Journal articles: 'Geometric finiteness'

1

Lück, Wolfgang. "The Geometric Finiteness Obstruction." Proceedings of the London Mathematical Society s3-54, no. 2 (March 1987): 367–84. http://dx.doi.org/10.1112/plms/s3-54.2.367.

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2

Swarup, G. A. "Geometric finiteness and rationality." Journal of Pure and Applied Algebra 86, no. 3 (May 1993): 327–33. http://dx.doi.org/10.1016/0022-4049(93)90107-5.

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3

Tuschmann, Wilderich. "Geometric diffeomorphism finiteness in low dimensions and homotopy group finiteness." Mathematische Annalen 322, no. 2 (February 2002): 413–20. http://dx.doi.org/10.1007/s002080100281.

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4

Scott, G. P., and G. A. Swarup. "Geometric finiteness of certain Kleinian groups." Proceedings of the American Mathematical Society 109, no. 3 (March 1, 1990): 765. http://dx.doi.org/10.1090/s0002-9939-1990-1013981-6.

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5

Grove, Karsten, Peter Petersen, and Jyh-Yang Wu. "Geometric finiteness theorems via controlled topology." Inventiones Mathematicae 99, no. 1 (December 1990): 205–13. http://dx.doi.org/10.1007/bf01234417.

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Kapovich, Michael, and Beibei Liu. "Geometric finiteness in negatively pinched Hadamard manifolds." Annales Academiae Scientiarum Fennicae Mathematica 44, no. 2 (June 2019): 841–75. http://dx.doi.org/10.5186/aasfm.2019.4444.

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7

Torroba, Gonzalo. "Finiteness of flux vacua from geometric transitions." Journal of High Energy Physics 2007, no. 02 (February 21, 2007): 061. http://dx.doi.org/10.1088/1126-6708/2007/02/061.

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8

Proctor, Emily. "Orbifold homeomorphism finiteness based on geometric constraints." Annals of Global Analysis and Geometry 41, no. 1 (May 24, 2011): 47–59. http://dx.doi.org/10.1007/s10455-011-9270-4.

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9

Durumeric, Oguz C. "Geometric finiteness in large families in dimension 3." Topology 40, no. 4 (July 2001): 727–37. http://dx.doi.org/10.1016/s0040-9383(99)00080-4.

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Grove, Karsten, Peter Petersen V, and Jyh-Yang Wu. "Erratum to Geometric finiteness theorems via controlled topology." Inventiones mathematicae 104, no. 1 (December 1991): 221–22. http://dx.doi.org/10.1007/bf01245073.

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11

Taylor, Edward C. "Geometric finiteness and the convergence of Kleinian groups." Communications in Analysis and Geometry 5, no. 3 (1997): 497–533. http://dx.doi.org/10.4310/cag.1997.v5.n3.a5.

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12

Babych, Vyacheslav, and Nataliya Golovashchuk. "Galois coverings of one-sided bimodule problems." Proceedings of the International Geometry Center 14, no. 2 (August 30, 2021): 93–116. http://dx.doi.org/10.15673/tmgc.v14i2.1768.

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Applying geometric methods of 2-dimensional cell complex theory, we construct a Galois covering of a bimodule problem satisfying some structure, triangularity and finiteness conditions in order to describe the objects of finite representation type. Each admitted bimodule problem A is endowed with a quasi multiplicative basis. The main result shows that for a problem from the considered class having some finiteness restrictions and the schurian universal covering A', either A is schurian, or its basic bigraph contains a dotted loop, or it has a standard minimal non-schurian bimodule subproblem.

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13

Chen, Xiaoyang, and Frederico Xavier. "Finiteness of prescribed fibers of local biholomorphisms: a geometric approach." Mathematische Annalen 362, no. 3-4 (December 2, 2014): 1001–19. http://dx.doi.org/10.1007/s00208-014-1148-x.

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14

Lari-Lavassani, Ali. "A tangent space characterisation of the equivalence of germs for geometric subgroups of and." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 125, no. 3 (1995): 587–93. http://dx.doi.org/10.1017/s0308210500032698.

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It is shown that under the action of a geometric subgroup of and , for a germ f satisfying a certain finiteness condition, given a germ p, if the tangent spaces of f and f + p are equal for all t ∈ [0, 1], then f and f + p are -equivalent.

