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Journal articles on the topic 'Homogenization algebra'

Author: Grafiati

Published: 28 June 2021

Last updated: 5 February 2022

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1

Heckenberger, István, and Volkmar Welker. "A Deformation of the Orlik-Solomon Algebra." MATHEMATICA SCANDINAVICA 118, no. 2 (June 9, 2016): 183. http://dx.doi.org/10.7146/math.scand.a-23686.

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A deformation of the Orlik-Solomon algebra of a matroid $\mathfrak{M}$ is defined as a quotient of the free associative algebra over a commutative ring $R$ with $1$. It is shown that the given generators form a Gröbner basis and that after suitable homogenization the deformation and the Orlik-Solomon have the same Hilbert series as $R$-algebras. For supersolvable matroids, equivalently fiber type arrangements, there is a quadratic Gröbner basis and hence the algebra is Koszul.

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2

Ballico, E., B. Callander, and E. Gasparim. "Compactifications of adjoint orbits and their Hodge diamonds." Journal of Algebra and Its Applications 17, no. 06 (May 23, 2018): 1850099. http://dx.doi.org/10.1142/s0219498818500998.

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A recent theorem of [E. Gasparim, L. Grama and L. A. B. San Martin, Lefschetz fibrations on adjoint orbits, Forum Math. 28(5) (2016) 967–980.] showed that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We investigate the behavior of their fiberwise compactifications. Expressing adjoint orbits and fibers as affine varieties in their Lie algebra, we compactify them to projective varieties via homogenization of the defining ideals. We find that their Hodge diamonds vary wildly according to the choice of homogenization, and that extensions of the potential to the compactification must acquire degenerate singularities.

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3

MARTÍNEZ-VILLA, ROBERTO. "ON THE HOMOGENIZED ENVELOPING ALGEBRA OF THE LIE ALGEBRA Sℓ(2,ℂ) II." Glasgow Mathematical Journal 59, no. 1 (June 10, 2016): 189–219. http://dx.doi.org/10.1017/s0017089516000112.

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AbstractIn a previous paper, we studied the homogenized enveloping algebra of the Lie algebrasℓ(2,ℂ) and the homogenized Verma modules. The aim of this paper is to study the homogenization$\mathcal{O}$Bof the Bernstein–Gelfand–Gelfand category$\mathcal{O}$of sℓ(2,ℂ), and to apply the ideas developed jointly with J. Mondragón in our work on Groebner basis algebras, to give the relations between the categories$\mathcal{O}$Band$\mathcal{O}$as well as, between the derived categories$\mathcal{D}$b($\mathcal{O}$B) and$\mathcal{D}$b($\mathcal{O}$).

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4

Woukeng, Jean Louis. "Homogenization in algebras with mean value." Banach Journal of Mathematical Analysis 9, no. 2 (2015): 142–82. http://dx.doi.org/10.15352/bjma/09-2-12.

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5

Zhikov, V. V. "Connectedness and homogenization. Examples of fractal conductivity." Sbornik: Mathematics 187, no. 8 (August 31, 1996): 1109–47. http://dx.doi.org/10.1070/sm1996v187n08abeh000150.

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6

Sandrakov, G. V. "Homogenization of variational inequalities for obstacle problems." Sbornik: Mathematics 196, no. 4 (April 30, 2005): 541–60. http://dx.doi.org/10.1070/sm2005v196n04abeh000891.

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7

Gadyl'shin, R. R. "Analogues of the Helmholtz resonator in homogenization theory." Sbornik: Mathematics 193, no. 11 (December 31, 2002): 1611–38. http://dx.doi.org/10.1070/sm2002v193n11abeh000691.

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8

Pastukhova, S. E. "Homogenization of elasticity problems on periodic composite structures." Sbornik: Mathematics 196, no. 7 (August 31, 2005): 1033–73. http://dx.doi.org/10.1070/sm2005v196n07abeh000947.

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9

Sirazhudinov, M. M. "G-convergence and homogenization of generalized Beltrami operators." Sbornik: Mathematics 199, no. 5 (June 30, 2008): 755–86. http://dx.doi.org/10.1070/sm2008v199n05abeh003941.

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10

Zhikov, V. V., and S. E. Pastukhova. "Homogenization for elasticity problems on periodic networks of critical thickness." Sbornik: Mathematics 194, no. 5 (June 30, 2003): 697–732. http://dx.doi.org/10.1070/sm2003v194n05abeh000735.

