Number Of Combinations Research Articles

A Hyper-Parameter Optimizer Algorithm Based on Conditional Opposition Local-Based Learning Forbidden Redundant Indexes Adaptive Artificial Bee Colony Applied to Regularized Extreme Learning Machine

Finding the best configuration of a neural network’s hyper-parameters may take too long to be feasible using an exhaustive search, especially when the cardinality of the search space has a big combinatorial number of possible solutions with various hyper-parameters. This problem is aggravated when we also need to optimize the parameters of the neural network, such as the weight of the hidden neurons and biases. Extreme learning machines (ELMs) are part of the random weights neural network family, in which parameters are randomly initialized, and the solution, unlike gradient-descent-based algorithms, can be found analytically. This ability is especially useful for metaheuristic analysis due to its reduced training times allowing a faster optimization process, but the problem of finding the best hyper-parameter configuration is still remaining. In this paper, we propose a modification of the artificial bee colony (ABC) metaheuristic to act as parameterizers for a regularized ELM, incorporating three methods: an adaptive mechanism for ABC to balance exploration (global search) and exploitation (local search), an adaptation of the opposition-based learning technique called opposition local-based learning (OLBL) to strengthen exploitation, and a record of access to the search space called forbidden redundant indexes (FRI) that allow us to avoid redundant calculations and track the explored percentage of the search space. We set ten parameterizations applying different combinations of the proposed methods, limiting them to explore up to approximately 10% of the search space, with results over 98% compared to the maximum performance obtained in the exhaustive search in binary and multiclass datasets. The results demonstrate a promising use of these parameterizations to optimize the hyper-parameters of the R-ELM in datasets with different characteristics in cases where computational efficiency is required, with the possibility of extending its use to other problems with similar characteristics with minor modifications, such as the parameterization of support vector machines, digital image filters, and other neural networks, among others.

Powers of the operator $u(z)\frac{d}{dz}$ and their connection with some combinatorial numbers

In this paper the operator $A = u(z)\frac{d}{dz}$ is considered, where $u$ is an entire or \linebreak meromorphic function in the complex plane. The expansion of $A^{k}$ ($k\geq1$) with the help of the powers of the differential operator $D=\frac{d}{dz}$ is obtained, and it is shown that this expansion depends on special numbers. Connections between these numbers and known combinatorial numbers are given. Some special cases of the operator $A$, corresponding to $u(z) = z, u(z) = e^{z}, u(z) = \frac{1}{z},$ are considered.

A dynamic importance function for accidental scenarios generation by RESTART in the computational risk assessment of cyber-physical infrastructures

The Computational Risk Assessment (CRA) of Cyber-Physical Systems (CPSs) calls for the analysis of accidental scenarios emerging from the complexities and interdependencies typical of CPSs. Generating these scenarios via crude Monte Carlo Simulation (MCS) is impractical due to the high computational demand of simulation codes of CPSs, considering the combinatorial number of possible scenarios. In this paper, we tailor the use of Repetitive Simulation Trials After Reaching Thresholds (RESTART), a rare-event simulation method of literature, to efficiently generate relevant accidental scenarios. The tailored RESTART is guided by a dynamic Importance Function (IF) originally introduced here to dynamically characterize the relevance of the scenarios with reference to the current topology of the CPS and the susceptibility of its components. Two case studies of increasing complexity are considered: a single power grid and a CPS consisting of an Integrated Power and Telecommunication (IP&TLC) infrastructure. Results show that RESTART mines out more relevant scenarios than crude MCS for a number of different IFs based on vulnerability metrics of literature, and thus particularly efficiently when the novel IF is adopted.

Foundations of iterated star maps and their use in combinatorics

We develop a framework for nonstandard analysis that gives foundations to the interplay between external and internal iterations of the star map, and we present a few examples to show the strength and flexibility of such a nonstandard technique for applications in combinatorial number theory.

Practical Integer-Constrained Cone Construction for Conformal Parameterizations.

We propose a practical method to construct sparse integer-constrained cone singularities with low distortion constraints for conformal parameterizations. Our solution for this combinatorial problem is a two-stage procedure that first enhances sparsity for generating an initialization and then optimizes to reduce the number of cones and the parameterization distortion. Central to the first stage is a progressive process to determine the combinatorial variables, i.e., numbers, locations, and angles of cones. The second stage iteratively conducts adaptive cone relocations and merges close cones for optimization. We extensively test our method on a data set containing 3885 models, demonstrating practical robustness and performance. Our method achieves fewer cone singularities and lower parameterization distortion than state-of-the-art methods.

