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1
Blyth, T. S., and M. H. Almeida Santos. "On weakly multiplicative inverse transversals." Proceedings of the Edinburgh Mathematical Society 37, no. 1 (February 1994): 91–99. http://dx.doi.org/10.1017/s001309150001871x.
Повний текст джерелаАнотація:
We show that an inverse transversal of a regular semigroup is multiplicative if and only if it is both weakly multiplicative and a quasi-ideal. Examples of quasi-ideal inverse transversals that are not multiplicative are known. Here we give an example of a weakly multiplicative inverse transversal that is not multiplicative. An interesting feature of this example is that it also serves to show that, in an ordered regular semigroup in which every element x has a biggest inverse x0, the mapping x↦x00 is not in general a closure; nor is x↦x** in a principally ordered regular semigroup.
2
Cherubini, A., and A. Varisco. "Rings satisfying certain conditions either on subsemigroups or on endomorphisms." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 40, no. 2 (April 1986): 194–202. http://dx.doi.org/10.1017/s1446788700027178.
Повний текст джерелаАнотація:
AbstractWe characterize rings whose multiplicative subsemigroups containing 0 and the additive inverse of each element are subrings. In addition we consider commutative rings for which every non-constant multiplicative endormorphism that preserves additive inverses is a ring endomorphism, and we show that they belong to one of three easily-described classes of rings.
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Saito, Tatsuhiko. "Naturally ordered regular semigroups with maximum inverses." Proceedings of the Edinburgh Mathematical Society 32, no. 1 (February 1989): 33–39. http://dx.doi.org/10.1017/s001309150000688x.
Повний текст джерелаАнотація:
Let S be a regular semigroup. An inverse subsemigroup S° of S is called an inverse transversal if S° contains a unique inverse of each element of S. An inverse transversal S° of S is called multiplicative if x°xyy° is an idempotent of S° for every x, y∈S, where x° denotes the unique inverse of x∈S in S°. In Section 1, we obtain a necessary and sufficient condition in order for inverse transversals to be multiplicative.
4
Blyth, T. S., and M. H. Almeida Santos. "A simplistic approach to inverse transversals." Proceedings of the Edinburgh Mathematical Society 39, no. 1 (February 1996): 57–69. http://dx.doi.org/10.1017/s0013091500022781.
Повний текст джерелаАнотація:
An inverse transversal of a regular semigroup S is an inverse subsemigroup that contains precisely one inverse of each element of S. In the literature there are three known types of inverse transversal, namely those that are multiplicative, those that are weakly multiplicative, and those that form quasi-ideals. Here, by considering natural ways in which certain words can be simplified, we reveal four new types of inverse transversal. All of these can be illustrated nicely in examples that are based on 2 × 2 matrices.
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Sheibani, Abdolyousefi. "P-Hirano inverses in rings." Filomat 34, no. 13 (2020): 4473–82. http://dx.doi.org/10.2298/fil2013473s.
Повний текст джерелаАнотація:
We introduce and study a new class of generalized inverses in rings. An element a in a ring R has p-Hirano inverse if there exists b ? R such that bab = b,b ? comm2(a),(a2-ab)k ? J(R) for some k ? N. We prove that a ? R has p-Hirano inverse if and only if there exists p = p2 ? comm2(a) such that (a2-p)k ? J(R) for some k ? N. Multiplicative and additive properties for such generalized inverses are thereby obtained. We then completely determine when a 2 x 2 matrix over local rings has p-Hirano inverse.
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Saito, Tatsuhiko. "Construction of regular semigroups with inverse transversals." Proceedings of the Edinburgh Mathematical Society 32, no. 1 (February 1989): 41–51. http://dx.doi.org/10.1017/s0013091500006891.
Повний текст джерелаАнотація:
Let S be a regular semigroup. An inverse subsemigroup S° of S is an inverse transversal if |V(x)∩S°| = 1 for each x∈S, where V(x) denotes the set of inverses of x. In this case, the unique element of V(x)∩S° is denoted by x°, and x°° denotes (x°)–1. Throughout this paper S denotes a regular semigroup with an inverse transversal S°, and E(S°) = E° denotes the semilattice of idempotents of S°. The sets {e∈S:ee° = e} and {f∈S:f°f=f} are denoted by Is and Λs, respectively, or simply I and Λ. Though each element of these sets is idempotent, they are not necessarily sub-bands of S. When both I and Λ are sub-bands of S, S° is called an S-inverse transversal. An inverse transversal S° is multiplicative if x°xyy°∈E°, and S° is weakly multiplicative if (x°xyy°)°∈E° for every x, y∈S. A band B is left [resp. right] regular if e f e = e f [resp. e f e = f e], and B is left [resp. right] normal if e f g = e g f [resp. e f g = f e g] for every e, f, g∈B. A subset Q of S is a quasi-ideal of S if QSQ ⊆ S.
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Grošek, Otokar, and Tomáš Fabšič. "Computing multiplicative inverses in finite fields by long division." Journal of Electrical Engineering 69, no. 5 (September 1, 2018): 400–402. http://dx.doi.org/10.2478/jee-2018-0059.
Повний текст джерелаАнотація:
Abstract We study a method of computing multiplicative inverses in finite fields using long division. In the case of fields of a prime order p, we construct one fixed integer d(p) with the property that for any nonzero field element a, we can compute its inverse by dividing d(p) by a and by reducing the result modulo p. We show how to construct the smallest d(p) with this property. We demonstrate that a similar approach works in finite fields of a non-prime order, as well. However, we demonstrate that the studied method (in both cases) has worse asymptotic complexity than the extended Euclidean algorithm.
