Distinct Complex Numbers Research Articles

A particular of equivalent of picard's little theorem

It is shown that for any three distinct collinear complex numbers p,q,t, the statement “An enite function which misses the values p and q but assumes the value t, at a preassigned point, is a constant function” is equivalent to Picard's Little Theorem. It is hoped that this particular version of Picard's Little Theorem could be used to give a simpler proof of Picard's Little Theorem.

A structure theorem for sum form functional equations

It is proved that, iffij:]0, 1[→ C (i = 1, ⋯,k;j = 1, ⋯,l) are measurable, satisfy the equation (1) (with some functionsgit, hjt:]0, 1[→ C), then eachfij is in a linear space (called Euler space) spanned by the functionsx → xλj(logx)k (x ∈]0, 1[;j = 1, ⋯,M;k = 0, ⋯,mj − 1), whereλ1, ⋯, λM are distinct complex numbers andm1, ⋯, mM natural numbers. The dimension of this linear space is bounded by a linear function ofN.

On Doubly Symmetric Tridiagonal Forms for Complex Matrices and Tridiagonal Inverse Eigenvalue Problems

It is well known that for any distinct real numbers $\lambda _1 , \cdots ,\lambda _n $ there exists a doubly symmetric (i.e., symmetric and persymmetric or, equivalently, symmetric and centrosymmetric) tridiagonal real $n \times n$ matrix T with the $\lambda $’s as its eigenvalues. Such a matrix can be constructed finitely using only arithmetic operations and square roots. We prove in this paper that the analogous assertions hold for any distinct complex numbers $\lambda _1 , \cdots ,\lambda _n $ with T being a complex matrix. It follows that any complex $n \times n$ matrix with distinct eigenvalues is similar to a doubly symmetric tridiagonal matrix. The condition that the eigenvalues be distinct is essential: we show that the tridiagonal form above does not exist or is trivial (depending on the Jordan structure imposed) if $\lambda _1 = \lambda _2 = \lambda _3 = 0$.

Exact pole assignment with regularity by output feedback in descriptor systems—Part I

In this paper, the first of two, we examine the problem of modal control of linear, time-invariant, multivariable descriptor systems. In particular, for systems satisfying a certain inequality we will give necessary and sufficient conditions for the existence of an output feedback matrix (possibly complex) which exactly assigns an arbitrary set of r (the number of independent semi-state variables of the system) distinct complex numbers to a generalized eigenspectrum of the closed loop system while ensuring that the resulting system has a unique smooth solution for each admissible initial condition. Furthermore, it is shown that if the set of proposed poles contains only real elements, the assignment can be carried out using real feedback. Next, we present conditions that are necessary and sufficient for pole assignability by real feedback of a self-conjugate set of complex numbers provided that the system satisfies a further inequality.

On the eigenvalues of principal submatrices of normal, hermitian and symmetric matrices

Let λ1…λ n be distinct complex numbers and let λ i1…λ in−1 ,i= 1,…n, be complex numbers, not necessarily distinct. We show that the problem of determining when a normal n × n matrix A exists satisfying σ(A) ={λ1…λ n } and σ(Ai )={λ i1…λ in−1 },i=1,…,n where Ai is the (n−1) × (n−1) principal submatrix obtained from A by deleting its ith row and column, is equivalent to determining whether a certain n×n matrix constructed from the λ i ,λ ij is unistochastic. This result, coupled with the characterization of unistochastic 3 × 3 matrices in [2], allows a characterization of the possible eigenvalues of the three 2 × 2 principal submatrices of a normal, Hermitian or real symmetric 3 × 3 matrix. Further results obtained involve the possible eigenvalues of a single (n−1) × (n−1) principal submatrix of a normal n × n matrix and the possible eigenvalues of a pair of (n−1) × (n−1) principal submatrices of a normal, Hermitian or real symmetric n × n matrix. Some of these results are extended to the case where the λ i are not distinct.

A construction of q-analogue of Dedekind sums

If one looks back the classical proof (cf. Carlitz [4]) of the reciprocity law for Dedekind sums in order to construct q-analogue of Dedekind sums which also have the reciprocity law, one can soon see that the following elementary equation is essential in the proof:for any distinct complex numbers u and v, where means the generating function of Euler numbers associated to u. So we must extend the above equation to the generating function of q-Euler numbers for our purpose.

Stabilizability of linear systems using discrete feedback

The eigenvalue assignment problem of a controllable continuous linear system using continuous and discrete feedback loops is discussed. It is shown that for a given set of distinct self-conjugate complex numbers included inside of the unit circle, and for an arbitrarily given value of the sampling frequency (in the discrete feedback loop), it is possible to choose constant feedback matrices such that eigenvalues of the matrix of dynamics of the closed-loop system are located sufficiently near to appropriate numbers of the given set of complex numbers. The obtained result may immediately be applied in the case of discrete linear systems with sampling of the input signal and continuous state feedbacks.

Nuclear proteins from the rat pituitary gland bind to regulatory sequences of the thyrotrophin-beta gene.

