Riemann Zeroes and Phase Transitions via the Spectral Operator on Fractal Strings

The spectral operator was introduced by M. L. Lapidus and M. van Frankenhuijsen [La-vF3] in their reinterpretation of the earlier work of M. L. Lapidus and H. Maier [LaMa2] on inverse spectral problems and the Riemann hypothesis. In essence, it is a map that sends the geometry of a fractal string onto its spectrum. In this survey paper, we present the rigorous functional analytic framework given by the authors in [HerLa1] and within which to study the spectral operator. Furthermore, we also give a necessary and sufficient condition for the invertibility of the spectral operator (in the critical strip) and therefore obtain a new spectral and operator-theoretic reformulation of the Riemann hypothesis. More specifically, we show that the spectral operator is invertible (or equivalently, that zero does not belong to its spectrum) if and only if the Riemann zeta function zeta(s) does not have any zeroes on the vertical line Re(s)=c. Hence, it is not invertible in the mid-fractal case when c=1/2, and it is invertible everywhere else (i.e., for all c in(0,1) with c not equal to 1/2) if and only if the Riemann hypothesis is true. We also show the existence of four types of (mathematical) phase transitions occurring for the spectral operator at the critical fractal dimension c=1/2 and c=1 concerning the shape of the spectrum, its boundedness, its invertibility as well as its quasi-invertibility

Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function

We survey some of the universality properties of the Riemann zeta function $\zeta(s)$ and then explain how to obtain a natural quantization of Voronin's universality theorem (and of its various extensions). Our work builds on the theory of complex fractal dimensions for fractal strings developed by the second author and M. van Frankenhuijsen in \cite{La-vF4}. It also makes an essential use of the functional analytic framework developed by the authors in \cite{HerLa1} for rigorously studying the spectral operator $\mathfrak{a}$ (mapping the geometry onto the spectrum of generalized fractal strings), and the associated infinitesimal shift $\partial$ of the real line: $\mathfrak{a}=\zeta(\partial)$. In the quantization (or operator-valued) version of the universality theorem for the Riemann zeta function $\zeta(s)$ proposed here, the role played by the complex variable $s$ in the classical universality theorem is now played by the family of `truncated infinitesimal shifts' introduced in \cite{HerLa1} to study the invertibility of the spectral operator in connection with a spectral reformulation of the Riemann hypothesis as an inverse spectral problem for fractal strings. This latter work provided an operator-theoretic version of the spectral reformulation obtained by the second author and H. Maier in \cite{LaMa2}. In the long term, our work (along with \cite{La5, La6}), is aimed in part at providing a natural quantization of various aspects of analytic number theory and arithmetic geometry

Learning-by-Doing or Habit Formation?

In a recent paper, Chang, Gomes, and Schorfheide (2002) extend the standard real business cycle (RBC) model to allow for a learning-by-doing (LBD) mechanism whereby current labour supply affects future productivity. They show that this feature magnifies the propagation of shocks and improves the matching performance of the standard RBC model. In this paper, the authors show that the LBD model is nearly observationally equivalent to an RBC model with habit formation in labour (or, equivalently, in leisure). Under the same calibration of the parameters, the two models share the same equilibrium paths of output, consumption, and investment, but have different implications for hours worked. Using Bayesian techniques, the authors investigate which of the LBD and habit models fits the U.S. data best. Their results suggest that the habit specification is more strongly supported by the data.Business fluctuations and cycles; Labour markets; Economic models; Econometric and statistical methods

Fluctuations in the Foreign Exchange Market: How Important are Monetary Policy Shocks?

We study the effects of U.S. monetary policy shocks on the bilateral exchange rate between the U.S. and each of the G7 countries. We also estimate deviations from uncovered interest rate parity and exchange rate pass-through conditional on these shocks. The analysis is based on a structural vector autoregression in which monetary policy shocks are identified through the conditional heteroscedasticity of the structural disturbances. Unlike earlier work in this area, our empirical methodology avoids making arbitrary assumptions about the relevant policy indicator or transmission mechanism in order to achieve identification. At the same time, it allows us to assess the implications of imposing invalid identifying restrictions. Our results indicate that the nominal exchange rate exhibits delayed overshooting in response to a monetary expansion, depreciating for roughly ten months before starting to appreciate. The shock also leads to large and persistent departures from uncovered interest rate parity, and to a prolonged period of incomplete pass-through. Variance-decomposition results indicate that monetary policy shocks account for a non-trivial proportion of exchange rate fluctuations.Conditions heteroscedasticity, delayed overshooting, exchange rate pass-through, identification, structural vector autoregression, uncovered interest rate parity

Fractal Complex Dimensions, Riemann Hypothesis and Invertibility of the Spectral Operator

