Stability of uncertain discrete systems

The uncertain system

$x_{n + 1} = A_n x_n , n = 0,1,2, \ldots ,$

is considered, where the coefficients a _ij (n) of the m×m matrix A _n are functionals of any nature subject to the constraints

$\begin{array}{*{20}c} {\left| {a_{i,i} (n)} \right| \leqslant \alpha _ * < 1,} \\ {\left| {a_{i,j} (n)} \right| \leqslant \alpha _0 for j \geqslant i + 1,} \\ {\left| {a_{i,j} (n)} \right| \leqslant \delta for j < i.} \\ \end{array} $

Such systems include, in particular, switched-type systems, whose matrix A can take values in a given finite set.By using a special Lyapunov function, a bound δ ≤ δ(α₀,α_*) ensuring the global asymptotic stability of the system is found. In particular, the system is stable if the last inequality is replaced by a _i,j (n) = 0 for j < i.It is shown that pulse-width modulated systems reduce to the uncertain systems under consideration; moreover, in the case of a pulse-width modulation of the first kind, the coefficients of the matrix A are functions of x(n), and in the case of a modulation of the second kind, they are functionals.     

Nonaffine differential-algebraic curves do not exist

The paper outlines why the spectrum of maximal ideals Spec _? A of a countable-dimensional differential ?-algebra A of transcendence degree 1 without zero divisors is locally analytic, which means that for any ?-homomorphism ψ _M: A → ? (M ∈ Spec _? A) and any a ∈ A the Taylor series \(\widetilde {{\psi _M}}{\left( a \right)^{\underline{\underline {def}} }}\sum\limits_{m = 0}^\infty {\psi M\left( {{a^{\left( m \right)}}} \right)} \frac{{{z^m}}}{{m!}}\) has nonzero radius of convergence depending on the element a ∈ A.     

A Schur complement approach to preconditioning sparse linear least-squares problems with some dense rows

The effectiveness of sparse matrix techniques for directly solving large-scale linear least-squares problems is severely limited if the system matrix A has one or more nearly dense rows. In this paper, we partition the rows of A into sparse rows and dense rows (A_s and A_d) and apply the Schur complement approach. A potential difficulty is that the reduced normal matrix AsTA_s is often rank-deficient, even if A is of full rank. To overcome this, we propose explicitly removing null columns of A_s and then employing a regularization parameter and using the resulting Cholesky factors as a preconditioner for an iterative solver applied to the symmetric indefinite reduced augmented system. We consider complete factorizations as well as incomplete Cholesky factorizations of the shifted reduced normal matrix. Numerical experiments are performed on a range of large least-squares problems arising from practical applications. These demonstrate the effectiveness of the proposed approach when combined with either a sparse parallel direct solver or a robust incomplete Cholesky factorization algorithm.     

Singular values of some modular functions

For an integer N greater than 5 and a triple \({\mathfrak{a}}=[a_{1},a_{2},a_{3}]\) of integers with the properties 0<a _i ≤N/2 and a _i ≠a _j for i≠j, we consider a modular function \(W_{\mathfrak{a}}(\tau)=\frac{\wp (a_{1}/N;L_{\tau})-\wp (a_{3}/N;L_{\tau})}{\wp (a_{2}/N;L_{\tau})-\wp(a_{3}/N;L_{\tau})}\) for the modular group Γ ₁(N), where ?(z;L _τ ) is the Weierstrass ?-function relative to the lattice L _τ generated by 1 and a complex number τ with positive imaginary part. For a pair of such triples \({\mathfrak{A}}=[{\mathfrak{a}},{\mathfrak{b}}]\) and a pair of non-negative integers F=[m,n], we define a modular function \(T_{{\mathfrak{A}},F}\) for the group Γ ₀(N) as the trace of the product \(W_{\mathfrak{a}}^{m}W_{\mathfrak{b}}^{n}\) to the modular function field of Γ ₀(N). In this article, we study the integrality of singular values of the functions \(W_{\mathfrak{a}}\) and \(T_{{\mathfrak{A}},F}\) by using their modular equations. We prove that the functions \(T_{{\mathfrak{A}},F}\) for suitably chosen \({\mathfrak{A}}\) and F generate the modular function field of Γ ₀(N), and from Shimura reciprocity and Gee–Stevenhagen method we obtain that singular values \(T_{{\mathfrak{A}},F}(\tau)\) for suitably chosen \({\mathfrak{A}}\) and F generate ring class fields. Further, we study the class polynomial of \(T_{{\mathfrak{A}},F}\) for Schertz N-system.     

