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Journal articles on the topic 'Arithmetical rank'

Author: Grafiati
Published: 4 June 2021
Last updated: 10 February 2022

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1

Katsabekis, Anargyros. "Arithmetical rank of binomial ideals." Archiv der Mathematik 109, no. 4 (August 3, 2017): 323–34. http://dx.doi.org/10.1007/s00013-017-1071-y.

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2

Thoma, A. "On the binomial arithmetical rank." Archiv der Mathematik 74, no. 1 (January 2000): 22–25. http://dx.doi.org/10.1007/pl00000405.

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3

Kimura, Kyouko, Naoki Terai, and Ken-ichi Yoshida. "Arithmetical rank of squarefree monomial ideals of small arithmetic degree." Journal of Algebraic Combinatorics 29, no. 3 (May 29, 2008): 389–404. http://dx.doi.org/10.1007/s10801-008-0142-3.

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4

Kimura, Kyouko, and Paolo Mantero. "Arithmetical rank of strings and cycles." Journal of Commutative Algebra 9, no. 1 (March 2017): 89–106. http://dx.doi.org/10.1216/jca-2017-9-1-89.

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5

Eto, Kazufumi. "Binomial arithmetical rank of lattice ideals." manuscripta mathematica 109, no. 4 (December 1, 2002): 455–63. http://dx.doi.org/10.1007/s00229-002-0317-5.

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6

Lyubeznik, Gennady. "On the arithmetical rank of monomial ideals." Journal of Algebra 112, no. 1 (January 1988): 86–89. http://dx.doi.org/10.1016/0021-8693(88)90133-0.

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7

Mohammadi, Fatemeh, and Dariush Kiani. "On the Arithmetical Rank of the Edge Ideals of Some Graphs." Algebra Colloquium 19, spec01 (October 31, 2012): 797–806. http://dx.doi.org/10.1142/s1005386712000685.

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In this paper, we compute the projective dimension of the edge ideals of graphs consisting of some cycles and lines which are joint in a common vertex. Moreover, we show that for such graphs, the arithmetical rank equals the projective dimension. As an application, we can compute the arithmetical rank for some homogenous monomial ideals.
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8

Varbaro, Matteo. "On the arithmetical rank of certain segre embeddings." Transactions of the American Mathematical Society 364, no. 10 (October 1, 2012): 5091–109. http://dx.doi.org/10.1090/s0002-9947-2012-05435-3.

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9

Katsabekis, Anargyros. "Arithmetical rank of toric ideals associated to graphs." Proceedings of the American Mathematical Society 138, no. 09 (September 1, 2010): 3111. http://dx.doi.org/10.1090/s0002-9939-2010-10335-0.

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10

Kimura, Kyouko, and Naoki Terai. "Binomial arithmetical rank of edge ideals of forests." Proceedings of the American Mathematical Society 141, no. 6 (January 2, 2013): 1925–32. http://dx.doi.org/10.1090/s0002-9939-2013-11473-5.

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11

BARILE, MARGHERITA, DARIUSH KIANI, FATEMEH MOHAMMADI, and SIAMAK YASSEMI. "ARITHMETICAL RANK OF THE CYCLIC AND BICYCLIC GRAPHS." Journal of Algebra and Its Applications 11, no. 02 (April 2012): 1250039. http://dx.doi.org/10.1142/s0219498811005634.

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We show that for the edge ideals of the graphs consisting of one cycle or two cycles of any length connected through a vertex, the arithmetical rank equals the projective dimension of the corresponding quotient ring.
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12

Katsabekis, Anargyros. "On the binomial arithmetical rank of toric ideals." Journal of Algebra and Its Applications 13, no. 07 (May 2, 2014): 1450030. http://dx.doi.org/10.1142/s0219498814500303.

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In this paper, we investigate sufficient conditions which ensure that the minimal number of generators of a toric ideal IA is equal to its binomial arithmetical rank. We study particularly the case where IA is the toric ideal of a graph.
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13

Morales, Marcel. "Simplicial ideals, 2-linear ideals and arithmetical rank." Journal of Algebra 324, no. 12 (December 2010): 3431–56. http://dx.doi.org/10.1016/j.jalgebra.2010.08.025.

