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In a previous paper, the second and third named authors introduced the concept of the complete cycle index and discussed a relation with the complete weight enumerator in coding theory. In this paper, we introduce the concept of the complete joint cycle index and the average complete joint cycle index, and discuss a relation with the complete joint weight enumerator and the average complete joint weight enumerator, respectively, in coding theory. Moreover, the notion of the average intersection numbers is given, and we discuss a relation with the average intersection numbers in coding theory. [ABSTRACT FROM AUTHOR]

Copyright of China Petroleum Machinery is the property of China Petroleum Machinery Editorial Department and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Abstract The long‐term cyclic gas injection and production mode of the gas storage will lead to accelerated deformation of the surrounding rock of the old cavity, and even the instability and failure of the cavity. Considering the geological conditions and cavity measurement data of the old cavity of the horizontal butted‐well in China, the deformation and failure characteristics of the long‐term cyclic injection‐production process of the surrounding rock of the old cavity under different internal pressure difference and injection‐production frequency are studied. The long‐term stability of the gas storage is predicted by six indexes: the deformation of surrounding rock, dilatancy safety factor, volume shrinkage rate, plastic zone volume, equivalent strain, and no tensile stress. The results show that with the increase of operation duration and cycle times, the deformation, volume shrinkage rate, and plastic zone volume of the surrounding rock gradually increase after the old cavity is reconstructed into the gas storage and the difference of results under different UGS and CAES gradually expands. The stability of the surrounding rock of the old cavity under the cycle action of CAES will be lower. The number of cycles has a limited impact on the stability of the surrounding rock of the old cavity, while the operation duration and injection‐production pressure are the main determinants of the stability of the surrounding rock of the old cavity. If the old cavity of the salt mine is reconstructed into CAES, the parameters such as the number of cycles, the operation duration, and the injection‐production pressure still need to be considered comprehensively. The research results can provide reference value for the reasonable design of the gas storage pressure range of the old cavity and the safety and stability evaluation of the surrounding rock.

The long‐term cyclic gas injection and production mode of the gas storage will lead to accelerated deformation of the surrounding rock of the old cavity, and even the instability and failure of the cavity. Considering the geological conditions and cavity measurement data of the old cavity of the horizontal butted‐well in China, the deformation and failure characteristics of the long‐term cyclic injection‐production process of the surrounding rock of the old cavity under different internal pressure difference and injection‐production frequency are studied. The long‐term stability of the gas storage is predicted by six indexes: the deformation of surrounding rock, dilatancy safety factor, volume shrinkage rate, plastic zone volume, equivalent strain, and no tensile stress. The results show that with the increase of operation duration and cycle times, the deformation, volume shrinkage rate, and plastic zone volume of the surrounding rock gradually increase after the old cavity is reconstructed into the gas storage and the difference of results under different UGS and CAES gradually expands. The stability of the surrounding rock of the old cavity under the cycle action of CAES will be lower. The number of cycles has a limited impact on the stability of the surrounding rock of the old cavity, while the operation duration and injection‐production pressure are the main determinants of the stability of the surrounding rock of the old cavity. If the old cavity of the salt mine is reconstructed into CAES, the parameters such as the number of cycles, the operation duration, and the injection‐production pressure still need to be considered comprehensively. The research results can provide reference value for the reasonable design of the gas storage pressure range of the old cavity and the safety and stability evaluation of the surrounding rock. [ABSTRACT FROM AUTHOR]

Finding the number of isomers of compounds with a given molecular geometry or the number of chain isomers of an alkane or alkyl group is analogous to finding the number of ways of colouring the vertices of a polyhedron (or the beads in a necklace) with a given number of colours. Pólya's enumeration theorem (PET), which involves the concept of permutation groups and cycle indices, is a powerful method for this purpose. The present article attempts to explain the 'permutation group' and 'cycle index' of a permutation from an elementary level of understanding and shows how PET works in finding the number of chain isomers of alkanes and alkanols and geometrical isomers of tetrahedral Mabcd, Mab3, Ma2b2 and octahedral Mabcdef, Mab5, Ma2b4 and Ma3b3 types of molecules. [ABSTRACT FROM AUTHOR]

