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Articles 1 - 7 of 7
Full-Text Articles in Harmonic Analysis and Representation
Fundamental Solutions To The Fractional Heat Operator, Jacob Flores
Fundamental Solutions To The Fractional Heat Operator, Jacob Flores
Math Theses
In this thesis, we are interested in showing the existence of a fundamental solution to the fractional heat operator. The fractional heat operator is a nonlocal linear operator used to model the time evolution of anomalous diffusion processes whose applicability arises in a wide variety of fields in the physical sciences, engineering, economics, and finance. Fundamental solutions to a partial differential operator are a class of generalized solutions formulated with rich mathematical analysis grounded in classes of well-behaved smooth functions referred to as test functions and their continuous linear functionals referred to as distributions. A primary tool that we use …
Cdt-1d Cnn Integration With Simpson-Sobolev Regularization For High-Frequency Options Trading: With Fem-Based Heston Option Pricing, Daniel M. Margolis, Johannes Tausch, Arthur K. Selender
Cdt-1d Cnn Integration With Simpson-Sobolev Regularization For High-Frequency Options Trading: With Fem-Based Heston Option Pricing, Daniel M. Margolis, Johannes Tausch, Arthur K. Selender
Mathematics Theses and Dissertations
This dissertation presents a computational framework for high-frequency options trading that combines Cross-Data-Type 1-D Convolutional Neural Networks (CDT-1D CNN) with Simpson-Sobolev regularization for directional prediction, and finite element methods (FEM) for realistic option pricing during backtesting. The core innovation lies in developing a mathematically rigorous regularization approach that maintains the adaptability of modern deep learning while enabling accurate evaluation through stochastic volatility models. The primary contribution is the Simpson-Sobolev regularization scheme, which extends traditional Sobolev regularization by incorporating Simpson’s rule for numerical integration. This approach achieves higher-order accuracy in approximating the Sobolev norms that control function smoothness. Simpson’s rule attains …
Constructing Orthonormal Bases With The Residuals Of Successive Approximations, An Introduction To Multiresolution Analysis, Elijah J. Guptill
Constructing Orthonormal Bases With The Residuals Of Successive Approximations, An Introduction To Multiresolution Analysis, Elijah J. Guptill
Master's Theses
Wavelets and wavelet analysis are used in the study of signal processing, quantum field theory, functional analysis, multifractal analysis, and various other areas of mathematics. Multiresolution analysis provides a framework for building a wavelet basis of $\mathcal{L}^{2}(\mathbb{R})$ from a scaling function $\phi$, whose dyadic dilations and translations, $\{2^{j /2}\phi(2^{j}x-k):j,k\in \mathbb{Z}\}$, approximate $\mathcal{L}^{2}(\mathbb{R})$. One of the key properties of $\phi$ is that it must satisfy $\phi(x)=\sum_{k\in \mathbb{Z}}{p_{k}2^{j /2}\phi(2^{j}x-k)}$ with respect to the norm on $\mathcal{L}^{2}(\mathbb{R})$. This equation is called a two-scale difference equation. Such equations enforce a regularity on the ordinary generating function $2^{-1 /2}\sum_{k\in \mathbb{Z}}{p_{k}z^{k}}$, known as the quadrature condition. …
Quiver Of Affine Monoid Of A Vector Space Over Finite Field, James Junie Chen Cleary
Quiver Of Affine Monoid Of A Vector Space Over Finite Field, James Junie Chen Cleary
Dissertations, Theses, and Capstone Projects
In this paper, we study the quiver of the complex monoid algebra CAFF(n, q). There are n + 1 maximal subgroups of AFF(n, q), each isomorphic to AGL(k, q) for some 0 ≤ k ≤ n. Every irreducible representation of CAFF(n, q) arises from a character of CAGL(k, q) for a suitable k. Thus, we study two different approaches to classifying the characters of CAGL(k, q). Next, we compute the full quiver Q(CAFF(n, q)). Finally, we show that this quiver is a disjoint union of straight-line paths and that its basic algebra has radical square zero. Hence, it has finite …
The Isoperimetric Inequality And Wirtinger’S Inequality, Mason Neal
The Isoperimetric Inequality And Wirtinger’S Inequality, Mason Neal
Honors Theses
In this thesis, we present Hurwitz’s proof of the Isoperimetric Inequality, which roughly states that the area enclosed by a simple closed curve is always less than or equal to the area of a circle with the same perimeter. Hurwitz’s proof relies on Wirtinger’s Inequality. We survey results about periodic functions and Fourier series, and we use them to provide a proof of Wirtinger’s Inequality. We then give a new proof of a variant of Wirtinger’s Inequality due to Alzer and generalize this variant to higher powers.
Numerical And Harmonic Analysis Of Simplex Number Parity, Hunter Dm Hannula
Numerical And Harmonic Analysis Of Simplex Number Parity, Hunter Dm Hannula
All NMU Master's Theses
The primary object of this thesis is the study of periodicity in the parity of simplex numbers by number-theoretic and harmonic methods. The regular d-simplex numbers are introduced geometrically, arithmetically, and combinatorially. We demonstrate that all sequences indexing even d-simplex numbers are defined by finitely many congruences in a single modulus, and are thus quasiperiodic. We therefore show that these "even index-sequences" are particular elements in an affine space of functions, providing a natural decomposition result. We then introduce the discrete Fourier transform to construct the periodic parts of each index sequence, enabling the development of explicit forms for the …
Fourier Analysis Of Electronic Synthesizer Waveforms, Jiayan Ling, Jack W. Maseberg
Fourier Analysis Of Electronic Synthesizer Waveforms, Jiayan Ling, Jack W. Maseberg
SACAD: Scholarly Activities
We compute Fourier series coefficients for some standard electronic synthesizer waveforms and provide plots of our results.