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14

KNIGHT, J. F., and J. MILLAR. "COMPUTABLE STRUCTURES OF RANK $\omega_{1}^{{\rm CK}}$." Journal of Mathematical Logic 10, no. 01n02 (June 2010): 31–43. http://dx.doi.org/10.1142/s0219061310000912.

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For countable structure, "Scott rank" provides a measure of internal, model-theoretic complexity. For a computable structure, the Scott rank is at most [Formula: see text]. There are familiar examples of computable structures of various computable ranks, and there is an old example of rank [Formula: see text]. In the present paper, we show that there is a computable structure of Scott rank [Formula: see text]. We give two different constructions. The first starts with an arithmetical example due to Makkai, and codes it into a computable structure. The second re-works Makkai's construction, incorporating an idea of Sacks.
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15

Seyyedi, Seyyede, and Farhad Rahmati. "Cohen Macaulayness and Arithmetical Rank of Generalized Theta Graphs." Mathematics 4, no. 3 (June 29, 2016): 43. http://dx.doi.org/10.3390/math4030043.

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16

Barile, Margherita. "On the Arithmetical Rank of an Intersection of Ideals." Algebra Colloquium 15, no. 04 (December 2008): 681–88. http://dx.doi.org/10.1142/s1005386708000643.

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17

Kimura, Kyouko. "Lyubeznik resolutions and the arithmetical rank of monomial ideals." Proceedings of the American Mathematical Society 137, no. 11 (November 1, 2009): 3627. http://dx.doi.org/10.1090/s0002-9939-09-09950-x.

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18

Kimura, Kyouko, Giancarlo Rinaldo, and Naoki Terai. "Arithmetical Rank of Squarefree Monomial Ideals Generated by Five Elements or with Arithmetic Degree Four." Communications in Algebra 40, no. 11 (November 2012): 4147–70. http://dx.doi.org/10.1080/00927872.2011.602781.

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19

Clark, David. "An Arithmetical Function Associated With the Rank of Elliptic Curves." Canadian Mathematical Bulletin 34, no. 2 (June 1, 1991): 181–85. http://dx.doi.org/10.4153/cmb-1991-029-4.

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AbstractWe define an arithmetical function, f(n), which gives a lower bound for the rank of elliptic curves, y2 = x3 + nx, n square-free. Thus, if f(n) is unbounded for square-free values of n, then there are elliptic curves of arbitrarily large rank. We show that f(n) is unbounded as n ranges over all integers.
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20

Katsabekis, Anargyros. "Projections of cones and the arithmetical rank of toric varieties." Journal of Pure and Applied Algebra 199, no. 1-3 (July 2005): 133–47. http://dx.doi.org/10.1016/j.jpaa.2004.12.009.

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21

Barile, Margherita, and Antonio Macchia. "The arithmetical rank of the edge ideals of cactus graphs." Communications in Algebra 45, no. 12 (March 31, 2017): 5407–19. http://dx.doi.org/10.1080/00927872.2017.1310873.

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22

Barile, Margherita. "On the Arithmetical Rank of the Edge Ideals of Forests." Communications in Algebra 36, no. 12 (December 9, 2008): 4678–703. http://dx.doi.org/10.1080/00927870802161220.

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23

Katsabekis, Anargyros, and Apostolos Thoma. "Specializations of Multigradings and the Arithmetical Rank of Lattice Ideals." Communications in Algebra 38, no. 5 (April 26, 2010): 1904–18. http://dx.doi.org/10.1080/00927870903034821.

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24

Kimura, Kyouko, and Naoki Terai. "Arithmetical rank of Gorenstein squarefree monomial ideals of height three." Journal of Algebra 422 (January 2015): 11–32. http://dx.doi.org/10.1016/j.jalgebra.2014.09.005.

