Genetic Code and Stochastic Matrices

2010 ◽  
pp. 91-114
Author(s):  
Sergey Petoukhov ◽  
Matthew He
Keyword(s):  
Genetic Code ◽  
Fractal Structure ◽  
Hamming Distance ◽  
Stochastic Matrix ◽  
Building Blocks ◽  
Stochastic Matrices ◽  
Frequency Distributions ◽  
Decomposition Formula ◽  
The Matrix ◽  
G 11

In this chapter, we first use the Gray code representation of the genetic code C = 00, U = 10, G = 11, and A = 01 (C pairs with G, A pairs with U) to generate a sequence of genetic code-based matrices. In connection with these code-based matrices, we use the Hamming distance to generate a sequence of numerical matrices. We then further investigate the properties of the numerical matrices and show that they are doubly stochastic and symmetric. We determine the frequency distributions of the Hamming distances, building blocks of the matrices, decomposition and iterations of matrices. We present an explicit decomposition formula for the genetic code-based matrix in terms of permutation matrices. Furthermore, we establish a relation between the genetic code and a stochastic matrix based on hydrogen bonds of DNA. Using fundamental properties of the stochastic matrices, we determine explicitly the decomposition formula of genetic code-based biperiodic table. By iterating the stochastic matrix, we demonstrate the symmetrical relations between the entries of the matrix and DNA molar concentration accumulation. The evolution matrices based on genetic code were derived by using hydrogen bondsbased symmetric stochastic (2x2)-matrices as primary building blocks. The fractal structure of the genetic code and stochastic matrices were illustrated in the process of matrix decomposition, iteration and expansion in corresponding to the fractal structure of the biperiodic table introduced by Petoukhov (2001a, 2001b, 2005).

Regular stochastic matrices and digraphs

1973 ◽  
Vol 10 (01) ◽  
pp. 241-243
Author(s):  
Elizabeth Berman
Keyword(s):  
Stochastic Matrix ◽  
Main Diagonal ◽  
Stochastic Matrices ◽  
The Matrix ◽  
Strongly Connected ◽  
Positive Entry

This paper presents an algorithm to determine whether a stochastic matrix is regular. The main theorem is the following. Hypothesis: An n-by-n stochastic matrix has at least one positive entry off the main diagonal in every row and column. There is at most one row with n — 1 zeros and at most one column with n — 1 zeros. There are no j-by-k submatrices consisting entirely of zeros, where j and k are integers greater than 1, with j + k = n. Conclusion: The matrix is regular. Similar results hold for strongly connected digraphs.

Log-Concavity and the Spectral Gap of Stochastic Matrices

1997 ◽  
Vol 6 (3) ◽  
pp. 371-379
Author(s):  
EKKEHARD WEINECK
Keyword(s):  
Characteristic Polynomial ◽  
Spectral Gap ◽  
Stochastic Matrix ◽  
Identity Matrix ◽  
Combinatorial Approach ◽  
Stochastic Matrices ◽  
Largest Eigenvalue ◽  
The Matrix ◽  
Second Largest Eigenvalue

Let Q be a stochastic matrix and I be the identity matrix. We show by a direct combinatorial approach that the coefficients of the characteristic polynomial of the matrix I−Q are log-concave. We use this fact to prove a new bound for the second-largest eigenvalue of Q.

scholarly journals The Inequalities of Merris and Foregger for Permanents

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1782
Author(s):  
Divya K. Udayan ◽  
Kanagasabapathi Somasundaram
Keyword(s):  
Symmetric Group ◽  
Linear Algebra ◽  
Symmetric Function ◽  
Stochastic Matrix ◽  
Elementary Symmetric Function ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Classes Of Matrices ◽  
The Matrix ◽  
Element Set

