Analytical Proof Research Articles

Analytic approach to S1–equivariant Morse inequalities

It is well known that the cohomology groups of a closed manifold $M$ can be reconstructed using the gradient dynamical of a Morse-Smale function $f\colon M\to \R$. A direct result of this construction are Morse inequalities that provide lower bounds for the number of critical points of $f$ in term of Betti numbers of $M$. These inequalities can be deduced through a purely analytic method by studying the asymptotic behaviour of the deformed Laplacian operator. This method was introduced by E. Witten and has inspired a numbers of great achievements in Geometry and Topology in few past decades. In this paper, adopting the Witten approach, we provide an analytic proof for; the so called; equivariant Morse inequalities when the underlying manifold is acted on by the Lie group $G=S^1$ and the Morse function $f$ is invariant with respect to this action.

Full quantum theory of nonequilibrium phonon condensation and phase transition

Fr\"olich condensation is a room-temperature nonequilibrium phenomenon which is expected to occur in many physical and biological systems. Though predicted theoretically a half century ago, the nature of such condensation remains elusive. In this Letter, we derive a full quantum theory of Fr\"ohlich condensation from the Wu-Austin Hamiltonian and present for the first time an analytical proof that a second-order phase transition induced by nonequilibrium and nonlinearity emerges in the large-$D$ limit with and without decorrelation approximation. This critical behavior cannot be witnessed if external sources are treated classically. We show that the phase transition is accompanied by large fluctuations in the statistical distribution of condensate phonons and that the Mandel-Q factor which characterizes fluctuations becomes negative in the limit of excessive external energy input. In contrast with the cold atom equilibrium BEC, the Fr\"ohlich condensate is a result of the nonequilibrium driving where the pump plays a role of setting the number of particles, and the medium plays a role of setting the temperature. Hence, BEC can either arise by reducing the medium temperature at fixed pump (equilibrium case), or by increasing the pump at fixed medium temperature (nonequilibrium case).

Random matrix theory and moments of moments of L-functions

In this paper, we give an analytic proof of the asymptotic behavior of the moments of moments of the characteristic polynomials of random symplectic and orthogonal matrices. We therefore obtain alternate, integral expressions for the leading order coefficients previously found by Assiotis, Bailey and Keating. We also discuss the conjectures of Bailey and Keating for the corresponding moments of moments of [Formula: see text]-functions with symplectic and orthogonal symmetry. Specifically, we show that these conjectures follow from the shifted moments conjecture of Conrey, Farmer, Keating, Rubinstein and Snaith.

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In this paper, we prove the security of the LM05’s two-way quantum key distribution (TWQKD) under the assumption that detector devices in both Alice’s and Bob’s sides are controlled by eavesdroppers. Using a relatively simple method, more practical, we give an analytical proof that LM05’s TWQKD is measurement-device-independent (MDI) security in both the quantum forward channel and the quantum backward channel. This is the first work shows that TWQKD has fully MDI characteristic, thus makes it immune to all possible detector-device-attacks in two-quantum-way in actual implementations.

The continuation method and the real analyticity of the accessory parameters: the parabolic case

We give the proof of the real analyticity of the accessory parameters in Liouville field theory as a function of the position of the sources in the case in which in addition to elliptic sources, parabolic sources are present. The method is a non trivial extension of the elliptic case as it requires in an intermediate step the introduction of a regulator. The treatment holds also in the case of the torus. A discussion is given of the extension to higher genus surfaces.

Galileo’s Swiftest Descent Problem Revisited: Complete Analytic Solution

Nearly four centuries ago, Galileo Galilei proved that descent along any two-chord path in the lower quadrant of a circle is faster than descent along the straight one-chord path from the same starting point to the same endpoint, if the initial speed is zero, and the paths end in the lowest point of the circle. Moreover, Galileo posed two conjectures: (1) the main conclusion remains the same if the body initially at rest falls to a starting point and thus the speed there is not zero; (2) descent along the circular arc itself is faster than along any broken line of chords. In this article, analytical proofs are given that confirm the validity of both of these conjectures.

