Discrete Functions Research Articles

Perceptual effects of lexical competition on Cantonese tone categories

Listeners use lexical information and the speech signal to categorize sounds and recognize words despite substantial acoustic-phonetic variation in natural speech. In diachronic mergers, where systematic variation acts to neutralize lexical contrasts, the role of the lexicon becomes less clear. We examined how lexical competition structures phonetic variability of (merging) lexical tone categories in Cantonese using three experiments. Listeners categorized tokens from lexical tone continua generated from minimal pairs (Experiment 1: Word identification) and categorized tokens from tone continua generated from word-nonword pairs (Experiment 2: Lexical decision). The presence of a lexical competitor at both continuum endpoints in Experiment 1 maintained more discrete categorization functions for non-merging tone pairs than in Experiment 2 where only one endpoint was a word. In the merging tone pairs, categorization was less discrete and the effect of lexical competition was different. Exploratory data from a goodness rating task, Experiment 3, suggest that lexical competition affects internal category structure for merging tones, but not non-merging tones. Overall, these data provide evidence that tone mergers affect phonetic category boundaries and internal category structure in the lexicon, and that, for non-merging tones, the range of acceptable phonetic variation is constrained by the presence of a lexical competitor.

Abstract 2229: Cell state transitions shape the intratumoral composition of small cell lung carcinoma

Abstract Introduction: Small cell lung carcinoma (SCLC) is characterized by rapid growth, early metastases, and initial response followed by almost invariable resistance to therapy. Studies to date have not determined the extent that diverse transcriptional programs drive SCLC and contribute to its lethality. We sought to characterize the intra-tumoral transcriptional heterogeneity of SCLC. We identify multivalent, distinct, and commutable transcriptional states that confer discrete functions in individual SCLC tumors. Methods: We developed a biorepository of patient-derived xenografts (PDX) (n = 64) and matched PDX-derived ex vivo lines. We used multi-omic profiling (RNAseq, scRNAseq, and ATAC seq), single-cell fluorescence tracking of fate-defining transcription factor (TF)-driven states, and mathematical and statistical models (Markov chain) to study the topology of the SCLC transcriptional landscape and its plasticity. Human tumor material and associated clinical data were obtained after informed written consent on an IRB-approved prospective registry. Results: We show that individual SCLC tumors are more heterogenous than previously appreciated, displaying distinctive equilibria in the proportion of cells within well-delimited cellular states (ASCL1, NEUROD1 and YAP1). We also show that transcriptional states undergo transitions, which we identified as a mechanism for maintaining cell state diversity. We measured the kinetics of state transitions using single-cell fluorescence tracking of ex vivo cultures and found that these measure were associated significantly with transition estimates using stochastic transition theory (i.e. Markov chains). ATAC-seq profiling indicated a role for the epigenome in the state diversity of SCLC. Namely, there was preferential promoter accessibility to Ascl1, NeuroD1, and Yap1 in a manner consistent with gene and protein expression in the respective subpopulations. Our results indicate that the transition rates between cell types in individual tumors were largely governed by tendencies to reach an equilibrium state that are critical for configuring intratumoral cell state proportions. Conclusion: In conclusion, we demonstrate that TF driven cell states can transition to maintain an equilibrium in cell state proportions. Our work advances a model of cellular states and program diversity in SCLC and nominates new therapeutic strategies designed to limit the plasticity of this lethal cancer. Citation Format: Priyanka Gopal, Aaron Petty, Kevin Rogacki, Titas Bera, Rohan Bareja, Craig Peacock, Mohamed Abazeed. Cell state transitions shape the intratumoral composition of small cell lung carcinoma [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2023; Part 1 (Regular and Invited Abstracts); 2023 Apr 14-19; Orlando, FL. Philadelphia (PA): AACR; Cancer Res 2023;83(7_Suppl):Abstract nr 2229.

Forecasting extreme negative returns in gold and silver: A discrete‐duration approach to POT models

AbstractSudden and massive drops in precious metal prices severely impact the investment risk on these commodity markets. In this study, we examine the dynamics of extreme negative returns on gold and silver, as well as propose the discrete‐duration version of the autoregressive conditional duration peaks‐over‐threshold (ACD‐POT) model for measuring market risk. The model is tailored for the dynamics of extreme events in precious metal markets; this is because it can exhibit both the strong clustering of extreme‐event days, and the serial correlation in the magnitudes of extreme losses. From an econometric perspective, our approach complements the existing dynamic versions of POT models; this is so that the time interval (i.e., duration) between the extreme negative returns is treated not as a continuous variable, but as a discrete one. The discrete hazard function in our model represents the daily extreme loss event probability and it can be used to derive one‐day‐ahead forecasts of both the value at risk and expected shortfall for investments in gold and silver. Formal backtesting methods show that the discretized version of the POT model has a superior in‐sample fit and forecasting performance than the continuous‐duration POT models.

