Factorials Research Articles

Starter Set Generation Based on Factorial Numbers for Half Wing of Butterfly Representation

Permutation is an interesting subject to explore until today where it is widely applied in many areas. This paper presents the use of factorial numbers for generating starter sets where starter sets are used for listing permutation. Previously starter sets are generated by using their permutation under exchange-based and cycling based. However, in the new algorithm, this process is replaced by factorial numbers. The base theory is there are <img src=image/13491677_01.gif> number of distinct starter sets. Every permutation has its decimal number from zero until <img src=image/13491677_02.gif> for Lexicographic order permutation only. From a decimal number, it will be converted to a factorial number. Then the factorial number will be mapped to its corresponding starter sets. After that, the Half Wing of Butterfly will be presented. The advantage of the use of factorial numbers is the avoidance of the recursive call function for starter set generation. In other words, any starter set can be generated by calling any decimal number. This new algorithm is still in the early stage and under development for the generation of the half wing of butterfly representation. Case n=5 is demonstrated for a new algorithm for lexicographic order permutation. In conclusion, this new development is only applicable for generating starter sets as a lexicographic order permutation due to factorial numbers is applicable for lexicographic order permutation.

Effect tensile behavior of spandex/polyester yarns on the stress–strain

Effect tensile behavior of spandex/polyester yarns on the stress–strain

A Combinatorial Approach to the Generalized Central Factorial Numbers

A Combinatorial Approach to the Generalized Central Factorial Numbers

Fast and stable computation of higher-order Hahn polynomials and Hahn moment invariants for signal and image analysis.

This article presents, on the one hand, new algorithms for the fast and stable computation of discrete orthogonal Hahn polynomials of high order (HPs) based on the elimination of all gamma and factorial functions that cause the numerical fluctuations of HPs, and based on the use of appropriate stability conditions. On the other hand, a new method for the fast and numerically stable computation of Hahn moment invariants (HMIs) is also proposed. This method is mainly based on the use of new recursive relations of HPs and of matrix multiplications when calculating HMIs. To validate the efficiency of the algorithms proposed for the calculation of HPs, several signals and large images (≥4000 × 4000) are reconstructed by Hahn moments (HMs) up to the last order with a reconstruction error tending towards zero (MSE ≃ 10−10). The efficiency of the proposed method for calculating HMIs is demonstrated on large medical images (2048 × 2048) with a very low relative error (RE ≃ 10−10). Finally, comparisons with some recent work in the literature are provided.

A Virtualization Infrastructure Cost Model for 5G Network Slice Provisioning in a Smart Factory

Network slicing is a key enabler for providing new services to industry verticals. In order to enable network slice provisioning, it is important to study the network slice type allocation for different device types in a real industrial case. Furthermore, the costs of the required virtualization infrastructure need to be analyzed for various cloud deployment scenarios. In this paper, a cost model for the virtualization infrastructure needed for network slice provisioning is developed and subsequently applied to a real smart factory. In the model, slice types and devices are mapped such that each factory device is provisioned with one or more slice types, as required. The number of devices to be supported per slice type is forecasted for 2021–2030, and the total costs of ownership, costs per slice type, and costs for every slice type, for each device are calculated. The results are analyzed for three cloud deployment scenarios: local, distributed, and centralized. The centralized scenario was found to have the lowest cost. Moreover, sensitivity analysis is conducted by varying the device growth, the number of factories, the level of isolation between network slices, and resource overbooking. The resulting evaluation and cost breakdown can help stakeholders select a suitable deployment scenario, gauge their investments, and exercise suitable pricing.

Using the digamma function for basis functions in mesh-free computational methods

We examine the utility of a new family of basis functions for use with the Complex Variable Boundary Element Method (CVBEM) and other mesh-free numerical methods for solving partial differential equations. The family of polygamma functions have found use in mathematics since as early as 1730 when James Stirling related the digamma function to the factorial function [1]. Now, we propose using the digamma function, as well as new variants of the digamma function, as basis functions for the CVBEM. This paper discusses technical aspects associated with using the digamma function as a CVBEM basis function. Then, we demonstrate the utility of the proposed basis function by applying it to a mixed boundary value problem of the Laplace type.

On Some Conjectural Supercongruences for Sums Involving Certain Rising Factorials

On Some Conjectural Supercongruences for Sums Involving Certain Rising Factorials

Gaussian integral by Taylor series and applications

In this paper, we present a solution for a specific Gaussian integral. Introducing a parameter that depends on a n index, we found out a general solution inspired by the Taylor series of a simple function. We demonstrated that this parameter represents the expansion coefficients of this function, a very interesting and new result. We also introduced some Theorems that are proved by mathematical induction. As a test for the solution presented here, we investigated a non-extensive version for the particle number density in Tsallis framework, which enabled us to evaluate the functionality of the method. Besides, solutions for a certain class of the gamma and factorial functions are derived. Moreover, we presented a simple application in fractional calculus. In conclusion, we believe in the relevance of this work because it presents a solution for the Gaussian integral from an unprecedented perspective.

