In mathematics, singular integral operators on closed curves arise in problems in analysis, in particular complex analysis and harmonic analysis. The two...

T^{t}(b_{2}),} is in BMO, where Tt is the transpose operator of T. Singular integral operators on closed curves Stein, Elias (1993). "Harmonic Analysis". Princeton...

the integrals used are all proper integrals. Singular or weakly singular: An integral equation is called singular or weakly singular if the integral is...

that the plane curve Y 2 = X 3 {\displaystyle Y^{2}=X^{3}} has a singularity at the origin. Another domain that is not integrally closed is A = Z [ 5 ]...

}}{\Big )}} Singular integral operators on closed curves (account of the Sokhotski–Plemelj theorem for the unit circle and a closed Jordan curve) Kramers–Kronig...

that is, a function F such that F′ = f on the interval. Provided the integrand and integral have no singularities on the path of integration, by the fundamental...

the curve decreases as the curve converges to a circular shape, before collapsing to a singularity. If two disjoint simple smooth closed curves evolve...

integral theorem, one can show that the integral over C (or the closed rectifiable curve) is equal to the same integral taken over an arbitrarily small circle...

support, the duality between Jordan curves and 1-forms implies that the integral of ω around any closed Jordan curve is zero and hence that ω is exact....

)\subset \mathbb {R} .\end{aligned}}} The curves γ x {\displaystyle \gamma _{x}} are called integral curves or trajectories (or less commonly, flow lines)...

These two singularities correspond to two points, rather than two curves. In this example, the other integral curves are all simple closed curves. Many flows...

theorem is that a Riemannian metric on a closed manifold M determines a real-valued inner product on the integral cohomology of M modulo torsion. It follows...

line integral around the boundary curve ∂Σ, with the loop indicating the curve is closed. ∬ Σ {\displaystyle \iint _{\Sigma }} is a surface integral over...

extending the notion of Fourier transformation to include Fourier integral operators, some non-linear equations as well. The Fourier transform is also...

of class inclusions: rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃...

family of bounded operators on X. Recall the Cauchy integral formula from classical function theory. Let f : C → C be holomorphic on some open set D ⊂...

Cauchy's integral formula. The Poincaré lemma generally fails for singular spaces. For example, if one considers algebraic differential forms on a complex...

geometry) Riemann's theorem on removable singularities (complex analysis) Riemann–Roch theorem (Riemann surfaces, algebraic curves) Rouché's theorem (complex...

smooth. By contrast, in the direction normal to the curve, the function has a singularity. To decide on whether the function is smooth in another direction...

∈ U {\displaystyle u_{0}\in U} if it preserves angles between directed curves through u 0 {\displaystyle u_{0}} , as well as preserving orientation. Conformal...

cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an...

over the disk into the integral over its boundary. Let Ω be a region bounded by a simple closed curve. Given a smooth function f on the closure of Ω its...

include the closed convex curves (the boundaries of bounded convex sets), the smooth curves that are convex, and the strictly convex curves, which have...

show that, if M is a closed three-dimensional smooth manifold, then finite-time singularities of the Ricci flow on M are modeled on complete gradient shrinking...

inequality for curves in the Euclidean plane is also valid on the sphere or in the hyperbolic plane: namely he showed that among all closed curves bounding...

and zeros of either the closed-loop or open-loop system (although the number of each type of right-half-plane singularities must be known). As a result...

One-dimensional manifolds include lines and circles, but not self-crossing curves such as a figure 8. Two-dimensional manifolds are also called surfaces....

case where the boundary is just a Lipschitz curve was constructed by Calderón using singular integral operators and generalized by Stein (1970). It is sufficient...

of curved surfaces of 1827 and 1825". [Princeton] The Princeton university library. 1902. Goldman, R. (2005). "Curvature formulas for implicit curves and...

curves, the total curvature of a smooth plane curve is the amount it turns anticlockwise, in radians, from start to finish, computed as the integral of...