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Journal articles on the topic 'Embedding dimension'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Sivakumar, B., R. Berndtsson, J. Olsson, K. Jinno, and A. Kawamura. "Dynamics of monthly rainfall-runoff process at the Gota basin: A search for chaos." Hydrology and Earth System Sciences 4, no. 3 (September 30, 2000): 407–17. http://dx.doi.org/10.5194/hess-4-407-2000.

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Abstract. Sivakumar et al. (2000a), by employing the correlation dimension method, provided preliminary evidence of the existence of chaos in the monthly rainfall-runoff process at the Gota basin in Sweden. The present study verifies and supports the earlier results and strengthens such evidence. The study analyses the monthly rainfall, runoff and runoff coefficient series using the nonlinear prediction method, and the presence of chaos is investigated through an inverse approach, i.e. identifying chaos from the results of the prediction. The presence of an optimal embedding dimension (the embedding dimension with the best prediction accuracy) for each of the three series indicates the existence of chaos in the rainfall-runoff process, providing additional support to the results obtained using the correlation dimension method. The reasonably good predictions achieved, particularly for the runoff series, suggest that the dynamics of the rainfall-runoff process could be understood from a chaotic perspective. The predictions are also consistent with the correlation dimension results obtained in the earlier study, i.e. higher prediction accuracy for series with a lower dimension and vice-versa, so that the correlation dimension method can indeed be used as a preliminary indicator of chaos. However, the optimal embedding dimensions obtained from the prediction method are considerably less than the minimum dimensions essential to embed the attractor, as obtained by the correlation dimension method. A possible explanation for this could be the presence of noise in the series, since the effects of noise at higher embedding dimensions could be significantly greater than that at lower embedding dimensions. Keywords: Rainfall-runoff; runoff coefficient; chaos; phase-space; correlation dimension; nonlinear prediction; noise

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2

Aleksić, Zoran. "Estimating the embedding dimension." Physica D: Nonlinear Phenomena 52, no. 2-3 (September 1991): 362–68. http://dx.doi.org/10.1016/0167-2789(91)90132-s.

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3

Grines, V. Z., E. Ya Gurevich, and O. V. Pochinka. "On Embedding of the Morse-Smale Diffeomorphisms in a Topological Flow." Contemporary Mathematics. Fundamental Directions 66, no. 2 (December 15, 2020): 160–81. http://dx.doi.org/10.22363/2413-3639-2020-66-2-160-181.

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This review presents the results of recent years on solving of the Palis problem on finding necessary and sufficient conditions for the embedding of Morse-Smale cascades in topological flows. To date, the problem has been solved by Palis for Morse-Smale diffeomorphisms given on manifolds of dimension two. The result for the circle is a trivial exercise. In dimensions three and higher new effects arise related to the possibility of wild embeddings of closures of invariant manifolds of saddle periodic points that leads to additional obstacles for Morse-Smale diffeomorphisms to embed in topological flows. The progress achieved in solving of Paliss problem in dimension three is associated with the recently obtained complete topological classification of Morse-Smale diffeomorphisms on three-dimensional manifolds and the introduction of new invariants describing the embedding of separatrices of saddle periodic points in a supporting manifold. The transition to a higher dimension requires the latest results from the topology of manifolds. The necessary topological information, which plays key roles in the proofs, is also presented in the survey.

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4

HAMBLY, B. M., and T. KUMAGAI. "ASYMPTOTICS FOR THE SPECTRAL AND WALK DIMENSION AS FRACTALS APPROACH EUCLIDEAN SPACE." Fractals 10, no. 04 (December 2002): 403–12. http://dx.doi.org/10.1142/s0218348x02001270.

