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Journal articles on the topic 'Projective Lattice'

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Published: 6 September 2023

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1

de Pagter, Ben, and Anthony W. Wickstead. "Free and projective Banach lattices." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 145, no. 1 (January 30, 2015): 105–43. http://dx.doi.org/10.1017/s0308210512001709.

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We define and prove the existence of free Banach lattices in the category of Banach lattices and contractive lattice homomorphisms, and establish some of their fundamental properties. We give much more detailed results about their structure in the case when there are only a finite number of generators, and give several Banach lattice characterizations of the number of generators being, respectively, one, finite or countable. We define a Banach lattice P to be projective if, whenever X is a Banach lattice, J is a closed ideal in X, Q : X → X/J is the quotient map, T : P → X/J is a linear lattice homomorphism and ε > 0, there exists a linear lattice homomorphism : P → X such that T = Q º and ∥∥ ≤ (1 + ε)∥T∥. We establish the connection between projective Banach lattices and free Banach lattices, describe several families of Banach lattices that are projective and prove that some are not.

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2

Symonds, Peter. "Relative characters for H-projective RG-lattices." Mathematical Proceedings of the Cambridge Philosophical Society 104, no. 2 (September 1988): 207–13. http://dx.doi.org/10.1017/s0305004100065397.

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If G is a group with a subgroup H and R is a Dedekind domain, then an H-projective RG-lattice is an RG-lattice that is a direct summand of an induced lattice for some RH-lattice N: they have been studied extensively in the context of modular representation theory. If H is the trivial group these are the projective lattices. We define a relative character χG/H on H-projective lattices, which in the case H = 1 is equivalent to the Hattori–Stallings trace for projective lattices (see [5, 8]), and in the case H = G is the ordinary character. These characters can be used to show that the R-ranks of certain H-projective lattices must be divisible by some specified number, generalizing some well-known results: cf. Corollary 3·6. If for example we take R = ℤ, then |G/H| divides the ℤ-rank of any H-projective ℤG-lattice.

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3

Pfeiffer, Thorsten, and Stefan E. Schmidt. "Projective mappings between projective lattice geometries." Journal of Geometry 54, no. 1-2 (November 1995): 105–14. http://dx.doi.org/10.1007/bf01222858.

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4

RUMP, WOLFGANG. "CHARACTERIZATION OF PROJECTIVE QUANTALES." Journal of the Australian Mathematical Society 100, no. 3 (January 8, 2016): 403–20. http://dx.doi.org/10.1017/s1446788715000506.

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It is proved that a quantale is projective if and only if it is isomorphic to a derived tensor quantale over a completely distributive sup-lattice. Furthermore, an intrinsic characterization of projectivity is given in terms of inertial sup-lattices and derivations of quantales.

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5

Jakubík, Ján. "On projective intervals in a modular lattice." Mathematica Bohemica 117, no. 3 (1992): 293–98. http://dx.doi.org/10.21136/mb.1992.126283.

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6

Uchino, K. "Arnold's Projective Plane and -Matrices." Advances in Mathematical Physics 2010 (2010): 1–9. http://dx.doi.org/10.1155/2010/956128.

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We will explain Arnold's 2-dimensional (shortly, 2D) projective geometry (Arnold, 2005) by means of lattice theory. It will be shown that the projection of the set of nontrivial triangular -matrices is the pencil of tangent lines of a quadratic curve on Arnold's projective plane.

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7

Zhuravlev, V., and I. Tsyganivska. "Projective lattices of tiled orders." Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics, no. 4 (2018): 16–19. http://dx.doi.org/10.17721/1812-5409.2018/4.2.

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Tiled orders over discrete valuation ring have been studied since the 1970s by many mathematicians, in particular, by Yategaonkar V.A., Tarsy R.B., Roggenkamp K.W, Simson D., Drozd Y.A., Zavadsky A.G. and Kirichenko V.V. Yategaonkar V.A. proved that for every n > 2, there is, up to an isomorphism, a finite number of tiled orders over a discrete valuation ring O of finite global dimension which lie in $M_n(K)$ where K is a field of fractions of a commutatively discrete valuation ring O. The articles by R.B. Tarsy, V.A. Yategaonkar, H. Fujita, W. Rump and others are devoted to the study of the global dimension of tiled orders. H. Fujita described the reduced tiled orders in Mn(D) of finite global dimension for n = 4; 5. V.M. Zhuravlev and D.V. Zhuravlev described reduced tiled orders in Mn(D) of finite global dimension for n = 6: This paper examines the necessary condition for the finiteness of the global dimension of the tile order. Let A be a tiled order. The kernel of the projective resolvent of an irreducible lattice has the form M1f1 +M2f2 + ::: +Msfs, where Mi is irreducible lattice, fi is some vector. If the tile order has a finite global dimension, then there is a projective lattice that is the intersection of projective lattices. This condition is the one explored in the paper.

