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Journal articles on the topic 'Lattice naturale'

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Published: 9 March 2023

Last updated: 10 March 2023

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1

Grabowski, Adam. "Stone Lattices." Formalized Mathematics 23, no. 4 (December 1, 2015): 387–96. http://dx.doi.org/10.1515/forma-2015-0031.

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Summary The article continues the formalization of the lattice theory (as structures with two binary operations, not in terms of ordering relations). In the paper, the notion of a pseudocomplement in a lattice is formally introduced in Mizar, and based on this we define the notion of the skeleton and the set of dense elements in a pseudocomplemented lattice, giving the meet-decomposition of arbitrary element of a lattice as the infimum of two elements: one belonging to the skeleton, and the other which is dense. The core of the paper is of course the idea of Stone identity $$a^* \sqcup a^{**} = {\rm{T}},$$ which is fundamental for us: Stone lattices are those lattices L, which are distributive, bounded, and satisfy Stone identity for all elements a ∈ L. Stone algebras were introduced by Grätzer and Schmidt in [18]. Of course, the pseudocomplement is unique (if exists), so in a pseudcomplemented lattice we defined a * as the Mizar functor (unary operation mapping every element to its pseudocomplement). In Section 2 we prove formally a collection of ordinary properties of pseudocomplemented lattices. All Boolean lattices are Stone, and a natural example of the lattice which is Stone, but not Boolean, is the lattice of all natural divisors of p 2 for arbitrary prime number p (Section 6). At the end we formalize the notion of the Stone lattice B [2] (of pairs of elements a, b of B such that a ⩽ b) constructed as a sublattice of B 2, where B is arbitrary Boolean algebra (and we describe skeleton and the set of dense elements in such lattices). In a natural way, we deal with Cartesian product of pseudocomplemented lattices. Our formalization was inspired by [17], and is an important step in formalizing Jouni Järvinen Lattice theory for rough sets [19], so it follows rather the latter paper. We deal essentially with Section 4.3, pages 423–426. The description of handling complemented structures in Mizar [6] can be found in [12]. The current article together with [15] establishes the formal background for algebraic structures which are important for [10], [16] by means of mechanisms of merging theories as described in [11].

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Alvarado-García, Alejandro, César Cejudo-Castilla, Hugo Alberto Rincón-Mejía, and Ivan Fernando Vilchis-Montalvo. "Pseudocomplements and strong pseudocomplements in lattices of module classes." Journal of Algebra and Its Applications 17, no. 01 (January 2018): 1850016. http://dx.doi.org/10.1142/s0219498818500160.

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In this work, we consider the existence and construction of pseudocomplements in some lattices of module classes. The classes of modules belonging to these lattices are defined via closure under operations such as taking submodules, quotients, extensions, injective hulls, direct sums or products. We characterize the rings for which the lattices [Formula: see text]-tors (of hereditary torsion classes), [Formula: see text]-nat (the lattice of natural classes) and [Formula: see text]-conat (the lattice of conatural classes) coincide.

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3

Chajda, Ivan, and Helmut Länger. "Left residuated lattices induced by lattices with a unary operation." Soft Computing 24, no. 2 (November 1, 2019): 723–29. http://dx.doi.org/10.1007/s00500-019-04461-x.

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Abstract In a previous paper, the authors defined two binary term operations in orthomodular lattices such that an orthomodular lattice can be organized by means of them into a left residuated lattice. It is a natural question if these operations serve in this way also for more general lattices than the orthomodular ones. In our present paper, we involve two conditions formulated as simple identities in two variables under which this is really the case. Hence, we obtain a variety of lattices with a unary operation which contains exactly those lattices with a unary operation which can be converted into a left residuated lattice by use of the above mentioned operations. It turns out that every lattice in this variety is in fact a bounded one and the unary operation is a complementation. Finally, we use a similar technique by using simpler terms and identities motivated by Boolean algebras.

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4

BÉTERMIN, LAURENT. "ON A LATTICE GENERALISATION OF THE LOGARITHM AND A DEFORMATION OF THE DEDEKIND ETA FUNCTION." Bulletin of the Australian Mathematical Society 102, no. 1 (February 20, 2020): 118–25. http://dx.doi.org/10.1017/s000497272000012x.

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We consider a deformation $E_{L,\unicode[STIX]{x1D6EC}}^{(m)}(it)$ of the Dedekind eta function depending on two $d$-dimensional simple lattices $(L,\unicode[STIX]{x1D6EC})$ and two parameters $(m,t)\in (0,\infty )$, initially proposed by Terry Gannon. We show that the minimisers of the lattice theta function are the maximisers of $E_{L,\unicode[STIX]{x1D6EC}}^{(m)}(it)$ in the space of lattices with fixed density. The proof is based on the study of a lattice generalisation of the logarithm, called the lattice logarithm, also defined by Terry Gannon. We also prove that the natural logarithm is characterised by a variational problem over a class of one-dimensional lattice logarithms.

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5

Longstaff, W. E., J. B. Nation, and Oreste Panaia. "Abstract reflexive sublattices and completely distributive collapsibility." Bulletin of the Australian Mathematical Society 58, no. 2 (October 1998): 245–60. http://dx.doi.org/10.1017/s0004972700032226.

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There is a natural Galois connection between subspace lattices and operator algebras on a Banach space which arises from the notion of invariance. If a subspace lattice ℒ is completely distributive, then ℒ is reflexive. In this paper we study the more general situation of complete lattices for which the least complete congruence δ on ℒ such that ℒ/δ is completely distributive is well-behaved. Our results are purely lattice theoretic, but the motivation comes from operator theory.