Interactions between DNA sequences and nuclear proteins are important in the regulation of gene expression. We have applied a gel retardation method to examine the binding of nuclear proteins from the rat pituitary gland to regulatory sequences of the rat TSH beta-subunit gene. Binding between nuclear protein and DNA is demonstrated by an alteration in electrophoretic mobility of the DNA. A 0.4 M NaCl fraction of pituitary nuclear proteins bound to a fragment of DNA containing the promoter region of the TSH-beta gene, but not to plasmid DNA or insulin cDNA of comparable size. This nuclear protein fraction also bound to fragments of DNA containing sequences from the 5'-flanking region of the gene; binding of nuclear proteins was evident as far as 1 kb 5' to the structural gene. Each of these regions of the TSH-beta gene formed a number of distinct complexes with the nuclear protein extract, distinguished by their differing binding affinities in the presence of poly(dI-dC), and by their electrophoretic mobility. These findings suggest that transcriptional regulation of the rat TSH-beta gene may be mediated by interaction of a number of nuclear proteins with regulatory sequences of DNA up to 1 kb upstream of the transcriptional start site of the gene.

Eigenvalue placement for generalized linear systems

This paper deals with some aspects of eigenvalue placement by state feedback for generalized linear systems described by E = Ax + Bu , where E is a singular map. It is shown that controllability of the infinite eigenvalues of the pencil ( sE − A ) is equivalent to the existence of a state feedback map which assigns those eigenvalues to pre-specified complex numbers. A procedure for the assignment of all eigenvalues to distinct complex numbers is also discussed.

On exponential bases for the Sobolev spaces over an interval

We suppose that K is a countable index set and that Λ = {λ k¦ k ϵ K} is a sequence of distinct complex numbers such that E(Λ) = {e λ k t ¦ λ k ϵ Λ} forms a Riesz (strong) basis for L 2[ a, b], a < b. Let Σ = { σ 1, σ 2,…, σ m } consist of m complex numbers not in Λ. Then, with p( λ) = Π k = 1 m ( λ − σ k ), E(Σ ∪ Λ) = {e σ 1 t …, e σ m t } ∪ { e λ k t p(λ k) ¦ k ϵ K} forms a Riesz (strong) bas Sobolev space H m [ a, b]. If we take σ 1, σ 2,…, σ m to be complex numbers already in Λ, then, defining p( λ) as before, E(Λ − Σ) = {p(λ k) e λ k t ¦ k ϵ K, λ k ≠ σ j = 1,…, m} forms a Riesz (strong) basis for the space H − m [ a, b]. We also discuss the extension of these results to “generalized exponentials” t n e λ k t .

ASYMPTOTICS OF HADAMARD DETERMINANTS AND THE CONVERGENCE OF ROWS OF PADÉ APPROXIMANTS FOR SUMS OF EXPONENTIALS

The asymptotics of Hadamard determinants (of dimension ) for arbitrary fixed and for the function are studied, where are distinct complex numbers with unit modulus. A theorem on the convergence of the th row of the Pade table for the function ( are arbitrary distinct complex numbers) in the topology of holds for arbitrary natural numbers and equal to the number of with modulus that is maximal among the , .Bibliography: 8 titles.

INTERPOLATION PROBLEMS, NONTRIVIAL EXPANSIONS OF ZERO, AND REPRESENTING SYSTEMS

Let be a convex domain with support function , and let be distinct complex numbers. In this paper the author determines when the system is absolutely representing in the space of functions analytic in , with the topology of uniform convergence on compact sets. In particular he proves the Theorem. Let be an exponential function with indicator and simple zeros . For the system to be absolutely representing in it is necessary and sufficient that either of the following two conditions hold: 1) The system has a nontrivial expansion of zero in , i.e. for every . 2) is a function of completely regular growth and there exists a function of class such that Bibliography: 16 titles.

Divided Differences and Systems of Nonharmonic Fourier Series

Suppose that ${\omega _{n,0}},{\omega _{n,1}}, \ldots ,{\omega _{n,k}}$ are distinct complex numbers with $|n - {\omega _{n,j}}| \leqslant \delta$ for all $n \in {\mathbf {Z}},j = 0,1, \ldots ,k$. We show that if $\delta > 0$ is small enough then, given complex numbers ${c_{n,j}}(n \in {\mathbf {Z}},j = 0,1, \ldots ,k)$ there exists $f \in {L^2}( - (k + 1)\pi ,(k + 1)\pi )$ with \[ \int _{ - (k + 1)\pi }^{(k + 1)\pi } {f(t){e^{ - it{\omega _{n,j}}}}} dt = {c_{n,j}}\quad {\text {for}}\;n \in {\mathbf {Z}},j = 0,1, \ldots ,k\] if and only if certain “divided differences” involving the ${c_{n,j}}$’s and the ${\omega _{n,j}}$’s are square summable. This extends a classical theorem of Paley and Wiener, which is equivalent to the case $k = 0$ above.

