1. Basic convexity 2. Boundary structure 3. Minkowski addition 4. Curvature measure and quermass integrals 5. Mixed volumes 6. Inequalities for mixed volumes 7. Selected applications Appendix.

/pdf/convex-bodies-the-brunn-minkowski-theory-3raynvod45.pdf

Convex bodies : the Brunn-Minkowski theory

Consider the function defined for $\alpha _{1},\ \alpha _{2} \in \mathbb{R}$ (α 1 2 +α 2 2 ≠ 0) and $\beta _{1},\beta _{2} \in \mathbb{C}$ by the series 

$$\displaystyle{ E_{\alpha _{1},\beta _{1};\alpha _{2},\beta _{2}}(z) \equiv \sum _{k=0}^{\infty } \frac{z^{k}} {\varGamma (\alpha _{1}k +\beta _{1})\varGamma (\alpha _{2}k +\beta _{2})}\ \ (z \in \mathbb{C}). }$$

(6.1.1)

Such a function with positive α 1 > 0, α 2 > 0 and real $\beta _{1},\beta _{2} \in \mathbb{R}$ was introduced by Dzherbashian [Dzh60].

Multi-index Mittag-Leffler Functions

We investigate the growth of the Nevanlinna characteristic of f(z+η) for a fixed η∈C in this paper. In particular, we obtain a precise asymptotic relation between T(r,f(z+η)) and T(r,f), which is only true for finite order meromorphic functions. We have also obtained the proximity function and pointwise estimates of f(z+η)/f(z) which is a discrete version of the classical logarithmic derivative estimates of f(z). We apply these results to give new growth estimates of meromorphic solutions to higher order linear difference equations. This also allows us to solve an old problem of Whittaker (Interpolatory Function Theory, Cambridge University Press, Cambridge, 1935) concerning a first order difference equation. We show by giving a number of examples that all of our results are best possible in certain senses. Finally, we give a direct proof of a result in Ablowitz, Halburd and Herbst (Nonlinearity 13:889–905, 2000) concerning integrable difference equations.

/pdf/on-the-nevanlinna-characteristic-of-f-z-e-and-difference-3hx04cyhj0.pdf

On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane

The Uncertainty Principle (up) as understood in this lecture is the following informal assertion: a non-zero “object” (a function, distribution, hyperfunction) and its Fourier image cannot be too small simultaneously. “The smallness” is understood in a very broad sense meaning fast decay (at infinity or at a point, bilateral or unilateral), perforated (or bounded, or semibounded) support etc. The UP becomes a theorem for many “smallnesses” and has a multitude of quite concrete quantitative forms. It plays a fundamental role as one of the major themes of classical Fourier analysis (and neighboring parts of analysis), but also in applications to physics and engineering. The lecture is a review of facts and techniques related to the UP; connections with local and non-local shift invariant operators are discussed at the end of the lecture (including some topical problems of potential theory). The lecture is intended for the general audience acquainted with basic facts of Fourier analysis on the line and circle, and rudiments of complex analysis.

/pdf/the-uncertainty-principle-in-harmonic-analysis-1je8xgk79l.pdf

The uncertainty principle in harmonic analysis

Normal forms and desingularization Singular points of planar analytic vector fields Local and global theory of linear systems Functional moduli of analytic classification of resonant germs and their applications Global properties of complex polynomial foliations Appendix. First aid Bibliography Index.

/pdf/lectures-on-analytic-differential-equations-1b5sepa58b.pdf

B. I︠a︡. Levin

Papers

Distribution of zeros of entire functions