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6

Restall, Greg, and Francesco Paoli. "The geometry of non-distributive logics." Journal of Symbolic Logic 70, no. 4 (December 2005): 1108–26. http://dx.doi.org/10.2178/jsl/1129642117.

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AbstractIn this paper we introduce a new natural deduction system for the logic of lattices, and a number of extensions of lattice logic with different negation connectives. We provide the class of natural deduction proofs with both a standard inductive definition and a global graph-theoretical criterion for correctness, and we show how normalisation in this system corresponds to cut elimination in the sequent calculus for lattice logic. This natural deduction system is inspired both by Shoesmith and Smiley's multiple conclusion systems for classical logic and Girard's proofnets for linear logic.

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7

Chen, Zai Liang, Luo Hong Deng, and Cong Jing. "Dynamic Performance and Structural Optimization of Large-Scale Machine Table." Applied Mechanics and Materials 615 (August 2014): 313–16. http://dx.doi.org/10.4028/www.scientific.net/amm.615.313.

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Designed new table for large floor boring and milling machine, used ANSYS to optimize the structure of the table as a whole. According to the contours of removable material the materials which can be removed, obtained the inner ribs layout of table and the sand holes location of rib plate. Dynamic optimization variables on basic ribs cell, studied the effect of steel lattice structure parameters influenced on the natural frequency of the lattices and the related parameter of lattices influenced on whole table, to get the ideal rib lattice structure after optimizing again. Optimized bench can reduce quality, increase rigidity and dynamic performance.

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8

Várilly-Alvarado, Anthony, and David Zywina. "Arithmetic E8 Lattices with Maximal Galois Action." LMS Journal of Computation and Mathematics 12 (2009): 144–65. http://dx.doi.org/10.1112/s1461157000001479.

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AbstractWe construct explicit examples of E8 lattices occurring in arithmetic for which the natural Galois action is equal to the full group of automorphisms of the lattice, i.e., the Weyl group of E8. In particular, we give explicit elliptic curves over Q(t) whose Mordell-Weil lattices are isomorphic to E8 and have maximal Galois action.Our main objects of study are del Pezzo surfaces of degree 1 over number fields. The geometric Picard group, considered as a lattice via the negative of the intersection pairing, contains a sublattice isomorphic to E8. We construct examples of such surfaces for which the action of Galois on the geometric Picard group is maximal.

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9

ADARICHEVA, KIRA, and J. B. NATION. "LATTICES OF QUASI-EQUATIONAL THEORIES AS CONGRUENCE LATTICES OF SEMILATTICES WITH OPERATORS: PART I." International Journal of Algebra and Computation 22, no. 07 (November 2012): 1250065. http://dx.doi.org/10.1142/s0218196712500658.

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We show that for every quasivariety 𝒦 of structures (where both functions and relations are allowed) there is a semilattice S with operators such that the lattice of quasi-equational theories of 𝒦 (the dual of the lattice of sub-quasivarieties of 𝒦) is isomorphic to Con(S, +, 0, 𝒡. As a consequence, new restrictions on the natural quasi-interior operator on lattices of quasi-equational theories are found.

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10

Niederle, Josef. "Natural generalization of some lattice theory concepts to partially ordered sets." Czechoslovak Mathematical Journal 41, no. 2 (1991): 297–99. http://dx.doi.org/10.21136/cmj.1991.102463.

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11

Kirsanov, M. N. "Analytical Estimation of the Natural Oscillation Frequency of a Planar Lattice." Advanced Engineering Research 22, no. 4 (January 9, 2023): 315–22. http://dx.doi.org/10.23947/2687-1653-2022-22-4-315-322.

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Introduction. A new scheme of a flat statically determinate regular lattice is proposed. The lattice rods are hinged. The study aims at deriving a formula for the dependence on the number of panels of the first natural oscillation frequency of nodes endowed with masses, each of which has two degrees of freedom in the lattice plane. The rigidity of all rods is assumed to be the same, the supports (movable and fixed hinges) — nondeformable. Another objective of the study is to find the dependence of the stresses in the most compressed and stretched rods on the number of panels in an analytical form. Materials and Methods. An approximate Dunkerley’s method was used to determine the lower bound for the lattice natural frequency. The lattice rigidity was found in analytical form according to Maxwell-Mohr formula. The rod stresses and the reactions of the supports were determined from the equilibrium equations compiled for all lattice nodes. Generalization of the result to an arbitrary number of panels was performed by induction using Maple symbolic math operators for analytical solutions to a number of problems for lattices with different number of panels. Results. The lower analytical estimate of the first oscillation frequency was in good agreement with the numerical solution for the minimum frequency of the oscillation spectrum of the structure. Formulas were found for the stresses in four most compressed and stretched rods and their linear asymptotics. All required transformations were made in the system of Maple symbolic math. Discussion and Conclusions. The obtained dependence of the first frequency of lattice oscillations on the number of panels, mass and dimensions of the structure has a compact form and can be used as a test problem for numerical solutions and optimization of the structure.

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12

Kroc, Jiří. "Influence of Lattice Anisotropy on Models Formulated by Cellular Automata in Presence of Grain Boundary Movement: A Case Study." Materials Science Forum 482 (April 2005): 195–98. http://dx.doi.org/10.4028/www.scientific.net/msf.482.195.

