Vendor | : | HP |

Exam Code | : | HP0-460 |

Exam Name | : | Implementing HP XP12000/10000 Solution Fundamentals |

Questions and Answers | : | 75 Q & A |

Updated On | : | December 12, 2018 |

PDF Download Mirror | : | HP0-460 Brain Dump |

Get Full Version | : | Pass4sure HP0-460 Full Version |

HP0-460 exam Dumps Source : Implementing HP XP12000/10000 Solution Fundamentals

Test Code : HP0-460

Test Name : Implementing HP XP12000/10000 Solution Fundamentals

Vendor Name : HP

Q&A : 75 Real Questions

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The "Low HP Tractor Market in India - industry Outlook and Forecast 2019-2024" file has been introduced to ResearchAndMarkets.com's offering.

The India low HP tractor market is estimated to provide volumes of over 1019 thousand units with the aid of 2024, transforming into at a CAGR of more than 6% throughout 2018-2024.

Key manufacturers are offering machines on hire to small farmers who can not have enough money expensive tractors. for instance, Mahindra has launched a tractor rental services application, Trringo within the Indian market. The increasing adoption of farm mechanization practices that support to enrich the crop production will force transformation out there over the following couple of years.

The India low HP tractor market is pushed via the growing demand for agriculture equipment from the important and japanese components of the nation. The implementation of subsidies on fertilizers, electricity, and gasoline and the govt imposing schemes similar to Rashtriya Krishi Vikas Yojna and KCC will propel the increase of the Indian market. The market analysis document provides in-depth market analysis and segmental analysis of the India low HP tractor market with the aid of HP, wheel pressure, and zones.

The record considers the existing state of affairs of the India low HP tractor market and its market dynamics for the duration 2018-2024. It considers all sub-segments of horsepower including 0-20 hp, 20-30 hp, 30-forty ho, 40-50 hp, 50-60 hp, and 60+ hp. It covers a detailed overview of a variety of market growth enablers, restraints, and traits. It also profiles and analyzes main agencies and a number of other well-known groups operating in the market.

Low HP Tractor Market in India - Dynamics

The rising number of public and private sector businesses investing within the agricultural business is propelling the increase of the low HP tractor market in India. The increasing profits degrees of inhabitants in establishing states, inequitable inhabitants increase, and an agricultural output coupled with climatic alterations affecting the agricultural output are one of the most simple factors attributing to the demand for meals in the Indian market. Such high demand is encouraging farmers to undertake sensible tech solutions out there. Two of the highest promoting manufacturers in the Indian market are Mahindra and Sonalika.

the combination of applied sciences similar to facts analytics, telematics, far off sensing, GPS, and mobile technology will make farming extra actual and complex out there. These resourceful machines will outcome in circulate with precision to reduce gas, fertilizers, and seed wastage and make a contribution to the revenue within the low HP tractor market in India.

The transforming into concerns over environmental toxins are fuelling the demand for biodiesel tractors in the low HP tractor market in India. The low working fees and low upkeep of biodiesel machines will boost the sale of those products in the Indian market. leading producers are studying and developing option fuels corresponding to bio-diesel and electrical or battery-operated machines to benefit a bigger customer group in the low HP tractor market in India.

The major vendors in the India low HP tractor market are:

The different well known avid gamers within the India low HP tractor market

Key themes lined:

1 analysis Methodology

2 research aims

3 analysis method

four record insurance

four.1 Market Definition

4.2 Base 12 months

four.3 Scope of analyze

5 file Assumptions & Caveats

5.1 Key Caveats

5.2 Inclusions

5.3 Exclusions

5.four forex Conversion

5.5 Market Derivation

6 Market at a look

7 Introduction

7.1 Overview

7.1.1 Agriculture state of affairs of India

7.1.2 Land keeping buildings and Cropping depth: How Farmers domesticate vegetation

7.1.three Agricultural Exports

7.1.four role of Tractors in Indian Agriculture

8 Market Dynamics

eight.1 Market growth Enablers

eight.1.1 elevated fiscal assist from the Indian executive

8.1.2 boom in Agricultural productiveness and Exports

eight.1.three increased Adoption of Farm Mechanization

8.1.four Migration to Cities leading to Labor scarcity

8.2 Market increase Restraints

8.2.1 upward thrust in Tractor apartment classes

eight.2.2 Lack of awareness about Agriculture know-how

8.2.3 Marginal and Fragmented Farmlands

8.2.4 distinctive Soil conditions and Low entry to credit score in Rural Areas

eight.three Market opportunities & tendencies

8.three.1 Emergence of Precision Farming Practices

eight.3.2 usage of Alternate and Non-well-known gasoline in Tractors

9 international Low HP Tractor Market

9.1 Market Overview

9.2 Market measurement and Forecast

10 Low HP Tractor Market In India

10.1 Market Overview

10.2 Market dimension and Forecast

10.three Porter's 5 Forces evaluation

eleven with the aid of Wheel force

11.1 Market Overview

eleven.1.1 Market with the aid of Wheel drive

11.2 2WD Tractor Market

eleven.three 4 wheel drive Tractor Market

12 Market by HP

12.1 Market Overview

12.2 0-20 HP

12.three 20-30 HP

12.four 30-40 HP

12.5 40-50 HP

12.6 50-60 HP

12.7 60+ HP

13 Market by using Zone

13.1 Market Overview

14 competitive panorama

14.1 Market Share evaluation

14.2 brand Loyalty

14.three earnings and Exports

15 Key company Profiles

For extra counsel about this record seek advice from https://www.researchandmarkets.com/research/nqsqmn/low_hp_tractor?w=4

View supply version on businesswire.com: https://www.businesswire.com/information/home/20181127005524/en/

supply: research and Markets

ResearchAndMarkets.comLaura wood, Senior Press Managerpress@researchandmarkets.comFor E.S.T workplace Hours call 1-917-300-0470For U.S./CAN Toll Free call 1-800-526-8630For GMT workplace Hours call +353-1-416-8900Related issues: Tractors

Copyright company Wire 2018

MCLEAN, Va., November 26, 2018 (Newswire.com) - HumanTouch LLC welcomes Laurie Chidlow, PMP, as application director for health. Ms. Chidlow’s background has depth and breadth as a senior counsel technology professional with 18 years of management consulting. She is a certified task management expert and Scrum grasp with confirmed management capabilities in the Federal IT space, including strategic planning and enterprise construction.

