Monomial Sequences Research Articles

Pair correlation of the fractional parts of $\alpha n^{\theta}$

Fix \alpha,\theta >0 and consider the sequence (\alpha n^{\theta}\; \mathrm{mod}\; 1)_{n\ge 1} . Since the seminal work of Rudnick–Sarnak (1998) and due to the Berry–Tabor conjecture in quantum chaos, the fine-scale properties of these dilated monomial sequences have been intensively studied. In this paper, we show that for \theta <14/41 and \alpha>0 , the pair correlation function is Poissonian. While (for a given \theta \neq 1 ) this strong pseudo-randomness property has been proven for almost all values of \alpha , there are next-to-no instances where this has been proven for explicit \alpha . Our result holds for all \alpha>0 and relies solely on classical Fourier analytic techniques. This addresses (in the sharpest possible way) a problem posed by Aistleitner–El-Baz–Munsch (2021).

Model subspaces techniques to study Fourier expansions in L2 spaces associated to singular measures

Let μ be a probability measure on T that is singular with respect to the Haar measure. In this paper we study Fourier expansions in L2(T,μ) using techniques from the theory of model subspaces of the Hardy space. Since the sequence of monomials {zn}n∈N is effective in L2(T,μ), it has a Parseval frame associated via the Kaczmarz algorithm. Our first main goal is to identify the aforementioned frame with boundary values of the frame Pφ(zn) for the model subspace H(φ)=H2⊖φH2, where Pφ is the orthogonal projection from the Hardy space H2 onto H(φ). The study of Fourier expansions in L2(T,μ) also leads to consider positive kernels in the Hardy space. Our second main goal is to study the set of measures μ which reproduce a kernel contained in a model subspace. We completely characterize this set when the kernel is the reproducing kernel of a model subspace, and we study the consequences of this characterization.

Bases in the spaces of homogeneous polynomials and multilinear operators on Banach spaces

For Banach spaces $E_1, \dots ,E_m$, $E$ and $F$ with their bases, we show that a particular monomial sequence forms a basis of $\mathcal {P}(^mE; F)$, the space of continuous $m$-homogeneous polynomials from $E$ to $F$ (resp.\ a basis of $\mathcal {L}(E_1,\dots ,E_m;F)$, the space of continuous $m$-linear operators from $E_1\times \cdots \times E_m$ to $F$) if and only if the basis of $E$ (resp. the basis of $E_1,\dots ,E_m$) is a shrinking basis and every $P \in \mathcal {P}(^mE; F)$ (resp.\ every $T \in \mathcal {L}(E_1,\dots ,E_m;F)$) is weakly continuous on bounded sets.

Bases in the space of regular multilinear operators on Banach lattices

For Banach lattices E1,…,Em and F with 1-unconditional bases, we show that the monomial sequence forms a 1-unconditional basis of Lr(E1,…,Em;F), the Banach lattice of all regular m-linear operators from E1×⋯×Em to F, if and only if each basis of E1,…,Em is shrinking and every positive m-linear operator from E1×⋯×Em to F is weakly sequentially continuous. As a consequence, we obtain necessary and sufficient conditions for which the m-fold Fremlin projective tensor product E1⊗ˆ|π|⋯⊗ˆ|π|Em (resp. the m-fold positive injective tensor product E1⊗ˇ|ϵ|⋯⊗ˇ|ϵ|Em) has a shrinking basis or a boundedly complete basis.

Chebyshev polynomials via quadratic and cubic decompositions of the canonical sequence

In this paper we give general quadratic and cubic decompositions of the sequence of monomials and prove how the correspondent component sequences are connected with Chebyshev polynomial sequences (PSs). The two-way street that joins together the polynomial sequences brought by quadratic or cubic decomposition and those famous PSs is established by the reversed polynomials and initial perturbations. In the cubic decomposition case, the relation is given through a 2-orthogonal Chebyshev PS.

SOME MONOMIAL SEQUENCES ARISING FROM GRAPHS

s-sequences and d-sequences are fundamental sequences intensively studied in many fields of algebra. In this paper we are interested in dealing with monomial sequences associated to graphs in order to establish conditions for which they are s-sequences and/or d-sequences.

On the graphic realization of certain monomial sequences

d-sequences and s-sequences are important monomial sequences. In this paper, graphs that have edge ideals with generators forming d-sequences or s-sequences are examined. Two equivalent conditions are given.

On weighted Bochner-Martinelli residue currents

We study the weighted Bochner-Martinelli residue current $R^p(f)$ associated with a sequence $f=(f_1,\dots,f_m)$ of holomorphic germs at $0\in{\mathsf C}^n$, whose common zero set equals the origin, and $p=(p_1,\ldots, p_m)\in\mathsf{N}^m$. Our main results are a description of $R^p(f)$ in terms of the Rees valuations of the ideal generated by $(f_1^{p_1},\ldots, f_m^{p_m})$ and an explicit description of $R^p(f)$ when $f$ is monomial. For a monomial sequence $f$ we show that $R^p(f)$ is independent of $p$ if and only if $f$ is a regular sequence.

