Search results for: squeeze theorem

Authors: Septimia Sarbu

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The entropy rate of a stochastic process is a fundamental concept in information theory. It provides a limit to the amount of information that can be transmitted reliably over a communication channel, as stated by Shannon's coding theorems. Recently, researchers have focused on developing new measures of information that generalize Shannon's classical theory. The aim is to design more efficient information encoding and transmission schemes. This paper continues the study of generalized entropy rates, by deriving a closed-form solution to the Sharma-Mittal entropy rate for Gaussian processes. Using the squeeze theorem, we solve the limit in the definition of the entropy rate, for different values of alpha and beta, which are the parameters of the Sharma-Mittal entropy. In the end, we compare it with Shannon and Rényi's entropy rates for Gaussian processes.

Keywords: generalized entropies, Sharma-Mittal entropy rate, Gaussian processes, eigenvalues of the covariance matrix, squeeze theorem

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Authors: Mohamed Ben Amar, Najib Souissi, Chedly Bradai

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A Taguchi design investigation has been made into the relationship between the ductility and process variables in a squeeze cast 2017A wrought aluminium alloy. The considered process parameters were: squeeze pressure, melt temperature and die preheating temperature. An orthogonal array (OA), main effect, signal-to-noise (S/N) ratio, and the analysis of variance (ANOVA) are employed to analyze the effect of casting parameters. The results have shown that the selected parameters significantly affect the ductility of 2017A wrought Al alloy castings. Optimal squeeze cast process parameters were provided to illustrate the proposed approach and the results were proven to be trustworthy through practical experiments.

Keywords: Taguchi method, squeeze casting, process parameters, ductility, microstructure

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Authors: Fahad Suleiman, Sammani Abdullahi

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The study depicts that a Wedderburn Artin – theorem for rings is considered to be a semisimple ring R which is isomorphic to a product of finitely many mi by mi matrix rings over division rings Di, for some integers n_i, both of which are uniquely determined up to permutation of the index i. It has been concluded that when R is simple the Wedderburn – Artin theorem is known as Wedderburn’s theorem.

Keywords: Commutativity, Wedderburn theorem, Semisimple ring, R module

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Authors: Emmanuel Kamal Aziz Saba, Gihan Abd El-Lateif Younis El-Tantawi, Mohammed Hamdy Zahran, Ibrahim Khalil Ibrahim, Mohammed Abd El-Salam Shehata, Hussein Al-Moghazy Sultan, Medhat Mohamed Anwar

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Examination of the external anal sphincter muscle strength of voluntary contraction is essential in initial assessment and assessment of efficacy of rehabilitation of patients with faecal incontinence (FI) and obstructed defecation (OD). The present study was conducted to evaluate the digital assessment of the external anal sphincter muscle strength of voluntary contraction by using Modified Oxford Scale (MOS) in comparison to anal manometry squeeze pressure. The present cross-sectional study included 65 patients. There were 40 patients (61.5 %) with FI and 25 patients (38.5 %) with OD. All patients were subjected to history taking, clinical examination including assessment of the external anal sphincter muscle strength of voluntary contraction by using MOS and anal manometry (mean squeeze pressure and maximal squeeze pressure). There was a statistically significant positive correlation between MOS and anal manometry squeeze pressures including mean squeeze pressure and maximal squeeze pressure among FI group and OD group. In conclusion, assessment of the external anal sphincter muscle strength of voluntary contraction by using MOS is a valid method and can substitute anal manometry assessment.

Keywords: anal manometry, external anal sphincter muscle, Modified Oxford Scale, muscle strength

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Authors: Ahmed Almurshedi, Mark Atherton, Mares Cristinel, Tadeusz Stolarski, Masaaki Miyatake

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Transportation and handling of delicate and lightweight objects is a significant issue in some industries. Two levitation prototype designs, a horn transducer design and surface-mounted piezoelectric actuator vibrating plate design, are compared. Both designs are based on the method of squeeze-film levitation (SFL) and the aim of this study is to evaluate the characteristics and performance of each. To this end, physical experiments are conducted and are demonstrated that the horn-type transducer prototype design produces better levitation performance but it design complexity and operating characteristics make it less suitable than the vibrating plate design for practical applications.