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This paper continues in the previous research focussed to two simple questions. The first one reads: ”What is the influence of anisotropy of computational lattice on simulations of boundary movement?” where grain boundary movement typically appears in simulations of grain boundary migration and static/dynamic recrystallization. The second question reads: ”How is the computational anisotropy related to natural anisotropy of the material lattice itself?” This study is focussed on the influence of change of the computational algorithm and/or lattice on the grain boundary movement. Two algorithms, the majority rule and the simple modification of the Monte Carlo method for two different lattices – namely square and hexagonal one – are used.

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13

Grätzer, G., H. Lakser, and E. T. Schmidt. "Congruence Representations of Join-homomorphisms of Finite Distributive Lattices: Size and Breadth." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 68, no. 1 (February 2000): 85–103. http://dx.doi.org/10.1017/s1446788700001592.

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AbstractLet K and L be lattices, and let ϕ be a homomorphism of K into L.Then ϕ induces a natural 0-preserving join-homomorphism of Con K into Con L.Extending a result of Huhn, the authors proved that if D and E are finite distributive lattices and ψ is a 0-preserving join-homomorphism from D into E, then D and E can be represented as the congruence lattices of the finite lattices K and L, respectively, such that ψ is the natural 0-preserving join-homomorphism induced by a suitable homomorphism ϕ: K → L. Let m and n denote the number of join-irreducible elements of D and E, respectively, and let k = max (m, n). The lattice L constructed was of size O(22(n+m)) and of breadth n+m.We prove that K and L can be constructed as ‘small’ lattices of size O(k5) and of breadth three.

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14

Couch, W. E., M. Surovy, and R. J. Torrence. "Some singular motions of non-Abelian Toda lattices." Canadian Journal of Physics 78, no. 2 (March 2, 2000): 99–112. http://dx.doi.org/10.1139/p00-025.

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Motions of finite Toda lattices are known to be associated with linear wave equations whose general solutions can be expressed in terms of progressing waves, and this association is known to generalize to finite non-Abelian Toda lattices of n x n matrices and systems of n coupled linear wave equations. We present a nontrivial family of non-Abelian Toda lattice motions that can be specialized to ones that are not finite, but not infinitely extendible either, as they contain nonvanishing but singular matrices of rank (n s). In these cases we give a natural continuation of the lattice dynamics by means of nonsingular matrices of dimension (n s) x (n s), and describe how to find s progressing wave solutions of the associated system of n coupled linear wave equations.PACS No.: 5.45-a

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15

de Viola-Prioli, Ana M., and Jorge E. Viola-Prioli. "Asymmetry in the lattice of kernel functors." Glasgow Mathematical Journal 33, no. 1 (January 1991): 95–97. http://dx.doi.org/10.1017/s0017089500008089.

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Much of the research done by different authors on the lattice of kernel functors (equivalently, linear topologies) has been summarized by Golan in [2]. More recently, the rings whose lattices of kernel functors are linearly ordered were introduced in [3] as a categorical generalization of valuation rings in the non-commutative case. Results (and examples) in [3] show that there is an abundance of non-commutative rings R whose lattices (R), both in Mod-R and R-Mod, are simultaneously linearly ordered; however, the question of the symmetry of this condition remained open. Here we will prove that, for every natural number n≥3, there exists a ring Rn such that (Mod-Rn) is a linearly ordered lattice of n elements, whereas (Rn-Mod) is not linearly ordered.

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16

Chajda, Ivan, and Helmut Länger. "When does a semiring become a residuated lattice?" Asian-European Journal of Mathematics 09, no. 04 (November 28, 2016): 1650088. http://dx.doi.org/10.1142/s1793557116500881.

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It is an easy observation that every residuated lattice is in fact a semiring because multiplication distributes over join and the other axioms of a semiring are satisfied trivially. This semiring is commutative, idempotent and simple. The natural question arises if the converse assertion is also true. We show that the conversion is possible provided the given semiring is, moreover, completely distributive. We characterize semirings associated to complete residuated lattices satisfying the double negation law where the assumption of complete distributivity can be omitted. A similar result is obtained for idempotent residuated lattices.

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17

BUNIMOVICH, LEONID A. "MANY-DIMENSIONAL LORENTZ CELLULAR AUTOMATA AND TURING MACHINES." International Journal of Bifurcation and Chaos 06, no. 06 (June 1996): 1127–35. http://dx.doi.org/10.1142/s0218127496000618.

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We study the class of cellular automata that generalizes the Lorentz lattice gases in statistical mechanics, the models of industrious ants in the theory of an artificial life and the so-called Tur-mites (many-dimensional Turing machines). We prove that on the square lattice ℤd, d = 2, the existence of a bounded orbit of a particle (ant, machine) determines all nondegenerate local scattering rules (states of a machine). For higher dimensional (d ≥ 3) cubic lattices we show that under some natural conditions all possible bounded orbits (vortices) can live only in some “vortex sheets” that have a dimension strictly less than d.

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18

GEERSE, C. P. M., and A. HOF. "LATTICE GAS MODELS ON SELF-SIMILAR APERIODIC TILINGS." Reviews in Mathematical Physics 03, no. 02 (June 1991): 163–221. http://dx.doi.org/10.1142/s0129055x91000072.

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We discuss lattice gas models on the vertices of tilings in arbitrary dimension that are self-similar in the way Penrose tilings of the plane are self-similar. Among these, there are systems that fundamentally lack translation invariance. Under natural hypotheses on the interactions and the states, we prove the existence of thermodynaraic functions — the mean pressure, the mean energy and the mean entropy — and derive the variational principle. The relation between Gibbs states and tangent functionals to the mean pressure is investigated. Generalizations to quantum systems are also discussed. Our work extends results known for lattice gas models on periodic lattices.

