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Автор: Grafiati

Опубліковано: 14 березня 2023

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1

Dey, Papri. "Definite determinantal representations of multivariate polynomials." Journal of Algebra and Its Applications 19, no. 07 (July 23, 2019): 2050129. http://dx.doi.org/10.1142/s0219498820501297.

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Анотація:

In this paper, we consider the problem of representing a multivariate polynomial as the determinant of a definite (monic) symmetric/Hermitian linear matrix polynomial (LMP). Such a polynomial is known as determinantal polynomial. Determinantal polynomials can characterize the feasible sets of semidefinite programming (SDP) problems that motivates us to deal with this problem. We introduce the notion of generalized mixed discriminant (GMD) of matrices which translates the determinantal representation problem into computing a point of a real variety of a specified ideal. We develop an algorithm to determine such a determinantal representation of a bivariate polynomial of degree [Formula: see text]. Then we propose a heuristic method to obtain a monic symmetric determinantal representation (MSDR) of a multivariate polynomial of degree [Formula: see text].

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2

Kyrchei, Ivan I. "Determinantal Representations of the Core Inverse and Its Generalizations with Applications." Journal of Mathematics 2019 (October 1, 2019): 1–13. http://dx.doi.org/10.1155/2019/1631979.

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In this paper, we give the direct method to find of the core inverse and its generalizations that is based on their determinantal representations. New determinantal representations of the right and left core inverses, the right and left core-EP inverses, and the DMP, MPD, and CMP inverses are derived by using determinantal representations of the Moore-Penrose and Drazin inverses previously obtained by the author. Since the Bott-Duffin inverse has close relation with the core inverse, we give its determinantal representation and its application in finding solutions of the constrained linear equations that is an analog of Cramer’s rule. A numerical example to illustrate the main result is given.

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3

Marcus, Marvin. "Determinantal Loci." College Mathematics Journal 23, no. 1 (January 1992): 44. http://dx.doi.org/10.2307/2686198.

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4

Fulton, William. "determinantal formulas." Duke Mathematical Journal 65, no. 3 (March 1992): 381–420. http://dx.doi.org/10.1215/s0012-7094-92-06516-1.

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5

Beauville, Arnaud. "Determinantal hypersurfaces." Michigan Mathematical Journal 48, no. 1 (2000): 39–64. http://dx.doi.org/10.1307/mmj/1030132707.

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6

Marcus, Marvin. "Determinantal Loci." College Mathematics Journal 23, no. 1 (January 1992): 44–47. http://dx.doi.org/10.1080/07468342.1992.11973433.

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7

Kimura, Kenichiro, Shun-ichi Kimura, and Nobuyoshi Takahashi. "Motivic zeta functions in additive monoidal categories." Journal of K-theory 9, no. 3 (December 8, 2011): 459–73. http://dx.doi.org/10.1017/is011011006jkt174.

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Анотація:

AbstractLet C be a pseudo-abelian symmetric monoidal category, and X a Schur-finite object of C. We study the problem of rationality of the motivic zeta function ζx(t) of X. Since the coefficient ring is not a field, there are several variants of rationality — uniform, global, determinantal and pointwise rationality. We show that ζx(t) is determinantally rational, and we give an example of C and X for which the motivic zeta function is not uniformly rational.

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8

Stanimirović, Predrag S., and Milan Lj Zlatanović. "Determinantal Representation of Outer Inverses in Riemannian Space." Algebra Colloquium 19, spec01 (October 31, 2012): 877–92. http://dx.doi.org/10.1142/s1005386712000740.

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Анотація:

Starting from a known determinantal representation of outer inverses, we derive their determinantal representation in terms of the inner product in the Euclidean space. We define the double inner product of two miscellaneous tensors of rank 2 in a Riemannian space. The corresponding determinantal representation as well as the general representation of outer inverses in the Riemannian space are derived. A non-zero {2}-inverse X of a given tensor A obeying ρ(X) = s with 1 ≤ s ≤ r = ρ(A) is expressed in terms of the double inner product involving compound tensors with minors of order s, extracted from A and appropriate tensors.

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9

Conca, Aldo. "Symmetric ladders." Nagoya Mathematical Journal 136 (December 1994): 35–56. http://dx.doi.org/10.1017/s0027763000024958.

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Анотація:

In this paper we define and study ladder determinantal rings of a symmetric matrix of indeterminates. We show that they are Cohen-Macaulay domains. We give a combinatorial characterization of their h-vectors and we compute the a-invariant of the classical determinantal rings of a symmetric matrix of indeterminates.

