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Charge conjugation symmetry In quantum theory Charge_conjugation_symmetry > Informal overview > In quantum theory The creation and annihilation operators obey the canonical commutation relations, in that the one operator "undoes" what the other "creates". This implies that any given solution u ( p → , σ , n ) {\displaystyle u\left({\vec {p}},\sigma ,n\right)} must be paired with its "anti-solution" v ( p → , σ , n ) {\displaystyle v\left({\vec {p}},\sigma ,n\right)} so that one undoes or cancels out the other. The pairing is to be performed so that all symmetries are preserved.
DC distribution system (ship propulsion) Benefits DC_distribution_system_(ship_propulsion) > Benefits In addition to boosting efficiency by up to 20 percent, other benefits include space and weight savings of up to 30 percent and flexible placement of electrical equipment. This allows for significantly more cargo space and a more functional vessel layout where the electrical system is designed around the vessel functions and not vice versa. The efficiency improvement is mainly achieved from the system no longer being locked at a specific frequency (usually 60 Hz on ships), even though a 60 Hz power source can also be connected to the grid. This new freedom of being able to control each power source totally independently opens up numerous ways of optimizing fuel consumption.
Anisidine value Safety Anisidine_value > Safety p-Anisidine is a relatively toxic compound with a permissible exposure limit of 0.5 mg/m3.
Matrix representation Basic mathematical operations Matrix_representation > Basic mathematical operations An m × n (read as m by n) order matrix is a set of numbers arranged in m rows and n columns. Matrices of the same order can be added by adding the corresponding elements. Two matrices can be multiplied, the condition being that the number of columns of the first matrix is equal to the number of rows of the second matrix.
Genetic variations Post-Darwinian concepts of heritable variation Genetic_variations > History of genetic variation > Post-Darwinian concepts of heritable variation 1918 - Variance: In a seminal paper entitled "The correlation between relatives on the supposition of Mendelian inheritance", R.A. Fisher introduced the statistical concept of variance; the average of squared deviations of a collection of observations from their mean ( σ 2 = 1 I ∑ i = 1 I ( x i − μ ) 2 {\textstyle \sigma ^{2}={\frac {1}{I}}\sum _{i=1}^{I}(x_{i}-\mu )^{2}} ), where σ 2 {\displaystyle \sigma ^{2}} is the variance and μ {\displaystyle \mu } is the mean of the population from which the observations x i {\displaystyle x_{i}} are drawn).
Pulmonary heart disease Epidemiology Pulmonary_heart_disease > Epidemiology The epidemiology of pulmonary heart disease (cor pulmonale) accounts for 7% of all heart disease in the U.S According to Weitzenblum, et al., the mortality that is related to cor pulmonale is not easy to ascertain, as it is a complication of COPD.
Signal transition graphs Asynchronous Circuit Synthesis Signal_transition_graphs > Asynchronous Circuit Synthesis Various problems in the synthesis of asynchronous circuits from STG specification have been investigated. One of the ways for their classification is based on the analysis approaches used to represent the state space of the STG specification, such as explicit state spaces, unfolding of the underlying Petri net, structural analysis of Petri nets and direct mapping (syntax-direct translation) of STGs. These approaches are usually linked with the complexity of the algorithms of synthesis and, hence, run-time of the tools. On the other hand, some of these techniques impose certain constraints on the class of the Petri nets. For example, explicit state space based methods typically work for an arbitrary Petri net type, whereas some structural methods require that the underlying Petri net is a marked graph or a free-choice net.
Outline of electronics Microcontrollers Outline_of_electronics > General concepts > Microcontrollers Features Analog-to-digital converter Central processing unit (CPU) Clock generator (Quartz timing crystal, resonator or RC circuit) Debugging support Digital-to-analog converters Discrete input and output bits In-circuit programming Non-volatile memory (ROM, EPROM, EEPROM or Flash) Peripherals (Timers, event counters, PWM generators, and watchdog) Serial interface (Input/output such as serial ports (UARTs)) Serial communications (I²C, Serial Peripheral Interface and Controller Area Network) Volatile memory (RAM) 8-bit microcontroller families:AVR - PIC - COP8 - MCS-48 - MCS-51 - Z8 - eZ80 - HC08 - HC11 - H8 - PSoC Some notable suppliers: ARM Atmel Cypress Semiconductor Freescale Intel MIPS Microchip Technology NXP Semiconductors Parallax Propeller PowerPC Rabbit 2000 Renesas RX, V850 Silicon Laboratories STMicroelectronics Texas Instruments Toshiba TLCS
Velvet worm Respiration Velvet_worms > Anatomy > Respiration Oxygen uptake occurs to an extent via simple diffusion through the entire body surface, with the coxal vesicles on the legs possibly being involved in some species. However, of most importance is gas exchange via fine unbranched tubes, the tracheae, which draw oxygen from the surface deep into the various organs, particularly the heart. The walls of these structures, which are less than three micrometers thick in their entirety, consist only of an extremely thin membrane through which oxygen can easily diffuse. The tracheae originate at tiny openings, the spiracles, which themselves are clustered together in dent-like recesses of the outer skin, the atria.
Telomeric repeat-containing RNAs Factors Regulating Transcription Telomeric_Repeat-Containing_RNA_(TERRA) > Structure & Biogenesis > Factors Regulating Transcription A number of proteins shown to associate with the ends of telomeres are involved in modulating TERRA transcription levels within cells. Factors involved in modulating the heterochromatic state at telomeres, such as histone deacetylase, SUV39H1 H3K9 histone methyltransferase, and DNA methyltransferase 3b (DNMT3B) in human cells, play an important role in regulating total TERRA expression. Additionally, the Shelterin component TRF1 has been shown to support TERRA transcription.
