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We describe the **offset logarithm function** and illustrate its applications to physics. The offset log can be considered an extension of the logarithm function, and it can also be considered as a generalization of the Lambert W function.
The offset log function shows up in a variety of physical and biological models, such as the mean field model (Weiss model) of ferromagnetism, and models of carbon nanoribbons. It provides a convenient framework for describing phenomena in which the behaviour of a physical variable at one location or time, has an exponential or logarithmic relationship to the value of attribute at a displaced location or time.

The offset logarithm function of order $k$ is the multivalued function $w = L_k(z)$ of the complex variable $z$, which satisfies the equation
\begin{equation} \frac{ w-k }{w+k}e^w = z \qquad(1) \end{equation}
Each of $k$, $z$, and $w$ can be in general a complex number. However, for this, we consider $k$, $z$ and $w$ as real numbers. Approximate solutions to Equation (1) can be calculated numerically, but a study of the analytic properties of that equation can provide insight into the system being described. 

The offset log function is multi-branch. Each branch of $L_k(z)$ is an analytic function for certain regions of $ k$ and $z$ values. We present the main characteristics of the $L_k(z)$ function. For each domain of the $k$ parameter, the properties of $L_k$ in terms of the branches are given. We show that the number of solutions of Equation (1) depends upon the domains of $k$ and $z$. For these domains we give the number of solutions and we give the interval where each solution lies. We identify the cases where a solution is a multiple solution. We give a symbolic formula for the multiple solutions.

CAP Virtual Congress 2021

The offset logarithm function and some applications in physics: A generalization of the Lambert W function

offset logarithm function

generalized lambert function

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technical paper

Transitions between fractionalized and conventional quantum phases of matter in 2+1 dimensions are conceptually best understood within the framework of parton gauge theories, whereby the confinement of fractionalized excitations and spontaneous breaking of global symmetries in conventional phases is argued to result from the proliferation of gauge monopoles/instantons. To complement recent studies of the quantum numbers and scaling dimensions of monopole operators in parton gauge theories of frustrated quantum antiferromagnets, we provide an explicit semiclassical derivation of instanton contributions to the low-energy effective parton gauge theory for hardcore bosons on the honeycomb lattice using methods originally developed by 't Hooft in the solution of the U(1) problem in quantum chromodynamics. While the symmetry-breaking effect of instantons is typically associated with massless fermions and the Atiyah-Singer index theorem on compact manifolds, we show that the spontaneous breaking of the U(1) boson number conservation symmetry in the superfluid phase results here from Euclidean zero modes of massive Dirac fermions bound to instantons in noncompact spacetime.

Symmetry Breaking Effects of Instantons in Parton Gauge Theories

The creation and annihilation of particles is a fundamental feature of relativistic quantum fields. A famous example of this is provided by Schwinger’s 1951 prediction that the vacuum is unstable to particle-antiparticle production if a static electric field is applied to it. In this talk we examine the classical field limit of Schwinger pair production by mapping the Klein-Gordon equation with an electric field onto the non-relativistic Schrödinger equation with a 1/r^2 potential. This latter problem has two regimes depending on the depth of the potential: a sub-critical regime where PT symmetry is preserved and a supercritical regime where PT symmetry is broken and one finds ``fall-to-the-centre’’ where probability is absorbed at the origin. Schwinger pair production occurs in the latter regime and can be described in terms of non-Hermitian quantum mechanics.

Schwinger pair production as a non-Hermitian problem

More than 30 million individuals worldwide are living with Alzheimer’s Disease. To further the current understanding on this neurodegenerative disease, we developed a technique to create amyloid peptide clusters in synthetic, brain-like membranes, which mimic the senile plaques found in the brains of Alzheimer's patients. I compared the molecular functioning of homotaurine, a peptic anti-aggregant that binds to amyloid peptides directly, and curcumin, a non-peptic molecule that can inhibit aggregation by changing membrane properties. By using microscopy, x-ray diffraction, and UV-vis spectroscopy, we found that both curcumin and homotaurine significantly reduce the number of small, nanoscopic amyloid aggregates and the corresponding β- and cross-β sheet signals. This research shows that membrane active drugs can be as efficient as peptide targeting drugs in inhibiting amyloid aggregation in-vitro [1]. The findings can open new pathways for the developments of drugs to slow down first occurrence and progression of the disease.

