Regular Problems Research Articles

Optimal time for the excess of loss reinsurance with fixed costs

We consider the optimal excess of loss reinsurance for an insurance company facing a constant fixed cost when reinsurance contract is signed. To maximize the survival probability, the company needs to determine the starting time of reinsurance and the claims covered by insurer. It concludes a combination of optimal stopping and regular stochastic control problems, and we solve it by employing the dynamic programming principle. The results indicate that the maximum amount of loss plays an important role for the optimal timing of excess of loss reinsurance. When the claims have a long tail, the insurer will launch an excess of loss reinsurance no matter how expensive of reinsurance is. When the claims have a finite maximum loss, the insurer will not take the excess of loss reinsurance if the reinsurance is too expensive. In addition, the fixed cost only affects the timing of excess of loss reinsurance, but the relative price between reinsurer and insurer affects both the timing and the retained amount of risks. To explain the results, we also give some numerical examples.

Eigenvalues of Sturm-Liouville problems with distributional potentials and eigenparameter-dependent boundary conditions

In this paper, regular Sturm-Liouville problems with distributional potentials and eigenparameter-dependent boundary conditions are investigated. We obtain that the eigenvalues of the problems depend not only continuously but also smoothly on the parameters of the problem: the coefficients, the boundary conditions, and the endpoints, in particular, the eigenparameter-dependent boundary condition matrix. Moreover, we find the differential expressions for each parameters.

Physicochemical Characterization of Natural Rocks and Their Applications for Wastewater Treatment

Arid countries such as Arabian Gulf countries are suffering from a water shortage, especially with the recent high-water demand. The best solution for this shortage is the management of currently available water resources, through the reuse of treated wastewater for irrigation purposes. This solution also solves the problem of regularization of wastewater discharge, with positive impacts on the environment. This study aimed to apply an innovative, advanced method for treating wastewater with a favorable environment, low economic cost, and less energy consumption. The research investigated the possibility of using natural rocks such as volcanic and zeolite for advanced treatment of wastewater effluent. The research methodology relied on an experimental work in the lab scales and applied on materials available in Saudi Arabia. The experiments included a leaching batch test to first examine what leaches out from these rocks into water. Then, the materials were tested with wastewater effluent. The main mechanism of treatment was based on the absorption process. The results exhibited significant improvement in the water quality of treated wastewater. On the other hand, the results of the leaching tests showed many ions being dissolved from both rocks into water; thus, it is recommended to soak and flush these solid materials with clean water before using them for the treatment process. Further research is required to determine the best pretreated methods to be applied on these rocks to improve their performance as absorbents.

Regular matching problems for infinite trees

We study the matching problem of regular tree languages, that is, "$\exists \sigma:\sigma(L)\subseteq R$?" where $L,R$ are regular tree languages over the union of finite ranked alphabets $\Sigma$ and $\mathcal{X}$ where $\mathcal{X}$ is an alphabet of variables and $\sigma$ is a substitution such that $\sigma(x)$ is a set of trees in $T(\Sigma\cup H)\setminus H$ for all $x\in \mathcal{X}$. Here, $H$ denotes a set of "holes" which are used to define a "sorted" concatenation of trees. Conway studied this problem in the special case for languages of finite words in his classical textbook "Regular algebra and finite machines" published in 1971. He showed that if $L$ and $R$ are regular, then the problem "$\exists \sigma \forall x\in \mathcal{X}: \sigma(x)\neq \emptyset\wedge \sigma(L)\subseteq R$?" is decidable. Moreover, there are only finitely many maximal solutions, the maximal solutions are regular substitutions, and they are effectively computable. We extend Conway's results when $L,R$ are regular languages of finite and infinite trees, and language substitution is applied inside-out, in the sense of Engelfriet and Schmidt (1977/78). More precisely, we show that if $L\subseteq T(\Sigma\cup\mathcal{X})$ and $R\subseteq T(\Sigma)$ are regular tree languages over finite or infinite trees, then the problem "$\exists \sigma \forall x\in \mathcal{X}: \sigma(x)\neq \emptyset\wedge \sigma_{\mathrm{io}}(L)\subseteq R$?" is decidable. Here, the subscript "$\mathrm{io}$" in $\sigma_{\mathrm{io}}(L)$ refers to "inside-out". Moreover, there are only finitely many maximal solutions $\sigma$, the maximal solutions are regular substitutions and effectively computable. The corresponding question for the outside-in extension $\sigma_{\mathrm{oi}}$ remains open, even in the restricted setting of finite trees.

