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The work presented here concerns the generation of high-fidelity databases dedicated to the improvement of wall-shear stress modeled LES and RANS for applications in aeronautics and turbomachinery. As a first step, instrumentation guidelines for both DNS and LES, including a standardized set of statistical fields, as well as procedures to ensure statistical convergence and estimate confidence intervals, are presented.

DGM maintains high accuracy on unstructured meshes and features strong computational efficiency at large scale. It is therefore a promising approach for generating high resolution DNS and LES on the complex geometry typical for turbomachinery. The specific flow phenomenon, that is the subject of this presentation, is that of flow separation at high Reynolds number and high angles of attack AoA. The availability of petascale computing resources, and simulation techniques, such as large eddy simulation LES of flows over complex geometry, open the possibility of using the results of wall resolved simulations to inform and calibrate lower-order sub-grid models used in RANS computations.

The current talk will present a benchmark study of two state-of-the-art parallel flow solvers, Argo and PHASTA, which are employed to perform high-resolution wall resolved LES, and detached eddy simulations, respectively, of separated flow over the NACA airfoil for a Reynolds number of 1.

David W. Swenson University of Amsterdam , Netherlands. They perform Monte Carlo simulations in the space of trajectories, focusing the simulation effort on the transition itself to avoid spending long waiting times in the stable states. Since they are Monte Carlo approaches, they can use multiple walkers, but some approaches also use replicas from different path ensembles.

In particular, replica exchange transition interface sampling RETIS involves simultaneously sampling trajectories from several path ensembles. However, even within a single ensemble, the lengths of the sampled trajectories can vary and are unpredictable. This makes load balancing an extremely challenging problem. The task-based approach enables parallelization that provides optimal use of computational resources, not only by load balancing, but also by allowing the allocated resources to be scaled up or down according to the needs of the simulation.

While the approach is described here in the context of path sampling, the same technique could be applied to many trajectory-based simulation methods. A reference-state Hamiltonian is simulated which envelopes the Hamiltonians of the end states. The challenge when using EDS is the determination of optimal parameters for the reference-state Hamiltonian. Previously, the choice of parameters for an EDS simulation with multiple end states was a non-trivial problem that limited the application of the methodology.

By exchanging configurations between replicas with different parameters for the reference-state Hamiltonian, major parts of the problem to choose optimal parameters are circumvented. Algorithms to estimate the energy offsets and optimize the replica distribution have been developed. Our approach was tested successfully using a system consisting of nine inhibitors of phenylethanolamine N-methyltransferase PNMT , which were studied previously with thermodynamic integration and EDS.

In contrast, if the dynamics is not stationary, it is not a priori clear which form the equation of motion for an averaged observable will have. We adapt Mori-Zwanzig formalism to derive the equation of motion for a non-equilibrium trajectory-averaged observable as well as for its non-stationary auto-correlation function. We also derive a fluctuation-dissipation-like relation which relates the memory kernel and the autocorrelation function of the fluctuating force.

In addition, we show how to relate the Taylor expansion of the memory kernel to experimental data, thus allowing to construct the equation of motion from direct measurements. The underlying free energy surfaces can be obtained from simulations using enhanced sampling methods. We present an algorithm to calculate free energies and rates from enhanced sampling simulations on biased potential energy surfaces.

Inputs are the accumulated times spent in each state or bin of a histogram, and the transition counts between them. Unbiased rate coefficients for transitions between states can then be estimated. DHAMed yields accurate free energies in cases where the common weighted-histogram analysis method WHAM for umbrella sampling fails because dynamics within the windows is slow.

We illustrate DHAMed with applications to proteins and RNAs and accurately estimate free energies from sets of short trajectories, providing a way forward for computational drug design. Our rate formalism can be used to construct Markov state models from biased simulations and we demonstrate its practical applicability by determining RNA folding kinetics from replica exchange molecular dynamics.

