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The fourth eigenvalue, which is usually stable (very negative), represents the castoring behavior of the front wheel, as it tends to turn towards the direction in which the bike is traveling. Note that this idealized model does not exhibit the wobble or shimmy and rear wobble instabilities described above. They are see... | Wikipedia - Bike physics | null | null | null |
It is possible to calculate the optimal rotation R based on the matrix formula R = ( H T H ) 1 2 H − 1 {\displaystyle R=\left(H^{\mathsf {T}}H\right)^{\frac {1}{2}}H^{-1}} but implementing a numerical solution to this formula becomes complicated when all special cases are accounted for (for example, the case of H not h... | Wikipedia - Kabsch algorithm | null | null | null |
It is possible to calculate étale cohomology groups with constant coefficients of order divisible by the characteristic in a similar way, using the Artin–Schreier sequence 0 → Z / p Z → K → x ↦ x p − x K → 0 {\displaystyle 0\to \mathbf {Z} /p\mathbf {Z} \to K\ {\xrightarrow {{} \atop x\mapsto x^{p}-x}}\ K\to 0} instead... | Wikipedia - Étale cohomology | null | null | null |
It is possible to combine different materials, or different heat treatments, to produce desirable qualities in a blade. For example, the finest Japanese swords were routinely made of up to seven sections of metals and even poorer quality swords were often made of two. These would include soft irons that could absorb th... | Wikipedia - Sword blade | null | null | null |
Pattern welding involved forging together twisted bars of soft (bendable) low carbon and hard (brittle) higher carbon iron. This was done because furnaces of the time were typically able to produce only one grade or the other, and neither was well suited for more than a very limited use blade. The ability of modern ste... | Wikipedia - Sword blade | null | null | null |
Acid etching and polishing blades made of different grades of steel can be used to produce decorative or artistic effects. Japanese sword makers developed the technique of differential hardening by covering their sword blades in different thicknesses of clay before quenching. Thinner clay allowed the heated metal to co... | Wikipedia - Sword blade | null | null | null |
It is possible to control the sample size by multiplying the acceptance probability with a constant c {\displaystyle c} . For a larger sample size, pick c > 1 {\displaystyle c>1} and adjust the acceptance probability to min ( c a ( x i , y i ) , 1 ) {\displaystyle \min(ca(x_{i},y_{i}),1)} . For a smaller sample size, t... | Wikipedia - Local case-control sampling | null | null | null |
It is possible to create IBIs for use by minimally trained monitoring personnel, however the precision obtainable is lower than that conducted by trained professionals. Safeguards to assure robustness in spite of potential misidentifications or protocol variations require careful testing. Ongoing quality control by est... | Wikipedia - Index of biological integrity | null | null | null |
EPA has published guidance to assist volunteer programs in formulating IBIs and related findings. While IBIs from such programs are legally admissible in US courts, defending the validity of conclusions based solely on such results is unlikely to be feasible. | Wikipedia - Index of biological integrity | null | null | null |
Agreement among multiple IBIs from data collected by established professionals can be more conclusive. A case in point is the phenomenon that stream IBI scores indicate significant impairment, or partial ecological collapse where more than 10 to 15 percent of the immediately surrounding watershed is impervious due to u... | Wikipedia - Index of biological integrity | null | null | null |
It is possible to define another version of propositional calculus, which defines most of the syntax of the logical operators by means of axioms, and which uses only one inference rule. | Wikipedia - Sentential logic | null | null | null |
It is possible to define the notion of "tending to infinity" in the domain of f {\displaystyle f} , In this expression, the infinity is considered to be signed: either + ∞ {\displaystyle +\infty } or − ∞ {\displaystyle -\infty } . The "limit of f as x tends to positive infinity" is defined as follows. It is a real numb... | Wikipedia - Limit (mathematics) | null | null | null |
Equivalently, for any sequence x n → + ∞ {\displaystyle x_{n}\rightarrow +\infty } , we have f ( x n ) → L {\displaystyle f(x_{n})\rightarrow L} . It is also possible to define the notion of "tending to infinity" in the value of f {\displaystyle f} , The definition is given as follows. Given any real number M > 0 {\dis... | Wikipedia - Limit (mathematics) | null | null | null |
