QUANTITATIVE EQUI-UNIFORM APPROXIMATION PROCESSES OF INTEGRAL OPERATORS IN BANACH SPACES

In the present paper, we introduce q-analogue of the Jakimovski-Leviatan operators with the help of q-Appell polynomials. We establish some moments and auxiliary results by using q-derivatives and then prove a basic convergence theorem. Also, the Voronovskaja-type asymptotic formula and some direct results for the above operators are discussed. Moreover, the rate of convergence and weighted approximation by these operators in terms of modulus of continuity are studied.

In this paper, we introduce the q-analogue of the Jakimovski-Leviatan type modified operators introduced by Atakut with the help of the q-Appell polynomials.We obtain some approximation results via the well-known Korovkin’s theorem for these operators.We also study convergence properties by using the modulus of continuity and the rate of convergence of the operators for functions belonging to the Lipschitz class. Moreover,we study the rate of convergence in terms of modulus of continuity of these operators in aweighted space.

In this paper we give a further development of the results of the paper [1] and apply it to convolution operators in the spaces Lp(x) . We consider the question of extendability of the Young theorem: well known for constant p and q, to the case when they may be variable. We also potential type operators with the Kernel . In section 1 we develop some estimates for Lp(x) -norms of power functions of distance truncated to exterior of a ball of radius . Section 2 deals with convolution operators in the spaces Lp(x) and Section 3 is devoted to potential type operators.

The Cauchy problem for a parabolic system of integro-dierential equations with an operator of Volterra-Fredholm type is considered. A fundamental matrix of solutions of the problem in classical Holder spaces is constructed, the estimates for the matrix and its derivatives are established. This makes it possible to prove the correctness theorem.

Applying the theory of pseudodifferential and Calderón-Zygmund operators, we study the compactness of commutators of multiplication operators aI and convolution operators \( W^{0}(b) \) on weighted Lebesgue spaces \( L^{p}(\mathbb{R},w)\;{\rm with}\; p \in (1,\infty)\) and Muckenhoupt weights ?? for some classes of piecewise slowly oscillating functions \( a \in PSO^{\diamond} \; and b \in PSO^{\diamond}_{p,w}\) on the real line ℝ. Then we study the Banach algebra \( Z_{p,w} \) generated by the operators \( aW^{0}(b)\) with functions \( a \in SO^{\diamond} \; and b \in SO^{\diamond}_{p,w}\) admitting slowly oscillating discontinuities at every point \( lambda \in \mathbb{R} \cup \left\{\infty \right\} \). Applying the method of limit operators under some condition on Muckenhoupt weights w, we describe the maximal ideal space of the commutative quotient Banach algebra \( Z^{\pi}_{p,w}\;=\;Z_{p,w}/K_{p,w}{\rm where}\;K_{p,w}\) is the ideal of compact operators on \( L^{p}(\mathbb{R,w}) \) define the Gelfand transform for \( Z^{\pi}_{p,w}) \) and establish the Fredholmness for the operators \( A\in Z_{p,w}\)

This work includes some new aspects of motion estimation by the optic flow method in scale spaces. The usual techniques for motion estimation are limited to the application of coarse to fine strategies. The coarse to fine strategies can be successful only if there is enough information in every scale. In this work we investigate the motion estimation in the scale space more basically. The wavelet choice for scale space decomposition of image sequences is discussed in the first part of this work. We make use of the continuous wavelet transform with rotationally symmetric wavelets. Bandpass decomposed sequences allow the replacement of the structure tensor by the phase invariant energy operator. The structure tensor is computationally more expensive because of its spatial or spatio-temporal averaging. The energy operator needs in general no further averaging. The numerical accuracy of the motion estimation with the energy operator is compared to the results of usual techniques, based on the structure tensor. The comparison tests are performed on synthetic and real life sequences. Another practical contribution is the accuracy measurement for motion estimation by adaptive smoothed tensor fields. The adaptive smoothing relies on nonlinear anisotropic diffusion with discontinuity and curvature preservation. We reached an accuracy gain under properly chosen parameters for the diffusion filter. A theoretical contribution from mathematical point of view is a new discontinuity and curvature preserving regularization for motion estimation. The convergence of solutions for the isotropic case of the nonlocal partial differential equation is shown. For large displacements between two consecutive frames the optic flow method is systematically corrupted because of the violence of the sampling theorem. We developed a new method for motion analysis by scale decomposition, which allows to circumvent the systematic corruption without using the coarse to fine strategy. The underlying assumption is, that in a certain neighborhood the grey value undergoes the same displacement. If this is fulfilled, then the same optic flow should be measured in all scales. If there arise inconsistencies in a pixel across the scale space, so they can be detected and the scales containing this inconsistencies are not taken into account.

