Let u t = ∇ 2 u − q(x)u := Lu in D × [0, ∞), where D ⊂ R 3 is a bounded domain with a smooth connected boundary S, and q(x) ∈ L 2 (S) is a real-valued function with compact support in D. is the outer normal to S, one can find q(x) uniquely. A similar result is obtained for the heat equation u t = Lu := ∇ · (a∇u). These results are based on new versions of Property C.

In this paper, we give a short proof of the Lorentz estimates for gradients very weak solutions to linear parabolic equations with Muckenhoupt class A q -weights u t − div ( x , ) ∇ = F in bounded domain Ω × 0 T ⊂ R N + 1 where has small mean oscillation, and is Lipchistz Lipschitz constant.

We study the question of whether for a natural Hamiltonian system on a two-dimensional compact configuration manifold, a single trajectory of sufficiently high energy is almost surely enough to reconstruct a real analytic potential. Consider a compact configuration manifold M equipped with a finite Borel measure (essentially we deal with the dimension n = 2) and a natural Hamiltonian system the...

We study the domain number and size distributions in the one-dimensional Ising and q-state Potts models subject to zero-temperature Glauber dynamics. The survival probability of a domain S(t);t and an unreacted domain Q1(t);t 2d are characterized by two independent nontrivial exponents. For the Ising case, we find c50.126 and d51.27 using numerical simulations. We develop an independent interva...

1. The automorphism group T(P) of a partial order P is the collection of all order preserving permutations (automorphisms) of P, a subgroup of the symmetric group on P. If P and g are partial orders then P x g becomes a partial order by reverse lexicography: (p, q) < (p', q') if q < q' or q = q' and p < p'. If ƒ is a function whose domain contains the element a, we use af to denote the image of...

Let Ω ⊂ R, n ≥ 2, be a bounded domain with smooth boundary. Consider utt −4xu+ q(x, t)u = 0 in Ω× [0, T ] u(x, 0) = 0, ut(x, 0) = 0 if x ∈ Ω u(x, t) = f(x, t) on ∂Ω× [0, T ] We show that if u and f are known on ∂Ω× [0, T ], for all f ∈ C∞ 0 (∂Ω× [0, T ]), then q(x, t) may be reconstructed on C = { (x, t) : x∈Ω, 0 < t < T, x− tω & x+ (T − t)ω 6∈ Ω ∀ ω ∈ R, |ω| = 1 } provided q is known at all po...

The paper deals with the controllability of finite-dimensional linear difference delay equations, i.e., dynamics for which state at a given time $t$ is obtained as combination control evaluated and finitely many previous instants $t-\Lambda_1,\dots,t-\Lambda_N$. Based on realization theory developed by Y.Yamamoto general infinite-dimensional dynamical systems, we obtain necessary sufficient con...