Keywords: floating, levitation, piezoelectric, squeeze-film, transducer

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Authors: A. Kraiem, A. Ayadi

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The squeeze flow tests were studied by many authors to measure the rheological properties of fluid. Experimental squeezing flow test with constant area between two parallel disks of bitumen is investigated in the present work. The effect of the temperature, the process of preparing the sample and the gap between the discs were discussed. The obtained results were compared with the theoretical models. The behavior of bitumen depends on the viscosity and the yield stress. Thus, the bitumen was presented as a power law for a small power law exponent and as a biviscous fluid when the viscosity ratio was smaller than one. Also, the influence of the ambient temperature is required for the compression test. Therefore, for a high temperature the yield stress decrease.

Keywords: bitumen, biviscous fluid, squeeze flow, viscosity, yield stress

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Authors: Ayubi Wirara, Ardya Suryadinata

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Improper application of the RSA algorithm scheme can cause vulnerability to attacks. The attack utilizes the relationship between broadcast messages sent to the user with some fixed polynomial functions that belong to each user. Scheme attacks carried out by applying the Chinese Remainder Theorem to obtain a general polynomial equation with the same modulus. The formation of the general polynomial becomes a first step to get back the original message. Furthermore, to solve these equations can use Coppersmith's theorem.

Keywords: RSA algorithm, broadcast message, Chinese Remainder Theorem, Coppersmith’s theorem

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Authors: Mohammed Husein Mohammad Rashid

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For a bounded linear operator T acting on a complex infinite dimensional Hilbert space ℋ, we say that T is ∗-class A operator (abbreviation T∈A*) if |T²|≥ |T*|². In this article, we prove the following assertions:(i) we establish some conditions which imply the normality of ∗-class A; (ii) we consider ∗-class A operator T ∈ ℬ(ℋ) with reducing kernel such that TX = XS for some X ∈ ℬ(K, ℋ) and prove the Fuglede-Putnam type theorem when adjoint of S ∈ ℬ(K) is dominant operators; (iii) furthermore, we extend the asymmetric Putnam-Fuglede theorem the class of ∗-class A operators.

Keywords: fuglede-putnam theorem, normal operators, ∗-class a operators, dominant operators

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Authors: Mohamed Ben Amar, Henda Barhoumi, Hokia Siala, Foued Elhalouani

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The present contribution consists of a purely experimental investigation on the effect of Squeeze casting on the micro structural and mechanical propriety of Al-Si alloys destined to automotive industry. Accordingly, we have proceeding, by ourselves, to all the thermal treatment consisting of solution treatment at 540°C for 8h and aging at 160°C for 4h. The various thermal treatment, have been carried out in order to monitor the processes of formation and dissolution accompanying the solid state phase transformations as well as the resulting changes in the mechanical proprieties. The examination of the micrographs of the aluminum alloys reveals the dominant presence of dendrite. Concerning the mechanical characteristic the Vickers micro-hardness curve an increase as a function of the pressure. As well as the heat treatment increase mechanical propriety such that pressure and micro hardness. The curves have been explained in terms of structural hardening resulting from the various compounds formation.

Keywords: squeeze casting, process parameters, heat treatment, ductility, microstructure

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Authors: Sina Modares Ahmadi, Mohamad Reza Ghazavi, Mandana Sheikhzad

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Squeeze film damper systems (SFD) are often used in machines with high rotational speed to reduce non-periodic behavior by creating external damping. These types of systems are frequently used in aircraft gas turbine engines. There are some structural parameters which are of great importance in designing these kinds of systems, such as oil film thickness, C, and outer race mass, mo. Moreover, there is a crucial parameter associated with manufacturing process, under the title of waviness. Geometric imperfections are often called waviness if its wavelength is much longer than Hertzian contact width which is a considerable source of vibration in ball bearings. In this paper, a system of a flexible rotor and two ball bearings with floating ring squeeze film dampers and consideration of waviness has been modeled and solved by a numerical integration method, namely Runge-Kutta method to investigate the dynamic response of the system. The results show that by increasing the number of wave lobes, which is due to inappropriate manufacturing, non- periodic and chaotic behavior increases. This result reveals the importance of manufacturing accuracy. Moreover, as long as C< 1.5×10-4 m, by increasing the oil film thickness, unwanted vibrations and non-periodic behavior of the system have been reduced, On the other hand, when C>1.5×10-4 m, increasing the outer oil film thickness results in the increasing chaotic and non-periodic responses. This result shows that although the presence of oil film results in reduction the non-periodic and chaotic behaviors, but the oil film has an optimal thickness. In addition, with increasing mo, the disc displacement amplitude increases. This result reveals the importance of utilizing light materials in manufacturing the squeeze film dampers.