8
Gould, Victoria. "Semigroups of left quotients—the uniqueness problem." Proceedings of the Edinburgh Mathematical Society 35, no. 2 (June 1992): 213–26. http://dx.doi.org/10.1017/s0013091500005496.
Повний текст джерелаАнотація:
Let S be a subsemigroup of a semigroup Q. Then Q is a semigroup of left quotients of S if every element of Q can be written as a*b, where a lies in a group -class of Q and a* is the inverse of a in this group; in addition, we insist that every element of S satisfying a weak cancellation condition named square-cancellable lie in a subgroup of Q.J. B. Fountain and M. Petrich gave an example of a semigroup having two non-isomorphic semigroups of left quotients. More positive results are available if we restrict the classes of semigroups from which the semigroups of left quotients may come. For example, a semigroup has at most one bisimple inverse ω-semigroup of left quotients. The crux of the matter is the restrictions to a semigroup S of Green's relations ℛ and ℒ in a semigroup of quotients of S. With this in mind we give necessary and sufficient conditions for two semigroups of left quotients of S to be isomorphic under an isomorphism fixing S pointwise.The above result is then used to show that if R is a subring of rings Q1 and Q2 and the multiplicative subsemigroups of Q1 and Q2 are semigroups of left quotients of the multiplicative semigroup of R, then Ql and Q2 are isomorphic rings.
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Dey, Sankhanil, and Ranjan Ghosh. "Mathematical Method to Search for Monic Irreducible Polynomials with Decimal Equivalents of Polynomials over Galois Field GF(pq)." Circulation in Computer Science 2, no. 11 (December 20, 2017): 17–22. http://dx.doi.org/10.22632/ccs-2017-252-68.
Повний текст джерелаАнотація:
Substitution boxes or S-boxes play a significant role in encryption and decryption of bit level plaintext and cipher-text respectively. Irreducible Polynomials (IPs) have been used to construct 4-bit or 8-bit substitution boxes in many cryptographic block ciphers. In Advance Encryption Standard the 8-bit the elements S-box have been obtained from the Multiplicative Inverse (MI) of elemental polynomials (EPs) of the 1st IP over Galois field GF(28) by adding an additive element. In this paper a mathematical method and the algorithm of the said method with the discussion of the execution time of the algorithm, to obtain monic IPs over Galois field GF(pq) have been illustrated with example. The method is very similar to polynomial multiplication of two polynomials over Galois field GF(pq) but has a difference in execution. The decimal equivalents of polynomials have been used to identify Basic Polynomials (BPs), EPs, IPs and Reducible polynomials (RPs). The monic RPs have been determined by this method and have been cancelled out to produce monic IPs. The non-monic IPs have been obtained with multiplication of α where α GF(pq) and assume values from 2 to (p-1) to monic IPs.
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Zheleznyak, Аlexander V. "Multiplicative property of series used in the Nevanlinna-Pick problem." Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 9, no. 1 (2022): 37–45. http://dx.doi.org/10.21638/spbu01.2022.104.
Повний текст джерелаАнотація:
In the paper we obtained substantially new sufficient condition for negativity of coefficients of power series inverse to series with positive ones. It has been proved that element-wise product of power series retains this property. In particular, it gives rise to generalization of the classical Hardy theorem about power series. These results are generalized for cases of series with multiple variables. Such results are useful in Nevanlinna – Pick theory. For example, if function k(x, y) can be represented as power series Pn≥0 an(x¯y)n, an > 0, and reciprocal function 1/k(x, y) can be represented as power series Pn≥0 bn(x¯y)n such that bn < 0, n > 0, then k(x, y) is a reproducing kernel function for some Hilbert space of analytic functions in the unit disc D with Nevanlinna – Pick property. The reproducing kernel 1/(1 − x¯y) of the classical Hardy space H2(D) is a prime example for our theorems.
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Brown, Elizabeth M., and Elizabeth Jones. "Using Clock Arithmetic to Teach Algebra Concepts." Mathematics Teaching in the Middle School 11, no. 2 (September 2005): 104–9. http://dx.doi.org/10.5951/mtms.11.2.0104.
Повний текст джерелаАнотація:
Learning algebra concepts can be difficult for middle school students. One reason may be because we work in only one number system, the set of real numbers. Students have only one frame of reference to provide examples of abstract concepts, such as the additive and multiplicative identities, additive and multiplicative inverses, and connections among the operations. These concepts are essential in solving equations. For example, we can think of an equation like 3x + 4 = 7 in the following way: Begin with a number, multiply it by 3, and add 4. If the answer is 7, what number did we start with? To solve this type of equation algebraically, we just undo what was done to the original number, that is, we add -4, the additive inverse of 4, and multiply the result by 1/3, the multiplicative inverse of 3. We can also think of this as subtracting 4 and dividing by 3, because addition and subtraction are inverse operations as are multiplication and division. The ideas of inverse operations and inverse elements are, therefore, central to algebra. Many students understand these ideas well enough to do simple problems like 3x + 4 = 7 but get confused with more difficult problems, such as 3(x + 6) + 2 = 5x + 5, where it is not as easy to see the order in which to undo these operations. We use finite systems to help students understand these key concepts in algebra, including additive and multiplicative identities, additive and multiplicative inverses, closure, and the relationships between addition and subtraction and multiplication and division.
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Mahzoon, Hesam. "Weighted Dual Covariance Moore-Penrose Inverses with respect to an Invertible Element inC*-Algebras." Journal of Applied Mathematics 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/918107.
