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Let $mathscr{X}$ be the set of positive real sequences $x=(x_n)$ such that the series $sum_n x_n$ is divergent. For each $x in mathscr{X}$, let $mathcal{I}_x$ be the collection of all $Asubseteq mathbf{N}$ such that the subseries $sum_{n in A}x_n$ is convergent. Moreover, let $mathscr{A}$ be the set of sequences $x in mathscr{X}$ such that $lim_n x_n=0$ and $mathcal{I}_x eq mathcal{I}_y$ for all sequences $y=(y_n) in mathscr{X}$ with $liminf_n y_{n+1}/y_n>0$. We show that $mathscr{A}$ is comeager and that contains uncountably many sequences $x$ which generate pairwise nonisomorphic ideals $mathcal{I}_x$. This answers, in particular, an open question recently posed by M. Filipczak and G. Horbaczewska.
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Assume that $mathcal{I}$ is an ideal on $mathbb{N}$, and $sum_n x_n$ is a divergent series in a Banach space $X$. We study the Baire category, and the measure of the set $A(mathcal{I}):=left{t in {0,1}^{mathbb{N}} colon sum_n t(n)x_n textrm{ is } mathcal{I}textrm{-convergent}right}$. In the category case, we assume that $mathcal{I}$ has the Baire property and $sum_n x_n$ is not unconditionally convergent, and we deduce that $A(mathcal{I})$ is meager. We also study the smallness of $A(mathcal{I})$ in the measure case when the Haar probability measure $lambda$ on ${0,1}^{mathbb{N}}$ is considered. If $mathcal{I}$ is analytic or coanalytic, and $sum_n x_n$ is $mathcal{I}$-divergent, then $lambda(A(mathcal{I}))=0$ which extends the theorem of Dindov{s}, v{S}alat and Toma. Generalizing one of their examples, we show that, for every ideal $mathcal{I}$ on $mathbb{N}$, with the property of long intervals, there is a divergent series of reals such that $lambda(A(Fin))=0$ and $lambda(A(mathcal{I}))=1$.
This is the translation of Leonhard Eulers paper De Seriebus divergentibus written in Latin into English. Leonhard Euler defines and discusses divergent series. He is especially interested in the example $1!-2!+3!-text{etc.}$ and uses different methods to sum it. He finds a value of about $0.59...$.
Let $f$ be a band-limited function in $L^2({mathbb{R}})$. Fix $T >0$ and suppose $f^{prime}$ exists and is integrable on $[-T, T]$. This paper gives a concrete estimate of the error incurred when approximating $f$ in the root mean square by a partial sum of its Hermite series. Specifically, we show, for $K=2n, quad n in Z_+,$ $$ left[frac{1}{2T}int_{-T}^T[f(t)-(S_Kf)(t)]^2dtright]^{1/2}leq left(1+frac 1Kright)left(left[ frac{1}{2T}int_{|t|> T}f(t)^2dtright]^{1/2} +left[frac{1}{2T} int_{|omega|>N}|hat f(omega)|^2domegaright]^{1/2} right) +frac{1}{K}left[frac{1}{2T}int_{|t|leq T}f_N(t)^2dtright]^{1/2} +frac{1}{pi}left(1+frac{1}{2K}right)S_a(K,T), $$ in which $S_Kf$ is the $K$-th partial sum of the Hermite series of $f, hat f $ is the Fourier transform of $f$, $displaystyle{N=frac{sqrt{2K+1}+% sqrt{2K+3}}{2}}$ and $f_N=(hat f chi_{(-N,N)})^vee(t)=frac{1}{pi}int_{-infty}^{infty}frac{sin (N(t-s))}{t-s}f(s)ds$. An explicit upper bound is obtained for $S_{a}(K,T)$.
In this paper we prove existence, uniqueness and regularity of certain perturbed (subsonic--supersonic) transonic potential flows in a two-dimensional Riemannian manifold with convergent-divergent metric, which is an approximate model of the de Laval nozzle in aerodynamics. The result indicates that transonic flows obtained by quasi-one-dimensional flow model in fluid dynamics are stable with respect to the perturbation of the velocity potential function at the entry (i.e., tangential velocity along the entry) of the nozzle. The proof is based upon linear theory of elliptic-hyperbolic mixed type equations in physical space and a nonlinear iteration method.
The conditional distribution of the next outcome given the infinite past of a stationary process can be inferred from finite but growing segments of the past. Several schemes are known for constructing pointwise consistent estimates, but they all demand prohibitive amounts of input data. In this paper we consider real-valued time series and construct conditional distribution estimates that make much more efficient use of the input data. The estimates are consistent in a weak sense, and the question whether they are pointwise consistent is still open. For finite-alphabet processes one may rely on a universal data compression scheme like the Lempel-Ziv algorithm to construct conditional probability mass function estimates that are consistent in expected information divergence. Consistency in this strong sense cannot be attained in a universal sense for all stationary processes with values in an infinite alphabet, but weak consistency can. Some applications of the estimates to on-line forecasting, regression and classification are discussed.
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