Discrete diffusion-type equation on regular graphs and its applications

ABSTRACT We derive an explicit formula for the fundamental solution to the discrete-time diffusion equation on the -regular tree in terms of the discrete I-Bessel function. We then use the formula to derive an explicit expression for the fundamental solution to the discrete-time diffusion equation on any -regular graph X. Going further, we develop three applications. The first one is to derive a general trace formula that relates the spectral data on X to its topological data. Though we emphasize the results in the case when X is finite, our method also applies when X has a countably infinite number of vertices. As a second application, we obtain a closed-form expression for the return time probability distribution of the uniform random walk on any -regular graph. The expression is obtained by relating to the uniform random walk on a -regular graph. We then show that if is a sequence of -regular graphs whose number of vertices goes to infinity and which satisfies a certain natural geometric condition, then the limit of the return time probability distributions from is equal to the return time probability distribution on the tree . As a third application, we derive formulas which express the number of distinct closed irreducible walks without tails on a finite graph X in terms of moments of the spectrum of its adjacency matrix.

Unimodular Perfect and Nearly Perfect Sequences: a Variation of Björck’s Scheme

Constant Amplitude (CA), Zero Auto Correlation (ZAC) sequences (or CAZAC sequences, aka perfect sequences) have numerous applications. We generalize the CAZAC notion to what we term as CASAC by permitting small autocorrelations (SAC). We extend Björck’s classification result of two-valued CAZAC sequences by providing a complete classification of all almost 2-valued (i.e., two-valued except for the first position which uses a third value) CASAC sequences. While Björck’s original work dealt only with primes p, we extend his ideas to any abelian group of order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$v\equiv 1\pmod {4}$ </tex-math></inline-formula> , as opposed to restricting just to the prime fields GF(p). Björck sequences have better ambiguity function than Zadoff-Chu sequences, making them suitable for radar and communications applications in the presence of high Doppler shifts. In fact, the discrete narrow band ambiguity function has an optimal bound in case of Björck sequences (as opposed to Gauss sequences). A one-parameter infinite family of CASAC we construct would have applications in Multiple-Input Multiple-Output (MIMO) areas. Toward MIMO applications, we introduce a performance measure we term as cross merit factor to study cross correlation behavior, generalizing the well-known notion of Golay Merit Factor (GMF).

Photometric Monitoring of Blazar 3C 66A with the Yunnan University Astronomical Observatory 1 m Telescope

3C 66A is one of our first batches of photometric monitoring objects with the 1 m optical telescope at Yunnan University Astronomical Observatory. In the present work, the observational campaign was performed from 2021 November 1 to 2022 February 27 in the Johnson–Morgan system V and R bands. The average magnitudes in each band were mag and mag. The overall variability amplitudes were ΔV = 0.ͫ74, Amp = 70.27% and ΔR = 0.ͫ72, Amp = 68.56%, respectively. Most of the intraday variabilities (IDVs) occurred in 2021 December and 2022 February. The minimal rise/decay timescale was about 6 minutes (5.82 ± 2.74 minutes and 6.18 ± 2.81 minutes on 2022 February 11, 6.99 ± 3.70 minutes and 6.17 ± 2.91 minutes on 2022 February 12). Durations of these rapid variabilities were from 11.99 to 179.67 minutes. The discrete correlation function analyses between V and R bands showed significantly correlated variability. Color index analysis of IDVs showed that the spectrums do not change with variabilities.