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We discuss the behavior of the dynamic dimension exponents for families of fractals based on the Sierpinski gasket and carpet. As the length scale factor for the family tends to infinity, the lattice approximations to the fractals look more like the tetrahedral or cubic lattice in Euclidean space and the fractal dimension converges to that of the embedding space. However, in the Sierpinski gasket case, the spectral dimension converges to two for all dimensions. In two dimensions, we prove a conjecture made in the physics literature concerning the rate of convergence. On the other hand, for natural families of Sierpinski carpets, the spectral dimension converges to the dimension of the embedding Euclidean space. In general, we demonstrate that for both cases of finitely and infinitely ramified fractals, a variety of asymptotic values for the spectral dimension can be achieved.

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5

SATHER-WAGSTAFF, SEAN. "EMBEDDING MODULES OF FINITE HOMOLOGICAL DIMENSION." Glasgow Mathematical Journal 55, no. 1 (August 2, 2012): 85–96. http://dx.doi.org/10.1017/s0017089512000353.

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AbstractThis paper builds on work of Hochster and Yao that provides nice embeddings for finitely generated modules of finite G-dimension, finite projective dimension or locally finite injective dimension. We extend these results by providing similar embeddings in the relative setting, that is, for certain modules of finite GC-dimension, finite C-projective dimension, locally finite C-injective dimension or locally finite C-injective dimension where C is a semidualizing module. Along the way, we extend some results for modules of finite homological dimension to modules of locally finite homological dimension in the relative setting.

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6

Chapman, S. T., P. A. García-Sánchez, D. Llena, and J. Marshall. "Elements in a Numerical Semigroup with Factorizations of the Same Length." Canadian Mathematical Bulletin 54, no. 1 (March 1, 2011): 39–43. http://dx.doi.org/10.4153/cmb-2010-068-3.

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AbstractQuestions concerning the lengths of factorizations into irreducible elements in numerical monoids have gained much attention in the recent literature. In this note, we show that a numerical monoid has an element with two different irreducible factorizations of the same length if and only if its embedding dimension is greater than two. We find formulas in embedding dimension three for the smallest element with two different irreducible factorizations of the same length and the largest element whose different irreducible factorizations all have distinct lengths. We show that these formulas do not naturally extend to higher embedding dimensions.

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7

Gács, Peter. "Clairvoyant embedding in one dimension." Random Structures & Algorithms 47, no. 3 (June 13, 2014): 520–60. http://dx.doi.org/10.1002/rsa.20551.

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8

Lequain, Yves. "Embedding dimension in local rings." Communications in Algebra 18, no. 11 (January 1990): 3923–31. http://dx.doi.org/10.1080/00927879008824117.

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9

Rosales, J. C., and P. A. Garc�a-S�anchez. "Numerical semigroups with embedding dimension three." Archiv der Mathematik 83, no. 6 (December 2004): 488–96. http://dx.doi.org/10.1007/s00013-004-1149-1.

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10

Mees, A. I., P. E. Rapp, and L. S. Jennings. "Singular-value decomposition and embedding dimension." Physical Review A 36, no. 1 (July 1, 1987): 340–46. http://dx.doi.org/10.1103/physreva.36.340.

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11

Anick, David J. "Thin algebras of embedding dimension three." Journal of Algebra 100, no. 1 (April 1986): 235–59. http://dx.doi.org/10.1016/0021-8693(86)90076-1.

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12

Anwar, Imran, and Dorin Popescu. "Stanley conjecture in small embedding dimension." Journal of Algebra 318, no. 2 (December 2007): 1027–31. http://dx.doi.org/10.1016/j.jalgebra.2007.06.005.

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13

ABARBANEL, HENRY D. I., and MIKHAIL M. SUSHCHIK. "LOCAL OR DYNAMICAL DIMENSIONS OF NONLINEAR SYSTEMS INFERRED FROM OBSERVATIONS." International Journal of Bifurcation and Chaos 03, no. 03 (June 1993): 543–50. http://dx.doi.org/10.1142/s0218127493000428.

