Heuristic lumped Stirling of mathematics, cardinality, induction, and the axiom of choice at Hilbert's paradox of the Grand Hotel
A heuristic technique, often called simply a heuristic, is any approach to problem solving, learning, or discovery that employs a practical method, not guaranteed to be optimal, perfect, logical, or rational, but instead sufficient for reaching an immediate goal.
Good hobbit, bad hobbit math theory
In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result. Commutative diagrams play the role in category theory that equations play in algebra .
In mathematics, a hierarchy is a set-theoretical object, consisting of a preorder defined on a set. This is often referred to as an ordered set, though that is an ambiguous term that many authors reserve for partially ordered sets or totally ordered sets. The term pre-ordered set is unambiguous, and is always synonymous with a mathematical hierarchy. The term hierarchy is used to stress a hierarchical relation among the elements.
In mathematics, and more specifically in naive set theory, the range of a function refers to either the codomain or the image of the function, depending upon usage. Modern usage almost always uses range to mean image.
In mathematics, an image is the subset of a function's codomain which is the output of the function from a subset of its domain.
Evaluating a function at each element of a subset X of the domain, produces a set called the image of X under or through the function. The inverse image or preimage of a particular subset S of the codomain of a function is the set of all elements of the domain that map to the members of S.
Image and inverse image may also be defined for general binary relations, not just functions.
For Lipschitz continuity sake, let us consider the Slutsky equation.
From, Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving, Book by Sanjoy Mahajan, Ch. 3 Lumping,
Full Width at Half Maximum FWHM heuristic, Stirling's Aproximation are discussed.
In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other.
Anaconda Python or Excel statistical analysis of mass convergence in a Debye torodial moment with lepton kinetic enthalpy
"Injective, Surjective and Bijective" tells us about how a function behaves.
The lumped element model (also called lumped parameter model, or lumped component model) simplifies the description of the behaviour of spatially distributed physical systems into a topology consisting of discrete entities that approximate the behaviour of the distributed system under certain assumptions. It is useful in electrical systems (including electronics), mechanical multibody systems, heat transfer, acoustics, etc.
Mathematically speaking, the simplification reduces the state space of the system to a finite dimension, and the partial differential equations (PDEs) of the continuous (infinite-dimensional) time and space model of the physical system into ordinary differential equations (ODEs) with a finite number of parameters.
1.2.4 Lumping
Mathematical Modelling and Computers in Endocrinology
By Rosalind McIntosh
The Sexy prime Freudenthal–Tits magic square Lie Algebra
What does the phrase like it or lump it mean?
like it or lump it. informal. If you tell someone to like it or lump it, you mean that person must accept a situation they do not like, because it cannot be changed: Like it or lump it, romantic fiction is read regularly by thousands.
Stirling's Approximation for n!
What is a take it or leave it contract?
A standard form contract (sometimes referred to as a contract of adhesion, a leonine contract, a take-it-or-leave-it contract, or a boilerplate contract) is a contract between two parties, where the terms and conditions of the contract are set by one of the parties, and the other party has little or no ability to negotiate more favorable terms and is thus placed in a "take it or leave it" position.
The electron phonon interaction that forms Cooper pairs in low-Tc superconductors (type-I superconductors) can also be modeled as a polaron, and two opposite spin electrons may form a bipolaron sharing a phonon cloud.
Valleytronics is a portmanteau combining the terms valley and electronics. Certain semiconductors present multiple "valleys" in the first Brillouin zone, and are known as multivalley semiconductors. The term refers to the technology of control over the valley degree of freedom, a local maximum/minimum on the valence/conduction band, of such multivalley semiconductors.
The term was coined in analogy to the blooming field of spintronics. While in spintronics the internal degree of freedom of spin is harnessed to store, manipulate and read out bits of information, the proposal for valleytronics is to perform similar tasks using the multiple extrema of the band structure, so that the information of 0s and 1s would be stored as different discrete values of the crystal momentum.
The term is often used as an umbrella term to other forms of quantum manipulation of valleys in semiconductors, including quantum computation with valley-based qubits,valley blockade and other forms of quantum electronics. First experimental evidence of valley blockade predicted in Ref. (which completes the set of Coulomb charge blockade and Pauli spin blockade) has been observed in a single atom doped silicon transistor.
Several theoretical proposals and experiments were performed in a variety of systems, such as graphene, few-layer phosphorene, some transition metal dichalcogenide monolayers, diamond, bismuth, silicon, carbon nanotubes, aluminium arsenide and silicene.
Debye toroidal moment of surface plasmons as SBIR ESCO model
By David Hirsch
In mathematics, Stirling's approximation is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though it was first stated by Abraham de Moivre.
Stirling's Formula
In mathematics, particularly in combinatorics, a Stirling number of the second kind or Stirling partition number is the number of ways to partition a set of n objects into k non-empty subsets and is denoted by S ( n , k ) or n/k. Stirling numbers of the second kind occur in the field of mathematics called combinatorics and the study of partitions.
