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Articles 1 - 9 of 9
Full-Text Articles in Harmonic Analysis and Representation
(Si15-084) Application Of Differentiation Matrices Corresponding To Scaling And Wavelet Functions To Integral Equations, Viresh Kumar, Rakesh Kumar
(Si15-084) Application Of Differentiation Matrices Corresponding To Scaling And Wavelet Functions To Integral Equations, Viresh Kumar, Rakesh Kumar
Applications and Applied Mathematics: An International Journal (AAM)
In the present paper, we have studied the technique by which a differentiation matrix corresponding to scaling and wavelet function is helpful in representing an integral equation in discrete form. In this technique, only finite times continuously differentiable scaling and wavelet functions were used for the differentiation matrix. This method explores the applications of compactly supported wavelets in discretization of Fredholm integral equation and provides us with a bridge for the solution of integral equations via compactly supported finitely differentiable wavelets.
(Si15-010) On Perturbations Of Gabor Frames, Jamkhongam Touthang
(Si15-010) On Perturbations Of Gabor Frames, Jamkhongam Touthang
Applications and Applied Mathematics: An International Journal (AAM)
Stability plays a crucial role in frame theory and its applications. The present paper studies the interaction between Gabor frames and perturbations, presenting perturbation results related to small changes of the frame parameters and the window functions both in regular and irregular settings. Examples are provided for illustration. Additionally, the paper briefly discusses algorithms pertinent to Gabor frames under perturbations and highlights challenging areas in the field.
Fourier Analyses Of Optical Profilometry As An Inferential Measurement For Impact Coverage., Langdon Feltner, Paul Mort
Fourier Analyses Of Optical Profilometry As An Inferential Measurement For Impact Coverage., Langdon Feltner, Paul Mort
15th International Conference on Shot Peening
A critical consideration in peening process design is achieving sufficient impact coverage. Conventional methods for assessing coverage rely on manual inspection, which is time-consuming and poorly suited for automated control. In this work, we investigate the use of frequency-domain analysis to quantify surface modification in peened samples using optical profilometry (OP) data. Three-dimensional surface maps of Almen strips were acquired using a high-resolution OP system and analyzed via fast Fourier transform (FFT) to compute spatial power spectral densities (PSDs). PSD maps and radially averaged profiles reveal consistent amplification of harmonic components similar to the nominal particle size, with increasing intensity …
Property Testing Ai: An Efficient Frontier, Paul Sopher Lintilhac
Property Testing Ai: An Efficient Frontier, Paul Sopher Lintilhac
Dartmouth College Ph.D Dissertations
In this dissertation, we take a step towards addressing the major problem of a lack of standardized and rigorous approaches to testing and evaluation of AI systems. Taking inspiration from both the fields of Property Testing and Property Based Testing (for programs), we develop a novel taxonomy of partially overlapping classes of properties of AI systems, including simple properties, compound properties, higher order properties, data relation properties, and architecture-utility properties. We argue that this taxonomy categorizes a diverse set of AI traits -- including accuracy, fairness, robustness, monotonicity, point-wise and global privacy properties, sensitivity, and more -- according to the …
Langlands Reciprocity And The Splitting Behavior Of Primes In Number Fields, Skip E. Moses
Langlands Reciprocity And The Splitting Behavior Of Primes In Number Fields, Skip E. Moses
Master's Theses
This thesis explores the evolution of reciprocity laws in number theory in order to provide a conceptual bridge between the classical ideas of quadratic reciprocity and the modern framework of the Langlands program. We develop the necessary algebraic background to understand how the splitting behavior of primes in number fields reflects deep arithmetic structure in ℚ. Starting with quadratic fields and cyclotomic extensions, we motivate the development of the Kronecker–Weber theorem and the characterization of abelian extensions of ℚ. We then introduce Artin reciprocity and show how it generalizes quadratic reciprocity through the formalism of Frobenius elements and Artin L-functions. …
Banach Algebras And The Gelfand Theory Of Group Algebras On Locally Compact Abelian Groups, James Gabriel Bonvanie
Banach Algebras And The Gelfand Theory Of Group Algebras On Locally Compact Abelian Groups, James Gabriel Bonvanie
Master's Theses
A Banach algebra is a complex algebra that is simultaneously a Banach space in which the norm is submultiplicative. Notably, $L^1(\mathbb{R})$ with the convolutional product is an Abelian, non-unital Banach algebra that admits an approximate identity. We rectify $L^1(\mathbb{R})$ lacking a unit via the unitization $L^1(\mathbb{R})\times\mathbb{C}$ with identity $(0,1)$. Unitization opens the discussion to the spectrum $\sigma(x)$ of a Banach algebra element, in which the spectrum is a nonempty, compact subset of the complex plane. The spectrum of an Abelian Banach algebra is fully characterized with multiplicative linear functionals, and we prove that the Fourier transform is the unique multiplicative …
The Spectral Asymptotics Of Toeplitz Operators On Hilbert Spaces Of Analytic Functions, Trevor Camper
The Spectral Asymptotics Of Toeplitz Operators On Hilbert Spaces Of Analytic Functions, Trevor Camper
All Dissertations
Many physical systems, whether they are ocean waves or particles moving through space, can be described using the mathematical language of “partial differ- ential equations.” In many circumstances, it is useful to study how these equations amplify an input to the equation, in which case the amplification factor is called an “eigenvalue.” The usefulness of these amplification factors is that they can be used to describe properties of the physical system. In this dissertation, I have studied this amplification factor for a related set of equations called “Toeplitz operators.” In par- ticular, I have studied eigenvalues using statistical techniques. The …
Stability Criteria For The Generalized El Borhamy-Rashad-Sobhy Equation, Mohamed El-Borhamy Assoc.Prof, Essam Eddin Rashad Prof., Fathi Mousa Dr., Mai Hamouda
Stability Criteria For The Generalized El Borhamy-Rashad-Sobhy Equation, Mohamed El-Borhamy Assoc.Prof, Essam Eddin Rashad Prof., Fathi Mousa Dr., Mai Hamouda
Journal of Engineering Research
This article is concerned with the study of stability criteria for one of the generalization form of El Borhamy-Rashad Sobhy equation, which is a linear second-order ordinary differential equation with periodically time varying coefficients. Many engineering applications can be represented by this generalization, for instance, including the modeling of RLC circuit with time varying inductance, resistance and capacitance, and the vibration of a stretched string, whose mass per unit length is periodic, under a periodic motion. An approximate solution is derived by using the Wenhl-Kramers-Brillonin (WKB) approach. A method of constructing Liapunov function is employed to derive extra conditions for …
Local Limit Theorems On Finitely Generated Abelian Groups, Yutong Yan
Local Limit Theorems On Finitely Generated Abelian Groups, Yutong Yan
Honors Theses
In this thesis, we classify the pointwise behavior of finite-range random walks on finitely generated abelian groups in terms of local limit theorems. Random walks are central objects of research in probability theory, and the theory has found applications in statistics, physics, and even card shuffling. One significant topic in this line of study is random walks on finitely generated groups. Starting from the pioneering work of G. Pólya and H. Kesten, random walks on finitely generated groups have been studied extensively. However, many notable results on the subject (local limit theorems, for example) make assumptions about periodicity and irreducibility …