Invertibility Properties Research Articles

Unlocking thin sand potential: a data-driven approach to reservoir characterization and pore pressure mapping

The gradual decline of production of the major fields of the Indus Basin has influenced the field pressure. Therefore, the prospects need to be assessed by the petro-elastic relationship. This study highlights the value-addition for identifying gas pockets within the E-Sand of the Lower Goru Formation through estimating and populating petro-elastic properties. A probabilistic neural network (PNN) was employed to develop the relationship of pore pressure to the inverted properties. The outcomes will help identify the high-pressure areas in accordance with the petro-elastic relationship for optimizing production strategy. In order to evaluate the petro-elastic relationship, stochastic seismic inversion was employed to holistically capture the reservoir’s variability. Pore pressure volumetric was estimated from inverted attributes using a PNN, a non-linear algorithm that was more suited to heterogeneous reservoirs. The main objective of the research is achieved via enhanced resolution of elastic properties attained through the integration of stochastic inversion and PNN processes. The high-resolution elastic properties including P-impedance, S-impedance, and Vp/Vs ratio can delineate the heterogeneities more profoundly. The petrophysical volumes obtained via enhanced inverted properties combined with pore-pressure through PNN illuminated the potential facies and the correlations in predicting the properties. The procedure provides a linkage of elastic, petrophysical, and geomechanical properties that is helpful for professionals covering a broad field of the petroleum sector. The developed workflow can be followed globally where problems occur regarding heterogeneities and early depletion.

Core of an ideal in Prüfer domains

This paper contributes to the study of the core of an ideal in integral domains. Our aim is to develop explicit formulas for the core in various classes of Prüfer domains. We pay particular attention to relevant ideal-theoretic notions such as stability, invertibility, and h-local property. We also provide decomposition results for the core of an ideal in integral domains with effectual ramifications to Prüfer domains. All main results are illustrated with original examples, where we explicitly compute the core. We also provide counter-examples to test the limits of the assumptions used in the main results.

Construction of Quasigroups with Invertibility Properties

Construction of Quasigroups with Invertibility Properties

The Mathematical Formulas for Inverting Plithogenic Matrices of Special Orders Between 20 and 24

This paper is dedicated to study the Invertibility properties of all plithogenic square matrices of high orders between 20 and 24, where we present the mathematical conditions and formulas for computing the inverses of all plithogenic square matrices with real plithogenic entries. This goal will be completed by proving many theorems that describe the Invertibility of all classic matrix parts of the corresponding symbolic m-plithogenic matrix.

On the fractional space-time Fourier transforms

ABSTRACT The notion of a fractional space-time Fourier transform (FSFT) is outlined in this paper, and the properties of invertibility, linearity, Plancherel and others are derived. By establishing the relationship between the FSFT and space-time Fourier transform, a directional uncertainty principle (UP) and its specialization to coordinates are proved. Moreover, the Hardy UP of the FSFT is also obtained. The fractional Mustard convolution for space-time valued signals is introduced and written in terms of the standard convolution. Finally, the FSFT is employed in solving a partial differential equation in space-time analysis.

Enhancing plane-wave emission with a combination device based on acoustic metamaterial

Generating high sound-intensity plane waves typically requires transducer arrays with complex synchronizing systems and high-output amplifiers. This work proposes a simple combination device to convert fan-shaped waves into elevated sound-pressure plane waves. The proposed device primarily relies on phase reconstruction of near-zero index metamaterials and the superposition of reflected waves in the reverberation field. The compact device, which is easy to manufacture, consists of a short waveguide equipped with two types of converters. One of the converters makes use of inverted cone's sound-guiding properties to convert fan-shaped waves into cylindrical waves over a broad frequency band. The other converter is comprised of rigid cylinders with triangular lattice arrays, effectively acting as the near-zero index metamaterial to transform cylindrical waves into plane waves. Both the simulated and experimental results show that the emitted plane waves can be linearly amplified by increasing the number of point sources that are separated by integer multiples of the wavelength in the vertical cavity. This study presents a straightforward and efficient approach to generate plane waves with a low-cost array assembly design, thereby enabling the widespread use of acoustic testing and underwater detection.