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25

Barile, Margherita. "On Ideals Generated by Monomials and One Binomial." Algebra Colloquium 14, no. 04 (December 2007): 631–38. http://dx.doi.org/10.1142/s1005386707000582.

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26

Kimura, Kyouko. "Arithmetical rank of Cohen-Macaulay squarefree monomial ideals of height two." Journal of Commutative Algebra 3, no. 1 (March 2011): 31–46. http://dx.doi.org/10.1216/jca-2011-3-1-31.

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27

Macchia, Antonio. "The arithmetical rank of the edge ideals of graphs with whiskers." Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 56, no. 1 (April 12, 2014): 147–58. http://dx.doi.org/10.1007/s13366-014-0198-2.

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28

Calvert, Wesley, Julia F. Knight, and Jessica Millar. "Computable trees of Scott rank ω1CK, and computable approximation." Journal of Symbolic Logic 71, no. 1 (March 2006): 283–98. http://dx.doi.org/10.2178/jsl/1140641175.

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AbstractMakkai [10] produced an arithmetical structure of Scott rank ω1CK. In [9], Makkai's example is made computable. Here we show that there are computable trees of Scott rank ω1CK. We introduce a notion of “rank homogeneity”. In rank homogeneous trees, orbits of tuples can be understood relatively easily. By using these trees, we avoid the need to pass to the more complicated “group trees” of [10] and [9], Using the same kind of trees, we obtain one of rank ω1CK that is “strongly computably approximable”. We also develop some technology that may yield further results of this kind.
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29

Barile, Margherita, and Antonio Macchia. "The arithmetical rank of the edge ideals of graphs with pairwise disjoint cycles." Journal of Algebra and Its Applications 15, no. 07 (July 22, 2016): 1650120. http://dx.doi.org/10.1142/s0219498816501206.

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We prove that, for the edge ideal of a graph whose cycles are pairwise vertex-disjoint, the arithmetical rank is bounded above by the sum of the number of cycles and the maximum height of its associated primes.
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30

Young, Matthew P. "Moments of the Critical Values of Families of Elliptic Curves, with Applications." Canadian Journal of Mathematics 62, no. 5 (October 1, 2010): 1155–81. http://dx.doi.org/10.4153/cjm-2010-049-5.

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AbstractWe make conjectures on the moments of the central values of the family of all elliptic curves and on themoments of the first derivative of the central values of a large family of positive rank curves. In both cases the order of magnitude is the same as that of the moments of the central values of an orthogonal family of L-functions. Notably, we predict that the critical values of all rank 1 elliptic curves is logarithmically larger than the rank 1 curves in the positive rank family.Furthermore, as arithmetical applications, we make a conjecture on the distribution of ap's amongst all rank 2 elliptic curves and show how the Riemann hypothesis can be deduced from sufficient knowledge of the first moment of the positive rank family (based on an idea of Iwaniec).
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31

Ramananantoandro, Tahiana, Pierre Larricq, and Olivier Eterradossi. "Relationships between 3D roughness parameters and visuotactile perception of surfaces of maritime pinewood and MDF." Holzforschung 68, no. 1 (January 1, 2014): 93–101. http://dx.doi.org/10.1515/hf-2012-0208.

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Abstract The objective measurement and subjective estimation of the surface quality by the consumers are important issues for furniture production and marketing. The objective of the present study is to find suitable (objective) roughness parameters that are linked to human sensation. Maritime pinewood and medium-density fiberboard (MDF) made from the same species are in focus. The roughness was measured by means of a 3D confocal profilometer and 13 independent 3D roughness parameters were computed. The quality of surfaces was modified through sanding, whereas the grit size of sand papers ranged between P60 and P320. The applied pressure and sanding time were controlled. The subjective estimation of the surface aspects was performed by several individuals through sensorial analysis. Biplots of several roughness parameters versus tactile rank revealed in the case of MDF a significant correlation to arithmetical average roughness Sa and valley material component Sr2. For pinewood, the parameters arithmetical average roughness Sa and texture aspect ratio Str correlated best to tactile rank.
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32

Kimura, Kyouko, Naoki Terai, and Ken-ichi Yoshida. "Arithmetical Rank of a Squarefree Monomial Ideal whose Alexander Dual is of Deviation Two." Acta Mathematica Vietnamica 40, no. 3 (June 10, 2015): 375–91. http://dx.doi.org/10.1007/s40306-015-0136-x.