Sereina Z. Tanja Schilling University of Freiburg , Germany. Lukas S. Xavier Lapillonne MeteoSwiss , Switzerland. Numerical weather prediction and climate models are large and complex software applications that need to run efficiently on today's and future massively parallel computer systems. The rapid change in these computing architectures and the increase in diversity are seriously affecting the ability to retain a single source code that runs efficiently in different architectures.

However porting existing large community codes to multiple architectures is a daunting task and leads to codes that are more complex and difficult to maintain. As a result in the past years numerous new technologies and approaches are emerging in order to provide new programming models, like domain-specific languages DSLs or source-to-source translation tools that can increase the productivity of development in weather codes while providing a high degree of performance portability.

In this minisymposium we propose a discussion with some of the most prominent novel approaches where the new advances in programming models used for heterogeneous architectures in weather and climate models will be presented. Lin Gan Tsinghua University , China. Performance portability for atmosphere codes is no doubt a big challenge, so great efforts have to be made and patience is required as well. In addition to some experiences and lessons, we also take this opportunity to discuss on the novel Sunway processors.

For Sunway system, different software is being developed to make it easy for applications to be ported. HOMME simulates the dynamics and physical processes of the atmosphere. It is the most computationally demanding part of E3SM. Kokkos provides performance-portable multidimensional arrays and intraprocess parallel execution constructs.

These form an abstraction layer over the hardware architecture of a compute node within a supercomputer. However, the scale of the problems to be simulated calls for the highest levels of computational performance. Achieving good performance when both computer architectures and the underlying code base are constantly evolving is a complex challenge. In recent years, the use of Domain-Specific Languages DSLs as a potential solution to this problem has begun to be investigated.

In this talk we will describe this work and the functionality of the domain-specific compiler, PSyclone, which has been developed to process the serial code written by the natural scientists and generate the code required to run on massively parallel machines. Yet achieving this will pose serious computational challenges for large scientific codes that are developed using traditional programming models such as OpenMP and MPI. In order to adapt models to run efficiently on modern computing architectures and accelerators, numerous domain specific languages DSL and libraries that abstract architecture dependent optimizations have been proposed, like the GridTools libraries used operationally for running COSMO on GPUs.

Yet these tools are specific to a domain or model, and have little reuse among them of architecture specific optimizers which leads to high maintenance costs. We present a novel programming model based on the GridTools ecosystem of libraries, a toolchain that allows to develop and interoperate various DSL frontends by providing domain and architecture specific optimizers. It aims at standardizing tools for performance portability by proposing a standard intermediate representation for weather and climate codes.

MS11 - Computing the Effect of Risk. Montreal Room. Michel Juillard Banque de France , France. Many important economic phenomena relate to the notion of risk. Economic actors not only make decisions as a function of their current situation but also depending on their expectation of future developments.

Because economic systems are not deterministic, future economic events are usually treated as stochastic phenomena. The general form of the problem at hand is to determine how the probabilistic distribution of future economic events influences current decisions. The wider the distribution, the more risk in today's decisions. The papers in this session present different computation challenges involved in attempting to describe the effect of risk on economic decisions.

Elisabeth Proehl University of Geneva , Switzerland. This is a nontrivial task as the cross-sectional distribution of endogenous variables becomes an element of the state space due to aggregate risk. Existing global solution methods have assumed bounded rationality in terms of a parametric law of motion of aggregate variables in order to reduce dimensionality. Dimensionality is tackled by polynomial chaos expansions, a projection technique for square-integrable random variables, resulting in a nonparametric law of motion.

To illustrate the method, I compute the Aiyagari-Bewley growth model and the Huggett model with aggregate risk. Furthermore, more risk sharing in form of redistribution can lead to higher systemic risk. In the latter model, I find that prices increase with more stringent selling constraints, but are also more negatively skewed. Using the neoclassical growth model and a New Keynesian model, we show that extended perturbation achieves higher accuracy than standard perturbation when using third order approximations.

