Open Mathematical Problems Research Articles

Science Popularization by Organizing Training Activities Within the Electronic Game Laboratories

The article describes the history of the knowledge popularization in Russia and the current situation in this domain with the impact of rapidly developing computer technologies. The article also introduces the Construct, Test, Explore (CTE) contest aimed to take students into the scientific phenomena through the game laboratories illustrating the unsolved and difficult problems of mathematics, informatics, and physics. The CTE School project based on the contest ideas is devoted to learning basic algorithms from the theory of information or discrete mathematics through training experiments.

Mathematical Model of Telomere Length Maintenance in Mitochondrial DNA of Yeast

The terminal structures of linear mitochondrial DNA in yeast consist of repetitive long tandem units of different lengths (telomeres). Besides these linear telomeres other cyclic configurations as telomeric circles and telomeric loops were observed experimentally and are suspected to play an important role in the alternative mechanism of the telomere length maintenance in the absence of telomerase (ALT).In this work we construct a mathematical model that captures biophysical interactions of various telomeric structures on a short time scale and that is able to reproduce experimental measurements in C. parapsilosis, P. philodendra and C. salmanticensis. We identify the key factors influencing the length distribution of telomeric circles, linear telomeres and telomeric loops using numerical simulations for the model we have constructed. Moreover, the model opens up a couple of interesting open mathematical problems in quasi-steady state approximation and discrete coagulation-fragmentation dynamical systems.

Mathematical Challenges in Stochastic Networks

The workshop was devoted to the discussion of recent progress in modern stochastic network theory and to the exploration of open mathematical challenging problems in the field. The workshop covered a wide range of mathematical topics; while being centered around applied probability, it also included a substantial amount of graph theory and (combinatorial) optimization.

Potential-flow models for channelled two-dimensional premixed flames around near-circular obstacles

The dynamics of two-dimensional thin premixed flames is addressed in the framework of mathematical models where the flow field on either side of the front is piecewise incompressible and vorticity free. Flames confined in channels with asymptotically straight impenetrable walls are considered. Besides a few free propagations along straight channels, attention is focused on flames propagating against high-speed flows and positioned near a round central obstacle or near two symmetric bumps protruding inward. Combining conformal maps and Green's functions, a regularized generalization of Frankel's integro-differential equation for the instantaneous front shape in each configuration is derived and solved numerically. This produces a variety of real looking phenomena: steady fronts (symmetric or not), noise-induced subwrinkles, flashback events, and breathing fronts in pulsating flows. Perspectives and open mathematical and physical problems are finally evoked.

The number of eigenstates: counting function and heat kernel

The main aim of this paper is twofold: (1) revealing a relation between the counting function N(lambda) (the number of the eigenstates with eigenvalue smaller than a given number) and the heat kernel K(t), which is still an open problem in mathematics, and (2) introducing an approach for the calculation of N(lambda), for there is no effective method for calculating N(lambda) beyond leading order. We suggest a new expression of N(lambda) which is more suitable for practical calculations. A renormalization procedure is constructed for removing the divergences which appear when obtaining N(lambda) from a nonuniformly convergent expansion of K(t). We calculate N(lambda) for D-dimensional boxes, three-dimensional balls, and two-dimensional multiply-connected irregular regions. By the Gauss-Bonnet theorem, we generalize the simply-connected heat kernel to the multiply-connected case; this result proves Kac's conjecture on the two-dimensional multiply-connected heat kernel. The approaches for calculating eigenvalue spectra and state densities from N(lambda) are introduced.

Tripartite Entanglement Transformations and Tensor Rank

A basic question regarding quantum entangled states is whether one can be probabilistically converted to another through local operations and classical communication exclusively. While the answer for bipartite systems is known, we show that for tripartite systems, this question encodes some of the most challenging open problems in mathematics and computer science. In particular, we show that there is no easy general criterion to determine the feasibility, and in fact, the problem is NP hard. In addition, we find obtaining the most efficient algorithm for matrix multiplication to be precisely equivalent to determining the maximum rate to convert the Greenberger-Horne-Zeilinger state to a triangular distribution of three EPR states. Our results are based on connections between multipartite entanglement and tensor rank (also called Schmidt rank), a key concept in algebraic complexity theory.