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8

Day, Alan. "Doubling Constructions in Lattice Theory." Canadian Journal of Mathematics 44, no. 2 (April 1, 1992): 252–69. http://dx.doi.org/10.4153/cjm-1992-017-7.

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AbstractThis paper examines the simultaneous doubling of multiple intervals of a lattice in great detail. In the case of a finite set of W-failure intervals, it is shown that there in a unique smallest lattice mapping homomorphically onto the original lattice, in which the set of W-failures is removed. A nice description of this new lattice is given. This technique is used to show that every lattice that is a bounded homomorphic image of a free lattice has a projective cover. It is also used to give a sufficient condition for a fintely presented lattice to be weakly atomic and shows that the problem of which finitely presented lattices are finite is closely related to the problem of characterizing those finite lattices with a finite W-cover.

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9

Hirai, Hiroshi. "Uniform modular lattices and affine buildings." Advances in Geometry 20, no. 3 (July 28, 2020): 375–90. http://dx.doi.org/10.1515/advgeom-2020-0007.

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AbstractA simple lattice-theoretic characterization for affine buildings of type A is obtained. We introduce a class of modular lattices, called uniform modular lattices, and show that uniform modular lattices and affine buildings of type A constitute the same object. This is an affine counterpart of the well-known equivalence between projective geometries (≃ complemented modular lattices) and spherical buildings of type A.

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10

Osofsky, Barbara L. "Projective dimension is a lattice invariant." Journal of Pure and Applied Algebra 161, no. 1-2 (July 2001): 205–17. http://dx.doi.org/10.1016/s0022-4049(00)00090-6.

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11

Greferath, Marcus, and Stefan E. Schmidt. "On point-irreducible projective lattice geometries." Journal of Geometry 50, no. 1-2 (July 1994): 73–83. http://dx.doi.org/10.1007/bf01222664.

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12

Nolze, Gert, and Aimo Winkelmann. "Crystallometric and projective properties of Kikuchi diffraction patterns." Journal of Applied Crystallography 50, no. 1 (February 1, 2017): 102–19. http://dx.doi.org/10.1107/s1600576716017477.

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Kikuchi diffraction patterns can provide fundamental information about the lattice metric of a crystalline phase. In order to improve the possible precision and accuracy of lattice parameter determination from the features observed in Kikuchi patterns, some useful fundamental relationships of geometric crystallography are reviewed, which hold true independently of the actual crystal symmetry. The Kikuchi band positions and intersections and the Kikuchi band widths are highly interrelated, which is illustrated by the fact that all lattice plane trace positions of the crystal are predetermined by the definition of only four traces. If, additionally, the projection centre of the gnomonic projection is known, the lattice parameter ratios and the angles between the basis vectors are fixed. A further definition of one specific Kikuchi band width is sufficient to set the absolute sizes of all lattice parameters and to predict the widths of all Kikuchi bands. The mathematical properties of the gnomonic projection turn out to be central to an improved interpretation of Kikuchi pattern data, emphasizing the importance of the exact knowledge of the projection centre.

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13

Greferath, Marcus. "Global-Affine Morphisms of Projective Lattice Geometries." Results in Mathematics 24, no. 1-2 (August 1993): 76–83. http://dx.doi.org/10.1007/bf03322318.

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14

Parsapour, A., and KH Ahmad Javaheri. "On the projective comaximal graphs of lattices." Discrete Mathematics, Algorithms and Applications 08, no. 04 (November 8, 2016): 1650072. http://dx.doi.org/10.1142/s1793830916500725.

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15

Brower, Richard C., Claudio Rebbi, and Ettore Vicari. "Non-Abelian projective multigrid for lattice gauge theory." Physical Review Letters 66, no. 10 (March 11, 1991): 1263–66. http://dx.doi.org/10.1103/physrevlett.66.1263.

