Search results for: primes

Authors: Yingxu Wang

Abstract:

The base 6 structure and properties of mirror primes are discovered in this work towards the proof of Goldbach Conjecture. This paper reveals a fundamental pattern on pairs of mirror primes adjacent to any even number nₑ > 2 with symmetrical distances on both sides determined by a methodology of Mirror Prime Decomposition (MPD). MPD leads to a formal proof of the Goldbach conjecture, which states that the conjecture holds because any pivot even number, nₑ > 2, is a sum of at least an adjacent pair of primes divided by 2. This work has not only revealed the analytic pattern of base 6 primes but also proven the infinitive validation of the Goldbach conjecture.

Keywords: number theory, primes, mirror primes, double recursive patterns, Goldbach conjecture, formal proof, mirror-prime decomposition, applications

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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: Benjamin Lee Warren

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Many problems exist in additive number theory which is essential to determine the primes that are the sum of two elements from a given single-variable polynomial sequence, and most of them are unattackable in the present day. Here, we determine solutions for this problem to a few certain sequences (certain binomial coefficients and power sums) using only elementary algebra and some algebraic factoring methods (as well as Euclid’s Lemma and Faulhaber’s Formula). In particular, we show that there are finitely many primes as sums of two of these types of elements. Several cases are fully illustrated, and bounds are presented for the cases not fully illustrated.

Keywords: binomial coefficients, power sums, primes, algebra

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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: Chanon Promsakon, Sayan Panma

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Let S be a finite semigroup, A a subset of S and f an endomorphism on S. The endo-Cayley digraph of a semigroup S corresponding to a connecting set A and an endomorphism f, denoted by endo − Cayf (S, A) is a digraph whose vertex set is S and a vertex u is adjacent to a vertex v if and only if v = f(u)a for some a ∈ A. A digraph D is called undirected if any edge uv in D, there exists an edge vu in D. We consider the undirectedness of an endo-Cayley of a cyclic group of order prime, Zp. In this work, we investigate conditions for connecting sets and endomorphisms to make endo-Cayley digraphs of cyclic groups of order primes be undirected. Moreover, we give some conditions for an undirected endo-Cayley of cycle group of any order.

Keywords: endo-Cayley graph, undirected digraphs, cyclic groups, endomorphism

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Authors: Ali Al Moussaoui

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An experiment was designed to assess the performance of 24 Lebanese adults (mean age 29:5 years) in a lexical decision making (LDM) task to find out how the facilitatory effect of phonological priming (PP) affects the speed of visual word recognition in Arabic as lexicality (wordhood) and phonological overlap positions (POP) vary. The experiment falls in line with previous research on phonological priming in the light of the cohort theory and in relation to visual word recognition. The experiment also departs from the research on the Arabic language in which the importance of the consonantal root as a distinct morphological unit is confirmed. Based on previous research, it is hypothesized that (1) PP has a facilitating effect in LDM with words but not with nonwords and (2) final phonological overlap between the prime and the target is more facilitatory than initial overlap. An LDM task was programmed on PsychoPy application. Participants had to decide if a target (e.g., bayn ‘between’) preceded by a prime (e.g., bayt ‘house’) is a word or not. There were 4 conditions: no PP (NP), nonwords priming nonwords (NN), nonwords priming words (NW), and words priming words (WW). The conditions were simultaneously controlled for word length, wordhood, and POP. The interstimulus interval was 700 ms. Within the PP conditions, POP was controlled for in which there were 3 overlap positions between the primes and the targets: initial (e.g., asad ‘lion’ and asaf ‘sorrow’), final (e.g., kattab ‘cause to write’ 2sg-mas and rattab ‘organize’ 2sg-mas), or two-segmented (e.g., namle ‘ant’ and naħle ‘bee’). There were 96 trials, 24 in each condition, using a within-subject design. The results show that concerning (1), the highest average reaction time (RT) is that in NN, followed firstly by NW and finally by WW. There is statistical significance only between the pairs NN-NW and NN-WW. Regarding (2), the shortest RT is that in the two-segmented overlap condition, followed by the final POP in the first place and the initial POP in the last place. The difference between the two-segmented and the initial overlap is significant, while other pairwise comparisons are not. Based on these results, PP emerges as a facilitatory phenomenon that is highly sensitive to lexicality and POP. While PP can have a facilitating effect under lexicality, it shows no facilitation in its absence, which intersects with several previous findings. Participants are found to be more sensitive to the final phonological overlap than the initial overlap, which also coincides with a body of earlier literature. The results contradict the cohort theory’s stress on the onset overlap position and, instead, give more weight to final overlap, and even heavier weight to the two-segmented one. In conclusion, this study confirms the facilitating effect of PP with words but not when stimuli (at least the primes and at most both the primes and targets) are nonwords. It also shows that the two-segmented priming is the most influential in LDM in Arabic.

