Symmetric Series Research Articles

Involution symmetry quantification using recurrences.

Symmetries are ubiquitous in science, aiding theoretical comprehension by discerning patterns in mathematical models and natural phenomena. This work introduces a method for assessing the extent of symmetry within a time series. We explore both microscopic and macroscopic features extracted from a recurrence plot. By analyzing the statistics of small recurrence matrices, our approach delves into microscale dynamics, facilitating the identification of symmetric time series segments through diagonal macroscale structures on a recurrence plot. We validate our approach by successfully quantifying involution symmetries for three-dimensional dynamical models, specifically, order-2 rotational symmetry in the Lorenz '63 model, and inversion symmetry in the Chua circuit. Our quantifier also detects symmetry breaking in the modified Lorenz model for El Niño phenomenon. The method can be applied in a versatile manner, not only to three-dimensional trajectories but also to univariate time series. Symmetry quantification in time series is promising for enhancing dynamical system modeling and profiling.

Coumarin Schiff base derivatives with alkyl 4’-formyl-biphenyl-4-carboxylate: design, synthesis, mesomorphic studies and DFT calculations

ABSTRACT Symmetric and unsymmetric homologous series of coumarin Schiff base derivatives with alkyl 4’-formyl-biphenyl-4-carboxylate have been synthesised and studied for their mesomorphic behaviour. In the symmetric homologous series of Schiff base derivatives BPES(6–16), alkoxy chain length was varied at both ends of the molecules with n = 6,8,10,12,14,16, while in the unsymmetric homologous series of Schiff base derivatives 10BPEU(2–12), chain length was varied at coumarin end only (n = 2–8,10,12). In symmetric homologous series, compounds BPES(6–10) (n = 6,8,10) exhibited enantiotropic smectic A (SmA) mesophase, while smectic C (SmC) mesophase in cooling cycle only. SmC mesophase became prominent on increasing the chain length n = 12,14,16 in BPES(12–16). In the unsymmetric homologous series 10BPEU(2–12), compound with n = 2 showed monotropic nematic mesophase and highly stable (~165 °C) SmA mesophase. Higher analogues in unsymmetric series were turned biphasic and showed SmA as well as SmC mesophase during cooling scan only. Compound with n = 12 showed stable enantiotropic SmC mesophase (~92 °C). VT-XRD analysis of symmetric compound with n = 12, showed the tilted SmC mesophase. Further DFT calculations were investigated for un/symmetric series along with the literature molecules with similar variations and compared for various structural (length, aspect ratio) and electronic (dipole moment, polarisability) parameters to establish correlation with observed mesomorphic behaviour.

Symmetry versus asymmetry game in vaporization enthalpies of imidazolium-based ionic liquids

The vaporization enthalpy as a function of alkyl chain length was studied using both experimental techniques and molecular dynamics simulations for imidazolium based ionic liquids (ILs) with symmetric and asymmetric imidazolium substitution. It was found that the vaporization enthalpy increases with increasing alkyl chain length due to the increased van der Waals (vdW) interaction. In addition, the vaporization enthalpies of the symmetric ILs ([CnCnIm][X], X = NTf2-, Cl-, Br-, I-) series were found to be systematically lower (by 5–10 kJ mol−1) in comparison with the corresponding asymmetric imidazolium-based ILs ([CmC1Im][X]). This energetic difference was attributed to the intramolecular dispersion interactions between the two alkyl side chains existing in the symmetric cation series.

Scaling Analysis of Time-Reversal Asymmetries in Fully Developed Turbulence

In fully developed turbulence, there is a flux of energy from large to small scales in the inertial range until the dissipation at small scales. It is associated with irreversibility, i.e., a breaking of the time reversal symmetry. Such turbulent flows are characterized by scaling properties, and we consider here how irreversibility depends on the scale. Indicators of time-reversal symmetry for time series are tested involving triple correlations in a non-symmetric way. These indicators are built so that they are zero for a time-reversal symmetric time series, and a departure from zero is an indicator of irreversibility. We study these indicators applied to two fully developed turbulence time series, from flume tank and wind tunnel databases. It is found that irreversibility occurs in the inertial range and has scaling properties with slopes close to one. A maximum value is found around the injection scale. This confirms that the irreversibility is associated with the turbulent cascade in the inertial range and shows that the irreversibility is maximal at the injection scale, the largest scale of the turbulent cascade.

Lan(n+1)+xNin(n+5)+ySi(n+1)(n+2)-z: A Symmetric Mirror Homologous Series in the La-Ni-Si System.

