Steady capillary-gravity waves on deep water

The properties of capillary-gravity waves of permanent form on deep water are studied. Two different formulations to the problem are given. The theory of simple bifurcation is reviewed. For small amplitude waves a formal perturbation series is used. The Wilton ripple phenomenon is reexamined and shown to be associated with a bifurcation in which a wave of permanent form can double its period. It is shown further that Wilton's ripples are a special case of a more general phenomenon in which bifurcation into subharmonics and factorial higher harmonics can occur. Numerical procedures for the calculation of waves of finite amplitude are developed. Bifurcation and limit lines are calculated. Pure and combination waves are continued to maximum amplitude. It is found that the height is limited in all cases by the surface enclosing one or more bubbles. Results for the shape of gravity waves are obtained by solving an integra-differential equation. It is found that the family of solutions giving the waveheight or equivalent parameter has bifurcation points. Two bifurcation points and the branches emanating from them are found specifically, corresponding to a doubling and tripling of the wavelength. Solutions on the new branches are calculated.

Liquid silicate equation of state : using shock waves to understand the properties of the deep Earth

The equations of state (EOS) of several geologically important silicate liquids have been constrained via preheated shock wave techniques. Results on molten Fe₂SiO₄ (fayalite), Mg₂SiO₄ (forsterite), CaFeSi₂O₆ (hedenbergite), an equimolar mixture of CaAl₂Si₂O₈-CaFeSi₂O₆ (anorthite-hedenbergite), and an equimolar mixture of CaAl₂Si₂O₈-CaFeSi₂O₆-CaMgSi₂O₆(anorthite-hedenbergite-diopside) are presented. This work represents the first ever direct EOS measurements of an iron-bearing liquid or of a forsterite liquid at pressures relevant to the deep Earth (> 135 GPa). Additionally, revised EOS for molten CaMgSi₂O₆ (diopside), CaAl₂Si₂O₈ (anorthite), and MgSiO₃ (enstatite), which were previously determined by shock wave methods, are also presented.

The liquid EOS are incorporated into a model, which employs linear mixing of volumes to determine the density of compositionally intermediate liquids in the CaO-MgO-Al₂O₃-SiO₂-FeO major element space. Liquid volumes are calculated for temperature and pressure conditions that are currently present at the core-mantle boundary or that may have occurred during differentiation of a fully molten mantle magma ocean.

The most significant implications of our results include: (1) a magma ocean of either chondrite or peridotite composition is less dense than its first crystallizing solid, which is not conducive to the formation of a basal mantle magma ocean, (2) the ambient mantle cannot produce a partial melt and an equilibrium residue sufficiently dense to form an ultralow velocity zone mush, and (3) due to the compositional dependence of Fe2+ coordination, there is a threshold of Fe concentration (molar X_Fe ≤ 0.06) permitted in a liquid for which its density can still be approximated by linear mixing of end-member volumes.

The Role of Fucose in Learning and Memory: The Identification of a Fucosylprotein, Synapsin I, in the Rat Brain and the Characterization of its Fucosylation

Previous studies have shown that the glycoproteins containing the fucose moiety are involved in neuronal communication phenomena such as long-term potentiation and memory formation. These results imply that fucose containing glycoproteins might play an important role in learning and memory. To understand the role of fucose in neuronal communication, and the mechanisms by which fucose may be involved in information storage, the identification of fucosylproteins is essential. This report describes the identification and characterization of fucosylproteins in the brain, which will provide new insights into the role of the fucose involved molecular interactions.