Keywords: lexicality, phonological overlap positions, phonological priming, visual word recognition

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Authors: Prittesh Patel, Rushabh Shah, Krishnamurthy Ramar, Vakulbhushan Bhaskar

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The present research work aims at finding genetic differences in the genomes of sugarcane red rot isolates Colletotrichum falcatum Went using Random Amplified Polymorphic DNA (RAPD) and interspersed simple sequence repeat (ISSR) molecular markers. Ten isolates of C. falcatum isolated from different red rot infected sugarcane cultivars stalk were used in present study. The amplified bands were scored across the lanes obtained in 15 RAPD primes and 21 ISSR primes successfully. The data were analysed using NTSYSpc 2.2 software. The results showed 80.6% and 68.07% polymorphism in RPAD and ISSR analysis respectively. Based on the RAPD analysis, ten genotypes were grouped into two major clusters at a cut-off value of 0.75. Geographically distant C. falcatum isolate cfGAN from south Gujarat had a level of similarity with Coimbatore isolate cf8436 presented on separate clade of bootstrapped dendrograms. First and second cluster consisted of five and three isolates respectively, indicating the close relation among them. The 21 ISSR primers produced 119 distinct and scorable loci in that 38 were monomorphic. The number of scorable loci for each primer varied from 2 (ISSR822) to 8 (ISSR807, ISSR823 and ISSR15) with an average of 5.66 loci per primer. Primer ISSR835 amplified the highest number of bands (57), while only 16 bands were obtained by primers ISSR822. Four primers namely ISSR830, ISSR845, ISSR4 and ISSR15 showed the highest value of percentage of polymorphism (100%). The results indicated that both of the marker systems RAPD and ISSR, individually can be effectively used in determination of genetic relationship among C falcatum accessions collected from different parts of south Gujarat.

Keywords: Colletotrichum falcatum, ISSR, RAPD, Red Rot

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Authors: Liang Ming Zhong, Yu Zhong, Wen Zhong, Fei Fei Yin

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Trinary affinity is a description of existence: every object exists as it is known and spoken of, in a system of 2 differences (denoted dif1, dif₂) and 1 similarity (Sim), equivalently expressed as dif₁ / Sim / dif₂ and kn / 0 / tkn (kn = the known, tkn = the 'to be known', 0 = the zero point of knowing). They are mathematically verified and illustrated in this paper by the arrangement of all integers onto 3 columns, where each number exists as a difference in relation to another number as another difference, and the 2 difs as arbitrated by a third number as the Sim, resulting in a trinary affinity or trinity of 3 numbers, of which one is the known, the other the 'to be known', and the third the zero (0) from which both the kn and tkn are measured and specified. Consequently, any number is horizontally specified either as 3n, or as '3n – 1' or '3n + 1', and vertically as 'Cn + c', so that any number seems to occur at the intersection of its X and Y axes and represented by its X and Y coordinates, as any point on Earth’s surface by its latitude and longitude. Technically, i) primes are viewed and treated as progenitors, and composites as descending from them, forming families of composites, each capable of being measured and specified from its own zero called in this paper the realistic zero (denoted 0r, as contrasted to the mathematic zero, 0m), which corresponds to the constant c, and the nature of which separates the composite and prime numbers, and ii) any number is considered as having a magnitude as well as a position, so that a number is verified as a prime first by referring to its descriptive formula and then by making sure that no composite number can possibly occur on its position, by dividing it with factors provided by the composite number formulas. The paper consists of 3 parts: 1) a brief explanation of the trinary affinity of things, 2) the 8 formulas that represent ALL the primes, and 3) families of composite numbers, each represented by a formula. A composite number family is described as 3n + f₁‧f₂. Since there are an infinitely large number of composite number families, to verify the primality of a great probable prime, we have to have it divided with several or many a f₁ from a range of composite number formulas, a procedure that is as laborious as it is the surest way to verifying a great number’s primality. (So, it is possible to substitute planned division for trial division.)

Keywords: trinary affinity, difference, similarity, realistic zero

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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: 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: Joserra Díez, Ainhoa Meñika, Iñaki Sanz-Azkue, Arritokieta Ortuzar

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Biodiversity provides humans with a great range of ecosystemic services; it is therefore an indispensable resource and a legacy to coming generations. However, in the last decades, the increasing exploitation of the Planet has caused a great loss of biodiversity and its acquaintance has decreased remarkably; especially in urbanized areas, due to the decreasing attachment of humans to nature. Yet, the Primary Education curriculum primes the identification of flora and fauna to guarantee the knowledge of children on their surroundings, so that they care for the environment as well as for themselves. In order to produce effective didactic material that meets the needs of both teachers and pupils, it is fundamental to diagnose the current situation. In the present work, the knowledge on biodiversity of 3^rd cycle Primary Education students in Biscay (n=98) and its relation to the size of the town/city of their school is discussed. Two tests have been used with such aim: one for tree identification and the other one so that the students enumerated the species of trees and animals they knew. Results reveal that knowledge of students on tree identification is scarce regardless the size of the city/town and of their school. On the other hand, animal species are better known than tree species.

