On concentrated loads and Green's functions in elastostatics

This investigation is concerned with the notion of concentrated loads in classical elastostatics and related issues. Following a limit treatment of problems involving concentrated internal and surface loads, the orders of the ensuing displacements and stress singularities, as well as the stress resultants of the latter, are determined. These conclusions are taken as a basis for an alternative direct formulation of concentrated-load problems, the completeness of which is established through an appropriate uniqueness theorem. In addition, the present work supplies a reciprocal theorem and an integral representation-theorem applicable to singular problems of the type under consideration. Finally, in the course of the analysis presented here, the theory of Green's functions in elastostatics is extended.

An arithmetical theorem for partially ordered sets

The simplest multiplicative systems in which arithmetical ideas can be defined are semigroups. For such systems irreducible (prime) elements can be introduced and conditions under which the fundamental theorem of arithmetic holds have been investigated (Clifford (3)). After identifying associates, the elements of the semigroup form a partially ordered set with respect to the ordinary division relation. This suggests the possibility of an analogous arithmetical result for abstract partially ordered sets. Although nothing corresponding to product exists in a partially ordered set, there is a notion similar to g.c.d. This is the meet operation, defined as greatest lower bound. Thus irreducible elements, namely those elements not expressible as meets of proper divisors can be introduced. The assumption of the ascending chain condition then implies that each element is representable as a reduced meet of irreducibles. The central problem of this thesis is to determine conditions on the structure of the partially ordered set in order that each element have a unique such representation.

Part I contains preliminary results and introduces the principal tools of the investigation. In the second part, basic properties of the lattice of ideals and the connection between its structure and the irreducible decompositions of elements are developed. The proofs of these results are identical with the corresponding ones for the lattice case (Dilworth (2)). The last part contains those results whose proofs are peculiar to partially ordered sets and also contains the proof of the main theorem.

Neural representation of auditory temporal structure

Neurons in the songbird forebrain nucleus HVc are highly sensitive to auditory temporal context and have some of the most complex auditory tuning properties yet discovered. HVc is crucial for learning, perceiving, and producing song, thus it is important to understand the neural circuitry and mechanisms that give rise to these remarkable auditory response properties. This thesis investigates these issues experimentally and computationally.

Extracellular studies reported here compare the auditory context sensitivity of neurons in HV c with neurons in the afferent areas of field L. These demonstrate that there is a substantial increase in the auditory temporal context sensitivity from the areas of field L to HVc. Whole-cell recordings of HVc neurons from acute brain slices are described which show that excitatory synaptic transmission between HVc neurons involve the release of glutamate and the activation of both AMPA/kainate and NMDA-type glutamate receptors. Additionally, widespread inhibitory interactions exist between HVc neurons that are mediated by postsynaptic GABA_A receptors. Intracellular recordings of HVc auditory neurons in vivo provides evidence that HV c neurons encode information about temporal structure using a variety of cellular and synaptic mechanisms including syllable-specific inhibition, excitatory post-synaptic potentials with a range of different time courses, and burst-firing, and song-specific hyperpolarization.

The final part of this thesis presents two computational approaches for representing and learning temporal structure. The first method utilizes comput ational elements that are analogous to temporal combination sensitive neurons in HVc. A network of these elements can learn using local information and lateral inhibition. The second method presents a more general framework which allows a network to discover mixtures of temporal features in a continuous stream of input.

Convex Relaxation for Low-Dimensional Representation: Phase Transitions and Limitations

There is a growing interest in taking advantage of possible patterns and structures in data so as to extract the desired information and overcome the curse of dimensionality. In a wide range of applications, including computer vision, machine learning, medical imaging, and social networks, the signal that gives rise to the observations can be modeled to be approximately sparse and exploiting this fact can be very beneficial. This has led to an immense interest in the problem of efficiently reconstructing a sparse signal from limited linear observations. More recently, low-rank approximation techniques have become prominent tools to approach problems arising in machine learning, system identification and quantum tomography.

