Representation Of Function Research Articles

Numerical version of complex scaling Green operator method

Abstract A new treatment of scattering problems is formulated by improving the complex scaling Green operator (CSGO) method, which can calculate the transition or scattering matrix from the spectral representation of the complex-scaled Green function and the asymptotic solution expressed in terms of the analytic function. In the present formulation, the CSGO method is extended to handle the general scattering problem in the charged particle systems by employing the Coulomb wave function numerically calculated under the complex-scaled asymptotic Hamiltonian. We call the new method the CSGO-numerical (CSGON) method, and the CSGON method is applied to the scattering problem of the 4He + proton system to check its validity. The method of the continuum level density (CLD), which is another method to solve the scattering problem, is also applied to the same system. The numerical precision and the applicability of the CSGON method is discussed by comparing with the CLD and CSGO methods.

Solution of Schrödinger Equation for Simple Diatomic Molecules Using One‐Parameter 1s Slater‐Type Orbitals Wave Function

ABSTRACTThe Schrödinger equation for simple homonuclear and heteronuclear diatomic molecules is analytically solved using one‐parameter Slater‐type orbitals (STOs) to approximate the electronic wave function within a molecular orbital (MO)‐like approach. The resulting total energies, equilibrium bond lengths, potential curves, and electron densities are presented in detail. Calculations using a selected orbital exponent accurately reproduce results from standard methods. Furthermore, the optimization of the orbital exponent allows for a more accurate representation of the electronic wave function, leading to the improved results of the total energy and the equilibrium bond length, as well as minimal computational cost. Seen in the heteronuclear diatomic molecule, the use of the one‐parameter STOs allows the transformation of the heteronuclear problem into the homonuclear one, revealing the electron repulsion effect on the orbital exponent parameter.

About Filtration in a Geophysical Bridge with a Seepage Site

Based on the representation of a modular elliptic function in the form of a combination of simple algebraic formulas that conformally map the area of the velocity hodograph (curved triangle) onto a half-plane, a direct definition of filtration rates in a geophysical bridge is given. For the first time, a family of isotope lines of equal filtration rates was constructed for the inner area of the bridge in the absence of water in the downstream. For special cases, the values of the proposed formulas almost completely coincide with the hydromechanical solution of Masket M. (for 4 cases) and the numerical calculations of Khairullin Z.E. (for 2 cases). The well-known Nelson-Skornyakov F.B. formula for the output filtration rates through the lower face of the jumper, adopted by analogy with the flow outlet from the base of the flatbed into the horizontal drainage, gives an underestimation of the results by up to 45%, and therefore cannot be recommended for practical use.

A new Marshall-Olkin generalized-k family of distributions with applications

Abstract In this article, we describe a new Marshall-Olkin family of distributions based on the Marshall-Olkin family of distributions by employing a different generator. We have derived several statistical properties of the new Marshall-Olkin family, including survival function, hazard rate function, quantile function, moments, and linear representation. We have taken the Weibull distribution as a baseline distribution and found that the new Marshall-Olkin Weibull distribution exhibits a variety of shapes for its hazard rates and densities. We have tested the performance of various estimation methods for the new Marshall-Olkin family of distributions, including eight different estimation methods. Additionally, we conducted extensive simulations to assess the robustness of the estimation methods across various scenarios and determine which methods are most effective for varying sample sizes. To further validate the new Marshall-Olkin Weibull distribution, we have used two different data sets to compare the fitted new Marshall-Olkin Weibull distribution with the well-known comparative models. By testing the performance of different estimation methods and using multiple data sets, we can demonstrate the versatility and usefulness of the new Marshall-Olkin family of distributions for modeling a wide range of phenomena.

Initial tensor construction and dependence of the tensor renormalization group on initial tensors

We propose a method to construct a tensor network representation of partition functions without singular value decompositions nor series expansions. The approach is demonstrated for one- and two-dimensional Ising models and we study the dependence of the tensor renormalization group (TRG) on the form of the initial tensors and their symmetries. We further introduce variants of several tensor renormalization algorithms. Our benchmarks reveal a significant dependence of various TRG algorithms on the choice of initial tensors and their symmetries. However, we show that the boundary TRG technique can eliminate the initial tensor dependence for all TRG methods. The numerical results of TRG calculations can thus be made significantly more robust with only a few changes in the code. Furthermore, we study a three-dimensional Z2 gauge theory without gauge fixing and confirm the applicability of the initial tensor construction. Our method can straightforwardly be applied to systems with longer range and multisite interactions, such as the next-nearest neighbor Ising model. Published by the American Physical Society 2024

