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Journal articles on the topic 'Arithmetic applications'

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Published: 30 May 2022

Last updated: 31 May 2022

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1

de Figueiredo, Luiz Henrique, and Jorge Stolfi. "Affine Arithmetic: Concepts and Applications." Numerical Algorithms 37, no. 1-4 (December 2004): 147–58. http://dx.doi.org/10.1023/b:numa.0000049462.70970.b6.

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2

Milea, Suzana, Christopher D. Shelley, and Martin H. Weissman. "Arithmetic of arithmetic Coxeter groups." Proceedings of the National Academy of Sciences 116, no. 2 (December 26, 2018): 442–49. http://dx.doi.org/10.1073/pnas.1809537115.

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In the 1990s, J. H. Conway published a combinatorial-geometric method for analyzing integer-valued binary quadratic forms (BQFs). Using a visualization he named the “topograph,” Conway revisited the reduction of BQFs and the solution of quadratic Diophantine equations such as Pell’s equation. It appears that the crux of his method is the coincidence between the arithmetic group PGL2(Z) and the Coxeter group of type (3,∞). There are many arithmetic Coxeter groups, and each may have unforeseen applications to arithmetic. We introduce Conway’s topograph and generalizations to other arithmetic Coxeter groups. This includes a study of “arithmetic flags” and variants of binary quadratic forms.

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3

Wood, Eric F. "Applications: Self-Checking Codes—An Application of Modular Arithmetic." Mathematics Teacher 80, no. 4 (April 1987): 312–16. http://dx.doi.org/10.5951/mt.80.4.0312.

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In the teaching of mathematics or computer science in secondary school, it is sometimes difficult to find material to present to the students that is interesting, comprehensible, and, at the same time, representative of modern achievement in the discipline. In this article I present two very relevant applications of mathematics and computer science that all students can relate to: International Standard Book Numbers (ISBN) that appear in textbooks and universal product codes that appear on grocery products.

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4

Harris, Michael. "Arithmetic applications of the Langlands program." Japanese Journal of Mathematics 5, no. 1 (April 2010): 1–71. http://dx.doi.org/10.1007/s11537-010-0945-6.

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5

Polonsky, S. V., Jao Ching Lin, and A. V. Rylyakov. "RSFQ arithmetic blocks for DSP applications." IEEE Transactions on Appiled Superconductivity 5, no. 2 (June 1995): 2823–26. http://dx.doi.org/10.1109/77.403179.

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6

Audeh, Wasim. "Applications of Arithmetic Geometric Mean Inequality." Advances in Linear Algebra & Matrix Theory 07, no. 02 (2017): 29–36. http://dx.doi.org/10.4236/alamt.2017.72004.

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7

Marschall, Tobias, Inke Herms, Hans-Michael Kaltenbach, and Sven Rahmann. "Probabilistic Arithmetic Automata and Their Applications." IEEE/ACM Transactions on Computational Biology and Bioinformatics 9, no. 6 (November 2012): 1737–50. http://dx.doi.org/10.1109/tcbb.2012.109.

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8

Rodríguez-Villegas, E., M. J. Avedillo, J. M. Quintana, G. Huertas, and A. Rueda. "νMOS-based Sorter for Arithmetic Applications." VLSI Design 11, no. 2 (January 1, 2000): 129–36. http://dx.doi.org/10.1155/2000/57240.

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The capabilities of the conceptual link between threshold gates and sorting networks are explored by implementing some arithmetic demonstrators. In particular, both an (8 × 8)-multiplier and a (15, 4) counter which use a sorter as the main building block have been implemented. Traditional disadvantages of binary sorters such as their hardware intensive nature are avoided by using νMOS circuits. It allows both an improving of previous results for multipliers based on a similar architecture, and to obtain a new type of counter which shows a reduced delay when compared to a conventional implementation.

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9

Kim, Dae San, Taekyun Kim, Seog-Hoon Rim, and Sang Hun Lee. "Hermite Polynomials and their Applications Associated with Bernoulli and Euler Numbers." Discrete Dynamics in Nature and Society 2012 (2012): 1–13. http://dx.doi.org/10.1155/2012/974632.