Divided differences and systems of nonharmonic Fourier series

Suppose that ω n , 0 , ω n , 1 , … , ω n , k {\omega _{n,0}},{\omega _{n,1}}, \ldots ,{\omega _{n,k}} are distinct complex numbers with | n − ω n , j | ⩽ δ |n - {\omega _{n,j}}| \leqslant \delta for all n ∈ Z , j = 0 , 1 , … , k n \in {\mathbf {Z}},j = 0,1, \ldots ,k . We show that if δ &gt; 0 \delta &gt; 0 is small enough then, given complex numbers c n , j ( n ∈ Z , j = 0 , 1 , … , k ) {c_{n,j}}(n \in {\mathbf {Z}},j = 0,1, \ldots ,k) there exists f ∈ L 2 ( − ( k + 1 ) π , ( k + 1 ) π ) f \in {L^2}( - (k + 1)\pi ,(k + 1)\pi ) with \[ ∫ − ( k + 1 ) π ( k + 1 ) π f ( t ) e − i t ω n , j d t = c n , j for n ∈ Z , j = 0 , 1 , … , k \int _{ - (k + 1)\pi }^{(k + 1)\pi } {f(t){e^{ - it{\omega _{n,j}}}}} dt = {c_{n,j}}\quad {\text {for}}\;n \in {\mathbf {Z}},j = 0,1, \ldots ,k \] if and only if certain “divided differences” involving the c n , j {c_{n,j}} ’s and the ω n , j {\omega _{n,j}} ’s are square summable. This extends a classical theorem of Paley and Wiener, which is equivalent to the case k = 0 k = 0 above.

On fundamental sequences of translates

We find Müntz-type theorems for sequences of the form { f ( t + c n ) } \{ f(t + {c_n})\} or { exp ⁡ ( − c n t ) f ( t ) } \{ \exp ( - {c_n}t)f(t)\} on [ 0 , ∞ ) [0,\infty ) .

A note on simultaneous polynomial approximation of exponential functions

Let α1, …, αmbe distinct complex numbers and τ(1), …, τ(m) be non-negative integers. We obtain conditions under which the functionsform a perfect system, that is, for every set ρ(1), …, ρ(m) of non-negative integers, there are polynomialsa1(z), …,am(z) with respective degrees exactly ρ(1)−1, …, ρ(m)−1, such that the functionhas a zero of order at least ρ(1) + … + ρ(m)−1 at the origin. Moreover, subject to the evaluation of certain determinants, we give explicit formulae for the approximating polynomialsa1(z), …,am(z).

Interpolation in a classical Hilbert space of entire functions

Let H denote the Paley-Wiener space of entire functions of exponential type π \pi which belong to L 2 ( − ∞ , ∞ ) {L^2}( - \infty ,\infty ) on the real axis. A sequence { λ n } \{ {\lambda _n}\} of distinct complex numbers will be called an interpolating sequence for H if T H ⊃ l 2 TH \supset {l^2} , where T is the mapping defined by T f = { f ( λ n ) } Tf = \{ f({\lambda _n})\} . If in addition { λ n } \{ {\lambda _n}\} is a set of uniqueness for H, then { λ n } \{ {\lambda _n}\} is called a complete interpolating sequence. The following results are established. If Re ⁡ ( λ n + 1 ) − Re ⁡ ( λ n ) ≥ γ &gt; 1 \operatorname {Re} ({\lambda _{n + 1}}) - \operatorname {Re} ({\lambda _n}) \geq \gamma &gt; 1 and if the imaginary part of λ n {\lambda _n} is sufficiently small, then { λ n } \{ {\lambda _n}\} is an interpolating sequence. If | Re ⁡ ( λ n ) − n | ≤ L ≤ ( log ⁡ 2 ) / π ( − ∞ &gt; n &gt; ∞ ) |\operatorname {Re} ({\lambda _n}) - n| \leq L \leq (\log 2)/\pi \;( - \infty &gt; n &gt; \infty ) and if the imaginary part of λ n {\lambda _n} is uniformly bounded, then { λ n } \{ {\lambda _n}\} is a complete interpolating sequence and { e i λ n t } \{ {e^{i{\lambda _n}t}}\} is a basis for L 2 ( − π , π ) {L^2}( - \pi ,\pi ) . These results are used to investigate interpolating sequences in several related spaces of entire functions of exponential type.

A note on absolute summability

sents a real circle with a positive radius having the origin in its interior (because the constant term in this equation is negative); and when c, d are not both zero, the straight line represented by the equation (4) meets this circle in two real and distinct points. We can, therefore, always find (at least) two distinct complex numbers zs such that z =zk satisfy the equation (1). This proves slightly more than what we set out to prove. For real inner product spaces X we may similarly reduce the proof of the corresponding theorem to showing that a certain quadratic equation with real coefficients has real roots. The author wishes to thank Professor V. Ganapathy Iyer for his encouragement and guidance.

On a class of entire functions

where the ~v are distinct complex numbers, and the coefficients A,v are arbitrary complex constants. In the present paper I continue his investigations a little further and prove a general theorem which may have some interest in itself. In order to make the paper self-contained, I have repeated some o f Gelfond's proofs. 1. Let ~o, ~ . . . . . ~,-1 be finitely many distinct complex numbers; let mo, ml . . . . . m,_a be an equal number of positive integers; and let