Genetic analysis using genomic representations

Analysis of the genetic changes in human tumors is often problematical because of the presence of normal stroma and the limited availability of pure tumor DNA. However, large amounts of highly reproducible “representations” of tumor and normal genomes can be made by PCR from nanogram amounts of restriction endonuclease cleaved DNA that has been ligated to oligonucleotide adaptors. We show here that representations are useful for many types of genetic analyses, including measuring relative gene copy number, loss of heterozygosity, and comparative genomic hybridization. Representations may be prepared even from sorted nuclei from fixed and archived tumor biopsies.

Invariance between subjects of brain wave representations of language

In three experiments, electric brain waves of 19 subjects were recorded under several different experimental conditions for two purposes. One was to test how well we could recognize which sentence, from a set of 24 or 48 sentences, was being processed in the cortex. The other was to study the invariance of brain waves between subjects. As in our earlier work, the analysis consisted of averaging over trials to create prototypes and test samples, to both of which Fourier transforms were applied, followed by filtering and an inverse transformation to the time domain. A least-squares criterion of fit between prototypes and test samples was used for classification. In all three experiments, averaging over subjects improved the recognition rates. The most significant finding was the following. When brain waves were averaged separately for two nonoverlapping groups of subjects, one for prototypes and the other for test samples, we were able to recognize correctly 90% of the brain waves generated by 48 different sentences about European geography.

Invariance of brain-wave representations of simple visual images and their names

In two experiments, electric brain waves of 14 subjects were recorded under several different conditions to study the invariance of brain-wave representations of simple patches of colors and simple visual shapes and their names, the words blue, circle, etc. As in our earlier work, the analysis consisted of averaging over trials to create prototypes and test samples, to both of which Fourier transforms were applied, followed by filtering and an inverse transformation to the time domain. A least-squares criterion of fit between prototypes and test samples was used for classification. The most significant results were these. By averaging over different subjects, as well as trials, we created prototypes from brain waves evoked by simple visual images and test samples from brain waves evoked by auditory or visual words naming the visual images. We correctly recognized from 60% to 75% of the test-sample brain waves. The general conclusion is that simple shapes such as circles and single-color displays generate brain waves surprisingly similar to those generated by their verbal names. These results, taken together with extensive psychological studies of auditory and visual memory, strongly support the solution proposed for visual shapes, by Bishop Berkeley and David Hume in the 18th century, to the long-standing problem of how the mind represents simple abstract ideas.

On representations of finite type

Representations of the (infinite) canonical anticommutation relations and the associated operator algebra, the fermion algebra, are studied. A “coupling constant” (in (0,1]) is defined for primary states of “finite type” of that algebra. Primary, faithful states of finite type with arbitrary coupling are constructed and classified. Their physical significance for quantum thermodynamical systems at high temperatures is discussed. The scope of this study is broadened to include a large class of operator algebras sharing some of the structural properties of the fermion algebra.

Adjoint modular Galois representations and their Selmer groups

In the last 15 years, many class number formulas and main conjectures have been proven. Here, we discuss such formulas on the Selmer groups of the three-dimensional adjoint representation ad(φ) of a two-dimensional modular Galois representation φ. We start with the p-adic Galois representation φ0 of a modular elliptic curve E and present a formula expressing in terms of L(1, ad(φ0)) the intersection number of the elliptic curve E and the complementary abelian variety inside the Jacobian of the modular curve. Then we explain how one can deduce a formula for the order of the Selmer group Sel(ad(φ0)) from the proof of Wiles of the Shimura–Taniyama conjecture. After that, we generalize the formula in an Iwasawa theoretic setting of one and two variables. Here the first variable, T, is the weight variable of the universal p-ordinary Hecke algebra, and the second variable is the cyclotomic variable S. In the one-variable case, we let φ denote the p-ordinary Galois representation with values in GL2(Zp[[T]]) lifting φ0, and the characteristic power series of the Selmer group Sel(ad(φ)) is given by a p-adic L-function interpolating L(1, ad(φk)) for weight k + 2 specialization φk of φ. In the two-variable case, we state a main conjecture on the characteristic power series in Zp[[T, S]] of Sel(ad(φ) ⊗ ν−1), where ν is the universal cyclotomic character with values in Zp[[S]]. Finally, we describe our recent results toward the proof of the conjecture and a possible strategy of proving the main conjecture using p-adic Siegel modular forms.

Deforming semistable Galois representations

Let V be a p-adic representation of Gal(Q̄/Q). One of the ideas of Wiles’s proof of FLT is that, if V is the representation associated to a suitable autromorphic form (a modular form in his case) and if V′ is another p-adic representation of Gal(Q̄/Q) “closed enough” to V, then V′ is also associated to an automorphic form. In this paper we discuss which kind of local condition at p one should require on V and V′ in order to be able to extend this part of Wiles’s methods.

On degree 2 Galois representations over

We discuss proofs of some new special cases of Serre’s conjecture on odd, degree 2 representations of Gℚ.

Culture as shared cognitive representations.

Culture consists of shared cognitive representations in the minds of individuals. This paper investigates the extent to which English speakers share the "same" semantic structure of English kinship terms. The semantic structure is defined as the arrangement of the terms relative to each other as represented in a metric space in which items judged more similar are placed closer to each other than items judged as less similar. The cognitive representation of the semantic structure, residing in the mind of an individual, is measured by judged similarity tasks involving comparisons among terms. Using six independent measurements, from each of 122 individuals, correspondence analysis represents the data in a common multidimensional spatial representation. Judged by a variety of statistical procedures, the individuals in our sample share virtually identical cognitive representations of the semantic structure of kinship terms. This model of culture accounts for 70-90% of the total variability in these data. We argue that our findings on kinship should generalize to all semantic domains--e.g., animals, emotions, etc. The investigation of semantic domains is important because they may reside in localized functional units in the brain, because they relate to a variety of cognitive processes, and because they have the potential to provide methods for diagnosing individual breakdowns in the structure of cognitive representations typical of such ailments as Alzheimer disease.

Numerical representations in primates.

Research has demonstrated that human infants and nonhuman primates have a rudimentary numerical system that enables them to count objects or events. More recently, however, studies using a preferential looking paradigm have suggested that preverbal human infants are capable of simple arithmetical operations, such as adding and subtracting a small number of visually presented objects. These findings implicate a relatively sophisticated representational system in the absence of language. To explore the evolutionary origins of this capacity, we present data from an experiment with wild rhesus monkeys (Macaca mulatta) that methodologically mirrors those conducted on human infants. Results suggest that rhesus monkeys detect additive and subtractive changes in the number of objects present in their visual field. Given the methodological and empirical similarities, it appears that nonhuman primates such as rhesus monkeys may also have access to arithmetical representations, although alternative explanations must be considered for both primate species.