It is possible to devise abstract algebras whose models include formal languages with quantification, but progress has been slow and interest in such algebra has been limited. Three approaches have been devised to date: Relation algebra, invented by Augustus De Morgan, and developed by Charles Sanders Peirce, Ernst Sch... | Wikipedia - Range of quantification | null | null | null |
It is possible to distinguish four different categories of optimal MAPF solvers: Extensions of A*: algorithms in this category employ modified versions of the A* approach. Increasing Cost Tree Search: a novel formalization of the MAPF problem is proposed, that comprehends an increasing search tree and the corresponding... | Wikipedia - Multi-agent pathfinding | null | null | null |
It is possible to distinguish some common characteristics among the stress–strain curves of various groups of materials and, on this basis, to divide materials into two broad categories; namely, the ductile materials and the brittle materials. : 51 | Wikipedia - Stress-strain curve | null | null | null |
It is possible to enrich melanin with selenium instead of sulphur. This selenium analogue of pheomelanin has been successfully synthesized through chemical and biosynthetic routes using selenocystine as a feedstock. Due selenium's higher atomic number, the obtained selenomelanin can be expected to provide better protec... | Wikipedia - Catechol melanin | null | null | null |
It is possible to explore versions of kappa calculus with substructural types such as linear, affine, and ordered types. These extensions require eliminating or restricting the ! τ {\displaystyle !_{\tau }} expression. In such circumstances the × type operator is not a true cartesian product, and is generally written ⊗... | Wikipedia - Kappa calculus | null | null | null |
It is possible to express the property that a set T {\displaystyle T} of edges with a choice of root vertex r {\displaystyle r} forms a Trémaux tree, in the monadic second-order logic of graphs, and more specifically in the form of this logic called MSO2, which allows quantification over both vertex and edge sets. This... | Wikipedia - Normal tree | null | null | null |
This can be expressed logically as the statement that there does not exist a nonempty subset C {\displaystyle C} of T {\displaystyle T} for which each vertex is incident to either zero or two edges of C {\displaystyle C} . Every edge e {\displaystyle e} not in T {\displaystyle T} connects an ancestor-descendant pair of... | Wikipedia - Normal tree | null | null | null |
It can be expressed logically as the statement that, for all edges e {\displaystyle e} , there exists a subset P {\displaystyle P} of T {\displaystyle T} such that exactly two vertices, one of them r {\displaystyle r} , are incident to a single edge of P {\displaystyle P} , and such that both endpoints of e {\displayst... | Wikipedia - Normal tree | null | null | null |
It is possible to extend the definition of temperature even to systems of few particles, like in a quantum dot. The generalized temperature is obtained by considering time ensembles instead of configuration-space ensembles given in statistical mechanics in the case of thermal and particle exchange between a small syste... | Wikipedia - Static temperature | null | null | null |
It is possible to find a corresponding computational branch for every major field in physics: Computational mechanics consists of computational fluid dynamics (CFD), computational solid mechanics and computational contact mechanics.Computational electrodynamics is the process of modeling the interaction of electromagne... | Wikipedia - Computational biophysics | null | null | null |
It is possible to find a low rank approximation to an LU decomposition using a randomized algorithm. Given an input matrix A {\textstyle A} and a desired low rank k {\textstyle k} , the randomized LU returns permutation matrices P , Q {\textstyle P,Q} and lower/upper trapezoidal matrices L , U {\textstyle L,U} of size ... | Wikipedia - LU factorization | null | null | null |
It is possible to find an st-numbering, and a bipolar orientation, of a given graph with designated vertices s and t, in linear time using depth-first search. The algorithm of Tarjan (1986) uses a depth-first search that starts at vertex s and first traverses edge st. As in the depth-first-search based algorithm for te... | Wikipedia - Bipolar orientation | null | null | null |