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In time delay reconstruction of the phase space of a system from observed scalar data, one requires a time lag and an integer embedding dimension. The minimum embedding dimension, dE, may be larger than the actual local dimension of the underlying dynamics, dL. The embedding theorem only guarantees that the attractor of the system is unfolded in the integer dE greater than 2dA with dA being the attractor dimension. We present two methods for determining the dimension, dL≤dE, of the underlying dynamics. The first relies on the local Lyapunov exponents of the dynamics, and the second seeks an optimum dimension for prediction of the time series for steps forward and then backward in time. We demonstrate these methods on several examples. Model building of the dynamics should take place in the dL-dimensional space.

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14

Olberding, Bruce. "Generic formal fibers and analytically ramified stable rings." Nagoya Mathematical Journal 211 (September 2013): 109–35. http://dx.doi.org/10.1017/s0027763000010801.

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AbstractLetAbe a local Noetherian domain of Krull dimensiond. Heinzer, Rotthaus, and Sally have shown that if the generic formal fiber ofAhas dimensiond– 1, thenAis birationally dominated by a 1-dimensional analytically ramified local Noetherian ring having residue field finite over the residue field ofA. We explore further this correspondence between prime ideals in the generic formal fiber and 1-dimensional analytically ramified local rings. Our main focus is on the case where the analytically ramified local rings are stable, and we show that in this case the embedding dimension of the stable ring reflects the embedding dimension of a prime ideal maximal in the generic formal fiber, thus providing a measure of how far the generic formal fiber deviates from regularity. A number of characterizations of analytically ramified local stable domains are also given.

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15

GALKA, ANDREAS, and GERD PFISTER. "DYNAMICAL CORRELATIONS ON RECONSTRUCTED INVARIANT DENSITIES AND THEIR EFFECT ON CORRELATION DIMENSION ESTIMATION." International Journal of Bifurcation and Chaos 13, no. 03 (March 2003): 723–32. http://dx.doi.org/10.1142/s0218127403006881.

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We investigate the structure of dynamical correlations on reconstructed attractors which were obtained by time-delay embedding of periodic, quasi-periodic and chaotic time series. Within the specific sampling of the invariant density by a finite number of vectors which results from embedding, we identify two separate levels of sampling, corresponding to two different types of dynamical correlations, each of which produces characteristic artifacts in correlation dimension estimation: the well-known trajectory bias and a characteristic oscillation due to periodic sampling. For the second artifact we propose random sampling as a new correction method which is shown to provide improved sampling and to reduce dynamical correlations more efficiently than it has been possible by the standard Theiler correction. For accurate numerical analysis of correlation dimension in a bootstrap framework both corrections should be combined. For tori and the Lorenz attractor we also show how to construct time-delay embeddings which are completely free of any dynamical correlations.

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16

Moreno-Frias, Maria Angeles, and Jose Carlos Rosales. "Perfect numerical semigroups with embedding dimension three." Publicationes Mathematicae Debrecen 97, no. 1-2 (July 1, 2020): 77–84. http://dx.doi.org/10.5486/pmd.2020.8699.

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17

Garcı́a-Garcı́a, Juan Ignacio, Daniel Marı́n-Aragón, María Ángeles Moreno-Frías, José Carlos Rosales, and Alberto Vigneron-Tenorio. "Semigroups with fixed multiplicity and embedding dimension." Ars Mathematica Contemporanea 17, no. 2 (November 4, 2019): 397–417. http://dx.doi.org/10.26493/1855-3974.1937.5ea.

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18

Pan, Yaozhang, Shuzhi Sam Ge, and Abdullah Al Mamun. "Weighted locally linear embedding for dimension reduction." Pattern Recognition 42, no. 5 (May 2009): 798–811. http://dx.doi.org/10.1016/j.patcog.2008.08.024.

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19

Söderberg, Jonas. "Artinian level modules of embedding dimension two." Journal of Pure and Applied Algebra 207, no. 2 (October 2006): 417–32. http://dx.doi.org/10.1016/j.jpaa.2005.10.002.

