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On the arithmetic sums of Cantor sets
, 2008
"... Let Cλ and Cγ be two affine Cantor sets in R with similarity dimensions dλ and dγ, respectively. We define an analog of the BandtGraf condition for selfsimilar systems and use it to give necessary and sufficient conditions for having H d λ+dγ (Cλ + Cγ)> 0 where Cλ + Cγ denotes the arithmetic su ..."
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Cited by 1 (0 self)
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Let Cλ and Cγ be two affine Cantor sets in R with similarity dimensions dλ and dγ, respectively. We define an analog of the BandtGraf condition for selfsimilar systems and use it to give necessary and sufficient conditions for having H d λ+dγ (Cλ + Cγ)> 0 where Cλ + Cγ denotes the arithmetic
THE EVALUATION OF CERTAIN ARITHMETIC SUMS
"... where ^k = aki + ~ ' + akmk and where the sum is over all the a'%, each ranging from zero to some positive integer. We also consider analogous sums for min. For example we obtain, from some general results which we establish, the formula r,] P max(a + h,c+d) = 22 ^ \ ) + 170 ( r 2) + 42o ..."
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where ^k = aki + ~ ' + akmk and where the sum is over all the a'%, each ranging from zero to some positive integer. We also consider analogous sums for min. For example we obtain, from some general results which we establish, the formula r,] P max(a + h,c+d) = 22 ^ \ ) + 170 ( r 2) + 42
Graphics applications, the arithmetic SumofProducts, Shifters
"... Abstract — In modern Digital Signal Processing (DSP) and ..."
Moreira’s theorem on the arithmetic sum of dynamically defined Cantor sets
 ANDREW FERGUSON, MATHEMATICS INSTITUTE, ZEEMAN BUILDING, UNIVERSITY OF WARWICK
, 2008
"... We present a complete proof of a theorem of C.G. Moreira. Under mild checkable conditions, the theorem asserts that the Hausdorff dimension of the arithmetic sum of two dynamically defined Cantor subsets of the real line, equals either the sum of the dimensions or 1, whichever is smaller. ..."
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Cited by 2 (1 self)
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We present a complete proof of a theorem of C.G. Moreira. Under mild checkable conditions, the theorem asserts that the Hausdorff dimension of the arithmetic sum of two dynamically defined Cantor subsets of the real line, equals either the sum of the dimensions or 1, whichever is smaller.
MENON’S IDENTITY AND ARITHMETICAL SUMS REPRESENTING FUNCTIONS OF SEVERAL VARIABLES
"... Abstract. We generalize Menon’s identity by considering sums representing arithmetical functions of several variables. As an application, we give a formula for the number of cyclic subgroups of the direct product of several cyclic groups of arbitrary orders. We also point out extensions of Menon’s i ..."
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Cited by 14 (8 self)
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Abstract. We generalize Menon’s identity by considering sums representing arithmetical functions of several variables. As an application, we give a formula for the number of cyclic subgroups of the direct product of several cyclic groups of arbitrary orders. We also point out extensions of Menon’s
REMARKS OF CONGRUENT ARITHMETIC SUMS OF THETA FUNCTIONS DERIVED FROM DIVISOR FUNCTIONS
"... Abstract. In this paper, we study a distinction the two generating functions: ϕk(q) = n=0 rk(n)q n and ϕ∗,k(q) = ϕk(q) − ϕk(q2) (k = 2, 4, 6, 8, 10, 12, 16), where rk(n) is the number of representations of n as the sum of k squares. We also obtain some congruences of representation numbers and di ..."
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Abstract. In this paper, we study a distinction the two generating functions: ϕk(q) = n=0 rk(n)q n and ϕ∗,k(q) = ϕk(q) − ϕk(q2) (k = 2, 4, 6, 8, 10, 12, 16), where rk(n) is the number of representations of n as the sum of k squares. We also obtain some congruences of representation numbers
Good ErrorCorrecting Codes based on Very Sparse Matrices
, 1999
"... We study two families of errorcorrecting codes defined in terms of very sparse matrices. "MN" (MacKayNeal) codes are recently invented, and "Gallager codes" were first investigated in 1962, but appear to have been largely forgotten, in spite of their excellent properties. The ..."
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Cited by 741 (23 self)
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. The decoding of both codes can be tackled with a practical sumproduct algorithm. We prove that these codes are "very good," in that sequences of codes exist which, when optimally decoded, achieve information rates up to the Shannon limit. This result holds not only for the binarysymmetric channel
A Fast Algorithm for Particle Simulations
, 1987
"... this paper to the case where the potential (or force) at a point is a sum of pairwise An algorithm is presented for the rapid evaluation of the potential and force fields in systems involving large numbers of particles interactions. More specifically, we consider potentials of whose interactions a ..."
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Cited by 1145 (19 self)
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this paper to the case where the potential (or force) at a point is a sum of pairwise An algorithm is presented for the rapid evaluation of the potential and force fields in systems involving large numbers of particles interactions. More specifically, we consider potentials of whose interactions
Results 1  10
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181,676