Deep tissue fluorescence imaging with time-reversed light

Advances in optical techniques have enabled many breakthroughs in biology and medicine. However, light scattering by biological tissues remains a great obstacle, restricting the use of optical methods to thin ex vivo sections or superficial layers in vivo. In this thesis, we present two related methods that overcome the optical depth limit—digital time reversal of ultrasound encoded light (digital TRUE) and time reversal of variance-encoded light (TROVE). These two techniques share the same principle of using acousto-optic beacons for time reversal optical focusing within highly scattering media, like biological tissues. Ultrasound, unlike light, is not significantly scattered in soft biological tissues, allowing for ultrasound focusing. In addition, a fraction of the scattered optical wavefront that passes through the ultrasound focus gets frequency-shifted via the acousto-optic effect, essentially creating a virtual source of frequency-shifted light within the tissue. The scattered ultrasound-tagged wavefront can be selectively measured outside the tissue and time-reversed to converge at the location of the ultrasound focus, enabling optical focusing within deep tissues. In digital TRUE, we time reverse ultrasound-tagged light with an optoelectronic time reversal device (the digital optical phase conjugate mirror, DOPC). The use of the DOPC enables high optical gain, allowing for high intensity optical focusing and focal fluorescence imaging in thick tissues at a lateral resolution of 36 µm by 52 µm. The resolution of the TRUE approach is fundamentally limited to that of the wavelength of ultrasound. The ultrasound focus (~ tens of microns wide) usually contains hundreds to thousands of optical modes, such that the scattered wavefront measured is a linear combination of the contributions of all these optical modes. In TROVE, we make use of our ability to digitally record, analyze and manipulate the scattered wavefront to demix the contributions of these spatial modes using variance encoding. In essence, we encode each spatial mode inside the scattering sample with a unique variance, allowing us to computationally derive the time reversal wavefront that corresponds to a single optical mode. In doing so, we uncouple the system resolution from the size of the ultrasound focus, demonstrating optical focusing and imaging between highly diffusing samples at an unprecedented, speckle-scale lateral resolution of ~ 5 µm. Our methods open up the possibility of fully exploiting the prowess and versatility of biomedical optics in deep tissues.

Anatomical and developmental patterns of interleukin-2 gene expression in the mouse: analysis of IL-2 expressing cells and partial characterization of interactions involved in mediating IL-2 gene expression in vivo

Interleukin-2 (IL-2) is an important mediator in the vertebrate immune system. IL-2 is a potent growth factor that mature T lymphocytes use as a proliferation signal and the production of IL-2 is crucial for the clonal expansion of antigen-specific T cells in the primary immune response. IL-2 driven proliferation is dependent on the interaction of the lymphokine with its cognate multichain receptor. IL-2 expression is induced only upon stimulation and transcriptional activation of the IL-2 gene relies extensively on the coordinate interaction of numerous inducible and constitutive trans-acting factors. Over the past several years, thousands of papers have been published regarding molecular and cellular aspects of IL-2 gene expression and IL-2 function. The vast majority of these reports describe work that has been carried out in vitro. However, considerably less is known about control of IL-2 gene expression and IL-2 function in vivo.

To gain new insight into the regulation of IL-2 gene expression in vivo, anatomical and developmental patterns of IL-2 gene expression in the mouse were established by employing in situ hybridization and immunohistochemical staining methodologies to tissue sections generated from normal mice and mutant animals in which T -cell development was perturbed. Results from these studies revealed several interesting aspects of IL-2 gene expression, such as (1) induction of IL-2 gene expression and protein synthesis in the thymus, the primary site of T-cell development in the body, (2) cell-type specificity of IL-2 gene expression in vivo, (3) participation of IL-2 in the extrathymic expansion of mature T cells in particular tissues, independent of an acute immune response to foreign antigen, (4) involvement of IL-2 in maintaining immunologic balance in the mucosal immune system, and (5) potential function of IL-2 in early events associated with hematopoiesis.

Extensive analysis of IL-2 mRNA accumulation and protein production in the murine thymus at various stages of development established the existence of two classes of intrathymic IL-2 producing cells. One class of intrathymic IL-2 producers was found exclusively in the fetal thymus. Cells belonging to this subset were restricted to the outermost region of the thymus. IL-2 expression in the fetal thymus was highly transient; a dramatic peak ofiL-2 mRNA accumulation was identified at day 14.5 of gestation and maximal IL-2 protein production was observed 12 hours later, after which both IL-2 mRNA and protein levels rapidly decreased. Significantly, the presence of IL-2 expressing cells in the day 14-15 fetal thymus was not contingent on the generation of T-cell receptor (TcR) positive cells. The second class of IL-2 producing cells was also detectable in the fetal thymus (cells found in this class represented a minority subset of IL-2 producers in the fetal thymus) but persist in the thymus during later stages of development and after birth. Intrathymic IL-2 producers in postnatal animals were located in the subcapsular region and cortex, indicating that these cells reside in the same areas where immature T cells are consigned. The frequency of IL-2 expressing cells in the postnatal thymus was extremely low, indicating that induction of IL-2 expression and protein synthesis are indicative of a rare activation event. Unlike the fetal class of intrathymic IL-2 producers, the presence of IL-2 producing cells in the postnatal thymus was dependent on to the generation of TcR+ cells. Subsequent examination of intrathymic IL-2 production in mutant postnatal mice unable to produce either αβ or γδ T cells showed that postnatal IL-2 producers in the thymus belong to both αβ and γδ lineages. Additionally, further studies indicated that IL-2 synthesis by immature αβ -T cells depends on the expression of bonafide TcR αβ-heterodimers. Taken altogether, IL-2 production in the postnatal thymus relies on the generation of αβ or γδ-TcR^+ cells and induction of IL-2 protein synthesis can be linked to an activation event mediated via the TcR.

