Open Mathematical Problems Research Articles

Correspondence principle in plane and axisymmetric mixed boundary-value problems of elasticity

This paper advances the relations by A. Ja. Aleksandrov between axisymmetric and plane strain states to the case of mixed boundary-value problems. We revise two models of a strip-shaped and a circular stamp indented into a half-plane and a half-space, respectively, when the normal and tangential traction components are unknown a priori. Also, two models of a strip-shaped and a penny-shaped interfacial crack are considered. By using the theory of Abelian integral operators and the Riemann-Hilbert problem on a segment we derive the solutions to the model problems and establish the relations between the governing systems of integral equations associated with the four models and their solutions. These relations can be interpreted as mappings between (i) plane and axisymmetric contact problems, (ii) plane and axisymmetric fracture models, (iii) plane contact and fracture problems, and (iv) axisymmetric contact and fracture problems. The mappings enable us to write down the governing systems of integral equations and the solutions to any three models by making use of the governing system and the solution to the fourth problem. The transformations are specified in the scalar cases when there is no friction in the contact zone and when a crack is in a homogeneous elastic medium. By considering the contact frictionless problem of an annulus stamp it is shown that, although an exact solution to the plane strain frictionless contact problem of two stamps is available, a transformation of the plane to the axisymmetric solution in this case is not possible to obtain, and derivation of a closed-form solution to the annulus stamp model is still an open mathematical problem.

A Complete Analytical Solution for the On and Off Dynamic Equations of a TaO Memristor

In this brief we provide a complete analytical model for the time evolution of the state of a real-world memristor under any dc stimulus and for all initial conditions. The analytical dc model is derived through the application of mathematical techniques to Strachan’s accurate mathematical description of a tantalum oxide nano-device from Hewlett Packard Labs. Under positive dc inputs the state equation of the Strachan model can be solved analytically, providing a closed-form expression for the device memory state response. However, to the best of our knowledge, the analytical integration of the state equation of the Strachan model under dc inputs of negative polarity is an unsolved mathematical problem. In order to bypass this issue, the state evolution function is first expanded in a series of Lagrange polynomials, which reproduces accurately the original model predictions on the device off-switching kinetics. The solution to the resulting state equation approximation may then be computed analytically by applying methods from the field of mathematics. Our full analytical model matches both qualitatively and quantitatively the tantalum oxide memristor response captured by the original differential algebraic equation set to typical stimuli of interest such as symmetric and asymmetric pulse excitations. It is further insensitive to the convergence issues that typically arise in the numerical integration of the original model, and may be easily integrated into software programs for circuit synthesis, providing designers with a reliable tool for exploratory studies on the capability of a certain circuit topology to satisfy given design specifications.

Discrete and Discontinuous Increase in the Micellar Aggregation Number: Effects of the Alkyl Chain Length on Platonic Micelles.

Micelles with perfect monodispersity in terms of the aggregation number ( Nagg) have recently been discovered, whose values of Nagg interestingly always coincide with the vertex or face number of regular polyhedral structures (i.e., Platonic solids). Owing to the monodispersity of the micelles, named Platonic micelles, we could expect them to exhibit unprecedented aggregation behavior. In this study, the effects of alkyl chain length on micellar aggregation behavior were characterized using small-angle scattering techniques such as small-angle X-ray scattering and asymmetrical flow field-flow fractionation coupled with multi-angle light scattering, as well as analytical ultracentrifugation measurements. The Nagg of Platonic micelles discretely and discontinuously increased when increasing the alkyl chain length, which differs markedly from the findings for conventional micelles. This aggregation behavior could be reasonably explained by the relationship between the thermodynamic stability of the micelles and the coverage density defined by one of the unsolved mathematical problems: the Tammes problem.