Continuum reverberation mapping of MCG 08-11-011

Aims. We report the results from a photometric reverberation mapping campaign carried out with the C18 telescope at the Wise Observatory from 2019 to 2020, targeting the active galactic nucleus (AGN) MCG 08-11-011. The monitoring was conducted on a daily basis with specially designed narrow-band filters, spanning from optical to near-infrared wavelengths (∼4000 to 8000 Å) and avoiding prominent broad emission lines. We aim to measure inter-band continuum time lags, determine the size–wavelength relation, and estimate the host-subtracted AGN luminosity for this system. Methods. We used the point-spread function photometry to extract the continuum light curves and measure the inter-band time lags using several methods, including the interpolated cross-correlation function, the z-transformed discrete correlation function, a von Neumann estimator, JAVELIN (in spectroscopic and photometric mode), MICA, and a multivariate correlation function. Results. We find wavelength-dependent lags, τ(λ), up to ∼7 days between the multiband light curves of MCG 08-11-011. The observed lags are larger than predictions based on standard thin-disk theory by a factor of ∼3 − 7. We discern a significantly steeper (τ ∝ λ4.74) size-wavelength relation than the τ ∝ λ4/3 expected for a geometrically thin and optically thick accretion disk, which may result from the contribution of diffuse continuum emission to the flux. These results are similar to those found by previous continuum reverberation mapping campaigns.

Transition Path Theory for Langevin Dynamics on Manifolds: Optimal Control and Data-Driven Solver

We present a data-driven point of view for rare events, which represent conformational transitions in biochemical reactions modeled by overdamped Langevin dynamics on manifolds in high dimensions. We first reinterpret the transition state theory and the transition path theory from the optimal control viewpoint. Given a point cloud probing the manifold, we construct a discrete Markov chain with a -matrix computed from an approximated Voronoi tesselation via the point cloud. We use this -matrix to compute a discrete committor function whose level set automatically orders the point cloud. Then based on the committor function, an optimally controlled random walk on point clouds is constructed and utilized to efficiently sample transition paths, which become an almost sure event in time instead of a rare event in the original reaction dynamics. To compute the mean transition path efficiently, a local averaging algorithm based on the optimally controlled random walk is developed, which adapts the finite temperature string method to the controlled Monte Carlo samples. Numerical examples on sphere/torus including a conformational transition for the alanine dipeptide in vacuum are conducted to illustrate the data-driven solver for the transition path theory on point clouds. The mean transition path obtained via the controlled Monte Carlo simulations highly coincides with the computed dominant transition path in the transition path theory.

Structural insights into the multifunctionality of rabies virus P3 protein

Viruses form extensive interfaces with host proteins to modulate the biology of the infected cell, frequently via multifunctional viral proteins. These proteins are conventionally considered as assemblies of independent functional modules, where the presence or absence of modules determines the overall composite phenotype. However, this model cannot account for functions observed in specific viral proteins. For example, rabies virus (RABV) P3 protein is a truncated form of the pathogenicity factor P protein, but displays a unique phenotype with functions not seen in longer isoforms, indicating that changes beyond the simple complement of functional modules define the functions of P3. Here, we report structural and cellular analyses of P3 derived from the pathogenic RABV strain Nishigahara (Nish) and an attenuated derivative strain (Ni-CE). We identify a network of intraprotomer interactions involving the globular C-terminal domain and intrinsically disordered regions (IDRs) of the N-terminal region that characterize the fully functional Nish P3 to fluctuate between open and closed states, whereas the defective Ni-CE P3 is predominantly open. This conformational difference appears to be due to the single mutation N226H in Ni-CE P3. We find that Nish P3, but not Ni-CE or N226H P3, undergoes liquid-liquid phase separation and this property correlates with the capacity of P3 to interact with different cellular membrane-less organelles, including those associated with immune evasion and pathogenesis. Our analyses propose that discrete functions of a critical multifunctional viral protein depend on the conformational arrangements of distant individual domains and IDRs, in addition to their independent functions.

Witten Complex of Transitive Digraph and Its Convergence

Digraphs are generalization of graphs in which each edge is given one or two directions. For each digraph, there exists a transitive digraph containing it. Moreover, all the formal linear combinations of allowed elementary paths form a basis of the path complex for a transitive digraph. Hence, the study of discrete Morse theory on transitive digraphs is very important for the further study of discrete Morse theory on general digraphs. As we know, the definition of discrete Morse function on a digraph is different from that on a simplical complex or a cell complex: each discrete Morse function on a digraph is a discrete flat Witten-Morse function. In this paper, we deform the usual boundary operator, replacing it with a boundary operator with parameters and consider the induced Laplace operators. In addition, we consider the eigenvectors of the eigenvalues of the Laplace operator that approach to zero when the parameters approach infinity, define the generation space of these eigenvectors, and further give the Witten complex of digraphs. Finally, we prove that for a transitive digraph, Witten complex approaches to its Morse complex. However, for general digraphs, the structure of Morse complex is not as simple as that of transitive digraphs and the critical path is not directly related to the eigenvector with zero eigenvalue of Laplace operator. This is explained in the last part of the paper.

