Existence theoremIn mathematics, an existence theorem is a theorem which asserts the existence of a certain object. It might be a statement which begins with the phrase "there exist(s)", or it might be a universal statement whose last quantifier is existential (e.g., "for all x, y, ... there exist(s) ..."). In the formal terms of symbolic logic, an existence theorem is a theorem with a prenex normal form involving the existential quantifier, even though in practice, such theorems are usually stated in standard mathematical language.
Uniqueness quantificationIn mathematics and logic, the term "uniqueness" refers to the property of being the one and only object satisfying a certain condition. This sort of quantification is known as uniqueness quantification or unique existential quantification, and is often denoted with the symbols "∃!" or "∃=1". For example, the formal statement may be read as "there is exactly one natural number such that ". The most common technique to prove the unique existence of a certain object is to first prove the existence of the entity with the desired condition, and then to prove that any two such entities (say, and ) must be equal to each other (i.
Peano existence theoremIn mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy, is a fundamental theorem which guarantees the existence of solutions to certain initial value problems. Peano first published the theorem in 1886 with an incorrect proof. In 1890 he published a new correct proof using successive approximations.
CounterexampleA counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "student John Smith is not lazy" is a counterexample to the generalization “students are lazy”, and both a counterexample to, and disproof of, the universal quantification “all students are lazy.” In mathematics, the term "counterexample" is also used (by a slight abuse) to refer to examples which illustrate the necessity of the full hypothesis of a theorem.
Weak solutionIn mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precisely defined sense. There are many different definitions of weak solution, appropriate for different classes of equations. One of the most important is based on the notion of distributions.
Stokes' theoremStokes' theorem, also known as the Kelvin–Stokes theorem after Lord Kelvin and George Stokes, the fundamental theorem for curls or simply the curl theorem, is a theorem in vector calculus on . Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. The classical theorem of Stokes can be stated in one sentence: The line integral of a vector field over a loop is equal to the flux of its curl through the enclosed surface.
Adversarial machine learningAdversarial machine learning is the study of the attacks on machine learning algorithms, and of the defenses against such attacks. A survey from May 2020 exposes the fact that practitioners report a dire need for better protecting machine learning systems in industrial applications. To understand, note that most machine learning techniques are mostly designed to work on specific problem sets, under the assumption that the training and test data are generated from the same statistical distribution (IID).
Mean value theoremIn mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It is one of the most important results in real analysis. This theorem is used to prove statements about a function on an interval starting from local hypotheses about derivatives at points of the interval.
Virial theoremIn statistical mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by a conservative force (forces characterized exclusively by their work), with that of the total potential energy of the system. Mathematically, the theorem states where T is the total kinetic energy of the N particles, Fk represents the force on the kth particle, which is located at position rk, and angle brackets represent the average over time of the enclosed quantity.
LausanneLausanne (loʊˈzæn , USalsoloʊˈzɑːn , lɔzan; Losena lɔˈzəna) is the capital and largest city of the Swiss French-speaking canton of Vaud. It is a hilly city situated on the shores of Lake Geneva, about halfway between the Jura Mountains and the Alps, and facing the French town of Évian-les-Bains across the lake. Lausanne is located northeast of Geneva, the nearest major city. The municipality of Lausanne has a population of about 140,000, making it the fourth largest city in Switzerland after Basel, Geneva, and Zurich, with the entire agglomeration area having about 420,000 inhabitants (as of January 2019).
Lausanne DistrictLausanne District (District de Lausanne) is a district in the canton of Vaud, Switzerland. The seat of the district is the city of Lausanne. Lausanne has an area, , of . Of this area, or 27.3% is used for agricultural purposes, while or 30.9% is forested. Of the rest of the land, or 41.6% is settled (buildings or roads) and or 0.1% is unproductive land. Lausanne has a population () of . Most of the population () speaks French (159,559 or 79.6%), with German being second most common (8,534 or 4.