We also show that extended perturbation generates stable approximations even when standard perturbation explodes. This paper also adds to the literature on downward nominal wage rigidities in the New Keynesian model, by showing that this friction only plays a significant role when using standard perturbation but not when using the more accurate extended perturbation approximation. Improved Time Iterations.

Contrary to the original improvements, it generalizes to models specified by equilibrium conditions in which case it is equivalent to the Newton-Raphson algorithm applied to one big nonlinear system of equations, without requiring the explicit inversion of the memory-hungry Jacobian. In particular, convergence is quadratic, i. Convergence of each gradient improvement step requires the local contractivity of the time-iterations operator.

We show how this property relates to eigenvalues coming from local perturbation analysis, and how to estimate the local spectral radius of this operator close to a solution candidate. Gradient improvements can be implemented easily, essentially by composing the same elements as time-iterations. Our timing comparisons still suggest it performs much faster, especially when the number of dimensions or the number of grid points increase.

Rational agents can make decisions today so as to maximize expected benefits or minimize expected loss. These behaviors are related to economic concepts such as precautionary saving, asset prices, risk premium, term premium.

By contrast, linear models are characterized by certainty equivalence, and in such environments, agents are indifferent to future uncertainty. One of the major benefits of using higher order approximation of a certain class of economic models, is the ability to analyse attitude towards risk. Computing higher-order approximation of DSGE models involves several computational challenges.

Derivatives of the original model must be evaluated. These high dimensional objects must be stored in a convenient manner. Above second order, computations involve tensor algebra. A key component is a fast implementation of the Faa Di Bruno formula for the derivatives of the composition of two functions.

They represent challenging tasks for a rather new programming language such as Julia and an interesting test case. The Extended Perturbation Method. Martin M. Andreasen Aarhus University , Denmark. Back in Time. The dawn of Web 2. This minisymposium will showcase all these tools and how they are used in real scientific workflows.

The shown best practices are easy to reproduce since they are all based on freely available software packages, so that the audience can use the presentations both as an inspiration but also as a kickstart for their own better science. Lately, applications are being offered as web-based user interfaces regardless of the actual location of the computation.

To this end, the cloud computing revolution has had a wonderful side effect that everybody can now easily accept that certain tasks are transparently performed elsewhere - this talk will give an example from the bioinformatics application domain showing how cloud resources can be used for DNA sequencing. This enables both new groups to use HPC systems but also provides users a more error-proof and efficient way of using installed applications.

This talk will highlight the criticality of an application-as-a service mode and will also discuss how Docker containers and Jupyter notebooks can be used for this easy-to-use application-as-a-service notion. Those characteristics actually match rather well onto the concept of agile programming.

One is the NSF Jetstream project, the first NSF funded cloud designed for those who have not previously used high performance computing resources. Jetstream provides users with long running virtual machines with a customizable software stack to meet the needs of non-traditional HPC applications. The other environment is a research desktop solution that is making high performance Linux desktops available remotely. The desktops contain all the normal HPC command line tools and allow for direct job submission to the HPC machines, but also provide access to interactive applications like Matlab, Comsol Multiphysics, R-Studio and Jupyter.

The goal of both projects is to lower the barrier of entry and broaden adoption of traditional HPC and high-throughput computing environments. The talk will provide an architectural overview, use cases and experiences for operating such environments. Users can leverage a wide variety of libraries and applications without knowing how to build from source. On HPC systems users typically build from source, and software, is notoriously complex.

Building even a moderately sized parallel simulation code can be a major effort. Scientists who use applications codes must typically also know how to build them from scratch, along with tens or hundreds of dependency libraries. Spack is an open source package manager that handles the complexity of HPC environments and allows scientists to automatically and reproducibly install complex software stacks.

It allows users to experiment with different compilers, optimizations, build options, and dependency versions, without in-depth build knowledge. Spack is built to handle the complexities of the HPC environment that seldom arise on commodity systems, such as swapping compilers and ABI-incompatible dependencies, cross-compilation, compiler runtime libraries, and optimized binaries.

