It is well known that Itô’s formula is an essential tool in stochastic analysis. But it cannot be used for general stochastic Volterra integral equations SVIEs . In this paper, we first introduce the concept of quasi-Itô process which is a generalization of well-known Itô process. And then we extend Itô’s formula to a more general form applicable to some kinds of SVIEs. Furthermore, the stabili...

I proposed rate tolerance and discussed its relation to rate distortion in my book “A Generalized Information Theory” published in 1993. Recently, I examined the structure function and the complexity distortion based on Kolmogorov’s complexity theory. It is my understanding now that complexity-distortion is only a special case of rate tolerance while constraint sets change from fuzzy sets into ...

In this paper, we find the greatest values [Formula: see text] and the smallest values [Formula: see text] such that the double inequalities [Formula: see text] and [Formula: see text] hold for all [Formula: see text] with [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text] are the arithmetic-geometric, Toader and generalized logarithmic means of two posi...

The [Formula: see text] and [Formula: see text]-approximation schemes are introduced to approximate the popular generalized Cornu spirals with the help of the parametric rational cubic trigonometric Bézier curves. The [Formula: see text]-approximation scheme has two free parameters whereas [Formula: see text]-approximation scheme has four free parameters. To approximate the generalized Cornu sp...

Using fish-covering model, this paper intuitively explains how to extend Hartley’s information formula to the generalized information formula step by step for measuring subjective information: metrical information (such as conveyed by thermometers), sensory information (such as conveyed by color vision), and semantic information (such as conveyed by weather forecasts). The pivotal step is to di...

Many weighted polynomial inequalities, such as the Bernstein, Marcinkiewicz, Schur, Remez, Nikolskii inequalities, with doubling weights were proved by Mastroianni and Totik for the case [Formula: see text], and by Tamás Erdélyi for [Formula: see text]. In this paper we extend such polynomial inequalities to those for generalized trigonometric polynomials. We also prove the large sieve for gene...

The Krĕın-Langer integral representation of a matrix-valued generalized Nevanlinna function arises in problems of spectral theory and interpolation. A version of this formula which is suitable for such problems, and a corresponding Stieltjes inversion formula, are derived. Some classes of generalized Nevanlinna functions which are defined in terms of behavior at infinity are characterized in te...

In this letter, we compute the corrections to the Cardy-Verlinde formula of d−dimensional Schwarzschild black hole. These corrections stem from the generalized uncertainty principle. Then we show, one can taking into account the generalized uncertainty principle corrections of the Cardy-Verlinde entropy formula by just redefining the Virasoro operator L 0 and the central charge c.

The Krĕın-Langer integral representation of a matrix-valued generalized Nevanlinna function arises in problems of spectral theory and interpolation. A version of this formula which is suitable for such problems, and a corresponding Stieltjes inversion formula, are derived. Some classes of generalized Nevanlinna functions which are defined in terms of behavior at infinity are characterized in te...

A simple formula for computing the generalized Hubbell radiation rectangular source integral [formula in text] is introduced. Tables are given to compare the numerical values derived from our approximation formula with those given earlier in the literature.