The choice of a thermodynamic formulation dramatically affects modelled chemical zoning in minerals

Quantifying natural processes that shape our planet is a key to understanding the geological observations. Many phenomena in the Earth are not in thermodynamic equilibrium. Cooling of the Earth, mantle convection, mountain building are examples of dynamic processes that evolve in time and space and are driven by gradients. During those irreversible processes, entropy is produced. In petrology, several thermodynamic approaches have been suggested to quantify systems under chemical and mechanical gradients. Yet, their thermodynamic admissibility has not been investigated in detail. Here, we focus on a fundamental, though not yet unequivocally answered, question: which thermodynamic formulation for petrological systems under gradients is appropriate—mass or molar? We provide a comparison of both thermodynamic formulations for chemical diffusion flux, applying the positive entropy production principle as a necessary admissibility condition. Furthermore, we show that the inappropriate solution has dramatic consequences for understanding the key processes in petrology, such as chemical diffusion in the presence of pressure gradients.

Shock waves in Euler equations for compressible medium

Shock waves of arbitrary strength in the Euler equations for compressible media are studied. The admissibility condition for a shock wave is shown to be equivalent to its formation according to the entropy production criterion. The Riemann problem with large data has a unique admissible solutions. These quantitative results are based on the exact global expressions for the basic physical variables as the states move along the Hugoniot and wave curves.

A Discrete Morse Perspective on Knot Projections and a Generalised Clock Theorem

We obtain a simple and complete characterisation of which matchings on the Tait graph of a knot diagram induce a discrete Morse function (dMf) on the two sphere, extending a construction due to Cohen. We show these dMfs are in bijection with certain rooted spanning forests in the Tait graph. We use this to count the number of such dMfs with a closed formula involving the graph Laplacian. We then simultaneously generalise Kauffman's Clock Theorem and Kenyon-Propp-Wilson's correspondence in two different directions; we first prove that the image of the correspondence induces a bijection on perfect dMfs, then we show that all perfect matchings, subject to an admissibility condition, are related by a finite sequence of click and clock moves. Finally, we study and compare the matching and discrete Morse complexes associated to the Tait graph, in terms of partial Kauffman states, and provide some computations.

Linear canonical curvelet transform and the associated Heisenberg-type inequalities

In this paper, we have introduced a novel integral transform namely linear canonical curvelet transform (LCCT). Firstly, we established basic properties of LCCT including admissibility condition, Moyals principle, inversion formula and range theorem. Toward the culmination of the paper, we formulate a couple of Heisenberg-type inequalities associated with LCCT.

The choice of a thermodynamic formulation dramatically affects modelled chemical zoning in minerals

<p>Quantifying natural processes that shape our planet is a key to understanding the geological observations. Many phenomena in the Earth are not in thermodynamic equilibrium. Cooling of the Earth, mantle convection, mountain building are examples of dynamic processes that evolve in time and space and are driven by gradients. During those irreversible processes, entropy is produced. In petrology, several thermodynamic approaches have been suggested to quantify systems under chemical and mechanical gradients. Yet, their thermodynamic admissibility has not been investigated in detail. Here, we focus on a fundamental, though not yet unequivocally answered, question: which thermodynamic formulation for petrological systems under gradients is appropriate – mass or molar?  We provide a comparison of both thermodynamic formulations for chemical diffusion flux, applying the positive entropy production principle as a necessary admissibility condition. Furthermore, we show that the inappropriate solution has dramatic consequences for understanding the key processes in petrology, such as chemical diffusion in the presence of stress gradients.</p>

Admissibility Analysis and Controller Design for Discrete Singular Time-Delay Systems Embracing Uncertainties in the Difference and Systems’ Matrices

This paper mainly investigates the admissibility analysis and the admissibilizing controller design for the uncertain discrete singular system with delayed state. Based on Lyapunov–Krasovskii stability theory, an original admissibility condition for the nominal singular delay system is first presented. By involving the uncertainties in both difference and system matrices simultaneously, we devote to analyzing the robust admissibility for the regarded uncertain discrete singular system with delayed state. Furthermore, by hiring the state feedback control law, we further discuss the admissibilizing controller design for the resulting closed-loop system. Since all the derived criteria are expressed in terms of strict linear matrix inequalities (LMIs) or parametric LMIs, we thus can handily verify them via current LMI solvers. Finally, two numerical examples are given to illustrate the effectiveness and validity of the proposed approach.

Variable order, directional ℋ2-matrices for Helmholtz problems with complex frequency

The sparse approximation of high-frequency Helmholtz-type integral operators has many important physical applications such as problems in wave propagation and wave scattering. The discrete system matrices are huge and densely populated; hence, their sparse approximation is of outstanding importance. In our paper, we will generalize the directional $\mathcal{H}^{2}$-matrix techniques from the ‘pure’ Helmholtz operator $\mathcal{L}u=-\varDelta u+\zeta ^{2}u$ with $\zeta =-\operatorname *{i}k$, $k\in \mathbb{R}$ to general complex frequencies $\zeta \in \mathbb{C}$ with $\operatorname{Re}\zeta\geq0$. In this case, the fundamental solution decreases exponentially for large arguments. We will develop a new admissibility condition that contains $\operatorname{Re}\zeta $ in an explicit way, and introduces the approximation of the integral kernel function on admissible blocks in terms of frequency-dependent directional expansion functions. We develop an error analysis that is explicit with respect to the expansion order and with respect to $\operatorname{Re}\zeta $ and $\operatorname{Im}\zeta $. This allows for choosing the variable expansion order in a quasi-optimal way, depending on $\operatorname{Re}\zeta $, but independent of, possibly large, $\operatorname{Im}\zeta $. The complexity analysis is explicit with respect to $\operatorname{Re}\zeta $ and $\operatorname{Im}\zeta $, and shows how higher values of $\operatorname{Re} \zeta $ reduce the complexity. In certain cases, it even turns out that the discrete matrix can be replaced by its nearfield part. Numerical experiments illustrate the sharpness of the derived estimates and the efficiency of our sparse approximation.

Performance of reconstruction factors for a class of new complex continuous wavelets

In this paper, we determine the factors necessary for the reconstruction of the signal from its continuous wavelet transform performed using the new complex continuous wavelet family by making use of admissible conditions and studied some of its properties. We also make a comparative assessment of its performance with the existing complex continuous wavelets, such as Morlet, Paul and DOG, in terms of reconstruction capability. The reconstruction was performed on three data sets, namely, a signal with a mixture of low and high frequencies, a non-stationary signal (synthetic) and an ECG signal. The results show that the proposed family of wavelets reconstruction capability is comparable with Morlet, Paul and DOG wavelets. Further, we investigate an alternate reconstruction formula without making use of admissibility condition and compare its efficiency of reconstruction with the standard (restricted) Morlet wavelet.