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Complex networks have been a prominent topic of research for several years, spanning a wide range of fields from mathematics to computer science and also to social and biological sciences. The eigenvalues of the Seidel matrix, Seidel Signless Laplacian matrix, Seidel energy, Seidel Signless Laplacian energy, Maximum and Minimum energy, Degree Sum energy and Distance Degree energy of the Unitary Cayley graphs [UCG] have been calculated. Low-power devices must be able to transfer data across long distances with low delay and reliability. To overcome this drawback a small-world network depending on the unitary Cayley graph is proposed to decrease the delay and increase the reliability and is also used to create and analyze network communication. Small-world networks based on the Cayley graph have a basic construction and are highly adaptable. The simulation result shows that the small-world network based on unitary Cayley graphs has a shorter delay and is more reliable. Furthermore, the maximum delay is lowered by 40%. [ABSTRACT FROM AUTHOR]

The symmetric algorithm is a variant of the well-known Euler-Seidel method which has proven useful in the study of linearly recurrent sequences. In this paper, we introduce a multivariate generalization of the symmetric algorithm which reduces to it when all parameters are unity. We derive a general explicit formula via a combinatorial argument and also an expression for the row generating function. Several applications of our algorithm to the q-Fibonacci and q-hyper-Fibonacci numbers are discussed. Among our results is an apparently new recursive formula for the Carlitz Fibonacci polynomials. Finally, a (p, q)-analogue of the algorithm is introduced and an explicit formula for it in terms of the (p, q)-binomial coefficient is found. [ABSTRACT FROM AUTHOR]

A class of non-oriented simple graphs is called Seidel switching self-complementary (s.s.c. for short) if the complement of any representing graph is in the same equivalence class. Relating the Seidel adjacency matrix of a graph with a Gram matrix, we introduce the 3-signature ( s , t ) of a switching class of n -vertex graphs. The numbers s and t are the numbers of positive and negative triples within any representing graph of the class. It appears that, for any s.s.c. class of n -vertex graphs, these numbers are equal, yielding ( n 3 ) even. Consequently if n ≡ 3 ( mod 4 ) then there is no s.s.c. class of n -vertex graphs. We also prove that this 3-signature depends only on the spectrum of the adjacency matrix of the graph. We then consider the switching classes of Paley conference graphs with 4 k + 2 vertices, 4 k + 1 = p α , p an odd prime and α a positive integer. We reprove that these classes are s.s.c. Moreover, it is proven that all 4 k -vertex graphs contained in a ( 4 k + 2 ) -vertex Paley conference graph are switching equivalent and their class is still a s.s.c. class. In addition, the 3-signature is generalized in view of obtaining a complete invariant of switching classes up to order 8. [ABSTRACT FROM AUTHOR]

We present simple analytical formulae for the calculation of the spectral phase and residual angular dispersion of an ultrashort pulse propagating through the Offner stretcher. Based on these formulae, we show that the radii of curvature of both convex and concave mirrors in the Offner triplet can be adapted to tune the fourth order dispersion term of the spectral phase of the pulse. As an example, a single-grating Offner stretcher design suitable for the suppression of material dispersion in the Ti:Sa PALS laser system is proposed. The results obtained by numerical raytracing well match those calculated from the analytical formulae. [ABSTRACT FROM AUTHOR]

In this paper Seidel matrices are studied, and their spectrum and several related algebraic properties are determined for order n ≤ 13 . Based on this Seidel matrices with exactly three distinct eigenvalues of order n ≤ 23 are classified. One consequence of the computational results is that the maximum number of equiangular lines in R 12 with common angle 1 ∕ 5 is exactly 20 . [ABSTRACT FROM AUTHOR]

In his Opticks, published in 1704, Newton thought he had demonstrated the impossibility to eliminate chromatic aberration for lens systems. However, in 1748 Euler proposed a theory supposed to eliminate both chromatic and spherical aberrations in telescopes' objectives. A controversy soon ensued between Dollond who built the first achromatic glasses in 1758 and Euler whose theory appeared to be problematical because it was based on a wrong law of light scattering. In the early 1760s Clairaut and D'Alembert developed in competition a theory of optical aberrations that addresses both the chromatic aberration and geometric aberrations, including off‐axis, which appeared in four of the five so‐called Seidel aberrations. However several sociological, technological, and scientific factors precluded its application to the manufacturing of optical instruments. Subsequent scholars thus ignored these contributions until almost a century later. This article examines the work of Clairaut and D'Alembert, which undoubtedly constitutes one of the major progresses in theoretical optics in the age of Enlightenment, and intends to offer a new light on this period, which is often considered as an era of stagnation in optics. [ABSTRACT FROM AUTHOR]