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20

Iglésias, Laura, and Ana Margarida Neto. "Densities of maximal embedding dimension numerical semigroups." Communications in Algebra 46, no. 6 (December 15, 2017): 2730–37. http://dx.doi.org/10.1080/00927872.2017.1399405.

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21

Bates, Jonathan. "The embedding dimension of Laplacian eigenfunction maps." Applied and Computational Harmonic Analysis 37, no. 3 (November 2014): 516–30. http://dx.doi.org/10.1016/j.acha.2014.03.002.

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22

Kishimoto, Takashi. "Abhyankar-Sathaye embedding problem in dimension three." Journal of Mathematics of Kyoto University 42, no. 4 (2002): 641–69. http://dx.doi.org/10.1215/kjm/1250283832.

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23

Witsenhausen, H. S. "Minimum dimension embedding of finite metric spaces." Journal of Combinatorial Theory, Series A 42, no. 2 (July 1986): 184–99. http://dx.doi.org/10.1016/0097-3165(86)90089-0.

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24

Rosales, J. C., and P. A. García-Sánchez. "On Numerical Semigroups with High Embedding Dimension." Journal of Algebra 203, no. 2 (May 1998): 567–78. http://dx.doi.org/10.1006/jabr.1997.7341.

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25

Kalton, N. J., A. Koldobsky, V. Yaskin, and M. Yaskina. "The Geometry of L0." Canadian Journal of Mathematics 59, no. 5 (October 1, 2007): 1029–49. http://dx.doi.org/10.4153/cjm-2007-044-0.

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AbstractSuppose that we have the unit Euclidean ball in ℝn and construct new bodies using three operations — linear transformations, closure in the radial metric, and multiplicative summation defined by We prove that in dimension 3 this procedure gives all origin-symmetric convex bodies, while this is no longer true in dimensions 4 and higher. We introduce the concept of embedding of a normed space in L0 that naturally extends the corresponding properties of Lp-spaces with p ≠ 0, and show that the procedure described above gives exactly the unit balls of subspaces of L0 in every dimension. We provide Fourier analytic and geometric characterizations of spaces embedding in L0, and prove several facts confirming the place of L0 in the scale of Lp-spaces.

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26

Buse, O., and R. Hind. "Ellipsoid embeddings and symplectic packing stability." Compositio Mathematica 149, no. 5 (March 4, 2013): 889–902. http://dx.doi.org/10.1112/s0010437x12000826.

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AbstractWe prove packing stability for rational symplectic manifolds. This will rely on a general symplectic embedding result for ellipsoids which assumes only that there is no volume obstruction and that the domain is sufficiently thin relative to the target. We also obtain easily computable bounds for the Embedded Contact Homology capacities which are sufficient to imply the existence of some symplectic volume filling embeddings in dimension 4.

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27

Steinbach, Anja, and Hendrik Van Maldeghem. "Regular Embeddings of Generalized Hexagons." Canadian Journal of Mathematics 56, no. 5 (October 1, 2004): 1068–93. http://dx.doi.org/10.4153/cjm-2004-048-3.

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AbstractWe classify the generalized hexagons which are laxly embedded in projective space such that the embedding is flat and polarized. Besides the standard examples related to the hexagons defined over the algebraic groups of type G2,3D4and6D4(and occurring in projective dimensions 5, 6, 7), we find new examples in unbounded dimension related to the mixed groups of type G2.

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28

Xia, Jie Yun, and Shuai Bin Lian. "The Globally Linear Embedding Algorithm." Advanced Materials Research 756-759 (September 2013): 2682–86. http://dx.doi.org/10.4028/www.scientific.net/amr.756-759.2682.