“we're longing for Ms. Chidlow to convey her technology company administration knowledge to HumanTouch. Our federal purchasers will benefit from her ability to optimize and deliver in programs support, cybersecurity and statistics management. Laurie is an expert in helping groups put in force approaches that are efficient and supply demonstrated price,” said Rick Hill, senior vice president of HumanTouch.

Ms. Chidlow become most lately at eGlobalTech as a task manager in the business’s regularly occurring capabilities administration account. Working within the workplace of government-wide coverage, she led a program and venture administration group that supported the implementation of expertise policy courses and initiatives across executive. She stood up a portfolio management office including concepts, tools and governance processes to superior control investments and projects.

Ms. Chidlow has also worked as account supervisor for Hewlett-Packard commercial enterprise features (HP); Encore II program manager for both HP and Booz Allen Hamilton; and as senior administration advisor for digital information techniques (EDS). She has an MBA from the school of William & Mary, Mason college of business, and a Bachelor of Science degree in psychology from Mary Washington faculty.

HumanTouch LLC provides solutions for challenges within the areas of cybersecurity, infrastructure engineering, operations administration, solutions engineering, approach/ modernization and extra. For over 20 years, we have used our crew of specialists to Infuse innovation, management and the human contact to sync with our client's mission. The hallmark of HumanTouch is the potential to give a boost to our clients’ effectiveness within the industry with the aid of bolstering their expertise of the industry, waiting for and responding to increase and demands, and decreasing chance within the digital age. We fit our expertise with integrity, acumen and execution.

For extra assistance, press contact Rebecca Churchill, news@humantouchllc.com, 917-518-9789.

Headquarters: 7918 Jones department force, Suite 800, McLean VA 22102Atlanta workplace: 2971 vegetation Rd. S, Suite 213, Atlanta GA 30341E-mail: information@humantouchllc.com | telephone: (703) 910-5090 DUNS#: 843919197 / CAGE Code: 1XS43

supply: HumanTouch LLC

HP0-460 exam Dumps Source : Implementing HP XP12000/10000 Solution Fundamentals

Test Code : HP0-460

Test Name : Implementing HP XP12000/10000 Solution Fundamentals

Vendor Name : HP

Q&A : 75 Real Questions

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It seems that the split from HP Inc (HPQ) in Nov 2015 has been a boon for the shareholders of Hewlett Packard Enterprise Co (HPE). This is because it allowed a customized approach to two different businesses, which was not possible while they operated as a single entity. HPE stock has been clocking solid returns since then and has gained approximately 59.7%.

The major part of the rally was witnessed last year mainly driven by a series of restructuring initiatives, which includes trimming down its core businesses and lowering costs. In 2016, the stock gained 52.2%, outperforming the Zacks categorized Computer-Integrated Systems industry’s return of 28.7% during the same time frame.

Driving Factors for Hewlett Packard Enterprises

After the split, Hewlett Packard made it clear that it will focus on restructuring and realigning its businesses to drive long-term sustainable growth and improve margins. In keeping with this effort, the company divested its stake in Mphasis Limited, an IT service provider in Bangalore, India.

Apart from this, Hewlett Packard decided to spin-off its Software and IT Services businesses last year and entered into deals to merge these with Micro Focus International Plc and Computer Sciences Corporation (CSC), respectively.

The primary motive behind such a massive restructuring drive is to reassure investors of the company’s sustained focus on improving profitability and returning value to shareholders in the form of dividend and share repurchases.

Also, by trimming its size, the company intends to focus more on fast growing and high margin businesses such as high performance computing (HPC), private cloud, all-flash arrays and hyper-converged computing.

We believe that the company’s divestment strategy has provided it with enough cash to make investments in the aforementioned fast growing businesses. It should be noted that Hewlett Packard bought Silicon Graphics in November last year, which provides HPC services such as servers, storage, and data center solutions to clients in the cloud computing, oil & gas, e-commerce, social networking, and other industries.

Furthermore, it seems that Hewlett Packard views Industrial Internet of Things (IoT) as the next major market as evident from its recent partnership with GE Digital, a unit of General Electric Company (GE).

Per market research firm IC Insights, Industrial IoT implementation revenues are likely to grow 19% year over year and reach $18.4 billion in 2016. It further forecasts that implementation revenues will approximately be doubled to $29.6 billion by 2019 from $15.4 billion in 2015. We believe that the partnership with GE Digital will help Hewlett Packard Enterprise to better tap the growing opportunity in the space.

Bottom Line: HPE’s Strong Valuation is a Boon for Investors

We believe Hewlett Packard’s massive restructuring moves will complement its focus on core businesses and enable it to compete with players like Oracle Corporation (ORCL), Cisco Systems, Inc. (CSCO) and NetApp Inc. (NTAP) as well as the new entrant, Dell going forward.

Moreover, the company’s traction in the cloud, security and Big Data segments will enhance its growth trajectory, going forward. Also, its strategic divestments and initiatives to return value to shareholders in the form of dividend and share repurchases bode well.

On the valuation front too, Hewlett Packard looks very impressive. The stock currently trades at a trailing twelve months (ttm) P/E multiple of 12.0x, lower than the Zacks categorized Computer-Integrated Systems industry average of 13.2x.

