We present a one dimensional semilinear parabolic equation for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. In our example the derivative blows up in the interior of the space interval rather than at the boundary, as in earlier examples. In the case of monotone solutions we show that gradient blow-up occurs at a single...

Coupled oscillators are shown to experience amplitude death for a much larger set of parameter values when they are connected with time delays distributed over an interval rather than concentrated at a point. Distributed delays enlarge and merge death islands in the parameter space. Furthermore, when the variance of the distribution is larger than a threshold, the death region becomes unbounded...

We extend a result of Dubins (Ann Probab 3:89–99, 1975) from bounded to unbounded random variables. Dubins showed that a finitely additive expectation over the collection of bounded random variables can be written as an integral of conditional expectations (disintegrability) if and only if the marginal expectation is always within the smallest closed interval containing the conditional expectat...

In this note we prove that for equilibrium states of axiom A systems the time τB(x) needed for a typical point x to enter for the first time in a typical ball B with radius r scales as τB(x) ∼ r d where d is the local dimension of the invariant measure at the center of the ball. A similar relation is proved for a full measure set of interval excanges. Some applications to Birkoff averages of un...

The large-time behavior of solutions to the initial and initial boundary value problems for a one-dimensional viscous polytropic ideal gas in unbounded domains is investigated. Using a special cut-oo function to localize the problem, we derive a local representation for the speciic volume. With the help of the local representation, and certain new estimates for the temperature and the stress, a...

In this paper we study the convergence of the numerical path integration method which produces approximate probability density functions for the solutions of stochastic differential equations. This method enjoys several advantages compared to path-wise approximation schemes. We prove that the approximate probability densities converge strongly in L1 to the probability density of the stochastic ...

We study the homogenization of Lagrangian functionals of Hamilton-Jacobi equations (HJ) with quadratic nonlinearity and unbounded stationary ergodic random potential in R, d≥1. We show that homogenization holds if and only if the potential is bounded from above. When the potential is unbounded from above, homogenization breaks down, due to the almost sure growth of the running maxima of the ran...

In the standard theorems on weak convergence of stochastic processes, it is invariably assumed that the parameter set is a bounded interval. The object of this paper is to indicate that analogues of these theorems for unbounded intervals are also valid. We shall confine our attention to the results of Skorohod [l], and in particular to those results concerning his Ji topology. Let £ be a comple...

Given a word w over a nite alphabet and a set of ordered pairs of letters which de ne adjacencies we construct a graph which we call the letter graph of w The lettericity of a graph G is the least size of alphabet permitting to obtain G as a letter graph The set of letter graphs consists of threshold graphs unbounded interval graphs and their complements We determine the lettericity of cycles a...

Motivated by the equivalent definition of a continuous operator between Banach spaces in terms weakly null nets, we introduce unbounded operators replacing weak convergence with absolutely (uaw-convergence) lattices. We characterize order lattices and reflexive these operators. Moreover, motivated characterizing lattice Cauchy sequences, consider pre-unbounded which maps uaw-Cauchy sequences to...