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16

Kalkreuter, Thomas. "Projective block spin transformations in lattice gauge theories." Nuclear Physics B 376, no. 3 (June 1992): 637–60. http://dx.doi.org/10.1016/0550-3213(92)90122-r.

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17

CABRER, LEONARDO, and DANIELE MUNDICI. "RATIONAL POLYHEDRA AND PROJECTIVE LATTICE-ORDERED ABELIAN GROUPS WITH ORDER UNIT." Communications in Contemporary Mathematics 14, no. 03 (June 2012): 1250017. http://dx.doi.org/10.1142/s0219199712500174.

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An ℓ-groupG is an abelian group equipped with a translation invariant lattice-order. Baker and Beynon proved that G is finitely generated projective if and only if it is finitely presented. A unital ℓ-group is an ℓ-group G with a distinguished order unit, i.e. an element 0 ≤ u ∈ G whose positive integer multiples eventually dominate every element of G. Unital ℓ-homomorphisms between unital ℓ-groups are group homomorphisms that also preserve the order unit and the lattice structure. A unital ℓ-group (G, u) is projective if whenever ψ : (A, a) → (B, b) is a surjective unital ℓ-homomorphism and ϕ : (G, u) → (B, b) is a unital ℓ-homomorphism, there is a unital ℓ-homomorphism θ : (G, u) → (A, a) such that ϕ = ψ ◦ θ. While every finitely generated projective unital ℓ-group is finitely presented, the converse does not hold in general. Classical algebraic topology (à la Whitehead) is combined in this paper with the Włodarczyk–Morelli solution of the weak Oda conjecture for toric varieties, to describe finitely generated projective unital ℓ-groups.

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18

GARBER, DAVID, MINA TEICHER, and UZI VISHNE. "CLASSES OF WIRING DIAGRAMS AND THEIR INVARIANTS." Journal of Knot Theory and Its Ramifications 11, no. 08 (December 2002): 1165–91. http://dx.doi.org/10.1142/s0218216502002190.

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Wiring diagrams usually serve as a tool in the study of arrangements of lines and pseudolines. Here we go in the opposite direction, using known properties of line arrangements to motivate certain equivalence relations and actions on sets of wiring diagrams, which preserve the incidence lattice and the fundamental groups of the affine and projective complements of the diagrams. These actions are used in [GTV] to classify real arrangements of up to 8 lines and show that in this case, the incidence lattice determines both the affine and the projective fundamental groups.

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19

Bouc, Serge, and Jacques Thévenaz. "Simple and projective correspondence functors." Representation Theory of the American Mathematical Society 25, no. 9 (April 2, 2021): 224–64. http://dx.doi.org/10.1090/ert/564.

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A correspondence functor is a functor from the category of finite sets and correspondences to the category of k k -modules, where k k is a commutative ring. We determine exactly which simple correspondence functors are projective. We also determine which simple modules are projective for the algebra of all relations on a finite set. Moreover, we analyze the occurrence of such simple projective functors inside the correspondence functor F F associated with a finite lattice and we deduce a direct sum decomposition of F F .

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20

Katsov, Yefim, Tran Giang Nam, and Jens Zumbrägel. "On simpleness of semirings and complete semirings." Journal of Algebra and Its Applications 13, no. 06 (April 20, 2014): 1450015. http://dx.doi.org/10.1142/s0219498814500157.

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In this paper, we investigate various classes of semirings and complete semirings regarding the property of being ideal-simple, congruence-simple, or both. Among other results, we describe (complete) simple, i.e. simultaneously ideal- and congruence-simple, endomorphism semirings of (complete) idempotent commutative monoids; we show that the concepts of simpleness, congruence-simpleness, and ideal-simpleness for (complete) endomorphism semirings of projective semilattices (projective complete lattices) in the category of semilattices coincide iff those semilattices are finite distributive lattices; we also describe congruence-simple complete hemirings and left artinian congruence-simple complete hemirings. Considering the relationship between the concepts of "Morita equivalence" and "simpleness" in the semiring setting, we obtain the following further results: The ideal-simpleness, congruence-simpleness, and simpleness of semirings are Morita invariant properties; a complete description of simple semirings containing the infinite element; the "Double Centralizer Property" representation theorem for simple semirings; a complete description of simple semirings containing a projective minimal one-sided ideal; a characterization of ideal-simple semirings having either an infinite element or a projective minimal one-sided ideal; settling a conjecture and a problem as published by Katsov in 2004 for the classes of simple semirings containing either an infinite element or a projective minimal left (right) ideal, showing, respectively, that semirings of those classes are not perfect and that the concepts of "mono-flatness" and "flatness" for semimodules over semirings of those classes are the same. Finally, we give a complete description of ideal-simple, artinian additively idempotent chain semirings, as well as of congruence-simple, lattice-ordered semirings.

