Infinite Values Research Articles

Linear programming with infinite, finite, and infinitesimal values in the right-hand side

The goal of this work is to propose a new type of constraint for linear programs: inequalities having infinite, finite, and infinitesimal values in the right-hand side. Because of the nature of such constraints, the feasible region polyhedron becomes more complex, since its vertices can be represented by non-purely finite coordinates, and so is the optimum of the problem. The introduction of such constraints enlarges the class of linear programs, where those described by finite values only become a special case. To tackle optimization problems over such polyhedra, there is a need for an ad-hoc solving routine: this work proposes a generalization of the Simplex algorithm, which is able to solve common linear programs as corner cases. Finally, the study presents three relevant applications that can benefit from the use of these novel constraints, making the use of the extended Simplex algorithm essential. For each application, an exemplifying benchmark is solved, showing the effectiveness of the proposed routine.

Singularity properties of the entropy in an enclosed system characterized by accumulated noise stochastic processes

Due to the irreversibility of processes in an enclosed system, the randomness in the system increases in time and can be represented by an accumulated noise stochastic process. In contrast to the thermodynamics theory, the analysis of the system is based on the information theory point of view. The system is defined by two states: a time state and a timeless state. Based on the central limit theorem, due to the additivity of the noise process, the time state is characterized by the time-dependent Gaussian stochastic process. In contrast to Shannon's theory, precise expressions for the probability density, information, and entropy functions of the random variables defining the time-dependent process are derived and analyzed for a finite, infinite, and zero value of the related time-dependent variance. It has been proven that the entropy rate is not necessarily inversely proportional to time, as presented in previous works. Furthermore, mathematical proofs are presented, showing that the system entropy is a singularity function that increases towards infinity in the time state and then drops to zero value when the system enters the timeless state. The timeless state is characterized by the undefined pdf function, infinite values of information, and zero entropy.

Optimal design of diversion dams based on Upgraded Multi-Objective Particle Swarm Optimization (UMOPSO)

In the present research, the application of the Upgraded Multi-Objective Particle Swarm Optimization (UMOPSO) algorithm is investigated with the aim of reducing concreting costs through minimizing the diversion dam weight function. At the time of using optimization algorithms, regarding the nature of such a;gorithms and the research problem, making changes in algorithm running steps is necessary. Hence, in the current study, making some changes in the UMOPSO algorithm develops it to solve multi-objective optimization problems by considering their problem restrictions. In this paper, the study diversion dam is the Nazelian dam located in Kermanshah, Province, Iran. The decision variables in the optimization problem include the height of upstream and downstream cut off walls, the length and the thickness of the upstream concrete bed and the thickness of the stilling basin. By considering the initial population (n = 15), the optimal answer is achieved in iteration 100 and after algorithm run 1500. In this study, considering that the objective function should be evaluated for five decision variables with infinite possible values on a discrete range for each variable, the general result of the present study emphasizes on the desirable efficiency and speed of the UMOPSO algorithm in finding the optimal answer of the diversion dam optimal design problem which shows the strength of this algorithm.

Systematic evaluation with practical guidelines for single-cell and spatially resolved transcriptomics data simulation under multiple scenarios

BackgroundSingle-cell RNA sequencing (scRNA-seq) and spatially resolved transcriptomics (SRT) have led to groundbreaking advancements in life sciences. To develop bioinformatics tools for scRNA-seq and SRT data and perform unbiased benchmarks, data simulation has been widely adopted by providing explicit ground truth and generating customized datasets. However, the performance of simulation methods under multiple scenarios has not been comprehensively assessed, making it challenging to choose suitable methods without practical guidelines.ResultsWe systematically evaluated 49 simulation methods developed for scRNA-seq and/or SRT data in terms of accuracy, functionality, scalability, and usability using 152 reference datasets derived from 24 platforms. SRTsim, scDesign3, ZINB-WaVE, and scDesign2 have the best accuracy performance across various platforms. Unexpectedly, some methods tailored to scRNA-seq data have potential compatibility for simulating SRT data. Lun, SPARSim, and scDesign3-tree outperform other methods under corresponding simulation scenarios. Phenopath, Lun, Simple, and MFA yield high scalability scores but they cannot generate realistic simulated data. Users should consider the trade-offs between method accuracy and scalability (or functionality) when making decisions. Additionally, execution errors are mainly caused by failed parameter estimations and appearance of missing or infinite values in calculations. We provide practical guidelines for method selection, a standard pipeline Simpipe (https://github.com/duohongrui/simpipe; https://doi.org/10.5281/zenodo.11178409), and an online tool Simsite (https://www.ciblab.net/software/simshiny/) for data simulation.ConclusionsNo method performs best on all criteria, thus a good-yet-not-the-best method is recommended if it solves problems effectively and reasonably. Our comprehensive work provides crucial insights for developers on modeling gene expression data and fosters the simulation process for users.

