Integer Entries Research Articles

Parse-matrix evolution for symbolic regression

Data-driven model is highly desirable for industrial data analysis in case the experimental model structure is unknown or wrong, or the concerned system has changed. Symbolic regression is a useful method to construct the data-driven model (regression equation). Existing algorithms for symbolic regression such as genetic programming and grammatical evolution are difficult to use due to their special target programming language (i.e., LISP) or additional function parsing process. In this paper, a new evolutionary algorithm, parse-matrix evolution (PME), for symbolic regression is proposed. A chromosome in PME is a parse-matrix with integer entries. The mapping process from the chromosome to the regression equation is based on a mapping table. PME can easily be implemented in any programming language and free to control. Furthermore, it does not need any additional function parsing process. Numerical results show that PME can solve the symbolic regression problems effectively.

A universal nilpotent group of [formula omitted] diffeomorphisms of the interval

Let Nn be the group of n×n lower-triangular matrices with integer entries and ones in the diagonal. In this work we are interested in constructing a C1 action on the interval of a specific and somewhat universal nilpotent group denoted by N, which is the union of all of them. This yields a unified version of a construction due to B. Farb and J. Franks.

Cayley’s hyperdeterminant: A combinatorial approach via representation theory

Cayley’s hyperdeterminant is a homogeneous polynomial of degree 4 in the 8 entries of a 2×2×2 array. It is the simplest (nonconstant) polynomial which is invariant under changes of basis in three directions. We use elementary facts about representations of the 3-dimensional simple Lie algebra sl2(C) to reduce the problem of finding the invariant polynomials for a 2×2×2 array to a combinatorial problem on the enumeration of 2×2×2 arrays with non-negative integer entries. We then apply results from linear algebra to obtain a new proof that Cayley’s hyperdeterminant generates all the invariants. In the last section we discuss the application of our methods to general multidimensional arrays.

Kronecker products and the RSK correspondence

A matrix M with nonnegative integer entries is minimal if the nonincreasing sequence of its entries (called π-sequence) is minimal, in the dominance order of partitions, among all nonincreasing sequences of entries of matrices with nonnegative integers that have the same 1-marginals as A.The starting point for this work is an identity that relates the number of minimal matrices that have fixed 1-marginals and π-sequence to a linear combination of Kronecker coefficients. In this paper we provide a bijection that realizes combinatorially this identity. From this bijection we obtain an algorithm that to each minimal matrix associates a minimal component, with respect to the dominance order, in a Kronecker product, and a combinatorial description of the corresponding Kronecker coefficient in terms of minimal matrices and tableau insertion. Our bijection follows from a generalization of the dual RSK correspondence to 3-dimensional binary matrices, which we state and prove. With the same tools we also obtain a generalization of the RSK correspondence to 3-dimensional integer matrices.

Multipartite entanglement arising from dense Euclidean lattices in dimensions 4–24

The group of automorphisms of Euclidean (embedded in ) dense lattices such as the root lattices D4 and E8, the Barnes–Wall lattice BW16, the unimodular lattice D12+ and the Leech lattice Λ24 may be generated by entangled quantum gates of the corresponding dimension. These (real) gates/lattices are useful for quantum error correction: for instance, the two- and four-qubit real Clifford groups are the automorphism groups of the lattices D4 and BW16, respectively, and the three-qubit real Clifford group is maximal in the Weyl group W(E8). Technically, the automorphism group Aut(Λ) of the lattice Λ is the set of orthogonal matrices B such that, following the conjugation action by the generating matrix of the lattice, the output matrix is unimodular (of determinant ±1, with integer entries). When the degree n is equal to the number of basis elements of Λ, Aut(Λ) also acts on basis vectors and is generated with matrices B such that the sum of squared entries in a row is 1, i.e. B may be seen as a quantum gate. For the dense lattices listed above, maximal multipartite entanglement arises. In particular, one finds a balanced tripartite entanglement in E8 (the two- and three-tangles have the same magnitude 1/4) and a Greenberger–Horne–Zeilinger-type entanglement in BW16. In this paper, we also investigate the entangled gates from D12+ and Λ24, by seeing them as systems coupling a qutrit to two- and three-qubits, respectively. In addition to quantum computing, the work may be related to particle physics in the spirit of Planat et al (2011 Rep. Math. Phys.66 39–51).

Some Smooth Compactly Supported Tight Framelets

For any dilation matrix with integer entries, we construct a family of smooth compactly supported tight wavelet frames in $L^2(\mathbb{R}^d)$, $d\ge 1$. Estimates for the degrees of smoothness of these framelets are given. Our construction involves some compactly supported refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stockler.

