Paranormal operator on a Hilbert space

In this thesis an extensive study is made of the set P of all paranormal operators in B(H), the set of all bounded endomorphisms on the complex Hilbert space H. T ϵ B(H) is paranormal if for each z contained in the resolvent set of T, d(z, σ(T))//(T-zI)^-1 = 1 where d(z, σ(T)) is the distance from z to σ(T), the spectrum of T. P contains the set N of normal operators and P contains the set of hyponormal operators. However, P is contained in L, the set of all T ϵ B(H) such that the convex hull of the spectrum of T is equal to the closure of the numerical range of T. Thus, N≤P≤L.

If the uniform operator (norm) topology is placed on B(H), then the relative topological properties of N, P, L can be discussed. In Section IV, it is shown that: 1) N P and L are arc-wise connected and closed, 2) N, P, and L are nowhere dense subsets of B(H) when dim H ≥ 2, 3) N = P when dimH ˂ ∞ , 4) N is a nowhere dense subset of P when dimH ˂ ∞ , 5) P is not a nowhere dense subset of L when dimH ˂ ∞ , and 6) it is not known if P is a nowhere dense subset of L when dimH ˂ ∞.

The spectral properties of paranormal operators are of current interest in the literature. Putnam [22, 23] has shown that certain points on the boundary of the spectrum of a paranormal operator are either normal eigenvalues or normal approximate eigenvalues. Stampfli [26] has shown that a hyponormal operator with countable spectrum is normal. However, in Theorem 3.3, it is shown that a paranormal operator T with countable spectrum can be written as the direct sum, N ⊕ A, of a normal operator N with σ(N) = σ(T) and of an operator A with σ(A) a subset of the derived set of σ(T). It is then shown that A need not be normal. If we restrict the countable spectrum of T ϵ P to lie on a C²-smooth rectifiable Jordan curve G_o, then T must be normal [see Theorem 3.5 and its Corollary]. If T is a scalar paranormal operator with countable spectrum, then in order to conclude that T is normal the condition of σ(T) ≤ G_o can be relaxed [see Theorem 3.6]. In Theorem 3.7 it is then shown that the above result is not true when T is not assumed to be scalar. It was then conjectured that if T ϵ P with σ(T) ≤ G_o, then T is normal. The proof of Theorem 3.5 relies heavily on the assumption that T has countable spectrum and cannot be generalized. However, the corollary to Theorem 3.9 states that if T ϵ P with σ(T) ≤ G_o, then T has a non-trivial lattice of invariant subspaces. After the completion of most of the work on this thesis, Stampfli [30, 31] published a proof that a paranormal operator T with σ(T) ≤ G_o is normal. His proof uses some rather deep results concerning numerical ranges whereas the proof of Theorem 3.5 uses relatively elementary methods.

A study of the reproductive biology of dace in the middle Ob [introduction , summary and table captions only] [Translation from: Trudy tomsk. gos. Univ. 125, pp.77-91]

The Siberian Dace (Leuciscus leuciscus baicalensis (Dyb)is an important trade fish in Siberian waters. In the Ob basin more than 30,000 centners are produced annually. Catches of dace fluctuate significantly both between different rivers and between years in the Tomsk region. Defining the stocks of dace in the waters of the Tomsk region and explaining the fluctuations over time seems to be a very important and relevant question for the workers of the fishing industry. An answer, however, requires an accurate knowledge of the biology of dace; its reproductive, feeding and migration habits and the conditions of wintering etc. In the following we examine one of the above questions i.e. the biology of the reproduction of dace. The study was carried out in the Middle Ob in May 1951. This tranlations provides the introduction, summary and table captions only of the original article.

Changes in feeding biology of Nile tilapia, Oreochromis niloticus (L.), after invasion of water hyacinth, Eichhornia crassipes (Mart.) Solms, in Lake Victoria, Kenya

Oreochrimis niloticus (L.) was introduced to Lake victoria in the 1950s. It remained relatively uncommon in catches until 1965, when the numbers began to increase dramatically. It is now the third most important commercial fish species after the Nile perch, Lates niloticus (L.) and Rastrineobola argentea (Pellegrin). Oreochromis niloticus is considered a herbivore, feeding mostly on algae and plant material. The diet now appears to be more diversified , with insects, fish, algae and plant materials all being important food items. Fish smaller than 5 cm TL have a diverse diet but there is a decline in the importance of zooplankton, the preferred food item of small fish, as fish get larger. The shift in diet could be due to changes which have occurred in the lake. Water hyacinth, Eichhornia crassipes (Mart.) Solms, which harbours numerous insects in its root balls, now has extensively coverage over the lake. The native fish species which preyed on these insects (e.g. haplochromines) have largely been eliminated and O. niloticus could be filling niches previously occupied by these cichlids and non cichlid fishes. The change in diet could also be related to food availability and abundance where the fish is feeding on the most readily available food items.

