Open Mathematical Problems Research Articles

Some open mathematical problems in electronic structure models and calculations

It is a very challenging topic to carry out electronic structure calculations of large systems fast and efficiently. The mathematical foundations and properties of electronic structure models play an important role in understanding, analyzing and designing computational methods for first-principles electronic structure calculations. In this paper, we shall present some open mathematical problems in the mathematical foundation of electronic structure models and the method and theory of electronic structure calculations.

On the weak coupling limit of quantum many-body dynamics and the quantum Boltzmann equation

The rigorous derivation of the Uehling-Uhlenbeck equation from more fundamental quantum many-particle systems is a challenging open problem in mathematics. In this paper, we exam the weak coupling limit of quantum N-particle dynamics. We assume the integral of the microscopic interaction is zero and we assume W^{4,1} per-particle regularity on the coressponding BBGKY sequence so that we can rigorously commute limits and integrals. We prove that, if the BBGKY sequence does converge in some weak sense, then this weak-coupling limit must satisfy the infinite quantum Maxwell-Boltzmann hierarchy instead of the expected infinite Uehling-Uhlenbeck hierarchy, regardless of the statistics the particles obey. Our result indicates that, in order to derive the Uehling-Uhlenbeck equation, one must work with per-particle regularity bound below W^{4,1}.

A model of riots dynamics: Shocks, diffusion and thresholds

We introduce and analyze several variants of a system of differential equations which model the dynamics of social outbursts, such as riots. The systems involve the coupling of an explicit variable representing the intensity of rioting activity and an underlying (implicit) field of social tension. Our models include the effects of exogenous and endogenous factors as well as various propagation mechanisms. From numerical and mathematical analysis of these models we show that the assumptions made on how different locations influence one another and how the tension in the system disperses play a major role on the qualitative behavior of bursts of social unrest. Furthermore, we analyze here various properties of these systems, such as the existence of traveling wave solutions, and formulate some new open mathematical problems which arise from our work.

A model of riots dynamics: Shocks, diffusion and thresholds

We introduce and analyze several variants of a system of differential equations which model the dynamics of social outbursts, such as riots.The systems involve the coupling of an explicit variable representing the intensity of rioting activity and an underlying (implicit) field of social tension.Our models include the effects ofexogenous and endogenous factors as well as various propagation mechanisms. From numerical and mathematical analysisof these models we show that the assumptions made on how different locations influence one another and how the tension inthe system disperses play a major role on the qualitative behavior of bursts of social unrest.Furthermore, we analyze here various properties of these systems, such as the existence of traveling wave solutions, and formulatesome new open mathematical problems which arise from our work.

Outline of Hadronic Mathematics, Mechanics and Chemistry as Conceived by R. M. Santilli

In this paper, we outline the various branches of hadronic mathematics and their applications to corresponding branches of hadronic mechanics and chemistry as conceived by the Italian-American scientist Ruggero Maria Santilli. According to said conception, hadronic mathematics comprises the following branches for the treatment of matter in conditions of increasing complexity: 1) 20th century mathematics based on Lie’s theory; 2) IsoMathematics based on Santilli’s isotopies of Lie’s theory; 3) GenoMathematics based on Santilli’s formulation of Albert’s Lie-admissibility; 4) HyperMathematics based on a multi-valued realization of genomathematics with classical operations; and 5) HyperMathematics based on Vougiouklis Hv hyperstructures expressed in terms of hyperoperations. Additionally, hadronic mathematics comprises the anti-Hermitean images (called isoduals) of the five preceding mathematics for the description of antimatter also in conditions of increasing complexity. The outline presented in this paper includes the identification of represented physical or chemical systems, the main mathematical structure, and the main dynamical equations per each branch. We also show the axiomatic consistency of various branches of hadronic mathematics as sequential coverings of 20th century mathematics; and indicate a number of open mathematical problems. Novel physical and chemical applications permitted by hadronic mathematics are presented in subsequent collections.

PROFIL BERPIKIR KREATIF SISWA KELAS AKSELERASI DALAM MEMECAHKAN MASALAH MATEMATIKA TERBUKA

The background of the research is the needs of human resources that are able to think creatively in order to utilize and develop science and technology. One of the ways to overcome it is to optimize the improvement of students’ ability to think creatively in accelerated classes. Accelerated classes are accelerated learning program. Thus, the need of creative human resources can be immediately fulfilled. The aim of the research are to determine the students’ creativity and describe the process of students’ creative thinking in solving open mathematical problem. Creative thinking process is traced by referring to the stage of creative thinking by Wallas, namely preparation, incubation, illumination, and verification. This research is qualitative- descriptive. The research instruments used in the research were a problem-solving task sheet and in-depth interviews. The subjects in this research were students in accelerated classes in Jambi. Analysis of data used data analysis according to Miles and Huberman which consists of data reduction, exposure data and drawing conclusions. The results show that there are 8% of creative students from 25 students in accelerated class in solving open mathematical problems. Then 72% are less creative and 20% are not creative. In solving open mathematical problems, creative and less creative students pass through the four stages of creative thinking process by Wallas. Students who are not creative only pass through the three stages of creative thinking process which are preparation, incubation, and illumination.