Point mutation Repeat-induced point mutation Point_mutations > Repeat-induced point mutation The RIP mutations do not seem to be limited to repeated sequences. Indeed, for example, in the phytopathogenic fungus L. maculans, RIP mutations are found in single copy regions, adjacent to the repeated elements. These regions are either non-coding regions or genes encoding small secreted proteins including avirulence genes.
Enzyme denaturation Loss of activity due to heavy metals and metalloids Protein_denaturation > Protein denaturation > How denaturation occurs at levels of protein structure > Loss of activity due to heavy metals and metalloids These heavy metals can form a complex with the functional side chain groups present in a protein or form bonds to free thiols. Heavy metals also play a role in oxidizing amino acid side chains present in protein. Along with this, when interacting with metalloproteins, heavy metals can dislocate and replace key metal ions. As a result, heavy metals can interfere with folded proteins, which can strongly deter protein stability and activity.
Info-gap decision theory Engineering Info-gap_decision_theory > Applications > Engineering Another example is the structural design of a building subject to uncertain loads such as from wind or earthquakes. The response of the structure depends strongly on the spatial and temporal distribution of the loads. However, storms and earthquakes are highly idiosyncratic events, and the interaction between the event and the structure involves very site-specific mechanical properties which are rarely known.
Unimolecular nucleophilic substitution Summary SN1_reaction Instead, the rate equation may be more accurately described using steady-state kinetics. The reaction involves a carbocation intermediate and is commonly seen in reactions of secondary or tertiary alkyl halides under strongly basic conditions or, under strongly acidic conditions, with secondary or tertiary alcohols.
Electronic filters Classification by topology Electronic_filters > Classification by topology Electronic filters can be classified by the technology used to implement them. Filters using passive filter and active filter technology can be further classified by the particular electronic filter topology used to implement them. Any given filter transfer function may be implemented in any electronic filter topology. Some common circuit topologies are: Cauer topology – passive Sallen–Key topology – active Multiple feedback topology – active State variable topology – active Biquadratic topology – active
Simulated Annealing The neighbours of a state Deterministic_annealing > Overview > The neighbours of a state Optimization of a solution involves evaluating the neighbours of a state of the problem, which are new states produced through conservatively altering a given state. For example, in the travelling salesman problem each state is typically defined as a permutation of the cities to be visited, and the neighbors of any state are the set of permutations produced by swapping any two of these cities. The well-defined way in which the states are altered to produce neighboring states is called a "move", and different moves give different sets of neighboring states.
Mark Diesendorf Grey literature Mark_Diesendorf > Publications > Grey literature The Conversation 9 May 2016; reprinted in RenewEconomy Renewable energy versus nuclear: dispelling the myths. The Ecologist, 19 April 2016, reprinted in RenewEconomy 22 April 2016. Dispelling the nuclear 'baseload' myth: nothing renewables can't do better!
Permanent magnets Magnetic moment Magnetic_polarity > Physics > Magnetic moment The strength of the magnetic field it produces is at any given point proportional to the magnitude of its magnetic moment. In addition, when the magnet is put into an external magnetic field, produced by a different source, it is subject to a torque tending to orient the magnetic moment parallel to the field. The amount of this torque is proportional both to the magnetic moment and the external field. A magnet may also be subject to a force driving it in one direction or another, according to the positions and orientations of the magnet and source. If the field is uniform in space, the magnet is subject to no net force, although it is subject to a torque.A wire in the shape of a circle with area A and carrying current I has a magnetic moment of magnitude equal to IA.
Weighted fair queueing WFQ as a GPS approximation Weighted_fair_queuing > WFQ as a GPS approximation WFQ, under the name PGPS, has been designed as "an excellent approximation to GPS", and it has been proved that it approximates GPS "to within one packet transmission time, regardless of the arrival patterns. "Since WFQ implementation is similar to fair queuing, it has the same O(log(n)) complexity, where n is the number of flows. This complexity comes from the need to select the queue with the smallest virtual finish time each time a packet is sent. After WFQ, several other implementations of GPS have been defined.
Post–Hartree–Fock Details Post–Hartree–Fock > Details In general, the SCF procedure makes several assumptions about the nature of the multi-body Schrödinger equation and its set of solutions: For molecules, the Born–Oppenheimer approximation is inherently assumed. The true wavefunction should also be a function of the coordinates of each of the nuclei. Typically, relativistic effects are completely neglected. The momentum operator is assumed to be completely nonrelativistic.
Dimensional stability (fabric) Significance Dimensional_stability_(fabric) > Significance The tests are conducted with provided specifications of buyers imitating the same conditions like washing cycle time, temperature and water ratio and fabric load and sometimes top loading and front loading washing machines are chosen to authenticate the test and assurance of the results. This procedure provides standard and alternate home laundering conditions using an automatic washing machine. While the procedure includes several options, it is not possible to include every existing combination of laundering parameters.
Proton pump inhibitors Mechanism of action Proton-pump_inhibitor > Mechanism of action Proton pump inhibitors act by irreversibly blocking the hydrogen/potassium adenosine triphosphatase enzyme system (the H+/K+ ATPase, or, more commonly, the gastric proton pump) of the gastric parietal cells. The proton pump is the terminal stage in gastric acid secretion, being directly responsible for secreting H+ ions into the gastric lumen, making it an ideal target for inhibiting acid secretion. Because the H,K-ATPase is the final step of acid secretion, an inhibitor of this enzyme is more effective than receptor antagonists in suppressing gastric acid secretion. All of these drugs inhibit the gastric H,K-ATPase by covalent binding, so the duration of their effect is longer than expected from their levels in the blood.Targeting the terminal step in acid production, as well as the irreversible nature of the inhibition, results in a class of medications that are significantly more effective than H2 antagonists and reduce gastric acid secretion by up to 99%.Decreasing the acid in the stomach can aid the healing of duodenal ulcers and reduce the pain from indigestion and heartburn.
