Binomial Coefficients Research Articles

Maximizing weighted sums of binomial coefficients using generalized continued fractions

Let $m,\,r\in {\mathbb {Z}}$ and $\omega \in {\mathbb {R}}$ satisfy $0\leqslant r\leqslant m$ and $\omega \geqslant 1$ . Our main result is a generalized continued fraction for an expression involving the partial binomial sum $s_m(r) = \sum _{i=0}^r\binom{m}{i}$ . We apply this to create new upper and lower bounds for $s_m(r)$ and thus for $g_{\omega,m}(r)=\omega ^{-r}s_m(r)$ . We also bound an integer $r_0 \in \{0,\,1,\,\ldots,\,m\}$ such that $g_{\omega,m}(0)<\cdots < g_{\omega,m}(r_0-1)\leqslant g_{\omega,m}(r_0)$ and $g_{\omega,m}(r_0)>\cdots >g_{\omega,m}(m)$ . For real $\omega \geqslant \sqrt 3$ we prove that $r_0\in \{\lfloor \frac {m+2}{\omega +1}\rfloor,\,\lfloor \frac {m+2}{\omega +1}\rfloor +1\}$ , and also $r_0 =\lfloor \frac {m+2}{\omega +1}\rfloor$ for $\omega \in \{3,\,4,\,\ldots \}$ or $\omega =2$ and $3\nmid m$ .

Equivalent static wind load based on displacement mode and load combination for high‐rise buildings

AbstractUnder the action of the fluctuating wind load, the low frequency part produces background response to the structures, while the high frequency part produces resonance response to the structures. In structural design, equivalent static wind load is usually used to equate the fluctuating wind load. Although there are various methods of evaluating the equivalent static wind load, they did not consider the correlation between modal responses or considered them insufficiently. Therefore, in this paper, the correlation between modal response is considered to evaluate the equivalent static wind load, the displacement response is decomposed by proper orthogonal decomposition (POD) method, the correlation between modal displacement is removed, the equivalent static wind load is expressed in the form of displacement mode, and then the correlation between modal response is fully considered in the extreme value combination. This paper combines the equivalent static wind loads in order to make the wind resistance design more reasonable for high‐rise buildings. First, the formulas of equivalent static wind loads expressed by displacement modes of background and resonant response are deduced based on modal decomposition and POD method. Second, the combination formulas of square‐root‐of‐sum‐square (SRSS) and complete‐quadratic‐combination (CQC) rules for the equivalent static wind loads considering the mean wind loads are proposed. Both the linear combination formula of SRSS for the equivalent static wind load and the weighting factor expressions of background and resonance equivalent static wind load are given. Third, the accuracy and validity of the formulas of equivalent static wind load are verified by a wind tunnel pressure test of a high‐rise building. Finally, the simplified combination coefficient formulas for the equivalent static wind loads are proposed, and the combination of the high‐rise building base's fluctuating equivalent static wind loads of along‐wind direction, across‐wind direction, and torsional direction is analyzed.

Fractal Geometry, Fibonacci Numbers, Golden Ratios, And Pascal Triangles as Designs

Fractal geometry is a part of mathematics that discusses the shape of fractals or any form that is self-similarity. A fractal can be broken down into parts that are all similar to the original fractal. Fractals have infinite detail and can have self-similar structures at different magnifications. In many cases, a fractal can be generated by repeating a pattern, which is usually in a recursive or iterative process. In mathematics and art, two values are considered to be a golden ratio relationship if the ratio between the sum of the two values to the large value is equal to the ratio between the large value to the small value. A Fibonacci sequence is a sequence of numbers that has a unique shape. The first term of this sequence of numbers is 1, then the second term is also 1, then for the third term it is determined by adding the two previous terms so that a sequence of numbers with a certain pattern is obtained. Pascal's triangle is a geometric rule of binomial coefficients in a triangle. Shapes that resemble fractal geometry, golden ratios and Fibonacci numbers are found in many places in this realm, for example the shapes of various plants, animals, as well as in humans themselves, even the universe. This fractal geometry can also be used to design batik motif creations, architectural design or musical art. This paper describes how a fractal object can be made with geometric transformations that can be used to develop batik motif creations. And also look at the relationship between fractal geometry, the golden ratio, Fibonacci numbers, and Pascal's triangles and the shapes of these shapes that exist in this nature.

