منابع مشابه
Restricted partitions
We prove a known partitions theorem by Bell in an elementary and constructive way. Our proof yields a simple recursive method to compute the corresponding Sylvester polynomials for the partition. The previous known methods to obtain these polynomials are in general not elementary. 1. Proof of Bell's theorem. The main purpose of this section is to prove the following theorem, originally proved b...
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In [1] certain restricted partitions were studied in an elementary way. In this article we study other problem about restricted partitions. We prove the following theorem in a direct way and using elementary methods ( combinatory and mathematical induction). Theorem 1. Let n be a positive integer, and let A1, A2, . . . , An be n positive integers. Then, for all positive integers N = MK + r, the...
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Have you ever wondered how many ways the total of 5 dollars can be made with coins having cent values {1, 2, 5, 10, 20, 50, 100, 200}? Would you like to take a guess at how many ways there are? Would you be surprised if I told you there were 6295434? Maybe we should tone it down a bit and ask how many ways there are to make 50 cents. There are still 451! Lets step back once more and determine h...
متن کاملRandom partitions with restricted part sizes
For a subsetS of positive integers let (n,S)be the set of partitions ofn into summands that are elements of S. For every λ ∈ (n,S), let Mn(λ) be the number of parts, with multiplicity, that λ has. Put a uniform probability distribution on (n,S), and regardMn as a random variable. In this paper the limiting density of the (suitably normalized) random variableMn is determined for sets that are su...
متن کاملTraining Restricted Boltzmann Machines with Overlapping Partitions
Restricted Boltzmann Machines (RBM) are energy-based models that are successfully used as generative learning models as well as crucial components of Deep Belief Networks (DBN). The most successful training method to date for RBMs is the Contrastive Divergence method. However, Contrastive Divergence is inefficient when the number of features is very high and the mixing rate of the Gibbs chain i...
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ژورنال
عنوان ژورنال: Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences
سال: 1969
ISSN: 0098-8979
DOI: 10.6028/jres.073b.032