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We derive some interesting identities and arithmetic properties of Bernoulli and Euler polynomials from the orthogonality of Hermite polynomials. LetPn={p(x)∈ℚ[x]∣deg p(x)≤n}be the(n+1)-dimensional vector space overℚ. Then we show that{H0(x),H1(x),…,Hn(x)}is a good basis for the spacePnfor our purpose of arithmetical and combinatorial applications.

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10

Wu, X. M., Gui Xian Li, De Bin Shan, and G. B. Yu. "RBF Neural Network Arithmetic and Applications in Surface Interpolation Reconstruction." Key Engineering Materials 460-461 (January 2011): 575–80. http://dx.doi.org/10.4028/www.scientific.net/kem.460-461.575.

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Aiming at problems such as: surface interpolation reconstruction of points cloud data，surface hole filling and two simple surface connection, a neural network arithmetic was employed. Based on radial basis function neural network, simulated annealing was employed to adjust the network weights. The new arithmetic can approach any nonlinear function by arbitrary precision, and also keep the network from getting into local minimum for global optimization feature of simulated annealing. MATLAB program was compiled, experiments on points cloud data have been done employing this arithmetic, the result shows that this arithmetic can efficiently approach the surface with 10-4 mm error precision, and also the learning speed is quick and reconstruction surface is smooth.

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11

Markov, Svetoslav, Rene Alt, and Jean-Luc Lamotte. "Stochastic Arithmetic: s-spaces and Some Applications." Numerical Algorithms 37, no. 1-4 (December 2004): 275–84. http://dx.doi.org/10.1023/b:numa.0000049474.51465.41.

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12

Litvin, M. E., and S. Mourad. "Self-reset logic for fast arithmetic applications." IEEE Transactions on Very Large Scale Integration (VLSI) Systems 13, no. 4 (April 2005): 462–75. http://dx.doi.org/10.1109/tvlsi.2004.842921.

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13

Bohlender, Gerd, Arnold Kaufmann, and Madan M. Gupta. "Introduction to Fuzzy Arithmetic, Theory and Applications." Mathematics of Computation 47, no. 176 (October 1986): 762. http://dx.doi.org/10.2307/2008199.

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14

Zhdanov, D. A. "Filtered Arithmetic Mean Measure and Its Applications." Theory of Probability & Its Applications 53, no. 2 (January 2009): 368–79. http://dx.doi.org/10.1137/s0040585x97983596.

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15

Kahrobaei, Delaram, and Keivan Mallahi-Karai. "Some applications of arithmetic groups in cryptography." Groups Complexity Cryptology 11, no. 1 (May 1, 2019): 25–33. http://dx.doi.org/10.1515/gcc-2019-2002.

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AbstractIn this paper, we will offer a new symmetric-key cryptographic scheme which is based on the existence of exponentially distorted subgroups in arithmetic groups. Aside from this, we will also provide new examples of distorted subgroups in {\mathrm{SL}_{n}(\mathbb{Z}[x])} which can be utilized for the same purpose.

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16

Berenstein, Arkady, and Alek Vainshtein. "Concavity of weighted arithmetic means with applications." Archiv der Mathematik 69, no. 2 (August 1, 1997): 120–26. http://dx.doi.org/10.1007/s000130050101.

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17

Chen, N. X., F. Y. Zhu, and Y. H. Ku. "Introduction to fuzzy arithmetic—theory and applications." Journal of the Franklin Institute 321, no. 3 (March 1986): 189–90. http://dx.doi.org/10.1016/0016-0032(86)90009-8.

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18

Eastman, Caroline M. "Introduction to fuzzy arithmetic: Theory and applications." International Journal of Approximate Reasoning 1, no. 1 (January 1987): 145–46. http://dx.doi.org/10.1016/0888-613x(87)90010-7.

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19

Eastman, Caroline M. "Introduction to fuzzy arithmetic: Theory and applications." International Journal of Approximate Reasoning 1, no. 1 (January 1987): 141–43. http://dx.doi.org/10.1016/0888-613x(87)90009-0.

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20

Sousa, Leonel. "Nonconventional Computer Arithmetic Circuits, Systems and Applications." IEEE Circuits and Systems Magazine 21, no. 1 (2021): 6–40. http://dx.doi.org/10.1109/mcas.2020.3027425.

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21

Ramya Rani, N. "Implementation of Embedded Floating Point Arithmetic Units on FPGA." Applied Mechanics and Materials 550 (May 2014): 126–36. http://dx.doi.org/10.4028/www.scientific.net/amm.550.126.

