On Quantitative Substitutional Analysis

Young, Alfred F.

doi:10.1112/plms/s2-28.1.255

Cited by 96 publications

(111 citation statements)

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“…These coefficients are well-known objects in the combiThis paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14). The full collection is available at https://www.emis.de/journals/SIGMA/OPSFA2017.html natorics of partitions, namely the number of standard Young tableaux [31,32]. Hence, this work leads us to believe that the use of partitions in the description of exceptional orthogonal polynomials is indeed the natural setting.…”

Section: Introductionmentioning

confidence: 99%

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Recurrence Relations for Wronskian Hermite Polynomials

Bonneux¹,

Stevens²

2018

SIGMA

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Abstract. We consider polynomials that are defined as Wronskians of certain sets of Hermite polynomials. Our main result is a recurrence relation for these polynomials in terms of those of one or two degrees smaller, which generalizes the well-known three term recurrence relation for Hermite polynomials. The polynomials are defined using partitions of natural numbers, and the coefficients in the recurrence relation can be expressed in terms of the number of standard Young tableaux of these partitions. Using the recurrence relation, we provide another recurrence relation and show that the average of the considered polynomials with respect to the Plancherel measure is very simple. Furthermore, we show that some existing results in the literature are easy corollaries of the recurrence relation.

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Section: Introductionmentioning

confidence: 99%

“…. , λ r ), we draw a diagram of r rows, where the i'th row consists of λ i boxes [31,32]. For example, for the partition λ = (4, 2, 1), the Young diagram looks like:…”

Section: Partitions and Standard Young Tableauxmentioning

confidence: 99%

Recurrence Relations for Wronskian Hermite Polynomials

Bonneux¹,

Stevens²

2018

SIGMA

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“…The idea of majorisation was adopted to chemistry for the first time when Ruch together with Schönhofer introduced a greater relation for Young diagrams [44][45][46][47] in order to answer questions in connection with the theory of chirality functions. Later on, Ruch generalised this concept in his articles on diagram lattices as structural principle 48 , on the principles of increasing mixing 49 and on information extent, and information distance 50 .…”

Section: Introductionmentioning

confidence: 99%

Discrete Vector Models for Catalysis and Autocatalysis

Haß

Sauerbrei

Plath

2008

Discrete Dynamics in Nature and Society

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Based on Ruch's concept of diagram lattices formed by Young diagrams we investigated the possibility to transform incomparable diagrams into comparable ones by means of vector catalysis. Ruch's diagram lattices allow a very general description of comparing frequency distributions by their mixing-character as an order relation which is equivalent to majorisation in the mathematical theory of inequalities. Dealing with Young diagrams or vectors containing only integer components, respectively, vector catalysis is strongly related to entanglement catalysis in quantum informatics. In a very systematic way the diagram lattices of the partitions up to the number n 20 have been searched for incomparable pairs which can be catalysed. This concept opens the opportunity for regarding vector catalysis as a universal phenomenon which is not restricted to the quantum mechanical idea of entanglement catalysis. Such a general approach offers the possibility to compare vector catalysis with chemical ideas of catalysis and autocatalysis in a very fundamental sense. We emphasize that vector catalysis is a universally valid procedure for classification purposes, where incomparable sequences of symbols are transformed into comparable ones in a much higher dimensional space ignoring any physical interpretation of these symbols.

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“…On the level of bracket algebra, a geometric theorem prover can be implemented using the straightening algorithm (Young, 1928;Doubilet et al, 1974). The main idea behind this approach is to rewrite the projective incidence statement as a term in Grassmann algebra which vanishes if and only if the statement is true.…”

Section: Introductionmentioning

confidence: 99%

Automated Theorem Proving in Incidence Geometry — A Bracket Algebra Based Elimination Method

2001

Automated Deduction in Geometry

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Abstract.In this paper we propose a bracket algebra based elimination method for automated generation of readable proofs for theorems in incidence geometry. This method is based on two techniques, the first being some heuristic elimination rules which improve the performance of the area method of without introducing signed length ratios, the second being a simplification technique called contraction, which reduces the size of bracket polynomials. More than twenty theorems in incidence geometry have been proved, for which short proofs are produced swiftly. An interesting phenomenon is that a proof composed of polynomials of at most two terms can always be found for any of these theorems, similar to that by the final biquadratic polynomial method of Richter-Gebert (1995).

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On Quantitative Substitutional Analysis

Cited by 96 publications

References 0 publications

Recurrence Relations for Wronskian Hermite Polynomials

Recurrence Relations for Wronskian Hermite Polynomials

Discrete Vector Models for Catalysis and Autocatalysis

Automated Theorem Proving in Incidence Geometry — A Bracket Algebra Based Elimination Method

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