Nucleophilic acyl substitution Carboxylic acids Nucleophilic_acyl_substitution > Reactions of acyl derivatives > Carboxylic acids Phosphorus(III) chloride (PCl3) and phosphorus(V) chloride (PCl5) will also convert carboxylic acids to acid chlorides, by a similar mechanism. One equivalent of PCl3 can react with three equivalents of acid, producing one equivalent of H3PO3, or phosphorus acid, in addition to the desired acid chloride. PCl5 reacts with carboxylic acids in a 1:1 ratio, and produces phosphorus(V) oxychloride (POCl3) and hydrogen chloride (HCl) as byproducts.
Visual attention Clinical model Visual_attention > Alternative topics and discussions > Clinical model Sustained attention (vigilance and concentration): The ability to maintain a consistent behavioral response during continuous and repetitive activity. Selective attention: The ability to maintain a behavioral or cognitive set in the face of distracting or competing stimuli. Therefore, it incorporates the notion of "freedom from distractibility." Alternating attention: The ability of mental flexibility that allows individuals to shift their focus of attention and move between tasks having different cognitive requirements. Divided attention: This refers to the ability to respond simultaneously to multiple tasks or multiple task demands.This model has been shown to be very useful in evaluating attention in very different pathologies, correlates strongly with daily difficulties and is especially helpful in designing stimulation programs such as attention process training, a rehabilitation program for neurological patients of the same authors.
Atomic-force microscopy Forces as a function of tip geometry Scanning_force_microscopy > Forces as a function of tip geometry The forces between the tip and the sample strongly depend on the geometry of the tip. Various studies were exploited in the past years to write the forces as a function of the tip parameters. Among the different forces between the tip and the sample, the water meniscus forces are highly interesting, both in air and liquid environment. Other forces must be considered, like the Coulomb force, van der Waals forces, double layer interactions, solvation forces, hydration and hydrophobic forces.
Inductive probability Universal artificial intelligence Inductive_probability > History > Universal artificial intelligence The theory of universal artificial intelligence applies decision theory to inductive probabilities. The theory shows how the best actions to optimize a reward function may be chosen. The result is a theoretical model of intelligence.It is a fundamental theory of intelligence, which optimizes the agents behavior in, Exploring the environment; performing actions to get responses that broaden the agents knowledge. Competing or co-operating with another agent; games.
Real-Time Messaging Protocol Handshake Real-Time_Messaging_Protocol > Protocol > Handshake It follows straight with C1 without waiting for S0 to be received first which contains 1536 bytes, with the first four representing the epoch timestamp, the second four all being 0, and the rest being random (and which can be set to 0 in third party implementations). C2 and S2 are an echo of S1 and C1 respectively, except with the second four bytes being the time the respective message was received (instead of 0). After C2 and S2 are received, the handshake is considered complete.
Nonlinear system Nonlinear algebraic equations Non-linear_differential_equations > Nonlinear algebraic equations For a single polynomial equation, root-finding algorithms can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of algebraic geometry, a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see Hilbert's Nullstellensatz). Nevertheless, in the case of the systems with a finite number of complex solutions, these systems of polynomial equations are now well understood and efficient methods exist for solving them.
Market anomaly Summary Market_anomaly Relatedly, return predictability by itself does not disprove the efficient market hypothesis, as one needs to show predictability over and above that implied by a particular model of risk.The four primary explanations for market anomalies are (1) mispricing, (2) unmeasured risk, (3) limits to arbitrage, and (4) selection bias. Academics have not reached a consensus on the underlying cause, with prominent academics continuing to advocate for selection bias, mispricing, and risk-based theories.Anomalies can be broadly categorized into time-series and cross-sectional anomalies. Time-series anomalies refer to predictability in the aggregate stock market, such as the often-discussed Cyclically Adjusted Price-Earnings (CAPE) predictor.
Alan Turing University and work on computability Alan_Turing > Early life and education > University and work on computability In this paper, Turing reformulated Kurt Gödel's 1931 results on the limits of proof and computation, replacing Gödel's universal arithmetic-based formal language with the formal and simple hypothetical devices that became known as Turing machines. The Entscheidungsproblem (decision problem) was originally posed by German mathematician David Hilbert in 1928.
Hydrogen isotope biogeochemistry Fossil fuels Hydrogen_isotope_biogeochemistry > Applications > Fossil fuels Marine and lacustrine environments are characterized by distinctly different δD values. Many studies have tried to relate measured δD with source types.
Deductive reasoning Probability logic Deductive_reasoning > In various fields > Probability logic Particular consequences will differ in their probability. However, the goal is to state a condition under which this attribute is ensured, regardless of which consequence one draws, and fulfilment of that condition is required to complete the task. This principle can be demonstrated in a moderately clear way.
Trajectory Physics of trajectories Trajectory > Physics of trajectories In the gravitational field of a point mass or a spherically-symmetrical extended mass (such as the Sun), the trajectory of a moving object is a conic section, usually an ellipse or a hyperbola. This agrees with the observed orbits of planets, comets, and artificial spacecraft to a reasonably good approximation, although if a comet passes close to the Sun, then it is also influenced by other forces such as the solar wind and radiation pressure, which modify the orbit and cause the comet to eject material into space. Newton's theory later developed into the branch of theoretical physics known as classical mechanics.
Pythagorean school Music Pythagoreanism > Philosophy > Music This ratio, also known as the "pure" perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear and because of importance attributed to the integer 3. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions. "The fact that mathematics could explain the human sentimental world had a profound impact on the Pythagorean philosophy.
Lebesgue–Rokhlin probability space A naive white noise Lebesgue–Rokhlin_probability_space > Examples of non-standard probability spaces > A naive white noise However, the integral of a white noise function from 0 to 1 should be a random variable distributed N(0, 1). In contrast, the integral (from 0 to 1) of f ∈ ( R , γ ) R {\displaystyle \textstyle f\in \textstyle (\mathbb {R} ,\gamma )^{\mathbb {R} }} is undefined. ƒ also fails to be almost surely measurable, and the probability of ƒ being measurable is undefined. Indeed, if X is a random variable distributed (say) uniformly on (0, 1) and independent of ƒ, then ƒ(X) is not a random variable at all (it lacks measurability).