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11

Sandrakov, G. V. "Homogenization of variational inequalities and equations defined by pseudomonotone operators." Sbornik: Mathematics 199, no. 1 (February 28, 2008): 67–98. http://dx.doi.org/10.1070/sm2008v199n01abeh003911.

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12

Nguetseng, Gabriel, Hubert Nnang, and Nils Svanstedt. "G-convergence and homogenization of monotone damped hyperbolic equations." Banach Journal of Mathematical Analysis 4, no. 1 (2010): 100–115. http://dx.doi.org/10.15352/bjma/1272374674.

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13

Waurick, Marcus. "On the homogenization of partial integro-differential-algebraic equations." Operators and Matrices, no. 2 (2016): 247–83. http://dx.doi.org/10.7153/oam-10-15.

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14

Redman, Irmgard T. "The homogenization of the three dimensional skew polynomial algebras of type I." Communications in Algebra 27, no. 11 (January 1999): 5587–602. http://dx.doi.org/10.1080/00927879908826775.

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15

Vasilevskaya, E. S. "A periodic parabolic Cauchy problem: Homogenization with corrector." St. Petersburg Mathematical Journal 21, no. 1 (November 4, 2009): 1–41. http://dx.doi.org/10.1090/s1061-0022-09-01083-8.

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16

Veniaminov, N. "Homogenization of periodic differential operators of high order." St. Petersburg Mathematical Journal 22, no. 5 (October 1, 2011): 751–75. http://dx.doi.org/10.1090/s1061-0022-2011-01166-5.

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17

CAPRIZ, G., and G. MAZZINI. "A σ-ALGEBRA AND A CONCEPT OF LIMIT FOR BODIES." Mathematical Models and Methods in Applied Sciences 10, no. 06 (August 2000): 801–13. http://dx.doi.org/10.1142/s0218202500000410.

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Recent developments in mechanics of continua (the search for optimal shapes of bodies, homogenization theory, the study of the trabecular structure of bones, the dynamics of immiscible mixtures, etc.) render some of the introductory axioms of continuum mechanics inadequate. Not only does one need to give meaning to the join and meet of two bodies, but also to extend the consequent algebra so as to encompass the result of a countable sequence of operations of join or meet; and one should also be able to define the limit of a sequence of bodies. To achieve this goal we propose here to define a body ab initio through the assignment of a probability measure dπ. We realize that π leaves, generally, too much of the texture of the body unspecified; to make up for this deficiency, we suggest the use of appropriate texture measures, reminescent of Tartar's H-measures.9

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18

Skrypnik, I. V. "Homogenization of non-linear Dirichlet problems in perforated domains of general type." Sbornik: Mathematics 187, no. 8 (August 31, 1996): 1229–60. http://dx.doi.org/10.1070/sm1996v187n08abeh000154.

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19

Nazarov, Sergei A., Guido H. Sweers, and Andrey S. Slutskij. "Homogenization of a thin plate reinforced with periodic families of rigid rods." Sbornik: Mathematics 202, no. 8 (August 31, 2011): 1127–68. http://dx.doi.org/10.1070/sm2011v202n08abeh004181.

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20

Suslina, T. A. "Homogenization with corrector for a stationary periodic Maxwell system." St. Petersburg Mathematical Journal 19, no. 3 (March 21, 2008): 455–95. http://dx.doi.org/10.1090/s1061-0022-08-01006-6.

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21

Pastukhova, S. E. "The Neumann problem for elliptic equations with multiscale coefficients: operator estimates for homogenization." Sbornik: Mathematics 207, no. 3 (March 31, 2016): 418–43. http://dx.doi.org/10.1070/sm8486.

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22

Monzner, Alexandra, Nicolas Vichery, and Frol Zapolsky. "Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization." Journal of Modern Dynamics 6, no. 2 (2012): 205–49. http://dx.doi.org/10.3934/jmd.2012.6.205.

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23

Dorodnyĭ, M. A. "Homogenization of periodic Schrödinger-type equations, with lower order terms." St. Petersburg Mathematical Journal 31, no. 6 (October 27, 2020): 1001–54. http://dx.doi.org/10.1090/spmj/1632.

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24

Sakamoto, Kunimochi. "Spatial homogenization and internal layers in a reaction-diffusion system." Hiroshima Mathematical Journal 30, no. 3 (2000): 377–402. http://dx.doi.org/10.32917/hmj/1206124605.

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25

Ming, Pingbing, and Pingwen Zhang. "Analysis of the heterogeneous multiscale method for parabolic homogenization problems." Mathematics of Computation 76, no. 257 (January 1, 2007): 153–78. http://dx.doi.org/10.1090/s0025-5718-06-01909-0.