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21

Akhtar, Mohammad E., and Alexander M. Kasprzyk. "Mutations of Fake Weighted Projective Planes." Proceedings of the Edinburgh Mathematical Society 59, no. 2 (June 10, 2015): 271–85. http://dx.doi.org/10.1017/s0013091515000115.

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AbstractIn previous work by Coates, Galkin and the authors, the notion of mutation between lattice polytopes was introduced. Such mutations give rise to a deformation between the corresponding toric varieties. In this paper we study one-step mutations that correspond to deformations between weighted projective planes, giving a complete characterization of such mutations in terms ofT-singularities. We also show that the weights involved satisfy Diophantine equations, generalizing results of Hacking and Prokhorov.

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22

Brower, Richard C., Claudio Rebbi, and Ettore Vicari. "Projective multigrid method for propagators in lattice gauge theory." Physical Review D 43, no. 6 (March 15, 1991): 1965–73. http://dx.doi.org/10.1103/physrevd.43.1965.

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23

Jaber, Jamel. "The Fremlin projective tensor product of Banach lattice algebras." Journal of Mathematical Analysis and Applications 488, no. 2 (August 2020): 123993. http://dx.doi.org/10.1016/j.jmaa.2020.123993.

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24

Saliola, Franco V. "The Face Semigroup Algebra of a Hyperplane Arrangement." Canadian Journal of Mathematics 61, no. 4 (August 1, 2009): 904–29. http://dx.doi.org/10.4153/cjm-2009-046-2.

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Abstract.This article presents a study of an algebra spanned by the faces of a hyperplane arrangement. The quiver with relations of the algebra is computed and the algebra is shown to be a Koszul algebra. It is shown that the algebra depends only on the intersection lattice of the hyperplane arrangement. A complete systemof primitive orthogonal idempotents for the algebra is constructed and other algebraic structure is determined including: a description of the projective indecomposablemodules, the Cartan invariants, projective resolutions of the simple modules, the Hochschild homology and cohomology, and the Koszul dual algebra. A new cohomology construction on posets is introduced, and it is shown that the face semigroup algebra is isomorphic to the cohomology algebra when this construction is applied to the intersection lattice of the hyperplane arrangement.

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25

SEMENOVA, MARINA, and FRIEDRICH WEHRUNG. "SUBLATTICES OF LATTICES OF ORDER-CONVEX SETS, III: THE CASE OF TOTALLY ORDERED SETS." International Journal of Algebra and Computation 14, no. 03 (June 2004): 357–87. http://dx.doi.org/10.1142/s021819670400175x.

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For a partially ordered set P, let Co(P) denote the lattice of all order-convex subsets of P. For a positive integer n, we denote by [Formula: see text] (resp., SUB(n)) the class of all lattices that can be embedded into a lattice of the form [Formula: see text] where <Ti|i∈I> is a family of chains (resp., chains with at most n elements). We prove the following results: (1) Both classes [Formula: see text] and SUB(n), for any positive integer n, are locally finite, finitely based varieties of lattices, and we find finite equational bases of these varieties. (2) The variety [Formula: see text] is the quasivariety join of all the varieties SUB(n), for 1≤n<ω, and it has only countably many subvarieties. We classify these varieties, together with all the finite subdirectly irreducible members of [Formula: see text]. (3) Every finite subdirectly irreducible member of [Formula: see text] is projective within [Formula: see text], and every subquasivariety of [Formula: see text] is a variety.

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26

Symonds, Peter. "The Reduction of an RG–Lattice Modulo pn." Canadian Journal of Mathematics 42, no. 2 (April 1, 1990): 342–64. http://dx.doi.org/10.4153/cjm-1990-019-0.

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We define the cover of an RG-module V to consist of an RG lattice Ṽ and a homomorphism π : Ṽ→ V such that π induces an isomorphism on Ext*RG(M, —) for any RG-lattice M. Here G is a finite group and, for simplicity in this introduction, R is a complete discrete valuation ring of characteristic zero with prime element p and perfect valuation class field. Let pn(G) be the highest power of p that divides |G| and, given an RG-lattice M, let pn(M) be the smallest power of p such that pn(M) idM : M→M factors through a projective lattice: n(M)≦n(G).