Adaptive Statistical Inference

This workshop in mathematical statistics highlights recent advances in adaptive methods for statistical estimation, testing and confidence sets. Related open mathematical problems are discussed with potential impact on the development of computationally efficient algorithms of data processing under prior uncertainty. Parcticular emphasis is on high dimensional models, inverse problems and discrtete structures.

A High-Dimensional Pricing Framework for Fi nancial Instruments Valuation

We present in this paper a method allowing to consider valuations of complex OTC products, including american exercising, written on a large number of underlyings. The motivation is to provide a framework able to quickly valuate big investment bank portfolios with more complex pricing algorithms, as those coming from risk measures, and we think particularly to CVA (counterpart risk valuation) ones, leading to consider high dimensional Optimal Stopping Problems.Indeed, it has already been noticed that current numerical methods devoted to pricing algorithms face an open mathematical problem, the curse of dimensionality (see for instance [5], [12], [4]), particularly in the context of CVA computations. We notice also that the econometric problem, consisting in fi nding a risk-neutral dynamic of the underlyings, faces the very same open problem. Indeed, the CVA approach, as others existing risk measures, although vertuous, can not be credible as long as they will be cursed.The framework introduced in this paper has two purposes: firstly to provide a fast calibration algorithm to determine a risk-neutral econometric, using market prices, able to capture correlation structures between underlyings. Secondly, to design a fast backwardation algorithms upon this econometry. Both algorithms break the curse of dimensionality wall. This is achieved in this paper in a particular context, given by martingales processes, a classical assumption in mathematical finance. From a mathematical point of view, these results are obtained blending together stochastic, partial differential equations, and optimal transport theories, and are a generalization to the multi-dimensional case of this paper. We believe that these tools should be the right candidates to provide a standardized framework for large portfolios valuations of risk measures or investment strategies, as well as a pedestal to tackle credible systemic risk measures.

Hilbert's 6th Problem and Axiomatic Quantum Field Theory

��� 1. The Main Claims This paper has two parts, a historical and a systematic. In the historical part it is argued that the two major axiomatic approaches to relativistic quantum aeld theory, the Wightman and Haag-Kastler axiomatizations, are realizations of the program of axiomatization of physical theories announced by Hilbert in his 6th of the 23 problems discussed in his famous 1900 Paris lecture on open problems in mathematics, if axiomatizing physical theories is interpreted in a soft and opportunistic sense suggested in 1927 by Hilbert, Nordheim, and von Neumann. To show this, Section 2 recalls arst some of Hilbert’s views about the axiomatic approach to physical theories as these were formulated in his 6th problem. It will be seen that Hilbert did not specify in his Paris lecture what precisely the axiomatic approach in physics is supposed to be. It will be seen in Section 3 that he was more explicit in 1927 however: The joint paper (Hilbert et al. 1927) addresses the issue of the axiomatic approach in physics explicitly and embraces a soft notion of axiomatization, which is assumed to be carried out in an opportunistic way only in physics. Axiomatic aeld theories are axiomatizations in this sense. The systematic part of the paper isolates and comments on some characteristic features of the axiomatic quantum aeld theories, elements of which are recalled brieoy in Section 4. The features will be of two types: general-methodological and speciac-physical. The arst category includes what we call here ‘intended non-categoricity” and “large internal room,” discussed in Section 5; the speciac-physical we discuss (in Section 6) are “ontological silence” and “causal completeness.” “Intended Research supported in part by the Hungarian Scientiac Research Found (OTKA). Contract number: 100715

Stoichiometric producer-grazer models with varying nitrogen pools and ammonia toxicity

We formulate and analyze a stoichiometric model of producer-grazer systems with excess nutrient recycling (waste) that may inhibit grazer survival and growth. Specifically, we model the intoxication dynamics caused by accumulation of grazer waste and dead biomass decay. This system has a range of applications, but we focus on those in which the producers are microalgae and the limiting nutrient is nitrogen. High levels of ammonia (and to a lesser extent nitrite) have been observed to increase grazer death, especially in aquaculture systems. We assume that all nitrification is due to nitrogen uptake and assimilation by the producer; therefore, the model explores systems in which the producer serves the dual role of grazer food and water treatment. The model exhibits three equilibria corresponding to total extinction, grazer-only extinction, and coexistence. While a sufficient condition is found under which grazer extinction equilibrium is globally stable, we propose a conjecture for neccessary and sufficient conditions, which remains an open mathematical problem. Local stability of grazer extinction equilibrium is ensured under a sharp necessary and sufficient condition. Local stability for the coexistence equilibrium is studied algebraically and numerically. Bifurcation diagrams with respect to total nitrogen and its implications are also presented.

