Recursive Methods for Construction of Balanced N-ary Block Designs

2000 Mathematics Subject Classification: Primary 05B05; secondary 62K10.

Lq estimates for rearrangements of Fourier series

This work is concerned with estimating the upper envelopes S* of the absolute values of the partial sums of rearranged trigonometric sums. A.M. Garsia [Annals of Math. 79 (1964), 634-9] gave an estimate for the L₂ norms of the S*, averaged over all rearrangements of the original (finite) sum. This estimate enabled him to prove that the Fourier series of any function in L₂ can be rearranged so that it converges a.e. The main result of this thesis is a similar estimate of the L_q norms of the S*, for all even integers q. This holds for finite linear combinations of functions which satisfy a condition which is a generalization of orthonormality in the L₂ case. This estimate for finite sums is extended to Fourier series of L_q functions; it is shown that there are functions to which the Men’shov-Paley Theorem does not apply, but whose Fourier series can nevertheless be rearranged so that the S* of the rearranged series is in L_q.

Study of and design procedure for dual circularly polarized waveguide slot arrays

For the past sixty years, waveguide slot radiator arrays have played a critical role in microwave radar and communication systems. They feature a well-characterized antenna element capable of direct integration into a low-loss feed structure with highly developed and inexpensive manufacturing processes. Waveguide slot radiators comprise some of the highest performance—in terms of side-lobe-level, efficiency, etc. — antenna arrays ever constructed. A wealth of information is available in the open literature regarding design procedures for linearly polarized waveguide slots. By contrast, despite their presence in some of the earliest published reports, little has been presented to date on array designs for circularly polarized (CP) waveguide slots. Moreover, that which has been presented features a classic traveling wave, efficiency-reducing beam tilt. This work proposes a unique CP waveguide slot architecture which mitigates these problems and a thorough design procedure employing widely available, modern computational tools. The proposed array topology features simultaneous dual-CP operation with grating-lobe-free, broadside radiation, high aperture efficiency, and good return loss. A traditional X-Slot CP element is employed with the inclusion of a slow wave structure passive phase shifter to ensure broadside radiation without the need for performance-limiting dielectric loading. It is anticipated this technology will be advantageous for upcoming polarimetric radar and Ka-band SatCom systems. The presented design methodology represents a philosophical shift away from traditional waveguide slot radiator design practices. Rather than providing design curves and/or analytical expressions for equivalent circuit models, simple first-order design rules – generated via parametric studies — are presented with the understanding that device optimization and design will be carried out computationally. A unit-cell, S-parameter based approach provides a sufficient reduction of complexity to permit efficient, accurate device design with attention to realistic, application-specific mechanical tolerances. A transparent, start-to-finish example of the design procedure for a linear sub-array at X-Band is presented. Both unit cell and array performance is calculated via finite element method simulations. Results are confirmed via good agreement with finite difference, time domain calculations. Array performance exhibiting grating-lobe-free, broadside-scanned, dual-CP radiation with better than 20 dB return loss and over 75% aperture efficiency is presented.

The length-weight relationship in Sardinella sirm Wal from the Andaman Sea

Preliminary details are given of studies conducted regarding the length-weight relationship of Sardinella sirm from the Andaman Sea, a species of economic importance in the area. Results show that this species grows in proportion to length, weight and girth; the group of fish studied was composed of the same population and did not reveal any subspecies of sub-variety.