Conjectures on permanents are well-known unsettled conjectures in linear algebra. Let A be an n×n matrix and Sn be the symmetric group on n element set. The permanent of A is defined as perA=∑σ∈Sn∏i=1naiσ(i). The Merris conjectured that for all n×n doubly stochastic matrices (denoted by Ωn), nperA≥min1≤i≤n∑j=1nperA(j|i), where A(j|i) denotes the matrix obtained from A by deleting the jth row and ith column. Foregger raised a question whether per(tJn+(1−t)A)≤perA for 0≤t≤nn−1 and for all A∈Ωn, where Jn is a doubly stochastic matrix with each entry 1n. The Merris conjecture is one of the well-known conjectures on permanents. This conjecture is still open for n≥4. In this paper, we prove the Merris inequality for some classes of matrices. We use the sub permanent inequalities to prove our results. Foregger’s inequality is also one of the well-known inequalities on permanents, and it is not yet proved for n≥5. Using the concepts of elementary symmetric function and subpermanents, we prove the Foregger’s inequality for n=5 in [0.25, 0.6248]. Let σk(A) be the sum of all subpermanents of order k. Holens and Dokovic proposed a conjecture (Holen–Dokovic conjecture), which states that if A∈Ωn,A≠Jn and k is an integer, 1≤k≤n, then σk(A)≥(n−k+1)2nkσk−1(A). In this paper, we disprove the conjecture for n=k=4.

Regular stochastic matrices and digraphs

1973 ◽  
Vol 10 (1) ◽  
pp. 241-243
Author(s):  
Elizabeth Berman
Keyword(s):  
Stochastic Matrix ◽  
Main Diagonal ◽  
Stochastic Matrices ◽  
The Matrix ◽  
Strongly Connected ◽  
Positive Entry

This paper presents an algorithm to determine whether a stochastic matrix is regular. The main theorem is the following. Hypothesis: An n-by-n stochastic matrix has at least one positive entry off the main diagonal in every row and column. There is at most one row with n — 1 zeros and at most one column with n — 1 zeros. There are no j-by-k submatrices consisting entirely of zeros, where j and k are integers greater than 1, with j + k = n. Conclusion: The matrix is regular. Similar results hold for strongly connected digraphs.

An expansion for the permanent of a doubly stochastic matrix

1973 ◽  
Vol 15 (4) ◽  
pp. 504-509
Author(s):  
R. C. Griffiths
Keyword(s):  
Symmetric Group ◽  
Line Segment ◽  
Convex Set ◽  
Stochastic Matrix ◽  
Weighted Average ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Survey Article ◽  
The Matrix ◽  
Image Position

The permanent of an n-square matrix A = (aij) is defined by where Sn is the symmetric group of order n. Kn will denote the convex set of all n-square doubly stochastic matrices and K0n its interior. Jn ∈ Kn will be the matrix with all elements equal to 1/n. If M ∈ K0n, then M lies on a line segment passing through Jn and another B ∈ Kn — K0n. This note gives an expansion for the permanent of such a line segment as a weighted average of permanents of matrices in Kn. For a survey article on permanents the reader is referred to Marcus and Mine [3].

Exploration Study of Multifunctional Metallic Nanocomposite Utilizing Single-Walled Carbon Nanotubes for Micro/Nano Devices

2005 ◽  
Vol 219 (2) ◽  
pp. 67-72 ◽  
Author(s):  
Quanfang Chen ◽  
Guang Chai ◽  
Bo Li
Keyword(s):  
Carbon Nanotubes ◽  
Energy Transport ◽  
Metallic Matrix ◽  
Building Blocks ◽  
Copper Matrix ◽  
Interfacial Bonding ◽  
Single Walled Carbon ◽  
The Matrix ◽  
Raman Scattering Spectra ◽  
Multifunctional Properties

Carbon nanotubes (CNTs) are excellent multifunctional materials in terms of mechanical robustness, thermal, and electrical conductivities. These multifunctional properties, as well as the small size of the structures, make CNTs ideal building blocks in developing nanocomposites. However, the matrix materials and the fabrication processes are critical in achieving the expected multifunctional properties of a CNT-reinforced nanocomposite. This paper has proved that electrochemical co-deposition of a metallic nanocomposite is a good approach for achieving good interfacial bonding between CNTs and a metallic matrix. Good interfacial bonding between a single-walled carbon nanotube (SWCNT) and a copper matrix has been verified by enhanced fracture toughness (increased stickiness) and a shift in the Raman scattering spectra. For the Cu/SWCNT nanocomposite, the radial breath mode (RBM) has disappeared and the tangential or G-band has shifted and widened, which is an indication of better energy transport.

