Zeros Of Derivatives Research Articles

On the zeros of functions of the class

Let be meromorphic in and of finite order. We construct some angular regions for the sequence (z ν) such that f has infinitely many zeros. We treat the cases where the order is an integer value or not in different ways. The results have applications to zeros of derivatives of meromorphic functions.

On simple zeros of derivatives of the Riemann -function

We get a lower bound for the number of simple zeros of the function on the critical line, where .

Proof of a conjecture of Pólya on the zeros of successive derivatives of real entire functions

We prove Pólya’s conjecture of 1943: For a real entire function of order greater than 2 with finitely many non-real zeros, the number of non-real zeros of the nth derivative tends to infinity, as $n\to\infty$. We use the saddle point method and potential theory, combined with the theory of analytic functions with positive imaginary part in the upper half-plane.

Non-real zeros of higher derivatives of real entire functions of infinite order

Letf be a real meromorphic function of infinite order in the plane such thatf has finitely many poles. Then for eachk≥3, at least one off andf (k) has infinitely many non-real zeros. Together with a result of Edwards and Hellerstein, this establishes the analogue for higher derivatives of a conjecture going back to Wiman around 1911.

Simultaneous Spectrophotometric Multicomponent Determination of Folic Acid, Thiamine, Riboflavin, and Pyridoxal by Using Double Divisor–Ratio Spectra Derivative–Zero Crossing Method

A graphical method based on the double divisor–ratio spectra derivative–zero crossing was proposed for the spectrophotometric multicomponent analysis of synthetic mixtures containing folic acid (vitamin B0), thiamine (vitamin B1), riboflavin (vitamin B2), and pyridoxal (vitamin B6) in the presence of strongly overlapping signals without any chemical separation. The graphical method is based on the use of derivative signals of the ratio spectra by using double divisor. The selected data of ultraviolet‐visible absorption spectra consisted of points corresponding to the spectral region 225–475 nm. The linear determination ranges were 1–26 µg mL−1 of B0, 4–50 of µg mL−1 of B1, 1–28 µg mL−1 of B2, 6–42 µg mL−1 of B6 in a buffer solution at (pH 5.8). The proposed method applied to the simultaneous determination of four vitamins by using synthetic quaternary mixtures. The obtained results were statistically compared with each other.

Non-oscillating Paley-Wiener functions

A non-oscillating Paley-Wiener function is a real entire functionf of exponential type belonging toL 2(R) and such that each derivativef (n),n=0, 1, 2,…, has only a finite number of real zeros. It is established that the class of such functions is non-empty and contains functions of arbitrarily fast decay onR allowed by the convergence of the logarithmic integral. It is shown that the Fourier transform of a non-oscillating Paley-Wiener function must be infinitely differentiable outside the origin. We also give close to best possible asymptotic (asn→∞) estimates of the number of real zeros of then-th derivative of a functionf of the class and the size of the smallest interval containing these zeros.

Use of Cartesian Rotation Vectors in Brownian Dynamics Algorithms: Theory and Simulation Results

AbstractSummary: We have shown that the components of Cartesian rotation vectors can be used successfully as generalized coordinates describing angular orientation in Brownian dynamics simulations of non‐spherical nanoparticles. For this particular choice of generalized coordinates, we rigorously derived the conformation‐space diffusion equations from kinetic theory for both free nanoparticles and nanoparticles interconnected by springs or holonomic constraints into polymer chains. The equivalent stochastic differential equations were used as a foundation for the Brownian dynamics algorithms. These new algorithms contain singularities only for points in the conformation‐space where both the probability density and its first coordinate derivative equal zero (weak singularities). In addition, the coordinate values after a single Brownian dynamics time step are throughout the conformation‐space equal to the old coordinate values plus the respective increments. For some parts of the conformation‐space these features represent a major improvement compared to the situation when Eulerian angles describe rotational dynamics. The presented simulation results of the equilibrium probability density for free nanoparticles are in perfect agreement with the results from kinetic theory.Simulation of p(eq)(Φ) for free nanoparticles.imageSimulation of p(eq)(Φ) for free nanoparticles.

Zeros of higher order logarithmic derivatives

We extend a result of Langley and Shea [J.K. Langley and D. Shea (1998). On multiple points of meromorphic functions. J. London Math. Soc., 57(2), 371–384.] concerning the distribution of zeros of the logarithmic derivative f ′/f to higher order logarithmic derivatives of the form f (k)/f,.

Unicity of meromorphic functions in a certain family

Let f and g be two meromorphic functions in family which contain all transcendental meromorphic functions having little poles and zeros. By estimating the zeros of the kth derivative of the function we study the relationship between f and g, and obtain some results which are extensions of some known results. As an application, we prove a uniqueness theorem on meromorphic functions that share three values counting multiplicity.

Cubic Spline Population Density Functions and Satellite City Delimitation: The Case of Barcelona

The presence of satellite cities within large metropolitan areas cannot be captured by an exponential function. Cubic spline functions seem more appropriate to depict the polycentric pattern of modern urban systems. Using data from the Barcelona Metropolitan Region, two possible population satellite city delimitation procedures using cubic spline density functions are discussed: one, taking an estimated derivative equal to zero; the other, a density gradient equal to zero. It is argued that a delimitation strategy based on derivatives is more appropriate than one based on gradients because the estimated density can be negative in sections with very low densities and few observations, leading to sudden changes in estimated gradients. It is also argued that delimiting satellite cities using a second derivative with a zero value permits the capture of a more restricted area than using a first derivative zero. This methodology can also be used for intermediate ring delimitation.