Piaget's Theory in the Development of Creative Thinking

Piaget’s revolutionary study on the cognitive development of children has focused on the development of logic. Logical operations and a variety of classifications based on the set of accepted rules involve convergent thinking. Children and adults have logical and creative thinking which deal with a reality of thinking. This study aims to examine a cognitive structure of students, which is closely related to the Piaget’s cognitive development theories of students when creative thinking. Students were given an open mathematical problem and were expected to be able to take ad-vantage of sensitivity, fluency, flexibility, originality, and elaboration which can be seen as clearly of their structure cognitive.

Layman's method to verify Goldbach's conjectures

Goldbach conjecture of prime numbers is one of the unsolved mathematical problems. Many trial solutions appeared in the literature, but so far none has been accepted by the mathematics societies. This paper describes a graphical method devised by me to explain the mystery of the said conjecture. My method based on the teachings of analytical geometry using a rectangular coordinate frame with even numbers as ordinates and prime numbers as abscissas. Straight lines with 45 degree slop and intercepets of varying prime numbers on the ordinate are drawn to meet all the vertical straight draw grom the abscissas. These diagonal lines are designated as separation lines and identified by its intercept number. The intersection of vertical abscissa line, the separation line and a horizontal line drawn from the ordinates shows the relationship of an even number and its pair of prime numbers. These intersections vividly appear on the horizontal even number line and can be easily seen. This method is a graphical version of binary combination of prime numbers and can locate the prime-pairs of any even nuber by drawing a family of separation lines.

Where Are the Zeros of xi of t? [On the Lighter Side

Now that Andrew Wiles has proved the Taniyama-Shimura conjecture, inter alia proving Fermat's last theorem, and Grigory Perelman has proved the Poincare conjecture in three dimensions, thus completing the earlier work of Steve Smale (who proved the generalized Poincare conjecture in all dimensions greater than or equal to five) and Michael Freedman (who proved the generalized Poincare conjecture in four dimensions), the Riemann hypothesis is perhaps the most famous open problem in mathematics. This hypothesis is contained in Riemann's presentation upon being elected to the Berlin Academy in 1859. Like many fascinating problems in mathematics, the hypothesis can be stated very simply.

Special issue on Multidimensional systems theory and control

Multidimensional systems and signal processing is, todaymore than ever before, a very lively research area, which attracts a great deal of attention and research interest, both in the control and signal processing communities. Indeed, an increasingly wider research community devotes much of its efforts to the solution of control, estimation and signal processing problems involving multidimensional systems. This is undoubtedly motivated by the fact that this area plays a crucial role in a variety of applications, ranging from remote sensing for environmental monitoring and geological mapping, to medical imaging and the automatic control of industrial processes. The impact that this topic has shown in recent years in events such as the International Symposium on Mathematical Theory of Networks and Systems (MTNS), and the importance of other events entirely devoted to this topic, such as the International Workshop onMultidimensional Systems (nDS), are a clear evidence of the increasing interest that this area attracts within the control, signal processing and mathematics communities. The broad range of application areas for multidimensional systems and signal processing, the great need to tackle unsolved fundamental mathematical problems within this area, and the ever increasing capacity of computing technologies, provide substantial motivation for further development of the underlying mathematical tools. Motivated by these facts, this Special Issue aims at providing an overview of several recent developments in this wide area. The contributions selected for this Special Issue clearly display the broad scope of multidimensional systems, from open mathematical

