Related papers: Farey determinants matrix
We construct infinite families of graphs that are determined by their generalized spectrum. This construction is based on new formulae for the determinant of the walk matrix of a graph. The graphs constructed here all satisfy a lower…
We derive identities for the determinants of matrices whose entries are (rising) powers of (products of) polynomials that satisfy a recurrence relation. In particular, these results cover the cases for Fibonacci polynomials, Lucas…
We prove that all correlations of the sequence of Farey fractions exist and provide formulas for the correlation measures.
We introduce a method for calculating individual elements of matrix functions. Our technique makes use of a novel series expansion for the action of matrix functions on basis vectors that is memory efficient even for very large matrices. We…
Matrices over the ring of formal power series are considered. Normal forms with respect to various sub-groups of the two-sided transformations are constructed. The construction is based on the special property of the action: it induces a…
A real vector space combined with an inverse for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale. The choice of sign for different components of the vector inverse…
Two square matrices of (arbitrary) order N are introduced. They are defined in terms of N arbitrary numbers z_{n}, and of an arbitrary additional parameter (a respectively q), and provide finite-dimensional representations of the two…
The article considers arrowhead and diagonal-plus-rank-one matrices in F^(nxn) where F in R,C or H. H is a non-commutative field of quaternions. We give unified formulas for fast matrix-vector multiplications, determinants, and inverses for…
In this paper, we introduce a new method for calculating fractional integrals and differentials. The method involves an equation that we have obtained from infinite applied integration by parts. The equation works for special class of…
It is shown how to define difference operators and equations on particular lattices $\{x_n\}$, $2n\in\mathbb{Z}$, such that the divided difference operator $(\mathcal{D}f)(x_{n+1/2})= (f(x_{n+1})-f(x_n))/(x_{n+1}-x_n)$ has the property that…
Matrix factorization exploits the idea that, in complex high-dimensional data, the actual signal typically lies in lower-dimensional structures. These lower dimensional objects provide useful insight, with interpretability favored by sparse…
We consider properties of the box polynomials, a one variable polynomial defined over all integer partitions $\lambda$ whose Young diagrams fit in an $m$ by $n$ box. We show that these polynomials can be expressed by the finite difference…
We present a simple and accessible method which uses contour integration methods to derive formulae for functional determinants. To make the presentation as clear as possible, the general idea is first illustrated on the simplest case: a…
Fractional gradient descent has been studied extensively, with a focus on its ability to extend traditional gradient descent methods by incorporating fractional-order derivatives. This approach allows for more flexibility in navigating…
In a series of recent papers we have shown how the dynamical behavior of certain classical systems can be analyzed using operators evolving according to Heisenberg-like equations of motions. In particular, we have shown that raising and…
Convenient parameterizations of matrices in terms of vectors transform (certain classes of) matrix equations into covariant (hence rotation-invariant) vector equations. Certain recently introduced such parameterizations are tersely…
Let $(P,\preceq)$ be a lattice and $f$ a complex-valued function on $P$. We define meet and join matrices on two arbitrary subsets $X$ and $Y$ of $P$ by $(X,Y)_f=(f(x_i\wedge y_j))$ and $[X,Y]_f=(f(x_i\vee x_j))$ respectively. Here we…
The Farey sequence is the sequence of all rational numbers in the real unit interval, stratified by increasing denominators. A classical result by Hall says that its normalized gap distribution is the same as the distribution of the random…
A Redheffer--type matrix with Fibonacci entries is defined, and the determinant and spectral properties of this matrix are studied. Also, more general Redheffer--type matrices are considered and intriguing number-theoretic examples are…
The paper deals with a fractional derivative introduced by means of the Fourier transform. The explicit form of the kernel of general derivative operator acting on the functions analytic on a curve in complex plane is deduced and the…