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10

Hardy, Adrien, and Mylène Maïda. "Determinantal Point Processes." EMS Newsletter 2019-6, no. 112 (June 6, 2019): 8–15. http://dx.doi.org/10.4171/news/112/3.

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11

Osogami, Takayuki, and Rudy Raymond. "Determinantal Reinforcement Learning." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 4659–66. http://dx.doi.org/10.1609/aaai.v33i01.33014659.

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We study reinforcement learning for controlling multiple agents in a collaborative manner. In some of those tasks, it is insufficient for the individual agents to take relevant actions, but those actions should also have diversity. We propose the approach of using the determinant of a positive semidefinite matrix to approximate the action-value function in reinforcement learning, where we learn the matrix in a way that it represents the relevance and diversity of the actions. Experimental results show that the proposed approach allows the agents to learn a nearly optimal policy approximately ten times faster than baseline approaches in benchmark tasks of multi-agent reinforcement learning. The proposed approach is also shown to achieve the performance that cannot be achieved with conventional approaches in partially observable environment with exponentially large action space.

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12

Watanabe, Junzo, and Kohji Yanagawa. "Vandermonde determinantal ideals." MATHEMATICA SCANDINAVICA 125, no. 2 (October 19, 2019): 179–84. http://dx.doi.org/10.7146/math.scand.a-114906.

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Анотація:

We show that the ideal generated by maximal minors (i.e., $k+1$-minors) of a $(k+1) \times n$ Vandermonde matrix is radical and Cohen-Macaulay. Note that this ideal is generated by all Specht polynomials with shape $(n-k,1, …,1)$.

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13

Bufetov, Alexander I. "Infinite determinantal measures." Electronic Research Announcements in Mathematical Sciences 20 (February 2013): 12–30. http://dx.doi.org/10.3934/era.2013.20.12.

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14

Boralevi, Ada, Jasper van Doornmalen, Jan Draisma, Michiel E. Hochstenbach, and Bor Plestenjak. "Uniform Determinantal Representations." SIAM Journal on Applied Algebra and Geometry 1, no. 1 (January 2017): 415–41. http://dx.doi.org/10.1137/16m1085656.

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15

Ene, Viviana, Jürgen Herzog, Takayuki Hibi, and Fatemeh Mohammadi. "Determinantal facet ideals." Michigan Mathematical Journal 62, no. 1 (March 2013): 39–57. http://dx.doi.org/10.1307/mmj/1363958240.

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16

Hüttenhain, Jesko, and Christian Ikenmeyer. "Binary determinantal complexity." Linear Algebra and its Applications 504 (September 2016): 559–73. http://dx.doi.org/10.1016/j.laa.2016.04.027.

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17

Loonis, V., and X. Mary. "Determinantal sampling designs." Journal of Statistical Planning and Inference 199 (March 2019): 60–88. http://dx.doi.org/10.1016/j.jspi.2018.05.005.

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18

Lyons, Russell. "Determinantal probability measures." Publications mathématiques de l'IHÉS 98, no. 1 (December 2003): 167–212. http://dx.doi.org/10.1007/s10240-003-0016-0.

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19

Mond, B., Josip Pečarić, and B. Tepeš. "Some Determinantal Inequalities." Mathematical Inequalities & Applications, no. 2 (2005): 331–36. http://dx.doi.org/10.7153/mia-08-30.

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20

Pirinççiog̃lu, Nurettin. "Determinantal invariant gravity." Modern Physics Letters A 34, no. 15 (May 20, 2019): 1950117. http://dx.doi.org/10.1142/s0217732319501177.

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Анотація:

Einstein–Hilbert action with a determinantal invariant has been considered. The obtained field equation contains the inverse Ricci tensor, [Formula: see text]. The linearized solution of invariant has been examined, and it is concluded that the constant curvature spacetime metric solution of the field equation gives different curvature constant for each values of [Formula: see text]. The [Formula: see text] gives a trivial solution for constant curvature, [Formula: see text].

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21

Lenagan, T. H., and K. R. Goodearl. "Quantum determinantal ideals." Duke Mathematical Journal 103, no. 1 (May 2000): 165–90. http://dx.doi.org/10.1215/s0012-7094-00-10318-3.

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22

Conca, Aldo. "Ladder determinantal rings." Journal of Pure and Applied Algebra 98, no. 2 (January 1995): 119–34. http://dx.doi.org/10.1016/0022-4049(94)00039-l.