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26

Birman, M. Sh, and T. A. Suslina. "Operator error estimates in the homogenization problem for nonstationary periodic equations." St. Petersburg Mathematical Journal 20, no. 6 (October 1, 2009): 873–928. http://dx.doi.org/10.1090/s1061-0022-09-01077-2.

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27

KRONE, ROBERT. "NUMERICAL ALGORITHMS FOR DUAL BASES OF POSITIVE-DIMENSIONAL IDEALS." Journal of Algebra and Its Applications 12, no. 06 (May 9, 2013): 1350018. http://dx.doi.org/10.1142/s0219498813500187.

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An ideal of a local polynomial ring can be described by calculating a standard basis with respect to a local monomial ordering. However the usual standard basis algorithms are not numerically stable. A numerically stable approach to describing the ideal is by finding the space of dual functionals that annihilate it, which reduces the problem to one of linear algebra. There are several known algorithms for finding the truncated dual up to any specified degree, which is useful for describing zero-dimensional ideals. We present a stopping criterion for positive-dimensional cases based on homogenization that guarantees all generators of the initial monomial ideal are found. This has applications for calculating Hilbert functions.

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28

Pastukhova, S. E., and R. N. Tikhomirov. "On operator-type homogenization estimates for elliptic equations with lower order terms." St. Petersburg Mathematical Journal 29, no. 5 (July 26, 2018): 841–61. http://dx.doi.org/10.1090/spmj/1518.

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29

Sango, M. "Homogenization of singular numbers for a non self-adjoint elliptic problem in a perforated domain." Integral Equations and Operator Theory 43, no. 2 (June 2002): 177–88. http://dx.doi.org/10.1007/bf01200252.

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30

Zhikov, V. V. "Estimates of Nash-Aronson type for a diffusion equation with asymmetric matrix and their applications to homogenization." Sbornik: Mathematics 197, no. 12 (December 31, 2006): 1775–804. http://dx.doi.org/10.1070/sm2006v197n12abeh003822.

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31

HORIE, Kazuo, and Hitoshi ISHII. "Simultaneous Effects of Homogenization and Vanishing Viscosity in Fully Nonlinear Elliptic Equations." Funkcialaj Ekvacioj 46, no. 1 (2003): 63–88. http://dx.doi.org/10.1619/fesi.46.63.

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32

Senik, N. N. "Homogenization for a periodic elliptic operator in a strip with various boundary conditions." St. Petersburg Mathematical Journal 25, no. 4 (June 5, 2014): 647–97. http://dx.doi.org/10.1090/s1061-0022-2014-01311-8.

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33

Zhikov, V. V., and S. E. Pastukhova. "Homogenization and two-scale convergence in the Sobolev space with an oscillating exponent." St. Petersburg Mathematical Journal 30, no. 2 (February 14, 2019): 231–51. http://dx.doi.org/10.1090/spmj/1540.

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34

Pakhnin, M. A., and T. A. Suslina. "Operator error estimates for homogenization of the elliptic Dirichlet problem in a bounded domain." St. Petersburg Mathematical Journal 24, no. 6 (September 23, 2013): 949–76. http://dx.doi.org/10.1090/s1061-0022-2013-01274-x.

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35

Meshkova, Yu M., and T. A. Suslina. "Homogenization of the first initial boundary-value problem for parabolic systems: operator error estimates." St. Petersburg Mathematical Journal 29, no. 6 (September 4, 2018): 935–78. http://dx.doi.org/10.1090/spmj/1521.

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36

Dorodnyi, M., and T. Suslina. "Homogenization of hyperbolic equations with periodic coefficients in ℝ^{𝕕}: Sharpness of the results." St. Petersburg Mathematical Journal 32, no. 4 (July 9, 2021): 605–703. http://dx.doi.org/10.1090/spmj/1664.