Keywords: squeeze-film damper, waviness, ball bearing, bifurcation

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Authors: Ahmed Lotfy, Andrey V. Pozdniakov, Vadim C. Zolotorevskiy

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The purpose of this research is to provide a competitive alternative to aluminum silicon alloys used in automotive applications. This alternative was created by developing three types of composites Al-5%Cu- (B₄C, BN or Si₃N₄) particulates with a low coefficient of thermal expansion. Stir casting was used to synthesis composites containing 2, 5 and 7 wt. % of B₄C, Si₃N₄ and 2, 5 of BN followed by squeeze casting. The squeeze casting process decreased the porosity of the final composites. The composites exhibited a fairly uniform particle distribution throughout the matrix alloy. The microstructure and XRD results of the composites suggested a significant reaction occurred at the interface between the particles and alloy. Increasing the aging temperature from 200 to 250°C decreased the hardness values of the matrix and the composites and decreased the time required to reach the peak. Turner model was used to calculate the expected values of thermal expansion coefficient CTE of matrix and its composites. Deviations between calculated and experimental values of CTE were not exceeded 10%. Al-5%Cu-B₄C composites experimentally showed the lowest values of CTE (17-19)·10-6 °С-1 and (19-20) ·10-6 °С-1 in the temperature range 20-100 °С and 20-200 °С respectively.

Keywords: aluminum matrix composites, coefficient of thermal expansion, X-ray diffraction, squeeze casting, electron microscopy,

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Authors: Zhiyan Meng, Dan Liu, Jintao Meng

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Dependable encrypted traffic classification is crucial for improving cybersecurity and handling the growing amount of data. Large language models have shown that learning from large datasets can be effective, making pre-trained methods for encrypted traffic classification popular. However, attention-based pre-trained methods face two main issues: their large neural parameters are not suitable for low-computation environments like mobile devices and real-time applications, and they often overfit by getting stuck in local minima. To address these issues, we developed a lightweight transformer model, which reduces the computational parameters through lightweight vocabulary construction and Squeeze-and-Excitation Block. We use sharpness-aware optimization to avoid local minima during pre-training and capture temporal features with relative positional embeddings. Our approach keeps the model's classification accuracy high for downstream tasks. We conducted experiments on four datasets -USTC-TFC2016, VPN 2016, Tor 2016, and CICIOT 2022. Even with fewer than 18 million parameters, our method achieves classification results similar to methods with ten times as many parameters.

Keywords: sharpness-aware optimization, encrypted traffic classification, squeeze-and-excitation block, pretrained model

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Authors: Yu Lu

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Fixed point theory is a basic tool for the study of the existence of Nash equilibria in game theory. This paper presents a significant generalization of the Veinott-Zhou fixed point theorem for increasing correspondences, which serves as an essential framework for investigating the existence of Nash equilibria in supermodular and quasisupermodular games. To establish our proofs, we explore different conceptions of multivalued increasingness and provide comprehensive results concerning the existence of the largest/least fixed point. We provide two distinct approaches to the proof, each offering unique insights and advantages. These advancements not only extend the applicability of the Veinott-Zhou theorem to a broader range of economic scenarios but also enhance the theoretical framework for analyzing equilibrium behavior in complex game-theoretic models. Our findings pave the way for future research in the development of more sophisticated models of economic behavior and strategic interaction.

Keywords: fixed-point, Tarski’s fixed-point theorem, Nash equilibrium, supermodular game

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Authors: Kilari Nikhil, Ankur Tibrewal, Srinivas Kruthiventi S. S.