基于信息重建技术的地震信号处理方法研究

The dissertation addressed the problems of signals reconstruction and data restoration in seismic data processing, which takes the representation methods of signal as the main clue, and take the seismic information reconstruction (signals separation and trace interpolation) as the core. On the natural bases signal representation, I present the ICA fundamentals, algorithms and its original applications to nature earth quake signals separation and survey seismic signals separation. On determinative bases signal representation, the paper proposed seismic dada reconstruction least square inversion regularization methods, sparseness constraints, pre-conditioned conjugate gradient methods, and their applications to seismic de-convolution, Radon transformation, et. al. The core contents are about de-alias uneven seismic data reconstruction algorithm and its application to seismic interpolation. Although the dissertation discussed two cases of signal representation, they can be integrated into one frame, because they both deal with the signals or information restoration, the former reconstructing original signals from mixed signals, the later reconstructing whole data from sparse or irregular data. The goal of them is same to provide pre-processing methods and post-processing method for seismic pre-stack depth migration. ICA can separate the original signals from mixed signals by them, or abstract the basic structure from analyzed data. I surveyed the fundamental, algorithms and applications of ICA. Compared with KL transformation, I proposed the independent components transformation concept (ICT). On basis of the ne-entropy measurement of independence, I implemented the FastICA and improved it by covariance matrix. By analyzing the characteristics of the seismic signals, I introduced ICA into seismic signal processing firstly in Geophysical community, and implemented the noise separation from seismic signal. Synthetic and real data examples show the usability of ICA to seismic signal processing and initial effects are achieved. The application of ICA to separation quake conversion wave from multiple in sedimentary area is made, which demonstrates good effects, so more reasonable interpretation of underground un-continuity is got. The results show the perspective of application of ICA to Geophysical signal processing. By virtue of the relationship between ICA and Blind Deconvolution , I surveyed the seismic blind deconvolution, and discussed the perspective of applying ICA to seismic blind deconvolution with two possible solutions. The relationship of PC A, ICA and wavelet transform is claimed. It is proved that reconstruction of wavelet prototype functions is Lie group representation. By the way, over-sampled wavelet transform is proposed to enhance the seismic data resolution, which is validated by numerical examples. The key of pre-stack depth migration is the regularization of pre-stack seismic data. As a main procedure, seismic interpolation and missing data reconstruction are necessary. Firstly, I review the seismic imaging methods in order to argue the critical effect of regularization. By review of the seismic interpolation algorithms, I acclaim that de-alias uneven data reconstruction is still a challenge. The fundamental of seismic reconstruction is discussed firstly. Then sparseness constraint on least square inversion and preconditioned conjugate gradient solver are studied and implemented. Choosing constraint item with Cauchy distribution, I programmed PCG algorithm and implement sparse seismic deconvolution, high resolution Radon Transformation by PCG, which is prepared for seismic data reconstruction. About seismic interpolation, dealias even data interpolation and uneven data reconstruction are very good respectively, however they can not be combined each other. In this paper, a novel Fourier transform based method and a algorithm have been proposed, which could reconstruct both uneven and alias seismic data. I formulated band-limited data reconstruction as minimum norm least squares inversion problem where an adaptive DFT-weighted norm regularization term is used. The inverse problem is solved by pre-conditional conjugate gradient method, which makes the solutions stable and convergent quickly. Based on the assumption that seismic data are consisted of finite linear events, from sampling theorem, alias events can be attenuated via LS weight predicted linearly from low frequency. Three application issues are discussed on even gap trace interpolation, uneven gap filling, high frequency trace reconstruction from low frequency data trace constrained by few high frequency traces. Both synthetic and real data numerical examples show the proposed method is valid, efficient and applicable. The research is valuable to seismic data regularization and cross well seismic. To meet 3D shot profile depth migration request for data, schemes must be taken to make the data even and fitting the velocity dataset. The methods of this paper are used to interpolate and extrapolate the shot gathers instead of simply embedding zero traces. So, the aperture of migration is enlarged and the migration effect is improved. The results show the effectiveness and the practicability.

An investigation into formal and functional characteristics of qualifications in legislative writing and its application to English for academic legal purposes.

