Let F be a distribution inD′ and let f be a locally summable function. The composition F f x of F and f is said to exist and be equal to the distribution h x if the limit of the sequence {Fn f x } is equal to h x , where Fn x F x ∗ δn x for n 1, 2, . . . and {δn x } is a certain regular sequence converging to the Dirac delta function. In the ordinary sense, the composition δ s sinh−1x r does no...

We establish the existence and uniqueness of the Dipole Solution of the porous medium equation (PME), i.e. the solution of u t = (juj m?1 u) ; in Q = f(x; t) : x = (x i) 2 R N ; t > 0g ; with m > 1 and initial value u(x; 0) = M @ @x 1 (x) ; where is Dirac delta function in R N and M 6 = 0. It has the self-similar form u(x; t) = t ? U(xt ?) ; with exponents and determined from dimensional consid...

The weak coupling asymptotics, to order r c 2 ( ) , of the ground state energy of the delta-function Bose gas is derived. Here  c 2 0 is the delta-function potential amplitude and ρ the density of the gas in the thermodynamic limit. The analysis uses the electrostatic interpretation of the Lieb–Liniger integral equation.

Learning and data clustering with an RBF-based spiking neuron network Natacha Gueorguieva a , Iren Valova b & Georgi Georgiev c a Computer Science , CSI/City University of New York , 2800 Victory Boulevard, Staten Island, NY 10314, USA b Computer Science , University of Massachusetts Dartmouth , 285 Old Westport Road, N. Dartmouth, MA 02747, USA c Computer Science , University of Wisconsin Oshk...

Impulse The word ’impulse’ is used in many areas in different ways. In classical mechanics, the impulse is the integral of force with respect to time. It is also used to refer to a fast-acting force, which is often idealized by a step function or a delta function. In this text, it is used to represent any functional form of pressure increase, either static or transient, which can generate obser...

A theoretical study of the potential energy surface and bound states is performed for the ground state of the NeI(2) van der Waals (vdW) complex. The three-dimensional interaction energies are obtained from ab initio coupled-cluster, coupled-cluster single double (triple)/complete basis set, calculations using large basis sets, of quadruple- through quintuple-zeta quality, in conjunction with r...

We consider a one-dimensional fractional diffusion equation: ∂α t u(x, t) = ∂ ∂x ( p(x) ∂u ∂x (x, t) ) , 0 < x < `, where 0 < α < 1 and ∂α t denotes the Caputo derivative in time of order α. We attach the homogeneous Neumann boundary condition at x = 0, ` and the initial value given by the Dirac delta function. We prove that α and p(x), 0 < x < `, are uniquely determined by data u(0, t), 0 < t ...

A new paradigm is proposed for sorting spikes in multi-electrode data using ratios of transfer functions between cells and electrodes. It is assumed that for every cell and electrode there is a stable linear relation. These are dictated by the properties of the tissue, the electrodes and their relative geometries. The main advantage of the method is that it is insensitive to variations in the s...

By using minimax methods and critical point theory, we obtain infinitely many periodic solutions for a second-order nonautonomous Hamiltonian systems, when the gradient of potential energy does not exceed linear growth.

Evidence for human learning without awareness of what is learned has been sought in serial reaction time (SRT) tasks in which, unknown to participants, the locations of stimuli follow a particular rule or sequence (Willingham, Nissen & Bullemer, 1989). A number of criticisms have been levelled at such tasks, including a lack of adequate control for sequential effects and a discrepancy in sensit...