Reconstruction of Planar Domains from Partial Integral Measurements

We consider the problem of reconstruction of planar domains from their moments. Specifically, we consider domains with boundary which can be represented by a union of a finite number of pieces whose graphs are solutions of a linear differential equation with polynomial coefficients. This includes domains with piecewise-algebraic and, in particular, piecewise-polynomial boundaries. Our approach is based on one-dimensional reconstruction method of [Bat]* and a kind of "separation of variables" which reduces the planar problem to two one-dimensional problems, one of them parametric. Several explicit examples of reconstruction are given. Another main topic of the paper concerns "invisible sets" for various types of incomplete moment measurements. We suggest a certain point of view which stresses remarkable similarity between several apparently unrelated problems. In particular, we discuss zero quadrature domains (invisible for harmonic polynomials), invisibility for powers of a given polynomial, and invisibility for complex moments (Wermer's theorem and further developments). The common property we would like to stress is a "rigidity" and symmetry of the invisible objects. * D.Batenkov, Moment inversion of piecewise D-finite functions, Inverse Problems 25 (2009) 105001Comment: Proceedings of Complex Analysis and Dynamical Systems V, 201

Accuracy of reconstruction of spike-trains with two near-colliding nodes

We consider a signal reconstruction problem for signals $F$ of the form $F(x)=\sum_{j=1}^{d}a_{j}\delta\left(x-x_{j}\right),$ from their moments $m_k(F)=\int x^kF(x)dx.$ We assume $m_k(F)$ to be known for $k=0,1,\ldots,N,$ with an absolute error not exceeding $\epsilon > 0$. We study the "geometry of error amplification" in reconstruction of $F$ from $m_k(F),$ in situations where two neighboring nodes $x_i$ and $x_{i+1}$ near-collide, i.e $x_{i+1}-x_i=h \ll 1$. We show that the error amplification is governed by certain algebraic curves $S_{F,i},$ in the parameter space of signals $F$, along which the first three moments $m_0,m_1,m_2$ remain constant

Economic Cycles of Carnot Type

Originally, the Carnot cycle was a theoretical thermodynamic cycle that provided an upper limit on the efficiency that any classical thermodynamic engine can achieve during the conversion of heat into work, or conversely, the efficiency of a refrigeration system in creating a temperature difference by the application of work to the system. The first aim of this paper is to introduce and study the economic Carnot cycles concerning Roegenian economics, using our thermodynamic–economic dictionary. These cycles are described in both a Q−P diagram and a E−I diagram. An economic Carnot cycle has a maximum efficiency for a reversible economic “engine”. Three problems together with their solutions clarify the meaning of the economic Carnot cycle, in our context. Then we transform the ideal gas theory into the ideal income theory. The second aim is to analyze the economic Van der Waals equation, showing that the diffeomorphic-invariant information about the Van der Waals surface can be obtained by examining a cuspidal potential

On Oscillations in a Gene Network with Diffusion

We consider one system of partial derivative equations of the parabolic type as a model of a simple 3D gene network in the presence of diffusion of its three components. Using discretization of the phase portrait of this system, comparison theorems, and other methods of the qualitative theory of differential equations, we show uniqueness of the equilibrium solution to this system and find conditions of instability of this equilibrium. Then, we obtain sufficient conditions of existence of at least one oscillating functioning regime of this gene network. An estimate of lower and upper bounds for periods of these oscillations is given as well. In quite a similar way, these results on the existence of cycles in 3D gene networks can be extended to higher-dimensional systems of parabolic or other evolution equations in order to construct mathematical models of more complicated molecular–genetic systems