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Gupta, Kumar S., and Siddhartha Sen. "Geometric finiteness and non-quasinormal modes of the BTZ black hole." Physics Letters B 618, no. 1-4 (July 2005): 237–42. http://dx.doi.org/10.1016/j.physletb.2005.05.049.

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16

AUBRY, E. "Finiteness of π1π1 and geometric inequalities in almost positive Ricci curvature." Annales Scientifiques de l’École Normale Supérieure 40, no. 4 (July 2007): 675–95. http://dx.doi.org/10.1016/j.ansens.2007.07.001.

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17

Ling, Shiqing. "On the probabilistic properties of a double threshold ARMA conditional heteroskedastic model." Journal of Applied Probability 36, no. 3 (September 1999): 688–705. http://dx.doi.org/10.1239/jap/1032374627.

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Following Tweedie (1988), this paper constructs a special test function which leads to sufficient conditions for the stationarity and finiteness of the moments of a general non-linear time series model, the double threshold ARMA conditional heteroskedastic (DTARMACH) model. The results are applied to two well-known special cases, the GARCH and threshold ARMA (TARMA) models. The condition for the finiteness of the moments of the GARCH model is simple and easier to check than the condition given by Milhøj (1985) for the ARCH model. The condition for the stationarity of the TARMA model is identical to the condition given by Brockwell et al. (1992) for a special case, and verifies their conjecture that the moving average component does not affect the stationarity of the model. Under an additional irreducibility assumption, the geometric ergodicity of the GARCH and TARMA models is also established.

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Ling, Shiqing. "On the probabilistic properties of a double threshold ARMA conditional heteroskedastic model." Journal of Applied Probability 36, no. 03 (September 1999): 688–705. http://dx.doi.org/10.1017/s0021900200017502.

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Following Tweedie (1988), this paper constructs a special test function which leads to sufficient conditions for the stationarity and finiteness of the moments of a general non-linear time series model, the double threshold ARMA conditional heteroskedastic (DTARMACH) model. The results are applied to two well-known special cases, the GARCH and threshold ARMA (TARMA) models. The condition for the finiteness of the moments of the GARCH model is simple and easier to check than the condition given by Milhøj (1985) for the ARCH model. The condition for the stationarity of the TARMA model is identical to the condition given by Brockwell et al. (1992) for a special case, and verifies their conjecture that the moving average component does not affect the stationarity of the model. Under an additional irreducibility assumption, the geometric ergodicity of the GARCH and TARMA models is also established.

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19

Hong, Sungbok, and Darryl McCullough. "Ubiquity of geometric finiteness in mapping class groups of Haken 3-manifolds." Pacific Journal of Mathematics 188, no. 2 (March 1, 1999): 275–301. http://dx.doi.org/10.2140/pjm.1999.188.275.

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20

Kołasiński, Sławomir, Paweł Strzelecki, and Heiko von der Mosel. "Compactness and isotopy finiteness for submanifolds with uniformly bounded geometric curvature energies." Communications in Analysis and Geometry 26, no. 6 (2018): 1251–316. http://dx.doi.org/10.4310/cag.2018.v26.n6.a2.

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21

Yang, YanHong. "The Fixed Point Locus of the Verschiebung on ℳX(2, 0) for Genus-2 Curves X in Charateristic 2." Canadian Mathematical Bulletin 57, no. 2 (June 14, 2014): 439–48. http://dx.doi.org/10.4153/cmb-2013-019-1.

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Abstract.We prove that for every ordinary genus-2 curve X over a finite field κ of characteristic 2 with Aut(X/κ) = ℤ/2ℤ × S3 there exist SL(2; κ[[s]])-representations of π1(X) such that the image of π1(X̄) is infinite. This result produces a family of examples similar to Y. Laszlo’s counterexample to A. J. de Jong’s question regarding the finiteness of the geometric monodromy of representations of the fundamental group.

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22

McCarthy, Corrine. "Modeling morphological variation and development." Linguistic Approaches to Bilingualism 2, no. 1 (February 10, 2012): 25–53. http://dx.doi.org/10.1075/lab.2.1.02mcc.

