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Partitions of Zn into arithmetic progressions

2009
*
European journal of combinatorics (Print)
*

We introduce the notion

doi:10.1016/j.ejc.2008.09.027
fatcat:ogsvo6c5fjht3mki3myn7yojze
*of**arithmetic**progression*blocks or m-APblocks*of*Z n , which can be represented as sequences*of*the form Then we consider the problem*of**partitioning*Z n*into*m-AP-blocks. ... We show that subject to a technical condition, the number*of**partitions**of*Z n*into*m-AP-blocks*of*a given type is independent*of*m, and is equal to the cyclic multinomial coefficient which has occurred ... This work was supported by the 973 Project, the PCSIRT Project*of*the Ministry*of*Education, the Ministry*of*Science and Technology, and the National Science Foundation*of*China. ...##
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Enumeration of monochromatic three term arithmetic progressions in two-colorings of cyclic groups
[article]

2014
*
arXiv
*
pre-print

One

arXiv:1408.1058v4
fatcat:nokdeyg6n5cc5aqmulhiuc7aoi
*of*the toughest problems in Ramsey theory is to determine the existence*of*monochromatic*arithmetic**progressions*in groups whose elements have been colored. ... We study the harder problem to not only determine the existence*of*monochromatic*arithmetic**progressions*, but to also count them. ... It follows that the minimum number*of*monochromatic*arithmetic**progression**of*length 3 in a 2-coloring*of*Z n is R(3, Z n , 2) = min x∈{−1,1} n {a,b,c} is an A.P. in*Zn*p(x a , x b , x c ). ...##
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Discrepancy in modular arithmetic progressions
[article]

2021
*
arXiv
*
pre-print

We asymptotically determine the logarithm

arXiv:2104.03929v1
fatcat:jqjygnctubh5jfffuum26kc4ai
*of*the discrepancy*of**arithmetic**progressions*in ℤ_n for all positive integer n. ... Celebrated theorems*of*Roth and*of*Matoušek and Spencer together show that the discrepancy*of**arithmetic**progressions*in the first n positive integers is Θ(n^1/4). ... Since each Z p α i i is*partitioned**into*S (t) i for 0 ≤ t ≤ α i , their product is*partitioned**into*Y r for r divides n, so Z n is*partitioned**into*X r for r divides n. ...##
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A new, fast, and efficient image codec based on set partitioning in hierarchical trees

1996
*
IEEE transactions on circuits and systems for video technology (Print)
*

Moreover, we present a new and different implementation based on set

doi:10.1109/76.499834
fatcat:mqpuwranwbe3ffpudaxsm4uhzi
*partitioning*in hierarchical trees (SPIHT), which provides even better performance than our previously reported extension*of*EZW that ... In addition, the new coding and decoding procedures are extremely fast, and they can be made even faster, with only small loss in performance, by omitting entropy coding*of*the bit stream by*arithmetic*... If C(z, J ) is significant, then it is*partitioned**into*the four sets D ( k , I ) , with ( k , I ) E O(z, J ) . ...##
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Chebyshev-Vandermonde Systems

1991
*
Mathematics of Computation
*

Fast

doi:10.2307/2938712
fatcat:yucrxy53argqzhmulkqwsf73o4
*progressive*algorithms for the solution*of*the Chebyshev-Vandermonde systems are described. These algorithms are closely related to methods recently presented by Higham. ... We present a*progressive*scheme for allocating distinct nodes zk on the boundary*of*the ellipse such that the Chebyshev-Vandermonde matrices obtained are reasonably well-conditioned. ... The product (3.13) is*partitioned**into*subproducts, each*of*which contains 2 nodes zk that are distributed roughly like the first 2 nodes (1.9). Such a*partitioning*is described by [6, Lemma 2.3] . ...##
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Chebyshev-Vandermonde systems

1991
*
Mathematics of Computation
*

Fast

doi:10.1090/s0025-5718-1991-1094957-9
fatcat:rl4yxixhzfad5f6gccmavyfmqa
*progressive*algorithms for the solution*of*the Chebyshev-Vandermonde systems are described. These algorithms are closely related to methods recently presented by Higham. ... We present a*progressive*scheme for allocating distinct nodes zk on the boundary*of*the ellipse such that the Chebyshev-Vandermonde matrices obtained are reasonably well-conditioned. ... The product (3.13) is*partitioned**into*subproducts, each*of*which contains 2 nodes zk that are distributed roughly like the first 2 nodes (1.9). Such a*partitioning*is described by [6, Lemma 2.3] . ...##
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Author Index

1984
*
Mathematics of Computation
*

Explicit Estimates for the Error Term in the Prime Number
Theorem for

doi:10.1090/s0025-5718-84-99808-9
fatcat:mzjl4g2yffbhljdgyokyx4opoa
*Arithmetic**Progressions*. 265 McCurley, Kevin S. Explicit Estimates for 9(x;3,l) and i/>(x;3,Z). 287 Major, R. D. ... On the Sharpness*of*Certain Local Estimates for H Projections*into*Finite Element Spaces: Influence*of*a Reen-Difference Method for a Singular Boundary Value Problem*of*Second Order. 441 Weiss, Richard ...##
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Approaching optimality in spatially scalable video coding: From resampling and prediction to quantization and entropy coding