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LLE is a very effective non-linear dimension reduction algorithm and widely explored in machine learning, pattern recognition, data mining and etc. Locally linear, Globally non-linear has always been regarded as the features and advantages of LLE. However, the theoretical derivation presented in this paper shows that when the size of neighborhood is larger than the dimension of the space in which the data are presented, LLE is no longer global nonlinear and almost has the same effect as PCA in dimensionality reduction. At present, a lot of literatures on LLE verify their results on Swiss Roll, Punctured Sphere, Twin Peaks, etc. These manifolds are presented in the three-dimensional Euclidean space and the size of neighborhood is always larger than three to prevent too small to be effective. But in these cases, LLE cannot play its advantage of nonlinearity.

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29

Yuan, Ye, Zhi Qiang Huang, and Ze Min Cai. "Classification of Multi-Types of EEG Time Series Based on Embedding Dimension Characteristic Parameter." Key Engineering Materials 474-476 (April 2011): 1987–92. http://dx.doi.org/10.4028/www.scientific.net/kem.474-476.1987.

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We have studied the detection of epileptic seizure by EEG signals based on embedding dimension as the input characteristic parameter of artificial neural networks has been studied in the research before. The results of the experiments showed that the overall accuracy as high as 100% can be achieved for distinguishing normal and epileptic EEG time series. In this paper, classification of multi-types of EEG time series based on embedding dimension as input characteristic parameter of artificial neural network will be studied, and the probabilistic neural network (PNN) will be also employed as the classifier for comparing the results with those obtained before. Cao’s method is also applied for computing the embedding dimension of normal and epileptic EEG time series. The results show that different types of EEG time series can be classified using the embedding dimension of EEG time series as characteristic parameter when the number of feature points exceed some value, however, the accuracy were not satisfied up to now, some work need to be done to improve the classification accuracy.

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30

Schoutens, Hans. "Dimension and singularity theory for local rings of finite embedding dimension." Journal of Algebra 386 (July 2013): 1–60. http://dx.doi.org/10.1016/j.jalgebra.2013.04.009.

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31

Gupta, Purvi, and Rasul Shafikov. "Polynomially convex embeddings of odd-dimensional closed manifolds." Journal für die reine und angewandte Mathematik (Crelles Journal) 2021, no. 777 (May 13, 2021): 273–99. http://dx.doi.org/10.1515/crelle-2021-0021.

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Abstract It is shown that any smooth closed orientable manifold of dimension 2 ⁢ k + 1 {2k+1} , k ≥ 2 {k\geq 2} , admits a smooth polynomially convex embedding into ℂ 3 ⁢ k {\mathbb{C}^{3k}} . This improves by 1 the previously known lower bound of 3 ⁢ k + 1 {3k+1} on the possible ambient complex dimension for such embeddings (which is sharp when k = 1 {k=1} ). It is further shown that the embeddings produced have the property that all continuous functions on the image can be uniformly approximated by holomorphic polynomials. Lastly, the same technique is modified to construct embeddings whose images have nontrivial hulls containing no nontrivial analytic disks. The distinguishing feature of this dimensional setting is the appearance of nonisolated CR-singularities, which cannot be tackled using only local analytic methods (as done in earlier results of this kind), and a topological approach is required.

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32

Oymak, Samet, and Joel A. Tropp. "Universality laws for randomized dimension reduction, with applications." Information and Inference: A Journal of the IMA 7, no. 3 (November 17, 2017): 337–446. http://dx.doi.org/10.1093/imaiai/iax011.

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Abstract Dimension reduction is the process of embedding high-dimensional data into a lower dimensional space to facilitate its analysis. In the Euclidean setting, one fundamental technique for dimension reduction is to apply a random linear map to the data. This dimension reduction procedure succeeds when it preserves certain geometric features of the set. The question is how large the embedding dimension must be to ensure that randomized dimension reduction succeeds with high probability. This article studies a natural family of randomized dimension reduction maps and a large class of data sets. It proves that there is a phase transition in the success probability of the dimension reduction map as the embedding dimension increases. For a given data set, the location of the phase transition is the same for all maps in this family. Furthermore, each map has the same stability properties, as quantified through the minimum RSV. These results can be viewed as new universality laws in high-dimensional stochastic geometry. Universality laws for randomized dimension reduction have many applications in applied mathematics, signal processing and statistics. They yield design principles for numerical linear algebra algorithms, for compressed sensing measurement ensembles and for random linear codes. Furthermore, these results have implications for the performance of statistical estimation methods under a large class of random experimental designs.

