相关论文: Smooth planar $r$-splines of degree $2r$

TD-Delaunay graphs, where TD stands for triangular distance, is a variation of the classical Delaunay triangulations obtained from a specific convex distance function. Bonichon et. al. noticed that every triangulation is the TD-Delaunay…

We prove a Steiner formula for regular surfaces with no characteristic points in 3D contact sub-Riemannian manifolds endowed with an arbitrary smooth volume. The formula we obtain, which is equivalent to a half-tube formula, is of local…

Graded skew-commutative rings occur often in practice. Here are two examples: 1) The cohomology ring of a compact three-dimensional manifold. 2) The cohomology ring of the complement of a hyperplane arrangement (the Orlik-Solomon algebra).…

We introduce W-spin structures on a Riemann surface and give a precise definition to the corresponding W-spin equations for any quasi-homogeneous polynomial W. Then, we construct examples of nonzero solutions of spin equations in the…

Moser's Bernstein theorem \cite{moser61} says that an entire minimal graph of codimension 1 with bounded slope must be a hyperplane. An analogous result for arbitrary codimension is not true, by an example of Lawson-Osserman. Here, we show…

For distinct complex numbers $z_1,...,z_{2N}$, we give a polynomial $P(y_1,...,y_{2N})$ in the variables $y_1,...,y_{2N}$, which is homogeneous of degree $N$, linear with respect to each variable, $sl_2$-invariant with respect to a natural…

We prove that a logarithmic foliation corresponding to a generic line arrangement of $d+1 \geq 3$ lines in the complex plane, with pairwise natural and co-prime residues, is a smooth point of the center set of plane foliations (vector…

Let $\mathbb{F}_q$ be the field of $q$ elements and let $A=\mathbb{F}_q[t]$ be the polynomial ring over $\mathbb{F}_q$. Let $\mathfrak{n}\in A\setminus \mathbb{F}_q$ be a monic polynomial with a prime factor of degree prime to $q-1$. Let…

We prove Strichartz estimates on general flat d-torus for arbitrary d. Using these estimates, we prove local wellposedness for the cubic nonlinear Schr\"odinger equations in appropriate Sobolev spaces. In dimensions 2 and 3, we prove…

We give geometric descriptions of the category C_k(n,d) of rational polynomial representations of GL_n over a field k of degree d for d less than or equal to n, the Schur functor and Schur-Weyl duality. The descriptions and proofs use a…

This survey gives an overview of several fundamental algebraic constructions which arise in the study of splines. Splines play a key role in approximation theory, geometric modeling, and numerical analysis, their properties depend on…

Graded skew-commutative rings occur often in practice. Here are two examples: 1) The cohomology ring of a compact three-dimensional manifold. 2) The cohomology ring of the complement of a hyperplane arrangement (the Orlik-Solomon algebra).…

We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d $\mathcal{N}=2$ theories where such structures a priori are not manifest. These modular structures…

We establish a generalization of Littlewood's criterion on $L^\alpha$-flatness by proving that there is no $L^\alpha$-flat polynomials, $\alpha>0$, within the class of analytic polynomials on the unit circle of the form $…

We present computational methods for constructing orthogonal/orthonormal polynomials over arbitrary polygonal domains in $\mathbb{R}^2$ using bivariate spline functions. Leveraging a mature MATLAB implementation which generates spline…

We use techniques of Alper-Hall-Rydh to prove a local structure theorem for smooth morphisms between smooth stacks around points with linearly reductive stabilizers. This implies that the good moduli space of a smooth stack over a base has…

The moduli space of Riemann surfaces of genus $g\geq 2$ is (up to a finite \'etale cover) a complex manifold and so it makes sense to speak of its Dolbeault cohomological dimension. The conjecturally optimal bound is $g-2$. This expectation…

We investigate the complex geometry of D=10 pure spinor space. The K\"ahler structure and the corresponding metric giving rise to the desired Calabi-Yau property are determined, and an explicit covariant expression for the Laplacian is…

We provide several ingredients towards a generalization of the Littlewood-Richardson rule from Chow groups to algebraic cobordism. In particular, we prove a simple product-formula for multiplying classes of smooth Schubert varieties with…

We consider the class of all conformal mappings from a compact Riemann surface into the threedimensional or fourdimensional Euclidean space. A sequence in this class with bounded Willmore functional is shown to have a sequence of conformal…