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Interfaces divide all phases of matter and yet in most practical settings it is tempting to ignore their energies and the associated implications. There are many reasons for this, not the least of which is the introduction of a new pair of canonically conjugate variables—interfacial energy and its counterpart the surface area. A key set of questions surrounding the treatment of multiphase flows concerns how and when we must account for such effects. I begin this discussion with an abbreviated review of the basic theory of lower-dimensional phase transitions and describe a range of situations in which the bulk behaviour of a two-phase (and in some cases two-component) system is dominated by surface effects. Then I discuss a number of settings in which the bulk and surface behaviour can interact on equal footing. These can include the dynamic and thermodynamic behaviour of floating sea ice, the freezing and drying of colloidal suspensions (such as soil) and the mechanisms of protoplanetesimal formation by inter-particle collisions in accretion discs. This article is part of the theme issue ‘The physics and chemistry of ice: scaffolding across scales, from the viability of life to the formation of planets’.

Inspired by the Conjugate Variables Theorem in physics, we provide a general expectation identity for univariate continuous random variables by utilizing integration by parts. We then apply the general expectation identity to some common univariate continuous random variables (normal, gamma (including chi-square and exponential), beta, double exponential, F, inverse gamma, logistic, lognormal, Pareto, t, uniform, and Weibul) and obtain their specific expectation identities from the general expectation identity. After that, we use the specific expectation identities to derive high-order moments of the corresponding univariate continuous random variables.

Originally devised as an extension of von Neumann measurement Hamiltonian to joint measurement of conjugate variables, the Arthurs–Kelly Hamiltonian has been found to have many other practical applications. I summarize in particular, experimental bounds on von Neumann entropy, noiseless quantum tracking of conjugate observables, remote tomography, entanlement swapping and exact measurement of correlation between conjugate observables.

The complex variable conjugate approach has been derived analyticaly for derivative computation. Computational results are then used in calculating the amplitude of analytic signal. It is the square root of the square of the total magnetic field anomaly derivative. The total magnetic fields are generated by the upper and lower parts of a 2D finite prism, and subtraction of both parts yields the magnetic field anomaly. While the approach is obtained by truncating the Taylor series expansion of the total magnetic field function in argument of the complex conjugate in the h2 order-term. Truncating the series does not significantly affect the computational results. This is because when step-size h get smaller, and at h≤10−2, then the errors due to the series truncation became 0 (Kp→0). For derivative computation, the approach has precision on the order of 10−17 to 10−12 towards analytical settlement. On the order of h≤10−2, the approach is insensitive to the selection of step-size h for small numbers, so it can be done arbitrarily without any particular treatment or requires a complicated combination of numbers. The computational results of the analytic signal amplitude show that the positive and negative polarity on the magnetic profile is transformed into a positive profile only. This can facilitate the interpretation of actual magnetic data, especially in determining the causative source position of anomalies.

This book explores the most recent developments in the theory of planar quasiconformal mappings with a particular focus on the interactions with partial differential equations and nonlinear analysis. It gives a thorough and modern approach to the classical theory and presents important and compelling applications across a spectrum of mathematics: dynamical systems, singular integral operators, inverse problems, the geometry of mappings, and the calculus of variations. It also gives an account of recent advances in harmonic analysis and their applications in the geometric theory of mappings. The book explains that the existence, regularity, and singular set structures for second-order divergence-type equations--the most important class of PDEs in applications--are determined by the mathematics underpinning the geometry, structure, and dimension of fractal sets; moduli spaces of Riemann surfaces; and conformal dynamical systems. These topics are inextricably linked by the theory of quasiconformal mappings. Further, the interplay between them allows the authors to extend classical results to more general settings for wider applicability, providing new and often optimal answers to questions of existence, regularity, and geometric properties of solutions to nonlinear systems in both elliptic and degenerate elliptic settings.

A new microsystem designed to detect and measure in real time the enthalpy of mixing of two fluid constituents is presented. A preliminary approach to quantify the enthalpy of dilution values or mixing is first discussed. Then, a coherent rationale leading to structure devices operating in real time is formulated, considering the straightforward assessment of heat-flux transducers (HFTs) capability. Basic thermodynamic observations regarding the analogy between thermal and electrical systems are highlighted prior consideration of practical examples involving mixing water and alcohols. Fundamentals about HFT design are highlighted before presenting an adequate way to integrate both functions of mixing and measuring the entailed heat exchange as two continuously flowing fluids interact with one another. Thereby, the development of a relevant prototype of such a dedicated microsystem is discussed. Its design, fabrication and implementation under real operating conditions are presented together with its assessed performance and limits so as to highlight the advantages and shortcomings of the concept.

In this paper, the (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation is investigated to acquire the complexiton solutions by the Hirota direct method. It is essential to transform the equation into Hirota bi-linear form and to build N-compilexiton solutions by pairs of conjugate wave variables.

This study aimed at the development of an algorithm for the computational optimization of free-piston Stirling engines. The design algorithm includes an optimization method and two compatible strategies. The optimization method is an improved version of traditional conjugate gradient method and is named the variable-step simplified conjugate gradient method (VSCGM). The free-piston Stirling engine is operable only in narrow-bounded parameter regions. Using the present approach, the operable variable combinations can be found efficiently. Two compatible strategies, the wake-up and backward-comparison strategies, are integrated with the VSCGM. The present design algorithm can handle multiple-parameter optimization with more flexible objective function definitions. Meanwhile, it features faster convergence as compared with the traditional conjugate gradient methods. Moreover, the feasibility of the VSCGM and the two compatible strategies is demonstrated in two test cases. It was found that the present approach can optimize the ten designed variables simultaneously, and the optimal designs can be yielded in a finite number of iterations. The results show that the inoperable initial designs were successfully optimized to reach a high power output.

The phase-sensitive optical time-domain reflectometer (φ-OTDR) is commonly used in various industries such as oil and gas pipelines, power communication networks, safety maintenance, and perimeter security. However, one challenge faced by the φ-OTDR system is low pattern recognition accuracy. To overcome this issue, a Dendrite Net (DD)-based pattern recognition method is proposed to differentiate the vibration signals detected by the φ-OTDR system, and normalize the differential signals with the original signals for feature extraction. These features serve as input for the pattern recognition task. To optimize the DD for the pattern recognition of the feature vectors, the Variable Three-Term Conjugate Gradient (VTTCG) is employed. The experimental results demonstrate the effectiveness of the proposed method. The classification accuracy achieved using this method is 98.6%, which represents a significant improvement compared to other techniques. Specifically, the proposed method outperforms the DD, Support Vector Machine (SVM), and Extreme Learning Machine (ELM) by 7.5%, 8.6%, and 1.5% respectively. The findings of this research paper indicate that the pattern recognition method based on DD and optimized using the VTTCG can greatly enhance the accuracy of the φ-OTDR system. This improvement has important implications for various applications in industries such as pipeline monitoring, power communication networks, safety maintenance, and perimeter security.