Solution to Euler’s elastic problem

Euler’s problem on stationary configurations of elastic rod with fixed endpoints and tangents at the endpoints is considered. The corresponding optimal control problem is reduced to several systems of algebraic equations in Jacobi’s functions. An algorithm and software for solving the optimal control problem are constructed.     

Speech-based solution to Parkinson’s disease management

Parkinson’s disease (PD) is a neurological disorder marked by decreased dopamine levels in the brain. Persons suffering from PD, exhibits vocal symptoms such as dysphonia and dysarthria. Speech impairments in PD are grouped together and called as hypokinetic dysarthria. Traditional PD management is based on a patient’s clinical history and through physical examination as there are currently no known biomarkers for its diagnosis. Automatic analysis techniques aid clinicians in diagnosis and monitoring patients using speech and provide frequent, cost effective and objective assessment. This paper presents pilot experiment to detect presence of dysarthria in speech and detect level of severity based on deep learning approach. Automated feature extraction and classification using convolutional neural network shows 77.48% accuracy on test samples of TORGO database with five fold validation. Using transfer learning, system performance is further analyzed for gender specific performance as well as in detection of severity of disease.

     

Some problems for Clark’s model. II. A solution for Merton’s portfolio problem1

The problem of finding the optimal control over the portfolio for an investor in (B, S)-market is considered. Clark’s model is taken as a model for the stock price evolution. The cases of risk-loving, risk-neutral, and risk-averse investors are considered.     

Introduction to the Geometer’s Angle

Geometer Rachel Fletcher introduces the Geometer’s Angle, a column in the Nexus Network Journal dedicated to geometrical constructions, the procedures for drawing them, their use, significance and symbolism.     

Optimal navigation in planar time-varying flow: Zermelo��s problem revisited

This paper is concerned with time-optimal navigation for flight vehicles in a planar, time-varying flow-field, where the objective is to find the fastest trajectory between initial and final points. The primary contribution of the paper is the observation that in a point-symmetric flow, such as inside vortices or regions of eddie-driven upwelling/downwelling, the rate of the steering angle has to be equal to one-half of the instantaneous vertical vorticity. Consequently, if the vorticity is zero, then the steering angle is constant. The result can be applied to find the time-optimal trajectories in practical control problems, by reducing the infinite-dimensional continuous problem to numerical optimization involving at most two unknown scalar parameters.     

Semi-tensor product of matrices and its application to Morgen’s problem

This paper proposes a new matrix product, namely, semi-tensor product. It is a generalization of the conventional matrix product. Meanwhile, it is also closely related to Kronecker (tensor) product of matrices. The purpose of introducing this product is twofold: (i) treat multi-dimensional data; (ii) treat nonlinear problems in a linear way. Then the computer and numerical methods can be easily used for solving nonlinear problems. Properties and formulas are deduced. As an application, the Morgen’s problem for control systems is formulated as a numerically solvable problem.     

From Boltzmann to Euler: Hilbert’s 6th problem revisited

This article addresses the hydrodynamic limit of the Boltzmann equation, namely the compressible Euler equations of gas dynamics. An exact summation of the Chapman–Enskog expansion originally given by Gorban and Karlin is the key to the analysis. An appraisal of the role of viscosity and capillarity in the limiting process is then given where the analogy is drawn to the limit of the Korteweg–de Vries–Burgers equations as a small parameter tends to zero.     

DYRE: a DYnamic REconfigurable solution to increase GPGPU’s reliability

The Journal of Supercomputing - General-purpose graphics processing units (GPGPUs) are extensively used in high-performance computing. However, it is well known that these devices’...     

Algorithm for the discrete Weber’s problem with an accuracy estimate

We consider a relaxation of the quadratic assignment problem without the constraint on the number of objects assigned to a specific position. This problem is NP-hard in the general case. To solve the problem, we propose a polynomial algorithm with guaranteed posterior accuracy estimate; we distinguish a class of problems with special assignment cost functions where the algorithm is 2-approximate. We show that if the graph in question contains one simple loop, and the set of assignment positions is a metric space, the proposed algorithm is 2-approximate and guaranteed to be asymptotically exact. We conduct a computational experiment in order to analyze the algorithm’s errors and evaluate its accuracy.     

