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Let p_[1,r;t] be defined by ∑n=0∞p[1,r;t](n)qn=(E1Er)t, where t is a non-zero rational number, r ≥ 1 is an integer and Er=∏n=0∞(1−qr(n+1)) for |q| < 1. The function p_[1,r;t](n) is the generalisation of the two-colour partition function p_[1,r;−1](n). In this paper, the authors prove some new congruences modulo odd prime ℓ by taking r = 5, 7, 11 and 13, and non-integral rational values of t.

Using q-series expansion/identities, the authors established general congruence modulo prime number for two-colour partition function.

In the paper, the authors study congruence properties of two-colour partition function for fractional values. The authors also give some particular cases as examples.

The partition functions for fractional value is studied in 2019 by Chan and Wang for Ramanujan's general partition function and then extended by Xia and Zhu in 2020. In 2021, Baruah and Das also proved some congruences related to fractional partition functions previously investigated by Chan and Wang. In this sequel, some congruences are proved for two-colour partitions in this paper. The results presented in the paper are original.

The space fractional PDEs (SFPDEs) play an important role in the fractional calculus field. Proposing a high-order, stable and flexible numerical procedure for solving SFPDEs is the main aim of most researchers. This paper devotes to developing a novel spectral algorithm to solve the FitzHugh–Nagumo models with space fractional derivatives.

The fractional derivative is defined based upon the Riesz derivative. First, a second-order finite difference formulation is used to approximate the time derivative. Then, the Jacobi spectral collocation method is employed to discrete the spatial variables. On the other hand, authors assume that the approximate solution is a linear combination of special polynomials which are obtained from the Jacobi polynomials, and also there exists Riesz fractional derivative based on the Jacobi polynomials. Also, a reduced order plan, such as proper orthogonal decomposition (POD) method, has been utilized.

A fast high-order numerical method to decrease the elapsed CPU time has been constructed for solving systems of space fractional PDEs.

The spectral collocation method is combined with the POD idea to solve the system of space-fractional PDEs. The numerical results are acceptable and efficient for the main mathematical model.

The purpose of this paper is to discuss how brassiere manufacturers develop new designs for bra products, suitable for individual consumers, through consumer codesign.

New product design that relies on conjoint analysis algorithms can depict multidimensional attribute profiles, such that consumers' choice behavior reflects their preferences and overall judgment of the profiles. This statistical technique provides a means to codesign and customize bra products and thereby enhance the overall bra design process.

Bra products codesign suggests goals such as attractive appearance, shoulder strap style, vivid/mild color, elegance/sexy lace, comfort/practicality, fabric, lining, comfort/attractive appearance, neckline design, comfort/excellent function cut, sewn cups, and generous quantities. The most preferred combination of attributes for all respondents is a cotton/cotton blend fabric, seamless bra that offers a detachable shoulder strap, lavender color, a two‐strap style, lace details, and a low‐cut plunge neckline. The paper illustrates consumers' bra awareness attributes, codesign approach, and individual optimum individualized bra designs.

The results provide a useful source of information for product managers, who should consider the use of codesign to design the best products for individual consumers and decrease the risk of design failure, as well as promote consumer loyalty and satisfaction toward the product.

The paper provides a unique method to understand the new product codesign structure and make bra product design decisions that integrate optimum individualized design.

The purpose of this paper is to test the existence of stylized facts, such as the volatility clustering, heavy tails seen on financial series, long-term dependence and multifractality on the returns of four real estate indexes using different types of indexes: conventional and Islamic by comparing pre and during COVID-19 pandemic.

Firstly, the authors examined the characteristics of the indexes. Secondly, the authors estimated the parameters of the stable distribution. Then, the long memory is detected via the estimation of the Hurst exponents. Afterwards, the authors determine the graphs of the multifractal detrended fluctuation analysis (MF-DFA). Finally, the authors apply the WTMM method.

The results suggest that the real estate indexes are far from being efficient and that the lowest level of multifractality was observed for Islamic indexes.

The inefficiency behavior of real estate indexes gives us an idea about the prediction of the behavior of future returns in these markets on the basis of past informations. Similarly, market participants would do well to reassess their investment and risk management framework to mitigate new and somewhat higher levels of risk of their exposures during the turbulent period.

To the authors’ knowledge, this is the first real estate market study employing STL decomposition before applying the MF-DFA in the context of the COVID-19 crisis. Likewise, the study is the first investigation that focuses on these four indexes.

High-efficiency video coding (HEVC) is the latest video coding standard that has better coding efficiency than the H.264/advanced video coding (AVC) standard. The purpose of this paper is to design and develop an effective block search mechanism for the video compression-HEVC standard such that the developed compression standard is applied for the communication applications.

