Search results for: prime field

Authors: Tun Myat Aung, Ni Ni Hla

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This paper begins by describing basic properties of finite field and elliptic curve cryptography over prime field and binary field. Then we discuss the discrete logarithm problem for elliptic curves and its properties. We study the general common attacks on elliptic curve discrete logarithm problem such as the Baby Step, Giant Step method, Pollard’s rho method and Pohlig-Hellman method, and describe in detail experiments of these attacks over prime field and binary field. The paper finishes by describing expected running time of the attacks and suggesting strong elliptic curves that are not susceptible to these attacks.c

Keywords: discrete logarithm problem, general attacks, elliptic curve, prime field, binary field

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Authors: Shai Sarussi

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The union of a nonempty chain of prime ideals in a noncommutative ring is not necessarily a prime ideal. An ideal is called union-prime if it is a union of a nonempty chain of prime ideals but is not a prime. In this paper, some relations between chains of prime ideals and the induced chains of union-prime ideals are shown; in particular, the cardinality of such chains and the cardinality of the sets of cuts of such chains are discussed. For a ring R and a nonempty full chain of prime ideals C of R, several characterizations for the property of immediate neighbors in C are given.

Keywords: prime ideals, union-prime ideals, immediate neighbors, Kaplansky's conjecture

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Authors: Anthony Overmars, Sitalakshmi Venkatraman

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This paper presents a method for determining all of the co-prime right angle triangles in the Euclidean field by looking at the intersection of the Pythagorean and Platonic right angle triangles and the corresponding lattice that this produces. The co-prime properties of each lattice point representing a unique right angle triangle are then considered. This paper proposes a conjunction between these two ancient disparaging theorists. This work has wide applications in information security where cryptography involves improved ways of finding tuples of prime numbers for secure communication systems. In particular, this paper has direct impact in enhancing the encryption and decryption algorithms in cryptography.

Keywords: Pythagorean triples, platonic triples, right angle triangles, co-prime numbers, cryptography

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Authors: Jerry Hu

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Let G = (V, E) with V = {1, 2, ..., k} be a graph, the k positive integers a₁, a₂, ..., ak are G-wise relatively prime if (aᵢ, aⱼ ) = 1 for {i, j} ∈ E. We use an inductive approach to give an asymptotic formula for the number of k-tuples of integers that are G-wise relatively prime. An exact formula is obtained for the probability that k positive integers are G-wise relatively prime. As a corollary, we also provide an exact formula for the probability that k positive integers have exactly r relatively prime pairs.

Keywords: graph, independent set, G-wise relatively prime, probability

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Authors: Ying Yi Wu

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The study of finite groups is a central topic in algebraic structures, and one of the most fundamental questions in this field is the classification of finite groups up to isomorphism. In this paper, we investigate the cyclic property of groups of prime order, which is a crucial result in the classification of finite abelian groups. We prove the following statement: If p is a prime, then every group G of order p is cyclic. Our proof utilizes the properties of group actions and the class equation, which provide a powerful tool for studying the structure of finite groups. In particular, we first show that any non-identity element of G generates a cyclic subgroup of G. Then, we establish the existence of an element of order p, which implies that G is generated by a single element. Finally, we demonstrate that any two generators of G are conjugate, which shows that G is a cyclic group. Our result has significant implications in the classification of finite groups, as it implies that any group of prime order is isomorphic to the cyclic group of the same order. Moreover, it provides a useful tool for understanding the structure of more complicated finite groups, as any finite abelian group can be decomposed into a direct product of cyclic groups. Our proof technique can also be extended to other areas of group theory, such as the classification of finite p-groups, where p is a prime. Therefore, our work has implications beyond the specific result we prove and can contribute to further research in algebraic structures.

Keywords: group theory, finite groups, cyclic groups, prime order, classification.