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This paper proposes a model of morphological variation and development grounded in feature-geometric theory. It tests two hypotheses that follow from this model on a corpus of data from speakers of Spanish as a second language (L2). First, variation is systematic; default, underspecified feature values are adopted when errors occur. This hypothesis is supported for person, number, and finiteness, as 3rd, singular, and nonfinite defaults surface in place of 1st, plural, and finite verbs. Second, developmental trends are observed as nodes are added to the geometry; the unmarked/less specified feature value is successfully produced prior to the marked/more specified one. This hypothesis is partially supported, as accuracy in 3rd person emerges prior to 1st. However, no developmental pattern is found for number. Errors in finiteness are limited to lower-proficiency speakers, whereas intermediate speakers favor 3rd person, finite defaults. Together, these results suggest systematic variation and gradual development in the morphology.

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23

Platonov, V. P., G. V. Fedorov, and V. S. Zhgoon. "ON THE FINITENESS OF THE SET OF GENERALIZED JACOBIANS WITH NONTRIVIAL TORSION POINTS OVER ALGEBRAIC NUMBER FIELDS." Доклады Российской академии наук. Математика, информатика, процессы управления 513, no. 1 (September 1, 2023): 66–70. http://dx.doi.org/10.31857/s2686954323700285.

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For a smooth projective curve $\mathcal{C}$ defined over algebraic number field k, we investigate the question of finiteness of the set of generalized Jacobians ${{J}_{\mathfrak{m}}}$ of a curve $\mathcal{C}$ associated with modules $\mathfrak{m}$ defined over k such that a fixed divisor representing a class of finite order in the Jacobian J of the curve $\mathcal{C}$ provides the torsion class in the generalized Jacobian ${{J}_{\mathfrak{m}}}$. Various results on the finiteness and infiniteness of the set of generalized Jacobians with the above property are obtained depending on the geometric conditions on the support of $\mathfrak{m}$, as well as on the conditions on the field $k$. These results were applied to the problem of the periodicity of a continuous fraction decomposition constructed in the field of formal power series $k((1{\text{/}}x))$, for the special elements of the field of functions $k(\tilde {\mathcal{C}})$ of the hyperelliptic curve $\tilde {\mathcal{C}}:{{y}^{2}} = f(x)$.

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24

Matsuzaki, Katsuhiko. "Geometric finiteness, quasiconformal stability and surjectivity of the Bers map for Kleinian groups." Tohoku Mathematical Journal 43, no. 3 (1991): 327–36. http://dx.doi.org/10.2748/tmj/1178227457.

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25

Canary, Richard Douglas, and Sa'ar Hersonsky. "Ubiquity of geometric finiteness in boundaries of deformation spaces of hyperbolic 3-manifolds." American Journal of Mathematics 126, no. 6 (2004): 1193–220. http://dx.doi.org/10.1353/ajm.2004.0042.

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26

Gupta, Kumar S., E. Harikumar, Siddhartha Sen, and M. Sivakumar. "Geometric finiteness, holography and quasinormal modes for the warped AdS 3 black hole." Classical and Quantum Gravity 27, no. 16 (July 8, 2010): 165012. http://dx.doi.org/10.1088/0264-9381/27/16/165012.

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27

Shimizu, Koji. "Finiteness of Frobenius Traces of a Sheaf on a Flat Arithmetic Scheme." International Mathematics Research Notices 2020, no. 9 (June 20, 2018): 2864–80. http://dx.doi.org/10.1093/imrn/rny145.

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Abstract For a lisse $\ell $-adic sheaf on a scheme flat and of finite type over $\mathbb{Z}$, we consider the field generated over $ \mathbb{Q}$ by Frobenius traces of the sheaf at closed points. Assuming conjectural properties of geometric Galois representations of number fields and the Generalized Riemann Hypothesis, we prove that the field is finite over $\mathbb{Q}$ when the sheaf is de Rham at $\ell $ pointwise. This is a number field analog of Deligne’s finiteness result about Frobenius traces in the function field case.

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28

Orr, Martin, and Alexei N. Skorobogatov. "Finiteness theorems for K3 surfaces and abelian varieties of CM type." Compositio Mathematica 154, no. 8 (July 18, 2018): 1571–92. http://dx.doi.org/10.1112/s0010437x18007169.

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We study abelian varieties and K3 surfaces with complex multiplication defined over number fields of fixed degree. We show that these varieties fall into finitely many isomorphism classes over an algebraic closure of the field of rational numbers. As an application we confirm finiteness conjectures of Shafarevich and Coleman in the CM case. In addition we prove the uniform boundedness of the Galois invariant subgroup of the geometric Brauer group for forms of a smooth projective variety satisfying the integral Mumford–Tate conjecture. When applied to K3 surfaces, this affirms a conjecture of Várilly-Alvarado in the CM case.

