Dedekind Sums, the Building Blocks of Lattice-Point Enumeration

2015 ◽  
pp. 149-165
Author(s):  
Matthias Beck ◽  
Sinai Robins
Keyword(s):  
Lattice Point ◽  
Building Blocks ◽  
Dedekind Sums ◽  
Lattice Point Enumeration

scholarly journals Franel integrals of order four

1996 ◽  
Vol 60 (2) ◽  
pp. 192-203 ◽  
Author(s):  
Richard J. McIntosh
Keyword(s):  
Lattice Point ◽  
Elementary Proof ◽  
Unit Interval ◽  
Dedekind Sums ◽  
Dedekind Sum ◽  
Elementary Method ◽  
The Third ◽  
Reciprocity Law ◽  
Order Four

AbstractLet ((x)) =x−⌊x⌋−1/2 be the swatooth function. Ifa, b, cand e are positive integeral, then the integral or ((ax)) ((bx)) ((cx)) ((ex)) over the unit interval involves Apolstol's generalized Dedekind sums. By expressing this integral as a lattice-point sum we obtain an elementary method for its evaluation. We also give an elementary proof of the reciprocity law for the third generalized Dedekind sum.

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.

Photochemical Evolution of Interstellar/Precometary Organic Material

1997 ◽  
Vol 161 ◽  
pp. 23-47 ◽  
Author(s):  
Louis J. Allamandola ◽  
Max P. Bernstein ◽  
Scott A. Sandford
Keyword(s):  
Origin Of Life ◽  
Building Blocks ◽  
Complex Species ◽  
Infrared Observations ◽  
Interstellar Ice ◽  
Polycyclic Aromatic ◽  
Ultraviolet Photolysis ◽  
The Origin Of Life ◽  
Simple Molecules ◽  
Complex Organic

AbstractInfrared observations, combined with realistic laboratory simulations, have revolutionized our understanding of interstellar ice and dust, the building blocks of comets. Since comets are thought to be a major source of the volatiles on the primative earth, their organic inventory is of central importance to questions concerning the origin of life. Ices in molecular clouds contain the very simple molecules H2O, CH3OH, CO, CO2, CH4, H2, and probably some NH3and H2CO, as well as more complex species including nitriles, ketones, and esters. The evidence for these, as well as carbonrich materials such as polycyclic aromatic hydrocarbons (PAHs), microdiamonds, and amorphous carbon is briefly reviewed. This is followed by a detailed summary of interstellar/precometary ice photochemical evolution based on laboratory studies of realistic polar ice analogs. Ultraviolet photolysis of these ices produces H2, H2CO, CO2, CO, CH4, HCO, and the moderately complex organic molecules: CH3CH2OH (ethanol), HC(= O)NH2(formamide), CH3C(= O)NH2(acetamide), R-CN (nitriles), and hexamethylenetetramine (HMT, C6H12N4), as well as more complex species including polyoxymethylene and related species (POMs), amides, and ketones. The ready formation of these organic species from simple starting mixtures, the ice chemistry that ensues when these ices are mildly warmed, plus the observation that the more complex refractory photoproducts show lipid-like behavior and readily self organize into droplets upon exposure to liquid water suggest that comets may have played an important role in the origin of life.

Laboratory and in-situ analysis of comet dust

1988 ◽  
Vol 46 ◽  
pp. 8-9
Author(s):  
D.E. Brownlee ◽  
A.L. Albee
Keyword(s):  
Electron Microscopy ◽  
Solar System ◽  
Laboratory Study ◽  
Isotopic Analysis ◽  
Building Blocks ◽  
Sem Analysis ◽  
Early Solar System ◽  
Cometary Material ◽  
Comet Dust

Comets are primitive, kilometer-sized bodies that formed in the outer regions of the solar system. Composed of ice and dust, comets are generally believed to be relic building blocks of the outer solar system that have been preserved at cryogenic temperatures since the formation of the Sun and planets. The analysis of cometary material is particularly important because the properties of cometary material provide direct information on the processes and environments that formed and influenced solid matter both in the early solar system and in the interstellar environments that preceded it.The first direct analyses of proven comet dust were made during the Soviet and European spacecraft encounters with Comet Halley in 1986. These missions carried time-of-flight mass spectrometers that measured mass spectra of individual micron and smaller particles. The Halley measurements were semi-quantitative but they showed that comet dust is a complex fine-grained mixture of silicates and organic material. A full understanding of comet dust will require detailed morphological, mineralogical, elemental and isotopic analysis at the finest possible scale. Electron microscopy and related microbeam techniques will play key roles in the analysis. The present and future of electron microscopy of comet samples involves laboratory study of micrometeorites collected in the stratosphere, in-situ SEM analysis of particles collected at a comet and laboratory study of samples collected from a comet and returned to the Earth for detailed study.

Cryo-TEM of amphiphilic polymer and amphiphile/polymer solutions

1993 ◽  
Vol 51 ◽  
pp. 876-877
Author(s):  
Yeshayahu Talmon
Keyword(s):  
Microstructural Characterization ◽  
Building Blocks ◽  
Quantitative Information ◽  
Physical Models ◽  
Specimen Preparation ◽  
Time Resolved ◽  
Temperature Changes ◽  
Temperature And Humidity ◽  
Microstructured Fluids ◽  
Air Streams

To achieve complete microstructural characterization of self-aggregating systems, one needs direct images in addition to quantitative information from non-imaging, e.g., scattering or Theological measurements, techniques. Cryo-TEM enables us to image fluid microstructures at better than one nanometer resolution, with minimal specimen preparation artifacts. Direct images are used to determine the “building blocks” of the fluid microstructure; these are used to build reliable physical models with which quantitative information from techniques such as small-angle x-ray or neutron scattering can be analyzed.To prepare vitrified specimens of microstructured fluids, we have developed the Controlled Environment Vitrification System (CEVS), that enables us to prepare samples under controlled temperature and humidity conditions, thus minimizing microstructural rearrangement due to volatile evaporation or temperature changes. The CEVS may be used to trigger on-the-grid processes to induce formation of new phases, or to study intermediate, transient structures during change of phase (“time-resolved cryo-TEM”). Recently we have developed a new CEVS, where temperature and humidity are controlled by continuous flow of a mixture of humidified and dry air streams.

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