An Indian solution to ‘incompleteness’

Kurt Gödel’s Incompleteness theorem is well known in Mathematics/Logic/Philosophy circles. Gödel was able to find a way for any given P (UTM), (read as, “P of UTM” for “Program of Universal Truth Machine”), actually to write down a complicated polynomial that has a solution iff (=if and only if), G is true, where G stands for a Gödel-sentence. So, if G’s truth is a necessary condition for the truth of a given polynomial, then P (UTM) has to answer first that G is true in order to secure the truth of the said polynomial. But, interestingly, P (UTM) could never answer that G was true. This necessarily implies that there is at least one truth a P (UTM), however large it may be, cannot speak out. Daya Krishna and Karl Potter’s controversy regarding the construal of India’s Philosophies dates back to the time of Potter’s publication of “Presuppositions of India’s Philosophies” (1963, Englewood Cliffs Prentice-Hall Inc.) In attacking many of India’s philosophies, Daya Krishna appears to have unwittingly touched a crucial point: how can there be the knowledge of a ‘non-cognitive’ mok?a? [‘mok?a’ is the final state of existence of an individual away from Social Contract]—See this author’s Indian Social Contract and its Dissolution (2008) mok?a does not permit the knowledge of one’s own self in the ordinary way with threefold distinction, i.e., subject–knowledge-object or knower–knowledge–known. But what is important is to demonstrate whether such ‘knowledge’ of non-cognitive mok?a state can be logically shown, in a language, to be possible to attain, and that there is no contradiction involved in such demonstration, because, no one can possibly express the ‘experience-itself’ in language. Hence, if such ‘knowledge’ can be shown to be logically not impossible in language, then, not only Daya Krishna’s arguments against ‘non-cognitive mok?a’ get refuted but also it would show the possibility of achieving ‘completeness’ in its truest sense, as opposed to Gödel’s ‘Incompleteness’. In such circumstances, man would himself become a Universal Truth Machine. This is because the final state of mok?a is construed as the state of complete knowledge in Advaita. This possibility of ‘completeness’ is set in this paper in the backdrop of ?rī ?a?karācārya’s Advaitic (Non-dualistic) claim involved in the mahāvākyas (extra-ordinary propositions). (Mahāvākyas that ?a?kara refers to are basically taken from different Upani?ads. For example, “Aham Brahmāsmi” is from B?hadāra?yaka Upanisad, and “Tattvamasi” is from Chāndogya Upani?ad. ?rī ?a?karācārya has written extensively. His main works include his Commentary on Brahma-Sūtras, on major Upani?ads, and on ?rīmadBhagavadGītā, called Bhā?yas of them, respectively. Almost all these works are available in English translation published by Advaita Ashrama, 5 Dehi Entally Road, Calcutta, 700014.) On the other hand, the ‘Incompleteness’ of Gödel is due to the intervening G-sentence, which has an adverse self-referential element. Gödel’s incompleteness theorem in its mathematical form with an elaborate introduction by R.W. Braithwaite can be found in Meltzer (Kurt Gödel: on formally undecidable propositions of principia mathematica and related systems. Oliver &; Boyd, Edinburgh, 1962). The present author believes first that semantic content cannot be substituted by any amount of arithmoquining, (Arithmoquining or arithmatization means, as Braithwaite says,—“Gödel’s novel metamathematical method is that of attaching numbers to the signs, to the series of signs (formulae) and to the series of series of signs (“proof-schemata”) which occur in his formal system…Gödel invented what might be called co-ordinate metamathematics…”) Meltzer (1962 p. 7). In Antone (2006) it is said “The problem is that he (Gödel) tries to replace an abstract version of the number (which can exist) with the concept of a real number version of that abstract notion. We can state the abstraction of what the number needs to be, [the arithmoquining of a number cannot be a proof-pair and an arithmoquine] but that is a concept that cannot be turned into a specific number, because by definition no such number can exist.”.), especially so where first-hand personal experience is called for. Therefore, what ultimately rules is the semanticity as in a first-hand experience. Similar points are voiced, albeit implicitly, in Antone (Who understands Gödel’s incompleteness theorem, 2006). (“…it is so important to understand that Gödel’s theorem only is true with respect to formal systems—which is the exact opposite of the analogous UTM (Antone (2006) webpage 2. And galatomic says in the same discussion chain that “saying” that it ((is)) only true for formal systems is more significant… We only know the world through “formal” categories of understanding… If the world as it is in itself has no incompleteness problem, which I am sure is true, it does not mean much, because that is not the world of time and space that we experience. So it is more significant that formal systems are incomplete than the inexperiencable ‘World in Itself’ has no such problem.—galatomic”) Antone (2006) webpage 2. Nevertheless galatomic certainly, but unwittingly succeeds in highlighting the possibility of experiencing the ‘completeness’ Second, even if any formal system including the system of Advaita of ?a?kara is to be subsumed or interpreted under Gödel’s theorem, or Tarski’s semantic unprovability theses, the ultimate appeal would lie to the point of human involvement in realizing completeness since any formal system is ‘Incomplete’ always by its very nature as ‘objectual’, and fails to comprehend the ‘subject’ within its fold.     

