The purpose of this paper is to study pseudo-Riemannian algebras, which are algebras with pseudo-Riemannian non-degenerate symmetric bilinear forms. We nd that pseudo-Riemannian algebras whose left centers are isotropic play a curial role and show that the decomposition of pseudo-Riemannian algebras whose left centers are isotropic into indecomposable non-degenerate ideals is unique up to a special automorphism. Furthermore, if the left center equals the center, the orthogonal decomposition of any pseudo-Riemannian algebra into indecomposable non-degenerate ideals is unique up to an isometry.

In this paper, a pair of Wolfe type nondifferentiable sec-ond order symmetric minimax mixed integer dual problems is formu-lated. Symmetric and self-duality theorems are established under $\eta_1$-bonvexity/$\eta_2$-boncavity assumptions. Several known results are obtained as special cases. Examples of such primal and dual problems are also given.

In this paper, we study the fundamental group and orbits of cohomogeneity two Riemannian manifolds of constant negative curvature.

The notion of the generalized Fibonacci matrix $\mathcal{F}_n^{(a,b,s)}$ of type s, whose nonzero elements are generalized Fibonacci numbers, is introduced in the paper [23]. Regular case s = 0 is investigated in [23]. In the present article we consider singular case s = -1. Pseudoinverse of the generalized Fibonacci matrix $\mathcal{F}_n^{(a,b,-1)}$ is derived. Correlations between the matrix $\mathcal{F}_n^{(a,b,-1)}$ and the Pascal matrices are considered. Some combinatorial identities involving generalized Fibonacci numbers are derived. A class of test matrices for computing the Moore-Penrose inverse is presented in the last section.

Let D be an integral domain with quotient field K, X be a nonempty set of indeterminates over D, * be a star operation on D, $N_*$={f $\in$ D[X]|c(f)$^*$= D}, $*_w$ be the star operation on D defined by $I^{*_w}$ = ID[X]${_N}_*$ $\cap$ K, and [*] be the star operation on D[X] canonically associated to * as in Theorem 2.1. Let $A^g$ (resp., $A^{[*]g}$, $A^{[*]g}$) be the global (resp.,*-global, [*]-global) transform of a ring A. We show that D is a $*_w$-Noetherian domain if and only if D[X] is a [*]-Noetherian domain. We prove that $D^{*g}$[X]${_N}_*$ = (D[X]${_N}_*$)$^g$ = (D[X])$^{[*]g}$; hence if D is a $*_w$-Noetherian domain, then each ring between D[X]${_N}_*$ and $D^{*g}$[X]${_N}_*$ is a Noetherian domain. Let $\tilde{D}$ = $\cap${$D_P$|P $\in$ $*_w$-Max(D) and htP $\geq$2}. We show that $D\;\subseteq\;\tilde{D}\;\subseteq\;D^{*g}$ and study some properties of $\tilde{D}$ and $D^{*g}$.

On the hyperbolic space $D^n$, codimension-one totally geodesic foliations of class $C^k$ are classified. Except for the unique parabolic homogeneous foliation, the set of all such foliations is in one-one correspondence (up to isometry) with the set of all functions z : [0, $\pi$] $\rightarrow$ $S^{n-1}$ of class $C^{k-1}$ with z(0) = $e_1$ = z($\pi$) satisfying |z'(r)| ${\leq}1$ for all r, modulo an isometric action by O(n-1) ${\times}\mathbb{R}{\times}\mathbb{Z}_2$. Since 1-dimensional metric foliations on $D^n$ are always either homogeneous or flat (that is, their orthogonal distributions are integrable), this classifies all 1-dimensional metric foliations as well. Equations of leaves for a non-trivial family of metric foliations on $D^2$ (called "fifth-line") are found.

In this note, compound-commuting additive maps on matrix spaces are studied. We show that compound-commuting additive maps send rank one matrices to matrices of rank less than or equal to one. By using the structural results of rank-one nonincreasing additive maps, we characterize compound-commuting additive maps on four types of matrices: triangular matrices, square matrices, symmetric matrices and Hermitian matrices.

