scholarly journals Distinct partitions and overpartitions

2021 ◽  
Vol 38 (1) ◽  
pp. 149-158
Author(s):  
MIRCEA MERCA ◽  
Keyword(s):  
Partition Function ◽  
Recurrence Relation ◽  
Recurrence Relations ◽  
Convergent Series ◽  
The Other ◽  
Large Integer ◽  
Linear Recurrence ◽  
Fast Computation ◽  
Real Numbers ◽  
Very High

In 1963, Peter Hagis, Jr. provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the partition function $Q(n)$ which counts partitions of $n$ into distinct parts. Computing $Q(n)$ by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we investigate new connections between partitions into distinct parts and overpartitions and obtain a surprising recurrence relation for the number of partitions of $n$ into distinct parts. By particularization of this relation, we derive two different linear recurrence relations for the partition function $Q(n)$. One of them involves the thrice square numbers and the other involves the generalized octagonal numbers. The recurrence relation involving the thrice square numbers provide a simple and fast computation of the value of $Q(n)$. This method uses only (large) integer arithmetic and it is simpler to program. Infinite families of linear inequalities involving partitions into distinct parts and overpartitions are introduced in this context.

scholarly journals On a nonlinear relation for computing the overpartition function

2021 ◽  
Vol 71 (3) ◽  
pp. 535-542
Author(s):  
Mircea Merca
Keyword(s):  
Recurrence Relation ◽  
High Precision ◽  
Convergent Series ◽  
New Method ◽  
Large Integer ◽  
Fast Computation ◽  
Real Numbers ◽  
Integer Arithmetic ◽  
Nonlinear Relation ◽  
Very High

Abstract In 1939, H. S. Zuckerman provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the overpartition function p (n). Computing p (n) by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we provide a formula to compute the values of p (n) that requires only the values of p (k) with k ≤ n/2. This formula is combined with a known linear homogeneous recurrence relation for the overpartition function p (n) to obtain a simple and fast computation of the value of p (n). This new method uses only (large) integer arithmetic and it is simpler to program.

scholarly journals On Sequences of Numbers and Polynomials Defined by Linear Recurrence Relations of Order 2

2009 ◽  
Vol 2009 ◽  
pp. 1-21 ◽  
Author(s):  
Tian-Xiao He ◽  
Peter J.-S. Shiue
Keyword(s):  
Recurrence Relation ◽  
Chebyshev Polynomials ◽  
Recurrence Relations ◽  
Linear Recurrence ◽  
Lucas Numbers ◽  
Polynomial Sequences ◽  
Fibonacci And Lucas Numbers ◽  
Linear Recurrence Relation ◽  
Humbert Polynomials ◽  
General Method

Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer-Humbert polynomials are also discussed. The derived idea provides a general method to construct identities of number or polynomial sequences defined by linear recurrence relations. The applications using the method to solve some algebraic and ordinary differential equations are presented.

scholarly journals Sequences of Non-Gegenbauer-Humbert Polynomials Meet the Generalized Gegenbauer-Humbert Polynomials

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-18 ◽  
Author(s):  
Tian-Xiao He ◽  
Peter J.-S. Shiue
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Linear Recurrence ◽  
Polynomial Sequences ◽  
Linear Recurrence Relation ◽  
The Relationship ◽  
Humbert Polynomials

Here, we present a connection between a sequence of polynomials generated by a linear recurrence relation of order 2 and sequences of the generalized Gegenbauer-Humbert polynomials. Many new and known transfer formulas between non-Gegenbauer-Humbert polynomials and generalized Gegenbauer-Humbert polynomials are given. The applications of the relationship to the construction of identities of polynomial sequences defined by linear recurrence relations are also discussed.

scholarly journals Simultaneous Linear Recurrence Relations with Variable Coefficients

1958 ◽  
Vol 9 (4) ◽  
pp. 183-206
Author(s):  
H. D. Ursell
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Standard Form ◽  
Variable Coefficients ◽  
Linear Recurrence ◽  
Integer Variable ◽  
Self Consistent ◽  
Fixed Order ◽  
Image Position ◽  
Linear Recurrence Relation

Our subject is a set of equationswhere the uj(n)(j = 1, 2, …, k) are k “unknown” functions of the integer variable n, the zi(n) (i = 1, 2, … h) are h “known” functions of n, and the Aij(n) are hk “known” operatorswhich are polynomials in E, each of fixed order pij but with coefficients which may vary with n. E is the usual operator defined byOur first task is to determine whether the equations (1) are self-consistent. Secondly, if they are self-consistent, we ask what follows from them for a given subset of the unknowns, e.g. for (uj+1, …, uk) in other words we wish to eliminate (u1, …, uj). In particular we wish to eliminate all the variables but one, say uk. We shall in fact find that either uk is arbitrary or else that it has only to satisfy a single linear recurrence relation : and the order of that relation is of interest to us. Thirdly, we ask that reduction to standard form is possible, with or without a transformation of the unknowns themselves.