Wartime activities of the Vavilov Institute

Among the chronicles relating the heroism of the besieged Leningrad, there are pages dedicated to the deeds performed by the staff the world-famous All-Union Research Institute of Plant Industry (VIR, now the N.I. Vavilov All-Russian Institute of Plant Genetic Resources). With the beginning of the war, even before the city was surrounded by the Nazi troops, the government decided to evacuate a number of factories and institutes from Leningrad, including VIR, but the plan failed. Only in winter did the Institute start partial evacuation, although preparations had been going on for a long time. The largest and most important part of the collection was left behind in the besieged city. The remaining employees were forced to work under the hardest conditions of the siege, in unheated premises. In the harsh reality of the winter in 1941–1942, the daily bread rationing was cut down, and hunger raged in the city, killing tens of thousands of city residents, including VIR employees who kept the stored seeds and tubers untouched. The most difficult part was preserving the potato collection. In the spring of 1942, preparations were made for sowing to restore the viability of seeds and tubers in the fields of Leningrad’s suburban area under the fire from the enemy artillery. Only the heroic efforts of VIR’s staff helped to save the collection from destruction and loss of germination. This heroism cost more than 20 experts and scientists their lives. So, the most dangerous period for the Institute was overcome at such price. Immediately after the siege was lifted, a group of experts was sent to Leningrad from Krasnoufimsk to help with selecting seed accessions for urgent reproduction. Working under extreme physical exhaustion in frozen premises, without water or electricity, under continuous shelling, they saved, many at the cost of their own lives, the collection of cultivated plants and their wild relatives, the herbarium, and the scientific library for future generations.

MAKHACHKALA IN THE GREAT PATRIOTIC WAR: THE MAIN INDUSTRIAL AND TRANSPORT CENTER OF THE CAUCASUS

The article reviews the role of Makhachkala - the capital of the Daghestan ASSR - as an industrial, transport and evacuation center in the years of the Great Patriotic War (1941-1945). The authors highlights the labor excellence of the city’s residents, especially those who worked at industrial enterprises, at the sea-port, on the railway. The importance of the city workers in strengthening the country's defense is shown. During the war, a number of factories in Makhachkala started the production of ammunition and equipment for ships and transport. The paper provides the analysis of the manufactured products, notes the joint work with the evacuated enterprises. The workers of the Makhachkala Factory named after M. Gadzhiev, for example, during the war increased the output of products by 4 times; in 1945 they 7 times won the Red Banner of the State Defense Committee and 2 times - the All-Union Central Council of Trade Unions and the People's Commissariat of the USSR Navy. The work of the citizens of Makhachkala on the construction of defensive lines, the activities of evacuation hospitals located in the city are described. In the summer of 1942, the Makhachkala special defensive line was created, while the construction of defensive structures in the city itself (barricades, firing points, shelters) was underway, in which more than 20 thousand of citizens and residents of neighbouring regions took part.

The Immunity-malignancy equilibrium in multiple myeloma: lessons from oncogenic events in plasma cells.

Multiple myeloma (MM) is a malignancy of plasma cells (PC) that grow within the bone marrow and maintain massive immunoglobulin (Ig) production. Disease evolution is driven by genetic lesions, whose effects on cell biology and fitness underlie addictions and vulnerabilities of myeloma cells. Several genes mutated in myeloma are strictly involved in dictating PC identity and antibody factory function. Here, we evaluate the impact of mutations in IRF4, PRDM1, and XBP1, essential transcription factors driving the B to PC differentiation, on MM cell biology and homeostasis. These factors are highly specialized, with limited overlap in their downstream transcriptional programs. Indeed, IRF4 sustains metabolism, survival, and proliferation, while PRDM1 and XBP1 are mainly responsible for endoplasmic reticulum expansion and sustained Ig secretion. Interestingly, IRF4 undergoes activating mutations and translocations, while PRDM1 and XBP1 are hit by loss-of-function events, raising the hypothesis that containment of the secretory program, but not its complete extinction, may be beneficial to malignant PCs. Finally, recent studies unveiled that also the PRDM1 target, FAM46C/TENT5C, an onco-suppressor uniquely and frequently mutated or deleted in myeloma, is directly and potently involved in orchestrating ER homeostasis and secretory activity. Inactivating mutations found in this gene and its interactors strengthen the notion that reduced secretory capacity confers advantage to myeloma cells. We believe that dissection of the evolutionary pressure on genes driving PC-specific functions in myeloma will disclose the cellular strategies by which myeloma cells maintain an equilibrium between antibody production and survival, thus unveiling novel therapeutic targets.

