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Thesis (Ph. D.)--University of Wisconsin--Madison, 1969. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliography.

Overloading of the back can cause instability such that buttressing the instability is a primary objective of many of the leading edge therapeutic approaches. However, a challenge lies in determining the location of the instability or the least stable vertebral joint. A mathematical analysis, based on a commonly used approach in engineering for determining structural stability, has been developed for the lumbar spine. The purpose of this investigation was to determine the feasibility of a method for mathematically locating potential areas of instability within a computer-based model of the lumbar spine. To validate this method, the eigenvector from the stability analysis was compared to the output from a geometric equation that approximated individual vertebral joint rotational stiffness with the idea that the entry in the eigenvector with the largest absolute value would correspond to the vertebral joint and axis with the lowest stiffness. Validation of the eigenvector was not possible due to computational similarities between the stability analysis and the geometric rotational stiffness method. However, it has been previously demonstrated that the eigenvector can be useful for locating instability, and thus warrants future study. Determining the least stable vertebral joint and axis can be used to guide proper motor pattern training as a clinical intervention. It was also shown in this investigation that an even distribution of fascicle force and stiffness generated stability. This supports the idea that well-coordinated efforts of muscle activation are beneficial for improving stability of the lumbar spine.

Overloading of the back can cause instability such that buttressing the instability is a primary objective of many of the leading edge therapeutic approaches. However, a challenge lies in determining the location of the instability or the least stable vertebral joint. A mathematical analysis, based on a commonly used approach in engineering for determining structural stability, has been developed for the lumbar spine. The purpose of this investigation was to determine the feasibility of a method for mathematically locating potential areas of instability within a computer-based model of the lumbar spine. To validate this method, the eigenvector from the stability analysis was compared to the output from a geometric equation that approximated individual vertebral joint rotational stiffness with the idea that the entry in the eigenvector with the largest absolute value would correspond to the vertebral joint and axis with the lowest stiffness. Validation of the eigenvector was not possible due to computational similarities between the stability analysis and the geometric rotational stiffness method. However, it has been previously demonstrated that the eigenvector can be useful for locating instability, and thus warrants future study. Determining the least stable vertebral joint and axis can be used to guide proper motor pattern training as a clinical intervention. It was also shown in this investigation that an even distribution of fascicle force and stiffness generated stability. This supports the idea that well-coordinated efforts of muscle activation are beneficial for improving stability of the lumbar spine.

The inverse spectral problem is the problem of understanding the potential function of the Sturm-Liouville operator from the set of eigenvalues plus some additional spectral data. The theory of transformation operators, first introduced by Marchenko, and then reinforced by Gelfand and Levitan, is a powerful method to deal with the different stages of the inverse spectral problem: uniqueness, reconstruction, stability and existence. In this thesis, we shall give a survey on the theory of transformation operators. In essence, the theory says that the transformation operator $X$ mapping the solution of a Sturm-Liouville operator $varphi$ to the solution of a Sturm-Liouville operator, can be written as $$Xvarphi=varphi(x)+int_{0}^{x}K(x,t)varphi(t)dt,$$ where the kernel $K$ satisfies the Goursat problem $$K_{xx}-K_{tt}-(q(x)-q_{0}(t))K=0$$ plus some initial boundary conditions. Furthermore, $K$ is related by a function $F$ defined by the spectral data ${(lambda_{n},alpha_{n})}$ where $alpha_{n}=(int_{0}^{pi}|varphi_{n}(t)|^{2})^{frac{1}{2}}$ through the famous Gelfand-Levitan equation $$K(x,y)+F(x,y)+int_{o}^{x}K(x,t)F(t,y)dt=0.$$ Furthermore, all the above relations are bilateral, that is $$qLeftrightarrow KLeftrightarrow FLeftarrow {(lambda_{n},alpha_{n})}.$$ hspace*{0.25in}We shall give a concise account of the above theory, which involves Riesz basis and order of entire functions. Then, we also report on some recent applications on the uniqueness result of the inverse spectral problem.

A non-linear singularity-preserving solution to the least-squares seismic imaging problem with sparseness and continuity constraints is proposed. The applied formalism explores curvelets as a directional frame that, by their sparsity on the image, and their invariance under the imaging operators, allows for a stable recovery of the amplitudes. Our method is based on the estimation of the normal operator in the form of an ’eigenvalue’ decomposition with curvelets as the ’eigenvectors’. Subsequently, we propose an inversion method that derives from estimation of the normal operator and is formulated as a convex optimization problem. Sparsity in the curvelet domain as well as continuity along the reflectors in the image domain are promoted as part of this optimization. Our method is tested with a reverse-time ’wave-equation’ migration code simulating the acoustic wave equation.

