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For 0 < q < 1, the series converges to a function F(x) on an interval (0,A] if |f(x)x α | is bounded on the interval (0, A] for some 0 ≤ α < 1. The q-integral is a Riemann–Stieltjes integral with respect to a step function having infinitely many points of increase at the points q j..The jump at the point q j is q j. Calling this step ...
Can the rotation distance between two binary trees be computed in polynomial time? Can graphs of bounded clique-width be recognized in polynomial time? [1] Can one find a simple closed quasigeodesic on a convex polyhedron in polynomial time? [2] Can a simultaneous embedding with fixed edges for two given graphs be found in polynomial time? [3]
One particle: N particles: One dimension ^ = ^ + = + ^ = = ^ + (,,) = = + (,,) where the position of particle n is x n. = + = = +. (,) = /.There is a further restriction — the solution must not grow at infinity, so that it has either a finite L 2-norm (if it is a bound state) or a slowly diverging norm (if it is part of a continuum): [1] ‖ ‖ = | |.
Quantum stochastic calculus is a generalization of stochastic calculus to noncommuting variables. [1] The tools provided by quantum stochastic calculus are of great use for modeling the random evolution of systems undergoing measurement , as in quantum trajectories.
In calculus, the trapezoidal rule (also known as the trapezoid rule or trapezium rule) [a] is a technique for numerical integration, i.e., approximating the definite integral: (). The trapezoidal rule works by approximating the region under the graph of the function f ( x ) {\displaystyle f(x)} as a trapezoid and calculating its area.
The q-derivative of a function f(x) is defined as [1] [2] [3] () = ().It is also often written as ().The q-derivative is also known as the Jackson derivative.. Formally, in terms of Lagrange's shift operator in logarithmic variables, it amounts to the operator
Differential equations are an important area of mathematical analysis with many applications in science and engineering. Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. [1] [2]
It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. [40] Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used for advanced parts ...