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27

Vilela Mendes, R. "A consistent measure for lattice Yang–Mills." International Journal of Modern Physics A 32, no. 02n03 (January 25, 2017): 1750016. http://dx.doi.org/10.1142/s0217751x17500166.

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The construction of a consistent measure for Yang–Mills is a precondition for an accurate formulation of nonperturbative approaches to QCD, both analytical and numerical. Using projective limits as subsets of Cartesian products of homomorphisms from a lattice to the structure group, a consistent interaction measure and an infinite-dimensional calculus have been constructed for a theory of non-Abelian generalized connections on a hypercubic lattice. Here, after reviewing and clarifying past work, new results are obtained for the mass gap when the structure group is compact.

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28

Camere, Chiara. "Some remarks on moduli spaces of lattice polarized holomorphic symplectic manifolds." Communications in Contemporary Mathematics 20, no. 04 (May 20, 2018): 1750044. http://dx.doi.org/10.1142/s0219199717500444.

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We construct quasi-projective moduli spaces of [Formula: see text]-general lattice polarized irreducible holomorphic symplectic manifolds. Moreover, we study their Baily–Borel compactification and investigate a relation between one-dimensional boundary components and equivalence classes of rational Lagrangian fibrations defined on mirror manifolds.

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29

SHIMADA, ICHIRO. "Holes of the Leech lattice and the projective models of K3 surfaces." Mathematical Proceedings of the Cambridge Philosophical Society 163, no. 1 (September 9, 2016): 125–43. http://dx.doi.org/10.1017/s030500411600075x.

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AbstractUsing the theory of holes of the Leech lattice and Borcherds method for the computation of the automorphism group of a K3 surface, we give an effective bound for the set of isomorphism classes of projective models of fixed degree for certain K3 surfaces.

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30

Cogolludo-Agustín, José Ignacio, Jorge Martín-Morales, and Jorge Ortigas-Galindo. "Numerical adjunction formulas for weighted projective planes and lattice point counting." Kyoto Journal of Mathematics 56, no. 3 (September 2016): 575–98. http://dx.doi.org/10.1215/21562261-3600184.

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31

Owens, Brendan. "SMOOTH, NONSYMPLECTIC EMBEDDINGS OF RATIONAL BALLS IN THE COMPLEX PROJECTIVE PLANE." Quarterly Journal of Mathematics 71, no. 3 (June 30, 2020): 997–1007. http://dx.doi.org/10.1093/qmathj/haaa013.

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Abstract We exhibit an infinite family of rational homology balls, which embed smoothly but not symplectically in the complex projective plane. We also obtain a new lattice embedding obstruction from Donaldson’s diagonalization theorem and use this to show that no two of our examples may be embedded disjointly.

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32

Furukawa, Katsuhisa, and Atsushi Ito. "A combinatorial description of dual defects of toric varieties." Communications in Contemporary Mathematics 23, no. 01 (June 15, 2020): 2050001. http://dx.doi.org/10.1142/s0219199720500017.

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From a finite set in a lattice, we can define a toric variety embedded in a projective space. In this paper, we give a combinatorial description of the dual defect of the toric variety using the structure of the finite set as a Cayley sum with suitable conditions. We also interpret the description geometrically.

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33

Gruenberg, K. W. "Stably free resolutions of lattices over finite groups." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 49, no. 3 (December 1990): 364–85. http://dx.doi.org/10.1017/s1446788700032390.

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AbstractFor a ZG-lattice A, the nth partial free Euler characteristic εn(A) is defined as the infimum of all where F* varies over all free resolutions of A. It is shown that there exists a stably free resolution E* of A which realises εn(A) for all n≥0 and that the function n → εn(A) is ultimately polynomial no residue classes. The existence of E* is established with the help of new invariants σn(A) of A. These are elements in certain image groups of the projective class group of ZG. When ZG allows cancellation, E* is a minimal free resolution and is essentially unique. When A is periodic, E* is ultimately periodic of period a multiple of the projective period of A.

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34

Maddux, Roger D. "Identities Generalizing the Theorems of Pappus and Desargues." Symmetry 13, no. 8 (July 29, 2021): 1382. http://dx.doi.org/10.3390/sym13081382.