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:Floating point arithmetic plays a major role in scientific and embedded computing applications. But the performance of field programmable gate arrays (FPGAs) used for floating point applications is poor due to the complexity of floating point arithmetic. The implementation of floating point units on FPGAs consumes a large amount of resources and that leads to the development of embedded floating point units in FPGAs. Embedded applications like multimedia, communication and DSP algorithms use floating point arithmetic in processing graphics, Fourier transformation, coding, etc. In this paper, methodologies are presented for the implementation of embedded floating point units on FPGA. The work is focused with the aim of achieving high speed of computations and to reduce the power for evaluating expressions. An application that demands high performance floating point computation can achieve better speed and density by incorporating embedded floating point units. Additionally this paper describes a comparative study of the design of single precision and double precision pipelined floating point arithmetic units for evaluating expressions. The modules are designed using VHDL simulation in Xilinx software and implemented on VIRTEX and SPARTAN FPGAs.

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22

Yamashita, Yukihiko. "Arithmetic Coder." Journal of The Institute of Image Information and Television Engineers 66, no. 1 (2012): 65–67. http://dx.doi.org/10.3169/itej.66.65.

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23

Stankovic, Radomir, and Jaakko Astola. "Remarks on applications of arithmetic expressions for efficient implementation of elementary functions." Facta universitatis - series: Electronics and Energetics 20, no. 3 (2007): 295–308. http://dx.doi.org/10.2298/fuee0703295s.

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It has been recently shown in [1], that elementary mathematical functions (as trigonometric, logarithmic, square root, gaussian, sigmoid, etc) are compactly represented by the Arithmetic transform expressions and related Binary Moment Diagrams (BMDs). The complexity of the representations is estimated through the number of non-zero coefficients in arithmetic expressions and the number of nodes in BMDs. In this paper, we show that further optimization can be achieved when the method in [1] is combined with Fixed-polarity Arithmetic expressions (FPRAs). In addition, besides complexity measures used in [1], we also compared the number of bits and 1-bits required to represent arithmetic transform coefficients in zero polarity and optimal polarity arithmetic expressions. This is a complexity measure relevant for the alternative implementations of elementary functions suggested in [1]. Experimental results confirm that exploiting of FPARs may provide for considerable reduction in terms of the complexity measures considered.

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24

Collins, Pieter. "Computable analysis with applications to dynamic systems." Mathematical Structures in Computer Science 30, no. 2 (February 2020): 173–233. http://dx.doi.org/10.1017/s096012952000002x.

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AbstractNumerical computation is traditionally performed using floating-point arithmetic and truncated forms of infinite series, a methodology which allows for efficient computation at the cost of some accuracy. For most applications, these errors are entirely acceptable and the numerical results are considered trustworthy, but for some operations, we may want to have guarantees that the numerical results are correct, or explicit bounds on the errors. To obtain rigorous calculations, floating-point arithmetic is usually replaced by interval arithmetic and truncation errors are explicitly contained in the result. We may then ask the question of which mathematical operations can be implemented in a way in which the exact result can be approximated to arbitrary known accuracy by a numerical algorithm. This is the subject of computable analysis and forms a theoretical underpinning of rigorous numerical computation. The aim of this article is to provide a straightforward introduction to this subject that is powerful enough to answer questions arising in dynamic system theory.

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25

Zribi, Amin, Sonia Zaibi, Ramesh Pyndiah, and Ammar Bouallègue. "Chase-Like Decoding of Arithmetic Codes with Applications." International Journal of Computer Vision and Image Processing 1, no. 1 (January 2011): 27–40. http://dx.doi.org/10.4018/ijcvip.2011010103.

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Motivated by recent results in Joint Source/ Channel (JSC) coding and decoding, this paper addresses the problem of soft input decoding of Arithmetic Codes (AC). A new length-constrained scheme for JSC decoding of these codes is proposed based on the Maximum a posteriori (MAP) sequence estimation criterion. The new decoder, called Chase-like arithmetic decoder is supposed to know the source symbol sequence and the compressed bit-stream lengths. First, Packet Error Rates (PER) in the case of transmission on an Additive White Gaussian Noise (AWGN) channel are investigated. Compared to classical arithmetic decoding, the Chase-like decoder shows significant improvements. Results are provided for Chase-like decoding for image compression and transmission on an AWGN channel. Both lossy and lossless image compression schemes were studied. As a final application, the serial concatenation of an AC with a convolutional code was considered. Iterative decoding, performed between the two decoders showed substantial performance improvement through iterations.