This study addresses the problem of speech quality enhancement by adaptive and nonadaptive filtering algorithms. The well-known two-microphone forward blind source separation (TM-FBSS) structure has been largely studied in the literature. Several two-microphone algorithms combined with TM-FBSS have been recently proposed. In this study, we propose 2 contributions: In the first, a new two-microphone Gauss-Seidel pseudo affine projection (TM-GSPAP) algorithm is combined with TM-FBSS. In the second, we propose to use the new TM-GSPAP algorithm in speech enhancement. Furthermore, we show the efficiency of the proposed TM-GSPAP algorithm in speech enhancement when highly noisy observations are available. To validate the good performances of our algorithm, we have evaluated the adaptive filtering properties in computational complexity and convergence speed performance by system mismatch criteria. A fair comparison with adaptive and nonadaptive noise reduction algorithms are also presented. The adaptive algorithms are the well-known two-microphone normalized least mean square algorithm, and the recently published two-microphone pseudo affine projection algorithm. The nonadaptive algorithms are the one-microphone spectral subtraction and the two-microphone Wiener filter algorithm. We evalute the quality of the output speech signal in each algorithm by several objective and subjective criteria as the segmental signal-to-noise ratio, cepstral distance, perceptual evaluation of speech quality, and the mean opinion score. Finally, we validate the superior performances of the proposed algorithm with physically measured signals. [ABSTRACT FROM AUTHOR]

Considering that numerous sample data points are required in the probabilistic method, a non-probabilistic interval analysis method can be an alternative when the information is insufficient. In the paper, new strategies, which are iterative algorithm based interval uncertainty analysis methods (IA-IUAMs), are developed to acquire the bounds of the responses in multidisciplinary system. Two iterative processes, Jacobi iteration and Seidel iteration, are applied in the new methods respectively. The Jacobi iteration based interval uncertainty analysis method (JI-IUAM) utilizes the strategy of concurrent subsystem analysis to improve computational efficiency while the Seidel iteration based interval uncertainty analysis method (SI-IUAM) can accelerate convergence by utilizing the newest information. Both IA-IUAMs are able to evaluate the bounds of responses accurately and quickly. The presented methods are compared with general sensitivity analysis based interval uncertainty analysis method (SIUAM) and conventional Monte Carlo simulation approach (MCS). The validity and efficiency of the new methods are demonstrated by two numerical examples and two engineering examples. Results show that, on the one hand, IA-IUAMs are more efficient than MCS by avoiding hundreds of system analyses, on the other hand, IA-IUAMs are more accurate and have a wider range of application than SIUAM by avoiding linear approximation and global sensitivity calculation. [ABSTRACT FROM AUTHOR]

In this study defining Shifted Euler-Seidel matrices we generalize the Euler-Seidel matrices method. Owing to this generalization one can investigate any sequences (sn) which have two term linear recurrences as sm+n = αsm+n-1+ βsn-1 (α and β are real parameters and n,m ϵ Z+). By way of illustration, we give some examples related to the Fibonacci p-numbers. [ABSTRACT FROM AUTHOR]

In this paper, the Jacobi and Gauss–Seidel-type iteration methods are proposed for solving the matrix equation A X B = C , which are based on the splitting schemes of the matrices A and B . The convergence and computational cost of these iteration methods are discussed. Furthermore, we give the preconditioned Jacobi and Gauss–Seidel-type iteration methods. Numerical examples are given to demonstrate the efficiency of these methods proposed in this paper. [ABSTRACT FROM AUTHOR]

The Seidel method for solving a system of linear algebraic equations and an estimate of its convergence rate are considered. It is proposed to change the order of equations. It is shown that the method described in Faddeevs' book Computational Methods of Linear Algebra can deteriorate the convergence rate estimate rather than improve it. An algorithm for establishing the optimal order of equations is proposed, and its validity is proved. It is shown that the computational complexity of the reordering is 2 n additions and (12) n divisions. Numerical results for random matrices of order 100 are presented that confirm the proposed improvement. [ABSTRACT FROM AUTHOR]