Molecular simulation and related topics: some open mathematical problems

This paper presents some open mathematical questions in molecular simulation and related topics. Molecular simulation is a broad field, of major practical interest, which has received little attention from the mathematical community for years. The situation is rapidly changing. We review here some mathematical questions that we find the most challenging ones among the many that remain unsolved.

The 'Happy Ending Problem' of cardiac pacing? Cardiac resynchronization therapy for patients with atrial fibrillation and heart failure after atrioventricular junction ablation

The ‘Happy Ending Problem’ in mathematics is the following statement: Any set of five points in the plane in a general position has a subset of four points that form the vertices of a convex quadrilateral .A particular analogy can be drawn between this so-called unsolved problem in mathematics and the still unsolved problem of treatment of patients with heart failure (HF) associated with atrial fibrillation. Patients with HF and atrial fibrillation represent a subgroup of patients that encounters additional difficulties when compared with patients in sinus rhythm (SR). Most available studies, which include this subgroup of patients, suggest a beneficial effect of cardiac resynchronization therapy (CRT) on the outcome ( Table 1 ).1–8 However, most studies in the field of CRT excluded this particular patient population. Moreover, studies until now have contained 95% of the patients in SR. Despite this, a few well-conducted studies have addressed this problem. Unfortunately, some studies have caused a certain amount of confusion because of controversial data. This controversy arises as a result of errors in the study design, interpretation, and the low inclusion rates. The above-mentioned analogy in mathematics clearly points out another difficulty in decision-making in this multifactorial problem. The five points in this geometrical problem can represent at least five independent factors, which could influence future recommendations. Furthermore, these factors could have a potential effect on the outcome: the effect of atrioventricular (AV) node ablation and pacing, the effect of the pacing site on cardiac function, a diminished atrial function (lack of atrial kick), the effect of rhythm regularization, and the presence … *Corresponding author. Tel: +31 104633991; fax: +36 12151220 ext. 413. E-mail address : t.szilitorok{at}erasmusmc.nl

Blowup or no blowup? The interplay between theory and numerics

The question of whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data has been an outstanding open problem in fluid dynamics and mathematics. Recent studies indicate that the local geometric regularity of vortex lines can lead to dynamic depletion of vortex stretching. Guided by the local non-blowup theory, we have performed large scale computations of the 3D Euler equations on some of the most promising blowup candidates. Our results show that there is tremendous dynamic depletion of vortex stretching. The local geometric regularity of vortex lines and the anisotropic solution structure play an important role in depleting the nonlinearity dynamically and thus prevents a finite time blowup.

Some open issues associated with the new type of component: digital-to-frequency converter [Open Column

This paper presents seven unsolved mathematical problems which all relate to a new type of electronic component: digital-to-frequency converter. These problems are parts of the theoretical foundation of this new component. Even though theoretical in nature, these problems will have profound impact on future real electronic system design. These problems present themselves as an excellent example of demonstrating the significance of mathematical analysis in circuit design applications. They show the importance of fundamental research on directing the real world work.

On the Euler equations of incompressible fluids

Euler equations of incompressible fluids use and enrich many branches of mathematics, from integrable systems to geometric analysis. They present important open physical and mathematical problems. Examples include the stable statistical behavior of ill-posed free surface problems such as the Rayleigh-Taylor and Kelvin-Helmholtz instabilities. The paper describes some of the open problems related to the incompressible Euler equations, with emphasis on the blowup problem, the inviscid limit and anomalous dissipation. Some of the recent results on the quasigeostrophic model are also mentioned.

Data Distribution in a Peer to Peer Storage System

This article presents a data distribution algorithm for data storage in a P2P storage system, named Us. One of the Us aims is data durability. For scalability, data are distributing on thin peers using the well known Rabin dispersal technique. Unlike other systems such as OceanStore, where data are distributed on server peers, data are distributed on end user peers. In Us, when a peer fails, a reconstruction process rebuilds lost data with help from others peers. In a previous works, we showed that for data durability such system has to face a continuous large number of reconstructions to insure data durability. To minimize end user traffic due to the reconstruction process, distribution strategies must take into account a new measure: the maximum disturbance cost of a peer during the reconstruction process. The disturbance cost is indicated by the number of data communications which are requested from a single peer for rebuilding lost data. The main goal of this article is to define algorithm able to dilute the reconstruction process in the system. We show that this problem is similar to an open mathematical problem. Hence a algorithm is defined in order to distribute data and minimize the maximum disturbance cost for each peer. Finally, toe show that our distribution algorithm is close to the non-constructive theoretical optimal distribution.