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Conventional speaker identification systems often fall short of capturing the diverse variations present in speech data due to fixed-scale architectures. In this research, we propose a CNN-based architecture, USENet, designed to overcome these limitations. Leveraging two key techniques, our approach achieves superior performance on the VoxCeleb 1 Dataset without any pre-training. Firstly, we adopt a U-net-inspired design to extract features at multiple scales, empowering our model to capture speech characteristics effectively. Secondly, we introduce the squeeze and excitation block to enhance spatial feature learning. The proposed architecture showcases significant advancements in speaker identification, outperforming existing methods, and holds promise for future research in this domain.

Keywords: multi-scale feature extraction, squeeze and excitation, VoxCeleb1 speaker identification, mel-spectrograms, USENet

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Authors: Azad Hussain, Andrew Cobley

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The demand for lightweight parts with high mechanical strength(s) and integrity, in sectors such as the aerospace and automotive is ever increasing, motivated by the need for weight reduction in order to increase fuel efficiency with components usually manufactured using a high grade primary metal or alloy. For components manufactured using the squeeze casting process, this alloy is usually A356 aluminium (Al), it is one of the most versatile Al alloys; and is used extensively in castings for demanding environments. The A356 castings provide good strength to weight ratio making it an attractive option for components where strength has to be maintained, with the added advantage of weight reduction. In addition, the versatility in castabilitiy, weldability and corrosion resistance are other attributes that provide for the A356 cast alloy to be used in a large array of industrial applications. Conversely, it is rare to use remelted Al in these cases, due the nature of the applications of components in demanding environments, were material properties must be defined to meet certain specifications for example a known strength or ductility. However the use of remelted Al, especially primary grade Al such as A356, would offer significant cost and energy savings for manufacturers using primary alloys, provided that remelted aluminium can offer similar benefits in terms of material microstructure and mechanical properties. This study presents the results of the material microstructure and properties of 100% primary A356 Al and 100% remelt Al cast, manufactured via the direct squeeze cast method. The microstructures of the castings made from remelted A356 Al were then compared with the microstructures of primary A356 Al. The outcome of using remelting Al on the microstructure was examined via different analytical techniques, optical microscopy of polished and etched surfaces, and scanning electron microscopy. Microstructural analysis of the 100% remelted Al when compared with primary Al show similar α-Al phase, primary Al dendrites, particles and eutectic constituents. Mechanical testing of cast samples will elucidate further information as to the suitability of utilising 100% remelt for casting.

Keywords: A356, microstructure, remelt, squeeze casting

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Authors: M. H. M. Rashid

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Given H a Hilbert space and B(H) the algebra of bounded linear operator in H, let δAB denote the generalized derivation defined by A and B. The main objective of this article is to study Weyl type theorems for generalized derivation for (A,B) satisfying a couple of Fuglede.

Keywords: Fuglede Property, Weyl’s theorem, generalized derivation, Aluthge transform

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Authors: Li Kewen, Su Zhaoxin, Wang Xingmou, Zhu Jian Bing

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Conventional reservoir prediction methods ar not sufficient to explore the implicit relation between seismic attributes, and thus data utilization is low. In order to improve the predictive classification accuracy of reservoir lithology, this paper proposes a deep learning lithology prediction method based on ResNet (Residual Neural Network) and SENet (Squeeze-and-Excitation Neural Network). The neural network model is built and trained by using seismic attribute data and lithology data of Shengli oilfield, and the nonlinear mapping relationship between seismic attribute and lithology marker is established. The experimental results show that this method can significantly improve the classification effect of reservoir lithology, and the classification accuracy is close to 70%. This study can effectively predict the lithology of undrilled area and provide support for exploration and development.

Keywords: convolutional neural network, lithology, prediction of reservoir, seismic attributes

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Authors: Tze Jin Wong, Lee Feng Koo, Pang Hung Yiu

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Lenstra’s attack uses Chinese remainder theorem as a tool and requires a faulty signature to be successful. This paper reports on the security responses of fourth and sixth order Lucas based (LUC_4,6) cryptosystem under the Lenstra’s attack as compared to the other two Lucas based cryptosystems such as LUC and LUC₃ cryptosystems. All the Lucas based cryptosystems were exposed mathematically to the Lenstra’s attack using Chinese Remainder Theorem and Dickson polynomial. Result shows that the possibility for successful Lenstra’s attack is less against LUC_4,6 cryptosystem than LUC₃ and LUC cryptosystems. Current study concludes that LUC_4,6 cryptosystem is more secure than LUC and LUC₃ cryptosystems in sustaining against Lenstra’s attack.