Theorems for Near Stable Points on a Near Banach Space Furnished with Graph

The paper introduces a novel method for defining the graph associated with a near Banach space. In mathematics, a graph typically represents relationships between objects. Here, it seems the graph is being defined in the context of a near Banach space, which is a generalization of Banach spaces allowing the norm to take infinite values. An iteration function is utilized to define the subgraph of the graph associated with the near Banach space. This subgraph likely captures specific properties or relationships within the original graph. The paper presents near-fixed point theorems by well-known math- ematicians such as Banach, Kannan, Chatterja, and Ciric [2][18][7][9]. These theorems deal with the existence of points that are approximately fixed under certain mappings or operations. The near-fixed point theorems mentioned are obtained or derived using the new approach introduced for defining the graph and its subgraph associated with the near Banach space [20]. This suggests that the new approach is effective in providing a framework for proving these theorems or extending their applicability to near Banach spaces. The paper discusses a fresh method for defining the graph of a near Banach space, employs an iteration function to define its subgraph, and then demonstrates the utility of this approach by deriving near-fixed point theorems by eminent mathemati- cians in the field [14][15[16].

Statistical Verification using Surrogate Models and Conformal Inference and a Comparison with Risk-Aware Verification

Uncertainty in safety-critical cyber-physical systems can be modeled using a finite number of parameters or parameterized input signals. Given a system specification in Signal Temporal Logic (STL), we would like to verify that for all (infinite) values of the model parameters/input signals, the system satisfies its specification. Unfortunately, this problem is undecidable in general. Statistical model checking (SMC) offers a solution by providing guarantees on the correctness of CPS models by statistically reasoning on model simulations. We propose a new approach for statistical verification of CPS models for user-provided distribution on the model parameters. Our technique uses model simulations to learn surrogate models , and uses conformal inference to provide probabilistic guarantees on the satisfaction of a given STL property. Additionally, we can provide prediction intervals containing the quantitative satisfaction values of the given STL property for any user-specified confidence level. We compare this prediction interval with the interval we get using risk estimation procedures. We also propose a refinement procedure based on Gaussian Process (GP)-based surrogate models for obtaining fine-grained probabilistic guarantees over sub-regions in the parameter space. This in turn enables the CPS designer to choose assured validity domains in the parameter space for safety-critical applications. Finally, we demonstrate the efficacy of our technique on several CPS models.

Topology of Conical Intersection Seams and the Geometric Phase.

In this paper, we demonstrate two topological properties of crossing seams, that is, the sets of points in the N-dimensional space of nuclear coordinates where two electronic eigenstates are degenerate. We shall examine the typical case of states of the same spin with accidental degeneracies, whereby the crossing seam is of dimension N - 2. The first property we demonstrate is that a crossing seam has no boundary, therefore, it must either extend asymptotically to infinite values of one or more coordinates or wrap on itself. The second property is that two (or more) crossing seams can intersect each other but in such a way that neither of them ends at the intersection. When N = 3, the crossing seam is a line in a 3D space; this is so in triatomic molecules but also in reduced dimensionality treatments of larger polyatomics. The above-mentioned rules then mean that the crossing seam is a line of infinite length or a closed loop and can split into three branches but not in two. The example of the first two excited 1A' states of H2Cl+ illustrates these rules and shows their usefulness for computational search and characterization of crossing seams.