Considering Hewlett Packard’s strong fundamentals along with impressive P/E ratio and the Zacks VGM Style Score of “A”, we believe that the stock is worth retaining in one’s portfolio. The stock carries a Zacks Rank #3 (Hold). You can see the complete list of today’s Zacks #1 Rank (Strong Buy) stocks here.

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More From InvestorPlaceEinstein’s theory of general relativity affords an enormously successful description of gravity. The theory encodes the gravitational interaction in the metric, a tensor field on spacetime that satisfies partial differential equations known as the Einstein equations. This review introduces some of the fundamental concepts of numerical relativity—solving the Einstein equations on the computer—in simple terms. As a primary example, we consider the solution of the general relativistic two-body problem, which features prominently in the new field of gravitational wave astronomy.

The basic equations of general relativity are the Einstein equations, first published in 1915 (1). However, even today there are large gaps in our understanding of the physics implied by the Einstein equations. Stated in general terms, a major goal of research in general relativity is to solve the Einstein equations for the physical situations of interest. Fundamental analytic solutions of the Einstein equations include the flat Minkowski spacetime known from special relativity, the Schwarzschild and Kerr spacetimes describing single black holes, and the simple Big Bang cosmologies. Also predicted by general relativity are gravitational waves, which for weak fields can be obtained as analytic solutions of the linearized Einstein equations. However, the few known analytic solutions describe only very special situations, and approximation methods fail in the regime where the nonlinear, strong-field effects of relativity play a crucial role. If we are interested in the truly relativistic regime, we must turn to computer simulations to obtain numerical solutions to the full Einstein equations.

Solving the full Einstein equations on the computer is the subject of numerical relativity, which could also be called computational general relativity. Computers also play a role in algebraic computations and in approximation schemes, and such calculations are important topics in numerical relativity. But the distinguishing feature of numerical relativity is that, in principle, the Einstein equations in full generality can and must be solved numerically.

Numerical relativity spans a large range of different topics including mathematical general relativity, astrophysics, numerical methods for partial differential equations, computer programming, and simulation science. Current research in numerical relativity is in a transition from a self-contained topic in theoretical physics to a physical theory with numerous connections to observational astronomy (2, 3). Gravitational wave astronomy holds much promise for the future, as recognized by the 2017 Nobel Prize in Physics, and numerical relativity is providing key theoretical predictions and analysis tools for the ongoing gravitational wave observations (4).

The general relativistic two-body problemAs a primary application of numerical relativity, we consider the gravitational two-body problem. The two-body problem in Newtonian gravitational physics can be formulated for two point masses moving in their mutual gravitational field. A particular solution of the Newtonian two-body problem is a Keplerian elliptical orbit. However, in Einsteinian gravity, such orbital motion generates gravitational waves that carry away energy and momentum. Binary orbits therefore decay, and the motion of the two bodies follows an inward spiral that eventually terminates with the collision and merger of the two objects. In most astrophysical situations, the energy loss due to the emission of gravitational waves is so small that a binary orbit decays only on time scales of millions or billions of years. However, for compact objects such as neutron stars or black holes in very tight binaries, general relativistic effects such as gravitational wave emission play a major role (5).

Research in this field seeks to provide a theoretical framework for the physics of binary black holes, neutron stars, and gravitational waves. Such an endeavor must rely on numerical simulations in general relativity and general relativistic hydrodynamics. But a reasonably complete framework still requires substantial progress in numerical relativity and related fields. Currently there are serious limitations in our ability to model the entire range of relevant physics, from the nuclear physics of neutron star matter to the large-scale, strong-gravity effects encountered in binary neutron star mergers (6). The different dynamical phases of the binary evolution—known as the inspiral, the merger, and the evolution of the remnant—are accompanied by characteristic gravitational wave signatures (Fig. 1). For binaries involving at least one neutron star, depending on the specifics of the system, key features include the disruption of the star(s) before merger, the formation of a hypermassive neutron star, the prompt or delayed collapse to a black hole, the dynamics of the accretion torus plus the central merged object, and the creation of unbound material, the ejecta. Before discussing simulations of these systems, we introduce the mathematical foundation of numerical relativity.

Fig. 1 Binary neutron star mergers emit gravitational waves.The waves reveal unique information about extreme gravity and extreme matter—information that can be unraveled with the help of numerical relativity. Shown is a waveform and snapshots of the neutron star matter for the inspiral, merger, and remnant. The amplitude of the gravitational wave is plotted versus time. The merger occurs at t = 0.

IMAGE: COURTESY OF T. DIETRICH, BASED ON (78) Mathematical foundationCombining space and time into spacetime can be considered a triumph of human thought, allowing us to perceive the true nature of relativistic and gravitational physics (7). However, this does not mean that we cannot or need not consider space and time separately. Somewhat ironically, after working hard to unify space and time, the mathematical setup of numerical relativity starts by splitting spacetime again into space and time and by making gauge (coordinate) choices (8) in order to reformulate the Einstein equations as a well-posed mathematical problem.