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26

Darmon, Henri. "Integration on ℋ p × ℋ and Arithmetic Applications." Annals of Mathematics 154, no. 3 (November 2001): 589. http://dx.doi.org/10.2307/3062142.

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27

Aguirre-Hernandez, Mariano, and Monico Linares-Aranda. "CMOS Full-Adders for Energy-Efficient Arithmetic Applications." IEEE Transactions on Very Large Scale Integration (VLSI) Systems 19, no. 4 (April 2011): 718–21. http://dx.doi.org/10.1109/tvlsi.2009.2038166.

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28

Elstrodt, J., F. Grunewald, and J. Mennicke. "Arithmetic Applications of the Hyperbolic Lattice Point Theorem." Proceedings of the London Mathematical Society s3-57, no. 2 (September 1988): 239–83. http://dx.doi.org/10.1112/plms/s3-57.2.239.

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29

Rajagopal, S., and J. R. Cavallaro. "Truncated Online Arithmetic with Applications to Communication Systems." IEEE Transactions on Computers 55, no. 10 (October 2006): 1240–12529. http://dx.doi.org/10.1109/tc.2006.168.

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30

Choi, So Young, and Ja Kyung Koo. "Estimation of genus of arithmetic curves and applications." Ramanujan Journal 15, no. 1 (December 18, 2007): 1–17. http://dx.doi.org/10.1007/s11139-007-9063-3.

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31

Strobach, Peter. "Efficient covariance ladder algorithms for finite arithmetic applications." Signal Processing 13, no. 1 (July 1987): 29–70. http://dx.doi.org/10.1016/0165-1684(87)90111-3.

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32

Bettin, Sandro, and Sary Drappeau. "Two arithmetic applications of perturbations of composition operators." Journal d'Analyse Mathématique 144, no. 1 (December 2021): 335–49. http://dx.doi.org/10.1007/s11854-021-0184-1.

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33

Yuan, Xinyi. "On volumes of arithmetic line bundles." Compositio Mathematica 145, no. 6 (November 2009): 1447–64. http://dx.doi.org/10.1112/s0010437x0900428x.

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AbstractWe show an arithmetic generalization of the recent work of Lazarsfeld–Mustaţǎ which uses Okounkov bodies to study linear series of line bundles. As applications, we derive a log-concavity inequality on volumes of arithmetic line bundles and an arithmetic Fujita approximation theorem for big line bundles.

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34

Zilinskas, Julius, and Ian David Lockhart Bogle. "Balanced random interval arithmetic." Computers & Chemical Engineering 28, no. 5 (May 2004): 839–51. http://dx.doi.org/10.1016/j.compchemeng.2004.02.020.

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35

Mehrabani, Yavar Safaei, Mona Parsapour, Mona Moradi, and Mehdi Bagherizadeh. "A Novel Efficient CNFET-Based Inexact Full Adder Design for Image Processing Applications." International Journal of Nanoscience 20, no. 02 (January 22, 2021): 2150016. http://dx.doi.org/10.1142/s0219581x21500162.

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Employing inexact arithmetic circuits in error-resilient applications results in reduction of hardware-level metrics such as power consumption, delay and occupied area. These criteria are very important in portable applications because they are battery limited. Full Adder cell is as a building block of many arithmetic circuits. Therefore, it can influence the performance of the entire digital system. This paper presents a novel low-power and high-speed design of one-bit inexact full adder cell based on 32-nm (CNFET) technology for error resilient applications. This design technique can be utilized in various applications particularly in image processing. The presented design employs capacitive threshold logic (CTL) approach which significantly reduces the number of transistors. The peak signal-to-noise ratio (PSNR) is considered to evaluate accuracy of circuits at application level. Then extensive simulations regarding various power supplies, temperatures and loads at transistor level are performed to measure power consumption and propagation delay criteria. Moreover, some new metrics are introduced to trade-off between application and hardware level parameters. Comprehensive simulations demonstrate the supremacy of the proposed cell than others.