Mole (sauce) Popularity Mole_(sauce) > Popularity However, northern versions are far less complex and generally used to make enchiladas. The consumption of mole is strongly associated with celebrations. In Mexico, to say "to go to a mole" (ir a un mole) means to go to a wedding.
Protein synthesis Post-transcriptional modifications Protein_biosynthesis > Transcription > Post-transcriptional modifications These distinct mRNA modifications enable the cell to detect that the full mRNA message is intact if both the 5' cap and 3' tail are present.This modified pre-mRNA molecule then undergoes the process of RNA splicing. Genes are composed of a series of introns and exons, introns are nucleotide sequences which do not encode a protein while, exons are nucleotide sequences that directly encode a protein.
Momentum vector Objects of variable mass Momentum > Newtonian > Objects of variable mass The correct equation is where u is the velocity of the ejected/accreted mass as seen in the object's rest frame. This is distinct from v, which is the velocity of the object itself as seen in an inertial frame. This equation is derived by keeping track of both the momentum of the object as well as the momentum of the ejected/accreted mass (dm). When considered together, the object and the mass (dm) constitute a closed system in which total momentum is conserved.
Genetic representation Redundancy Genetic_representation > Relationships between search space and problem space > Redundancy Thus, depending on the use of the genetic operators, there may be phenotypically unchanged offspring, which can lead to unnecessary fitness determinations, among other things. Since the evaluation in real-world applications usually accounts for the lion's share of the computation time, it can slow down the optimization process. In addition, this can cause the population to have higher genotypic diversity than phenotypic diversity, which can also hinder evolutionary progress.
Quantity equation Siano's extension: orientational analysis Quantity_equation > Geometry: position vs. displacement > Siano's extension: orientational analysis Siano distinguishes between geometric angles, which have an orientation in 3-dimensional space, and phase angles associated with time-based oscillations, which have no spatial orientation, i.e. the orientation of a phase angle is 1 0 {\displaystyle 1_{0}} . The assignment of orientational symbols to physical quantities and the requirement that physical equations be orientationally homogeneous can actually be used in a way that is similar to dimensional analysis to derive more information about acceptable solutions of physical problems. In this approach, one solves the dimensional equation as far as one can.
Feynman rules Monte Carlo Feynman_rules > Path integral formulation > Monte Carlo For real scalar fields, the algorithm must generate only one of each pair φ(k), φ(−k), and make the second the complex conjugate of the first. To find any correlation function, generate a field again and again by this procedure, and find the statistical average: ⟨ ϕ ( x 1 ) ⋯ ϕ ( x n ) ⟩ = lim \| C \| → ∞ ∑ C ϕ C ( x 1 ) ⋯ ϕ C ( x n ) \| C \| {\displaystyle \left\langle \phi (x_{1})\cdots \phi (x_{n})\right\rangle =\lim _{\|C\|\rightarrow \infty }{\frac {\displaystyle \sum _{C}\phi _{C}(x_{1})\cdots \phi _{C}(x_{n})}{\|C\|}}} where \|C\| is the number of configurations, and the sum is of the product of the field values on each configuration. The Euclidean correlation function is just the same as the correlation function in statistics or statistical mechanics.
Absolute probability judgement Background Absolute_probability_judgement > Background Absolute probability judgement is an expert judgement-based approach which involves using the beliefs of experts (e.g. front-line staff, process engineers etc.) to estimate HEPs. There are two primary forms of the technique; Group Methods and Single Expert Methods i.e. it can be done either as a group or as an individual exercise. Group methods tend to be the more popular and widely used as they are more robust and are less subject to bias. Moreover, within the context of use, it is unusual for a single individual to possess all the required information and expertise to be able to solely estimate, in an accurate manner, the human reliability in question. In the group approach, the outcome of aggregating individual knowledge and opinions is more reliable.
Mathematical abstraction Description Abstraction_(mathematics) > Description Finally Felix Klein's "Erlangen program" identified the underlying theme of all of these geometries, defining each of them as the study of properties invariant under a given group of symmetries. This level of abstraction revealed connections between geometry and abstract algebra.In mathematics, abstraction can be advantageous in the following ways: It reveals deep connections between different areas of mathematics. Known results in one area can suggest conjectures in another related area.
Mechanism of action Omics based methods Mechanism_of_action > Determination > Omics based methods Omics based methods use omics technologies, such as chemoproteomics, reverse genetics and genomics, transcriptomics, and proteomics, to identify the potential targets of the compound of interest. Reverse genetics and genomics approaches, for instance, uses genetic perturbation (e.g. CRISPR-Cas9 or siRNA) in combination with the compound to identify genes whose knockdown or knockout abolishes the pharmacological effect of the compound. On the other hand, transcriptomics and proteomics profiles of the compound can be used to compare with profiles of compounds with known targets. Thanks to computation inference, it is then possible to make hypotheses about the mechanism of action of the compound, which can subsequently be tested.
Psychometric software ResidPlots-2 Psychometric_software > Additional item response theory software > ResidPlots-2 ResidPlots-2 is a free program for IRT graphical residual analysis. It was developed by Tie Liang, Kyung (Chris) T. Han, and Ronald K. Hambleton at the University of Massachusetts Amherst.
Holevo's theorem Proof Holevo's_theorem > Proof The initial state of the system, where Alice prepares the state ρ x {\displaystyle \rho _{x}} with probability p x {\displaystyle p_{x}} , is described by ρ X Q := ∑ x = 1 n p x \| x ⟩ ⟨ x \| ⊗ ρ x . {\displaystyle \rho ^{XQ}:=\sum _{x=1}^{n}p_{x}\|x\rangle \langle x\|\otimes \rho _{x}.} Afterwards, Alice sends the quantum state to Bob.
