Arithmetic of cyclotomic fields and Fermat-type equations

Let l be any odd prime, and ζ a primitive l-th root of unity. Let C_l be the l-Sylow subgroup of the ideal class group of Q(ζ). The Teichmüller character w : Z_l → Z^*_l is given by w(x) = x (mod l), where w(x) is a p-1-st root of unity, and x ∈ Z_l. Under the action of this character, C_l decomposes as a direct sum of C^((i))_l, where C^((i))_l is the eigenspace corresponding to w^i. Let the order of C^((3))_l be l^h_3). The main result of this thesis is the following: For every n ≥ max( 1, h_3 ), the equation x^(ln) + y^(ln) + z^(ln) = 0 has no integral solutions (x,y,z) with l ≠ xyz. The same result is also proven with n ≥ max(1,h_5), under the assumption that C_l^((5)) is a cyclic group of order l^h_5. Applications of the methods used to prove the above results to the second case of Fermat's last theorem and to a Fermat-like equation in four variables are given.

The proof uses a series of ideas of H.S. Vandiver ([Vl],[V2]) along with a theorem of M. Kurihara [Ku] and some consequences of the proof of lwasawa's main conjecture for cyclotomic fields by B. Mazur and A. Wiles [MW]. In [V1] Vandiver claimed that the first case of Fermat's Last Theorem held for l if l did not divide the class number h^+ of the maximal real subfield of Q(e^(2πi/i)). The crucial gap in Vandiver's attempted proof that has been known to experts is explained, and complete proofs of all the results used from his papers are given.

Learning and using taxonomies for visual and olfactory classification

Humans are able of distinguishing more than 5000 visual categories even in complex environments using a variety of different visual systems all working in tandem. We seem to be capable of distinguishing thousands of different odors as well. In the machine learning community, many commonly used multi-class classifiers do not scale well to such large numbers of categories. This thesis demonstrates a method of automatically creating application-specific taxonomies to aid in scaling classification algorithms to more than 100 cate- gories using both visual and olfactory data. The visual data consists of images collected online and pollen slides scanned under a microscope. The olfactory data was acquired by constructing a small portable sniffing apparatus which draws air over 10 carbon black polymer composite sensors. We investigate performance when classifying 256 visual categories, 8 or more species of pollen and 130 olfactory categories sampled from common household items and a standardized scratch-and-sniff test. Taxonomies are employed in a divide-and-conquer classification framework which improves classification time while allowing the end user to trade performance for specificity as needed. Before classification can even take place, the pollen counter and electronic nose must filter out a high volume of background “clutter” to detect the categories of interest. In the case of pollen this is done with an efficient cascade of classifiers that rule out most non-pollen before invoking slower multi-class classifiers. In the case of the electronic nose, much of the extraneous noise encountered in outdoor environments can be filtered using a sniffing strategy which preferentially samples the visensor response at frequencies that are relatively immune to background contributions from ambient water vapor. This combination of efficient background rejection with scalable classification algorithms is tested in detail for three separate projects: 1) the Caltech-256 Image Dataset, 2) the Caltech Automated Pollen Identification and Counting System (CAPICS) and 3) a portable electronic nose specially constructed for outdoor use.

Electrical and optical interconnects for high-performance computing

Technology scaling has enabled drastic growth in the computational and storage capacity of integrated circuits (ICs). This constant growth drives an increasing demand for high-bandwidth communication between and within ICs. In this dissertation we focus on low-power solutions that address this demand. We divide communication links into three subcategories depending on the communication distance. Each category has a different set of challenges and requirements and is affected by CMOS technology scaling in a different manner. We start with short-range chip-to-chip links for board-level communication. Next we will discuss board-to-board links, which demand a longer communication range. Finally on-chip links with communication ranges of a few millimeters are discussed.