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Authors: Yingxu Wang

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This work presents basic research on the recursive structures and dual sequential patterns of primes for the formal proof of the Twin-Prime Conjecture (TPC). A rigorous methodology of Twin-Prime Decomposition (TPD) is developed in MATLAB to inductively verify potential twins in the dual sequences of primes. The key finding of this basic study confirms that the dual sequences of twin primes are not only symmetric but also infinitive in the unique base 6 cycle, except a limited subset of potential pairs is eliminated by the lack of dual primality. Both theory and experiments have formally proven that the infinity of twin primes stated in TPC holds in the P x P space.

Keywords: number theory, primes, twin-prime conjecture, dual primes (P x P), twin prime decomposition, formal proof, algorithm

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Authors: Elvira Kusniyanti, Hanni Garminia, Pudji Astuti

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This research studies an interconnection between finitely generated uniform modules and Dedekind prime rings. The characterization of modules over Dedekind prime rings that will be investigated is an adoption of Noetherian and hereditary concept. Dedekind prime rings are Noetherian and hereditary rings. This property of Dedekind prime rings is a background of the idea of adopting arises. In Noetherian area, it was known that a ring R is Noetherian ring if and only if every finitely generated R-module is a Noetherian module. Similar to that result, a characterization of the hereditary ring is related to its projective modules. That is, a ring R is hereditary ring if and only if every projective R-module is a hereditary module. Due to the above two results, we suppose that characterization of a Dedekind prime ring can be analyzed from finitely generated modules over it. We propose a conjecture: a ring R is a Dedekind prime ring if and only if every finitely generated uniform R-module is a Dedekind module. In this article, we will generalize a concept of the Dedekind module for non-commutative ring case and present a part of the above conjecture.

Keywords: dedekind domains, dedekind prime rings, dedekind modules, uniform modules

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Authors: Nidal Ali

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Consider an algebraic number field K of degree n, A0 K is its ring of integers and a prime number p inert in K. Let F(u1, . . . , un, x) be the generic polynomial of integers of K. We will study in advance the stability of this polynomial and then, we will apply it in order to obtain all the monic irreducible polynomials in Fp[x] of degree d dividing n.

Keywords: generic polynomial, irreducibility, iteration, stability, inert prime, totally ramified

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Authors: Shai Sarussi

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Suppose F is a field with valuation v and valuation domain Oᵥ, and R is an Oᵥ-algebra. It is known that there exists a filter quasi-valuation on R; the existence of a quasi-valuation yields several important connections between Oᵥ and R, in particular with respect to their prime spectra. In this paper, the notion of a module valuation is introduced. It is shown that any torsion-free module over Oᵥ has an induced module valuation. Moreover, several results connecting the filter quasi-valuation and module valuations are presented.

Keywords: valuations, quasi-valuations, prime spectrum, algebras over valuation domains

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Authors: Hashem Sazegar

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In 1937, Vinograd of Russian Mathematician proved that each odd large number can be shown by three primes. In 1973, Chen Jingrun proved that each odd number can be shown by one prime plus a number that has maximum two primes. In this article, we state one proof for Goldbach’conjecture. Introduction: Bertrand’s postulate state for every positive integer n, there is always at least one prime p, such that n < p < 2n. This was first proved by Chebyshev in 1850, which is why postulate is also called the Bertrand-Chebyshev theorem. Legendre’s conjecture states that there is a prime between n2 and (n+1)2 for every positive integer n, which is one of the four Landau’s problems. The rest of these four basic problems are; (i) Twin prime conjecture: There are infinitely many primes p such that p+2 is a prime. (ii) Goldbach’s conjecture: Every even integer n > 2 can be written asthe sum of two primes. (iii) Are there infinitely many primes p such that p−1 is a perfect square? Problems (i), (ii), and (iii) are open till date.