Loschmidt's paradox, also known as the reversibility paradox, irreversibility paradox or Umkehreinwand,is the objection that it should not be possible to deduce an irreversible process from time-symmetric dynamics. This puts the time reversal symmetry of (almost) all known low-level fundamental physical processes at odds with any attempt to infer from them the second law of thermodynamics which describes the behaviour of macroscopic systems. Both of these are well-accepted principles in physics, with sound observational and theoretical support, yet they seem to be in conflict; hence the paradox.
Connective
A function, or the symbol representing a function, which corresponds to English conjunctions such as "and," "or," "not," etc. that takes one or more truth values as input and returns a single truth value as output. The terms "logical connective" and "propositional connective" (Mendelson 1997, p. 13) are also used.
A derangement is a permutation with no fixed points. That is, a derangement of a set leaves no element in its original place. For example, the derangements of {1,2,3} are {2, 3, 1} and {3, 1, 2}, but {3,2, 1} is not a derangement of {1,2,3} because 2 is a fixed point.
In combinatorial mathematics, the rencontres numbers are a triangular array of integers that enumerate permutations of the set { 1, ..., n } with specified numbers of fixed points: in other words, partial derangements. (Rencontre is French for encounter. By some accounts, the problem is named after a solitaire game.) For n = 0 and 0 = k = n, the rencontres number Dn, k is the number of permutations of { 1, ..., n } that have exactly k fixed points.
Rencontrer Steep Chandelle, Snoopy's Red Barron Antworten for Isle of Dogs
For example, if seven presents are given to seven different people, but only two are destined to get the right present, there are D7, 2 = 924 ways this could happen. Another often cited example is that of a dance school with 7 couples, where after tea-break the participants are told to randomly find a partner to continue, and there are D7, 2 = 924 possibilities once more, now, that 2 previous couples meet again just by chance.
QCD: Quantum Chromodynamics
An exergonic reaction refers to a reaction where energy is released. Because the reactants lose energy (G decreases), Gibbs free energy (?G) is negative under constant temperature and pressure. However, some exergonic reactions do not occur spontaneously and require a small input of energy to start the reaction. This input of energy is called activation energy. Once the activation energy requirement is fulfilled by an outside source, the reaction proceeds to break bonds and form new bonds and energy is released as the reaction takes place.
QED: Photons -- Corpuscles of Light -- Richard Feynman (1/4)
Entropy (arrow of time)
Entropy is the only quantity in the physical sciences (apart from certain rare interactions in particle physics; see below) that requires a particular direction for time, sometimes called an arrow of time. As one goes "forward" in time, the second law of thermodynamics says, the entropy of an isolated system can increase, but not decrease. Hence, from one perspective, entropy measurement is a way of distinguishing the past from the future. However, in thermodynamic systems that are not closed, entropy can decrease with time: many systems, including living systems, reduce local entropy at the expense of an environmental increase, resulting in a net increase in entropy. Examples of such systems and phenomena include the formation of typical crystals, the workings of a refrigerator and living organisms, used in thermodynamics.
Much like temperature, despite being an abstract concept, everyone has an intuitive sense of the effects of entropy. For example, it is often very easy to tell the difference between a video being played forwards or backwards. A video may depict a wood fire that melts a nearby ice block, played in reverse it would show that a puddle of water turned a cloud of smoke into unburnt wood and froze itself in the process. Surprisingly, in either case the vast majority of the laws of physics are not broken by these processes, a notable exception is the second law of thermodynamics. When a law of physics applies equally when time is reversed it is said to show T-symmetry, in this case entropy is what allows one to decide if the video described above is playing forwards or in reverse as intuitively we identify that only when played forwards the entropy of the scene is increasing. Because of the second law of thermodynamics entropy prevents macroscopic processes showing T-symmetry.
The Fourth Phase of Water: Dr. Gerald Pollack at TEDxGuelphU
Human consciousness is LINKED to the laws of the UNIVERSE
THE human consciousness is linked to the chaotic nature of the universe, according to an astonishing new theory.
By Sean Martin
Making Clean Water with a Spark of Electricity | Korneel Rabaey | TEDxBrussels
Roy's identity (named for French economist René Roy) is a major result in microeconomics having applications in consumer choice and the theory of the firm. The lemma relates the ordinary (Marshallian) demand function to the derivatives of the indirect utility function.
Power Sets
What Is Happening In Saudi Arabia? - Marwa Osman on The Corbett Report
Opinion
If a Prince Murders a Journalist, That’s Not a Hiccup
In the end, Saudi Arabia played Kushner, Trump and his other American acolytes for suckers.
By Nicholas Kristof
Turbulence, the oldest unsolved problem in physics
The flow of water through a pipe is still in many ways an unsolved problem.
