Additional Assumptions Research Articles

Maximum principles for elliptic operators in unbounded Riemannian domains

The necessity of a Maximum Principle arises naturally when one is interested in the study of qualitative properties of solutions to partial differential equations. In general, to ensure the validity of these kinds of principles one has to consider some additional assumptions on the ambient manifold or on the differential operator. The present work aims to address, using both of these approaches, the problem of proving Maximum Principles for second order, elliptic operators acting on unbounded Riemannian domains under Dirichlet boundary conditions. Hence there is a natural division of this article in two distinct and standalone sections.

The Born rule: Axiom or result?

The Born rule is part of the collapse axioms in the standard version of quantum theory, as presented by most textbooks on the subject. We show here that its signature quadratic dependence on the initial wavefunction's projection onto the measured outcome state follows from a single additional assumption beyond the other axioms. We give two examples of such an assumption, with a separate derivation for each, and we discuss their relationship with existing derivations. Our presentation is suitable for advanced undergraduates or graduate students who have taken a standard course in quantum theory. It does not depend on any particular interpretation of the theory.

Block‐iterative schemes for the split common fixed point problem and applications

In this paper, based on the extrapolated Landweber‐type operators, we present new strongly convergent block‐iterative schemes for solving the split common fixed point problem (SCFPP) with demiclosed strongly quasi‐nonexpansive operators on Hilbert spaces. The strong convergence is proved without the additional assumptions such as the boundedly regular condition and the closedness property of the range of the transformation operator, assumed recently in the literature for the problem. A necessary and sufficient condition that ensures that a th iterate is a solution is given. An application of our results to solve the multiple‐sets split convex feasibility problem (MSSCFP) is showed with computational experiments for illustration.

Analytical models of the electric field and carrier transit time in the base of bipolar junction transistors

The conventional theory of the bipolar junction transistor (BJT) does not clarify the electric field and its effect on the carrier transit in the uniformly doped base region, although the electric field is considered to be a second-order effect at high-level injection. In this study, analytical models of the carrier distribution, electric field, and carrier transit time in the base region are developed without making additional assumptions. It is found that the electric field may change direction, from forward to reverse, as long as the injection level reaches a criterion, affecting the carrier transport behavior. The models are based on the renovated p–n junction theory, provide insight into the carrier transit and are able to cover all levels of injection without the need to handle the low- and high-level injection separately. These models hold significance in retrieving the theoretical integrity of bipolar junction devices, and can be useful for further investigation in structure design, capacitance control, and AC performance optimization of the p–n junction-based device.

An application of the graph approach to life-cycle optimisation of vehicle electrification

Abstract Although durable goods with low energy consumption are being promoted to achieve a decarbonised society, from the perspective of life-cycle assessment, the choice of new durable goods may increase CO2 emissions. To address this problem, research has been conducted on product replacement based on life-cycle optimisation (LCO), a method for identifying a replacement life span that minimises life-cycle CO2 emissions. However, several additional assumptions complicate the analysis of replacement patterns of products and conditional formulas because cumulative emissions do not increase linearly when considering energy mix and technology improvement, and it is difficult to extend the model to optimisation methods in previous LCO studies. This study developed a new LCO approach by applying the shortest path problem to graph theory. Our methodology can contribute to the following: (i) it is computationally inexpensive; (ii) it is intuitively easy to add complex conditions, such as various policy scenarios and parameter changes; and (iii) once the graph of replacement patterns is defined, the optimal solution can be derived using existing solution methods, such as the Dijkstra algorithm. As a case study, we focused on vehicle replacement, which is a major source of CO2 emissions and is being electrified. In particular, we identified vehicle switching paths that minimise life-cycle CO2 emissions by considering changes in Japan’s energy mix and alternative fuel vehicle (AFV) characteristics. We determined that the optimal vehicle replacement path method to reduce CO2 emissions is to switch first to plug-in hybrid electric vehicles (PHEVs) and then to battery electric vehicles (BEVs). Thus, we suggest that the transition to electric vehicles requires a step-by-step process. This methodology is not only conducive to AFV deployment for decarbonisation but can also be applied to other products, such as air conditioners and lighting. Thus, various transition policies could be formulated using our methodology.