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33

Saremi, Hero, and Amir Mafi. "Unmixedness and arithmetic properties of matroidal ideals." Archiv der Mathematik 114, no. 3 (November 15, 2019): 299–304. http://dx.doi.org/10.1007/s00013-019-01402-w.

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AbstractLet $$R=k[x_1,\ldots ,x_n]$$R=k[x1,…,xn] be the polynomial ring in n variables over a field k and let I be a matroidal ideal of degree d. In this paper, we study the unmixedness properties and the arithmetical rank of I. Moreover, we show that $$ara(I)=n-d+1$$ara(I)=n-d+1. This answers the conjecture made by Chiang-Hsieh (Comm Algebra 38:944–952, 2010, Conjecture).
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34

ZHU, Guangjun, Feng SHI, and Yan GU. "Arithmetical rank of the edge ideals of some $n$-cyclic graphs with a common edge." TURKISH JOURNAL OF MATHEMATICS 39 (2015): 112–23. http://dx.doi.org/10.3906/mat-1407-39.

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35

KABLE, ANTHONY C. "AN ARITHMETICAL INVARIANT OF ORBITS OF AFFINE ACTIONS AND ITS APPLICATION TO SIMILARITY CLASSES OF QUADRATIC SPACES." International Journal of Number Theory 06, no. 06 (September 2010): 1215–53. http://dx.doi.org/10.1142/s1793042110003460.

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Given an action of an affine algebraic group on an affine variety and a relatively invariant regular function, all defined over the ring of integers of a number field and having suitable additional properties, an invariant of the rational orbits of the action is defined. This invariant, the reduced replete Steinitz class, takes its values in the reduced replete class group of the number field. The general framework is then applied to obtain an invariant of similarity classes of non-degenerate quadratic spaces of even rank. The invariant is related to more familiar invariants. It is shown that if the similarity classes are weighted by the volume of an associated automorphism group then their reduced replete Steinitz classes are asymptotically uniformly distributed with respect to a natural parameter.
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36

Zhuang, Zheng-Yun, Chang-Ching Lin, Chih-Yung Chen, and Chia-Rong Su. "Rank-Based Comparative Research Flow Benchmarking the Effectiveness of AHP–GTMA on Aiding Decisions of Shredder Selection by Reference to AHP–TOPSIS." Applied Sciences 8, no. 10 (October 18, 2018): 1974. http://dx.doi.org/10.3390/app8101974.

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The AHP–GTMA (analytic hierarchy process and graph theory and matrix approach) has been applied to select the best paper shredder before a company was making a bulk purchase order. However, there is a question as to whether one such relatively recent approach is effective to aid the selection decision problems in industrial/commercial practice. In this paper, a novel multi-measure, rank-based comparative research flow is proposed. The real decision problem case mentioned above is solved using the AHP–GTMA and the AHP–TOPSIS methods, respectively, with relevant datasets sourced. Several measures in the proposed flow, i.e., the arithmetical, geometrical, or even statistical ones, are multiplexed and used to validate the similarity between the rank order vectors (ROVs) (and thus between the final preferential orders determined over the alternatives) that are obtained using these two different methods. While AHP–TOPSIS is a confident multi-attribute decision-making (MADM) approach which has been successfully applied to many other fields, the similarity validated between these individual results using the proposed method is used to confirm the efficacy of the AHP–GTMA approach and to determine its applicability in practice. In addition, along with this study, some contributable points are also rendered for implementing the decision models, e.g., the optimized recursive implementation in R to compute the permanent value of a square ASAM (alternative selection attribute matrix, which is the computational basis required by AHP–GTMA) of any dimension. The proposed methodological flow to confirm the similarity based on the ordinal rank information is not only convenient in operational practice with ubiquitous tool supports (e.g., the vector-based R statistical platform), but also generalizable (to verify between another pair of results obtained using any other MADM methods). This gives options for future research.
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37