General relativity is the theory of a metric tensor on a four-dimensional manifold, plus matter described by additional tensor fields. A manifold ℳ endowed with a metric gab is called a spacetime (ℳ, gab). The metric measures lengths, here in four dimensions. The infinitesimal line element

(1)provides a generalization of the Pythagorean theorem. Repeated indices are summed over, following the Einstein summation convention. The metric is symmetric (gab = gba), has a Lorentz signature of – + + +, and there exists an inverse metric gab defined by , where is the identity matrix. A special example is the Minkowski line element ds2 = –c2dt2 + dx2 + dy2 + dz2, where c is the speed of light, t is the time coordinate, and x, y, and z are spatial coordinates. The components of the Minkowski metric are constants, but in general gab is a field with nonconstant components.The field equations of general relativity are the Einstein equations,

(2)where Gab is the Einstein tensor, which depends on the metric and its first and second derivatives, and Tab is the stress-energy tensor constructed from the matter fields Φ and in general also from the metric. For numerical implementations, the first step is to write the Einstein equations as a well-posed system of partial differential equations (PDEs) for the metric. Equation 2 represents 10 coupled, nonlinear PDEs for the 10 independent components of the metric, but without further adjustments these equations are in no known sense hyperbolic (i.e., well-posed as an initial value problem).The differential operator acting on the metric in the Einstein equations is given by the Ricci tensor,

(3)The first term by itself, gcd∂c∂dgab, which is often denoted g□gab (where □ is the d’Alembert operator), has the form of the principal part of a simple hyperbolic wave equation, but note that the metric appears in two places: as the wave field gab and as the inverse metric gcd in the wave operator. The other second-derivative terms are not standard wave operators. The best we can say about the complete principal part in Eq. 3 is that it is quasi-linear in the metric; that is, it is linear in the highest-order derivatives but with coefficients that depend (nonlinearly) on the variable itself. The lower-order terms are quite involved as well, with typical terms of the form g–1g–1∂g∂g. Approaching the problem in this way makes it difficult to recognize that these equations are describing the simple geometric concept of curvature and that there is a time evolution being defined. Further, their well-posedness properties are quite unclear.The so-called 3+1 decomposition—for example, in the form of Arnowitt, Deser, and Misner (ADM) (8)—assumes that the manifold (at least locally) allows a split into time and space, ℳ = R × Σ. Physics is then describable by time-dependent tensors on three-dimensional hypersurfaces Σ, which correspond to t = constant slices of ℳ, resulting in a “foliation” of spacetime in terms of three-dimensional spaces. Geometrically, we obtain a normal vector na to Σ that allows the decomposition of tensors in directions normal and tangential to the hypersurfaces. These are the time-like and space-like directions, respectively. For concreteness, we can assume coordinates xa = (t, xi) = (t, x, y, z) with a time coordinate x0 = t and spatial coordinates xi, where i = 1, 2, 3 and a = 0, 1, 2, 3.

Decomposing the Einstein equations is accomplished by projecting Gab and Rab in the directions parallel and orthogonal to na. We discover that the differential operator Eq. 3 leads to two types of equations: (i) evolution equations containing time derivatives, and (ii) four constraint equations that are essentially elliptic equations, highlighting the indeterminate type of Eq. 3. The constraints are the Hamiltonian constraint and the momentum constraints. The latter are reminiscent of the Gauss law constraint of electrodynamics, where the divergence of the electric field gives the charge density.

Given the evolution and constraint equations as PDEs, we still must choose a spatial and temporal domain with boundary conditions. For problems in astrophysics such as the two-body problem of black holes and neutron stars, we consider isolated systems where at large distances gravitational fields become weak and spacetime becomes asymptotically flat (in contrast to typical cosmological models). Because gravity is universally attractive and long-range, it is not natural to restrict a system to a finite box, especially given that the goal is to compute waves traveling to infinity. Nonetheless, a typical configuration for numerical simulations is a finite-size spatial domain (e.g., a sphere) with boundary conditions at some finite radius that implement the proper fall-off of the fields and an outgoing-wave boundary (9).

Features unique to numerical relativity are various aspects of black hole spacetimes, in particular the causal structure associated with black hole event horizons and the possibility of spacetime singularities. This latter aspect can be viewed as the problem of specifying additional boundaries that represent black holes within the simulation domain.

Building blocks of numerical relativityTo define a particular strategy to solve the Einstein equations, we consider the following building blocks that define the anatomy of a numerical relativity simulation, with a focus on the compact binary problem. The following items are certainly relevant to any evolution problem in computational physics: initial data, evolution, analysis, and numerics. We must specify the initial conditions, integrate the equations of motion to obtain the evolved data, and perform an analysis of the evolved data to extract physical information. The numerical treatment of each item may require the implementation of specific numerical techniques.

EvolutionFormulation: Choose one of many inequivalent formulations (i.e., choose variables and rewrite the Einstein equations to obtain a well-posed initial value problem). Choose the order of time and space derivatives; make structural choices about the gauge and the constraints.

Reformulating the Einstein equations as a well-posed initial value problem has been the subject of much research (10, (11). To give an example, the result of the generalized harmonic gauge (GHG) formulation (12) can be cast in a standard first-order PDE form as

(4)Here, the state vector uμ collects all 10 components gab, the 40 first derivatives ∂cgab, and a few additional fields depending on the formulation. In addition, there may be variables for the matter fields. Equation 4 for GHG is strongly and even symmetric hyperbolic (10, (11). To give an example, the result of the generalized harmonic gauge (GHG) formulation (12) and is well suited for numerical implementation. Another standard way to proceed is the classic ADM formulation that makes the geometry of the 3+1 decomposition in time and space more explicit. Basic variables are the 3-metric gij and the extrinsic curvature Kij, which is essentially the first time derivative of the metric (8). The ADM equations are only weakly hyperbolic and are not suitable for numerics. However, closely related systems, the so-called BSSN and Z4c formulations (10, 11), are strongly hyperbolic. Most current simulations in numerical relativity rely on either the GHG or BSSN/Z4c families of formulations.Constraint propagation: Maintain the constraints during evolution. Perform free evolutions and monitor the convergence of the constraints; use, for example, constraint damping to maintain the constraints explicitly.

