scholarly journals Crystal structures and continued fractions

2021 ◽  
Vol 2099 (1) ◽  
pp. 012012
Author(s):  
L V Pekhtereva ◽  
V A Seleznev
Keyword(s):  
Crystal Structure ◽  
Crystal Structures ◽  
Continued Fractions ◽  
Matrix Representation ◽  
Integer Lattice ◽  
Natural Numbers ◽  
Matrix Representations ◽  
The Matrix

Abstract In this paper, we consider the properties of a flat crystal structure associated with the matrix representation of finite continued fractions generating unimodular morphisms of a flat integer lattice. The used matrix representations of the continued fractions and their properties are obtained in [1]. The constructed model allows us to explain the existing limitations of the sets of Weiss parameters (the rational ratio of the lengths of the edges of the forming cell) of crystals by the Gauss-Kuzmin distribution of natural numbers in the representation of continued fractions.

scholarly journals Hashing Based Answer Selection

2020 ◽  
Vol 34 (05) ◽  
pp. 9330-9337
Author(s):  
Dong Xu ◽  
Wu-Jun Li
Keyword(s):  
Question Answering ◽  
Matrix Representation ◽  
Matrix Representations ◽  
Binary Matrix ◽  
Large Memory ◽  
The Matrix ◽  
Novel Method ◽  
Online Prediction ◽  
Memory Cost ◽  
Vector Representations

Answer selection is an important subtask of question answering (QA), in which deep models usually achieve better performance than non-deep models. Most deep models adopt question-answer interaction mechanisms, such as attention, to get vector representations for answers. When these interaction based deep models are deployed for online prediction, the representations of all answers need to be recalculated for each question. This procedure is time-consuming for deep models with complex encoders like BERT which usually have better accuracy than simple encoders. One possible solution is to store the matrix representation (encoder output) of each answer in memory to avoid recalculation. But this will bring large memory cost. In this paper, we propose a novel method, called hashing based answer selection (HAS), to tackle this problem. HAS adopts a hashing strategy to learn a binary matrix representation for each answer, which can dramatically reduce the memory cost for storing the matrix representations of answers. Hence, HAS can adopt complex encoders like BERT in the model, but the online prediction of HAS is still fast with a low memory cost. Experimental results on three popular answer selection datasets show that HAS can outperform existing models to achieve state-of-the-art performance.

Matrix representations of right ample semigroups

2015 ◽  
Vol 08 (03) ◽  
pp. 1550042 ◽  
Author(s):  
Junying Guo ◽  
Xiaojiang Guo ◽  
K. P. Shum
Keyword(s):  
Matrix Representation ◽  
Matrix Representations ◽  
Ample Semigroup ◽  
The Matrix

The properties of right ample semigroups have been extensively considered and studied by many authors. In this paper, we concentrate on the matrix representations of right ample semigroups. The (left; right) uniform matrix representation is initially defined. After some properties of left uniform matrix representations of a right ample semigroup are given, we prove that any irreducible left uniform representations of a right ample semigroup can be obtained by using an irreducible left uniform representation of some primitive right ample semigroup. In particular, a construction theorem of prime left uniform representation of right ample semigroups is established.

scholarly journals Eigenvalues of matrices related to the octonions

4open ◽  
2019 ◽  
Vol 2 ◽  
pp. 16
Author(s):  
Rogério Serôdio ◽  
Patricia Beites ◽  
José Vitória
Keyword(s):  
Matrix Representation ◽  
Real Matrix ◽  
Matrix Representations ◽  
Estimation Algorithms ◽  
The Matrix

A pseudo real matrix representation of an octonion, which is based on two real matrix representations of a quaternion, is considered. We study how some operations defined on the octonions change the set of eigenvalues of the matrix obtained if these operations are performed after or before the matrix representation. The established results could be of particular interest to researchers working on estimation algorithms involving such operations.

scholarly journals Isomorphism and matrix representation of point groups

2019 ◽  
Vol 15 (1) ◽  
pp. 88-92
Author(s):  
Wan Heng Fong ◽  
Aqilahfarhana Abdul Rahman ◽  
Nor Haniza Sarmin
Keyword(s):  
Group Theory ◽  
Matrix Representation ◽  
Point Group ◽  
Multiplication Table ◽  
Matrix Representations ◽  
The Matrix ◽  
Point Groups ◽  
Complete Set ◽  
Symmetry Of Molecules ◽  
Symmetry Operations

In chemistry, point group is a type of group used to describe the symmetry of molecules. It is a collection of symmetry elements controlled by a form or shape which all go through one point in space, which consists of all symmetry operations that are possible for every molecule. Next, a set of number or matrices which assigns to the elements of a group and represents the multiplication of the elements is said to constitute representation of a group. Here, each individual matrix is called a representative that corresponds to the symmetry operations of point groups, and the complete set of matrices is called a matrix representation of the group. This research was aimed to relate the symmetry in point groups with group theory in mathematics using the concept of isomorphism, where elements of point groups and groups were mapped such that the isomorphism properties were fulfilled. Then, matrix representations of point groups were found based on the multiplication table where symmetry operations were represented by matrices. From this research, point groups of order less than eight were shown to be isomorphic with groups in group theory. In addition, the matrix representation corresponding to the symmetry operations of these point groups wasis presented. This research would hence connect the field of mathematics and chemistry, where the relation between groups in group theory and point groups in chemistry were shown.