Keywords: Bertrand-Chebyshev theorem, Landau’s problems, twin prime, Legendre’s conjecture, Oppermann’s conjecture

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Authors: Gurjit Kaur, Shashank Johri, Arpit Mehrotra

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In this paper we have analyzed and compared the performance of various coding schemes. The basic ID prime sequence codes are unique in only dimension i.e. time slots whereas 2D coding techniques are not unique by their time slots but with their wavelengths also. In this research we have evaluated and compared the performance of 1D and 2D coding techniques constructed using prime sequence coding pattern for OCDMA system on a single platform. Results shows that 1D Extended Prime Code (EPC) can support more number of active users compared to other codes but at the expense of larger code length which further increases the complexity of the code. Modified Prime Code (MPC) supports lesser number of active users at λc=2 but it has a lesser code length as compared to 1D prime code. Analysis shows that 2D prime code supports lesser number of active users than 1D codes but they are having large code family and are the most secure codes compared to other codes. The performance of all these codes is analyzed on basis of number of active users supported at a Bit Error Rate (BER) of 10-9.

Keywords: CDMA, OCDMA, BER, OOC, PC, EPC, MPC, 2-D PC/PC, λc, λa

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Authors: A. I. Augie, Y. Alhassan, U. Z. Magawata

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Geophysical investigation was carried out at wacot rice factory Argungu north-western Nigeria, using the 2D electrical resistivity method. The area falls between latitude 12˚44′23ʺN to 12˚44′50ʺN and longitude 4032′18′′E to 4032′39′′E covering a total area of about 1.85 km. Two profiles were carried out with Wenner configuration using resistivity meter (Ohmega). The data obtained from the study area were modeled using RES2DIVN software which gave an automatic interpretation of the apparent resistivity data. The inverse resistivity models of the profiles show the high resistivity values ranging from 208 Ωm to 651 Ωm. These high resistivity values in the overburden were due to dryness and compactness of the strata that lead to consolidation, which is an indication that the area is free from leachate contaminations. However, from the inverse model, there are regions of low resistivity values (1 Ωm to 18 Ωm), these zones were observed and identified as clayey and the most contaminated zones. The regions of low resistivity thereby indicated the leachate plume or the highly leachate concentrated zones due to similar resistivity values in both clayey and leachate. The regions of leachate are mainly from the factory into the surrounding area and its groundwater. The maximum leachate infiltration was found at depths 1 m to 15.9 m (P1) and 6 m to 15.9 m (P2) vertically, as well as distance along the profiles from 67 m to 75 m (P1), 155 m to 180 m (P1), and 115 m to 192 m (P2) laterally.

Keywords: contaminant, leachate, soil, groundwater, electrical, resistivity

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Authors: Abdelkarim Boua, Abdelhakim Chillali

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The study of non-associative structures in algebraic structures has become a separate entity; for, in the case of groups, their corresponding non-associative structure i.e. loops is dealt with separately. Similarly there is vast amount of research on the nonassociative structures of semigroups i.e. groupoids and that of rings i.e. nonassociative rings. However it is unfortunate that we do not have a parallel notions or study of non-associative near-rings. In this work we shall attempt to generalize a few known results and study the commutativity of Jordan ideal in 3-prime near-rings satisfying certain identities involving the Jordan ideal. We study the derivations satisfying certain differential identities on Jordan ideals of 3-prime near-rings. Moreover, we provide examples to show that hypothesis of our results are necessary. We give some new results and examples concerning the existence of Jordan ideal and derivations in near-rings. These near-rings can be used to build a new codes.

Keywords: 3-prime near-rings, near-rings, Jordan ideal, derivations

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Authors: S. Dey, P. Sibanda, S. Gupta, A. Chakraborty

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The occurrence of high nocturnal surface ozone over a tropical urban area (23̊ 32′16.99″ N and 87̊ 17′ 38.95″ E) is analyzed in this paper. Five incidences of nocturnal ozone maxima are recorded during the observational span of two years (June, 2013 to May, 2015). The maximum and minimum values of the surface ozone during these five occasions are 337.630 μg/m³ and 13.034 μg/m³ respectively. HYSPLIT backward trajectory analyses and wind rose diagrams support the horizontal transport of ozone from distant polluted places. Planetary boundary layer characteristics, concentration of precursor (NO₂) and meteorology are found to play important role in the horizontal and vertical transport of surface ozone during nighttime.

