Behavior of the Solutions to a Functional Equation Which Equates a Function\u27s Inverse to its Reciprocal

This thesis explores the behavior of solutions of the functional equation f-1(x)= 1/f(x) for x ε Dom(f), where f is a real-valued function of a real variable. It is quite common to mistake the notation f-1, which means the inverse of f with respect to composition with the inverse of f with respect to multiplication, usually denoted by 1/f. This thesis shows that although f-1 and 1/f are usually different function, they do indeed sometimes represent the same function. This thesis will also provide methods of generating solutions of the functional equation f-1(x)= 1/f(x) for x ε Dom(f)

Coreset Clustering on Small Quantum Computers

Many quantum algorithms for machine learning require access to classical data in superposition. However, for many natural data sets and algorithms, the overhead required to load the data set in superposition can erase any potential quantum speedup over classical algorithms. Recent work by Harrow introduces a new paradigm in hybrid quantum-classical computing to address this issue, relying on coresets to minimize the data loading overhead of quantum algorithms. We investigate using this paradigm to perform $k$-means clustering on near-term quantum computers, by casting it as a QAOA optimization instance over a small coreset. We compare the performance of this approach to classical $k$-means clustering both numerically and experimentally on IBM Q hardware. We are able to find data sets where coresets work well relative to random sampling and where QAOA could potentially outperform standard $k$-means on a coreset. However, finding data sets where both coresets and QAOA work well--which is necessary for a quantum advantage over $k$-means on the entire data set--appears to be challenging

Niche separation between different functional types of mixoplankton: results from NPZ-style N-based model simulations

Protist plankton comprise phytoplankton (incapable of phagotrophy), protozooplankton (incapable of phototrophy) and mixoplankton (capable of phototrophy and phagotrophy). Of these, only phytoplankton and zooplankton are typically described in models. Over the last decade, however, the importance of mixoplankton across all marine biomes has risen to prominence. We thus need descriptions of mixoplankton within marine models. Here we present a simple yet flexible N-based model describing any one of the five basic patterns of protist plankton: phytoplankton, protozooplankton, and the three functional groups of mixoplankton: general non-constitutive mixoplankton (GNCM), specialist non-constitutive mixoplankton (SNCM), and constitutive mixoplankton (CM). By manipulation of a few input switch values, the same model can be used to describe any of these patterns, while adjustment of salient features, such as the percent of C-fixation required for mixotrophic growth, and the rate of phototrophic prey ingestion required to enable growth of GNCM and SNCM types, readily provides fine tuning. Example outputs are presented showing how the performance of these different protist configurations accords with expectations (set against empirical evidence). Simulations demonstrate clear niche separations between these protist functional groups according to nutrient, prey and light resource availabilities. This addition to classic NPZ plankton models provides for the exploration of the implications of mixoplankton activity in a simple yet robust fashion.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Product states optimize quantum pp-spin models for large pp

We consider the problem of estimating the maximal energy of quantum $p$-local spin glass random Hamiltonians, the quantum analogues of widely studied classical spin glass models. Denoting by $E^*(p)$ the (appropriately normalized) maximal energy in the limit of a large number of qubits $n$, we show that $E^*(p)$ approaches $\sqrt{2\log 6}$ as $p$ increases. This value is interpreted as the maximal energy of a much simpler so-called Random Energy Model, widely studied in the setting of classical spin glasses. Our most notable and (arguably) surprising result proves the existence of near-maximal energy states which are product states, and thus not entangled. Specifically, we prove that with high probability as $n\to\infty$, for any $E<E^*(p)$ there exists a product state with energy $\geq E$ at sufficiently large constant $p$. Even more surprisingly, this remains true even when restricting to tensor products of Pauli eigenstates. Our approximations go beyond what is known from monogamy-of-entanglement style arguments -- the best of which, in this normalization, achieve approximation error growing with $n$. Our results not only challenge prevailing beliefs in physics that extremely low-temperature states of random local Hamiltonians should exhibit non-negligible entanglement, but they also imply that classical algorithms can be just as effective as quantum algorithms in optimizing Hamiltonians with large locality -- though performing such optimization is still likely a hard problem. Our results are robust with respect to the choice of the randomness (disorder) and apply to the case of sparse random Hamiltonian using Lindeberg's interpolation method. The proof of the main result is obtained by estimating the expected trace of the associated partition function, and then matching its asymptotics with the extremal energy of product states using the second moment method.Comment: Added a disclaimer about error in current draf

Endoscopic and Robotic Stapes Surgery: Review with Emphasis on Recent Surgical Refinements

Purpose of Review Stapes surgery has been established as the gold standard for surgical treatment of conductive hearing loss in otosclerosis. Excellent outcomes with very low complication rate are reported for this surgery. Recent advances to improve surgical outcome have modified the surgical technique with endoscopes, and recent studies report development of robotical assistance. This article reviews the use of endoscopes and robotical assistance for stapes surgery. Recent Findings While different robotic models have been developed, 2 models for stapes surgery have been used in the clinical setting. These can be used concomitant to an endoscope or microscope. Endoscopes are used on a regular base regarding stapes surgery with similar outcomes as microscopes. Endoscopic stapes surgery shows similar audiological results to microscopic technique with an advantage of less postoperative dysgeusia and pain. Its utility in cases of revision surgery or malformation is emphasized. Summary Endoscopic stapes surgery is used on a regular basis with excellent outcomes similar to the microscopic approach, while reducing surgical morbidity. Robotic technology is increasingly being developed in the experimental setting, and first applications are reported in its clinical use