The Cambridge controversies in the theory of capital: contributions from the complex plane

A controversy in capital theory concerns reswitching. When two production techniques are compared, reswitching occurs when one technique is cheapest at low interest rates, switches to being more expensive at higher rates, and then reswitches to being cheapest at yet higher rates. Some believe this inconsistency undermines neoclassical economics. The time-value-of-money (TVM) equation is at the core of the puzzle. The equation is a polynomial having n roots, implying n interest rates. In most analyses, including reswitching, one interest rate is used and the remaining rates are ignored. This analysis demonstrates that every TVM equation has a ‘dual’ form employing all interest rates. The dual of the reswitching equation explains the puzzle

Constraints on two-lepton, two quark operators

Physics from beyond the Standard Model, such as leptoquarks, can induce four fermion operators involving a quark, an anti-quark, a lepton and an anti-lepton. We update the (flavour dependent) constraints on the coefficients of such interactions, arising from collider searches for contact interactions, meson decays and other rare processes. We then make naive estimates for the magnitude of the coefficients, as could arise in texture models or from inverse hierarchies in the kinetic term coefficients. These estimates suggest that rare Kaon decays could be a good place to look for such operators.Comment: Modified presentation (to constrain coefficients of four fermion interactions, rather than gauge invariant operators), some numbers corrected, and minor changes. 28 pages, 18 table

Differential Recursion Relations for Laguerre Functions on Symmetric Cones

Let $\Omega$ be a symmetric cone and $V$ the corresponding simple Euclidean Jordan algebra. In \cite{ado,do,do04,doz2} we considered the family of generalized Laguerre functions on $\Omega$ that generalize the classical Laguerre functions on $\mathbb{R}^+$. This family forms an orthogonal basis for the subspace of $L$-invariant functions in $L^2(\Omega,d\mu_\nu)$, where $d\mu_\nu$ is a certain measure on the cone and where $L$ is the group of linear transformations on $V$ that leave the cone $\Omega$ invariant and fix the identity in $\Omega$. The space $L^2(\Omega,d\mu_\nu)$ supports a highest weight representation of the group $G$ of holomorphic diffeomorphisms that act on the tube domain $T(\Omega)=\Omega + iV.$ In this article we give an explicit formula for the action of the Lie algebra of $G$ and via this action determine second order differential operators which give differential recursion relations for the generalized Laguerre functions generalizing the classical creation, preservation, and annihilation relations for the Laguerre functions on $\mathbb{R}^+$

A Framework for Designing MIMO systems with Decision Feedback Equalization or Tomlinson-Harashima Precoding

We consider joint transceiver design for general Multiple-Input Multiple-Output communication systems that implement interference (pre-)subtraction, such as those based on Decision Feedback Equalization (DFE) or Tomlinson-Harashima precoding (THP). We develop a unified framework for joint transceiver design by considering design criteria that are expressed as functions of the Mean Square Error (MSE) of the individual data streams. By deriving two inequalities that involve the logarithms of the individual MSEs, we obtain optimal designs for two classes of communication objectives, namely those that are Schur-convex and Schur-concave functions of these logarithms. For Schur-convex objectives, the optimal design results in data streams with equal MSEs. This design simultaneously minimizes the total MSE and maximizes the mutual information for the DFE-based model. For Schur-concave objectives, the optimal DFE design results in linear equalization and the optimal THP design results in linear precoding. The proposed framework embraces a wide range of design objectives and can be regarded as a counterpart of the existing framework of linear transceiver design.Comment: To appear in ICASSP 200

Properties of developmental gene regulatory networks

The modular components, or subcircuits, of developmental gene regulatory networks (GRNs) execute specific developmental functions, such as the specification of cell identity. We survey examples of such subcircuits and relate their structures to corresponding developmental functions. These relations transcend organisms and genes, as illustrated by the similar structures of the subcircuits controlling the specification of the mesectoderm in the Drosophila embryo and the endomesoderm in the sea urchin, even though the respective subcircuits are composed of nonorthologous regulatory genes

Multipliers of embedded discs

We consider a number of examples of multiplier algebras on Hilbert spaces associated to discs embedded into a complex ball in order to examine the isomorphism problem for multiplier algebras on complete Nevanlinna-Pick reproducing kernel Hilbert spaces. In particular, we exhibit uncountably many discs in the ball of $\ell^2$ which are multiplier biholomorphic but have non-isomorphic multiplier algebras. We also show that there are closed discs in the ball of $\ell^2$ which are varieties, and examine their multiplier algebras. In finite balls, we provide a counterpoint to a result of Alpay, Putinar and Vinnikov by providing a proper rational biholomorphism of the disc onto a variety $V$ in $\mathbb B_2$ such that the multiplier algebra is not all of $H^\infty(V)$. We also show that the transversality property, which is one of their hypotheses, is a consequence of the smoothness that they require.Comment: 34 pages; the earlier version relied on a result of Davidson and Pitts that the fibre of the maximal ideal space of the multiplier algebra over a point in the open ball consists only of point evaluation. This result fails for $d = \infty$, and has necessitated some changes; to appear in Complex Analysis and Operator Theor