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# Topic: Arithmetic lattice

###### In the News (Sun 19 May 13)

 Distributed Arithmetic FIR (DA-FIR) The Lattice Distributed Arithmetic Finite Impulse Response (DA-FIR) Filter Generator IP core implements a highly-configurable multi-channel DA-FIR Filter, using distributed arithmetic algorithms implemented in FPGA Look-Up Tables (LUTs) or Embedded Block Memory (EBR) to efficiently support the sum-of-product calculations required to perform the filter function. As a result, the DA-FIR Filter Generator IP core is extremely useful for implementing custom DSP blocks in Lattice FPGAs. Please refer to the user's guide to determine which cores are available for each device family. www.latticesemi.com /products/intellectualproperty/ipcores/distributedarithmeticfird.cfm   (328 words)

 Lattice (order) - Wikipedia, the free encyclopedia Lattices constitute one of the most prominent representatives of a series of "lattice-like" structures which admit order-theoretic as well as algebraic descriptions, such as semilattices, Heyting algebras, or Boolean algebras. The set of compact elements of an arithmetic complete lattice is a lattice with a least element, where the lattice operations are given by restricting the respective operations of the arithmetic lattice. These conditions basically amount to saying that there is a functor from the category of sets and functions to the category of lattices and lattice homomorphisms which is left adjoint to the forgetful functor from lattices to their underlying sets. en.wikipedia.org /wiki/Lattice_(order)   (2281 words)

 Lattice (order) - free-definition   (Site not responding. Last check: 2007-10-15) In mathematics, a lattice is a partially ordered set in which all nonempty finite subsets have both a supremum (join) and an infimum (meet). This lattice has the empty set as least element, but it will only contain a greatest element if A itself is finite. The lattice of all equivalence relations on a set; the equivalence relation ~ is considered to be smaller (or "finer") than ≈ if x~y always implies x≈y. www.free-definition.com /Lattice-%28order%29.html   (2682 words)

 Other Algorithms Lattice multiplication and variations of today's standard long multiplication were introduced into Europe by Fibonacci (whose correct name is Leonardo of Pisa). In lattice multiplication, the partial products are laid out in a lattice and adding along the diagonals gives the answer to the multiplication. As 28 and 57 have two digits each a lattice is set out with two columns and two rows. online.edfac.unimelb.edu.au /485129/wnproj/multiply/lattice.htm   (875 words)

 Lattice (order) - Wikipedia, the free encyclopedia ), a homomorphism of lattices is a function f : L → M with the properties that For example, the lattice of submodules of a module and the lattice of normal subgroups of a group have this special property. One of the consequences of this statement is that the free lattice of a three element set of generators is already infinite. en.wikipedia.org /wiki/Meet   (2281 words)

 Multiplication of Sets A space on which two operations are defined in a way that reminds us of the intersection and union of sets is known as a lattice. Lattices have been introduced by the German mathematician J.W.R.Dedekind(1831-1916) along with his invention of ideals in rings. The word "lattice" was first circulated by the american G.D.Birkhoff (1884-1944) in 1930s. www.cut-the-knot.com /do_you_know/mul_set.shtml   (415 words)

 The Jewel of Arithmetic: Quadratic Reciprocity To prove this theorem, consider the rectangular region of the xy plane where x ranges from 1/2 to p/2 and y ranges from 1/2 to q/2. At each integer value of x from 1 to p-1 the region in between these two lines has a height of 1, so it contains precisely one lattice point whose y component is the nearest integer to qx/p. It's clear that this has a distinct value on each of these lattice points, because if there was another lattice point (X,Y[X]) in the same region such that py-qx www.mathpages.com /home/kmath075.htm   (1007 words)

 Space group classification: Massimo Nespolo's Research Themes - LCM3B UHP Nancy 1   (Site not responding. Last check: 2007-10-15) A space group is characterized not only by the translational and point symmetry, but also by the metric of the lattice (in other words, the cell parameters enter in the definition of space group). An arithmetic crystal class is indicated by the symbol of the corresponding crystallographic point group, followed by the symbol of the lattice. But when the metric symmetry of the lattice is accidentally higher than the symmetry normally corresponding to the space group, we lack one information, i.e. www.lcm3b.uhp-nancy.fr /lcm3b/Pages_Perso/Nespolo/spacegroups.htm   (1110 words)