Abu Al-Nadi, Maram Fuad. "The Degree Represent of the Faculty Members in Official Jordanian Universities for the Human Relation Amongst them (University Collegiality) as a Professional Ethical Behavior." Modern Applied Science 12, no. 9 (August 28, 2018): 209. http://dx.doi.org/10.5539/mas.v12n9p209.

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This study aims at identifying the degree represent of the faculty members in official Jordanian Universities for the human relation among them (University Collegiality) as a professional ethical behavior, the survey methodology has been used in this study, and developed questionnaire consists of (26) paragraphs, and the tool has been verified its validity and reliability. The study sample consisted of official Jordanian universities’ members represented by the universities of Jordan, Yarmouk and Mutah, a total of (300) member, were selected in a simple random method, and the study concluded the following results:The Arithmetical averages and standard deviations showed that the degree of represent the faculty members in official Jordanian Universities for the human relation among them (University Collegiality) as a professional ethical behavior it was a medium degree.The result showed statistically significant differences at the significance level (p=0.05) in the estimates of the study sample members for the degree represent of the faculty members in official Jordanian Universities for the human relation among them (University Collegiality) as a professional ethical behavior, it referred to gender variable, and it was in favor of males, the results also showed statistically significant differences for the faculty class and it was in favor of sciences faculties, also showed statistically significant differences for the academic rank and it was in favor of associate Professor rank.The study recommended to focus on the subject of the University Collegiality among the faculty members and develop an agreement of the university collegiality in participation of official and unofficial Jordanian universities.
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38

Golub, Y. I., and F. V. Starovoitov. "Digital image contrast assessment based on the Weibull distribution parameters." «System analysis and applied information science», no. 2 (August 19, 2021): 4–13. http://dx.doi.org/10.21122/2309-4923-2021-2-4-13.

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The goal of the studies described in the paper is to find a quantitative assessment that maximally correlates with the subjective assessment of the contrast image quality in the absence of reference image. As a result of the literature analysis, 16 functions were selected that are used for no-refernce image quality assessment: BEGH, BISH, BREN, CMO, CURV, FUS, HELM, EBCM, KURT, LAPD, LAPL, LAPM, LOCC, LOEN, SHAR, WAVS. They all use the arithmetical mean of the local contrast quality. As an alternative to averaging local estimates (since the mean is one of two parameters of the normal distribution), it is proposed to use one of two parameters of the Weibull distribution of the same data – scale or shape.For the experiments, digital images with nonlinear contrast distortion from the available CCID2014 database were used. It contains 15 original images with a size of 768x512 pixels and 655 versions with modified contrast. This database of images contains the average visual quality assessment (Mean Opinion Score, briefly MOS) of each image. Spearman’s rank correlation coefficient was used to determine the correspondence between the visual MOS scores and the studied quantitative measures.As a result of the research, a new quality assessment measure of contrast images in the absence of references is presented. To obtain the estimate, local quality values are calculated by the BREN measure, their set is described by the Weibull distribution, and the scale parameter of the distribution serves as the best numerical estimate of the quality of contrast images. This conclusion is confirmed experimentally, and the proposed measure correlates better than other variants with the subjective assessments of experts.
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39

Göral, Ramazan, and Simge Tokay. "Online Customer Reviews on Restaurants and Expert Opinions: An Integrated Approach." European Journal of Interdisciplinary Studies 1, no. 2 (August 30, 2015): 9. http://dx.doi.org/10.26417/ejis.v1i2.p9-19.