Arithmetic of Hecke L-functions of quadratic extensions of totally real fields

Deep work by Shintani in the 1970's describes Hecke L-functions associated to narrow ray class group characters of totally real fields F in terms of what are now known as Shintani zeta functions. However, for [F:Q]=n≥3, Shintani's method was ineffective due to its crucial dependence on abstract fundamental domains for the action of totally positive units of F on R+n, so-called Shintani sets. These difficulties were recently resolved in independent work of Charollois, Dasgupta, and Greenberg and Diaz y Diaz and Friedman. For those narrow ray class group characters whose conductor is an inert rational prime in a totally real field F with narrow class number 1, we obtain a natural combinatorial description of these sets, allowing us to obtain a simple description of the associated Hecke L-functions. As a consequence, we generalize earlier work of Girstmair, Hirzebruch, and Zagier, that offer combinatorial class number formulas for imaginary quadratic fields, to real and imaginary quadratic extensions of totally real number fields F with narrow class number 1. For CM quadratic extensions of F, our work may be viewed as an effective affirmative answer to Hecke's Conjecture that the relative class number has an elementary arithmetic expression in terms of the relative discriminant.

On the Hausdorff dimension of invariant measures of piecewise smooth circle homeomorphisms

AbstractWe show that, generically, the unique invariant measure of a sufficiently regular piecewise smooth circle homeomorphism with irrational rotation number and zero mean nonlinearity (e.g. piecewise linear) has zero Hausdorff dimension. To encode this generic condition, we consider piecewise smooth homeomorphisms as generalized interval exchange transformations (GIETs) of the interval and rely on the notion of combinatorial rotation number for GIETs, which can be seen as an extension of the classical notion of rotation number for circle homeomorphisms to the GIET setting.

Structural characterization of methylation-independent PP2A assembly guides alphafold2Multimer prediction of family-wide PP2A complexes

Dysregulation of phosphorylation-dependent signaling is a hallmark of tumorigenesis. Protein phosphatase 2 (PP2A) is an essential regulator of cell growth. One scaffold subunit (A) binds to a catalytic subunit (C) to form a core AC heterodimer, which together with one of many regulatory (B) subunits forms the active trimeric enzyme. The combinatorial number of distinct PP2A complexes is large, which results in diverse substrate specificity and subcellular localization. The detailed mechanism of PP2A assembly and regulation remains elusive and reports about an important role of methylation of the carboxy terminus of PP2A C are conflicting. A better understanding of the molecular underpinnings of PP2A assembly and regulation is critical to dissect PP2A function in physiology and disease. Here, we combined biochemical reconstitution, mass spectrometry, X-ray crystallography and functional assays to characterize the assembly of trimeric PP2A. In vitro studies demonstrated that methylation of the carboxy-terminus of PP2A C was dispensable for PP2A assembly in vitro. To corroborate these findings, we determined the X-ray crystal structure of the unmethylated PP2A Aα-B56ε-Cα trimer complex to 3.1 Å resolution. The experimental structure superimposed well with an Alphafold2Multimer prediction of the PP2A trimer. We then predicted models of all canonical PP2A complexes providing a framework for structural analysis of PP2A. In conclusion, methylation was dispensable for trimeric PP2A assembly and integrative structural biology studies of PP2A offered predictive models for all canonical PP2A complexes.

Polynomial extension of some symmetric partition regular structures

The study of symmetric structures is a new trend in combinatorial number theory. Recently in [9], Di Nasso proved symmetrized versions of some classical theorems like Hindman's theorem, Van der Waerden's theorem and Deuber's theorem. This opens the question of which other classical theorems can be symmetrized, as well as if other symmetric operations allow for such generalizations. Here we give positive answers to both questions, by showing that symmetrization of the polynomial extension of Van der Waerden's and Deuber's Theorem is possible.

Generating functions for polynomials derived from central moments involving bernstein basis functions and their applications

The main objective of this article is to construct generating functions for central moments involving Bernstein basis functions. We give some alternating generating functions of these functions. We also give derivative formulas and a recurrence relation of central moments with the help of their generating functions. We also establish new relations between combinatorial numbers and polynomials, and also central moments. Furthermore, by applying Euler operator and Laplace transformation to central moments, we derive some important results. Finally, we give further remarks, observations and comments related to the content of this paper.

Novel real number representations in Ising machines and performance evaluation: Combinatorial random number sum and constant division.