Residual (statistics) Summary Statistical_error In statistics and optimization, errors and residuals are two closely related and easily confused measures of the deviation of an observed value of an element of a statistical sample from its "true value" (not necessarily observable). The error of an observation is the deviation of the observed value from the true value of a quantity of interest (for example, a population mean). The residual is the difference between the observed value and the estimated value of the quantity of interest (for example, a sample mean). The distinction is most important in regression analysis, where the concepts are sometimes called the regression errors and regression residuals and where they lead to the concept of studentized residuals. In econometrics, "errors" are also called disturbances.
Counter machine Final conditions Counter_machine > Example: COPY the count from register #2 to #3 > Final conditions The program HALTs with the contents of register #2 at its original count and the contents of register #3 equal to the original contents of register #2, i.e., = .
Sampling (statistics) Errors in sample surveys Sampling_(statistics) > Errors in sample surveys Survey results are typically subject to some error. Total errors can be classified into sampling errors and non-sampling errors. The term "error" here includes systematic biases as well as random errors.
Parsing algorithm Parser Natural_language_parser > Computer languages > Parser Programming languages tend to be specified in terms of a deterministic context-free grammar because fast and efficient parsers can be written for them. For compilers, the parsing itself can be done in one pass or multiple passes – see one-pass compiler and multi-pass compiler. The implied disadvantages of a one-pass compiler can largely be overcome by adding fix-ups, where provision is made for code relocation during the forward pass, and the fix-ups are applied backwards when the current program segment has been recognized as having been completed.
Orthoherpesviridae MHC downregulation Orthoherpesviridae > Immune system evasions > MHC downregulation As discussed above, one way is by a viral chemokine homolog such as IL-10. Another mechanism to down regulate MHCs is to encode viral proteins that detain the newly formed MHC in the endoplasmic reticulum (ER). The MHC cannot reach the cell surface and therefore cannot activate the T cell response.
Epi-convergence Properties Epi-convergence > Properties ( f n ) {\displaystyle (f_{n})} epi-converges to f {\displaystyle f} if and only if ( − f n ) {\displaystyle (-f_{n})} hypo-converges to − f {\displaystyle -f} . ( f n ) {\displaystyle (f_{n})} epi-converges to f {\displaystyle f} if and only if ( epi ⁡ f n ) {\displaystyle (\operatorname {epi} f_{n})} converges to epi ⁡ f {\displaystyle \operatorname {epi} f} as sets, in the Painlevé–Kuratowski sense of set convergence. Here, epi ⁡ f {\displaystyle \operatorname {epi} f} is the epigraph of the function f {\displaystyle f} . If f n {\displaystyle f_{n}} epi-converges to f {\displaystyle f} , then f {\displaystyle f} is lower semi-continuous.
Distance formula Squared Euclidean distance Pythagorean_distance > Squared Euclidean distance As an equation, the squared distance can be expressed as a sum of squares: Beyond its application to distance comparison, squared Euclidean distance is of central importance in statistics, where it is used in the method of least squares, a standard method of fitting statistical estimates to data by minimizing the average of the squared distances between observed and estimated values, and as the simplest form of divergence to compare probability distributions. The addition of squared distances to each other, as is done in least squares fitting, corresponds to an operation on (unsquared) distances called Pythagorean addition. In cluster analysis, squared distances can be used to strengthen the effect of longer distances.Squared Euclidean distance does not form a metric space, as it does not satisfy the triangle inequality.
Singular boundary method History of the singular boundary method Singular_boundary_method > History of the singular boundary method The second approach is to employ a regularization technique to cancel the singularities of the fundamental solution and its derivatives. Consequently, the origin intensity factors can be determined directly without using any sample nodes. This scheme makes the method more stable, accurate, efficient, and extends its applicability.
Renfrey Potts Journal articles Renfrey_Potts > Publications > Journal articles 745–763 Abstract 1982 "Differential and Difference Equations", Renfrey B. Potts, The American Mathematical Monthly, Vol. 89, No. 6 (Jun. – Jul.
Cantilevered wing Aircraft Cantilever > Aircraft The end fixed rigidly to the central fuselage is known as the root and the far end as the tip. In flight, the wings generate lift and the spars carry this load through to the fuselage. To resist horizontal shear stress from either drag or engine thrust, the wing must also form a stiff cantilever in the horizontal plane.
Neutron Summary Neutronium-1 Neutrons are produced copiously in nuclear fission and fusion. They are a primary contributor to the nucleosynthesis of chemical elements within stars through fission, fusion, and neutron capture processes. The neutron is essential to the production of nuclear power.
Polyhalogen ions Bonding Polyhalogen_ions > Bonding The bonding in polyhalogen ions mostly invoke the predominant use of p-orbitals. Significant d-orbital participation in the bonding is improbable as much promotional energy will be required, while scant s-orbital participation is expected in iodine-containing species due to the inert-pair effect, suggested by data from Mössbauer spectroscopy. However, no bonding model has been capable of reproducing such wide range of bond lengths and angles observed so far.As expected from the fact that an electron is removed from the antibonding orbital when X2 is ionized to +, the bond order as well as the bond strength in + gets higher, consequently the interatomic distances in the molecular ion is less than those in X2.