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29

Bibbona, Enrico, Jinsu Kim, and Carsten Wiuf. "Stationary distributions of systems with discreteness-induced transitions." Journal of The Royal Society Interface 17, no. 168 (July 2020): 20200243. http://dx.doi.org/10.1098/rsif.2020.0243.

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We provide a theoretical analysis of some autocatalytic reaction networks exhibiting the phenomenon of discreteness-induced transitions. The family of networks that we address includes the celebrated Togashi and Kaneko model. We prove positive recurrence, finiteness of all moments and geometric ergodicity of the models in the family. For some parameter values, we find the analytic expression for the stationary distribution and discuss the effect of volume scaling on the stationary behaviour of the chain. We find the exact critical value of the volume for which discreteness-induced transitions disappear.

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Sadovnichii, V. A., Ya T. Sultanaev, and A. M. Akhtyamov. "The finiteness of the spectrum of boundary value problems defined on a geometric graph." Transactions of the Moscow Mathematical Society 80 (April 1, 2020): 123–31. http://dx.doi.org/10.1090/mosc/293.

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31

Verdet, Cécile, Yannick Anguy, Colette Sirieix, Rémi Clément, and Cécile Gaborieau. "On the effect of electrode finiteness in small-scale electrical resistivity imaging." GEOPHYSICS 83, no. 6 (November 1, 2018): EN39—EN52. http://dx.doi.org/10.1190/geo2018-0074.1.

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Electrical resistivity tomography (ERT) profiles including finite steel-rod electrodes have been widely used in assuming a surface-node current injection for inversion. This hypothesis was shown by others to be safe for ratios of electrode embedment to electrode spacing smaller than 20%. Relying on the conductive cell model (CCM), we took into account the complete electrodes in the DC forward problem. We found that an electrode effect is included in resistivity sections inverted with a surface point electrode model. Several synthetic examples indicated that this unwanted effect is particularly developed when the electrode spacing does not meet a double constraint from the characteristic size of a shallow heterogeneity and from the electrode embedment. This effect deserved correction. A point approximation for a finite electrode referred to as the equivalent electrode point (EEP) was sought by placing a point-source current in the ground along the electrode length. The appropriate EEP depth was the one for which the CCM and a buried point source minimized a systematic geometric error; i.e., the relative change of the geometric factors obtained with the CCM and with an EEP. An EEP placed at 73% of the electrode length was declared as a suitable point approximation for an electrode. Use of this point assumption for inversion remedied efficiently the electrode effect subject to conditions. More precisely, the electrode spacing should stay within a lower bound equal to twice the electrode embedment and an upper bound equal to the shallow heterogeneity characteristic size divided by 0.75. The interest of such a metrological appraisal of the suitable acquisition layout to be used on the field was illustrated by a small-scale ERT field survey. This case study permitted us to understand more reliably the impact from fires upon a centimetric shallow layer in a calcareous wall.

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Fuhrmann, Gabriel, and Maik Gröger. "Constant length substitutions, iterated function systems and amorphic complexity." Mathematische Zeitschrift 295, no. 3-4 (November 22, 2019): 1385–404. http://dx.doi.org/10.1007/s00209-019-02426-2.

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AbstractWe show how geometric methods from the general theory of fractal dimensions and iterated function systems can be deployed to study symbolic dynamics in the zero entropy regime. More precisely, we establish a dimensional characterization of the topological notion of amorphic complexity. For subshifts with discrete spectrum associated to constant length substitutions, this characterization allows us to derive bounds for the amorphic complexity by interpreting the subshift as the attractor of an iterated function system in a suitable quotient space. As a result, we obtain the general finiteness and positivity of amorphic complexity in this setting and provide a closed formula in case of a binary alphabet.

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HAUCOURT, EMMANUEL, and NICOLAS NININ. "Unique decomposition of homogeneous languages and application to isothetic regions." Mathematical Structures in Computer Science 29, no. 5 (October 10, 2018): 681–730. http://dx.doi.org/10.1017/s0960129518000294.

