Riemann zeros, prime numbers and fractal potentials

Using two distinct inversion techniques, the local one-dimensional potentials for the Riemann zeros and prime number sequence are reconstructed. We establish that both inversion techniques, when applied to the same set of levels, lead to the same fractal potential. This provides numerical evidence that the potential obtained by inversion of a set of energy levels is unique in one-dimension. We also investigate the fractal properties of the reconstructed potentials and estimate the fractal dimensions to be $D=1.5$ for the Riemann zeros and $D = 1.8$ for the prime numbers. This result is somewhat surprising since the nearest-neighbour spacings of the Riemann zeros are known to be chaotically distributed whereas the primes obey almost poisson-like statistics. Our findings show that the fractal dimension is dependent on both the level-statistics and spectral rigidity, $\Delta_3$, of the energy levels.Comment: Five postscript figures included in the text. To appear in Phys. Rev.

Environment mediated multipartite and multidimensional entanglement

Quantum entanglement is usually considered a fragile quantity and decoherence through coupling to an external environment, such as a thermal reservoir, can quickly destroy the entanglement resource. This doesn't have to be the case and the environment can be engineered to assist in the formation of entanglement. We investigate a system of qubits and higher dimensional spins interacting only through their mutual coupling to a reservoir. We explore the entanglement of multipartite and multidimensional system as mediated by the bath and show that at low temperatures and intermediate coupling strengths multipartite entanglement may form between qubits and between higher spins, i.e., qudits. We characterise the multipartite entanglement using an entanglement witness based upon the structure factor and demonstrate its validity versus the directly calculated entanglement of formation, suggesting possible experiments for its measure.Comment: 9 pages, 10 figure

Acromegaly, Mr Punch and caricature.

The origin of Mr Punch from the Italian Pulcinella of the Commedia dell'arte is well known but his feature, large hooked nose, protruding chin, kyphosis and sternal protrusion all in an exaggerated form also suggest the caricature of an acromegalic. This paper looks at the physical characteristics of acromegaly, the origin of Mr Punch and the development of caricature linking them together in the acromegalic caricature that now has a life of its own

Euler buckling in red blood cells: An optically driven biological micromotor

We investigate the physics of an optically-driven micromotor of biological origin. A single, live red blood cell, when placed in an optical trap folds into a rod-like shape. If the trapping laser beam is circularly polarized, the folded RBC rotates. A model based on the concept of buckling instabilities captures the folding phenomenon; the rotation of the cell is simply understood using the Poincar\`e sphere. Our model predicts that (i) at a critical intensity of the trapping beam the RBC shape undergoes large fluctuations and (ii) the torque is proportional to the intensity of the laser beam. These predictions have been tested experimentally. We suggest a possible mechanism for emergence of birefringent properties in the RBC in the folded state

Obliquity-driven expansion of North Atlantic sea ice during the last glacial

Author Posting. © American Geophysical Union, 2015. This article is posted here by permission of American Geophysical Union for personal use, not for redistribution. The definitive version was published in Geophysical Research Letters 42 (2015): 10,382–10,390, doi:10.1002/2015GL066344.North Atlantic late Pleistocene climate (60,000 to 11,650 years ago) was characterized by abrupt and extreme millennial duration oscillations known as Dansgaard-Oeschger (D-O) events. However, during the Last Glacial Maximum (LGM) 23,000 to 19,000 cal years ago (23 to 19 ka), no D-O events are observed in the Greenland ice cores. Our new analysis of the Greenland δ18O record reveals a switch in the stability of the climate system around 30 ka, suggesting that a critical threshold was passed. Climate system modeling suggests that low axial obliquity at this time caused vastly expanded sea ice in the Labrador Sea, shifting Northern Hemisphere westerly winds south and reducing the strength of meridional overturning circulation. The results suggest that these feedbacks tipped the climate system into full glacial conditions, leading to maximum continental ice growth during the LGM.Australian Research Council2016-06-1

Quantum mechanical potentials related to the prime numbers and Riemann zeros

Prime numbers are the building blocks of our arithmetic, however, their distribution still poses fundamental questions. Bernhard Riemann showed that the distribution of primes could be given explicitly if one knew the distribution of the non-trivial zeros of the Riemann $\zeta(s)$ function. According to the Hilbert-P{\'o}lya conjecture there exists a Hermitean operator of which the eigenvalues coincide with the real part of the non-trivial zeros of $\zeta(s)$. This idea encourages physicists to examine the properties of such possible operators, and they have found interesting connections between the distribution of zeros and the distribution of energy eigenvalues of quantum systems. We apply the Mar{\v{c}}henko approach to construct potentials with energy eigenvalues equal to the prime numbers and to the zeros of the $\zeta(s)$ function. We demonstrate the multifractal nature of these potentials by measuring the R{\'e}nyi dimension of their graphs. Our results offer hope for further analytical progress.Comment: 7 pages, 5 figures, 2 table

Photoacoustic imaging of the human placental vasculature

Minimally invasive fetal interventions require accurate imaging from inside the uterine cavity. Twin‐to‐twin transfusion syndrome (TTTS), a condition considered in this study, occurs from abnormal vascular anastomoses in the placenta that allow blood to flow unevenly between the fetuses. Currently, TTTS is treated fetoscopically by identifying the anastomosing vessels, and then performing laser photocoagulation. However, white light fetoscopy provides limited visibility of placental vasculature, which can lead to missed anastomoses or incomplete photocoagulation. Photoacoustic (PA) imaging is an alternative imaging method that provides contrast for hemoglobin, and in this study, two PA systems were used to visualize chorionic (fetal) superficial and subsurface vasculature in human placentas. The first system comprised an optical parametric oscillator for PA excitation and a 2D Fabry‐Pérot cavity ultrasound sensor; the second, light emitting diode arrays and a 1D clinical linear‐array ultrasound imaging probe. Volumetric photoacoustic images were acquired from ex vivo normal term and TTTS‐treated placentas. It was shown that superficial and subsurface branching blood vessels could be visualized to depths of approximately 7 mm, and that ablated tissue yielded negative image contrast. This study demonstrated the strong potential of PA imaging to guide minimally invasive fetal therapies

Physics of the Riemann Hypothesis

Physicists become acquainted with special functions early in their studies. Consider our perennial model, the harmonic oscillator, for which we need Hermite functions, or the Laguerre functions in quantum mechanics. Here we choose a particular number theoretical function, the Riemann zeta function and examine its influence in the realm of physics and also how physics may be suggestive for the resolution of one of mathematics' most famous unconfirmed conjectures, the Riemann Hypothesis. Does physics hold an essential key to the solution for this more than hundred-year-old problem? In this work we examine numerous models from different branches of physics, from classical mechanics to statistical physics, where this function plays an integral role. We also see how this function is related to quantum chaos and how its pole-structure encodes when particles can undergo Bose-Einstein condensation at low temperature. Throughout these examinations we highlight how physics can perhaps shed light on the Riemann Hypothesis. Naturally, our aim could not be to be comprehensive, rather we focus on the major models and aim to give an informed starting point for the interested Reader.Comment: 27 pages, 9 figure