Natural gas is a viable oil displacement agent in ultra-low-permeability reservoirs due to its good fluidity. It can also cause gas channeling during continuous injection, which limits its oilfield application. In order to relieve gas channeling during natural gas flooding, the injection mode should be changed. The use of intermittent natural gas injection (IGI) after the continuous natural gas injection in an ultra-low-permeability reservoir is proposed, and optimization of the injection parameters is discussed. The results show that IGI can be divided into three stages, the gas injection stage, the well shutting stage and the oil production stage. With the increase in injection time, the oil recovery enhances obviously as a result of IGI because the gas fingering can be controlled at the well shutting stage, and the gas/liquid ratio grows slowly because the gas breakthrough can be reduced at the oil production stage. The oil recovery improves with the increase in cycle time of IGI, while the increase rate reduces evidently after the cycle time reaches 360 min. The oil recovery increment is low if the cycle index exceeds 3 in the ultra-low-permeability reservoir. Thus, the optimal cycle time for each round and the appropriate cycle index of IGI are 360 min and three rounds. [ABSTRACT FROM AUTHOR]

An algorithm is developed for calculating the number of double cosets P ﹨ S n / P , where P is a Sylow- p -subgroup of the symmetric group S n. Several examples of its use are given. [ABSTRACT FROM AUTHOR]

In order to investigate the appropriate value of one of the simulation parameters, cycle index in the wear prognosis of main bearing, the method of dynamics simulation and lubrication analysis were applied in the simulation of crankshaft system of internal combustion engine. The coupling model of multi-body dynamics and lubrication of the crankshaft system of a certain diesel engine in service were established. Based on Archard model of wear theory, several groups of cycle indexes were set up for the wear calculation, and a portion of the maximum wear depths and shell profiles of main bearings were selected to be compared and evaluated. The variation law of the calculation results of maximum wear depth with the increase of cycle index was summarized. The results show that index 3 is supposed to be selected in the wear simulation condition in consideration of the cost and reliability of calculation. This study provides guidance and suggestions for the optimization of wear prognosis. [ABSTRACT FROM AUTHOR]

The convex hull of n + 1 affinely independent vertices of the unit n-cube In is called a 0/1-simplex. It is nonobtuse if none its dihedral angles is obtuse, and acute if additionally none of them is right. In terms of linear algebra, acute 0/1-simplices in In can be described by nonsingular 0/1-matrices P of size n × n whose Gramians G = PTP have an inverse that is strictly diagonally dominant, with negative off-diagonal entries [6, 7]. The first part of this paper deals with giving a detailed description of how to efficiently compute, by means of a computer program, a representative from each orbit of an acute 0/1-simplex under the action of the hyperoctahedral group Bn [17] of symmetries of In. A side product of the investigations is a simple code that computes the cycle index of Bn, which can in explicit form only be found in the literature [11] for n ≤ 6. Using the computed cycle indices for B3, . . . ,B11 in combination with Pólya’s theory of enumeration shows that acute 0/1-simplices are extremely rare among all 0/1-simplices. In the second part of the paper, we study the 0/1-matrices that represent the acute 0/1-simplices that were generated by our code from a mathematical perspective. One of the patterns observed in the data involves unreduced upper Hessenberg 0/1-matrices of size n × n, block-partitioned according to certain integer compositions of n. These patterns will be fully explained using a so-called One Neighbor Theorem [4]. Additionally, we are able to prove that the volumes of the corresponding acute simplices are in one-to-one correspondence with the part of Kepler’s Tree of Fractions [1, 24] that enumerates ℚ ⋂ (0, 1). Another key ingredient in the proofs is the fact that the Gramians of the unreduced upper Hessenberg matrices involved are strictly ultrametric [14, 26] matrices.

In this paper, we introduce the concept of the complete cycle index and discuss a relation with the complete weight enumerator in coding theory. This work was motivated by Cameron's lecture notes "Polynomial aspects of codes, matroids and permutation groups." [ABSTRACT FROM AUTHOR]