Lippmann-Schwinger equation representation of Green's function and its preconditioned generalized over-relaxation iterative solution in wavelet domain

The calculation of Green's function is the core of seismic forward and inverse methods based on integral operators. When the Lippmann-Schwinger (L-S) equation is used to calculate Green's function in strongly scattering media, both the Born scattering series and the numerical iterative method encounter issues of slow convergence or divergence. Although the renormalization method derived from quantum mechanics can effectively address the convergence problem of Born scattering series in strong scattering problems, it is acknowledgeed that the convergence conditions and rates of convergence of different reformulation series may vary, and no universal convergence reformulation scattering series exists. Numerical methods for solving integral equations tend to be more general and mathematically robust. In this work, we focus on the numerical solution method of L-S equations. By using a wavelet-domain preconditioner to a reformulated or equivalent Lippmann-Schwinger (L-S) equation, we present an iterative method for numerically solving the equivalent L-S equation aimed at improving the rate of convergence and iteration efficiency in strongly inhomogeneous media. Following Jakobsen et al. (2020), we first introduce a small imaginary component into the background wave number，then rewrite the L-S equation to derive the equivalent complex wave number L-S equation. This reformulation ensures that the coefficient matrix exhibits a banded structure after numerical discretization, allowing the wavelet coefficient matrix to maintain good sparsity. We employ a multi-level fill-in incomplete LU (ILU) factorization method along with a block ILU-based algebraic recursive multilevel solve (ARMS) method in the wavelet domain to generate sparse approximate inverses as preconditioning operators, thereby accelerating the convergence of the generalized successive over-relaxation (GSOR) iterative method. This method is applied to compute numerical Green's functions in strongly inhomogeneous media. Numerical results demonstrate that our method yields simulation outcomes consistent with those obtained from the direct method for solving the original real wave number L-S equation. By testing various preconditioners, we find that the ARMS preconditioner offers significant advantages in operator generation efficiency and non-zero element filling ratio, effectively accelerating the convergence of the GSOR iterative method while achieving higher computational efficiency.

Application of Clifford’s Algebra to Describe the Early Universe

This article is a shortened review of previous results obtained by the author. The advantages of describing the geometric nature of the physical properties of the early universe using the Clifford algebra approach are demonstrated. A geometric representation of the wave function of the early universe is used, and a new mechanism of spontaneous symmetry breaking with different degrees of freedom is proposed. A possible supersymmetry is revealed, and it is shown that the energy of the initial vacuum can be considered equal to zero. The origin of baryonic asymmetry and the nature of dark matter can be explained using a geometric representation of the wave function of the early universe.

Examining high school students' conception of the function concept from a theoretical thinking perspective

This study explores the conceptions of functions among high school students through the lens of theoretical thinking, encompassing three core components: reflective, systemic and analytical thinking. A case study methodology was employed in this research. The data were gathered through interviews with 11 senior high school students and analysed using deductive content analysis techniques. The findings reveal that students needing more reflective thinking tend to resort to practical methods and quickly gave up when facing challenges. Those who showed limited systemic thinking often relied on incorrect or inadequate criteria to determine if an expression constituted a function and struggled to connect the function concept with other concepts. Students needing more analytical thinking faced challenges in employing mathematical language and representations for functions, verifying if an expression is a function or transforming representations and frequently overlooked quantifiers or logical connectors related to variables in definitional properties. In this study, the theoretical thinking framework, mainly used for university-level mathematical concepts, provided a broad perspective in determining high-school students' understanding of the concept of function. Future research should extend this approach to various educational levels and concepts to enhance our understanding of students' conceptions and enrich the theoretical thinking framework.

Extremal Functions and Calderon’s Formulas for the Riemann-Liouville Two-wavelet Transform

The Riemann-Liouville operator has been extensively investigated and his witnessed a remarkable development in numerous fields of harmonic analysis. Knowing the fact of the study of the time-frequency analysis are both theoritically interesting and pratically useful, we investigated several problems for this subject on the setting of the Riemann-Liouville wavelet transform. Firstly, we introduce the notion of Riemann-Liouville two-wavelet and we present generalized version of Parseval’s, Plancherel’s, inversion and Calderon’s reproducing formulas. Next, using the theory of reproducing kernels, we give best estimates and an integral representation of the extremal functions related to the Riemann-Liouville wavelet transform on weighted Sobolev spaces.