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A language is said to be homogeneous when all its words have the same length. Homogeneous languages thus form a monoid under concatenation. It becomes freely commutative under the simultaneous actions of every permutation group on the collection of homogeneous languages of length n ∈ ℕ. One recovers the isothetic regions from (Haucourt 2017, to appear (online since October 2017)) by considering the alphabet of connected subsets of the space |G|, viz the geometric realization of a finite graph G. Factoring the geometric model of a conservative program amounts to parallelize it, and there exists an efficient factoring algorithm for isothetic regions. Yet, from the theoretical point of view, one wishes to go beyond the class of conservative programs, which implies relaxing the finiteness hypothesis on the graph G. Provided that the collections of n-dimensional isothetic regions over G (denoted by |G|) are co-unital distributive lattices, the prime decomposition of isothetic regions is given by an algorithm which is, unfortunately, very inefficient. Nevertheless, if the collections |G| satisfy the stronger property of being Boolean algebras, then the efficient factoring algorithm is available again. We relate the algebraic properties of the collections |G| to the geometric properties of the space |G|. On the way, the algebraic structure |G| is proven to be the universal tensor product, in the category of semilattices with zero, of n copies of the algebraic structure |G|.

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Padmanabhan, T., and Hamsa Padmanabhan. "Quantum gravity at Hubble scales determines the cosmological constant and the amplitude of primordial perturbations." International Journal of Modern Physics D 26, no. 12 (October 2017): 1743002. http://dx.doi.org/10.1142/s0218271817430027.

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Gravity controls the amount of information that is accessible to any specific observer. We quantify the notion of cosmic information (‘CosmIn’) for the case of an eternal observer in the universe. Demanding the finiteness of CosmIn requires the universe to have a late-time accelerated expansion phase. Combined with some generic features of the quantum structure of spacetime, this leads to the determination of (i) the numerical value of the cosmological constant, as well as (ii) the amplitude of the primordial, scale invariant perturbation spectrum in terms of a single free parameter, which specifies the energy scale at which the universe makes a transition from a pre-geometric phase to the classical phase. This formalism also shows that the quantum gravitational information content of spacetime can be tested by using precision cosmology.

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35

Durango Idárraga, Sebastián, Mariline C. Delgado Martínez, César A. Álvarez Vargas, Rubén D. Flórez Hurtado, and Manuel A. Flórez Ruiz. "Diseño cinemático de un robot paralelo 2-PRR." Scientia et Technica 25, no. 3 (September 30, 2020): 372–79. http://dx.doi.org/10.22517/23447214.24161.

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In civil construction Abstract— This paper presents a dimensional synthesis for a 2-PRR planar parallel robot with a structural plane of symmetry. This robot can achieve the translation of the moving platform without changing the orientation, being useful for applications that require controlled positions with high rigidity. Because the performance of parallel robots is highly sensitive to their geometric parameters, many methodologies to state the dimensional synthesis has been developed. We used the method of Parameter - Finiteness Normalization Method (PFNM) to state the dimensional synthesis using Global Condition Index (GCI) and workspace ( ) design atlases. For the two, GCI and , designed atlases, it is not possible to maximize one of the indexes without diminishing the other one, which represents a design compromise. Also, we remark singular configurations that are coming from specific geometry or limit positions. The complete dimensional synthesis is also presented.

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36

Badger, Matthew, and Raanan Schul. "Multiscale Analysis of 1-rectifiable Measures II: Characterizations." Analysis and Geometry in Metric Spaces 5, no. 1 (March 16, 2017): 1–39. http://dx.doi.org/10.1515/agms-2017-0001.

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Abstract A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean space for all n ≥ 2 in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an L2 gauge the extent to which μ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the classical theorems of Besicovitch, Morse and Randolph, and Moore, we do not assume an a priori relationship between μ and 1-dimensional Hausdorff measure H1. We also characterize purely 1-unrectifiable Radon measures, i.e. locally finite measures that give measure zero to every finite length curve. Characterizations of this form were originally conjectured to exist by P. Jones. Along the way, we develop an L2 variant of P. Jones’ traveling salesman construction, which is of independent interest.

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Cais, Bryden. "The geometry of Hida families II: -adic -modules and -adic Hodge theory." Compositio Mathematica 154, no. 4 (March 8, 2018): 719–60. http://dx.doi.org/10.1112/s0010437x17007680.