By Lee Phillips
Chetaev instability theorem
Trump Panama Building A Magnet For Dirty Money Laundering | On Assignment with Richard Engel | MSNBC
Trump says U.S. would be 'punishing' itself if it halts Saudi arms sales
In mathematics, and in particular, in the mathematical background of string theory, the Goddard–Thorn theorem (also called the no-ghost theorem) is a theorem describing properties of a functor that quantizes bosonic strings. The name "no-ghost theorem" stems from the fact that in the original statement of the theorem, the natural inner product induced on the output vector space is positive definite. Thus, there were no so-called ghosts (Pauli–Villars ghosts), or vectors of negative norm.
In theoretical physics, a no-go theorem is a theorem that states that a particular situation is not physically possible. Specifically, the term describes results in quantum mechanics like Bell's theorem and the Kochen–Specker theorem that constrain the permissible types of hidden variable theories which try to explain the apparent randomness of quantum mechanics as a deterministic model featuring hidden states.
Gog and Magog
The bosonic string quantization functors described here can be applied to any conformal vertex algebra of central charge 26, and the output naturally has a Lie algebra structure. The Goddard–Thorn theorem can then be applied to concretely describe the Lie algebra in terms of the input vertex algebra.
Perhaps the most spectacular case of this application is Borcherds's proof of the Monstrous Moonshine conjecture, where the unitarizable Virasoro representation is the Monster vertex algebra (also called "Moonshine module") constructed by Frenkel, Lepowsky, and Meurman. By taking a tensor product with the vertex algebra attached to a rank 2 hyperbolic lattice, and applying quantization, one obtains the monster Lie algebra, which is a generalized Kac–Moody algebra graded by the lattice. By using the Goddard–Thorn theorem, Borcherds showed that the homogeneous pieces of the Lie algebra are naturally isomorphic to graded pieces of the Moonshine module, as representations of the monster simple group.
Earlier applications include Frenkel's determination of upper bounds on the root multiplicities of the Kac-Moody Lie algebra whose Dynkin diagram is the Leech lattice, and Borcherds's construction of a generalized Kac-Moody Lie algebra that contains Frenkel's Lie algebra and saturates Frenkel's 1/∆ bound.
quantum physics
Famous Experiment Dooms Alternative to Quantum Weirdness
Oil droplets guided by “pilot waves” have failed to reproduce the results of the quantum double-slit experiment, crushing a century-old dream that there exists a single, concrete reality.
By Natalie Wolchover
We lump together the attractive forces between molecules, atoms, and ions as intermolecular forces (IMF). IMF is a general term for the attractions between discrete chemical units that we toss together under the general term "molecular."
All of these forces are driven by electrostatics, the forces felt between charged particles. In molecules these forces arise due to the electrons in the molecules.
We have three broad types of IMFs that we will deal with.
Dipole-Dipole forces. These are forces between polar molecules (a molecule with a permanent dipole moment, µ).
Hydrogen Bonding. This is a special case of extreme dipole-dipole forces that occurs when a hydrogen atom is bonded to a highly electronegative atom. The highly electronegative atom has to be N, O, or F.
Dispersion Forces. This one gets several names: Induced dipole-induced dipole, Dispersion forces, London forces, or van der waals forces. Dispersion Forces are often called the "weakest" but it is actually the most important since it is ubiquitous. Every molecule has dispersion forces. Not only do they have these forces, but they can add up across the whole molecule to be substantial. The degree to which a molecule has dispersion forces is measured via its polarizability, a. Polarizability (a) is to non-polar compounds, what dipole moment (µ) is to polar compounds.
In mathematics, the cardinality of a set is a measure of the "number of elements of the set". There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is also called its size, when no confusion with other notions of size is possible.
In quantum mechanics, bra–ket notation is a standard notation for describing quantum states. It can also be used to denote abstract vectors and linear functionals in mathematics. The notation uses angle brackets (the ? and ? symbols) and a vertical bar (the | symbol), to denote the scalar product of vectors or the action of a linear functional on a vector in a complex vector space.
The proton–proton chain reaction is one of two known sets of nuclear fusion reactions by which stars convert hydrogen to helium. It dominates in stars with masses less than or equal to that of the Sun's, whereas the CNO cycle, the other known reaction, is suggested by theoretical models to dominate in stars with masses greater than about 1.3 times that of the Sun's.
In general, proton–proton fusion can occur only if the kinetic energy (i.e. temperature) of the protons is high enough to overcome their mutual electrostatic or Coulomb repulsion.
Matthew 6:3
But when you give to someone in need, don't let your left hand know what your right hand is doing.
The triple-alpha process is a set of nuclear fusion reactions by which three helium-4 nuclei (alpha particles) are transformed into carbon.
The story of the Prodigal Son, also known as the Parable of the Lost Son, follows immediately after the parables of the Lost Sheep and the Lost Coin. The Prodigal Son was a young man who asked his father for his inheritance and then left home for “a far country, and there wasted his substance with riotous living.” The Parable of the Two Sons is a parable told by Jesus in the New Testament, found in Matthew (Matthew 21:28–32). It contrasts the tax collectors and prostitutes who accepted the message taught by John the Baptist and with the "religious" people who did not.
