Narrow Results

We consider the rate of convergence of solutions of spatially inhomogeneous Boltzmann equations, with hard-sphere potentials, to some equilibriums, called Maxwellians. Maxwellians are spatially homogeneous static Maxwell velocity distributions with different temperatures and mean velocities. We study solutions in weighted space $L^1({ℝ}^3\times {𝕋}^3)$. The result is that, assuming the solution is sufficiently localized and sufficiently smooth, then the solution, in $L^1$-space, converges to a Maxwellian, exponentially fast in time.

In this paper, we prove that four-dimensional gradient Yamabe solitons must have a Yamabe metric, provided that an asymptotic condition holds. The $n$-dimensional gradient Yamabe solitons are also considered.

The main features of obtaining the asymptotic behavior of the electric structure function $A(p)$ at large values of the transmitted momentum are analyzed. The asymptotic behavior of the structure function $A(p)$ was determined to take into account the asymptotic behavior of the deuteron form factors and the original dipole approximation for the nucleon form factors. Asymptotic values of $A(p)$ were obtained for the nucleon–nucleon potential Reid93 and compared with the calculations for different nucleon form factor models and their approximations. In the broad momentum range up to 12.5 fm${}^{-1}$, the basic forms of the asymptotic behavior of the electric structure function are demonstrated and compared with the experimental data of the modern collaborations. As the analysis shows in most cases considered, the asymptotic for $A(p)$ is represented in the form of the power function $p^{-n}$.

Numerical modeling of the deuteron wave function in the coordinate representation for the phenomenological nucleon–nucleon potential Argonne v18 has been performed. For this purpose, the asymptotic behavior of the radial wave function has been taken into account near the origin of coordinates and at infinity. The charge deuteron form factor $G_C(p)$, depending on the transmitted momentums up to $p=22{\mathrm{\text{fm}}}^{-1}$, has been calculated employing five models for the deuteron wave function. A characteristic difference in calculations of $G_C$ is observed near the positions of the first and second zero. The difference between the obtained values for $G_C$ form factor has been analyzed using the values of the ratios and differences for the results. Obtained outcomes for charge deuteron form factor at large momentums may be a prediction for future experimental data.

In this paper, we study the convergence of a European lookback option with floating strike evaluated with the binomial model of Cox–Ross–Rubinstein to its evaluation with the Black–Scholes model. We do the same for its delta. We confirm that these convergences are of order . For this, we use the binomial model of Cheuk–Vorst which allows us to write the price of the option using a double sum. Based on an improvement of a lemma of Lin–Palmer, we are able to give the precise value of the term in in the expansion of the error; we also obtain the value of the term in 1/n if the risk free interest rate is nonzero. This modelization will also allow us to determine the first term in the expansion of the delta.

In this paper, we set up an abstract theory of Murata densities, well tailored to general arithmetical semigroups. In [On certain densities of sets of primes, Proc. Japan Acad. Ser. A Math. Sci.56(7) (1980) 351–353; On some fundamental relations among certain asymptotic densities, Math. Rep. Toyama Univ.4(2) (1981) 47–61], Murata classified certain prime density functions in the case of the arithmetical semigroup of natural numbers. Here, it is shown that the same density functions will obey a very similar classification in any arithmetical semigroup whose sequence of norms satisfies certain general growth conditions. In particular, this classification holds for the set of monic polynomials in one indeterminate over a finite field, or for the set of ideals of the ring of S-integers of a global function field (S finite).

An integer $n\ge 1$ is said to be practical if every natural number $m\le n$ can be expressed as a sum of distinct positive divisors of $n$. The number of practical numbers up to $x$ is asymptotic to $cx/logx$, where $c$ is a constant. In this paper, we show that $c=1.33607\dots .$

We obtain a full asymptotic for the sum of ${\zeta }^{(n)}(\rho )X^{\rho }$, where ${\zeta }^{(n)}(s)$ denotes the nth derivative of the Riemann zeta function, X is a positive real number, and $\rho$ denotes a nontrivial zero of the Riemann zeta function. The sum is over the zeros with imaginary parts up to a height T, as $T\to \infty$.

We also specify what the asymptotic formula becomes when X is a positive integer, highlighting the differences in the asymptotic expansions as X changes its arithmetic nature.

In this paper we formulate a Laplace-transform multiple scale expansion procedure to develop asymptotic solution of weakly non-linear partial differential equation. The method is applied to some general nonlinear wave and diffusion equations.

In pricing an option on multiple assets, volatilities of individual assets and the correlations among different assets are necessary inputs. In practice, implied volatility surface for individual underlying asset can be used to calibrate the marginal distribution, but information on correlations is generally difficult to obtain. By regarding the correlation between two assets as an ambiguous parameter, we obtain the worst case price to option seller. The corresponding pricing problem is formulated as a stochastic optimal control problem in which the uncertain correlation takes the role as a control function. Consequently, the Black-Scholes equation is replaced by an HJB equation for deriving of the option price bounds. We solve this problem using the framework of stochastic volatility asymptotics and explain how volatility surfaces of individual assets can be robustly pulled together to estimate the price bounds of an option on multiple assets. Empirical study with foreign exchange option data illustrates the practical use of the proposed approach.