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The 1.5 inch pipe rolling machine utilizes an electric motor to drive the pipe holder and a 2ton jack as a pipe press. Although the roller machine is widely used in conventional workshops by using human power, it causes the pipe rolling time to be long and the machine user gets tired quickly. This is due to lack of resources to think about developing and improving the performance of these machines. Therefore, the focus of this research is the design of a 1.5 inch pipe rolling machine using an electric motor as a drive and a 2 ton jack as a pipe press. This research aims to produce a design and machine for 1.5 inch pipe rolling and to help accelerate the pipe rolling process and reduce worker fatigue. This pipe rolling machine consists of several components, namely, the engine frame, electric motor, gearbox reducer, chain-sprocket, shaft, pipe drive wheel, hydraulic pump and so on. The working process of this machine begins by preparing a 1.5 inch x 3 meter pipe then the pipe is positioned above the drive wheel, lowers the upper pressure wheel to the desired depth with a 2ton jack power, turn on the electric motor and press the button to roll the pipe back and forth and so on. up to pipe roll size as needed. The stages in making this machine are design, detailed drawing, manufacturing process, quality control, assembly and testing, data retrieval and data analysis. The research resulted in drawings of the design and manufacture of the machine, the calculation of the rotational speed of the machine for optimal rolling obtained 20.46 rpm and an average pipe rolling time of 6.80 minutes for one workpiece. This machine can still be optimized by adding a workpiece movement limiter when rolling.

An autonomous vertical take-off and landing (VTOL) must be supported with an accurate positioning system, especially for autonomous take-off, landing, and other tasks in small area. This paper presents a novel method of small local outdoor positioning system for localizing the area of dropping and landing of autonomous VTOL by utilizing the low-cost precision ultra-wide band (UWB) ranging system. We compared symmetrical single sided-two way ranging (SSS-TWR), symmetrical double sided-two way ranging (SDS-TWR), and asymmetrical double sided-two way ranging (ADS-TWR) methods to get precision ranging measurement on UWB radio module. ADS-TWR was superior to others by resulting in minimum distance error. The ADS-TWR average error was 1.38 % (35.88 cm), SDS-TWR average error was 1.83 % (47.58 cm), and SSS-TWR average error was 2.73 % (70.98 cm). Furthermore, the trilateration method was utilized to obtain the local position of the autonomous VTOL. The trilateration method successfully implemented autonomous VTOL quadcopter positioning in a small local outdoor area (20 m x 30 m). Autonomous VTOL has been able to drop seven payloads in seven areas (2 m x 2 m) and landed in the home position (3 m x 3 m) successfully.

Combinatorial t-designs have nice applications in coding theory, finite geometries and engineering areas. t-designs can be constructed from image sets of a fixed size of some special polynomials. This paper constructs t-designs from the quadratic polynomial $$x^{4}+x^{3}$$ over $$\mathrm {GF}(2^{n})$$ and determine their parameters. We yield 2- $$\left( 2^n,3\cdot 2^{n-2},3\cdot 2^{n-2}\left( 3\cdot 2^{n-2}-1 \right) \right) $$ designs for n even and 3- $$\left( 2^n,2^{n-1},2^{n-1}\left( 2^{n-2}-1 \right) \right) $$ designs for n odd.

Let Dm be an elliptic curve over ℚ of the form y2 = x3 − m2x + m2, where m is an integer. In this paper we prove that the two points P−1 = (−m, m) and P0 = (0, m) on Dm can be extended to a basis for Dm(ℚ) under certain conditions described explicitly.

In the present paper, we consider the complete asymptotic expansion of certain exponential-type operators connected with $$2x^{3/2}$$ 2 x 3 / 2 . Also, a modification of such exponential-type operators is provided, which preserve the function $$\mathrm{e}^{Ax}$$ e Ax .