In sparse and low-rank estimation problems, the challenge is the inherent intractability of the objective function, and one needs efficient methods to capture the low-dimensionality of these models. Convex optimization is often a promising tool to attack such problems. An intractable problem with a combinatorial objective can often be "relaxed" to obtain a tractable but almost as powerful convex optimization problem. This dissertation studies convex optimization techniques that can take advantage of low-dimensional representations of the underlying high-dimensional data. We provide provable guarantees that ensure that the proposed algorithms will succeed under reasonable conditions, and answer questions of the following flavor:

More specifically, i) Focusing on linear inverse problems, we generalize the classical error bounds known for the least-squares technique to the lasso formulation, which incorporates the signal model. ii) We show that intuitive convex approaches do not perform as well as expected when it comes to signals that have multiple low-dimensional structures simultaneously. iii) Finally, we propose convex relaxations for the graph clustering problem and give sharp performance guarantees for a family of graphs arising from the so-called stochastic block model. We pay particular attention to the following aspects. For i) and ii), we aim to provide a general geometric framework, in which the results on sparse and low-rank estimation can be obtained as special cases. For i) and iii), we investigate the precise performance characterization, which yields the right constants in our bounds and the true dependence between the problem parameters.

Studies of the representation of interatomic distances in molecules and crystal as sum of atomic radii

The electron diffraction investigation of the following compounds has been carried out: sulfur, sulfur nitride, realgar, arsenic trisulfide, spiropentane, dimethyltrisulfide, cis and trans lewisite, methylal, and ethylene glycol.

The crystal structures of the following salts have been determined by x-ray diffraction: silver molybdateand hydrazinium dichloride.

Suggested revisions of the covalent radii for B, Si, P, Ge, As, Sn, Sb, and Pb have been made, and values for the covalent radii of Al, Ga, In, Ti, and Bi have been proposed.

The Schomaker-Stevenson revision of the additivity rule for single covalent bond distances has been used in conjunction with the revised radii. Agreement with experiment is in general better with the revised radii than with the former radii and additivity.

The principle of ionic bond character in addition to that present in a normal covalent bond has been applied to the observed structures of numerous molecules. It leads to a method of interpretation which is at least as consistent as the theory of multiple bond formation.

The revision of the additivity rule has been extended to double bonds. An encouraging beginning along these lines has been made, but additional experimental data are needed for clarification.

The conjugate representation of signal envelope

The problem of the representation of signal envelope is treated, motivated by the classical Hilbert representation in which the envelope is represented in terms of the received signal and its Hilbert transform. It is shown that the Hilbert representation is the proper one if the received signal is strictly bandlimited but that some other filter is more appropriate in the bandunlimited case. A specific alternative filter, the conjugate filter, is proposed and the overall envelope estimation error is evaluated to show that for a specific received signal power spectral density the proposed filter yields a lower envelope error than the Hilbert filter.

Symmetric representation of an integral domain over a subdomain

Let F(θ) be a separable extension of degree n of a field F. Let Δ and D be integral domains with quotient fields F(θ) and F respectively. Assume that Δ ᴝ D. A mapping φ of Δ into the n x n D matrices is called a Δ/D rep if (i) it is a ring isomorphism and (ii) it maps d onto dI_n whenever d ϵ D. If the matrices are also symmetric, φ is a Δ/D symrep.

Every Δ/D rep can be extended uniquely to an F(θ)/F rep. This extension is completely determined by the image of θ. Two Δ/D reps are called equivalent if the images of θ differ by a D unimodular similarity. There is a one-to-one correspondence between classes of Δ/D reps and classes of Δ ideals having an n element basis over D.

The condition that a given Δ/D rep class contain a Δ/D symrep can be phrased in various ways. Using these formulations it is possible to (i) bound the number of symreps in a given class, (ii) count the number of symreps if F is finite, (iii) establish the existence of an F(θ)/F symrep when n is odd, F is an algebraic number field, and F(θ) is totally real if F is formally real (for n = 3 see Sapiro, “Characteristic polynomials of symmetric matrices” Sibirsk. Mat. Ž. 3 (1962) pp. 280-291), and (iv) study the case D = Z, the integers (see Taussky, “On matrix classes corresponding to an ideal and its inverse” Illinois J. Math. 1 (1957) pp. 108-113 and Faddeev, “On the characteristic equations of rational symmetric matrices” Dokl. Akad. Nauk SSSR 58 (1947) pp. 753-754).