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In L 2 ( R d ; C n ) L_2(\mathbb {R}^d;\mathbb {C}^n) , a selfadjoint strongly elliptic second order differential operator A ε \mathcal {A}_\varepsilon is considered. It is assumed that the coefficients of A ε \mathcal {A}_\varepsilon are periodic and depend on x / ε \mathbf {x}/\varepsilon , where ε > 0 \varepsilon >0 is a small parameter. We find approximations for the operators cos ⁡ ( A ε 1 / 2 τ ) \cos (\mathcal {A}_\varepsilon ^{1/2}\tau ) and A ε − 1 / 2 sin ⁡ ( A ε 1 / 2 τ ) \mathcal {A}_\varepsilon ^{-1/2}\sin (\mathcal {A}_\varepsilon ^{1/2}\tau ) in the norm of operators acting from the Sobolev space H s ( R d ) H^s(\mathbb {R}^d) to L 2 ( R d ) L_2(\mathbb {R}^d) (with suitable s s ). We also find approximation with corrector for the operator A ε − 1 / 2 sin ⁡ ( A ε 1 / 2 τ ) \mathcal {A}_\varepsilon ^{-1/2}\sin (\mathcal {A}_\varepsilon ^{1/2}\tau ) in the ( H s → H 1 ) (H^s \to H^1) -norm. The question about the sharpness of the results with respect to the type of the operator norm and with respect to the dependence of estimates on τ \tau is studied. The results are applied to study the behavior of the solutions of the Cauchy problem for the hyperbolic equation ∂ τ 2 u ε = − A ε u ε + F \partial _\tau ^2 \mathbf {u}_\varepsilon = -\mathcal {A}_\varepsilon \mathbf {u}_\varepsilon + \mathbf {F} .

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37

Belyaev, A. G., and G. A. Chechkin. "Homogenization of a mixed boundary-value problem for the Laplace operator in the case of an insoluble 'limit' problem." Sbornik: Mathematics 186, no. 4 (April 30, 1995): 511–25. http://dx.doi.org/10.1070/sm1995v186n04abeh000029.

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38

Nazarov, S. A. "Homogenization of Kirchhoff plates joined by rivets which are modeled by the Sobolev point conditions." St. Petersburg Mathematical Journal 32, no. 2 (March 2, 2021): 307–48. http://dx.doi.org/10.1090/spmj/1649.

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39

Bertone, Cristina, Francesca Cioffi, and Margherita Roggero. "Macaulay-like marked bases." Journal of Algebra and Its Applications 16, no. 05 (April 12, 2017): 1750100. http://dx.doi.org/10.1142/s0219498817501006.

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We define marked sets and bases over a quasi-stable ideal [Formula: see text] in a polynomial ring on a Noetherian [Formula: see text]-algebra, with [Formula: see text] a field of any characteristic. The involved polynomials may be non-homogeneous, but their degree is bounded from above by the maximum among the degrees of the terms in the Pommaret basis of [Formula: see text] and a given integer [Formula: see text]. Due to the combinatorial properties of quasi-stable ideals, these bases behave well with respect to homogenization, similarly to Macaulay bases. We prove that the family of marked bases over a given quasi-stable ideal has an affine scheme structure, is flat and, for large enough [Formula: see text], is an open subset of a Hilbert scheme. Our main results lead to algorithms that explicitly construct such a family. We compare our method with similar ones and give some complexity results.

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40

Cardone, G., A. Corbo Esposito, and S. A. Nazarov. "Homogenization of the mixed boundary-value problem for a formally selfadjoint elliptic system in a periodically punched domain." St. Petersburg Mathematical Journal 21, no. 4 (May 20, 2010): 601–34. http://dx.doi.org/10.1090/s1061-0022-2010-01108-7.

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41

Meshkova, Yu M. "Homogenization of periodic parabolic systems in the $L_2(\mathbb {R}^d)$-norm with the corrector taken into account." St. Petersburg Mathematical Journal 31, no. 4 (June 11, 2020): 675–718. http://dx.doi.org/10.1090/spmj/1619.

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42

Sokołowski, Damian, Marcin Kamiński, and Artur Wirowski. "Energy Fluctuations in the Homogenized Hyper-Elastic Particulate Composites with Stochastic Interface Defects." Energies 13, no. 8 (April 17, 2020): 2011. http://dx.doi.org/10.3390/en13082011.