Mrzonki o pokoju. Wojna nemezis ludzkości

1970 ◽  
Vol 23 (301) ◽  
pp. 175-191
Author(s):  
Justyna Grażyna Otto
Keyword(s):  
Political Philosophy ◽  
Genetic Code ◽  
Human Life ◽  
Building Blocks ◽  
Human Being ◽  
Human Race ◽  
Political Sciences ◽  
The Federalist ◽  
History Of ◽  
Research Goal

The main thesis of the article is as follows: war shares a lasting and unbreakable bond with human life and state politics and, despite the Utopian dreams of never-ending peace, conflicts are the sine qua non building blocks of a country’s policy and the development of the human race. Why? Because a state is an imperfect, yet the most perfect among the civilisationally achievable means of organising the lives of people who, for various reasons, display war inclinations. This paper is a political sciences analysis of the problem from the perspective of history of political philosophy. To achieve the research goal, the author first analyses the Utopian visions of peace ruling over war and the designs of eternal peace, putting forward subtheses on the primate of peace over war and the human drive towards peace which was to be determined by 1). religion, 2). proper upbringing and education as well as the new organisation of a society and its reflection in the federalist project. These theses are debunked and proven to be wishful thinking as each human being and each state have violence in their genetic code.

Symmetries of the Degeneracy of the Vertebrate Mitochondrial Genetic Code in the Matrix Form

2010 ◽  
pp. 31-49
Author(s):  
Sergey Petoukhov ◽  
Matthew He
Keyword(s):  
Mathematical Modeling ◽  
Genetic Code ◽  
Matrix Form ◽  
Black And White ◽  
Mitochondrial Genetic Code ◽  
The Matrix

Symmetries of the degeneracy of the vertebrate mitochondrial genetic code in the mosaic matrix form of its presentation are described in this chapter. The initial black-and-white genomatrix of this code is reformed into a new mosaic matrix when internal positions in all triplets are permuted simultaneously. It is revealed unexpectedly that for all six variants of positional permutations in triplets (1-2-3, 2-3-1, 3-1-2, 1-3-2, 2-1-3, 3-2-1) the appropriate genetic matrices possess symmetrical mosaics of the code degeneracy. Moreover the six appropriate mosaic matrices in their binary presentation have the general non-trivial property of their “tetra-reproduction,” which can be utilized in particular for mathematical modeling of the phenomenon of the tetra-division of gametal cells in meiosis. Mutual interchanges of the genetic letters A, C, G, U in the genomatrices lead to new mosaic genomatrices, which possess similar symmetrical and tetra-reproduction properties as well.

The genetics of Huntington’s disease

2020 ◽  
pp. 44-53
Author(s):  
Oliver Quarrell
Keyword(s):  
Amino Acids ◽  
Genetic Code ◽  
Building Blocks ◽  
Repeat Expansion ◽  
Cag Repeat ◽  
Dna Molecule ◽  
Cag Repeat Expansion ◽  
Males And Females ◽  
Chemical Messenger ◽  
Hd Gene

This chapter describes the nature of the genetic mistake. The genetic code, or DNA molecule, is wound up onto structures called chromosomes. The gene for HD is located on chromosome 4. As we have two copies of our genes the chromosomes are in pairs. Only one copy of the HD has to be abnormal to cause the condition. This results in a pattern of inheritance called autosomal dominant and both males and females can be affected. Genes code for proteins; the protein encoded by the HD gene is called huntingtin. Proteins are made of building blocks called amino acids. The gene for HD has an expansion of the genetic code for glutamine. Therefore, abnormal huntingtin has an expansion of the number of glutamines. The genetic code for glutamine is CAG so the mistake in the gene is sometimes called a CAG repeat expansion disorder or in referring to the protein it is called a polyglutamine repeat expansion. The gene is in one part of the cell and the protein-making machinery is in another part of the cell so a chemical messenger is required which is called RNA. Explaining this is important for understanding some current clinical trials

Sign in / Sign up

Export Citation Format

Share Document