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It is possible to have a packing by translates of a cube that is maximal (i.e.\ no other cube can be added without overlapping) but does not form a tiling. In the long running analogy of packing and tiling to orthogonality and completeness…
We suggest an upper bound on binomial coefficients that holds over the entire parameter range and whose form repeats the form of the de Moivre-Laplace approximation of the symmetric binomial distribution. Using the bound, we estimate the…
We prove new mean value theorems for primes in arithmetic progressions to moduli larger than $x^{1/2}$. Our main result shows that the primes are equidistributed for a fixed residue class over all moduli of size $x^{1/2+\delta}$ with a…
We give an optimal necessary and sufficient condition for the quotient polynomial and remainder in the division algorithm to have positive coefficients.
In a previous paper, the sup-interpretation method was proposed as a new tool to control memory resources of first order functional programs with pattern matching by static analysis. Basically, a sup-interpretation provides an upper bound…
Maximal sets of mutually unbiased bases are useful throughout quantum physics, both in a foundational context and for applications. To date, it remains unknown if complete sets of mutually unbiased bases exist in Hilbert spaces of…
We study integer coefficient polynomials of fixed degree and maximum height $H$, that are irreducible by Dumas's criterion. We call such polynomials Dumas polynomials. We derive upper bounds on the number of Dumas polynomials, as $H$…
Statistical inference in high-dimensional settings is challenging when standard unregularized methods are employed. In this work, we focus on the case of multiple correlated proportions for which we develop a Bayesian inference framework.…
For a nonempty compact set D of R we determine the maximal possible dimension of a subspace X of polynomial functions of degree at most m which possesses a positive bases (where positivity is understood on D). The exact value of this…
Models often need to be constrained to a certain size for them to be considered interpretable. For example, a decision tree of depth 5 is much easier to understand than one of depth 50. Limiting model size, however, often reduces accuracy.…
We introduce a new type of reduction of inversive difference polynomials that is associated with a partition of the basic set of automorphisms $\sigma$ and uses a generalization of the concept of effective order of a difference polynomial.…
The main goal of this paper is to size up the minimal graded free resolution of a homogeneous ideal in terms of its generating degrees. By and large, this is too ambitious an objective. As understood, sizing up means looking closely at the…
Bit addition arises virtually everywhere in digital circuits: arithmetic operations, increment/decrement operators, computing addresses and table indices, and so on. Since bit addition is such a basic task in Boolean circuit synthesis, a…
Several upper bounds on the size of quantum codes are derived using the linear programming approach. These bounds are strengthened for the linear quantum codes.
High-dimensional quantum systems are vital for quantum technologies and are essential in demonstrating practical quantum advantage in quantum computing, simulation and sensing. Since dimensionality grows exponentially with the number of…
We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. From this algorithm, we derive a new…
For a fixed integer $k \ge 0$, consider representations of positive integers as sums of binomial coefficients of the form $\binom{n}{k}$. While exact minimal bounds for the number of required summands are known only in a few low-dimensional…
We give a new complexity bound for calculating the complex dimension of an algebraic set. Our algorithm is completely deterministic and approaches the best recent randomized complexity bounds. We also present some new, significantly sharper…
In this paper, we prove that a binary definite quadratic form over F_q[t], where q is odd, is completely determined up to equivalence by the polynomials it represents up to degree 3m-2, where m is the degree of its discriminant. We also…
We provide an upper bound as a random variable for the functions of estimators in high dimensions. This upper bound may help establish the rate of convergence of functions in high dimensions. The upper bound random variable may converge…