In a petrochemical plant, training new operators to operate plant safely and consistent is a need. Conventional training for operators relies on senior operators' experience. When experience operator leaves or resigns, it will take longer duration to train new personnel to gain the equivalent experience of previous or senior operators. The operators control the plant by using Distributed Control System (DCS) as an interface and ensure the reliability and productivity of plant sustains. The more challenges they face, the more experience they gain. Most of the operators are trained by real time events they faced on the job but errors or plant upsets should be avoided. It is estimated that in the United States alone, 10 billion USD is lost annually in the process industry caused by errors when faced with plant upsets. Operator training plan is therefore the crucial step to reduce the losses as well as the risk of operating the plant. Operator Training Simulator (OTS) is an alternative solution to safely provide the operators with various realistic operational conditions of the plant, including start-up, shutdown, and emergency conditions. This article discusses on the methodology of building an OTS for Ethylene plant, that are beneficial for educating operator response when faced with critical process upsets and succeeding quicker start-up. A dynamic process model was developed to give accurate dynamic behaviour of the plant process mainly during start-up, shutdown and upset conditions. Testing and verification processes of the models such as Model Acceptance Test (MAT), Factory Acceptance Test (FAT) and Site Acceptance Test (SAT) were deliberated. Few issues or problems faced during the testing were also discussed. Based on the experience of developing the simulator, process model was developed using mass and energy balance method does not reflect the actual plant conditions. It might be suitable for commissioning a new plant. However, once the plant is started all the parameters need to be realigned to meet the actual plant data. Process model for existing plant should always apply the actual plant data as a primary data. Especially during MAT, FAT and SAT processes, experience of the panel man is the best input data to get the best output of the simulations.

Invertible map equivalences are approximations of graph isomorphism that refine the well-known Weisfeiler-Leman method. They are parameterized by a number k and a set Q of primes. The intuition is that two equivalent graphs G equiv^IM_{k, Q} H cannot be distinguished by means of partitioning the set of k-tuples in both graphs with respect to any linear-algebraic operator acting on vector spaces over fields of characteristic p, for any p in Q. These equivalences have first appeared in the study of rank logic, but in fact they can be used to delimit the expressive power of any extension of fixed-point logic with linear-algebraic operators. We define {LA^{k}}(Q), an infinitary logic with k variables and all linear-algebraic operators over finite vector spaces of characteristic p in Q and show that equiv^IM_{k, Q} is the natural notion of elementary equivalence for this logic. The logic LA^{omega}(Q) = Cup_{k in omega} LA^{k}(Q) is then a natural upper bound on the expressive power of any extension of fixed-point logics by means of Q-linear-algebraic operators. By means of a new and much deeper algebraic analysis of a generalized variant, for any prime p, of the CFI-structures due to Cai, Furer, and Immerman, we prove that, as long as Q is not the set of all primes, there is no k such that equiv^IM_{k, Q} is the same as isomorphism. It follows that there are polynomial-time properties of graphs which are not definable in LA^{omega}(Q), which implies that no extension of fixed-point logic with linear-algebraic operators can capture PTIME, unless it includes such operators for all prime characteristics. Our analysis requires substantial algebraic machinery, including a homogeneity property of CFI-structures and Maschke's Theorem, an important result from the representation theory of finite groups.