Invertible Koopman Network and its application in data-driven modeling for dynamic systems

Koopman operator, acting on an infinite-dimensional Hilbert space of the observables, provides a global systematic linear representation of nonlinear systems, which is a leading candidate for data-driven modeling. In practical applications, the observable function is required to have an invertible property to predict or control the original system. However, two approaches for constructing the invertible observable functions in current research, introducing the full-state observables and adopting the auto-encoder, have drawbacks such as overly strong assumptions and lossy reconstruction. To address these issues, we propose the Invertible Koopman Network (IKN) to approximate the Koopman operator. With special invertible blocks, IKN achieves structure-level reversibility while retaining sufficient nonlinear mapping capability. This network results in a transformation from the complex dynamic system’s states to the observables with simpler dynamics and affords lossless reconstruction via its invertible process. Then, we use the IKN as a dynamical embedding model and adopt the Transformer based on an autoregressive task to learn the evolution pattern of the observables. Through the greedy decoding strategy, the trained IKN and Transformer model enable data-driven prediction where only initial value is provided. The method is tested on three typical chaotic dynamic systems (Lorenz system, Chen system, and Kuramoto-Sivashinsky equation) and is shown to capture the systems’ intrinsic evolution patterns, such as the structure of the strange attractor. Comparative experiments with alternative data-driven modeling approaches illustrate that our proposed method can model a longer evolutionary process with less error, further demonstrating its effectiveness and superiority.

Efficient Algebraic Method for Testing the Invertibility of Finite State Machines

The emergence of new embedded system technologies, such as IoT, requires the design of new lightweight cryptosystems to meet different hardware restrictions. In this context, the concept of Finite State Machines (FSMs) can offer a robust solution when using cryptosystems based on finite automata, known as FAPKC (Finite Automaton Public Key Cryptosystems), introduced by Renji Tao. These cryptosystems have been proposed as alternatives to traditional public key cryptosystems, such as RSA. They are based on composing two private keys, which are two FSMs M1 and M2 with the property of invertibility with finite delay to obtain the composed FSM M=M1oM2, which is the public key. The invert process (factorizing) is hard to compute. Unfortunately, these cryptosystems have not really been adopted in real-world applications, and this is mainly due to the lack of profound studies on the FAPKC key space and a random generator program. In this paper, we first introduce an efficient algebraic method based on the notion of a testing table to compute the delay of invertibility of an FSM. Then, we carry out a statistical study on the number of invertible FSMs with finite delay by varying the number of states as well as the number of output symbols. This allows us to estimate the landscape of the space of invertible FSMs, which is considered a first step toward the design of a random generator.

Invertibility properties of operator matrices on Hilbert spaces

Denote by $$T_n^d(A)$$ an upper triangular operator matrix of dimension n whose diagonal entries are given and the others are unknown. In this article, we provide necessary and sufficient conditions for various types of Fredholm and Weyl completions of $$T_n^d(A)$$ . As consequences, we recover many known results of Zhang and Wu (J Math Anal Appl 392(2):103–110, 2012) for the case $$n=2$$ already existing in the literature, as well as some perturbation results of Wu and Huang (Acta Math Sin (Engl Ser) 36(7):783–796, 2020; Ann Funct Anal 11(3):780–798, 2020) given for the case of arbitrary $$n\ge 2$$ .

Calibration reinforcement regularizations for optimized snapshot spectral imaging.