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33

Arene, Christophe, David Kohel, and Christophe Ritzenthaler. "Complete addition laws on abelian varieties." LMS Journal of Computation and Mathematics 15 (September 1, 2012): 308–16. http://dx.doi.org/10.1112/s1461157012001027.

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AbstractWe prove that under any projective embedding of an abelian variety A of dimension g, a complete set of addition laws has cardinality at least g+1, generalizing a result of Bosma and Lenstra for the Weierstrass model of an elliptic curve in ℙ2. In contrast, we prove, moreover, that if k is any field with infinite absolute Galois group, then there exists for every abelian variety A/k a projective embedding and an addition law defined for every pair of k-rational points. For an abelian variety of dimension 1 or 2, we show that this embedding can be the classical Weierstrass model or the embedding in ℙ15, respectively, up to a finite number of counterexamples for ∣k∣≤5 .

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34

Repovš, Dušan. "Peripheral acyclicity in 3-manifolds." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 42, no. 3 (June 1987): 312–21. http://dx.doi.org/10.1017/s1446788700028597.

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AbstractWe study weak and strong peripheral 1-acyclicity, a homology version of D. R. McMillan, Jr.'s weak cellularity criterion and cellularity criterion, for embeddings of compacta in 3-manifolds. In contrast with the two cellularity criteria we prove that the two peripheral acyclicities are equivalent and moreover, for compacta of dimension at most 1, independent of the embedding. We also give some results concerning regular neighborhoods of compact polyhedra in 3-manifolds.

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35

Schröer, Stefan. "On fibrations whose geometric fibers are nonreduced." Nagoya Mathematical Journal 200 (December 2010): 35–57. http://dx.doi.org/10.1017/s0027763000010163.

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AbstractWe give a bound on embedding dimensions of geometric generic fibers in terms of the dimension of the base, for fibrations in positive characteristic. This generalizes the well-known fact that for fibrations over curves, the geometric generic fiber is reduced. We illustrate our results with Fermat hypersurfaces and genus 1 curves.

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36

Schröer, Stefan. "On fibrations whose geometric fibers are nonreduced." Nagoya Mathematical Journal 200 (December 2010): 35–57. http://dx.doi.org/10.1215/00277630-2010-011.

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AbstractWe give a bound on embedding dimensions of geometric generic fibers in terms of the dimension of the base, for fibrations in positive characteristic. This generalizes the well-known fact that for fibrations over curves, the geometric generic fiber is reduced. We illustrate our results with Fermat hypersurfaces and genus 1 curves.

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37

Luo, Bei Er, Jun Xing Wang, and Ying Ying Zhang. "Researches on the Chaotic Characteristics of Fluctuating Pressure in Slit-Type Energy Dissipater." Advanced Materials Research 1025-1026 (September 2014): 1150–59. http://dx.doi.org/10.4028/www.scientific.net/amr.1025-1026.1150.

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On the basis of research on energy dissipation scheme of Changheba hydropower station in model tests, the chaotic characteristics of the fluctuating pressure in the slit-type contraction section were studied with the employment of chaotic theory. According to Takens’ embedding theorem, we performed a phase space reconstruction on the measured fluctuating series, where the optimal delay time was determined with the average mutual information (AMI) method, the optimal embedding dimension was determined with the averaged false nearest neighbor (AFN) method. Calculated from the testing results, the optimal delay time was ranging from 7 to 10 while the optimal embedding dimension was ranging from 12 to 14. With the optimal embedding parameters obtained, the correlation dimension D2 and the largest Lyaponov exponent λ1 was calculated, with the obtained D2 varying from 7.626 to 8.821 and λ1 varying from 0.091 to 0.302. Conclusively, the flow characteristics on the floor were more complex than those on the sidewall, while the flow structure exhibited no essential difference. Moreover, the distribution law of correlation dimension indicated that the effects of contraction on the flow around the floor were more remarkable; however, the calculated largest Lyapunov exponent could only be served as a qualitative indicator of a chaotic system, without any instruction to the degree of chaos.

