Smooth sets for borel equivalence relations and the covering property for σ-ideals of compact sets

This thesis is divided into three chapters. In the first chapter we study the smooth sets with respect to a Borel equivalence realtion E on a Polish space X. The collection of smooth sets forms σ-ideal. We think of smooth sets as analogs of countable sets and we show that an analog of the perfect set theorem for Σ¹₁ sets holds in the context of smooth sets. We also show that the collection of Σ¹₁ smooth sets is ∏¹₁ on the codes. The analogs of thin sets are called sparse sets. We prove that there is a largest ∏¹₁ sparse set and we give a characterization of it. We show that in L there is a ∏¹₁ sparse set which is not smooth. These results are analogs of the results known for the ideal of countable sets, but it remains open to determine if large cardinal axioms imply that ∏¹₁ sparse sets are smooth. Some more specific results are proved for the case of a countable Borel equivalence relation. We also study I(E), the σ-ideal of closed E-smooth sets. Among other things we prove that E is smooth iff I(E) is Borel.

In chapter 2 we study σ-ideals of compact sets. We are interested in the relationship between some descriptive set theoretic properties like thinness, strong calibration and the covering property. We also study products of σ-ideals from the same point of view. In chapter 3 we show that if a σ-ideal I has the covering property (which is an abstract version of the perfect set theorem for Σ¹₁ sets), then there is a largest ∏¹₁ set in I^int (i.e., every closed subset of it is in I). For σ-ideals on 2^ω we present a characterization of this set in a similar way as for C₁, the largest thin ∏¹₁ set. As a corollary we get that if there are only countable many reals in L, then the covering property holds for Σ¹₂ sets.

ABCD matrix for reflection and refraction of Gaussian beams at the surface of a parabola of revolution

We report the formulation of an ABCD matrix for reflection and refraction of Gaussian light beams at the surface of a parabola of revolution that separate media of different refractive indices based on optical phase matching. The equations for the spot sizes and wave-front radii of the beams are also obtained by using the ABCD matrix. With these matrices, we can more conveniently design and evaluate some special optical systems, including these kinds of elements. (c) 2005 Optical Society of America

提高能量密度的超衍射极限激光光束相位补偿技术

定义了激光光束衍射远场光斑压缩前后的能量比以及能量密度比来衡量超衍射极限激光光束的质量。通过利用反向传递算法设计了合适的补偿相位板，不但对准直放大的单一横模激光光束进行小于光学衍射极限的发散度的压缩，同时又保证光束能量集中于压缩后的远场衍射主瓣中，使压缩后的远场衍射光斑的能量密度增加。给出了相应的实例。这一结论不但解决了光学超分辨中光束压缩与能量损失不可避免这一矛盾，而且为发散度小且能量密度高的超衍射极限激光光束的实验工作以及该类光束的实际应用提供了理论基础。

Formal methods in the foundations of science

The intent of this study is to provide formal apparatus which facilitates the investigation of problems in the methodology of science. The introduction contains several examples of such problems and motivates the subsequent formalism.

A general definition of a formal language is presented, and this definition is used to characterize an individual’s view of the world around him. A notion of empirical observation is developed which is independent of language. The interplay of formal language and observation is taken as the central theme. The process of science is conceived as the finding of that formal language that best expresses the available experimental evidence.

To characterize the manner in which a formal language imposes structure on its universe of discourse, the fundamental concepts of elements and states of a formal language are introduced. Using these, the notion of a basis for a formal language is developed as a collection of minimal states distinguishable within the language. The relation of these concepts to those of model theory is discussed.

An a priori probability defined on sets of observations is postulated as a reflection of an individual’s ontology. This probability, in conjunction with a formal language and a basis for that language, induces a subjective probability describing an individual’s conceptual view of admissible configurations of the universe. As a function of this subjective probability, and consequently of language, a measure of the informativeness of empirical observations is introduced and is shown to be intuitively plausible – particularly in the case of scientific experimentation.

The developed formalism is then systematically applied to the general problems presented in the introduction. The relationship of scientific theories to empirical observations is discussed and the need for certain tacit, unstatable knowledge is shown to be necessary to fully comprehend the meaning of realistic theories. The idea that many common concepts can be specified only by drawing on knowledge obtained from an infinite number of observations is presented, and the problems of reductionism are examined in this context.

A definition of when one formal language can be considered to be more expressive than another is presented, and the change in the informativeness of an observation as language changes is investigated. In this regard it is shown that the information inherent in an observation may decrease for a more expressive language.

