We consider the problem of determining a pair of functions (u, f) satisfying the two-dimensional backward heat equation ut −∆u = φ(t)f(x, y), t ∈ (0, T ), (x, y) ∈ (0, 1)× (0, 1), u(x, y, T ) = g(x, y) with a homogeneous Cauchy boundary condition, where φ and g are given approximately. The problem is severely ill-posed. Using an interpolation method and the truncated Fourier series, we construc...

We prove a linear operator T acting between lp-type spaces attains its norm if, and only if, there exists a not weakly null maximizing sequence for T . For 1 < p 6= q we show that any not weakly null maximizing sequence for a norm attaining operator T : lp → lq has a norm-convergent subsequence. We also prove that for any fixed x0 in lp, the set of operators T : lp → lq that attain their norm a...

Abstract. In this paper, a linearized backward Euler method is discussed for the equations of motion arising in the Oldroyd model of viscoelastic fluids. Some new a priori bounds are obtained for the solution under realistically assumed conditions on the data. Further, the exponential decay properties for the exact as well as the discrete solutions are established. Finally, a priori error estim...

Neural-symbolic approaches have recently gained popularity to inject prior knowledge into a learner without requiring it induce this from data. These can potentially learn competitive solutions with significant reduction of the amount supervised A large class neural-symbolic is based on First-Order Logic represent knowledge, relaxed differentiable form using fuzzy logic. This paper shows that l...

Let A be a type t artinian level algebra with h-vector h =) be the h-vector of the generic type c level quotient of A having the same socle degree e. The main result of this paper is an (in general sharp) lower-bound for the h-vector H H c,gen u ≥ 1 t 2 − 1 ((t − c)h e−u + (ct − 1)h u). This result generalizes a recent theorem of Iarrobino (which treats the case t = 2). Finally, we begin to obt...