Let A be the classical, singly-graded Steenrod algebra over prime order field F2 and let P⊗h:=F2[t1,…,th] denote polynomial on h generators, each of degree 1, viewed as a module A. Write GLh for usual general linear group rank F2. As well known, (mod 2) cohomology groups algebra, ExtAh,h+⁎(F2,F2) is still largely mysterious all homological degrees h≥6. The h-th algebraic transferTrhA:(F2⊗GLhAnn...

abstract background: athletes’ recovery is important in improving their performance. nutritional strategies can be effective in enhancing recovery rate. choosing the best food items in appropriate intervals can play effective roles in resynthesis of fuels and recovery of muscle injury. beverage micro and macronutrient content are helpful in fuel restoration. in this study, we assess the effects...

(1) Let F , F1, F2 be finite sequences of elements of R and given r1, r2. If F1 = 〈r1〉 F or F1 = F a 〈r1〉 and if F2 = 〈r2〉 F or F2 = F a 〈r2〉, then ∑(F1−F2) = r1− r2. (2) Let F1, F2 be finite sequences of elements of R. If lenF1 = lenF2, then len(F1 +F2) = lenF1 and len(F1−F2) = lenF1 and ∑(F1 +F2) = ∑F1 +∑F2 and ∑(F1−F2) = ∑F1−∑F2. (3) Let F1, F2 be finite sequences of elements of R. If lenF1 ...

Let G be a graph. We say that is perfectly divisible if for each induced subgraph H of G, V(H) can partitioned into A and B such H[A] perfect ω(H[B])<ω(H). use Pt Ct to denote path cycle on t vertices, respectively. For two disjoint graphs F1 F2, we F1∪F2 the graph with vertex set V(F1)∪V(F2) edge E(F1)∪E(F2), F1+F2 E(F1)∪E(F2)∪{xy|x∈V(F1) y∈V(F2)}. In this paper, prove (i) (P5,C5,K2,3)-free ar...

(3) (δh[f ])(x) = (∇h 2 [f ])(x)− (∇−h2 [f ])(x). (4) (~ ∆h[r f1 + f2])(n+ 1)(x) = r · (~ ∆h[f1])(n+ 1)(x) + (~ ∆h[f2])(n+ 1)(x). (5) (~ ∆h[f1 + r f2])(n+ 1)(x) = (~ ∆h[f1])(n+ 1)(x) + r · (~ ∆h[f2])(n+ 1)(x). (6) (~ ∆h[r1 f1 − r2 f2])(n+ 1)(x) = r1 · (~ ∆h[f1])(n+ 1)(x)− r2 · (~ ∆h[f2])(n+ 1)(x). (7) (~ ∆h[f ])(1) = ∆h[f ]. (8) (~ ∇h[r f1 + f2])(n+ 1)(x) = r · (~ ∇h[f1])(n+ 1)(x) + (~ ∇h[f2])(...