Reproductive biology of Rastrineobola argentea (Pellegrin) in the northern waters of Lake Victoria

Size at first maturity, breeding periods and condition factor were determined for the small pelagic cyprinid Rastrineobola argentea (Pellegrin) in the Jinja waters of Lake Victoria in 1996-1997. Females showed a reduced size at maturity compared to ten years earlier when exploitation of the species was minimal. The males, however, have changed little. Although the species breeds throughout the year, two breeding peaks were observed during the drier months of August and December-January. Minimal breeding was observed in the rainy months of April-May and October-November. Fish from the open water station at Bugaia showed a higher proportion of breeding individuals than those from inshore areas. The mean monthly condition factor of fish from Napoleon Gulf confirmed breeding peaks obtained from examination of gonad development.

Reproductive biology of Oreochromis niloticus (L.) in the Nyanza Gulf of Lake Victoria

The reproduction of Nile tilapia, Oreochromis niloticus (L.), in the Nyanza Gulf of Lake Victoria was studied from June 1998 to May 1999. Length at maturity ranged from 28-30 cm TL for females and from 32-34 cm TL for males. Males were more abundant in all length classes longer than 36 cm TL. Relative condition factor was above unity, except in August, October and May for males, and October for females. Gonadosomatic index (GSI) was low during the post spawning period (July to October) and high during the protracted breeding period (December-June).

Phase-space analysis of wavefront coding imaging systems

We explore the use of the Radon-Wigner transform, which is associated with the fractional Fourier transform of the pupil function, for determining the point spread function (PSF) of an incoherant defocused optical system. Then we introduce these phase-space tools to analyse the wavefront coding imaging system. It is shown that the shape of the PSF for such a system is highly invarient to the defocous-related aberrations except for a lateral shift. The optical transfer function of this system is also investigated briefly from a new understanding of ambiguity function.

Some generalizations of commutativity for linear transformations on a finite dimensional vector space

Let L be the algebra of all linear transformations on an n-dimensional vector space V over a field F and let A, B, ƐL. Let A_i+1 = A_iB - BA_i, i = 0, 1, 2,…, with A = A_o. Let f_k (A, B; σ) = A_2K+1 - ^σ1^A2K-1 ^{+ σ}2^A2K-3 -… +(-1)^Kσ_KA₁ where σ = (σ₁, σ₂,…, σ_K), σ_i belong to F and K = k(k-1)/2. Taussky and Wielandt [Proc. Amer. Math. Soc., 13(1962), 732-735] showed that f_n(A, B; σ) = 0 if σ_i is the i^th elementary symmetric function of (β₄- β_s)², 1 ≤ r ˂ s ≤ n, i = 1, 2, …, N, with N = n(n-1)/2, where β₄ are the characteristic roots of B. In this thesis we discuss relations involving f_k(X, Y; σ) where X, Y Ɛ L and 1 ≤ k ˂ n. We show: 1. If F is infinite and if for each X Ɛ L there exists σ so that f_k(A, X; σ) = 0 where 1 ≤ k ˂ n, then A is a scalar transformation. 2. If F is algebraically closed, a necessary and sufficient condition that there exists a basis of V with respect to which the matrices of A and B are both in block upper triangular form, where the blocks on the diagonals are either one- or two-dimensional, is that certain products X₁, X₂…X_r belong to the radical of the algebra generated by A and B over F, where X_i has the form f₂(A, P(A,B); σ), for all polynomials P(x, y). We partially generalize this to the case where the blocks have dimensions ≤ k. 3. If A and B generate L, if the characteristic of F does not divide n and if there exists σ so that f_k(A, B; σ) = 0, for some k with 1 ≤ k ˂ n, then the characteristic roots of B belong to the splitting field of g_k(w; σ) = w^2K+1 - σ₁w^2K-1 + σ₂w^2K-3 - …. +(-1)^K σ_Kw over F. We use this result to prove a theorem involving a generalized form of property L [cf. Motzkin and Taussky, Trans. Amer. Math. Soc., 73(1952), 108-114]. 4. Also we give mild generalizations of results of McCoy [Amer. Math. Soc. Bull., 42(1936), 592-600] and Drazin [Proc. London Math. Soc., 1(1951), 222-231].