In set theory, the kernel of a function f may be taken to be either the equivalence relation on the function's domain that roughly expresses the idea of "equivalent as far as the function f can tell", or the corresponding partition of the domain.
"Those who live in glass houses should not throw stones" is a proverb used in several European countries. It means that one should not criticize others, because everybody has faults of one kind or another. Another less common but accurate meaning is: "One who is vulnerable to criticism should not criticize others."
Georg Cantor published his first set theory article in 1874, and it contains the first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is "Cantor's revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite. This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument. The title of the article, "On a Property of the Collection of All Real Algebraic Numbers" ("Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"), refers to its first theorem: the set of real algebraic numbers is countable.
Cantor's article also contains a proof of the existence of transcendental numbers. As early as 1930, mathematicians have disagreed on whether this proof is constructive or non-constructive. Books as recent as 2014 and 2015 indicate that this disagreement has not been resolved. Since Cantor's proof either constructs transcendental numbers or does not, an analysis of his article can determine whether his proof is constructive or non-constructive. Cantor's correspondence with Richard Dedekind shows the development of his ideas and reveals that he had a choice between two proofs, one that uses the uncountability of the real numbers and one that does not.
Historians of mathematics have examined Cantor's article and the circumstances in which it was written. For example, they have discovered that Cantor was advised to leave out his uncountability theorem in the article he submitted; he added it during proofreading. They have traced this and other facts about the article to the influence of Karl Weierstrass and Leopold Kronecker. Historians have also studied Dedekind's contributions to the article, including his contributions to the theorem on the countability of the real algebraic numbers. In addition, they have looked at the article's legacy, which includes the impact that the uncountability theorem and the concept of countability have had on mathematics.
Religion
Pope Accepts Resignation Of D.C. Archbishop Donald Wuerl Amid Sex Abuse Crisis
Mathematical induction is a mathematical proof technique. It is essentially used to prove that a property P(n) holds for every natural number n, i.e. for n = 0, 1, 2, 3, and so on. Metaphors can be informally used to understand the concept of mathematical induction, such as the metaphor of falling dominoes or climbing a ladder:
Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one (the step). —?Concrete Mathematics, page 3 margins.
The method of induction requires two cases to be proved. The first case, called the base case (or, sometimes, the basis), proves that the property holds for the number 0. The second case, called the induction step, proves that, if the property holds for one natural number n, then it holds for the next natural number n + 1. These two steps establish the property P(n) for every natural number n = 0, 1, 2, 3, ... The base step need not begin with zero. Often it begins with the number one, and it can begin with any natural number, establishing the truth of the property for all natural numbers greater than or equal to the starting number.
The method can be extended to prove statements about more general well-founded structures, such as trees; this generalization, known as structural induction, is used in mathematical logic and computer science. Mathematical induction in this extended sense is closely related to recursion. Mathematical induction, in some form, is the foundation of all correctness proofs for computer programs.
1 Maccabees 13
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if the collection is infinite.
Investing
The Big Problem With Machine Learning Algorithms
The potential for tapping new data sets is enormous, but the track record is mixed.
By Jon Asmundsson
In mathematics, a "helping theorem" or lemma (plural lemmas or lemmata) is a proven proposition which is used as a stepping stone to a larger result rather than as a statement of interest by itself. The word derives from the Ancient Greek ??µµa ("anything which is received, such as a gift, profit, or a bribe").
The 8-dimensional space that must be searched for alien life
In the theory of ordinary differential equations (ODEs), Lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov, Lyapunov functions (also called the Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory. A similar concept appears in the theory of general state space Markov chains, usually under the name Foster–Lyapunov functions.
For certain classes of ODEs, the existence of Lyapunov functions is a necessary and sufficient condition for stability. Whereas there is no general technique for constructing Lyapunov functions for ODEs, in many specific cases the construction of Lyapunov functions is known. For instance, quadratic functions suffice for systems with one state; the solution of a particular linear matrix inequality provides Lyapunov functions for linear systems; and conservation laws can often be used to construct Lyapunov functions for physical systems.
A triangular number or triangle number counts objects arranged in an equilateral triangle, as in the diagram on the right. The nth triangular number is the number of dots in the triangular arrangement with n dots on a side, and is equal to the sum of the n natural numbers from 1 to n. The triangular number is a figurate number that can be represented in the form of a triangular grid of points where the first row contains a single element and each subsequent row contains one more element than the previous one.
In mathematics, the law of trichotomy states that every real number is either positive, negative, or zero.
In plane geometry, a chord is the line segment joining two points on a curve.
In mathematics, a ternary operation is an n-ary operation with n = 3. A ternary operation on a set A takes any given three elements of A and combines them to form a single element of A. An example of a ternary operation is the product in a heap.