Statistical moments of the solution of the random Burgers–Riemann problem

We solve Burgers’ equation with random Riemann initial conditions. The closed solution allows simple expressions for its statistical moments. Using these ideas we design an efficient algorithm to calculate the statistical moments of the solution. Our methodology is an alternative to the Monte Carlo method. The present approach does not demand a random numbers generator as does the Monte Carlo method. Computational tests are added to validate our approach.     

Inverse problem for Chaplygin’s nonholonomic systems

Chaplygin’s nonholonomic systems are familiar mechanical systems subject to unintegrable linear constraints, which can be reduced into holonomic nonconservative systems in a subspace of the original state space. The inverse problem of the calculus of variations or Lagrangian inverse problem for such systems is analyzed by making use of a reduction of the systems into new ones with time reparametrization symmetry and a genotopic transformation related with a conformal transformation. It is evident that the L...     

Stochastic optimality in the portfolio tracking problem involving investor’s temporal preferences

We consider an optimal portfolio selection problem to track a riskless reference portfolio. Portfolio management strategies are compared taking into account the investor’s temporal preferences. We investigate stochastic optimality of the strategy that minimizes the expected long-run cost, deriving an asymptotical upper (almost sure) estimate for the difference between the values of the objective functional corresponding to the optimal strategy and for any admissible control.     

A meshless method to the numerical solution of an inverse reaction–diffusion–convection problem

In this paper, the determination of the source term in a reaction–diffusion convection problem is investigated. First with suitable transformations, the problem is reduced, then a new meshless method based on the use of the heat polynomials as basis functions is proposed to solve the inverse problem. Due to the ill-posed inverse problem, the Tikhonov regularization method with a generalized cross-validation criterion is employed to obtain a numerical stable solution. Finally, some numerical examples are presented to show the accuracy and effectiveness of the algorithm.     

Viscous Taylor droplets in axisymmetric and planar tubes: from Bretherton’s theory to empirical models

The aim of this study is to derive accurate models for quantities characterizing the dynamics of droplets of non-vanishing viscosity in capillaries. In particular, we propose models for the uniform-film thickness separating the droplet from the tube walls, for the droplet front and rear curvatures and pressure jumps, and for the droplet velocity in a range of capillary numbers, Ca, from \(10^{-4}\) to 1 and inner-to-outer viscosity ratios, \(\lambda\), from 0, i.e. a bubble, to high-viscosity droplets. Theoretical asymptotic results obtained in the limit of small capillary number are combined with accurate numerical simulations at larger Ca. With these models at hand, we can compute the pressure drop induced by the droplet. The film thickness at low capillary numbers (\(Ca<10^{-3}\)) agrees well with Bretherton’s scaling for bubbles as long as \(\lambda <1\). For larger viscosity ratios, the film thickness increases monotonically, before saturating for \(\lambda>10^3\) to a value \(2^{2/3}\) times larger than the film thickness of a bubble. At larger capillary numbers, the film thickness follows the rational function proposed by Aussillous and Quéré (Phys Fluids 12(10):2367–2371, 2000) for bubbles, with a fitting coefficient which is viscosity-ratio dependent. This coefficient modifies the value to which the film thickness saturates at large capillary numbers. The velocity of the droplet is found to be strongly dependent on the capillary number and viscosity ratio. We also show that the normal viscous stresses at the front and rear caps of the droplets cannot be neglected when calculating the pressure drop for \(Ca>10^{-3}\).     

The original tls solution sets of the multidimensional TLS problem ∗

The original TLS solution set of the multidimensional TLS problem and the expressions of its mini-F-norm TLS solution are established in this paper.