Smallest Part Research Articles

RAMANUJAN’S SPT-CRANK FOR MARKED OVERPARTITIONS

In 1916, Ramanujan’s showed the spt-crank for marked overpartitions. The corresponding special functions , and are found in Ramanujan’s notebooks, part 111.
 In 2009, Bingmann, Lovejoy and Osburn defined the generating functions for ,
 and . In 2012, Andrews, Garvan, and Liang defined the in terms of partition pairs. In this article the number of smallest parts in the overpartitions of n with smallest part not overlined, not overlined and odd, not overlined and even are discussed, and the vector partitions and - partitions with 4 components, each a partition with certain restrictions are also discussed. The generating functions , , , , are shown with the corresponding results in terms of modulo 3, where the generating functions , are collected from Ramanujan’s notebooks, part 111. This paper shows how to prove the Theorem 1 in terms of ,Theorem 2 in terms of and Theorem 3 in terms of respectively with the numerical examples, and shows how to prove the Theorems 4,5 and 6 with the help of in terms of partition pairs. In 2014, Garvan and Jennings-Shaffer are able to defined the for marked overpartitions. This paper also shows another results with the help of 6 -partition pairs of 3, help of 20 -partition pairs of 5 and help of 15 -partition pairs of 8 respectively.

Interpreting the Truncated Pentagonal Number Theorem using Partition Pairs

In 2012 Andrews and Merca gave a new expansion for partial sums of Euler's pentagonal number series and expressed \[\sum_{j=0}^{k-1}(-1)^j(p(n-j(3j+1)/2)-p(n-j(3j+5)/2-1))=(-1)^{k-1}M_k(n)\] where $M_k(n)$ is the number of partitions of $n$ where $k$ is the least integer that does not occur as a part and there are more parts greater than $k$ than there are less than $k$. We will show that $M_k(n)=C_k(n)$ where $C_k(n)$ is the number of partition pairs $(S, U)$ where $S$ is a partition with parts greater than $k$, $U$ is a partition with $k-1$ distinct parts all of which are greater than the smallest part in $S$, and the sum of the parts in $S \cup U$ is $n$. We use partition pairs to determine what is counted by three similar expressions involving linear combinations of pentagonal numbers. Most of the results will be presented analytically and combinatorially.

CONGRUENCES FOR PARTITION PAIRS WITH CONDITIONS

We prove congruences for the number of partition pairs $(\pi_1,\pi_2)$ such that $\pi_1$ is non-empty, $s(\pi_1)\le s(\pi_2)$, and $\ell(\pi_2)< 2s(\pi_1)$ where $s(\pi)$ is the smallest part and $\ell(\pi)$ is the largest part of a partition. The proofs use Bailey's Lemma and a generalized Lambert series identity of Chan. We also discuss how a partition pair crank gives combinatorial refinements of these congruences.

Partitions with fixed differences between largest and smallest parts

We study the number p(n, t) of partitions of n with difference t between largest and smallest parts. Our main result is an explicit formula for the generating function Pt(q) := ∑ n≥1 p(n, t) q . Somewhat surprisingly, Pt(q) is a rational function for t > 1; equivalently, p(n, t) is a quasipolynomial in n for fixed t > 1. Our result generalizes to partitions with an arbitrary number of specified distances. Enumeration results on integer partitions form a classic body of mathematics going back to at least Euler, including numerous applications throughout mathematics and some areas of physics; see, e.g., [2]. A partition of a positive integer n is, as usual, an integer k-tuple λ1 ≥ λ2 ≥ · · · ≥ λk > 0, for some k, such that n = λ1 + λ2 + · · ·+ λk . The integers λ1, λ2, . . . , λk are the parts of the partition. We are interested in the counting function p(n, t) := #partitions of n with difference t between largest and smallest parts. It is immediate that p(n, 0) = d(n) where d(n) denotes the number of divisors of n. Charmingly, p(n, 1) equals the number of nondivisors of n: p(n, 1) = n− d(n) , which can be explained bijectively by the fact that the partitions counted by p(n, 0)+p(n, 1) contain exactly one sample with k parts, for each k = 1, 2, . . . , n [1, Sequence A049820], or by the generating function identity∑ n≥1 p(n, 1) q = ∑ m≥1 qm 1− qm qm+1 1− qm+1 = q (1− q)2 − ∑ m≥1 qm 1− qm . (The last equation follows from a few elementary operations on rational functions). An even less obvious instance of our partition counting function is (1) p(n, 2) = (⌊n 2 ⌋ 2 ) , as observed by Reinhard Zumkeller in 2004 [1, Sequence A008805]. (It is not clear to us where in the literature this formula first appeared, though specific values of p(n, k) are well represented in Date: 14 July 2014. 2010 Mathematics Subject Classification. Primary 11P84; Secondary 05A17.

