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The book "Number Theory Through the Eyes of Sophie Germain: An Inquiry Course" by David Pengelley explores the correspondence between Sophie Germain and renowned mathematicians Joseph-Louis Lagrange, Adrien-Marie Legendre, and Carl Friedrich Gauss as they attempted to find a proof for Fermat's Last Theorem. Although Germain's plan was ultimately unsuccessful, it represented significant progress on the problem and provided insights into the theorem. The book takes an inquiry-based approach, using primary source materials to teach undergraduate students elementary number theory. It offers a unique and engaging perspective on the subject, allowing students to discover and prove results themselves. [Extracted from the article]

A professor and her students were working through a routine problem on isomorphisms that required finding the multiplicative inverse of 43 modulo 50. The professor, however, simply asked for "the inverse" of 43 modulo 50 and got a surprising response. One student quickly answered that the inverse of 43 modulo 50 was 7. Thinking that the student was responding with the additive inverse, the professor restated her question and asked for the multiplicative inverse. The student checked her work and said that 7 is the multiplicative inverse of 43 mod 50. The observation that the additive and multiplicative inverse of an element could be the same led to a research project into the questions of when, why, and how frequently this phenomenon occurs. Answering these questions led to tools such asthe Chinese remainder theorem, the fundamental theorem of cyclic groups, and quadratic residues. [ABSTRACT FROM AUTHOR]

In this paper we confirm three conjectures of Z.-W. Sun on determinants. We first show that any odd integer n>3 divides the determinant (i 2 + d j 2) i 2 + d j 2 n 0 ≤ i , j ≤ (n − 1) / 2 , where d is any integer and (⋅ n) is the Jacobi symbol. Then we prove some divisibility results concerning |(i + dj)n|0≤i,j≤n−1 and |(i2 + dj2)n|0≤i,j≤n−1, where d ≠ 0 and n>2 are integers. Finally, for any odd prime p and integers c and d with p ∤ c d , we determine completely the Legendre symbol S c (d , p) p , where S c (d , p) := i 2 + d j 2 + c p 1 ≤ i , j ≤ (p − 1) / 2 . [ABSTRACT FROM AUTHOR]

If this were not the case, then we would have two points I P SB i sb i and I P SB j sb i that have the same distance from line Graph HT ht and are on the same side of that line. Now consider an element Graph HT ht . Determine all points I X i for which the area of triangle Graph HT ht is minimized. A variable point I X i is chosen on minor arc Graph HT ht of I i , and segments Graph HT ht and Graph HT ht meet at I D i . [Extracted from the article]

Let be distinct primes. Let , where at least two of αi' s are nonzero positive integers. In this article, it is shown that with the help of λ-mapping it is easy to count the number of q-cyclotomic cosets modulo m. The explicit expressions of primitive idempotents are obtained in the semisimple ring Rm over Fq taking q as a primitive root (or a quadratic residue) modulo pi under the condition that , where denote the order of q modulo pi. [ABSTRACT FROM AUTHOR]

For an odd prime p we determine four kinds of patterns concerning the distribution of quadratic residues depending on three parameters, according to the class of p modulo 8. [ABSTRACT FROM AUTHOR]

It is well known that when a prime p is congruent to 1 modulo 4, the sum of the quadratic residues equals the sum of the quadratic nonresidues. In this note, we give elementary proofs of V.-A. Lebesgue's analogous results for the case where p is congruent to 3 modulo 4 [ABSTRACT FROM AUTHOR]

Influences of the chitin-rich residues on growth and yield of Butterhead lettuce (Lactuca sativaL.) were investigated. The production and postharvest quality of lettuce were enhanced through the sole application of 5% shrimp shell powder (SSP), 20% fermented chitinous material (FCM) and 10 ml of chitinase-producingBacillus licheniformisstrain SK-1 (SK-1) or 5% SSP in combination with SK-1 and 2.5% SSP in combination with 2.5% FCM. Lettuces were cultivated during three successive crop seasons. SSP and/or FCM treatments resulted in significant increases in yield in all crop seasons. When applied in the first and third crop seasons, lettuces grown with the presence of 20% FCM showed the highest increase in yield as evaluated in terms of fresh weight, dry weight, leaf number, and leaf width and length. Weight losses after storage at 8°C and 60% relative humidity for 2 weeks were significantly reduced in all treatments with SSP or FCM. The results suggest that the application of 20% FCM is the best all-year-round supplement for Butterhead lettuce cultivation. [ABSTRACT FROM AUTHOR]

