Igusa-type functions associated to finite formed spaces and their functional equations

We study symmetries enjoyed by the polynomials enumerating non-degenerate flags in finite vector spaces, equipped with a non-degenerate alternating bilinear, hermitian or quadratic form. To this end we introduce Igusa-type rational functions encoding these polynomials and prove that they satisfy certain functional equations. Some of our results are achieved by expressing the polynomials in question in terms of what we call parabolic length functions on Coxeter groups of type $A$. While our treatment of the orthogonal case exploits combinatorial properties of integer compositions and their refinements, we formulate a precise conjecture how in this situation, too, the polynomials may be described in terms of parabolic length functions.Comment: Slightly revised version, to appear in Trans. Amer. Math. Soc

Logistic model for stock market bubbles and anti-bubbles

Log-periodic power laws often occur as signatures of impending criticality of hierarchical systems in the physical sciences. It has been proposed that similar signatures may be apparent in the price evolution of financial markets as bubbles and the associated crashes develop. The features of such market bubbles have been extensively studied over the past 20 years, and models derived from an initial discrete scale invariance assumption have been developed and tested against the wealth of financial data with varying degrees of success. In this paper, the equations that form the basis for the standard log-periodic power law model and its higher extensions are compared to a logistic model derived from the solution of the Schroder equation for the renormalization group with nonlinear scaling function. Results for the S&P 500 and Nikkei 225 indices studied previously in the literature are presented and compared to established models, including a discussion of the apparent frequency shifting observed in the S&P 500 index in the 1980s. In the particular case of the Nikkei 225 anti-bubble between 1990 and 2003, the logistic model appears to provide a better description of the large-scale observed features over the whole 13-year period, particularly near the end of the anti-bubble

Semiclassical Approach to Heterogeneous Vacuum Decay

We derive the decay rate of an unstable phase of a quantum field theory in the presence of an impurity in the thin-wall approximation. This derivation is based on the how the impurity changes the (flat spacetime) geometry relative to case of pure false vacuum. Two examples are given that show how to estimate some of the additional parameters that enter into this heterogeneous decay rate. This formalism is then applied to the Higgs vacuum of the Standard Model (SM), where baryonic matter acts as an impurity in the electroweak Higgs vacuum. We find that the probability for heterogeneous vacuum decay to occur is suppressed with respect to the homogeneous case. That is to say, the conclusions drawn from the homogeneous case are not modified by the inclusion of baryonic matter in the calculation. On the other hand, we show that Beyond the Standard Model physics with a characteristic scale comparable to the scale that governs the homogeneous decay rate in the SM, can in principle lead to an enhanced decay rate.Comment: v3: version published in JHEP, very minor changes from v

That's Where the Money Was: Foreign Bias and English Investment Abroad, 1866-1907

Why did Victorian Britain invest so much capital abroad? We collect over 500,000 monthly returns of British and foreign securities trading in London and the United States between 1866 and 1907. These heretofore-unknown data allow us to better quantify the historical benefits of international diversification and revisit the question of whether British Victorian investor bias starved new domestic industries of capital. We find no evidence of bias. A British investor who increased his investment in new British industry at the expense of foreign diversification would have been worse off. The addition of foreign assets significantly expanded the mean-variance frontier and resulted in utility gains equivalent to a meaningful increase in lifetime consumption.Capital markets, Home bias, History, Victorian overseas investment

Learning the Right Lessons from Iraq

Foreign policy experts and policy analysts are misreading the lessons of Iraq. The emerging conventional wisdom holds that success could have been achieved in Iraq with more troops, more cooperation among U.S. government agencies, and better counterinsurgency doctrine. To analysts who share these views, Iraq is not an example of what not to do but of how not to do it. Their policy proposals aim to reform the national security bureaucracy so that we will get it right the next time. The near-consensus view is wrong and dangerous. What Iraq demonstrates is a need for a new national security strategy, not better tactics and tools to serve the current one. By insisting that Iraq was ours to remake were it not for the Bush administration's mismanagement, we ignore the limits on our power that the war exposes and in the process risk repeating our mistake. The popular contention that the Bush administration's failures and errors in judgment can be attributed to poor planning is also false. There was ample planning for the war, but it conflicted with the Bush administration's expectations. To the extent that planning failed, therefore, the lesson to draw is not that the United States national security establishment needs better planning, but that it needs better leaders. That problem is solved by elections, not bureaucratic tinkering. The military gives us the power to conquer foreign countries, but not the power to run them. Because there are few good reasons to take on missions meant to resuscitate failed governments, terrorism notwithstanding, the most important lesson from the war in Iraq should be a newfound appreciation for the limits of our power