Keywords: nocturnal ozone, planetary boundary layer, horizontal transport, meteorology, urban area

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Authors: Gurjit Kaur, Neena Gupta

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In this paper, we have analyzed and compared the performance of various coding schemes. The basic ID prime sequence codes are unique in only dimension, i.e. time slots, whereas 2D coding techniques are not unique by their time slots but with their wavelengths also. In this research, we have evaluated and compared the performance of 1D and 2D coding techniques constructed using prime sequence coding pattern for Optical Code Division Multiple Access (OCDMA) system on a single platform. Analysis shows that 2D prime code supports lesser number of active users than 1D codes, but they are having large code family and are the most secure codes compared to other codes. The performance of all these codes is analyzed on basis of number of active users supported at a Bit Error Rate (BER) of 10-9.

Keywords: CDMA, OCDMA, BER, OOC, PC, EPC, MPC, 2-D PC/PC, λc, λa

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Authors: Aaminah Hassan

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Advertisements carry a great potential to influence our lives because they are crafted to meet particular ends. Stereotypical representation in advertisements is capable of forming unconscious attitudes among people towards any gender and their abilities. This study focuses on gender representation in Pakistani prime time advertisements. For this purpose, 13 advertisements were selected from three different categories of foods and beverages, cosmetics, cell phones and cellular networks from the prime time slots of one of the leading Pakistani entertainment channel, ‘Urdu 1’. Both quantitative and qualitative analyses are carried out for range of variables like gender, age, roles, activities, setting, appearance and voice overs. The results revealed that gender representation in advertisements is stereotypical. Moreover, in few instances, the portrayal of women is not only culturally inappropriate but is demeaning to the image of women as well. Their bodily charm is used to promote products. Comparing different entertainment channels for their prime time advertisements and broadening the scope of this research will yield greater implications for the researchers who want to carry out the similar research. It is hoped that the current study would help in the promotion of media literacy among the viewers and media authorities in Pakistan.

Keywords: Advertisements, Content Analysis, Gender, Prime time

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Authors: Hashem Sazegar

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Legendre’s conjecture states that there is a prime number between n^2 and (n + 1)^2 for every positive integer n. In this paper we prove that every composite number between n2 and (n + 1)2 can be written u^2 − v^2 or u^2 − v^2 + u − v that u > 0 and v ≥ 0. Using these result as well as induction and residues (modq) we prove Legendre’s conjecture.

Keywords: bertrand-chebyshev theorem, landau’s problems, goldbach’s conjecture, twin prime, ramanujan proof

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Authors: Saurabh Rawat, Anushree Sah

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K Maps are generally and ideally, thought to be simplest form for obtaining solution of Boolean equations. Cubical Representation of Boolean equations is an alternate pick to incur a solution, otherwise to be meted out with Truth Tables, Boolean Laws, and different traits of Karnaugh Maps. Largest possible k- cubes that exist for a given function are equivalent to its prime implicants. A technique of minimization of Logic functions is tried to be achieved through cubical methods. The main purpose is to make aware and utilise the advantages of cubical techniques in minimization of Logic functions. All this is done with an aim to achieve minimal cost solution.r

Keywords: K-maps, don’t care conditions, Boolean equations, cubes

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Authors: Shiji Hu, Lei Li, Wanting Zhou, DaoHong Yang

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The data encryption, is the foundation of today’s communication. On this basis, how to improve the speed of data encryption and decryption is always a problem that scholars work for. In this paper, we proposed an elliptic curve crypto processor architecture based on SM2 prime field. In terms of hardware implementation, we optimized the algorithms in different stages of the structure. In finite field modulo operation, we proposed an optimized improvement of Karatsuba-Ofman multiplication algorithm, and shorten the critical path through pipeline structure in the algorithm implementation. Based on SM2 recommended prime field, a fast modular reduction algorithm is used to reduce 512-bit wide data obtained from the multiplication unit. The radix-4 extended Euclidean algorithm was used to realize the conversion between affine coordinate system and Jacobi projective coordinate system. In the parallel scheduling of point operations on elliptic curves, we proposed a three-level parallel structure of point addition and point double based on the Jacobian projective coordinate system. Combined with the scalar multiplication algorithm, we added mutual pre-operation to the point addition and double point operation to improve the efficiency of the scalar point multiplication. The proposed ECC hardware architecture was verified and implemented on Xilinx Virtex-7 and ZYNQ-7 platforms, and each 256-bit scalar multiplication operation took 0.275ms. The performance for handling scalar multiplication is 32 times that of CPU(dual-core ARM Cortex-A9).