Machine learning-based inverse design of auxetic metamaterial with zero Poisson's ratio

The inverse design from property to microstructure is more urgent in practical engineering than the regular design from microstructure to property. In this paper, a data-driven machine learning (ML) model based on the combination of artificial back-propagation neural network (BPNN) and genetic algorithm (GA) is developed for designing auxetic metamaterial with specific Poisson's ratio, i.e. zero Poisson’s ratio. Different to topology optimization, the ML model can optimize auxetic metamaterials with higher computational efficiency, lower requirement of deep knowledge of mathematics and physical model. In the ML model, the data set prepared by solving a large number of regular design problems using finite element simulation are used to train the BPNN to establish the underlying mapping relationships from the microstructure parameters to the Poisson’s ratio, and through which the GA optimization is conducted to globally seek optimal solution of the microstructure parameters related to the specific Poisson’s ratio. The effectiveness of the ML model is demonstrated by comparing to the tensile experiment and the finite element simulation of the structure designed with the given prediction. The results show the ML-based method offers an efficient pathway to design the microstructure of auxetic metamaterials with arbitrary specific Poisson’s ratio.

A primal–dual penalty-interior-point method for solving the reactive optimal power flow problem with discrete control variables

The Optimal Reactive Power Flow (ORPF) problem has been used as an important computational tool for power system planning and operation. Its mixed-discrete version (DORPF) is formulated as a non-convex, non-linear optimization problem with discrete and continuous variables, which is aimed at minimizing the transmission losses while meeting the power demand and enforcing operational and physical limits of the system. Although the DORPF problem has been solved by a myriad of methods, some of them present regularization problems associated with poor matrix-conditioning in the optimal solution, some do not provide for rapid infeasibility detection and some do not provide effective ways for handling the discrete nature of the control variables. Although some of these complicating issues have been tackled separately in the literature by various studies, a method that addresses all these issues has not yet been proposed for solving the DORPF problem. In this paper, an integration of optimization approaches is proposed for handling all such complicating issues. The basis of this approach is a primal–dual penalty-interior-point method, which integrates the good properties of penalty methods (e.g. regularization effects on the constraints and rapid infeasibility detection) and interior-point methods (e.g. scalability and good convergence behavior), without suffering from their disadvantages. In the proposed approach, this method is integrated with a sinusoidal penalty function method for handling the discrete nature of the control variables of the DORPF problem, together with a specific inertia correction strategy designed to avoid local maximizers associated with such a penalty function. Numerical tests carried out with the IEEE 30-, 57-, 118- and 300-bus systems focus on showing that all the complicating issues have been addressed by the proposed method. Comparisons with the results obtained by interior-point methods are also provided.

On regularity of Max-CSPs and Min-CSPs

We study the approximability of regular constraint satisfaction problems, i.e., CSPs where each variable in an instance has the same number of occurrences. In particular, we show that for any CSP Λ, the existence of an α-approximation algorithm for unweighted regular Max-CSP Λ implies the existence of an (α−o(1))-approximation algorithm for weighted Max-CSP Λ for which the regularity of instances is not imposed. We also give an analogous result for Min-CSPs, and therefore show that up to an arbitrarily small error it is sufficient to conduct the study of the approximability of CSPs only on regular unweighted instances.

Bounded solutions to the axially symmetric Navier Stokes equation in a cusp region

A domain in R3 that touches the x3 axis at one point is found with the following property. For any initial value in a C2 class, the axially symmetric Navier Stokes equations with Navier slip boundary condition have a finite energy solution that stays bounded for any given time, i.e. no finite time blow up of the fluid velocity occurs. The result seems to be the first case where the Navier-Stokes regularity problem is solved beyond dimension 2.