Peculiarities for domain walls in Taub coordinates

In the present letter the infinite domain wall geometry in GR (Vilenkin in Phys Rev D 23:852, 1981; Vilenkin in Phys Lett B 133:177, 1983; Ipser and Sikivie in Phys Rev D 30:712, 1984) is reconsidered in Taub coordinates (Taub in Ann Math 53:472, 1951). The use of these coordinates makes explicit that the regions between the horizons and the wall and the outer ones are flat. By use of these coordinates, it is suggested that points inside the horizon and outside never communicate each other. The wall is seen on the left and the right side as contracting and expanding portions of spheres and a plane singularity, which is the imprint the contracting and expanding domain wall. Particles of each region will never reach this imprint. In addition, at some point during the evolution of the system, four curious holes inside the space time appear, growing at the speed of light. This region is not parameterized by the standard Taub coordinates, and the boundary of this hole adsorbs all the particles that intersect it. The boundary of these holes are composed by points which in the coordinates of [1–3] are asymptotic, in the sense that they correspond to trajectories tending to infinite values of the time or space like coordinates, while the proper time elapsed for the travel is in fact finite. This is not paradoxical, as the coordinates [1–3] are not to be identified with the true lengths or proper time on the space time. The correct interpretation of the boundary is particularity relevant when studying scattering of quantum fields approaching the domain wall. A partial analysis about this issue is done in the last section.

A novel way to build expert systems with infinite-valued attributes

<abstract><p>An expert system is a computer program that uses the knowledge of an expert to solve problems in a specific domain. Expert systems are used in a wide variety of fields, such as medicine, financial diagnosis and engineering. The attributes of an expert system are the characteristics of the problems that the system can solve. In traditional expert systems, attributes typically have a finite number of possible values. However, in scenarios where an attribute can assume a value from an infinite (or significantly large finite) set, the expert system cannot be represented using propositional logic. Until now, no method had been identified to implement such a system on a Computer Algebra System. Here, we break new ground by presenting a model that not only addresses this gap but also provides a fresh perspective on previous results. In fact, these prior results can be viewed as specific instances within the broader framework of our proposed solution. In this paper, we put forth an algebraic approach for the development of expert systems capable of handling attributes with infinite values, thereby expanding the problem-solving capacity of these systems.</p></abstract>

Локальная предельная теорема для ветвящихся случайных систем Гальтона–Ватсона с иммиграцией и бесконечной дисперсией

The paper considers a stochastic population growth model, which is the critical Galton-Watson stochastic branching system allowing immigration. We consider the case when the mathematical expectation of the law of influx of immigrant particles and the variance of the law of reproduction of aboriginal particles have infinite values. The asymptotic properties of transition probabilities are studied in the case when the states of the system are irreversible. For this case, a local limit theorem has been established.

A New Optimization Approach for the Tallest Column Design

Motivated by the classical brachistochrone we present a new pattern of finding the shape of a vertical column that attains the maximum height if its material and volume are prescribed. It comprises the optimal control problem with a free end point. Besides, the constitutive equation of the column is such that it can suffer flexure, compression and shear. The critical load of a heavy compressed column for the finite values of shear and extensional rigidity and the novel use of the Pontryagin maximum principle with the corresponding first integral yielding the non-vanishing optimal cross section as a solution of a quadratic equation are main novelties of this work. The classical solution for the tallest column under selfweight is covered as a special case for infinite values of shear and extensional rigidity.

Holographic RG flows from four-dimensional N = 2 gauged supergravities

In quantum field theories, the examination of particle interactions is facilitated through the application of Feynman diagrams. The computation of loop Feynman diagrams often yields divergent or infinite values. To tackle this issue, a method known as renormalization is employed, which entails the removal of these infinities. The phenomenon of a physical system undergoing changes under different scales gives rise to a concept referred to as the renormalization group. Certain renormalization group trajectories, denoted as RG flows, describe transformations of a conformal field theory (CFT) into other conformal or non-conformal theories. These transformations result in deformations of a UV conformal fixed point into another fixed point or into a non-conformal phase within the infrared (IR) regime. In this study, we investigate holographic RG flows originating from N = 2 gauged supergravity with an SO(2) × SO(6) gauge group. These solutions delineate RG flows from the N = 2 CFT to a three-dimensional non-conformal field theory driven by mass deformations. This transformation is elucidated through the AdS/CFT correspondence, also known as AdS/CFT holography.

