We show that a noncompact manifold with bounded sectional curvature, whose ends are sufficiently collapsed, has a finite dimensional space of square-integrable harmonic forms. In the special case of a finite-volume manifold with pinched negative sectional curvature, we show that the essential spectrum of the p-form Laplacian is the union of the essential spectra of a collection of ordinary diff...

The theory of conjugate points in the calculus of variations is reconsidered with a perspective emphasizing the connection to finite-dimensional optimization. The object of central importance is the spectrum of the second-variation operator, analogous to the eigenvalues of the Hessian matrix in finite dimensions. With a few basic properties of this spectrum, one can gain a new perspective on th...

I discuss the zeromode spectrum of lattice chiral fermions in the domain wall model suggested recently. In particular I give the critical momenta where the fermions cease to be chiral and show that the spectrum is directly related to the behaviour of the Chern-Simons current on the lattice. First results for the domain wall model on the finite lattice indicate that the relevant features of the ...

We investigate the effect of drag force on the enstrophy cascade of two-dimensional Navier-Stokes turbulence. We find a power law decrease of the energy wave number (k) spectrum that is faster than the classical (no-drag) prediction of k(-3). It is shown that the enstrophy cascade with drag can be analyzed by making use of a previous theory for finite lifetime passive scalars advected by a Lagr...

We determine the finite-volume corrections to the spectrum and matrix elements of twohadron states in a moving frame, i.e. one in which the total momentum of the twohadrons is non-zero. The analysis is performed entirely within field theory and the results are accurate up to exponential corrections in the volume. Our results for the spectrum are equivalent to those of Rummukainen and Gottlieb w...

We show that a nonlinear dynamical system in Poincaré-Dulac normal form (in R) can be seen as a constrained linear system; the constraints are given by the resonance conditions satisfied by the spectrum of (the linear part of) the system and identify a naturally invariant manifold for the flow of the “parent” linear system. The parent system is finite dimensional if the spectrum satisfies only ...

For n ≥ 9, we construct maximal partial line spreads for non-singular quadrics of PG(n, q) for every size between approximately (cn + d)(qn−3 + qn−5) log 2q and qn−2, for some small constants c and d. These results are similar to spectrum results on maximal partial line spreads in finite projective spaces by Heden, and by Gács and Szőnyi. These results also extend spectrum results on maximal pa...

We discuss results where the discrete spectrum (or partial information on the discrete spectrum) and partial information on the potential q of a one-dimensional Schrödinger operator H = − d2 dx2 + q determine the potential completely. Included are theorems for finite intervals and for the whole line. In particular, we pose and solve a new type of inverse spectral problem involving fractions of ...

We study the spectrum of the Dirac operator on hyperbolic manifolds of finite volume. Depending on the spin structure it is either discrete or the whole real line. For link complements in S we give a simple criterion in terms of linking numbers for when essential spectrum can occur. We compute the accumulation rate of the eigenvalues of a sequence of closed hyperbolic 2or 3-manifolds degenerati...

The instability mechanisms, related to the implementation of distributed delay controllers in the context of finite spectrum assignment, were studied in detail in the past few years. In this note we introduce a distributed delay control law that assigns a finite closed-loop spectrum and whose implementation with a sum of point-wise delays is safe. This property is obtained by implicitly includi...