We begin with establishing properties of the unit revenue function, g(·). Lemma 6. 1. g(·) is a non-negative, continuous, non-decreasing, and concave function on R+, with g(0) = 0. 2. yg(1/y) is non-decreasing and concave on R++. 3. g(y)/y is non-increasing on R+. 4. If u, v > 0, then g(u) g(v) ≥ min( u v , 1), 1 u ∫ u 0 g(v)dv ≤ g(u/2). Proof. 1. That g(·) is non-negative, continuous and non-d...

r p c h v t … h p q p f q r t „ i b h b c q v r „ r v q p q b t c t … ‡ t w g f v c f p q b t c f ¹ p c w p c p a g f b f t … … p ˆ b a g … ¸ c h q b t c b c µ t … p f q i ˆ p q b h „ p q b v c q f † Ä c q v r c p q b t c p a Å v ¸ r t „ f g h i b p q r b h Ä c q v r ¶ b v ‚ ¹ e t a b f i ¶ v r f b t c AE † Ç † Ç † ¹ e p c b h p c w ‰ µ t r p „ i t ‡ b p à h p a v È e ‰ Ã É ¹ ´ i v Ê t w g à v ...

Let $R$ be a prime ring of characteristic not equal to $2$, $U$ the Utumi quotient and $C$ extended centroid $R$. $G$ $F$ two generalized derivations on $L$ non-central Lie ideal If $F\Big(G(u)\Big)u = G(u^{2})$ for all $u \in L$, then one following holds: (1) $G=0$. (2) There exist $p,q U$ such that $G(x)=p x$, $F(x)=qx$ $x R$ with $qp=p$. (3) satisfies $s_4$.

The present work continues the study of the stability of the functional equations of the type f(pr, qs) + f(ps, qr) = f(p, q) f(r, s) namely (i) f(pr, qs)+f(ps, qr) = g(p, q) g(r, s), and (ii) f(pr, qs)+f(ps, qr) = g(p, q) h(r, s) for all p, q, r, s ∈ G, where G is an abelian group. These functional equations arise in the characterization of symmetrically compositive sumform distance measures.

The knowledge e n g i n e e r p r a c t i c e s t h e a r t o f b r i n g i n g the p r i n c i p l e s and t o o l s o f A I r e s e a r c h t o bear o n d i f f i c u l t a p p l i c a t i o n s p rob lems r e q u i r i n g e x p e r t s ' knowledge f o r t h e i r s o l u t i o n . The t e c h n i c a l i s s u e s o f a c q u i r i n g t h i s k n o w l e d g e , r e p r e s e n t i n g i t...

t i o n s h i p , a s i n , t h e p o o l ' s c l a r i t y o r t h e c l a r i t y o f t h e p o o l . A n d f i n a l l y , o b E X P R E S S I O N O F P O S S E S S I O N B Y L A T I N O C H I L D R E N 1 6 7 j e c t i v e p o s s e s s i v e s d o n o t s i g n a l a c t u a l p o s s e s s i o n , b u t t h e t w o n o u n s a r e n o n e t h e l e s s c l o s e l y r e l a t e d , a s i n...

d e v e l o p m e n t p r o c e s s . A n u n d e r l y i n g a s p e c t i s t h e e f f e c t i v e n e s s o f f i x e s : i f a f a i r n u m b e r o f f i x e d b u g s a r e r e o p e n e d , i t c o u l d i n d i c a t e i n s t a b i l i t y i n t h e s o f t w a r e s y s t e m . T o t h e b e s t o f o u r k n o w l e d g e t h e r e h a s b e e n o n l i t t l e p r i o r w o r k o n...