Analytically, the constraints propagate; that is, if they are satisfied initially, they remain satisfied during a well-posed evolution. Numerically, even small rounding errors can trigger divergence from the constraint-satisfying solution, which can lead to a catastrophic failure of the simulation. How the constraints are controlled is a distinguishing feature of each formulation. A key ingredient in stable binary black hole evolutions (13) is the constraint-damping scheme (14). The Z4c formulation improves BSSN in the way the Hamiltonian constraint is treated, which leads to improved conservation of mass for neutron star simulations (15). Apart from instabilities, constraint violations in 3+1 relativity signify a problem with four-dimensional covariance. The 3+1 decomposition breaks covariance of the full theory by choosing a foliation, but the constraints ensure that four-dimensional covariance is maintained.

Gauge: Choose a coordinate condition—for example, in terms of lapse and shift or in terms of gauge source functions. Construct coordinates that avoid physical and coordinate singularities and are suitable for the black hole problem.

The main point about the gauge choice is that not only do we have the freedom to choose coordinates, but it is necessary to choose nontrivial coordinates. For example, even for the simplest black hole spacetimes, a foliation can fail by running into the physical singularity, and the hypersurface (or slice) may become badly distorted by slice stretching when points start falling into the black hole. The topic of how to dynamically construct good coordinates that lead to stable evolutions, cover spacetime with a regular foliation, avoid coordinate singularities, and avoid physical singularities inside black holes has become its own area of interest. In that context, the 3+1 decomposition is about “spacetime engineering” because we not only evolve the metric variables, but also build up the spacetime slice by slice in coordinates that are constructed dynamically during the evolution. The GHG formulation relies on the harmonic gauge to obtain hyperbolicity (12). For BSSN, the moving puncture gauge is essential to obtain long-term black hole evolutions, preventing slice stretching (16) and allowing the black hole punctures to move freely (17, 18).

Boundary conditions: Specify outer boundary conditions appropriate for outgoing waves and asymptotic flatness. Specify inner boundaries for black holes; choose between black hole excision and black hole punctures. Handle coordinate patch boundaries.

Although approximate boundary conditions are possible, for a clean treatment, strong or symmetric hyperbolicity is required for well-posedness (11). We can then specify boundary conditions in terms of the ingoing and outgoing characteristic fields. The outer boundary conditions in numerical relativity tend to be substantially more complicated than the Einstein equations themselves, because outgoing-wave boundaries are typically constructed by taking additional derivatives of the right sides of the equations (10). The Einstein equations share with other nonlinear wave equations the feature that there is backscattering by waves off themselves (and, for binary systems, also due to the gradient in the gravitational well). This is a fundamental problem for boundaries at finite radius, because in principle we must account for all future backscattering from outside the domain. Consistent boundaries at finite radius have only been addressed quite recently, considering the long history of the Einstein equations (19, 20).

Initial dataFormulation: Rewrite the constraints as elliptic equations, identifying suitable free and dependent variables.

To give an indication of what the formulation of the constraints entails (8), consider a conformal rescaling of the metric,

, which conveniently transforms the Hamiltonian constraint into a scalar elliptic equation for ψ. We have the freedom to specify a conformal metric, , that is not physical because it does not solve the constraints, but by solving the elliptic equation for ψ we find a physical solution gab that solves the constraints. The conformal transverse-traceless decomposition (8) is widely used for the full set of constraints; for neutron star initial data in particular, the conformal thin-sandwich construction (21, 22) is used, where typically an additional elliptic equation is added to initialize the gauge condition.Physical content: Solve the constraints for data that contain multiple black holes or neutron stars with arbitrary mass, spin, and momentum.

Because the constraints are nonlinear, we cannot simply “add up” the metric tensors of, for example, two Schwarzschild black holes to obtain binary data, although that can be a useful approximate initial guess. As a result, some aspects of the initial data construction are indirect. For example, we can start with two single black hole solutions for particular masses, which will be combined to form a binary. But solving the constraints for the binary leads to a change in the individual masses of the black holes because of the conformal rescaling. In some cases we have to perform evolutions to determine whether the initial data were constructed appropriately for a particular dynamical situation.

There is a growing variety of initial data constructions for binaries that correspond to the variety of physical configurations. For black holes, there are excision-type data, where the interior of black holes is removed (23, 24). Alternatively, black hole puncture data handle the black hole interior with a coordinate singularity at a point (25), which sometimes is called automatic excision. The thin-sandwich formulation is well suited for quasi-equilibrium data of black holes and/or neutron stars, which, for example, can approximate the state of a binary system during a quasi-circular inspiral (26). Only quite recently have methods been developed for neutron stars that generalize the quasi-equilibrium, quasi-circular construction to eccentric orbits (27) and to neutron stars with spin (28) (Fig. 2). Solving the constraints for electromagnetic field configurations is another recent topic of investigation (29).

Fig. 2 Binary neutron star evolution with spin and precession.As a result of the general relativistic frame-dragging effect, a binary of neutron stars with (unaligned) spin will not move within a fixed orbital plane. (A) The orbital motion, indicated by two different colors for the two stars. (B) The angular momentum. Both can show precession and nutation effects, which will also be visible in the gravitational wave signal. The axes indicate spatial coordinates (A) and vector components of the spin (B).

IMAGE: ADAPTED BY C. BICKEL FROM FIGURE 16 OF (79) AnalysisBlack holes and neutron stars: Determine all physical parameters during evolution. Find horizons of black holes. Analyze the rich phenomenology of neutron star mergers with the remnant, torus, jets, and ejecta. Connect to multi-messenger astronomy.

In any binary simulation, a wide range of detailed information is of interest, especially when matter is involved. The “relativity” in general relativity means, however, that many quantities have no direct physical meaning. In general, any tensor component (such as gtt or gxy) is not meaningful by itself; we have to construct proper gauge-invariant quantities. For example, mass and spin must be carefully defined because their local meaning at a point is problematic. For black holes, special methods are required to find the event horizon, which is a global concept in spacetime and therefore expensive to compute. See Fig. 3 for examples. Instead, black hole excision relies on the apparent horizon [e.g., (13)].