A series of four homologous silicides have been discovered during systematic explorations in the central part of the La-Ni-Si system at 1000 °C. All compounds La12.5Ni28.0Si18.3 (n = 3; a = 28.8686(8), c = 4.0737(2) Å, Z = 3), La22.1Ni39.0Si27.8 (n = 4; a = 20.9340(6), c = 4.1245(2) Å, Z = 1), La32.9Ni49.8Si39.3 (n = 5; a = 24.946(1), c = 4.1471(5) Å, Z = 1), and La44.8Ni66.1Si53.4 (n = 6; a = 28.995(5), c = 4.158(1) Å, Z = 1) crystallize in the hexagonal space group P63/m and can be generalized according to Lan(n+1)+xNin(n+5)+ySi(n+1)(n+2)-z with n = 3-6. Their crystal structures are based on AlB2-type building blocks, fused La-centered Ni6Si6 hexagonal prisms, yielding larger oligomeric equilateral domains with the edge size equal to n. The domains extend along the c axis and show checkered ordering of the cationic and anionic parts, while all their atoms are located on mirror planes. Lan(n+1)+xNin(n+5)+ySi(n+1)(n+2)-z can be considered as a mirror series to the La-rich La(n+1)(n+2)Nin(n-1)+2Sin(n+1), where an exchange of the formal cationic and anionic sites, i.e., La and Si, occurs. The La-Ni-Si system is the first system where two such analogous series have been observed.

Three Symmetric Double Series by Telescoping

The telescoping method is skillfully employed to evaluate double series expressions for and , as well as the Catalan constant G.

Uniformly convergent Fourier series with universal power parts on closed subsets of measure zero

Given a closed subset E of Lebesgue measure zero on the unit circle T there is a function f on T with uniformly convergent symmetric Fourier series Sn(f,ζ)=∑k=−nnfˆ(k)ζk⇉Tf(ζ),such that for every continuous function g on E, there is a subsequence of partial power sums Sn+(f,ζ)=∑k=0nfˆ(k)ζkof f, which converges to g uniformly on E. Here fˆ(k)=∫Tζ̄kf(ζ)dm(ζ),and m is the normalized Lebesgue measure on T.

Prime Limiting Function

Among all number series, prime number series is most mysterious. Even after years of work, this series never fail to amaze us. Another amazing fact is realized and reciprocated with this research is that any three consecutive prime numbers can be reduced to the first prime number present in the series. Also, using this limiting function a question is imposed on the existence of negative prime numbers. This series can also be viewed as a symmetric series about 2 i.e. the first prime number

SYMARFIMA: A dynamical model for conditionally symmetric time series with long range dependence mean structure

In this work we introduce a dynamical model for conditionally symmetric time series accommodating a long range dependent structure for the conditional mean. More specifically, the proposed model specify the underlying distribution of the time series, conditionally to its past, to be symmetric. The conditional mean is specified to accommodate a long range dependent structure, following an ARFIMA-like design, as well as a (possibly time dependent) set of regressors. We provide conditions for the existence and stationarity of the proposed model as well as closed formulas for its unconditional mean, variance and covariance structure. Parameter estimation is carried out via partial likelihood. The score vector and Hessian are obtained in closed forms. A finite sample study of the proposed partial likelihood estimation is carried out. To show the usefulness of the proposed model, we present an application to a real data set related to wind speed in certain locations in Brazil.

Super-Resolution Ultrasound Imaging Scheme Based on a Symmetric Series Convolutional Neural Network.

In this paper, we propose a symmetric series convolutional neural network (SS-CNN), which is a novel deep convolutional neural network (DCNN)-based super-resolution (SR) technique for ultrasound medical imaging. The proposed model comprises two parts: a feature extraction network (FEN) and an up-sampling layer. In the FEN, the low-resolution (LR) counterpart of the ultrasound image passes through a symmetric series of two different DCNNs. The low-level feature maps obtained from the subsequent layers of both DCNNs are concatenated in a feed forward manner, aiding in robust feature extraction to ensure high reconstruction quality. Subsequently, the final concatenated features serve as an input map to the latter 2D convolutional layers, where the textural information of the input image is connected via skip connections. The second part of the proposed model is a sub-pixel convolutional (SPC) layer, which up-samples the output of the FEN by multiplying it with a multi-dimensional kernel followed by a periodic shuffling operation to reconstruct a high-quality SR ultrasound image. We validate the performance of the SS-CNN with publicly available ultrasound image datasets. Experimental results show that the proposed model achieves a high-quality reconstruction of the ultrasound image over the conventional methods in terms of peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM), while providing compelling SR reconstruction time.

Erratum to “The Symmetric Series of Multiples of Primes” [Advances in Pure Mathematics, 12 (2022) 160-177

The original online version of this article (Doroszlai, P. and Keller, H. (2022) The Symmetric Series of Multiples of Primes, Advances in Pure Mathematics, Vol. 12, 160-177. http://doi.org/10.4236/apm.2022.123014) unfortunately contains some small mistakes. The author discovered the possible problem in Equations (1.6), (1.7) and in the last equation of the proof of lemma 2.3 and the numbering of the chapters 3 and 4 should be changed. To fix the problem, the author wishes to change these equations and titles.

