Divergence Of Series Research Articles

Necessary and sufficient conditions for the stabilization of a solution to the Dirichlet problem for parabolic equation

We obtain necessary and sufficient conditions for the stabilization to zero of the solution to the Dirichlet problem for a nondivergence parabolic equation with measurable coefficients. The conditions are formulated in terms of the divergence of Wiener series containing the heat capacity introduced by E. M. Landis. In the case of the heat equation, these conditions yield the stabilization criterion. Bibliography: 3 titles.

The Limit Comparison Test Needs Positivity

SummaryThe limit comparison test for positive series does not extend to general series. An example is given. In a certain sense, this is the only possible example. Given a conditionally convergent series, there exists a termwise much smaller series so that the sum of the two series diverges. Given a divergent series with terms tending to zero, there exists a convergent but termwise much bigger series.

Lineability of Universal Divergence of Fourier Series

It is proved that large vector spaces of continuous complex functions on the unit circle exist such that all their nonzero members satisfy that, for many small subsets E of the circle, the partial sums of their Fourier series approximate any prescribed function on E. This completes or improves a number of results by several authors.

On Divergence of Fourier Series by Some Methods of Summability

A new summability method of series is introduced and studied. The particular cases of this method are, for example, variable-order Cesaro and Riesz methods. Applications to divergence problem of Fourier series are given. An extension of Kolmogorov, Schipp, and Bočkarev’s well-known theorems on divergence of Fourier trigonometric, Walsh, and orthonormal series is established.

Multifractal analysis of the divergence of Fourier series

A famous theorem of Carleson says that, given any function $f\in L^p(\TT)$, $p\in(1,+\infty)$, its Fourier series $(S_nf(x))$ converges for almost every $x\in \mathbb T$. Beside this property, the series may diverge at some point, without exceeding $O(n^{1/p})$. We define the divergence index at $x$ as the infimum of the positive real numbers $\beta$ such that $S_nf(x)=O(n^\beta)$ and we are interested in the size of the exceptional sets $E_\beta$, namely the sets of $x\in\mathbb T$ with divergence index equal to $\beta$. We show that quasi-all functions in $L^p(\TT)$ have a multifractal behavior with respect to this definition. Precisely, for quasi-all functions in $L^p(\mathbb T)$, for all $\beta\in[0,1/p]$, $E_\beta$ has Hausdorff dimension equal to $1-\beta p$. We also investigate the same problem in $\mathcal C(\mathbb T)$, replacing polynomial divergence by logarithmic divergence. In this context, the results that we get on the size of the exceptional sets are rather surprizing.

On absolute divergence of Fourier-Haar series of characteristic functions

The paper studies some questions related to almost everywhere, absolute divergence of the series in Haar system. It is constructed a measurable set E ⊂ [0, 1] such that the Fourier-Haar series of the characteristic function of the set E absolutely diverges almost everywhere.

Another proof on the existence of Mertens’s constant

Let $\mathcal{P}$ denote the set of all prime numbers, and let $p_{k}$ denothe the $k$th prime. In 1873 Mertens presented a quantitative proof of the divergence of the series $\sum_{p\in \mathcal{P}}\frac{1}{p}$ by showing the limit $B:= \lim_{x\to\infty}(\sum_{p\le x}\frac{1}{p} - \log\log x)$ exists with $B=0.26149\ldots$. In this paper we give another proof on the divergence of the above series. We prove the following Theorem The sequence $f(n):= \sum_{k=1}^{n}\frac{1}{p_{k}} - \log\log n$, $n=2,3, \ldots$, is decreasing and bounded from below, and its limit equals the Mertens’s constant $B$. In proofs of the first two conditions we use only classical estimations for $p_{k}$, obtained in 1939 and 1962 by Rosser and Schoenfeld.

Generalized Hardy identity and relations to Gibbs–Wilbraham and Pinsky phenomena

We consider the Fourier series of the indicator functions of several dimensional balls. For the spherical partial sum of the Fourier series, we extract the Gibbs–Wilbraham (or Gibbs), Pinsky and the third phenomena as an extension of Hardy's identity. The third phenomenon has been shown by Kuratsubo recently. We prove the Gibbs–Wilbraham phenomenon for all dimensions and give another proof of the Pinsky phenomenon. Pinsky gave the order of the divergence for the Fourier inversion at the origin. We give the order of the divergence of the Fourier series at the origin and show that both orders coincide. We also investigate the uniform convergence for the Fourier series and the Fourier inversion.

Ranks in Families of Jacobian Varieties of Twisted Fermat Curves

AbstractIn this paper, we prove that the unboundedness of ranks in families of Jacobian varieties of twisted Fermat curves is equivalent to the divergence of certain infinite series.

