A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics

We present a formal geometric framework for the study of adiabatic quantum mechanics for arbitrary finite-dimensional non-degenerate Hamiltonians. This framework generalizes earlier holonomy interpretations of the geometric phase to non-cyclic states appearing for non-Hermitian Hamiltonians. We start with an investigation of the space of non-degenerate operators on a finite-dimensional state space. We then show how the energy bands of a Hamiltonian family form a covering space. Likewise, we show that the eigenrays form a bundle, a generalization of a principal bundle, which admits a natural connection yielding the (generalized) geometric phase. This bundle provides in addition a natural generalization of the quantum geometric tensor and derived tensors, and we show how it can incorporate the non-geometric dynamical phase as well. We finish by demonstrating how the bundle can be recast as a principal bundle, so that both the geometric phases and the permutations of eigenstates can be expressed simultaneously by means of standard holonomy theory.

Competitiveness of Malaysian Fisheries Exports: A Constant Market Share Analysis

Malaysia, one of the global major fish producers, has highly traded fisheries products given its many water bodies. Nonetheless, it faces a serious fish trade deficit, implying that the Malaysian fisheries sector might lose its competitiveness in the global market. This paper adopts a modified constant market share (CMS) analysis, which incorporates a net-share approach index and geometric framework, to measure the export competitiveness of the Malaysian fisheries sector. The findings reveal that half of the fisheries products exhibit optimistic export competitiveness. Malaysia reflects the strongest competitiveness in exporting frozen fish and the least competitiveness in the export of crustaceans. Additional effort and attention on those less competitive groups of aquatic invertebrates, live fish and crustaceans are required to improve the export performance. Application of the modified approach is highly proposed as it is not only a simple measurement that gives relatively more accurate results but also succeeds to overcome inconsistency in the traditional approach. The findings provide evidence of unrealised fish export potential regarding product categories, which helps policymakers, traders and marketers to develop their long-term strategic plans and enhance the export competitiveness of the fisheries sector in Malaysia.

Generalized compact star models with conformal symmetry

We generate a new generalized regular charged anisotropic exact model that admits conformal symmetry in static spherically symmetric spacetime. Our model was examined for physical acceptability as realistic stellar models. The regularity is not violated, the energy conditions are satisfied, the physical forces balanced at equilibrium, the stability is satisfied via adiabatic index, and the surface red shift and mass–radius ratio are within the required bounds. Our conformal charged anisotropic exact solution contains models generated by Finch–Skea, Vaidya–Tikekar and Schwarzschild. Also, some recent charged or neutral and anisotropic or isotropic conformally symmetric models are found as special cases of our exact model. Our approach using a conformal symmetry provides a generalized geometric framework for studying compact objects.

Nutrigonometry I: using right-angle triangles to quantify nutritional trade-offs in multidimensional performance landscapes

Animals regulate their diet in order to maximise the expression of fitness traits that often have different nutritional needs. These nutritional trade-offs have been experimentally uncovered using the Geometric framework for nutrition (GF). However, current analytical methods to measure such responses rely on either visual inspection or complex models applied to multidimensional performance landscapes, making these approaches subjective, or conceptually difficult, computationally expensive, and in some cases inaccurate. This limits our ability to understand how animal nutrition evolved to support life-histories within and between species. Here, we introduce a simple trigonometric model to measure nutritional trade-offs in multidimensional landscapes ("Nutrigonometry"). Nutrigonometry is both conceptually and computationally easier than current approaches, as it harnesses the trigonometric relationships of right-angle triangles instead of vector calculations. Using landmark GF datasets, we first show how polynomial (Bayesian) regressions can be used for precise and accurate predictions of peaks and valleys in performance landscapes, irrespective of the underlying structure of the data (i.e., individual food intakes vs fixed diet ratios). Using trigonometric relationships, we then identified the known nutritional trade-off between lifespan and reproductive rate both in terms of nutrient balance and concentration. Nutrigonometry enables a fast, reliable and reproducible quantification of nutritional trade-offs in multidimensional performance landscapes, thereby broadening the potential for future developments in comparative research on the evolution of animal nutrition.