A programmable logic array (PLA) is a kind of programmable logic device used to implement combinational logic circuits. The PLA has a set of programmable AND gate planes, which link to a set of programmable OR gate planes, which can then be conditionally complemented to produce an output. It has 2N AND gate s for N input variables and for M outputs from PLA, there should be M OR gates, each with programmable inputs from all of the AND gates. This layout allows for many logic functions to be synthesized in the sum of products canonical forms. The Programmable Logic Array (PLA) has a programmable AND array followed by a programmable OR array. Programmable Logic Array (PLA) circuit is built by using AND gates and OR gates. The 3x 4 bits data can be stored in this circuit. The large storage data bits of PLA circuit store by a using large AND-OR array with lots of inputs and product terms, and programmable connections. Programmable Logic Array circuit functions as ROM circuit.

In this paper, we investigate the relative essential spectra of a 3 x 3 block matrix operator with unbounded entries and with domain consisting of vectors satisfying certain relations between their components. Our results are formulated in term of relative generalized weak demicompactness and measure of non-strict-singularity.

The purpose of this paper is to introduce a system that uses the combination of working aid blind techniques to help blind people to have better navigation against objects, also to be aware of road and traffic lights signs using real-time image-processing. This study aims at improving the processing speed of object detection by introducing the latest Raspberry Pi 4 module, which is more powerful than the previous versions. As a result, the Single-Shot Multibox Detector MobileNet v2 convolutional neural network on Raspberry Pi 4 using TensorFlow Lite 2, is employed for object detection. A model called SSD MobileNet v2 320x320, which is trained on the COCO dataset from the TensorFlow model zoo, which is used in this system. Also, the traffic light signs classification is achieved via machine learning techniques. With all modules attached on the hat, video frames captured from the piCamera module are analyzed and searched for known objects by the Raspberry Pi module and playing the pre-recorded sound of the detected object(s) at an approximate distance away from the user. MobileNet uses 3 x 3 depthwise separable convolutions, which uses between 8 to 9 times lesser computing resources compared to the standard convolution at a small cost in accuracy. Thus, this makes the device suitable for the resource constraint devices such as the Raspberry Pi. The model achieves about 5 frames per second on the Raspberry P 4.

We find an upper bound for the number of limit cycles, bifurcating from the 8-loop of the Duffing oscillator $x"= x-x^{3}$ under the special cubic perturbation $$ x"= x-x^{3}+\lambda_{1}y+\lambda_{2}x^{2}+\lambda_{3}xy+\lambda_{4}x^{2}y . $$, Comment: 4 figures, minor corrections, references added. arXiv admin note: text overlap with arXiv:1306.2340

At THz frequencies, the limited penetration through interior walls resulting from significant transmission attenuation losses likely confines the propagation mechanisms to inside rooms. Distributed antenna systems (DAS) being a choice to maintain reliable 100 Gbit/s wireless links are employed in this paper to study the THz coverage in the presence of multiple humans. The study environment is a non-cubic 3D office room model of 7 x 7 x 3 m 3 at our Institute of Digital Signal Processing (DSV) as shown in Fig. 1 (a) . For this purpose, we employ up to five 3D models of both male (1.76 m high) and female (1.70 m high) clothed bodies designed in a 3D computer graphic software tool and then imported as an object (computation intensive with 48,090 polygons/faces) to ray-tracing tool. Upon entering the office room, the human models follow certain trajectories till they reach their respective workplaces. Moreover, the gait motion phases (in a total walking cycle) along with a predefined step size for each human body model is taken into account. Assessing the coverage, the transmitting access point antenna is placed at different locations (i.e., middle and all four corners) in the room and the coverage maps are then generated at two chosen heights of the receiving antenna grid as depicted in Fig. 1 (b) . The chosen RXPs are 0.25 m from the floor when taking into account the worst-case coverage whereas in case of table top terminals these possess a height of 0.75 m. There are 72,900 receiver points (RXPs) covering the entire office room, required to study the frequency selective fading. The details of the human body models together with the antenna positions in addition to simulation parameters as well as material parameters at 300 GHz will be conferred in the full paper.