Cardiovascular exercise Potential health effects Cardiovascular_exercise > Potential health effects Among the possible health benefits of regular aerobic exercise are: May improve mood Strengthens and enlarges the heart muscle, to improve its pumping efficiency and reduce the resting heart rate, known as aerobic conditioning May improve circulation efficiency and reduce blood pressure May help maintain independence in later life Increases the total number of red blood cells in the body, facilitating transport of oxygen Improves mental health, including reducing stress and lowering the incidence of depression, as well as increased cognitive capacity. Slightly reduced depression may also be observed, especially if aerobic exercises are used as additional treatment for patients with a hematological malignancy Reduces the risk for diabetes (One meta-analysis has shown, from multiple conducted studies, that aerobic exercise does help lower Hb A1Clevels for type 2 diabetics.) Moderates the risk of death due to cardiovascular problems Promotes weight loss Reduces the risk of osteoporosis May improve episodic memory
Term (logic) Motivation Term_(logic) > Related concepts > Lambda terms > Motivation Mathematical notations as shown in the table do not fit into the scheme of a first-order term as defined above, as they all introduce an own local, or bound, variable that may not appear outside the notation's scope, e.g. t ⋅ ∫ a b sin ⁡ ( k ⋅ t ) d t {\displaystyle t\cdot \int _{a}^{b}\sin(k\cdot t)\;dt} doesn't make sense. In contrast, the other variables, referred to as free, behave like ordinary first-order term variables, e.g. k ⋅ ∫ a b sin ⁡ ( k ⋅ t ) d t {\displaystyle k\cdot \int _{a}^{b}\sin(k\cdot t)\;dt} does make sense. All these operators can be viewed as taking a function rather than a value term as one of their arguments. For example, the lim operator is applied to a sequence, i.e. to a mapping from positive integer to e.g. real numbers.
Pro-inflammatory cytokine Summary Proinflammatory_cytokines Dysregulation has also been linked to depression and other neurological diseases. A balance between proinflammatory and anti-inflammatory cytokines is necessary to maintain health. Aging and exercise also play a role in the amount of inflammation from the release of proinflammatory cytokines. Therapies to treat inflammatory diseases include monoclonal antibodies that either neutralize inflammatory cytokines or their receptors.
Apostatic selection Summary Apostatic_selection Apostatic selection is a form of negative frequency-dependent selection. It describes the survival of individual prey animals that are different (through mutation) from their species in a way that makes it more likely for them to be ignored by their predators. It operates on polymorphic species, species which have different forms. In apostatic selection, the common forms of a species are preyed on more than the rarer forms, giving the rare forms a selective advantage in the population.
Quantum sensor Characteristics Quantum_sensing > Characteristics The basic working mechanism typically relies on optical states of light, often involving quantum mechanical properties such as squeezing or two-mode entanglement. These states are sensitive to physical transformations that are detected by interferometric measurements.Quantum sensing can also be utilized in non-photonic areas such as spin qubits, trapped ions, flux qubits, and nanoparticles.
Polymer capacitor Comparison of electrical parameters Polymer_electrolytic > Comparison of the polymer families > Comparison of electrical parameters Electrical properties of polymer capacitors can best be compared, using consistent capacitance, rated voltage and dimensions. The values for the ESR and the ripple current are the most important parameters for the use of for polymer capacitors in electronic equipment. The leakage current is significant, because it is higher than that of e-caps with non-polymer electrolytes. The respective values of Ta-e-caps with MnO2 electrolyte and wet Al-e-caps are included.
Inductive probability The frequentist approach applied to possible worlds Inductive_probability > Probability and frequency > The frequentist approach applied to possible worlds In the frequentist approach, probabilities are defined as the ratio of the number of outcomes within an event to the total number of outcomes. In the possible world model each possible world is an outcome, and statements about possible worlds define events. The probability of a statement being true is the number of possible worlds where the statement is true divided by the total number of possible worlds.
Titer Summary Titer Titres are sometimes expressed by the denominator only, for example 1:256 is written 256.The term also has two other, conflicting meanings. In titration, the titer is the ratio of actual to nominal concentration of a titrant, e.g. a titer of 0.5 would require 1/0.5 = 2 times more titrant than nominal. This is to compensate for possible degradation of the titrant solution. Second, in textile engineering, titre is also a synonym for linear density.
Trophic dynamics Complexity and stability Food_webs > Quantitative food webs > Complexity and stability Food webs are extremely complex. Complexity is a term that conveys the mental intractability of understanding all possible higher-order effects in a food web. Sometimes in food web terminology, complexity is defined as product of the number of species and connectance., though there have been criticisms of this definition and other proposed methods for measuring network complexity. Connectance is "the fraction of all possible links that are realized in a network".
Map symbolization Visual variables Map_symbol > Visual variables Maps often use hue to differentiate categories of nominal variables, such as land cover types or geologic layers, or for its psychological connotations, such as red implying heat or danger and blue implying cold or water.Color value or lightness, how light or dark an object appears. Value effectively connotes "more" and "less," an ordinal measure; this makes it a very useful form of symbology in thematic maps, especially choropleth maps. Value also contributes strongly to Visual hierarchy; elements that contrast most with the value of the background tend to stand out most (e.g., black on a white sheet of paper, white on a black computer screen).
Kernel embedding of distributions Distribution regression Kernel_embedding_of_distributions > Applications > Distribution regression In distribution regression, the goal is to regress from probability distributions to reals (or vectors). Many important machine learning and statistical tasks fit into this framework, including multi-instance learning, and point estimation problems without analytical solution (such as hyperparameter or entropy estimation). In practice only samples from sampled distributions are observable, and the estimates have to rely on similarities computed between sets of points. Distribution regression has been successfully applied for example in supervised entropy learning, and aerosol prediction using multispectral satellite images.Given ( { X i , n } n = 1 N i , y i ) i = 1 ℓ {\displaystyle {\left(\{X_{i,n}\}_{n=1}^{N_{i}},y_{i}\right)}_{i=1}^{\ell }} training data, where the X i ^ := { X i , n } n = 1 N i {\displaystyle {\hat {X_{i}}}:=\{X_{i,n}\}_{n=1}^{N_{i}}} bag contains samples from a probability distribution X i {\displaystyle X_{i}} and the i th {\displaystyle i^{\text{th}}} output label is y i ∈ R {\displaystyle y_{i}\in \mathbb {R} } , one can tackle the distribution regression task by taking the embeddings of the distributions, and learning the regressor from the embeddings to the outputs.
