Characteristic functions of almost solvable extensions of Hermitian operators

The characteristic operator-functions W() are studied of the almost solvable extensions of an Hermitian operator. The inverse problem is solved, a multiplication theorem is proved, and a formula is derived expressing W() in terms of the Weyl function and the boundary operator. Characteristic functions are computed of various differential and difference operators, with the help of which are proved theorems of the completeness of the systems of proper and adjoint vectors.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 4, pp. 435–459, April, 1992.     

Simulations of radiative effects on the Rayleigh–Taylor instability using the CRASH code

Future experiments at the National Ignition Facility will be able to generate diagnosable Rayleigh–Taylor instability growth in the presence of locally generated, high radiation fluxes. This interplay of radiative energy transfer and hydrodynamic instability is relevant to many astrophysical systems, such as core-collapse red supergiant supernovae. Previous simulations of high-energy-density Rayleigh–Taylor instabilities in the presence of a hot environment near a radiative shock demonstrate behavior that differs from that found in non-radiative cases. However, these simulations considered only 1D or single wavelength cases. Here we report simulations of an entire experimental system using the CRASH code. These simulations lead to modified predictions, attributed to the effects of radial energy losses.     

On the Deficiency Indices of Block Jacobi Matrices Related to Dirac Operators with Point Interactions

Mathematical Notes -     

Trace formulas for additive and non-additive perturbations

Trace formulas for pairs of self-adjoint, maximal dissipative and accumulative as well as other types of resolvent comparable operators are obtained. In particular, the existence of a complex-valued spectral shift function for a pair {H^′,H}$\{H^{\prime },H\}$ of maximal accumulative operators has been proved. We investigate also the existence of a real-valued spectral shift function. Moreover, we treat in detail the case of additive trace class perturbations. Assuming that H   and H^′=H+V$H^{\prime }=H+V$ are maximal accumulative and V is trace class, we prove the existence of a summable   complex-valued spectral shift function. We also obtain trace formulas for pairs {H,H^?}$\{H,H^{?}\}$assuming only that H and  H^?$H^{?}$are resolvent comparable. In this case the determinant of the characteristic function of H is involved in trace formulas.     

Nonmyopic optimal portfolios in viable markets

We provide a representation for the nonmyopic optimal portfolio of an agent consuming only at the terminal horizon when the single state variable follows a general diffusion process and the market consists of one risky asset and a risk-free asset. The key term of our representation is a new object that we call the “rate of macroeconomic fluctuation” whose properties are fundamental for the portfolio dynamics. We show that, under natural cyclicality conditions, (i) the agent’s hedging demand is positive (negative) when the product of his prudence and risk tolerance is below (above) two and (ii) the portfolio weights decrease in risk aversion. We apply our results to study a general continuous-time capital asset pricing model and show that under the same cyclicality conditions, the market price of risk is countercyclical and the price of the risky asset exhibits excess volatility.     

Landslides, forest fires, and earthquakes: examples of self-organized critical behavior

Per Bak conceived self-organized criticality as an explanation for the behavior of the sandpile model. Subsequently, many cellular automata models were found to exhibit similar behavior. Two examples are the forest-fire and slider-block models. Each of these models can be associated with a serious natural hazard: the sandpile model with landslides, the forest-fire model with actual forest fires, and the slider-block model with earthquakes. We examine the noncumulative frequency–area statistics for each natural hazard, and show that each has a robust power-law (fractal) distribution. We propose an inverse-cascade model as a general explanation for the power-law frequency–area statistics of the three cellular-automata models and their ‘associated’ natural hazards.