Electrical signaling is a natural choice for chip-to-chip communication due to efficient integration and low cost. IO data rates have increased to the point where electrical signaling is now limited by the channel bandwidth. In order to achieve multi-Gb/s data rates, complex designs that equalize the channel are necessary. In addition, a high level of parallelism is central to sustaining bandwidth growth. Decision feedback equalization (DFE) is one of the most commonly employed techniques to overcome the limited bandwidth problem of the electrical channels. A linear and low-power summer is the central block of a DFE. Conventional approaches employ current-mode techniques to implement the summer, which require high power consumption. In order to achieve low-power operation we propose performing the summation in the charge domain. This approach enables a low-power and compact realization of the DFE as well as crosstalk cancellation. A prototype receiver was fabricated in 45nm SOI CMOS to validate the functionality of the proposed technique and was tested over channels with different levels of loss and coupling. Measurement results show that the receiver can equalize channels with maximum 21dB loss while consuming about 7.5mW from a 1.2V supply. We also introduce a compact, low-power transmitter employing passive equalization. The efficacy of the proposed technique is demonstrated through implementation of a prototype in 65nm CMOS. The design achieves up to 20Gb/s data rate while consuming less than 10mW.

An alternative to electrical signaling is to employ optical signaling for chip-to-chip interconnections, which offers low channel loss and cross-talk while providing high communication bandwidth. In this work we demonstrate the possibility of building compact and low-power optical receivers. A novel RC front-end is proposed that combines dynamic offset modulation and double-sampling techniques to eliminate the need for a short time constant at the input of the receiver. Unlike conventional designs, this receiver does not require a high-gain stage that runs at the data rate, making it suitable for low-power implementations. In addition, it allows time-division multiplexing to support very high data rates. A prototype was implemented in 65nm CMOS and achieved up to 24Gb/s with less than 0.4pJ/b power efficiency per channel. As the proposed design mainly employs digital blocks, it benefits greatly from technology scaling in terms of power and area saving.

As the technology scales, the number of transistors on the chip grows. This necessitates a corresponding increase in the bandwidth of the on-chip wires. In this dissertation, we take a close look at wire scaling and investigate its effect on wire performance metrics. We explore a novel on-chip communication link based on a double-sampling architecture and dynamic offset modulation technique that enables low power consumption and high data rates while achieving high bandwidth density in 28nm CMOS technology. The functionality of the link is demonstrated using different length minimum-pitch on-chip wires. Measurement results show that the link achieves up to 20Gb/s of data rate (12.5Gb/s/$\mu$m) with better than 136fJ/b of power efficiency.

Development of Microfluidic Devices with the Use of Nanotechnology to Aid in the Analysis of Biological Systems Including Membrane Protein Separation, Single Cell Analysis, and Genetic Markers

Computation technology has dramatically changed the world around us; you can hardly find an area where cell phones have not saturated the market, yet there is a significant lack of breakthroughs in the development to integrate the computer with biological environments. This is largely the result of the incompatibility of the materials used in both environments; biological environments and experiments tend to need aqueous environments. To help aid in these development chemists, engineers, physicists and biologists have begun to develop microfluidics to help bridge this divide. Unfortunately, the microfluidic devices required large external support equipment to run the device. This thesis presents a series of several microfluidic methods that can help integrate engineering and biology by exploiting nanotechnology to help push the field of microfluidics back to its intended purpose, small integrated biological and electrical devices. I demonstrate this goal by developing different methods and devices to (1) separate membrane bound proteins with the use of microfluidics, (2) use optical technology to make fiber optic cables into protein sensors, (3) generate new fluidic devices using semiconductor material to manipulate single cells, and (4) develop a new genetic microfluidic based diagnostic assay that works with current PCR methodology to provide faster and cheaper results. All of these methods and systems can be used as components to build a self-contained biomedical device.