Romanel-sur-LausanneRomanel-sur-Lausanne (ʁɔmanɛl syʁ lɔzan, literally Romanel on Lausanne; Romanél) is a municipality in the canton of Vaud in Switzerland, located in the district of Lausanne. Romanel-sur-Lausanne is first mentioned in 1184 as Romanes. Romanel-sur-Lausanne has an area, , of (depending on calculation method). Of this area, or 59.7% is used for agricultural purposes, while or 3.8% is forested. Of the rest of the land, or 37.8% is settled (buildings or roads). Of the built up area, industrial buildings made up 5.
Le Mont-sur-LausanneLe Mont-sur-Lausanne (lə mɔ̃ syʁ lɔzan, literally Le Mont on Lausanne; Lo Mont) is a municipality in the district of Lausanne in the canton of Vaud in Switzerland. It is a suburb of the city of Lausanne. Le Mont-sur-Lausanne is first mentioned in 1237 as Monte super Lausannam. Le Mont-sur-Lausanne has an area, , of (depending on calculation method). Of this area, or 51.0% is used for agricultural purposes, while or 18.2% is forested. Of the rest of the land, or 30.0% is settled (buildings or roads).
Cheseaux-sur-LausanneCheseaux-sur-Lausanne (ʃəzo syʁ lɔzan, literally Cheseaux on Lausanne; Chesâls) is a municipality in the district of Lausanne in the canton of Vaud in Switzerland. It is a suburb of the city of Lausanne. Cheseaux-sur-Lausanne is first mentioned in 1228 as Chesaus. Cheseaux-sur-Lausanne has an area, , of . Of this area, or 60.3% is used for agricultural purposes, while or 14.6% is forested. Of the rest of the land, or 24.0% is settled (buildings or roads), or 0.7% is either rivers or lakes.
Belmont-sur-LausanneBelmont-sur-Lausanne (bɛlmɔ̃ syʁ lɔzan, literally Belmont on Lausanne) is a municipality in the district of Lavaux-Oron in the canton of Vaud in Switzerland. It is a suburb of the city of Lausanne. Belmont-sur-Lausanne is first mentioned in 1228 as Belmunt sowie apud bellum Montem. Belmont-sur-Lausanne has an area, , of . Of this area, or 35.1% is used for agricultural purposes, while or 29.8% is forested. Of the rest of the land, or 33.2% is settled (buildings or roads). Of the built up area, housing and buildings made up 23.
DataIn common usage and statistics, data (USˈdætə; UKˈdeɪtə) is a collection of discrete or continuous values that convey information, describing the quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted formally. A datum is an individual value in a collection of data. Data is usually organized into structures such as tables that provide additional context and meaning, and which may themselves be used as data in larger structures.
ExistenceExistence is the ability of an entity to interact with reality. In philosophy, it refers to the ontological property of being. The term existence comes from Old French existence, from Medieval Latin existentia/exsistentia, from Latin existere, to come forth, be manifest, ex + sistere, to stand. Materialism holds that the only things that exist are matter and energy, that all things are composed of material, that all actions require energy, and that all phenomena (including consciousness) are the result of the interaction of matter.
Existence of GodThe existence of God (or more generally, the existence of deities) is a subject of debate in theology, philosophy of religion and popular culture. A wide variety of arguments for and against the existence of God or deities can be categorized as logical, empirical, metaphysical, subjective or scientific. In philosophical terms, the question of the existence of God or deities involves the disciplines of epistemology (the nature and scope of knowledge) and ontology (study of the nature of being or existence) and the theory of value (since some definitions of God include "perfection").
Big dataBig data primarily refers to data sets that are too large or complex to be dealt with by traditional data-processing application software. Data with many entries (rows) offer greater statistical power, while data with higher complexity (more attributes or columns) may lead to a higher false discovery rate. Though used sometimes loosely partly because of a lack of formal definition, the interpretation that seems to best describe big data is the one associated with a large body of information that we could not comprehend when used only in smaller amounts.
Data analysisData analysis is the process of inspecting, cleansing, transforming, and modeling data with the goal of discovering useful information, informing conclusions, and supporting decision-making. Data analysis has multiple facets and approaches, encompassing diverse techniques under a variety of names, and is used in different business, science, and social science domains. In today's business world, data analysis plays a role in making decisions more scientific and helping businesses operate more effectively.