Keywords: Lucas sequence, Dickson polynomial, faulty signature, corresponding signature, congruence

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Authors: Mohamed Houas, Maamar Benbachir

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In this paper, under weak assumptions, we study the existence and uniqueness of solutions for a nonlinear fractional boundary value problem. New existence and uniqueness results are established using Banach contraction principle. Other existence results are obtained using scheafer and krasnoselskii's fixed point theorem. At the end, some illustrative examples are presented.

Keywords: caputo derivative, boundary value problem, fixed point theorem, local conditions

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Authors: Essam Shalaby, Alexander Churyumov, Malak Abou El-Khair, Atef Daoud

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A comparative study of the thermal and mechanical behavior of squeezed A359 composites containing 5, 10 and 15 wt.% SiC, (SiC+ Si3N4) and AlN particulates was investigated. Stir followed by squeeze casting techniques are used to produce A359 composites. It was noticed that, A359/AlN composites have high thermal conductivity as compared to A359 alloy and even to A359/SiC or A359/(SiC+Si3N4) composites. Microstructures of the composites have shown homogeneous and even distribution of reinforcements within the matrix. Interfacial reactions between particles and matrix were investigated using X-ray diffraction and energy dispersive X-ray analysis. The presence of particles led not only to increase peak hardness of the composites but also to accelerate the aging kinetics. As compared with A359 matrix alloy, compression test of the composites has exhibited a significant increase in the yield and the ultimate compressive strengths with a relative reduction in the failure strain. Those light weight composites have a high potential to be used for automotive and aerospace applications.

Keywords: metal-matrix composite, squeeze, microstructure, thermal conductivity, compressive properties

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Authors: Thanin Sitthiwirattham, Jiraporn Reunsumrit

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We study the existence of positive solutions to the three points difference summation boundary value problem. We show the existence of at least one positive solution if f is either superlinear or sublinear by applying the fixed point theorem due to Krasnoselskii in cones.

Keywords: positive solution, boundary value problem, fixed point theorem, cone

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Authors: Paola Cattabriga

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According to Gödel’s first incompleteness argument it is possible to construct a formally undecidable proposition in Principia mathematica, a statement that, although true, turns out to be neither provable nor refutable for the system, making therefore incomplete any formal system suitable for the arithmetic of integers. Its features and limitation effects are today widespread basics throughout whole scientific thought. This article brings Gödel’s achievement into question by the definition of the refutability predicate as a number-theoretical statement. We develop proof of invalidity of Theorem VI in Gödel’s 1931, the so-called Gödel’s first incompleteness theorem, in two steps: defining refutability within the same recursive status as provability and showing that as a consequence propositions (15) and (16), derived from definition 8.1 in Gödel’s 1931, are false and unacceptable for the system. The achievement of their falsity blocks the derivation of Theorem VI, which turns out to be therefore invalid, together with all the depending theorems. This article opens up thus new perspectives for mathematical research and for the overall scientific reasoning.

Keywords: Gödel numbering, incompleteness, provability predicate, refutability predicate

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Authors: Hao-Hsi Tseng, Hsin-Yun Lee

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The Most Advantageous Tender (MAT) has been criticized for its susceptibility to dictatorial situations and for its processing of same score, same rank issues. This study applies the four criteria from Arrow's Impossibility Theorem to construct a mechanism for revealing illegitimate scores in scoring methods. While commonly be used to improve on problems resulting from extreme scores, ranking methods hide significant defects, adversely affecting selection fairness. To address these shortcomings, this study relies mainly on the overall evaluated score method, using standardized scores plus normal cumulative distribution function conversion to calculate the evaluation of vender preference. This allows for free score evaluations, which reduces the influence of dictatorial behavior and avoiding same score, same rank issues. Large-scale simulations confirm that this method outperforms currently used methods using the Impossibility Theorem.