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In recent years, online customer reviews have been widespread and it has been taken into account in the point of the decision by the people. Resources which create a virtual environment interaction are developing with the user reviews and ratings and it is increasingly gaining importance for both customers and tourism businesses. Customers, who are evaluating food and beverage consumption as an experience and looking at truelife experiences when deciding which experience to lives, are more increasingly using online resources serving sector. These platforms provide the customers the opportunity of dialoguing among them and provide information while they provide feedback for the improvements for the service providers they need to do. Because of these properties, researchers argue online customer reviews and they express that services restaurant could offer and the quality of them could be compared better by using customer feedback. However, customer review system has been criticized. The point critics pointed is the difficulty of verification reviews and deficiencies in impartiality point. An integrated approach is proposed in this study. Online customer judgments assessing the food and beverage business and expert opinions underlie the integrated approach. For this purpose, the data obtained from TripAdvisor website and expert reviews will be integrated with Analytical Hierarch Process (AHP) method. In the study, restaurants which take place in Turkey’s biggest tourist areas in four provinces and locating in the top 10 according to customer rewievs on TripAdvisor website are evaluated. Then, arithmetical average of the aggregated customer ratings relating to these restaurants and the integration of assessment of experts providing training in food and beverage area are ensured. Consequently, these four provinces’ rank is determined by an integrated approach in terms of restaurants.
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40

Göral, Ramazan, and Simge Tokay. "Online Customer Reviews on Restaurants and Expert Opinions: An Integrated Approach." European Journal of Interdisciplinary Studies 2, no. 1 (August 30, 2015): 9. http://dx.doi.org/10.26417/ejis.v2i1.p9-19.

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In recent years, online customer reviews have been widespread and it has been taken into account in the point of the decision by the people. Resources which create a virtual environment interaction are developing with the user reviews and ratings and it is increasingly gaining importance for both customers and tourism businesses. Customers, who are evaluating food and beverage consumption as an experience and looking at truelife experiences when deciding which experience to lives, are more increasingly using online resources serving sector. These platforms provide the customers the opportunity of dialoguing among them and provide information while they provide feedback for the improvements for the service providers they need to do. Because of these properties, researchers argue online customer reviews and they express that services restaurant could offer and the quality of them could be compared better by using customer feedback. However, customer review system has been criticized. The point critics pointed is the difficulty of verification reviews and deficiencies in impartiality point. An integrated approach is proposed in this study. Online customer judgments assessing the food and beverage business and expert opinions underlie the integrated approach. For this purpose, the data obtained from TripAdvisor website and expert reviews will be integrated with Analytical Hierarch Process (AHP) method. In the study, restaurants which take place in Turkey’s biggest tourist areas in four provinces and locating in the top 10 according to customer rewievs on TripAdvisor website are evaluated. Then, arithmetical average of the aggregated customer ratings relating to these restaurants and the integration of assessment of experts providing training in food and beverage area are ensured. Consequently, these four provinces’ rank is determined by an integrated approach in terms of restaurants.
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41

Pakhomov, Fedor, and Alexander Zapryagaev. "Multi-dimensional Interpretations of Presburger Arithmetic in Itself." Journal of Logic and Computation 30, no. 8 (September 22, 2020): 1681–93. http://dx.doi.org/10.1093/logcom/exaa050.

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Abstract Presburger arithmetic is the true theory of natural numbers with addition. We study interpretations of Presburger arithmetic in itself. The main result of this paper is that all self-interpretations are definably isomorphic to the trivial one. Here we consider interpretations that might be multi-dimensional. We note that this resolves a conjecture by Visser (1998, An overview of interpretability logic. Advances in Modal Logic, pp. 307–359). In order to prove the result, we show that all linear orderings that are interpretable in $({\mathbb{N}},+)$ are scattered orderings with the finite Hausdorff rank and that the ranks are bounded in the terms of the dimensions of the respective interpretations.
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42

Fesenko, Ivan. "Analysis on arithmetic schemes. II." Journal of K-theory 5, no. 3 (May 20, 2010): 437–557. http://dx.doi.org/10.1017/is010004028jkt103.