Model-free prediction of time series: a nonparametric approach

We propose a novel approach for model-free time series forecasting. Unlike most existing methods, the proposed method does not rely on parametric error distributions nor assume parametric forms of the mean function, leading to broad applicability. We achieve such generality by establishing a simple but powerful representation of a time series { X t ; t ∈ Z } with sup t E | X t | < ∞ , that is, X t has a solution which is a linear combination of infinite past values. Then using the obtained solution a prediction algorithm is presented, with large sample theoretical guarantees. Simulation studies show favourable performance of the proposed method compared with popular parametric and neural networks methods, and suggest its superiority when the sample size is small. An application to practical time series is discussed.

Γ-convergence of nonconvex unbounded integrals in strongly connected sets

We study Γ-convergence of nonconvex integrals of the calculus of variations in strongly connected sets when the integrands do not have polynomial growth and can take infinite values. Applications to homogenization of unbounded integrals in strongly perforated sets are also developed.

Influence of temperature and anion type on thermophysical properties of aqueous solutions of morpholine based amino acid ionic liquids

Densities and sound velocities of aqueous solutions of N-butyl-N-methylmorpholine based amino acid ionic liquids (AAILs), including N-acetyl-L-alanine, N-acetyl-L-valinate, N-acetyl-L-leucinate, and N-acetyl-L-izoleucinate anions were measured at a temperature from 293.15 to 313.15 K at 5 K intervals and atmospheric pressure. These data were used to calculate the apparent molar volumes and the apparent molar compressibilities in the concentration range of (0.02 to 0.3) mol.kg−1. Infinite dilution values for the volumetric and acoustic properties were evaluated by an extrapolation procedure based on Redlich – Meyer type equations. The limiting apparent molar expansibility coefficients were also obtained from the slopes of the apparent molar volume at infinite dilution versus temperature. Using the Passynski method, the hydration numbers for AAILs in an aqueous medium were calculated. The results indicate that in the studied solutions, the ion–solvent interactions are determined by the hydrophobic effect, which results in the decrease of the limiting apparent molar compressibilities and an increase of limiting apparent molar expansibility coefficients in the order: [Mor1,4][N-Ac-L-Ala], [Mor1,4][N-Ac-L-Val], [Mor1,4][N-Ac-L-Leu], [Mor1,4][N-Ac-L-Izoleu]. The empirical slopes in Redlich –Rosenfeld – Meyer equations show that the hydrophobicity of amino acid anions is also responsible for the ion-ion interactions.

Thermodynamic insights into Trans-Aconitate interactions with H+, Cd2+, Mn2+, and Pb2+: Equilibrium constants, enthalpy changes and sequestering ability

The thermodynamics of interaction of trans‐aconitate (L3-) with proton, sodium and potassium cations was studied by means of potentiometric titration performed at different temperatures, ionic strengths, and aqueous ionic media (NaCl, KCl and (C2H5)4NI). Three protonation constants and corresponding enthalpy changes are reported at infinite dilution together with their dependence parameters on temperature and ionic strength according to van’t Hoff and Extended Debye Hückel equations, respectively. Enthalpy changes resulted slightly endothermic at infinite dilution and values decrease with increasing ionic strength. Proton binding resulted always entropic in nature. Weak association constants of Na+/L3- and K+/L3- species were determined using the pure water model approach. Formation constants of trans‐aconitate with Cd2+, Pb2+ and Mn2+ were determined in KCl(aq) at different ionic strength values and at 298.15 K. Three complex species were found with all metal cations (ML-, MHL0(aq), MH2L+), whose formation constant values at infinite dilution and at T = 298.15 K were log β = 4.54, 9.94, 13.77 for cadmium, 5.00, 10.57, 14.81 for manganese and 4.97, 10.44, 14.13 for lead. The as found values resulted unexpectedly high. The sequestering ability of trans‐aconitate towards M2+ was evaluated by determining pL0.5 (the ligand total concentration required to bind 50% of the metal cation), and the results show that, throughout the investigated pH range, trans‐aconitate shows the highest sequestering ability towards Mn2+. Data reported in this paper were critically compared to other tricarboxylic acid, namely citric and tricarballylic acid.