Comparison of steam-gasification characteristics of coal char and petroleum coke char in drop tube furnace

The steam-gasification reaction characteristics of coal and petroleum coke (PC) were studied in the drop tube furnace (DTF). The effects of various factors such as types of carbonaceous material, gasification temperature (1100–1400°C) and mass ratio of steam to char (0.4:1, 0.6:1 and 1:1 separately) on gasification gas or solid products were investigated. The results showed that for all carbonaceous materials studied, H2 content exhibited the largest part of gasification gaseous products and CH4 had the smallest part. For the two petroleum cokes, CO2 content was higher than CO, which was similar to Zun-yi char. When the steam/char ratio was constant, the carbon conversion of both Shen-fu and PC chars increased with increasing temperature. When the gasification temperature was constant, the carbon conversions of all char samples increased with increasing steam/char ratio. For all the steam/char ratios, compared to water gas shift reaction, char-H2O and char-CO2 reaction were further from the thermodynamic equilibrium due to a much lower char gasification rate than that of water gas shift reaction rate. Therefore, kinetic effects may play a more important role in a char gasification step than thermodynamic effects when the gasification reaction of char was held in DTF. The calculating method for the equilibrium shift in this study will be a worth reference for analysis of the gaseous components in industrial gasifier. The reactivity of residual cokes decreased and the crystal layer (L002/d002) numbers of residual cokes increased with increasing gasification temperature. Therefore, L002/d002, the carbon crystallite structure parameter, can be used to evaluate the reactivity of residual cokes.

DEVELOPMENT OF FOKKINK-FOKKINK-WANG’S GENERATING FUNCTION FOR FFW(n)

In 1995, R. Fokkink, W. Fokkink and Wang defined the in terms of , where is the smallest part of partition . In 2008, Andrews obtained the generating function for . In 2013, Andrews, Garvan and Liang extended the FFW-function and obtained the similar expressions for the spt-function and then defined the spt-crank generating functions. They also defined the generating function for in various ways. This paper shows how to find the number of partitions of n into distinct parts with certain conditions and shows how to prove the Theorem 1 by induction method. This paper shows how to prove the Theorem 2 with the help of two generating functions.

Dynamics of assembly production flow

Despite recent developments in management theory, maintaining a manufacturing schedule remains difficult because of production delays and fluctuations in demand and supply of materials. The response of manufacturing systems to such disruptions to dynamic behavior has been rarely studied. To capture these responses, we investigate a process that models the assembly of parts into end products. The complete assembly process is represented by a directed tree, where the smallest parts are injected at leaves and the end products are removed at the root. A discrete assembly process, represented by a node on the network, integrates parts, which are then sent to the next downstream node as a single part. The model exhibits some intriguing phenomena, including overstock cascade, phase transition in terms of demand and supply fluctuations, nonmonotonic distribution of stockout in the network, and the formation of a stockout path and stockout chains. Surprisingly, these rich phenomena result from only the nature of distributed assembly processes. From a physical perspective, these phenomena provide insight into delay dynamics and inventory distributions in large-scale manufacturing systems.

DEVELOPMENT OF ANDREWS-GARVAN-LIANG’S SELF-CONJUGATE S-PARTITIONS

In 2013, Andrews, Garvan and Liang defined Self-conjugate S-partitions. In 2011, Andrews stated the definition of spt(n). This paper shows how to find the Self-conjugate S-partitions of 4 and 5 respectively, and proves the Corollary-1 that is ‘The number of Self-conjugate S-partitions counted according to the weight w is congruent to the total number of smallest parts in all the partitions of n modulo 2’. This paper shows how to generate the generating functions for Msc(n) in different ways. This paper shows how to find the number of partitions of n with an odd number of smallest parts and a total number of even (respectively odd) parts, and shows how to find the number of partitions of n in odd parts without gaps, and also shows how to generate the generating functions for and respectively. This paper shows how to prove the further three Corollaries with the help of examples, and shows how to prove the three Theorems by easy algebraic method.