A chemometric class modelling strategy (unequal dispersed classes – UNEQ) was applied for the first time as a possible screening method to monitor the abuse of growth promoters in veal calves. Five serum biomarkers, known to reflect the exposure to classes of compounds illegally used as growth promoters, were determined from 50 untreated animals in order to design a model of controls, representing veal calves reared under good, safe and highly standardised breeding conditions. The class modelling was applied to 421 commercially bred veal calves to separate them into ‘compliant’ and ‘non-compliant’ with respect to the modelled controls. Part of the non-compliant animals underwent further histological and chemical examinations to confirm the presence of either alterations in target tissues or traces of illegal substances commonly administered for growth-promoting purposes. Overall, the congruence between the histological or chemical methods and the UNEQ non-compliant outcomes was approximately 58%, likely underestimated due to the blindness nature of this examination. Further research is needed to confirm the validity of the UNEQ model in terms of sensitivity in recognising untreated animals as compliant to the controls, and specificity in revealing deviations from ideal breeding conditions, for example due to the abuse of growth promoters. [ABSTRACT FROM AUTHOR]

In this paper we consider the following two equivalence relations on the lattice 𝒞(S) of all congruences on a semigroupSwith the set of idempotentsES:where kerρ = {x ∈ S: (x,x2) ∈ ρ} and trρ = ρ ∩ (ES × ES) (ρ ∈ 𝒞(S)). We show that ifSis idempotent-surjective, then every κ-class ρκ (ρ ∈ 𝒞(S)) coincides with some interval [ρκ, ρκ]. We study idempotent pure andE-disjunctive congruences on an arbitrary idempotent-surjective semigroup. In particular, we show that if τ is the greatest idempotent pure congruence on an idempotent-surjective semigroupS, thenS/τ isE-disjunctive (see Section 2). In Section 3 we investigate the trace class ρθ (ρ ∈ 𝒞(S)) of an arbitrary idempotent-surjective semigroupS. Also, we study the connections between ρ ∈ 𝒞(S) and four associated (with ρ) congruences ρκ, ρκ, ρθ, ρθ. In particular, using a conjunction between the least Clifford congruence ξ (on an eventually regular semigroup) and the congruences ξκ, ξθ, we give the necessary and sufficient conditions for ξ to be idempotent pure (see Section 4). In Section 5 we describe allE-unitary congruences on an arbitrary idempotent-surjective semigroup. Finally, some remarks concerning stronglyE-reflexive congruences on an eventually regular semigroup are given (see Section 6). [ABSTRACT FROM AUTHOR]

Among other results on subtractive semirings, we present complete descriptions of subtractive artinian ideal-simple (congruence-simple) semirings as well as subtractive semisimple semirings. Also, solving Problem 2 of [12] for semisimple semirings, we show that matrix semirings over a semisimple semiring R are subtractive iff R is a ring. Finally, confirming a conjecture of [11] for the classes of additively regular semisimple and additively regular subtractive artinian semirings, we show that perfect semirings in those classes are just perfect rings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a class of generalized regular languages, namely PS-regular languages, and give some characterizations of such generalized regular languages. As applications of the results, we obtain some characterizations of regular languages. Also, we consider the closure properties of the class of PS-regular languages, and the relationship among PS-regular languages, context-free languages and context-sensitive languages. [ABSTRACT FROM AUTHOR]

The article focuses on the Fermat's spiral, the distinctive yin-yang line in the class of curves algebraic in polar coordinates. It says that yin-yang is a concept that came from ancient Chinese philosophy. Hence, to prove Fermat's spiral, D denote a disk of unit area wherein set A ⊂ D if it measures ½ and the maximum symmetric subset of A measure ¼. It adds that β divides into two congruent set, it intersects each concentric of D twice and in each radius of D once.

