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Perfectness and Imperfectness of the kth Power of Lattice Graphs
"... Abstract. Given a pair of nonnegative integers m and n, S(m, n) denotes a square lattice graph with a vertex set {0, 1, 2,...,m − 1} × {0, 1, 2,...,n − 1}, where a pair of two vertices is adjacent if and only if the distance is equal to 1. A triangular lattice graph T (m, n) has a vertex set {(xe1 ..."
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+ ye2)  x ∈{0, 1, 2,...,m − 1}, y∈{0, 1, 2,...,n − 1}} where def. def. e1 = (1, 0), e2 = (1/2, 3/2), and an edge set consists of a pair of vertices with unit distance. Let S k (m, n) andT k (m, n) bethekth power of the graph S(m, n) andT (m, n), respectively. Given an undirected graph G =(V,E) and a
www.elsevier.com/locate/jnt Values of the Euler function free of kth powers
, 2005
"... Communicated by C. Pomerance We establish an asymptotic formula for the number of positive integers n � x for which ϕ(n) is free of kth powers. ..."
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Communicated by C. Pomerance We establish an asymptotic formula for the number of positive integers n � x for which ϕ(n) is free of kth powers.
On the parity of kth powers mod p: A generalization of a problem of Lehmer
 In preparation
"... Abstract. Let p be an odd prime, k, A ∈ Z, p ∤ A, d = (p − 1, k), d1 = (p − 1, k − 1), s = (p − 1)/d, t = (p − 1)/d1, E the set of even residues in Zp = Z/(p), O the set of odd residues, and Nk = #{x ∈ E: Axk ∈ O}. We give several estimates of Nk, each of different strength depending on the size and ..."
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Cited by 1 (1 self)
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and parity of s, t and k. In particular we show that Nk ∼ p provided that k 4 is even and the set of all kth powers is uniformly distributed, or k is odd and d1 = o(p). We also show that if t and A  are both small odd numbers then Nk ∼ (1 − 1 p) At 4.
DETERMINANTS INVOLVING Kth POWERS FROM SECOND ORDER SEQUENCES
"... Let a be a sequence of complex numbers satisfying the difference equation d! a, = a a.. (3 a for n = 0, 1,..., n+2 n+1 n ..."
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Let a be a sequence of complex numbers satisfying the difference equation d! a, = a a.. (3 a for n = 0, 1,..., n+2 n+1 n
ScaleSpace Theory in Computer Vision
, 1994
"... A basic problem when deriving information from measured data, such as images, originates from the fact that objects in the world, and hence image structures, exist as meaningful entities only over certain ranges of scale. "ScaleSpace Theory in Computer Vision" describes a formal theory fo ..."
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Cited by 625 (21 self)
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for representing the notion of scale in image data, and shows how this theory applies to essential problems in computer vision such as computation of image features and cues to surface shape. The subjects range from the mathematical foundation to practical computational techniques. The power of the methodology
New bounds for Gauss sums derived from kth powers, and for Heilbronn’s exponential sum’, Quart
 J. Math
"... This paper is concerned with the Gauss sums and with Heilbronn’s sum G(a) = Gp(a, k) = H(a) = Hp(a) = ..."
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Cited by 58 (3 self)
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This paper is concerned with the Gauss sums and with Heilbronn’s sum G(a) = Gp(a, k) = H(a) = Hp(a) =
Finding integers k for which a given Diophantine Equation has no solution in kth powers of integers
"... : For a given polynomial f we use `local' methods to find exponents k for which there are no nontrivial integer solutions x 1 ; x 2 ; : : : ; xn to the Diophantine equation f(x k 1 ; x k 2 ; : : : ; x k n ) = 0 1. Introduction For a given polynomial f(X 1 ; X 2 ; : : : ; Xn ) 2 Z[X 1 ; ..."
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: For a given polynomial f we use `local' methods to find exponents k for which there are no nontrivial integer solutions x 1 ; x 2 ; : : : ; xn to the Diophantine equation f(x k 1 ; x k 2 ; : : : ; x k n ) = 0 1. Introduction For a given polynomial f(X 1 ; X 2 ; : : : ; Xn ) 2 Z[X 1 ; X 2 ; : : : ; Xn ] we shall investigate the set T (f) of exponents k for which the Diophantine equation (1) f(x k 1 ; x k 2 ; : : : ; x k n ) = 0 has solutions in nonzero integers x 1 ; x 2 ; : : : ; xn . For homogenous diagonal f of degree one, Davenport and Lewis showed that k 2 T (f) whenever (n \Gamma 1) 1=2 k 18; however, Ankeny and Erdos [AE] showed that T (f) has zero density in the set of all positive integers provided that all distinct subsets of the set of coefficients of f have different sums. For general polynomials f , Ribenboim [R] showed that certain values of k cannot belong to T (f ), and the result of Ankeny and Erdos shows that T (f) has zero density, under the sam...
Results 1  10
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444