The stability method is very useful for obtaining exact solutions of many extremal graph problems. Its key step is to establish the stability property which, roughly speaking, states that any two almost optimal graphs of the same order n can be made isomorphic by changing o(n2) edges. Here we show how the recently developed theory of graph limits can be used to give an analytic approach to stab...

This paper extends the singular value decomposition to a path of matrices E(t). An analytic singular value decomposition of a path of matrices E(t) is an analytic path of factorizations E(t) = X(t)S(t)Y (t)T where X(t) and Y (t) are orthogonal and S(t) is diagonal. To maintain di erentiability the diagonal entries of S(t) are allowed to be either positive or negative and to appear in any order....

The double product integral →← ∏ (1+λ(dA†⊗dA−dA⊗dA†)) is constructed as a family of unitary operators in double Fock space satisfying quantum stochastic differential equations with unbounded operator coefficients.

With the widespread awareness of Big Data, mission leaders now expect that the data available to their organizations will be immediately relevant to their missions. However, the continuing onslaught of the ”data tsunami”, with data becoming more diverse, changing nature more quickly, and growing in volume and speed, confounds all but the simplest analysis and the most capable organizations. The...

Harmonic and analytic functions have natural discrete analogues. Harmonic functions can be defined on every graph, while analytic functions (or, more precisely, holomorphic forms) can be defined on graphs embedded in orientable surfaces. Many important properties of the “true” harmonic and analytic functions can be carried over to the discrete setting.

ANALYTIC COMBINATORICS is primarily a book about combinatorics, that is, the study of finite structures built according to a finite set of rules. Analytic in the title means that we concern ourselves with methods from mathematical analysis, in particular complex and asymptotic analysis. The two fields, combinatorial enumeration and complex analysis, are organized into a coherent set of methods ...

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In the treatment of superficial lesions with 8-13 MeV electrons, lead shields are often used to protect the underlying tissue. Measurements were made with film and ion chamber to analyse various aspects of external and internal shielding in electron beam therapy. Data were obtained on the thickness of lead required for shielding, the effect of blocking on dose-rate, electron-backscattering from...