Correlating Thermal Machines and the Second Law at the Nanoscale

Thermodynamics at the nanoscale is known to differ significantly from its familiar macroscopic counterpart: The possibility of state transitions is not determined by free energy alone but by an infinite family of free-energy-like quantities; strong fluctuations (possibly of quantum origin) allow one to extract less work reliably than what is expected from computing the free-energy difference. However, these known results rely crucially on the assumption that the thermal machine is not only exactly preserved in every cycle but also kept uncorrelated from the quantum systems on which it acts. Here, we lift this restriction: We allow the machine to become correlated with the microscopic systems on which it acts while still exactly preserving its own state. Surprisingly, we show that this possibility restores the second law in its original form: Free energy alone determines the possible state transitions, and the corresponding amount of work can be invested or extracted from single systems exactly and without any fluctuations. At the same time, the work reservoir remains uncorrelated from all other systems and parts of the machine. Thus, microscopic machines can increase their efficiency via clever “correlation engineering” in a perfectly cyclic manner, which is achieved by a catalytic system that can sometimes be as small as a single qubit (though some setups require very large catalysts). Our results also solve some open mathematical problems on majorization which may lead to further applications in entanglement theory.Received 31 August 2018Revised 17 November 2018DOI:https://doi.org/10.1103/PhysRevX.8.041051Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasNonequilibrium statistical mechanicsQuantum statistical mechanicsQuantum thermodynamicsResource theoriesQuantum InformationStatistical Physics

Multistationarity and Bistability for Fewnomial Chemical Reaction Networks

Bistability and multistationarity are properties of reaction networks linked to switch-like responses and connected to cell memory and cell decision making. Determining whether and when a network exhibits bistability is a hard and open mathematical problem. One successful strategy consists of analyzing small networks and deducing that some of the properties are preserved upon passage to the full network. Motivated by this, we study chemical reaction networks with few chemical complexes. Under mass action kinetics, the steady states of these networks are described by fewnomial systems, that is polynomial systems having few distinct monomials. Such systems of polynomials are often studied in real algebraic geometry by the use of Gale dual systems. Using this Gale duality, we give precise conditions in terms of the reaction rate constants for the number and stability of the steady states of families of reaction networks with one non-flow reaction.

Mitigating Interference in Content Delivery Networks by Spatial Signal Alignment: The Approach of Shot-Noise Ratio

Multimedia content, especially videos, is expected to dominate data traffic in next-generation mobile networks. Caching popular content at the network edge, namely content helpers (base stations and access points), has emerged as a solution for low-latency content delivery. Compared with traditional wireless communication, content delivery has a key characteristic that many signals coexisting in the air carry identical popular content. However, they can interfere with each other at a receiver if their modulation-and-coding (MAC) schemes are adapted to individual channels following the classic approach. To address this issue, we present a novel idea of content adaptive MAC (CAMAC) where adapting MAC schemes to content ensures that all signals carrying identical content are encoded using an identical MAC scheme to achieve spatial MAC alignment. Consequently, interference can be harnessed as signals to improve the reliability of wireless delivery. In the remaining part of the paper, we focus on quantifying the gain that CAMAC can bring to a content-delivery network by using a stochastic-geometry model. Specifically, content helpers are distributed as a Poisson point process and each of them transmits a file from a content database based on a given popularity distribution. Given a fixed threshold on the signal-to-interference ratio for successful transmission, it is discovered that the successful content-delivery probability is closely related to the distribution of the ratio of two independent shot noise processes, named a shot-noise ratio . The distribution itself is an open mathematical problem that we tackle in this work. Using stable-distribution theory and tools from stochastic geometry, the distribution function is derived in closed form. Extending the result in the context of content-delivery networks with CAMAC yields the content-delivery probability in different closed forms. In addition, the gain in the probability due to CAMAC is shown to grow with the level of skewness in the content popularity distribution.

Beyond the BKM criterion for the 2D resistive magnetohydrodynamic equations

The question of whether the two-dimensional (2D) magnetohydrodynamic (MHD) equations with only magnetic diffusion can develop a finite time singularity from smooth initial data is a challenging open problem in fluid dynamics and mathematics. In this paper, we derive a regularity criterion less restrictive than the Beale-Kato-Madja (BKM) regularity criterion type, namely any solution $(u,b) \in C([0,T[;H^r(\R^2))$ with $r>2$ remains in $H^r(\R^2)$ up to time $T$ under the assumption that $\dsp \int_0^T \frac{\|\nabla u(t)\|_\infty^{\frac{1}{2}}}{\log (e+\|\nabla u(t)\|_\infty)} dt 2$ remains in $H^r(\R^2)$ up to time $T$. Indeed, in virtue of $\nabla u = {\rm R} ((\nabla \times u)\, {\rm Id})$ with ${\rm R}$ Riesz transform on matrix-valued functions, we expect that the blow-up rate at a time $T$ of $\|\nabla u(t)\|_\infty$ be the same as the one of $\|\nabla \times u(t)\|_\infty$. Furthermore, our result applies also to a class of equations arising in hydrodynamics and studied in \cite{EM} for their $L^\infty$ ill-posedness.

