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We present a dimension formula for spaces of vector-valued modular forms of integer weight in case the associated multiplier system has finite image, and discuss the weight distribution of the module generators of holomorphic and cusp…
Digital computers store information in the form of bits that can take on one of two values 0 and 1, while quantum computers are based on qubits that are described by a complex wavefunction, whose squared magnitude gives the probability of…
We consider partially observed multiscale diffusion models that are specified up to an unknown vector parameter. We establish for a very general class of test functions that the filter of the original model converges to a filter of reduced…
We generalize the differential dimension polynomial from prime differential ideals to characterizable differential ideals. Its computation is algorithmic, its degree and leading coefficient remain differential birational invariants, and it…
Polynomial approximations of functions are widely used in scientific computing. In certain applications, it is often desired to require the polynomial approximation to be non-negative (resp. non-positive), or bounded within a given range,…
Embedding representations power machine intelligence in many applications, including recommendation systems, but they are space intensive -- potentially occupying hundreds of gigabytes in large-scale settings. To help manage this outsized…
The problem of evaluating potential integrals on planar triangular elements has been addressed using a polar coordinate decomposition. The resulting formulae are general, exact, easily implemented, and have only one special case, that of a…
In this paper we give a new and simple algorithm to put any multivariate polynomial into a normal determinant form in which each entry has the form , and in each column the same variable appears. We also apply the algorithm to obtain a…
Information about intrinsic dimension is crucial to perform dimensionality reduction, compress information, design efficient algorithms, and do statistical adaptation. In this paper we propose an estimator for the intrinsic dimension of a…
For a finite partially ordered set we calculate the dimension of the variety of its subspace representations having fixed dimension vector. The dimension is given in terms of the Euler quadratic form associated with a partially ordered set,…
We establish new mean value theorems for primes of size $x$ in arithmetic progressions to moduli as large as $x^{3/5-\epsilon}$ when summed with suitably well-factorable weights. This extends well-known work of Bombieri, Friedlander and…
Robust estimation is much more challenging in high dimensions than it is in one dimension: Most techniques either lead to intractable optimization problems or estimators that can tolerate only a tiny fraction of errors. Recent work in…
Let $E\subset [0,1]$ be a set that supports a probability measure $\mu$ with the property that $|\widehat{\mu}(t)|\ll (\log |t|)^{-A}$ for some constant $A>2.$ Let $\mathcal{A}=(q_n)_{n\in \N}$ be a positive, real-valued, lacunary sequence.…
Algorithms are proposed for the computation of set-valued quantiles and the values of the lower cone distribution function for bivariate data sets. These new objects make data analysis possible involving an order relation for the data…
Several recent papers construct auxiliary polynomials to bound the Weil height of certain classes of algebraic numbers from below. Following these techniques, the author gave a general method for introducing auxiliary polynomials to…
The Hilbert function of a module over a positively graded algebra is of quasi-polynomial type (Hilbert--Serre). We derive an upper bound for its grade, i.e. the index from which on its coefficients are constant. As an application, we give a…
We observe structure in the sequences of quotients and remainders of the Euclidean algorithm with two families of inputs. Analyzing the remainders, we obtain new algorithms for computing modular inverses and representating prime numbers by…
We study maximal subalgebras of an arbitrary finite dimensional algebra over a field, and obtain full classification/description results of such algebras. This is done by first obtaining a complete classification in the semisimple case, and…
The aim of the paper is twofold. Firstly, by using the constant rank level set theorem from differential geometry, we establish sharp upper bounds for the dimensions of the solution sets of polynomial variational inequalities under mild…
In this paper we give an upper bound, in characteristic 0, for the cohomological dimension of a graded ideal in a polynomial ring such that the quotient has depth at least 3. In positive characteristic the same bound holds true by a…