A direct attempt to goldbach problem

The so-called principal term of Goldbach problem, i.e., the principal order of the number of representations of an even number as a sum of two primes, is evaluated here in a way quite different from the classical circle method. It is amazing that the author did this with the knowledge just at the PNT (Prime Number Theorem) level, even without using high-level theorems in analytic number theory as usual, e.g., Siegel-Walfisz Theorem.     

An inverse problem by eigenvalues of four spectra

Certain parts of the Dirichlet–Dirichlet, Neumann–Dirichlet, Dirichlet–Neumann and Neumann–Neumann spectra are used to find the potential of the Sturm–Liouville equation on a finite interval. This problem possesses a unique solution. Conditions are found necessary and sufficient for four sequences to be the corresponding parts of the four spectra.     

A digest of the four color theorem

A major event in 1976 was the announcement that the Four Color Conjecture (4CC) had at long last become the Four Color Theorem (4CT). The proof by W. Haken, K. Appel, and J. Koch is published in the Illinois Journal of Mathematics, and their two-part article outlines the nature and reliability of the solution. The first section is a readable and informative historical survey. The reminder will appeal chiefly to specialists in graph theory. Although the logic of attack is relatively simple, the need to examine an immense number of individual cases is frustrating. Hopefully this first breakthrough will pave the way for a short elegant proof. For the second section, 1200 hours of computer time was required to verify the 4-color reducibility of nearly 1900 configurations. At this time there is no good way to condense the proof. In this digest we offer an exposition of the main ideas. The first and the last parts are intended for a general audience, but the intermediate sections assume more knowledge of graph theory proper. The usual statement of the 4CC goes as follows: “All maps on the sphere or plane can be colored with four colors so that neighboring regions are never colored alike.” The form in which the 4CT was proved is “There exists an unavoidable set of reducible configurations, relative to triangulations of the plane.” The initial part of our task is to explain how the 4CC comes to be expressed in such jargon. The next step is to show how one finds simple finite sets of unavoidable configurations. Then comes the question of how to prove reducibility, followed by a consideration of the known obstacles to reduction. Our concluding remarks and criticisms include a consideration of prospects for the future.     

Signed permutations and the four color theorem

Let σ₁,σ₂ be two permutations in the symmetric group S_n. Among the many sequences of elementary transpositions τ₁,…,τ_r transforming σ₁ into σ₂=τ_rτ₁σ₁, some of them may be signable, a property introduced in this paper. We show that the four color theorem in graph theory is equivalent to the statement that, for any n≥2 and any σ₁,σ₂S_n, there exists at least one signable sequence of elementary transpositions from σ₁ to σ₂. This algebraic reformulation rests on a former geometric one in terms of signed diagonal flips, together with a codification of the triangulations of a convex polygon on n+2 vertices by permutations in S_n.     

The four color proof suffices

The Mathematical Intelligencer - Since no one else has communicated any other errors in the published unavoidability proof since 1976 we assume that a misunderstanding of the nature of...     

Absolute planar retracts and the four color conjecture

The Four Color Conjecture (4CC) is shown to be equivalent with the existence of absolute retracts in the class $\text{P}$ of all planar graphs and also with some other conditions on the class $\text{P}$. Assuming the 4CC is true we determine all absolute planar retracts. A possible argument for the decidability¹ of the 4CC is discussed and proved to be equivalent to the 4CC itself.     

An equivalence to the Gleason problem

In this article we study the Gleason problem locally. A new method for solving the Gleason A problem is presented. This is done by showing an equivalent statement to the Gleason A problem. In order to prove this statement, necessary and a sufficient conditions for a bounded domain to have the Gleason A property are found. Also an example of a bounded but not smoothly-bounded domain in Cn is given, which satisfies the sufficient condition at the origin, and hence has the Gleason A property there.     

A note on the three color problem

It is shown that a planar graph withouti-circuits, 4 i 9, is 3-colorable. This result strengthens the result obtained by H.L. Abbott and B. Zhou.The author's research was partially supported by the Office of Naval Research, Grant number N00014-92-J-1965.     

An attempt to interpret the Weil explicit formula from Beurling's spectral theory

A. Beurling introduced the celebrated problem of spectral synthesis. Roughly speaking, it is a problem whether functions belonging to a certain Banach space have a possibility to be approximated by trigonometric polynomials in the appropriate topology. For this problem Beurling introduced the concept of spectral sets whose elements are regarded as exponents of trigonometric polynomials. In the Weil explicit formula we can see a certain phenomenon which may be related to Beurling's spectral sets. The purpose of this paper is to study the phenomenon.     

An answer to Li's problem

In this paper, an answer is given to a problem proposed by K.C. Li. This problem comes from the variable transformation for fitting a regression model.     

An effective computational attempt in DDEA

Dynamic DEA (DDEA) is a mathematical programming-based technique which assesses the performance of decision making units (DMUs) in the presence of time factor. This paper provides a new technique for reducing the computational complexity of some recently introduced DDEA models.     

Toward a language theoretic proof of the four color theorem

This paper considers the problem of showing that every pair of binary trees with the same number of leaves parses a common word under a certain simple grammar. We enumerate the common parse words for several infinite families of tree pairs and discuss several ways to reduce the problem of finding a parse word for a pair of trees to that for a smaller pair. The statement that every pair of trees has a common parse word is equivalent to the statement that every planar graph is four-colorable, so the results are a step toward a language theoretic proof of the four color theorem.     

An addendum to the hierarchical network design problem

The hierarchical network design problem is the problem to find a spanning tree of minimum total weight, when the edges of the path between two given nodes are weighted by an other cost function than the tree edges not on this path. We point out that a dynamic programming oriented heuristic can already be found in literature. Further we report on possible extensions and improvements.     

An asymptotic-numerical approach to the coronal loop problem

The coronal loop problem is characterized by mixed boundary conditions and the loop length condition, which is global. Using singular perturbation methods one can identify and construct two boundary layers at the base of the loop. Extending this to a combined asymptotic-numerical treatment it is possible to construct two static solutions satisfying the same conditions; this unusual feature arises from the presence of the first boundary layer, which corresponds with a steep temperature gradient and an energy balance dominated by conduction and radiation losses.     

On four color monochromatic sets with nondecreasing diameter

Let m and r be positive integers. Define f(m,r) to be the least positive integer N such that for every coloring of the integers 1,…,N with r colors there exist monochromatic subsets B₁ and B₂ (not necessarily of the same color), each having m elements, such that (a) max(B₁)-min(B₁)max(B₂)-min(B₂), and (b) max(B₁)B₂). We improve previous upper bounds to determine that f(m,4)=12m-9.