Remarks on the existence of minimal models of log canonical generalized pairs

Given an NQC log canonical generalized pair (X,B+M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(X,B+M)$$\\end{document} whose underlying variety X is not necessarily Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}$$\\end{document}-factorial, we show that one may run a (KX+B+M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(K_X+B+M)$$\\end{document}-MMP with scaling of an ample divisor which terminates, provided that (X,B+M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(X,B+M)$$\\end{document} has a minimal model in a weaker sense or that KX+B+M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_X+B+M$$\\end{document} is not pseudo-effective. We also prove the existence of minimal models of pseudo-effective NQC log canonical generalized pairs under various additional assumptions, for instance when the boundary contains an ample divisor.

Gradient Flow of the Sinai–Ruelle–Bowen Entropy

Motivated by an extension to Gallavotti–Cohen Chaotic Hypothesis, we study local and global existence of a gradient flow of the Sinai–Ruelle–Bowen entropy functional in the space of transitive Anosov maps. For the space of expanding maps on the unit circle, we equip it with a Hilbert manifold structure using a Sobolev norm in the tangent space of the manifold. Under the additional measure-preserving assumption and a slightly modified metric, we show that the gradient flow exists globally and every trajectory of the flow converges to a unique limiting map where the SRB entropy attains the maximal value. In a simple case, we obtain an explicit formula for the flow’s ordinary differential equation representation. This gradient flow has close connection to a nonlinear partial differential equation, a gradient-dependent diffusion equation.

Bilinear optimal control of a one‐dimensional degenerate parabolic equation with a nonlocal term

In this paper, a one‐dimensional degenerate parabolic equation involving a multiplicative control with a nonlocal space term represented by an integral kernel is considered. After some well‐posedness results, we derive several properties of the control‐to‐state mapping. We establish the existence of at least one optimal control and derive the first and the second conditions for optimality, giving the local uniqueness of an optimal control. Finally, we prove that we may expect the global uniqueness of the optimal control under additional assumptions.

Testing the postulates of quantum mechanics with coherent states of light and homodyne detection

Abstract Quantum mechanics has withstood every experimental test thus far. However, it relies on ad-hoc postulates which require experimental verification. Over the past decade there has been a great deal of research testing these postulates, with numerous tests of Born’s rule for determining probabilities and the complex nature of the Hilbert space being carried out. Although these tests are yet to reveal any significant deviation from textbook quantum theory, it remains important to conduct such tests in different configurations and using different quantum states. Here we perform the first such test using coherent states of light in a three-arm interferometer combined with homodyne detection. Our proposed configuration requires additional assumptions, but importantly allows us to use quantum states which exist in a larger Hilbert space compared to previous tests. For testing Born’s rule, we find that the third order interference is bounded to be κ = 0.002 ± 0.004 and for testing whether quantum mechanics is complex or not we find a Peres parameter of F = 1.0000 ± 0.0003 (F = 1 corresponds to the expected complex quantum mechanics). We also design and implement a test of Glauber’s theory of optical coherence.

Breast density quantification in dual-energy mammography using virtual anthropomorphic phantoms.