Global regularity for the 2D micropolar Rayleigh–Bénard problem with partial dissipation

This paper is devoted to the study of the global regularity problem to the 2D micropolar Rayleigh–Bénard problem with only partial velocity dissipation and micro-rotational dissipation in R2. By fully exploiting the special structure of the system and using the delicate energy methods, we establish the global well-posedness of classical solutions for this system.

A Model-Inspired Approach With Transformers for Hyperspectral Pansharpening

Hyperspectral pansharpening is a method of integrating a low-resolution hyperspectral image with a high-resolution panchromatic image to produce a high-resolution hyperspectral image. In recent years, a number of hyperspectral pansharpening methods have been developed by using convolutional neural networks. However, these methods only consider local information due to limitations on the size of convolution kernels in the convolution operation. In this paper, we design a model-inspired approach with transformers for hyperspectral pansharpening. Due to the representation ability of transformers and the algorithmic explanatory ability of optimization models, our method is able to explore intrinsic relationships at a global scale on both spectral and spatial features. First, we formulate an optimization model consisting of a fidelity term and a regularization term. Then, this optimization model is solved by a half quadratic splitting algorithm, and thus divided into two sub-optimization problems, the fidelity and regularization problems. Finally, the algorithm is implemented by a transformer-based deep structure. Specifically, the fidelity problem is solved by a gradient descent algorithm further, and then implemented through a convolutional network. The regularization problem is depicted by an approximation operation and implemented via a transformer network. The experimental results on different satellite datasets demonstrate the effectiveness of the proposed method.

Generalization of hyperbolic smoothing approach for non-smooth and non-Lipschitz functions

<p style='text-indent:20px;'>In this study, we concentrate on the hyperbolic smoothing technique for some sub-classes of non-smooth functions and introduce a generalization of hyperbolic smoothing technique for non-Lipschitz functions. We present some useful properties of this generalization of hyperbolic smoothing technique. In order to illustrate the efficiency of the proposed smoothing technique, we consider the regularization problems of image restoration. The regularization problem is recast by considering the generalization of hyperbolic smoothing technique and a new algorithm is developed. Finally, the minimization algorithm is applied to image restoration problems and the numerical results are reported.</p>

Discussion of Correspondence and Regularity Problem in Deriving and Applying Distribution Fraction Calculations

Discussion of Correspondence and Regularity Problem in Deriving and Applying Distribution Fraction Calculations

Seismic Data Interpolation by Shannon Entropy-Based Shaping

The undersampled seismic data may surfer from degraded quality and pose negative impacts on subsequent processing procedures. Seismic data interpolation is a cost-saving technique to obtain regular and high-density data in modern seismological community. In this study, I present a seismic data interpolation technique which is based on the Shannon entropy and shaping regularization. I consider seismic data interpolation problem as a process of improving the orderliness of a system. A seismic section with clean and completely-sampled signals can be treated as an orderly data system while the added noise or decimation will destruct such orderliness. Therefore, we can recover the missing signals as the orderliness of the seismic data section is improved. The wave field is predicted by a regularized least-squares matching with the assumption of local plane-wave. The orderliness-improving process is packaged as a shaper and incorporated into the shaping regularization framework to iteratively solve the seismic interpolation problem. Unlike the conventional methods, such as prediction-based, sparsity-promoting-based or rank-reduction-based methods, which generally assume the observed data are regularly or irregularly undersampled, the presented Shannon entropy shaping-based method has no requirement on the distribution pattern of missing traces and hence can be applied to interpolate both regular and irregular seismic data. I use both synthetic and field seismic data to test the performance of the proposed algorithm on both regular and irregular seismic data interpolation problems. The results demonstrate that the presented approach can achieve a superior performance compared with the widely-used techniques.