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The Theorems of Pappus and Desargues (for the projective plane over a field) are generalized here by two identities involving determinants and cross products. These identities are proved to hold in the three-dimensional vector space over a field. They are closely related to the Arguesian identity in lattice theory and to Cayley-Grassmann identities in invariant theory.

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35

Kasparian, Azniv. "Projective Embeddings of Ball Quotients, Birational to a Bi-elliptic Surface." Proceedings of the Bulgarian Academy of Sciences 76, no. 1 (January 30, 2023): 3–11. http://dx.doi.org/10.7546/crabs.2023.01.01.

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For a neat lattice $$\Gamma < SU(1,2)$$, whose quotient $${\mathbb B} / \Gamma$$ is birational to a bi-elliptic surface, we compute the dimensions of the cuspidal $$\Gamma$$-modular forms $$[ \Gamma,n]_o$$ and all modular forms $$[ \Gamma, n]$$ of weight $$n \geq 2. $$ The work provides a sufficient condition for a subspace $$V \subset [ \Gamma, n]$$ to determine a regular projective embedding of the Baily-Borel compactification $$\widehat{ {\mathbb B} / \Gamma}$$ and applies this criterion to a specific example.

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36

Pacifici, Emanuele. "On tensor factorisation for representations of finite groups." Bulletin of the Australian Mathematical Society 69, no. 1 (February 2004): 161–71. http://dx.doi.org/10.1017/s0004972700034365.

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We prove that, given a quasi-primitive complex representation D for a finite group G, the possible ways of decomposing D as an inner tensor product of two projective representations of G are parametrised in terms of the group structure of G. More explicitly, we construct a bijection between the set of such decompositions and a particular interval in the lattice of normal subgroups of G.

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37

MA, SHOUHEI. "FOURIER–MUKAI PARTNERS OF A K3 SURFACE AND THE CUSPS OF ITS KAHLER MODULI." International Journal of Mathematics 20, no. 06 (June 2009): 727–50. http://dx.doi.org/10.1142/s0129167x09005510.

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Using lattice theory, we establish a one-to-one correspondence between the set of Fourier–Mukai partners of a projective K3 surface and the set of 0-dimensional standard cusps of its Kahler moduli. We also study the relation between twisted Fourier–Mukai partners and general 0-dimensional cusps, and the relation between Fourier–Mukai partners with elliptic fibrations and certain 1-dimensional cusps.

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38

Soulé, Christophe. "Linear projections and successive minima." Nagoya Mathematical Journal 197 (March 2010): 45–57. http://dx.doi.org/10.1017/s0027763000009867.

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LetXbe an arithmetic surface, and letLbe a line bundle onX. Choose a metrichon the lattice Λ of sections ofLoverX. When the degree of the generic fiber ofXis large enough, we get lower bounds for the successive minima of (Λ,h) in terms of the normalized height ofX. The proof uses an effective version (due to C. Voisin) of a theorem of Segre on linear projections and Morrison’s proof that smooth projective curves of high degree are Chow semistable.

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39

Silverman, M. P., and W. Strange. "Projective imaging of periodic objects: recognition and retrieval of lattice structures in real space." Optics Communications 152, no. 4-6 (July 1998): 385–92. http://dx.doi.org/10.1016/s0030-4018(98)00169-2.

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40

Atkinson, James. "On the lattice-geometry and birational group of the six-point multi-ratio equation." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 471, no. 2173 (January 2015): 20140612. http://dx.doi.org/10.1098/rspa.2014.0612.

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The inherent self-consistency properties of the six-point multi-ratio equation allow it to be considered on a domain associated with a T-shaped Coxeter–Dynkin diagram. This extends the Kadomtsev–Petviashvili lattice, which has A N symmetry, and incorporates also Korteweg–de Vries-type dynamics on a sub-domain with D N symmetry, and Painlevé dynamics on a sub-domain with E ~ 8 symmetry. More generally, it can be seen as a distinguished representation of Coble’s Cremona group associated with invariants of point sets in projective space.

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41

Di Nola, Antonio, Giacomo Lenzi, and Tran Giang Nam. "Ultramatricial algebras over commutative chain semirings and application to MV-algebras." Forum Mathematicum 32, no. 2 (March 1, 2020): 287–305. http://dx.doi.org/10.1515/forum-2019-0056.