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23

Harnad, J., and A. Yu Orlov. "Determinantal identity for multilevel ensembles and finite determinantal point processes." Analysis and Mathematical Physics 2, no. 2 (March 9, 2012): 105–21. http://dx.doi.org/10.1007/s13324-012-0029-2.

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24

Kyrchei, Ivan I. "Explicit Determinantal Representation Formulas ofW-Weighted Drazin Inverse Solutions of Some Matrix Equations over the Quaternion Skew Field." Mathematical Problems in Engineering 2016 (2016): 1–13. http://dx.doi.org/10.1155/2016/8673809.

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Анотація:

By using determinantal representations of theW-weighted Drazin inverse previously obtained by the author within the framework of the theory of the column-row determinants, we get explicit formulas for determinantal representations of theW-weighted Drazin inverse solutions (analogs of Cramer’s rule) of the quaternion matrix equationsWAWX=D,XWBW=D, andW1AW1XW2BW2=D.

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25

Lin, Minghua. "On a determinantal inequality arising from diffusion tensor imaging." Communications in Contemporary Mathematics 19, no. 05 (June 14, 2016): 1650044. http://dx.doi.org/10.1142/s0219199716500449.

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Анотація:

In comparing geodesics induced by different metrics, Audenaert formulated the following determinantal inequality [Formula: see text] where [Formula: see text] are [Formula: see text] positive semidefinite matrices. We complement his result by proving [Formula: see text] Our proofs feature the fruitful interplay between determinantal inequalities and majorization relations. Some related questions are mentioned.

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26

Kyrchei, Ivan I. "Determinantal Representations of Solutions and Hermitian Solutions to Some System of Two-Sided Quaternion Matrix Equations." Journal of Mathematics 2018 (November 1, 2018): 1–12. http://dx.doi.org/10.1155/2018/6294672.

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Анотація:

Within the framework of the theory of quaternion row-column determinants previously introduced by the author, we derive determinantal representations (analogs of Cramer’s rule) of solutions and Hermitian solutions to the system of two-sided quaternion matrix equations A1XA1⁎=C1 and A2XA2⁎=C2. Since the Moore-Penrose inverse is a necessary tool to solve matrix equations, we use determinantal representations of the Moore-Penrose inverse previously obtained by the theory of row-column determinants.

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27

Liu, Xiaoji, Guangyan Zhu, Guangping Zhou, and Yaoming Yu. "An Analog of the Adjugate Matrix for the Outer InverseAT,S(2)." Mathematical Problems in Engineering 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/591256.

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Анотація:

We investigate the determinantal representation by exploiting the limiting expression for the generalized inverseAT,S(2). We show the equivalent relationship between the existence and limiting expression ofAT,S(2)and some limiting processes of matrices and deduce the new determinantal representations ofAT,S(2), based on some analog of the classical adjoint matrix. Using the analog of the classical adjoint matrix, we present Cramer rules for the restricted matrix equationAXB=D, R(X)⊂T, N(X)⊃S∼.

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28

Blose, Alexander, Patricia Klein, Owen Mcgrath, and A. N. D. Jackson Morris. "Toric double determinantal varieties." Communications in Algebra 49, no. 7 (February 24, 2021): 3085–93. http://dx.doi.org/10.1080/00927872.2021.1887882.

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29

Shannon, G. P. "78.8 A Determinantal Howler." Mathematical Gazette 78, no. 481 (March 1994): 64. http://dx.doi.org/10.2307/3619437.

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30

Chen, Justin, and Papri Dey. "Computing symmetric determinantal representations." Journal of Software for Algebra and Geometry 10, no. 1 (February 6, 2020): 9–15. http://dx.doi.org/10.2140/jsag.2020.10.9.

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31

Kleppe, Jan O., and Rosa M. Miró-Roig. "Families of determinantal schemes." Proceedings of the American Mathematical Society 139, no. 11 (November 1, 2011): 3831–43. http://dx.doi.org/10.1090/s0002-9939-2011-10802-5.

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32

Piontkowski, Jens. "Linear symmetric determinantal hypersurfaces." Michigan Mathematical Journal 54, no. 1 (April 2006): 117–46. http://dx.doi.org/10.1307/mmj/1144437441.

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33

Weyman, Jerzy. "Book Review: Determinantal rings." Bulletin of the American Mathematical Society 22, no. 2 (April 1, 1990): 357–62. http://dx.doi.org/10.1090/s0273-0979-1990-15911-7.