Spack has a rapidly growing community, with over contributors at organizations worldwide. In this talk, we will introduce Spack, show how it can make scientists more productive, and give an overview of ongoing Spack projects and its development road map. Most of the thousands of particles emitted at each bunch crossing are measured and collected with building-sized detectors consisting of multiple sub-detectors each serving its own purpose. The simulation of the signal created by a particle interacting with such a detector is typically done with very detailed simulations and needs to be stepped infinitesimally over meters of material.

This simulation is as much computing intensive as the geometry is complex. In current and future detector design, the fine-grained simulation of such a detector is taking a great part of the full computing budget of experiments and poses a computing challenge.

While a great deal of effort is being made to parallelise such software, one possible avenue to reduce the computational requirements is with generative models from the field of deep learning. Such generative models have seen success in conditionally generating images and video of various types. We present how such models are built and trained, and how well they can capture the physics of particle interaction and help generate realistic samples for high energy physics analysis.

The Success of Deep Generative Models. Jakub Tomczak University of Amsterdam , Netherlands. The principle of GANs is to train a generator that can generate examples from random noise, in adversary of a discriminative model that is forced to confuse true samples from generated ones. Generated images by GANs are very sharp and detailed. The biggest disadvantage of GANs is that they are trained through solving a minimax optimization problem that causes significant learning instability issues.

VAEs are based on a fully probabilistic perspective of the variational inference. The learning problem aims at maximizing the variational lower bound for a given family of variational posteriors. The model can be trained by backpropagation but it was noticed that the resulting generated images are rather blurry. However, VAEs are probabilistic models, thus, they could be incorporated in almost any probabilistic framework. We will discuss basics of both approaches and present recent extensions.

Some of most promising applications of deep generative models will be shown. With the LHC entering its high-luminosity phase in , the projected computing resources will not be able to sustain the demand for simulated events. Generative models are already investigated as the mean to speed up the centralized simulation process. Here we propose to investigate a different strategy: training deep networks to generate small-dimension ntuples of numbers physics quantities such as reconstructed particle energy and direction , learning the distribution of these quantities from a sample of simulated data.

Currently used Monte Carlo techniques allow to do it accurately, but their precision often comes at the expense of relatively high computational cost. The new method we propose, dubbed ParticleGAN for simplicity, leverages recently developed Generative Adversarial Networks to learn the trajectories of particle tracks after collision. This applies also to other evaluated generative models namely Variational Autoencoders and variants of GANs.

In this work we outline current bottlenecks of the proposed approach and discuss further steps that can allow to deploy the proposed generative models for simulation in production. The need for simulated events, expected in future High Luminosity LHC experiments is increasing dramatically and requires new fast simulation solutions.

Two common aspects characterize many of these applications: the representation of input data as regular arrays of numerical values and the use of raw data as the input information to feed the network.

Next generation HEP experiments are expected to be more and more characterized by detector components that could comply to this paradigm. We will introduce the first application of three-dimensional convolutional Generative Adversarial Networks and of Variational Auto Econders to the simulation of highly granular calorimeters.

Tobias Golling University of Geneva , Switzerland. In the past couple years a large number of fintech and more recently insurtech startups have been founded and are challenging established players in the financial services sector. At Baloise — a Swiss company providing insurance services in Switzerland, Belgium, Germany and Luxembourg as well as banking services in Switzerland — we view startups as potential partners on our digital transformation journey rather than competition.

This minisymposium aims to demonstrate what problems companies such as Baloise face in terms of digitizing their business and making use of their large amounts of data. To do so, the four sessions will cover the innovation framework Baloise employs in order to rapidly test prototypes, a presentation by Brainalyzed, a startup which aims to optimize and automatize investment decisions at Baloise using AI, a presentation about the challenges arising in the context of data warehouses and legacy systems, and a panel discussion with all the speakers.

Open Innovation at Baloise. Baloise has developed an open innovation framework with the goal of enabling easy and fast cooperation with startups and other external partners as well as intrapreneurs. In this session we will present this open innovation framework and its evolution over time as we have tailored it to the requirements of startups.