The present investigation is based on a linguistic analysis of the 'Housing Act 1980' and attempts to examine the role of qualifications in the structuring of the legislative statement. The introductory chapter isolates legislative writing as a "sub-variety “of legal language and provides an overview of the controversies surrounding the way it is written and the problems it poses to its readers. Chapter two emphasizes the limitations of the available work on the description of language-varieties for the analysis of legislative writing and outlines the approach adopted for the present analysis. This chapter also gives some idea of the information-structuring of legislative provisions and establishes qualification as a key element in their textualisation. The next three chapters offer a detailed account of the ten major qualification-types identified in the corpus, concentrating on the surface form they take, the features of legislative statements they textualize and the syntactic positions to which they are generally assigned in the statement of legislative provisions. The emerging hypotheses in these chapters have often been verified through a specialist reaction from a Parliamentary Counsel, largely responsible for the writing of the ‘Housing Act 1980’• The findings suggest useful correlations between a number of qualificational initiators and the various aspects of the legislative statement. They also reveal that many of these qualifications typically occur in those clause-medial syntactic positions which are sparingly used in other specialist discourse, thus creating syntactic discontinuity in the legislative sentence. Such syntactic discontinuities, on the evidence from psycholinguistic experiments reported in chapter six, create special problems in the processing and comprehension of legislative statements. The final chapter converts the main linguistic findings into a series of pedagogical generalizations, offers indications of how this may be applied in EALP situations and concludes with other considerations of possible applications.

Some results on orbifold quotients and related objects

The quotient of a finite-dimensional Euclidean space by a finite linear group inherits different structures from the initial space, e.g. a topology, a metric and a piecewise linear structure. The question when such a quotient is a manifold leads to the study of finite groups generated by reflections and rotations, i.e. by orthogonal transformations whose fixed point subspace has codimension one or two. We classify such groups and thereby complete earlier results by M. A. Mikhaîlova from the 70s and 80s. Moreover, we show that a finite group is generated by reflections and) rotations if and only if the corresponding quotient is a Lipschitz-, or equivalently, a piecewise linear manifold (with boundary). For the proof of this statement we show in addition that each piecewise linear manifold of dimension up to four on which a finite group acts by piecewise linear homeomorphisms admits a compatible smooth structure with respect to which the group acts smoothly. This solves a challenge by Thurston and confirms a conjecture by Kwasik and Lee. In the topological category a counterexample to the above mentioned characterization is given by the binary icosahedral group. We show that this is the only counterexample up to products. In particular, we answer the question by Davis of when the underlying space of an orbifold is a topological manifold. As a corollary of our results we generalize a fixed point theorem by Steinberg on unitary reflection groups to finite groups generated by reflections and rotations. As an application thereof we answer a question by Petrunin on quotients of spheres.

Non-linear dynamic analysis of rotors by finite element method

Non-linear natural vibration characteristics and the dynamic response of hingeless and fully articulated rotors of rectangular cross-section are studied by using the finite element method. In the formulation of response problems, the global variables are augmented with appropriate additional variables, facilitating direct determination of sub-harmonic response. Numerical results are given showing the effect of the geometric non-linearity on the first three natural frequencies. Response analysis of typical rotors indicates a possibility of substantial sub-harmonic response especially in the fully articulated rotors widely adopted in helicopters.

Some generalizations of commutativity for linear transformations on a finite dimensional vector space

Let L be the algebra of all linear transformations on an n-dimensional vector space V over a field F and let A, B, ƐL. Let A_i+1 = A_iB - BA_i, i = 0, 1, 2,…, with A = A_o. Let f_k (A, B; σ) = A_2K+1 - ^σ1^A2K-1 ^{+ σ}2^A2K-3 -… +(-1)^Kσ_KA₁ where σ = (σ₁, σ₂,…, σ_K), σ_i belong to F and K = k(k-1)/2. Taussky and Wielandt [Proc. Amer. Math. Soc., 13(1962), 732-735] showed that f_n(A, B; σ) = 0 if σ_i is the i^th elementary symmetric function of (β₄- β_s)², 1 ≤ r ˂ s ≤ n, i = 1, 2, …, N, with N = n(n-1)/2, where β₄ are the characteristic roots of B. In this thesis we discuss relations involving f_k(X, Y; σ) where X, Y Ɛ L and 1 ≤ k ˂ n. We show: 1. If F is infinite and if for each X Ɛ L there exists σ so that f_k(A, X; σ) = 0 where 1 ≤ k ˂ n, then A is a scalar transformation. 2. If F is algebraically closed, a necessary and sufficient condition that there exists a basis of V with respect to which the matrices of A and B are both in block upper triangular form, where the blocks on the diagonals are either one- or two-dimensional, is that certain products X₁, X₂…X_r belong to the radical of the algebra generated by A and B over F, where X_i has the form f₂(A, P(A,B); σ), for all polynomials P(x, y). We partially generalize this to the case where the blocks have dimensions ≤ k. 3. If A and B generate L, if the characteristic of F does not divide n and if there exists σ so that f_k(A, B; σ) = 0, for some k with 1 ≤ k ˂ n, then the characteristic roots of B belong to the splitting field of g_k(w; σ) = w^2K+1 - σ₁w^2K-1 + σ₂w^2K-3 - …. +(-1)^K σ_Kw over F. We use this result to prove a theorem involving a generalized form of property L [cf. Motzkin and Taussky, Trans. Amer. Math. Soc., 73(1952), 108-114]. 4. Also we give mild generalizations of results of McCoy [Amer. Math. Soc. Bull., 42(1936), 592-600] and Drazin [Proc. London Math. Soc., 1(1951), 222-231].

