Pick's Theorem Research Articles

An Euler-MacLaurin formula for polygonal sums

We prove an Euler-Maclaurin formula for double polygonal sums and, as a corollary, we obtain approximate quadrature formulas for integrals of smooth functions over polygons with integer vertices. Our Euler-Maclaurin formula is in the spirit of Pick's theorem on the number of integer points in an integer polygon and involves weighted Riemann sums, using tools from Harmonic analysis. Finally, we also exhibit a classical trick, dating back to Huygens and Newton, to accelerate convergence of these Riemann sums.

Comparing Two Generalized Noncommutative Nevanlinna–Pick Theorems

We explore the relationship between two noncommutative generalizations of the classical Nevanlinna–Pick theorem: one proved by Constantinescu and Johnson in 2003 and the other proved by Muhly and Solel in 2004. To make the comparison, we generalize Constantinescu and Johnson’s theorem to the context of \(W^*\)-correspondences and Hardy algebras. After formulating the so-called displacement equation in this context, we are able to follow Constantinescu and Johnson’s line of reasoning in our proof. Though our result is similar in appearance to Muhly and Solel’s, closer inspection reveals differences. Nevertheless, when the given data lie in the center of the dual correspondence, the theorems are essentially the same.

Pick's Theorem in two-dimensional subspace of ℝ3.

In the Euclidean space ℝ3, denote the set of all points with integer coordinate by ℤ3. For any two-dimensional simple lattice polygon P, we establish the following analogy version of Pick's Theorem, k(I(P) + (1/2)B(P) − 1), where B(P) is the number of lattice points on the boundary of P in ℤ3, I(P) is the number of lattice points in the interior of P in ℤ3, and k is a constant only related to the two-dimensional subspace including P.

Functions of exponential type in a half-plane

Let f be of exponential type τ in the open upper half-plane and continuous in the closed upper half-plane. Assuming that | f(x)| ≤ M on the real axis, we find the sharp upper bound for | f′(z 0)| at any given point z 0 of the open upper half-plane. Our main tool is the classical Schwarz–Pick theorem.

Algorithms for optimal control with invariant zeros on the extended imaginary axis

We transform an ℋ∞ control problem with invariant zeros on the extended imaginary axis into a regular problem. Then using the Nevanlinna–Pick theorem, we find necessary and sufficient existence conditions for the original ℋ∞ control problem. An algorithm and examples with invariant zeros on the extended imaginary axis are given.By the same approach, we also solve the singular ℋ2 control problem.

Pick interpolation in several variables

We investigate the Pick problem for the polydisk and unit ball using dual algebra techniques. Some factorization results for Bergman spaces are used to describe a Pick theorem for any bounded region in C d \mathbb {C}^d .

Absolute Continuity, Interpolation and the Lyapunov Order

We extend our Nevanlinna–Pick theorem for Hardy algebras and their representations to cover interpolation at the absolutely continuous points of the boundaries of their discs of representations. The Lyapunov order plays a crucial role in our analysis.

A formal proof of Pick's Theorem

Pick's Theorem relates the area of a simple polygon with vertices at integer lattice points to the number of lattice points in its inside and boundary. We describe a formal proof of this theorem using the HOL Light theorem prover. As sometimes happens for highly geometrical proofs, the formalisation turned out to be more work than initially expected. The difficulties arose mostly from formalising the triangulation process for an arbitrary polygon.

Systematic approaches to experimentation: The case of Pick's theorem

In this paper two 10th graders having an accumulated experience on problem-solving ancillary to the concept of area confronted the task to find Pick's formula for a lattice polygon's area. The formula was omitted from the theorem in order for the students to read the theorem as a problem to be solved. Their working is examined and emphasis is given to highlighting the students’ range of systematic approaches to experimentation in the context of problem solving and aspects of control that are reflected in these approaches.

A Colorful Proof of Pick's Theorem

A Colorful Proof of Pick's Theorem

On Cantor's First Uncountability Proof, Pick's Theorem, and the Irrationality of the Golden Ratio

In Cantor's original proof of the uncountability of the reals (not the diagonalization argument), he constructs, given any countable sequence of real numbers, a real number not in the sequence. When we apply this argument to a certain standard enumeration of the rationals, the real number we produce will necessarily be irrational. Using some planar geometry, including Pick's theorem on the number of lattice points enclosed within certain polygonal regions, we show that this number is the reciprocal of the golden ratio, whence follows the well-known fact that the golden ratio is irrational.

A Schwarz–Pick Theorem for higher-order hyperbolic derivatives

Using the hyperbolic divided differences introduced by Baribeau, Rivard and Wegert, we generalize the notion of hyperbolic derivative and we give the analogue of a Schwarz–Pick Theorem for these derivatives.

Drawing a Triangle on the Thurston Model of Hyperbolic Space

SummaryIn looking at a common physical model of the hyperbolic plane, the authors encountered surprising difficulties in drawing a large triangle. Understanding these difficulties leads to an intriguing exploration of the geometry of the Thurston model of the hyperbolic plane. In this exploration we encounter topics ranging from combinatorics and Pick's Theorem to differential geometry and the Gauss-Bonnet Theorem.

An Exploration of Pick's Theorem in Space

(2008). An Exploration of Pick's Theorem in Space. Mathematics Magazine: Vol. 81, No. 2, pp. 106-115.

91.70 An elementary proof of Pick's theorem

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Pick's Theorem via Minkowski's Theorem

(2007). Pick's Theorem via Minkowski's Theorem. The American Mathematical Monthly: Vol. 114, No. 8, pp. 732-736.

Gaps in semigroups

In this paper we investigate the behaviour of the gaps in numerical semigroups. We will give an explicit formula for the i th gap of a semigroup generated by k + 1 consecutive integers (generalizing a result due to Brauer) as well as for a special numerical semigroup of three generators. It is also proved that the number of gaps of the numerical semigroup generated by integers p and q with g . c . d . ( p , q ) = 1 , in the interval [ pq - ( k + 1 ) ( p + q ) , … , pq - k ( p + q ) ] is equals to 2 ( k + 1 ) + kq p + kp q for each k = 1 , … , pq p + q - 1 . We actually give two proofs of the latter result, the first one uses the so-called Apery sets and the second one is an application of the well-known Pick's theorem.

Polynomials and spatial Pick-type theorems

Pick's theorem about the area of a simple lattice planar polygon has many extensions and generalizations even in the planar case. The theorem has also higher-dimensional generalizations, which are not as commonly known as the 2-dimensional case. The aim of the paper is, on one hand, to give a few new higher-dimensional generalizations of Pick's theorem and, on the other hand, collect known ones. We also study some relationships between lattice points in a lattice polyhedron which lead to some new Pick-type formulae. Another purpose of this paper is to pose several problems related to the subject of higher-dimensional Pick-type theorems. We hope that the paper may popularize the idea of determining the volume of a lattice polyhedron P by reading information contained in a lattice and the tiling of the space generated by the lattice.

Euler's characteristics and Pick's theorem

Euler's characteristics and Pick's theorem

Pick's Theorem and Greatest Common Divisors

Pick's Theorem and Greatest Common Divisors