Breast density is a significant risk factor for breast cancer and can impact the sensitivity of screening mammography. Area-based breast density measurements may not provide an accurate representation of the tissue distribution, therefore volumetric breast density (VBD) measurements are preferred. Dual-energy mammography enables volumetric measurements without additional assumptions about breast shape. In this work we evaluated the performance of a dual-energy decomposition technique for determining VBD by applying it to virtual anthropomorphic phantoms. The dual-energy decomposition formalism was used to quantify VBD on simulated dual-energy images of anthropomorphic virtual phantoms with known tissue distributions. We simulated 150 phantoms with volumes ranging from 50 to 709mL and VBD ranging from 15% to 60%. Using these results, we validated a correction for the presence of skin and assessed the method's intrinsic bias and variability. As a proof of concept, the method was applied to 14 sets of clinical dual-energy images, and the resulting breast densities were compared to magnetic resonance imaging (MRI) measurements. Virtual phantom VBD measurements exhibited a strong correlation (Pearson's ) with nominal values. The proposed skin correction eliminated the variability due to breast size and reduced the bias in VBD to a constant value of -2%. Disagreement between clinical VBD measurements using MRI and dual-energy mammography was under 10%, and the difference in the distributions was statistically non-significant. VBD measurements in both modalities had a moderate correlation (Spearman's =0.68). Our results in virtual phantoms indicate that the material decomposition method can produce accurate VBD measurements if the presence of a third material (skin) is considered. The results from our proof of concept showed agreement between MRI and dual-energy mammography VBD. Assessment of VBD using dual-energy images could provide complementary information in dual-energy mammography and tomosynthesis examinations.

A novel application of data-consistent inversion to overcome spurious inference in genome-wide association studies.

The genome-wide association studies (GWAS) typically use linear or logistic regression models to identify associations between phenotypes (traits) and genotypes (genetic variants) of interest. However, the use of regression with the additive assumption has potential limitations. First, the normality assumption of residuals is the one that is rarely seen in practice, and deviation from normality increases the Type-I error rate. Second, building a model based on such an assumption ignores genetic structures, like, dominant, recessive, and protective-risk cases. Ignoring genetic variants may result in spurious conclusions about the associations between a variant and a trait. We propose an assumption-free model built upon data-consistent inversion (DCI), which is a recently developed measure-theoretic framework utilized for uncertainty quantification. This proposed DCI-derived model builds a nonparametric distribution on model inputs that propagates to the distribution of observed data without the required normality assumption of residuals in the regression model. This characteristic enables the proposed DCI-derived model to cover all genetic variants without emphasizing on additivity of the classic-GWAS model. Simulations and a replication GWAS with data from the COPDGene demonstrate the ability of this model to control the Type-I error rate at least as well as the classic-GWAS (additive linear model) approach while having similar or greater power to discover variants in different genetic modes of transmission.

Reliability-based design optimization for a vertical-type breakwater with multiple limit-state equations under Korean marine environments varying from sea to sea

In this study, as part of basic research aimed at enhancing the accuracy of load and resistance coefficients to ensure their suitability for practical design and promoting the application of underutilized reliability-based design in Korea, the author conducts optimal design based on the reliability analysis of a vertical-type breakwater in the seas off of Haeundae, Yeosu, Mokpo, Gunsan, and Incheon—representative ports in Korea. In doing so, the author utilized the double-loop approach, which simultaneously addresses a reliability problem nested within an optimization process, employing the Polak–He optimization algorithm. To mitigate the substantial numerical effort required by the double-loop approach based on the Polak–He optimization algorithm, which necessitates the gradients of both cost and constraint functions, the subset simulation method was employed. In this process, the author deliberately refrained from using design waves of a specific return period and linear probabilistic models such as the Gaussian distribution, especially concerning wave and lift forces, often viewed as barriers to the widespread application of reliability-based design in Korea. Instead, the author focused on characterizing the uncertainties associated with the wave force, lift force, and overturning moment—variables that significantly impact the integrity of vertical-type breakwaters—by developing probabilistic models for these random variables directly from long-term in situ wave data. These models capture the varied characteristics of the Korean marine environment from sea to sea. In this way, the need for additional assumptions concerning the interrelationship between significant wave and maximum wave heights, along with the wave period, can be eliminated. Following Occam's razor principle, which suggests that explanations constructed with the smallest possible set of assumptions are superior, the reliability-based design optimization of a vertical-type breakwater presented in this study demonstrates promise in terms of simplicity and practicality. The limit state of the vertical-type breakwater was defined to encompass sliding, overturning, and collapse failures, and the strong interrelations between the wave force, lift force, and overturning moment were described using the Nataf joint distribution. As anticipated, simulation results show that solely considering sliding failure, as in the current reliability-based design platform in Korea, leads to an underestimated failure probability. Furthermore, ensuring a consistent failure probability for vertical-type breakwaters using design waves with a specific return period, as in past studies, is not feasible. In contrast, this study demonstrated that breakwaters optimally designed to meet the reliability index requirement of β = 3.5–4 consistently maintain a target failure probability in all sea areas.