Applications of Differential-Difference Algebra in Discrete Calculus

Discrete calculus deals with developing the concepts and techniques of differential and integralcalculus in a discrete setting, often using difference equations and discrete function spaces. This paperexplores how differential-difference algebra can provide an algebraic framework for advancing discretecalculus. Differential-difference algebra studies algebraic structures equipped with both differential anddifference operators. These hybrid algebraic systems unify continuous and discrete analogues ofderivatives and shifts. This allows the development of general theorems and properties that cover bothsettings. In particular, we construct differential-difference polynomial rings and fields over discretefunction spaces. We define discrete derivatives and shifts algebraically using these operators. We thenstudy integration, summation formulas, fundamental theorems, and discrete analogues of multivariatecalculus concepts from an algebraic perspective. A key benefit is being able to state unified theorems indifferential-difference algebra that simultaneously yield results for both the continuous and discretecases. This provides new tools and insights for discrete calculus using modern algebraic techniques.We also discuss applications of representing discrete calculus problems in differential-differencealgebras. This allows bringing to bear algebraic methods and software tools for their solution. Specificexamples are provided in areas such as numerical analysis of discrete dynamical systems definedthrough difference equations. The paper aims to demonstrate the capabilities of differential-differencealgebra as a unifying framework for further developing the foundations and applications of discretecalculus. Broader connections to algebraic modeling of discrete physical systems are also discussed.

Prediction Intervals: A Geometric View

This article provides a review of the approaches to the construction of prediction intervals. To increase the reliability of prediction, point prediction methods are replaced by intervals for many aims. The interval prediction generates a pair as future values, including the upper and lower bounds for each prediction point. That is, according to historical data, which include a graph of a continuous and discrete function, two functions will be obtained as a prediction, i.e., the upper and lower bounds of estimation. In this case, the prediction boundaries should provide guaranteed probability of the location of the true values inside the boundaries found. The task of building a model from a time series is, by its very nature, incorrect. This means that there is an infinite set of equations whose solution is close to the time series for machine learning. In the case of interval use, the inverse problem of dynamics allows us to choose from the entire range of modeling methods, using confidence intervals as solutions, or intervals of a given width, or those chosen as a solution to the problems of multi-criteria optimization of the criteria for evaluating interval solutions. This article considers a geometric view of the prediction intervals and a new approach is given.

Phase and wave dependent analysis of health expenditure efficiency: A sample of OECD evidence.

Health expenditures are a factor that reflects the government's public health policy and contributes to the protection of national health. Therefore, this study focuses on measuring the effectiveness of health expenditures in order to evaluate and improve the public health system and policy during the pandemic period. In order to examine the effectiveness of health expenditures, the behaviors of the pandemic process were analyzed in two stages. The number of daily cases is analyzed in the first stage by dividing it into waves and phases according to the transmission coefficient (R). For this classification, the discrete cumulative Fourier function estimation is used. In the second stage, the unit root test method was used to estimate the stationarity of the number of cases in order to examine whether the countries made effective health expenditures according to waves and phases. The series being stationary indicates that the cases are predictable and that health expenditure is efficient. Data consists of daily cases from February 2020 to November 2021 for 5 OECD countries. The general results are shown that cases cannot be predicted, especially in the first stage of the pandemic. In the relaxation phase and at the beginning of the second wave, the countries that were seriously affected by the epidemic started to control the number of cas es by taking adequate measures, thus increasing the efficiency of their health systems. The common feature of all the countries we examined is that phase 1, which represents the beginning of the waves, is not stationary. After the waves fade, it can be concluded that the stationary number of health cases cannot be sustainable in preventing new waves' formation. It is seen that countries cannot make effective health expenditures for each wave and stage. According to these findings, the periods in which countries made effective health expenditures during the pandemic are shown. The study aims to help countries make effective short- and long-term decisions about pandemics. The research provides a view of the effectiveness of health expenditures on the number of cases per day in 5 OECD countries during the COVID-19 Pandemic.

Dimer model and holomorphic functions on t‐embeddings of planar graphs

AbstractWe introduce the framework of discrete holomorphic functions on t‐embeddings of weighted bipartite planar graphs; t‐embeddings also appeared under the name Coulomb gauges in a recent paper (Kenyon, Lam, Ramassamy, and Russkikh, Dimers and circle patterns, 2018). We argue that this framework is particularly relevant for the analysis of scaling limits of the height fluctuations in the corresponding dimer models. In particular, it unifies both Kenyon's interpretation of dimer observables as derivatives of harmonic functions on T‐graphs and the notion of s‐holomorphic functions originated in Smirnov's work on the critical Ising model. We develop an a priori regularity theory for such functions and provide a meta‐theorem on convergence of the height fluctuations to the Gaussian Free Field. We also discuss how several more standard discretizations of complex analysis fit this general framework.