The given graph will be biconnected (and will have a bipolar orientation) if and only if t is the only child of s in the depth-first search tree and low(v) < pre(v) for all vertices v other than s. Once these numbers have been computed, Tarjan's algorithm performs a second traversal of the depth-first search tree, main... | Wikipedia - Bipolar orientation | null | null | null |
As Tarjan shows, the vertex ordering resulting from this procedure gives an st-numbering of the given graph.Alternatively, efficient sequential and parallel algorithms may be based on ear decomposition. While the DFS-based algorithms above depend inherently on the special open ear decomposition caused by the underlying... | Wikipedia - Bipolar orientation | null | null | null |
An open ear decomposition exists if and only if the graph formed from the given graph by adding an edge st is biconnected (the same condition as the existence of a bipolar orientation), and it can be found in linear time. An st-orientation (and thus also an st-numbering) may be obtained easily by directing each ear in ... | Wikipedia - Bipolar orientation | null | null | null |
Whenever an ear is added, the endpoints of this ear must be checked on reachability, or, equivalently for the st-numbering, which vertex comes first in the preliminary st-numbering before. This obstacle can be solved in worst-case constant time by using the (somewhat involved) order data structure, or by more direct me... | Wikipedia - Bipolar orientation | null | null | null |
It is possible to find the hyperplanes guaranteed by the above theorems in O(Nd) steps. Also, if the 2d lists of the lower and upper endpoints of the intervals defining the boxes's ith coordinates are pre-sorted, then the best such hyperplane (according to a wide variety of optimality measures) may be found in O(Nd) st... | Wikipedia - Geometric separator | null | null | null |
It is possible to find the maximum clique, or the clique number, of an arbitrary n-vertex graph in time O(3n/3) = O(1.4422n) by using one of the algorithms described above to list all maximal cliques in the graph and returning the largest one. However, for this variant of the clique problem better worst-case time bound... | Wikipedia - Maximum clique problem | null | null | null |
Jian (1986) improved the time to O(20.304n) = O(1.2346n), and Robson (1986) improved it to O(20.276n) = O(1.2108n) time, at the expense of greater space usage. Robson's algorithm combines a similar backtracking scheme (with a more complicated case analysis) and a dynamic programming technique in which the optimal solut... | Wikipedia - Maximum clique problem | null | null | null |
These partial solutions are used to shortcut the backtracking recursion. The fastest algorithm known today is a refined version of this method by Robson (2001) which runs in time O(20.249n) = O(1.1888n).There has also been extensive research on heuristic algorithms for solving maximum clique problems without worst-case... | Wikipedia - Maximum clique problem | null | null | null |
It is possible to formalise PRA in such a way that it has no logical connectives at all—a sentence of PRA is just an equation between two terms. In this setting a term is a primitive recursive function of zero or more variables. Curry (1941) gave the first such system. The rule of induction in Curry's system was unusua... | Wikipedia - Primitive recursive arithmetic | null | null | null |
A later refinement was given by Goodstein (1954). The rule of induction in Goodstein's system is: F ( 0 ) = G ( 0 ) F ( S ( x ) ) = H ( x , F ( x ) ) G ( S ( x ) ) = H ( x , G ( x ) ) F ( x ) = G ( x ) . | Wikipedia - Primitive recursive arithmetic | null | null | null |
{\displaystyle {F(0)=G(0)\quad F(S(x))=H(x,F(x))\quad G(S(x))=H(x,G(x)) \over F(x)=G(x)}.} Here x is a variable, S is the successor operation, and F, G, and H are any primitive recursive functions which may have parameters other than the ones shown. The only other inference rules of Goodstein's system are substitution ... | Wikipedia - Primitive recursive arithmetic | null | null | null |
{\displaystyle {F(x)=G(x) \over F(A)=G(A)}\qquad {A=B \over F(A)=F(B)}\qquad {A=B\quad A=C \over B=C}.} Here A, B, and C are any terms (primitive recursive functions of zero or more variables). Finally, there are symbols for any primitive recursive functions with corresponding defining equations, as in Skolem's system ... | Wikipedia - Primitive recursive arithmetic | null | null | null |
In this way the propositional calculus can be discarded entirely. Logical operators can be expressed entirely arithmetically, for instance, the absolute value of the difference of two numbers can be defined by primitive recursion: P ( 0 ) = 0 P ( S ( x ) ) = x x − ˙ 0 = x x − ˙ S ( y ) = P ( x − ˙ y ) | x − y | = ( x −... | Wikipedia - Primitive recursive arithmetic | null | null | null |
