Connected sequences of functors whose domain, is the category of morphisms of an arbitrary abelian category$A$and whose range category$B$is also abelian are compared with the composition functors of Eckmann and Hilton acting between the same categories Sequences of functors of both types are obtained from any half-exact functor$A→B$if$A$has enough injectives and projectives.

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We study minimizers of the functionalwhere $B_{1}^{{\mathchoice {\raise .17ex\hbox {$\scriptstyle +$}} {\raise .17ex\hbox {$\scriptstyle +$}} {\raise .1ex\hbox {$\scriptscriptstyle +$}} {\scriptscriptstyle +}}}=\{x\in B_{1}: x_{1}>0\}$ ,$u$=0 on {$x$∈$B$_{1}:$x$_{1}=0}, $\lambda^{{\mathchoice {\raise .17ex\hbox {$\scriptstyle \pm $}} {\raise .17ex\hbox {$\scriptstyle \pm $}} {\raise .1ex\hbox {$\scriptscriptstyle \pm $}} {\scriptscriptstyle \pm }}}$ are two positive constants and 0<$p$<1. In two dimensions, we prove that the free boundary is a uniform$C$^{1}graph up to the flat part of the fixed boundary and also that two-phase points cannot occur on this part of the fixed boundary. Here, the free boundary refers to the union of the boundaries of the sets {$x$:±$u$($x$)>0}.

In this paper, we study the decomposition of sets of consecutive integers, i.e., how to express the set {0, 1, . . . ,mn−1} as the Minkovski sum of two sets, one with m terms and the other with n terms. First, we translate the problem into the language of polynomials. Then we use the properties of cyclotomic polynomials to determine the structure of all the solutions. Then, we deduce formulas for the number of such expressions. Finally, we give an algorithm to calculate the number of such expressions and analyze its computational complexity.

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