Synthesis of structures is a very difficult task even with only a small number of components that form a system; yet it is the catalyst of innovation. Molecular structures and nanostructures typically have a large number of similar components but different connections, which manifests a more challenging task for their synthesis. <p> This thesis presents a novel method and its related algorithms and computer programs for the synthesis of structures. This novel method is based on several concepts: (1) the structure is represented by a graph and further by the adjacency matrix; and (2) instead of only exploiting the eigenvalue of the adjacency matrix, both the eigenvalue and the eigenvector are exploited; specifically the components of the eigenvector have been found very useful in algorithm development. This novel method is called the Eigensystem method.<p> The complexity of the Eigensystem method is equal to that of the famous program called Nauty in the combinatorial world. However, the Eigensystem method can work for the weighted and both directed and undirected graph, while the Nauty program can only work for the non-weighted and both directed and undirected graph. The cause for this is the different philosophies underlying these two methods. The Nauty program is based on the recursive component decomposition strategy, which could involve some unmanageable complexities when dealing with the weighted graph, albeit no such an attempt has been reported in the literature. It is noted that in practical applications of structure synthesis, weighted graphs are more useful than non-weighted graphs for representing physical systems. <p> Pivoted at the Eigensystem method, this thesis presents the algorithms and computer programs for the three fundamental problems in structure synthesis, namely the isomorphism/automorphism, the unique labeling, and the enumeration of the structures or graphs.

Synthesis of structures is a very difficult task even with only a small number of components that form a system; yet it is the catalyst of innovation. Molecular structures and nanostructures typically have a large number of similar components but different connections, which manifests a more challenging task for their synthesis. <p> This thesis presents a novel method and its related algorithms and computer programs for the synthesis of structures. This novel method is based on several concepts: (1) the structure is represented by a graph and further by the adjacency matrix; and (2) instead of only exploiting the eigenvalue of the adjacency matrix, both the eigenvalue and the eigenvector are exploited; specifically the components of the eigenvector have been found very useful in algorithm development. This novel method is called the Eigensystem method.<p> The complexity of the Eigensystem method is equal to that of the famous program called Nauty in the combinatorial world. However, the Eigensystem method can work for the weighted and both directed and undirected graph, while the Nauty program can only work for the non-weighted and both directed and undirected graph. The cause for this is the different philosophies underlying these two methods. The Nauty program is based on the recursive component decomposition strategy, which could involve some unmanageable complexities when dealing with the weighted graph, albeit no such an attempt has been reported in the literature. It is noted that in practical applications of structure synthesis, weighted graphs are more useful than non-weighted graphs for representing physical systems. <p> Pivoted at the Eigensystem method, this thesis presents the algorithms and computer programs for the three fundamental problems in structure synthesis, namely the isomorphism/automorphism, the unique labeling, and the enumeration of the structures or graphs.

Let $l^2(Bbb Z)$ be the Hilbert space of square summable double sequences of complex numbers with standard basis ${e_n:ninBbb Z}$, and let us consider a bounded matrix $A$ on $l^2(Bbb Z)$ satisfying the following system of equations egin{itemize} item[1.] $lan Ae_{2j},e_{2i} an=p_{ij}+alan Ae_{j},e_i an$; item[2.] $lan Ae_{2j},e_{2i-1} an=q_{ij}+blan Ae_{j},e_{i} an$; item[3.] $lan Ae_{2j-1},e_{2i} an=v_{ij}+clan Ae_{j},e_{i} an$; item[4.] $lan Ae_{2j-1},e_{2i-1} an=w_{ij}+dlan Ae_{j},e_{i} an$ end{itemize} for all $i,j$, where $P=(p_{ij})$, $Q=(q_{ij})$, $V=(v_{ij})$, $W=(w_{ij})$ are bounded matrices on $l^2(Bbb Z)$ and $a,b,c,dinBbb C$. This type dyadic recurrent system arises in the study of bounded operators commuting with the slant Toeplitz operators, i.e., the class of operators ${{cal T}_vp:vpin L^infty(Bbb T)}$ satisfying $lan {cal T}_vp e_j,e_i an=c_{2i-j}$, where $c_n$ is the $n$-th Fourier coefficient of $vp$. It is shown in [10] that the solutions of the above system are closely related to the bounded solution $A$ for the operator equation [ phi(A)=S^*AS=lambda A+B, ] where $B$ is fixed, $lambdainBbb C$ and $S$ the shift given by ${cal T}_{arzeta+arxi z}^*$ (with $zetaxi ot=0$ and $|zeta|^2+|xi|^2=1$). In this paper, we shall characterize the ``eigenvectors" for $phi$ for the eigenvalue $lambda$ with $|lambda|leq1$, in terms of dyadic recurrent systems similar to the one above.

A considerable amount of evidence has shown that for every prime p &neq; N observed, a simultaneous eigenvector v_0 of Hecke operators T(l,i), i=1,2, in H^3(Γ_0(N),F(0,0,0)) has a “lift” v in H^3(Γ_0(N),F(p−1,0,0)) — i.e., a simultaneous eigenvector v of Hecke operators having the same system of eigenvalues that v_0 has. For each prime p>3 and N=11 and 17, we construct a vector v that is in the cohomology group H^3(Γ_0(N),F(p−1,0,0)). This is the first construction of an element of infinitely many different cohomology groups, other than modulo p reductions of characteristic zero objects. We proceed to show that v is an eigenvector of the Hecke operators T(2,1) and T(2,2) for p>3. Furthermore, we demonstrate that in many cases, v is a simultaneous eigenvector of all the Hecke operators.