Quantum annealing machines are next-generation computers for solving combinatorial optimization problems. Although physical simulations are one of the most promising applications of quantum annealing machines, a method how to embed the target problem into the machines has not been developed except for certain simple examples. In this study, we focus on a method of representing real numbers using binary variables, or quantum bits. One of the most important problems for conducting physical simulation by quantum annealing machines is how to represent the real number with quantum bits. The variables in physical simulations are often represented by real numbers but real numbers must be represented by a combination of binary variables in quantum annealing, such as quadratic unconstrained binary optimization (QUBO). Conventionally, real numbers have been represented by assigning each digit of their binary number representation to a binary variable. Considering the classical annealing point of view, we noticed that when real numbers are represented in binary numbers, there are numbers that can only be reached by inverting several bits simultaneously under the restriction of not increasing a given Hamiltonian, which makes the optimization very difficult. In this work, we propose three new types of real number representation and compared these representations under the problem of solving linear equations. As a result, we found experimentally that the accuracy of the solution varies significantly depending on how the real numbers are represented. We also found that the most appropriate representation depends on the size and difficulty of the problem to be solved and that these differences show a consistent trend for two annealing solvers. Finally, we explain the reasons for these differences using simple models, the minimum required number of simultaneous bit flips, one-way probabilistic bit-flip energy minimization, and simulation of ideal quantum annealing machine.

On signs of certain Toeplitz-Hessenberg determinants whose elements involve Bernoulli numbers

In the paper, by virtue of Wronski's formula and Kaluza's theorem related to a power series and its reciprocal, by means of Cahill and Narayan's recursive relation, and with the aid of the logarithmic convexity of the sequence of the Bernoulli numbers, the author presents the signs of certain Toeplitz-Hessenberg determinants whose elements involve the Bernoulli numbers and combinatorial numbers. Moreover, with the help of a derivative formula for the ratio of two differentiable functions, the author provides an alternative proof of Wronski's formula.

LP-Based Row Generation Using Optimization-Based Sorting Method for Solving Budget Allocation with a Combinatorial Number of Constraints

A novel approach was developed that combined LP-based row generation with optimization-based sorting to tackle computational challenges posed by budget allocation problems with combinatorial constraints. The proposed approach dynamically generated constraints using row generation and prioritized them using optimization-based sorting to ensure a high-quality solution. Computational experiments and case studies revealed that as the problem size increased, the proposed approach outperformed simplex solutions in terms of solution search time. Specifically, for a problem with 50 projects (N = 50) and 2,251,799,813,685,250 constraints, the proposed approach found a solution in just 1.4 s, while LP failed due to the problem size. The proposed approach demonstrated enhanced computational efficiency and solution quality compared to traditional LP methods.

Enabling discoveries of Solar System objects in large alert data streams

Context. With the advent of large-scale astronomical surveys, such as the Zwicky Transient Facility (ZTF) and the forthcoming Vera C. Rubin Observatory’s Legacy Survey of Space and Time (LSST), the number of alerts generated by transient, variable, and moving astronomical objects is growing rapidly, reaching millions of alerts per night. Concerning the minor planets of the Solar System, their identification requires linking the alerts for many observations over a potentially lengthy period of time, leading to a very large combinatorial number. Aims. The goal is to demonstrate how a third-party module dedicated to the identification of new minor planets of the Solar System can be integrated with the Fink alert broker real-time operations, which deals with massive alert data streams produced by large-scale surveys. Methods. Our analysis takes advantage of the scientific surplus brought on by the Fink alert broker classification capabilities to first reduce the 111 275 131 processed alerts from ZTF between November 2019 and December 2022 (755 observation nights) to only 389 530 new Solar System alert candidates over the same period. We implemented a simple, yet pedagogical linking algorithm called Fink-FAT to create trajectory candidates in real time from alert data and extract orbital parameters. The analysis was validated on ZTF alert packets linked to confirmed Solar System objects from the Minor Planet Center (MPC) database. Finally, the candidates were confronted with follow-up observations. Results. Between November 2019 and December 2022, Fink-FAT extracted 327 new orbits from candidate Solar System objects at the time of the observations, of which 65 had still remained unreported in the MPC database as of March 2023. After two late follow-up observation campaigns of six orbit candidates, four were associated with known minor planets of the Solar System, and two still remain unknown. In terms of performance, Fink-FAT took under 3 h to link alerts into trajectory candidates and to extract the orbital elements over the three years of Fink data, using a modest hardware configuration. Conclusions. Despite a much lower efficiency than present linking algorithms, Fink-FAT reaches a high level of purity in reconstructing orbits and it runs fast, making it suitable for the real-time discovery of new minor planets. Fink-FAT is deployed in the Fink broker and analyzes, in real time, the alert data from the ZTF survey by regularly extracting new candidates for Solar System objects. Tests of scalability also show that Fink-FAT is capable of handling the even larger volume of alert data that will be sent by the Rubin Observatory’s real-time difference image analysis processing.