Contrasting X-ray/UV time-lags in Seyfert 1 galaxies NGC 4593 and NGC 7469 usingAstroSatobservations

ABSTRACTWe study accretion disc–corona connection in Seyfert 1 galaxies using simultaneous UV/X-ray observations of NGC 4593 (2016 July 14–18) and NGC 7469 (2017 October 15–19) performed with AstroSat. We use the X-ray (0.5–7.0 keV) data acquired with the Soft X-ray Telescope (SXT) and the UV (FUV: 130–180 nm, NUV: 200–300 nm) data obtained with the Ultra-Violet Imaging Telescope (UVIT). We also use the contemporaneous Swift observations of NGC 4593 and demonstrate AstroSat’s capability for X-ray/UV correlation studies. We performed UV/X-ray cross-correlation analysis using the interpolated and the discrete cross-correlation functions and found similar results. In the case of NGC 4593, we found that the variations in the X-rays lead to those in the FUV and NUV bands by ∼38 ks and ∼44 ks, respectively. These UV lags favour the disc-reprocessing model; they are consistent with the previous results within uncertainties. In contrast, we found an opposite trend in NGC 7469 where the soft X-ray variations lag those in the FUV and NUV bands by ∼41 ks and ∼49 ks, respectively. The hard lags in NGC 7469 favour the thermal Comptonization model. Our results may provide direct observational evidence for the variable intrinsic UV emission from the accretion disk, which acts as the seed for thermal Comptonization in a hot corona in a lamp-post-like geometry. The non-detection of disk reverberation photons in NGC 7469, using AstroSat data, is most likely due to a high accretion rate resulting in a hot accretion disc and large intrinsic emission.

A Non-Equilibrium Interpolation Scheme for IB-LBM Optimized by Approximate Force

A non-equilibrium scheme and an optimized approximate force are proposed for the immersed boundary–lattice Boltzmann method (IB-LBM) to solve the fluid–structure interaction (FSI) equations. This new IB-LBM uses the discrete velocity distribution function and non-equilibrium distribution function to establish the interpolation operator and the spread operator at the mesoscopic scale. In the interpolation operator, we use the force model of LBM to derive a direct force with a simple form. In the spread operator, we give a theoretical proof with local second-order accuracy of the spread process using the non-equilibrium theory from the LBM. A non-iterative explicit force approximation scheme optimizes the direct force in that the streamlines have no penetration phenomenon, and the no-slip condition is strictly satisfied. Different from other schemes for the IB-LBM, we try to apply the non-equilibrium theory from the LBM to the IB-LBM and obtain good results. The explicit force obtained using the non-equilibrium scheme and then optimized via the non-iterative streamline correction equation simplifies the explicit direct force scheme and the original implicit scheme previously proposed but obtains a similar streamline correction result compared with the implicit method. Numerical tests prove the applicability and accuracy of this method in the simulation of complex conditions such as moving rigid bodies and deforming flexible bodies.

Conceptual Bases of Gray Transformations and Discrete Vilenkin-Crestenson Functions Systems

The article is devoted to the theoretical foundations of the construction of generalized Gray codes. These include the classical "left-side" and the proposed "right-side" Gray's codes. Left-side codes develop from left to right and right-side codes from right to left. Left- and right-handed Gray's codes augmented with reverse permutation operators form a subset of composite Gray's codes. Such codes have found constructive application in solving synthesis and analysis problems of discrete systems of Vilenkin-Crestenson functions (VCF). Particular cases of VCF systems are systems of discrete exponential functions and Walsh functions. The so-called indicator matrices (IM), bijective connected to the VCF systems, form the basis for VCF systems synthesis. The order of IMs is determined by the logarithmic dependence on the order of the VCF system. Unique Walsh-Cooley function systems, the only ones in the set of Walsh function systems that provide linear connectivity of the frequency scales of DFT processors, are developed. Examples of trees of VCF systems give. The orders of IMs are related by logarithmic dependence with the orders of VCF systems. Using IM: (1) estimates of the number of symmetric VCF systems in an unbounded range of changes of their parameters are obtained, (2) structures are determined and (3) rules of the interconnection of the systems established. The fundamental axioms and lemmas corresponding to the VCF systems formulating and directions for further research are outlined.

