476 research outputs found
New global stability estimates for the Gel'fand-Calderon inverse problem
We prove new global stability estimates for the Gel'fand-Calderon inverse
problem in 3D. For sufficiently regular potentials this result of the present
work is a principal improvement of the result of [G. Alessandrini, Stable
determination of conductivity by boundary measurements, Appl. Anal. 27 (1988),
153-172]
New global stability estimates for monochromatic inverse acoustic scattering
We give new global stability estimates for monochromatic inverse acoustic
scattering. These estimates essentially improve estimates of [P. Hahner, T.
Hohage, SIAM J. Math. Anal., 33(3), 2001, 670-685] and can be considered as a
solution of an open problem formulated in the aforementioned work
On an inverse problem for anisotropic conductivity in the plane
Let be a bounded domain with smooth
boundary and a smooth anisotropic conductivity on .
Starting from the Dirichlet-to-Neumann operator on
, we give an explicit procedure to find a unique domain
, an isotropic conductivity on and the boundary
values of a quasiconformal diffeomorphism which
transforms into .Comment: 9 pages, no figur
Formulas and equations for finding scattering data from the Dirichlet-to-Neumann map with nonzero background potential
For the Schrodinger equation at fixed energy with a potential supported in a
bounded domain we give formulas and equations for finding scattering data from
the Dirichlet-to-Neumann map with nonzero background potential. For the case of
zero background potential these results were obtained in [R.G.Novikov,
Multidimensional inverse spectral problem for the equation
-\Delta\psi+(v(x)-Eu(x))\psi=0, Funkt. Anal. i Ego Prilozhen 22(4), pp.11-22,
(1988)]
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
Kripke Semantics for Martin-L\"of's Extensional Type Theory
It is well-known that simple type theory is complete with respect to
non-standard set-valued models. Completeness for standard models only holds
with respect to certain extended classes of models, e.g., the class of
cartesian closed categories. Similarly, dependent type theory is complete for
locally cartesian closed categories. However, it is usually difficult to
establish the coherence of interpretations of dependent type theory, i.e., to
show that the interpretations of equal expressions are indeed equal. Several
classes of models have been used to remedy this problem. We contribute to this
investigation by giving a semantics that is standard, coherent, and
sufficiently general for completeness while remaining relatively easy to
compute with. Our models interpret types of Martin-L\"of's extensional
dependent type theory as sets indexed over posets or, equivalently, as
fibrations over posets. This semantics can be seen as a generalization to
dependent type theory of the interpretation of intuitionistic first-order logic
in Kripke models. This yields a simple coherent model theory, with respect to
which simple and dependent type theory are sound and complete
Initial-boundary value problems for discrete evolution equations: discrete linear Schrodinger and integrable discrete nonlinear Schrodinger equations
We present a method to solve initial-boundary value problems for linear and
integrable nonlinear differential-difference evolution equations. The method is
the discrete version of the one developed by A. S. Fokas to solve
initial-boundary value problems for linear and integrable nonlinear partial
differential equations via an extension of the inverse scattering transform.
The method takes advantage of the Lax pair formulation for both linear and
nonlinear equations, and is based on the simultaneous spectral analysis of both
parts of the Lax pair. A key role is also played by the global algebraic
relation that couples all known and unknown boundary values. Even though
additional technical complications arise in discrete problems compared to
continuum ones, we show that a similar approach can also solve initial-boundary
value problems for linear and integrable nonlinear differential-difference
equations. We demonstrate the method by solving initial-boundary value problems
for the discrete analogue of both the linear and the nonlinear Schrodinger
equations, comparing the solution to those of the corresponding continuum
problems. In the linear case we also explicitly discuss Robin-type boundary
conditions not solvable by Fourier series. In the nonlinear case we also
identify the linearizable boundary conditions, we discuss the elimination of
the unknown boundary datum, we obtain explicitly the linear and continuum limit
of the solution, and we write down the soliton solutions.Comment: 41 pages, 3 figures, to appear in Inverse Problem
The Escherichia coli transcriptome mostly consists of independently regulated modules
Underlying cellular responses is a transcriptional regulatory network (TRN) that modulates gene expression. A useful description of the TRN would decompose the transcriptome into targeted effects of individual transcriptional regulators. Here, we apply unsupervised machine learning to a diverse compendium of over 250 high-quality Escherichia coli RNA-seq datasets to identify 92 statistically independent signals that modulate the expression of specific gene sets. We show that 61 of these transcriptomic signals represent the effects of currently characterized transcriptional regulators. Condition-specific activation of signals is validated by exposure of E. coli to new environmental conditions. The resulting decomposition of the transcriptome provides: a mechanistic, systems-level, network-based explanation of responses to environmental and genetic perturbations; a guide to gene and regulator function discovery; and a basis for characterizing transcriptomic differences in multiple strains. Taken together, our results show that signal summation describes the composition of a model prokaryotic transcriptome
Compressive Inverse Scattering I. High Frequency SIMO Measurements
Inverse scattering from discrete targets with the
single-input-multiple-output (SIMO), multiple-input-single-output (MISO) or
multiple-input-multiple-output (MIMO) measurements is analyzed by compressed
sensing theory with and without the Born approximation. High frequency analysis
of (probabilistic) recoverability by the -based
minimization/regularization principles is presented. In the absence of noise,
it is shown that the -based solution can recover exactly the target of
sparsity up to the dimension of the data either with the MIMO measurement for
the Born scattering or with the SIMO/MISO measurement for the exact scattering.
The stability with respect to noisy data is proved for weak or widely separated
scatterers. Reciprocity between the SIMO and MISO measurements is analyzed.
Finally a coherence bound (and the resulting recoverability) is proved for
diffraction tomography with high-frequency, few-view and limited-angle
SIMO/MISO measurements.Comment: A new section on diffraction tomography added; typos fixed; new
figures adde
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