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36

Jeřábek, Emil. "Approximate counting in bounded arithmetic." Journal of Symbolic Logic 72, no. 3 (September 2007): 959–93. http://dx.doi.org/10.2178/jsl/1191333850.

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AbstractWe develop approximate counting of sets definable by Boolean circuits in bounded arithmetic using the dual weak pigeonhole principle (dWPHP(PV)), as a generalization of results from [15]. We discuss applications to formalization of randomized complexity classes (such as BPP, APP, MA, AM) in PV1 + dWPHP(PV).

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37

Santiago, Regivan, Flaulles Bergamaschi, Humberto Bustince, Graçaliz Dimuro, Tiago Asmus, and José Antonio Sanz. "On the Normalization of Interval Data." Mathematics 8, no. 11 (November 23, 2020): 2092. http://dx.doi.org/10.3390/math8112092.

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The impreciseness of numeric input data can be expressed by intervals. On the other hand, the normalization of numeric data is a usual process in many applications. How do we match the normalization with impreciseness on numeric data? A straightforward answer is that it is enough to apply a correct interval arithmetic, since the normalized exact value will be enclosed in the resulting “normalized” interval. This paper shows that this approach is not enough since the resulting “normalized” interval can be even wider than the input intervals. So, we propose a pair of axioms that must be satisfied by an interval arithmetic in order to be applied in the normalization of intervals. We show how some known interval arithmetics behave with respect to these axioms. The paper ends with a discussion about the current paradigm of interval computations.

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38

Grangetto, Marco, Enrico Magli, and Gabriella Olmo. "Distributed Arithmetic Coding." IEEE Communications Letters 11, no. 11 (November 2007): 883–85. http://dx.doi.org/10.1109/lcomm.2007.071172.

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39

Neumann, B., T. von Sydow, H. Blume, and T. G. Noll. "Design and quantitative analysis of parametrisable eFPGA-architectures for arithmetic." Advances in Radio Science 4 (September 6, 2006): 251–57. http://dx.doi.org/10.5194/ars-4-251-2006.

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Abstract. Future SoCs will feature embedded FPGAs (eFPGAs) to enable flexible and efficient implementations of high-throughput digital signal processing applications. Current research projects on and emerging products containing FPGAs are mainly based on "standard FPGA"-architectures that are optimised for a very wide range of applications. The implementation costs of these FPGAs are dominated by a very complex interconnect network. This paper presents a method to improve the efficiency of eFPGAs by tailoring them for a certain application domain using a parametrisable architecture template derived from the results of a systematic evaluation of the requirements of the application domain. Two different architectures are discussed, a reference architecture to illustrate the methodology and possible optimisation measures as well as a specialised arithmetic-oriented eFPGA for applications like correlators, decoders, and filters. For the arithmetic-oriented architecture, a novel logic element (LE) and a special interconnect architecture that was designed with respect to the connectivity characteristics of regular datapaths, are presented. For both architecture templates, physically optimised implementations based on an automatic design approach have been created. As a first cost comparison of these implementations with standard FPGAs, the LE-density (number of logic elements per mm2) is evaluated. For the arithmetic-oriented architecture, the LE-density could be increased by an order of magnitude compared to standard architectures.

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40

Keenan, Alexandra, Robert Schweller, Michael Sherman, and Xingsi Zhong. "Fast arithmetic in algorithmic self-assembly." Natural Computing 15, no. 1 (November 17, 2015): 115–28. http://dx.doi.org/10.1007/s11047-015-9512-7.

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41

König, Daniel, and Markus Lohrey. "Parallel identity testing for skew circuits with big powers and applications." International Journal of Algebra and Computation 28, no. 06 (September 2018): 979–1004. http://dx.doi.org/10.1142/s0218196718500431.

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Powerful skew arithmetic circuits are introduced. These are skew arithmetic circuits with variables, where input gates can be labeled with powers [Formula: see text] for binary encoded numbers [Formula: see text]. It is shown that polynomial identity testing for powerful skew arithmetic circuits belongs to [Formula: see text], which generalizes a corresponding result for (standard) skew circuits. Two applications of this result are presented: (i) Equivalence of higher-dimensional straight-line programs can be tested in [Formula: see text]; this result is even new in the one-dimensional case, where the straight-line programs produce words. (ii) The compressed word problem (or circuit evaluation problem) for certain wreath products of finitely generated abelian groups belongs to [Formula: see text]. Using the Magnus embedding, it follows that the compressed word problem for a free metabelian group belongs to [Formula: see text].