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Chajda, Ivan, and Helmut Länger. "Adjoint Operations in Twist-Products of Lattices." Symmetry 13, no. 2 (February 2, 2021): 253. http://dx.doi.org/10.3390/sym13020253.

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Given an integral commutative residuated lattices L=(L,∨,∧), its full twist-product (L2,⊔,⊓) can be endowed with two binary operations ⊙ and ⇒ introduced formerly by M. Busaniche and R. Cignoli as well as by C. Tsinakis and A. M. Wille such that it becomes a commutative residuated lattice. For every a∈L we define a certain subset Pa(L) of L2. We characterize when Pa(L) is a sublattice of the full twist-product (L2,⊔,⊓). In this case Pa(L) together with some natural antitone involution ′ becomes a pseudo-Kleene lattice. If L is distributive then (Pa(L),⊔,⊓,′) becomes a Kleene lattice. We present sufficient conditions for Pa(L) being a subalgebra of (L2,⊔,⊓,⊙,⇒) and thus for ⊙ and ⇒ being a pair of adjoint operations on Pa(L). Finally, we introduce another pair ⊙ and ⇒ of adjoint operations on the full twist-product of a bounded commutative residuated lattice such that the resulting algebra is a bounded commutative residuated lattice satisfying the double negation law, and we investigate when Pa(L) is closed under these new operations.

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Conrad, Jonathan, Jens Eisert, and Francesco Arzani. "Gottesman-Kitaev-Preskill codes: A lattice perspective." Quantum 6 (February 10, 2022): 648. http://dx.doi.org/10.22331/q-2022-02-10-648.

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We examine general Gottesman-Kitaev-Preskill (GKP) codes for continuous-variable quantum error correction, including concatenated GKP codes, through the lens of lattice theory, in order to better understand the structure of this class of stabilizer codes. We derive formal bounds on code parameters, show how different decoding strategies are precisely related, propose new ways to obtain GKP codes by means of glued lattices and the tensor product of lattices and point to natural resource savings that have remained hidden in recent approaches. We present general results that we illustrate through examples taken from different classes of codes, including scaled self-dual GKP codes and the concatenated surface-GKP code.

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WEHRUNG, FRIEDRICH. "FROM JOIN-IRREDUCIBLES TO DIMENSION THEORY FOR LATTICES WITH CHAIN CONDITIONS." Journal of Algebra and Its Applications 01, no. 02 (June 2002): 215–42. http://dx.doi.org/10.1142/s0219498802000148.

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For a finite lattice L, the congruence lattice Con L of L can be easily computed from the partially ordered set J (L) of join-irreducible elements of L and the join-dependency relation DL on J (L). We establish a similar version of this result for the dimension monoid Dim L of L, a natural precursor of Con L. For L join-semidistributive, this result takes the following form: Theorem 1. Let L be a finite join-semidistributive lattice. Then Dim L is isomorphic to the commutative monoid defined by generators Δ(p), for p ∈ J(L), and relations [Formula: see text] As a consequence of this, we obtain the following results: Theorem 2. Let L be a finite join-semidistributive lattice. Then L is a lower bounded homomorphic image of a free lattice iff Dim L is strongly separative, iff it satisfies the axiom [Formula: see text] Theorem 3. Let A and B be finite join-semidistributive lattices. Then the box product A □ B of A and B is join-semidistributive, and the following isomorphism holds: [Formula: see text] where ⊗ denotes the tensor product of commutative monoids.

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Rump, Wolfgang. "Von Neumann algebras, L-algebras, Baer *-monoids, and Garside groups." Forum Mathematicum 30, no. 4 (July 1, 2018): 973–95. http://dx.doi.org/10.1515/forum-2017-0108.

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AbstractIt is shown that the projection lattice of a von Neumann algebra, or more generally every orthomodular latticeX, admits a natural embedding into a group{G(X)}with a lattice ordering so that{G(X)}determinesXup to isomorphism. The embedding{X\hookrightarrow G(X)}appears to be a universal (non-commutative) group-valued measure onX, while states ofXturn into real-valued group homomorphisms on{G(X)}. The existence of completions is characterized by a generalized archimedean property which simultaneously applies toXand{G(X)}. By an extension of Foulis’ coordinatization theorem, the negative cone of{G(X)}is shown to be the initial object among generalized Baer{{}^{\ast}}-semigroups. For finiteX, the correspondence betweenXand{G(X)}provides a new class of Garside groups.

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Xu, Xiaoping. "Theta Series of Unimodular Lattices, Combinatorial Identities and Weighted Symmetric Polynomials." Algebra Colloquium 13, no. 01 (March 2006): 67–86. http://dx.doi.org/10.1142/s1005386706000101.

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Hecke proved that the theta series of a positive definite even unimodular lattice is a polynomial of the well-known Essenstein series E4(z) and the Ramanujan series Δ24(z). A natural question is what kind of polynomials in E4(z) and Δ24(z) could be the theta series of positive definite even unimodular lattices. In this paper, we find two combinatorial identities on the theta series of the root lattices of the finite-dimensional simple Lie algebras of type D4n and the cosets in their integral duals, in terms of E4(z) and Δ24(z). Using these two identities, we prove that three families of weighted symmetric polynomials of two fixed families of polynomials of E4(z) and Δ24(z) are the theta series of certain positive definite even unimodular lattices, obtained by gluing finitely many copies of the root lattices of the finite-dimensional simple Lie algebras of type D2n. The results also show that the full permutation groups are the hidden symmetry of the theta series of certain unimodular lattices.

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Semenov, Alexei, and Sergei Soprunov. "Automorphisms and Definability (of Reducts) for Upward Complete Structures." Mathematics 10, no. 20 (October 12, 2022): 3748. http://dx.doi.org/10.3390/math10203748.