Anisotropic Geodesic Fluid in Non-Comoving Spherical Coordinates

We start with a recently introduced spherically symmetric geodesic fluid model (arXiv: 1601.07030) whose energy-momentum tensor in the comoving frame is dust-like with nontrivial energy flux. In the non-comoving energy frame (vanishing energy flux) the same EMT contains besides dust only radial pressure. We present Einstein's equations together with the matter equations in static spherically symmetric coordinates. These equations are self-contained (four equations for four unknowns). We solve them analytically except for a resulting nonlinear ordinary differential equation (ODE) for the gravitational potential. This ODE can be rewritten as a Lienard differential equation which, however, may be transformed into a rational Abel differential equation of the first kind. Finally we list some open mathematical problems and outline possible physical applications (galactic halos, dark energy stars) and related open problems.

Open Problems in Mathematics with John Nash

Open Problems in Mathematics with John Nash

A Cognitive Analysis of Students’ Mathematical Problem Solving Ability on Geometry

The purpose of this research is to analyze of mathematical problem solving ability of students in one of secondary school on geometry. This research was conducted by using quantitative approach with descriptive method. Population in this research was all students of that school and the sample was twenty five students that was chosen by purposive sampling technique. Data of mathematical problem solving were collected through essay test. The results showed the percentage of achievement of mathematical problem solving indicators of students were: 1) solve closed mathematical problems with context in math was 50%; 2) solve the closed mathematical problems with the context beyond mathematics was 24%; 3) solving open mathematical problems with contexts in mathematics was 35%; And 4) solving open mathematical problems with contexts outside mathematics was 44%. Based on the percentage, it can be concluded that the level of achievement of mathematical problem solving ability in geometry still low. This is because students are not used to solving problems that measure mathematical problem solving ability, weaknesses remember previous knowledge, and lack of problem solving framework. So the students’ ability of mathematical problems solving need to be improved with implement appropriate learning strategy.

Global solutions to the electrodynamic two-body problem on a straight line

The classical electrodynamic two-body problem has been a long standing open problem in mathematics. For motion constrained to the straight line, the interaction is similar to that of the two-body problem of classical gravitation. The additional complication is the presence of unbounded state-dependent delays in the Coulomb forces due to the finiteness of the speed of light. This circumstance renders the notion of local solutions meaningless, and therefore, straight-forward ODE techniques can not be applied. Here, we study the time-symmetric case, i.e., the Fokker-Schwarzschild-Tetrode (FST) equations, comprising both advanced and retarded delays. We extend the technique developed in \cite{DirkGuenter}, where existence of FST solutions was proven on the half-line, to ensure global existence -- a result that had been obtained by Bauer \cite{Bauer} in 1997. Due to the novel technique, the presented proof is shorter and more transparent but also relies on the idea to employ asymptotic data to characterize solutions.

One Way Functions –Conjecture, Status, Applications and Future Research Scope

The conjecture that one way function exists is an open problem, the resolution of which holds the key for the solution of many unsolved problems in mathematics and cryptography. This paper presents the introduction of one way functions from complexity & modern cryptography theory viewpoint and their significance in cryptographic applications and research. This paper presents the features and limitations of proposed candidate functions, and the implications of proof of one way functions conjecture.

The secret life of 1/n: A journey far beyond the decimal point

The decimal expansions of the numbers 1/n (such as 1/3 = .03333..., 1/7 = 0.142857...) are most often viewed as tools for approximating quantities to a desired degree of accuracy. The aim of this exposition is to show how these modest expressions in fact have much more to offer, particularly in the case when the expansions are infinitely long. First we discuss how simply asking about the period (that is, the length of the repeating sequence of digits) of the decimal expansion of 1/n naturally leads to more sophisticated ideas from elementary number theory, as well as to unsolved mathematical problems. Then we describe a surprising theorem of K. Girstmair showing that the digits of the decimal expansion of 1/p, for certain primes p, secretly contain deep facts that have long delighted algebraic number theorists.

Kinetic Transition Networks for the Thomson Problem and Smale's Seventh Problem.

The Thomson problem, arrangement of identical charges on the surface of a sphere, has found many applications in physics, chemistry and biology. Here, we show that the energy landscape of the Thomson problem for N particles with N=132, 135, 138, 141, 144, 147, and 150 is single funneled, characteristic of a structure-seeking organization where the global minimum is easily accessible. Algorithmically, constructing starting points close to the global minimum of such a potential with spherical constraints is one of Smale's 18 unsolved problems in mathematics for the 21st century because it is important in the solution of univariate and bivariate random polynomial equations. By analyzing the kinetic transition networks, we show that a randomly chosen minimum is, in fact, always "close" to the global minimum in terms of the number of transition states that separate them, a characteristic of small world networks.