The smallest singular value of random rectangular matrices with no moment assumptions on entries

Let δ > 1 and β > 0 be some real numbers. We prove that there are positive u, v, N₀ depending only on β and δ with the following property: for any N,n such that N ≥ max(N₀, δ_n), any N × n random matrix A = (a_ij) with i.i.d. entries satisfying \({\sup _{\lambda \in \mathbb{R}}}P\left\{ {\left| {{a_{11}} - \lambda } \right| \leqslant 1} \right\} \leqslant 1 - \beta \) and any non-random N × n matrix B, the smallest singular value s_n of A + B satisfies \(P\left\{ {{s_n}\left( {A + B} \right) \leqslant u\sqrt N } \right\} \leqslant \exp \left( { - vN} \right)\). The result holds without any moment assumptions on the distribution of the entries of A.     

Acceleration of convergence of general linear sequences by the Shanks transformation

The Shanks transformation is a powerful nonlinear extrapolation method that is used to accelerate the convergence of slowly converging, and even diverging, sequences {A _n }. It generates a two-dimensional array of approximations \({A^{(j)}_n}\) to the limit or anti-limit of {A _n } defined as solutions of the linear systems

$A_l=A^{(j)}_n +\sum^{n}_{k=1}\bar{\beta}_k(\Delta A_{l+k-1}),\ \ j\leq l\leq j+n,$

where \({\bar{\beta}_{k}}\) are additional unknowns. In this work, we study the convergence and stability properties of \({A^{(j)}_n}\) , as j → ∞ with n fixed, derived from general linear sequences {A _n }, where \({{A_n \sim A+\sum^{m}_{k=1}\zeta_k^n\sum^\infty_{i=0} \beta_{ki}n^{\gamma_k-i}}}\) as n → ∞, where ζ _k  ≠ 1 are distinct and |ζ ₁| = ... = |ζ _m | = θ, and γ _k  ≠ 0, 1, 2, . . .. Here A is the limit or the anti-limit of {A _n }. Such sequences arise, for example, as partial sums of Fourier series of functions that have finite jump discontinuities and/or algebraic branch singularities. We show that definitive results are obtained with those values of n for which the integer programming problems

$\begin{array}{ll}{\quad\quad\quad\quad\max\limits_{s_1,\ldots,s_m}\sum\limits_{k=1}^{m}\left[(\Re\gamma_k)s_k-s_k(s_k-1)\right],}\\ {{\rm subject\,to}\,\, s_1\geq0,\ldots,s_m\geq0\quad{\rm and}\quad \sum\limits_{k=1}^{m} s_k = n,}\end{array}$

have unique (integer) solutions for s ₁, . . . , s _m . A special case of our convergence result concerns the situation in which \({{\Re\gamma_1=\cdots=\Re\gamma_m=\alpha}}\) and n = mν with ν = 1, 2, . . . , for which the integer programming problems above have unique solutions, and it reads \({A^{(j)}_n-A=O(\theta^j\,j^{\alpha-2\nu})}\) as j → ∞. When compared with A _j ? A = O(θ ^j j ^α ) as j → ∞, this result shows that the Shanks transformation is a true convergence acceleration method for the sequences considered. In addition, we show that it is stable for the case being studied, and we also quantify its stability properties. The results of this work are the first ones pertaining to the Shanks transformation on general linear sequences with m > 1.