In this paper the problem of convergence of the important type of a multidimensional generalization of continued fractions, the branched continued fractions with independent variables, is considered. This fractions are an efficient apparatus for the approximation of multivariable functions, which are represented by multiple power series. When variables are fixed these fractions are called the branched continued fractions of the special form. Their structure is much simpler then the structure of general branched continued fractions. It has given a possibility to establish the necessary and sufficient conditions of convergence of branched continued fractions of the special form with the positive elements. The received result is the multidimensional analog of Seidel's criterion for the continued fractions. The condition of convergence of investigated fractions is the divergence of series, whose elements are continued fractions. Therefore, the sufficient condition of the convergence of this fraction which has been formulated by the divergence of series composed of partial denominators of this fraction, is established. Using the established criterion and Stieltjes-Vitali Theorem the parabolic theorems of branched continued fractions of the special form with complex elements convergence, is investigated. The sufficient conditions gave a possibility to make the condition of convergence of the branched continued fractions of the special form, whose elements lie in parabolic domains. [ABSTRACT FROM AUTHOR]

We examine the propagation of a complex-source-point spherical wave via a concave non-spherical grating. First we trace the paraxial paths of a chief ray propagating from a complex source point through the system to complex image points of different orders. Then we represent the wave function of the light in terms of Seidel-type aberrations, while the terms of up to fourth order in aperture variables are taken into account. Based on the derived formula, we also evaluate the quality of a Gaussian beam degraded by the system. We show how the beam qualities in the sagittal and tangential sections are influenced by the conic constant (or the aspheric coefficient of fourth order) and the positions of source and stop. [ABSTRACT FROM AUTHOR]

We use quilted Floer theory to generalize Seidel's long exact sequence in symplectic Floer theory to fibered Dehn twists. We then apply the sequence to construct versions of the Floer and Khovanov-Rozansky exact triangles in Lagrangian Floer theory of moduli spaces of bundles. [ABSTRACT FROM AUTHOR]

We propose a correspondence approximation approach between temporally adjacent frames for motion analysis. First, energy map is established to represent image spatial features on multiple scales using Gaussian convolution. On this basis, energy flow at each layer is estimated using Gauss-Seidel iteration according to the energy invariance constraint. More specifically, at the core of energy invariance constraint is "energy conservation law" assuming that the spatial energy distribution of an image does not change significantly with time. Finally, energy flow field at different layers is reconstructed by considering different smoothness degrees. Due to the multiresolution origin and energy-based implementation, our algorithm is able to quickly address correspondence searching issues in spite of background noise or illumination variation. We apply our correspondence approximation method to motion analysis, and experimental results demonstrate its applicability. [ABSTRACT FROM AUTHOR]

This paper introduces a new dictionary learning strategy based on atoms obtained by translating the composition of $$K$$ convolutions with $$S$$ -sparse kernels of known support. The dictionary update step associated with this strategy is a non-convex optimization problem. We propose a practical formulation of this problem and introduce a Gauss-Seidel type algorithm referred to as alternative least square algorithm for its resolution. The search space of the proposed algorithm is of dimension $$KS$$ , which is typically smaller than the size of the target atom and much smaller than the size of the image. Moreover, the complexity of this algorithm is linear with respect to the image size, allowing larger atoms to be learned (as opposed to small patches). The conducted experiments show that we are able to accurately approximate atoms such as wavelets, curvelets, sinc functions or cosines for large values of K. The proposed experiments also indicate that the algorithm generally converges to a global minimum for large values of $$K$$ and $$S$$ . [ABSTRACT FROM AUTHOR]