The case D = Z and n = 2 is studied in detail. Let Δ’ be an integral domain also having quotient field F(θ) and such that Δ’ ᴝ Δ. Let φ be a Δ/Z symrep. A method is given for finding a Δ’/Z symrep ʘ such that the Δ’ ideal class corresponding to the class of ʘ is an extension to Δ’ of the Δ ideal class corresponding to the class of φ. The problem of finding all Δ/Z symreps equivalent to a given one is studied.

Rings faithfully represented on their left socle

In 1964 A. W. Goldie [1] posed the problem of determining all rings with identity and minimal condition on left ideals which are faithfully represented on the right side of their left socle. Goldie showed that such a ring which is indecomposable and in which the left and right principal indecomposable ideals have, respectively, unique left and unique right composition series is a complete blocked triangular matrix ring over a skewfield. The general problem suggested above is very difficult. We obtain results under certain natural restrictions which are much weaker than the restrictive assumptions made by Goldie.

We characterize those rings in which the principal indecomposable left ideals each contain a unique minimal left ideal (Theorem (4.2)). It is sufficient to handle indecomposable rings (Lemma (1.4)). Such a ring is also a blocked triangular matrix ring. There exist r positive integers K₁,..., K_r such that the i,j^th block of a typical matrix is a K_i x K_j matrix with arbitrary entries in a subgroup D_ij of the additive group of a fixed skewfield D. Each D_ii is a sub-skewfield of D and D_ri = D for all i. Conversely, every matrix ring which has this form is indecomposable, faithfully represented on the right side of its left socle, and possesses the property that every principal indecomposable left ideal contains a unique minimal left ideal.

The principal indecomposable left ideals may have unique composition series even though the ring does not have minimal condition on right ideals. We characterize this situation by defining a partial ordering ρ on {i, 2,...,r} where we set iρj if D_ij ≠ 0. Every principal indecomposable left ideal has a unique composition series if and only if the diagram of ρ is an inverted tree and every D_ij is a one-dimensional left vector space over D_ii (Theorem (5.4)).

We show (Theorem (2.2)) that every ring A of the type we are studying is a unique subdirect sum of less complex rings A₁,...,A_s of the same type. Namely, each A_i has only one isomorphism class of minimal left ideals and the minimal left ideals of different A_i are non-isomorphic as left A-modules. We give (Theorem (2.1)) necessary and sufficient conditions for a ring which is a subdirect sum of rings A_i having these properties to be faithfully represented on the right side of its left socle. We show ((4.F), p. 42) that up to technical trivia the rings A_i are matrix rings of the form

[...]. Each Q_j comes from the faithful irreducible matrix representation of a certain skewfield over a fixed skewfield D. The bottom row is filled in by arbitrary elements of D.

In Part V we construct an interesting class of rings faithfully represented on their left socle from a given partial ordering on a finite set, given skewfields, and given additive groups. This class of rings contains the ones in which every principal indecomposable left ideal has a unique minimal left ideal. We identify the uniquely determined subdirect summands mentioned above in terms of the given partial ordering (Proposition (5.2)). We conjecture that this technique serves to construct all the rings which are a unique subdirect sum of rings each having the property that every principal-indecomposable left ideal contains a unique minimal left ideal.

Multiplication in Riesz spaces

A.G. Vulih has shown how an essentially unique intrinsic multiplication can be defined in certain types of Riesz spaces (vector lattices) L. In general, the multiplication is not universally defined in L, but L can always be imbedded in a large space L^# in which multiplication is universally defined.

If ф is a normal integral in L, then ф can be extended to a normal integral on a large space L₁(ф) in L^#, and L₁(ф) may be regarded as an abstract integral space. A very general form of the Radon-Nikodym theorem can be proved in L₁(ф), and this can be used to give a relatively simple proof of a theorem of Segal giving a necessary and sufficient condition that the Radon-Nikodym theorem hold in a measure space.

In another application, the multiplication is used to give a representation of certain Riesz spaces as rings of operators on a Hilbert space.