Similarity recognition approach to identify zero-added MSG soy sauce using stable isotopes and amino acid profiles

Seasonings such as naturally fermented soy sauce without added monosodium glutamate (MSG), are currently a growth market in China. However, fraudulent and mislabeled zero-added MSG soy sauce may cause a risk of excessive MSG intake, increasing food safety issues for consumers. This study investigates stable carbon and nitrogen isotopes and 16 amino acids in typical Chinese in-market soy sauces and uses a similarity method to establish criteria to authenticate MSG addition claims. Results reveal most zero-added MSG soy sauces had lower δ13C values (−25.2 ‰ to −17.7 ‰) and glutamic acid concentrations (8.97 mg mL−1 to 34.76 mg mL−1), and higher δ15N values (−0.27 ‰ +0.95 ‰) and other amino acid concentrations than added-MSG labeled samples. A combined approach, using isotopes, amino acids, similarity coefficients and uncertainty values, was evaluated to rapidly and accurately identify zero-added MSG soy sauces from MSG containing counterparts.

New procedure for evaluation of U(3) coupling and recoupling coefficients

A simple method to calculate Wigner coupling coefficients and Racah recoupling coefficients for U(3) in two group–subgroup chains is presented. While the canonical U(3)⊃U(2)⊃U(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rm U(3)\\supset U(2)\\supset U(1)$$\\end{document} coupling and recoupling coefficients are applicable to any system that respects U(3) symmetry, the U(3)⊃SO(3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rm U(3) \\supset SO(3)$$\\end{document} coupling coefficients are more specific to nuclear structure studies. This new procedure precludes the use of binomial coefficients and alternating sums which were used in the 1973 formulation of Draayer and Akiyama, and in so doing provides a faster and more accurate determination of any and all required results. The resolution of the outer multiplicity is based on the null space concept of the U(3) generators proposed by Alex et al., whereas the inner multiplicity in the angular momentum subgroup chain is obtained from the dimension of the null space of the SO(3) raising operator. It is anticipated that a C++ library will ultimately be available for determining generic coupling and recoupling coefficients associated with both the canonical and the physical group–subgroup chains of U(3).

Parameterized Finite Binomial Sums

We offer intriguing new insights into parameterized finite binomial sums, revealing elegant identities such as ∑k=0,k≠nm+n(−1)kn−km+nk=(−1)nm+nn(Hm−Hn), where n,m are non-negative integers and Hn is the harmonic number. These formulas beautifully capture the intricate relationship between harmonic numbers and binomial coefficients, providing a fresh and captivating perspective on combinatorial sums.

Hypergeometric series and generalized harmonic numbers

By computing the higher-order derivatives of some well-known hypergeometric summation theorems, we evaluate many different infinite series containing generalized harmonic functions and central binomial coefficients.

Multi-objective meta-learning

Meta-learning has arisen as a powerful tool for many machine learning problems. With multiple factors to be considered when designing learning models for real-world applications, meta-learning with multiple objectives has attracted much attention recently. However, existing works either linearly combine multiple objectives into one objective or adopt evolutionary algorithms to handle it, where the former approach needs to pay high computational cost to tune the combination coefficients while the latter approach is computationally heavy and incapable to be integrated into gradient-based optimization. To alleviate those limitations, in this paper, we aim to propose a generic gradient-based Multi-Objective Meta-Learning (MOML) framework with applications in many machine learning problems. Specifically, the MOML framework formulates the objective function of meta-learning with multiple objectives as a Multi-Objective Bi-Level optimization Problem (MOBLP) where the upper-level subproblem is to solve several possibly conflicting objectives for the meta-learner. Different from those existing works, in this paper, we propose a gradient-based algorithm to solve the MOBLP. Specifically, we devise the first gradient-based optimization algorithm by alternately solving the lower-level and upper-level subproblems via the gradient descent method and the gradient-based multi-objective optimization method, respectively. Theoretically, we prove the convergence property and provide a non-asymptotic analysis of the proposed gradient-based optimization algorithm. Empirically, extensive experiments justify our theoretical results and demonstrate the superiority of the proposed MOML framework for different learning problems, including few-shot learning, domain adaptation, multi-task learning, neural architecture search, and reinforcement learning. The source code of MOML is available at https://github.com/Baijiong-Lin/MOML.