Sorting a heap of numbers from the ground lump complex vector bundles of triple alpha process, may be done with Folkman's theorem, Ken Brown's lemma, and Solving Recurrences in Double Free set Chern Class Degrees of Freedom.
As in computer science, a ternary operator is an operator that takes three arguments. In computer science, heapsort is a comparison-based sorting algorithm. Heapsort can be thought of as an improved selection sort: like that algorithm, it divides its input into a sorted and an unsorted region, and it iteratively shrinks the unsorted region by extracting the largest element and moving that to the sorted region.
Hilbert's paradox of the Grand Hotel (colloquial: Infinite Hotel Paradox or Hilbert's Hotel) is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated that a fully occupied hotel with infinitely many rooms may still accommodate additional guests, even infinitely many of them, and this process may be repeated infinitely often. The idea was introduced by David Hilbert in a 1924 lecture "Über das Unendliche", reprinted in (Hilbert 2013, p.730), and was popularized through George Gamow's 1947 book One Two Three... Infinity.
In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result. Commutative diagrams play the role in category theory that equations play in algebra .
In mathematics, a hierarchy is a set-theoretical object, consisting of a preorder defined on a set. This is often referred to as an ordered set, though that is an ambiguous term that many authors reserve for partially ordered sets or totally ordered sets. The term pre-ordered set is unambiguous, and is always synonymous with a mathematical hierarchy. The term hierarchy is used to stress a hierarchical relation among the elements.
In mathematics, and more specifically in naive set theory, the range of a function refers to either the codomain or the image of the function, depending upon usage. Modern usage almost always uses range to mean image.
In mathematics, an image is the subset of a function's codomain which is the output of the function from a subset of its domain.
Evaluating a function at each element of a subset X of the domain, produces a set called the image of X under or through the function. The inverse image or preimage of a particular subset S of the codomain of a function is the set of all elements of the domain that map to the members of S.
Image and inverse image may also be defined for general binary relations, not just functions.
For Lipschitz continuity sake, let us consider the Slutsky equation.
From, Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving, Book by Sanjoy Mahajan, Ch. 3 Lumping,
Full Width at Half Maximum FWHM heuristic, Stirling's Aproximation are discussed.
In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other.
Anaconda Python or Excel statistical analysis of mass convergence in a Debye torodial moment with lepton kinetic enthalpy
"Injective, Surjective and Bijective" tells us about how a function behaves.
The lumped element model (also called lumped parameter model, or lumped component model) simplifies the description of the behaviour of spatially distributed physical systems into a topology consisting of discrete entities that approximate the behaviour of the distributed system under certain assumptions. It is useful in electrical systems (including electronics), mechanical multibody systems, heat transfer, acoustics, etc.
Mathematically speaking, the simplification reduces the state space of the system to a finite dimension, and the partial differential equations (PDEs) of the continuous (infinite-dimensional) time and space model of the physical system into ordinary differential equations (ODEs) with a finite number of parameters.
1.2.4 Lumping
Mathematical Modelling and Computers in Endocrinology
By Rosalind McIntosh
The Sexy prime Freudenthal–Tits magic square Lie Algebra
What does the phrase like it or lump it mean?
like it or lump it. informal. If you tell someone to like it or lump it, you mean that person must accept a situation they do not like, because it cannot be changed: Like it or lump it, romantic fiction is read regularly by thousands.
Stirling's Approximation for n!
What is a take it or leave it contract?
A standard form contract (sometimes referred to as a contract of adhesion, a leonine contract, a take-it-or-leave-it contract, or a boilerplate contract) is a contract between two parties, where the terms and conditions of the contract are set by one of the parties, and the other party has little or no ability to negotiate more favorable terms and is thus placed in a "take it or leave it" position.
The electron phonon interaction that forms Cooper pairs in low-Tc superconductors (type-I superconductors) can also be modeled as a polaron, and two opposite spin electrons may form a bipolaron sharing a phonon cloud.
Experiments hint at a new type of electronics: valleytronics
Crystal structure of molybdenum disulfide creates valleys for flow of charge.
By Matthew FrancisValleytronics is a portmanteau combining the terms valley and electronics. Certain semiconductors present multiple "valleys" in the first Brillouin zone, and are known as multivalley semiconductors. The term refers to the technology of control over the valley degree of freedom, a local maximum/minimum on the valence/conduction band, of such multivalley semiconductors.
The term was coined in analogy to the blooming field of spintronics. While in spintronics the internal degree of freedom of spin is harnessed to store, manipulate and read out bits of information, the proposal for valleytronics is to perform similar tasks using the multiple extrema of the band structure, so that the information of 0s and 1s would be stored as different discrete values of the crystal momentum.
The term is often used as an umbrella term to other forms of quantum manipulation of valleys in semiconductors, including quantum computation with valley-based qubits,valley blockade and other forms of quantum electronics. First experimental evidence of valley blockade predicted in Ref. (which completes the set of Coulomb charge blockade and Pauli spin blockade) has been observed in a single atom doped silicon transistor.