Toward Understanding Astrophysical Phenomena

Fast radio bursts (FRBs), a novel type of radio pulse, whose physics is not yet understood at all. Only a handful of FRBs had been detected when we started this project. Taking account of the scant observations, we put physical constraints on FRBs. We excluded proposals of a galactic origin for their extraordinarily high dispersion measures (DM), in particular stellar coronas and HII regions. Therefore our work supports an extragalactic origin for FRBs. We show that the resolved scattering tail of FRB 110220 is unlikely to be due to propagation through the intergalactic plasma. Instead the scattering is probably caused by the interstellar medium in the FRB's host galaxy, and indicates that this burst sits in the central region of that galaxy. Pulse durations of order $\ms$ constrain source sizes of FRBs implying enormous brightness temperatures and thus coherent emission. Electric fields near FRBs at cosmological distances would be so strong that they could accelerate free electrons from rest to relativistic energies in a single wave period. When we worked on FRBs, it was unclear whether they were genuine astronomical signals as distinct from `perytons', clearly terrestrial radio bursts, sharing some common properties with FRBs. Recently, in April 2015, astronomers discovered that perytons were emitted by microwave ovens. Radio chirps similar to FRBs were emitted when their doors opened while they were still heating. Evidence for the astronomical nature of FRBs has strengthened since our paper was published. Some bursts have been found to show linear and circular polarizations and Faraday rotation of the linear polarization has also been detected. I hope to resume working on FRBs in the near future. But after we completed our FRB paper, I decided to pause this project because of the lack of observational constraints.

The pulsar triple system, J0733+1715, has its orbital parameters fitted to high accuracy owing to the precise timing of the central $\ms$ pulsar. The two orbits are highly hierarchical, namely $P_{\mathrm{orb,1}}\ll P_{\mathrm{orb,2}}$, where 1 and 2 label the inner and outer white dwarf (WD) companions respectively. Moreover, their orbital planes almost coincide, providing a unique opportunity to study secular interaction associated purely with eccentricity beyond the solar system. Secular interaction only involves effect averaged over many orbits. Thus each companion can be represented by an elliptical wire with its mass distributed inversely proportional to its local orbital speed. Generally there exists a mutual torque, which vanishes only when their apsidal lines are parallel or anti-parallel. To maintain either mode, the eccentricity ratio, $e_1/e_2$, must be of the proper value, so that both apsidal lines precess together. For J0733+1715, $e_1\ll e_2$ for the parallel mode, while $e_1\gg e_2$ for the anti-parallel one. We show that the former precesses $\sim 10$ times slower than the latter. Currently the system is dominated by the parallel mode. Although only a little anti-parallel mode survives, both eccentricities especially $e_1$ oscillate on $\sim 10^3\yr$ timescale. Detectable changes would occur within $\sim 1\yr$. We demonstrate that the anti-parallel mode gets damped $\sim 10^4$ times faster than its parallel brother by any dissipative process diminishing $e_1$. If it is the tidal damping in the inner WD, we proceed to estimate its tidal quantity parameter ($Q$) to be $\sim 10^6$, which was poorly constrained by observations. However, tidal damping may also happen during the preceding low-mass X-ray binary (LMXB) phase or hydrogen thermal nuclear flashes. But, in both cases, the inner companion fills its Roche lobe and probably suffers mass/angular momentum loss, which might cause $e_1$ to grow rather than decay.

Several pairs of solar system satellites occupy mean motion resonances (MMRs). We divide these into two groups according to their proximity to exact resonance. Proximity is measured by the existence of a separatrix in phase space. MMRs between Io-Europa, Europa-Ganymede and Enceladus-Dione are too distant from exact resonance for a separatrix to appear. A separatrix is present only in the phase spaces of the Mimas-Tethys and Titan-Hyperion MMRs and their resonant arguments are the only ones to exhibit substantial librations. When a separatrix is present, tidal damping of eccentricity or inclination excites overstable librations that can lead to passage through resonance on the damping timescale. However, after investigation, we conclude that the librations in the Mimas-Tethys and Titan-Hyperion MMRs are fossils and do not result from overstability.

