scholarly journals Thirty-six Full Matrix Forms of the Pascal Triangle: Derivation and Symmetry Relations

2021 ◽  
pp. e00932
Author(s):  
Prosper K. Doh ◽  
Kondo H. Adjallah ◽  
Babiga Birregah
Keyword(s):  
Full Matrix ◽  
Pascal Triangle ◽  
Symmetry Relations

PENERAPAN METODE SEGITIGA PASCAL ALJABAR UNTUK MEMAKSIMALKAN TEKNIK PENYANDIAN ALGORITMA WORD AUTO KEY ENCRYPTION (WAKE)

2019 ◽  
Vol 3 (1) ◽  
Author(s):  
Kaldius Ndruru ◽  
Putri Ramadhani
Keyword(s):  
Data Security ◽  
Important Data ◽  
Pascal Triangle ◽  
Absolute Requirement ◽  
Cryptographic Algorithms ◽  
Secure Data ◽  
Triangle Method ◽  
Symmetric Key

Security of data stored on computers is now an absolute requirement, because every data has a high enough value for the user, reader and owner of the data itself. To prevent misuse of the data by other parties, data security is needed. Data security is the protection of data in a system against unauthorized authorization, modification, or destruction. The science that explains the ways of securing data is known as cryptography, while the steps in cryptography are called critical algorithms. At this time, there are many cryptographic algorithms whose keys are weak especially the symmetric key algorithm because they only have one key, the key for encryption is the same as the decryption key so it needs to be modified so that the cryptanalysts are confused in accessing important data. The cryptographic method of Word Auto Key Encryption (WAKE) is one method that has been used to secure data where in this case the writer wants to maximize the encryption key and description of the WAKE algorithm that has been processed through key formation. One way is to apply the algebraic pascal triangle method to maximize the encryption key and description of the WAKE algorithm, utilizing the numbers contained in the columns and rows of the pascal triangle to make shifts on the encryption key and the description of the WAKE algorithm.Keywords: Cryptography, WAKE, pascal

scholarly journals Ramanujan’s function k(τ)=r(τ)r 2(2τ) and its modularity

2020 ◽  
Vol 18 (1) ◽  
pp. 1727-1741
Author(s):  
Yoonjin Lee ◽  
Yoon Kyung Park
Keyword(s):  
Continued Fraction ◽  
Quadratic Field ◽  
Singular Values ◽  
Modular Function ◽  
Imaginary Quadratic Field ◽  
Modular Equations ◽  
Positive Rational Number ◽  
Congruence Relations ◽  
Symmetry Relations ◽  
Ray Class Field

Abstract We study the modularity of Ramanujan’s function k ( τ ) = r ( τ ) r 2 ( 2 τ ) k(\tau )=r(\tau ){r}^{2}(2\tau ) , where r ( τ ) r(\tau ) is the Rogers-Ramanujan continued fraction. We first find the modular equation of k ( τ ) k(\tau ) of “an” level, and we obtain some symmetry relations and some congruence relations which are satisfied by the modular equations; these relations are quite useful for reduction of the computation cost for finding the modular equations. We also show that for some τ \tau in an imaginary quadratic field, the value k ( τ ) k(\tau ) generates the ray class field over an imaginary quadratic field modulo 10; this is because the function k is a generator of the field of the modular function on Γ 1 ( 10 ) {{\mathrm{\Gamma}}}_{1}(10) . Furthermore, we suggest a rather optimal way of evaluating the singular values of k ( τ ) k(\tau ) using the modular equations in the following two ways: one is that if j ( τ ) j(\tau ) is the elliptic modular function, then one can explicitly evaluate the value k ( τ ) k(\tau ) , and the other is that once the value k ( τ ) k(\tau ) is given, we can obtain the value k ( r τ ) k(r\tau ) for any positive rational number r immediately.

scholarly journals Divergent ⇒ complex amplitudes in two dimensional string theory

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Ashoke Sen
Keyword(s):  
Scattering Amplitudes ◽  
String Theory ◽  
Matrix Model ◽  
Two Dimensional ◽  
Full Matrix ◽  
Instanton Contribution ◽  
The Matrix ◽  
Theory Result ◽  
The One ◽  
Remarkable Agreement