Active Information in Metabiology

Metabiology is a fascinating intellectual romp in the surreal world of the mathematics of algorithmic information theory. In this world, halting oracles hunt for busy beaver numbers and busy beaver numbers unearth Chaitin’s number, knowledge of which in turn allows resolution of numerous unsolved mathematical problems, many of whose solutions would earn large cash bounties. All this, despite the fact that halting oracles can’t be implemented on a computer, a computer can never make a list of busy beaver numbers, and Chaitin’s number, always a positive real number less than one, is proven to be unknowable. The fun of metabiology is the application of these ideas to illustrate Darwinian evolution. When metabiology’s evolutionary process is stripped of the glitter of algorithmic information theory, however, what remains is a procedure similar to that used in other attempts to model Darwinian evolution, like the ev and AVIDA computer programs. Metabiology, like ev and AVIDA, succeeds because available sources of knowledge about the solution being sought can be mined. We show the mining of information from a halting oracle has striking similarities to mining information from a simple Hamming oracle. Unlike a halting oracle, however, Hamming oracles can be implemented on a computer. We demonstrate that for both oracles, information can be mined by search strategies that are analogous in some respects even though the methods differ; in both cases the search strategy used greatly influences the result. Because metabiology’s process relies on unknowable numbers and infinite resources, its reported relative performance measures can only be expressed asymptotically. That is, the results of metabiology are only proven to be true on the largest possible scale. In fact, simple simulations using bounded resources suggest the asymptote is not always approached quickly, indicating that metabiology results may only hold for scales larger than any practical system.

The Logic and Fundamental Concepts of the Grand Unified Theory

The grand unified theory (GUT) originated in mathematics with this question: why are there long standing unsolved problems in mathematics, e.g., Fermat’s conjecture (also known as Fermat’s last theorem (FLT))? The answer came quickly: its underlying fields—foundations and the real number system—are defective. In particular, formal logic is inapplicable to mathematics (language of science) and the real number system is inconsistent. Critique-rectification of these fields was undertaken leading to a new mathematical methodology and the consistent new real number system that provides the main mathematics of GUT. Similar question was posed in physics: why are there long standing problems, e.g., the gravitational n-body and turbulence problems? The answer: the present methodology, quantitative modeling is inadequate and the remedy is a new methodology—qualitative mathematics and modeling that solved these problems and provided the initial formulation of GUT. This paper presents the basic logic of GUT and its fundamental concepts, particularly, the superstring or fundamental building block of matter.

New Practical Algebraic Public-Key Cryptosystem and Some Related Algebraic and Computational Aspects

The most popular present-day public-key cryptosystems are RSA and ElGamal cryptosystems. Some practical algebraic generalization of the ElGamal cryptosystem is considered-basic modular matrix cryptosystem (BMMC) over the modular matrix ring M2(Zn). An example of computation for an artificially small number n is presented. Some possible attacks on the cryptosystem and mathematical problems, the solution of which are necessary for implementing these attacks, are studied. For a small number n, computational time for compromising some present-day public-key cryptosystems such as RSA, ElGamal, and Rabin, is compared with the corresponding time for the ВММС. Finally, some open mathematical and computational problems are formulated.

Reverse-engineering Reverse Mathematics

An important open problem in Reverse Mathematics (Montalbán, 2011 [16]; Simpson, 2009 [25]) is the reduction of the first-order strength of the base theory from IΣ1 to IΔ0+exp. The system ERNA, a version of Nonstandard Analysis based on the system IΔ0+exp, provides a partial solution to this problem. Indeed, weak Königʼs lemma and many of its equivalent formulations from Reverse Mathematics can be ‘pushed down’ into ERNA, while preserving the equivalences, but at the price of replacing equality with ‘≈’, i.e. infinitesimal proximity (Sanders, 2011 [19]). The logical principle corresponding to weak Königʼs lemma is the universal transfer principle from Nonstandard Analysis. Here, we consider the intermediate and mean value theorem and their uniform generalizations. We show that ERNAʼs Reverse Mathematics mirrors the situation in classical Reverse Mathematics. This further supports our claim from Sanders (2011) [19] that the Reverse Mathematics of ERNA plus universal transfer is a copy up to infinitesimals of that of WKL0. We discuss some of the philosophical implications of our results.

Extracting Dynamical Equations from Experimental Data is NP Hard

The behavior of any physical system is governed by its underlying dynamical equations. Much of physics is concerned with discovering these dynamical equations and understanding their consequences. In this Letter, we show that, remarkably, identifying the underlying dynamical equation from any amount of experimental data, however precise, is a provably computationally hard problem (it is NP hard), both for classical and quantum mechanical systems. As a by-product of this work, we give complexity-theoretic answers to both the quantum and classical embedding problems, two long-standing open problems in mathematics (the classical problem, in particular, dating back over 70years).

Open mathematical problems regarding non-Newtonian fluids

We present three open problems in the mathematical modelling of the flow of non-Newtonian fluids. The first problem is rather long standing: a discontinuity in the dependence of the rise velocity of a gas bubble on its volume. This is very well characterized experimentally but not, so far, fully reproduced either numerically or analytically. The other two are both instabilities. The first is observed experimentally but never predicted analytically or numerically. In the second instability, numerical studies reproduce the experimental observations but there is as yet no analytical or semi-analytical prediction of the linear instability which must be present.