Fig. 3 The twisted pair of pants.(A and B) Spacetime plot of the event horizon of two inspiraling black holes that merge and ring down: equal (A) and unequal (B) masses, time t running up, horizontal x-y slices of the event horizon (80). (C) Pair of pants computed numerically in the 1990s for axisymmetric, head-on collisions, time t running up, horizontal slices in z-ρ coordinates (81).

IMAGE: ADAPTED BY C. BICKELGravitational waves: Compute gravitational wave emission; control numerical and systematic errors. Produce gravitational wave templates in a form that is ready to use for gravitational wave detectors. Treat both waveform prediction and waveform analysis.

Gravitational waves are propagating variations in the metric tensor, and the challenge is to separate the physical waves from various coordinate effects. In the weak-field limit, we can define gravitational waves as small perturbations around a background metric, and a first-order gauge-invariant formalism can be used to eliminate leading-order gauge effects (30). Such methods are applicable because we assume that the detectors are located far from the source where an asymptotically flat background is available. In simulations, the numerical grids often include extra patches for the far zone [e.g., (9)], possibly at lower resolution (see below).

A major effort in numerical relativity is directed toward obtaining accurate waveforms with controlled error bars for long time intervals. For the signal-to-noise ratio of current observations, a sufficiently accurate waveform model may begin with a post-Newtonian approximation (assuming nonrelativistic speeds) for the initial inspiral, matched to 10 to 20 orbits up to and including the merger from numerical simulations of the full Einstein equations. Initially, the goal was to filter the signal out of the noise by matching against theoretical waveforms. However, as the quality of the signals is improving, the main goal of gravitational wave astronomy is to estimate unknown source parameters. For example, we need detailed waveform models to distinguish black hole mergers from neutron star mergers, determine masses and spins, etc. The first detection of gravitational waves by Advanced LIGO (2) was accompanied by a theory paper describing how the properties of GW150914 were deduced from the observational data (4). Only by combining data with theory was it possible to arrive at the interpretation of GW150914 as the signature of a binary black hole merger, with specific parameters and credibility intervals. Two families of models were used, the EOBNR and Phenom families of waveforms (2) (Fig. 4). To analyze the data stream from the detectors, various parametrized waveform models are being developed for high-speed template matching (e.g., reduced-order surrogate models) (31).

Fig. 4 Numerical waveform catalogs anticipated the first gravitational wave observations.Shown are examples for template construction for gravitational waves from binary black hole mergers. (A) Various numerical waveforms computed by different research groups forming an international collaboration. (B) Combining post-Newtonian models for the inspiral with numerical relativity. In (A) and (B), the amplitude of the gravitational wave is plotted versus time. The merger occurs at t = 0. In (B), the numerical waveform is preceded by a post-Newtonian waveform to cover more orbits of the inspiral. Such waveforms, which were purely theoretical, became real with the first observation of gravitational waves in 2015 [compare to figure 1 of (2)], making it possible to interpret the first signals as the merger events of two black holes.

IMAGES: (A) ADAPTED BY C. BICKEL FROM FIGURE 1 OF (82); (B) ADAPTED BY C. BICKEL FROM FIGURE 2 OF (83) NumericsDiscretization: Choose a discretization in space and time. Introduce adaptive mesh refinement (AMR) in space and time to efficiently represent different physical length scales. Choose coordinate patches and transformations to adapt coordinates to the underlying physics.

Once a suitably hyperbolic form of the PDEs of general relativity has been derived, we have access to several standard discretizations from applied mathematics. The recent trend has been toward high-order discretizations, with different choices for the geometry and the matter fields. In vacuum or where the matter is smooth, the geometry is smooth as well. For smooth metrics, fourth- to eighth-order finite differencing in space is applied routinely, as well as pseudospectral methods for exponential convergence. Neutron star matter is represented by general relativistic fluids, and handling relativistic shocks becomes important. Several high-resolution shock-capturing (HRSC) fifth-order methods are available (6), as is work on smoothed particle hydrodynamics (32, 33).

The physics of a binary involves several physical scales. The wavelength of gravitational waves near merger is about 100 times the size of the black holes, and the simulation domain is typically chosen to be at least 1000 times the size of the black holes to accommodate several wave cycles. Simulations in three spatial dimensions therefore become several orders of magnitude more efficient with AMR, often of the Berger-Oliger type with refinements not just in space, but also in time. Many codes use several coordinate patches to transition from two (or more) central objects to spherical shells near the outer boundary.

Scientific computing: Implement parallel algorithms for high-performance computing. Invest in professional software engineering for a collaborative computational infrastructure.

Numerical relativity has been very successful with the hybrid MPI (message passing interface) plus OpenMP (open multiprocessing) or a similar parallelization strategy. Still, a typical numerical relativity simulation for a binary coalescence, representing just a single data point in a template catalog, may take roughly 1 month on 1000 to 10,000 cores of a supercomputer. The numerical relativity community is working on improving the efficiency of these methods, including spectral methods and improved AMR schemes, which tend to be a bottleneck for massive parallelism. Most efforts in numerical relativity are group efforts with a long-term investment in an evolving code base. These efforts include SpEC (34), SACRA (35), Whisky/THC (36), Pretorius (37), HAD (38), BAM (39), and the community code Einstein Toolkit (40). Some codes approximate general relativity but provide more advanced neutron star physics (32, 33). Although similar in some regards—after all, the same or similar physics is studied—the different projects vary greatly in the range and the specifics of the physics modules, the flexibility and extensibility of the codes, the level of software optimization, and the collaboration and code-sharing models.

The main challenge common to all these projects is that they are implementing a “moving target,” as formulations and basic equations are still changing and more physics is added to the simulations. Simultaneously, they must handle the trend in technology toward massively parallel computers and heterogeneous hardware, which is challenging given the complex algorithms required for numerical relativity.