The general problem of induction and its relation to the scientific method are discussed. Two hypotheses concerning an individual’s selection of an optimal language for a particular domain of discourse are presented and specific examples from the introduction are examined.

The Molecular Basis of Lysine Acetylation: Addition, Removal, and Recognition

Acetyltransferases and deacetylases catalyze the addition and removal, respectively, of acetyl groups to the epsilon-amino group of protein lysine residues. This modification can affect the function of a protein through several means, including the recruitment of specific binding partners called acetyl-lysine readers. Acetyltransferases, deacetylases, and acetyl-lysine readers have emerged as crucial regulators of biological processes and prominent targets for the treatment of human disease. This work describes a combination of structural, biochemical, biophysical, cell-biological, and organismal studies undertaken on a set of proteins that cumulatively include all steps of the acetylation process: the acetyltransferase MEC-17, the deacetylase SIRT1, and the acetyl-lysine reader DPF2. Tubulin acetylation by MEC-17 is associated with stable, long-lived microtubule structures. We determined the crystal structure of the catalytic domain of human MEC-17 in complex with the cofactor acetyl-CoA. The structure in combination with an extensive enzymatic analysis of MEC-17 mutants identified residues for cofactor and substrate recognition and activity. A large, evolutionarily conserved hydrophobic surface patch distal to the active site was shown to be necessary for catalysis, suggesting that specificity is achieved by interactions with the alpha-tubulin substrate that extend outside of the modified surface loop. Experiments in C. elegans showed that while MEC-17 is required for touch sensitivity, MEC-17 enzymatic activity is dispensible for this behavior. SIRT1 deacetylates a wide range of substrates, including p53, NF-kappaB, FOXO transcription factors, and PGC-1-alpha, with roles in cellular processes ranging from energy metabolism to cell survival. SIRT1 activity is uniquely controlled by a C-terminal regulatory segment (CTR). Here we present crystal structures of the catalytic domain of human SIRT1 in complex with the CTR in an apo form and in complex with a cofactor and a pseudo-substrate peptide. The catalytic domain adopts the canonical sirtuin fold. The CTR forms a beta-hairpin structure that complements the beta-sheet of the NAD^+-binding domain, covering an essentially invariant, hydrophobic surface. A comparison of the apo and cofactor bound structures revealed conformational changes throughout catalysis, including a rotation of a smaller subdomain with respect to the larger NAD^+-binding subdomain. A biochemical analysis identified key residues in the active site, an inhibitory role for the CTR, and distinct structural features of the CTR that mediate binding and inhibition of the SIRT1 catalytic domain. DPF2 represses myeloid differentiation in acute myelogenous leukemia. Finally, we solved the crystal structure of the tandem PHD domain of human DPF2. We showed that DPF2 preferentially binds H3 tail peptides acetylated at Lys14, and binds H4 tail peptides with no preference for acetylation state. Through a structural and mutational analysis we identify the molecular basis of histone recognition. We propose a model for the role of DPF2 in AML and identify the DPF2 tandem PHD finger domain as a promising novel target for anti-leukemia therapeutics.

Phase-step calibration technique based on a two-run-times-two-frame phase-shift method

A novel phase-step calibration technique is presented on the basis of a two-run-times-two-frame phase-shift method. First the symmetry factor M is defined to describe the distribution property of the distorted phase due to phase-shifter miscalibration; then the phase-step calibration technique, in which two sets of two interferograms with a straight fringe pattern are recorded and the phase step is obtained by calculating M of the wrapped phase map, is developed. With this technique, a good mirror is required, but no uniform illumination is needed and no complex mathematical operation is involved. This technique can be carried out in situ and is applicable to any phase shifter, whether linear or nonlinear. (c) 2006 Optical Society of America.

Optimum linear coding for additive noise systems using information feedback

The feedback coding problem for Gaussian systems in which the noise is neither white nor statistically independent between channels is formulated in terms of arbitrary linear codes at the transmitter and at the receiver. This new formulation is used to determine a number of feedback communication systems. In particular, the optimum linear code that satisfies an average power constraint on the transmitted signals is derived for a system with noiseless feedback and forward noise of arbitrary covariance. The noisy feedback problem is considered and signal sets for the forward and feedback channels are obtained with an average power constraint on each. The general formulation and results are valid for non-Gaussian systems in which the second order statistics are known, the results being applicable to the determination of error bounds via the Chebychev inequality.