Maximal sets of convergence and unbounded divergence of multiple fourier series with J κ -lacunary sequence of partial sums

Maximal sets of convergence and unbounded divergence of multiple fourier series with J κ -lacunary sequence of partial sums

On Divergence of Fourier Series with Respect to Multiplicative Systems on the Sets of Measure Zero

Abstract For any multiplicative system of bounded type and any set of measure zero there exists a bounded measurable function whose Fourier series with respect to this system diverges on this set.

Summation Characterization of the Recessive Solution for Half-Linear Difference Equations

We show that the recessive solution of the second-order half-linear difference equation , , , where are real-valued sequences, is closely related to the divergence of the infinite series .

Kergin approximation in Banach spaces

We explore the convergence of Kergin interpolation polynomials of holomorphic functions in Banach spaces, which need not be of bounded type. We also investigate a case where the Kergin series diverges.

The Hilbert problem: The case of infinitely many discontinuity points of coefficients

We obtain a solution to the Hilbert boundary value problem in the theory of analytic functions on the half-plane in the case that the coefficients of the boundary condition have countably many discontinuity points of the first kind. We elaborate the two substantially different situations: the series consisting of the jumps of the argument of the coefficient function and the increments of its continuous part converges and this series diverges. Accordingly, Hilbert problems with finite and infinite indices result. We derive formulas for the general solution and investigate the pictures of solvability of these problems.

On the theory of cavities with point-like perturbations: part I. General theory

The theoretical interpretation of measurements of ‘wavefunctions’ and spectra in electromagnetic cavities excited by antennas is considered. Assuming that the characteristic wavelength of the field inside the cavity is much larger than the radius of the antenna, we describe antennas as ‘point-like perturbations’. This approach strongly simplifies the problem reducing the whole information on the antenna to four effective constants. In the framework of this approach we overcame the divergency of series of the phenomenological scattering theory and justify assumptions lying at the heart of ‘wavefunction measurements’. This self-consistent approach allowed us to go beyond the one-pole approximation, in particular, to treat the experiments with degenerate states. The central idea of the approach is to introduce the ‘renormalized’ Green function, which contains the information on boundary reflections and has no singularity inside the cavity.

The dielectric virial expansion and the models of dipolar hard-sphere fluid

The virial expansion technique to determine the dielectric constant epsilon of dipolar hard-sphere fluid is developed. It is shown that the formalism allows to bring into agreement the results of Debye's, Onsager's, and Langevin's to the problem. The third virial coefficient of epsilon is considered as a series over dipolar parameter lambda=m(2)d(3)kT. The terms up to O(lambda(11)) are calculated analytically providing a correct description of the third virial coefficient for small and intermediate values of lambda (0<or=lambda<or=4). The results of the dielectric virial series are compared with the Monte Carlo data for epsilon found by Matyushov and Ladanyi [J. Chem. Phys. 110, 994 (1999)]. The theory is in agreement with simulations only at small values of lambda<or=2. At higher polarities, the virial series diverges. Realization of the renormalization procedure permits to enlarge the range of applicability of the virial series. In this way, the new expression for the dielectric constant as a function of two dipolar parameters, lambda and y=4 pi nm(2)9kT, has been found explicitly. The expression gives a perfect upper bound of the dielectric constant and is more reliable for determination of epsilon than the previously known ones.

The Divergence of Balanced Harmonic-like Series

Carl V. Lutzer (cvlsma@rit.edu) is an Assistant Professor of Mathematics and Statistics at Rochester Institute of Technology. He received his Ph.D. in 2000 under the direction of Peter Hislop at the University of Kentucky. His research interests include PDE (mixtures of eye lens proteins that undergo liquid-liquid phase separation), and applied dynamical systems (micro-electro-mechanical systems). In addition to mathematics and teaching, he enjoys fencing (sabre) and writing fiction.

The Divergence of Balanced Harmonic-like Series

The Divergence of Balanced Harmonic-like Series

Divergence of geometrical optics series at the boundary of its applicability: An analytical example in elementary functions (2D Gaussian beam in free space)

Abstract An analytical example in elementary functions is presented (2D Gaussian beam diffraction in free space), which demonstrates the divergence of the geometrical optics (GO) series when the conditions for its applicability are violated. This example shows that accounting for higher terms in GO power series leads to divergence and therefore becomes completely useless beyond the boundaries of GO applicability.

Building Blocks Problem Related to Harmonic Series

In this discussion I give an explanation of the divergence and convergence of infinite series through the building blocks problem and at the same time I touch on the fact that mathematics is not just about manipulating complicated numerical formulas but also a field in which logical ways of thought are learnt. I emphasize that in order to overcome the aversion of university students to mathematics, teachers must pour their energies into developing study materials taken from topics relevant to the students.