Let f be a function which assigns a positive integer f(v) to each vertex v $\in$ V (G), let r, s and t be non-negative integers. An f-coloring of G is an edge-coloring of G such that each vertex v $\in$ V (G) has at most f(v) incident edges colored with the same color. The minimum number of colors needed to f-color G is called the f-chromatic index of G and denoted by ${\chi}'_f$(G). An [r, s, t; f]-coloring of a graph G is a mapping c from V(G) $\bigcup$ E(G) to the color set C = {0, 1, $\ldots$; k - 1} such that |c($v_i$) - c($v_j$ )| $\geq$ r for every two adjacent vertices $v_i$ and $v_j$, |c($e_i$ - c($e_j$)| $\geq$ s and ${\alpha}(v_i)$ $\leq$ f($v_i$) for all $v_i$ $\in$ V (G), ${\alpha}$ $\in$ C where ${\alpha}(v_i)$ denotes the number of ${\alpha}$-edges incident with the vertex $v_i$ and $e_i$, $e_j$ are edges which are incident with $v_i$ but colored with different colors, |c($e_i$)-c($v_j$)| $\geq$ t for all pairs of incident vertices and edges. The minimum k such that G has an [r, s, t; f]-coloring with k colors is defined as the [r, s, t; f]-chromatic number and denoted by ${\chi}_{r,s,t;f}$ (G). In this paper, we present some general bounds for [r, s, t; f]-coloring firstly. After that, we obtain some important properties under the restriction min{r, s, t} = 0 or min{r, s, t} = 1. Finally, we present some problems for further research.

By developing a linear algebra program involving many different structures associated to a three-graded H*-algebra, it is shown that if L is a Lie triple automorphism of an infinite-dimensional topologically simple associative H*-algebra A, then L is either an automorphism, an anti-automorphism, the negative of an automorphism or the negative of an anti-automorphism. If A is finite-dimensional, then there exists an automorphism, an anti-automorphism, the negative of an automorphism or the negative of an anti-automorphism F : A $\rightarrow$ A such that $\delta$:= F - L is a linear map from A onto its center sending commutators to zero. We also describe L in the case of having A zero annihilator.

We employ a hyperbolic tangent function to construct nonlinear transformations which are useful in numerical evaluation of weakly singular integrals and Cauchy principal value integrals. Results of numerical implementation based on the standard Gauss quadrature rule show that the present transformations are available for the singular integrals and, in some cases, give much better approximations compared with those of existing non-linear transformation methods.

In this paper, we consider a multi-dimensional diffusion process in a self-similar random environment and prove a limit theorem for the shape of the full trajectory of the diffusion by using the localization phenomenon.

We consider a curve $\alpha$= $\alpha$(s) in Minkowski 3-space $E_1^3$ and denote by {T, N, B} the Frenet frame of $\alpha$. We say that $\alpha$ is a slant helix if there exists a fixed direction U of $E_1^3$ such that the function is constant. In this work we give characterizations of slant helices in terms of the curvature and torsion of $\alpha$. Finally, we discuss the tangent and binormal indicatrices of slant curves, proving that they are helices in $E_1^3$.

In this paper, we find character analogues of infinite series identities which come from a certain modular transformation formula given by B. C. Berndt.

The object of the present paper is to obtain a more general condition for univalence of analytic functions in the open unit disk U. The significant relationships and relevance with other results are also given. A number of known univalent conditions would follow upon specializing the parameters involved in our main results.

It is well known that for a Krull domain R, the divisor class group of R is a torsion group if and only if every subintersection of R is a ring of quotients. Thus a natural question is that under what conditions, for a non-Krull domain R, every (t-)subintersection (resp., t-linked overring) of R is a ring of quotients or every (t-)subintersection (resp., t-linked overring) of R is at. To address this question, we introduce the notions of *-compact packedness and *-coprime packedness of (an ideal of) an integral domain R for a star operation * of finite character, mainly t or w. We also investigate the t-theoretic analogues of related results in the literature.

Let R be a commutative ring and let M be a GV -torsionfree R-module. Then M is said to be a $\omega$-module if $Ext_R^1$(R/J, M) = 0 for any J $\in$ GV (R), and the w-envelope of M is defined by $M_{\omega}$ = {x $\in$ E(M) | Jx $\subseteq$ M for some J $\in$ GV (R)}. In this paper, $\omega$-modules over commutative rings are considered, and the theory of $\omega$-operations is developed for arbitrary commutative rings. As applications, we give some characterizations of $\omega$-Noetherian rings and Krull rings.