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AbstractIn this paper, we give a complete description of strongly projective semimodules over a semiring which is a finite direct product of matrix semirings over commutative chain semirings. We then classify ultramatricial algebras over commutative chain semirings by their ordered {\mathrm{SK}_{0}}-groups. Consequently, we get that there is a one-one correspondence between isomorphism classes of ultramatricial algebras A whose {\mathrm{SK}_{0}(A)} is lattice-ordered over a given commutative chain semiring and isomorphism classes of countable MV-algebras.

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42

El Machkouri, Mohamed, and Davide Giraudo. "Orthomartingale-coboundary decomposition for stationary random fields." Stochastics and Dynamics 16, no. 05 (July 15, 2016): 1650017. http://dx.doi.org/10.1142/s0219493716500179.

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We provide a new projective condition for a stationary real random field indexed by the lattice [Formula: see text] to be well approximated by an orthomartingale in the sense of Cairoli (1969). Our main result can be viewed as a multidimensional version of the martingale-coboundary decomposition method which the idea goes back to Gordin (1969). It is a powerful tool for proving limit theorems or large deviations inequalities for stationary random fields when the corresponding result is valid for orthomartingales.

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43

Ye, Qing, Mingxing Hu, Guangxuan Chen, and Panke Qin. "An Improved Encryption Scheme for Traitor Tracing from Lattice." International Journal of Digital Crime and Forensics 10, no. 4 (October 2018): 21–35. http://dx.doi.org/10.4018/ijdcf.2018100102.

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This article first describes a paper by Ling, Phan, and Stehle at the CRYPTO 2014 which presented the first encryption scheme for traitor tracing from lattice, and the scheme is almost as efficient as the learning with errors (LWE) encryption. However, their scheme is not constructed on an efficient trapdoor, that is, the trapdoor generation and preimage sampling algorithms are rather complex and not suitable for practice. This article is considered to use the MP12 trapdoor to construct an improved traitor tracing scheme. First, by using batch execution method, this article proposes an improved extracting algorithm for the user's key. Then, this article combines that with multi-bit encryption system to construct an efficient one-to-many encryption scheme. Furthermore, it is presented that a novel projective sampling family has very small hidden constants. Finally, a comparative analysis shows that the parameters of the scheme such as lattice dimension, trapdoor size, and ciphertext expansion rate, etc., all decrease in some degree, and the computational cost is reduced.

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De Falco, Maria, Francesco de Giovanni, and Carmela Musella. "Modular chains in infinite groups." Archiv der Mathematik 117, no. 6 (October 20, 2021): 601–12. http://dx.doi.org/10.1007/s00013-021-01665-2.

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AbstractThe aim of this paper is to investigate the behaviour of projective images of the groups which are finite over a term of their upper central series. In particular, we prove that for any positive integer k, the class of finitely generated groups in which the k-th term of the upper central series has finite index can be described in terms of lattice invariants, and so it is invariant under projectivities. In this context, we also study groups that have only finitely many maximal subgroups which are not permodular.

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Soulé, Christophe. "Linear projections and successive minima." Nagoya Mathematical Journal 197 (March 2010): 45–57. http://dx.doi.org/10.1215/00277630-2009-002.

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Let X be an arithmetic surface, and let L be a line bundle on X. Choose a metric h on the lattice Λ of sections of L over X. When the degree of the generic fiber of X is large enough, we get lower bounds for the successive minima of (Λ,h) in terms of the normalized height of X. The proof uses an effective version (due to C. Voisin) of a theorem of Segre on linear projections and Morrison’s proof that smooth projective curves of high degree are Chow semistable.

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Coppola, M., and D. Karevski. "Some speculations about local thermalization of nonequilibrium extended quantum systems." Condensed Matter Physics 26, no. 1 (2023): 13502. http://dx.doi.org/10.5488/cmp.26.13502.

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We discuss the possibility of defining an emergent local temperature in extended quantum many-body systems evolving out of equilibrium. For the most simple case of free-fermionic systems, we give an explicit formula for the effective temperature in the case of, not necessarily unitary, Gaussian preserving dynamics. In this framework, we consider the hopping fermions on a one-dimensional lattice submitted to randomly distributed projective measurements of the local occupation numbers. We show from the average over many quantum trajectories that the effective temperature relaxes exponentially towards infinity.

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CHEN, NING, and NANNAN LUO. "CONSTRUCTION OF SPHERICAL PATTERNS FROM PLANAR DYNAMIC SYSTEMS." Fractals 21, no. 01 (March 2013): 1350005. http://dx.doi.org/10.1142/s0218348x13500059.