Keywords: Arrow’s impossibility theorem, cumulative normal distribution function, most advantageous tender, scoring method

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Authors: A. Almurshedi, M. Atherton, C. Mares, T. Stolarski, M. Miyatake

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Transportation and handling of delicate and lightweight objects is currently a significant issue in some industries. Two common contactless movement prototype designs, ultrasonic transducer design and vibrating plate design, are compared. Both designs are based on the method of squeeze-film levitation, and this study aims to identify the limitations, and challenges of each. The designs are evaluated in terms of levitation capabilities, and characteristics. To this end, theoretical and experimental explorations are made. It is demonstrated that the ultrasonic transducer prototype design is better suited to the terms of levitation capabilities. However, the design has some operating and mechanical designing difficulties. For making accurate industrial products in micro-fabrication and nanotechnology contexts, such as semiconductor silicon wafers, micro-components and integrated circuits, non-contact oil-free, ultra-precision and low wear transport along the production line is crucial for enabling. One of the designs (design A) is called the ultrasonic chuck, for which an ultrasonic transducer (Langevin, FBI 28452 HS) comprises the main part. Whereas the other (design B), is a vibrating plate design, which consists of a plain rectangular plate made of Aluminium firmly fastened at both ends. The size of the rectangular plate is 200x100x2 mm. In addition, four rounded piezoelectric actuators of size 28 mm diameter with 0.5 mm thickness are glued to the underside of the plate. The vibrating plate is clamped at both ends in the horizontal plane through a steel supporting structure. In addition, the dynamic of levitation using the designs (A and B) has been investigated based on the squeeze film levitation (SFL). The input apparatus that is used with designs consist of a sine wave signal generator connected to an amplifier type ENP-1-1U (Echo Electronics). The latter has to be utilised to magnify the sine wave voltage that is produced by the signal generator. The measurements of the maximum levitation for three different semiconductor wafers of weights 52, 70 and 88 [g] for design A are 240, 205 and 187 [um], respectively. Whereas the physical results show that the average separation distance for a disk of 5 [g] weight for design B reaches 70 [um]. By using the methodology of squeeze film levitation, it is possible to hold an object in a non-contact manner. The analyses of the investigation outcomes signify that the non-contact levitation of design A provides more improvement than design B. However, design A is more complicated than design B in terms of its manufacturing. In order to identify an adequate non-contact SFL design, a comparison between two common such designs has been adopted for the current investigation. Specifically, the study will involve making comparisons in terms of the following issues: floating component geometries and material type constraints; final created pressure distributions; dangerous interactions with the surrounding space; working environment constraints; and complication and compactness of the mechanical design. Considering all these matters is essential for proficiently distinguish the better SFL design.

Keywords: ANSYS, floating, piezoelectric, squeeze-film

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Authors: Mukhtari Hussaini Muhammad, Abba Adamu

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The paper will examine the approaches and solutions that will be offered by Senior Secondary School II Students (Demonstration Secondary School, Azare Bauchi State Northern Nigeria – Hausa/ Fulani predominant area) toward solving exercises related to the circle theorem. The angle that an arc of a circle subtends at the center is twice that which it subtends at any point on the remaining part of the circumference. The Students will be divided in to 2 groups by given them numbers 1, 2; 1, 2; 1, 2, then all 1s formed group I and all 2s formed group II. Group I will be considered as control group in which the traditional method will be applied during instructions. Thus, the researcher will revise the concept of circle, state the theorem, prove the theorem and then solve examples. Group II, experimental group in which the concept of circle will be revised to the students and then the students will be asked to draw different circles, mark arcs, draw angle at the center, angle at the circumference then measure the angles constructed. The students will be asked to explain what they can infer/deduce from the angles measured and lastly, examples will be solved. During the next contact day, both groups will be subjected to solving exercises in the classroom related to the theorem. The angle that an arc of a circle subtends at the center is twice that which it subtends at any point on the remaining part of circumference. The solution to the exercises will be marked, the scores compared/analysed using relevant statistical tool. It is expected that group II will perform better because of the method/ technique followed during instructions is more learner-centered. By exploiting the talents of the individual learners through listening to the views and asking them how they arrived at a solution will really improve learning and understanding.