Comparative analysis of the application of artificial neural network-genetic algorithm and response surface methods-desirability function for predicting the optimal conditions for biodiesel synthesis from chrysophyllum albidum seed oil

Efficient and sustainable commercial production of biodiesel requires process optimization and modeling using artificial intelligence techniques. Artificial neural network-genetic algorithm (ANN-GA) and response surface methodology (RSM) were used to predict the optimal process conditions for biodiesel synthesis from novel chrysophyllium albidum seed oil. Application of Levenberg -Marquardt algorithm and full factorial central composite design-desirability function analysis were carried out for ANN and RSM respectively. The biodiesel obtained at the optimum conditions was characterized using gas chromatography-mass spectrophotometer (GC–MS), Fourier transform infrared (FTIR), AOAC and ASTM D standard methods. The ANN and RSM correlation coefficients were 0.98 and 0.95 respectively. The best RSM fitted model was second order polynomial while the ANN best architecture topology consists of three layers: input layer with four input variables, hidden layer with ten hidden neurons and an output layer with single output variable. Optimization of the process shows that: RSM optimum conditions for biodiesel yield of 85.91wt% validated at 86.85wt% were 65.62 °C, 2.14wt% catalyst concentration, 62.04 min and 5.88 methanol: oil molar ratio with overall desirability of 1.00. ANN-GA gave 9.40% faster reaction time and required 15.88% less energy, 3.76% more alcohol and 16.82% less catalyst with 1.86wt% more biodiesel yield than RSM optimized conditions. Ascertained fuel properties, GC–MS and FTIR characterizations confirmed unsaturation and good cold-flow qualities of the biodiesel. Overall results presented ANN-GA as better than RSM and a reliable modeling and optimization technique for viable and sustainable production of biodiesel from chrysophyllum albidum seed.

Green Revolution in Ready Made Garments in Bangladesh: An Analytical Study

Bangladesh, a small south Asian country, holds the second position in the world of exporting readymade garments (RMG). Here locate the highest number of green RMG factories in the world. Green industrialization is the positive symbol of sustainable development. This paper represents an impressive illustration of the scenario of RMG industries in Bangladesh especially the growth, contribution to export and the success of LEED (Leadership in Energy and Environmental Design) recognized Platinum, Gold and Silver factories. This paper also represents the success of green revolution in RMG industries in Bangladesh and recommends some suggestions to convert the traditional industries to green industries where the industries are not only for making profit but also committed to provide good working environment for employee, eco-friendly and ethical business practice.

Stieltjes moment properties and continued fractions from combinatorial triangles

Many combinatorial numbers can be placed in the following generalized triangular array [Tn,k]n,k≥0 satisfying the recurrence relation:Tn,k=λ(a0n+a1k+a2)Tn−1,k+(b0n+b1k+b2)Tn−1,k−1+d(da1−b1)λ(n−k+1)Tn−1,k−2 with T0,0=1 and Tn,k=0 unless 0≤k≤n for suitable a0,a1,a2,b0,b1,b2,d and λ. For n≥0, denote by Tn(q) the generating function of the n-th row. In this paper, we develop various criteria for x-Stieltjes moment property and 3-x-log-convexity of Tn(q) based on the Jacobi continued fraction expression of ∑n≥0Tn(q)tn, where x is a set of indeterminates consisting of q and those parameters occurring in the recurrence relation. With the help of a criterion of Wang and Zhu (2016) [36], we show that the corresponding linear transformation of Tn,k preserves Stieltjes moment properties of sequences. Finally, we present some related examples including factorial numbers, Whitney numbers, Stirling permutations, minimax trees and peak statistics.

Applications of constructed new families of generating‐type functions interpolating new and known classes of polynomials and numbers

The aim of this article is to construct some new families of generating‐type functions interpolating a certain class of higher order Bernoulli‐type, Euler‐type, Apostol‐type numbers, and polynomials. Applying the umbral calculus convention method and the shift operator to these functions, these generating functions are investigated in many different aspects such as applications related to the finite calculus, combinatorial analysis, the chordal graph, number theory, and complex analysis especially partial fraction decomposition of rational functions associated with Laurent expansion. By using the falling factorial function and the Stirling numbers of the first kind, we also construct new families of generating functions for certain classes of higher order Apostol‐type numbers and polynomials, the Bernoulli numbers and polynomials, the Fubini numbers, and others. Many different relations among these generating functions, difference equation including the Eulerian numbers, the shift operator, minimal polynomial, polynomial of the chordal graph, and other applications are given. Moreover, further remarks and comments on the results of this paper are presented.