On the number of monochromatic solutions of integer linear systems on Abelian groups

Abstract Let G be a finite cyclic group an let r be a positive integer. Let A be a k × m matrix with integer entries. We show that, if A satisfies some natural conditions, then the homogeneous linear system A x = 0 has Ω ( | G N | m − k ) monochromatic solutions for each r-coloring of G N { 0 } and sufficiently large N. Density versions of this counting result are also addressed.

A note on the (regularizing) preconditioning of [formula omitted]-Toeplitz sequences via [formula omitted]-circulants

For a given nonnegative integer g, a matrix An of size n is called g-Toeplitz if its entries obey the rule An=[ar−gs]r,s=0n−1. Analogously, a matrix An again of size n is called g-circulant if An=[a(r−gs)modn]r,s=0n−1. In a recent work we studied the asymptotic properties, in terms of spectral distribution, of both g-circulant and g-Toeplitz sequences in the case where {ak} can be interpreted as the sequence of Fourier coefficients of an integrable function f over the domain (−π,π). Here we are interested in the preconditioning problem which is well understood and widely studied in the last three decades in the classical Toeplitz case, i.e., for g=1. In particular, we consider the generalized case with g≥2 and the nontrivial result is that the preconditioned sequence {Pn}={Pn−1An}, where {Pn} is the sequence of preconditioner, cannot be clustered at 1 so that the case of g=1 is exceptional. However, while a standard preconditioning cannot be achieved, the result has a potential positive implication since there exist choices of g-circulant sequences which can be used as basic preconditioning sequences for the corresponding g-Toeplitz structures. Generalizations to the block and multilevel case are also considered, where g is a vector with nonnegative integer entries. A few numerical experiments, related to a specific application in signal restoration, are presented and critically discussed.

Quotients of hypersurfaces in weighted projective space

Abstract In [Bini, van Geemen, Kelly, Mirror quintics, discrete symmetries and Shioda maps, 2009] some quotients of one-parameter families of Calabi–Yau varieties are related to the family of Mirror Quintics by using a construction due to Shioda. In this paper, we generalize this construction to a wider class of varieties. More specifically, let A be an invertible matrix with non-negative integer entries. We introduce varieties XA and in weighted projective space and in , respectively. The variety turns out to be a quotient of a Fermat variety by a finite group. As a by-product, XA is a quotient of a Fermat variety and is a quotient of XA by a finite group. We apply this construction to some families of Calabi–Yau manifolds in order to show their birationality.

Cyclage, catabolism, and the affine Hecke algebra

We identify a subalgebra H ˆ n + of the extended affine Hecke algebra H ˆ n of type A. The subalgebra H ˆ n + is a u-analogue of the monoid algebra of S n ⋉ Z ⩾ 0 n and inherits a canonical basis from that of H ˆ n . We show that its left cells are naturally labeled by tableaux filled with positive integer entries having distinct residues mod n, which we term positive affine tableaux (PAT). We then exhibit a cellular subquotient R 1 n of H ˆ n + that is a u-analogue of the ring of coinvariants C [ y 1 , … , y n ] / ( e 1 , … , e n ) with left cells labeled by PAT that are essentially standard Young tableaux with cocharge labels. Multiplying canonical basis elements by a certain element π ∈ H ˆ n + corresponds to rotations of words, and on cells corresponds to cocyclage. We further show that R 1 n has cellular quotients R λ that are u-analogues of the Garsia–Procesi modules R λ with left cells labeled by (a PAT version of) the λ-catabolizable tableaux. We give a conjectural description of a cellular filtration of H ˆ n + , the subquotients of which are isomorphic to dual versions of R λ under the perfect pairing on R 1 n . This turns out to be closely related to the combinatorics of the cells of H ˆ n worked out by Shi, Lusztig, and Xi, and we state explicit conjectures along these lines. We also conjecture that the k-atoms of Lapointe, Lascoux and Morse (2003) [9] and the R-catabolizable tableaux of Shimozono and Weyman (2000) [20] have cellular counterparts in H ˆ n + . We extend the idea of atom copies from Lapointe, Lascoux and Morse (2003) [9] to positive affine tableaux and give descriptions, mostly conjectural, of some of these copies in terms of catabolizability.