Wind-Load Calculation Program for Rectangular Buildings Based on Wind Tunnel Experimental Data for Preliminary Structural Designs

In this study, we developed a wind load calculation program (WCP) capable of predicting wind loads with relative precision during the preliminary design phase. First, wind tunnel tests were conducted to identify the essential factors necessary for calculating wind loads and the variables influencing these factors. Square building shapes were considered, and the wind force coefficients and power spectral density were measured by combining four ground roughness values, eleven side ratios (D/B), four aspect ratios (H/BD), and wind directions ranging from 0° to 90°. The wind power coefficient and the spectral coefficient were formulated so that the wind load could be calculated according to various conditions. The WCP computations were based on the calculation of the load combination coefficient using the resonant wind load. Finally, the wind loads obtained from the wind tunnel tests were compared with those predicted by the WCP using an actual project model (inner-core (A) and outer-core (B) types). Building A yielded similar WCP and wind tunnel experimental responses when subjected to wind and laminar wind loads. Additionally, Building B yielded a larger error than that of Building A, but similar results were obtained when buildings were subjected to combination and laminar wind loads.

Triple Symmetric Sums of Circular Binomial Products

By employing the generating function approach, 16 triple sums for circular binomial products of binomial coefficients are examined. Recurrence relations and generating functions are explicitly determined. These symmetric sums may find potential applications in the analysis of algorithms, symbolic calculus, and computations in theoretical physics.

Accurate approximations of classical and generalized binomial coefficients

Several sharp approximations of the generalized-binomial-coefficient function having real arguments are presented on the basis of Stirling’s approximation formula for Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document} function.

In-depth modeling of tilt-to-length coupling in LISA’s interferometers and TDI Michelson observables

We present first-order models for tilt-to-length (TTL) coupling in Laser Interferometer Space Antenna (LISA), both for the individual interferometers, as well as in the time-delay interferometry (TDI) Michelson observables. These models include the noise contributions from angular and lateral jitter coupling of the six test masses, six movable optical subassemblies, and three spacecraft. We briefly discuss which terms are considered to be dominant and reduce the TTL model for the second-generation TDI Michelson X observable to these primary noise contributions to estimate the resulting noise level. We show that the expected TTL noise will initially violate the entire mission displacement noise budget, resulting in the known necessity to fit and subtract TTL noise in data postprocessing. By comparing the noise levels for different assumptions prior to subtraction, we show why noise mitigation by realignment prior to subtraction is favorable. We then discuss that the TTL coupling in the individual interferometers will have noise contributions that will not be present in the TDI observables. Models for TTL coupling noise in TDI and in the individual interferometers are therefore different, and commonly made assumptions are valid as such only for TDI, but not for the individual interferometers. Finally, we analyze what implications can be drawn from the presented models for the subsequent fit-and-subtraction in postprocessing. We show that noise contributions from the test mass and intersatellite interferometers are indistinguishable, such that only the combined coefficients can be fit and used for subtraction. However, a distinction is considered not necessary. Additionally, we show a correlation between coefficients for transmitter and receiver jitter couplings in each individual TDI Michelson observable. This full correlation, however, can be resolved by using all three Michelson observables for fitting the TTL coefficients. Published by the American Physical Society 2024

Automatically Detecting Pancreatic Cysts in Autosomal Dominant Polycystic Kidney Disease on MRI Using Deep Learning.