Structural/Property Relationships for Optical Materials

1993 ◽  
Vol 329 ◽  
Author(s):  
Vivien D.
Keyword(s):  
Crystal Structure ◽  
Optical Properties ◽  
Electronic Structure ◽  
Chemical Composition ◽  
Field Strength ◽  
Optical Materials ◽  
Site Occupancy ◽  
Laser Materials ◽  
Crystal Field Strength ◽  
The Matrix

AbstractIn this paper the relationships between the crystal structure, chemical composition and electronic structure of laser materials, and their optical properties are discussed. A brief description is given of the different laser activators and of the influence of the matrix on laser characteristics in terms of crystal field strength, symmetry, covalency and phonon frequencies. The last part of the paper lays emphasis on the means to optimize the matrix-activator properties such as control of the oxidation state and site occupancy of the activator and influence of its concentration.

Effect of solution acidity on the crystallization of polychromates in uranyl-bearing systems: synthesis and crystal structures of Rb2[(UO2)(Cr2O7)(NO3)2] and two new polymorphs of Rb2Cr3O10

2021 ◽  
Vol 236 (1-2) ◽  
pp. 11-21
Author(s):  
Evgeny V. Nazarchuk ◽  
Oleg I. Siidra ◽  
Dmitry O. Charkin ◽  
Stepan N. Kalmykov ◽  
Elena L. Kotova
Keyword(s):  
Crystal Structure ◽  
Aqueous Solutions ◽  
Crystal Structures ◽  
Hydrothermal Treatment ◽  
Ambient Conditions ◽  
Screw Axis ◽  
3D Framework ◽  
Solution Acidity

Abstract Three new rubidium polychromates, Rb2[(UO2)(Cr2O7)(NO3)2] (1), γ-Rb2Cr3O10 (2) and δ-Rb2Cr3O10 (3) were prepared by combination of hydrothermal treatment at 220 °C and evaporation of aqueous solutions under ambient conditions. Compound 1 is monoclinic, P 2 1 / c $P{2}_{1}/c$ , a = 13.6542(19), b = 19.698(3), c = 11.6984(17) Å, β = 114.326(2)°, V = 2867.0(7) Å3, R 1 = 0.040; 2 is hexagonal, P 6 3 / m $P{6}_{3}/m$ , a = 11.991(2), c = 12.828(3) Å, γ = 120°, V = 1597.3(5) Å3, R 1 = 0.031; 3 is monoclinic, P 2 1 / n $P{2}_{1}/n$ , a = 7.446(3), b = 18.194(6), c = 7.848(3) Å, β = 99.953(9)°, V = 1047.3(7) Å3, R 1 = 0.037. In the crystal structure of 1, UO8 bipyramids and NO3 groups share edges to form [(UO2)(NO3)2] species which share common corners with dichromate Cr2O7 groups producing novel type of uranyl dichromate chains [(UO2)(Cr2O7)(NO3)2]2−. In the structures of new Rb2Cr3O10 polymorphs, CrO4 tetrahedra share vertices to form Cr3O10 2− species. The trichromate groups are aligned along the 63 screw axis forming channels running in the ab plane in the structure of 2. The Rb cations reside between the channels and in their centers completing the structure. The trichromate anions are linked by the Rb+ cations into a 3D framework in the structure of 3. Effect of solution acidity on the crystallization of polychromates in uranyl-bearing systems is discussed.

scholarly journals Crystal Chemical Relations in the Shchurovskyite Family: Synthesis and Crystal Structures of K2Cu[Cu3O]2(PO4)4 and K2.35Cu0.825[Cu3O]2(PO4)4

Crystals ◽  
2021 ◽  
Vol 11 (7) ◽  
pp. 807
Author(s):  
Ilya V. Kornyakov ◽  
Sergey V. Krivovichev
Keyword(s):  
Crystal Structure ◽  
Crystal Structures ◽  
Structural Complexity ◽  
Crystal Chemical ◽  
X Ray Diffraction ◽  
Complexity Measures ◽  
X Ray ◽  
The Family ◽  
Similarities And Differences ◽  
Related Compounds

Single crystals of two novel shchurovskyite-related compounds, K2Cu[Cu3O]2(PO4)4 (1) and K2.35Cu0.825[Cu3O]2(PO4)4 (2), were synthesized by crystallization from gaseous phase and structurally characterized using single-crystal X-ray diffraction analysis. The crystal structures of both compounds are based upon similar Cu-based layers, formed by rods of the [O2Cu6] dimers of oxocentered (OCu4) tetrahedra. The topologies of the layers show both similarities and differences from the shchurovskyite-type layers. The layers are connected in different fashions via additional Cu atoms located in the interlayer, in contrast to shchurovskyite, where the layers are linked by Ca2+ cations. The structures of the shchurovskyite family are characterized using information-based structural complexity measures, which demonstrate that the crystal structure of 1 is the simplest one, whereas that of 2 is the most complex in the family.

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