A B-spline approach for empirical mode decompositions

We propose an alternative B-spline approach for empirical mode decompositions for nonlinear and nonstationary signals. Motivated by this new approach, we derive recursive formulas of the Hilbert transform of B-splines and discuss Euler splines as spline intrinsic mode functions in the decomposition. We also develop the Bedrosian identity for signals having vanishing moments. We present numerical implementations of the B-spline algorithm for an earthquake signal and compare the numerical performance of this approach with that given by the standard empirical mode decomposition. Finally, we discuss several open mathematical problems related to the empirical mode decomposition.

Computer supported mathematics with Ω mega

Classical automated theorem proving of today is based on ingenious search techniques to find a proof for a given theorem in very large search spaces—often in the range of several billion clauses. But in spite of many successful attempts to prove even open mathematical problems automatically, their use in everyday mathematical practice is still limited. The shift from search based methods to more abstract planning techniques however opened up a paradigm for mathematical reasoning on a computer and several systems of that kind now employ a mix of interactive, search based as well as proof planning techniques. The Ω mega system is at the core of several related and well-integrated research projects of the Ω mega research group, whose aim is to develop system support for a working mathematician as well as a software engineer when employing formal methods for quality assurance. In particular, Ω mega supports proof development at a human-oriented abstract level of proof granularity. It is a modular system with a central proof data structure and several supplementary subsystems including automated deduction and computer algebra systems. Ω mega has many characteristics in common with systems like NuPrL, CoQ, Hol, Pvs, and Isabelle. However, it differs from these systems with respect to its focus on proof planning and in that respect it is more similar to the proof planning systems Clam and λ Clam at Edinburgh.

Singular behavior of vacuum states for compressible fluids

In this survey paper, we will present the recent work on the study of the compressible fluids with vacuum states by illustrating its interesting and singular behavior through some systems of fluid dynamics, that is, Euler equations, Euler–Poisson equations and Navier–Stokes equations. The main concern is the well-posedness of the problem when vacuum presents and the singular behavior of the solution near the interface separating the vacuum and the gas. Furthermore, the relation of the solutions for the gas dynamics with vacuum to those of the Boltzmann equation will also be discussed. In fact, the results obtained so far for vacuum states are far from being complete and satisfactory. Therefore, this paper can only be served as an introduction to this interesting field which has many open and challenging mathematical problems. Moreover, the problems considered here are limited to the author's interest and knowledge in this area.

