\mathbf P(t<\tau_{a,b}^{(\nu)}<\infty)$ is asymptotically equal to $\kappa_1^{(\nu)}t^{-\nu}+\kappa_2^{(\nu)} t^{-\nu-1}$ as $t\to\infty$ for some explicit constants $\kappa_1^{(\nu)}$ and $\kappa_2^{(\nu)}$. The constant $\kappa_1^{(\nu)}$ is known and the aim is to get $\kappa_2^{(\nu)}$. Combining our result with the known facts, we obtain the precise asymptotic formula for every index $\nu$.

Let $\mu$ be a non-negative Radon measure on $\mathbb{R}^{d}$ which may be a non-doubling measure. In this paper, the authors prove that the Littlewood-Paley $g^{\ast}_{\lambda,\mu}$-function is bounded on the generalized Morrey space $\mathcal{L}^{p,\phi}(\mu)$, and also obtain that the commutator $g^{\ast}_{\lambda,\mu,b}$ generated by the Littlewood-Paley function $g^{\ast}_{\lambda,\mu}$ and the regular bounded mean oscillation space $(=$RBMO$)$, which is due to X. Tolsa, is bounded on $\mathcal{L}^{p,\phi}(\mu)$. As a corollary, the authors prove that the commutator $g^{\ast}_{\lambda,\mu,b}$ is bounded on the Morrey space $\mathcal{M}^{p}_{q}(\mu)$ defined by Sawano and Tanaka when we take $\phi(t)=t^{1-\frac{p}{q}}$ with $1<p<q<\infty$.