COMPARISON OF c AND d-SEQUENCES

In our previously published papers [Monomial sequence of linear type, Illinois J. Math.52 (2008) 1213–1221; Sequences between d-sequences and sequences of linear type, Comment. Math. Univ. Carolin.50 (2009) 1–9; Two-generated ideals of linear type, Acta Math. Univ. Comenian. (N.S.)78 (2009) 97–102] we have introduced and discussed the notion of a c-sequence in a commutative ring. In this paper we compare this notion with the notion of a d-sequence. The ideal generated by any of these sequences is of linear type. We show that there are c-sequences that are not d-sequences and that c-sequences satisfy analogs of sharp Artin–Rees properties that hold for d-sequences. We also discuss closeness of c and d-sequences.

Quasi‐ordinary singularities, essential divisors and Poincaré series

We define Poincar\'e series associated to a toric or analytically irreducible quasi-ordinary hypersurface singularity, (S,0), by a finite sequence of monomial valuations, such that at least one of them is centered at the origin 0. This involves the definition of a multi-graded ring associated to the analytic algebra of the singularity by the sequence of valuations. We prove that the Poincar\'e series is a rational function with integer coefficients, which can be defined also as an integral with respect of the Euler characteristic, over the projectivization of the analytic algebra of the singularity, of a function defined by the valuations. In particular, the Poincar\'e series associated to the set of divisorial valuations associated to the essential divisors, considered both over the singular locus and over the point 0, is an analytic invariant of the singularity. In the quasi-ordinary hypersurface case we prove that this Poincar\'e series determines and it is determined by the normalized sequence of characteristic monomials. These monomials in the analytic case define a complete invariant of the embedded topological type of the hypersurface singularity.

Monomial sequences of linear type

Monomial sequences of linear type

Orthogonal Laurent polynomials on the unit circle and snake-shaped matrix factorizations

Let there be given a probability measure μ on the unit circle T of the complex plane and consider the inner product induced by μ . In this paper we consider the problem of orthogonalizing a sequence of monomials { z r k } k , for a certain order of the r k ∈ Z , by means of the Gram–Schmidt orthogonalization process. This leads to a sequence of orthonormal Laurent polynomials { ψ k } k . We show that the matrix representation with respect to { ψ k } k of the operator of multiplication by z is an infinite unitary or isometric matrix allowing a ‘snake-shaped’ matrix factorization. Here the ‘snake shape’ of the factorization is to be understood in terms of its graphical representation via sequences of little line segments, following an earlier work of S. Delvaux and M. Van Barel. We show that the shape of the snake is determined by the order in which the monomials { z r k } k are orthogonalized, while the ‘segments’ of the snake are canonically determined in terms of the Schur parameters for μ . Isometric Hessenberg matrices and unitary five-diagonal matrices (CMV matrices) follow as a special case of the presented formalism.

On certain monomial sequences

We give equivalent conditions for a monomial sequence to be a d-sequence or a proper sequence, and a sufficient condition for a monomial sequence to be an s-sequence in order to compute invariants of the symmetric algebra of the ideal generated by it.

Taylor complexes for Koszul boundaries

For all boundary modules of the Koszul complex of a monomial sequence we construct complexes, which we call Taylor complexes. For a monomial d-sequences these complexes provide free resolutions of the boundary modules. Let M be the ideal generated by a monomial d-sequence. We use the Taylor complexes to construct minimal free resolutions of the Rees algebra and the associated graded ring of M.

A simple approach for construction of algebraic-geometric codes from affine plane curves

The current algebraic-geometric (AG) codes are based on the theory of algebraic-geometric curves. In this paper we present a simple approach for the construction of AG codes, which does not require an extensive background in algebraic geometry. Given an affine plane irreducible curve and its set of all rational points, we can find a sequence of monomials x/sup i/y/sup j/ based on the equation of the curve. Using the first r monomials as a basis for the dual code of a linear code, the designed minimum distance d of the linear code, called the AG code, can be easily determined. For these codes, we show a fast decoding procedure with a complexity O(n/sup 7/3/), which can correct errors up to [(d-1/2]. For this approach it is neither necessary to know the genus of curve nor the basis of a differential form. This approach can be easily understood by most engineers.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Weakly uniform distributions mod $m$ for certain recursive sequences and for monomial sequences

Weakly uniform distributions mod $m$ for certain recursive sequences and for monomial sequences

Note on Orthogonal Polynomials inv-Variables

We consider the polynomials in v-variables orthogonal with respect to a weight $w({\bf x})$ that need not be decomposable into a product of functions of one variable and obtained by applying the orthogonalization process to an ordered sequence of monomials in v-variables. We derive recursion relations, an expression for the Christoffel–Darboux kernel and a system of partial differential equations for these polynomials.