Undecidable Cases of Model Checking Probabilistic Temporal-Epistemic Logic (Extended Abstract)

We investigate the decidability of model-checking logics of time, knowledge and probability, with respect to two epistemic semantics: the clock and synchronous perfect recall semantics in partially observed discrete-time Markov chains. Decidability results are known for certain restricted logics with respect to these semantics, subject to a variety of restrictions that are either unexplained or involve a longstanding unsolved mathematical problem. We show that mild generalizations of the known decidable cases suffice to render the model checking problem definitively undecidable. In particular, for a synchronous perfect recall, a generalization from temporal operators with finite reach to operators with infinite reach renders model checking undecidable. The case of the clock semantics is closely related to a monadic second order logic of time and probability that is known to be decidable, except on a set of measure zero. We show that two distinct extensions of this logic make model checking undecidable. One of these involves polynomial combinations of probability terms, the other involves monadic second order quantification into the scope of probability operators. These results explain some of the restrictions in previous work.

Meaning of temperature in different thermostatistical ensembles.

Depending on the exact experimental conditions, the thermodynamic properties of physical systems can be related to one or more thermostatistical ensembles. Here, we survey the notion of thermodynamic temperature in different statistical ensembles, focusing in particular on subtleties that arise when ensembles become non-equivalent. The 'mother' of all ensembles, the microcanonical ensemble, uses entropy and internal energy (the most fundamental, dynamically conserved quantity) to derive temperature as a secondary thermodynamic variable. Over the past century, some confusion has been caused by the fact that several competing microcanonical entropy definitions are used in the literature, most commonly the volume and surface entropies introduced by Gibbs. It can be proved, however, that only the volume entropy satisfies exactly the traditional form of the laws of thermodynamics for a broad class of physical systems, including all standard classical Hamiltonian systems, regardless of their size. This mathematically rigorous fact implies that negative 'absolute' temperatures and Carnot efficiencies more than 1 are not achievable within a standard thermodynamical framework. As an important offspring of microcanonical thermostatistics, we shall briefly consider the canonical ensemble and comment on the validity of the Boltzmann weight factor. We conclude by addressing open mathematical problems that arise for systems with discrete energy spectra.

The Mathematical Representation of Turbulent Heat Fluxes Using Reynold’s Decomposition

From a mathematical perspective, it is fundamental to develop a rigorous background upon which to study the physical quantities of a turbulent flow. The first problem in the mathematical theory is related to the deterministic nature of chaotic systems assumed in dynamical system theory and believed to hold inturbulence. This has actually not been proved for the Navier-Stokes equations. It is in fact one of the most outstanding open problems in mathematics to determine whether given an initial condition for the velocity field there exists, in some sense, a unique solution of the Navier-Stokes equations starting with this initial condition and valid for all later times. In a turbulent atmosphere, a turbulent stress term, the Reynolds stress, must be applied. All the terms in the horizontal motion equations are the order of 10﹣4 - 10﹣3 m·s2. Under certain condition, some terms are very small and can be neglected for example, the rotational term is insignificant in the equation of vertical motion and has been omitted, instead gravitational acceleration term appears in the equation for vertical motion [1]; for steady flow, the tendency can be neglected; in the centre of high and low pressure, gradient force can be neglected; at the equator of for small scale processes, the Coriolis force can be neglected, and above the atmospheric boundary layer the stress terms can be neglected.

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.

Some mathematical problems in liquid crystal theories

Liquid crystals are a kind of soft matter with broad applications, and they are widely studied in physics, chemistry and material sciences. This survey summarizes various mathematical models for liquid crystals and recent progresses on these models and the connections between them. In addition, we list some mathematical open problems.

Free boundary problems in biology.

In this paper, I review several free boundary problems that arise in the mathematical modelling of biological processes. The biological topics are quite diverse: cancer, wound healing, biofilms, granulomas and atherosclerosis. For each of these topics, I describe the biological background and the mathematical model, and then proceed to state mathematical results, including existence and uniqueness theorems, stability and asymptotic limits, and the behaviour of the free boundary. I also suggest, for each of the topics, open mathematical problems.