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The principle aim of this study is to analyze deformation energy of hyper-elastic particulate composites, which is the basis for their further probabilistic homogenization. These composites have some uncertain interface defects, which are modelled as small semi-spheres with random radius and with bases positioned on the particle-matrix interface. These defects are smeared into thin layer of the interphase surrounding the reinforcing particle introduced as the third component of this composite. Matrix properties are determined from the experimental tests of Laripur LPR 5020 High Density Polyurethane (HDPU). It is strengthened with the Carbon Black particles of spherical shape. The Arruda–Boyce potential has been selected for numerical experiments as fitting the best stress-strain curves for the matrix behavior. A homogenization procedure is numerically implemented using the cubic Representative Volume Element (RVE). Spherical particle is located centrally, and computations of deformation energy probabilistic characteristics are carried out using the Iterative Stochastic Finite Element Method (ISFEM). This ISFEM is implemented in the algebra system MAPLE 2019 as dual approach based upon the stochastic perturbation method and, independently, upon a classical Monte-Carlo simulation, and uniform uniaxial deformations of this RVE are determined in the system ABAQUS and its 20-noded solid hexahedral finite elements. Computational experiments include initial deterministic numerical error analysis and the basic probabilistic characteristics, i.e., expectations, deviations, skewness and kurtosis of the deformation energy. They are performed for various expected values of the defects volume fraction. We analyze numerically (1) if randomness of homogenized deformation energy can correspond to the normal distribution, (2) how variability of the interface defects volume fraction affects the deterministic and stochastic characteristics of composite deformation energy and (3) whether the stochastic perturbation method is efficient in deformation energy computations (and in FEM analysis) of hyper-elastic media.

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43

Suslina, T. A. "Homogenization in the Sobolev class $H^{1}(\mathbb R^{d})$ for second order periodic elliptic operators with the inclusion of first order terms." St. Petersburg Mathematical Journal 22, no. 1 (February 1, 2011): 81. http://dx.doi.org/10.1090/s1061-0022-2010-01135-x.

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44

Rossi, Michele, and Lea Terracini. "Toric varieties and Gröbner bases: the complete $$\mathbb {Q}$$-factorial case." Applicable Algebra in Engineering, Communication and Computing 31, no. 5-6 (July 22, 2020): 461–82. http://dx.doi.org/10.1007/s00200-020-00452-w.

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Abstract We present two algorithms determining all the complete and simplicial fans admitting a fixed non-degenerate set of vectors V as generators of their 1-skeleton. The interplay of the two algorithms allows us to discerning if the associated toric varieties admit a projective embedding, in principle for any values of dimension and Picard number. The first algorithm is slower than the second one, but it computes all complete and simplicial fans supported by V and lead us to formulate a topological-combinatoric conjecture about the definition of a fan. On the other hand, we adapt the Sturmfels’ arguments on the Gröbner fan of toric ideals to our complete case; we give a characterization of the Gröbner region and show an explicit correspondence between Gröbner cones and chambers of the secondary fan. A homogenization procedure of the toric ideal associated to V allows us to employing GFAN and related software in producing our second algorithm. The latter turns out to be much faster than the former, although it can compute only the projective fans supported by V. We provide examples and a list of open problems. In particular we give examples of rationally parametrized families of $$\mathbb {Q}$$ Q -factorial complete toric varieties behaving in opposite way with respect to the dimensional jump of the nef cone over a special fibre.

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45

Frid, Hermano, and Jean Silva. "Homogenization of Nonlinear PDEs in the Fourier–Stieltjes Algebras." SIAM Journal on Mathematical Analysis 41, no. 4 (January 2009): 1589–620. http://dx.doi.org/10.1137/080737022.

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46

Gaddis, Jason. "PBW deformations of Artin–Schelter regular algebras." Journal of Algebra and Its Applications 15, no. 04 (February 19, 2016): 1650064. http://dx.doi.org/10.1142/s021949881650064x.

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We consider properties and extensions of PBW deformations of Artin–Schelter regular algebras. PBW deformations in global dimension two are classified and the geometry associated to the homogenizations of these algebras is exploited to prove that all simple modules are one-dimensional in the non-PI case. It is shown that this property is maintained under tensor products and certain skew polynomial extensions.

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47

Frid, Hermano, and Jean Silva. "Homogenization of degenerate porous medium type equations in ergodic algebras." Advances in Mathematics 246 (October 2013): 303–50. http://dx.doi.org/10.1016/j.aim.2013.07.005.

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48

Roch, Steffen, and Pedro A. Santos. "Two points, one limit: Homogenization techniques for two-point local algebras." Journal of Mathematical Analysis and Applications 391, no. 2 (July 2012): 552–66. http://dx.doi.org/10.1016/j.jmaa.2012.02.054.

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49

Frid, Hermano, Jean Silva, and Henrique Versieux. "Homogenization of a generalized Stefan problem in the context of ergodic algebras." Journal of Functional Analysis 268, no. 11 (June 2015): 3232–77. http://dx.doi.org/10.1016/j.jfa.2015.03.021.

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50

Wang, Juan, and Jie Zhao. "Convergence rates of nonlinear Stokes problems in homogenization." Boundary Value Problems 2019, no. 1 (May 24, 2019). http://dx.doi.org/10.1186/s13661-019-1209-x.

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