Dyadic bivariate wavelet multipliers in L 2(ℝ2)

The single 2 dilation wavelet multipliers in one-dimensional case and single A-dilation (where A is any expansive matrix with integer entries and |detA| = 2) wavelet multipliers in twodimensional case were completely characterized by Wutam Consortium (1998) and Li Z., et al. (2010). But there exist no results on multivariate wavelet multipliers corresponding to integer expansive dilation matrix with the absolute value of determinant not 2 in L 2(ℝ2). In this paper, we choose $2I_2 = \left( {\begin{array}{*{20}c} 2 & 0 \\ 0 & 2 \\ \end{array} } \right)$ as the dilation matrix and consider the 2I 2-dilation multivariate wavelet Φ = {ψ 1, ψ 2, ψ 3}(which is called a dyadic bivariate wavelet) multipliers. Here we call a measurable function family f = {f 1, f 2, f 3} a dyadic bivariate wavelet multiplier if $\Psi _1 = \left\{ {\mathcal{F}^{ - 1} \left( {f_1 \widehat{\psi _1 }} \right),\mathcal{F}^{ - 1} \left( {f_2 \widehat{\psi _2 }} \right),\mathcal{F}^{ - 1} \left( {f_3 \widehat{\psi _3 }} \right)} \right\}$ is a dyadic bivariate wavelet for any dyadic bivariate wavelet Φ = {ψ 1, ψ 2, ψ 3}, where $\hat f$ and F −1 denote the Fourier transform and the inverse transform of function f respectively. We study dyadic bivariate wavelet multipliers, and give some conditions for dyadic bivariate wavelet multipliers. We also give concrete forms of linear phases of dyadic MRA bivariate wavelets.

Fibonacci numbers and orthogonal polynomials

We prove that the sequence ( 1 / F n + 2 ) n ⩾ 0 of reciprocals of the Fibonacci numbers is a moment sequence of a certain discrete probability measure and we identify the orthogonal polynomials as little q-Jacobi polynomials with q = 1 - 5 / 1 + 5 . We prove that the corresponding kernel polynomials have integer coefficients, and from this we deduce that the inverse of the corresponding Hankel matrices ( 1 / F i + j + 2 ) have integer entries. We prove analogous results for the Hilbert matrices.

Number theory of matrix semigroups

Unlike factorization theory of commutative semigroups which are well-studied, very little literature exists describing factorization properties in noncommutative semigroups. Perhaps the most ubiquitous noncommutative semigroups are semigroups of square matrices and this article investigates the factorization properties within certain subsemigroups of M n ( Z ) , the semigroup of n × n matrices with integer entries. Certain important invariants are calculated to give a sense of how unique or non-unique factorization is in each of these semigroups.

The centro-invertible matrix: A new type of matrix arising in pseudo-random number generation

This paper defines a new type of matrix (which will be called a centro-invertible matrix) with the property that the inverse can be found by simply rotating all the elements of the matrix through 180 degrees about the mid-point of the matrix. Centro-invertible matrices have been demonstrated in a previous paper to arise in the analysis of a particular algorithm used for the generation of uniformly-distributed pseudo-random numbers.An involutory matrix is one for which the square of the matrix is equal to the identity. It is shown that there is a one-to-one correspondence between the centro-invertible matrices and the involutory matrices. When working in modular arithmetic this result allows all possible k by k centro-invertible matrices with integer entries modulo M to be enumerated by drawing on existing theoretical results for involutory matrices.Consider the k by k matrices over the integers modulo M. If M takes any specified finite integer value greater than or equal to two then there are only a finite number of such matrices and it is valid to consider the likelihood of such a matrix arising by chance. It is possible to derive both exact expressions and order-of-magnitude estimates for the number of k by k centro-invertible matrices that exist over the integers modulo M. It is shown that order (√N) of the N=M(k2) different k by k matrices modulo M are centro-invertible, so that the proportion of these matrices that are centro-invertible is order (1/√N).

Expressing Polynomials as the Permanent of low rank Square Matrices

It is known that the problem of computing the permanent of a given matrix is #P hard. However, Alexander I. Barvinok has proven that if we fix the rank of the matrix then its permanent can be computed in strongly polynomial time. Barvinok's algorithm [Barvinok, A. I., Two algorithmic results for the traveling salesman problem, Math. Oper. Res. 21 (1996), 65–84.] computes the permanent of square matrices of fixed rank by constructing polynomials. We study the problem of expressing polynomials as the permanent of low rank square matrices and vice versa. We prove that the permanent of a square matrix with rank 1 is a monomial and the permanent of a square matrix (with integer entries) that has not full rank, is a polynomial with even coefficients. We also prove that, for a polynomial f∈k[x], there exist a square matrix of rank 2, whose permanent is the polynomial f. Our results contribute in computing the permanent of a square matrix efficiently.

The path-connectivity of MRA wavelets in $L^2(\mathbb {R}^{d})$

We show that for any $d\times d$ expansive matrix $A$ with integer entries and $|\det(A)|=2$, the set of all $A$-dilation MRA wavelets is path-connected under the $L^2(\mathbb{R}^d)$ norm topology. We do this through the application of $A$-dilation wavelet multipliers, namely measurable functions $f$ with the property that the inverse Fourier transform of $(f\widehat{\psi})$ is an $A$-dilation wavelet for any $A$-dilation wavelet $\psi$ (where $\widehat{\psi}$ is the Fourier transform of $\psi$). In this process, we have completely characterized all $A$-dilation wavelet multipliers for any integral expansive matrix $A$ with $|\det(A)|=2$.