[1] Xingyuan Zou, Sebastian Himbert, Janos Juhasz, Samantha Ros, Harald D. H. Stover, and Maikel C. Rheinstädter, “Curcumin and homotaurine suppress amyloid-b25-35 aggregation in synthetic brain membranes”, under review with ACS Chemical Neuroscience, Manuscript ID: cn-2021-00057r

Curcumin and homotaurine suppress amyloid-b25-35 aggregation in synthetic brain membranes

Positron Emission Tomography (PET) Imaging of the brain might become the most effective imaging technique to predict Alzheimer's disease. However, the definition of the brain in PET images is low and the lesion area is not easy to define, so the accuracy of traditional machine learning algorithms in predicting Alzheimer's disease from PET images is low. Deep learning algorithms can effectively improve the accuracy of prediction. Here, a deep learning model based on multi-attention mechanisms is constructed for the prediction of Alzheimer's disease. Firstly, in order to improve the images and reduce the loss of spatial information caused by convolution, a soft-attention mechanism was introduced to embed non-local modules and CBAM modules into the prediction model, which effectively solved the problems of the lack of detail information and the lack of connection between channels in PET images after deep convolution. Secondly, a split-attention mechanism was introduced, and the influence of different parameters of feature images with different sizes on the prediction accuracy was enhanced by using a packet network. Finally, head movement correction, image registration, craniocerebral separation, Gaussian smoothing and other image preprocessing were performed on the acquired PET images to effectively improve the image features of Alzheimer's disease focal areas in brain PET images. The experimental results showed that the prediction accuracy, sensitivity and specificity of the model in the ADNI database for Alzheimer's disease were 90.5%, 86.1% and 93.5% respectively, which could provide more accurate diagnostic results compared with the existing methods.

Prediction of Alzheimer's Disease Based on Multi-Attentional Mechanisms using Brain PET Image

The behaviour of confined polymeric systems has been the topic of much recent research. The information from these studies may help guide the development and optimization of nanofluidic devices and understand the segregation of DNA chromosomes in prokaryotic cells. Some recent experiments examined two DNA molecules confined to a box-like cavity with strong confinement in one dimension. The researchers looked at the effect of using an anisotropic cross-sectional box geometry and of having non-identical polymers. In this study, we use computer simulations to examine the effect of such anisotropies on the behaviour of the confined-polymer systems. We employ Metropolis Monte Carlo simulations where the polymers are modelled as bead spring chains. We calculate probability distributions for the centre-of-mass positions of either polymer and for the centre-of-mass displacement between the polymers. These distributions are used to examine the effect of anisotropies in polymer length for two polymers confined to a square box-like cavity and anisotropies in box dimensions for two identical polymers confined to a box-like cavity. The information obtained from this study can then be used to have a better understanding of the trends seen in the experiments.

Simulation of Two Polymers Confined to a Box-Like Cavity: The Effect of Anisotropies in Polymer Length and Confinement Dimensions

We elucidate the vector space (twisted relative cohomology) that is Poincar\'e dual to the vector space of Feynman integrals (twisted cohomology) in general dimension. These spaces are paired via an inner product called the intersection number -- an invariant which can be computed algebraically. In this language, reduction of an integrand modulo integration-by-part identities is simply its projection onto a chosen basis. The dual-forms turn out to be far simpler than their Feynman counterparts; they are uplifts of lower-dimensional forms supported on the maximal cuts of various sub-topologies (boundaries). The intersection numbers are then computed by taking residues around cuts where various subsets of propagators become on-shell, giving a systematic approach to generalized unitarity. Moreover, the dual integrals satisfy the transposed differential equation to their Feynman counterparts. Since the dual-forms are localized to cut the transposed differential equation is a much simpler to construct.

Generalized Unitarity and the Poincaré Duals of Feynman Integrals

Laser-induced rotational wavepacket of hydrogen molecules has been experimentally observed in real time by using two sequential 25-fs laser pulses (pump-probe scheme) and a COLTRIMs spectrometer. By measuring the time-dependent yield of the above-threshold dissociation and the enhanced ionization of the molecule, we observed a few-femtosecond time delay in between the two dissociation pathways for both H2 and D2. The delay was understood and interpreted by a classical model considering enhanced ionization requires extra interaction with the laser field. We demonstrate the molecule rotational clock in hydrogen molecule is a straightforward method for timing ultrafast molecular dissociation dynamics.