A spectral reformulation of the Riemann hypothesis was obtained in [LaMa2] by the second author and H. Maier in terms of an inverse spectral problem for fractal strings. This problem is related to the question "Can one hear the shape of a fractal drum?" and was shown in [LaMa2] to have a positive answer for fractal strings whose dimension is c\in(0,1)-\{1/2} if and only if the Riemann hypothesis is true. Later on, the spectral operator was introduced heuristically by M. L. Lapidus and M. van Frankenhuijsen in their theory of complex fractal dimensions [La-vF2, La-vF3] as a map that sends the geometry of a fractal string onto its spectrum. We focus here on presenting the rigorous results obtained by the authors in [HerLa1] about the invertibility of the spectral operator. We show that given any $c\geq0$, the spectral operator $\mathfrak{a}=\mathfrak{a}_{c}$, now precisely defined as an unbounded normal operator acting in a Hilbert space $\mathbb{H}_{c}$, is `quasi-invertible' (i.e., its truncations are invertible) if and only if the Riemann zeta function $\zeta=\zeta(s)$ does not have any zeroes on the line $Re(s)=c$. It follows that the associated inverse spectral problem has a positive answer for all possible dimensions $c\in (0,1)$, other than the mid-fractal case when $c=1/2$, if and only if the Riemann hypothesis is true.Comment: To appear in: "Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics", Part 1 (D. Carfi, M. L. Lapidus, E. P. J. Pearse and M. van Frankenhuijsen, eds.), Contemporary Mathematics, Amer. Math. Soc., Providence, RI, 2013. arXiv admin note: substantial text overlap with arXiv:1203.482

Has Exchange Rate Pass-Through Really Declined in Canada?

Several empirical studies suggest that exchange rate pass-through has declined in recent years in industrialized countries. Results for Canada also indicate that, in the 1990s, import and consumer prices became less responsive to exchange rate movements. These findings are based on reducedform regressions that are typically motivated by partial-equilibrium models of pricing. Bouakez and Rebei instead use a structural, general-equilibrium approach to test the premise that exchange rate pass-through has decreased in Canada. Their approach consists in estimating a dynamic stochastic general-equilibrium model for Canada over two subsamples, which cover the periods before and after the Bank's adoption of inflation targeting. The authors then use impulse-response analysis to assess the stability of exchange rate pass-through across the two subsamples. Their results indicate that pass-through to Canadian import prices has been rather stable, while passthrough to Canadian consumer prices has declined in recent years. Counterfactual experiments reveal that the change in monetary policy regime is largely responsible for this decline.Business fluctuations and cycles; Economic models; Exchange rates; Inflation and prices; International topics

Exchange Rate Pass-Through in Industrialized Countries

Economists' long-standing interest in the degree to which exchange rate movements are reflected in prices was rekindled in the 1970s by a combination of rising inflation and the adoption of more flexible exchange rate regimes in many industrialized countries. Specifically, there were concerns that a large currency depreciation could degenerate into an inflationary spiral. Such fears were curtailed in the 1980s and early 1990s as industrialized countries began to reduce and stabilize their inflation rates. The low-inflation period most industrialized countries entered approximately a decade ago coincided with significant exchange rate depreciations that had much smaller effects on consumer prices than expected. This led to a belief that the extent to which exchange rate movements are passed through to consumer prices has declined. In this article, the authors examine why pass-through could be incomplete and review empirical estimates to determine whether pass-through has indeed declined, suggesting possible reasons for this decline and discussing the implications for monetary policy.

Factors affecting farmers’ willingness to adopt salt-tolerant forage crops in south-eastern Tunisia

This paper analyzed the factors that affect farmers’ willingness to adopt salt-tolerant forage for livestock, using a Tobit model. The data used for the empirical analysis was obtained from a survey of 97 farmers in southeastern Tunisia. The results of this study show that variables related to age, education level, the salinity level of water and membership in a farmers’ association do not significantly influence the degree to which salt-tolerant forage production is adopted. It did, however, find a positive relationship between off-farm income availability and adoption. In addition, the flock size variable, expressed in Standard Livestock Units, has a significant and positive relationship with adoption. This indicates farmers’ need to cover their forage deficit. Agricultural extension services should ensure that the requisite forage species are made available, and work with farmers to encourage them to adopt salt-tolerant forage species and pass on their knowledge to other farmers.Farmers’ willingness to adopt, Salt-tolerant forage, Tobit model, Tunisia, Crop Production/Industries,

Fiscal Policy and External Adjustment: New Evidence

Relatively little empirical evidence exists about countries’ external adjustment to changes in fiscal policy and, in particular, to changes in taxes. This paper addresses this question by measuring the effects of tax and government spending shocks on the current account and the real exchange rate in a sample of four industrialized countries. Our analysis is based on a structural vector autoregression in which the interaction of fiscal variables and macroeconomic aggregates is left unrestricted. Identification is instead achieved by exploiting the heteroscedasticity of the structural disturbances. Three main findings emerge: (i) the data provide little support for the twin-deficit hypothesis, (ii) the estimated effects of unexpected tax cuts are generally inconsistent with the predictions of standard economic models, except for the US, and (iii) the puzzling real depreciation triggered by an expansionary public spending shock is substantially larger in magnitude than predicted by traditional identification approaches.Government spending, current account, exchange rate, taxes, structural vector auto-regression, twin deficits