Related papers: Differential theory of zero-dimensional schemes

We prove that to each real singularity $f: (\mathbb{R}^{n}, 0) \to (\mathbb{R}^k, 0)$ with $k\geq 2$ one can associate systems of differential equations $\mathfrak{g}^{k}_f$ which are pushforwards in the category of $\mathcal{D}$-modules…

Regularity estimates for an integral operator with a symmetric continuous kernel on a convex bounded domain are derived. The covariance of a mean-square continuous random field on the domain is an example of such an operator. The estimates…

We classify all complete projective special real manifolds with reducible cubic potential, obtaining four series. For two of the series the manifolds are homogeneous, for the two others the respective automorphism group acts with…

The $N$-dimensional quantum Hamiltonian $ \hat{H} = -\frac{\hbar^2 {|\mathbf{q} } | }{2(\eta +| {\mathbf{q}} |)} {\mathbf{\nabla}}^2 - \frac{k}{\eta + |{\mathbf{q}} |} $ is shown to be exactly solvable for any real positive value of the…

In this work we develop a novel fully discrete version of the plates complex, an exact Hilbert complex relevant for the mixed formulation of fourth-order problems. The derivation of the discrete complex follows the discrete de Rham…

We establish basic facts about the varieties of homogeneous polynomials divisible by powers of linear forms, and explain consequences for geometric complexity theory. This includes quadratic set-theoretic equations, a description of the…

We develop explicit formulas and algorithms for arithmetic in radical function fields K/k(x) over finite constant fields. First, we classify which places of k(x) whose local integral bases have an easy monogenic form, and give explicit…

A deformed differential calculus is developed based on an associative star-product. In two dimensions the Hamiltonian vector fields model the algebra of pseudo-differential operator, as used in the theory of integrable systems. Thus one…

For a compact complex manifold, we introduce holomorphic foliations associated with certain abelian subgroups of the automorphism group. Such foliations are generalizations of holomorphic principal torus bundles. If there exists a…

A quantum system can be entirely described by the K\"ahler structure of the projective space P(H) associated to the Hilbert space H of possible states; this is the so-called geometrical formulation of quantum mechanics. In this paper, we…

We generalize Roe's Index Theorem for operators of Dirac type on open manifolds to elliptic pseudodifferential operators. To this end we introduce a class of pseudodifferential operators on manifolds of bounded geometry which is more…

In this paper, we investigate $\mathbb T^d$-invariant Hilbert modules $\mathscr H$ over the polynomial ring $\mathbb C[z_1, \ldots, z_d]$ and their quotients, with primary emphasis on the classification of subnormal quotient modules of the…

Let S be a smooth projective surfaces and S^[n] the Hilbert scheme of zero-dimensional subschemes of S of length n. We proof that the class of S^[n] in the complex cobordism ring depends only on the class of the surface itself. Moreover, we…

We develop differential algebraic K-theory of regular arithmetic schemes. Our approach is based on a new construction of a functorial, spectrum level Beilinson regulator using differential forms. We construct a cycle map which represents…

For a given algebraically closed field $k$ of characteristic $p>0$ we consider the set ${\mathcal C}_k$, of graded isomorphism classes of {\em standard graded pairs} $(R, I)$, where $R$ is a standard graded ring over the field and $I$ is a…

In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces $L^{2}(\mu)$, with $\mu$ a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix to the…

We consider the theory of regression on a manifold using reproducing kernel Hilbert space methods. Manifold models arise in a wide variety of modern machine learning problems, and our goal is to help understand the effectiveness of various…

In this note, we continue the investigation of a projective K\"ahler manifold $M$ of semi-negative holomorphic sectional curvature $H$. We introduce a new differential geometric numerical rank invariant which measures the number of linearly…

We investigate projection constants for spaces of bihomogeneous harmonic and bihomogeneous polynomials on the unit sphere in finite-dimensional complex Hilbert spaces. Using averaging techniques, we demonstrate that the minimal norm…

We develop a Helmholtz-like theorem for differential forms in Euclidean space $E_{n}$ using a uniqueness theorem similar to the one for vector fields. We then apply it to Riemannian manifolds, $R_{n}$, which, by virtue of the…