On the computation of lattice sums without translational invariance

This paper introduces a new method for the efficient computation of oscillatory multidimensional lattice sums in geometries with boundaries. Such sums are ubiquitous in both pure and applied mathematics and have immediate applications in condensed matter and topological quantum physics. The challenge in their evaluation results from the combination of singular long-range interactions with the loss of translational invariance caused by the boundaries, rendering standard tools like Ewald summation ineffective. Our work shows that these lattice sums can be generated from a generalization of the Riemann zeta function to multidimensional non-periodic lattice sums. We put forth a new representation of this zeta function together with a numerical algorithm that ensures exponential convergence across an extensive range of geometries. Notably, our method’s runtime is influenced only by the complexity of the considered geometries and not by the number of particles, providing the foundation for efficient and precise simulations of macroscopic condensed matter systems. We showcase the practical utility of our method by computing interaction energies in a three-dimensional crystal structure with 3 × 10 23 3\times 10^{23} particles. Our method’s accuracy is demonstrated through extensive numerical experiments. A reference implementation is provided online along with this article.

Integrity of central nervous autonomous tracts is associated with cardiac function, cardiovascular risk factors, organ damage and major adverse cardiovascular events

Abstract Background Alterations in the central autonomic nervous system have been linked to the pathogenesis of cardiovascular (CV) risk and disease. Moreover, these conditions are treatable by pharmacological or interventional modification of the activity of the autonomic nervous system. Purpose The UK Biobank enables the investigation of the relationship between changes in the CAN and CV function or disease within a large population. Microstructural integrity of the CAN was assessed for associations with cardiac function, CV risk and damage representations, and major adverse cardiovascular events (MACE). Method Microstructural diffusion metrics for 7 CAN tracts were estimated using a Bayesian approach, which determines the three components of a standard white matter-based tissue model: 1. the free water fraction (CSF); 2. the fraction within neuronal processes, i.e. axons and dendrites (INTRA); and 3. the fraction outside of axons or dendrites (EXTRA), representing the cellular compartment and the extracellular matrix. Volume fractions of the three microstructural components were analysed for associations with cardiac index, blood pressure, CV target organ damage (arterial stiffness, left ventricular mass, and wall thickness), and MACE, using covariate-adjusted regression models that corrected for multiple comparisons. Results Analysis of cardiac function in 26,302 participants revealed significant associations between cardiac index and CAN regions in the cortical and subcortical networks. Arterial stiffness was significantly associated with markers of microstructural integrity in the cortical networks only (n=22,491), while systolic (n=34,114) and diastolic blood pressure (n=34,130), left ventricular mass (n=22,491) and wall thickness (n=22,491) showed extensive associations with the cortical and subcortical CAN. MACE (n=33,444) was associated with cortical markers of microstructural integrity of the CAN in limbic pathways (significant CANs all p<0.002) and the CAN pathways connecting cortical areas (significant CANs left all <0.001, right=0.001). The observed pattern of microstructural metrics suggests that the integrity of neural connections (intra-fraction) positively correlates with adverse outcomes, while an increase in free fluid (CSF) volume was typically associated with a negative impact. Conclusion In this analysis, markers of CAN integrity and degradation showed extensive associations with indicators of CV function, risk factors, damage parameters, and outcomes. This evidence supports the crucial role of autonomic regulation in cardiovascular disease. Future research is needed to determine whether these associations indicate harmful processes, such as neuroinflammation, contributing to cardiovascular (CV) risk and disease development, or if they are consequences of pre-existing elevated risk factors or events.CANs associated with MACEAssociations of CAN with CV organ damage

Toden-E: Topology-Based and Density-Based Ensembled Clustering for the Development of Super-PAG in Functional Genomics using PAG Network and LLM.