It is possible to generalize the definition of a formal language from a set of finite sequences over a finite basis to include many other sets of mathematical structures, so long as they are built up by finitary means from finite materials. What's more, many of these families of formal structures are especially well-su... | Wikipedia - Truth-functional propositional calculus | null | null | null |
The exigencies of practical computation on formal languages frequently demand that text strings be converted into pointer structure renditions of parse graphs, simply as a matter of checking whether strings are well-formed formulas or not. Once this is done, there are many advantages to be gained from developing the gr... | Wikipedia - Truth-functional propositional calculus | null | null | null |
It is possible to give a formula for the volume of distinguished arithmetic surfaces from the arithmetic data with which it was constructed. Let O {\displaystyle {\mathcal {O}}} be a maximal order in the quaternion algebra A {\displaystyle A} of discriminant D A {\displaystyle D_{A}} over the field F {\displaystyle F} ... | Wikipedia - Arithmetic Fuchsian group | null | null | null |
The reader can check that if O = M 2 ( Z ) {\displaystyle {\mathcal {O}}=M_{2}(\mathbb {Z} )} the output of this formula recovers the well-known result that the hyperbolic volume of the modular surface equals π / 3 {\displaystyle \pi /3} . Coupled with the description of maximal subgroups and finiteness results for num... | Wikipedia - Arithmetic Fuchsian group | null | null | null |
It is possible to give a precise meaning to the claim that SN is the universe of ordinary mathematics; it is a model of Zermelo set theory, the axiomatic set theory originally developed by Ernst Zermelo in 1908. Zermelo set theory was successful precisely because it was capable of axiomatising "ordinary" mathematics, f... | Wikipedia - Universe (mathematics) | null | null | null |
The final step, forming S as an infinitary union, requires the axiom of replacement, which was added to Zermelo set theory in 1922 to form Zermelo–Fraenkel set theory, the set of axioms most widely accepted today. So while ordinary mathematics may be done in SN, discussion of SN goes beyond the "ordinary", into metamat... | Wikipedia - Universe (mathematics) | null | null | null |
Going back to X = {}, the empty set, and introducing the (standard) notation Vi for Si{}, V0 = {}, V1 = P{}, and so on as before. But what used to be called "superstructure" is now just the next item on the list: Vω, where ω is the first infinite ordinal number. This can be extended to arbitrary ordinal numbers: V i :=... | Wikipedia - Universe (mathematics) | null | null | null |
It is possible to interrupt or abort the queued stream instead of waiting for the stream to finish. This is done by specifying the data as urgent. This marks the transmission as out-of-band data (OOB) and tells the receiving program to process it immediately. When finished, TCP informs the application and resumes the s... | Wikipedia - TCP connection | null | null | null |
An example is when TCP is used for a remote login session where the user can send a keyboard sequence that interrupts or aborts the remotely-running program without waiting for the program to finish its current transfer.The urgent pointer only alters the processing on the remote host and doesn't expedite any processing... | Wikipedia - TCP connection | null | null | null |
It is possible to model the progress of most infectious diseases mathematically to discover the likely outcome of an epidemic or to help manage them by vaccination. This field tries to find parameters for various infectious diseases and to use those parameters to make useful calculations about the effects of a mass vac... | Wikipedia - Computational biomodeling | null | null | null |
It is possible to model the wheel mathematically with a system of constraint equations, and then prove that that system is nonholonomic. First, we define the configuration space. The wheel can change its state in three ways: having a different rotation about its axle, having a different steering angle, and being at a d... | Wikipedia - Non-holonomic system | null | null | null |
Thus, the configuration space is: We must now relate these variables to each other. We notice that as the wheel changes its rotation, it changes its position. The change in rotation and position implying velocities must be present, we attempt to relate angular velocity and steering angle to linear velocities by taking ... | Wikipedia - Non-holonomic system | null | null | null |