A vertex-based finite volume method applied to non-linear material problems in computational solid mechanics

A vertex-based finite volume (FV) method is presented for the computational solution of quasi-static solid mechanics problems involving material non-linearity and infinitesimal strains. The problems are analysed numerically with fully unstructured meshes that consist of a variety of two- and threedimensional element types. A detailed comparison between the vertex-based FV and the standard Galerkin FE methods is provided with regard to discretization, solution accuracy and computational efficiency. For some problem classes a direct equivalence of the two methods is demonstrated, both theoretically and numerically. However, for other problems some interesting advantages and disadvantages of the FV formulation over the Galerkin FE method are highlighted.

Linear stern waves in finite depth channels

This paper formulates an analytically tractable problem for the wake generated by a long flat bottom ship by considering the steady free surface flow of an inviscid, incompressible fluid emerging from beneath a semi-infinite rigid plate. The flow is considered to be irrotational and two-dimensional so that classical potential flow methods can be exploited. In addition, it is assumed that the draft of the plate is small compared to the depth of the channel. The linearised problem is solved exactly using a Fourier transform and the Wiener-Hopf technique, and it is shown that there is a family of subcritical solutions characterised by a train of sinusoidal waves on the downstream free surface. The amplitude of these waves decreases as the Froude number increases. Supercritical solutions are also obtained, but, in general, these have infinite vertical velocities at the trailing edge of the plate. Consideration of further terms in the expansions suggests a way of canceling the singularity for certain values of the Froude number.

Sub-modelling for the ratchetting failure of insulated rail joints

Insulated rail joints are critical for train safety as they control electrical signalling systems; unfortunately they exhibit excessive ratchetting of the railhead near the endpost insulators. This paper reports a three-dimensional global model of these joints under wheel–rail contact pressure loading and a sub-model examining the ratchetting failures of the railhead. The sub-model employs a non-linear isotropic–kinematic elastic–plastic material model and predicts stress/strain levels in the localised railhead zone adjacent to the endpost which is placed in the air gap between the two rail ends at the insulated rail joint. The equivalent plastic strain plot is utilised to capture the progressive railhead damage adequately. Associated field and laboratory testing results of damage to the railhead material suggest that the simulation results are reasonable.

Numerical estimation of 3D mechanical forces exerted by cells on non-linear materials

The exchange of physical forces in both cell-cell and cell-matrix interactions play a significant role in a variety of physiological and pathological processes, such as cell migration, cancer metastasis, inflammation and wound healing. Therefore, great interest exists in accurately quantifying the forces that cells exert on their substrate during migration. Traction Force Microscopy (TFM) is the most widely used method for measuring cell traction forces. Several mathematical techniques have been developed to estimate forces from TFM experiments. However, certain simplifications are commonly assumed, such as linear elasticity of the materials and/or free geometries, which in some cases may lead to inaccurate results. Here, cellular forces are numerically estimated by solving a minimization problem that combines multiple non-linear FEM solutions. Our simulations, free from constraints on the geometrical and the mechanical conditions, show that forces are predicted with higher accuracy than when using the standard approaches.