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34

Conca, Aldo. "Gorenstein Ladder Determinantal Rings." Journal of the London Mathematical Society 54, no. 3 (December 1996): 453–74. http://dx.doi.org/10.1112/jlms/54.3.453.

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35

Docampo, Roi. "Arcs on determinantal varieties." Transactions of the American Mathematical Society 365, no. 5 (September 18, 2012): 2241–69. http://dx.doi.org/10.1090/s0002-9947-2012-05564-4.

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36

Kalantari, Bahman, and Thomas H. Pate. "A determinantal lower bound." Linear Algebra and its Applications 326, no. 1-3 (March 2001): 151–59. http://dx.doi.org/10.1016/s0024-3795(00)00287-1.

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37

Mühlbach, G. "On extending determinantal identities." Linear Algebra and its Applications 132 (April 1990): 145–62. http://dx.doi.org/10.1016/0024-3795(90)90060-p.

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38

Brändén, Petter. "Obstructions to determinantal representability." Advances in Mathematics 226, no. 2 (January 2011): 1202–12. http://dx.doi.org/10.1016/j.aim.2010.08.003.

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39

Drury, Stephen, and Minghua Lin. "Reversed Fischer determinantal inequalities." Linear and Multilinear Algebra 62, no. 8 (June 14, 2013): 1069–75. http://dx.doi.org/10.1080/03081087.2013.804919.

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40

Hough, J. Ben, Manjunath Krishnapur, Yuval Peres, and Bálint Virág. "Determinantal Processes and Independence." Probability Surveys 3 (2006): 206–29. http://dx.doi.org/10.1214/154957806000000078.

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41

Soshnikov, A. "Determinantal random point fields." Russian Mathematical Surveys 55, no. 5 (October 31, 2000): 923–75. http://dx.doi.org/10.1070/rm2000v055n05abeh000321.

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42

Busé, Laurent. "Resultants of determinantal varieties." Journal of Pure and Applied Algebra 193, no. 1-3 (October 2004): 71–97. http://dx.doi.org/10.1016/j.jpaa.2004.02.010.

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43

Gonciulea, Nicolae, and Claudia Miller. "Mixed Ladder Determinantal Varieties." Journal of Algebra 231, no. 1 (September 2000): 104–37. http://dx.doi.org/10.1006/jabr.2000.8358.

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44

Zhao, Xu’an, Hongzhu Gao, and Xiaole Su. "Decompositions of determinantal varieties." Frontiers of Mathematics in China 2, no. 4 (October 2007): 623–37. http://dx.doi.org/10.1007/s11464-007-0038-x.

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45

Kummer, Mario. "Determinantal representations and Bézoutians." Mathematische Zeitschrift 285, no. 1-2 (June 16, 2016): 445–59. http://dx.doi.org/10.1007/s00209-016-1715-9.

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46

Ahmed, Imran, and Maria Aparecida Soares Ruas. "Determinacy of determinantal varieties." manuscripta mathematica 159, no. 1-2 (June 14, 2018): 269–78. http://dx.doi.org/10.1007/s00229-018-1041-0.

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47

Csavinszky, P. "Determinantal inequalities among ?rn?" International Journal of Quantum Chemistry 48, S27 (March 13, 1993): 377–84. http://dx.doi.org/10.1002/qua.560480838.

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48

Steffen, Frauke. "Generic determinantal schemes and the smoothability of determinantal schemes of codimension 2." Manuscripta Mathematica 82, no. 1 (December 1994): 417–31. http://dx.doi.org/10.1007/bf02567711.

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49

Bebiano, Natalia, and Joao da Providencia. "Determinantal inequalities for J-accretive dissipative matrices." Studia Universitatis Babes-Bolyai Matematica 62, no. 1 (March 1, 2017): 119–25. http://dx.doi.org/10.24193/subbmath.2017.0009.

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50

Barone, R. C. A., and R. K. Kaul. "A Factorization Algorithm for Wave Propagation in Periodic Structures with Application to Torsional Waves in an Infinite Cylinder." Journal of Mechanics 16, no. 1 (March 2000): 9–14. http://dx.doi.org/10.1017/s1727719100001714.

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Анотація:

ABSTRACTIn this paper we introduce a fundamentally new concept in the field of wave-propagation in periodic structures. We show that a phenomenal amount of simplification can be achieved by using symmetry arguments. Problems which ordinarily lead to (2n × 2n) determinantal eigenvalue equations, can effectively be reduced to two (n × n) determinantal equations. We state the final result in the form of a factorization theorem and then with the help of a simple problem, we show the superiority of this new result over the traditional methods of solution.

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