A generalized Hausdorff dimension for functions and sets

Let E be a compact subset of the n-dimensional unit cube, 1_n, and let C be a collection of convex bodies, all of positive n-dimensional Lebesgue measure, such that C contains bodies with arbitrarily small measure. The dimension of E with respect to the covering class C is defined to be the number

d_C(E) = sup(β:H_{β, C}(E) > 0),

where H_{β, C} is the outer measure

inf(Ʃm(C_i)^β:UC_i Ↄ E, C_i ϵ C) .

Only the one and two-dimensional cases are studied. Moreover, the covering classes considered are those consisting of intervals and rectangles, parallel to the coordinate axes, and those closed under translations. A covering class is identified with a set of points in the left-open portion, 1’_n, of 1_n, whose closure intersects 1_n - 1’_n. For n = 2, the outer measure H_{β, C} is adopted in place of the usual:

Inf(Ʃ(diam. (C_i))^β: UC_i Ↄ E, C_i ϵ C),

for the purpose of studying the influence of the shape of the covering sets on the dimension d_C(E).

If E is a closed set in 1₁, let M(E) be the class of all non-decreasing functions μ(x), supported on E with μ(x) = 0, x ≤ 0 and μ(x) = 1, x ≥ 1. Define for each μ ϵ M(E),

d_C(μ) = lim/c → inf/0 log ∆μ(c)/log c , (c ϵ C)

where ∆μ(c) = v/x (μ(x+c) – μ(x)). It is shown that

d_C(E) = sup (d_C(μ):μ ϵ M(E)).

This notion of dimension is extended to a certain class Ӻ of sub-additive functions, and the problem of studying the behavior of d_C(E) as a function of the covering class C is reduced to the study of d_C(f) where f ϵ Ӻ. Specifically, the set of points in 1₁,

(*) {d_B(F), d_C(f)): f ϵ Ӻ}

is characterized by a comparison of the relative positions of the points of B and C. A region of the form (*) is always closed and doubly-starred with respect to the points (0, 0) and (1, 1). Conversely, given any closed region in 1₂, doubly-starred with respect to (0, 0) and (1, 1), there are covering classes B and C such that (*) is exactly that region. All of the results are shown to apply to the dimension of closed sets E. Similar results can be obtained when a finite number of covering classes are considered.

In two dimensions, the notion of dimension is extended to the class M, of functions f(x, y), non-decreasing in x and y, supported on 1₂ with f(x, y) = 0 for x · y = 0 and f(1, 1) = 1, by the formula

d_C(f) = lim/s · t → inf/0 log ∆f(s, t)/log s · t , (s, t) ϵ C

where

∆f(s, t) = V/x, y (f(x+s, y+t) – f(x+s, y) – f(x, y+t) + f(x, t)).

A characterization of the equivalence d_C1(f) = d_C2(f) for all f ϵ M, is given by comparison of the gaps in the sets of products s · t and quotients s/t, (s, t) ϵ C_i (I = 1, 2).

Changes in the coherence of quasi-monochromatic electromagnetic Gaussian Schell-model beams propagating through turbulent atmosphere

Based on the extended Huygens-Fresnel principle, the mutual coherence function of quasi-monochromatic electromagnetic Gaussian Schell-model (EGSM) beams propagating through turbulent atmosphere is derived analytically. By employing the lateral and the longitudinal coherence length of EGSM beams to characterize the spatial and the temporal coherence of the beams, the behavior of changes in the spatial and the temporal coherence of those beams is studied. The results show that with a fixed set of beam parameters and under particular atmospheric turbulence model, the lateral coherence of an EGSM beam reaches its maximum value as the beam propagates a certain distance in the turbulent atmosphere, then it begins degrading and keeps decreasing along with the further distance. However, the longitudinal coherence length of an EGSM beam keeps unchanging in this propagation. Lastly, a qualitative explanation is given to these results. (c) 2007 Elsevier B.V. All rights reserved.

Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation

An analytical formula for the cross-spectral density matrix of the electric field of anisotropic electromagnetic Gaussian-Schell model beams propagating in free space is derived by using a tensor method. The effects of coherence on those beams are studied. It is shown that two anisotropic stochastic electromagnetic beams that propagate from the source plane z = 0 into the half-space z > 0 may have different beam shapes (i.e., spectral density) and states of polarization in the half-space, even though they have the same beam shape and states of polarization in the source plane. This fact is due to a difference in the coherence properties of the field in the source plane. (C) 2007 Optical Society of America.