Copyright of Journal of the Chinese Society of Rare Earths is the property of Editorial Department of Journal of the Chinese Society of Rare Earths and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Hierarchical enumerations of octahedral derivatives are conducted in accord with the hierarchy of groups for characterizing an octahedral skeleton, i.e., point groups (O and Oh; orders 24 and 48) ⊂RS-stereoisomeric group (Ohσ~I^; order 96) ⊂ stereoisomeric group (O~hσ~I^(=SσI^[6]); order 1440) = isoskeletal group (O~~hσ~I^(=SσI^[6]); order 1440). The corresponding cycle indices with chirality fittingness (CI-CFs) are calculated by using combined-permutation representations (Fujita in MATCH Commun Math Comput Chem 76:379-400, 2016). Then, a set of three ligand-inventory functions for 3D enumeration is introduced into the CI-CFs for the enumerations under the point groups and the RS-stereoisomeric group, while a single ligand-inventory function for 2D (graph) enumeration is introduced into the CI-CFs for the enumerations under the stereoisomeric group and the isoskeletal group. The expansion of the resulting equations gives generating functions, in which the coefficients of respective terms show the numbers of octahedral derivatives. They are discussed by drawing isomer-classification diagrams after they are categorized into octahedral derivatives with achiral proligands, those with achiral and chiral proligands, and those with chiral proligands. Type-I and type-V stereoisograms are drawn to demonstrate the conceptual distinction between RS-stereogenicity and stereogenicity, where the combination of configuration indices and C/A-descriptors is discussed. [ABSTRACT FROM AUTHOR]

With the increasing of the complexity of a system, there is a variety of indeterminacy in the practical applications of graph theory. We focus on uncertain random graph, in which some edges exist with degrees in probability measure and others exist with degrees in uncertain measure. In this paper, the chance theory is applied to construct the cycle index of an uncertain random graph. Then a method to calculate the cycle index of an uncertain random graph is presented. We also discuss some properties of the cycle index. [ABSTRACT FROM AUTHOR]

Information inequalities can be used to derive the fundamental limits of information systems. Many information inequalities and problem-specific constraints are linear equalities or inequalities of joint entropies, and thus, outer bounding the fundamental limits can be viewed as and in principle computed through linear programming. However, for many practical engineering problems, the resultant linear program (LP) is very large, rendering such a computational approach almost completely inapplicable in practice. It was shown recently that symmetry can be used to effectively reduce the scale of the LP; however, the precise amount of reduction was not well understood. In this paper, we provide a method to pinpoint this reduction by counting the number of orbits induced by the symmetry on the set of the LP variables and the LP constraints, respectively. The Pólya counting theorem is a powerful tool for such counting task, which requires identifying the cycle index function. We propose a generic three-layer decomposition of the group structures for the quantities in typical information systems to facilitate such a calculation. Three problems are studied using this approach: extremal pairwise cyclically symmetric entropy inequalities, the regenerating code problem, and the caching problem, for which explicit formulas are provided for the cycle indices of the induced permutations. [ABSTRACT FROM AUTHOR]

This paper expounds a technique for calculating unsteady gas temperatures, air and heat transfer elements of the corrugated plate of a recuperative heat exchanger with complanar channels for start and shut-off modes of the regenerative turbine engine. Under consideration are the distribution of thermal stresses in the plate, heat exchanger service life under stationary conditions, and the allowable number of dynamic 'start-shut-off' cycles. [ABSTRACT FROM AUTHOR]

A derangement is a permutation that has no fixed points. In this paper, we are interested in the proportion of derangements of the finite affine general linear groups. We prove a remarkably simple and explicit formula for this proportion. We also give a formula for the proportion of derangements of prime power order. Both formulae rely on a result of independent interest on partitions: we determine the generating function for the partitions with m parts and with the kth largest part not k, for every $$k\in \mathbb {N}$$ . [ABSTRACT FROM AUTHOR]

Hetero-fullerenes are fullerenes in which some of the carbon atoms are replaced by other atoms. This paper uses the Pólya theorem to count the number of their possible positional isomers and chiral isomers. To do this, the computer algebra system GAP was applied to compute this number for a class of IPR hetero-fullerenes with Ih point group symmetry. These fullerenes were constructed by means of the leapfrog principle.

We consider the group action of the automorphism group Un = Aut(Zn) on the set Zn, that is the set of residue classes modulo n. Clearly, this group action provides a representation of Un as a permutation group acting on n points. One problem to be solved regarding this group action is to find its cycle index. Once it is found, there appears a vast class of related enumerative and computational problems with interesting applications. We provide the cycle index of specified group action in two ways. One of them is more abstract and hence compact, while another one is basically a procedure of composing the cycle index from some building blocks. However, those building blocks are also well explained and finally presented in a very detailed fashion. [ABSTRACT FROM AUTHOR]

In 1937, George Pólya wrote what is considered one of the most significant papers in 20th-century mathematics. The article contained one theorem and 100 pages of applications. It introduced a combinatorial method which led to unexpected applications to diverse problems in science like the enumeration of isomers of chemical compounds. Pólya's theory of enumeration was discussed in detail in [1] by Shriya Anand, a summer student of the author. In what follows, we briefly recall the theory and complement the earlier article by adding some other applications not discussed there. [ABSTRACT FROM AUTHOR]