Computationally Secure Semi‐Quantum All‐Or‐Nothing Oblivious Transfer from Dihedral Coset States

AbstractThe quest for perfect quantum oblivious transfer (QOT) with information‐theoretic security remains a challenge, necessitating the exploration of computationally secure QOT as a viable alternative. Unlike the unconditionally secure quantum key distribution (QKD), the computationally secure QOT relies on specific quantum‐safe computational hardness assumptions, such as the post‐quantum hardness of learning with errors (LWE) problem and quantum‐hard one‐way functions. This raises an intriguing question: Are there additional efficient quantum hardness assumptions that are suitable for QOT? In this work, leveraging the dihedral coset state derived from the dihedral coset problem (DCP), a basic variant of OT, known as the all‐or‐nothing OT, is studied in the semi‐quantum setting. Specifically, the DCP originates from the dihedral hidden subgroup problem (DHSP), conjectured to be challenging for any quantum polynomial‐time algorithms. First, a computationally secure quantum protocol is presented for all‐or‐nothing OT, which is then simplified into a semi‐quantum OT protocol with minimal quantumness, where the interaction needs merely classical communication. To efficiently instantiate the dihedral coset state, a powerful cryptographic tool called the LWE‐based noisy trapdoor claw‐free functions (NTCFs) is used. The construction requires only a three‐message interaction and ensures perfect statistical privacy for the receiver and computational privacy for the sender.

Primary superconductivity in regular rotating electrically charged compact objects

In this paper, we refer to the early hypotheses on superconductivity in the models of the electron as an extended particle, and show how Nonlinear Electrodynamics minimally coupled to Gravity (NED-GR) predicts, without additional assumptions, the existence of primary superconductivity as the universal property of regular electrically charged rotating black holes and electromagnetic spinning solitons, replacing naked singularities and including that with the parameters of the electron. NED-GR dynamical equations describe basic generic properties of these objects in a self-consistent and the model-independent way, and establish the existence of the superconducting ring current in their deep interiors, which presents the non-dissipative source of the electromagnetic field of an object and provides its practically unlimited life time.

First-time Sensitive Homeomorphisms

AbstractWe introduce first-time sensitivity for a homeomorphism of a compact metric space, that is a condition on the first increasing times of open balls of the space. Continuum-wise expansive homeomorphisms, the shift map on the Hilbert cube, and also some partially hyperbolic diffeomorphisms satisfy this condition. We prove the existence of local unstable continua satisfying similar properties with the local unstable continua of continuum-wise expansive homeomorphisms, but assuming first-time sensitivity. As a consequence we prove that first-time sensitivity (with some additional technical assumptions) implies positive topological entropy.