The integrative analysis of gene sets, networks, and pathways is pivotal for deciphering omics data in translational biomedical research. To significantly increase gene coverage and enhance the utility of pathways, annotated gene lists, and gene signatures from diverse sources, we introduced pathways, annotated gene lists, and gene signatures (PAGs) enriched with metadata to represent biological functions. Furthermore, we established PAG-PAG networks by leveraging gene member similarity and gene regulations. However, in practice, high similarity in functional descriptions or gene membership often leads to redundant PAGs, hindering the interpretation from a fuzzy enriched PAG list. In this study, we developed todenE (topology-based and density-based ensemble) clustering, pioneering in integrating topology-based and density-based clustering methods to detect PAG communities leveraging the PAG network and Large Language Models (LLM). In computational genomics annotation, the genes can be grouped/clustered through the gene relationships and gene functions via guilt by association. Similarly, PAGs can be grouped into higher-level clusters, forming concise functional representations called Super-PAGs. TodenE captures PAG-PAG similarity and encapsulates functional information through LLM, in characterizing network-based functional Super-PAGs. In synthetic data, we introduced a metric called the Disparity Index (DI), measuring the connectivity of gene neighbors to gauge clusterability. We compared multiple clustering algorithms to identify the best method for generating performance-driven clusters. In non-simulated data (Gene Ontology), by leveraging transfer learning and LLM, we formed a language-based similarity embedding. TodenE utilizes this embedding together with the topology-based embedding to generate putative Super-PAGs with superior performance in semantic and gene member inclusiveness.

Optogenetic stimulation recruits cortical neurons in a morphology-dependent manner.

Single-photon optogenetics enables precise, cell-type-specific modulation of neuronal circuits, making it a crucial tool in neuroscience. Its miniaturization in the form of fully implantable wide-field stimulator arrays enables long-term interrogation of cortical circuits and bares promise for Brain-Machine Interfaces for sensory and motor function restoration. However, achieving selective activation of functional cortical representations poses a challenge, as studies show that targeted optogenetic stimulation results in activity spread beyond one functional domain. While recurrent network mechanisms contribute to activity spread, here we demonstrate with detailed simulations of isolated pyramidal neurons from cat of unknown sex that already neuron morphology causes a complex spread of optogenetic activity at the scale of one cortical column. Since the shape of a neuron impacts its optogenetic response, we find that a single stimulator at the cortical surface recruits a complex spatial distribution of neurons that can be inhomogeneous and vary with stimulation intensity and neuronal morphology across layers. We explore strategies to enhance stimulation precision, finding that optimizing stimulator optics may offer more significant improvements than preferentially somatic expression of the opsin through genetic targeting. Our results indicate that, with the right optical setup, single-photon optogenetics can precisely activate isolated neurons at the scale of functional cortical domains spanning several hundred micrometers.Significance Statement Sensory features, such as the position or orientation of a visual stimulus, are mapped onto the surface of cortex as functional domains. Their selective activation, that may enable eliciting complex percepts, is intensively pursued for basic science and clinical applications. However, delivery of light into one functional domain in optogenetically transfected cortex results in complex, widespread neuronal activity, spreading beyond the targeted domain. Our computational study reveals that neuron morphology contributes to this diffuse response in a cortical-layer and intensity-dependent manner. We show that enhancing the stimulator optics is more effective than soma-targeting of the opsin in increasing spatial precision of stimulation. Our simulations provide insights for designing optogenetic stimulation protocols and hardware to achieve selective activation of functional domains.

Overcoming the streetlight effect: Shining light on the foundations of learning and development in early childhood.

Developmental theory has long emphasized a range of skills that young children need for healthy development across the life course. Nevertheless, most evaluations of early childhood programs and policies have focused on measuring a somewhat limited set of competencies. In this article, we explore this "streetlight effect" in early childhood intervention research and propose an initial set of skills that we argue should be prioritized alongside traditionally measured outcomes as targets of intervention during the preschool period (i.e., between ages 3 and 5 years). These skills, which we call the foundations of learning and development (FOLD) skills, include both well-studied and emerging constructs such as curiosity, creativity, self-regulation and executive function, critical thinking, perspective taking, and internal representations of self. To better understand FOLD skills' potential as more practical, effective, and inclusive targets of early childhood programs and policies, we review research regarding each skill's malleability, measurability, predictive validity, and universality. We end with a set of future directions for the field, including the need to (a) formulate a more inclusive taxonomy of FOLD skills that incorporates currently omitted competencies relevant to marginalized populations, (b) measure these skills in scalable and actionable ways, and (c) enhance or modify intervention strategies to optimize the development of these FOLD skills in the preschool period. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Harmonic flow-field representations of quantum bits and gates