With regard to tissue specificity of IL-2 gene expression in vivo, analysis of whole body sections obtained from normal neonatal mouse pups by in situ hybridization demonstrated that IL-2 mRNA^+ cells were found in both lymphoid and nonlymphoid tissues with which T cells are associated, such as the thymus (as described above), dermis and gut. Tissues devoid of IL-2 mRNA^+ cells included brain, heart, lung, liver, stomach, spine, spinal cord, kidney, and bladder. Additional analysis of isolated tissues taken from older animals revealed that IL-2 expression was undetectable in bone marrow and in nonactivated spleen and lymph nodes. Thus, it appears that extrathymic IL-2 expressing cells in nonimmunologically challenged animals are relegated to particular epidermal and epithelial tissues in which characterized subsets of T cells reside and thatinduction of IL-2 gene expression associated with these tissues may be a result of T-cell activation therein.

Based on the neonatal in situ hybridization results, a detailed investigation into possible induction of IL-2 expression resulting in IL-2 protein synthesis in the skin and gut revealed that IL-2 expression is induced in the epidermis and intestine and IL-2 protein is available to drive cell proliferation of resident cells and/or participate in immune function in these tissues. Pertaining to IL-2 expression in the skin, maximal IL-2 mRNA accumulation and protein production were observed when resident Vγ_3^+ T-cell populations were expanding. At this age, both IL-2 mRNA^+ cells and IL-2 protein production were intimately associated with hair follicles. Likewise, at this age a significant number of CD3ε^+ cells were also found in association with follicles. The colocalization of IL-2 expression and CD3ε^+ cells suggests that IL-2 expression is induced when T cells are in contact with hair follicles. In contrast, neither IL-2 mRNA nor IL-2 protein were readily detected once T-cell density in the skin reached steady-state proportions. At this point, T cells were no longer found associated with hair follicles but were evenly distributed throughout the epidermis. In addition, IL-2 expression in the skin was contingent upon the presence of mature T cells therein and induction of IL-2 protein synthesis in the skin did not depend on the expression of a specific TcR on resident T cells. These newly disclosed properties of IL-2 expression in the skin indicate that IL-2 may play an additional role in controlling mature T-cell proliferation by participating in the extrathymic expansion of T cells, particularly those associated with the epidermis.

Finally, regarding IL-2 expression and protein synthesis in the gut, IL-2 producing cells were found associated with the lamina propria of neonatal animals and gut-associated IL-2 production persisted throughout life. In older animals, the frequency of IL-2 producing cells in the small intestine was not identical to that in the large intestine and this difference may reflect regional specialization of the mucosal immune system in response to enteric antigen. Similar to other instances of IL-2 gene expression in vivo, a failure to generate mature T cells also led to an abrogation of IL-2 protein production in the gut. The presence of IL-2 producing cells in the neonatal gut suggested that these cells may be generated during fetal development. Examination of the fetal gut to determine the distribution of IL-2 producing cells therein indicated that there was a tenfold increase in the number of gut-associated IL-2 producers at day 20 of gestation compared to that observed four days earlier and there was little difference between the frequency of IL-2 producing cells in prenatal versus neonatal gut. The origin of these fetally-derived IL-2 producing cells is unclear. Prior to the immigration of IL-2 inducible cells to the fetal gut and/or induction of IL-2 expression therein, IL-2 protein was observed in the fetal liver and fetal omentum, as well as the fetal thymus. Considering that induction of IL-2 protein synthesis may be an indication of future functional capability, detection of IL-2 producing cells in the fetal liver and fetal omentum raises the possibility that IL-2 producing cells in the fetal gut may be extrathymic in origin and IL-2 producing cells in these fetal tissues may not belong solely to the T lineage. Overall, these results provide increased understanding of the nature of IL-2 producing cells in the gut and how the absence of IL-2 production therein and in fetal hematopoietic tissues can result in the acute pathology observed in IL-2 deficient animals.