Several theoretical proposals and experiments were performed in a variety of systems, such as graphene, few-layer phosphorene, some transition metal dichalcogenide monolayers, diamond, bismuth, silicon, carbon nanotubes, aluminium arsenide and silicene.
Debye toroidal moment of surface plasmons as SBIR ESCO model
By David Hirsch
In mathematics, Stirling's approximation is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though it was first stated by Abraham de Moivre.
Stirling's Formula
In mathematics, particularly in combinatorics, a Stirling number of the second kind or Stirling partition number is the number of ways to partition a set of n objects into k non-empty subsets and is denoted by S ( n , k ) or n/k. Stirling numbers of the second kind occur in the field of mathematics called combinatorics and the study of partitions.
Loschmidt's paradox, also known as the reversibility paradox, irreversibility paradox or Umkehreinwand,is the objection that it should not be possible to deduce an irreversible process from time-symmetric dynamics. This puts the time reversal symmetry of (almost) all known low-level fundamental physical processes at odds with any attempt to infer from them the second law of thermodynamics which describes the behaviour of macroscopic systems. Both of these are well-accepted principles in physics, with sound observational and theoretical support, yet they seem to be in conflict; hence the paradox.
A function, or the symbol representing a function, which corresponds to English conjunctions such as "and," "or," "not," etc. that takes one or more truth values as input and returns a single truth value as output. The terms "logical connective" and "propositional connective" (Mendelson 1997, p. 13) are also used.
James Woods deletes tweet calling Clinton bomb threat a 'political stunt'
By Aris Folley - 10/24/18A derangement is a permutation with no fixed points. That is, a derangement of a set leaves no element in its original place. For example, the derangements of {1,2,3} are {2, 3, 1} and {3, 1, 2}, but {3,2, 1} is not a derangement of {1,2,3} because 2 is a fixed point.
In combinatorial mathematics, the rencontres numbers are a triangular array of integers that enumerate permutations of the set { 1, ..., n } with specified numbers of fixed points: in other words, partial derangements. (Rencontre is French for encounter. By some accounts, the problem is named after a solitaire game.) For n = 0 and 0 = k = n, the rencontres number Dn, k is the number of permutations of { 1, ..., n } that have exactly k fixed points.
Rencontrer Steep Chandelle, Snoopy's Red Barron Antworten for Isle of Dogs
For example, if seven presents are given to seven different people, but only two are destined to get the right present, there are D7, 2 = 924 ways this could happen. Another often cited example is that of a dance school with 7 couples, where after tea-break the participants are told to randomly find a partner to continue, and there are D7, 2 = 924 possibilities once more, now, that 2 previous couples meet again just by chance.
QCD: Quantum Chromodynamics
An exergonic reaction refers to a reaction where energy is released. Because the reactants lose energy (G decreases), Gibbs free energy (?G) is negative under constant temperature and pressure. However, some exergonic reactions do not occur spontaneously and require a small input of energy to start the reaction. This input of energy is called activation energy. Once the activation energy requirement is fulfilled by an outside source, the reaction proceeds to break bonds and form new bonds and energy is released as the reaction takes place.
Entropy (arrow of time)
Entropy is the only quantity in the physical sciences (apart from certain rare interactions in particle physics; see below) that requires a particular direction for time, sometimes called an arrow of time. As one goes "forward" in time, the second law of thermodynamics says, the entropy of an isolated system can increase, but not decrease. Hence, from one perspective, entropy measurement is a way of distinguishing the past from the future. However, in thermodynamic systems that are not closed, entropy can decrease with time: many systems, including living systems, reduce local entropy at the expense of an environmental increase, resulting in a net increase in entropy. Examples of such systems and phenomena include the formation of typical crystals, the workings of a refrigerator and living organisms, used in thermodynamics.
Much like temperature, despite being an abstract concept, everyone has an intuitive sense of the effects of entropy. For example, it is often very easy to tell the difference between a video being played forwards or backwards. A video may depict a wood fire that melts a nearby ice block, played in reverse it would show that a puddle of water turned a cloud of smoke into unburnt wood and froze itself in the process. Surprisingly, in either case the vast majority of the laws of physics are not broken by these processes, a notable exception is the second law of thermodynamics. When a law of physics applies equally when time is reversed it is said to show T-symmetry, in this case entropy is what allows one to decide if the video described above is playing forwards or in reverse as intuitively we identify that only when played forwards the entropy of the scene is increasing. Because of the second law of thermodynamics entropy prevents macroscopic processes showing T-symmetry.
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Roy's identity (named for French economist René Roy) is a major result in microeconomics having applications in consumer choice and the theory of the firm. The lemma relates the ordinary (Marshallian) demand function to the derivatives of the indirect utility function.