A Class of Semilinear Parabolic Problems and Analytic Semigroups

(1) Background: This paper is devoted to the study of a class of semilinear initial boundary value problems of parabolic type. (2) Methods: We make use of fractional powers of analytic semigroups and the interpolation theory of compact linear operators due to Lions–Peetre. (3) Results: We give a functional analytic proof of the C2 compactness of a bounded regular solution orbit for semilinear parabolic problems with Dirichlet, Neumann and Robin boundary conditions. (4) Conclusions: As an application, we study the dynamics of a population inhabiting a strongly heterogeneous environment that is modeled by a class of diffusive logistic equations with Dirichlet and Neumann boundary conditions.

ON THE NUMBER OF NEARLY SELF-CONJUGATE PARTITIONS

AbstractA partition $\lambda $ of n is said to be nearly self-conjugate if the Ferrers graph of $\lambda $ and its transpose have exactly $n-1$ cells in common. The generating function of the number of such partitions was first conjectured by Campbell and recently confirmed by Campbell and Chern (‘Nearly self-conjugate integer partitions’, submitted for publication). We present a simple and direct analytic proof and a combinatorial proof of an equivalent statement.

Machine Learning Feedback Control Approach Based on Symbolic Regression for Robotic Systems

A control system of an autonomous robot produces a control signal based on feedback. This type of control implies the control of an object according to its state that is mathematically the control synthesis problem. Today there are no universal analytical methods for solving the general synthesis problem, and it is solved by certain particular approaches depending on the type of control object. In this paper, we propose a universal numerical approach to solving the problem of optimal control with feedback using machine learning methods based on symbolic regression. The approach is universal and can be applied to various objects. However, the use of machine learning methods imposes two aspects. First, when using them, it is necessary to reduce the requirements for optimality. In machine learning, optimization algorithms are used, but strictly optimal solutions are not sought. Secondly, in machine learning, analytical proofs of the received properties of solutions are not required. In machine methods, a set of tests is carried out and it is shown that this is sufficient to achieve the required properties. Thus, in this article, we initially introduce the fundamentals of machine learning control, introduce the basic concepts, properties and machine criteria for application of this technique. Then, with regard to the introduced notations, the feedback optimal control problem is considered and reformulated in order to add to the problem statement that such a property adjusts both the requirements of stability and optimality. Next, a description of the proposed approach is presented, theoretical formulations are given, and its efficiency is demonstrated on the computational examples in mobile robot control tasks.

Another Proof of a Sakhanenko Theorem

We give an analytic proof of Sakhanenko's theorem on the strong law of large numbers. Our arguments are based on the method of characteristic functions: under the Lindeberg-type condition, the expectation of the absolute value of the sum of independent random variables (r.v.'s) tends to zero. In our proof, we represent the expectation of the absolute value of an r.v. in terms of the corresponding characteristic function.

Error estimates for total-variation regularized minimization problems with singular dual solutions

Recent quasi-optimal error estimates for the finite element approximation of total-variation regularized minimization problems using the Crouzeix–Raviart element require the existence of a Lipschitz continuous dual solution, which is not generally given. We provide analytic proofs showing that the Lipschitz continuity of a dual solution is not necessary, in general. Using the Lipschitz truncation technique, we, in addition, derive error estimates that depend directly on the Sobolev regularity of a given dual solution.

A Study on the Proofs Used by Primary Education Teacher Candidates in Circumference Problem Solutions and Instructional Explanations

This study aims to determine the proof schemes used by primary education teacher candidates when solving a given problem and in their instructional explanations. Having a qualitative nature, the study utilized data collected from 277 third-year teacher candidates studying at the primary education department of the education faculty of a state university in Ankara. The study group was selected via criterion sampling. The data were tested by using an open-ended problem case and analyzed via document analysis. The proofs used by students in solving the problem and in instructional explanations were categorized as Proof A, Proof B ... Proof H, and the data were evaluated in these categories. The results obtained were discussed in line with the proof schemes outlined by Harel and Sowder (1998). However, as certain proofs fell into several categories, they could not be evaluated in only one group. The primary education teacher candidates were found to use authoritative, habitual and symbolic proof schemes, albeit to a little extent, in the external proof scheme category as they solved the given problem situation and explained the solution to their students. The majority of the candidates used the empirical proof scheme known as the perceptual proof scheme, and included some sample-based proofs. Analytical proof schemes were used less frequently than others.