Keywords: Elliptic curve cryptosystems, SM2, modular multiplication, point multiplication.

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Authors: William Wong, Zakaria Alomari, Hon Ching Lai, Zhida Li

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Modern-day information security depends on implementing Diffie-Hellman, which requires the generation of prime numbers. Because the number of primes is infinite, it is impractical to store prime numbers for use, and therefore, primality tests are indispensable in modern-day information security. A primality test is a test to determine whether a number is prime or composite. There are two types of primality tests, which are deterministic tests and probabilistic tests. Deterministic tests are adopting algorithms that provide a definite answer whether a given number is prime or composite. While in probabilistic tests, a probabilistic result would be provided, there is a degree of uncertainty. In this paper, we review three probabilistic tests: the Fermat Primality Test, the Miller-Rabin Test, and the Baillie-PSW Test, as well as one deterministic test, the Agrawal-Kayal-Saxena (AKS) Test. Furthermore, we do an analysis of these tests. All of the reviews discussed are not based on the Elliptic Curve. The analysis demonstrates that, in the majority of real-world scenarios, the Baillie- PSW test’s favorability stems from its typical operational complexity of O(log 3n) and its capacity to deliver accurate results for numbers below 2^64.

Keywords: primality tests, Fermat’s primality test, Miller-Rabin primality test, Baillie-PSW primality test, AKS primality test

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Authors: Mohammad Java Esmaeily

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Background: We evaluate the effects of tissue and organ perfusion during and after coronary artery bypass graft surgery with either colloid (Voluven) or crystalloid (Lactated ringers) as a prime solution. Materials and Methods: In this prospective randomized-controlled trial study, 70 patients undergoing on-pump coronary artery bypass graft surgery were randomly assigned to receive either colloid (Voluven) or crystalloid (Lactated ringer's) as a prime solution for initiation of cardiopulmonary bypass machine procedure. Tissue and organ perfusion markers, including lactate, troponin I, liver and renal function tests and electrolytes, were measured sequentially before induction (T1) to the second days after surgery (T5). Results: With the exception of chloride and potassium levels, no significant differences were detected in other measurements, and laboratory results were identical entirely in the two groups. Conclusion: Voluven® (hydroxyethyl starch, HES 130/0.4) has a not significant difference in comparison with crystalloid (Lactated ringer's) as priming solution on the basis of organ and tissue perfusion tests assessment.

Keywords: prime, colloid, crystalloid, lactate, troponin, hydroxyethyl starch

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Authors: Shai Sarussi

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Let S be an integral domain which is not a field, let F be its field of fractions, and let A be an F-algebra. An S-subalgebra R of A is called S-nice if R ∩ F = S and F R = A. A topological space whose set of open sets is closed under arbitrary intersections is called an Alexandroff space. Inspired by the well-known Zariski-Riemann space and the Zariski topology on the set of prime ideals of a commutative ring, we define a topology on the set of all S-nice subalgebras of A. Consequently, we get an interplay between Algebra and topology, that gives us a better understanding of the S-nice subalgebras of A. It is shown that every irreducible subset of S-nice subalgebras of A has a supremum; and a characterization of the irreducible components is given, in terms of maximal S-nice subalgebras of A.