Abstract In a recent paper, Balthazar, Rodriguez and Yin found remarkable agreement between the one instanton contribution to the scattering amplitudes of two dimensional string theory and those in the matrix model to the first subleading order. The comparison was carried out numerically by analytically continuing the external energies to imaginary values, since for real energies the string theory result diverges. We use insights from string field theory to give finite expressions for the string theory amplitudes for real energies. We also show analytically that the imaginary parts of the string theory amplitudes computed this way reproduce the full matrix model results for general scattering amplitudes involving multiple closed strings.

scholarly journals An explicit construction of the dimension-9 operator basis in the standard model effective field theory

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Yi Liao ◽  
Xiao-Dong Ma
Keyword(s):  
Field Theory ◽  
Standard Model ◽  
Effective Field Theory ◽  
Equations Of Motion ◽  
Baryon Number ◽  
Explicit Construction ◽  
Effective Field ◽  
Starting Point ◽  
The Standard Model ◽  
Symmetry Relations

Abstract We investigate systematically dimension-9 operators in the standard model effective field theory which contains only standard model fields and respects its gauge symmetry. With the help of the Hilbert series approach to classifying operators according to their lepton and baryon numbers and their field contents, we construct the basis of operators explicitly. We remove redundant operators by employing various kinematic and algebraic relations including integration by parts, equations of motion, Schouten identities, Dirac matrix and Fierz identities, and Bianchi identities. We confirm counting of independent operators by analyzing their flavor symmetry relations. All operators violate lepton or baryon number or both, and are thus non-Hermitian. Including Hermitian conjugated operators there are $$ {\left.384\right|}_{\Delta B=0}^{\Delta L=\pm 2}+{\left.10\right|}_{\Delta B=\pm 2}^{\Delta L=0}+{\left.4\right|}_{\Delta B=\pm 1}^{\Delta L=\pm 3}+{\left.236\right|}_{\Delta B=\pm 1}^{\Delta L=\mp 1} $$ 384 Δ B = 0 Δ L = ± 2 + 10 Δ B = ± 2 Δ L = 0 + 4 Δ B = ± 1 Δ L = ± 3 + 236 Δ B = ± 1 Δ L = ∓ 1 operators without referring to fermion generations, and $$ {\left.44874\right|}_{\Delta B=0}^{\Delta L=\pm 2}+{\left.2862\right|}_{\Delta B=\pm 2}^{\Delta L=0}+{\left.486\right|}_{\Delta B=\pm 1}^{\Delta L=\pm 3}+{\left.42234\right|}_{\Delta B=\mp 1}^{\Delta L=\pm 1} $$ 44874 Δ B = 0 Δ L = ± 2 + 2862 Δ B = ± 2 Δ L = 0 + 486 Δ B = ± 1 Δ L = ± 3 + 42234 Δ B = ∓ 1 Δ L = ± 1 operators when three generations of fermions are referred to, where ∆L, ∆B denote the net lepton and baryon numbers of the operators. Our result provides a starting point for consistent phenomenological studies associated with dimension-9 operators.

Studies in aryltin chemistry. Part 6. Crystal and molecular structures of tetra(p-methylsulphonylphenyl)tin(IV) and tetra(p-methylsulphinylphenyl)tin(IV) monohydrate

1990 ◽  
Vol 68 (8) ◽  
pp. 1277-1282 ◽  
Author(s):  
Ivor Wharf ◽  
Michel G. Simard ◽  
Henry Lamparski
Keyword(s):  
Least Squares ◽  
Crystal Structures ◽  
Direct Method ◽  
Molecular Structures ◽  
Water Molecules ◽  
Monoclinic Space Group ◽  
Molecular Symmetry ◽  
Asymmetric Unit ◽  
Full Matrix ◽  
Space Group