 [No title] The results of free lattices also mean that an "infinity" of cases may be ignored in applying probability measure, and for relational datbase querying (as in Oracle). The distributive lattice was created by Richard Dedekind (1831-1916) to explicate arithmetical factor theory of (positive) integers, since the complemented distributive lattice cannot. The complement in a distributive lattice is isomorphic to exception in logic, since, for example, to say that 4 an 15 are mutually complementary in the lattice on factors of 60, is to say that each contains 60 except for the other. members.fortunecity.com /jonhays/mixmatch.htm   (1845 words)

 Number Theory at the University of Natural Resources (Vienna)   (Site not responding. Last check: 2007-10-15) Nowadays, there is a small but efficient group working on topics like the theory of lattice points in large regions, the asymptotic behaviour of arithmetic functions, estimations of exponential sums, uniform distribution of sequences, Diophantine approximation, etc.: His research concentrates on the theory of lattice points in large regions and on the asymptotic theory of arithmetic functions. Primitive lattice points in a thin strip along the boundary of a convex planar domain (with E. Krätzel). www.boku.ac.at /math/nth.html   (807 words)

 FPGA-Based FIR Filters using Distributed Arithmetic   (Site not responding. Last check: 2007-10-15) Cascade and lattice structures present several interesting properties such as low quantification error and high-stability in the filter coefficients. The goal of this article is to implement FPGA-based direct-form, cascade, and lattice high-order FIR filters using bit-serial DA. You use the recursive equations that describe the lattice cell structures (Equation 7) to obtain cascade implementations of M cells. www.techonline.com /community/related_content/20135?print   (1291 words)

 Trinary Logic Geometric Series The arithmetic value of the horizontal leg b = 1 + [2(a - 1)/3] and that of the hypotenuse c = b + [(a - b)/2]. The constant ratio of the total arithmetic value of the perimeter of the infinitely expanding triangle to the total number of logical digits occupying it is, at the point defined by the longest edge of each successive edge-sharing triangle of 1's, At infinity the ratio of the total arithmetic value of any two larger and smaller successive groups of digits lying between edge-sharing triangles of 1's is 16 to 1, the ratio of the successive edge-sharing triangles of 1's beginning with 1. dgleahy.com /dgl/p05.html   (3450 words)

 [No title] The LLL algorithm takes a basis of a lattice and returns a new basis of the lattice which is em LLL-reduced which usually means that the vectors of the new basis have small norms. Enumeration of all shortest vectors of a lattice The genus of the 12-dimensional Coxeter-Todd lattice K_12 is enumerated in 16 seconds and has 16 classes of lattices and mass 4649359/4213820620800 approx 0.000001103359. www.math.niu.edu /help/math/magmahelp/text71.html   (2483 words)

 Damien Stehlé 's Homepage   (Site not responding. Last check: 2007-10-15) Given an integer d-dimensional lattice basis with vectors of norm less than B in an n-dimensional space, L^3 outputs a so-called L^3-reduced basis in polynomial time O(d^5 n log^3 B), using arithmetic operations on integers of bit-length O(d log B). This worst-case complexity is problematic for lattices arising in cryptanalysis where d or/and log B are often large. This is a 35-dimensional lattice for which NTL's LLL_FP (with delta=0.99) saves the calculations by using extended precision. www.loria.fr /~stehle/FPLLL.html   (255 words)

 Damien Stehlé 's Homepage   (Site not responding. Last check: 2007-10-15) We study a greedy lattice basis reduction algorithm for the Euclidean norm, which is arguably the most natural lattice basis reduction algorithm, because it is a straightforward generalization of the well-known two-dimensional Gaussian algorithm. However, as soon as the lattice dimension is strictly higher than four, the output basis may not even reach the first minimum. Our analysis, based on geometric properties of low-dimensional lattices and in particular Vorono\"{\i} cells, arguably simplifies Semaev's analysis in dimensions two and three, unifies the cases of dimensions two, three and four, but breaks down in dimension five. www.loria.fr /~stehle/LOWDIM.html   (175 words)