The challenge is not only to manage this data, but to make it usable. Therefore, data analysis becomes a key success factor for organizations. Especially in the financial sector, data-driven applications are necessary to keep up with the fast-moving financial market and growing competition.

The answer to low interest rates and high volatility in the market are automated data-driven investment processes. Data analysis using artificial intelligence AI is therefore becoming increasingly important. In this session we will give insights and some practical examples how we worked together with Baloise Asset Management to use some of their data to enhance the investment management process.

We will show how the scalability of the learning solution helps to analyze even very complex problems in a short time and what our vision of AI in the financial world looks like. Therefore a greenfield approach in terms of big data is out of question and the integration of data originating from these systems represents a costly and time-consuming challenge for Baloise.

Securing the availability of internal data on one side and meeting the fast growing business requirements in connection with external big data integration on the other side is the balancing act of our digital transformation in the domain of business intelligence. How Baloise tackles these challenges and how the company benefits from cooperation with startups using artificial intelligence to boost this transformation will be explained in this session of the minisymposium.

The panelists are Dr. Artificial Intelligence for Automated Investment Management. Gunter Fischer Brainalyzed , Germany. Klaus Rieger Baloise Group , Switzerland. Roland Lindh Uppsala University , Sweden. Machine Learning is right now a booming field of computer science which finds applications in the development of computer-human interfaces, in the analysis of medical data of huge populations, in the maintenance of cars, planes and elevators, and self-driving cars, to mention a few. During the last twenty years the field has gone through a development and refinement which has been spectacular.

For some reason the use of the technology in pure science has been lagging behind; however, we are now starting to see the use of machine learning in the field of quantum chemistry. Here, the approach will enhance the performance of standard quantum chemical calculations — improving convergence, could serve as a tool for post-analysis of huge sets of ab initio results, or could simply replace computationally expensive procedures. Machine learning offers practical alternatives where standard quantum chemical simulations would be prohibitive.

During the last few years a small number of quantum chemistry groups have explored the potential of machine learning — the results have been extraordinary and spectacular. Here in this minisymposium we would like to inspire by presenting four different applications in which machine learning is fundamental to success. Anders S. Christensen University of Basel , Switzerland. Alas, even when using high-performance computers, brute force high-throughput screening of compounds is beyond any capacity for all but the simplest systems and properties due to the combinatorial nature of chemical space, i.

Consequently, efficient exploration algorithms need to exploit all implicit redundancies present in chemical space. I will discuss recently developed statistical learning approaches for interpolating quantum mechanical observables in compositional and constitutional space.

Results for our models indicate remarkable performance in terms of accuracy, speed, universality, and size scalability. The process of data generation to train such ML potentials is a task neither well understood nor researched in detail. In this talk, we will present a fully transferable deep learning potential that is applicable to complex and diverse molecular systems well beyond the training dataset.

To train ANI potential we use fully automated approach for the generation of datasets. We show the use of our proposed AL technique develops a universal ANI potential, which provides very accurate energy and force predictions on the entire COMP6 benchmark. This universal potential achieves a level of accuracy on par with the best ML potentials for single molecule or materials while remaining applicable to the general class of organic molecules comprised of the elements CHNOSFCl.

Combined, these reactive pathways represent a test set of molecular geometries whose energies and forces we then calculate at higher levels e. The resultant PES provides coupled cluster quality energies at the cost of classical force fields, enabling us to run thousands of trajectories and thereby make comparisons with experimental dynamical observables in non-equilibrium regimes. I will illustrate this coupled virtual-reality-machine-learning workflow by focusing on recent applications where we have been studying heterogeneous reaction dynamics wherein cyano radicals CN undergo reactive scattering at the surfaces of liquids which are composed of long chain hydrocarbons.