Tetrakis(p-methylsulphonylphenyl)tin(IV) and tetrakis(p-methylsulphinylphenyl)tin(IV) n-hydrate have been prepared and their spectra (ir 1350–400 cm−1; nmr, 1H, 13C, 119Sn) and X-ray crystal structures are reported. The first compound is monoclinic, space group C2/c, Z = 4, with a = 21.589(6), b = 6.207(3), c = 22.861(11) Å, β = 93.80(3)° (22 °C); the structure was solved by the direct method and refined by full-matrix least squares calculations to R = 0.043 for 2755 observed reflections. It has 2 molecular symmetry with the methyl group and one oxygen atom completely disordered in both CH3S(O2) groups in the asymmetric unit. The second compound is tetragonal, space group P42/n, Z = 2, with a = b = 15.408(6), c = 6.379(2) Å (−100 °C); the structure was solved by the Patterson method and refined by full-matrix least squares calculations to R = 0.060 for 1209 observed reflections. It has [Formula: see text] molecular symmetry with the whole asymmetric unit disordered. Water molecules occupy positions on parallel 42 axes but molecular packing requirements prevent all sites having 100% occupancy giving n ~ 1 for the hydrate. Keywords: Tetra-aryltins, crystal structures, sulphone, sulphoxide, hydrogen-bonding.

The crystal structure of arsenic telluroxide AsTe0.5O2

1993 ◽  
Vol 205 (1) ◽  
Author(s):  
A. C. Stergiou
Keyword(s):  
Crystal Structure ◽  
Least Squares ◽  
Single Crystals ◽  
Direct Methods ◽  
Anomalous Dispersion ◽  
Thermal Parameters ◽  
Trigonal Prism ◽  
Population Parameter ◽  
Full Matrix

AbstractSingle crystals of AsTeSolution of the structure was essentialy effected by direct methods combined with successive Fourier syntheses. The positional and anisotropic thermal parameters were refined by full-matrix least-squares calculations. Absorption and anomalous dispersion corrections were applied to all atoms. The finalThe As atom is coordinated by six O atoms forming a right trigonal prism. The Te atom site is partially occupied by Te atoms with a population parameter 0.5 and surrounded by six O atoms also forming a right trigonal prism. The structure looks like that of NiAs. Each of the AsO

Refinement of the crystal structure of metatorbernite

1993 ◽  
Vol 205 (1) ◽  
pp. 1-7 ◽  
Author(s):  
A. C. Stergiou ◽  
P. J. Rentzeperis ◽  
S. Sklavounos
Keyword(s):  
Crystal Structure ◽  
Least Squares ◽  
Anomalous Dispersion ◽  
Thermal Parameters ◽  
Water Molecules ◽  
Full Matrix ◽  
Absorption Correction ◽  
Atom Position ◽  
Oxygen Atoms

AbstractThe crystal structure of metatorbernite with composition CuThe positional and thermal parameters were refined by full-matrix least-squares calculations. Absorption correction and correction for anomalous dispersion, for all atoms, were applied. The finalThe structure is essentially similar to that described by M. Ross, H. Evans and D. Appleman (1964) for metatorbernite, with a difference in the Cu atom position, which here is 1/4 1/4 0.31 instead of 1/4 1/4 0.80. The U atoms are six-coordinated by two O atoms (uranyl group) and four phosphate – oxygen atoms forming an asymmetrical tetragonal dipyramid. The Cu atoms are six-coordinated by two oxygen atoms of two different uranyl groups and four water molecules forming also an asymmetrical tetragonal dipyramid. The four water molecules form squares Cu(H

Structural Systematics of Rare Earth Complexes. VII. Crystal Structure of Bis(2,2'/6',2␛-Terpyridinium) Octaaquaterbium(III) Heptachloride Hydrate

1994 ◽  
Vol 47 (2) ◽  
pp. 391 ◽  
Author(s):  
CJ Kepert ◽  
BW Skeleton ◽  
AH White
Keyword(s):  
Crystal Structure ◽  
Rare Earth ◽  
Single Crystal ◽  
Structural Characterization ◽  
Title Compound ◽  
Room Temperature ◽  
Chloride Ions ◽  
Full Matrix ◽  
X Ray

The room-temperature single-crystal X-ray structural characterization of the title compound (tpyH2)2[Tb(OH2)8]Cl7.~2⅓H2O is recorded. Crystals are triclinic, Pī , a 17.063(5), b 16.243(3), c 7.878(3) Ǻ, α 84.78(2), β 84.39(3), γ 87.81(2)°, Z = 2 formula units; 3167 'observed' diffractometer reflections were refined by full-matrix least-squares procedures to a residual of 0.057. Notable features of interest of the compound are the 'chelation' of chloride ions by the terpyridinium cations , and the existence of a free [Tb(OH2)8]2+ cation in the presence of an abundance of chloride ions.

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