The Symmetric Series of Multiples of Primes

The union of the straight and over the point of reflection—reflected series of the arithmetic progression of primes results in the double density of occupation of integer positions. It is shown that the number of free positions left by the double density of occupation has a lower limit function, which is growing to infinity. The free positions represent equidistant primes to the point of reflection: in case the point of reflection is an even number, they satisfy Goldbach’s conjecture. The double density allows proving as well that at any distance from the origin large enough—the distance between primes is smaller, than the square root of the distance to the origin. Therefore, the series of primes represent a continuum and may be integrated. Furthermore, it allows proving that the largest gap between primes is growing to infinity with the distance and that the number of any two primes, with a given even number as the distance between them, is unlimited. Thus, the number of twin primes is unlimited as well.

Symmetry vs. asymmetry – Enthalpic differences in imidazolium-based ionic liquids

Enthalpies of ionic liquids [CnCnIm][X] synthesis reactions were measured with help of differential scanning calorimetry. These results in combination with quantum-chemical calculations were used for estimation of vaporization enthalpies of symmetric imidazolium based ionic liquids with X = Cl, Br, and I. Vaporization enthalpies of the symmetric ([CnCnIm][X]) series were found to be systematically lower (of 5–8 kJ mol−1) in comparison with the corresponding asymmetric imidazolium-based ILs ([CnC1Im][X]). This energetic difference was attributed to the intensity of dispersion interactions.

Thioether-linked azobenzene-based liquid crystal dimers exhibiting the twist-bend nematic phase over a wide temperature range

ABSTRACT Two homologous series of sulfur-containing thioether-linked 4-cyanoazobenzene-based liquid crystal (LC) dimers were developed: a symmetric series based on 4-cyanoazobenzene arms [(CN)AzoSnSAzo(CN)] and a non-symmetric series based on 4-cyanoazobenzene and 4-cyanobiphenyl arms [(CN)AzoSnSCB]. They contained different flexible oligomethylene spacers (C n H2 n ) consisting of odd numbers of carbon atoms (n = 3, 5, 7, 9, and 11). The symmetric (CN)AzoSnSAzo(CN) dimers with n = 5, 7, 9, and 11 showed an elusive monotropic twist–bend nematic (NTB) phase, while all the non-symmetric (CN)AzoSnSCB dimers exhibited the NTB phase. Notably, the NTB phase observed for non-symmetric (CN)AzoS7SCB could be supercooled to room temperature, leading to a glassy NTB phase. (CN)AzoS9SCB showed a similar tendency accompanied by partial crystallisation. These are the first reported examples of azobenzene-based LC dimers exhibiting a broad temperature range of NTB phases supercooled to room temperature. Such prominent stability of the NTB phase is due to the supercooling effect assisted by the synergy of molecular bend arising from thioether linkages and non-symmetric mesogenic arms. The phase-transition temperatures were overall higher for symmetric (CN)AzoSnSAzo(CN) than for non-symmetric (CN)AzoSnSCB. The difference in the mesogenic arm structures hardly influenced their molecular geometry or anisotropy in terms of entropy changes.

On approximations for functions in the space of uniformly convergent Fourier series

This paper studies the possibility of approximating functions in the space of all uniformly convergent symmetric and non-symmetric Fourier series from finitely many samples of the given function. It is shown that no matter what approximation method is chosen, there always exists a residual subset such that the approximation method diverges for all functions from this subset. This general result implies that there exists no method to effectively calculate the Fourier series expansion on a digital computer for all functions from the space of uniformly convergent Fourier series. In particular, there exists no Turing computable approximation method in these spaces.

Probabilistic Modeling of Monthly Temperature Historical Series in Mossoró, Northeastern Brazil

We fitted the following seven distribution probabilities to the data of monthly average temperature in Mossoró, northeastern Brazil: Normal, Log-Normal, Beta, Gamma, Log-Pearson (Type III), Gumbel, and Weibull. To assess the goodness of fit the empirical distributions to the theoretical distribution, we applied the tests of Kolmogorov-Smirnov, Chi-square, Cramer-von Mises, Anderson-Darling, Kuiper, and Logarithm of Maximum Likelihood, at 10% of probability. The temperature series were obtained from 1970 to 2007. The Normal distribution provided the best fit to the historical series of average monthly temperature. Although the Kolmogorov-Smirnov test showed a very high level of approval, which generated some uncertainty regarding the test criteria, it is the more recommended to studies with approximately symmetric data and small series.

