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2. Determinants of matrices associated with incidence functions on posets
- Creator:
- Hong, Shaofang and Sun, Qi
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- meet-closed set, greatest-type lower, incidence function, determinant, and nonsingularity
- Language:
- English
- Description:
- Let $S=\lbrace x_1,\dots ,x_n\rbrace $ be a finite subset of a partially ordered set $P$. Let $f$ be an incidence function of $P$. Let $[f(x_i\wedge x_j)]$ denote the $n\times n$ matrix having $f$ evaluated at the meet $x_i\wedge x_j$ of $x_i$ and $x_j$ as its $i,j$-entry and $[f(x_i\vee x_j)]$ denote the $n\times n$ matrix having $f$ evaluated at the join $x_i\vee x_j$ of $x_i$ and $x_j$ as its $i,j$-entry. The set $S$ is said to be meet-closed if $x_i\wedge x_j\in S$ for all $1\le i,j\le n$. In this paper we get explicit combinatorial formulas for the determinants of matrices $[f(x_i\wedge x_j)]$ and $[f(x_i\vee x_j)]$ on any meet-closed set $S$. We also obtain necessary and sufficient conditions for the matrices $f(x_i\wedge x_j)]$ and $[f(x_i\vee x_j)]$ on any meet-closed set $S$ to be nonsingular. Finally, we give some number-theoretic applications.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
3. Integer matrices related to Liouville's function
- Creator:
- Oon, Shea-Ming
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- Liouville's function, determinant, and LU decomposition
- Language:
- English
- Description:
- In this note, we construct some integer matrices with determinant equal to certain summation form of Liouville's function. Hence, it offers a possible alternative way to explore the Prime Number Theorem by means of inequalities related to matrices, provided a better estimate on the relation between the determinant of a matrix and other information such as its eigenvalues is known. Besides, we also provide some comparisons on the estimate of the lower bound of the smallest singular value. Such discussion may be extended to that of Riemann hypothesis.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
4. Nested matrices and inverse M-matrices
- Creator:
- Stuart, Jeffrey L
- Format:
- print, bez média, and svazek
- Type:
- model:article and TEXT
- Subject:
- matematika, mathematics, nested matrix, tridiagonal matrix, inverse M-matrix, principal minor, determinant, QR-factorization, 13, and 51
- Language:
- English
- Description:
- Given a sequence of real or complex numbers, we construct a sequence of nested, symmetric matrices. We determine the LU- and QR-factorizations, the determinant and the principal minors for such a matrix. When the sequence is real, positive and strictly increasing, the matrices are strictly positive, inverse M-matrices with symmetric, irreducible, tridiagonal inverses., Jeffrey L. Stuart., and Obsahuje seznam literatury
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public