Victor Kasatkin | California Institute of Technology (original) (raw)
Address: Pasadena, California, United States
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Papers by Victor Kasatkin
Journal of Mathematical Sciences, 2011
We study the C * -algebra B generated in L 2 (R) by operators of multiplication by functions with... more We study the C * -algebra B generated in L 2 (R) by operators of multiplication by functions with finitely many discontinuities of the first kind and by convolution operators with the Fourier transforms of such functions. The algebra B is represented as the restricted direct sum A 1 ⊕ C A 2 . We express the spectrum of the restricted direct sum in terms of the spectra of its summands. This result is used to express the spectrum of the algebra B in terms of the spectra of A 1 and A 2 . We describe all equivalence classes of irreducible representations of the algebra B, the topology on the spectrum of this algebra, and solving composition series. We discuss the abstract index group of the quotient algebra B by the ideal of compact operators and by the ideal com B generated by the commutators of elements of the algebra B. Bibliography: 14 titles.
Bulletin of Mathematical Biology, 2008
Generating morphogenetic gradients during early development is a fundamental step of positional s... more Generating morphogenetic gradients during early development is a fundamental step of positional signaling, which ultimately results in patterning and cell specialization. Based on morphogens propagation from cells to cells, we have presented a biophysical model in Holcman et al. (in press), where gradients and boundaries between different morphogenetic regions can be generated. In that theory, morphogens are transcription factors which induce their own activation and at the same time propagate in a cell ensemble. We analyze here a variant version of the biophysical model proposed in Holcman et al. (in press), where now morphogens can form dimers. As a result, gradients are smoother and borders are much sharper. Because random perturbations of a gradient can affect the precise location of the boundary between two morphogenetic regions, we also analyze these fluctuations and in particular, we obtain an analytic expression for the variance of the boundary location as a function of the variance of the random perturbations. This formula can be used to study the noise intrinsic effect on the boundary position between morphogenetic regions, which can be at the origin of interindividual variations.
We study the C * -algebra B generated in L 2 (R) by operators of multiplication by functions with... more We study the C * -algebra B generated in L 2 (R) by operators of multiplication by functions with finitely many discontinuities of the first kind and by convolution operators with the Fourier transforms of such functions. The algebra B is represented as the restricted direct sum A 1 ⊕ C A 2 . We express the spectrum of the restricted direct sum in terms of the spectra of its summands. This result is used to express the spectrum of the algebra B in terms of the spectra of A 1 and A 2 . We describe all equivalence classes of irreducible representations of the algebra B, the topology on the spectrum of this algebra, and solving composition series. We discuss the abstract index group of the quotient algebra B by the ideal of compact operators and by the ideal com B generated by the commutators of elements of the algebra B. Bibliography: 14 titles.
Journal of Mathematical Sciences, 2011
We study the C * -algebra B generated in L 2 (R) by operators of multiplication by functions with... more We study the C * -algebra B generated in L 2 (R) by operators of multiplication by functions with finitely many discontinuities of the first kind and by convolution operators with the Fourier transforms of such functions. The algebra B is represented as the restricted direct sum A 1 ⊕ C A 2 . We express the spectrum of the restricted direct sum in terms of the spectra of its summands. This result is used to express the spectrum of the algebra B in terms of the spectra of A 1 and A 2 . We describe all equivalence classes of irreducible representations of the algebra B, the topology on the spectrum of this algebra, and solving composition series. We discuss the abstract index group of the quotient algebra B by the ideal of compact operators and by the ideal com B generated by the commutators of elements of the algebra B. Bibliography: 14 titles.
Bulletin of Mathematical Biology, 2008
Generating morphogenetic gradients during early development is a fundamental step of positional s... more Generating morphogenetic gradients during early development is a fundamental step of positional signaling, which ultimately results in patterning and cell specialization. Based on morphogens propagation from cells to cells, we have presented a biophysical model in Holcman et al. (in press), where gradients and boundaries between different morphogenetic regions can be generated. In that theory, morphogens are transcription factors which induce their own activation and at the same time propagate in a cell ensemble. We analyze here a variant version of the biophysical model proposed in Holcman et al. (in press), where now morphogens can form dimers. As a result, gradients are smoother and borders are much sharper. Because random perturbations of a gradient can affect the precise location of the boundary between two morphogenetic regions, we also analyze these fluctuations and in particular, we obtain an analytic expression for the variance of the boundary location as a function of the variance of the random perturbations. This formula can be used to study the noise intrinsic effect on the boundary position between morphogenetic regions, which can be at the origin of interindividual variations.
We study the C * -algebra B generated in L 2 (R) by operators of multiplication by functions with... more We study the C * -algebra B generated in L 2 (R) by operators of multiplication by functions with finitely many discontinuities of the first kind and by convolution operators with the Fourier transforms of such functions. The algebra B is represented as the restricted direct sum A 1 ⊕ C A 2 . We express the spectrum of the restricted direct sum in terms of the spectra of its summands. This result is used to express the spectrum of the algebra B in terms of the spectra of A 1 and A 2 . We describe all equivalence classes of irreducible representations of the algebra B, the topology on the spectrum of this algebra, and solving composition series. We discuss the abstract index group of the quotient algebra B by the ideal of compact operators and by the ideal com B generated by the commutators of elements of the algebra B. Bibliography: 14 titles.