On the invariance of wave operators

Let H, K be self-adjoint operators on a Hilbert space. Kato's Invariance Principle (T. Kato, “Perturbation Theory for Linear Operators,” Springer-Verlag, Berlin, 1966) states that under certain conditions W(φ(H), φ(K)) = s-lim_{t → + ∞}e^it(φ)ke^?it(φ)H exists and is independent of the monotone function φ whenever W(H, K) exists. The purpose of this paper is to present a new proof of Kato's result based upon a study of the variation of W(φ(H), φ(K)) with respect to φ. It is shown (Theorem 1) that this variation vanishes, and (Theorem 2) that the invariance principle holds provided that K-H belongs to a large subset of the Hilbert-Schmidt class of compact operators.     

A note on the definition of small overlap monoids

Small overlap conditions are simple and natural combinatorial conditions on semigroup and monoid presentations, which serve to limit the complexity of derivation sequences between equivalent words in the generators. They were introduced by J.H. Remmers, and more recently have been extensively studied by the present author. However, the definition of small overlap conditions hitherto used by the author was slightly more restrictive than that introduced by Remmers; this note eliminates this discrepancy by extending the recent methods and results of the author to apply to Remmers’ small overlap monoids in full generality.     

On extensions of the domain invariance theorem

If f maps continuously a compact subset X of Rⁿ into Rⁿ and x is a point whose distance from the boundary ∂X is greater than double diameter of the fibres of the points in f(∂X) then f(x) is in the interior of f(X). This theorem extends some results due to Borsuk and Sitnikov.     

On the invariance principle of scattering theory

The invariance principle of scattering theory is proved under certain rate-of-convergence conditions. As a consequence we obtain an invariance principle in the case of potential scattering with potentials V(x) satisfying (1+|x|)^δV(x)∈l₂(R³) for some δ > 0.     

On the reflection invariance of residuated chains

It is shown that, under certain conditions, a subset of the graph of a commutative residuated chain is invariant under a geometric reflection. This result implies that a certain part of the graph of the monoidal operation of a commutative residuated chain determines another part of the graph via the reflection on one hand, and tells us about the structure of continuity points of the monoidal operation on the other. Finally, these results are applied for the subdomains of uniqueness problem, yielding new results.     

On the combinatorial invariance of Kazhdan-Lusztig polynomials

In this paper, we solve the conjecture about the combinatorial invariance of Kazhdan-Lusztig polynomials for the first open cases, showing that it is true for intervals of length 5 and 6 in the symmetric group. We also obtain explicit formulas for the R-polynomials and for the Kazhdan-Lusztig polynomials associated with any interval of length 5 in any Coxeter group, showing in particular what they look like in the symmetric group.     

On the translation invariance of wavelet subspaces

An examination of the translation invariance of V₀ under dyadic rationals is presented, generating a new equivalence relation on the collection of wavelets. The equivalence classes under this relation are completely characterized in terms of the support of the Fourier transform of the wavelet. Using operator interpolation, it is shown that several equivalence classes are non-empty.     

On the frame invariance of linear elasticity theory

It is widely believed that classical linear elasticity theory does not conform to the Galilean frame invariance of general (non-relativistic) field theories. This view is traced here to an interpretation of the relationship between the deformation gradient and the displacement gradient which does not reflect the tensor character of the variables involved. Frame invariance is shown to follow if tensor character is imposed on this relationship at the outset.     

On the invariance of male optimal stable matching

The stable matching problem is that of matching two sets of agents in such a manner that no two unmatched agents prefer each other to their actual partners under the matching. In this paper, we present a set of sufficient conditions on the preference lists of any given stable matching instance, under which the optimality of the original male optimal stable matching is still preserved.     

On the frame invariance of linear elasticity theory

It is widely believed that classical linear elasticity theory does not conform to the Galilean frame invariance of general (non-relativistic) field theories. This view is traced here to an interpretation of the relationship between the deformation gradient and the displacement gradient which does not reflect the tensor character of the variables involved. Frame invariance is shown to follow if tensor character is imposed on this relationship at the outset.     

On intrinsic invariance of fractal dimensions

We show that intrinsic isometry does not have to preserve a fractal dimension.     

On expansion and topological overlap

We give a detailed and easily accessible proof of Gromov’s Topological Overlap Theorem. Let X be a finite simplicial complex or, more generally, a finite polyhedral cell complex of dimension d. Informally, the theorem states that if X has sufficiently strong higher-dimensional expansion properties (which generalize edge expansion of graphs and are defined in terms of cellular cochains of X) then X has the following topological overlap property: for every continuous map \(X\rightarrow \mathbb {R}^d\) there exists a point \(p\in \mathbb {R}^d\) that is contained in the images of a positive fraction \(\mu >0\) of the d-cells of X. More generally, the conclusion holds if \(\mathbb {R}^d\) is replaced by any d-dimensional piecewise-linear manifold M, with a constant \(\mu \) that depends only on d and on the expansion properties of X, but not on M.     

On the invariance principle for U-statistics

Let T_n be a U-statistic and S_n its projection (in the sense of Hájek). Limit theory for U-statistics is usually considered in two disjoint cases, termed degenerate and nondegenerate. The traditional method is to treat the cases separately, using different techniques in each to obtain a solution. Here we present a unified treatment based on a joint invariance principle for the vector (T_n, T_n – S_n), from which the invariance principles in both the degenerate and nondegenerate cases follow as easy corollaries.     

On the invariance principle for differentiable statistical functionals

Sup-norm differentiability of a statistical functional T is shown to be a sufficient condition for invariance principles for T. Application of the result to repeated significance testing is sketched.     

On the domain invariance of countably condensing vector fields

Using homotopy theory, we give the domain invariance theorem for countably condensing vector fields, where the notion of countably condensing maps is due to Väth. A starting point of this investigation is that there is a symmetric characteristic set for a countably condensing map.