Problems for combinatorial numbers satisfying a class of triangular arrays

Numbers satisfying a class of triangular arrays, defined by a bivariate first-order linear difference equation with linear coefficients, include a wide range of combinatorial numbers: binomial coefficients, Morgan numbers, Stirling numbers of the first and the second types, non-central Stirling numbers, Eulerian numbers, Lah numbers, and their generalizations. In this work, we derive the general analytic expression of the numbers satisfying a class of triangular arrays and propose problems (both teaching and unsolved ones) for undergraduates studying probability theory and analytical combinatorics subjects in the study programs of the fields of mathematics and computer science. Some of the unsolved challenges can also be used as the basis for a thesis.

Combining formal methods and Bayesian approach for inferring discrete-state stochastic models from steady-state data

Stochastic population models are widely used to model phenomena in different areas such as cyber-physical systems, chemical kinetics, collective animal behaviour, and beyond. Quantitative analysis of stochastic population models easily becomes challenging due to the combinatorial number of possible states of the population. Moreover, while the modeller easily hypothesises the mechanistic aspects of the model, the quantitative parameters associated to these mechanistic transitions are difficult or impossible to measure directly. In this paper, we investigate how formal verification methods can aid parameter inference for population discrete-time Markov chains in a scenario where only a limited sample of population-level data measurements—sample distributions among terminal states—are available. We first discuss the parameter identifiability and uncertainty quantification in this setup, as well as how the existing techniques of formal parameter synthesis and Bayesian inference apply. Then, we propose and implement four different methods, three of which incorporate formal parameter synthesis as a pre-computation step. We empirically evaluate the performance of the proposed methods over four representative case studies. We find that our proposed methods incorporating formal parameter synthesis as a pre-computation step allow us to significantly enhance the accuracy, precision, and scalability of inference. Specifically, in the case of unidentifiable parameters, we accurately capture the subspace of parameters which is data-compliant at a desired confidence level.

A Generalization of the Chevalley–Warning and Ax–Katz Theorems with a View Towards Combinatorial Number Theory

A Generalization of the Chevalley–Warning and Ax–Katz Theorems with a View Towards Combinatorial Number Theory

Bayesian Optimization of Sampling Densities in MRI

Data-driven optimization of sampling patterns in MRI has recently received a significant attention. Following recent observations on the combinatorial number of minimizers in off-the-grid optimization, we propose a framework to globally optimize the sampling densities using Bayesian optimization. Using a dimension reduction technique, we optimize the sampling trajectories more than 20 times faster than conventional off-the-grid methods, with a restricted number of training samples. This method – among other benefits – discards the need of automatic differentiation. Its performance is slightly worse than state-of-the-art learned trajectories since it reduces the space of admissible trajectories, but comes with significant computational advantages. Other contributions include: i) a careful evaluation of the distance in probability space to generate trajectories ii) a specific training procedure on families of operators for unrolled reconstruction networks and iii) a gradient projection based scheme for trajectory optimization.

Combinatorial Game and Number Triangle

By considering a triangular array of numbers for first six rows, we introduce a combinatorial game, whose solution depends on a number triangle. The conclusion brings us with a surprising consequence in deciding the result of the game. This paper analyzes the game and present the solution in detail using the number triangle which resembles the famous Leibniz Triangle of unit fractions.

Deep Learning-Aided Modulation Recognition for Non-Orthogonal Signals.

Automatic Modulation Recognition (AMR) can obtain the modulation mode of the received signal for subsequent processing without the assistance of the transmitter. Although the existing AMR methods have been mature for the orthogonal signals, these methods face challenges when deployed in non-orthogonal transmission systems due to the superimposed signals. In this paper, we aim to develop efficient AMR methods for both downlink and uplink non-orthogonal transmission signals using deep learning-based data-driven classification methodology. Specifically, for downlink non-orthogonal signals, we propose a Bi-directional Long Short-Term Memory (BiLSTM)-based AMR method that exploits long-term data dependence to automatically learn irregular signal constellation shapes. Transfer learning is further incorporated to improve recognition accuracy and robustness under varying transmission conditions. For uplink non-orthogonal signals, the combinatorial number of classification types explodes exponentially with the number of signal layers, which becomes the major obstacle to AMR. We develop a spatio-temporal fusion network based on the attention mechanism to efficiently extract spatio-temporal features, and network details are optimized according to the superposition characteristics of non-orthogonal signals. Experiments show that the proposed deep learning-based methods outperform their conventional counterparts in both downlink and uplink non-orthogonal systems. In a typical uplink scenario with three non-orthogonal signal layers, the recognition accuracy can approach 96.6% in the Gaussian channel, which is 19% higher than the vanilla Convolution Neural Network.