This paper focuses on the issue of how generalizations of continuous and leftcontinuous t-norms over linearly ordered sets should be from a logical point of view. Taking into account recent results in the scope of algebraic semantics for fuzzy logics over chains with a monoidal residuated operation, we advocate linearly ordered BL-algebras and MTL-algebras as adequate generalizations of continu...

The Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition is considered. Local well-posedness for data u0 in the space b H r (T), defined by the norms ‖u0‖ b Hs r (T) = ‖〈ξ〉 s b u0‖lr′ ξ , is shown in the parameter range s ≥ 1 2 , 2 > r > 4 3 . The proof is based on an adaptation of the gauge transform to the periodic setting and an appropriate varian...

We are concerned with the variety T of algebras of type (2,2,2,0,0) generated by the algebra (I,◦), where I = ([0,1],∧,∨,0,1) is the unit interval with minimum and maximum determined by the usual order and.◦ 6 = ∧ is a continuous t-norm. We have shown that a strict t-norm and a nilpotent t-norm, and in fact any continuous t-norm except minimum, generate the same variety. Moreover, this variety ...

Let u(x; t) be the solution to the free time-dependent Schrr odinger equation at the point (x; t) in space-time R n+1 with initial data f. We characterize the size of u in terms L p (L q)-estimates with power weights. Our bounds are given by norms of f in homogeneous Sobolev spaces _ H s (R n). Our methods include use of spherical harmonics, uniformity properties of Bessel functions and interpo...

Let u(x; t) be the solution to the free time-dependent Schrr odinger equation at the point (x; t) in space-time R n+1 with initial data f. We characterize the size of u in terms L p (L q)-estimates with power weights. Our bounds are given by norms of f in homogeneous Sobolev spaces _ H s (R n). Our methods include use of spherical harmonics, uniformity properties of Bessel functions and interpo...

We propose a new technique for analyzing the error of finite element approximations of parabolic problems with non-smooth initial data. For homogeneous equation we prove optimal L-error estimate of order O ( h/t ) for t > 0 when the given initial data is in L. Further, for non-homogeneous parabolic equation with zero initial data we establish an optimal error estimate of order O(h) in L. Thus, ...

We consider a fully discrete approximation of the H gradient flow of an energy integral where the energy density is given by the sum of a nonnegative multi-well potential term and a bending energy term. The spatial discretization is based on a Fourier spectral method, which is combined with an implicit Euler time discretization. The numerical method is shown to be stable and to exhibit optimal ...

In this paper, stability and error estimates for time discretizations of linear semilinear parabolic equations by the two-step backward differentiation formula (BDF2) method with variable step-sizes are derived. An affirmative answer is provided to question: whether upper bound step-size ratios $l^{\infty }(0,T;H)$ -stability BDF2 identical zero-stability. The }(0,T;V)$ also established under m...

We obtain an explicit asymptotic formula for the norms of the spectral projections of the non-self-adjoint harmonic oscillator H. We deduce that the spectral expansion of e−Ht is norm convergent if and only if t is greater than a certain explicit positive constant.

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