Pancreatic cysts in autosomal dominant polycystic kidney disease (ADPKD) correlate with PKD2 mutations, which have a different phenotype than PKD1 mutations. However, pancreatic cysts are commonly overlooked by radiologists. Here, we automate the detection of pancreatic cysts on abdominal MRI in ADPKD. Eight nnU-Net-based segmentation models with 2D or 3D configuration and various loss functions were trained on positive-only or positive-and-negative datasets, comprising axial and coronal T2-weighted MR images from 254 scans on 146 ADPKD patients with pancreatic cysts labeled independently by two radiologists. Model performance was evaluated on test subjects unseen in training, comprising 40 internal, 40 external, and 23 test-retest reproducibility ADPKD patients. Two radiologists agreed on 52% of cysts labeled on training data, and 33%/25% on internal/external test datasets. The 2D model with a loss of combined dice similarity coefficient and cross-entropy trained with the dataset with both positive and negative cases produced an optimal dice score of 0.7 ± 0.5/0.8 ± 0.4 at the voxel level on internal/external validation and was thus used as the best-performing model. In the test-retest, the optimal model showed superior reproducibility (83% agreement between scan A and B) in segmenting pancreatic cysts compared to six expert observers (77% agreement). In the internal/external validation, the optimal model showed high specificity of 94%/100% but limited sensitivity of 20%/24%. Labeling pancreatic cysts on T2 images of the abdomen in patients with ADPKD is challenging, deep learning can help the automated detection of pancreatic cysts, and further image quality improvement is warranted.

A reconstruction method of stress fields, bending moments and wave parameters for floating nuclear power plants through regular wave identification

Floating nuclear power plants (FNPPs) have the potential to supply abundant electricity, hot water, and freshwater resources for offshore islands and offshore oil and gas extraction platforms. However, the consequences of a nuclear reactor accident can be extremely serious. In addition, there may be many uncertain factors in the actual operation of FNPPs. These factors lead to higher structural safety requirements. So， developing more scientific structural monitoring methods is an effective way to enhance the structural safety of FNPP. Traditional hull monitoring schemes can only track bending and torsional forces on specific cross-sections, failing to identify the overall stress field of the entire vessel structure. The paper proposes a reconstruction method for the stress field, bending moment distribution, and wave parameters of FNPP using regular wave identification. Initially, irregular waves are decomposed into a linear combination of multiple regular waves with different amplitudes and frequencies. Subsequently, the strains induced by these regular waves on the FNPP are utilized as basic functions to establish a mathematical model for reconstructing the stress field. The stress field, bending moment distribution, and wave parameters of the FNPP can be reconstructed by identifying the combination coefficients of the regular wave loads. The ill-posedness of the mathematical model for stress field reconstruction has the potential to significantly impact the accuracy of the stress field reconstruction, which was addressed by optimizing the sensor installation position in this study. Meanwhile, the optimization of the sensor installation position greatly reduces the dimensionality of mathematical models. Finally, a numerical study, i.e. FNPP being designed by China, was used to validate the accuracy of the proposed stress field reconstruction algorithm. The reconstructed stress field of the FNPP closely matched the original stress distribution under various irregular wave conditions.

Test-Retest Reliability of a Physical Activity Behavior, Health and Wellbeing Questionnaire in Adolescents

Background The aim of this study was to examine the test-retest reliability of the physical activity behavior, health and wellbeing questionnaire, in adolescent populations, administered by teachers in school settings, in the Republic of Ireland. Methods A cross-sectional, mixed sample of 55 participants (45.5% males: Age, 13.94 (±.40) years) were included. The participants completed the questionnaire on two occasions (T1 and T2), on the same day and time, one week apart following identical procedures. Variables for testing included physical activity behavior (n=13), health (n=11) and wellbeing (n=2). Test-retest reliability of the questionnaire’s covariates, including family affluence and physical impairments were also examined. Results Systematic error (Bland-Altman plots) was found to be near to zero for each of the physical activity behavior, health and wellbeing variables. The combined mean coefficient of variation was lower for females (10.19%) in comparison to males (13.01%). The combined mean intraclass correlation coefficients were higher for females (0.901) than males (0.822). Similarly, the combined mean Cronbach alpha coefficient were higher for girls (0.908) than boys (0.821). Conclusions This study found the physical activity behavior, health and wellbeing questionnaire to be reliable for use in adolescent populations.