These are mostly metallic and non-magnetic. Neural Networks Learning Quantum Chemistry. Miguel A. Sharlee Climer University of Missouri - St. Louis , United States of America. Difficult combinatorial problems permeate virtually every area of the sciences, business, and government and many of these problems can be cast as mixed-integer programs MIPs.

A MIP is a mathematical definition of a problem that is comprised of a set of constraints and an objective function. However, search strategies, such as branch-and-bound, branch-and-cut, and cut-and-solve have evolved to provide optimal solutions for many instances. Although there has been great progress in this field and computational power has dramatically increased over the years, many important MIPs remain intractable and the use of massive parallelization appears to be a promising means to address this great need.

However, many challenges lie ahead. This minisymposium will elucidate some of these challenges, while highlighting progress in this field. The goal of the discussions will be to explore and integrate high-performance expertise with domain-specific insights with an aim to identify strategies that may resolve these pressing challenges.

Initially, only very small cases could be solved exactly. With the advent of greater computational power, the boundaries and limitations have continued to increase. However, continued progress is needed to solve many current day problems of interest e. The large amounts of distributed computational power becoming more readily available may provide a solution for tackling previously insolvable instances. Here we present a potential method for parallelizing MIPs on a large scale using cut-and-solve and demonstrate our approach for a combinatorial genetics problem.

The general-purpose solver SCIP-Jack can solve the classical Steiner tree problem as well as 11 related problems to optimality. Furthermore, the solver comes with shared and distributed parallelization extensions by means of the UG framework that allow the parallelization of its branch-and-bound search. A MIP is a mathematical definition of a problem that is comprised of a set of decision variables, some or all of which are required to have integral values; a linear objective function to be minimized or maximized; and a set of constraints, all of which are linear equalities or inequalities.

MIPs are generally NP-hard problems, yet progress in the field has led to limited successes in solving moderate to large size instances. The application of cutting planes is integral for state-of-the-art solvers that use Branch-and-Cut, but this application is inherently sequential. This round-table discussion will focus on the challenges faced when massively parallelizing computations for solving MIPs and explore strategies for circumventing these challenges.

Michael Chan University of Missouri - St. Computational Fluid Dynamics CFD is a natural driver for exascale computing both for academic and industrial cases, and has the potential for substantial societal impact, like reduced energy consumption, alternative sources of energy, improved health care, and improved climate models. This minisymposium focuses on algorithms and methods applicable on the way to exascale for CFD simulations.

Application cases were discussed in Part I, whereas in Part II we focus on some of the relevant methodological aspects. Basic electronic transport properties and associated electronic velocities are described in Sections 6 and 7. The new developments related to parallelization and speedup of the EPW software are presented in Section 8.

The code is currently tested on a wide range of compilers and architectures using a Buildbot test farm, as described in Section 9. Finally, we highlight in Section 10 the new capabilities of EPW through five physically relevant examples: spectral functions and linewidths of B-doped diamond; scattering rate of undoped Si; spectral function and electronic resistivity of Pb with and without spin-orbit coupling; electron-phonon matrix elements for the polar semiconductor GaN; and superconducting properties of MgB 2 based on the Migdal-Eliashberg theory.

The EPW software is a freely available Fortran90 code for periodic systems that relies on DFPT and MLWFs to compute properties related to the electron-phonon coupling on very fine electron k and phonon q wavevector grids.

EPW supports norm-conserving pseudopotentials with or without non-linear core-corrections Perdew , Troullier , Fuchs as well as Hamann multi-projector norm-conserving potentials Bachelet , Hamann Many different exchange-correlation functionals are supported in the framework of the local-density LDA Ceperley , Perdew or generalized-gradient approximation GGA Perdewa. The current version 4. The code is parallelized using a message passing interface MPI library.

Recent developments and new functionalities since EPW v. This enables calculations for semi-conductors and insulators in addition to metals;. A forum has been created;. The EPW name and logo see Fig. In this section we give a brief summary of the basic physical quantities that can be computed using the EPW software. Detailed derivation of the equations can be found in Refs. Giustino and Margine The nucleus masses are already included in the phonon eigenmodes.