For a standard Wiener process \(\{ {{W}_{t}}:t\varepsilon [0,1]\}\), denoting \(X(\rho ) = {{\sup }_{{|s - t| \leqslant \rho }}}|{{W}_{s}} - {{W}_{t}}|, g(x) = \sqrt {{2x\ln (\tfrac{1}{x})}}\) and \(h(x) = \sqrt {{2x\ln \ln \tfrac{1}{x}}}\) , we have proved the following two results: \(E {{\sup }_{{\rho \leqslant \delta }}}\tfrac{{X(\rho )}}{{g(\rho )}}\) converges down to 1 at least as fast as \(3.27/\sqrt {{\ln \tfrac{1}{\delta }}}\) and \({{\sup }_{{t \leqslant \delta }}}\tfrac{{{{W}_{t}}}}{{h(t)}}\) converges down to 1 at least as fast as \(3\ln \ln \ln \tfrac{1}{\delta }/\ln \ln \tfrac{1}{\delta }\)

With the problem of difficulty that a single filter to adapt to various complex changes in the tracking process, an adaptive multi-filter target tracking algorithm based on the efficient convolution operators for tracking is proposed. Spatial-temporal regularized filter, the consistency check filter and the correlation filter in the efficient convolution operator tracker, convolve with target features respectively, which obtains three detection scores. The training method of spatial-temporal regularized filter is to introduce temporal regularization into loss function. The consistency check filter is a filter that uses current filter to track the target of previous several frames and updates only when the error of forward and backward position is less than the threshold. Target position is estimated by the best filter detection score with the peak-to-side ratio is maximum. The improved algorithm is tested with the OTB-2015 dataset and UAV123 dataset. The experimental results show that the proposed algorithm can better adapt to the complex environment in tracking process, which has high precision and robustness.

The following results are obtained for the metric of a sign sensitive weight: necessary and sufficient conditions imposed on the weight in order to ensure the fulfillment of the complete analogue of Jackson's theorem on the estimate of the best approximation of an arbitrary continuous function by means of its modulus of continuity; the analogue of Bernstein's inequality on the estimate of the derivative of a trigonometric polynomial; the analogue of Stechkin's theorem on the connection between the modulus of continuity of a function and the rate of its approximation by polynomials; the analogue of Dolzhenko's inequality on the estimate of the variation of a rational function; and the analogue of Dolzhenko's theorem on the estimate of the variation of a function by means of its best rational approximation.

A three node two-dimensional laminated composite curved beam finite element formulation for linear static analysis is presented where the displacement approximation for the laminate is piecewise hierarchical and is derived based on p-version. The displacement approximation for the element is developed by establishing a hierarchical displacement approximation for each lamina of the laminate and then by imposing interlamina continuity conditions of displacement at the interfaces between the laminas. The approximation functions and the nodal variables for each lamina are derived directly from the Lagrange family of interpolation functions of order pξand pη. This is accomplished by first establishing one-dimensional hierarchical approximation functions and the corresponding nodal variable operators in the ξ and η directions for the three and one node equivalent configurations that correspond to pξ+1 and pη+1 equally spaced nodes in the ξ and η directions and then taking their products. The nodal variables for the entire laminate are derived from the nodal variables of the laminas and the interlamina continuity conditions of displacements. The element formulation ensures C0 continuity of displacement across the interelement as well as interlamina boundaries.

We investigate some new inequalities by estimating the rate of convergence by means of the complete modulus of continuity and a class of Lipschitz functions for the bivariate Bernstein-type operator including Bézier basis and present an example including numerical results comparing its rate of convergence. Moreover, we introduce its GBS (Generalized Boolean Sum) operator and obtain its rate of convergence with the help of the mixed modulus of continuity and Lipschitz class of Bögel continuous functions with exemplifying numerical results. Our research will demonstrate that the GBS operator possesses at least better numerical results than the bivariate Bernstein-type operator. All mentioned results point out the novelty of this study.