     

Simplified anti-Gauss quadrature rules with applications in linear algebra

The need to compute inexpensive estimates of upper and lower bounds for matrix functions of the form w ^T f(A)v with \(A\in {\mathbb {R}}^{n\times n}\) a large matrix, f a function, and \(v,w\in {\mathbb {R}}^{n}\) arises in many applications such as network analysis and the solution of ill-posed problems. When A is symmetric, u = v, and derivatives of f do not change sign in the convex hull of the spectrum of A, a technique described by Golub and Meurant allows the computation of fairly inexpensive upper and lower bounds. This technique is based on approximating v ^T f(A)v by a pair of Gauss and Gauss-Radau quadrature rules. However, this approach is not guaranteed to provide upper and lower bounds when derivatives of the integrand f change sign, when the matrix A is nonsymmetric, or when the vectors v and w are replaced by “block vectors” with several columns. In the latter situations, estimates of upper and lower bounds can be computed quite inexpensively by evaluating pairs of Gauss and anti-Gauss quadrature rules. When the matrix A is large, the dominating computational effort for evaluating these estimates is the evaluation of matrix-vector products with A and possibly also with A ^T . The calculation of anti-Gauss rules requires one more matrix-vector product evaluation with A and maybe also with A ^T than the computation of the corresponding Gauss rule. The present paper describes a simplification of anti-Gauss quadrature rules that requires the evaluation of the same number of matrix-vector products as the corresponding Gauss rule. This simplification makes the computational effort for evaluating the simplified anti-Gauss rule negligible when the corresponding Gauss rule already has been computed.     

Multiple recurrence and convergence for hardy sequences of polynomial growth

We study the limiting behavior of multiple ergodic averages involving sequences of integers that satisfy some regularity conditions and have polynomial growth. We show that for “typical” choices of Hardy field functions a(t) with polynomial growth, the averages

${1 \over N}\sum\nolimits_{n = 1}^N {{f_1}({T^{[a(n)]}}x) \cdots {f_\ell }({T^{\ell [a(n)]}}x)} $

converge in mean and we determine their limit. For example, this is the case if a(t) = t ^3/2, t log t, or t ² + (log t)². Furthermore, if {a ₁(t), …, a _? (t)} is a “typical” family of logarithmico-exponential functions of polynomial growth, then for every ergodic system, the averages

${1 \over N}\sum\nolimits_{n = 1}^N {{f_1}({T^{[{a_1}(n)]}}x) \cdots {f_\ell }({T^{[{a_\ell }(n)]}}x)} $

converge in mean to the product of the integrals of the corresponding functions. For example, this is the case if the functions a _i (t) are given by different positive fractional powers of t. We deduce several results in combinatorics. We show that if a(t) is a non-polynomial Hardy field function with polynomial growth, then every set of integers with positive upper density contains arithmetic progressions of the form {m,m + [a(n)], …, m + ?[a(n)]}. Under suitable assumptions, we get a related result concerning patterns of the form {m,m + [a ₁(n)], …,m + [a _? (n)]}.

     

Factorization of cp-rank-3 completely positive matrices

A symmetric positive semi-definite matrix A is called completely positive if there exists a matrix B with nonnegative entries such that A = BB^?. If B is such a matrix with a minimal number p of columns, then p is called the cp-rank of A. In this paper we develop a finite and exact algorithm to factorize any matrix A of cp-rank 3. Failure of this algorithm implies that A does not have cp-rank 3. Our motivation stems from the question if there exist three nonnegative polynomials of degree at most four that vanish at the boundary of an interval and are orthonormal with respect to a certain inner product.     

The permutation of integers with small least common multiple of two subsequent terms

Erd?s, Freud and Hegyvári [1] constructed a permutation a ₁,a ₂,… of positive integers with \([a_{i}, a_{i+1}]< i\exp \left\{c\sqrt{\log i}\log\log i\,\right\}\) for an absolute constant c>0 and all i≧3. In this note, we construct a permutation of all positive integers such that for any ε>0 there exists an i ₀ with \([a_{i}, a_{i+1}]\allowbreak < i\exp \left\{\left(2\sqrt{2}+\varepsilon\right) \sqrt{\log i\log\log i}\,\right\}\) for all i≧i ₀.     