Optical coding is a fundamental tool in snapshot computational spectral imaging for capturing encoded scenes that are then decoded by solving an inverse problem. Optical encoding design is crucial, as it determines the invertibility properties of the system sensing matrix. To ensure a realistic design, the optical mathematical forward model must match the physical sensing. However, stochastic variations related to non-ideal characteristics of the implementation exist; therefore, these variables are not known a priori and have to be calibrated in the laboratory setup. Thus, the optical encoding design leads to suboptimal performance in practice, even if an exhaustive calibration process is carried out. This work proposes an algorithm to speed up the reconstruction process in a snapshot computational spectral imaging, in which theoretically optimized coding design is distorted by the implementation process. Specifically, two regularizers are proposed that perform the gradient algorithm iterations of the distorted calibrated system in the direction of the originally, theoretically optimized system. We illustrate the benefits of the reinforcement regularizers for several state-of-the-art recovery algorithms. For a given lower bound performance, the algorithm converges in fewer iterations due to the effect of the regularizers. Simulation results show an improvement of up to 2.5dB of peak signal-to-noise ratio (PSNR) when fixing the number of iterations. Furthermore, the required number of iterations reduces up to 50% when the proposed regularizers are included to obtain a desired performance quality. Finally, the effectiveness of the proposed reinforcement regularizations was evaluated in a test-bed implementation, where a better spectral reconstruction was evidenced when compared with a non-regularized system's reconstruction.

Quadrature Methods for Singular Integral Equations of Mellin Type Based on the Zeros of Classical Jacobi Polynomials

In this paper we formulate necessary conditions for the stability of certain quadrature methods for Mellin type singular integral equations on an interval. These methods are based on the zeros of classical Jacobi polynomials, not only on the Chebyshev nodes. The method is considered as an element of a special C*-algebra such that the stability of this method can be reformulated as an invertibility problem of this element. At the end, the mentioned necessary conditions are invertibility properties of certain linear operators in Hilbert spaces. Moreover, for the proofs we need deep results on the zero distribution of the Jacobi polynomials.

3-Lie algebras, ternary Nambu-Lie algebras and the Yang-Baxter equation

We construct ternary self-distributive (TSD) objects from compositions of binary Lie algebras, 3-Lie algebras and, in particular, ternary Nambu-Lie algebras. We show that the structures obtained satisfy an invertibility property resembling that of racks. We prove that these structures give rise to Yang-Baxter operators in the tensor product of the base vector space and, upon defining suitable twisting isomorphisms, we obtain representations of the infinite (framed) braid group. We consider examples for low-dimensional Lie algebras, where the ternary bracket is defined by composition of the binary ones, along with simple 3-Lie algebras. We show that the Yang-Baxter operators obtained are not gauge equivalent to the transposition operator, and we consider the problem of deforming the operators to obtain new solutions to the Yang-Baxter equation. We discuss the applications of this deformation procedure to the construction of (framed) link invariants.

TRANSMUTATION OPERATORS METHOD FOR STURM-LIOUVILLE EQUATIONS IN IMPEDANCE FORM I: CONSTRUCTION AND ANALYTICAL PROPERTIES

The construction of integro-differential transmutation operators for the Sturm-Liouville equation in impedance form is presented. Their analytical properties of boundedness and invertibility in appropriate functional spaces are studied. A Fourier-Legendre series for the integral transmutation kernel in terms of the Legendre polynomials is obtained together with a representation of the solutions of the Sturm-Liouville equation in impedance form as Neumann series of Bessel functions.

Linear Operators and Equations with Partial Integrals

We consider linear operators and equations with partial integrals in Banach ideal spaces, spaces of vector functions, and spaces of continuous functions. We study the action, regularity, duality, algebras, Fredholm properties, invertibility, and spectral properties of such operators. We describe principal properties of linear equations with partial integrals. We show that such equations are essentially different compared to usual integral equations. We obtain conditions for the Fredholm alternative, conditions for zero spectral radius of the Volterra operator with partial integrals, and construct resolvents of invertible equations. We discuss Volterra–Fredholm equations with partial integrals and consider problems leading to linear equations with partial integrals.

Dual-Band General Toeplitz Operators

We relate dual-band general Toeplitz operators to block truncated Toeplitz operators and, via equivalence after extension, with Toeplitz operators with 4 times 4 matrix symbols. We discuss their norm, their kernel, Fredholmness, invertibility and spectral properties in various situations, focusing on the spectral properties of the dual-band shift, which turns out to be considerably complex, leading to new and nontrivial connections with the boundary behaviour of the associated inner function.

