Exact Proof Research Articles

How to Formulate an Exact Proof of the Riemann Hypothesis?

How to Formulate an Exact Proof of the Riemann Hypothesis?

Quantitatively Exact proof of the Euler-Goldbach hypothesis

Quantitatively Exact proof of the Euler-Goldbach hypothesis

Theorema 2.0: Computer-Assisted Natural-Style Mathematics

The Theorema project aims at the development of a computer assistant for the working mathematician. Support should be given throughout all phases of mathematical activity, from introducing new mathematical concepts by definitions or axioms, through first (computational) experiments, the formulation of theorems, their justification by an exact proof, the application of a theorem as an algorithm, until to the dissemination of the results in form of a mathematical publication, the build up of bigger libraries of certified mathematical content and the like. This ambitious project is exactly along the lines of the QED manifesto issued in 1994 (see e.g. http://www.cs.ru.nl/~freek/qed/qed.html) and it was initiated in the mid-1990s by Bruno Buchberger. The Theorema system is a computer implementation of the ideas behind the Theorema project. One focus lies on the natural style of system input (in form of definitions, theorems, algorithms, etc.), system output (mainly in form of mathematical proofs) and user interaction. Another focus is theory exploration, i.e. the development of large consistent mathematical theories in a formal frame, in contrast to just proving single isolated theorems. When using the Theorema system, a user should not have to follow a certain style of mathematics enforced by the system (e.g. basing all of mathematics on set theory or certain variants of type theory), rather should the system support the user in her preferred flavour of doing math. The new implementation of the system, which we refer to as Theorema 2.0, is open-source and available through GitHub.

Analysis and perturbation of degree correlation in complex networks

Degree correlation is an important topological property common to many real-world networks such as the protein-protein interactions and the metabolic networks. In the letter, the statistical measures for characterizing the degree correlation in networks are investigated analytically. We give an exact proof of the consistency for the statistical measures, and reveal the general linear relation in the degree correlation, which provides a simple and interesting perspective on the analysis of the degree correlation in complex networks. By using the general linear analysis, we investigate the perturbation of the degree correlation in complex networks caused by the simple structural variation such as the “rich club”. The results show that the assortativity of homogeneous networks such as the Erdős-Rényi graphs is easily strongly affected by the simple structural changes, while it has only a slight variation for heterogeneous networks with broad degree distribution such as the scale-free networks. Clearly, the homogeneous networks are more sensitive to the perturbation than the heterogeneous networks.

Translation of Ludwig Boltzmann’s Paper “On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium” Sitzungberichte der Kaiserlichen Akademie der Wissenschaften. Mathematisch-Naturwissen Classe. Abt. II, LXXVI 1877, pp 373-435 (Wien. Ber. 1877, 76:373-435). Reprinted in Wiss. Abhandlungen, Vol. II, reprint 42, p. 164-223, Barth, Leipzig, 1909

Translation of the seminal 1877 paper by Ludwig Boltzmann which for the first time established the probabilistic basis of entropy. Includes a scientific commentary.

Testing $$R_h=ct$$ R h = c t cosmology from fundamental considerations: Is the Friedmann universe intrinsically flat

Recently Melia and Shevchuk (Mon Not R Astron Soc 419:2579,2012) (MS) have proposed the so-called \(R_h =ct\) cosmology where the “Gravitational Horizon” of the universe \(R_h\) is equal to the distance travelled by light since “Big Bang”. Here we would like to see whether the basic claim \(R_h=ct\) is correct or not because MS have not given any cogent derivation for the same. Essentially we will compare the twin expressions for the Einstein energy momentum complex (EMC) of the Friedmann universe obtained by using an appropriate superpotential and also by a direct method. To enable a meaningful comparison of the twin expressions, both are computed by using the same quasi-Cartesian coordinates. We however do not claim that Einstein EMC is superior to many other routes of defining EM of a self-gravitating system. In fact, for static isolated spherical syatems, the idea of a coordinate independent field energy of Lynden-Bell and Katz (Mon Not R Astron Soc 213:21, 1985) might be quite physically significant. Yet, here, we use Einstein EMC because (i) our system is non-static and not isolated one (ii) our primary aim is not find any absolute value of EM, and, finally, (iii) only Einstein pseudo-tensor offers equivalent twin expressions for EM which one can be equated irrespective of any physical significance. Following such comparison of equivalent twin expressions of Einstein energy, we find an exact proof as to why Friedmann universe must be spatially flat even though, mathematically one can conceive of curved spaces in any dimension. Additionally, it follows that, apparently, the scale factor \(S(t) \propto t\) as insisted by \(R_h=ct\) proposition. Nonetheless, because of close similarity of this form, \(S(t) \propto t\), with the (vacuum) Milne metric, and also because of implied unphysical equation of state, \(R_h=ct\) cosmology is unlikely to represent the physical universe.