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The Svenonius theorem establishes the correspondence between definability of relations in a countable structure and automorphism groups of these relations in extensions of the structure. This may help in finding a description of the lattice constituted by all definability spaces (reducts) of the original structure. Results on definability lattices were previously obtained only for ω-categorical structures with finite signature. In our work, we introduce the concept of an upward complete structure and define the upward completion of a structure. For upward complete structures, the Galois correspondence between definability lattice and the lattice of closed supergroups of the automorphism group of the structure is an anti-isomorphism. We describe the natural class of structures which have upward completion, we call them discretely homogeneous graphs, present the explicit construction of their completion and automorphism groups of completions. We establish the general localness property of discretely homogeneous graphs and present examples of completable structures and their completions.

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HUANG, HAIBO, T. S. LEE, and C. SHU. "THERMAL CURVED BOUNDARY TREATMENT FOR THE THERMAL LATTICE BOLTZMANN EQUATION." International Journal of Modern Physics C 17, no. 05 (May 2006): 631–43. http://dx.doi.org/10.1142/s0129183106009059.

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In this paper, a recent curved non-slip wall boundary treatment for isothermal Lattice Boltzmann equation (LBE) [Z. Guo, C. Zheng and B. Shi, Phys. Fluids14(6) (2002)] is extended to handle the thermal curved wall boundary for a double-population thermal lattice Boltzmann equation (TLBE). The unknown distribution population at a wall node which is necessary to fulfill streaming step is decomposed into its equilibrium and non-equilibrium parts. The equilibrium part is evaluated according to Dirichlet and Neumann boundary constraints, and the non-equilibrium part is obtained using a first-order extrapolation from fluid lattices. To validate the thermal boundary condition treatment, we carry out numerical simulations of Couette flow between two circular cylinders, the natural convection in a square cavity, and the natural convection in a concentric annulus between an outer square cylinder and an inner circular cylinder. The results agree very well with analytical solution or available data in the literature. Our numerical results also demonstrate that the TLBE together with the present boundary scheme is of second-order accuracy.

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Gu, Anhui, and Peter E. Kloeden. "Asymptotic Behavior of a Nonautonomous p-Laplacian Lattice System." International Journal of Bifurcation and Chaos 26, no. 10 (September 2016): 1650174. http://dx.doi.org/10.1142/s0218127416501741.

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The existence of a pullback attractor for the nonautonomous [Formula: see text]-Laplacian type equations on infinite lattices is established under certain natural dissipative conditions. In particular, there is no restriction on the power index [Formula: see text] of the nonlinearity relative to the index [Formula: see text]. The forward limiting behavior is also discussed and, under suitable assumptions on the time dependent terms, the lattice system is shown to be asymptotically autonomous with its pullback attractor component sets converging upper semi-continuously to the autonomous global attractor of the limiting autonomous system.

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GONZÁLEZ-FÉREZ, JUAN, and LEANDRO MARÍN. "MONOMORPHISMS AND KERNELS IN THE CATEGORY OF FIRM MODULES." Glasgow Mathematical Journal 52, A (June 24, 2010): 83–91. http://dx.doi.org/10.1017/s0017089510000224.

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AbstractIn this paper we consider for a non-unital ring R, the category of firm R-modules for a non-unital ring R, i.e. the modules M such that the canonical morphism μM : R ⊗RM → M given by r ⊗ m ↦ rm is an isomorphism. This category is a natural generalization of the usual category of unitary modules for a ring with identity and shares many properties with it. The only difference is that monomorphisms are not always kernels. It has been proved recently that this category is not Abelian in general by providing an example of a monomorphism that is not a kernel in a particular case. In this paper we study the lattices of monomorphisms and kernels, proving that the lattice of monomorphisms is a modular lattice and that the category of firm modules is Abelian if and only if the composition of two kernels is a kernel.

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Revol, J. F. "Lattice resolution in natural polymers." Proceedings, annual meeting, Electron Microscopy Society of America 47 (August 6, 1989): 748–49. http://dx.doi.org/10.1017/s0424820100155712.

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Recent developments in low-dose electron microscopy have led to reports of crystal lattice fringes in natural polymers extremely sensitive to damage by electron irradiation. In one of these studies, lattice images of (β-chitin from the diatom Thalassiosira-fluviatilis were obtained and subsequent computer processing produced the molecular projection of this biopolymer crystal along its b-axis at 0.35 nm resolution. Similar direct structural studies have not been done yet on the α-chitin which is far the most abundant polymorphic form found in nature. The present work reports the direct imaging of crystallite lattice fringes of α-chitin microfibrils from lobster tendon, with a preliminary analysis of the results in terms of crystallite size and perfection.A typical preparation of α-chitin microfibrils is shown in Figure 1. The image, recorded by diffraction contrast, shows bundles of long microfibrils as well as smaller elements less aggregated and of different length. These smaller elements, composed of shorter crystallites, are the result of the combined action of the acid hydrolysis and of the ultrasound treatment which broke down the originally long microfibrils. When aggregated into larger ribbons, the width of the microfibrils cannot be detrmined due to overlapping, but when viewed individually, the microfibrils exhibit a lateral size of about 10 to 14 nm (see arrows).

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Graña, Manuel. "Lattice computing and natural computing." Neurocomputing 72, no. 10-12 (June 2009): 2065–66. http://dx.doi.org/10.1016/j.neucom.2008.11.021.

