In Section 1.2 we introduce some simple motivating examples of functional equations, such as the formula for the area of a rectangle, the simple interest, the sum of the internal angles of a polygon, or the associativity equation. In Section 1.3 we introduce some definitions and basic concepts...
Aoki, ‘‘Numerical analysis of a uniform flow of a rarefied gas past a sphere on the basis of the Boltzmann equation for hard-sphere molecules,’’ Phys. Fluids A 5, 716–737 (1993). Article MATH Google Scholar R. Barber and D. Emerson, Analytical Solution of Low Reynolds Number ...
B. M. Levitan, Expansion with Respect to the Eigenfunctions of a Second-Order Differential Equation [in Russian], Moscow-Leningrad (1950). B. M. Levitan, “On the expansions with respect to the eigenfunctions of the Laplace operator,” Dokl. Akad. Nauk SSSR,90, No. 2, 133–135 (1953...
Note that indeed the curl operator in the first equation of [25] can be replaced by (minus) the Laplacian operator −Δ, since divB=0. Actually, this formal similarity cannot be translated into mathematical arguments, simply because there is no pressure in [25]. In other terms, the ...
A mathematical discussion is provided on the relationship between Baxter''s adhesive hard sphere fluid and the hard sphere Yukawa fluid with sufficiently strong and short range attractive Yukawa tails. It is proved that the Percus-Yevick approximate solution of the Ornstein-Zernike equation for the...
Evolution EquationCliffordian FunctionsDouble-Complex Laplace OperatorMKdV Type EquationLax PairHeisenberg RelationGauss MapClifford Minimal HypersurfaceShallow Water EquationThis book contains the contributions by the participants in the nine of a series of workshops. Throughout the series of workshops, the ...
"reference_steps": [ "Let's call the price of a pencil p and the price of a jumbo eraser e. Then we can write two equations.", "We have $3p+e=1.24$ and $5p+e=1.82$.", "To solve this system, let's subtract the first equation from the second equation. This will eliminate e...
The defining equation of the metric is then nothing but the Pythagorean theorem applied to the differentials of the co-ordinates. In the special theory of relativity those co-ordinate changes (by transformation) are permitted for which also in the new co-ordinate system the quantity (cdt)2 ...
the tenth was to produce a method that could decide whether a multivariate polynomial equation over the integers has a solution. Subsequent work to resolve these problems shaped the direction of mathematical logic, as did the effort to resolve Hilbert's Entscheidungsproblem, posed in 1928. This ...
\small \int \int \left[ P(x,y) \log \frac{P(x,y)}{P(x)P(y)} + \mu P(x,y)\rho(x,y) + vP(x,y) \right] \mathrm{d}x \mathrm{d}y \\The variational equation (when we take the first variation on \small P(x,y) ) leads to 通过变分方程得到(首先求取 \small P(x,...