We describe a general procedure for mapping arbitrary n-qubit states to two-dimensional (2D) vector fields. The mappings use complex rational function representations of individual qubits, producing classical vector field configurations that can be interpreted in terms of 2D inviscid fluid flows or electric fields. Elementary qubits are identified with localized defects in 2D harmonic vector fields, and multiqubit states find natural field representations via complex superpositions of vector field products. In particular, separable states appear as highly symmetric flow configurations, making them both dynamically and visually distinct from entangled states. The resulting real-space representations of entangled qubit states enable an intuitive visualization of their transformations under quantum logic operations. We demonstrate this for the quantum Fourier transform and the period finding process underlying Shor's algorithm, along with other quantum algorithms. Due to its generic construction, the mapping procedure suggests the possibility of extending concepts such as entanglement or entanglement entropy to classical continuum systems, and thus may help guide new experimental approaches to information storage and nonstandard computation. Published by the American Physical Society 2024

Logics for at most countable first-order structures

Abstract In this paper we present two extensions of $\omega $-logic with infinitary inference rules, denoted $Arch$-$\omega $-logic and $non$-$Arch$-$\omega $-logic. We provide the corresponding Hilbert-style axiomatizations and prove their strong completeness with respect to countable Archimedean and non-Archimedean fields, respectively. Through several examples we illustrate a natural representation of various weight functions within the proposed framework and applications to non-monotonic reasoning and neuro-symbolic computing.

A note on two new integral representations of the Gauss hypergeometric function with an application

A note on two new integral representations of the Gauss hypergeometric function with an application

Herglotz's representation and Carathéodory's approximation

AbstractHerglotz's representation of holomorphic functions with positive real part and Carathéodory's theorem on approximation by inner functions are two well‐known classical results in the theory of holomorphic functions on the unit disc. We show that they are equivalent. On a multi‐connected domain , a version of Heglotz's representation is known. Carathéodory's approximation was not known. We formulate and prove it and then show that it is equivalent to the known form of Herglotz's representation. Additionally, it also enables us to prove a new Heglotz's representation in the style of Korányi and Pukánszky. Of particular interest is the fact that the scaling technique of the disc is replaced by Carathéodory's approximation theorem while proving this new form of Herglotz's representation. Carathéodory's approximation theorem is also proved for operator‐valued functions on a multi‐connected domain.

Neuronal threshold functions: Determining symptom onset in neurological disorders

Synaptic networks determine brain function. Highly complex interconnected brain synaptic networks provide output even under fluctuating or pathological conditions. Relevant to the treatment of brain disorders, understanding the limitations of such functional networks becomes paramount. Here we use the example of Parkinson’s Disease (PD) as a system disorder, with PD symptomatology emerging only when the functional reserves of neurons, and their interconnected networks, are unable to facilitate effective compensatory mechanisms. We have denoted this the “threshold theory” to account for how PD symptoms develop in sequence. In this perspective, threshold functions are delineated in a quantitative, synaptic, and cellular network context. This provides a framework to discuss the development of specific symptoms. PD includes dysfunction and degeneration in many organ systems and both peripheral and central nervous system involvement. The threshold theory accounts for and explains the reasons why parallel gradually emerging pathologies in brain and peripheral systems generate specific symptoms only when functional thresholds are crossed, like tipping points. New and mounting evidence demonstrate that PD and related neurodegenerative diseases are multisystem disorders, which transcends the traditional brain-centric paradigm. We believe that representation of threshold functions will be helpful to develop new medicines and interventions that are specific for both pre- and post-symptomatic periods of neurodegenerative disorders.

A new approach to MacMahon’s Partition Analysis and associated applications

We introduce a new elimination procedure for Partition Analysis to obtain generating functions free of Omega variables starting with a crude generating function containing or not factored polynomials. In contrast to Han’s approach (Ann Comb. 2003;7(4):467–480), we work with factored polynomials by judiciously constructing the quotient of a series modulo a certain prime ideal. Our results include diverse areas such as an Omega-based approach to compute the Hadamard product of power series and certain trigonometric integrals generalizing the main result of Kim (J Integer Seq. 2009;12:09.7.4) and recovering an integral identity of Boros and Moll (J Comput Appl Math. 1999;106(2):361–368), respectively. Furthermore, we construct a new Omega representation of analytic matrix functions which requires no spectral information and uses the matrix rank to simplify the calculations. By specializing to tridiagonal matrices, the main result of Kiliç (Appl Math Comput. 2008;197(1):345–357) is obtained.