Viewing the future value of a cash flow stream as a polynomial in the compounding factor (being one plus the interest rate), this article uses Ruffini's rule in the synthetic division of a polynomial to provide insight linking several discounted cash flow analysis concepts. We use the polynomial remainder theorem to relate the concepts of divisor, quotient, and remainder to concepts in cash flow analysis-future value, present value, interest rate, account balance, and internal rate(s) of return-establishing a remainder theorem for the future value of a cash flow. Our exposition suggests that future value analysis is perhaps more direct and mathematically more transparent than attention paid to examining present value. This article extends previous work on the resolution of multiple rates of return when the interest rate used in our analysis equals an internal rate of return of the cash flow. [ABSTRACT FROM AUTHOR]

Image encoding technique is an important issue to protect the security of an image and to prevent the image from being eavesdropped over the Internet. In 2001, Ammar et al. proposed an image encoding scheme based on residue number system (RNS). To overcome the disadvantages of Ammar et al.'s scheme, Wang et al. proposed a more efficient RNS image encoding scheme in 2004. However, these two image encoding schemes are insecure and inefficient. Thus, we propose two image encoding schemes using residue number system in this paper. Compared with the previous RNS image encoding schemes, the proposed schemes have the advantages of lower communication overhead, less moduli selection, higher security and fewer computation loads. Also, the experimental results show that the proposed two schemes are more secure, efficient and practical than the previous works. [ABSTRACT FROM AUTHOR]

Gas chromatograph-electron capture detector (GC-ECD) and gas chromatograph-negative chemical ionization mass spectrometry (GC-NCI-MS) were applied to determine the residues of 20 organochlorine pesticides (OCPs) in soil after Soxhlet extraction with hexane/acetone (v: v = 1: 1) followed by cleanup with Florisil solid phase extraction (SPE) column (Palo Alto, CA). The method parameters for two different detection systems, including linear range, precision, detection limit and recovery, were studied and compared. The detection limit was in the range from 0.027 to 0.098 μ g/kg detected by GC-ECD and from 1.5 × 10- 4-2.4 × 10- 3μ g/kg detected by GC-NCI-MS for 20 OCPs in soil. The average recoveries of spiked sample ranged from 70.0% to 127% for GC-ECD and from 73.0%-122% for GC-NCI-MS, respectively. The results indicated that the method of GC-NCI-MS was more sensitive and selective than GC-ECD for determination of OCPs in soil. [ABSTRACT FROM AUTHOR]

In this article, we study the stochastic orderings among residual record values as well as inactive record values in two-sample problems. Some of the results obtained here are extensions of the main result in Khaledi and Shojaei (2007). [ABSTRACT FROM AUTHOR]

For any n, the contour integral [image omitted] is associated with the differential equation [image omitted] Explicit solutions are calculated by residues for n = 0, 1, 2. Eigenvalues along with certain boundary conditions and the asymptotics of the solutions with Im (√λ) > 0 are explored for n = 1, 2. There is, in particular, one differential equation that does have a solution in terms of the trigonometric functions, which does not seem to have been explored, and it is also one of the purposes of this article to put it on record. [ABSTRACT FROM AUTHOR]

A more-than-5-year study directed toward the detection of trace residual halogenated solvents in circuit boards was recently completed. This work was undertaken to demonstrate the potential for reliable detection of solvents used during the fabrication of printed circuit boards found in a wide variety of electronic products. Residual solvent detection was demonstrated for spiked boards after an aging period of 5 months and for printed circuit boards removed from a variety of used (at least 5 years old) and new electronic products. The analytical system and standard operating procedure developed during this study provide the tools required to assess and confirm compliance with regulations associated with the use of ozone depleting chemicals. [ABSTRACT FROM AUTHOR]