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In mathematics, and in particular, in the mathematical background of string theory, the Goddard–Thorn theorem (also called the no-ghost theorem) is a theorem describing properties of a functor that quantizes bosonic strings. The name "no-ghost theorem" stems from the fact that in the original statement of the theorem, the natural inner product induced on the output vector space is positive definite. Thus, there were no so-called ghosts (Pauli–Villars ghosts), or vectors of negative norm.
In theoretical physics, a no-go theorem is a theorem that states that a particular situation is not physically possible. Specifically, the term describes results in quantum mechanics like Bell's theorem and the Kochen–Specker theorem that constrain the permissible types of hidden variable theories which try to explain the apparent randomness of quantum mechanics as a deterministic model featuring hidden states.
Gog and Magog
The bosonic string quantization functors described here can be applied to any conformal vertex algebra of central charge 26, and the output naturally has a Lie algebra structure. The Goddard–Thorn theorem can then be applied to concretely describe the Lie algebra in terms of the input vertex algebra.
Perhaps the most spectacular case of this application is Borcherds's proof of the Monstrous Moonshine conjecture, where the unitarizable Virasoro representation is the Monster vertex algebra (also called "Moonshine module") constructed by Frenkel, Lepowsky, and Meurman. By taking a tensor product with the vertex algebra attached to a rank 2 hyperbolic lattice, and applying quantization, one obtains the monster Lie algebra, which is a generalized Kac–Moody algebra graded by the lattice. By using the Goddard–Thorn theorem, Borcherds showed that the homogeneous pieces of the Lie algebra are naturally isomorphic to graded pieces of the Moonshine module, as representations of the monster simple group.
Earlier applications include Frenkel's determination of upper bounds on the root multiplicities of the Kac-Moody Lie algebra whose Dynkin diagram is the Leech lattice, and Borcherds's construction of a generalized Kac-Moody Lie algebra that contains Frenkel's Lie algebra and saturates Frenkel's 1/∆ bound.
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We lump together the attractive forces between molecules, atoms, and ions as intermolecular forces (IMF). IMF is a general term for the attractions between discrete chemical units that we toss together under the general term "molecular."
All of these forces are driven by electrostatics, the forces felt between charged particles. In molecules these forces arise due to the electrons in the molecules.
We have three broad types of IMFs that we will deal with.
Dipole-Dipole forces. These are forces between polar molecules (a molecule with a permanent dipole moment, µ).
Hydrogen Bonding. This is a special case of extreme dipole-dipole forces that occurs when a hydrogen atom is bonded to a highly electronegative atom. The highly electronegative atom has to be N, O, or F.
Dispersion Forces. This one gets several names: Induced dipole-induced dipole, Dispersion forces, London forces, or van der waals forces. Dispersion Forces are often called the "weakest" but it is actually the most important since it is ubiquitous. Every molecule has dispersion forces. Not only do they have these forces, but they can add up across the whole molecule to be substantial. The degree to which a molecule has dispersion forces is measured via its polarizability, a. Polarizability (a) is to non-polar compounds, what dipole moment (µ) is to polar compounds.
In mathematics, the cardinality of a set is a measure of the "number of elements of the set". There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is also called its size, when no confusion with other notions of size is possible.
In quantum mechanics, bra–ket notation is a standard notation for describing quantum states. It can also be used to denote abstract vectors and linear functionals in mathematics. The notation uses angle brackets (the ? and ? symbols) and a vertical bar (the | symbol), to denote the scalar product of vectors or the action of a linear functional on a vector in a complex vector space.
The proton–proton chain reaction is one of two known sets of nuclear fusion reactions by which stars convert hydrogen to helium. It dominates in stars with masses less than or equal to that of the Sun's, whereas the CNO cycle, the other known reaction, is suggested by theoretical models to dominate in stars with masses greater than about 1.3 times that of the Sun's.
In general, proton–proton fusion can occur only if the kinetic energy (i.e. temperature) of the protons is high enough to overcome their mutual electrostatic or Coulomb repulsion.
Matthew 6:3
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The triple-alpha process is a set of nuclear fusion reactions by which three helium-4 nuclei (alpha particles) are transformed into carbon.
The story of the Prodigal Son, also known as the Parable of the Lost Son, follows immediately after the parables of the Lost Sheep and the Lost Coin. The Prodigal Son was a young man who asked his father for his inheritance and then left home for “a far country, and there wasted his substance with riotous living.” The Parable of the Two Sons is a parable told by Jesus in the New Testament, found in Matthew (Matthew 21:28–32). It contrasts the tax collectors and prostitutes who accepted the message taught by John the Baptist and with the "religious" people who did not.
In set theory, the kernel of a function f may be taken to be either the equivalence relation on the function's domain that roughly expresses the idea of "equivalent as far as the function f can tell", or the corresponding partition of the domain.
"Those who live in glass houses should not throw stones" is a proverb used in several European countries. It means that one should not criticize others, because everybody has faults of one kind or another. Another less common but accurate meaning is: "One who is vulnerable to criticism should not criticize others."