Short history of binary simulationsThe first simulations of black holes in vacuum were attempted in 1964 (41). By the 1970s, many concepts of the 3+1 ADM formulation had been brought into numerical relativity (42), which led to the seminal numerical work on head-on (axisymmetric, 2+1-dimensional) black hole collisions and gravitational waves (43, 44). It took until the early 1990s (45, 46) to revisit the head-on collision with improved numerics, which confirmed the early results on gravitational waves (46). Numerical relativity in 3+1 dimensions began in 1995 with the evolution of a Schwarzschild black hole on a Cartesian grid (47) and the evolution of gravitational waves (48), followed by the first fully 3+1-dimensional simulation of a black hole binary (49, 50). All the early black hole simulations mentioned so far were numerically unstable, with barely enough evolution time to start with two separate black holes that promptly merged. The first full orbit was achieved in 2004 (51). In 2005–2006, the last missing ingredients for long-term stable black hole evolutions were found in two different approaches, one based on a harmonic gauge formulation and excision (13) and the other based on the BSSN formulation and black hole punctures (17, 18, 51). By 2010, the robustness and flexibility of these methods had been established. Improvements in the formulations, the boundary conditions, etc., are still ongoing today (11, 12, 51).

Neutron star simulations were pursued in parallel with the black hole simulations. The Valencia formalism of general relativistic hydrodynamics (GRHD), now the primary approach, was developed in the 1990s (52). The first fully general relativistic binary neutron star simulations were published in 2000 (53), with enormous progress in many groups since then. As far as the geometry of general relativity matters in these simulations, it turns out that the methods established for stable black hole simulations carry over to neutron star simulations (gauge, boundaries, initial data formulation, etc.). However, GRHD introduces its own challenge of relativistic shocks, and the range of different physics phenomena makes this a much more complex problem than black holes in vacuum.

OutlookNumerical relativity is developing rapidly in several directions, and we highlight a few representative examples.

High-order methodsHigh-order methods to address the ever-increasing demand for even more accurate and detailed simulations are a major topic of current research. Among the different high-order methods to solve partial differential equations, the discontinuous Galerkin (DG) method has emerged in recent years as a particularly successful general-purpose paradigm (54). It can be argued that the DG spectral-element method subsumes several of the key advantages of traditional finite-element and finite-volume methods. In particular, the DG method works with element-local stencils, which is a great advantage for parallelization and the construction of complicated grids. Furthermore, DG methods offer easy access to hp-adaptivity, where both the size of the computational elements (or cells) and the order of the polynomial approximation within each element can be adapted to the problem.

There are three major efforts to use DG methods for general relativity and/or GRHD (55–57). The first simulations of a single neutron star were achieved recently (55, 58), and simple binaries are a work in progress. With regard to high-order approximations, there is no doubt that if exponentially convergent spectral methods such as DG (or pseudospectral methods) are applicable, they will constitute a big improvement over finite-difference approximations, which give only polynomial convergence. High-order methods can provide breakthroughs by reaching accuracies that make new physics possible (e.g., for magnetic field amplification due to small-scale turbulence) or by reducing numerical errors to make gravitational wave analysis possible. Viewed differently, we can reach a given error criterion with much lower computational resources, making simulations feasible that are otherwise too computationally expensive.

Multi-physicsThe spectacular first observation of both gravitational waves (3) and electromagnetic radiation (59, 60) from a neutron star merger represents the beginning of multi-messenger astronomy including gravitational waves. To model such systems, we need to perform “multi-physics” simulations.

Modeling electromagnetic fields in GRHD can be accomplished by coupling the Maxwell equations to the GRHD equations, for which the prevalent approach has been ideal magnetohydrodynamics (IMHD). The assumption of IMHD is that the fluid has zero resistivity, but for the merger—and in particular for the fields surrounding the remnant with torus and ejecta—the quality of that approximation is unclear. Resistive magnetohydrodynamics (RMHD) is expected to be important for realistic models of plasma instabilities and magnetic reconnection. Apart from unknown physics, the mathematical character of the RMHD equations may be problematic (61, 62). There are only a few general relativistic simulations with RMHD [e.g., (61, 63, 64)]. Developing a proper theory of resistive relativistic plasmas is a large project in itself (65).

The microphysical equation of state of neutron stars remains unknown and is also a target for numerical models and for observations. Investigations may include 20 or more different equations of state in an attempt to cover all sensible proposals. Even determining just one parameter—the existence of neutron stars with 2.0 solar masses (66, 67)—provided a strong constraint. In principle, gravitational wave observations can do much better, gleaning information from the inspiral and the merger. Although inspiral signals will show rather systematic long-time effects (68–70), one of the grand challenges will be to disentangle the much more messy merger signal (71).

Standard merger models predict strong heating of the neutron star matter, which is expected to lead to an enormous amount of neutrino emission with luminosity on the order of 1054 erg s–1. This burst of energy plays a role in models of short gamma ray bursts (72) and also for the ejecta, which in turn affects heavy-element production and macro- or kilonovae (73). However, currently the high dimensionality of such radiative transport problems (3+1 spacetime plus 3 for the radiative transport) is prohibitive, leading to a wide array of approximations with variable applicability (74, 75). A coherent picture for neutrino physics in binary mergers is still lacking but should be a part of multi-messenger astrophysics.

Beyond current astrophysicsNumerical relativity has a large number of applications outside the area of compact binaries and gravitational waves (76, 77). Topics include gravitational collapse with surprising critical phenomena, boson stars and other exotic matter, and cosmological simulations. Going beyond classical general relativity, the field of numerical relativity for alternative gravity theories and gravity in higher dimensions is wide open.

