117 research outputs found

### Shapes of Quantum States

The shape space of k labelled points on a plane can be identified with the
space of pure quantum states of dimension k-2. Hence, the machinery of quantum
mechanics can be applied to the statistical analysis of planar configurations
of points. Various correspondences between point configurations and quantum
states, such as linear superposition as well as unitary and stochastic
evolution of shapes, are illustrated. In particular, a complete
characterisation of shape eigenstates for an arbitrary number of points is
given in terms of cyclotomic equations.Comment: Submitted to Proc. R. Statist. So

### Biorthogonal systems on unit interval and zeta dilation operators

An elementary 'quantum-mechanical' derivation of the conditions for a system
of functions to form a Reisz basis of a Hilbert space on a finite interval is
presented.Comment: 4 pages, 1 figur

### Modelling election dynamics and the impact of disinformation

Complex dynamical systems driven by the unravelling of information can be
modelled effectively by treating the underlying flow of information as the
model input. Complicated dynamical behaviour of the system is then derived as
an output. Such an information-based approach is in sharp contrast to the
conventional mathematical modelling of information-driven systems whereby one
attempts to come up with essentially {\it ad hoc} models for the outputs. Here,
dynamics of electoral competition is modelled by the specification of the flow
of information relevant to election. The seemingly random evolution of the
election poll statistics are then derived as model outputs, which in turn are
used to study election prediction, impact of disinformation, and the optimal
strategy for information management in an election campaign.Comment: 20 pages, 5 figure

### Asymptotic analysis on a pseudo-Hermitian Riemann-zeta Hamiltonian

The differential-equation eigenvalue problem associated with a
recently-introduced Hamiltonian, whose eigenvalues correspond to the zeros of
the Riemann zeta function, is analyzed using Fourier and WKB analysis. The
Fourier analysis leads to a challenging open problem concerning the formulation
of the eigenvalue problem in the momentum space. The WKB analysis gives the
exact asymptotic behavior of the eigenfunction

### Operator-valued zeta functions and Fourier analysis

The Riemann zeta function $\zeta(s)$ is defined as the infinite sum
$\sum_{n=1}^\infty n^{-s}$, which converges when ${\rm Re}\,s>1$. The Riemann
hypothesis asserts that the nontrivial zeros of $\zeta(s)$ lie on the line
${\rm Re}\,s= \frac{1}{2}$. Thus, to find these zeros it is necessary to
perform an analytic continuation to a region of complex $s$ for which the
defining sum does not converge. This analytic continuation is ordinarily
performed by using a functional equation. In this paper it is argued that one
can investigate some properties of the Riemann zeta function in the region
${\rm Re}\,s<1$ by allowing operator-valued zeta functions to act on test
functions. As an illustration, it is shown that the locations of the trivial
zeros can be determined purely from a Fourier series, without relying on an
explicit analytic continuation of the functional equation satisfied by
$\zeta(s)$.Comment: 8 pages, version to appear in J. Pays.

### Information of Interest

A pricing formula for discount bonds, based on the consideration of the
market perception of future liquidity risk, is established. An
information-based model for liquidity is then introduced, which is used to
obtain an expression for the bond price. Analysis of the bond price dynamics
shows that the bond volatility is determined by prices of certain weighted
perpetual annuities. Pricing formulae for interest rate derivatives are
derived.Comment: 12 pages, 3 figure

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