Pólya's Shire Theorem for Automorphic Functions

Generalizing a classic result by Polya concerning the zeros of successive derivatives of meromorphic functions in the complex plane, we study the accumulation set of zeros of successive derivatives of automorphic functions in the hyperbolic plane, as well as the connection between the automorphism group and the topology of this set in the Poincare disk model.

Non-real Zeros of Derivatives of Real Entire Functions and the Pólya-Wiman Conjectures

A function f is in the class $ V_2p $ iff $ f(z) = e^{-az^{2p+2}}g(z) $ where a S 0 and g is a constant multiple of a real entire function of genus h 2 p + 1 with only real zeros. The class $ U_2p $ is defined as follows: $ U_0 = V_0 $ , $ U_{2p} = V_{2p}-V_{2p-2} $ . Functions in the class $ U_{2p}^{*} $ are represented as $ g(z) = c(z)f(z) $ where $ f\\in U_{2p} $ and c is a real polynomial with no real zeros. Every real entire function g , of finite order with at most finitely many non-real zeros satisfies $ g\\in U_{2p}^{*} $ for a unique p . We show the exact number of non-real zeros of f" , for $ f\\in U_{2p} $ , in terms of the number of non-real zeros of f' and a geometrical condition on the components of Im Q ( z ) > 0, where $ \\displaystyle Q(z) = z-({f(z)}/{f'(z)}) $ . Further, for a subclass of $ f\\in U_{2p} $ , we show necessary and sufficient conditions for f" to have exactly 2 p non-real zeros. For a subclass of $ U_{2p}^{*} $ we show that if f' has only real zeros, then f" has exactly 2 p non-real zeros. For $ f\\in U_{2p}^{*} $ we show that 2 p is a lower bound for the number of non-real zeros of $ f^{(k)} $ for k S 2.

On differential polynomials, fixpoints and critical values of meromorphic functions

We prove some results concerning functions ƒ meromorphic of finite lower order in the plane, such that ƒƒ″−α(ƒ′)2 has few zeros, where α is a constant, using in part new estimates for the multipliers at fixpoints of certain functions. We go on to consider zeros of derivatives and the minimal lower growth of meromorphic functions with finitely many critical values.

On the Zeros of Higher Derivatives of Hardy'sZ-Function

Letkbe any positive integer andN0,k(T) the number of the zeros in the interval (0,T) ofZ(k)(t), thekth derivative of Hardy'sZ-function. We prove an inequality forN0,k(T) (Theorem 1), and also prove that it can be replaced by the equality under the Riemann hypothesis (Theorem 2). The key fact of the proof is the construction of a meromorphic functionηk(s), which satisfies an appropriate recurrence formula and a functional equation.

Zeros of derivatives of meromorphic functions with one pole *

Suppose that f (z) = g(z)/z n where g is a real entire function of finite order with g(0)≠ 0 and n is a positive integer, and that f f′, and f″ have only red zeros. We prove thatthen g is a polynomial of degree not exceeding n+1. This strengthens an earlier result of the author, when one had to consider derivatives up to an order depending on f and the order of the entire function gConversely, if f is of this form where g is a polynomial of degree at most n with only real zeros, then f(k) has only real zeros for all k≥ 0. If the degree of g is n+1 then f(k) has only real zeros for all k≥ 0 if, and only iff and f′ have only real zeros. The proof makes extensive use of complex dynamics

Convolution formulae for functions of Rayleigh type

N Kishore (1963 Proc. Am. Math. Soc. 14 527) considered the Rayleigh functions , , where the are the (non-zero) zeros of the Bessel function and provided a convolution-type sum formula for finding in terms of . Here we investigate corresponding expressions for sums of reciprocal powers of zeros of derivatives and other functions related to Bessel functions. It turns out that we can get results similar to Kishore's expressing in terms of and .

On Multiple Points of Meromorphic Functions

We establish lower bounds for the number of zeros of the logarithmic derivative of a meromorphic function of small growth with few poles.

An upper bound for the zeros of the derivative of Bessel functions

Letj vk ′ denotes thekth positive zero of the derivativeJ v ′ (x)=dJ v (x)/dx of Bessel functionJ v (x) fork=1, 2,…. We establish the upper bound $$j'_{\nu k}< \nu + a_k \left( {\nu + \frac{{{\rm A}_k^3 }}{{a_k^3 }}} \right)^{\frac{1}{3}} + \frac{3}{{10}}a_k^2 \left( {\nu + \frac{{A_k^3 }}{{a_k^3 }}} \right)^{\frac{1}{3}} , \nu \geqslant 0, k = 1,2, \ldots $$ whereA k =2a k √2a k /3,a k =x k ′ 2−1/3 andx k ′ is thekth positive zero the derivativeAi′(x) of the Airy functionAi(x). This bound is sharp for large values ofv and improves known results. Similar inequality holds for thekth positive zeroy vk ′ ofY v ′ (x), too, whereY v (x) denotes the Bessel function of second kind.

On the zeros of the second derivative

Suppose that f is meromorphic of finite order in the plane, and that f″ has only finitely many zeros. We prove a strong estimate for the frequency of distinct poles of f. In particular, if the poles of f have bounded multiplicities, then f has only finitely many poles.

On the differentiability of the coordinate functions of P�yla's space-filling curve

The coordinate functions of Polya's space-filling curve are nowhere differentiable if the smaller angle of the triangular target set is between 30° and 45°, limits included; are not differentiable almost everywhere and have derivative zero on a non-denumerable set if the angle is between 15° and 30°, lower limit included; and have derivative zero almost everywhere if the angle is less than 15°.