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A minor of an undirected graph G is any graph that may be obtained from G by a sequence of zero or more contractions of edges of G and deletions of edges and vertices of G.The minor relationship forms a partial order on the set of all distinct finite undirected graphs, as it obeys the three axioms of partial orders: it is reflexive (every graph is a minor of itself), transitive (a minor of a ...
In mathematical analysis, the Dirac delta function (or δ distribution), also known as the unit impulse, [1] is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one.
A graph G guarantees that Braess's paradox does not occur, iff it is a series-parallel graph. Milchtaich [ 16 ] analyzes the effect of network topology on the uniqueness of the PNE costs: A graph G guarantees that the PNE costs are unique iff G is a connection in series of one or more networks of several simple kinds.
It is also conjectured to be self-similar around generalized Feigenbaum points (e.g., −1.401155 or −0.1528 + 1.0397i), in the sense of converging to a limit set. [ 25 ] [ 26 ] The Mandelbrot set in general is quasi-self-similar, as small slightly different versions of itself can be found at arbitrarily small scales.
The dot product satisfies the properties [1] It is symmetric in x and y: x ⋅ y = y ⋅ x. It is linear in its first argument: (ax 1 + bx 2) ⋅ y = a(x 1 ⋅ y) + b(x 2 ⋅ y) for any scalars a, b, and vectors x 1, x 2, and y. It is positive definite: for all vectors x, x ⋅ x ≥ 0 , with equality if and only if x = 0.
The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics.It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.
The graph always lies above the x-axis, but becomes arbitrarily close to it for large negative x; thus, the x-axis is a horizontal asymptote. The equation d d x e x = e x {\displaystyle {\tfrac {d}{dx}}e^{x}=e^{x}} means that the slope of the tangent to the graph at each point is equal to its height (its y -coordinate) at that point.
The absolute value or modulus |x| of a real number x is the non-negative value of x without regard to its sign. Namely, |x| = x for a positive x, |x| = −x for a negative x (in which case −x is positive), and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3.