Higher order SPT functions for overpartitions, overpartitions with smallest part even, and partitions with smallest part even and without repeated odd parts

We consider the symmetrized moments of three ranks and cranks, similar to the work of Garvan in [17] for the rank and crank of a partition. By using Bailey pairs and elementary rearrangements, we are able to find useful expressions for these moments. We then deduce inequalities between the corresponding ordinary moments. In particular we prove that the crank moment for overpartitions is always larger than the rank moment for overpartitions, M¯2k(n)>N¯2k(n); with recent asymptotics this was known to hold for sufficiently large values of n for each fixed k. Lastly we provide higher order spt functions for overpartitions, overpartitions with smallest part even, and partitions with smallest part even and no repeated odds.

Notes on the $$\textit{spt}$$ spt function of George E. Andrews

Andrews defined \(\textit{spt}(n)\) to be the total number of appearances of the smallest parts in all of the partitions of \(n\). In this paper, we study the statistical distribution of \(\textit{spt}(\pi )\), the number of smallest parts in the partition \(\pi \) as \(\pi \) ranges over all partitions of \(n\). We also give a combinatorial proof of a conjecture of Hirschhorn, namely that $$\begin{aligned} p(0)+\ \cdots \ +p(n-1)<\textit{spt}(n)<p(0)+\ \cdots \ +p(n) \end{aligned}$$ for all \(n>1\), where \(p(n)\) is the number of partitions of \(n\).

Numerical simulation on the thermal performance of a solar molten salt cavity receiver

The receiver thermal performance directly affects the efficiency, reliability and safety of the entire solar tower power system. A model combining radiation, convection and conduction heat transfer modes was developed to model a 1 MW molten salt cavity receiver with determined basic design parameters. The coupled heat transfer model was used to predict the receiver's steady-state efficiency, temperature distribution and heat losses for various flow layouts and receiver aperture lip sizes.The results show that the fraction of reflective loss (18.78%) is the largest in all heat losses, followed by radiative and convective heat losses, with conduction heat losses as the smallest part which is usually negligible. The heat loss distribution is closely related to the external tube temperature field in the cavity receiver and the specific structure of the receiver. 10 cm lips on both the top and bottom of the receiver aperture reduced the reflective heat loss by 1%. The results for different flow layouts show that center-side flow layout has the best temperature homogeneity, with less possibility of overheating in the central area. This, however, comes at the expense of increased irreversibilities and decreased efficiency. The sides-center flow layout has the worst temperature homogeneity, but the receiver efficiency is 2.06% higher. These results provide a useful reference for designing and optimizing cavity receivers.

The status of photographs as library materials and their cataloguing through the perspective of Slovenian handbooks, reg

AbstractThe article presents an overview of Slovene handbooks, regulations and standards, which are, or have been, used in the cataloguing of library materials. The research is concentrated on their relation with photographic materials ie. with the smallest part, the photograph. The status of photographs is being looked at in the first Slovene handbook Knjižnice in knjižničarsko delo, the first Slovene regulations Abecedno imenski katalog and its second edition, and in Pravilnik i priručnik za izradbu abecednih kataloga (PPIAK). The International Standard Bibliographical Description for Non-Book Materials (ISBD(NBM)) and PREKAT also deal with the cataloguing of photographs. Greater attention is paid to the Pravila za katalogizacijo v grafičnih zbirkah, which is the work of professional committees of the Union of Associations of Slovene Librarians. Although some of the regulations are quite old, they are still very practical and usable. The author examines the handbooks from the point of view of how detailed and accurately they provide the regulations for cataloguing photographs.

The Emergence of Symbolic Principles: The Distribution of Mind in Early Sign Making

This paper considers the extent to which the earliest stages of learning about systems of inscription requires not just individual mental effort, but effort that is distributed across a wide physical and intellectual environment. It is particularly concerned with how children under the age of three learn about notational systems, including writing, and examines parallels with the evolution of written systems. It considers the position that children gain knowledge incrementally over the early months and years of life, supported by a commonly accepted view that learning is a mental activity in which knowledge accumulates hierarchically in the minds of individual children, starting with smallest parts of the system. The paper presents evidence to the contrary, suggesting that the early learning of inscriptional systems is associated not just with individual minds, but with social and cultural cognition that is dispersed across minds, bodies, tools, and material environments. It presents data and evidence from a small study of the sign-making of children under the age of three that indicates that children of this age already use notations purposefully in the construction of signs that are intentional, multimodal, and unbounded, and that already have features associated with conventional systems of inscription. In their early sign-making they use certain underlying principles of symbolic reference associated with conventional systems, including the use of ‘generic’ structures derived from social, bodily, and material experience. Central to this process are networks of interactions between co-participants, tools, materials, and the physical environment.

