whenever ‖~v‖V < δ. In this case the bounded linear operator dF~v0 is called the (Frechet) derivative of F at ~v0. Prove that if ‖ • ‖V is a norm on V which is equivalent to ‖ • ‖V , then F is differentiable at ~v9 with respect to ‖ • ‖V if and only if it is differentiable at ~v0 with respect to ‖ • ‖V , and in this case the derivatives are equal. Similarly, if ‖ • ‖W is a norm on W which is eq...

We describe a method of obtaining closed-form complete solutions of certain second-order linear partial differential equations with more than two independent variables. This method generalizes the classical method of Laplace transformations of second-order hyperbolic equations in the plane and is based on an idea given by Ulisse Dini in 1902.

The Hodge dual operator ∗ is one of the 3 basic operations on differential forms. (The other 2 are wedge product ∧ and exterior differentiation d.) However most treatments consider only positive-definite inner products, and there are at least 2 standard ways of generalizing this to inner products of arbitrary signature. We outline here a construction of the Hodge dual operator which works for a...

In this note we show that for analytic semigroups the so-called Weiss condition of uniform boundedness of the operators Re(λ) 1/2C(λ+A)−1, Re(λ) > 0 on the complex right half plane and weak Lebesgue L2,∞–admissibility are equivalent. Moreover, we show that the weak Lebesgue norm is best possible in the sense that it is the endpoint for the ’Weiss conjecture’ within the scale of Lorentz spaces L...

We give a complete characterisation of the sets in which Peano derivatives of functions, which are definable in an o-minimal expansion of a real closed field, are continuously respectively Fréchet differentiable.

• Banach-Alaoglu: compactness of polars • Variant Banach-Steinhaus/uniform boundedness • Second polars • Weak boundedness implies boundedness • Weak-to-strong differentiability The comparison of weak and strong differentiability is due to Grothendieck, although the original sources are not widely available.

in this study, different derivative spectrophotometric methods are proposed for the simultaneous determination of chlorpheniramine maleate (cp), phenylephrine hcl (pe) and phenylpropanolamine hcl (pp) in their ternary mixtures and in pharmaceutical dosage forms. spectra of single component and ternary mixtures of various concentrations and combinations from zero- to fourth-derivation were obtai...