Повний текст джерелаАнотація:
We study several algebraic properties of dual covariance and weighted dual covariance sets in rings with involution andC*-algebras. Moreover, we show that the weighted dual covariance set, seen as a multivalued map, has some kind of continuity. Also, we prove weighed dual covariance set invariant under the bijection multiplicative*-functions.
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Rabin, A. V., S. V. Michurin, and V. A. Lipatnikov. "DEVELOPMENT OF A CLASS OF SYSTEM AND RETURN SYSTEM MATRIXES PROVIDING INCREASE IN NOISE IMMUNITY OF SPECTRALLY EFFECTIVE MODULATION SCHEMES ON BASIS OF ORTHOGONAL CODING." Issues of radio electronics, no. 10 (October 20, 2018): 75–79. http://dx.doi.org/10.21778/2218-5453-2018-10-75-79.
Повний текст джерелаАнотація:
In work it is proposed in the digital systems of messages transmission for noise immunity's increase with the fixed code rate to use an additional coding called by the authors orthogonal. The way of a definition of orthogonal codes is presented, the synthesis algorithm of system and inverse system matrices of orthogonal codes is developed, and the main parameters of some matrices constructed by the offered algorithm are specified. Orthogonal coding as a special case of convolutional coding is defined by matrices, which elements are polynomials in the delay variable with integer coefficients. Code words are given by multiplication of an information polynomial by a system matrix, and decoding is performed by multiplication by an inverse system matrix. Basic ratios for orthogonal coding are given in article, and properties of system and inverse matrices are specified. Parameters of system and inverse system matrices assure additional gain in signal-to-noise ratio. This gain is got as a result of a more effective use of energy of transmitted signals. For transmission of one symbol energy of several symbols is accumulated.
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De Brauwer, Jolien, and Wim Fias. "The Representation of Multiplication and Division Facts in Memory." Experimental Psychology 58, no. 4 (February 1, 2011): 312–23. http://dx.doi.org/10.1027/1618-3169/a000098.
Повний текст джерелаАнотація:
Recently, using a training paradigm, Campbell and Agnew (2009) observed cross-operation response time savings with nonidentical elements (e.g., practice 3 + 2, test 5 − 2) for addition and subtraction, showing that a single memory representation underlies addition and subtraction performance. Evidence for cross-operation savings between multiplication and division have been described frequently (e.g., Campbell, Fuchs-Lacelle, & Phenix, 2006) but they have always been attributed to a mediation strategy (reformulating a division problem as a multiplication problem, e.g., Campbell et al., 2006). Campbell and Agnew (2009) therefore concluded that there exists a fundamental difference between addition and subtraction on the one hand and multiplication and division on the other hand. However, our results suggest that retrieval savings between inverse multiplication and division problems can be observed. Even for small problems (solved by direct retrieval) practicing a division problem facilitated the corresponding multiplication problem and vice versa. These findings indicate that shared memory representations underlie multiplication and division retrieval. Hence, memory and learning processes do not seem to differ fundamentally between addition-subtraction and multiplication-division.
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Ali-Pacha, Hana, Naima Hadj-Said, Adda Ali-Pacha, and Özen Özer. "Significant role of the specific prime number p = 257 in the improvement of cryptosystems." Notes on Number Theory and Discrete Mathematics 26, no. 4 (November 2020): 213–22. http://dx.doi.org/10.7546/nntdm.2020.26.4.213-222.
Повний текст джерелаАнотація:
Cryptology is the significant science which is inseparable from the means of communication of secrets. In a safe manner, it has the main objective of transmitting (potentially sensitive) information between two interlocutors. One distinguishes mainly two “dual” disciplines within cryptology: (a) cryptography, which is interested in the security of information. (b) cryptanalysis, which seeks to attack it. One have a starting set of 256 elements, we add a new element to this set to form a set of 257 elements. In this paper, we consider a finite field that contains 257 elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers modulo p when p is a prime number. For our case ℤ/pℤ, p = 257. We apply it to affine ciphers and show that this cipher looks like a permutation cipher. The idea based on this result, is to use the affine ciphers with the modulo 257 (as an initial permutation) in any specific algorithm of ciphering. Besides, one finishes with the decryption affine with the modulo 257 like an inverse permutation. This is to significantly increase the security of the specific encryption algorithm and to lengthen the 16-bits encryption key.
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Nurjayanto, Ery, Amrullah Amrullah, Arjudin Arjudin, and Sudi Prayitno. "Pembentukan Grup Matriks Singular 2×2." Griya Journal of Mathematics Education and Application 1, no. 3 (September 30, 2021): 403–11. http://dx.doi.org/10.29303/griya.v1i3.76.
Повний текст джерелаАнотація:
The study aims to determine the set of the singular matrix 2×2 that forms the group and describes its properties. The type of research was used exploratory research. Using diagonalization of the singular matrix S, whereas a generator matrix, pseudo-identity, and pseudo-inverse methods, we obtained a group singular matrix 2×2 with standard multiplication operations on the matrix, with conditions namely: (1) closed, (2) associative, (3) there was an element of identity, (4) inverse, there was (A)-1 so A x (A)-1 = (A)-1 x A = Is. The group was the abelian group (commutative group). In addition, in the group, Gs satisfied that if Ɐ A, X, Y element Gs was such that A x X = A x Y then X = Y and X x A = Y x A then X = Y. This show that the group can be applied the cancellation properties like the case in nonsingular matrix group. This research provides further research opportunities on the formation of singular matrix groups 3×3 or higher order.