Let p = 8k + 5, q = 8k + 3 be the twin prime pair for some nonnegative integer k. Assume that $$\left({{5 \over p}} \right) = - 1$$ or $$\left({{7 \over q}} \right) = - 1$$ . In this paper, we prove that the elliptic curve Epq: y2 = x3 + (pq − 12)x − 2(pq − 8) has unique integral point (2, 0).

Let p be a fixed odd prime. We combine some properties of quadratic and quartic Diophantine equations with elementary number theory methods to determine all integral points on the elliptic curve E: y2 = x3 − 4p2x. Further, let N(p) denote the number of pairs of integral points (x, ±y) on E with y > 0. We prove that if p ⩾ 17, then N(p) ⩽ 4 or 1 depending on whether p ≡ 1 (mod 8) or p ≡ −1 (mod 8).

The establishment of international standards for 5-axis control machining centers has been supported by the high interest of each country. Internationally, various accuracy inspection methods have been proposed and widely discussed. Accuracy measuring devices for these purposes have also been proposed. In 2014, inspection methods for 5-axis machines were published in ISO 10791-6 and 10791-7. In this research, we propose a test method to process 9 square faces as a new accuracy evaluation method. We simulate the influence of assembly error by the proposed square 3 × 3 machining method on the machined surface. By processing 9 square faces with different tool angle on the same plane, it was possible to evaluate the influence of assembly errors in the 5-axis machining center on the machined surface. Nine surfaces machined by the square 3 × 3 processing method cause differences in surface height due to alignment errors. In addition, nine machined surfaces become all diagonal not parallelism. The alignment errors of the 5-axis machining center is identified by evaluating the orientation of the machined surfaces. Specifically, we propose a newly method to measure the height difference of nine surfaces. Then, the possibility of identifying the alignment error of the 5-axis machining center using the measurement results is shown.

The present article deals with the approximation of certain exponential type operators defined by Ismail and May. We estimate the rate of convergence of these operators for functions of bounded variation.

The main aim of the present paper is to extensively study the γ-rigid Bohr Hamiltonian with anharmonic sextic oscillator potential for the variable β and γ = 0. For the corresponding spectral problem, a finite number of eigenvalues are explicitly found, by algebraic means, the so-called quasi-exact solvability (QES). The evolution of the spectral and electromagnetic properties by considering higher exact solvability orders is investigated, especially the approximate degeneracy of the ground and first two β bands at the critical point of the shape phase transition from a harmonic to an anharmonic prolate β-soft, as well as the shape evolution within an isotopic chain. The numerical results are given for 39 nuclei, namely, 98–108Ru, 100–102Mo, 116–130Xe, 180–196Pt, 172Os, 146–150Nd, 132–134Ce, 152–154Gd, 154–156Dy, 150–152Sm, 190Hg and 222Ra. Across this study, it seems that the higher QES order improves our results by decreasing the root mean square, mostly for deformed nuclei. The nuclei 100,104Ru, 118,120,126,128Xe, 148Nd and 172Os fall exactly at the critical point.

Due to the complexity of modeling the radiative transfer inside the accretion columns of neutron star binaries, their X-ray spectra are still commonly described with phenomenological models, for example, a cutoff power law. While the behavior of these models is well understood and they allow for a comparison of different sources and studying source behavior, the extent to which the underlying physics can be derived from the model parameters is very limited. During recent years, several physically motivated spectral models have been developed to overcome these limitations. Their application, however, is generally computationally much more expensive and they require a high number of parameters which are difficult to constrain. Previous works have presented an analytical solution to the radiative transfer equation inside the accretion column assuming a velocity profile that is linear in the optical depth. An implementation of this solution that is both fast and accurate enough to be fitted to observed spectra is available as a model in XSPEC. The main difficulty of this implementation is that some solutions violate energy conservation and therefore have to be rejected by the user. We propose a novel fitting strategy that ensures energy conservation during the $\chi^2$-minimization which simplifies the application of the model considerably. We demonstrate this approach as well a study of possible parameter degeneracies with a comprehensive Markov-chain Monte Carlo analysis of the complete parameter space for a combined NuSTAR and Swift/XRT dataset of Cen X-3. The derived accretion-flow structure features a small column radius of $\sim$63 m and a spectrum dominated by bulk-Comptonization of bremsstrahlung seed photons, in agreement with previous studies., Comment: 14 pages, 7 figures, A&A accepted