Draw it, if you can!

Draw it, if you can!

VAIKŲ KVĖPAVIMO SISTEMOS RODIKLIŲ FUNKCINIŲ RYŠIŲ ĮVERTINIMAS PRIKLAUSOMAI NUO LYTIES

Research aim was to identify functional relationships of the respiratory system for different gender childrens. There took part in research the children, which parents where informed about execution procedure and the parents accepted using of research results. Persons for research were detected according following criterions: age 7-8 years old, no motoric development problems, no infections invasions, ability for free physical action and freedom for activity after school time, no additional sporting after school time, no problems in case of breath and breath channels muscules. There were evaluated living capacity of lungs using pyrometer. Maximal exhalation speed measured using pneumotachometer. Power of breath muscules measured maximal inhalation (PImax) and exhalation (PEmax) pressure using pneumotonometer. The measurement device Micro Medical RPM used for maximal breath pressure measurement in this research. There were detected, that 39% of girls living capacity of lungs is normal limits, the rest 61% proportions are: 28% girls have less living capacity of lungs than the norms, and 33% have bigger than the norms are (p&lt;0,05). Data of boys during living capacity of lungs evaluation and comparing it to the norms have the following results: 17% have living capacity of lungs below norms, and it is the smallest part in this research. Data according norms limits are 42%, and results, what exceed‘s the norms, are 43% (p&lt;0,05). Conclusion. The boys usually have living capacity of lungs, maximal exhalation flow speed, maximal inhalation and exhalation pressure above norms and higher than that indexes of girls are. There exist functional relationships between functional indexes: if one of indexes is higher, other indexes increase in it‘s values too.

HECKE-TYPE CONGRUENCES FOR TWO SMALLEST PARTS FUNCTIONS

We prove infinitely many congruences modulo 3, 5, and powers of 2 for the overpartition function [Formula: see text] and two smallest parts functions: [Formula: see text] for overpartitions and M2spt(n) for partitions without repeated odd parts. These resemble the Hecke-type congruences found by Atkin for the partition function p(n) in 1966 and Garvan for the smallest parts function spt(n) in 2010. The proofs depend on congruences between the generating functions for [Formula: see text], [Formula: see text], and M2spt(n) and eigenforms for the half-integral weight Hecke operator T(ℓ2).