An in vitro biomolecular breadboard for prototyping synthetic biological circuits

Biomolecular circuit engineering is critical for implementing complex functions in vivo, and is a baseline method in the synthetic biology space. However, current methods for conducting biomolecular circuit engineering are time-consuming and tedious. A complete design-build-test cycle typically takes weeks' to months' time due to the lack of an intermediary between design ex vivo and testing in vivo. In this work, we explore the development and application of a "biomolecular breadboard" composed of an in-vitro transcription-translation (TX-TL) lysate to rapidly speed up the engineering design-build-test cycle. We first developed protocols for creating and using lysates for conducting biological circuit design. By doing so we simplified the existing technology to an affordable ($0.03/uL) and easy to use three-tube reagent system. We then developed tools to accelerate circuit design by allowing for linear DNA use in lieu of plasmid DNA, and by utilizing principles of modular assembly. This allowed the design-build-test cycle to be reduced to under a business day. We then characterized protein degradation dynamics in the breadboard to aid to implementing complex circuits. Finally, we demonstrated that the breadboard could be applied to engineer complex synthetic circuits in vitro and in vivo. Specifically, we utilized our understanding of linear DNA prototyping, modular assembly, and protein degradation dynamics to characterize the repressilator oscillator and to prototype novel three- and five-node negative feedback oscillators both in vitro and in vivo. We therefore believe the biomolecular breadboard has wide application for acting as an intermediary for biological circuit engineering.

Aspects of the biology of coarse fish in the Dorset Stour

Some of the results from an investigation of five species of coarse fish, in the Stour River, carried out from 1968-1978 are presented in this article. The species involved were: Rutilus rutilis, Leuciscus leuciscus, L. cephalus, Esox lucius and Perca fluviatilis : which are of particular interest to anglers. Although these species show some similarities, as in the shape of the annual and seasonal growth curves, in most other respects each species occupies a distinct niche in the ecosystem and has a life-history strategy peculiar to itself. In this study only 5 species were investigated. When all the species present are considered the relationships or diversities suggested here will therefore be made far more complex.

Fetntosecond pulse shaping with space-to-time conversion based on planar optics

We propose a novel structure of planar optical configuration for implementation of the space-to-time conversion for femtosecond pulse shaping. The previous apparatuses of femtosecond pulse shaping are 4f Fourier-transforming type system that is usually large, expensive, difficult to align. The planar integration of free-space optical systems on solid substrates is an optical module with the attractive advantages of compact, reliable and robust. This apparatus is analyzed in details and the design of the particular lens for femtosecond pulse shaping based on planar optics is presented. (c) 2006 Elsevier GmbH. All rights reserved.

Decomposition in lake sediments: bacterial action and interaction

This review discusses the processes involved in the decomposition of organic carbon derived initially from structural components of algae and other primary producers. It describes how groups of bacteria interact in time and space in a eutrophic lake. The relative importance of anaerobic and aerobic processes are discussed. The bulk of decomposition occurs within the sediment. The role of bacteria in the nitrogen cycle and the iron cycle, and in sulphate reduction and methanogenesis as the terminal metabolism of organic carbon are described.

The Riesz space structure of an Abelian W*-algebra

Let M be an Abelian W*-algebra of operators on a Hilbert space H. Let M₀ be the set of all linear, closed, densely defined transformations in H which commute with every unitary operator in the commutant M’ of M. A well known result of R. Pallu de Barriere states that if ɸ is a normal positive linear functional on M, then ɸ is of the form T → (Tx, x) for some x in H, where T is in M. An elementary proof of this result is given, using only those properties which are consequences of the fact that ReM is a Dedekind complete Riesz space with plenty of normal integrals. The techniques used lead to a natural construction of the class M₀, and an elementary proof is given of the fact that a positive self-adjoint transformation in M₀ has a unique positive square root in M₀. It is then shown that when the algebraic operations are suitably defined, then M₀ becomes a commutative algebra. If ReM₀ denotes the set of all self-adjoint elements of M₀, then it is proved that ReM₀ is Dedekind complete, universally complete Riesz spaces which contains ReM as an order dense ideal. A generalization of the result of R. Pallu de la Barriere is obtained for the Riesz space ReM₀ which characterizes the normal integrals on the order dense ideals of ReM₀. It is then shown that ReM₀ may be identified with the extended order dual of ReM, and that ReM₀ is perfect in the extended sense.