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RICHMAN, FRED. "A DIVISION ALGORITHM." Journal of Algebra and Its Applications 04, no. 04 (August 2005): 441–49. http://dx.doi.org/10.1142/s0219498805001289.
Повний текст джерелаАнотація:
A divisibility test of Arend Heyting, for polynomials over a field in an intuitionistic setting, may be thought of as a kind of division algorithm. We show that such a division algorithm holds for divisibility by polynomials of content 1 over any commutative ring in which nilpotent elements are zero. In addition, for an arbitrary commutative ring R, we characterize those polynomials g such that the R-module endomorphism of R[X] given by multiplication by g has a left inverse.
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Dixon, John D. "The Orbit-Stabilizer Problem for Linear Groups." Canadian Journal of Mathematics 37, no. 2 (April 1, 1985): 238–59. http://dx.doi.org/10.4153/cjm-1985-015-4.
Повний текст джерелаАнотація:
Let G be a subgroup of the general linear group GL(n, Q) over the rational field Q, and consider its action by right multiplication on the vector space Qn of n-tuples over Q. The present paper investigates the question of how we may constructively determine the orbits and stabilizers of this action for suitable classes of groups. We suppose that G is specified by a finite set {x1, …, xr) of generators, and investigate whether there exist algorithms to solve the two problems:(Orbit Problem) Given u, v ∊ Qn, does there exist x ∊ G such that ux = v; if so, find such an element x as a word in x1, …, xr and their inverses.(Stabilizer Problem) Given u, v ∊ Qn, describe all words in x1, …, xr and their inverses which lie in the stabilizer
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Martynova, Inna A. "SUBSTITUTION CHARACTERISTICS OF FACTORIAL SETS AND CRITERIA FOR CHOOSING THE SINGLE SUBSTITUTIONS." Автоматизация Процессов Управления 62, no. 4 (2020): 109–17. http://dx.doi.org/10.35752/1991-2927-2020-4-62-109-117.
Повний текст джерелаАнотація:
The substitution and permutation function, which are presented in the article as elements of a number of factorial sets, are the key functions of cryptographic systems that provide diffusion and mixing of information. A new scale of notation is proposed while analyzing this problem. This is the notation scale of a number of factorial sets. This scale of notation helps to index the elements of a number of factorial sets and establish a one-to-one correspondence between the number and a specific type of substitution. This allows analyzing substitutions characteristics systematically. This paper presents the basic concepts of a number of the factorial sets. It is noted that the permutations of the factorial sets form symmetric permutation groups, and specific permutations (when raised to a power) form cyclic groups. The group axioms are fulfilled for the permutations of a number of factorial sets. Also, the definition domain, the group operation of multiplication, and identical and inverse substitutions are given for them. The number of independent cycles, decrement, inverse, parity and sign are common characteristics of the substitutions of a number of factorial sets. The criteria for choosing single substitutions with the best characteristics are proposed.
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DIEKERT, VOLKER, NICOLE ONDRUSCH, and MARKUS LOHREY. "ALGORITHMIC PROBLEMS ON INVERSE MONOIDS OVER VIRTUALLY FREE GROUPS." International Journal of Algebra and Computation 18, no. 01 (February 2008): 181–208. http://dx.doi.org/10.1142/s0218196708004366.
Повний текст джерелаАнотація:
Let G be a finitely generated virtually-free group. We consider the Birget–Rhodes expansion of G, which yields an inverse monoid and which is denoted by IM (G) in the following. We show that for a finite idempotent presentation P, the word problem of a quotient monoid IM (G)/P can be solved in linear time on a RAM. The uniform word problem, where G and the presentation P are also part of the input, is EXPTIME-complete. With IM (G)/P we associate a relational structure, which contains for every rational subset L of IM (G)/P a binary relation, consisting of all pairs (x,y) such that y can be obtained from x by right multiplication with an element from L. We prove that the first-order theory of this structure is decidable. This result implies that the emptiness problem for Boolean combinations of rational subsets of IM (G)/P is decidable, which, in turn implies the decidability of the submonoid membership problem of IM (G)/P. These results were known previously for free groups, only. Moreover, we provide a new algorithmic approach for these problems, which seems to be of independent interest even for free groups. We also show that one cannot expect decidability results in much larger frameworks than virtually-free groups because the subgroup membership problem of a subgroup H in an arbitrary group G can be reduced to a word problem of some IM (G)/P, where P depends only on H. A consequence is that there is a hyperbolic group G and a finite idempotent presentation P such that the word problem is undecidable for some finitely generated submonoid of IM (G)/P. In particular, the word problem of IM (G)/P is undecidable.
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Mukhaiyar, Riki. "Generating a Cancellable Fingerprint using Matrices Operations and Its Fingerprint Processing Requirements." Asian Social Science 14, no. 6 (May 28, 2018): 1. http://dx.doi.org/10.5539/ass.v14n6p1.