scholarly journals Sequences of Numbers Meet the Generalized Gegenbauer-Humbert Polynomials

2011 ◽  
Vol 2011 ◽  
pp. 1-16
Author(s):  
Tian-Xiao He ◽  
Peter J.-S. Shiue ◽  
Tsui-Wei Weng
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Linear Recurrence ◽  
Linear Recurrence Relation ◽  
The Relationship ◽  
Humbert Polynomials

Here we present a connection between a sequence of numbers generated by a linear recurrence relation of order 2 and sequences of the generalized Gegenbauer-Humbert polynomials. Many new and known formulas of the Fibonacci, the Lucas, the Pell, and the Jacobsthal numbers in terms of the generalized Gegenbauer-Humbert polynomial values are given. The applications of the relationship to the construction of identities of number and polynomial value sequences defined by linear recurrence relations are also discussed.

Thrombotic Thrombocytopenic Purpura: Mechanism for Effectiveness of Plasmapheresis

1979 ◽  
Author(s):  
J. G. Kelton ◽  
P. B. Neame ◽  
I. Walker ◽  
A. G. Turpie ◽  
J. McBride ◽  
...  
Keyword(s):  
Platelet Count ◽  
Thrombotic Thrombocytopenic Purpura ◽  
The Other ◽  
Unknown Etiology ◽  
Normal Range ◽  
Antiplatelet Antibody ◽  
Thrombocytopenic Purpura ◽  
Serious Illness ◽  
Normal Platelet ◽  
Very High

Thrombotic thrombocytopenic purpura (TTP) is a rare but serious illness of unknown etiology. Treatment by plasmapheresis has been reported to be effective but the mechanism for benefit is unknown. We have investigated the effect of plasmapheresis in 2 patients with TTP by quantitating platelet associated IgG (PAIgG) levels prior to and following plasmapheresis. Both patients had very high levels of PAIgG at presentation (90 and A8 fg IgG/platelet respectively, normal 0-5). in both, the PAIgG levels progressively fell to within the normal range and the platelet count rose following plasmapheresis. One patient remained in remission with normal platelet counts and PAIgG levels. The other relapsed after plasmapheresis and the PAIgG level rose prior to the fall in platelet count. Plasmapheresis was repeated and resulted in normalization of both the platelet count and PAIgG level. It is suggested that plasmapheresis removes antiplatelet antibody or immune complexes which may be of etiological importance in this illness.

On Commutation Properties of the Composition Relation of Convergent and Divergent Permutations (Part I)

2014 ◽  
Vol 58 (1) ◽  
pp. 13-22
Author(s):  
Roman Wituła ◽  
Edyta Hetmaniok ◽  
Damian Słota
Keyword(s):  
Finite Order ◽  
Convergent Series ◽  
The Other ◽  
Other Hand ◽  
The Family ◽  
Composition Relation ◽  
The Continuum

Abstract In the paper we present the selected properties of composition relation of the convergent and divergent permutations connected with commutation. We note that a permutation on ℕ is called the convergent permutation if for each convergent series ∑an of real terms, the p-rearranged series ∑ap(n) is also convergent. All the other permutations on ℕ are called the divergent permutations. We have proven, among others, that, for many permutations p on ℕ, the family of divergent permutations q on ℕ commuting with p possesses cardinality of the continuum. For example, the permutations p on ℕ having finite order possess this property. On the other hand, an example of a convergent permutation which commutes only with some convergent permutations is also presented.

Growth Characteristics of Activated Sludges Acclimated to para-Nitrophenol in Batch and Continuous Modes

1994 ◽  
Vol 29 (7) ◽  
pp. 327-333
Author(s):  
Y. Matsui ◽  
F. Yamaguchi ◽  
Y. Suwa ◽  
Y. Urushigawa
Keyword(s):  
Growth Characteristics ◽  
The Other ◽  
Operational Mode ◽  
Continuous Mode ◽  
Relative Resistance ◽  
Batch Mode ◽  
The Given ◽  
Operational Modes ◽  
Very High ◽  
Sensitive Group

Activated sludges were acclimated to p-nitrophenol (PNP) in two operational modes, a batch and a continuous. The operational mode of the PNP acclimation of activated sludges strongly affected the physiological characteristics of predominant microorganisms responsible for PNP degradation. Predominant PNP degraders in the sludge in batch mode (Sludge B) had lower PNP affinity and were relatively insensitive to PNP concentration. Those of the sludge in continuous mode (Sludge C), on the other hand, had very high PNP affinity and were sensitive to PNP. MPN enumeration of PNP degraders in sludge B and C using media with different PNP concentrations (0.05, 0.2,0.5 and 2.0 mM) supported the above results. Medium with 0.2 mM of PNP did not recover PNP degraders in sludge C well, while it recovered PNP degraders in sludge B as well as the medium with 0.05 mM did. When switching from one operational mode to the other, the predominant population in sludge B shifted to the sensitive group, but that of sludge C did not shift at the given loading of PNP, showing relative resistance to inhibitive concentration.