Organochlorine pesticides (OCPs) and polychlorinated biphenyls (PCBs) in surface sediments from 10 sampling sites of the Songhuajiang River were analyzed to evaluate their potential pollution and hazards. Gas chromatography equipped with a 63Ni electron capture detector (GC-ECD) and gas chromatography/mass spectrometer (GC/MS) methods were used, respectively. The total concentrations of 13 OCPs varied from 4.26 to 18.45 ng g-1. ΣHCH (α-HCH, β-HCH, δ-HCH, γ-HCH), Σ-DDT (p, p'-DDE, p, p'-DDD, o,p'-DDT, p, p'-DDT) and ΣCyclodiene (Heptachlor, Aldrin, Heptachlor epoxide, Dieldrin, Endrin) ranged from 2.42 to 7.95, 0.14 to 5.12, 0.55 to 6.01 ng g-1, respectively. The most dominant pollutant among the OCPs was γ-HCH with the concentration of 3.4 (0.77 to 7.21) ng g-1. Total PCB concentrations ranged between 0.59 and 15.93 ng g-1. Low chlorinated biphenyls (tri- and tetra-) dominated in total PCBs. Positive correlations were observed between the ΣDDT, ΣCyclodiene, ΣOCP or ΣPCB and TOC of the sediments (P<0.05, n=10). No obvious correlation was observed between the ΣHCH and the TOC of the sediments. [ABSTRACT FROM AUTHOR]

The article focuses on density and polynomial congruencies. The author presents an algebraic problem for determining solutions of a congruence as the modulus varies. He discusses a mathematical analysis of the problem and offers solutions, incorporating Christopher Hooley's theorem on irreducible polynomials. A lemma on a polynomial with version of the remainder theorem also is presented.

The article presents a mathematical problem relating to lattice points. It presents the problem of a photographer who has to take pictures of a high school marching band while standing near them on the field, using the fewest number of photographs such that every person is visible and unobstructed in at least one of the photographs. Several answers to the problem using the Chinese Remainder Theorem as a key component also are presented.

The article examines the relationship between quadratic residues and the Frobenius coin problem. It presents a chart showing the non-representable Frobenius numbers for 5 and 7 and the squares of these numbers. It defines the terms quadratic residue and the coin problem of Frobenius. It provides an example which describes the case for five and seven-cent coins.

Using the group consisting of the eight Möbius transformations x, – x, 1 / x , − 1 / x , (x − 1) / (x + 1) , (x + 1) / (1 − x) , (x + 1) / (x − 1) , and (1 − x) / (x + 1) , we present an enumerative proof of the classical result for when the element 2 is a quadratic residue in the finite field Fq. [ABSTRACT FROM AUTHOR]

In asymptotic expansions of resolvent traces Tr( A ( P - λ) -1 ) for classical pseudodifferential operators on closed manifolds, the coefficient C 0 ( A , P ) of ( - λ) -1 is of special interest, since it is the first coefficient containing nonlocal elements from A ; moreover, it enters in index formulas. C 0 ( A , P ) also equals the zeta function value at zero when P is invertible. C 0 ( A , P ) is a trace modulo local terms, since C 0 ( A , P ) - C 0 ( A , P ′) and C 0 ([ A , A ′], P ) are local. By use of complex powers P s (or similar holomorphic families of order s ), Okikiolu, Kontsevich and Vishik, Melrose and Nistor showed formulas for these trace defects in terms of residues of operators defined from A , A ′, log P and log P ′. The present paper has two purposes. One is to show how the trace defect formulas can be obtained from the resolvents in a simple way without use of the complex powers of P as in the original proofs. We also give here a simple direct proof of a recent residue formula of Scott for C 0 ( I , P ). The other purpose is to establish trace defect residue formulas for operators on manifolds with boundary, where complex powers are not easily accessible; we do this using only resolvents. We also generalize Scott's formula to boundary problems. [ABSTRACT FROM AUTHOR]

In this article we classify all the smooth threefolds in [Popf]5 with an apparent quadruple point provided that the family of its 4-secant lines is an irreducible (first order) congruence. This is sufficient to conclude the classification of all the smooth codimension two varieties in [Popf]n with one apparent (n - 1)-point and with irreducible family of (n - 1)-secant lines. [ABSTRACT FROM AUTHOR]