Mathematical study of complex networks : brain, internet, and power grid

The dissertation is concerned with the mathematical study of various network problems. First, three real-world networks are considered: (i) the human brain network (ii) communication networks, (iii) electric power networks. Although these networks perform very different tasks, they share similar mathematical foundations. The high-level goal is to analyze and/or synthesis each of these systems from a “control and optimization” point of view. After studying these three real-world networks, two abstract network problems are also explored, which are motivated by power systems. The first one is “flow optimization over a flow network” and the second one is “nonlinear optimization over a generalized weighted graph”. The results derived in this dissertation are summarized below.

Brain Networks: Neuroimaging data reveals the coordinated activity of spatially distinct brain regions, which may be represented mathematically as a network of nodes (brain regions) and links (interdependencies). To obtain the brain connectivity network, the graphs associated with the correlation matrix and the inverse covariance matrix—describing marginal and conditional dependencies between brain regions—have been proposed in the literature. A question arises as to whether any of these graphs provides useful information about the brain connectivity. Due to the electrical properties of the brain, this problem will be investigated in the context of electrical circuits. First, we consider an electric circuit model and show that the inverse covariance matrix of the node voltages reveals the topology of the circuit. Second, we study the problem of finding the topology of the circuit based on only measurement. In this case, by assuming that the circuit is hidden inside a black box and only the nodal signals are available for measurement, the aim is to find the topology of the circuit when a limited number of samples are available. For this purpose, we deploy the graphical lasso technique to estimate a sparse inverse covariance matrix. It is shown that the graphical lasso may find most of the circuit topology if the exact covariance matrix is well-conditioned. However, it may fail to work well when this matrix is ill-conditioned. To deal with ill-conditioned matrices, we propose a small modification to the graphical lasso algorithm and demonstrate its performance. Finally, the technique developed in this work will be applied to the resting-state fMRI data of a number of healthy subjects.

Communication Networks: Congestion control techniques aim to adjust the transmission rates of competing users in the Internet in such a way that the network resources are shared efficiently. Despite the progress in the analysis and synthesis of the Internet congestion control, almost all existing fluid models of congestion control assume that every link in the path of a flow observes the original source rate. To address this issue, a more accurate model is derived in this work for the behavior of the network under an arbitrary congestion controller, which takes into account of the effect of buffering (queueing) on data flows. Using this model, it is proved that the well-known Internet congestion control algorithms may no longer be stable for the common pricing schemes, unless a sufficient condition is satisfied. It is also shown that these algorithms are guaranteed to be stable if a new pricing mechanism is used.

Electrical Power Networks: Optimal power flow (OPF) has been one of the most studied problems for power systems since its introduction by Carpentier in 1962. This problem is concerned with finding an optimal operating point of a power network minimizing the total power generation cost subject to network and physical constraints. It is well known that OPF is computationally hard to solve due to the nonlinear interrelation among the optimization variables. The objective is to identify a large class of networks over which every OPF problem can be solved in polynomial time. To this end, a convex relaxation is proposed, which solves the OPF problem exactly for every radial network and every meshed network with a sufficient number of phase shifters, provided power over-delivery is allowed. The concept of “power over-delivery” is equivalent to relaxing the power balance equations to inequality constraints.

Flow Networks: In this part of the dissertation, the minimum-cost flow problem over an arbitrary flow network is considered. In this problem, each node is associated with some possibly unknown injection, each line has two unknown flows at its ends related to each other via a nonlinear function, and all injections and flows need to satisfy certain box constraints. This problem, named generalized network flow (GNF), is highly non-convex due to its nonlinear equality constraints. Under the assumption of monotonicity and convexity of the flow and cost functions, a convex relaxation is proposed, which always finds the optimal injections. A primary application of this work is in the OPF problem. The results of this work on GNF prove that the relaxation on power balance equations (i.e., load over-delivery) is not needed in practice under a very mild angle assumption.