Multinomial distribution as part of valuation of the number of failures

Abstract. Aim. The paper aims to examine the application of a multinomial distribution as part of valuation of the number of an object’s failures. It is assumed that the valuation is “based on past experience” (a statistical sample of the number of an object’s failures accumulated over several preceding evaluation periods).Methods. The paper uses methods of system analysis, probability theory and mathematical statistics. The author analyses the primary indicators used to define the applied dependability indicators. It is noted that the valuation of such indicators based on statistical data for complex systems appears to be promising. The problem of valuation of the number of failures using a statistical sample is examined. The primary disadvantages of the used approaches are identified that are associated with errors in defining the average values of series, variation coefficients and asymmetry. It is shown that it is possible to solve the problem using the well-known combinatorics problem “on balls and boxes”, which leads to the use of a multinomial distribution. The paper examined the definition of probabilities for compositions and partitions of the number n into m parts, as well as the probabilities of a given number of balls being in a box with their maximum number. The author also considered formulas and algorithms that allow reducing the number of calculations in case of machine computation of the probabilities of a multinomial distribution. The feasibility of approximating a discrete distribution function by the Gumbel distribution is estimated. The paper demonstrates the feasibility of valuating the number of failures for a “segment” corresponding to a part (fraction) of an object’s dimension considered on a certain part (fraction) of the time interval. It also examines examples of valuating the number of failures for an object as a whole over a 1-month evaluation interval and for ½ of an object over a 12-month evaluation interval, while the total interval for which the statistical sample is accumulated is 72 months. The paper sets forth limitations on the application of the presented method and notes some of its possible advantages. In particular, it is noted that the statistical sample is only one implementation of the multinomial distribution, so it can be said that when applying the proposed method, the results of valuation are no longer affected by the presence of unlikely combinations of series values in the statistical sample. It is also noted that when applying the proposed method of valuating the number of failures, the obtained acceptable value will never be less than or equal to the average value of the statistical sample.Results. Formulas have been obtained for calculating, based on partitions, of the discrete density number and the maximum distribution function of a multinomial distribution. The paper presents the results of algorithm analysis for machine computation. The results of applying some algorithms are presented. A formula is proposed for approximating the distribution function of the maximum of a multinomial distribution using the Gumbel distribution (for the largest values) using the method of moments. The author recommends a range of values of the estimated interval, in which the proposed method provides acceptable reliability of the results. The task of further research is defined.

Study on Intelligent Analysis and Processing Technology of Computer Big Data Based on Clustering Algorithm

Aim: Clustering belongs to unsupervised learning, which divides the data objects into the data set into multiple clusters or classes, so that the objects in the same cluster have high similarity Background: The clustering of spatial data objects can be solved by optimization based on the clustering objective function. Objective: Study on intelligent analysis and processing technology of computer big data based on clustering algorithm. Methods: First, a new dynamic self-organizing feature mapping model is proposed, and the training algorithm of the model is given. Then, the spectral clustering technology and related concepts are introduced. The spectral clustering algorithm is studied and analyzed, and a spectral clustering algorithm that automatically determines the number of clusters is proposed. Furthermore, an algorithm for constructing a discrete Morse function to find the optimal solution is proposed, proving that the constructed function is the optimal discrete Morse function. At the same time, two optimization models based on the discrete Morse theory are constructed. Finally, the optimization model based on discrete Morse theory is applied to cluster analysis, and a density clustering algorithm based on the discrete Morse optimization model is proposed. Results: This study is focused on designing and implementing a partitional-based clustering algorithm based on big data, that is suitable for clustering huge datasets to meet low computational requirements. The experiments are conducted in terms of time and space complexity and it is observed that the measure of clustering quality and the run time is capable of running in very less time without negotiating the quality of clustering. The results show that the experiments are carried out on the artificial data set and the UCI data set. Conclusion: The efficiency and superiority of the new model, are verified by comparing it with the clustering results of the DBSCAN algorithm.

Revisiting the Coulomb problem: A novel representation of the confluent hypergeometric function as an infinite sum of discrete Bessel functions

We use the tridiagonal representation approach to solve the radial Schrödinger equation for the continuum scattering states of the Coulomb problem in a complete basis set of discrete Bessel functions. Consequently, we obtain a new representation of the confluent hypergeometric function as an infinite sum of Bessel functions, which is numerically very stable and more rapidly convergent than another well-known formula.