scholarly journals Lattice Point Enumeration on Block Reduced Bases

2015 ◽  
pp. 269-282 ◽  
Author(s):  
Michael Walter
Keyword(s):  
Lattice Point ◽  
Lattice Point Enumeration

scholarly journals Implementation of Direct Indexing and 2-V Golomb Coding of Lattice Vectors for Image Compression

2019 ◽  
Vol 8 (9) ◽  
pp. 1205-1210
Keyword(s):  
Image Compression ◽  
Vector Quantization ◽  
Compression Ratio ◽  
Lattice Point ◽  
Lattice Vector ◽  
Indexing Method ◽  
Index Assignment ◽  
Lattice Point Enumeration ◽  
Good Compression ◽  
Golomb Code

Indexing of code vectors is a most difficult task in lattice vector quantization. In this work we focus on the problem of efficient indexing and coding of indexes. Index assignment to the quantized lattice vectors is computed by direct indexing method, through which a vector can be represented by a scalar quantity which represents the index of that vector. This eliminates the need of calculating the prefix i.e. index of the radius ( R) or norm and suffix i.e. the index of the position of vector on the shell of radius R, also eliminates index assignment to the suffix based on lattice point enumeration or leader’s indexing . Two value golomb coding is used to enumerate indices of quantized lattice vectors. We use analytical means to emphasize the dominance of two value golomb code over one value golomb code. This method is applied to achieve image compression. Indexes of particular subband of test images like barbara, peppers and boat are coded using 2-value golomb coding (2-V GC) and compression ratio is calculated. We demonstrate the effectiveness of the 2-V GC while the input is scanned columnwise as compare to rowwise. Experimentally we also show that good compression ratio is achieved when only higher order bits of the indexes are encoded instead of complete bits

scholarly journals GOLDIE RANK OF PRIMITIVE QUOTIENTS VIA LATTICE POINT ENUMERATION

2013 ◽  
Vol 55 (A) ◽  
pp. 149-168
Author(s):  
JOANNA MEINEL ◽  
CATHARINA STROPPEL
Keyword(s):  
Lattice Point ◽  
Characteristic Zero ◽  
Differential Operators ◽  
Convex Geometry ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Weyl Algebras ◽  
Primitive Ideals ◽  
Invariant Differential ◽  
Lattice Point Enumeration

AbstractLet k be an algebraically closed field of characteristic zero. I. M. Musson and M. Van den Bergh (Mem. Amer. Math. Soc., vol. 136, 1998, p. 650) classify primitive ideals for rings of torus invariant differential operators. This classification applies in particular to subquotients of localized extended Weyl algebras $\mathcal{A}_{r,n-r}=k[x_1,\ldots,x_r,x_{r+1}^{\pm1}, \ldots, x_{n}^{\pm1},\partial_1,\ldots,\partial_n],$ where it can be made explicit in terms of convex geometry. We recall these results and then turn to the corresponding primitive quotients and study their Goldie ranks. We prove that the primitive quotients fall into finitely many families whose Goldie ranks are given by a common quasi-polynomial and then realize these quasi-polynomials as Ehrhart quasi-polynomials arising from convex geometry.

A New Condition of Formation and Stablity of All Crystalline Systems in a Good Agreement with Experimental Data

2015 ◽  
Vol 11 (3) ◽  
pp. 3224-3228
Author(s):  
Tarek El-Ashram
Keyword(s):  
Experimental Data ◽  
Brillouin Zone ◽  
Electron Concentration ◽  
Free Electron ◽  
Lattice Point ◽  
Valence Electron ◽  
Fermi Gas ◽  
Valence Electron Concentration ◽  
Fermi Sphere ◽  
Good Agreement

In this paper we derived a new condition of formation and stability of all crystalline systems and we checked its validity andit is found to be in a good agreement with experimental data. This condition is derived directly from the quantum conditionson the free electron Fermi gas inside the crystal. The new condition relates both the volume of Fermi sphere VF andvolume of Brillouin zone VB by the valence electron concentration VEC as ;𝑽𝑭𝑽𝑩= 𝒏𝑽𝑬𝑪𝟐for all crystalline systems (wheren is the number of atoms per lattice point).

On lattice point counting in $$\varDelta $$-modular polyhedra

2021 ◽  
Author(s):  
D. V. Gribanov ◽  
N. Yu. Zolotykh
Keyword(s):  
Lattice Point ◽  
Point Counting

scholarly journals Bounds on the Lattice Point Enumerator via Slices and Projections

2021 ◽  
Author(s):  
Ansgar Freyer ◽  
Martin Henk
Keyword(s):  
Convex Body ◽  
Lattice Point ◽  
Geometric Mean ◽  
Discrete Analogue ◽  
Lattice Points ◽  
General Context ◽  
General Bound

AbstractGardner et al. posed the problem to find a discrete analogue of Meyer’s inequality bounding from below the volume of a convex body by the geometric mean of the volumes of its slices with the coordinate hyperplanes. Motivated by this problem, for which we provide a first general bound, we study in a more general context the question of bounding the number of lattice points of a convex body in terms of slices, as well as projections.

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