Two innovative equivalent statements of the third law of thermodynamics

Abstract It is found from textbooks and literature that there are three different statements for the third law of thermodynamics, i.e., the Nernst theorem, unattainability statement of absolute zero temperature, and heat capacity statement. It is pointed out that such three statements correspond to three thermodynamic parameters, which are, respectively, the entropy, temperature, and heat capacity, and can be obtained by extrapolating the experimental results of different parameters at ultra-low temperatures to absolute zero. It is expounded that because there's no need for additional assumptions in the derivation of the Nernst equation, the Nernst theorem should be renamed as the Nernst statement. Moreover, it is proved that both the Nernst statement and the heat capacity statement are mutually deducible and equivalent, while the unattainability of absolute zero temperature is only a corollary of the Nernst statement or the heat capacity statement so that it is unsuitably referred to as one statement of the third law of thermodynamics. The conclusion is that the Nernst statement and the heat capacity statement are two equivalent statements of the third law of thermodynamics.

Why is the Cobb-Douglas production function so popular?

It is well known that, in continuous time, the Cobb-Douglas function can be derived from the underlying, data governing, accounting identity under some reasonable assumptions (factor shares are constant, and the weighted growth of the labour input price and the capital input price is constant). In this article these results are generalized in three ways: (1) the accounting identity contains a (pure) profit term; (2) continuous time is replaced by discrete time periods; (3) additional assumptions appear to be superfluous. The article also discusses extensions: from two to multiple inputs, from value added to gross output, and from a single production unit to an ensemble of those units.

Estimation of the mass density of biological matter from refractive index measurements

The quantification of physical properties of biological matter gives rise to novel ways of understanding functional mechanisms. One of the basic biophysical properties is the mass density (MD). It affects the dynamics in sub-cellular compartments and plays a major role in defining the opto-acoustical properties of cells and tissues. As such, the MD can be connected to the refractive index (RI) via the well known Lorentz-Lorenz relation, which takes into account the polarizability of matter. However, computing the MD based on RI measurements poses a challenge, as it requires detailed knowledge of the biochemical composition of the sample. Here we propose a methodology on how to account for assumptions about the biochemical composition of the sample and respective RI measurements. To this aim, we employ the Biot mixing rule of RIs alongside the assumption of volume additivity to find an approximate relation of MD and RI. We use Monte-Carlo simulations and Gaussian propagation of uncertainty to obtain approximate analytical solutions for the respective uncertainties of MD and RI. We validate this approach by applying it to a set of well-characterized complex mixtures given by bovine milk and intralipid emulsion and employ it to estimate the MD of living zebrafish (Danio rerio) larvae trunk tissue. Our results illustrate the importance of implementing this methodology not only for MD estimations but for many other related biophysical problems, such as mechanical measurements using Brillouin microscopy and transient optical coherence elastography.

On the coupling of Hamilton's principle and thermodynamic extremal principles

Extremal principles can generally be divided into two rather distinct classes. There are, on the one hand side, formulations based on the Lagrangian or Hamiltonian mechanics, respectively, dealing with time dependent problems, but essentially resting on conservation of energy and thus being not applicable to dissipative systems in a consistent way. On the other hand, there are formulations based essentially on maximizing the dissipation, working efficiently for the description of dissipative systems, but being not suitable for including inertia effects. Many attempts can be found in the literature to overcome this split into incompatible principles. However, essentially all of them possess an unnatural appearance. In this work, we suggest a solution to this dilemma resting on an additional assumption based on the thermodynamic driving forces involved. Applications to a simple dissipative structure and a material with varying mass demonstrate the capability of the proposed approach.

Infinite horizon LQG Graphon Mean Field Games: Explicit Nash values and local minima

In this study, we generalize the analysis of infinite horizon linear quadratic Gaussian (LQG) Mean Field Games within the framework of Graphon Mean Field Games (GMFG) introduced in Caines and Huang (2018) over finite horizons. Graphon Mean Field Games (GMFGs) are non-uniform generalizations of Mean Field Games where the non-uniformity of agents is characterized by the nodes on which they are located in a network. Under mild assumptions on the structure of the network and parameters of the game, we obtain for almost every node, an explicit analytical expression for the Nash values (i.e. the cost at equilibrium). With additional assumptions, we provide sufficient conditions for nodes to have locally minimal Nash values. We illustrate the results for the uniform attachment network.