Rubble piles are common in the solar system. Monolithic elements touch their neighbors in small localized areas. Voids occupy a significant fraction of the volume. In a fluid-free environment, heat cannot conduct through voids; only radiation can transfer energy across them. We model the effective thermal conductivity of a rubble pile and show that it is proportional the square root of the pressure, $P$, for $P\leq \epsy^3\mu$ where $\epsy$ is the material's yield strain and $\mu$ its shear modulus. Our model provides an excellent fit to the depth dependence of the thermal conductivity in the top $140\,\mathrm{cm}$ of the lunar regolith. It also offers an explanation for the low thermal inertias of rocky asteroids and icy satellites. Lastly, we discuss how rubble piles slow down the cooling of small bodies such as asteroids.

Electromagnetic (EM) follow-up observations of gravitational wave (GW) events will help shed light on the nature of the sources, and more can be learned if the EM follow-ups can start as soon as the GW event becomes observable. In this paper, we propose a computationally efficient time-domain algorithm capable of detecting gravitational waves (GWs) from coalescing binaries of compact objects with nearly zero time delay. In case when the signal is strong enough, our algorithm also has the flexibility to trigger EM observation {\it before} the merger. The key to the efficiency of our algorithm arises from the use of chains of so-called Infinite Impulse Response (IIR) filters, which filter time-series data recursively. Computational cost is further reduced by a template interpolation technique that requires filtering to be done only for a much coarser template bank than otherwise required to sufficiently recover optimal signal-to-noise ratio. Towards future detectors with sensitivity extending to lower frequencies, our algorithm's computational cost is shown to increase rather insignificantly compared to the conventional time-domain correlation method. Moreover, at latencies of less than hundreds to thousands of seconds, this method is expected to be computationally more efficient than the straightforward frequency-domain method.

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].

Studies of temperature sensitive mutants of polyoma virus

Polyoma virus can undergo two different types of interactions with susceptible cells; one type of interaction leads to the production of new infectious virus and eventual cell death while the other leads to a neoplastically transformed cell which is able to continue to divide under conditions that inhibit the multiplication of uninfected normal cells. In order to study the viral genes involved in both of these virus-cell interactions the isolation of temperature sensitive mutants of polyoma virus was undertaken.

Two strains (TS-a, TS-b) which were temperature sensitive in their plaque forming ability at 38.5˚C, but not at 31.5˚C, were isolated from a mutagenized stock of the polyoma wild type virus (PY). TS-a was studied in further detail.

TS-a grown at 31.5˚C was found to be indistinguishable from PY in a number of physical characteristics including the heat sensitivity of the completed viral components. TS-a was inhibited in its ability to produce infectious virus in mouse cells when incubated at 38.5˚C; this inhibition could be overcome by infection with high multiplicities.

The nature of the intracellular temperature sensitive step of TS-a was analysed to some degree. It was found that this step occurs after uncoating of the infecting virus particles and about the time of new viral DNA synthesis. New infectious viral DNA does not appear to be made at the nonpermissive temperature; in contrast noninfectious capsids are made at 38.5˚C, but in amounts smaller than a full yield, such as made by TS-a at 31.5˚C or by PY at both the high and low temperature.

TS-a has also been found to be temperature sensitive in its transforming ability in vitro. Cells transformed at 31.5˚C by TS-a retain their transformed characteristics upon cultivation at 38.5˚C. Thus the temperature sensitive function seems to be important for the initiation of transformation, but not essential for the maintenance of the transformed state. TS-a also appears to be temperature sensitive in the production of tumors in newborn hamsters.

Photonic and Device Design Principles for Ultrahigh-Efficiency (>50%), Spectrum-Splitting Photovoltaics