A finite analogue of a q-series identity of Bhoria, Eyyunni and Maji and its applications

Bhoria, Eyyunni and Maji recently obtained a four-parameter q-series identity which gives as special cases not only all five entries of Ramanujan on pages 354 and 355 of his second notebook but also allows them to obtain an analytical proof of a result of Bressoud and Subbarao. Here, we obtain a finite analogue of their identity which naturally gives finite analogues of Ramanujan's results. Using one of these finite analogues, we deduce an identity for a finite sum involving a ϕ12. This identity is then applied to obtain a generalization of the generating function version of Andrews' famous identity for the smallest parts function spt(n). The q-series which generalizes ∑n=1∞spt(n)qn is completely different from S(z,q) considered by Andrews, Garvan and Liang. Further applications of our identity are given. Lastly we generalize a result of Andrews, Chan and Kim which involves the first odd moments of rank and crank.

Влияние расширенных принципов бережливого производства на экономическую эффективность промышленных предприятий

The main purpose of this study is an analytical proof of the effect of the extended principles of lean manufacturing of economic products on the economic efficiency of industrial companies. To achieve this goal, the following tasks were performed: the tendency to increase the share of value-added operations was investigated, the analysis of existing cost accounting methods was carried out, it was proposed to consider the cost structure depending on the type of operation.
 The object of research is modern industrial enterprises using the concept of lean manufacturing and its analogues. The subject of the study is the effect of the extended principles of lean manufacturing in the creation of economic products.
 Methods of abstraction and comparative analysis were used to study the impact of the extended principles of lean manufacturing. In addition, methods of analysis and synthesis were used to study the existing methods of cost accounting. The historical method was also used in the work to analyze stable trends in the development of industry.
 Based on empirical data, there is a tendency to increase the share of operations that add value to the final product. It was found that non-value-added operations account for 84% of the cost of an economic product, which confirms the need to apply the expanded concept of lean manufacturing in modern realities, as it will significantly reduce this share by eliminating material and especially information losses. As a result of the study of existing methods of cost accounting, it is proposed to consider the cost of any economic product structurally consisting of material and information components for operations that add and do not add value to the final product.
 As a result of the conducted research, the influence of the expanded principles lean manufacturing of economic products on the economic efficiency of modern industrial enterprises was analytically proved and the expediency of applying the proposed concept of lean manufacturing was confirmed.

On the positivity of Coon amplitude in D = 4

The Coon amplitude is the unique solution to duality constraints with logarithmic Regge trajectories. A striking feature of this solution is that it interpolates between the Veneziano amplitude and a scalar particle amplitude. However, an analytic proof of unitarity of the amplitude is not yet known. In this short note, we explicitly compute the partial wave coefficients on the leading Regge trajectory in D = 4. We find that these coefficients always remain positive, even though their magnitude decreases with spin. Since the coefficients on the subleading trajectories are observed to be larger than those on the leading ones, our result indicates the positivity of the full Coon amplitude in D = 4.