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We investigated the generation of spherical continuous-tilings of the chaotic attractors or the filled-in Julia sets from the plane mappings. We build three plane mappings, which can be used to construct the continuous patterns on the surfaces of the hexahedron and the unit sphere. We discuss the coordinate transformation for a spatial point between the different coordinate systems and further discuss how to project a spherical point onto a surface of the inscribed hexahedron. We present a method of constructing a spherical pattern with the pattern of a square on the inscribed hexahedron from an arbitrary projective angle, and generate spherical patterns from the three plane mappings. The results show that we can construct a great number of spherical patterns of the chaotic attractors and the filled-in Julia sets from the plane mappings, which are based on the square lattice and meet the requirements of the continuity and the four rotation symmetries on the lattice's boundaries.

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48

Jain, Ranjana, and Ajay Kumar. "The Operator Space Projective Tensor Product: Embedding into the Second Dual and Ideal Structure." Proceedings of the Edinburgh Mathematical Society 57, no. 2 (September 5, 2013): 505–19. http://dx.doi.org/10.1017/s001309151300045x.

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AbstractWe prove that, for operator spaces V and W, the operator space V** ⊗hW** can be completely isometrically embedded into (V ⊗hW)**, ⊗h being the Haagerup tensor product. We also show that, for exact operator spaces V and W, a jointly completely bounded bilinear form on V × W can be extended uniquely to a separately w*-continuous jointly completely bounded bilinear form on V× W**. This paves the way to obtaining a canonical embedding of into with a continuous inverse, where is the operator space projective tensor product. Further, for C*-algebras A and B, we study the (closed) ideal structure of which, in particular, determines the lattice of closed ideals of completely.

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Mok, Ngaiming. "Zariski closures of images of algebraic subsets under the uniformization map on finite-volume quotients of the complex unit ball." Compositio Mathematica 155, no. 11 (September 19, 2019): 2129–49. http://dx.doi.org/10.1112/s0010437x19007577.

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We prove the analogue of the Ax–Lindemann–Weierstrass theorem for not necessarily arithmetic lattices of the automorphism group of the complex unit ball $\mathbb{B}^{n}$ using methods of several complex variables, algebraic geometry and Kähler geometry. Consider a torsion-free lattice $\unicode[STIX]{x1D6E4}\,\subset \,\text{Aut}(\mathbb{B}^{n})$ and the associated uniformization map $\unicode[STIX]{x1D70B}:\mathbb{B}^{n}\rightarrow \mathbb{B}^{n}/\unicode[STIX]{x1D6E4}=:X_{\unicode[STIX]{x1D6E4}}$. Given an algebraic subset $S\,\subset \,\mathbb{B}^{n}$ and writing $Z$ for the Zariski closure of $\unicode[STIX]{x1D70B}(S)$ in $X_{\unicode[STIX]{x1D6E4}}$ (which is equipped with a canonical quasi-projective structure), in some precise sense we realize $Z$ as a variety uniruled by images of algebraic subsets under the uniformization map, and study the asymptotic geometry of an irreducible component $\widetilde{Z}$ of $\unicode[STIX]{x1D70B}^{-1}(Z)$ as $\widetilde{Z}$ exits the boundary $\unicode[STIX]{x2202}\mathbb{B}^{n}$ by exploiting the strict pseudoconvexity of $\mathbb{B}^{n}$, culminating in the proof that $\widetilde{Z}\,\subset \,\mathbb{B}^{n}$ is totally geodesic. Our methodology sets the stage for tackling problems in functional transcendence theory for arbitrary lattices of $\text{ Aut}(\unicode[STIX]{x1D6FA})$ for (possibly reducible) bounded symmetric domains $\unicode[STIX]{x1D6FA}$.

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Momot, Aleksander. "On Modular Ball-Quotient Surfaces of Kodaira Dimension One." ISRN Geometry 2011 (June 19, 2011): 1–5. http://dx.doi.org/10.5402/2011/214853.

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Let be a lattice which is not co-ompact, of finite covolume with respect to the Bergman metric and acting freely on the open unit ball . Then the toroidal compactification is a projective smooth surface with elliptic compactification divisor . In this short note we discover a new class of unramifed ball quotients . We consider ball quotients with kod and . We prove that each minimal surface with finite Mordell-Weil group in the class described admits an étale covering which is a pull-back of . Here denotes the elliptic modular surface parametrizing elliptic curves with 6-torsion points which generate [6].

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