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30

Chajda, Ivan, and Helmut Länger. "Residuation in non-associative MV-algebras." Mathematica Slovaca 68, no. 6 (December 19, 2018): 1313–20. http://dx.doi.org/10.1515/ms-2017-0181.

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Abstract It is well known that every MV-algebra can be converted into a residuated lattice satisfying divisibility and the double negation law. In a previous paper the first author and J. Kühr introduced the concept of an NMV-algebra which is a non-associative modification of an MV-algebra. The natural question arises if an NMV-algebra can be converted into a residuated structure, too. Contrary to MV-algebras, NMV-algebras are not based on lattices but only on directed posets and the binary operation need not be associative and hence we cannot expect to obtain a residuated lattice but only an essentially weaker structure called a conditionally residuated poset. Considering several additional natural conditions we show that every NMV-algebra can be converted in such a structure. Also conversely, every such structure can be organized into an NMV-algebra. Further, we study an a bit more stronger version of an algebra where the binary operation is even monotone. We show that such an algebra can be organized into a residuated poset and, conversely, every residuated poset can be converted in this structure.

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31

Danso, Larry, and Eduard Karpov. "Reprogramming Static Deformation Patterns in Mechanical Metamaterials." Materials 11, no. 10 (October 20, 2018): 2050. http://dx.doi.org/10.3390/ma11102050.

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This paper discusses an x-braced metamaterial lattice with the unusual property of exhibiting bandgaps in their deformation decay spectrum, and, hence, the capacity for reprogramming deformation patterns. The design of polarizing non-local lattice arising from the scenario of repeated zero eigenvalues of a system transfer matrix is also introduced. We develop a single mode fundamental solution for lattices with multiple degrees of freedom per node in the form of static Raleigh waves. These waves can be blocked at the material boundary when the solution is constructed with the polarization vectors of the bandgap. This single mode solution is used as a basis to build analytical displacement solutions for any applied essential and natural boundary condition. Subsequently, we address the bandgap design, leading to a comprehensive approach for predicting deformation pattern behavior within the interior of an x-braced plane lattice. Overall, we show that the stiffness parameter and unit-cell aspect ratio of the x-braced lattice can be tuned to completely block or filter static boundary deformations, and to reverse the dependence of deformation or strain energy decay parameter on the Raleigh wavenumber, a behavior known as the reverse Saint Venant’s edge effect (RSV). These findings could guide future research in engineering smart materials and structures with interesting functionalities, such as load pattern recognition, strain energy redistribution, and stress alleviation.

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MUSCH, B. U., and A. PROKUDIN. "(BESSEL-)WEIGHTED ASYMMETRIES." International Journal of Modern Physics: Conference Series 04 (January 2011): 126–34. http://dx.doi.org/10.1142/s2010194511001632.

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Semi-inclusive deep inelastic scattering experiments allow us to probe the motion of quarks inside the proton in terms of so-called transverse momentum dependent parton distribution functions (TMD PDFs), but the information is convoluted with fragmentation functions (TMD FFs) and soft factors. It has long been known that weighting the measured event counts with powers of the hadron momentum before forming angular asymmetries de-convolutes TMD PDFs and TMD FFs in an elegant way, but this also entails an undesirable sensitivity to high momentum contributions. Using Bessel functions as weights, we find a natural generalization of weighted asymmetries that preserves the de-convolution property and features soft-factor cancellation, yet allows us to be less sensitive to high transverse momenta. The formalism also relates to TMD quantities studied in lattice QCD. We briefly show preliminary lattice results from an exploratory calculation of the Boer-Mulders shift using lattices generated by the MILC and LHP collaborations at a pion mass of 500 MeV.

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Aoki, Hideo, Takahiro Fukui, and Yasuhiro Hatsugai. "Topological Aspects of Quantum Hall Effect in Graphene." International Journal of Modern Physics B 21, no. 08n09 (April 10, 2007): 1133–39. http://dx.doi.org/10.1142/s0217979207042562.

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We study the recently observed quantum Hall effect (QHE) in graphene from a theoretical viewpoint of topological nature of the QHE to pose questions: (i) The zero-mass Dirac dispersion, which is the origin of the anomalous QHE, exists only around the zero gap, so a natural question is what happens to the QHE topological numbers over the entire energy spectrum. (ii) How the property that the bulk QHE topological number is equal to the edge QHE topological number, shown for the ordinary QHE, applies to the honeycomb lattice. We have shown that (a) the anomalous QHE ∝ (2N + 1) persists, surprisingly, all the way up to the van-Hove singularities, at which the normal behaviour abruptly takes over. (b) The edge-bulk correspondence persists as shown from the result for finite systems. All these properties hold for the entire sequence of lattice Hamiltonians that interpolate between square↔honeycomb↔ π-flux lattices, so the anomalous QHE is on a quantum critical line.

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34

Mahalank, Pushpalatha, Seyma Ozon Yildirim, Fikriye Ersoy Zihni, Bhairaba Kumar Majhi, and Ismail Naci Cangul. "Some topological indices of pentagonal double chains." ITM Web of Conferences 49 (2022): 01004. http://dx.doi.org/10.1051/itmconf/20224901004.

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In graph theory, lattices are used when some structural part of the graph repeats itself finitely or infinitely many times. They have applications in complex analysis and geometry in mathematics, and also natural applications in chemical graph theory. As a lattice can be taken as a graph, it is also possible to use them in the study of large networks. Recently, some topological graph indices of pentagonal chains is studied and here, we study some topological graph indices of pentagonal double chains similarly to that work. We make use of the vertex and edge partitions of these graphs and calculate their indices by means of these partitions and combinatorial methods.