 Lattices: Construction and Operations Dual of a lattice, dual quotient of a lattice Several interesting lattices are directly accessible inside Magma using standard constructions, e.g., root lattices and preimages of linear codes. For each lattice, a LLL-reduced basis for the lattice is computed and stored internally when necessary and subsequently used for many operations. magma.maths.usyd.edu.au /magma/Features/node167.html   (173 words)

 Arithmetic On Superelliptic Curves - Galbraith, Paulus, Smart (ResearchIndex)   (Site not responding. Last check: 2007-10-15) Abstract: This paper is concerned with algorithms for computing in the divisor class group of a nonsingular plane curve of the form y n = c(x) which has only one point at infinity. Divisors are represented as ideals and an ideal reduction algorithm based on lattice reduction is given. 3 Ideal arithmetic and infrastructure in purely cubic function.. citeseer.ist.psu.edu /332720.html   (559 words)

 Using Fuzzy Logic for Molecular Modeling Together, the arithmetic operators and trigonometric functions enable the modeling of chemical compounds and the calculation of fuzzy bond lengths and fuzzy bond angles. Typically, to model a chemical's structure, lattice parameters and a coordinates table (such as the Wyckoff coordinates for a chemical compound under examination) are needed. Additionally, by using fuzzy lattice parameters, it is possible to express other characteristics of the unit cell in imprecise terms, thereby providing researchers with possible ranges for characteristics of interest. www.tms.org /pubs/journals/JOM/9908/Ress/Ress-9908.html   (2109 words)

 cass 3   (Site not responding. Last check: 2007-10-15) a packing in which the discs are located on an arithmetic lattice) has density at most that of the hexagonal packing, as the figures illustrate. The density of a lattice packing is the ratio of the area of a disc to that of a fundamental parallelogram, and among all lattice packings with a given size disc the hexagonal lattice clearly minimizes the area of a suitable fundamental parallelogram. The fundamental parallelogram of this lattice is a square, and the density of the distribution of discs is the ratio of the area of a circle to its circumscribing square. www.math.sunysb.edu /~tony/whatsnew/column/pennies-1200/cass3.html   (454 words)

 [No title] It is convenient in the MILC implementation to have the result of message passing be a list of pointers to on-node data and data in a communication buffer. Subsequent lattice-wide arithmetic operations may then be done through pointers. All choices have the same names for entry points, so no changes should be required for the rest of the code when switching plug-ins. physics.bu.edu /~brower/SciDAC/doc/yatb.txt   (700 words)

 On Quaternions and Octonions, by John Conway and Derek Smith Three act on a lattice with square symmetry, and five act on a lattice with hexagonal symmetry. As a lattice, the resulting `Cayley integers' are just a reflected version of the Kirmse integers, so they are still well-packed. : the lattice of Kirmse integers is self-dual. math.ucr.edu /home/baez/octonions/conway_smith   (4511 words)

 [No title] New Haven, Connecticut, U.S.A. for his monumental contributions to algebra, in particular to the theory of lattices in semi-simple Lie groups, and striking applications of this to ergodic theory, representation theory, number theory, combinatorics, and measure theory, and At the center of Professor Gregory A. Margulis’s work lies his proof of the Selberg-Piatetskii-Shapiro Conjecture, affirming that lattices in higher rank Lie groups are arithmetic, a question whose origins date back to Poincaré. A third dramatic breakthrough came when Margulis showed that Kazhdan’s “Property T” (known to hold for rigid lattices) could be used in a single arithmetic lattice construction to solve two apparently unrelated problems. www.wolffund.org.il /full.asp?id=13   (596 words)

 Lattice %28order%29   (Site not responding. Last check: 2007-10-15) Fabbrica di materassi a molle, in lattice, ortopedici. Produzione e vendita di materassi in lattice e a molle. Azienda produttrice di materassi a molle e in lattice, presenta i prodotti realizzati e la mission aziendale. www.omniknow.com /common/wiki.php?in=en&term=Lattice_%28order%29   (730 words)

