We prove stability for a formally determined inverse problem for a hyperbolic PDE where the coefficients depend on space and time variables. The hyperbolic operator has constant wave speed and we study the recovery of zeroth order and first order coefficients and the space dimension can be one or higher. We use a modification of the Bukhgeim-Klibanov method to obtain our results.

In this paper, we investigate the relation between robustness of periodic orbits exhibited by systems with impulse effects and robustness of their corresponding Poincaré maps. In particular, we prove that input-to-state stability (ISS) of a periodic orbit under external excitation in both continuous and discrete time is equivalent to ISS of the corresponding zero-input fixed point of the associated forced Poincaré map. This result extends the classical Poincaré analysis for asymptotic stability of periodic solutions to establish orbital ISS of such solutions under external excitation. In our proof, we define the forced Poincaré map, and use it to construct ISS estimates for the periodic orbit in terms of ISS estimates of this map under mild assumptions on the input signals. As a consequence of the availability of these estimates, the equivalence between exponential stability (ES) of the fixed point of the zero-input (unforced) Poincaré map and the ES of the corresponding orbit is recovered. The results can be applied naturally to study the robustness of periodic orbits of continuous-time systems as well. Although our motivation for extending classical Poincaré analysis to address ISS stems from the need to design robust controllers for limit-cycle walking and running robots, the results are applicable to a much broader class of systems that exhibit periodic solutions.

We consider the problem of recovering the initial value, from the trace on the light cone, of the solution of an initial value problem for the wave equation. When the space is odd dimensional, we show that the map from the initial value to the traces of the (even or odd in time) solutions on the light cone is an isometry and we characterize the range of this map and construct its inverse. We do this by relating the problem to the recovery of a function from its spherical means over all spheres through the origin, which in turn is related to the Radon transform inversion via the inversion map on R^n.

In this paper we investigate the relation between robustness of periodic orbits exhibited by systems with impulse effects and robustness of their corresponding Poincar\'e maps. In particular, we prove that input-to-state stability (ISS) of a periodic orbit under external excitation in both continuous and discrete time is equivalent to ISS of the corresponding 0-input fixed point of the associated \emph{forced} Poincar\'e map. This result extends the classical Poincar\'e analysis for asymptotic stability of periodic solutions to establish orbital input-to-state stability of such solutions under external excitation. In our proof, we define the forced Poincar\'e map, and use it to construct ISS estimates for the periodic orbit in terms of ISS estimates of this map under mild assumptions on the input signals. As a consequence of the availability of these estimates, the equivalence between exponential stability (ES) of the fixed point of the 0-input (unforced) Poincar\'e map and ES of the corresponding orbit is recovered. The results can naturally be applied to continuous-time systems as well. Although our motivation for extending classical Poincar\'e analysis to address ISS stems from the need to design robust controllers for limit-cycle walking and running robots, the results are applicable to a much broader class of systems that exhibit periodic solutions.

We consider the inverse problem of recovering a potential by measuring the response at a point to a source located at the same point and then varying the point on the surface of a sphere. This is a similar to the inverse back-scattering problem. We show that if the angular derivatives of the difference of two potentials having the same data is controlled by the L^2 norm of the difference of the potentials they must be equal. In particular this shows injectivity of the inverse problem for radial potentials.

We consider the problem of recovering a smooth, compactly supported potential on R-3 from its backscattering data. We show that if two such potentials have the same backscattering data and the difference of the two potentials has controlled angular derivatives, then the two potentials are identical. In particular, if two potentials differ by a finite linear combination of spherical harmonics with radial coefficients and have the same backscattering data then the two potentials are identical.

We study a problem arising in oil exploration. We want to determine the properties of the interior of the earth and, since drilling is expensive, the interior is probed by acoustic waves. An explosion is set off on the surface of the earth, the acoustic signal travels into the interior of the earth and the earth’s response is measured on the surface of the earth. From this surface measurement we would like to determine the properties of the interior of the earth. Below B will represent the closed unit ball of radius 1 centered at the origin and S will denote its boundary.

We prove stability for a coefficient determination problem for a two velocity 2x2 system of hyperbolic PDEs in one space dimension.

The MIMO technology is one of the most important technologies to enhance the system performance to reach the challenge of IMT-advanced technical requirement ITU-R M.2134. To keep more antenna number while decrease physical size, dual-polarized antenna array is a potential type for MIMO application in the future IMT-advanced system. In this paper, a modified polarization model is proposed for the MIMO channel, with clear explanation on polarization parameters, XPD, polarization power loss, and channel correlation. Further, this proposed model is evaluated combined with MIMO antenna array and beam-forming algorithm.

We prove unique continuation of solutions of the wave equation along and across lower‐dimensional planes containing the t ‐axis. This is a sharpening and a generalization of a result of Cheng, Ding and Yamamoto as well as a simplification of the proof. Copyright © 2008 John Wiley & Sons, Ltd.