In recent years, with the increasing complexity of equipment, system design and safety analysis for the equipment are becoming more and more difficult. In this paper, we make a more comprehensive analysis of the hierarchical relationship of the system from the vertical and horizontal angles and put forward the 3 × 3 Hierarchical Model. Then, we propose a failure propagation analysis method based on 3 × 3 Hierarchical Model. At the same time, SysML is used to build the models, which is convenient for different people to learn and modify the models. Finally, a hydraulic transmission system is taken as an example to illustrate the feasibility of the method.

Let 𝓖 be the set of all 3 X 3 non-singular matrices ( a b c d e f g h i ) , where a, b, c, d, e, f, g, h, i are integers modulo p. Then 𝓖 is a group under matrix multiplication modulo p, of order (p n − 1) (p n -p) (p n -p 2) … … (p n -p n −1). Let G be the subgroup of 𝓖 defined by G = { ( a b c d e f g h i ) ∈ G : | a b c d e f g h i | = 1 } . Then G is of order ( p n − 1 ) ( p n − p ) ( p n − p 2 ) … … ( p n − p n − 1 ) p − 1 . Let L(G) be the lattice formed by all subgroups G. In this paper, we give the structure of the subgroups of order 21of L(G) in the case when P=2.

A chord diagram is a set of chords of a circle such that no pair of chords has a common endvertex. A chord diagram E is called nonintersecting if E contains no crossing. For a chord diagram E having a crossing S = { x 1 x 3 , x 2 x 4 } , the expansion of E with respect to S is to replace E with E 1 = ( E ∖ S ) ∪ { x 2 x 3 , x 4 x 1 } or E 2 = ( E ∖ S ) ∪ { x 1 x 2 , x 3 x 4 } . For a chord diagram E , let f ( E ) be the chord expansion number of E , which is defined as the cardinality of the multiset of all nonintersecting chord diagrams generated from E with a finite sequence of expansions. In this paper, it is shown that the chord expansion number f ( E ) equals the value of the Tutte polynomial at the point ( 2 , − 1 ) for the interlace graph G E corresponding to E . The chord expansion number of a complete multipartite chord diagram is also studied. An extended abstract of the paper was published (Nakamigawa and Sakuma, 2017) [ 13 ].

The aim of this paper is to study the varieties of ai-semirings satisfying \({x^{3}\approx x}\) . It is shown that the collection of all such varieties forms a distributive lattice of order 179. Also, all of them are finitely based and finitely generated. This generalizes and extends the main results obtained by Ghosh et al., Pastijn and Ren and Zhao.

We present an efficient approach for determining the cardinality of the set of points on each elliptic curve of the family $\mathcal{E}_p = \{E_a :y^2 = x^3 + a\,(mod\,p),a \in \mathbb{Z}_p^* \}$ by applying the famous Hasse’s bound together with an explicit formula for that cardinality reduced to modulo p which is derived by us. As a by-product it is shown that for fixed p ≡ 1 (mod 6) those cardinalities take exactly six distinct values. The latter permits us to provide a reasoning why the curve E 7 employed by Bitcoin system has an optimal resistance among the curves from ℰ p against the EC variant of Pohlig-Hellman algorithm when p is the actual modulo used in that system. This approach allows also to give another proof of the known result that in case p ≡ 2 (mod 3) the cardinalities of all curves from ℰ p are equal to p + 1.