Generalized Weighted Graphs: Motivated by power optimizations, this part aims to find a global optimization technique for a nonlinear optimization defined over a generalized weighted graph. Every edge of this type of graph is associated with a weight set corresponding to the known parameters of the optimization (e.g., the coefficients). The motivation behind this problem is to investigate how the (hidden) structure of a given real/complex valued optimization makes the problem easy to solve, and indeed the generalized weighted graph is introduced to capture the structure of an optimization. Various sufficient conditions are derived, which relate the polynomial-time solvability of different classes of optimization problems to weak properties of the generalized weighted graph such as its topology and the sign definiteness of its weight sets. As an application, it is proved that a broad class of real and complex optimizations over power networks are polynomial-time solvable due to the passivity of transmission lines and transformers.

Identification of fucose-α(1-2)-galactose binding proteins in the mammalian brain

Fucose-α(1-2)-galactose (Fucα(1-2)Gal) carbohydrates have been implicated in cognitive functions. However, the underlying molecular mechanisms that govern these processes are not well understood. While significant progress has been made toward identifying glycoconjugates bearing this carbohydrate epitope, a major challenge remains the discovery of interactions mediated by these sugars. Here, we employ the use of multivalent glycopolymers to enable the proteomic identification of weak affinity, low abundant Fucα(1-2)Gal-binding proteins (i.e. lectins) from the brain. End-biotinylated glycopolymers containing photoactivatable crosslinkers were used to capture and enrich potential Fucα(1-2)Gal-specific lectins from rat brain lysates. Candidate lectins were tested for their ability to bind Fucα(1-2)Gal, and the functional significance of the interaction was investigated for one such candidate, SV2a, using a knock-out mouse system. Our results suggest an important role for this glycan-lectin interaction in facilitating synaptic changes necessary for neuronal communication. This study highlights the use of glycopolymer mimetics to discover novel lectins and identify functional interactions between fucosyl carbohydrates and lectins in the brain.

Reduced deep-tissue image degradation in three-dimensional multiphoton microscopy with concentric two-color two-photon fluorescence excitation

We theoretically demonstrate that enhanced penetration depth in three-dimensional multiphoton microscopy can be achieved using concentric two-color two-photon (C2C2P) fluorescence excitation in which the two excitation beams are separated in space before reaching their common focal spot. Monte Carlo simulation shows that, in comparison with the one-color two-photon excitation scheme, the C2C2P fluorescence microscopy provides a significantly greater penetration depth for imaging into a highly scattering medium. (C) 2008 Optical Society of America.

Mechanisms of ultrafast laser-induced deep-subwavelength gratings on graphite and diamond

Deep-subwavelength gratings with periodicities of 170, 120, and 70 nm can be observed on highly oriented pyrolytic graphite irradiated by a femtosecond (fs) laser at 800 nm. Under picosecond laser irradiation, such gratings likewise can be produced. Interestingly, the 170-nm grating is also observed on single-crystal diamond irradiated by the 800-nm fs laser. In our opinion, the optical properties of the high-excited state of material surface play a key role for the formation of the deep-subwavelength gratings. The numerical simulations of the graphite deep-subwavelength grating at normal and high-excited states confirm that in the groove the light intensity can be extraordinarily enhanced via cavity-mode excitation in the condition of transverse-magnetic wave irradiation with near-ablation-threshold fluences. This field enhancement of polarization sensitiveness in deep-subwavelength apertures acts as an important feedback mechanism for the growth and polarization dependence of the deep-subwavelength gratings. In addition, we suggest that surface plasmons are responsible for the formation of seed deep-subwavelength apertures with a particular periodicity and the initial polarization dependence. Finally, we propose that the nanoscale Coulomb explosion occurring in the groove is responsible for the ultrafast nonthermal ablation mechanism.

The deep ocean density structure at the Last Glacial Maximum : What was it and why?