In the proposed method, an rate-distortion (RD) trade-off, named regressive RD trade-off is used based on the conditional autoregressive value at risk (CaViar) model. The motion estimation (ME) is based on the new block search mechanism, which is developed with the modification in the Ordered Tree-based Hex-Octagon (OrTHO)-search algorithm along with the chronological Salp swarm algorithm (SSA) based on deep recurrent neural network (deepRNN) for optimally deciding the shape of search, search length of the tree and dimension. The chronological SSA is developed by integrating the chronological concept in SSA, which is used for training the deep RNN for ME.

The competing methods used for the comparative analysis of the proposed OrTHO-search based RD + chronological-salp swarm algorithm (RD + C-SSA) based deep RNN are support vector machine (SVM), fast encoding framework, wavefront-based high parallel (WHP) and OrTHO-search based RD method. The proposed video compression method obtained a maximum peak signal-to-noise ratio (PSNR) of 42.9180 dB and a maximum structural similarity index measure (SSIM) of 0.9827.

In this research, an effective block search mechanism was developed with the modification in the OrTHO-search algorithm along with the chronological SSA based on deepRNN for the video compression-HEVC standard.

THE Fourth European Work Study Congress, held in Paris during the third week of May, was a well‐organized affair. A tribute is due to M. Loubert and his colleagues for the way in which they devised such well‐lubricated machinery for the convenience of their guests and for the imaginative touch of holding the official dinner aboard a bateau‐mouche as it sailed for two hours up and down the Seine.

For most people, especially those with fixed incomes, household budgets have to be balanced and sometimes the balance is precarious. With price rises of foods, there is a switch to a cheaper substitute within the group, or if it is a food for which there is no real substitute, reduced purchases follow. The annual and quarterly reviews of the National Food Survey over the years have shown this to be so; with carcase meat, where one meat is highly priced, housewives switch to a cheaper joint, and this is mainly the reason for the great increase in consumption of poultry; when recently the price of butter rose sharply, there was a switch to margarine. NFS statistics did not show any lessening of consumer preference for butter, but in most households, with budgets on a tight string, margarine had to be used for many purposes for which butter had previously been used. With those foods which have no substitute, and bread (also milk) is a classic example, to keep the sum spent on the food each week about the same, the amount purchased is correspondingly reduced. Again, NFS statistics show this to be the case, a practice which has been responsible for the small annual reductions in the amount of bread consumed per person per week over the last fifteen years or so; very small, a matter of an ounce or two, but adequate to maintain the balance of price/quantity since price rises have been relatively small, if fairly frequent. This artifice to absorb small price rises will not work, however, when price rises follow on one another rapidly and together are large. Bread is a case in point.

AT an international exhibition in Moscow during September a pavilion covering 6,500 sq. ft. displayed equipment designed to improve the organization of work. The unusual thing is that the equipment was office machinery and systems exhibited by the Business Equipment Trade Association of this country and five of its member firms. Until now the U.S.S.R. has been a negligible market for such goods, but the Soviet Union has now declared its intention to increase the automation of clerical work through installing modern equipment of the sort favoured by western countries. The report of a delegation of the Association's members put this point so strongly that, as a ‘joint under‐taking’ with the British Board of Trade, it was thought justifiable to attack this burgeoning market.

This paper addresses a significant research gap in the study of Rayleigh surface wave propagation within a piezoelectric medium characterized by piezoelectric properties, thermal effects and voids. Previous research has often overlooked the crucial aspects related to voids. This study aims to provide analytical solutions for Rayleigh waves propagating through a medium consisting of a nonlocal piezo-thermo-elastic material with voids under the Moore–Gibson–Thompson thermo-elasticity theory with memory dependencies.

The analytical solutions are derived using a wave-mode method, and roots are computed from the characteristic equation using the Durand–Kerner method. These roots are then filtered based on the decay condition of surface waves. The analysis pertains to a medium subjected to stress-free and isothermal boundary conditions.

Computational simulations are performed to determine the attenuation coefficient and phase velocity of Rayleigh waves. This investigation goes beyond mere calculations and examines particle motion to gain deeper insights into Rayleigh wave propagation. Furthermore, this investigates how kernel function and nonlocal parameters influence these wave phenomena.

The results of this study reveal several unique cases that significantly contribute to the understanding of Rayleigh wave propagation within this intricate material system, particularly in the presence of voids.

This investigation provides valuable insights into the synergistic dynamics among piezoelectric constituents, void structures and Rayleigh wave propagation, enabling advancements in sensor technology, augmented energy harvesting methodologies and pioneering seismic monitoring approaches.

This study formulates a novel governing equation for a nonlocal piezo-thermo-elastic medium with voids, highlighting the significance of Rayleigh waves and investigating the impact of memory.