Some secondary questions related to the Riesz space ReM are also studied. In particular it is shown that ReM is a perfect Riesz space, and that every integral is normal under the assumption that every decomposition of the identity operator has non-measurable cardinal. The presence of atoms in ReM is examined briefly, and it is shown that ReM is finite dimensional if and only if every order bounded linear functional on ReM is a normal integral.

On the Cesari fixed point method in a Banach space

In a paper published in 1961, L. Cesari [1] introduces a method which extends certain earlier existence theorems of Cesari and Hale ([2] to [6]) for perturbation problems to strictly nonlinear problems. Various authors ([1], [7] to [15]) have now applied this method to nonlinear ordinary and partial differential equations. The basic idea of the method is to use the contraction principle to reduce an infinite-dimensional fixed point problem to a finite-dimensional problem which may be attacked using the methods of fixed point indexes.

The following is my formulation of the Cesari fixed point method:

Let B be a Banach space and let S be a finite-dimensional linear subspace of B. Let P be a projection of B onto S and suppose Г≤B such that pГ is compact and such that for every x in PГ, P^-1x∩Г is closed. Let W be a continuous mapping from Г into B. The Cesari method gives sufficient conditions for the existence of a fixed point of W in Г.

Let I denote the identity mapping in B. Clearly y = Wy for some y in Г if and only if both of the following conditions hold:

(i) Py = PWy.

(ii) y = (P + (I - P)W)y.

Definition. The Cesari fixed paint method applies to (Г, W, P) if and only if the following three conditions are satisfied:

(1) For each x in PГ, P + (I - P)W is a contraction from P^-1x∩Г into itself. Let y(x) be that element (uniqueness follows from the contraction principle) of P^-1x∩Г which satisfies the equation y(x) = Py(x) + (I-P)Wy(x).

(2) The function y just defined is continuous from PГ into B.

(3) There are no fixed points of PWy on the boundary of PГ, so that the (finite- dimensional) fixed point index i(PWy, int PГ) is defined.

Definition. If the Cesari fixed point method applies to (Г, W, P) then define i(Г, W, P) to be the index i(PWy, int PГ).

The three theorems of this thesis can now be easily stated.

Theorem 1 (Cesari). If i(Г, W, P) is defined and i(Г, W, P) ≠0, then there is a fixed point of W in Г.

Theorem 2. Let the Cesari fixed point method apply to both (Г, W, P₁) and (Г, W, P₂). Assume that P₂P₁=P₁P₂=P₁ and assume that either of the following two conditions holds:

(1) For every b in B and every z in the range of P₂, we have that ‖b=P₂b‖ ≤ ‖b-z‖

(2)P₂Г is convex.

Then i(Г, W, P₁) = i(Г, W, P₂).

Theorem 3. If Ω is a bounded open set and W is a compact operator defined on Ω so that the (infinite-dimensional) Leray-Schauder index i_LS(W, Ω) is defined, and if the Cesari fixed point method applies to (Ω, W, P), then i(Ω, W, P) = i_LS(W, Ω).

Theorems 2 and 3 are proved using mainly a homotopy theorem and a reduction theorem for the finite-dimensional and the Leray-Schauder indexes. These and other properties of indexes will be listed before the theorem in which they are used.

From organism to population: the role of life-history theory

The role of life-history theory in population and evolutionary analyses is outlined. In both cases general life histories can be analysed, but simpler life histories need fewer parameters for their description. The simplest case, of semelparous (breed-once-then-die) organisms, needs only three parameters: somatic growth rate, mortality rate and fecundity. This case is analysed in detail. If fecundity is fixed, population growth rate can be calculated direct from mortality rate and somatic growth rate, and isoclines on which population growth rate is constant can be drawn in a ”state space” with axes for mortality rate and somatic growth rate. In this space density-dependence is likely to result in a population trajectory from low density, when mortality rate is low and somatic growth rate is high and the population increases (positive population growth rate) to high density, after which the process reverses to return to low density. Possible effects of pollution on this system are discussed. The state-space approach allows direct population analysis of the twin effects of pollution and density on population growth rate. Evolutionary analysis uses related methods to identify likely evolutionary outcomes when an organism's genetic options are subject to trade-offs. The trade-off considered here is between somatic growth rate and mortality rate. Such a trade-off could arise because of an energy allocation trade-off if resources spent on personal defence (reducing mortality rate) are not available for somatic growth rate. The evolutionary implications of pollution acting on such a trade-off are outlined.