Note that Eq. Instead, when studying metals and doped semiconductors, we compute the real part of the electron self energy as:. The code can also compute the Fermi surface nesting function defined as Bazhirov :. The nesting function is a property of the Fermi surface and is non-zero for wave-vectors that connect two points on the Fermi surface.

From this, the isotropic Eliashberg spectral function can be obtained via an average over the BZ:. The treatment of such divergence when performing Wannier interpolation has been very recently proposed Verdi , Sjakste The way to tackle this problem is to split the electron-phonon matrix elements into a short- S and a long-range L contribution Verdi :.

In Eq. However, due to the intrinsically localized nature of MLWFs, only the short range component can be treated in a Wannier interpolation scheme. Giustino is applied to the short-range component only, and finally iv the long-range component is added back to the interpolated short-range part for each arbitrary k and q -point. The overlap matrices in Eq. The Wannier rotation matrices U n m k can be obtained at arbitrary k and q -points through the interpolation of the electronic Hamiltonian Souza Ab-initio calculations of phonon-mediated superconducting properties are based on the Bardeen-Cooper-Schrieffer BCS theory Bardeen Approaches i and ii are implemented in the EPW code.

The critical temperature T c at which the phase transition occurs can be estimated with semi-empirical methods like the McMillan formula, later refined by Allen and Dynes Allen to account for strong electron-phonon coupling:. This function is nonzero below T c. The two components of Eq. The Pauli matrices are given by:. Inserting Eqs. Furthermore, due to the form of the order parameters in Eq. The Eliashberg theory is a generalization of the BCS theory to include retardation effects Eliashberg , Eliashberg Here V is the dynamically screened Coulomb interaction between electrons, and D is the dressed phonon propagator.

The two Feynman diagrams associated with Eq. Note that the two self-energies of Eqs. The approximation that allows one to neglect all other Feynman diagrams is called the Migdal theorem Migdal and rests on the observation that the neglected terms are of the order of the square-root of the electron to ion mass ratio.

We also rely on the band-diagonal approximation Allena , Chakrabortya , Pickett , that neglects band-mixing i. Since the superconducting energy pairing is very small, such approximation should be very good for non-degenerate bands. The dressed phonon propagator can be expressed in terms of its spectral representation as Allena , Marsiglio :.

Inserting Eq. The form of Eq. To compare with available experimental data, it is worthwhile to define Fermi-surface-averaged spectral functions:. The rationale for performing the averages around the Fermi surface is that the phonon energy is usually much smaller than the Fermi energy. Now, inserting Eqs. Direct comparison of the different Pauli matrix components of Eq.

This set of equations needs to be supplemented with an equation for the electron number N e in order to determine the Fermi energy Marsiglio :. The superconducting gap can then be obtained as the ratio between the order parameter and the renormalization function:. These four approximations lead to the two nonlinear coupled equations to be solved self-consistently Choi , Margine :.

An important observable related to superconductivity that can directly be computed is the superconducting specific heat. The specific heat can be obtained from the free energy difference between the superconducting and normal states Bardeen , Choi , Marsiglio :. From this, the specific heat difference can be obtained as:.

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Select the network also contains the. Ability to create staple in the. Giving up security family and friends. A vulnerability has or menu choice. Bridging organization theory and supply chain that lets you.To evaluate this function at every combination of points in the x and y vectors, you need to define a grid of values. For this task you should avoid using loops to iterate through the point combinations. In this example, x is a by-1 vector and y is a 1-by vector, so the operation produces a by matrix by expanding the second dimension of x and the first dimension of y.

In cases where you want to explicitly create the grids, you can use the meshgrid and ndgrid functions. A logical extension of the bulk processing of arrays is to vectorize comparisons and decision making. For example, suppose while collecting data from 10, cones, you record several negative values for the diameter. MATLAB can also compare two vectors with compatible sizes, allowing you to impose further restrictions.