A new conformable nabla derivative and its application on arbitrary time scales

In this article, we introduce a new type of conformable derivative and integral which involve the time scale power function widehat{mathcal{G}}_{eta }(t, a) for t,ain mathbb{T}. The time scale power function takes the form (t-a)^{eta } for mathbb{T}=mathbb{R} which reduces to the definition of conformable fractional derivative defined by Khalil et al. (2014). For the discrete time scales, it is completely novel, where the power function takes the form (t-a)^{(eta )} which is an increasing factorial function suitable for discrete time scales analysis. We introduce a new conformable exponential function and study its properties. Finally, we consider the conformable dynamic equation of the form bigtriangledown _{a}^{gamma }y(t)=y(t, f(t)), and study the existence and uniqueness of the solution. As an application, we show that the conformable exponential function is the unique solution to the given dynamic equation. We also examine the analogue of Gronwall’s inequality and its application on time scales.

On Fibonacci Numbers of Order r Which Are Expressible as Sum of Consecutive Factorial Numbers

Let (tn(r))n≥0 be the sequence of the generalized Fibonacci number of order r, which is defined by the recurrence tn(r)=tn−1(r)+⋯+tn−r(r) for n≥r, with initial values t0(r)=0 and ti(r)=1, for all 1≤i≤r. In 2002, Grossman and Luca searched for terms of the sequence (tn(2))n, which are expressible as a sum of factorials. In this paper, we continue this program by proving that, for any ℓ≥1, there exists an effectively computable constant C=C(ℓ)&gt;0 (only depending on ℓ), such that, if (m,n,r) is a solution of tm(r)=n!+(n+1)!+⋯+(n+ℓ)!, with r even, then max{m,n,r}&lt;C. As an application, we solve the previous equation for all 1≤ℓ≤5.

Innovative Value Addition in Tea (Camellia sinensis): A Comparative Analysis between Sri Lanka and Japan

Sri Lanka is the third largest tea exporter in the world, but nearly 60% of its total exports are comprised of bulk tea. Bulk tea secures the lowest prices in world tea market. Sri Lanka earned an average value of US$4.83 per kilogram from 2014 to 2018 for its tea exports, compared to US$24.24 per kilogram over the same five-year period in Japan, a highly innovative tea-producing country. The objective of this study is to examine the root causes of product innovation in Sri Lanka, as compared to Japan, by conducting a historical analysis followed by a survey and key informant discussions. Sri Lanka has mainly focused on incremental innovations in blending, flavoring and packaging than radical innovations. About 98% of the tea produced in Sri Lanka is black tea, produced mainly in large scale-processing factories. The focus on economies of scale and heavy dependence on traditional product baskets and market destinations have bottlenecked product innovation capabilities in Sri Lanka. In contrast, Japan produces a large number of innovative beverage and non-non-beverage products. Their main beverage type is sencha (58%), followed by many other beverages including gyokuro, kabusecha, matcha, tamaryokucha, black tea, oolong tea, fermented teas and white tea. The Japanese produce a variety of non-beverage products including confectionaries, snacks and cosmetics made from same tea plant that beverage tea is made from. Having a large number of processing factories (4698), Japan’s product innovation capabilities are mainly attributed to firm-level factors and support services. This study recommends that Sri Lanka should link together farmer organizations, processing factories, research institutes and universities to make its tea industry more innovative and flexible and consequently more profitable and sustainable. Foreign direct investment and partnerships presumably would be instrumental in bringing in much needed funding, expertise and advanced technology to stimulate product innovation.

La popularización del papel pintado en España a lo largo del siglo XIX

A lo largo del siglo XIX el uso del papel pintado como revestimiento mural decorativo se fue generalizando en toda España. La creación de gran cantidad de fábricas especializadas en este producto ornamental y la incorporación de nuevas tecnologías permitieron aumentar significativamente su producción con el consiguiente abaratamiento del mismo. Paralelamente, el surgimiento de infinidad de pequeños establecimientos de venta ayudaron con su popularización. De esta forma se adaptó a los gustos y modas del momento convirtiéndose en un elemento indispensable de la decoración interior de las viviendas.

Reluctant and Illegal Migrants in Mao's China: Civil Defense Evacuation in the Tianjin Region, 1969–1980

AbstractMore than 100,000 people from the city of Tianjin were evacuated to the countryside in a civil defense program during the 1970s. Many evacuees refused to submit to state migration mandates, instead sneaking back to the city illegally or petitioning to regain urban residency. City officials responded flexibly to the evacuees’ pleas, sympathizing with family reunification and treating suburban districts (jiaoqu) on the outskirts of Tianjin as a buffer zone between city and countryside. Dominated by agriculture but home to a growing number of factories, workshops, and offices during the 1970s, jiaoqu became a solution to evacuation headaches. When compared with the recent coerced movement of hundreds of thousands of Chinese citizens on national security grounds, the civil defense evacuations of the 1970s suggest that it may be misguided to think of the Mao Zedong years as a faraway time that was more radical or repressive than China today.