Dynamic modeling of chaos and turbulence

The series, Diophantus: Introduction to Mathematical Philosophy, Kalikasan, Manila, 1993; Nonlinear Anal. 30 (8) (1997) 5021–5032; Nonlinear Stud. 5 (2) (1998) 227–254; Nonlinear Anal., 35 (8) (1999) 259–285; Proc. Second Int. Conf. Tool. Math. Model., St. Petersburg, 4 (1999) 74–89; Proceedings of Third International Conference on Differential Equations, St. Petersburg, 2000 pp. 71–86; Probl. Nonlinear Anal. Eng. Syst. 7 (1) (2001) 56–78; Proceedings of Third International Conference on Dynamic Systems and Applications, Atlanta, 2001 pp. 201–208; Nonlinear Anal. 47 (2001) 5955–5966; Probl. Nonlinear Anal. Eng. Syst. 7 (2) (2001) 30–44; Indian J. Pure Appl. Math. 32 (10) (2001) 1539–1551; Appl. Math. Comput. 130 (1) (2001) 145–169; Indian J. Pure Appl. Math. 33 (1) (2003) 111–129; Appl. Math. Comput. 138 (2003) 127–149; Appl. Math. Comput. 139 (1) (2003) 23–36; Recent results, new inventions and new cosmology, accepted, Probl. Nonlinear Anal. Eng. Syst.; The new nonstandard analysis and intuitive calculus, submitted for publication; The philosophical and mathematical foundations of FLT's resolution, rectification and extension of underlying fields and applications, accepted, Int. J. Nonlinear Differential Equations; Extending the reach of computation, submitted for publication; The theory of learning: implications for math–science education and research, submitted for publications; Columbia: The crossroads for science, accepted, Int. J. Nonlinear Differential Equations; Dynamic modeling and applications, Proceedings of Fourth International Conference on Tools for Mathematical Modeling, 2003, St. Petersburg State Technical University; Nonlinear Anal. Phenomena, 1 (1) (2004) 1–23 and Nonlinear Anal. Phenomena, 2 (1) (2005) 15–30 introduced, embellished, developed, and elaborated dynamic modeling and its principal mathematical component—qualitative mathematics—and used it to resolve long-standing unsolved problems of mathematics and physics such as Fermat's last theorem, the gravitational n-body problem, Goldbach's Conjecture and the search for the basic constituent of matter. Among its major achievements is the development of the flux theory of gravitation (FTG) anchored on 42 laws of nature. The paper considers the questions of chaos and turbulence and shows that gravity is turbulence, specifically, the dynamics of vortex flux of dark matter consisting of its basic constituent, the superstring. FTG explains our universe as a super...super galaxy comprised of nested fractal sequences of vortices from our universe down through galaxy clusters, galaxies, stars, planets, moons and cosmic dust. As vortex, each term is self-similar to our universe. Beyond the cosmic dust is the domain of quantum gravity, forming nested fractal sequences of vortices each starting from a molecule, e.g., protein, down to the atom and primum. Crossing the boundary of visible matter we enter dark matter and find its basic constituent of nested fractal sequence of superstrings. The paper notes three new laws of nature discovered in the course of resolving the puzzles surrounding the disastrous final flight of the Columbia Space Shuttle (Int. J. Nonlinear Diff. Equations, Accepted). Finally using Hubble's law and guided by FTG, the rate of expansion of our universe and its acceleration are computed.

Open Circular Billiards and the Riemann Hypothesis

A comparison of escape rates from one and from two holes in an experimental container (e.g., a laser trap) can be used to obtain information about the dynamics inside the container. If this dynamics is simple enough one can hope to obtain exact formulas. Here we obtain exact formulas for escape from a circular billiard with one and with two holes. The corresponding quantities are expressed as sums over zeros of the Riemann zeta function. Thus we demonstrate a direct connection between recent experiments and a major unsolved problem in mathematics, the Riemann hypothesis.

Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

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The thermodynamic characterization of spontaneous electrochemical reactions

We consider whether the association of charged species of opposite parity in a chemical cell constitutes a spontaneous reaction. The initial distributions of the species are modeled as a steady-state phenomenon, characterized by a drift-diffusion system and two coupled constraints: (1) the electroneutrality of net system charge; and, (2) a coupled thermodynamic inequality constraint, reflecting the net decrease of the Gibbs' free energy in the closed system required for any spontaneous chemical reaction leading to uniform association of the species. A useful analytical technique of partial convexity allows the reformulation of thermodynamic compatibility. A control theory interpretation of the Dirichlet boundary conditions allows the selection of a trapping region for the range of the solution components which ensures that the reaction is spontaneous. A specific application is the production of hydrogen in an electrochemical cell. This is contained in a larger modeling context: reduction processes in electrochemistry. The final section describes extensions of the modeling in which an open mathematical problem and a pointer to the nonisothermal case are identified.

Structural characteristics of positionally disordered lattices: Relation to the first sharp diffraction peak in glasses

Positional disorder has been introduced into the atomic structure of certain crystalline lattices, and the orientationally-averaged structure factor S(k) and pair-correlation function g(r) of these disordered lattices have been studied. Analytical expressions for S(k) and g(r) for Gaussian positional disorder in 2D and 3D are confirmed with precise numerical simulations. These analytic results also have a bearing on the unsolved Gauss circle problem in mathematics. As the positional disorder increases, high-k peaks in S(k) are destroyed first, eventually leaving a single peak, that with the lowest-k value. The pair-correlation function for lattices with such high levels of positional disorder exhibits damped oscillations, with a period equal to the separation between the furthest-separated (lowest-k) lattice planes. The last surviving peak in S(k) is, for example for silicon and silica, at a wavevector nearly identical to that of the experimentally-observed first sharp diffraction peak (FSDP) in the amorphous phases of those materials. Thus, for these amorphous materials at least, the FSDP can be regarded as arising from scattering from atomic configurations equivalent to the single family of positionally-disordered local Bragg planes having the furthest separation.