Related papers: The mean king's problem: Prime degrees of freedom
We show how one can ascertain the values of four mutually complementary observables of a spin-1 degree of freedom.
It is shown how to ascertain the values of a complete set of mutually complementary observables of a prime power degree of freedom by generalizing the solution in prime dimensions given by Englert and Aharonov [Phys. Lett. A284, 1-5…
The Guide to the Expression of Uncertainty in Measurement advocates the use of an 'effective number of degrees of freedom' for the calculation of an interval of measurement uncertainty. However, it does not describe how this number is to be…
We present a Lagrangian approach to counting degrees of freedom in first-order field theories. The emphasis is on the systematic attainment of a complete set of constraints. In particular, we provide the first comprehensive procedure to…
We establish asymptotic formulae for the number of $k$-free values of polynmilas $F(x_1,\cdots,x_n)\in\mathbb{Z}[x_1,\cdots,x_n]$ of degree $d\geq 2$ for any $n\geq 1$, including when the variables are prime, as long as $k\geq (3d+1)/4$.…
In this work we discuss how to correctly obtain the number of degrees of freedom of a system constituted by n particles with fixed relative distances and which are immerse in a three-dimensional space. As a result of our analysis, we…
Assuming the Riemann Hypothesis we obtain asymptotic estimates for the mean value of the number of representations of an integer as a sum of two primes. By proving a corresponding Omega-term, we prove that our result is essentially the best…
We consider the question of counting the degrees of freedom in theoretical models, with an emphasis on theories of fields and gravity. Among the possible approaches, the Hamiltonian formulation remains one of the most systematic and robust…
We explicitly determine the associated primes of every power of a complementary edge ideal, prove that they satisfy the persistence property, and compute the $\text{v}$-function. In the course of the proofs, we completely describe the…
For physical theories, the degree of arbitrariness of a system is of great importance, and is often closely linked to the concept of degree of freedom, and for most systems this number is far from obvious. In this paper we present an easy…
We present the solution to the "mean king's problem" in the continuous variable setting. We show that in this setting, the outcome of a randomly-selected projective measurement of any linear combination of the canonical variables x and p…
We discuss several enumerative results for irreducible polynomials of a given degree and pairs of relatively prime polynomials of given degrees in several variables over finite fields. Two notions of degree, the {\em total degree} and the…
In this paper we study a sequence involving the prime numbers by deriving two asymptotic formulas and finding new upper and lower bounds, which improve the currently known estimates.
The Mean King's problem asks to determine the outcome of a measurement that is randomly selected from a set of complementary observables. We review this problem and offer a combinatorial solution. More generally, we show that whenever an…
We consider the problem of determining whether a set of primes, or, more generally, prime ideals in a number field, can be realized as a finite union of residue classes, or of Frobenius conjugacy classes. We give criteria for a set to be…
Degrees of freedom is a fundamental concept in statistical modeling, as it provides a quantitative description of the amount of fitting performed by a given procedure. But, despite this fundamental role in statistics, its behavior not…
We derive the degrees of freedom of the lasso fit, placing no assumptions on the predictor matrix $X$. Like the well-known result of Zou, Hastie and Tibshirani [Ann. Statist. 35 (2007) 2173-2192], which gives the degrees of freedom of the…
Extending a classical estimate of Mertens for the sum of the reciprocals of the first primes, we provide an explicit remainder formula for products of an arbitrary, but fixed, number of primes.
Assuming a $q$-variant of the prime $k$-tuple conjecture uniformly, we compute mixed moments of the number of primes in disjoint short intervals and progressions, respectively. This involves estimating the mean of singular series along…
The paper solves the problems of determining the asymptotics of the number of primes and the sums of functions of primes in a subset of the natural series that satisfies the conditions that the asymptotic density of the number of primes in…