Now we do some algebraic manipulation to transform the equation to Pfaffian form so it is possible to test whether it is holonomic, starting with: Then, let's separate the variables from their coefficients (left side of equation, derived from above). We also realize that we can multiply all terms by d t {\displaystyle ... | Wikipedia - Non-holonomic system | null | null | null |
However, we can use mathematical intuition to try our best to prove that the system is nonholonomic on the first test. Considering the test equation is: we can see that if any of the terms A α {\displaystyle A_{\alpha }} , A β {\displaystyle A_{\beta }} , or A γ {\displaystyle A_{\gamma }} were zero, that that part of ... | Wikipedia - Non-holonomic system | null | null | null |
It is possible to prove that the harmonic series diverges by comparing its sum with an improper integral. Specifically, consider the arrangement of rectangles shown in the figure to the right. Each rectangle is 1 unit wide and 1 n {\displaystyle {\tfrac {1}{n}}} units high, so if the harmonic series converged then the ... | Wikipedia - Alternating harmonic series | null | null | null |
However, the area under the curve is given by a divergent improper integral, Because this integral does not converge, the sum cannot converge either.In the figure to the right, shifting each rectangle to the left by 1 unit, would produce a sequence of rectangles whose boundary lies below the curve rather than above it.... | Wikipedia - Alternating harmonic series | null | null | null |
It is possible to reason about von Neumann ordinals in NFU. Recall that a von Neumann ordinal is a transitive set A such that the restriction of membership to A is a strict well-ordering. This is quite a strong condition in the NFU context, since the membership relation involves a difference of type. A von Neumann ordi... | Wikipedia - Formalized mathematics | null | null | null |
The only von Neumann ordinals which can be shown to exist in NFU without additional assumptions are the concrete finite ones. However, the application of a permutation method can convert any model of NFU to a model in which every strongly cantorian ordinal is the order type of a von Neumann ordinal. This suggests that ... | Wikipedia - Formalized mathematics | null | null | null |
It is possible to reduce the number of matrix additions by instead using the following form discovered by Winograd: = {\displaystyle {\begin{bmatrix}a&b\\c&d\end{bmatrix}}{\begin{bmatrix}A&C\\B&D\end{bmatrix}}={\begin{bmatrix}aA+bB&w+v+(a+b-c-d)D\\w+u+d(B+C-A-D)&w+u+v\end{bmatrix}}} where u = (c - a)(C - D), v = (c ... | Wikipedia - Strassen algorithm | null | null | null |
It is possible to relate the filling factor to the resistivity and hence, to the conductivity of the system. When ν {\displaystyle \nu } is an integer, the Fermi energy lies in between Landau levels where there are no states available for carriers, so the conductivity becomes zero (it is considered that the magnetic fi... | Wikipedia - Hall conductivity | null | null | null |
If the longitudinal resistivity is zero and transversal is finite, then det ρ ≠ 0 {\displaystyle \det \rho \neq 0} . Thus both, the longitudinal conductivity and resistivity become zero. Instead, when ν {\displaystyle \nu } is a half-integer, the Fermi energy is located at the peak of the density distribution of some L... | Wikipedia - Hall conductivity | null | null | null |
This means that the conductivity will have a maximum . This distribution of minimums and maximums corresponds to ¨quantum oscillations¨ called Shubnikov–de Haas oscillations which become more relevant as the magnetic field increases. Obviously, the height of the peaks are larger as the magnetic field increases since th... | Wikipedia - Hall conductivity | null | null | null |
It is possible to scrub sulfur dioxide by using a cold solution of sodium sulfite; this forms a sodium hydrogen sulfite solution. By heating this solution it is possible to reverse the reaction to form sulfur dioxide and the sodium sulfite solution. Since the sodium sulfite solution is not consumed, it is called a rege... | Wikipedia - Flue gas desulfurization | null | null | null |
The application of this reaction is also known as the Wellman–Lord process. In some ways this can be thought of as being similar to the reversible liquid–liquid extraction of an inert gas such as xenon or radon (or some other solute which does not undergo a chemical change during the extraction) from water to another p... | Wikipedia - Flue gas desulfurization | null | null | null |