Повний текст джерелаАнотація:
Cancellable fingerprint uses transformed or intentionally distorted biometric data instead of the original biometric data for identifying person. When a set of biometric data is found to be compromised, they can be discarded, and a new set of biometric data can be regenerated. This initial principal is identical with a non-invertible concept in matrices operations. In matrix domain, a matrix cannot be transformed into its original form if it meets several requirements such as non-square form matrix, consist of one zero row/column, and no row as multiple of another row. These conditions can be acquired by implementing three matrix operations using Kronecker Product (KP) operation, Elementary Row Operation (ERO), and Inverse Matrix (INV) operation. KP is useful to produce a non-square form matrix, to enlarge the size of matrix, to distinguish and disguise the element of matrix by multiplying each of elements of the matrix with a particular matrix. ERO can be defined as multiplication and addition force to matrix rows. INV is utilized to transform one matrix to another one with a different element or form as a reciprocal matrix of the original. These three matrix operations should be implemented together in generating the cancellable feature to robust image. So, if once three conditions are met by imposter, it is impossible to find the original image of the fingerprint. The initial aim of these operations is to camouflage the original look of the fingerprint feature into an abstract-look to deceive an un-authorized personal using the fingerprint irresponsibly. In this research, several fingerprint processing steps such as fingerprint pre-processing, core-point identification, region of interest, minutiae extration, etc; are determined to improve the quality of the cancellable feature. Three different databases i.e. FVC2002, FVC2004, and BRC are utilized in this work.
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Easdown, David, and Victoria Gould. "Orders in power semigroups." Glasgow Mathematical Journal 38, no. 1 (January 1996): 39–47. http://dx.doi.org/10.1017/s0017089500031232.
Повний текст джерелаАнотація:
In this paper we consider examples of orders in restricted power semigroups, where for any semigroup Sthe restricted power semigroup is given by with multiplication XY = {xy:x ∈ X, y ∈ Y} for all X, Y ∈ . We use the notion of order introduced by Fountain and Petrich in [2] which first appears in the form used here in [3]. If S is a subsemigroup of Q then S is an order in Q and Q is a semigroup of quotients of S if any q ∈ Q can be written as q = a*b = cd* where a, b, c, d ∈ S is the inverse of a(d) in a subgroup of Q, and in addition, all elements of S satisfying a weak cancellability condition called square-cancellability lie in a subgroup of Q.
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Van Tran, Nam, and Imme van den Berg. "An algebraic model for the propagation of errors in matrix calculus." Special Matrices 8, no. 1 (March 5, 2020): 68–97. http://dx.doi.org/10.1515/spma-2020-0008.
Повний текст джерелаАнотація:
AbstractWe assume that every element of a matrix has a small, individual error, and model it by an external number, which is the sum of a nonstandard real number and a neutrix, the latter being a convex (external) additive group. The algebraic properties of external numbers formalize common error analysis, with rules for calculation which are a sort of mellowed form of the axioms for real numbers.We model the propagation of errors in matrix calculus by the calculus of matrices with external numbers, and study its algebraic properties. Many classical properties continue to hold, sometimes stated in terms of inclusion instead of equality. There are notable exceptions, for which we give counterexamples and investigate suitable adaptations. In particular we study addition and multiplication of matrices, determinants, near inverses, and generalized notions of linear independence and rank.
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Amento, Brittanney, Martin Rotteler, and Rainer Steinwandt. "Quantum binary field inversion: improved circuit depth via choice of basis representation." Quantum Information and Computation 13, no. 1&2 (January 2013): 116–34. http://dx.doi.org/10.26421/qic13.1-2-7.
Повний текст джерелаАнотація:
Finite fields of the form ${\mathbb F}_{2^m}$ play an important role in coding theory and cryptography. We show that the choice of how to represent the elements of these fields can have a significant impact on the resource requirements for quantum arithmetic. In particular, we show how the use of Gaussian normal basis representations and of `ghost-bit basis' representations can be used to implement inverters with a quantum circuit of depth $\bigO(m\log(m))$. To the best of our knowledge, this is the first construction with subquadratic depth reported in the literature. Our quantum circuit for the computation of multiplicative inverses is based on the Itoh-Tsujii algorithm which exploits that in normal basis representation squaring corresponds to a permutation of the coefficients. We give resource estimates for the resulting quantum circuit for inversion over binary fields ${\mathbb F}_{2^m}$ based on an elementary gate set that is useful for fault-tolerant implementation.
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Han, Wei, Kan Feng, and Huagen Yang. "Phase Reversal Method for Damage Imaging in Composite Laminates Based on Data Fusion." Applied Sciences 12, no. 6 (March 11, 2022): 2894. http://dx.doi.org/10.3390/app12062894.
Повний текст джерелаАнотація:
This paper proposes a phase-reversal method (PRM) for damage imaging in plate structures. The PRM is a novel Lamb-wave-based method that mainly focuses on phase spectrum information of scattering waves reflected from a defect. The PRM reverses the phase angle along the propagation path by using the inverse Fourier transform first, and then the reversal reconstruction of the wave field in the frequency domain is performed for damage imaging. The proposed method analyzes the scattered wave field without using the baseline data and structural parameters. Moreover, dispersion characteristics and anisotropy are not involved in the process of damage positioning, thus making the PRM suitable for damage monitoring of composite laminates. To improve the PRM accuracy further, a combined addition and multiplication method of the correlation coefficient (CAMM) is proposed, which can reduce the effects of phase and noise artifacts and distortion. The results of the finite element simulations and experiments show that the combination of the PRM and CAMM methods can accurately locate damage in composite structures. Therefore, the PRM and CAMM methods have great application potential in damage imaging in composite laminates.
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Ren, Zihan, Peng Li, and Xin Wang. "An Implementation of Image Secret Sharing Scheme Based on Matrix Operations." Mathematics 10, no. 6 (March 9, 2022): 864. http://dx.doi.org/10.3390/math10060864.