Keywords: biodiversity, population, tree identification, animal identification

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Authors: John McDowall, Crysta Derham

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Broaden and build theory of emotion describes how experiencing positive emotions, such as happiness, broadens our ‘thought-action repertoire’ leading us to be more likely to go out and act on our positive emotions. This results in the building of new relationships, resources and skills, which we can draw on in times of need throughout life. In contrast, the experience of negative emotion is thought to narrow our ‘thought-action repertoire’, leading to specific actions to aid in survival. Three experiments aimed to explore the effect of briefly presented schematic faces (happy, sad, and neutral) on attentional scope using the flanker task. Based on the broaden and build theory it was hypothesised that there would be an increase in reaction time in trials primed with a happy face due to a broadening of attention, leading to increased flanker interference. A decrease in reaction time was predicted for trials primed with a sad face, due to a narrowing of attention leading to less flanker interference. Results lended partial support to the broaden and build hypothesis, with reaction times being slower following happy primes in incongruent flanker trials. Recent research is discussed in regards to potential mediators of the relationship between emotion and attention.

Keywords: emotion, attention, broaden and build, flanker task

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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: Michael Mihalicz, Aziz Guergachi

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Background: The science of modelling choice preferences has evolved over centuries into an interdisciplinary field contributing to several branches of Microeconomics and Mathematical Psychology. Early theories in Decision Science rested on the logic of rationality, but as it and related fields matured, descriptive theories emerged capable of explaining systematic violations of rationality through cognitive mechanisms underlying the thought processes that guide human behaviour. Cognitive limitations are not, however, solely responsible for systematic deviations from rationality and many are now exploring the effect of visceral factors as the more dominant drivers. The current study builds on the existing literature by exploring sleep deprivation, thermal comfort, stress, hunger, fear, anger and sadness as moderators to three distinct elements that define individual risk preference under Cumulative Prospect Theory. Methodology: This study is designed to compare the risk preference of participants experiencing an elevated affective or visceral state to those in a neutral state using nonparametric elicitation methods across three domains. Two experiments will be conducted simultaneously using different methodologies. The first will determine visceral states and risk preferences randomly over a two-week period by prompting participants to complete an online survey remotely. In each round of questions, participants will be asked to self-assess their current state using Visual Analogue Scales before answering a series of lottery-style elicitation questions. The second experiment will be conducted in a laboratory setting using psychological primes to induce a desired state. In this experiment, emotional states will be recorded using emotion analytics and used a basis for comparison between the two methods. Significance: The expected results include a series of measurable and systematic effects on the subjective interpretations of gamble attributes and evidence supporting the proposition that a portion of the variability in human choice preferences unaccounted for by cognitive limitations can be explained by interacting visceral states. Significant results will promote awareness about the subconscious effect that emotions and other drive states have on the way people process and interpret information, and can guide more effective decision making by informing decision-makers of the sources and consequences of irrational behaviour.

Keywords: decision making, emotions, prospect theory, visceral factors

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Authors: Iulianna Taritsa, Kuldeep Neote, Eric Fossel

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Lysosome-induced immunogenic cell death (LIICD) is a powerful mechanism of targeting cancer cells that kills circulating malignant cells and primes the host’s immune cells against future remission. Current immunotherapies for cancer are limited in preventing recurrence – a gap that can be bridged by training the immune system to recognize cancer neoantigens. Lysosomal leakage can be induced therapeutically to traffic antigens from dying cells to dendritic cells, which can later present those tumorigenic antigens to T cells. Previous research has shown that oxidative agents administered in the tumor microenvironment can initiate LIICD. We generated a fusion protein between an oxidative agent known as xanthine oxidase (XO) and a mini-antibody specific for EGFR/HER2-sensitive breast tumor cells. The anti-EGFR single domain antibody fragment is uniquely sourced from llama, which is functional without the presence of a light chain. These llama micro-antibodies have been shown to be better able to penetrate tissues and have improved physicochemical stability as compared to traditional monoclonal antibodies. We demonstrate that the fusion protein created is stable and can induce early markers of immunogenic cell death in an in vitro human breast cancer cell line (SkBr3). Specifically, we measured overall cell death, as well as surface-expressed calreticulin, extracellular ATP release, and HMGB1 production. These markers are consensus indicators of ICD. Flow cytometry, luminescence assays, and ELISA were used respectively to quantify biomarker levels between treated versus untreated cells. We also included a positive control group of SkBr3 cells dosed with doxorubicin (a known inducer of LIICD) and a negative control dosed with cisplatin (a known inducer of cell death, but not of the immunogenic variety). We looked at each marker at various time points after cancer cells were treated with the XO/antibody fusion protein, doxorubicin, and cisplatin. Upregulated biomarkers after treatment with the fusion protein indicate an immunogenic response. We thus show the potential for this fusion protein to induce an anticancer effect paired with an adaptive immune response against EGFR/HER2+ cells. Our research in human cell lines here provides evidence for the success of the same therapeutic method for patients and serves as the gateway to developing a new treatment approach against breast cancer.

Keywords: apoptosis, breast cancer, immunogenic cell death, lysosome

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