Keywords: Alexandroff topology, integral domains, Zariski-Riemann space, S-nice subalgebras

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Authors: Bingqing Xu, Wenxing Shuai

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This study investigates the priming effects of different Chinese compound words by native Mandarin speakers. There are lots of homonym, polysemy, and synonym in Chinese. However, it is unclear which kind of words have the biggest priming effect. Native Mandarin speakers were tested in a visual-word lexical decision experiment. The stimuli, which are all two-character compound words, consisted of two parts: primes and targets. Five types of relationships were used in all stimuli: morphologically related condition, in which the prime and the target contain the same morpheme; orthographically related condition, in which the target and the prime contain the different morpheme with the same form; phonologically related condition, in which the target and the prime contain the different morpheme with the same phonology; semantically related condition, in which the target and the prime contain the different morpheme with similar meanings; totally unrelated condition. The time since participants saw the target to respond was recorded. Analyses on reaction time showed that the average reaction time of morphologically related targets was much shorter than others, suggesting the morphological priming effect is the biggest. However, the reaction time of the phonologically related conditions was the longest, even longer than unrelated conditions. According to scatter plots analyses, 86.7% of participants had priming effects in morphologically related conditions, only 20% of participants had priming effects in phonologically related conditions. These results suggested that morphologically related conditions had the biggest priming effect. The orthographically and semantically related conditions also had priming effects, whereas the phonologically related conditions had few priming effects.

Keywords: priming effect, morphology, phonology, semantics, orthography

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Authors: Vedansh More, Lalatendu Bal, Ronak Panchal, Atharva Kulkarni

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The project aims at developing a cost-effective test bench capable of testing and validating the complete powertrain package of an electric vehicle. Emrax 228 high voltage synchronous motor was selected as the prime mover for study. A tandem type dynamometer comprising of two loading methods; inertial, using standard inertia rollers and absorptive, using a separately excited DC generator with resistive coils was developed. The absorptive loading of the prime mover was achieved by implementing a converter circuit through which duty of the input field voltage level was controlled. This control was efficacious in changing the magnetic flux and hence the generated voltage which was ultimately dropped across resistive coils assembled in a load bank with all parallel configuration. The prime mover and loading elements were connected via a chain drive with a 2:1 reduction ratio which allows flexibility in placement of components and a relaxed rating of the DC generator. The development will aid in determination of essential characteristics like torque-RPM, power-RPM, torque factor, RPM factor, heat loads of devices and battery pack state of charge efficiency but also provides a significant financial advantage over existing versions of dynamometers with its cost-effective solution.

Keywords: absorptive load, chain drive, chordal action, DC generator, dynamometer, electric vehicle, inertia rollers, load bank, powertrain, pulse width modulation, reduction ratio, road load, testbench

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Authors: Juraj Belan, Lenka Hurtalová, Eva Tillová, Alan Vaško, Milan Uhríčik

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High-pressure turbine (HPT) blades of DV–2 jet engines are made from Ni–base superalloy, a former Soviet Union production, specified as ŽS6K. For improving its high-temperature resistance are blades covered with Al–Si diffusion layer. A regular operation temperature of HPT blades vary from 705°C to 750°C depending on jet engine regime. An over-crossing working temperature range causes degradation of protective alitize layer as well as base material–gamma matrix and gamma prime particles what decreases turbine blade lifetime. High-temperature degradation has mainly diffusion mechanism and causes coarsening of strengthening phase gamma prime and protective alitize layer thickness growing. All changes have a significant influence on high-temperature properties of base material.

Keywords: alitize layer, gamma prime phase, high-temperature degradation, Ni–base superalloy ŽS6K, turbine blade

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Authors: Shai Sarussi

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Let S be an integral domain with field of fractions F and let A be an F-algebra. An S-subalgebra R of A is called S-nice if R∩F = S and the localization of R with respect to S \{0} is A. Denoting by W the set of all S-nice subalgebras of A, and defining a notion of open sets on W, one can view W as a T0-Alexandroff space. Thus, the algebraic structure of W can be viewed from the point of view of topology. It is shown that every nonempty open subset of W has a maximal element in it, which is also a maximal element of W. Moreover, a supremum of an irreducible subset of W always exists. As a notable connection with valuation theory, one considers the case in which S is a valuation domain and A is an algebraic field extension of F; if S is indecomposed in A, then W is an irreducible topological space, and W contains a greatest element.