An optimized liquid chromatographic system was developed to separate a mixture of Marfey's amino acid derivatives (1-fluoro-2,4-dinitrophenyl-5-L-alanine amide, FDAA). Alanine, glutamic acid, isoleucine, proline, and tyrosine were the five natural amino acid residues from the synthetic decapeptide, Suc-Tyr-Glu-Pro-Ile-Pro-Glu-Glu-Ala-Cha-Glu-OH, which was analyzed for optical purity as part of this study. Excellent chromatographic resolution of the five pairs of natural amino acid derivatives, including D-alanine and D-proline, within a 70 minute gradient elution using an ODS (C-18) column, was obtained using acetonitrile as the organic modifier and sodium acetate buffer. The effects of the mobile phase pH with respect to peak retention factor (k) and separation factor (α) were studied in this chromatographic system. [ABSTRACT FROM AUTHOR]

A non-linear curve-fitting model is presented which minimizes the sum of squares of relative residues, and expressions are derived for the fit parameters and their respective errors. A detailed comparison is made between the new general relative least squares model (GRLS) and other non-linear regression models available in the literature, using two sets of data representing fluid mechanics problems encountered in many engineering applications. The results showed that GRLS was the best model for fitting non-linear functions in the case of experimental data spanning several orders of magnitude, indicating its potential as a tool for data analysis. [ABSTRACT FROM AUTHOR]

A method was developed for the determination of the nitroimidazole compounds dimetridazole, metronidazole, and ronidazole in meat. Meat was extracted with acetonitrile and the organic fraction was separated from the aqueous with sodium chloride and dichloromethane. Water was added, the organic layer was evaporated, and fat was extracted with hexane. The water residue was injected into the LC/MS. The lower limit of quantification was 1, 2, and 4 ng/g for dimetridazole, metronidazole, and ronidazole, respectively. [ABSTRACT FROM AUTHOR]

To determine ciprofloxacin (CPFX) residues in pork, an ELISA and a high-performance liquid chromatography (HPLC) procedure were developed and compared on sensitivity, precision and accuracy. For ELISA, CPFX was converted by an active ester method into CFPX-human serum albumin (CPFX-HSA) and CFPX-bovine serum albumin (CPFX-BSA), and the latter allowed the production of CPFX-specific rabbit antisera for the development of an indirect competitive ELISA procedure. For HPLC, an ODS reverse phase column was used with ultraviolet detection, and triethylamine and acetonitrile (87:13) were employed as mobile phase. The limits of detection (LODs) were 0.32 ng ml[sup -1] for ELISA and 10 ng ml[sup -1] for HPLC. The linear detection ranges were 0.32-5000 ng ml[sup -1] for ELISA and 10-5120 ng ml[sup -1] for HPLC. The coefficients of variation were 4.51-11.50% for ELISA and 0.70-9.10% for HPLC over the range of CPFX concentrations of each method. The mean recovery of HPLC (80.58%) was higher than that of ELISA (66.18%). The results suggested that the ELISA, with its high detection throughput and excellent sensitivity and specificity, could be used as a screening method for CPFX residues in pork. The HPLC, for its great precision and accuracy, was an effective confirmatory method for CPFX residues in pork. [ABSTRACT FROM AUTHOR]

The article focuses on the solution to a mathematical problem on quadratic residues using modular hyperbolas. Topics discussed include different symmetries of modular hyperbolas being used to solve the problem, Euclidean distance being used for solving the problem, theorem by mathematician Joseph-Louis Lagrange related to polynomials being used, algebraic results not being used to study quadratic residues.