Повний текст джерелаАнотація:
The image secret sharing scheme shares a secret image as multiple shadows. The secret image can be recovered from shadow images that meet a threshold number. However, traditional image secret sharing schemes generally reuse the Lagrange’s interpolation in the recovery stage to obtain the polynomial in the sharing stage. Since the coefficients of the polynomial are the pixel values of the secret image, it is able to recover the secret image. This paper presents an implementation of the image secret sharing scheme based on matrix operations. Different from the traditional image secret sharing scheme, this paper does not use the method of Lagrange’s interpolation in the recovery stage, but first identifies the participants as elements to generate a matrix and calculates its inverse matrix. By repeating the matrix multiplication, the polynomial coefficients of the sharing stage are quickly derived, and then the secret image is recovered. By theoretical analysis and the experimental results, the implementation of secret image sharing based on matrix operation is higher than Lagrange’s interpolation in terms of efficiency.
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Alekseev, Aleksander. "CONTROL OF A COMPLEX OBJECTS, STATES OF WHICH ARE DESCRIBING BY THE MATRIX RATING MECHANISM." Applied Mathematics and Control Sciences, no. 1 (March 27, 2020): 114–39. http://dx.doi.org/10.15593/2499-9873/2020.1.08.
Повний текст джерелаАнотація:
The control problem of a multi-criteria object is considered. Controlled object that has several criteria that are significant for a decision maker. Each criterion characterizes a control object in terms of a particular result of activity or an efficiency indicator. To evaluate the effectiveness of the functioning of the managed facility as a whole, the rating matrix mechanism is used, taking into account all the criteria in the complex. The optimal control problem is formulated as a search for the values of aggregated criteria that provide a given value of a complex indicator with minimal costs for providing values of particular criteria. The generalized cost function was reduced to an equation with one variable. The analytical equation of the level line of the indicator aggregated as a result of the convolution of two criteria is obtained. The line equation is found for an arbitrary binary convolution matrix, including the elements of which are given continuous values. It is shown that the objective function is reduced to a fourth-order polynomial, which can be analytically solved using the Ferrari or Descartes-Euler methods. It is shown that the task of searching for the values of two particular criteria describing the state of the control object for which the complex indicator calculated using the additive-multiplicative approach to complex assessment is equal to the given value and the costs for their provision are minimal, has a solution in general form for arbitrary nondecreasing convolution matrix of two criteria. Particular solutions to the control problem are found using costly functions, which are the inverse function of the Cobb-Douglas production function. It was shown that the cost function of the aggregate indicator has additional terms and is described by an algebraic equation with nonzero coefficients for variables and an additional constant. Based on what it was concluded that the cost functions, which are the inverse function e of the Cobb-Douglas production function, can be applied to control objects that have only two criteria. A similar formulation of the control problem for an arbitrary non-decreasing convolution matrix of two criteria is considered when using the additive-multiplicative approach to aggregation and when using cost functions described by a second-order algebraic equation in general form. As a result of the study, it is shown that the form of the cost function for the aggregated indicator is preserved. Thus, using cost functions in the form of second-order equations, the control problem has a solution in the general form for any number of criteria.
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Hofmann, B., D. Düvelmeyer, and K. Krumbiegel. "APPROXIMATE SOURCE CONDITIONS IN TIKHONOV REGULARIZATION‐NEW ANALYTICAL RESULTS AND SOME NUMERICAL STUDIES." Mathematical Modelling and Analysis 11, no. 1 (March 31, 2006): 41–56. http://dx.doi.org/10.3846/13926292.2006.9637301.
Повний текст джерелаАнотація:
We present some new ideas and results for finding convergence rates in Tikhonov regularization for ill‐posed linear inverse problems with compact and non‐compact forward operators based on the consideration of approximate source conditions and corresponding distance functions. The new results and studies complement and extend in numerous points the recent papers [5, 7, 8, 10] that also exploit the distance functions originally introduced in [2] which measure the violation of a moderate source condition that works as a benchmark. In this context, we distinguish as in [8] logarithmic, power and exponential decay rates for the distance functions and their consequences. Under specific range inclusions the decay rate of distance functions is verified explicitly, whereas in [10] this result is also used but formulated only in an implicit manner. Applications to non‐compact multiplication operators are briefly reviewed from [8]. An important new result is that we can show for compact operators a one‐to‐one correspondence between the maximal power type decay rates for the distance functions and maximal exponents of Holder rates in Tikhonov regularization linked by the specific singular value expansion of the solution element. Some numerical studies on simple integration illustrate the compact operator case and the specific situation of discretized problems. Finally, some ideas of generalization are mentioned concerning the fact that the benchmark of the distance function can be shifted.
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Kliuchenia, V. V. "Architecture of the discrete sosine transformation processor for image compression systems on the losless-to-lossy circuit." Doklady BGUIR 19, no. 5 (August 26, 2021): 86–93. http://dx.doi.org/10.35596/1729-7648-2021-19-5-86-93.
Повний текст джерелаАнотація:
The hardware implementations of fixed-point DCT blocks, known as IntDCT [1] and BinDCT [2], require some solutions. One of the main issues is the choice between the implementation of the conversion on FPGA, or the implementation on a digital signal processor (Digital Signal Processor, DSP). Each of the implementations has its own pros and cons. One of the most important advantages of the DSP implementation is the presence of special instructions used in DSP, in particular, the ability to multiply two numbers in one clock cycle. Therefore, with the advent of DSP, the limitation on the number of multiplications in algorithms was removed. On the other hand, when implementing a block on an FPGA, we can limit not ourselves to the bitness of the data (within reasonable limits), we have the ability to parallelize all incoming data and implement specialized computing cores for various tasks. In fact, designing multimedia systems on FPGAs reminds the design of similar systems based on the logic of a small and medium degree of integration. Such an implementation has the same limitations: a relatively small amount of available memory, the need to design basic structural elements (multipliers, divisors), etc. It is the inequality of the addition and multiplication operations when they are implemented on FPGAs that caused the search for DCT algorithms with the smallest number of factors. However, even this is not enough, since the structure of the multiplier is many times more complex than the structure of the adder, which made it necessary to look for ways to transform without using multiplications at all. This article shows how, on the basis of integer direct and inverse DCT and distributed arithmetic, to create a new universal architecture of decorrelated transform on FPGAs without multiplication operations for image transformation coding systems that operate on the principle of lossless-to-lossy (L2L), and to obtain the best experimental results in terms of hardware resources compared to comparable compression systems.