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35

Choi, Seok-Ki, and Seong-O. Kim. "A STUDY ON THE CHOICE OF THERMAL MODELS IN THE COMPUTATION OF NATURAL CONVECTION WITH THE LATTICE BOLTZMANN METHOD." Journal of computational fluids engineering 16, no. 4 (December 31, 2011): 7–13. http://dx.doi.org/10.6112/kscfe.2011.16.4.007.

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36

Tang, Juan, Yong Fang, and Jian-Gang Tang. "The Lattice-Valued Turing Machines and the Lattice-Valued Type 0 Grammars." Mathematical Problems in Engineering 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/291870.

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Purpose.The purpose of this paper is to study a class of the natural languages called the lattice-valued phrase structure languages, which can be generated by the lattice-valued type 0 grammars and recognized by the lattice-valued Turing machines.Design/Methodology/Approach.From the characteristic of natural language, this paper puts forward a new concept of the l-valued Turing machine. It can be used to characterize recognition, natural language processing, and dynamic characteristics.Findings.The mechanisms of both the generation of grammars for the lattice-valued type 0 grammar and the dynamic transformation of the lattice-valued Turing machines were given.Originality/Value.This paper gives a new approach to study a class of natural languages by using lattice-valued logic theory.

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37

Gao, Niushan, and Cosimo Munari. "Surplus-Invariant Risk Measures." Mathematics of Operations Research 45, no. 4 (November 2020): 1342–70. http://dx.doi.org/10.1287/moor.2019.1035.

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This paper presents a systematic study of the notion of surplus invariance, which plays a natural and important role in the theory of risk measures and capital requirements. So far, this notion has been investigated in the setting of some special spaces of random variables. In this paper, we develop a theory of surplus invariance in its natural framework, namely, that of vector lattices. Besides providing a unifying perspective on the existing literature, we establish a variety of new results including dual representations and extensions of surplus-invariant risk measures and structural results for surplus-invariant acceptance sets. We illustrate the power of the lattice approach by specifying our results to model spaces with a dominating probability, including Orlicz spaces, as well as to robust model spaces without a dominating probability, where the standard topological techniques and exhaustion arguments cannot be applied.

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38

Kamozina, Olesia. "Boolean Lattices of n-multiply ωσ-fibered Fitting Classes." Bulletin of Irkutsk State University. Series Mathematics 40 (2022): 34–48. http://dx.doi.org/10.26516/1997-7670.2022.40.34.

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Let N be the set of all natural numbers. Consider all definitions and results taking into account the partitioning of the area for determining satellites and directions. An arbitrary Fitting class is considered a 0-multiply fibered Fitting class; for n equal to or greater than 1, a Fitting class is said to be n-multiply fibered if it has at least one satellite f, all non-empty values which are (n-1)-multiply fibered Fitting classes. The main result of this work is a description of n-multiply fibered Fitting classes, for which the lattice of all n-multiply fibered Fitting subclasses is Boolean. It is shown that such classes are representable in the form of a direct decomposition of lattice atoms. In this article, direct decompositions of n-multiply fibered Fitting classes are studied in detail. The direction of these classes is the main one, and is taken from the segment between the directions of the complete and local Fitting classes. Particular results for n-multiply complete and n-multiply local Fitting classes are obtained as corollaries of the corresponding theorems. When proving the statements, the methods of counter inclusions and mathematical induction were used. The results obtained can be used in the further study of Boolean lattices of n–multiply fibered Fitting classes with directions from other intervals, as well as Stone lattices of n-multiply fibered Fitting classes

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39

Penn, Michael, Christopher Sadowski, and Gautam Webb. "Principal subspaces of twisted modules for certain lattice vertex operator algebras." International Journal of Mathematics 30, no. 10 (September 2019): 1950048. http://dx.doi.org/10.1142/s0129167x19500484.

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This is the third in a series of papers studying the vertex-algebraic structure of principal subspaces of twisted modules for lattice vertex operator algebras. We focus primarily on lattices [Formula: see text] whose Gram matrix contains only non-negative entries. We develop further ideas originally presented by Calinescu, Lepowsky, and Milas to find presentations (generators and relations) of the principal subspace of a certain natural twisted module for the vertex operator algebra [Formula: see text]. We then use these presentations to construct exact sequences involving this principal subspace, which give a set of recursions satisfied by the multigraded dimension of the principal subspace and allow us to find the multigraded dimension of the principal subspace.

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40

Sameshima, T., G. S. Henderson, P. M. Black, and K. A. Rodgers. "X-ray diffraction studies of vivianite, metavivianite, and barićite." Mineralogical Magazine 49, no. 350 (March 1985): 81–85. http://dx.doi.org/10.1180/minmag.1985.049.350.11.

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AbstractVivianite specimens from various world localities yield X-ray powder patterns of two types: one corresponds with that shown by synthetic Fe3(PO4)2· 8H2O and is not readily distinguished from that of barićite; the second shows reflections of monoclinic vivianite and triclinic metavivianite along with reflections of a bobierrite-type phase. The triclinic phase occurs as two twin-related lattices with twin plane 110 being the structural equivalent of 010 in the monoclinic phase. The relationship of the bobierrite-type lattice to the other two has not been established. The ternary pattern is produced by some coarse-grained vivianites on natural oxidation. Finer grained vivianites oxidise to an X-ray amorphous state without passing through a triclinic intermediate.

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41

Dasari, M. Rao. "Lattice variations in natural magnesium calcites." Thermochimica Acta 101 (June 1986): 385–87. http://dx.doi.org/10.1016/0040-6031(86)80069-7.