Clocking enhanced ionization of hydrogen molecule using molecular rotational wavepackets

Currently, plasmonic nanofibers doped with semiconductor quantum dots, organic dye quantum emitters (QEs), and metallic nanoparticles (MNPs) have attracted much attention due to their wide range of applications including waveguides, light-sources, and optical sensors. These nanofibers doped with QEs and MNPs have been fabricated using a variety of metals and emitters. For example, Hu et al. [1] have studied the fabrication of a plasmonic random fiber from gold MNPs and pyrromethene dye molecules (QEs) embedded in the liquid core optical fiber. They found that a narrower and sharper photoluminescence (PL) spectrum can be more easily obtained when there is greater overlap between the plasmonic resonance of the gold-MNP and the dye molecules. Here we have developed a theory of photoluminescence for plasmonic nanofibers [2]. When probe light propagates inside the nanofiber, it induces surface plasmon polariton (SPPs) and electric dipoles in metallic nanoparticles. These dipoles interact with each other via the dipole-dipole interaction (DDI) [3]. The energy of photonic bound states in the presence of the SPP and DDI fields is then calculated. We have demonstrated that the number of bound states can be controlled by changing the strength of the SPP and DDI couplings. The expression of photoluminescence has been calculated using the density matrix method in the presence of the DDI coupling. We found that the intensity of the PL spectrum depends on the quality called quantum efficiency, which depends on the radiative and non-radiative decay rates. We have found that the quantum efficiency is enhanced when the exciton energy is in resonance with the bound photon energy. Further, we predicted that the PL intensity is also enhanced due to the DDI coupling. The enhancement of the PL spectrum can be used to fabricate plasmonic nanosensors. 

[1] Hu, Z et al., Gold nanoparticle-based plasmonic random fiber laser. J. Opt. 2015, 17, 35001.
[2] Singh, M. R.; Brassem, G.; Yastrebov, S. G. Optical quantum yield in plasmonic
Nanowaveguide. Annalen der Physik in press, 2021.
[3] Singh, M. R. The effect of the dipole–dipole interaction in electromagnetically induced transparency in polaritonic band gap materials. Journal of Modern Optics 2007, 54, 1739.

Study of photoluminescence in plasmonic nanoparticles

Graphene is an interesting system for condensed matter physicists because of many potential technological applications. It is interesting to theorists as an abelian strongly coupled system that has some of the interesting properties of QCD.

We are particularly interested in studying the effects of anisotropic strain of the graphene
lattice on the critical coupling, which determines the transition point for graphene
from a conductor to an insulator. We use a low energy effective theory and a Schwinger-Dyson approach. The full set of coupled equations is numerically very difficult to solve, and so previous calculations have made many simplifying assumptions in order to make the procedure tractable. These simplifications have led to disagreement about the effect anisotropy has on the critical coupling constant. We have studied the effect of some common approximations including polarization tensor approximations and vertex ansaetze. In this talk we discuss the motivation for several of these approximations, and show that in some cases their effect on the critical coupling is significant. We discuss more accurate ways to solve the non-perturbative low energy effective theory, and its ability to predict the critical coupling of the phase transition.

Semi-metal insulator phase transitions in graphene - the role of anisotropy

Organic molecules capable of forming strongly bound self-assembled monolayers (SAMs) offer a promising route to surface modification and functionalization. Technological advances including the protection and stabilization of nanoparticles and the development of lab-on-chip sensors have been largely realized by the adsorption of alkanethiols. *N*-heterocyclic carbenes (NHCs) were recently introduced as alternative anchors in the preparation of SAMs on metal surfaces and may be superior to thiol-based systems. NHCs have important advantages compared to thiol-based systems, most notably their chemical tunability and ability to form strongly bound monolayers with improved stability. Given their novelty in the field of materials science, many fundamental properties of NHC-based SAMs are not well-understood, including factors that control the adsorption and self-assembly processes.

By means of scanning tunneling microscopy and further supplemented by *ab initio* (DFT) and Monte Carlo simulations, we investigate the mechanisms by which NHCs bind and self-assemble on Au(111). Our investigation demonstrates a critical dependence of the binding mode on NHC structure, surface coverage, and substrate temperature. The dependence on these factors arises due to the minimization of molecule-surface steric interactions, the production of Au adatoms by the effect of NHC adsorption on the underlying gold substrate, and the reactivity and stability of the ligand, respectively. The main effect on the adsorption process is a competition between upright attachment geometries and the formation of flat-lying bis-NHC-metal complexes. These species form a rich variety of self-assembled structures, including in particular a zig-zag lattice formed by adatom-bound NHCs and a non-porous chiral Kagome lattice containing trapped NHC monomers. These lattices are analyzed within our three-pronged methodology, providing valuable insights into molecule--molecule and molecule-substate interactions, as well as the role of surface modifications and on-surface chemical reactions in the formation of NHC SAMs. Efforts to understand these processes were crucial in advancing the chemistry and applications of thiol-based SAMs, and will similarly offer critical strategies for the design of upright NHC SAMs and their applications.

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The offset logarithm function and some applications in physics: A generalization of the Lambert W function

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