The paper presents a new practical method of achieving a dramatic reduction in the size of protocols, to increase transmission rates. The method uses a new code subjected to matrix type-B code, called concatenated quasiquadratic or semiquadratic residue (SQR) matrix code. [ABSTRACT FROM AUTHOR]

A simple way of encouraging a nonlinear effect in a regression model to show itself in a partial residual (parres) plot (component-plus-residual plot) is to include a single quadratic term in the regression. It is computationally cheap to do this for each of the independent variables. Examples show that the resulting set of augmented partial residual plots gives insights that are not available from standard parres or added-variable plots. [ABSTRACT FROM AUTHOR]

Detection of outliers and other diagnostics based on residuals have gained widespread use in linear regression. Logistic regression has been both blessed and hindered by this development. Certainly logistic regression requires procedures to detect global and local model weaknesses. Thus the wealth of work done in linear regression provides guides and suggestions that may, with care and ingenuity, be applied to logistic regression. Several innovative attempts in this direction have been made by authors such as Tsiatis (1980), Pregibon (1979, 1981), Landwehr, Pregibon, and Shoemaker (1984), and Cook and Weisberg (1982). Unfortunately the similarities that allow such techniques to adapt to logistic regression seem, in addition, to hide many of the differences. Two such differences are discussed here. The first examines the effect of events of small probability on estimation in logistic regression. The simple conclusion is that to estimate the probability of success p in an area in which p is small, one must observe events of small probability. This suggests that the motives for outlier detection should be carefully considered. The second difference examined deals with the distribution of residuals. Let Y be the response variable and let x be a p x 1 vector of explanatory variable. In linear regression with normal errors, Y - E(Y | x) is normally distributed with mean 0 and variance sigma[sup 2]. Thus the distribution of this residual from the true model does not depend on the explanatory variables x. This is not true in logistic regression with binary data. There Y has a Bernoulli distribution with probability of success p(x) = Pr(Y = 1 | x). Thus any reasonable definition of a residual from the true model, for example, Y - p(x), has a two-point distribution that depends on x through p(x). That is, each residual... [ABSTRACT FROM AUTHOR]

The article offers information the power sum identity and its significance determining the binomial coefficient congruences. It mentions that the effectiveness of the power sum in finding and obtaining the mentioned congruences by Bernoulli and Stirling number theorems as well as the exponential Diophantine equations. Furthermore, the power sum was already proven by Blaise Pascal.

The calculus of residues is one of the many beautiful tools that comes out of the field of complex variables. The calculus of residues is applied, together with the inversion formula for characteristic functions, to compute the non-gamma probability density function for the sum of gamma variates with different shape parameters. The distribution of the sum of gamma variates is needed in problems in statistical inference, as well as stochastic processes. This derivation seems more elegant than previous methods for deriving the density function of such a sum. Furthermore, the numerical computation is straightforward, especially in any symbolic computer language. [ABSTRACT FROM AUTHOR]

The article presents a mathematical problem. It states that a girl going to a well with two jugs, one holding three pints and the other five and she is to come back with precisely four pints of water. The problem is to identify whether this is possible and to search the most efficient way to do it. It is said that the solution to the problem are Euclidean Algorithm, some applications of least residues and minimizing the number of stages through calculating the amount water and select the option with fewer stages.

Presents the standard elementary proof of the quadratic reciprocity law. Quadratic residue; Legendre symbol; Chinese remainder theorem; Permutation.

Focuses on determining which numbers between 0 and p/2 are quadratic residues modulo. Computing and plotting the points for twenty values of p; Information on a theorem; Calculation of quadratic residues.

The article focuses on the elementary proof of the quadratic character of two which assumes the existence of primitive root mod p which was first provided by Karl Gauss. It mentions the theorem stating that the number two is a quadratic residue of primes of the forms p = 8k + 1 and p = 8k + 7, as well as p = 8k + 3 and p = 8k + 5. Furthermore, a simple counting argument proves that number two is a quadratic residue mod p if p = 8k + 1 or p = 8k + 7 and not if p = 8k + 3 or p = 8k + 5.

The article provides information on the mathematical proof of P. G. Lejeune Dirichlet's theorem that in any positive integer N, there are infinitely many prime numbers. It states that with the use of the idea of Hillel Guachman and a little lemma, the theorem has been proven. Moreover, the theorem has also been proven using Fermat's little theorem, cq-1 ≡ 1.

Discusses the quadratic residues -1 and -3. Conditions in which the requirement for quadratic residues are met; Overview of the mathematical formula showing the results of the quadratic residues.