ConclusionThe next decade is sure to see numerical relativity grow in terms of computational power and applicability to different physical scenarios. The detailed theoretical models for black hole and neutron star binaries that are the target of research in numerical relativity are closely linked to the observation of gravitational waves. Numerical relativity, in combination with the highly anticipated future observations of gravitational waves, is expected to provide entirely new insights into extreme gravity and extreme matter.

References and NotesH. Minkowski, in The Principle of Relativity, H. A. Lorentz, A. Einstein, H. Minkowski, H. Weyl, Eds. (Dover, 1952), pp. 75–91.

J. W. York Jr., in Sources of Gravitational Radiation, L. Smarr, Ed. (Cambridge Univ. Press, 1979), pp. 83–126.

J. W. York, in Sources of Gravitational Radiation, L. L. Smarr, Ed. (Cambridge Univ. Press, 1979), pp. 83–126.

L. L. Smarr, thesis, University of Texas at Austin (1975).

K. R. Eppley, thesis, Princeton University (1975).

J. S. Hesthaven, T. Warburton, Nodal Discontinuous Galerkin Methods (Springer, 2008).

M. W. Choptuik, L. Lehner, F. Pretorius, in General Relativity and Gravitation: A Centennial Perspective, A. Ashtekar, B. K. Berger, J. Isenberg, M. MacCallum, Eds. (Cambridge Univ. Press, 2015), pp. 361–411.

M. Thierfelder, thesis, University of Jena (2008).

Acknowledgments: I gratefully acknowledge the joint work evident from the list of references. Without my collaborators, this review would not have been possible. Funding: Supported in part by DFG/NSF grant BR 2176/5-1. Author contributions: B.B. is responsible for the entire manuscript. Competing interests: None. Data and materials availability: There are no new data in this review.

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College-Board [2 Certification Exam(s) ]

CompTIA [76 Certification Exam(s) ]

ComputerAssociates [6 Certification Exam(s) ]

Consultant [2 Certification Exam(s) ]

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CPP-Institue [2 Certification Exam(s) ]

CPP-Institute [1 Certification Exam(s) ]

CSP [1 Certification Exam(s) ]

CWNA [1 Certification Exam(s) ]

CWNP [13 Certification Exam(s) ]

Dassault [2 Certification Exam(s) ]

DELL [9 Certification Exam(s) ]

DMI [1 Certification Exam(s) ]

DRI [1 Certification Exam(s) ]

ECCouncil [21 Certification Exam(s) ]

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FCTC [2 Certification Exam(s) ]

Filemaker [9 Certification Exam(s) ]

Financial [36 Certification Exam(s) ]

Food [4 Certification Exam(s) ]

Fortinet [12 Certification Exam(s) ]

Foundry [6 Certification Exam(s) ]

FSMTB [1 Certification Exam(s) ]

Fujitsu [2 Certification Exam(s) ]

GAQM [9 Certification Exam(s) ]

Genesys [4 Certification Exam(s) ]

GIAC [15 Certification Exam(s) ]

Google [4 Certification Exam(s) ]

GuidanceSoftware [2 Certification Exam(s) ]

H3C [1 Certification Exam(s) ]

HDI [9 Certification Exam(s) ]

Healthcare [3 Certification Exam(s) ]

HIPAA [2 Certification Exam(s) ]

Hitachi [30 Certification Exam(s) ]

Hortonworks [4 Certification Exam(s) ]

Hospitality [2 Certification Exam(s) ]

HP [746 Certification Exam(s) ]

HR [4 Certification Exam(s) ]

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Intel [2 Certification Exam(s) ]

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Magento [3 Certification Exam(s) ]

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McData [3 Certification Exam(s) ]

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Microsoft [368 Certification Exam(s) ]

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Military [1 Certification Exam(s) ]

Misc [1 Certification Exam(s) ]

Motorola [7 Certification Exam(s) ]

mySQL [4 Certification Exam(s) ]

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NCEES [2 Certification Exam(s) ]

NCIDQ [1 Certification Exam(s) ]

NCLEX [2 Certification Exam(s) ]

Network-General [12 Certification Exam(s) ]

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NI [1 Certification Exam(s) ]

NIELIT [1 Certification Exam(s) ]

Nokia [6 Certification Exam(s) ]

Nortel [130 Certification Exam(s) ]

Novell [37 Certification Exam(s) ]

OMG [10 Certification Exam(s) ]

Oracle [269 Certification Exam(s) ]

P&C [2 Certification Exam(s) ]

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PARCC [1 Certification Exam(s) ]

PayPal [1 Certification Exam(s) ]

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QAI [1 Certification Exam(s) ]

QlikView [1 Certification Exam(s) ]

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RACC [1 Certification Exam(s) ]

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RES [5 Certification Exam(s) ]

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SASInstitute [15 Certification Exam(s) ]

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See-Beyond [1 Certification Exam(s) ]

Siemens [1 Certification Exam(s) ]

Snia [7 Certification Exam(s) ]

SOA [15 Certification Exam(s) ]

Social-Work-Board [4 Certification Exam(s) ]

SpringSource [1 Certification Exam(s) ]

SUN [63 Certification Exam(s) ]

SUSE [1 Certification Exam(s) ]

Sybase [17 Certification Exam(s) ]

Symantec [134 Certification Exam(s) ]

Teacher-Certification [4 Certification Exam(s) ]

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Tibco [18 Certification Exam(s) ]

Trainers [3 Certification Exam(s) ]

Trend [1 Certification Exam(s) ]

TruSecure [1 Certification Exam(s) ]

USMLE [1 Certification Exam(s) ]

VCE [6 Certification Exam(s) ]

Veeam [2 Certification Exam(s) ]

Veritas [33 Certification Exam(s) ]

Vmware [58 Certification Exam(s) ]

Wonderlic [2 Certification Exam(s) ]

Worldatwork [2 Certification Exam(s) ]

XML-Master [3 Certification Exam(s) ]

Zend [6 Certification Exam(s) ]

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