Letter to the Editor: Avicenna's view on the prevention of thrombosis

We have greatly enjoyed reading the recently published manu-script by AA. Gooshah et al.[1] who related their view on Avicenna's(Ibn Sina) theory of humors on the prevention of thrombosis. It isinteresting to know that in one of the most prominent figures inthe history of medicine Avicenna himself had comments in hisAl-Qanun on the causes of obstructive cardiovascular diseases[2,3]. But, unfortunately, the inconsistent terms and conceptsfound in this manuscript could result in the miscommunication ofthe readers. In their article, the authors only emphasize Sauda andSafra which have important roles in blood coagulation in the circu-latory system — to decrease blood circulation and cause thrombusformation. In Avicenna's medicine, Sauda clinically divides intotwo categories — normal black Sauda and abnormal Sauda, whereinthe abnormal Sauda usually plays an important role in blood coagu-lation in the circulatory system [4–6].Although there is no concept of thrombosis in the Qanunof Medicine as such and the concept of thrombosis in modernmedicine distinguishes from the concept of obstruction and semi-obstruction by abnormal Sauda which is discussed in thefirstbook of Avicenna's Qanun, some important links could be foundbetween the old and the modern medicines' understanding.In Avicenna's medicine, the human body is composed of fourhumors (blood or Dam,phlegmorBalgham, yellow bile or Safra andblack bile orSauda). These four humors – yellow bile, blood, phlegmand black bile– mutually compensate, balance, control and supporteach other. In normal condition, the Blood is the humor most essen-tial and conducive to vitality and good health. The Blood plays a rolein hilit transportation and metabolism and that all the hilits con-stantly cycle in the blood. The Safra primarily functions to transportthe Balgham and Sauda in the Blood to the smallest parts of thehuman body by increasing the viscosity of the blood, facilitateblood flow in the vessels, inhibit the solidification of the Balgham,Blood and Sauda, and prevent blood stasis. One of the most impor-tant functions of the yellow bile is the suppression of the effects ofthe black bile [5]. The Balgham protects the hilits damaged by theSafra with hot character. It also transports the Sauda through thebloodtothebody'stissuesandorgans,suppliesbodyfluids,moistensandsoftenstissues,andreplacesblood.Itisthelasthilit,savda,whichhelps to maintain the shape of the organ and body.The cornerstone of health is the right ratio and specificbalanceof humors based on their quality and quantity. The disequilibriumof the humors quantitatively or qualitatively causes diseases [6].Avicenna suggests that all four humors are the important factors invessel obstruction (and then thrombus formation) in a differentstage of disease. Again, we should state that according to Avicenna'smedicine, all humors are taking part in the formation of thrombosis.The special treatment by the regulation of these humors could pre-vent thrombosis. Our research studies found that cardiovasculardisease risk factors such as metabolic abnormalities like abdominalobesity, high cholesterol, high blood pressure and high blood glucoseare the parts comprising the Balgham increase syndrome (abnormalBalgham) [7]. The increase of abnormal Balgham would decrease thequantity and quality of the Safra and thus enhance the Sauda's originalability to perform the formation, cohesion and thickening of the blood.The increase of blood viscosity and further atherosclerosis are under-stood as the deposition of abnormal Sauda in arterial walls which nar-rows the blood vessels, reduces circulation, and causes obstructionand semi-obstruction in the vessels [8,9].Itmustberememberedherethat conditions of deficient blood supply to the muscles and tissuesandjointsandtendonsareinfactaSaudacondition,becausewiththeab-senceofwarm,moistBlood,acold,drySaudaconditionwillbeproduced.And so, increasing the circulation and vital blood supply to an organ ortissue of the body is actually an anti-Sauda therapeutic approach.In our clinical studies of coronary heart disease cases and coagu-lation–anticoagulation and fibrinolytic system function disorders,theBalghamsyndromehasbeenpredominated.However,theseverestages of these diseases are clinically connected with abnormalSauda syndrome [5–7]. Based on Avicenna's view, the specific

Inventory size and complexity in the song of the American Robin

This paper discusses American Robin morning song and offers a component system to explain its complex structure and discuss the relationship between inventory size and syllable complexity. Previous research has noted that Robin song is combinatorial, meaning Robins are capable of creating different songs by concatenating elements called syllables. For each robin, all unique syllables produced during a 40 minute recording were grouped into a syllable inventory. Inspired by observations linguists have made about human phoneme inventories, a component system was devised that broke each syllable into its smallest parts. These parts were found by using minima in intensity contours to correspond to natural breaks in the robin's utterances. This component system included all unique syllable parts, 7 total, from 11 syllable inventories with between 8 and 24 different syllables. This system showed certain components occurred more often than others and robins with larger syllable inventories had more rare components on average. It also showed each token of the same syllable type had the same component structure. This component system will also be used to test for patterns in sequencing of components in syllable structure to determine where components can occur and constraints on syllable sequencing in overall song structure.

A Franklin type involution for squares

We find an involution as a combinatorial proof of Ramanujanʼs partial theta identity. Based on this involution, we obtain a Franklin type involution on the set of partitions into distinct parts with the smallest part being odd. Compared with the involution of Bessenrodt and Pak, our involution possesses a weight-preserving property that leads to a combinatorial proof of a weighted partition theorem derived by Alladi from Ramanujanʼs partial theta identity. This gives an indirect answer to a question of Berndt, Kim and Yee. Moreover, we obtain a partition theorem based on Andrewsʼ identity and provide a combinatorial proof via certain weight assignment for our involution. A specialization of this partition theorem is related to an identity of Andrews concerning partitions into distinct nonnegative parts with the smallest part being even. Finally, we give an extension of our partition theorem which corresponds to a generalization of Andrewsʼ identity.

Relation Between the Smallest and the Greatest Parts of the Partitions of n

In this paper the formulae for the number of smallest parts of partitions of $n\in N$ and relations between the $ith$ smallest parts and the $ith$ greatest parts are obtained.