Almost everywhere convergence of sequences of two-dimensional Walsh–Fejér means of integrable functions

The aim of this paper is to prove the a.e. convergence of sequences of the Fejér means of the Walsh–Fourier series of bivariate integrable functions. That is, let \(a = (a_{1}, a_{2})\:\mathbb{N} \to \mathbb{N}^{2}\) such that a _j (n+1)≧δsup _k≦n a _j (n) (j=1,2, n∈?) for some δ>0 and a ₁(+∞)=a ₂(+∞)=+∞. Then for each integrable function f∈L ¹(I ²) we have the a.e. relation \(\lim_{n\to\infty}\sigma_{a_{1}(n), a_{2}(n)}f = f\). It will be a straightforward and easy consequence of this result the cone restricted a.e. convergence of the two-dimensional Walsh–Fejér means of integrable functions which was proved earlier by the author and Weisz [3,8].     

Jacobi method for symmetric 4 × 4 matrices converges for every cyclic pivot strategy

The paper studies the global convergence of the Jacobi method for symmetric matrices of size 4. We prove global convergence for all 720 cyclic pivot strategies. Precisely, we show that inequality S(A ^[t+3]) ≤ γ S(A ^[t]), t ≥ 1, holds with the constant γ < 1 that depends neither on the matrix A nor on the pivot strategy. Here, A ^[t] stands for the matrix obtained from A after t full cycles of the Jacobi method and S(A) is the off-diagonal norm of A. We show why three consecutive cycles have to be considered. The result has a direct application on the J-Jacobi method.     

On $$(\ell,m)$$-regular partitions with distinct parts

Let \(a_{\ell ,m}(n)\) denote the number of \((\ell ,m)\)-regular partitions of a positive integer n into distinct parts, where \(\ell \) and m are relatively primes. In this paper, we establish several infinite families of congruences modulo 2 for \(a_{3,5}(n)\). For example,

$$\begin{aligned} a_{3, 5}\left(2^{6\alpha +4}5^{2\beta }n+\frac{ 2^{6\alpha +3}5^{2\beta +1}-1}{3}\right) \equiv 0 , \end{aligned}$$

where \(\alpha , \beta \ge 0\).

     

Inclusion results on statistical cluster points

We study the concepts of statistical cluster points and statistical core of a sequence for A _λ methods defined by deleting some rows from a nonnegative regular matrix A. We also relate A _λ-statistical convergence to A _μ-statistical convergence. Finally we give a consistency theorem for A-statistical convergence and deduce a core equality result.     

The real symmetric matrices of odd order with a P-set of maximum size

Suppose that A is a real symmetric matrix of order n. Denote by m_A(0) the nullity of A. For a nonempty subset α of {1, 2,..., n}, let A(α) be the principal submatrix of A obtained from A by deleting the rows and columns indexed by α. When m_A(α)(0) = m_A(0)+|α|, we call α a P-set of A. It is known that every P-set of A contains at most ?n/2? elements. The graphs of even order for which one can find a matrix attaining this bound are now completely characterized. However, the odd case turned out to be more difficult to tackle. As a first step to the full characterization of these graphs of odd order, we establish some conditions for such graphs G under which there is a real symmetric matrix A whose graph is G and contains a P-set of size (n ? 1)/2.     

A stability result using the matrix norm to bound the permanent

We prove a stability version of a general result that bounds the permanent of a matrix in terms of its operator norm. More specifically, suppose A is an n × n matrix over C (resp. R), and let P denote the set of n × n matrices over C (resp. R) that can be written as a permutation matrix times a unitary diagonal matrix. Then it is known that the permanent of A satisfies |per(A)| ≤ ||A|| ⁿ ₂ with equality iff A/||A||₂ ∈ P (where ||A||₂ is the operator 2-norm of A). We show a stability version of this result asserting that unless A is very close (in a particular sense) to one of these extremal matrices, its permanent is exponentially smaller (as a function of n) than ||A|| ⁿ ₂. In particular, for any fixed α, β > 0, we show that |per(A)| is exponentially smaller than ||A|| ⁿ ₂ unless all but at most αn rows contain entries of modulus at least ||A||₂(1?β).     