A nuclear operator T on a Banach space is an operator admitting a representation T = (SIGMA) T(,n) where each operator T(,n) has rank one and (SIGMA) (VERT.BAR)(VERT.BAR)(TAU)(,n)(VBAR)(VBAR) < (INFIN). A. Grothendieck proved that the sequence of eigenvalues, repeated according to their multiplicities, of such a nuclear operator must be square summable and that, in fact, two is the best possible exponent of summability. This work uses operators of diagonal type, i.e., operators which admit a diagonal representation with respect to a biorthogonal system, to construct examples of eigenvalue behavior. In particular we show that all of the known results concerning eigenvalue summability of nuclear operators can be obtained using operators of diagonal type. In Chapter 1, we present the terminology and notation to be used in this work. Chapter 2 gives a summary of results concerning nuclear (trace class) operators on Hilbert space. In Chapter 3, some of the important known results concerning eigenvalue behavior of nuclear operators on general Banach spaces are presented. The convolution operators on L(,p)(0,1), 1 < p < (INFIN), p (NOT=) 2, are diagonal operators since the trigonometric system (e('2(pi)inx)) is a conditional basis of these spaces. In Chapter 4 we use these important operators to exhibit extremal behavior of eigenvalue summability of nuclear operators on the L(,p)-spaces. In Chapter 5, we study nuclear cyclic diagonal operators. If X(,1),... ,X(,n) are sequence spaces, an operator D on (CRPLUS)X(,i) is a cyclic diagonal operator if its restriction to each X(,i) is a diagonal operator into X(,i+1) (if i = n into X(,1)). Certain eigenspaces of these operators have important applications to Banach space theory. Finally, in Chapter 6, we use a recently discovered space of G. Pisier to answer affirmatively a long outstanding question of. Pe l cynski and Saphar which is converse to Grothendieck's result: Given a (nonzero) sequence ((lamda)(,n)) in l(,2), is there a Banach space X and a nuclear operator on X whose eigenvalue sequence is ((lamda)(,n))?

Spinal degenerations can lead to segmental instability that is regarded as a major cause of back pain and is often an important factor in deciding on surgical fusion or prosthesis implant. The spinal kinematics analysis can provide useful information for diagnosis of instability and for the assessment of therapy and surgical treatment or for performance evaluation of disc prostheses. Digitized videofluoroscopy permits to analyze spinal motion during the full patient’s movement, with an acceptable low X-ray dose. By recognizing the vertebrae position on successive fluoroscopic images through manual selection or automated algorithms the relative kinematics between pairs of adjacent vertebrae (i.e. intervertebral kinematics) can be easily estimated. The application of fluoroscopy in the study of spinal kinematics is, however, limited because large errors can occur in the measurements. This thesis presents a comprehensive study of an innovative technique designed to provide a more accurate estimation of intervertebral kinematics. The recognition of vertebrae along the fluoroscopic sequence is implemented using an automated template-matching algorithm and involving a strong enhancement of the outline of vertebrae by resorting to derivative operators. Particular attention is devoted to fluoroscopic noise suppression and to edge-preserving filter design. Spline interpolation of the kinematic data extracted by videofluoroscopy is applied in order to obtain a more complete, continuous description of spinal kinematics and, more specifically, of instantaneous center of rotation. In the introductory part of the thesis (Chapter I and II) the motivation of the study and a survey of spinal measurement techniques are given. The feasibility of videofluoroscopic analysis of spinal motion is extensively discussed. In Chapter III common kinematic parameters (such as range of motion, center of rotation, etc.) utilized for describing intervertebral spinal behaviour are presented, providing particular emphasis on the difficulty to determine a “boundary” between normal and abnormal measures of segmental kinematics for the definition of spinal instability. An extensive review of recent proposals in analysis of segmental motion is reported. Manual recognition of anatomical landmarks in videofluoroscopy can be very problematic. It is also well-known that derivative operators, commonly used for automatic recognition, are highly sensitive to noise. Chapter IV attempts to address this issue: fluoroscopic noise model, also in presence of non-linear gray-level transformations for image enhancement, is presented; various denoising algorithms specifically designed for signal-dependent noise and AWGN are examined and a performance comparison among them is carried out. In Chapter V the proposed algorithm for automated vertebrae recognition is described and its performance is experimentally analyzed on fluoroscopic images of a calibration model. A comparison with a manual selection procedure and other automated algorithms on real lumbar fluoroscopic sequences is presented. In Chapter VI a continuous-time description of intervertebral motion by cubic smoothing spline interpolation is presented and the evaluation of instantaneous center of rotation of spinal motion segments by videofluoroscopy is discussed.