A palace is a grand residence a royal residence, or the home of a head of state or some other high-ranking dignitary, such as a bishop or archbishop. The word is derived from the Latin name Palātium, for Palatine Hill in Rome which housed the Imperial residences. Most European languages have a version of the term, many use it for a wider range of buildings than English. In many parts of Europe, the equivalent term is applied to large private houses in cities of the aristocracy. Many historic palaces are now put to other uses such as parliaments, hotels, or office buildings; the word is sometimes used to describe a lavishly ornate building used for public entertainment or exhibitions, such as a movie palace. The word palace comes from Old French palais, from Latin Palātium, the name of one of the seven hills of Rome; the original "palaces" on the Palatine Hill were the seat of the imperial power while the "capitol" on the Capitoline Hill was the religious nucleus of Rome. Long after the city grew to the seven hills the Palatine remained a desirable residential area.

Emperor Caesar Augustus lived there in a purposely modest house only set apart from his neighbours by the two laurel trees planted to flank the front door as a sign of triumph granted by the Senate. His descendants Nero, with his "Golden House", enlarged the house and grounds over and over until it took up the hill top; the word Palātium came to mean the residence of the emperor rather than the neighbourhood on top of the hill. Palace meaning "government" can be recognized in a remark of Paul the Deacon. AD 790 and describing events of the 660s: "When Grimuald set out for Beneventum, he entrusted his palace to Lupus". At the same time, Charlemagne was consciously reviving the Roman expression in his "palace" at Aachen, of which only his chapel remains. In the 9th century, the "palace" indicated the housing of the government too, the travelling Charlemagne built fourteen. In the early Middle Ages, the palas was that part of an imperial palace, that housed the Great Hall, where affairs of state were conducted.

In the Holy Roman Empire the powerful independent Electors came to be housed in palaces. This has been used as evidence that power was distributed in the Empire. In modern times, the term has been applied by archaeologists and historians to large structures that housed combined ruler and bureaucracy in "palace cultures". In informal usage, a "palace" can be extended to a grand residence of any kind. Early ancient palaces include the Assyrian palaces at Nimrud and Nineveh, the Minoan palace at Knossos, the Persian palaces at Persepolis and Susa. Palaces in East Asia, such as the imperial palaces of Japan, Vietnam, Thailand and large wooden structures in China's Forbidden City, consist of many low pavilions surrounded by vast, walled gardens, in contrast to the single building palaces of Medieval Western Europe; the Brazilian new capital, Brasília, hosts modern palaces, most designed by the city's architect Oscar Niemeyer. The Alvorada Palace is the official residence of Brazil's president; the Planalto Palace is the official workplace.

The Jaburu Palace is the official residence of Brazil's vice-president. Rio de Janeiro, the former capital of the Portuguese Empire and the Empire of Brazil, houses numerous royal and imperial palaces as the Imperial Palace of São Cristóvão, former official residence of the Brazil's Emperors, the Paço Imperial, its official workplace and the Guanabara Palace, former residence of Isabel, Princess Imperial of Brazil. Besides palaces of the nobility and aristocracy; the city of Petropolis, in the state of Rio de Janeiro, is known for its palaces of the imperial period such as the Petrópolis Palace and the Grão-Pará Palace. In Canada, Government House is a title given to the official residences of the Canadian monarchy and various viceroys. Though not universal, in most cases the title is the building's sole name; the use of the term Government House is an inherited custom from the British Empire, where there were and are many government houses. Rideau Hall is, since 1867, the official residence in Ottawa of both the Canadian monarch and his or her representative, the Governor General of Canada, has been described as "Canada's house".

It stands in Canada's capital on a 0.36 km2 estate at 1 Sussex Drive, with the main building consisting of 175 rooms across 9,500 m2, 27 outbuildings around the grounds. While the equivalent building in many countries has a prominent, central place in the national capital, Rideau Hall's site is unobtrusive within Ottawa, giving it more the character of a private home. Along with Rideau Hall, the Citadelle of Quebec known as La Citadelle, is an active military installation and official residence of both the Canadian monarch and the Governor General, it is located atop adjoining the Plains of Abraham in Quebec City, Quebec. The citadel is the oldest military building in Canada, forms part of the fortifications of Quebec City, one of only two cities in North America still surrounded by fortifications; the fortress is located within the historic district of Old Québec, which wa

Ray Grainger

Raymond "Ray" Grainger is the co-founder and CEO of the tech startup Mavenlink, a private cloud-based software-as-a-service company for business management software. Grainger was raised in California. After graduating high school he worked as a field assistant for the National Science Foundation, took part in two expeditions to the South Pole in 1979, he received an engineering degree from Harvey Mudd College in 1988. After graduating college he worked as an information technology consultant at Accenture, where he remained for 17 years becoming Global Managing Partner. During these years he invested in several companies, among others in InQuira, acquired by Oracle in 2012. In 2005 Grainger left Accenture and became an executive at InQuira, a startup that developes call-center management software, he was an Executive Vice President of the company. After three years he left InQuira to start Mavenlink. Grainger founded Mavenlink in 2010 and within five years had raised $18 million in venture capital funding.

The company is headquartered in California. It has three international offices, it now has more than 600,000 clients in 100 countries, including Salesforce, the Irvine Co. and Stanford University. The company completed a $48 million Series E financing round. In July 2019, Mavenlink had around 320 employees. Grainger has two grown children, a son and daughter. Grainger was a 2017 finalist for Orange County's Entrepreneur of the Year, he is an emeritus member of the Harvey Mudd College Board of Trustees and chairman of the Finance Committee

Discrete Laplace operator

For the discrete equivalent of the Laplace transform, see Z-transform. In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid. For the case of a finite-dimensional graph, the discrete Laplace operator is more called the Laplacian matrix; the discrete Laplace operator occurs in physics problems such as the Ising model and loop quantum gravity, as well as in the study of discrete dynamical systems. It is used in numerical analysis as a stand-in for the continuous Laplace operator. Common applications include image processing, where it is known as the Laplace filter, in machine learning for clustering and semi-supervised learning on neighborhood graphs. There are various definitions of the discrete Laplacian for graphs, differing by sign and scale factor; the traditional definition of the graph Laplacian, given below, corresponds to the negative continuous Laplacian on a domain with a free boundary.

Let G = be a graph with vertices V and edges E. Let ϕ: V → R be a function of the vertices taking values in a ring; the discrete Laplacian Δ acting on ϕ is defined by = ∑ w: d = 1 where d is the graph distance between vertices w and v. Thus, this sum is over the nearest neighbors of the vertex v. For a graph with a finite number of edges and vertices, this definition is identical to that of the Laplacian matrix; that is, ϕ can be written as a column vector. If the graph has weighted edges, that is, a weighting function γ: E → R is given the definition can be generalized to = ∑ w: d = 1 γ w v where γ w v is the weight value on the edge w v ∈ E. Related to the discrete Laplacian is the averaging operator: = 1 deg ⁡ v ∑ w: d = 1 ϕ. In addition to considering the connectivity of nodes and edges in a graph, mesh laplace operators take into account the geometry of a surface. For a manifold triangle mesh, the Laplace-Beltrami operator of a scalar function u at a vertex i can be approximated as i ≈ 1 2 A i Σ j, where the sum is over all adjacent vertices j of i, α i j and β i j are the two angles opposite of the edge i j, A i is the vertex area of i.

The above cotangent formula can be derived using many different methods among which are piecewise linear finite elements, finite volumes, discr