Entropy Stable Discontinuous Galerkin Finite Element Method with Multi-Dimensional Slope Limitation for Euler Equations

Abstract We present an entropy stable Discontinuous Galerkin (DG) finite element method to approximate systems of 2-dimensional symmetrizable conservation laws on unstructured grids. The scheme is constructed using a combination of entropy conservative fluxes and entropy-stable numerical dissipation operators. The method is designed to work on structured as well as on unstructured meshes. As solutions of hyperbolic conservation laws can develop discontinuities (shocks) in finite time, we include a multidimensional slope limitation step to suppress spurious oscillations in the vicinity of shocks. The numerical scheme has two steps: the first step is a finite element calculation which includes calculations of fluxes across the edges of the elements using 1-D entropy stable solver. The second step is a procedure of stabilization through a truly multi-dimensional slope limiter. We compared the Entropy Stable Scheme (ESS) versus Roe’s solvers associated with entropy corrections and Osher’s solver. The method is illustrated by computing solution of the two stationary problems: a regular shock reflection problem and a 2-D flow around a double ellipse at high Mach number.

Spiking Sparse Recovery With Non-Convex Penalties

Sparse recovery (SR) based on spiking neural networks has been shown to be computationally efficient with ultra-low power consumption. However, existing spiking-based sparse recovery (SSR) algorithms are designed for the convex <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{1}$</tex-math></inline-formula> -norm regularized SR problem, which often underestimates the true solution. This paper proposes an adaptive version of SSR, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i.e.</i> , A-SSR, to optimize a class of non-convex regularized SR problems and analyze its global asymptotic convergence. The superiority of A-SSR is validated with synthetic simulations and real applications, including image reconstruction and face recognition. Furthermore, it is shown that the proposed A-SSR essentially improves the recovery accuracy by avoiding systematic underestimation and obtains over 4 dB PSNR improvement in image reconstruction quality and around 5% improvement in recognition confidence. At the same time, the proposed A-SSR maintains energy efficiency in hardware implementation. When implemented on the neuromorphic Loihi chip, our method consumes only about 1% of the power of the iterative solver FISTA, enabling applications under energy-constrained scenarios.

Inhibition in proper Ser and in PLCγ2 synthesis are the main for causing Osteoarthritis, diabetes and C-lymphocytic leukemia diseases

Purpose of study: Understanding that the inhibition or mutation in S6K and in hydroponic acids ( Ser, Tyr, Pro... ), in BTK and in PLCγ1 will lead to inhibition in PLCγ2 and in Thromboxane-A 2 then will be the main reasons for chronic lymphocytic leukemia “CLL” disease, where proper S6K /BTK and PLCγ2 are main regulations for thromboxane-A synthesis and necessary for B-cells maturations and T-cells modulations. Also, it’s important to Understand main factors that cause and link the Osteoarthritis “OA” with diabetes which are the deficiency in Ser (hydrophobic) amino acids and the mutated S6K productions lead to deficiency or inhibition in Ser phosphorylation signaling which normally is the basis of Ser/Thr phosphorylation signalling which necessarily for Akt, for S6K1 synthesis and necessary for RORs and IFNs synthesis, and also necessary for proper PLCγ2 productions , where S6K is the main regulator for ATPase, for ribosomes, for OPA1 repair, and for proper PLCγ2 synthesis , that I have to note that the percentage of the shortages ratio of amino acids or in the increasing in positive linkages are the main ratio that can define the degree and type of specific disease which can differ from other diseases or can linked with the same Syndromes of other diseases. That also the shortage ratio between the beta Cytokines producions and the ratio of sudden high inflammations productions “and the type of its inflammatory molecules” have to be calculated and considered related to the patients ages (whether child, youth or old ages) and the duration of the chronic disease disease, that some can be confused to differentiate between auto-immune disease and regular disease problems diagnosis. That, There was a case of a child with 9-year-old who had a suspicion of loose of bone maturation and growth and has a sudden infection in the right lung and a lack of breathing with pain. It was found that there was a pulmonary abscesses in right lung and there was a development with the appearance of an air bag or “inflammatory fluid bag” surrounding respiratory cells in right side. The occurrence of sudden inflammations molecules and their growth was rapid enough faster than IFNs productions and faster than PLCγ2 productions due to to the age of the child, “Note some her regular treating doctors diagnosed her medical conditions as a type Autoimmune disease and she has weakened immunity due to sudden fast infection related to her young age”. Highlights: _increasing in PLCγ1 with Deficiency in Ser , deficiency in proper S6K, and decreasing in synthase activity with inhibition in PLCγ2 will reflect decreasing in anti-inflammatory processes, reflect Osteoarthritis and diabetes syndromes, and also inhibition in PLCγ2 will inhibit TXA2 synthesis and can be the main reason for CLL diseases. _proper healthy PLCγ2 are so necessary for increasing re-medulate immune efficiencies, and for re-modulate IgM and IgD antigen and T-cells functions , and also proper healthy PLCγ2 productions (which depend on PLCγ1 and BTK Biosynthesis) are so imp for recover osteoporosis and both Osteoarthritis and diabetes. _inhibition in PLCγ2 Bio-Synthesis can reflect decreasing or inhibition in Thromboxane-A 1 percentages and can lead to CLL diseases, Where, CLL characterized by inhibition in BTK which regulate PLCγ2 synthesis, inhibition in main antigen synthesis, and then inhibition in the proper normal Thromboxane-A synthesis (which regulated mainly by PLCγ1 and by BTK and then linked to IFNs productions and regulated by PLCγ2 proper producions). _Chronic lymphocytic leukemia (CLL) observed during treatment with B-cell receptor inhibitors pathway including inhibitor of Bruton’s tyrosine kinase-PLCγ2, where, CLL can be strongly linked to Osteoporosis “OA” and Linked to both Osteoarthritis and diabetes too.