Motivated by the recent result of Farhi we show that for each $n\equiv \pm 1\pmod{6}$ the title Diophantine equation has at least two solutions in integers. As a consequence, we get that each (even) perfect number is a sum of three cubes of integers. Moreover, we present some computational results concerning the considered equation and state some questions and conjectures.

We study the variation of the φ-Selmer groups of the elliptic curvesy2=x3−Dxunder quartic twists by square-free integers. We obtain a complete description of the distribution of the size of this group when the integerDis constrained to lie in a family for which the relative Tamagawa number of the isogeny φ is fixed.

In complete semigroups of unions Bx(D) defined by semilattices of class Σ2(X, 4), where |X| = 3, we select subsets of a certain type on which a binary equivalent relation is defined. By means of this relation, we describe irreducible generating sets of considered semigroups.

The aim of this paper is to study the varieties of semilattice-ordered Burnside semigroups satisfying $$x^3\approx x$$ and $$xy\approx yx.$$ It is shown that the collection of all such varieties forms a distributive lattice of order 9. Also, all of them are finitely based and finitely generated. This gives a generalization and expansion of the results obtained by McKenzie and Romanowska (Contrib Gen Algebra Proc Klagenf Conf 1978 1:213–218, 1979).

In smart factories, the entire manufacturing process from design to the final product is simulated in a virtual manufacturing environment and optimized before starting production. Suppliers and customers make decisions based on the simulation results. Therefore, effective rendering of the information of the virtual products to suppliers and customers is essential for this manufacturing paradigm. In this study, a method of rendering the surface roughness of the virtual products using a tactile display is presented. A tactile display device comprising a 3 x 3 array of individually controlled piezoelectric stack actuators is constructed. The surface topology of the virtual products is rendered directly by controlling the piezoelectric stack actuators. A series of experiments is performed to evaluate the performance of the tactile display device. An electrical discharge machined surface is rendered using the proposed method.

The parameter-independent (up to overall scale factors) predictions of the X(5)−β 2,X(5)−β 4, and X(3) models, which are variants of the X(5) critical point symmetry developed within the framework of the geometric collective model, are compared to two-parameter calculations in the framework of the interacting boson approximation (IBA) model. The results show that these geometricmodels coincide with IBA parameters consistentwith the phase/shape transition region of the IBA for boson numbers of physical interest (close to 10). 186Pt and 172Os are identified as good examples of X(3), while 146Ce, 174Os, and 158Er, 176Os are identified as good examples of X(5)−β 2 and X(5)−β 4 behavior, respectively

We developed a cubic keyboard to exploit the three-dimensional (3D) space of virtual reality (VR) environments. The user enters a word by drawing a stroke with the controller. The keyboard consists of 27 keys arranged in a 3 x 3 x 3 (vertical, horizontal, and depth) 3D array; all 26 letters of the alphabet are assigned to 26 keys; the center key is blank. The user moves the controller to the key of a letter of the word and then selects that key by slowing movement.

Building action recognition systems that are simultaneously fast, robust and requiring small memory space is very challenging. Though current state-of-the-art frameworks proposed the best performance on accuracy, speed or memory, they do not offer simultaneously best performance on all three metrics. Thus, it is still possible to achieve a good trade-off among all these three metrics. Using a compact patch-based pattern, this paper introduces a novel binary motion descriptor to efficiently describe motion in video. The descriptor, namely the Proximity Patches Motion (PPM), compares in two different ways a 3 x 3 central patch centered on a detected keypoint with other 24 patches compactly positioned around it between three consecutive frames. Experimental results on the Weizmann and KTH datasets show that the proposed method not only requires a small amount of memory, but is also faster and achieves competitive accuracy when compared to the state-of-the-art spatio-temporal binary descriptors.