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We construct the $\unicode[STIX]{x1D6EC}$-adic crystalline and Dieudonné analogues of Hida’s ordinary $\unicode[STIX]{x1D6EC}$-adic étale cohomology, and employ integral $p$-adic Hodge theory to prove $\unicode[STIX]{x1D6EC}$-adic comparison isomorphisms between these cohomologies and the $\unicode[STIX]{x1D6EC}$-adic de Rham cohomology studied in Cais [The geometry of Hida families I:$\unicode[STIX]{x1D6EC}$-adic de Rham cohomology, Math. Ann. (2017), doi:10.1007/s00208-017-1608-1] as well as Hida’s $\unicode[STIX]{x1D6EC}$-adic étale cohomology. As applications of our work, we provide a ‘cohomological’ construction of the family of $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules attached to Hida’s ordinary $\unicode[STIX]{x1D6EC}$-adic étale cohomology by Dee [$\unicode[STIX]{x1D6F7}$–$\unicode[STIX]{x1D6E4}$modules for families of Galois representations, J. Algebra 235 (2001), 636–664], and we give a new and purely geometric proof of Hida’s finiteness and control theorems. We also prove suitable $\unicode[STIX]{x1D6EC}$-adic duality theorems for each of the cohomologies we construct.

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38

Almeida, Kisnney. "The BNS-invariant for Artin groups of circuit rank 2." Journal of Group Theory 21, no. 2 (March 1, 2018): 189–228. http://dx.doi.org/10.1515/jgth-2017-0029.

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AbstractThe BNS-invariant or{\Sigma^{1}}-invariant is the first of a series of geometric invariants of finitely generated groups defined in the eighties that are deeply related to finiteness properties of their subgroups, although they are very hard to compute. Meier, Meinert and VanWyk have obtained a partial description of{\Sigma^{1}}of Artin groups, but the complete description of the general case is still an open problem. Let the circuit rank of an Artin group be the free rank of the fundamental group of its underlying graph. Meier, in a previous work, obtained a complete description for Artin groups of circuit rank 0, i.e., whose underlying graphs are trees. In a previous work we have proved, in joint work with Kochloukova, the same description to be true for Artin groups of circuit rank 1. In this paper we prove the description to be true for every Artin group of circuit rank 2.

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KIRBY, DAVID. "A sequence of complexes generated by a finite set of homogeneous polynomials." Mathematical Proceedings of the Cambridge Philosophical Society 124, no. 1 (July 1998): 81–96. http://dx.doi.org/10.1017/s0305004197002351.

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The usefulness of the Koszul complex in handling in an algebraic setting the two geometric notions of multiplicity and depth first became apparent with the work of Auslander and Buchsbaum [1] following a suggestion of Serre. Regarding the generators a1, …, an of the complex as a 1×n matrix first Eagon and Northcott [4] extended this work to a complex associated with an m×n matrix, then shortly afterwards a different extension was given by Buchsbaum and Rim [2, 3]. These two complexes are two of an infinite family [6] some of which inherit the depth sensitive property of the Koszul complex and all of which under a certain finiteness condition provide the same multiplicity as Euler–Poincaré characteristic [7].These two properties prove useful in geometric applications, see for example Lago and Rodicio [7] for depth sensitivity and Kirby [8] for the characteristic as multiplicity of intersection. During the course of these developments it became clear that it was most appropriate to regard the complexes as being generated by linear forms. From this point of view it is natural to ask if the linearity of the forms is necessary. In the present note we begin a response to this question by extending the work of [6] to complexes associated with forms of arbitrary positive degree. In a sequel we shall similarly extend the results of multiplicity in [7].

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Qian, Denghui, and Zhiyu Shi. "Using PWE/FE Method to Calculate the Band Structures of the Semi-Infinite PCs: Periodic in x–y Plane and Finite in z -direction." Archives of Acoustics 42, no. 4 (December 20, 2017): 735–42. http://dx.doi.org/10.1515/aoa-2017-0076.

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Abstract This paper introduces the concept of semi-infinite phononic crystal (PC) on account of the infinite periodicity in x-y plane and finiteness in z-direction. The plane wave expansion and finite element methods are coupled and formulized to calculate the band structures of the proposed periodic elastic composite structures based on the typical geometric properties. First, the coupled plane wave expansion and finite element (PWE/FE) method is applied to calculate the band structures of the Pb/rubber, steel/epoxy and steel/aluminum semi-infinite PCs with cylindrical scatters. Then, it is used to calculate the band structure of the Pb/rubber semi-infinite PC with cubic scatter. Last, the band structure of the rubbercoated Pb/epoxy three-component semi-infinite PC is calculated by the proposed method. Besides, all the results are compared with those calculated by the finite element (FE) method implemented by adopting COMSOL Multiphysics. Numerical results and further analysis demonstrate that the proposed PWE/FE method has strong applicability and high accuracy.