The sun has the potential to power the Earth's total energy needs, but electricity from solar power still constitutes an extremely small fraction of our power generation because of its high cost relative to traditional energy sources. Therefore, the cost of solar must be reduced to realize a more sustainable future. This can be achieved by significantly increasing the efficiency of modules that convert solar radiation to electricity. In this thesis, we consider several strategies to improve the device and photonic design of solar modules to achieve record, ultrahigh (> 50%) solar module efficiencies. First, we investigate the potential of a new passivation treatment, trioctylphosphine sulfide, to increase the performance of small GaAs solar cells for cheaper and more durable modules. We show that small cells (mm2), which currently have a significant efficiency decrease (~ 5%) compared to larger cells (cm2) because small cells have a higher fraction of recombination-active surface from the sidewalls, can achieve significantly higher efficiencies with effective passivation of the sidewalls. We experimentally validate the passivation qualities of treatment by trioctylphosphine sulfide (TOP:S) through four independent studies and show that this facile treatment can enable efficient small devices. Then, we discuss our efforts toward the design and prototyping of a spectrum-splitting module that employs optical elements to divide the incident spectrum into different color bands, which allows for higher efficiencies than traditional methods. We present a design, the polyhedral specular reflector, that has the potential for > 50% module efficiencies even with realistic losses from combined optics, cell, and electrical models. Prototyping efforts of one of these designs using glass concentrators yields an optical module whose combined spectrum-splitting and concentration should correspond to a record module efficiency of 42%. Finally, we consider how the manipulation of radiatively emitted photons from subcells in multijunction architectures can be used to achieve even higher efficiencies than previously thought, inspiring both optimization of incident and radiatively emitted photons for future high efficiency designs. In this thesis work, we explore novel device and photonic designs that represent a significant departure from current solar cell manufacturing techniques and ultimately show the potential for much higher solar cell efficiencies.

A hybrid-parallel framework for the nonlinear seismic analysis of very tall buildings

FRAME3D, a program for the nonlinear seismic analysis of steel structures, has previously been used to study the collapse mechanisms of steel buildings up to 20 stories tall. The present thesis is inspired by the need to conduct similar analysis for much taller structures. It improves FRAME3D in two primary ways.

First, FRAME3D is revised to address specific nonlinear situations involving large displacement/rotation increments, the backup-subdivide algorithm, element failure, and extremely narrow joint hysteresis. The revisions result in superior convergence capabilities when modeling earthquake-induced collapse. The material model of a steel fiber is also modified to allow for post-rupture compressive strength.

Second, a parallel FRAME3D (PFRAME3D) is developed. The serial code is optimized and then parallelized. A distributed-memory divide-and-conquer approach is used for both the global direct solver and element-state updates. The result is an implicit finite-element hybrid-parallel program that takes advantage of the narrow-band nature of very tall buildings and uses nearest-neighbor-only communication patterns.

Using three structures of varied sized, PFRAME3D is shown to compute reproducible results that agree with that of the optimized 1-core version (displacement time-history response root-mean-squared errors are ~〖10〗^(-5) m) with much less wall time (e.g., a dynamic time-history collapse simulation of a 60-story building is computed in 5.69 hrs with 128 cores—a speedup of 14.7 vs. the optimized 1-core version). The maximum speedups attained are shown to increase with building height (as the total number of cores used also increases), and the parallel framework can be expected to be suitable for buildings taller than the ones presented here.

PFRAME3D is used to analyze a hypothetical 60-story steel moment-frame tube building (fundamental period of 6.16 sec) designed according to the 1994 Uniform Building Code. Dynamic pushover and time-history analyses are conducted. Multi-story shear-band collapse mechanisms are observed around mid-height of the building. The use of closely-spaced columns and deep beams is found to contribute to the building's “somewhat brittle” behavior (ductility ratio ~2.0). Overall building strength is observed to be sensitive to whether a model is fracture-capable.

Part I. Regulation of development in the cellular slime mold Dictyostelium discoideum. Part II. Polysomes and RNA synthesis during early development of the surf clam Spisula solidissima

Part I. The cellular slime mold Dictyostelium discoideum is a simple eukaryote which undergoes a multi-cellular developmental process. Single cell myxamoebae divide vegetatively in the presence of a food source. When the food is depleted or removed, the cells aggregate, forming a migrating pseudoplasmodium which differentiates into a fruiting body containing stalk and spore cells. I have shown that during the developmental cycle glycogen phosphorylase, aminopeptidase, and alanine transaminase are developmentally regulated, that is their specific activities increased at a specific time in the developmental cycle. Phosphorylase activity is undetectable in developing cells until mid-aggregation whereupon it increases and reaches a maximum at mid-culmination. Thereafter the enzyme disappears. Actinomycin D and cycloheximide studies as well as studies with morphologically aberrant and temporally deranged mutants indicate that prior RNA and concomitant protein synthesis are necessary for the rise and decrease in activity and support the view that the appearance of the enzyme is regulated at the transcriptional level. Aminopeptidase and alanine transaminase increase 3 fold starting at starvation and reach maximum activity at 18 and 5 hours respectively.