Keywords: integral domains, Alexandroff topology, prime spectrum of a ring, valuation domains

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Authors: Nadya Inda Syartanti

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This research aims to identify, classify, and analyze descriptively the aspects of grammatical and lexical cohesion in the speech text of Japan’s Prime Minister Shinzo Abe entitled Nihon wa Modotte kimashita delivered in Washington DC, the United States on February 23, 2013, as a research data source. The method used is qualitative research, which uses descriptions through words that are applied by analyzing aspects of grammatical and lexical cohesion proposed by Halliday and Hasan (1976). The aspects of grammatical cohesion consist of references (personal, demonstrative, interrogative pronouns), substitution, ellipsis, and conjunction. In contrast, lexical cohesion consists of reiteration (repetition, synonym, antonym, hyponym, meronym) and collocation. Data classification is based on the 6 aspects of the cohesion. Through some aspects of cohesion, this research tries to find out the frequency of using grammatical and lexical cohesion in Shinzo Abe's speech text entitled Nihon wa Modotte kimashita. The results of this research are expected to help overcome the difficulty of understanding speech texts in Japanese. Therefore, this research can be a reference for learners, researchers, and anyone who is interested in the field of discourse analysis.

Keywords: cohesion, grammatical cohesion, lexical cohesion, speech text, Shinzo Abe

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Authors: Jawad Abuhlail, Hamza Hroub

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Let R be a commutative ring. Representation theory studies the representation of R-modules as (possibly finite) sums of special types of R-submodules. Here we are interested in a class of R-modules between the class of semisimple R-modules and the class of R-modules that can be written as (possibly finite) sums of secondary R-submodules (we know that every simple R-submodule is secondary). We investigate R-modules which can be written as (possibly finite) sums of second R-submodules (we call those modules second representable). Moreover, we investigate the class of (main) second attached prime ideals related to a module with such representation. We provide sufficient conditions for an R-module M to get a (minimal) second representation. We also found the collection of second attached prime ideals for some types of second representable R-modules, in particular within the class of injective R-modules. As we know that every simple R-submodule is second and every second R-submodule is secondary, we can see the importance of the second representable R-module.

Keywords: lifting modules, second attached prime ideals, second representations, secondary representations, semisimple modules, second submodules

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Authors: Hossein Pourali

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In this paper, we extend the concepts of primal and weakly primal ideals for posets. Further, the diameter of the zero divisor graph of a poset with respect to a non-primal ideal is determined. The relation between primary and primal ideals in posets is also studied.

Keywords: ‎associated prime ideal, ‎‎ideal, ‎‎primary ideal, primal ideal‎, prime‎ ‎ideal, semiprime ideal, ‎weakly primal ideal, zero divisors graph

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Authors: Djalal Boualili

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This article considers the problem of sizing prime movers for combined heating and power (CHP) systems operating at full load to satisfy a fraction of a facility's electric load, i.e. a base load. Prime mover sizing is examined using three criteria: operational cost, carbon dioxide emissions (CDE), and primary energy consumption (PEC). The sizing process leads to consider ratios of conversion factors applied to imported electricity to conversion factors applied to fuel consumed. These ratios are labelled RCost, R CDE, R PEC depending on whether the conversion factors are associated with operational cost, CDE, or PEC, respectively. Analytical results show that in order to achieve savings in operational cost, CDE, or PEC, the ratios must be larger than a unique constant R Min that only depends on the CHP components efficiencies. Savings in operational cost, CDE, or PEC due to CHP operation are explicitly formulated using simple equations. This facilitates the process of comparing the tradeoffs of optimizing the savings of one criterion over the other two – a task that has traditionally been accomplished through computer simulations. A hospital building, located in Chlef, Algeria, was used as an example to apply the methodology presented in this article.

Keywords: sizing, heating and power, ratios, energy consumption, carbon dioxide emissions

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