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Cardei, Petru. "Alternative Representations of Some Arithmetic Functions." PROOF 2 (May 9, 2022): 115–22. http://dx.doi.org/10.37394/232020.2022.2.14.
Повний текст джерелаАнотація:
This article presents some results of the attempt to simplify the writing of arithmetic functions on the computer so that users can apply them without additional operations, such as summing after a set whose elements must be calculated, such as the set of numbers prime. The important role of the remainder function in defining most arithmetic functions is highlighted. Defining algorithms for the prime factorization of natural numbers highlights the possibility of representing natural numbers in a basis as "natural" as possible for natural numbers, namely the basis of prime numbers. The disadvantage of this natural basis is, for the time being, that it is infinitely dimensional. For now, this representation provides advantages but also disadvantages. Among the arithmetic functions proposed in the article, there are also statistical characterizations of the distribution of prime numbers, given with the hope of helping a better knowledge of the set of prime numbers. The importance of the remainder function in the computational definitions of arithmetic functions leads to reflections on the importance of fundamental operations - addition and multiplication - of natural numbers and the importance of inverse functions - subtraction and division. In turn, these operations can be seen as functions of two variables on the set of natural numbers. From here, readers are invited to reflect on the problem of the origin of natural numbers, the origin based on revelation or the origin provided by set theory, although this may also be a revelation.
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Ruffato, Gianluca, Michele Massari, and Filippo Romanato. "Multiplication and division of the orbital angular momentum of light with diffractive transformation optics." Light: Science & Applications 8, no. 1 (December 2019). http://dx.doi.org/10.1038/s41377-019-0222-2.
Повний текст джерелаАнотація:
AbstractWe present a method to efficiently multiply or divide the orbital angular momentum (OAM) of light beams using a sequence of two optical elements. The key element is represented by an optical transformation mapping the azimuthal phase gradient of the input OAM beam onto a circular sector. By combining multiple circular-sector transformations into a single optical element, it is possible to multiply the value of the input OAM state by splitting and mapping the phase onto complementary circular sectors. Conversely, by combining multiple inverse transformations, the division of the initial OAM value is achievable by mapping distinct complementary circular sectors of the input beam into an equal number of circular phase gradients. Optical elements have been fabricated in the form of phase-only diffractive optics with high-resolution electron-beam lithography. Optical tests confirm the capability of the multiplier optics to perform integer multiplication of the input OAM, whereas the designed dividers are demonstrated to correctly split up the input beam into a complementary set of OAM beams. These elements can find applications for the multiplicative generation of higher-order OAM modes, optical information processing based on OAM beam transmission, and optical routing/switching in telecom.
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Witte, Julius, Daniel Arndt, and Guido Kanschat. "Fast Tensor Product Schwarz Smoothers for High-Order Discontinuous Galerkin Methods." Computational Methods in Applied Mathematics, November 11, 2020. http://dx.doi.org/10.1515/cmam-2020-0078.
Повний текст джерелаАнотація:
AbstractWe discuss the efficient implementation of powerful domain decomposition smoothers for multigrid methods for high-order discontinuous Galerkin (DG) finite element methods. In particular, we study the inversion of matrices associated to mesh cells and to the patches around a vertex, respectively, in order to obtain fast local solvers for additive and multiplicative subspace correction methods. The effort of inverting local matrices for tensor product polynomials of degree k is reduced from {\mathcal{O}(k^{3d})} to {\mathcal{O}(dk^{d+1})} by exploiting the separability of the differential operator and resulting low rank representation of its inverse as a prototype for more general low rank representations in space dimension d.
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Mohd Ali, Nor Muhainiah, Nur Azura Noor Azhuan, Nor Haniza Sarmin, and Farhana Johar. "The Computation of Some Properties of Additive and Multiplicative Groups of Integers Modulo n Using C++ Programming." Sains Humanika 9, no. 1-2 (January 2, 2017). http://dx.doi.org/10.11113/sh.v9n1-2.1100.
Повний текст джерелаАнотація:
This research is focused on two types of finite abelian groups which are the group of integers under addition modulo , and the group of integers under multiplication modulo , where is any positive integer at most 200. The computations of some properties of the group including the order of the group, the order and inverse of each element, the cyclic subgroups, the generators of the group, and the lattice diagrams get more complicated and time consuming as n increases. Therefore, a special program is needed in the computation of these properties. Thus in this research, a program has been developed by using Microsoft Visual C++ Programming. This program enables the user to enter any positive integer at most 200 to generate answers for the properties of the groups.
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He, Baochang, Jing Wang, Jing Lin, Jinfa Chen, Zhaocheng Zhuang, Yihong Hong, Lingjun Yan, et al. "Association Between Rare Earth Element Cerium and the Risk of Oral Cancer: A Case-Control Study in Southeast China." Frontiers in Public Health 9 (May 25, 2021). http://dx.doi.org/10.3389/fpubh.2021.647120.