The objective of this research was to determine the kinematic variables evolution in a sprint canoeing maximal test over 200 m, comparing women and men kayak paddlers and men canoeists. Speed evolution, cycle frequency, cycle length and cycle index were analysed each 50 m section in fifty-two young paddlers (20 male kayakers, 17 female kayakers and 15 male canoeists; 13-14 years-old). Recordings were taken from a boat which followed each paddler trial in order to measure the variables cited above. Kinematic evolution was similar in the three categories, the speed and cycle index decreased through the test after the first 50 m. Significant differences were observed among most of the sections in speed and the cycle index (p<0.05 and <0.01, respectively). Cycle length remained stable showing the lowest values in the first section when compared with the others (p<0.01). Cycle frequency progressively decreased along the distance. Significant differences were identified in the majority of the sections (p<0.01). Men kayakers attained higher values in all the variables than women kayakers and men canoeists, but only such variables as speed, cycle length and cycle index were observed to be significantly higher (p<0.01). Moreover, lower kinematic values were obtained from men canoeists. The study of the evolution of kinematic variables can provide valuable information for athletes and coaches while planning training sessions and competitions. [ABSTRACT FROM AUTHOR]

Enumeration of chemical compounds has been accomplished by various methods. The Polya-Redfield theorem has been a standard method for combinatorial enumerations of graphs, polyhedra, chemical compounds and so forth. Hetero-fullerenes are fullerene molecules in which one or more carbon atoms are replaced by hetero-atoms such as boron or nitrogen. In this paper, by using the Pólya theorem, we compute the number of permutational isomers of some fullerene graphs. [ABSTRACT FROM PUBLISHER]

Combinatorial analysis methods are employed to solve the problem of determining the number and form of X-substituted (X, XY, ... are some substituents) of carbometallic derivatives of boranes (CH)Co(CBH) C and (CH)Co(CBH) C (over vertices) based on Polya's theorem. formulas of Z symmetry and generating functions of the number of chiral and achiral substitution stereoisomers are determined. Family distributions of the isomers depending on the form and number of substituents and depending on the number m of possible substitution sites are found. Mono-, di-, and tri-X-substituted (X = CH, F, ...) isomers of (CH)Co(CBH) C are identified. Based on the partition of simple ( n) and triangular numbers ( K) of Pascal's triangle, additive schemes are obtained which take into account valence and non-valence pair interactions of atoms in the polyhedron framework and contain 2, 6, and 23 parameters for the calculation of properties of X-substituted borane (CH)Co(CBH) C. [ABSTRACT FROM AUTHOR]

We develop cycle index generating functions for orthogonal groups in even characteristic and give some enumerative applications. A key step is the determination of the values of the complex linear-Weil characters of the finite symplectic group, and their induction to the general linear group, at unipotent elements. We also define and study several natural probability measures on integer partitions. [ABSTRACT FROM AUTHOR]

The problem of determining the number and type of X-substituted (where X represents certain substituents) carbometallic derivatives of hydroborons [Co(CBH)] C and [Co(CBH)] D (from apexes) was solved on the basis of G. Pólya's theorem by means of combinatory analysis. The formulae of symmetry Z and generating functions of the number of achiral substitution isomers were determined. The family distributions of isomers (depending on the type and number of substituents) and the sites of possible substitution depending on the number m were determined. Mono and di-X-substituted isomers of [Co(CBH)] C, as well as mono and di-X-substituted isomers of [Co(CBH)] D, were identified. The procedure for plotting an additive model of calculating the properties of isomers of apical substitution of [Co(CBH)] D was described on the basis of splitting of polygonal numbers ( K, K, K, and so on) of Pascal triangle, upon use of which there is no randomization in the choice of parameters of calculation model. The additive schemes containing 7 and 25 parameters for the calculation of properties of X-substituted carbometallic derivative of hydroboron [Co(CBH)] D to various approximations were obtained. [ABSTRACT FROM AUTHOR]

Hetero-fullerenes are fullerenes in which some of the carbon atoms are replaced by other atoms. This paper uses the Pólya theorem to count the number of their possible positional isomers and chiral isomers. To do this, the computer algebra system GAP was applied to compute this number for a class of IPR hetero-fullerenes with Ih point group symmetry. These fullerenes were constructed by means of the leapfrog principle. [ABSTRACT FROM AUTHOR]