Keywords: circle theorem, control group, experimental group, traditional method

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Authors: Hemant Kumar Pathak

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In 1994, Matthews (S.G. Matthews, Partial metric topology, in: Proc. 8th Summer Conference on General Topology and Applications, in: Ann. New York Acad. Sci., vol. 728, 1994, pp. 183-197) introduced the concept of a partial metric as a part of the study of denotational semantics of data flow networks. He gave a modified version of the Banach contraction principle, more suitable in this context. In fact, (complete) partial metric spaces constitute a suitable framework to model several distinguished examples of the theory of computation and also to model metric spaces via domain theory. In this paper, we introduce the concept of almost partial Hausdorff metric. We prove a fixed point theorem for multi-valued mappings on partial metric space using the concept of almost partial Hausdorff metric and prove an analogous to the well-known Nadler’s fixed point theorem. In the sequel, we derive a homotopy result as an application of our main result.

Keywords: fixed point, partial metric space, homotopy, physical sciences

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Authors: Ti-Jun Xiao, Zhe Xu

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Controllability is one of the fundamental issues in control systems. In this paper, we study the controllability of second order evolutional control systems in Hilbert spaces with memory and boundary controls, which model dynamic behaviors of some viscoelastic materials. Transferring the control problem into a moment problem and showing the Riesz property of a family of functions related to Cauchy problems for some integrodifferential equations, we obtain a general boundary controllability theorem for these second order evolutional control systems. This controllability theorem is applicable to various concrete 1D viscoelastic systems and recovers some previous related results. It is worth noting that Riesz sequences can be used for numerical computations of the control functions and the identification of new Riesz sequence is of independent interest for the basis-function theory. Moreover, using the Riesz sequences, we obtain the existence and uniqueness of (weak) solutions to these second order evolutional control systems in Hilbert spaces. Finally, we derive the exact boundary controllability of a viscoelastic beam equation, as an application of our abstract theorem.

Keywords: evolutional control system, controllability, boundary control, existence and uniqueness

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Authors: D. S. Palimkar

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Random fixed point theory has received much attention in recent years, and it is needed for the study of various classes of random equations. The study of random fixed point theorems was initiated by the Prague school of probabilistic in the 1950s. The existence and uniqueness of fixed points for the self-maps of a metric space by altering distances between the points with the use of a control function is an interesting aspect in the classical fixed point theory. In a new category of fixed point problems for a single self-map with the help of a control function that alters the distance between two points in a metric space which they called an altering distance function. In this paper, we prove the results of existence of random common fixed point and its uniqueness for a pair of random mappings under weakly contractive condition for generalizing alter distance function in polish spaces using Random Common Fixed Point Theorem for Generalized Weakly Contractions.

Keywords: Polish space, random common fixed point theorem, weakly contractive mapping, altering function

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Authors: Eugene Benilov, Mikhail Benilov

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The Enskog-Vlasov (EV) equation is a widely used semi-phenomenological model of gas/liquid phase transitions. We show that it does not generally conserve energy, although there exists a restriction on its coefficients for which it does. Furthermore, if an energy-preserving version of the EV equation satisfies an H-theorem as well, it can be used to rigorously derive the so-called Maxwell construction which determines the parameters of liquid-vapor equilibria. Finally, we show that the EV model provides an accurate description of the thermodynamics of noble fluids, and there exists a version simple enough for use in applications.

Keywords: Enskog collision integral, hard spheres, kinetic equation, phase transition

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Authors: Ather Saeed, Arif Khan, Jeffrey Gosper

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Sensor devices are prone to errors and sudden node failures, which are difficult to detect in a timely manner when deployed in real-time, hazardous, large-scale harsh environments and in medical emergencies. Therefore, the loss of data can be life-threatening when the sensed phenomenon is not disseminated due to sudden node failure, battery depletion or temporary malfunctioning. We introduce a set of partial differential equations for localizing faults, similar to Green’s and Maxwell’s equations used in Electrostatics and Electromagnetism. We introduce a node organization and clustering scheme for self-stabilizing sensor networks. Green’s theorem is applied to regions where the curve is closed and continuously differentiable to ensure network connectivity. Experimental results show that the proposed GTFD (Green’s Theorem fault-detection and Self-stabilization) protocol not only detects faulty nodes but also accurately generates network stability graphs where urgent intervention is required for dynamically self-stabilizing the network.

Keywords: Green’s Theorem, self-stabilization, fault-localization, RSSI, WSN, clustering

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