Numerical investigation of excessive surge induced by wave overtopping in an inlet-bay system

During Hurricane Sandy, wave overtopping occurred along the barrier beach in front of the Rehoboth Bay, one of the Delaware Inland Bays, as indicated by field observations and the morphological profile model CSHORE. CSHORE estimated a total overtopping volume of about 9.7 ×107m3 which is close to the storm surge-induced water volume entering the bays through the inlet (about 1.3 ×108m3 ). In this study, we incorporated the wave overtopping result from CSHORE into the nearshore community model, NearCoM-TVD, to investigate the wave overtopping-induced excessive surge (OIS) in the Delaware Inland Bays. With the wave overtopping applied in the flux boundary condition, the model successfully reproduced the high water level measured at the station located in the Rehoboth Bay. Additional numerical experiments for the separate effects of wave forcing and local winds are also carried out. The results show that wave forcing has a minimal effect on the water level inside the bays. The adjustment of local wind forcing does not qualitatively improve the agreement between the model and data. The model with included wave overtopping predicted an excessive surge of about 20 cm in the Rehoboth Bay, agreeing better with the measured data. Finally, the responses of OIS to different overtopping conditions were performed using an idealized model setup. A strong asymmetry, represented by a much longer draining period versus a shorter period of overtopping increase to the OIS peak, was found in the OIS process for a given temporally symmetric time series of overtopping flux. The asymmetric feature is caused by the imbalance between the overtopping flux and the discharge through the inter-connecting channel inside the Inland Bays. Nonlinearity plays an important role in the process of OIS as demonstrated in modeling OIS-tide-surge interaction.

PM2.5 and ozone, indicators of air quality, and acute deaths in California, 2004–2007

Since the London Great Smog of 1952 was estimated to have killed over 4000 people, scientists have studied the relationship between air quality and acute mortality. Currently, the association between air quality and acute deaths is usually taken as evidence for causality. As air quality has markedly improved since 1952, do contemporary datasets support this view? We use a large dataset, eight air basins in California for the years 2004–2007, to examine the possible association of ozone and PM2.5 with acute deaths after statistically removing seasonal and weather effects. Our analysis dataset is available on request. We conducted a regression-corrected, case-crossover analysis for all non-accidental deaths age 75 and older. We used stepwise regression to examine three causes of death. After seasonal and weather adjustments, there was essentially no predictive power of ozone or PM2.5 for acute deaths. The case-crossover analysis produced odds ratio very close to 1.000 (no effect). The very narrow confidence limits indicated good statistical power. We study recent air quality in both time-stratified, symmetric, bidirectional case-crossover and time series regression and both give consistent results. There is no statistically significant association between either ozone or PM2.5 and acute human mortality. In the absence of an association, causality is in question.

Properties of Asymmetric Detrended Fluctuation Analysis in the time series of RR intervals

Heart rate asymmetry is a phenomenon by which the accelerations and decelerations of heart rate behave differently, and this difference is consistent and unidirectional, i.e. in most of the analyzed recordings the inequalities have the same directions. So far, it has been established for variance and runs based types of descriptors of RR intervals time series. In this paper we apply the newly developed method of Asymmetric Detrended Fluctuation Analysis, which so far has mainly been used with economic time series, to the set of 420 stationary 30 min time series of RR intervals from young, healthy individuals aged between 20 and 40. This asymmetric approach introduces separate scaling exponents for rising and falling trends. We systematically study the presence of asymmetry in both global and local versions of this method. In this study global means “applying to the whole time series” and local means “applying to windows jumping along the recording”. It is found that the correlation structure of the fluctuations left over after detrending in physiological time series shows strong asymmetric features in both magnitude, with α+<α−, where α+ is related to heart rate decelerations and α− to heart rate accelerations, and the proportion of the signal in which the above inequality holds. A very similar effect is observed if asymmetric noise is added to a symmetric self-affine function. No such phenomena are observed in the same physiological data after shuffling or with a group of symmetric synthetic time series.

Plethystic vertex operators and boson-fermion correspondences

We study the algebraic properties of plethystic vertex operators, introduced in (2010 J. Phys. A: Math. Theor. 43 405202), underlying the structure of symmetric functions associated with certain generalized universal character rings of subgroups of the general linear group, defined to stabilize tensors of Young symmetry type characterized by a partition of arbitrary shape π. Here we establish an extension of the well-known boson-fermion correspondence involving Schur functions and their associated (Bernstein) vertex operators: for each π, the modes generated by the plethystic vertex operators and their suitably constructed duals, satisfy the anticommutation relations of a complex Clifford algebra. The combinatorial manipulations underlying the results involve exchange identities exploiting the Hopf-algebraic structure of certain symmetric function series and their plethysms.