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AbstractWe construct adelic objects for rank two integral structures on arithmetic surfaces and develop measure and integration theory, as well as elements of harmonic analysis. Using the topological Milnor K2-delic and K1×K1-delic objects associated to an arithmetic surface, an adelic zeta integral is defined. Its unramified version is closely related to the square of the zeta function of the surface. For a proper regular model of an elliptic curve over a global field, a two-dimensional version of the theory of Tate and Iwasawa is derived. Using adelic analytic duality and a two-dimensional theta formula, the study of the zeta integral is reduced to the study of a boundary integral term. The work includes first applications to three fundamental properties of the zeta function: its meromorphic continuation and functional equation and a hypothesis on its mean periodicity; the location of its poles and a hypothesis on the permanence of the sign of the fourth logarithmic derivative of a boundary function; and its pole at the central point where the boundary integral explicitly relates the analytic and arithmetic ranks.
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43

Ballico, Edoardo, Giorgio Bolondi, and Rosa M. Mirò-Roig. "Numerical invariants of rank-2 arithmetically Buchsbaum sheaves." Journal of Pure and Applied Algebra 58, no. 2 (June 1989): 107–25. http://dx.doi.org/10.1016/0022-4049(89)90153-9.

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44

Tripathi, Amit. "Rank 3 arithmetically Cohen–Macaulay bundles over hypersurfaces." Journal of Algebra 478 (May 2017): 1–11. http://dx.doi.org/10.1016/j.jalgebra.2017.01.014.

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45

BURNS, DAVID. "ON THE GALOIS STRUCTURE OF ARITHMETIC COHOMOLOGY I: COMPACTLY SUPPORTED -ADIC COHOMOLOGY." Nagoya Mathematical Journal 239 (November 16, 2018): 294–321. http://dx.doi.org/10.1017/nmj.2018.41.

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We investigate the Galois structures of $p$-adic cohomology groups of general $p$-adic representations over finite extensions of number fields. We show, in particular, that as the field extensions vary over natural families the Galois modules formed by these cohomology groups always decompose as the direct sum of a projective module and a complementary module of bounded $p$-rank. We use this result to derive new (upper and lower) bounds on the changes in ranks of Selmer groups over extensions of number fields and descriptions of the explicit Galois structures of natural arithmetic modules.
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46

Börm, Steffen, and Knut Reimer. "Efficient arithmetic operations for rank-structured matrices based on hierarchical low-rank updates." Computing and Visualization in Science 16, no. 6 (December 2013): 247–58. http://dx.doi.org/10.1007/s00791-015-0233-3.

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47

Naldi, Simone. "Solving rank-constrained semidefinite programs in exact arithmetic." Journal of Symbolic Computation 85 (March 2018): 206–23. http://dx.doi.org/10.1016/j.jsc.2017.07.009.

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48

Singh, Sandip. "Coxeter groups are not higher rank arithmetic groups." Mathematical Research Letters 20, no. 3 (2013): 567–80. http://dx.doi.org/10.4310/mrl.2013.v20.n3.a13.

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49

Raz, Ran. "Tensor-Rank and Lower Bounds for Arithmetic Formulas." Journal of the ACM 60, no. 6 (November 2013): 1–15. http://dx.doi.org/10.1145/2535928.

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50

MEHRVARZ, ALI AKBAR, KAMAL BAHMANPOUR, and REZA NAGHIPOUR. "ARITHMETIC RANK, COHOMOLOGICAL DIMENSION AND FILTER REGULAR SEQUENCES." Journal of Algebra and Its Applications 08, no. 06 (December 2009): 855–62. http://dx.doi.org/10.1142/s0219498809003692.

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Abstract:
Let I be an ideal of a commutative Noetherian ring R such that ara (I) = t ≥ 2. The purpose of this article is to show that there exists an I-filter regular sequence y1, …, yt for R such that Rad (I) = Rad (y1, …, yt) and cd ((y1, …, yi), R) = i for all 1 ≤ i < t. Also, it is shown that ara (I) ≤ dim R + 1, which is a generalization of a nice result of Kronecker [14]. In addition, some applications are included.
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