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.

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.

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.