On the smallest singular value of symmetric random matrices

We show that for an $n\times n$ random symmetric matrix $A_n$ , whose entries on and above the diagonal are independent copies of a sub-Gaussian random variable $\xi$ with mean 0 and variance 1, \begin{equation*}\mathbb{P}[s_n(A_n) \le \epsilon/\sqrt{n}] \le O_{\xi}(\epsilon^{1/8} + \exp(\!-\Omega_{\xi}(n^{1/2}))) \quad \text{for all } \epsilon \ge 0.\end{equation*} This improves a result of Vershynin, who obtained such a bound with $n^{1/2}$ replaced by $n^{c}$ for a small constant c, and $1/8$ replaced by $(1/8) - \eta$ (with implicit constants also depending on $\eta > 0$ ). Furthermore, when $\xi$ is a Rademacher random variable, we prove that \begin{equation*}\mathbb{P}[s_n(A_n) \le \epsilon/\sqrt{n}] \le O(\epsilon^{1/8} + \exp(\!-\Omega((\!\log{n})^{1/4}n^{1/2}))) \quad \text{for all } \epsilon \ge 0.\end{equation*} The special case $\epsilon = 0$ improves a recent result of Campos, Mattos, Morris, and Morrison, which showed that $\mathbb{P}[s_n(A_n) = 0] \le O(\exp(\!-\Omega(n^{1/2}))).$ Notably, in a departure from the previous two best bounds on the probability of singularity of symmetric matrices, which had relied on somewhat specialized and involved combinatorial techniques, our methods fall squarely within the broad geometric framework pioneered by Rudelson and Vershynin, and suggest the possibility of a principled geometric approach to the study of the singular spectrum of symmetric random matrices. The main innovations in our work are new notions of arithmetic structure – the Median Regularized Least Common Denominator (MRLCD) and the Median Threshold, which are natural refinements of the Regularized Least Common Denominator (RLCD)introduced by Vershynin, and should be more generally useful in contexts where one needs to combine anticoncentration information of different parts of a vector.

Maternal Dietary Fatty Acid Composition and Newborn Epigenetic Aging—A Geometric Framework Approach

Background Maternal nutrition is associated with epigenetic and cardiometabolic risk factors in offspring. Research in humans has primarily focused on assessing the impact of individual nutrients. Objective We sought to assess the collective impact of maternal dietary monounsaturated (MUFA), polyunsaturated (PUFA), and saturated fat (SFA) on epigenetic aging and cardiometabolic risk markers in healthy newborn infants using a geometric framework approach. Design Body fatness (n = 162), aortic intima-media thickness (n = 131), heart rate variability (n = 118), and epigenetic age acceleration (n = 124) were assessed in newborn infants. Maternal dietary intake was cross-sectionally assessed in the immediate postpartum period via a validated 80-item self-administered food-frequency questionnaire. Generalized additive models were used to explore interactive associations of nutrient intake, with results visualized as response surfaces. Results After adjustment for total energy intake, maternal age, gestational age, and sex there was a 3-way interactive association of MUFA, PUFA, and SFA (P = 0.001) with newborn epigenetic aging. This suggests that the nature of each fat class association depends upon one another. Response surfaces revealed MUFA was positively associated with newborn epigenetic age acceleration only at proportionately lower intakes of SFA or PUFA. We also demonstrate a potential beneficial association of omega-3 PUFA with newborn epigenetic age acceleration (P = 0.008). There was no significant association of fat class with newborn aortic intima-media thickness, heart rate variability, or body fatness. Conclusions In this study, we demonstrate an association between maternal dietary fat class composition and epigenetic aging in newborns. Future research should consider other characteristics such as the source of maternal dietary fatty acids.

Protein-carbohydrate interaction effects on energy balance, FGF21, IGF-1 and hypothalamic genes expression in rats

Amino acids are involved in energy homeostasis, just as are carbohydrates and lipids. Therefore, mechanisms controlling protein intake should operate independently and in combination with systems controlling overall energy intake to coordinate appropriate metabolic and behavioural responses. The objective of this study was to quantify the respective roles of dietary protein and carbohydrate levels on energy balance, plasma FGF21 and IGF-1 concentrations, and hypothalamic neurotransmitters (POMC, NPY, AgRP and CART). In a simplified geometric framework, 7-week-old male Wistar rats were fed 12 diets containing 3 to 30% protein for 3 weeks, in which carbohydrates accounted for 30 to 75% of the carbohydrate and fat part of the diet. As a result of this study, most of the studied parameters (body composition, energy expenditure, plasma FGF21 and IGF-1 concentrations and Pomc/Agrp ratio) responded mainly to the protein content and to a lesser extent to the carbohydrate content in the diet.

New evidence for the design and urban integration of the Forum of Caesar, Forum of Augustus, Curia Julia, and Chalcidicum

This article presents a new hypothesis about the urban arrangement and function of the southeast end of the Forum of Caesar, together with the Forum of Augustus, the Curia Julia and the present-day church of SS. Luca e Martina, when the area was finalized in 2 BCE. The analysis focuses on the geometric framework of the extant structures and the topography of the site and, in relation to them, a new interpretation of the historical records and the archaeological data. It demonstrates that Augustus and his architects authored the urban composition and architectural forms of the entire area, including making significant changes to the original design of the Forum of Caesar. The article also shows that the site of the church of SS. Luca e Martina was the location of the Augustan Chalcidicum, which served as the monumental entrance to the first two Imperial fora.