A canonical basis for Garsia-Procesi modules

We identify a subalgebra $\widehat{\mathscr{H}}^+_n$ of the extended affine Hecke algebra $\widehat{\mathscr{H}}_n$ of type $A$. The subalgebra $\widehat{\mathscr{H}}^+_n$ is a u-analogue of the monoid algebra of $\mathcal{S}_n ⋉ℤ_≥0^n$ and inherits a canonical basis from that of $\widehat{\mathscr{H}}_n$. We show that its left cells are naturally labeled by tableaux filled with positive integer entries having distinct residues mod $n$, which we term positive affine tableaux (PAT). We then exhibit a cellular subquotient $\mathscr{R}_1^n$ of $\widehat{\mathscr{H}}^+_n$ that is a $u$-analogue of the ring of coinvariants $ℂ[y_1,\ldots,y_n]/(e_1, \ldots,e_n)$ with left cells labeled by PAT that are essentially standard Young tableaux with cocharge labels. Multiplying canonical basis elements by a certain element $*π ∈ \widehat{\mathscr{H}}^+_n$ corresponds to rotations of words, and on cells corresponds to cocyclage. We further show that $\mathscr{R}_1^n$ has cellular quotients $\mathscr{R}_λ$ that are $u$-analogues of the Garsia-Procesi modules $R_λ$ with left cells labeled by (a PAT version of) the $λ$ -catabolizable tableaux. On définit une sous-algèbre $\widehat{\mathscr{H}}^+_n$ de l'extension affine de l'algèbre de Hecke \$\widehat{\mathscr{H}}_n$ de type $A$. La sous-algèbre $\widehat{\mathscr{H}}^+_n$ est $u$-analogue à l'algèbre monoïde de $\mathcal{S}_n ⋉ℤ_≥0^n$ et hérite d'une base canonique de $\widehat{\mathscr{H}}_n$. On montre que ses cellules gauches sont naturellement classées par des tableaux remplis d'entiers naturels ayant chacun des restes différents modulo $n$, que l'on nomme Positive Affine Tableaux (PAT). On montre ensuite qu'un sous-quotient cellulaire $\mathscr{R}_1^n$ de $\widehat{\mathscr{H}}^+_n$ est une $u$-analogue de l'anneau des co-invariants $ℂ[y_1,\ldots,y_n]/(e_1, \ldots,e_n)$ avec des cellules gauches classées PAT qui sont essentiellement des tableaux de Young standards avec des labels cochargés. Multiplier les éléments de la base canonique par un certain élément $π ∈ \widehat{\mathscr{H}}^+_n$ correspond à des rotations de mots, et par rapport aux cellules cela correspond à un cocyclage. Plus loin, on montre que $\mathscr{R}_1^n$ a pour quotients cellulaires $\mathscr{R}_λ$ qui sont $u$- analogues aux modules de Garsia-Procesi $R_λ$ avec des cellules gauches définies par (une version PAT) des tableaux $λ$ -catabolisable.

Some counting questions for matrices with restricted entries

We obtain upper bounds for the number of arbitrary and symmetric matrices with integer entries in a given box (in an arbitrary location) and a given determinant. We then apply these bounds to estimate the number of matrices in such boxes which have an integer eigenvalues. Finally, we outline some open questions.

An upper bound on separable index of certain SL(3)-modules

Let X 3 be the set of all 3 × 3 matrices with non-negative integer entries whose column sum is fixed. We find an upper bound on the set which will turn out to be the maximum of S whenever it is prime. Using this and the Standard Monomial Theory, we get an upper bound of the separable index of certain highest weight representations of SL(3) which is sharper than the one given in V. Balaji & A.J. Parameswaran [Semistable principal bundles-II (positive characteristic), Transform. Groups, 8, (2003), pp. 3–36].

Multipliers, Phases and Connectivity of MRA Wavelets in L 2(ℝ2)

Let A be any 2×2 real expansive matrix. For any A-dilation wavelet ψ, let \(\widehat{\psi}\) be its Fourier transform. A measurable function f is called an A-dilation wavelet multiplier if the inverse Fourier transform of \((f\widehat{\psi})\) is an A-dilation wavelet for any A-dilation wavelet ψ. In this paper, we give a complete characterization of all A-dilation wavelet multipliers under the condition that A is a 2×2 matrix with integer entries and |{det }(A)|=2. Using this result, we are able to characterize the phases of A-dilation wavelets and prove that the set of all A-dilation MRA wavelets is path-connected under the L2(ℝ2) norm topology for any such matrix A.