The Gauss-Seidel limb scattering (GSLS) radiative transfer (RT) model simulates the transfer of solar radiation through the atmosphere and is imbedded in the retrieval algorithm used to process data from the Ozone Mapping and Profiler Suite (OMPS) limb profiler (LP), which was launched on the Suomi NPP satellite in October 2011. A previous version of this model has been compared with several other limb scattering RT models in previous studies, including Siro, MCC++, CDIPI, LIMBTRAN, SASKTRAN, VECTOR, and McSCIA. To address deficiencies in the GSLS radiance calculations revealed in earlier comparisons, several recent changes have been added that improve the accuracy and flexibility of the GSLS model, including 1. improved treatment of the variation of the extinction coefficient with altitude, both within atmospheric layers and above the nominal top of the atmosphere; 2. addition of multiple-scattering source function calculations at multiple solar zenith angles along the line of sight (LOS); 3. introduction of variable surface properties along the limb LOS, with minimal effort required to add variable atmospheric properties along the LOS as well; 4. addition of the ability to model multiple aerosol types within the model atmosphere. The model improvements 1 and 2 are verified by comparison to previously published results (using standard radiance tables whenever possible), demonstrating significant improvement in cases for which previous versions of the GSLS model performed poorly. The single-scattered radiance errors that were as high as 4 % in earlier studies are now generally reduced to 0.3%, while total radiance errors generally decline from 10 % to 1-3 %. In all cases, the tangent height dependence of the GSLS radiance error is greatly reduced. [ABSTRACT FROM AUTHOR]

Freeform surfaces enable imaginative optics by providing abundant degrees of freedom for an optical designer as compared to spherical surfaces. An off-axis two-mirror-based telescope design is presented, in which the primary mirror is a concave prolate spheroid and the secondary mirror is freeform surface-based. The off-axis configuration is employed here for removing the central obscuration problem which otherwise limits the central maxima in the point spread function. In this proposed design, an extended X - Y polynomial is used as a surface descriptor for the off-axis segment of the secondary mirror. The coefficients of this extended polynomial are directly related to the Seidel aberrations, and are thus optimized here for a better control of asymmetric optical aberrations at various field points. For this design, the aperture stop is located 500 mm before the primary mirror and the entrance pupil diameter is kept as 80 mm. The effective focal length is 439 mm and covers a full field of view of 2 deg. The image quality obtained here is near diffraction limited which can be inferred from metrics such as the spot diagram and modulation transfer function. [ABSTRACT FROM AUTHOR]

This Seidel Triangle Sequence Calculus makes it possible to derive several three-variate generating functions, in particular for the Bi-Entringer numbers, which count the alternating permutations according to their lengths, first and last letters. The paper has been motivated by this surprising observation: the number of alternating permutations, whose last letter has a prescribed value and is greater than its first letter, is equal to the Poupard number. [ABSTRACT FROM AUTHOR]

A 3-year-old neutered male Scottish Fold cat weighing 4.4 kg presented with a corneal laceration of the left eye. The cornea was not perforated, as determined by a Seidel test. Vision was normal. The defect was successfully repaired using deep lamellar keratectomy and application of a tectonic graft, using glycerin-preserved canine corneal stroma. Thirty-two days after surgery, corneal neovascularization surrounding the graft was evident. Forty-six days after surgery, part of the graft was slightly cloudy but the cornea was mostly clear. Therefore, canine glycerin-preserved corneal stroma may be suitable for use in tectonic heterologous grafting to repair deep lamellar keratectomy during keratoplasty in cats. [ABSTRACT FROM AUTHOR]

Optical blur due to lens aberrations and defocus has been demonstrated in some cases to be spatially varying across the image plane. However, existing models in the literature for the point-spread function (PSF) corresponding to this blur are either parameterized and spatially invariant or spatially varying but ad-hoc and discretely defined. A parameterized model is developed and presented for a spatially varying PSF due to lens aberrations and defocus in an imaging system. The model is motivated from an established theoretical framework in physics and is demonstrated to be able to unify a set of hundreds of PSF observations across the image plane into a single 10-parameter model. The accuracy of the model is demonstrated with simulations and measurement data collected by two separate research groups. [ABSTRACT FROM AUTHOR]

A Euler{Seidel matrix is determined by an infinite sequence whose elements are given by recursion. The recurrence relations are investigated for numbers and polynomials such as hyperharmonics, Lucas numbers, and Euler and Genocchi polynomials. Linear recurring sequences in finite fields are employed, for instance, in coding theory and in several branches of electrical engineering. In this work, we define the period of a Euler{Seidel matrix over a field 픽p with p elements, where p is a prime number. We give some results for the matrix whose initial sequence is {sr(n)}n=0∞, where sr(n) = ..., n ≥ 0 and r is a fixed positive number. The numbers sr(n) play an important role in combinatorics and number theory. These numbers are known as 픽ranel numbers for r = 3. [ABSTRACT FROM AUTHOR]