Повний текст джерелаАнотація:
Cerium (Ce), the most abundant of rare earth elements in the earth's crust, has received much health concerns due to its wide application in industry, agriculture, and medicine. The current study aims to evaluate whether there is an association between Ce exposures and the risk of developing oral cancer. Serum Ce level of 324 oral cancer patients and 650 matched healthy controls were measured by inductively coupled plasma mass spectrometry. Association between Ce level and the risk of oral cancer was estimated with an unconditional logistic regression model. Serum Ce concentrations in the oral cancer patients and controls were 0.57 (0.21–3.02) μg/L and 2.27 (0.72–4.26) μg/L, respectively. High level of Ce was associated with a decreased risk of oral cancer (OR: 0.60, 95% CI: 0.43–0.84). Stronger inverse associations between high level of Ce and oral cancer risk were observed among those with smoking (OR: 0.46, 95% CI: 0.27–0.79), drinking (OR: 0.50, 95% CI: 0.26–0.96), limited intake of leafy vegetables (OR: 0.40, 95% CI: 0.22–0.71) and fish (OR: 0.52, 95% CI: 0.33–0.83). There were significant multiplicative interactions between Ce level and alcohol drinking or intake of leafy vegetables and fish (all Pinteraction <0.05). This preliminary case-control study suggests an inverse association between high serum Ce level and the risk of oral cancer. Further prospective studies with a larger sample size are needed to confirm the findings.
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Bhardwaj, Kapil, and Mayank Srivastava. "Floating Memristor and Inverse Memristor Emulators with Electronic Tuning." Journal of Circuits, Systems and Computers, April 5, 2021, 2150224. http://dx.doi.org/10.1142/s0218126621502248.
Повний текст джерелаАнотація:
The work reports two different configurations to emulate the floating memristor and inverse memristor behavior. The presented circuits are based on a modified concept of active element VDTA (Voltage Differencing Transconductance Amplifier) termed as MVDTA. The reported floating memristor employs only a single MVDTA and single grounded capacitance. On the other end, the floating emulation circuit of inverse memristor emulator is based on two MVDTAs and single grounded capacitance. The behavior of the realized element for both the configurations can be tuned electronically through biasing voltage. Also, there is no employment of any commercial IC or external circuitry for multiplication of analogue voltages which is generally required to implement memristive elements. Along with the circuit implementations, mathematical properties of ideal memristor and inverse memristor considering both symmetric as well as nonsymmetric models are discussed. All the emulation circuits are verified by executing simulations using CMOS 0.18[Formula: see text]um process technique under PSPICE environment. The reported circuits are also realized using commercially available IC LM13700 and results are presented.
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Fan, Wei, Hui Ren, Ren Ju, and Weidong Zhu. "On the Approximation of the Full Mass Matrix in the Rotational-Coordinate-Based Beam Formulation." Journal of Computational and Nonlinear Dynamics 15, no. 4 (February 24, 2020). http://dx.doi.org/10.1115/1.4046245.
Повний текст джерелаАнотація:
Abstract A novel approach is developed to approximate the full mass matrix in the rotational-coordinate-based beam formulation, which can improve the efficiency of calculating its inverse in dynamic analyses. While the rotational-coordinate-based beam formulation can reduce numbers of elements and generalized coordinates, its mass matrix is a full matrix, such that corresponding Jacobian matrix is also full, and it is time-consuming to calculate its inverse. To increase efficiency of calculating its inverse, the full mass matrix is approximated in this work. Two approximations are adopted: (1) a double integral is approximated by a single integral; and (2) a full matrix is approximated by a sum of several rank-one matrices. Through this way, the approximate mass matrix can be decomposed as a band-diagonal sparse matrix and multiplication of low-rank matrices, and its inverse can be efficiently calculated using Sherman–Woodbury formula. Through this way, the approximate mass matrix can be efficiently calculated. Several numerical examples are presented to demonstrate the performance of the current approach, and its accuracy and efficiency are analyzed.
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Yin, Hao, Erol Lale, and Gianluca Cusatis. "GENERALIZED FORMULATION FOR THE BEHAVIOR OF GEOMETRICALLY CURVED AND TWISTED THREE-DIMENSIONAL TIMOSHENKO BEAMS AND ITS ISOGEOMETRIC ANALYSIS IMPLEMENTATION." Journal of Applied Mechanics, April 27, 2022, 1–25. http://dx.doi.org/10.1115/1.4054438.
Повний текст джерелаАнотація:
Abstract This paper presents a novel derivation for the governing equations of geometrically curved and twisted three-dimensional Timoshenko beams. The kinematic model of the beam was derived rigorously by adopting a parametric description of the axis of the beam, using the local Frenet-Serret reference system, and introducing the constraint of the beam cross-section planarity into the classical, first-order strain versus displacement relations for Cauchy&#39;s continua. The resulting beam kinematic model includes a multiplicative term consisting of the inverse of the Jacobian of the beam axis curve. This term is not included in classical beam formulations available in the literature; its contribution vanishes exactly for straight beams and is negligible only for curved and twisted beams with slender geometry. Furthermore, to simplify the description of complex beam geometries, the governing equations were derived with reference to a generic position of the beam axis within the beam cross-section. Finally, this study pursued the numerical implementation of the curved beam formulation within the conceptual framework of isogeometric analysis, which allows the exact description of the beam geometry. This avoids stress locking issues and the corresponding convergence problems encountered when classical straight beam finite elements are used to discretize the geometry of curved and twisted beams. Finally, the paper presents the solution of several numerical examples to demonstrate the accuracy and effectiveness of the proposed theoretical formulation and numerical implementation.