The search for reliable proxies of past deep ocean temperature and salinity has proved difficult, thereby limiting our ability to understand the coupling of ocean circulation and climate over glacial-interglacial timescales. Previous inferences of deep ocean temperature and salinity from sediment pore fluid oxygen isotopes and chlorinity indicate that the deep ocean density structure at the Last Glacial Maximum (LGM, approximately 20,000 years BP) was set by salinity, and that the density contrast between northern and southern sourced deep waters was markedly greater than in the modern ocean. High density stratification could help explain the marked contrast in carbon isotope distribution recorded in the LGM ocean relative to that we observe today, but what made the ocean's density structure so different at the LGM? How did it evolve from one state to another? Further, given the sparsity of the LGM temperature and salinity data set, what else can we learn by increasing the spatial density of proxy records?

We investigate the cause and feasibility of a highly and salinity stratified deep ocean at the LGM and we work to increase the amount of information we can glean about the past ocean from pore fluid profiles of oxygen isotopes and chloride. Using a coupled ocean--sea ice--ice shelf cavity model we test whether the deep ocean density structure at the LGM can be explained by ice--ocean interactions over the Antarctic continental shelves, and show that a large contribution of the LGM salinity stratification can be explained through lower ocean temperature. In order to extract the maximum information from pore fluid profiles of oxygen isotopes and chloride we evaluate several inverse methods for ill-posed problems and their ability to recover bottom water histories from sediment pore fluid profiles. We demonstrate that Bayesian Markov Chain Monte Carlo parameter estimation techniques enable us to robustly recover the full solution space of bottom water histories, not only at the LGM, but through the most recent deglaciation and the Holocene up to the present. Finally, we evaluate a non-destructive pore fluid sampling technique, Rhizon samplers, in comparison to traditional squeezing methods and show that despite their promise, Rhizons are unlikely to be a good sampling tool for pore fluid measurements of oxygen isotopes and chloride.

The autocorrelation of speckles in deep Fresnel diffraction region and characterizations of random self-affine fractal surfaces

This paper studies the correlation properties of the speckles in the deep Fresnel diffraction region produced by the scattering of rough self-affine fractal surfaces. The autocorrelation function of the speckle intensities is formulated by the combination of the light scattering theory of Kirchhoff approximation and the principles of speckle statistics. We propose a method for extracting the three surface parameters, i.e. the roughness w, the lateral correlation length xi and the roughness exponent alpha, from the autocorrelation functions of speckles. This method is verified by simulating the speckle intensities and calculating the speckle autocorrelation function. We also find the phenomenon that for rough surfaces with alpha = 1, the structure of the speckles resembles that of the surface heights, which results from the effect of the peak and the valley parts of the surface, acting as micro-lenses converging and diverging the light waves.

Microelectrode Implants for Spinal Cord Stimulation in Rats

Paralysis is a debilitating condition afflicting millions of people across the globe, and is particularly deleterious to quality of life when motor function of the legs is severely impaired or completely absent. Fortunately, spinal cord stimulation has shown great potential for improving motor function after spinal cord injury and other pathological conditions. Many animal studies have shown stimulation of the neural networks in the spinal cord can improve motor ability so dramatically that the animals can even stand and step after a complete spinal cord transaction.

This thesis presents work to successfully provide a chronically implantable device for rats that greatly enhances the ability to control the site of spinal cord stimulation. This is achieved through the use of a parylene-C based microelectrode array, which enables a density of stimulation sites unattainable with conventional wire electrodes. While many microelectrode devices have been proposed in the past, the spinal cord is a particularly challenging environment due to the bending and movement it undergoes in a live animal. The developed microelectrode array is the first to have been implanted in vivo while retaining functionality for over a month. In doing so, different neural pathways can be selectively activated to facilitate standing and stepping in spinalized rats using various electrode combinations, and important differences in responses are observed.

An engineering challenge for the usability of any high density electrode array is connecting the numerous electrodes to a stimulation source. This thesis develops several technologies to address this challenge, beginning with a fully passive implant that uses one wire per electrode to connect to an external stimulation source. The number of wires passing through the body and the skin proved to be a hazard for the health of the animal, so a multiplexed implant was devised in which active electronics reduce the number of wires. Finally, a fully wireless implant was developed. As these implants are tested in vivo, encapsulation is of critical importance to retain functionality in a chronic experiment, especially for the active implants, and it was achieved without the use of costly ceramic or metallic hermetic packaging. Active implants were built that retained functionality 8 weeks after implantation, and achieved stepping in spinalized rats after just 8-10 days, which is far sooner than wire-based electrical stimulation has achieved in prior work.