An Asymmetric Cryptosystem of Skewed Affine Cipher Over Elliptic Curves with Block Matrix

The retaining of confidentiality in data transmission with user authentication has become eminent for easy and secure communication. This is achieved by using the art of cryptology at its best. Various ciphers are developed balancing time, memory, and security. In particular, the invertible property is the basic requirement in constructing a secure cryptosystem. In this paper, an invertible block matrix is constructed over Fibonacci sequences. The sender and the recipient can derive the block matrix through the key exchanged over a secure channel for encryption and decryption. Also, an asymmetric cryptosystem is designed based on the skewed Affine cipher, which uses a bijective map over points on an elliptic curve. The developed cryptosystem is resistant to cryptanalytic attacks.

A note on invertibility and relative degree of MIMO LTI systems

The property of system invertibility is important both from a theoretical standpoint and an important consideration in left and right inversion problems that deal with input estimation and tracking control respectively. Nuances associated with the notion of relative degree in multi-input, multi-output (MIMO) systems on account of input–output delays pose challenges in system inversion problems and are more acute in the case of non-square systems. In this note, using the LTI discrete-time MIMO systems framework, we review existing notions of relative degree and develop new definitions and generalizations for relative degree to help address some of the gaps in system inversion problems. In doing so, we also show through simple but elegant results that existence of well-defined relative degree for an LTI MIMO system implies its invertibility. Furthermore, we also highlight the utility of these ideas in practical tracking control and input estimation situations and problems.

Naïve Bayesian classification of cumulative oil production from stochastic amplitude variation with angle inversion attributes: As applied to SEAM and a West Africa Field

ABSTRACTCumulative oil predictions are made from stochastically inverted earth attributes. The inverted attributes are from the SEAM Life of Field model and an offshore field in West Africa. To perform the prediction, we use a naïve Bayesian classifier for its transparency in methodology, computational efficiency and flexibility. The inverted properties we use for classification consist of porosity, P‐S wave velocity ratio (Vp/Vs), acoustic impedance and density which are extracted from within a set radius around the well‐path. The production from the wells in both SEAM and the West African field serve as the labels for the naïve Bayesian classifier, which are ultimately placed into two classes: the ‘high’ and ‘low’ producers. For calculating the accuracy of the classifier, we perform full cross‐validation with a set number of training wells. The cross‐validation accuracy is 78.0% and 83.3% for the SEAM model and the West African Field respectively. The classifier's sensitivity to the chosen radius around the wells and its inherent independence assumption are investigated and shown to change the bulk accuracy by less than 3.0%. We implement full reservoir classifications on SEAM to highlight potential drilling locations and show the capability of the classifier. Overall, we demonstrate how the naïve Bayesian classifier can efficiently and transparently synthesize the entire distribution of stochastic inversions of seismic data to predict oil production.

On a boundary-domain integral equation system for the Robin problem for the diffusion equation in non-homogeneous media

Abstract The Robin problem for the diffusion equation in non-homogeneous media partial differential equation is reduced to a system of direct segregated parametrix-based Boundary-Domain Integral Equations (BDIEs). We use a parametrix different from the one applied by Chkadua, Mikhailov, Natroshvili. in [4]. We prove the equivalence between the original BVP and the corresponding BDIE system. The invertibility and Fredholm properties of the boundary-domain integral operators are also analysed.

A Fast Power Allocation Strategy for Multibeam Tracking Multiple Targets in Clutter

A collocated multiple-input–multiple-output (MIMO) radar has the ability to track multiple targets in the simultaneous multibeam working mode, where the defocused beams are transmitted to illuminate the whole space, and focused beams are received to extract target information. A fast power allocation (FPA) strategy is put forward in this mode. The predicted posterior Cramer–Rao lower bound (PCRLB) in the cluttered background is slighted relaxed and utilized as the optimization metric. The optimization model is then established as minimizing the worst case of PCRLB under the constraint of total power budget. By using the monotonic decreasing and invertible properties of the relaxed PCRLB, a fast sequential relaxation-based solver is proposed for this problem solving. At each iteration, the proposed solver tests the power allocation (PA) to one target whether should be set as the minimum level. Simulation results show the effectiveness and efficiency of the proposed solver, compared with convex optimization-based methods. The results also imply that the target distance and reflectivity are two main factors that influence the PA in our strategy.