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42

Ehlen, Stephan, and Siddarth Sankaran. "On two arithmetic theta lifts." Compositio Mathematica 154, no. 10 (September 7, 2018): 2090–149. http://dx.doi.org/10.1112/s0010437x18007327.

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Our aim is to clarify the relationship between Kudla’s and Bruinier’s Green functions attached to special cycles on Shimura varieties of orthogonal and unitary type, which play a key role in the arithmetic geometry of these cycles in the context of Kudla’s program. In particular, we show that the generating series obtained by taking the differences of the two families of Green functions is a non-holomorphic modular form and has trivial (cuspidal) holomorphic projection. Along the way, we construct a section of the Maaß lowering operator for moderate growth forms valued in the Weil representation using a regularized theta lift, which may be of independent interest, as it in particular has applications to mock modular forms. We also consider arithmetic-geometric applications to integral models of $U(n,1)$ Shimura varieties. Each family of Green functions gives rise to a formal arithmetic theta function, valued in an arithmetic Chow group, that is conjectured to be modular; our main result is the modularity of the difference of the two arithmetic theta functions. Finally, we relate the arithmetic heights of the special cycles to special derivatives of Eisenstein series, as predicted by Kudla’s conjecture, and describe a refinement of a theorem of Bruinier, Howard and Yang on arithmetic intersections against CM points.

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43

Gorini, Catherine A. "Using Clock Arithmetic to Send Secret Messages." Mathematics Teacher 89, no. 2 (February 1996): 100–104. http://dx.doi.org/10.5951/mt.89.2.0100.

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Computers and electronic communications have influenced all aspects of our lives, from supermarket checkout to understanding the structure of DNA and the origins of the universe. These applications require the transmission, storage, and retrieval of large quantities of data. Two major concerns when working with these data are the transmission accuracy and security of the data, which are the subjects of coding theory and cryptology, respectively. Both use mathematical techniques extensively. This article describes a classroom activity in which students explore a real-life application of algebra by sending secret messages to each other. By so doing, students see the necessity of the laws of exponents for making calculator or computer calculations feasible. It is an excellent way for students to review the laws of exponents and to learn new ways to apply them.

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44

Jacobsen, Brian J. "Practical Applications of The Bad Arithmetic of Active Management." Practical Applications 5, no. 2 (September 1, 2017): 1.2–3. http://dx.doi.org/10.3905/pa.2017.5.2.233.

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45

Wang, Dong, Peng Cao, and Yang Xiao. "A parallel arithmetic array for accelerating compute-intensive applications." IEICE Electronics Express 11, no. 4 (2014): 20130981. http://dx.doi.org/10.1587/elex.11.20130981.

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46

Shen, Peiping, Kecun Zhang, and Yanjun Wang. "Applications of interval arithmetic in non-smooth global optimization." Applied Mathematics and Computation 144, no. 2-3 (December 2003): 413–31. http://dx.doi.org/10.1016/s0096-3003(02)00417-4.

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47

Bykovskii, V. A., and M. D. Monina. "Arithmetic identities associated with quadratic forms and their applications." Doklady Mathematics 87, no. 2 (March 2013): 202–4. http://dx.doi.org/10.1134/s1064562413020257.

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48

Jiang, Honglan, Francisco Javier Hernandez Santiago, Hai Mo, Leibo Liu, and Jie Han. "Approximate Arithmetic Circuits: A Survey, Characterization, and Recent Applications." Proceedings of the IEEE 108, no. 12 (December 2020): 2108–35. http://dx.doi.org/10.1109/jproc.2020.3006451.

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49

Aberth, Oliver. "Computation of topological degree using interval arithmetic, and applications." Mathematics of Computation 62, no. 205 (January 1, 1994): 171. http://dx.doi.org/10.1090/s0025-5718-1994-1203731-4.

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50

Harari, David, and Alexei Skorobogatov. "The Brauer group of torsors and its arithmetic applications." Annales de l’institut Fourier 53, no. 7 (2003): 1987–2019. http://dx.doi.org/10.5802/aif.1998.

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