Supersymmetric component actions via coset approach

We propose a method to construct the on-shell component actions for the theories with 1/2 partial breaking of global supersymmetry within the nonlinear realization (coset) approach. In contrast with the standard superfield approach in which unbroken supersymmetry plays the leading role, we have shifted the attention to the spontaneously broken supersymmetry. It turns out that in the theories in which half of supersymmetries is spontaneously broken, all physical fermions are just the fermions of the nonlinear realization. Moreover, the transformation properties of these fermions with respect to the broken supersymmetry are the same as in the famous Volkov–Akulov model. Just this fact completely fixed all possible appearances of the fermions in the component action: they can enter the action through the determinant of the vielbein (to compensate the transformation of the volume form) and the covariant derivatives, only. It is very important that in our parametrization of the coset the rest of physical components, i.e. all bosonic components, transform as “matter fields” with respect to the broken supersymmetry. Clearly, in such a situation the component action acquires the form of the Volkov–Akulov action for these “matter fields”. The complete form of the action can be further fixed by two additional requirements: (a) to reproduce the bosonic limit, which is explicitly known in many interesting cases, and (b) to have a proper linearized form, which has to be invariant with respect to the linearized unbroken supersymmetry. We supply the general consideration by a detailed example of the component action of N=1 supermembrane in D=4 constructed within our procedure. In this case we provide the exact proof of the invariance of the constructed component action with respect to both, broken and unbroken supersymmetries.

Partition Function of the Model of Perfect Gas of Clusters for Interacting Fluids

The last paper of A. Fronczak presents a new, combinatorial approach to the model of perfect gas of clusters of interacting fluids. In the paper the enumerative properties and combinatorial meaning of Bell polynomials have been used to reveal some properties of systems described by grand canonical ensemble . In this paper, an exact proof of one of the important assertions from the work mentioned above is given. The assertion states that for the system in which one can distinguish k non interacting clusters the canonical partition function is given by Bell polynomial of the derivatives of the grand-thermodynamic potential with respect to the fugacity. Some possible applications of the considered theorem are presented.

The Discrete-Time Framework of the Arbitrage-Free Nelson-Siegel Class of Term Structure Models

We derive the discrete-time arbitrage-free Nelson-Siegel class of term structure models with an exact solution and proof of uniqueness. We design a fast and reliable estimation procedure based on reduced-dimension optimization with multistep embedded regressions. After an analytical illustration, we also show empirically that arbitrage-free restrictions have a bounded advantage for in-sample fit and out-of-sample forecast, compared to its reduced-form counterpart. However, the arbitrage-free model is a powerful tool for analysing risk premia associated with Level, Slope and Curvature factors. Our empirical results have interesting implications for both the US bond yield conundrum of 2004-05 and the recent financial crisis.