Напружений стан гнучких пластин з отвором

The article is dedicated to solving of problems of nonlinear mechanics of plates and shells – problems about stress-deformed state of flexible plates with hole under action of moment loading on “infinity”. Solve of problems is offered to find with method of expansion of parameter of the external loading. During the solving was determined, that the value of bending and power descriptions unlimitedly increase at breaking from the edge of hole. For elaboration of regularization methods was conducted the survey and analysis of problems from different fields of mechanics. On the basis of this survey was formed the new approach to the solving of problem of regularization: it is offered to change the usual notions about particular sum of series and methods of their summing. It is created such methods of linear and nonlinear summing, when in summable functions the arbitrary parameters and functions enter. On basis of this method was solved the new geometrical-nonlinear problems of plates and shells mechanics in nonaxes-symmetrical axes-symmetrical arrangement about bending on “infinity” with moment loading of plates with hole. It is established, that the finding numeral data, diagrams do not conflict with the usual notions about stress-deformed conditions of plates with hole; definite mechanical effects are got. The methods of regularization are approved on test problems. It is grounded mathematically, that the got solutions to equilibrium equations with some asymptotical exactness and exactly to the linear limit conditions, if the operators of initial problem will be polylinear.

TDOA-Based Robust Sound Source Localization With Sparse Regularization in Wireless Acoustic Sensor Networks

Time difference of arrival (TDOA) measurements, which are contaminated by large values of error, known as outliers, would have a significant impact on the accuracy of sound source localization (SSL) in wireless acoustic sensor networks (WASNs). Few techniques are reported in the literature to tackle SSL in WASNs by taking TDOA outliers into consideration. To mitigate the effect of outliers on the accuracy of SSL, we propose outlier-resistant robust sound source localization (RSSL) algorithms based on sparse regularization using an unsynchronized network of microphone arrays. The TDOA errors are divided into two components: a) energy-bounded inliers and b) outliers. Assuming that outliers are sparse in the measurement set, we formulate the RSSL problem as that of minimizing the number of outliers, mathematically, a <inline-formula><tex-math notation="LaTeX">$\ell _{0}$</tex-math></inline-formula> (pseudo)-norm optimization problem with non-convex constraints. Five sub-optimal RSSL solvers are derived, among which the first two solvers are applicable to the scenario involving only outliers while the last three solvers concentrate on the scenario incorporating both outliers and inliers. In common, these solvers exploit a convex approximation technique called Concave Convex Procedure to dispose of the non-convex constraints. Differently, the first solver approximates the original <inline-formula><tex-math notation="LaTeX">$\ell _{0}$</tex-math></inline-formula> (pseudo)-norm cost function with the <inline-formula><tex-math notation="LaTeX">$\ell _{1}$</tex-math></inline-formula> norm while a concave surrogate function is adopted in the second solver to yield a tighter approximation to the <inline-formula><tex-math notation="LaTeX">$\ell _{0}$</tex-math></inline-formula> (pseudo)-norm. Apart from the application of these two approximation techniques, the third and fourth solvers relax the non-convex <inline-formula><tex-math notation="LaTeX">$\ell _{2}$</tex-math></inline-formula> norm constraint with the <inline-formula><tex-math notation="LaTeX">$\ell _{\infty }$</tex-math></inline-formula> norm. The fifth solver is dedicated to the <inline-formula><tex-math notation="LaTeX">$\ell _{1}$</tex-math></inline-formula> norm regularization problem with the Lasso formulation, which is equivalent to the M-estimator of Huber&#x2019;s function solved via the iteratively reweighted least squares paradigm. Experimental results validate the effectiveness and robustness of the proposed algorithms.

