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The sum number of d-partite complete hypergraphs

Hanns-Martin Teichert (1999)

Discussiones Mathematicae Graph Theory

A d-uniform hypergraph is a sum hypergraph iff there is a finite S ⊆ IN⁺ such that is isomorphic to the hypergraph d ( S ) = ( V , ) , where V = S and = v , . . . , v d : ( i j v i v j ) i = 1 d v i S . For an arbitrary d-uniform hypergraph the sum number σ = σ() is defined to be the minimum number of isolated vertices w , . . . , w σ V such that w , . . . , w σ is a sum hypergraph. In this paper, we prove σ ( n , . . . , n d d ) = 1 + i = 1 d ( n i - 1 ) + m i n 0 , 1 / 2 ( i = 1 d - 1 ( n i - 1 ) - n d ) , where n , . . . , n d d denotes the d-partite complete hypergraph; this generalizes the corresponding result of Hartsfield and Smyth [8] for complete bipartite graphs.

The tangent function and power residues modulo primes

Zhi-Wei Sun (2023)

Czechoslovak Mathematical Journal

Let p be an odd prime, and let a be an integer not divisible by p . When m is a positive integer with p 1 ( mod 2 m ) and 2 is an m th power residue modulo p , we determine the value of the product k R m ( p ) ( 1 + tan ( π a k / p ) ) , where R m ( p ) = { 0 < k < p : k is an m th power residue modulo p } . In particular, if p = x 2 + 64 y 2 with x , y , then k R 4 ( p ) 1 + tan π a k p = ( - 1 ) y ( - 2 ) ( p - 1 ) / 8 .

The total {k}-domatic number of digraphs

Seyed Mahmoud Sheikholeslami, Lutz Volkmann (2012)

Discussiones Mathematicae Graph Theory

For a positive integer k, a total k-dominating function of a digraph D is a function f from the vertex set V(D) to the set 0,1,2, ...,k such that for any vertex v ∈ V(D), the condition u N - ( v ) f ( u ) k is fulfilled, where N¯(v) consists of all vertices of D from which arcs go into v. A set f , f , . . . , f d of total k-dominating functions of D with the property that i = 1 d f i ( v ) k for each v ∈ V(D), is called a total k-dominating family (of functions) on D. The maximum number of functions in a total k-dominating family on D is the total k-domatic...

The tree of shapes of an image

Coloma Ballester, Vicent Caselles, P. Monasse (2003)

ESAIM: Control, Optimisation and Calculus of Variations

In [30], Kronrod proves that the connected components of isolevel sets of a continuous function can be endowed with a tree structure. Obviously, the connected components of upper level sets are an inclusion tree, and the same is true for connected components of lower level sets. We prove that in the case of semicontinuous functions, those trees can be merged into a single one, which, following its use in image processing, we call “tree of shapes”. This permits us to solve a classical representation...

The tree of shapes of an image

Coloma Ballester, Vicent Caselles, P. Monasse (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In [CITE], Kronrod proves that the connected components of isolevel sets of a continuous function can be endowed with a tree structure. Obviously, the connected components of upper level sets are an inclusion tree, and the same is true for connected components of lower level sets. We prove that in the case of semicontinuous functions, those trees can be merged into a single one, which, following its use in image processing, we call “tree of shapes”. This permits us to solve a classical representation problem...

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