Frequentist inference Bayesian inference Frequentist_statistics > Relationship with other approaches > Bayesian inference In contrast, a Bayesian approach allows probabilities to be associated with unknown parameters, where these probabilities can sometimes have a frequency probability interpretation as well as a Bayesian one. The Bayesian approach allows these probabilities to have an interpretation as representing the scientist's belief that given values of the parameter are true (see Bayesian probability - Personal probabilities and objective methods for constructing priors). The result of a Bayesian approach can be a probability distribution for what is known about the parameters given the results of the experiment or study. The result of a frequentist approach is either a decision from a significance test or a confidence interval.
Translational kinetic energy Relativistic kinetic energy Kinetic_Energy > Relativistic kinetic energy If a body's speed is a significant fraction of the speed of light, it is necessary to use relativistic mechanics to calculate its kinetic energy. In relativity, the total energy is given by the energy-momentum relation: Here we use the relativistic expression for linear momentum: p = m γ v {\displaystyle p=m\gamma v} , where γ = 1 / 1 − v 2 / c 2 {\textstyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}} . with m {\displaystyle m} being an object's (rest) mass, v {\displaystyle v} speed, and c the speed of light in vacuum. Then kinetic energy is the total relativistic energy minus the rest energy: At low speeds, the square root can be expanded and the rest energy drops out, giving the Newtonian kinetic energy.
Lebesgue integral Intuitive interpretation Lebesgue_integrable > Introduction > Intuitive interpretation Folland (1984) summarizes the difference between the Riemann and Lebesgue approaches thus: "to compute the Riemann integral of f, one partitions the domain into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of f. "For the Riemann integral, the domain is partitioned into intervals, and bars are constructed to meet the height of the graph. The areas of these bars are added together, and this approximates the integral, in effect by summing areas of the form f(x)dx where f(x) is the height of a rectangle and dx is its width. For the Lebesgue integral, the range is partitioned into intervals, and so the region under the graph is partitioned into horizontal "slabs" (which may not be connected sets). The area of a small horizontal "slab" under the graph of f, of height dy, is equal to the measure of the slab's width times dy: The Lebesgue integral may then be defined by adding up the areas of these horizontal slabs.
Skew-merged permutation Summary Skew-merged_permutation In the theory of permutation patterns, a skew-merged permutation is a permutation that can be partitioned into an increasing sequence and a decreasing sequence. They were first studied by Stankova (1994) and given their name by Atkinson (1998).
Enumeration Combinatorics Enumeration > Combinatorics In combinatorics, enumeration means counting, i.e., determining the exact number of elements of finite sets, usually grouped into infinite families, such as the family of sets each consisting of all permutations of some finite set. There are flourishing subareas in many branches of mathematics concerned with enumerating in this sense objects of special kinds. For instance, in partition enumeration and graph enumeration the objective is to count partitions or graphs that meet certain conditions.
Velocity stack Function Velocity_stack > Function The velocity stack is designed to: Allow smooth and even entry of air at high velocities into the intake tract with the flow stream adhering to the pipe walls known as laminar flow. Modify the dynamic tuning range of the intake tract by functioning as a resonating pipe which can adjust the frequency of pressure pulses based on its length within the tract.Modified engines often have the original air box and associated ducting removed and velocity stacks are installed as accessories. The length of the stack is known to have a direct effect on a particular engine's boosted power range. Most current aftermarket stacks are designed to be run "in" the airbox and a company that does research well will have some applications that have all the same length and some applications that have differing lengths of stacks on different cylinders.
Mølmer–Sørensen gate Summary Mølmer–Sørensen_gate In quantum computing, Mølmer–Sørensen gate scheme (or MS gate) refers to an implementation procedure for various multi-qubit quantum logic gates used mostly in trapped ion quantum computing. This procedure is based on the original proposition by Klaus Mølmer and Anders Sørensen in 1999-2000.This proposal was an alternative to the 1995 Cirac–Zoller controlled-NOT gate implementation for trapped ions, which requires that the system be restricted to the joint motional ground state of the ions.In an MS gate, entangled states are prepared by illuminating ions with a bichromatic light field. Mølmer and Sørensen identified two regimes in which this is possible: A weak-field regime, where single-photon absorption is suppressed and two-photon processes interfere in a way that makes internal state dynamics insensitive to the vibrational state A strong-field regime where the individual ions are coherently excited, and the motional state is highly entangled with the internal state until all undesirable excitations are deterministically removed toward the end of the interaction.In both regimes, a red and blue sideband interaction are applied simultaneously to each ion, with the red and blue tones symmetrically detuned by δ ′ {\displaystyle \delta '} from the sidebands. This results in laser detunings ± ( ω k + δ ′ ) {\displaystyle \pm (\omega _{k}+\delta ')} , where ω k {\displaystyle \omega _{k}} is the motional mode frequency.
V1 Saliency Hypothesis Neural mechanisms in V1 to generate the saliency map V1_Saliency_Hypothesis > Neural mechanisms in V1 to generate the saliency map For example, a neuron's response to a bar in its receptive field is higher when this bar is oriented in its preferred orientation. Analogously, many V1 neurons have their preferred colours. In this schematic, each input bar to the retina activates two (groups of) V1 neurons, one preferring its orientation and the other preferring its colour.
Body plethysmography Methodological approach Body_plethysmography > Organs studied > Lungs > Methodological approach While observing the so-called resistance loop (cabin pressure and flow), diseases can easily be recognized. If the resistance loop becomes planar, this shows a bad compliance of the lung. A COPD, for instance, can easily be discovered because of the unique shape of the corresponding resistance loop.