An improved compact finite difference scheme for solving an N‐carrier system with Neumann boundary conditions

AbstractRecently, we have developed a higher‐order and unconditionally stable compact finite difference scheme for solving a model of energy exchanges in an N‐carrier system with Neumann boundary conditions, which extends the concept of the well‐known parabolic two‐step model for microheat transfer. However, the combined compact finite difference scheme for the boundary is second‐order accurate. Unfortunately, our statement in (Dai and Tzou, Numer Methods Partial Differential Equation, Zhao et al., Numer Methods Partial Differential Equations 23 (2007), 949–959.), that it is a fourth‐order scheme is inaccurate, because the scheme was multiplied by Δx2 in the derivation. In this article, we develop a new combined compact finite difference scheme for the boundary, which is third‐order accurate. Using the exact same proof for stability analysis as in (Dai and Tzou, Numer Methods Partial Differential Equations), the new scheme is unconditionally stable with respect to the initial conditions and source terms. The improved compact scheme is then tested by a numerical example. Results show that the convergence rate with respect to the spatial variable from the new scheme is higher and the solution is much more accurate, when compared with those obtained using our previous compact scheme in (Dai and Tzou, Numer Methods Partial Differential Equations). © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011

Symbol-crunching the Harborth graph

The Harborth graph is widely regarded as the smallest known example of a 4-regular planar unit-distance graph. In this paper we sketch an exact proof of this property, and present some of the minimal polynomials of the coordinates of the vertices of its most prominent embedding, the calculations of which give an example for the heavy use of computer algebra in the area of geometric graph theory. Finally we use the particular algebraic structure of these polynomials to deduce some geometric properties of the Harborth graph.

Reply to: Consensus on unsolved issues of left ventricular hypertrabeculation/non-compaction is warranted

Abstract In reply to the letter by Finsterer and Stollberger entitled Consensus on unsolved issues of hypertrabeculation/noncompaction is warranted, the authors reaffirm the need for a concordant opinion on the unsolved issues which still loom over the diagnostic and clinical facets of left ventricular non-compaction. Subjects known to have ventricular hypertrabeculation and who subsequently experience a thromboembolic event should still be meticulously screened for other commoner and possibly co-existent embolic sources. In the absence of systolic dysfunction left ventricular non-compaction alone is not an indication for oral anticoagulation in so far as the primary prevention for thromboembolism is concerned. There exists no exact proof that the degree of inotropic dysfunction in hypertrabeculated hearts is directly and solely related to the extent of the non-compaction. Subendocardial perfusion deficits; diminished coronary blood flow reserve; trabecular fibrosis and aberrations at the cellular level may also be responsible for affecting ventricular systolic function. Early neurological referral is indicated following the diagnosis of non-compaction with the aim of screening for the many disorders known to be associated with this condition and genetic screening tests are best resorted to only if clinical examination fails to expose a relevant syndrome. The current cardiac magnetic resonance diagnostic criteria for non-compaction still have some important limitations which beckon a unifying consensus.

Thermodynamic consistency of energy and virial routes: An exact proof within the linearized Debye–Hückel theory

The linearized Debye-Hückel theory for liquid state is shown to provide thermodynamically consistent virial and energy routes for any potential and for any dimensionality. The importance of this result for bounded potentials is discussed.

Heat transport in low-dimensional systems

Recent results on theoretical studies of heat conduction in low-dimensional systems are presented. These studies are on simple, yet nontrivial, models. Most of these are classical systems, but some quantum-mechanical work is also reported. Much of the work has been on lattice models corresponding to phononic systems, and some on hard particle and hard disc systems. A recently developed approach, using generalized Langevin equations and phonon Green's functions, is explained and several applications to harmonic systems are given. For interacting systems, various analytic approaches based on the Green-Kubo formula are described, and their predictions are compared with the latest results from simulation. These results indicate that for momentum-conserving systems, transport is anomalous in one and two dimensions, and the thermal conductivity kappa, diverges with system size L, as kappa ~ L^alpha. For one dimensional interacting systems there is strong numerical evidence for a universal exponent alpha =1/3, but there is no exact proof for this so far. A brief discussion of some of the experiments on heat conduction in nanowires and nanotubes is also given.

Pathogenesis and treatment of pes adductus

Based on the literature and our own 143 patients the etiopathogenesis and treatment of metatarsus varus are discussed. The exact scientific proof is still lacking, whether there does exist, besides of the congenital mostly severer metatarsus varus, a slight exogen postnatal form with spontaneous healing, without or with a preceding permanent prone position of the infants. A fixed adductus position of the forefoot represents the indication for stepwise correction of the deformity by means of a plaster cast and additional wedging. It is necessary to produce a slight overcorrection including the typical valgus heel. Afterwards we are striving for strengthening of the foot abductors in order to gain an equilibrium of the musculature as in the treatment of club feet. Our own techniques of correction are demonstrated as well as the results.