Georg Cantor published his first set theory article in 1874, and it contains the first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is "Cantor's revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite. This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument. The title of the article, "On a Property of the Collection of All Real Algebraic Numbers" ("Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"), refers to its first theorem: the set of real algebraic numbers is countable.
Cantor's article also contains a proof of the existence of transcendental numbers. As early as 1930, mathematicians have disagreed on whether this proof is constructive or non-constructive. Books as recent as 2014 and 2015 indicate that this disagreement has not been resolved. Since Cantor's proof either constructs transcendental numbers or does not, an analysis of his article can determine whether his proof is constructive or non-constructive. Cantor's correspondence with Richard Dedekind shows the development of his ideas and reveals that he had a choice between two proofs, one that uses the uncountability of the real numbers and one that does not.
Historians of mathematics have examined Cantor's article and the circumstances in which it was written. For example, they have discovered that Cantor was advised to leave out his uncountability theorem in the article he submitted; he added it during proofreading. They have traced this and other facts about the article to the influence of Karl Weierstrass and Leopold Kronecker. Historians have also studied Dedekind's contributions to the article, including his contributions to the theorem on the countability of the real algebraic numbers. In addition, they have looked at the article's legacy, which includes the impact that the uncountability theorem and the concept of countability have had on mathematics.
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Mathematical induction is a mathematical proof technique. It is essentially used to prove that a property P(n) holds for every natural number n, i.e. for n = 0, 1, 2, 3, and so on. Metaphors can be informally used to understand the concept of mathematical induction, such as the metaphor of falling dominoes or climbing a ladder:
Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one (the step). —?Concrete Mathematics, page 3 margins.
The method of induction requires two cases to be proved. The first case, called the base case (or, sometimes, the basis), proves that the property holds for the number 0. The second case, called the induction step, proves that, if the property holds for one natural number n, then it holds for the next natural number n + 1. These two steps establish the property P(n) for every natural number n = 0, 1, 2, 3, ... The base step need not begin with zero. Often it begins with the number one, and it can begin with any natural number, establishing the truth of the property for all natural numbers greater than or equal to the starting number.
The method can be extended to prove statements about more general well-founded structures, such as trees; this generalization, known as structural induction, is used in mathematical logic and computer science. Mathematical induction in this extended sense is closely related to recursion. Mathematical induction, in some form, is the foundation of all correctness proofs for computer programs.
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In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if the collection is infinite.
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In mathematics, a "helping theorem" or lemma (plural lemmas or lemmata) is a proven proposition which is used as a stepping stone to a larger result rather than as a statement of interest by itself. The word derives from the Ancient Greek ??µµa ("anything which is received, such as a gift, profit, or a bribe").
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In the theory of ordinary differential equations (ODEs), Lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov, Lyapunov functions (also called the Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory. A similar concept appears in the theory of general state space Markov chains, usually under the name Foster–Lyapunov functions.
For certain classes of ODEs, the existence of Lyapunov functions is a necessary and sufficient condition for stability. Whereas there is no general technique for constructing Lyapunov functions for ODEs, in many specific cases the construction of Lyapunov functions is known. For instance, quadratic functions suffice for systems with one state; the solution of a particular linear matrix inequality provides Lyapunov functions for linear systems; and conservation laws can often be used to construct Lyapunov functions for physical systems.
A triangular number or triangle number counts objects arranged in an equilateral triangle, as in the diagram on the right. The nth triangular number is the number of dots in the triangular arrangement with n dots on a side, and is equal to the sum of the n natural numbers from 1 to n. The triangular number is a figurate number that can be represented in the form of a triangular grid of points where the first row contains a single element and each subsequent row contains one more element than the previous one.
In mathematics, the law of trichotomy states that every real number is either positive, negative, or zero.
In plane geometry, a chord is the line segment joining two points on a curve.
In mathematics, a ternary operation is an n-ary operation with n = 3. A ternary operation on a set A takes any given three elements of A and combines them to form a single element of A. An example of a ternary operation is the product in a heap.
Sorting a heap of numbers from the ground lump complex vector bundles of triple alpha process, may be done with Folkman's theorem, Ken Brown's lemma, and Solving Recurrences in Double Free set Chern Class Degrees of Freedom.
As in computer science, a ternary operator is an operator that takes three arguments. In computer science, heapsort is a comparison-based sorting algorithm. Heapsort can be thought of as an improved selection sort: like that algorithm, it divides its input into a sorted and an unsorted region, and it iteratively shrinks the unsorted region by extracting the largest element and moving that to the sorted region.
Hilbert's paradox of the Grand Hotel (colloquial: Infinite Hotel Paradox or Hilbert's Hotel) is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated that a fully occupied hotel with infinitely many rooms may still accommodate additional guests, even infinitely many of them, and this process may be repeated infinitely often. The idea was introduced by David Hilbert in a 1924 lecture "Über das Unendliche", reprinted in (Hilbert 2013, p.730), and was popularized through George Gamow's 1947 book One Two Three... Infinity.
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