Non-linear refined finite element elastic analysis of steel frames with generalized transverse member loads

In the finite element modelling of steel frames, external loads usually act along the members rather than at the nodes only. Conventionally, when a member is subjected to these transverse loads, they are converted to nodal forces which act at the ends of the elements into which the member is discretised by either lumping or consistent nodal load approaches. For a contemporary geometrically non-linear analysis in which the axial force in the member is large, accurate solutions are achieved by discretising the member into many elements, which can produce unfavourable consequences on the efficacy of the method for analysing large steel frames. Herein, a numerical technique to include the transverse loading in the non-linear stiffness formulation for a single element is proposed, and which is able to predict the structural responses of steel frames involving the effects of first-order member loads as well as the second-order coupling effect between the transverse load and the axial force in the member. This allows for a minimal discretisation of a frame for second-order analysis. For those conventional analyses which do include transverse member loading, prescribed stiffness matrices must be used for the plethora of specific loading patterns encountered. This paper shows, however, that the principle of superposition can be applied to the equilibrium condition, so that the form of the stiffness matrix remains unchanged with only the magnitude of the loading being needed to be changed in the stiffness formulation. This novelty allows for a very useful generalised stiffness formulation for a single higher-order element with arbitrary transverse loading patterns to be formulated. The results are verified using analytical stability function studies, as well as with numerical results reported by independent researchers on several simple structural frames.

Novel non-linear elastic structural analysis with generalised transverse element loads using a refined finite element

In the finite element modelling of structural frames, external loads such as wind loads, dead loads and imposed loads usually act along the elements rather than at the nodes only. Conventionally, when an element is subjected to these general transverse element loads, they are usually converted to nodal forces acting at the ends of the elements by either lumping or consistent load approaches. In addition, it is especially important for an element subjected to the first- and second-order elastic behaviour, to which the steel structure is critically prone to; in particular the thin-walled steel structures, when the stocky element section may be generally critical to the inelastic behaviour. In this sense, the accurate first- and second-order elastic displacement solutions of element load effect along an element is vitally crucial, but cannot be simulated using neither numerical nodal nor consistent load methods alone, as long as no equilibrium condition is enforced in the finite element formulation, which can inevitably impair the structural safety of the steel structure particularly. It can be therefore regarded as a unique element load method to account for the element load nonlinearly. If accurate displacement solution is targeted for simulating the first- and second-order elastic behaviour on an element on the basis of sophisticated non-linear element stiffness formulation, the numerous prescribed stiffness matrices must indispensably be used for the plethora of specific transverse element loading patterns encountered. In order to circumvent this shortcoming, the present paper proposes a numerical technique to include the transverse element loading in the non-linear stiffness formulation without numerous prescribed stiffness matrices, and which is able to predict structural responses involving the effect of first-order element loads as well as the second-order coupling effect between the transverse load and axial force in the element. This paper shows that the principle of superposition can be applied to derive the generalized stiffness formulation for element load effect, so that the form of the stiffness matrix remains unchanged with respect to the specific loading patterns, but with only the magnitude of the loading (element load coefficients) being needed to be adjusted in the stiffness formulation, and subsequently the non-linear effect on element loadings can be commensurate by updating the magnitude of element load coefficients through the non-linear solution procedures. In principle, the element loading distribution is converted into a single loading magnitude at mid-span in order to provide the initial perturbation for triggering the member bowing effect due to its transverse element loads. This approach in turn sacrifices the effect of element loading distribution except at mid-span. Therefore, it can be foreseen that the load-deflection behaviour may not be as accurate as those at mid-span, but its discrepancy is still trivial as proved. This novelty allows for a very useful generalised stiffness formulation for a single higher-order element with arbitrary transverse loading patterns to be formulated. Moreover, another significance of this paper is placed on shifting the nodal response (system analysis) to both nodal and element response (sophisticated element formulation). For the conventional finite element method, such as the cubic element, all accurate solutions can be only found at node. It means no accurate and reliable structural safety can be ensured within an element, and as a result, it hinders the engineering applications. The results of the paper are verified using analytical stability function studies, as well as with numerical results reported by independent researchers on several simple frames.