Triviality of Iwasawa module associated to some abelian fields of prime conductors

Let p be an odd prime number and \(\ell \) an odd prime number dividing \(p-1\). We denote by \(F=F_{p,\ell }\) the real abelian field of conductor p and degree \(\ell \), and by \(h_F\) the class number of F. For a prime number \(r \ne p,\,\ell \), let \(F_{\infty }\) be the cyclotomic \(\mathbb {Z}_r\)-extension over F, and \(M_{\infty }/F_{\infty }\) the maximal pro-r abelian extension unramified outside r. We prove that \(M_{\infty }\) coincides with \(F_{\infty }\) and consequently \(h_F\) is not divisible by r when r is a primitive root modulo \(\ell \) and r is smaller than an explicit constant depending on p.     

CLT for Random Walks of Commuting Endomorphisms on Compact Abelian Groups

Let \(\mathcal S\) be an abelian group of automorphisms of a probability space \((X, {\mathcal A}, \mu )\) with a finite system of generators \((A_1, \ldots , A_d).\) Let \(A^{{\underline{\ell }}}\) denote \(A_1^{\ell _1} \ldots A_d^{\ell _d}\), for \({{\underline{\ell }}}= (\ell _1, \ldots , \ell _d).\) If \((Z_k)\) is a random walk on \({\mathbb {Z}}^d\), one can study the asymptotic distribution of the sums \(\sum _{k=0}^{n-1} \, f \circ A^{\,{Z_k(\omega )}}\) and \(\sum _{{\underline{\ell }}\in {\mathbb {Z}}^d} {\mathbb {P}}(Z_n= {\underline{\ell }}) \, A^{\underline{\ell }}f\), for a function f on X. In particular, given a random walk on commuting matrices in \(SL(\rho , {\mathbb {Z}})\) or in \({\mathcal M}^*(\rho , {\mathbb {Z}})\) acting on the torus \({\mathbb {T}}^\rho \), \(\rho \ge 1\), what is the asymptotic distribution of the associated ergodic sums along the random walk for a smooth function on \({\mathbb {T}}^\rho \) after normalization? In this paper, we prove a central limit theorem when X is a compact abelian connected group G endowed with its Haar measure (e.g., a torus or a connected extension of a torus), \(\mathcal S\) a totally ergodic d-dimensional group of commuting algebraic automorphisms of G and f a regular function on G. The proof is based on the cumulant method and on preliminary results on random walks.     

Commutators of Riesz transforms of magnetic Schrödinger operators

Let \({A=-(\nabla-i{\vec a})\cdot (\nabla-i{\vec a}) +V}\) be a magnetic Schrödinger operator acting on \({L^2({\mathbb R}^n)}\), n ≥  1, where \({{\vec a}=(a_1, \ldots, a_n)\in L^2_{\rm loc}({\mathbb R}^n, {\mathbb R}^n)}\) and \({0\leq V\in L^1_{\rm loc}({\mathbb R}^n)}\). In this paper, we show that when a function \({b\in {\rm BMO}({\mathbb R}^n)}\), the commutators [b, T _k ]f = T _k (b f) ? b T _k f, k = 1, . . . , n, are bounded on \({L^p({\mathbb R}^n)}\) for all 1 < p < 2, where the operators T _k are Riesz transforms (?/?x _k  ? i a _k )A ^?1/2 associated with A.     

Globally convergent Jacobi methods for positive definite matrix pairs

The paper derives and investigates the Jacobi methods for the generalized eigenvalue problem A x = λ B x, where A is a symmetric and B is a symmetric positive definite matrix. The methods first “normalize” B to have the unit diagonal and then maintain that property during the iterative process. The global convergence is proved for all such methods. That result is obtained for the large class of generalized serial strategies from Hari and Begovi? Kova? (Trans. Numer. Anal. (ETNA) 47, 107–147, 2017). Preliminary numerical tests confirm a high relative accuracy of some of those methods, provided that both matrices are positive definite and the spectral condition numbers of Δ _A AΔ _A and Δ _B BΔ _B are small, for some nonsingular diagonal matrices Δ _A and Δ _B .