A broadband lumped-element quadrature hybrid is proposed and implemented for intermediate frequency (IF) applications. The proposed topology not only enables a wide fractional bandwidth (FBW) of 33% with only 7 elements, but also makes the required capacitance and inductance values lower, which is helpful for size reduction when establishing elements using interdigital capacitor and spiral inductor implementation instead of 0402 package element implementation. The size of the proposed quadrature hybrid is only 8 mm X 3 mm. Measured S11, S21, S31 and S41 are better than −14, −4.3, −3.6 and −13 dB from 50 to 70 MHz, respectively. And the measured inband phase difference between the coupled and direct ports is within 91°±1 0. Yield analysis, discussions and size comparisons are also implemented.

Speakman and Lee (2017) gave a formula for the volume of the convex hull of the graph of a trilinear monomial, y = x 1 x 2 x 3 , over a box in the nonnegative orthant, in terms of the upper and lower bounds on the variables. This was done in the context of using volume as a measure for comparing alternative convexifications to guide the implementation of spatial branch-and-bound for mixed integer nonlinear optimization problems. Here, we introduce an alternative method for computing this volume, making use of the rich theory of mixed volumes. This new method may lead to a natural approach for considering extensions of the problem.

We study the optimal time-decay and the L-2-stability of classical solutions to the two-species relativistic Landau-Maxwell system in the whole space R-x(3). The global existence of this system has been established by Yang and Yu [34] in the perturbative regime of global Maxwellian. Based on our previous works on the optimal time-decay for the Vlasov-Poisson-Boltzmann system, we prove that for this system and its simpler model, the relativistic Landau Poisson system, every order derivative of the solutions converges to the global Maxwellian at an optimal time decay rate. Moreover, the uniform L2-stability of the solutions in Yang and Yu [34] is also provided. (C) 2015 Elsevier Inc. All rights reserved.

Non-polynomial spline functions of the form Span{1, x, x 2 , x 3 , x 4 , x 5 , cos ( kx ) + e x } , where k can be real or pure imaginary, are used to find the numerical solution of linear fifth-order boundary value problems. The order of convergence of the method is observed to be of O ( h 2 ) . A fifth order convergent method is defined with the help of improved end-conditions. Three examples are considered to show the reliability and efficiency of the method. The numerical results, obtained, endorse the improved order of convergence of the method.

This paper presents a step towards the design of robust non-fragile power system stabilizers (PSSs) for single-machine infinite-bus systems. To ensure resiliency of a robust PSS, the proposed approach presents a characterization of all stabilizers that can guarantee robust stability (RS) over wide range of operating conditions. A three-term controller ( x 1 + x 2 s)/(1 + x 3 s) is considered to accomplish the design. Necessary and sufficient stability constraints for existing of such controller at certain operating point are derived via Routh–Hurwitz criterion. Continuous variation in the operating point is tackled by an interval plant model where RS problem is reduced to simultaneous stabilization of finite number of plants according to Kharitonov theorem. Controller triplets that can robustly stabilize vertex plants are characterized in a similar manner. The most resilient controller is computed at the center of maximum-area inscribed rectangle. Simulation results confirm robustness and resiliency of the proposed stabilizer.

Suppose that y 2 = x 3 +4px, y 2 = x 3 4px are elliptic curves where p is an odd prime. Then we treat ranks of these two curves according to the condition of prime number p. Mathematics Subject Classication: 11A41, 11G05

In this paper, a novel design of a broadband 3 x 3 Butler matrix is proposed. It is composed of three single-section coupled-line directional couplers and two Schiffman phase shifters and operates in the frequency range 2.5 GHz – 3.5 GHz. The fabricated 3 x 3 Butler matrix features very good frequency response in terms of transmission imbalance, isolation and impedance match.

