Abstract. Euler–Maclaurin and Poisson analogues of the summations ∑a<n≤b χ(n) f (n), ∑a<n≤b d(n) f (n), ∑a<n≤b d(n)χ(n) f (n) have been obtained in a unified manner, where (χ(n)) is a periodic complex sequence; d(n) is the divisor function and f (x) is a sufficiently smooth function on [a,b]. We also state a generalised Abel’s summation formula, generalised Euler’s summation formula and Euler’s...

the present note arises from the author's talk at the conference ``ischia group theory 2014''. for subgroups $fle n$ of a group $g$ denote by $lat(f,n)$ the set of all subgroups of $n$, containing $f$. let $d$ be a subgroup of $g$. in this note we study the lattice $ll=lat(d,g)$ and the lattice $ll'$ of subgroups of $g$, normalized by $d$. we say that $ll$ satisfies sandwich classification theo...

Erdős and Sós proposed a problem of determining the maximum number F (n) of rainbow triangles in 3-edge-colored complete graphs on n vertices. They conjectured that F (n) = F (a)+ F (b) +F (c) +F (d) + abc+ abd+ acd+ bcd, where a+ b+ c+ d = n and a, b, c, d are as equal as possible. We prove that the conjectured recurrence holds for sufficiently large n. We also prove the conjecture for n = 4 f...

T h e e x a c t M a x w e l l f B o l t z m a n n ( M B ) , B o s e f E i n s t e i n ( B E ) a n d F e r m i f D i r a c ( F D ) e n t r o p i e s a n d p r o b a b i l i s t i c d i s t r i b u t i o n s a r e d e r i v e d b y t h e c o m b i n a t o r i a l m e t h o d o f B o l t z m a n n , w i t h o u t S t i r l i n g ’ s a p p r o x i m a t i o n . T h e n e w e n t r o p y m e a s u r...

Let S be a degree six del Pezzo surface over an arbitrary field F . Motivated by the first author’s classification of all such S up to isomorphism [3] in terms of a separable F -algebra B×Q×F , and by his K-theory isomorphism Kn(S) ∼= Kn(B × Q × F ) for n ≥ 0, we prove an equivalence of derived categories D (cohS) ≡ D(modA) where A is an explicitly given finite dimensional F -algebra whose semi...

Let $D=(V,A)$ be a finite simple directed graph. A function$f:Vlongrightarrow {-1,0,1}$ is called a twin minus dominatingfunction (TMDF) if $f(N^-[v])ge 1$ and $f(N^+[v])ge 1$ for eachvertex $vin V$. The twin minus domination number of $D$ is$gamma_{-}^*(D)=min{w(f)mid f mbox{ is a TMDF of } D}$. Inthis paper, we initiate the study of twin minus domination numbersin digraphs and present some lo...

d a t a t y p e b o o l e a n _ b u s e s = ( * c o n s t r u c t o r s f o r e v e r y b u s t h a t s h o u l d b e a s s o c i a t e d w i t h t h e b o o l p r i m i t i v e t y p e * ) d a t a t y p e i n s _ c l a s s _ b u s e s = ( * c o n s t r u c t o r s f o r e v e r y b u s t h a t s h o u l d b e a s s o c i a t e d w i t h t h e c l a s s e s d a t a t y p e * ) d a t a t y p e i...