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The main goal of this paper is to introduce a set of conjectures on the relations in the tautological rings. In particular, the framework gives an efficient algorithm to calculate all tautological equations using only finite dimensional…
Quadrature formulas for spheres, the rotation group, and other compact, homogeneous manifolds are important in a number of applications and have been the subject of recent research. The main purpose of this paper is to study coordinate…
We consider the diffeological version of the Clifford algebra of a (diffeological) finite-dimensional vector space; we start by commenting on the notion of a diffeological algebra (which is the expected analogue of the usual one) and that…
We give a tutorial exposition of the analogue of the filtering equation for quantum systems focusing on the quantum probabilistic framework and developing the ideas from the classical theory. Quantum covariances and conditional expectations…
Building on recent advances in studying the co-homological properties of Feynman integrals, we apply intersection theory to the computation of Fourier integrals. We discuss applications pertinent to gravitational bremsstrahlung and deep…
There are quantum solutions for computational problems that make use of interference at some stage in the algorithm. These stages can be mapped into the physical setting of a single particle travelling through a many-armed interferometer.…
The work is devoted to the development of numerical methods for computing "formal solutions" of interval systems of linear algebraic equations. These solutions are found in Kaucher interval arithmetic, which extends and completes the…
Using algebraic methods, and motivated by the one variable case, we study a multipoint interpolation problem in the setting of several complex variables. The duality realized by the residue generator associated with an underlying Gorenstein…
An algebraic approach is presented for the valuative interpolation problem, which recovers and generalizes prior characterizations known in the complex analytic setting by the authors. We use the asymptotic Samuel function to give the…
An algorithm is presented that generates sets of size equal to the degree of a given variety defined by a homogeneous ideal. This algorithm suggests a versatile framework to study various problems in combinatorial algebraic geometry and…
Interpretation of the nonclassical total probability formula arising in some quantum experiments is provided based on stochastic models described by means of a sequence of random vectors changing in the measurement procedures.
In this paper we use the viewpoint of the formal calculus underlying vertex operator algebra theory to study certain aspects of the classical umbral calculus and we introduce and study certain operators generalizing the classical umbral…
In this article we cover the canonical problem formulation necessary to program the D-Wave quantum processing unit (QPU) and discuss how such a problem is compiled onto the QPU. We also cover recent joint work solving a problem from…
We study the operad of associative algebras equipped with a derivation. We show that it is determined by polynomials in several variables and substitution. Replacing polynomials by rational functions gives an operad which is isomorphic to…
We discuss the nature of structure-preserving maps of varies function algebras. In particular, we identify isomorphisms between special Colombeau algebras on manifolds with invertible manifold-valued generalized functions in the case of…
Computation of polynomial relative invariants is a classical tool in algebra. Relative differential invariants are central for the equivalence problem of geometric structures. We address the fundamental problem of finite generation of their…
Calibration$\unicode{x2014}$the problem of ensuring that predicted probabilities align with observed class frequencies$\unicode{x2014}$is a basic desideratum for reliable prediction with machine learning systems. Calibration error is…
We reveal a natural algebraic problem whose complexity appears to interpolate between the well-known complexity classes BQP and NP: (*) Decide whether a univariate polynomial with exactly m monomial terms has a p-adic rational root. In…
Given a probability measure $\mu$ on a set $\mathcal{X}$ and a vector-valued function $\varphi$, a common problem is to construct a discrete probability measure on $\mathcal{X}$ such that the push-forward of these two probability measures…
A classification of ordinary differential equations and finite-difference equations in one variable having polynomial solutions (the generalized Bochner problem) is given. The method used is based on the spectral problem for a polynomial…