A new look at quarks confinement

Quarks confinement is an experimental fact. ‘tHooft and later on Gross, Wilczek and Politzer have contributed in various ways to our present considerable theoretical understanding of the problem. However, an exact water proof theoretical derivation of the problem is at best still in progress. The present note argues that to understand quarks confinement, a deeper understanding of the Planck scale physics is indispensable and shows using analytical topological arguments that absolute confinement is a result of a phase transition of quantum spacetime at the Planck scale.

Image effects on the transport of intense nonaxisymmetric charged beams

The effect of conducting pipes on the equilibrium of intense nonaxisymmetric continuous beams of charged particles is investigated. For a cylindrical pipe and an elliptical beam, we obtain an exact closed form analytical expression for the electrostatic potential. Using a variational principle, we then explore the distortions that equilibrium beams suffer due to the conducting channel. Finally, we present an exact proof that despite the nonlinear forces acting on beams of arbitrary cross section inside conducting pipes of arbitrary shape, the density of these beams remains homogeneous and their cross sectional area remains the same as the one in free space.

Effective ac dielectric response of graded cylindrical composites

Under an external alternating current (ac) field, the effective ac dielectric response of graded composites consisting of the graded cylindrical inclusion having complex permittivity profiles has been investigated theoretically. A model that the dielectric function is assumed to be a constant while the conductivity has a power-law dependence on the radial variable r, namely ɛ i ( r ) = A + c r k / i ω , is studied and the local analytical potentials of the inclusion and the host regions are derived in terms of hyper-geometric function. In the dilute limit, the effective ac dielectric response is predicted. Meanwhile, we have given the exact proof of the differential effective dipole approximation (DEDA) method, which is suitable to arbitrary graded profiles. Furthermore, we have given the analytical potentials and the effective ac dielectric responses of coated graded cylindrical composites for two cases, case (a) graded core and case (b) graded coated layer, having the graded dielectric profiles, respectively.

A new algorithm for optimal multileaf collimator field segmentation

We present a new efficient leaf sequencing algorithm for the generation of intensity maps by a nonnegative combination of segments. Intensity maps describe the intensity modulation of beams in radiotherapy. We only study the static case (step and shoot). We exactly optimize the total number of monitor units and heuristically optimize the number of segments. We present a short exact proof for a formula giving the smallest total number of monitor units and describe a class of algorithms yielding this minimal value. A special member of this class provides a solution with a very small number of segments.

Extending cubic uniform B-splines by unified trigonometric and hyperbolic basis

In this paper, the trigonometric basis {sin t, cos t, t, 1} and the hyperbolic basis {sinh t, cosh t, t, 1} are unified by a shape parameter C (0 ≤ C < ∝). It yields the Functional B-splines (FB-splines) and its corresponding Subdivision B-splines (SB-splines). As well, a geometric proof of curvature continuity for SB-splines is provided. FB-splines and SB-splines inherited nearly all properties of B-splines, including the C 2 continuity, and can represent elliptic and hyperbolic arcs exactly. They are adjustable, and each control point b i can have its unique shape parameter C i . As C i increases from 0 to ∝, the corresponding breakpoint of b i on the curve is moved to the location of b i , and the curvature of this breakpoint is increased from 0 to ∝ too. For a set of control points and their shape parameters, SB-spline and FB-spline have the same position, tangent, and curvature on each breakpoint. If two adjacent control points in the set have identical parameters, their SB-spline and FB-spline segments overlap. However, in general cases, FB-splines have no simple subdivision equation, and SB-splines have no common evaluation function. Furthermore, FB-splines and SB-splines can generate shape adjustable surfaces. They can represent the quadric surfaces precisely for engineering applications. However, the exact proof of C 2 continuity for the general SB-spline surfaces has not been obtained yet.