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42

Salis, M. "Lattice defects in natural α-spodumene." Il Nuovo Cimento D 17, no. 6 (June 1995): 649–51. http://dx.doi.org/10.1007/bf02484367.

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43

Andréka, Hajnal, Steven Givant, and István Németi. "The lattice of varieties of representable relation algebras." Journal of Symbolic Logic 59, no. 2 (June 1994): 631–61. http://dx.doi.org/10.2307/2275414.

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AbstractWe shall show that certain natural and interesting intervals in the lattice of varieties of representable relation algebras embed the lattice of all subsets of the natural numbers, and therefore must have a very complicated lattice-theoretic structure.

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44

Van Der Sman, R. G. M. "Lattice-Boltzmann Scheme for Natural Convection in Porous Media." International Journal of Modern Physics C 08, no. 04 (August 1997): 879–88. http://dx.doi.org/10.1142/s0129183197000758.

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A lattice-Boltzmann scheme for natural convection in porous media is developed and applied to the heat transfer problem of a 1000 kg potato packaging. The scheme has features new to the field of LB schemes. It is mapped on a orthorhombic lattice instead of the traditional cubic lattice. Furthermore the boundary conditions are formulated with a single paradigm based upon the particle fluxes. Our scheme is well able to reproduce (1) the analytical solutions of simple model problems and (2) the results from cooling down experiments with potato packagings.

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45

Nespolo, Massimo, Giovanni Ferraris, and Hiroshi Takeda. "Identification of two allotwins of mica polytypes in reciprocal space through the minimal rhombus unit." Acta Crystallographica Section B Structural Science 56, no. 4 (August 1, 2000): 639–47. http://dx.doi.org/10.1107/s0108768100002044.

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The X-ray investigation (precession method) of the Ruiz Peak oxybiotite, which is well known for the occurrence of a large number of polytypes and twins, revealed two complex diffraction patterns, which cannot be identified as long-period polytypes. These patterns are analysed in terms of the minimal rhombus, a geometrical asymmetric unit in reciprocal space which permits the decomposition of the composite reciprocal lattice of a twin or allotwin into the reciprocal lattices of the individuals. Both the recorded patterns correspond to a 1M–2M 1 allotwin: the relative rotation between the individuals is 120° in one case and 60° in the other. The geometrical criteria for evaluating the presence of twinning or allotwinning are analysed through these two natural examples.

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46

Priestley, H. A. "Varieties of distributive lattices with unary operations I." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 63, no. 2 (October 1997): 165–207. http://dx.doi.org/10.1017/s144678870000063x.

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AbstractA unified study is undertaken of finitely generated varieties HSP () of distributive lattices with unary operations, extending work of Cornish. The generating algebra () is assusmed to be of the form (P; ∧, ∨, 0, 1, {fμ}), where each fμ is an endomorphism or dual endomorphism of (P; ∧, ∨, 0, 1), and the Priestly dual of this lattice is an ordered semigroup N whose elements act by left multiplication to give the maps dual to the operations fμ. Duality theory is fully developed within this framework, into which fit many varieties arising in algebraic logic. Conditions on N are given for the natural and Priestley dualities for HSP () to be essentially the same, so that, inter alia, coproducts in HSP () are enriched D-coproducts.

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47

Idczak, Eligiusz, and Tomasz Strek. "Vibration Transmission Loss of Auxetic Lattices." Applied Mechanics and Materials 797 (November 2015): 282–89. http://dx.doi.org/10.4028/www.scientific.net/amm.797.282.

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The auxetic lattices are structures, which have the negative Poisson’s ratio. When material has negative Poisson’s ratio, has also auxetic properties - during process of stretching, are made wider and during compressing are made narrower. This structures are cellular and negative Poisson’s ratio is depending on the geometry of single auxetic cell. When geometry of the cell is slightly changed also Poisson’s ratio is different. Auxetics have attracted attention of researchers because of their superior dynamic properties. The lattice auxetic structures at one of their natural frequencies exhibit the deformed geometry. It’s can be exploit as resonance to optimization of the power required for the occurrence localized deformations. The dynamic behavior of auxetic and their transmission of the vibration, which is circumscribed by the parameter VTL (Vibration Transmission Loss) will be analyzed in this article.

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48

Hall, T. E. "Identities for existence varieties of regular semigroups." Bulletin of the Australian Mathematical Society 40, no. 1 (August 1989): 59–77. http://dx.doi.org/10.1017/s000497270000349x.

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A natural concept of variety for regular semigroups is introduced: an existence variety (or e-variety) of regular semigroups is a class of regular semigroups closed under the operations H, Se, P of taking all homomorphic images, regular subsernigroups and direct products respectively. Examples include the class of orthodox semigroups, the class of (regular) locally inverse semigroups and the class of regular E-solid semigroups. The lattice of e-varieties of regular semigroups includes the lattices of varieties of inverse semigroups and of completely regular semigroups. A Birkhoff-type theorem is proved, showing that each e-variety is determined by a set of identities: such identities are then given for many e-varieties. The concept is meaningful in universal algebra, and as for regular semigroups could give interesting results for e-varieties of regular rings.

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Mondal, Bittagopal, and Xianguo Li. "Effect of volumetric radiation on natural convection in a square cavity using lattice Boltzmann method with non-uniform lattices." International Journal of Heat and Mass Transfer 53, no. 21-22 (October 2010): 4935–48. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2010.05.052.

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50

Zhou, Yiqiang. "The lattice of natural classes of modules." Communications in Algebra 24, no. 5 (January 1996): 1637–48. http://dx.doi.org/10.1080/00927879608825660.

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