The global optimization of phase-incoherent signals

The problem of global optimization of M phase-incoherent signals in N complex dimensions is formulated. Then, by using the geometric approach of Landau and Slepian, conditions for optimality are established for N = 2 and the optimal signal sets are determined for M = 2, 3, 4, 6, and 12.

The method is the following: The signals are assumed to be equally probable and to have equal energy, and thus are represented by points ṡ_i, i = 1, 2, …, M, on the unit sphere S₁ in C^N. If W_ik is the halfspace determined by ṡ_i and ṡ_k and containing ṡ_i, i.e. W_ik = {ṙϵC^N:| ≥ | ˂ṙ, ṡ_k˃|}, then the Ʀ_i = ∩/k≠i W_ik, i = 1, 2, …, M, the maximum likelihood decision regions, partition S₁. For additive complex Gaussian noise ṅ and a received signal ṙ = ṡ_ie^jϴ + ṅ, where ϴ is uniformly distributed over [0, 2π], the probability of correct decoding is P_C = 1/π^N ∞/ʃ/0 r^2N-1e^-(r2+1)U(r)dr, where U(r) = 1/M M/Ʃ/i=1 Ʀ_i ʃ/∩ S₁ I₀(2r | ˂ṡ, ṡ_i˃|)dσ(ṡ), and r = ǁṙǁ.

For N = 2, it is proved that U(r) ≤ ʃ/C_α I₀(2r|˂ṡ, ṡ_i˃|)dσ(ṡ) – 2K/M. h(1/2K [Mσ(C_α)-σ(S₁)]), where C_α = {ṡϵS₁:|˂ṡ, ṡ_i˃| ≥ α}, K is the total number of boundaries of the net on S₁ determined by the decision regions, and h is the strictly increasing strictly convex function of σ(C_α∩W), (where W is a halfspace not containing ṡ_i), given by h = ʃ/C_α∩W I₀ (2r|˂ṡ, ṡ_i˃|)dσ(ṡ). Conditions for equality are established and these give rise to the globally optimal signal sets for M = 2, 3, 4, 6, and 12.

Dynamical models for the Earth's geoid

The Earth's largest geoid anomalies occur at the lowest spherical harmonic degrees, or longest wavelengths, and are primarily the result of mantle convection. Thermal density contrasts due to convection are partially compensated by boundary deformations due to viscous flow whose effects must be included in order to obtain a dynamically consistent model for the geoid. These deformations occur rapidly with respect to the timescale for convection, and we have analytically calculated geoid response kernels for steady-state, viscous, incompressible, self-gravitating, layered Earth models which include the deformation of boundaries due to internal loads. Both the sign and magnitude of geoid anomalies depend strongly upon the viscosity structure of the mantle as well as the possible presence of chemical layering.

Correlations of various global geophysical data sets with the observed geoid can be used to construct theoretical geoid models which constrain the dynamics of mantle convection. Surface features such as topography and plate velocities are not obviously related to the low-degree geoid, with the exception of subduction zones which are characterized by geoid highs (degrees 4-9). Recent models for seismic heterogeneity in the mantle provide additional constraints, and much of the low-degree (2-3) geoid can be attributed to seismically inferred density anomalies in the lower mantle. The Earth's largest geoid highs are underlain by low density material in the lower mantle, thus requiring compensating deformations of the Earth's surface. A dynamical model for whole mantle convection with a low viscosity upper mantle can explain these observations and successfully predicts more than 80% of the observed geoid variance.

Temperature variations associated with density anomalies in the man tie cause lateral viscosity variations whose effects are not included in the analytical models. However, perturbation theory and numerical tests show that broad-scale lateral viscosity variations are much less important than radial variations; in this respect, geoid models, which depend upon steady-state surface deformations, may provide more reliable constraints on mantle structure than inferences from transient phenomena such as postglacial rebound. Stronger, smaller-scale viscosity variations associated with mantle plumes and subducting slabs may be more important. On the basis of numerical modelling of low viscosity plumes, we conclude that the global association of geoid highs (after slab effects are removed) with hotspots and, perhaps, mantle plumes, is the result of hot, upwelling material in the lower mantle; this conclusion does not depend strongly upon plume rheology. The global distribution of hotspots and the dominant, low-degree geoid highs may correspond to a dominant mode of convection stabilized by the ancient Pangean continental assemblage.