2013
*
2013 IEEE International Conference on Image Processing
*

The internal range

doi:10.1109/icip.2013.6738370
dblp:conf/icip/HanR13
fatcat:s6kjn4asnjfobgmb3zplyuoqeq
*of*the*arithmetic*coder is thus divided*into*(N − 1) subintervals, the i th*of*which has a length proportional toPi. ... INTRODUCTION Scalable video coding (SVC) consists*of*encoding a video sequence*into*a single bit-stream comprising multiple layers with*progressively*higher spatial, temporal, or quantization resolutions ...##
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On the reducibility of exact covering systems
[article]

2015
*
arXiv
*
pre-print

As a consequence, if all moduli

arXiv:1402.3957v2
fatcat:4pxkxxtaoja6thdsogf72y7jli
*of*an ECS A, are divisible by at most two distinct primes, then A is natural. That is, A can be formed by iteratively splitting the trivial ECS. ... These are ECS which are not a proper split*of*a coarser ECS. However, an ECS admiting a maximal modulus which is divisible by at most two distinct primes, primely splits a coarser ECS. ... Introduction An exact covering system (ECS) is a*partition**of*Z*into*finitely many*arithmetic**progressions*(1) A = {a s (n s )} k s=1 , where a(n) is the*arithmetic**progression*a +*Zn*. ...##
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Sum-free sets, coloured graphs and designs

1976
*
Journal of the Australian Mathematical Society
*

Embedding

doi:10.1017/s1446788700013343
fatcat:22f7ppsgsvckzp2rcsxvl756gm
*of*colourings is considered. Finally we illustrate a way*of*constructing colourings using block designs. ... Sum-free sets may be used to colour the edges*of*a complete graph in such a way as to avoid monochromatic triangles. We discuss the automorphism groups*of*such graphs. ... But T in*arithmetic**progression*implies that TI -Ti is also in*arithmetic**progression*; if | T) | = 5, then T) -T = S, which forces S f and hence Si to be in*arithmetic**progression*. ...##
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On arithmetic partitions of Zn

2009
*
European journal of combinatorics (Print)
*

Chen, Wang, and Zhang then studied the problem

doi:10.1016/j.ejc.2008.11.009
fatcat:id5mdzyuqfgordepzycgftj6e4
*of**partitioning*_n*into**arithmetical**progressions**of*a given type under some technical conditions. ... Generalizing a classical problem in enumerative combinatorics, Mansour and Sun counted the number*of*subsets*of*_n without certain separations. ... Acknowledgments We thank the referees for helpful comments on a previous version*of*this paper. ...##
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Ramsey's theorem for $n$-dimensional arrays

1969
*
Bulletin of the American Mathematical Society
*

Given integers k and r there exists an integer N(k t r) such that if n*

doi:10.1090/s0002-9904-1969-12202-0
fatcat:qes7zrtcfrbkzelbiawmyy2e4a
*zN*(k, r) and the set I n is*partitioned**into*r classes, then at least one class contains an*arithmetic**progression**of*length k. ...*of*the fe-parameter sets P k*into*r classes induces a*partition**of*(k -1) -parameter sets Pfc-i to which we can apply the induction hypothesis. ...##
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The remarkable effectiveness of ergodic theory in number theory

2009
*
Ensaios matemáticos
*

In this process, you get a

doi:10.21711/217504322009/em171
fatcat:wpa3cpqbmnfade45jd23kf4bai
*partition**of*the integer numbers*into*k disjoint subsets. ... . , N }*into*r*arithmetic**progressions**of*ratio r, e.g., {1 ≤ n ≤ N | n ≡ a(mod r)}, for each a with 0 ≤ a ≤ r − 1. ... In this direction, in view*of*the geometrical interpretation*of*F (see remark 3.3.1), we look at the lattices Λ ∈ E intersecting the triangle ∆ c−,c+*into*two points whose coordinates (w (1) 2 ∈ (c − , ...##
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Limestone doses affecting mineral contents in tropical grass forage

2005
*
Journal of Radioanalytical and Nuclear Chemistry
*

The statistical analysis showed a negative linear correlation

doi:10.1007/s10967-005-0670-4
fatcat:sn5rrbhilvfyxdgku5b63klx4y
*of*Br, Co, Cr, Mn and*Zn*contents in forage with the limestone doses, while the uptake*of*Mg was affected in a positive way. ... by the use*of*limestone and fertilizer. ...*Arithmetic*mean values and concentration ranges*of*the elements Br, Co, Cr, Mg, Mn and*Zn*obtained in Brachiaria decumbens forage*of*6 blocks are presented in Table 2 . ...##
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On the Statistical Ensemble for the Theory of Helix-Coil Transition of Copolymeric DNA

1967
*
Progress of theoretical physics
*

The treatment based on the Bernoulli ensemble leads to the

doi:10.1143/ptp.38.9
fatcat:fbj6nxv3kvcqjkqs6kmis5tt3e
*arithmetic*mean approximation for the statistical weight*of*bonded base pair and it fails to give the reasonable separate contributions*of*G-C ... Helix-coil transition*of*copolymeric DNA molecule is investigated hy calculating the*partition*functions averaged over two kinds*of*ensembles*of*'random sequences*of*base pairs, namely, the Bernoulli ensemble ... If we average the*partition*functions over the Bernoulli ensemble, it leads to the*arithmetic*mean approximation. ...
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