It is possible to set up equations correlating direct quantitative structure activity relationships with experimental properties, usually referred to as topological indices (TIs). Topological indices are used in the development of quantitative structure-activity relationships (QSARs) in which the biological activity or... | Wikipedia - Topology (chemistry) | null | null | null |
It is possible to simulate pointer behavior using an index to an (normally one-dimensional) array. Primarily for languages which do not support pointers explicitly but do support arrays, the array can be thought of and processed as if it were the entire memory range (within the scope of the particular array) and any in... | Wikipedia - Data pointer | null | null | null |
It is even theoretically possible, using the above technique, together with a suitable instruction set simulator to simulate any machine code or the intermediate (byte code) of any processor/language in another language that does not support pointers at all (for example Java / JavaScript). To achieve this, the binary c... | Wikipedia - Data pointer | null | null | null |
It is possible to store information using dot-like nanostructures created by the local oxidation of a surface. This storage uses the binary code considering the presence of a nanostructure as a 1 and its lack as a 0. In this way information can be stored in a small surface with a single SiO2 dot constituting a bit. In ... | Wikipedia - Local oxidation nanolithography | null | null | null |
It is possible to store the entire table precomputed, or just include the values n, y1, A, and an implementation of f −1(y) in the source code, and compute the remaining values when initializing the random number generator. As previously described, you can find xi = f −1(yi) and yi+1 = yi + A/xi. Repeat n − 1 times for... | Wikipedia - Ziggurat algorithm | null | null | null |
At the end, you should have yn = f(0). There will be some round-off error, but it is a useful sanity test to see that it is acceptably small. When actually filling in the table values, just assume that xn = 0 and yn = f(0), and accept the slight difference in layer n − 1's area as rounding error. | Wikipedia - Ziggurat algorithm | null | null | null |
It is possible to tessellate in non-Euclidean geometries such as hyperbolic geometry. A uniform tiling in the hyperbolic plane (that may be regular, quasiregular, or semiregular) is an edge-to-edge filling of the hyperbolic plane, with regular polygons as faces; these are vertex-transitive (transitive on its vertices),... | Wikipedia - Tiling (geometry) | null | null | null |
It is possible to test whether a given graph is a permutation graph, and if so construct a permutation representing it, in linear time.As a subclass of the perfect graphs, many problems that are NP-complete for arbitrary graphs may be solved efficiently for permutation graphs. For instance: the largest clique in a perm... | Wikipedia - Permutation graph | null | null | null |
It is possible to test whether a graph is bipartite, and to return either a two-coloring (if it is bipartite) or an odd cycle (if it is not) in linear time, using depth-first search. The main idea is to assign to each vertex the color that differs from the color of its parent in the depth-first search forest, assigning... | Wikipedia - Bipartite graphs | null | null | null |
If they do not, then the path in the forest from ancestor to descendant, together with the miscolored edge, form an odd cycle, which is returned from the algorithm together with the result that the graph is not bipartite. However, if the algorithm terminates without detecting an odd cycle of this type, then every edge ... | Wikipedia - Bipartite graphs | null | null | null |
It is possible to use IMC to perform metabolismic studies of living multicellular organisms—if they are small enough to be placed in IMC ampoules (Lamprecht & Becker 1988). IMC studies have been made of insect pupa metabolism during ventilating movements (Harak et al. 1996) and effects of chemical agents on pupal growt... | Wikipedia - Isothermal microcalorimetry (IMC) | null | null | null |
It is possible to use randomization in order to reduce the number of marks. The following randomized version of the recursive halving procedure achieves a proportional division using only O(n) mark queries on average. The idea is that, in each iteration, instead of asking all partners to make a half-value mark, only so... | Wikipedia - Even-Paz protocol | null | null | null |
