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In group theory, it is a fundamental question to determine what information about a group is necessary and sufficient tomake a group finitely presented. In this paper, we develop the theory of Σ-invariants due to Robert Bieri and Ralph Strebel [Bieri] whi provides an answer to this question for the class of metabelian groups. We begin our study of Σ-invariants with a digression on group extens...

Suppose that p,q,r≥1 satisfy the condition 1 p=1 q+1 r. The paper contains identification of best constants cp,q,r and Cp,q,r in estimates ‖g‖p,∞≤cp,q,r‖f‖q‖v∗‖ r,‖g‖p≤Cp,q,r‖f‖q‖v∗‖ r where f is an arbitrary Hilbert-space valued martingale, g its transform by a predictable sequence v, v∗ maximal function v. This extended to more general context differential subordination for continuous-time pr...

o r g a n i z a t i o n s i n a s u p p l y c h a i n a r e i n d e p e n d e n t e n t i t i e s. a l t h o u g h a c o m p l e t e l y i n t e g r a t e d s o l u t i o n m a y r e s u l t i n o p t i m a l s y s t e m p e r f o r m a n c e, s u p p l y c h a i n m e m b e r s a r e i n t e r e s t e d i n o p t i m i z i n g t h e i r o w n o b j e c t i v e r a t h e r t h a n t h a t o f t...

Let $G:\mathbb R\rightarrow \mathbb R$ be a continuous function. Under some assumptions on $G$, $s,\alpha ,p$ and $q$ we prove that $$\{G(f):f\in A_{p,q}^{s}(\mathbb R^{n},|\cdot |^{\alpha })\}\subset })$$ implie

u n d o u b t e d l y, i n t h e l a s t d e c a d e s, t h e a s s e t p r i c i n g m o d e l i so n e o f m o s t i m p o r t a n t a n d, a t t h e s a m e t i m e, a t t r a c t i v e, f i n a n c e a r e a s. c a p m h a s b e e n t h e d o m i n a n t m o d e l i n t h i s s e c t i o n f o r m o r e t h a n 40 y e a r s. o n e r e a s o n i s i t s s t r o n g b a s e, w h i c h h a s b...

In this paper we introduce remainder cordial labeling of graphs. Let $G$ be a $(p,q)$ graph. Let $f:V(G)rightarrow {1,2,...,p}$ be a $1-1$ map. For each edge $uv$ assign the label $r$ where $r$ is the remainder when $f(u)$ is divided by $f(v)$ or $f(v)$ is divided by $f(u)$ according as $f(u)geq f(v)$ or $f(v)geq f(u)$. The function$f$ is called a remainder cordial labeling of $G$ if $left| e_{...