A large number of computational intelligence algorithms are proposed every year and their performance is usually demonstrated on a cross-section of problems which may not provide an ideal comparative analysis. Hence it becomes important to independently evaluate the performance of such algorithms on complex optimization problems. In this work, we have used the benchmark suite of IEEE CEC2014 which has been extensively used in literature to study the performance of two recently proposed algorithms, viz., JAYA and Sine Cosine Algorithm. The performance is analyzed based on 9180 instances (30 functions X 3 dimensions X 51 runs X 2 algorithms) comprising of unimodal, multimodal, hybrid and composite benchmark functions involving up to 50 real variables.

A new nonconforming rectangle element with cubic convergence for the energy norm is introduced. The degrees of freedom (DOFs) are defined by the twelve values at the three Gauss points on each of the four edges. Due to the existence of one linear relation among the above DOFs, it turns out the DOFs are eleven. The nonconforming element consists of P2 � Span{x 3 y xy 3 }. We count the corresponding dimension for Dirichlet and Neumann boundary value problems of second-order elliptic problems. We also present the optimal error estimates in both broken energy and L2() norms. Finally, numerical examples match our theoretical results very well.

Modified physical and geometric parameters for the eclipsing x-ray binary system Cen X-3 are presented. The parameters were estimated by comparing synthetic photometric light curves with the observed ones in an iterative method until the best fit was achieved. The synthetic light curves were constructed in accordance with the Roche model, since Cen X-3 is likely to be powered by Roche-lobe overflow. We focused on the phenomenon of x-ray heating of the side of the optical component facing the compact object. The parameters and present status of this work are briefly discussed.

This article will describe a writing assignment designed for use in a liberal arts college whose mission stresses effective written communication both within and across disciplines. In this assignment, students write three separate solutions to the same mathematics problem: one for a mathematical peer, a second for a contemporary that does not necessarily have the mathematical background of the writer, and a third that can be understood by a general reader of the local newspaper. This paper will outline the framework of the assignment, provide examples of student work, detail assessment strategies, and give samples of student feedback.

This paper investigates the semi-online machine covering problem on two hierarchical machines where the jobs are correspondingly classified into two hierarchical classes. The objective is to maximize the minimum machine load. We show that if we only know the size of the largest job, no algorithm with a bounded competitive ratio exists. So we consider the case where we know both the size and the class of the largest job. If we know the size of the largest job and that it belongs to the higher class, then an optimal algorithm with a ( 1 + 2 2 ) -competitive ratio exists. If we know the size of the largest job and that it belongs to the lower class, we design an optimal algorithm with an α-competitive ratio, where α ≈ 2.48119 is the largest root of the equation x 3 − 2 x 2 − 2 x + 2 = 0 . For the case where the total size of all the jobs is known in advance, we show that the competitive ratio of an optimal algorithm is 2.

An orthogonal beam-forming network (BFN) is proposed for 4G pattern-diversity applications. Different from the traditional Butler beam-forming networks with 2 N orthogonal beams, the orthogonal BFN, composed of three 180 ◦ hybrids and a 90 ◦ phase shifter, provides three orthogonal beams. Design procedure of the orthogonal BFN based on the factorization of its transmission matrix is derived. Moreover, in order to implement the proposed orthogonal BFN with low insertion loss, a rat- race has been used to realize unequal power distribution between its two output ports. The measured scattering parameters of the orthogonal BFN are compared with the analytical and the simulated scattering parameters, validating the expected behavior. In addition, by varying the output power ratio of the non-equi-amplitude 180 ◦ hybrid, the performance of the orthogonal BFN is improved when the proposed orthogonal BFN is used in an antenna array.

In this paper we show the existence of three new families of spatial central configurations for the 7-body problem. More precisely, the position vectors x 1 , x 2 , x 3 , x 4 and x 5 are at the vertices of a square pyramid, the position vectors x 6 and x 7 are on a line connecting x 5 with the center of the opposite square.

In this paper, we calculate the number of points on elliptic curves y2 = x3 + Ax over Fpr modulo 24. This is a generalization of (8), (9) and (16).