Islamic hedging for pilgrimage funds: case of Indonesia

2019 ◽  
Vol 11 (3) ◽  
pp. 328-341
Author(s):  
Rifki Ismal ◽  
Nurul Izzati Septiana
Keyword(s):  
Exchange Rate ◽  
The Other ◽  
Exchange Rate Risk ◽  
Content Type ◽  
Other Hand ◽  
Rate Risk ◽  
Risk Identify ◽  
Hajj Season ◽  
The Right ◽  
Very High

Purpose The demand for Saudi Arabian real (SAR) is very high in the pilgrimage (hajj) season while the authority, unfortunately, does not hedge the hajj funds. As such, the hajj funds are potentially exposed to exchange rate risk, which can impact the value of hajj funds and generate extra cost to the pilgrims. The purpose of this paper is to conduct simulations of Islamic hedging for pilgrimage funds to: mitigate and minimize exchange rate risk, identify and recommend the ideal time, amount and tenors of Islamic hedging for hajj funds, estimate cost saving by pursuing Islamic hedging and propose technical and general recommendations for the authority. Design/methodology/approach Forward transaction mechanism is adopted to compute Islamic forward between SAR and Rupiah (Indonesian currency) or IDR. Findings – based on simulations, the paper finds that: the longer the Islamic hedging tenors, the better is the result of Islamic hedging, the decreasing of IDR/USD is the right time to hedge the hajj funds and, on the other hand, the IDR/SAR appreciation is not the right time to hedge the hajj funds. Findings Based on simulations, the paper finds that: the longer the Islamic hedging tenors, the better is the result of Islamic hedging, the decreasing of IDR/USD is the right time to hedge the hajj funds and, on the other hand, the IDR/SAR appreciation is not the right time to hedge the hajj funds. Research limitations/implications The research suggests the authority to (and not to) hedge the hajj fund, depending on economic conditions and market indicators. Even though the assessment is for the Indonesian case, other countries maintaining hajj funds might also learn from this paper. Originality/value To the best of author’s knowledge, this is the first paper in Indonesia that attempts to simulate the optimal hedging of hajj funds.

Rolling and Recrystallization Textures in High Purity Al Cold Rolled to Very High Rolling Reductions

2005 ◽  
Vol 495-497 ◽  
pp. 603-608 ◽  
Author(s):  
Atsushi Todayama ◽  
Hirosuke Inagaki
Keyword(s):  
Cold Rolling ◽  
High Purity ◽  
Work Hardening ◽  
Dynamic Recovery ◽  
The Other ◽  
Rolling Reduction ◽  
Theoretical Investigations ◽  
Theoretical Predictions ◽  
Cold Rolled ◽  
Very High

On the basis of Taylor-Bishop-Hill’s theory, many previous theoretical investigations have predicted that, at high rolling reductions, most of orientations should rotate along theβfiber from {110}<112> to {123}<634> and finally into the {112}<111> stable end orientations. Although some exceptions exist, experimental observations have shown, on the other hand, that the maximum on the β fiber is located still at about {123}<634> even after 97 % cold rolling. In the present paper, high purity Al containing 50 ppm Cu was cold rolled up to 99.4 % reduction in thickness and examined whether {112}<111> stable end orientation could be achieved experimentally. It was found that, with increasing rolling reduction above 98 %, {110}<112> decreased, while orientations in the range between {123}<634> and {112}<111> increased, suggesting that crystal rotation along the βfiber from {110}<112> toward {123}<634> and {112}<111> in fact took place. At higher rolling reductions, however, further rotation of this peak toward {112}<111> was extremely sluggish, and even at the highest rolling reduction, it could not arrive at {112}<111>. Such discrepancies between theoretical predictions and experimental observations should be ascribed to the development of dislocation substructures, which were formed by concurrent work hardening and dynamic recovery. Since such development of dislocation substructures are not taken into account in Taylor-Bishop-Hill’s theory, it seems that they can not correctly predict the development of rolling textures at very high rolling reductions, i. e. stable end orientations. On annealing specimens rolled above 98 % reduction in thickness, cube textures were very weak, suggesting that cube bands were almost completely rotated into other orientations during cold rolling. {325}<496>, which lay at an intermediate position between {123}<634> and {112}<111> along theβfiber, developed strongly in the recrystallization textures.

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