Maximum softly-penalized likelihood for mixed effects logistic regression

Maximum likelihood estimation in logistic regression with mixed effects is known to often result in estimates on the boundary of the parameter space. Such estimates, which include infinite values for fixed effects and singular or infinite variance components, can cause havoc to numerical estimation procedures and inference. We introduce an appropriately scaled additive penalty to the log-likelihood function, or an approximation thereof, which penalizes the fixed effects by the Jeffreys’ invariant prior for the model with no random effects and the variance components by a composition of negative Huber loss functions. The resulting maximum penalized likelihood estimates are shown to lie in the interior of the parameter space. Appropriate scaling of the penalty guarantees that the penalization is soft enough to preserve the optimal asymptotic properties expected by the maximum likelihood estimator, namely consistency, asymptotic normality, and Cramér-Rao efficiency. Our choice of penalties and scaling factor preserves equivariance of the fixed effects estimates under linear transformation of the model parameters, such as contrasts. Maximum softly-penalized likelihood is compared to competing approaches on two real-data examples, and through comprehensive simulation studies that illustrate its superior finite sample performance.

Exploring the Art of Folk Music Class in Northern Anhui Province by Taking "Zhoujiaban and Spinelim Horn" as an Example Cultural Value

The spinach horn has been inherited by the Zhou family class until now, and the Zhou family class has played a vital role in spreading the folk music culture of the spinach horn in northern Anhui. This paper explores the musical and cultural values and characteristics of the spinach horn through the investigation of the historical origin and development of the Zhou family class, so as to lay the foundation for the future inheritance and development of the spinach horn. In the past hundred years, the spinach horn has gradually changed from a single performance form to a diversified performance form under the inheritance of the Zhou family class, and nowadays the spinach horn is closely related to folklore and the society of the times; from a single fixed arrangement of musical instruments to the integration of modern instruments such as western instruments as auxiliary performance. The spinach forest horn is being understood, accepted and loved by more and more people. The spinach forest horn is the inheritance of regional music culture and the crystallization of national music culture, which contains infinite cultural values.

Numerical and analytical solutions for describing the mechanical response of segment joints for tunnels

In literature there are several analytical models for describing the mechanical behavior of non-bolted segment joints (longitudinal joints) of shield driven tunnels. These joints modify the structural response of the segmental lining and therefore they affect the global response of this kind of tunnels. The rotational stiffness of these joints is divided into linear and multi-linear models being two typical moment-rotation constitutive equations those exposed by Janssen and Gladwell. The first one considers a linear stress distribution in the longitudinal joint. However, this is not correct according to elasticity theory due to at the edge of the contact area stresses reach infinite values and non-linear stress distributions are produced at the joint. The second typical moment-rotation equation is based on elasticity theory. The analytical results obtained from these moment-rotation equations were compared to numerical results of a 3D model based on Finite Element Method (FEM) using isoparametric solid elements for modeling the concrete segments, whilst for simulating the joint were incorporated contact elements; achieving to confirm that the constitutive equation proposed by Gladwell provides results more precise of the mechanical response of these joints achieving a difference of 0.07% in the maximum capacity of bending moment between the analytical model proposed by Gladwell and 3D numerical model carried out in this study.

Synergistic determination of mode I stress intensity factors using photoelastic experimental hybrid method

AbstractEngineering components possess discontinuities such as cracks which develop during service while others exist due to structural and material defects. The physical discontinuities in mechanical structures lead to stress concentrations, which can trigger mechanical collapses and catastrophic failures of the structure. Stress field equations of fracture mechanics have been formulated to aid in determination of the critical fracture mechanics parameter namely; the stress intensity factors (SIFs) ahead of cracks. Mathematical formulations for fracture mechanics, however, are known to predict infinite values of stress near the crack tip because of the stress singularity at the crack tip and therefore, provide limited information. Further interpretations and a combination of results from different techniques are frequently required to obtain the complete stress condition. In order to accurately determine mode I stress intensity factors without reverting to supplemental techniques, photoelastic experimental hybrid method (PEHM) is utilized in this research. Hybrid methods synergizing mathematical, numerical and experimental data have been used with great success in experimental stress analysis. Its application in the current fracture mechanics problem shows that photoelastic experimental hybrid method stress intensity factors correspond well with those calculated from theoretical method. In addition, it is noted that varying the geometric condition 2a/W from 0.2 to 0.6 while maintaining constant load had the effect of changing K1/K0 from 1.1 to 1.5.