In [C. Benítez, Y. Sarantopoulos, A. Tonge, Lower bounds for norms of products of polynomials, Math. Proc. Cambridge Philos. Soc. 124 (3) (1998) 395–408] it was conjectured that for all unit vectors u1,…,ud in Rd,X(u1,…,ud):=supx∈Rd,|x|2=d∏i=1d〈x,ui〉2⩾1 with equality occurring iff u1,…,ud are orthonormal. We relate this to a conjecture about solutions of Ay=y−1, where A=(〈ui,uj〉), and show that if the conjecture fails then the u1,…,ud minimizing X must be linearly dependent. We also show X(u1,…,ud)⩾1 for certain families of u1,…,ud.

In the course of the development of a potent series of nitrofuranylamide anti-tuberculosis agents, we investigated if the exceptional activity resulted in part from the isoxazoline core and if it possessed any intrinsic anti-tuberculosis activity. This led to the discovery of an isoxazoline ester with appreciable anti-tuberculosis activity. In this study we explored the anti-tuberculosis structure–activity relationship of the isoxazoline ester compound through systematic modification of the 3,5-di-substituted isoxazoline core. Two approaches were used: (i) modification of the potentially metabolically labile ester functionality at the 3 position with acids, amines, amides, reverse amides, alcohols, hydrazides, and 1,3,4-oxadiazoles; (ii) substitution of the distal benzyl piperazine ring in the 5 position of the isoxazoline ring with piperazyl-ureas, piperazyl-carbamates, biaryl systems, piperidines and morpholine. Attempts to replace the ester group at C-3 position of isoxazoline with a variety of bioisosteric head groups led to significant loss of the tuberculosis inhibition indicating that an ester is required for anti-tuberculosis activity. Optimization of the isoxazoline C-5 position produced compounds with improved anti-tuberculosis activity, most notably the piperazyl-urea and piperazyl-carbamate analogs.

Consider a family of probability measures, indexed by partial derivative D, on a bounded open region D subset of R-d with a smooth boundary. For any starting point inside D, we run a standard d-dimensional Brownian motion in R-d until it first exits D, at which time it jumps to a point inside the domain D according to the jump measure at the exit point and starts a new Brownian motion. The same evolution is repeated independently each time the process reaches the boundary. We study the exponential rate at which the transition distribution of the process converges to its invariant measure, in terms of the spectral gap of the generator. In particular, we prove two conjectures of I. Ben-Ari and R. Pinsky for an interval (see J. Funct. Anal. 251 (2007), 122-140, and preprint (2007)) by studying when a combination of the sine and cosine transforms of probability measures on an interval has only real zeros.

We prove uniqueness for inverse problems for the operator partial derivative(2)(t) - Delta(x) - q(x) for data coming from a single coincident source - receiver pair. We prove uniqueness when q(1) >= q(2) or when q is a product of an unknown spherically symmetric function and a known function of the angular variables but the source - receiver pair is NOT at the center.

We establish inversion formulas of the so-called filtered back-projection type to recover a function supported in the ball in even dimensions from its spherical means over spheres centered on the boundary of the ball. We also find several formulas to recover initial data of the form $(f,0)$ (or $(0,g)$) for the free space wave equation in even dimensions from the trace of the solution on the boundary of the ball, provided that the initial data has support in the ball.

Nitrofuranyl isoxazolines with increased proteolytic stability over nitrofuranyl amides were designed and synthesized leading to discovery of several compounds with potent in vitro anti-tuberculosis activity. However, their in vivo activity was limited by high protein binding and poor distribution. Consequently, a series of non-nitrofuran containing isoxazolines were prepared to determine if the core had residual anti-tuberculosis activity. This led to the discovery of novel isoxazoline 12 as anti-tuberculosis agent with a MIC90 value of 1.56 mu g/mL. (c) 2007 Elsevier Ltd. All rights reserved.

Given a positive definite matrix A, we characterize the unique diagonal matrix D, D⩾A, with the smallest determinant. Equivalently, given an ellipsoid A, we characterize the unique ellipsoid of the largest volume contained in A, with principal axes parallel to the coordinate axes.

Let B represent the ball of radius ρ in Rn and S its boundary; consider the map , where represents the mean value of f on a sphere of radius r centered at p. We summarize and discuss the results concerning the injectivity of , the characterization of the range of , and the inversion of . There is a close connection between mean values over spheres and solutions of initial value problems for the wave equation. We also summarize the results for the corresponding wave equation problem.

Suppose n > 1 is an odd integer, f is a smooth function supported in a ball B with boundary S, and u is the solution of the initial value problemutt - Delta(x)u = 0, (x,t) is an element of R-n x [0,infinity);u(x, t = 0) = 0, u(t)(x,t = 0) = f (x), x is an element of R-n.We characterize the range of the map f -> u\S x[0,infinity) and give a stable scheme for the inversion of this map. This also characterizes the range of the map sending f to its mean values over spheres centred on S.

Suppose u is the solution of the initial value problemu(tt) - Delta(x)u = 0, (x, t) is an element of R-n x [0, infinity)u(x, t=0) = f(x), u(t)(x, t=0) = g(x), x is an element of R-nSuppose n >= 1 is odd, f and g are supported in a ball B with boundary S, and one of f or g is zero. We derive identities relating the norm of f or g to the norm of the trace of u on S x [0, infinity). These identities are derived using integral geometric and multiplier methods. Copyright (c) 2005 John Wiley & Sons, Ltd.