Then a cut is made, and each group of n/2 partners divides its half recursively.In the worst case we still need n-1 marks per iteration so the worst-case number of marks required is O(n log n). However, on average only O(log n) marks are required per iteration; by solving a recurrence formula it is possible to show tha... | Wikipedia - Even-Paz protocol | null | null | null |
It is possible to use solvents to prepare solids by precipitation or by evaporation. At times, the solvent is a hydrothermal that is under pressure at temperatures higher than the normal boiling point. A variation on this theme is the use of flux methods, which use a salt with a relatively low melting point as the solv... | Wikipedia - Solid-state chemistry | null | null | null |
It is possible to use the tensor algebra T(V) to describe the symmetric algebra S(V). In fact, S(V) can be defined as the quotient algebra of T(V) by the two-sided ideal generated by the commutators v ⊗ w − w ⊗ v . {\displaystyle v\otimes w-w\otimes v.} It is straightforward to verify that the resulting algebra satisfi... | Wikipedia - Symmetric square | null | null | null |
Because of the universal property of the tensor algebra, a linear map f from V to a commutative algebra A extends to an algebra homomorphism T ( V ) → A {\displaystyle T(V)\rightarrow A} , which factors through S(V) because A is commutative. The extension of f to an algebra homomorphism S ( V ) → A {\displaystyle S(V)\... | Wikipedia - Symmetric square | null | null | null |
It is possible to view the Ising model as a Markov chain, as the immediate probability Pβ(ν) of transitioning to a future state ν only depends on the present state μ. The Metropolis algorithm is actually a version of a Markov chain Monte Carlo simulation, and since we use single-spin-flip dynamics in the Metropolis alg... | Wikipedia - Spin lattice | null | null | null |
It is possible to write down relativistic quantum Boltzmann equations for relativistic quantum systems in which the number of particles is not conserved in collisions. This has several applications in physical cosmology, including the formation of the light elements in Big Bang nucleosynthesis, the production of dark m... | Wikipedia - Collisionless Boltzmann equation | null | null | null |
It is possible, in some theories of quantum gravity, to calculate the quantum corrections to ordinary, classical black holes. Contrarily to conventional black holes, which are solutions of gravitational field equations of the general theory of relativity, quantum gravity black holes incorporate quantum gravity effects ... | Wikipedia - Artificial black hole | null | null | null |
It is proposed to determine in a series of m {\displaystyle m} observations the limit of error, beyond which all observations involving so great an error may be rejected, provided there are as many as n {\displaystyle n} such observations. The principle upon which it is proposed to solve this problem is, that the propo... | Wikipedia - Outliers in statistics | null | null | null |
It is proved by Morton & Wassermann (1989) that the BMW algebra C n ( ℓ , m ) {\displaystyle \mathrm {C} _{n}(\ell ,m)} is isomorphic to the Kauffman's tangle algebra K T n {\displaystyle \mathrm {KT} _{n}} , the isomorphism ϕ: C n → K T n {\displaystyle \phi \colon \mathrm {C} _{n}\to \mathrm {KT} _{n}} is defined by ... | Wikipedia - Birman–Wenzl algebra | null | null | null |
It is quite likely that a large range of cognitive operations are necessary to freely press a button. Research at least suggests that our conscious self does not initiate all behavior. Instead, the conscious self is somehow alerted to a given behavior that the rest of the brain and body are already planning and perform... | Wikipedia - Neuroscience of free will | null | null | null |
Unconscious processes may play a larger role in behavior than previously thought.It may be possible, then, that our intuitions about the role of our conscious "intentions" have led us astray; it may be the case that we have confused correlation with causation by believing that conscious awareness necessarily causes the... | Wikipedia - Neuroscience of free will | null | null | null |
The discovery that humans possess a determined will would have implications for moral responsibility or lack thereof. Neuroscientist and author Sam Harris believes that we are mistaken in believing the intuitive idea that intention initiates actions. | Wikipedia - Neuroscience of free will | null | null | null |
In fact, Harris is even critical of the idea that free will is "intuitive": he says careful introspection can cast doubt on free will. Harris argues: "Thoughts simply arise in the brain. What else could they do? | Wikipedia - Neuroscience of free will | null | null | null |