Logical operators isinf and isnan exist to help perform logical tests for these special values. When vectorizing code, you often need to construct a matrix with a particular size or structure. Techniques exist for creating uniform matrices. The function repmat possesses flexibility in building matrices from smaller matrices or vectors. In many applications, calculations done on an element of a vector depend on other elements in the same vector. For example, a vector, x , might represent a set.

How to iterate through a set without a for or while loop is not obvious. The process becomes much clearer and the syntax less cumbersome when you use vectorized code. A number of different ways exist for finding the redundant elements of a vector. One way involves the function diff.

After sorting the vector elements, equal adjacent elements produce a zero entry when you use the diff function on that vector. Because diff x produces a vector that has one fewer element than x , you must add an element that is not equal to any other element in the set. NaN always satisfies this condition. Use the tic and toc functions if you want to measure the performance of each code snippet.

Rather than merely returning the set, or subset, of x , you can count the occurrences of an element in a vector. After the vector sorts, you can use the find function to determine the indices of zero values in diff x and to show where the elements change value. You can count the number of NaN and Inf values using the isnan and isinf functions. Choose a web site to get translated content where available and see local events and offers.

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A stack-buffer-overflow exists in libde v1. Those your five a paid account. You figured a Team What is Teams.Add a comment. Sorted by: Reset to default. Highest score default Trending recent votes count more Date modified newest first Date created oldest first. Improve this answer. Amro Amro k 25 25 gold badges silver badges bronze badges. Try adding the "scale" parameter, set to zero, to prevent automatic scaling, i. Thanks Martin for the scale argument. But the arrowheads in quiver3 doesn't look very nice as compared to arrow. Community Bot 1 1 1 silver badge. Shai k 35 35 gold badges silver badges bronze badges.

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Trending: A new answer sorting option. Linked 0. Now, collect information on 10, cones. The vectors D and H each contain 10, elements, and you want to calculate 10, volumes. Placing a period. Array operators also enable you to combine matrices of different dimensions. This automatic expansion of size-1 dimensions is useful for vectorizing grid creation, matrix and vector operations, and more.

Suppose that matrix A represents test scores, the rows of which denote different classes. You want to calculate the difference between the average score and individual scores for each class. A more direct way to do this is with A - mean A , which avoids the need of a loop and is significantly faster.

Even though A is a 7-by-3 matrix and mean A is a 1-by-3 vector, MATLAB implicitly expands the vector as if it had the same size as the matrix, and the operation executes as a normal element-wise minus operation. The size requirement for the operands is that for each dimension, the arrays must either have the same size or one of them is 1. If this requirement is met, then dimensions where one of the arrays has size 1 are expanded to be the same size as the corresponding dimension in the other array.

Another area where implicit expansion is useful for vectorization is if you are working with multidimensional data. Suppose you want to evaluate a function, F , of two variables, x and y. To evaluate this function at every combination of points in the x and y vectors, you need to define a grid of values.

For this task you should avoid using loops to iterate through the point combinations. In this example, x is a by-1 vector and y is a 1-by vector, so the operation produces a by matrix by expanding the second dimension of x and the first dimension of y. In cases where you want to explicitly create the grids, you can use the meshgrid and ndgrid functions.

A logical extension of the bulk processing of arrays is to vectorize comparisons and decision making. For example, suppose while collecting data from 10, cones, you record several negative values for the diameter. MATLAB can also compare two vectors with compatible sizes, allowing you to impose further restrictions. Logical operators isinf and isnan exist to help perform logical tests for these special values.

When vectorizing code, you often need to construct a matrix with a particular size or structure. Techniques exist for creating uniform matrices. The function repmat possesses flexibility in building matrices from smaller matrices or vectors. In many applications, calculations done on an element of a vector depend on other elements in the same vector. For example, a vector, x , might represent a set. How to iterate through a set without a for or while loop is not obvious. The process becomes much clearer and the syntax less cumbersome when you use vectorized code.

A number of different ways exist for finding the redundant elements of a vector. One way involves the function diff.

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