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Bibby, Christin, and Nir Gadish. "A generating function approach to new representation stability phenomena in orbit configuration spaces." Transactions of the American Mathematical Society, Series B 10, no. 9 (February 9, 2023): 241–87. http://dx.doi.org/10.1090/btran/130.

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As countless examples show, it can be fruitful to study a sequence of complicated objects all at once via the formalism of generating functions. We apply this point of view to the homology and combinatorics of orbit configuration spaces: using the notion of twisted commutative algebras, which essentially categorify algebras in exponential generating functions. This idea allows for a factorization of the orbit configuration space “generating function” into an infinite product, whose terms are surprisingly easy to understand. Beyond the intrinsic aesthetic of this decomposition and its quantitative consequences, it suggests a sequence of primary, secondary, and higher representation stability phenomena. Based on this, we give a simple geometric recipe for identifying new stabilization actions with finiteness properties in some cases, which we use to unify and generalize known stability results. We demonstrate our method by characterizing secondary and higher stability for configuration spaces on i i -acyclic spaces. For another application, we describe a natural filtration by which one observes a filtered representation stability phenomenon in configuration spaces on graphs.

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Iyudu, Natalia, and Agata Smoktunowicz. "Golod–Shafarevich-Type Theorems and Potential Algebras." International Mathematics Research Notices 2019, no. 15 (January 12, 2018): 4822–44. http://dx.doi.org/10.1093/imrn/rnx315.

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Abstract Potential algebras feature in the minimal model program and noncommutative resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Gröbner basis theory and generalized Golod–Shafarevich-type theorems for potential algebras, to determine finiteness conditions in terms of the potential. We consider two-generated potential algebras. Using Gröbner bases techniques and arguing in terms of associated truncated algebra we prove that they cannot have dimension smaller than 8. This answers a question of Wemyss [21], related to the geometric argument of Toda [17]. We derive from the improved version of the Golod–Shafarevich theorem, that if the potential has only terms of degree 5 or higher, then the potential algebra is infinite dimensional. We prove that potential algebra for any homogeneous potential of degree $n\geqslant 3$ is infinite dimensional. The proof includes a complete classification of all potentials of degree 3. Then we introduce a certain version of Koszul complex, and prove that in the class $\mathcal {P}_{n}$ of potential algebras with homogeneous potential of degree $n+1\geqslant 4$, the minimal Hilbert series is $H_{n}=\frac {1}{1-2t+2t^{n}-t^{n+1}}$, so they are all infinite dimensional. Moreover, growth could be polynomial (but nonlinear) for the potential of degree 4, and is always exponential for potential of degree starting from 5. For one particular type of potential we prove a conjecture by Wemyss, which relates the difference of dimensions of potential algebra and its abelianization with Gopakumar–Vafa invariants.

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Beraldo, Dario. "Sheaves of categories with local actions of Hochschild cochains." Compositio Mathematica 155, no. 08 (July 4, 2019): 1521–67. http://dx.doi.org/10.1112/s0010437x19007413.

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The notion of Hochschild cochains induces an assignment from $\mathsf{Aff}$ , affine DG schemes, to monoidal DG categories. We show that this assignment extends, under appropriate finiteness conditions, to a functor $\mathbb{H}:\mathsf{Aff}\rightarrow \mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat})$ , where the latter denotes the category of monoidal DG categories and bimodules. Any functor $\mathbb{A}:\mathsf{Aff}\rightarrow \mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat})$ gives rise, by taking modules, to a theory of sheaves of categories $\mathsf{ShvCat}^{\mathbb{A}}$ . In this paper, we study $\mathsf{ShvCat}^{\mathbb{H}}$ . Loosely speaking, this theory categorifies the theory of $\mathfrak{D}$ -modules, in the same way as Gaitsgory’s original $\mathsf{ShvCat}$ categorifies the theory of quasi-coherent sheaves. We develop the functoriality of $\mathsf{ShvCat}^{\mathbb{H}}$ , its descent properties and the notion of $\mathbb{H}$ -affineness. We then prove the $\mathbb{H}$ -affineness of algebraic stacks: for ${\mathcal{Y}}$ a stack satisfying some mild conditions, the $\infty$ -category $\mathsf{ShvCat}^{\mathbb{H}}({\mathcal{Y}})$ is equivalent to the $\infty$ -category of modules for $\mathbb{H}({\mathcal{Y}})$ , the monoidal DG category of higher differential operators. The main consequence, for ${\mathcal{Y}}$ quasi-smooth, is the following: if ${\mathcal{C}}$ is a DG category acted on by $\mathbb{H}({\mathcal{Y}})$ , then ${\mathcal{C}}$ admits a theory of singular support in $\operatorname{Sing}({\mathcal{Y}})$ , where $\operatorname{Sing}({\mathcal{Y}})$ is the space of singularities of ${\mathcal{Y}}$ . As an application to the geometric Langlands programme, we indicate how derived Satake yields an action of $\mathbb{H}(\operatorname{LS}_{{\check{G}}})$ on $\mathfrak{D}(\operatorname{Bun}_{G})$ , thereby equipping objects of $\mathfrak{D}(\operatorname{Bun}_{G})$ with singular support in $\operatorname{Sing}(\operatorname{LS}_{{\check{G}}})$ .