The truth about us is even stranger than we may suppose: The illusion of free will is itself an illusion". Neuroscientist Walter Jackson Freeman III, nevertheless, talks about the power of even unconscious systems and actions to change the world according to our intentions. He writes: "our intentional actions continual... | Wikipedia - Neuroscience of free will | null | null | null |
This dynamic system is the self in each of us, it is the agency in charge, not our awareness, which is constantly trying to keep up with what we do." To Freeman, the power of intention and action can be independent of awareness. An important distinction to make is the difference between proximal and distal intentions. | Wikipedia - Neuroscience of free will | null | null | null |
Proximal intentions are immediate in the sense that they are about acting now. For instance, a decision to raise a hand now or press a button now, as in Libet-style experiments. Distal intentions are delayed in the sense that they are about acting at a later point in time. For instance, deciding to go to the store late... | Wikipedia - Neuroscience of free will | null | null | null |
It is readily oxidized by atmospheric oxygen into formic acid. For this reason, commercial formaldehyde is typically contaminated with formic acid. Formaldehyde can be hydrogenated into methanol. | Wikipedia - CH2O | null | null | null |
It is recommended that polymorphs are identified (e.g. for ZnS where the two forms zincblende (cubic) and wurtzite (hexagonal)), as ZnS(c) and ZnS(h) respectively. == Notes and references == | Wikipedia - IUPAC nomenclature of inorganic chemistry 2005 | null | null | null |
It is recommended to read the proof of the Kruskal's algorithm first. The proof consists of two parts. First, it is proved that the edges that remain after the algorithm is applied form a spanning tree. Second, it is proved that the spanning tree is of minimal weight. | Wikipedia - Reverse-delete algorithm | null | null | null |
It is relatively easy to see that e v = e {\displaystyle e_{v}=e} if the signed message equals the verified message: r v = g s y e = g k − x e g x e = g k = r {\displaystyle r_{v}=g^{s}y^{e}=g^{k-xe}g^{xe}=g^{k}=r} , and hence e v = H ( r v ∥ M ) = H ( r ∥ M ) = e {\displaystyle e_{v}=H(r_{v}\parallel M)=H(r\parallel M... | Wikipedia - Schnorr signature | null | null | null |
It is routine to check that the real numbers R, the complex numbers C and the quaternions H are examples of associative Euclidean Hurwitz algebras with their standard norms and involutions. There are moreover natural inclusions R ⊂ C ⊂ H. Analysing such an inclusion leads to the Cayley–Dickson construction, formalized ... | Wikipedia - Euclidean Hurwitz algebra | null | null | null |
Pick a unit vector j in A orthogonal to B. Since (j, 1) = 0, it follows that j* = −j and hence j2 = −1. Let C be subalgebra generated by B and j. It is unital and is again a Euclidean Hurwitz algebra. It satisfies the following Cayley–Dickson multiplication laws: C = B ⊕ B j , ( a + b j ) ∗ = a ∗ − b j , ( a + b j ) ( ... | Wikipedia - Euclidean Hurwitz algebra | null | null | null |
{\displaystyle \displaystyle {C=B\oplus Bj,\,\,\,(a+bj)^{*}=a^{*}-bj,\,\,\,(a+bj)(c+dj)=(ac-d^{*}b)+(bc^{*}+da)j.}} B and B j are orthogonal, since j is orthogonal to B. If a is in B, then j a = a* j, since by orthogonal 0 = 2 (j, a*) = j a − a* j. The formula for the involution follows. To show that B ⊕ B j is closed ... | Wikipedia - Euclidean Hurwitz algebra | null | null | null |
(j c)b = j(b c) taking adjoints above. (b j)(c j) = −c* b since (b, c j) = 0, so that, for x in A, ((b j)(c j), x) = −((c j)x*, b j) = (b x*, (c j)j) = −(c* b, x).Imposing the multiplicativity of the norm on C for a + b j and c + d j gives: ( ‖ a ‖ 2 + ‖ b ‖ 2 ) ( ‖ c ‖ 2 + ‖ d ‖ 2 ) = ‖ a c − d ∗ b ‖ 2 + ‖ b c ∗ + d a... | Wikipedia - Euclidean Hurwitz algebra | null | null | null |
Hence d(a c) = (d a)c, so that B must be associative. This analysis applies to the inclusion of R in C and C in H. Taking O = H ⊕ H with the product and inner product above gives a noncommutative nonassociative algebra generated by J = (0, 1). This recovers the usual definition of the octonions or Cayley numbers. | Wikipedia - Euclidean Hurwitz algebra | null | null | null |
If A is a Euclidean algebra, it must contain R. If it is strictly larger than R, the argument above shows that it contains C. If it is larger than C, it contains H. If it is larger still, it must contain O. But there the process must stop, because O is not associative. In fact H is not commutative and a(b j) = (b a) j ... | Wikipedia - Euclidean Hurwitz algebra | null | null | null |
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