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38

Gorodetsky, Victor G. "Determination of Dimension of Embedding for Chaotic Attractors." Journal of Automation and Information Sciences 39, no. 10 (2007): 47–56. http://dx.doi.org/10.1615/jautomatinfscien.v39.i10.30.

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39

ROBLES-PEREZ, AURELIANO M., and JOSE CARLOS ROSALES. "Proportionally modular numerical semigroups with embedding dimension three." Publicationes Mathematicae Debrecen 84, no. 3-4 (May 1, 2014): 319–32. http://dx.doi.org/10.5486/pmd.2014.5357.

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40

Ge, Shuzhi Sam, Hongsheng He, and Chengyao Shen. "Geometrically local embedding in manifolds for dimension reduction." Pattern Recognition 45, no. 4 (April 2012): 1455–70. http://dx.doi.org/10.1016/j.patcog.2011.09.022.

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41

Broomhead, D. S., R. Jones, and Gregory P. King. "Comment on ‘‘Singular-value decomposition and embedding dimension’’." Physical Review A 37, no. 12 (June 1, 1988): 5004–5. http://dx.doi.org/10.1103/physreva.37.5004.

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42

谢, 华伦. "Document Embedding Dimension Homeomorphism—Government Work Report Analysis." Computer Science and Application 10, no. 06 (2020): 1194–208. http://dx.doi.org/10.12677/csa.2020.106124.

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43

Némethi, András, and Tomohiro Okuma. "The embedding dimension of weighted homogeneous surface singularities." Journal of Topology 3, no. 3 (2010): 643–67. http://dx.doi.org/10.1112/jtopol/jtq019.

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44

Bunte, Kerstin, Barbara Hammer, Thomas Villmann, Michael Biehl, and Axel Wismüller. "Neighbor embedding XOM for dimension reduction and visualization." Neurocomputing 74, no. 9 (April 2011): 1340–50. http://dx.doi.org/10.1016/j.neucom.2010.11.027.

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45

Srinivas, V. "On the embedding dimension of an affine variety." Mathematische Annalen 289, no. 1 (March 1991): 125–32. http://dx.doi.org/10.1007/bf01446563.

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46

Wang, Huibing, Lin Feng, Adong Kong, and Bo Jin. "Multi-view reconstructive preserving embedding for dimension reduction." Soft Computing 24, no. 10 (September 29, 2019): 7769–80. http://dx.doi.org/10.1007/s00500-019-04395-4.

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47

Lindenstrauss, Elon, and Masaki Tsukamoto. "Mean dimension and an embedding problem: An example." Israel Journal of Mathematics 199, no. 2 (July 19, 2013): 573–84. http://dx.doi.org/10.1007/s11856-013-0040-9.

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48

Dranišnikov, A. N., D. Repovš, and E. V. Ščepin. "On approximation and embedding problems for cohomological dimension." Topology and its Applications 55, no. 1 (January 1994): 67–86. http://dx.doi.org/10.1016/0166-8641(94)90065-5.

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49

Goldie, Alfred, and Günter Krause. "Embedding Rings with Krull Dimension in Artinian Rings." Journal of Algebra 182, no. 2 (June 1996): 534–45. http://dx.doi.org/10.1006/jabr.1996.0186.

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Kovalsky, M. G., and A. A. Hnilo. "LIGO series, dimension of embedding and Kolmogorov’s complexity." Astronomy and Computing 35 (April 2021): 100465. http://dx.doi.org/10.1016/j.ascom.2021.100465.

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