Harmonic analysis on symmetric Stein manifolds from the point of view of complex analysis

The classical theory of finite dimensional representations of compact and complex semisimple Lie groups is discussed from the perspective of multidimensional complex geometry and analysis. The key tool is the complex horospherical transform which establishes a duality between spaces of holomorphic functions on symmetric Stein manifolds and dual horospherical manifolds. Communicated by: Toshiyuki Kobayashi     

More odd graph theory from another point of view

The Kneser graph $K(n,k)$ has as vertices all $k$-element subsets of $\left[n\right]=\{1,2,\dots ,n\}$ and an edge between any two vertices that are disjoint. If $n=2k+1$, then $K(n,k)$ is called an odd graph. Let $n>4$ and $1<k<\frac{n}{2}$. In the present paper, we show that if the Kneser graph $K(n,k)$ is of even order where $n$ is an odd integer or both of the integers $n,k$ are even, then $K(n,k)$ is a vertex-transitive non Cayley graph. Although, these are special cases of Godsil [7], unlike his proof that uses some very deep group-theoretical facts, ours uses no heavy group-theoretic facts. We obtain our results by using some rather elementary facts of number theory and group theory. We show that ‘almost all’ odd graphs are of even order, and consequently are vertex-transitive non Cayley graphs. Finally, we show that if $k>4$ is an even integer such that $k$ is not of the form $k=2^t$ for some $t>2$, then the line graph of the odd graph $O_{k+1}$ is a vertex-transitive non Cayley graph.     

Variational field theory from the point of view of direct methods

In this paper we show that the classical field theory ofWeierstrass–Hilbert can be strengthen on applying direct methods. Concretely, given a field of extremals and an extremal that is an element of the field, we can show that the latter gives minimum in the class of Lipschitz functions with the same boundary data and with the graphs in the set covered by the field. We suggest the two proofs: a modern one (exploiting Tonelli’s Theorem on lower semicontinuity of integral functionals with respect to the weak convergence of admissible functions in W ^1,1) and the one based only on arguments available already in the 19th century.     

Homogeneous structures on manifolds: Differential geometry from the point of view of differential equations

Translated from Matematicheskie Zametki, Vol. 51, No. 4, pp. 54–68, April, 1992.     

A study of orthogonal arrays from the point of view of design theory

It is shown that several theorems in design theory are “translated” into those in orthogonal array theory.     

Affine normal curvature of hypersurfaces from the point of view of singularity theory

We consider smoothly embedded hypersurfaces under the action of the special affine group . We construct a differential invariant, called affine normal curvature, which assigns to a point and a tangent direction a number. We prove some of its nice properties which connect it with affine principal directions, affine umbilics, and affine mean curvature.      

Equations of hydrodynamic type from the point of view of the theory of webs

Translated from Matematicheskie Zametki, Vol. 50, No. 5, pp. 97–108, November, 1991.     

The algebraic regulator problem from the state-space point of view

We study the algebraic aspects of the regulator problem, using some new ideas in the state-space (“geometric”) approach to feedback design problems for linear multi- variable systems. Necessary and sufficient conditions are given for the solvability of a general version of this problem, requiring output stability, internal stability, and disturbance decoupling as well. An algorithm is given by which these conditions can be verified from the system parameters.     

Applications of proximal calculus to fixed point theory on Riemannian manifolds

We prove a general form of a fixed point theorem for mappings from a Riemannian manifold into itself which are obtained as perturbations of a given mapping by means of general operations which in particular include the cases of sum (when a Lie group structure is given on the manifold) and composition. In order to prove our main result we develop a theory of proximal calculus in the setting of Riemannian manifolds.     

Hamiltonian loops from the ergodic point of view

Let G be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold Y. A loop h:S¹→G is called strictly ergodic if for some irrational number α the associated skew product map T:S¹×Y→S¹×Y defined by T(t,y)=(t+α,h(t)y) is strictly ergodic. In the present paper we address the following question. Which elements of the fundamental group of G can be represented by strictly ergodic loops? We prove existence of contractible strictly ergodic loops for a wide class of symplectic manifolds (for instance for simply connected ones). Further, we find a restriction on the homotopy classes of smooth strictly ergodic loops in the framework of Hofer’s bi-invariant geometry on G. Namely, we prove that their asymptotic Hofer’s norm must vanish. This result provides a link between ergodic theory and symplectic topology. Received July 7, 1998 / final version received September 14, 1998     

An alternative point of view to the theory of fractional Fourier transform

The concept of the fractional Fourier transform is framed withinthe context of quantum evolution operators. This point of viewyields an extension of the above concept and greatly simplifiesthe underlying operational algebra. It is also proved that amultidimensional extension can be performed by using a biorthogonalmultiindex harmonic oscillator basis. It is finally shown thatmost of the proposed physical interpretations of the fractionalFourier transform are just trivial consequences of the analysisdeveloped in this paper.     

The Gibbs' phenomenon from a signal processing point of view

Recently, Fay and Kloppers gave two proofs to show that the well-known Gibbs' phenomenon for Fourier series at a jump discontinuity depends only on the size of the jump and is a multiple of the integral 1/π ∫₀ ^π (sin x / x) dx. We give another proof, based upon low-pass filtering of the Fourier transform, that uses the observation that a truncated Fourier series for a function ? (x) is ‘very nearly’ equal to the convolution integral 1/π ∫ _-∞ ^+∞ ? (x - t)(sin nt / t) dt.     

The SIR epidemic model from a PDE point of view

We present a derivation of the classical Susceptible-Infected-Removed (SIR) and Susceptible-Infected-Removed-Susceptible (SIRS) models through a mean-field approximation from a discrete version of SIR(S). We then obtain a hyperbolic forward Kolmogorov equation, and show that its projected characteristics recover the standard SIR(S) model. Moreover, for the SIRS model, we show that the long time limit of the SIRS model will be a Dirac measure supported on the corresponding isolated equilibria. For the SIR model, we show that the long time limit is a Radon measure supported in a segment of nonisolated equilibria.     

Perfect codes from the dual point of view I

The two concepts dual code and parity check matrix for a linear perfect 1-error correcting binary code are generalized to the case of non-linear perfect codes. We show how this generalization can be used to enumerate some particular classes of perfect 1-error correcting binary codes. We also use it to give an answer to a problem of Avgustinovich: whether or not the kernel of every perfect 1-error correcting binary code is always contained in some Hamming code.     

The utility of deductibles from the insurer's point of view

The relationship between the premiums for deductible cover and for full cover are analyzed with respect to the utility for the insurer. Theorem 1 shows that within natural bounds for the premiums neither form of coverage is generally preferred by all insurers. Theorems 2–4 indicate that in many realistic cases a deductible cover requires a higher risk loading factor by the insurer than a full insurance cover. Theorem 2(b) also contains an exception to this rule.