The cellular DNA s of D. discoideum were characterized by CsC1 buoyant density gradient centrifugation and by renaturation kinetics. Whole cell DNA exhibits three bands in CsCl: ρ = 1.676 g/cc (nuclear main band), 1.687 (nuclear satellite), and 1.682 (mitochondrial). Reassociation kinetics at a criterion of Tm -23°C indicates that the nuclear reiterated sequences make up 30% of the genome (Cot_1/2 (pure) 0.28) and the single-copy DNA 70% (Cot_1/2(pure) 70). The complexity of the nuclear genome is 30 x 10⁹ daltons and that of the mitochondrial DNA is 35-40 x 10⁶ daltons (Cot_1/2 0.15). rRNA cistrons constitute 2.2% of nuclear DNA and have a ρ = 1.682.

RNA extracted from 4 stages during developmental cycle of Dictyostelium was hybridized with purified single-copy nuclear DNA. The hybrids had properties indicative of single-copy DNA-RNA hybrids. These studies indicate that there are, during development, qualitative and quantitative changes in the portion of the single-copy of the genome transcribed. Overall, 56% of the genome is represented by transcripts between the amoeba and mid-culmination stages. Some 19% are sequences which are represented at all stages while 37% of the genome consists of stage specific sequences.

Part II. RNA and protein synthesis and polysome formation were studied during early development of the surf clam Spisula solidissima embryos. The oocyte has a small number of polysomes and a low but measurable rate of protein synthesis (leucine-³H incorporation). After fertilization, there is a continual increase in the percentage of ribosomes sedimenting in the polysome region. Newly synthesized RNA (uridine-5-³H incorporation) was found in polysomes as early as the 2-cell stage. During cleavage, the newly formed RNA is associated mainly with the light polysomes.

RNA extracted from polysomes labeled at the 4-cell stage is polydisperse, nonribosomal, and non-4 S. Actinomycin D causes a reduction of about 30% of the polysomes formed between fertilization and the 16-cell stage.

In the early cleavage stages the light polysomes are mostly affected by actinomycin.

Investigations of nonhistone chromosomal proteins

The major nonhistone chromosomal proteins (NHC proteins) are a group of 14-20 acidic proteins associated with DNA in eukaryotic chromatin. In comparisons by SDS gel electrophoresis (molecular weight sieving) one observes a high degree of homology among the NHC protein fractions of different tissues from a given species. Tissue-specific protein bands are also observed. The appearance of a new NHC protein, A, in the NHC proteins of rat liver stimulated to divide by partial hepatectomy and of rat ascites cells suggests that this protein may play a role in preparing the cell for division. The NHC proteins of the same tissue from different species are also very similar. Quantitative but not qualitative changes in the NHC proteins of rat uterus are observed on stimulation (in vivo) with estrogen. These observations suggest that the major NHC proteins play a general role in chromatin structure and the regulation of genome expression; several may be enzymes of nucleic acid and histone metabolism and/or structural proteins analogous to histones. One such enzyme, a protease which readily and preferentially degrades histones, can be extracted from chromatin with 0.7 N NaCl.

Although the NHC proteins readily aggregate, they can be separated from histone and fractionated by ion exchange chromatography on Sephadex SE C-25 resin in 10 M urea-25% formic acid (pH 2.5). Following further purification, four fractions of NHC protein are obtained; two of these are single purified proteins, and the other two contain 4-6 and 4-7 different proteins. These NHC proteins show a ratio of acidic to basic amino acids from 2.7 to 1.2 and isoelectric points from apparently less than 3.7 to 8.0. These isolated fractions appear more soluble and easier to work with than any whole NHC protein preparation.