 Publications of the SPACES team   (Site not responding. Last check: 2007-10-15) Its arithmetic complexity is polynomial in the complexity of evaluation of the input system, an intrinsic algebraic quantity and a combinatorial quantity. It is written in C and is based on the MPFR library for arbitrary precision floating-point arithmetic, which is in turn built upon the GMP library. The key idea is to precise the relations between the successive Sylvester matrix of A and B in one hand and of A and XB on the other hand, using the notion of G-remainder we introduce. www-calfor.lip6.fr /~safey/Spaces/publications.html   (13078 words)

 info: LATTICE (ORDER)   (Site not responding. Last check: 2007-10-15) ), a homomorphism of lattices is a function f : L ? Using the standard definition of universal algebra, a free lattice over a generating set S is a lattice L together with a function i:S? L, such that any function f from S to the underlying set of some lattice M can be factored uniquely through a lattice homomorphism f° from L to M. www.digital-innovations.net /Lattice_(order)   (2268 words)

 RR-5387 : Floating-Point LLL Revisited   (Site not responding. Last check: 2007-10-15) Given an integer \$d\$-dimensional lattice basis which vectors have norms smaller than \$B\$, LLL outputs a so-called LLL-reduced basis in time \$O(d^6 \log^3 B)\$, using arithmetic operations on integers of bit-length \$O(d \logB)\$. This worst-case complexity is problematic for lattices arising in cryptanalysis where \$d\$ or/and \$\logB\$ are often large. Unfortunately, this is known to be unstable in the worst-case: the usual floating-point LLL is not even guaranteed to terminate, and the output basis may not be LLL-reduced at all. www.inria.fr /rrrt/rr-5387.html   (530 words)

 [No title] A lattice ideal has received much attention in abstract algebra, but little in arithmetic (especially factor theory) and in logic. We open this omission and highlight it's implications for the Gödel numbers which provide the coding to be interpreted as undecidability or incompleteness in arithmetic. A lattice ideal can easily be described in, and graphed by, a Hasse Diagram of a lattice: members.fortunecity.com /jonhays/ideal.htm   (197 words)

 ispXPLD 5000MX 5000MX family represents a new class of devices from Lattice Semiconductor called eXpanded Programmable Logic Devices (XPLDs). These devices are built around a new building block, the Multi-Function Block (MFB). The Lattice ispXPLD Evaluation Board is a complete hardware kit that allows the user to program, evaluate, and de-bug a design for the Lattice ispXPLD architecture. www.latticesemi.com /products/cpldspld/ispxpld5000mx.cfm?source=topnav   (209 words)

 Lattice method   (Site not responding. Last check: 2007-10-15) This method apply to determinants whose entries are integers with a binary representation of at most b-bits. If the determinant has dimension dxd, its sign is obtained without error using exact arithmetic operations on integer numbers having at most a b+ ceil(log d)-2 bit long representation. Thus if b+ ceil(log d)-2 is less than 53 (the bit length of IEEE standard doubles) all computations can be performed using the fast floating point processor). www.inria.fr /prisme/personnel/yvinec/Determinants/lattice.html   (116 words)

 Bibliography of Pavel Pudlak [7] P. Pudlak, P.P. Palfy: Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups, Algebra Universalis, Vol.11, 1980, pp.22-27. [15] P. Pudlak: On congruence lattices of lattices, Algebra Universalis Vol.20, 1985, pp.96-114. In: Model Theory of Algebra and Arithmetic, LNM 843, Springer-Verlag, 1980, pp.174-185. www.math.cas.cz /~pudlak/moje-bib.html   (1934 words)

 Citations: A pseudorandom vector generator based on finite field arithmetic - Niederreiter (ResearchIndex) An Implementation of the Lattice and Spectral Tests for.. of the quality of this lattice: a q t close to 1 means that its points are more evenly distributed [2, 3, 11] The lattice structure implies that all the points (x n ; x n t Gamma1) lie on a relatively small familly of equidistant parallel hyperplanes, especially for large values of t A pseudorandom vector generator based on finite field arithmetic, Mathematica Japonica, 31, 759--774. citeseer.ist.psu.edu /context/516174/0   (322 words)

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