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Pereyra, Pedro. "Photonic Transmittance in Metallic and Left Handed Superlattices." Photonics 7, no. 2 (April 18, 2020): 29. http://dx.doi.org/10.3390/photonics7020029.

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We study the transmission of electromagnetic waves through layered structures of metallic and left-handed media. Resonant band structures of transmission coefficients are obtained as functions of the incidence angle, the geometric parameters, and the number of unit cells of the superlattices. The theory of finite periodic systems that we use is free of assumptions, the finiteness of the periodic system being an essential condition. We rederive the correct recurrence relation of the Chebyshev polynomials that carry the physical information of the coherent coupling of plasmon modes and interface plasmons and surface plasmons, responsible for the photonic bands and the resonant structure of the surface plasmon polaritons. Unlike the dispersion relations of infinite periodic systems, which at best predict the bandwidths, we show that the dispersion relation of this theory predicts not only the bands, but also the resonant plasmons’ frequencies, above and below the plasma frequency. We show that, besides the strong influence of the incidence angle and the characteristic low transmission of a single conductor slab for frequencies ω below the plasma frequency ω p , the coherent coupling of the bulk plasmon modes and the interface surface plasmon polaritons lead to oscillating transmission coefficients and, depending on the parity of the number of unit cells n of the superlattice, the transmission coefficient vanishes or amplifies as the conductor width increases. Similarly, the well-established transmission coefficient of a single left-handed slab, which exhibits optical antimatter effects, becomes highly resonant with superluminal effects in superlattices. We determine the space-time evolution of a wave packet through the λ / 4 photonic superlattice whose bandwidth becomes negligible, and the transmission coefficient becomes a sequence of isolated and equidistant peaks with negative phase times. We show that the space-time evolution of a Gaussian wave packet, with the centroid at any of these peaks, agrees with the theoretical predictions, and no violation of the causality principle occurs.

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Poénaru, Valentin, and Corrado Tanasi. "A group-theoretical finiteness theorem." Geometriae Dedicata 137, no. 1 (September 16, 2008): 1–25. http://dx.doi.org/10.1007/s10711-008-9279-4.

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Grunewald, Joachim. "Non-finiteness results for Nil-groups." Algebraic & Geometric Topology 7, no. 4 (December 18, 2007): 1979–86. http://dx.doi.org/10.2140/agt.2007.7.1979.

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Lustig, Martin, and Yoav Moriah. "A finiteness result for Heegaard splittings." Topology 43, no. 5 (September 2004): 1165–82. http://dx.doi.org/10.1016/j.top.2004.01.004.

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48

Wu, Bing-Le. "A finiteness theorem for isoparametric hypersurfaces." Geometriae Dedicata 50, no. 3 (May 1994): 247–50. http://dx.doi.org/10.1007/bf01267867.

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Fefferman, Charles, and Pavel Shvartsman. "Sharp Finiteness Principles For Lipschitz Selections." Geometric and Functional Analysis 28, no. 6 (September 14, 2018): 1641–705. http://dx.doi.org/10.1007/s00039-018-0467-6.

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Bárcenas, Noé, Dieter Degrijse, and Irakli Patchkoria. "Stable finiteness properties of infinite discrete groups." Journal of Topology 10, no. 4 (December 2017): 1169–96. http://dx.doi.org/10.1112/topo.12035.

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Journal articles on the topic 'Geometric finiteness'

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