Seismic Data Regularization on Nonequispaced Grid via a Joint Sparsity-Promotion Method

The seismic data regularization problem is vital to seismic data processing. We propose a joint sparsity-promotion method based on the compressive sensing theory named the curvelet-data-driven-tight-frame-based sparsity-promoting (CDSP) method. The CDSP method regularizes the seismic data directly on the nonequispaced grid along the spatial dimension. The joint sparsity is exploited in the curvelet and data-driven tight frame transform simultaneously on the projected regular spatial grid. The projection from the nonequispaced grid to the equispaced grid is achieved by the nonequispaced discretized Fourier transform. Comparing with the curvelet-sparsity-promotion-based (CSP) regularization method, CDSP combines the advantage of the predefined curvelet transform and adaptive-learned sparse transform into one single optimization model. An alternative directional method of multipliers (ADMM) is applied to solve the optimization problem. One synthetic and two field examples show that the CDSP method performs better on the preservation of the continuity of events and produces less artifacts than the CSP method.

Regularity of Lie groups

We solve the regularity problem for Milnor's infinite dimensional Lie groups in the $C^0$-topological context, and provide necessary and sufficient regularity conditions for the (standard) $C^k$-topological setting. We prove that the evolution map is $C^0$-continuous on its domain $\textit{iff}\hspace{1pt}$ the Lie group $G$ is locally $\mu$-convex. We furthermore show that if the evolution map is defined on all smooth curves, then $G$ is Mackey complete. Under the assumption that $G$ is locally $\mu$-convex, we show that each $C^k$-curve for $k\in \mathbb{N}_{\geq 1}\sqcup\{\mathrm{lip},\infty\}$ is integrable (contained in the domain of the evolution map) $\textit{iff}\hspace{1pt}$ $G$ is Mackey complete and $\mathrm{k}$-confined. The latter condition states that each $C^k$-curve in the Lie algebra $\mathfrak{g}$ of $G$ can be uniformly approximated by a special type of sequence that consists of piecewise integrable curves. A similar result is proven for the case $k\equiv 0$; and, we provide several mild conditions that ensure that $G$ is $\mathrm{k}$-confined for each $k\in \mathbb{N}\sqcup\{\mathrm{lip},\infty\}$. We finally discuss the differentiation of parameter-dependent integrals in the (standard) $C^k$-topological context. In particular, we show that if the evolution map is defined and continuous on $C^k([0,1],\mathfrak{g})$ for $k\in \mathbb{N}\sqcup\{\infty\}$, then it is smooth thereon $\textit{iff}\hspace{1pt}$ it is differentiable at zero $\textit{iff}\hspace{1pt}$ $\mathfrak{g}$ is $\hspace{0.2pt}$ Mackey$\hspace{1pt}/ \hspace{1pt}$integral$\hspace{1pt}$ complete for $k\in \mathbb{N}_{\geq 1}\sqcup\{\infty\}\hspace{1pt}/\hspace{1pt}k\equiv 0$. This result is obtained by calculating the directional derivatives explicitly, recovering the standard formulas that hold, e.g., in the Banach case.