Emerging Proof Productions of Freshmen in Euclidean Geometry Proof Tasks between Conjecturing and Proving

ABSTRACT This study was conducted to examine the effectiveness of proof tasks in the transition from conjecture to proof in Euclidean geometry on freshmen’s proof schemes. In line with this aim, the proof schemes of the freshmen who performed conjecture-proof and theorem-proof tasks were compared. The freshmen were composed of 109 pre-service middle school mathematics teachers who are enrolled in their first year of undergraduate education. The study was modeled as a multiple-case study. Fifty-three freshmen performed conjecture-proof tasks in Case-1, and fifty-six freshmen performed theorem-proof tasks in Case-2. The video recordings, including the written proof reports, reflection papers, and proof explanations of the freshmen, were used as data collection tools in the tasks. The proof schemes were used as construct maps to evaluate the proofs of freshmen and analyzed using Winsteps Rasch software. The proof schemes in freshmen’s proof tasks were evaluated by the Wright Map and supported with direct quotations from the proofs. In the study, it was observed that the proof schemes of freshmen who made proofs based on their own conjectures were mostly empirical and analytical proof schemes, while the proof schemes of freshmen who made proof of the presented theorem were generally external and empirical proof schemes.

N-band photonic Hopf insulators based on 2D microring lattices.

Hopf insulators are topological insulators whose topological behavior arises from the nontrivial mapping from a 3D sphere to a 2D sphere, known as the Hopf map. The Hopf map, typically encountered in the study of spinor and Skyrmion systems, is classified topologically by an integer invariant called the Hopf index. Here we show that, owing to the periodic circulation of light inside each microring, a 2D lattice of microring resonators can emulate an N-band photonic Hopf insulator with nontrivial Hopf index. In particular, we show by numerical computation and direct analytical proof that the N-band Hopf index of the microring lattice is identical to its winding number. The result shows that the Hopf index is an alternative topological invariant for classifying 2D microring photonic lattices and establishes a correspondence between the Hopf insulator phase and the anomalous Floquet insulator phase of the lattice. More generally, our work shows that 2D microring lattices can provide a versatile nanophotonic platform for studying non-Abelian topological photonic systems.

Local regularity estimates for general discrete dynamic programming equations

We obtain an analytic proof for asymptotic Hölder estimate and Harnack's inequality for solutions to a discrete dynamic programming equation. The results also generalize to functions satisfying Pucci-type inequalities for discrete extremal operators. Thus the results cover a quite general class of equations.

Mineral Raw Material Supply Chain Transparency and Traceability: Does Provenance Matter in the Supply Chain?

Raw material supply chains are complex systems. They build on the presence of economically mineable mineral commodities that will undergo several steps until they are finally being used as consumer products. Trustworthiness into the transparency of a supply chain is of increasing importance, both to upstream and downstream companies. Any deviation from best-practice and quality standards in mining, processing and production is critically looked at by consumers. Therefore, certification systems and proof of origin concepts have emerged within the past years, aiming at providing transparency to supply chains. The analytical proof of origin for mineral raw materials could be beneficial to certification schemes in several ways. It represents the least corruptible method of provenance analysis as it relates directly to the chemical composition of the raw material. Other methods, such as documents, tracers, or barcodes, can be outmanoeuvred in one way or another.

Basis-Free Formulas for Characteristic Polynomial Coefficients in Geometric Algebras

In this paper, we discuss characteristic polynomials in (Clifford) geometric algebras $$\mathcal {G}_{p,q}$$ of vector space of dimension $$n=p+q$$ . We present basis-free formulas for all characteristic polynomial coefficients in the cases $$n\le 6$$ , alongside with a method to obtain general form of these formulas. The formulas involve only the operations of geometric product, summation, and operations of conjugation. All the formulas are verified using computer calculations. We present an analytical proof of all formulas in the case $$n=4$$ , and one of the formulas in the case $$n=5$$ . We present some new properties of the operations of conjugation and grade projection and use them to obtain the results of this paper. We also present formulas for characteristic polynomial coefficients in some special cases. In particular, the formulas for vectors (elements of grade 1) and basis elements are presented in the case of arbitrary n, the formulas for rotors (elements of spin groups) are presented in the cases $$n\le 5$$ . The results of this paper can be used in different applications of geometric algebras in computer graphics, computer vision, engineering, and physics. The presented basis-free formulas for characteristic polynomial coefficients can also be used in symbolic computation.