The unknown vector estimation problem with bandwidth constrained wireless sensor network is considered. In such networks, sensor nodesmake distributed observations on the unknown vector and collaborate with a fusion center to generate a final estimate. Due to power and communication bandwidth limitations, each sensor nodemust compress its data and transmit to the fusion center. In this paper, both centralized and decentralized estimation frameworks are developed. The closed-form solution for the centralized estimation framework is proposed. The computational complexity of decentralized estimation problem is proven to be NP-hard and a Gauss-Seidel algorithm to search for an optimal solution is also proposed. Simulation results show the good performance of the proposed algorithms. [ABSTRACT FROM AUTHOR]

A Euler{Seidel matrix is determined by an infinite sequence whose elements are given by recursion. The recurrence relations are investigated for numbers and polynomials such as hyperharmonics, Lucas numbers, and Euler and Genocchi polynomials. Linear recurring sequences in finite fields are employed, for instance, in coding theory and in several branches of electrical engineering. In this work, we define the period of a Euler{Seidel matrix over a field 픽p with p elements, where p is a prime number. We give some results for the matrix whose initial sequence is {sr(n)}n=0∞, where sr(n) = ..., n ≥ 0 and r is a fixed positive number. The numbers sr(n) play an important role in combinatorics and number theory. These numbers are known as 픽ranel numbers for r = 3. [ABSTRACT FROM AUTHOR]

Abstract: This paper uses the third-order aberration theory to design a reflective zoom optical system. The advantage of this method is that, by solving a set of Seidel aberration coefficient functions, it allows the designer to achieve the initial construction parameters of the optical system, i.e., radii of curvature and separations between mirrors, etc. In order to verify the accuracy of this method, a prototype of a coaxial zoom optical system with three mirrors is designed. This paper introduces the manufacturing process of the prototype in detail, describes the procedures of aligning and experimenting, and analyzes the experiment results in the end. In this way, the feasibility of this method is confirmed. [Copyright &y& Elsevier]

Abstract: Recently it has been shown that all non-trivial closed permutation groups containing the automorphism group of the random poset are generated by two types of permutations: the first type are permutations turning the order upside down, and the second type are permutations induced by so-called rotations. In this paper we introduce rotations for finite posets, which can be seen as the poset counterpart of Seidel-switch for finite graphs. We analyze some of their combinatorial properties, and investigate in particular the question of when two finite posets are rotation-equivalent. We moreover give an explicit combinatorial construction of a rotation of the random poset whose image is again isomorphic to the random poset. As a corollary of our results on rotations of finite posets, we obtain that the group of rotating permutations of the random poset is the automorphism group of a homogeneous structure in a finite language. [Copyright &y& Elsevier]

The purpose of this paper is to revive interest in the problem of caustics with symmetry. Small imperfections in the point focus of a perfect lens are classified by the Seidel aberrations, while catastrophe theory deals satisfactorily with large ones. But the middle ground where an initial wave front is not perfectly spherical, and yet retains some symmetry, has proved more difficult to occupy. Caustics from water-drop lenses provide a motivation, but the analysis given is not restricted to such lenses. We start from a water-drop lens whose periphery has n-fold rotational symmetry combined with reflection symmetry (Dn). In 3D the unperturbed caustic is a flared and scalloped double cone with n rib lines. When n is even, the two cones are mutually rotated. Perturbing the drop by one of the terms suggested by catastrophe theory produces a sequence of caustics. It unfolds the original caustic. With parabolic umbilic singularities acting as off-stage organising centres, butterfly singularities and beak-to-beak events change the number of cusps in a cross-section until the caustic may eventually become n − 2 cusped triangles. We present such sequences for a number of low-order perturbations and find that identical sequences are produced if the perturbation is held fixed and the screen is moved. The two ways are related by power laws. For n = 6, a practical way of making a perturbation would be provided by the Bragg bubble model of a metal. Iluminating a single bubble in one of the hexagonally symmetric rafts would produce a hexagonally symmetric caustic that could be perturbed by distorting the raft elastically. The treatment can be extended to cover the chiral point groups Cn. [ABSTRACT FROM AUTHOR]