Harmonic Series with Multinomial Coefficient 4nn,n,n,n and Central Binomial Coefficient 2nn

Classical hypergeometric series are reformulated as analytic functions of their parameters (in both the numerator and the denominator). Then, the coefficient extraction method is applied to examine hypergeometric series transformations. Several new closed form evaluations are established for harmonic series containing multinomial coefficient 4nn,n,n,n and central binomial coefficient 2nn. These results exclusively concern the alternating series of convergence rate “−1/4”.

Proof of some congruences via the hypergeometric identities

In this paper, via the hypergeometric identities of F23, F34 and F45, we prove some congruences involving cubes of central binomial coefficients and harmonic numbers. We also obtained a congruence which generalizes a conjecture of Long, this conjecture has been confirmed by Swisher (Res. Math. Sci. 2:18, 2015).

Complex frequency divider

This work presents the Complex Frequency Divider Formula (CFDF), an explicit equation for the complex frequencies (CFs) of the voltage of every bus of a power system in terms of a linear combination of the CFs of the currents injected by the devices connected to the network. The coefficients of this linear combination depend on the topology of the system, voltages and currents. The CFDF carries valuable information on the impact of every device on the dynamic of the voltage and frequency of any particular bus of the system. The formulation is exact, as it does not require any simplification of the dynamic model of the system. The CFDF is implemented in two benchmark systems to show its potential in static and dynamic analysis in different scenarios. Finally, relevant notes on the applications of the proposed formula are outlined.

Some general summation results with applications to arctangent-type series

This article extends the telescopic summation and Abel summation by parts methods by introducing various tuning parameters and examining sequences of variable-dependent functions. In a substantial part, these general results are applied to arctangent-type series. We re-examine known results from a fresh viewpoint and establish numerous new ones, with an emphasis on establishing unified formulas applicable in various scenarios. In particular, some new arctangent-type series defined with the binomial coefficients and others with the Fibonacci sequence are demonstrated. Therefore, this work contributes to the advancement of telescopic techniques and enriches our understanding of arctangent-type series.

Kinetics of voids evolution during diffusion bonding of dissimilar metals on consideration of the realistic surface morphology: Modeling and experiments

A comprehensive model for analyzing the kinetics of void evolution and shrinkage during dissimilar diffusion bonding is proposed based on the sinusoidal profile. The model incorporates atom accumulation at void tip to determine the curvature radius of void neck, and integrates the local effect of surface diffusion to quantify the contribution of surface diffusion, thus avoiding the unrealistic void evolution observed in previous models. Furthermore, the model utilizes a combined diffusion coefficient for multicomponent alloys under the effect of stress and concentration gradients, to determine the interfacial diffusion coefficient for dissimilar joints. In contrast to previous models, the current model demonstrates significant improvements in predicted accuracy, efficiently depicting the realistic void evolution and interpreting the formation of various interfacial voids, which can be applied in various bonding cases, including for the dissimilar base metals with different surface conditions. Simulation results from bonding between austenitic steel and ferritic steel with different surface roughness reveal that interfacial voids predominantly form on the side of the base metal with the rougher bonding surface. Additionally, as surface roughness decreases, the prevailing surface source diffusion and interface source diffusion mechanisms promote void evolution and shrinkage, resulting in a significant reduction of the final bonding time. To achieve a high-quality bonding joint, it's suggested to place the smoother bonding surface on the higher creep-resistant base metal, which can accelerate the void closure process.