Gizzard (Scala framework) Summary Gizzard_(Scala_framework) Gizzard was an open source sharding framework to create custom fault-tolerant, distributed databases. It was initially used by Twitter and emerged from a wide variety of data storage problems. Gizzard operated as a middleware networking service that ran on the Java Virtual Machine. It managed partitioning data across arbitrary backend datastores, which allowed it to be accessed efficiently.
Mucin short variant S1 Preventing cell death Mucin_short_variant_S1 > Role in cancer > Preventing cell death MUC1 cytoplasmic tail has been shown to bind to p53. This interaction is increased by genotoxic stress. MUC1 and p53 were found to be associated with the p53 response element of the p21 gene promoter. This results in activation of p21 which results in cell cycle arrest.
Printed electronics Organic materials Printed_silicon_electronics > Materials > Organic materials Organic printed electronics integrates knowledge and developments from printing, electronics, chemistry, and materials science, especially from organic and polymer chemistry. Organic materials in part differ from conventional electronics in terms of structure, operation and functionality, which influences device and circuit design and optimization as well as fabrication method.The discovery of conjugated polymers and their development into soluble materials provided the first organic ink materials. Materials from this class of polymers variously possess conducting, semiconducting, electroluminescent, photovoltaic and other properties. Other polymers are used mostly as insulators and dielectrics.
S. P. L. Sørensen Work S._P._L._Sørensen > Work From 1901 to 1938, Sørensen was head of the prestigious Carlsberg Laboratory, Copenhagen. While working at the Carlsberg Laboratory he studied the effect of ion concentration on proteins and, because the concentration of hydrogen ions was particularly important, he introduced the pH-scale as a simple way of expressing it in 1909. The article in which he introduced the scale (using the notation p H {\displaystyle p_{\mathrm {H} }} ) was published in French and Danish as well as in German described two methods for measuring acidity which Sørensen and his students had refined. The first method was based on electrodes, whereas the second involved comparing the colours of samples and a preselected set of indicators.
James Prescott Joule Kinetic theory James_Prescott_Joule > Kinetic theory Kinetics is the science of motion. Joule was a pupil of Dalton and it is no surprise that he had learned a firm belief in the atomic theory, even though there were many scientists of his time who were still skeptical. He had also been one of the few people receptive to the neglected work of John Herapath on the kinetic theory of gases. He was further profoundly influenced by Peter Ewart's 1813 paper "On the measure of moving force".
Prion disease History Prion_diseases > History In the 18th and 19th centuries, exportation of sheep from Spain was observed to coincide with a disease called scrapie. This disease caused the affected animals to "lie down, bite at their feet and legs, rub their backs against posts, fail to thrive, stop feeding and finally become lame". The disease was also observed to have the long incubation period that is a key characteristic of transmissible spongiform encephalopathies (TSEs).
Telomerase reverse transcriptase Stem cells Telomerase_reverse_transcriptase > Stem cells hTERT is often up-regulated in cells that divide rapidly, including both embryonic stem cells and adult stem cells. It elongates the telomeres of stem cells, which, as a consequence, increases the lifespan of the stem cells by allowing for indefinite division without shortening of telomeres. Therefore, it is responsible for the self-renewal properties of stem cells. Telomerase are found specifically to target shorter telomere over longer telomere, due to various regulatory mechanisms inside the cells that reduce the affinity of telomerase to longer telomeres. This preferential affinity maintains a balance within the cell such that the telomeres are of sufficient length for their function and yet, at the same time, not contribute to aberrant telomere elongation.High expression of hTERT is also often used as a landmark for pluripotency and multipotency state of embryonic and adult stem cells. Over-expression of hTERT was found to immortalize certain cell types as well as impart different interesting properties to different stem cells.
Fokker periodicity block Definition of periodicity blocks Fokker_periodicity_blocks > Definition of periodicity blocks Examples of such lattices are the following (x, y, z and w are integers): In the one-dimensional case, the interval corresponding to a single step is generally taken to be a perfect fifth, with ratio 3/2, defining 3-limit just tuning. The lattice points correspond to the integers, with the point at position x being labeled with the pitch value 3x/2y for a number y chosen to make the resulting value lie in the range from 1 to 2. Thus, A(0) = 1, and surrounding it are the values... 128/81, 32/27, 16/9, 4/3, 1, 3/2, 9/8, 27/16, 81/64, ...In the two-dimensional case, corresponding to 5-limit just tuning, the intervals defining the lattice are a perfect fifth and a major third, with ratio 5/4.
Activation immunotherapy Immune tolerance Immunomodulatory_drug > Suppression immunotherapies > Immune tolerance The body naturally does not launch an immune system attack on its own tissues. Models generally identify CD4+ T-cells at the centre of the autoimmune response. Loss of T-cell tolerance then unleashes B-cells and other immune effector cells on to the target tissue. The ideal tolerogenic therapy would target the specific T-cell clones co-ordinating the autoimmune attack.Immune tolerance therapies seek to reset the immune system so that the body stops mistakenly attacking its own organs or cells in autoimmune disease or accepts foreign tissue in organ transplantation.
Disk Filing System Summary Disk_Filing_System The DFS shipped as a ROM and Disk Controller Chip fitted to the BBC Micro's motherboard. The filing system was of extremely limited functionality and storage capability, using a flat directory structure.
Polyvinyl siloxane Summary Polyvinyl_siloxane Polyvinyl siloxane (PVS), also called poly-vinyl siloxane, vinyl polysiloxane (VPS), or vinylpolysiloxane, is an addition-reaction silicone elastomer (an addition silicone). It is a viscous liquid that cures (solidifies) quickly into a rubber-like solid, taking the shape of whatever surface it was lying against while curing. As with two-part epoxy, its package keeps its two component liquids in separate tubes until the moment they are mixed and applied, because once mixed, they cure (harden) rapidly. Polyvinyl siloxane is widely used in dentistry as an impression material.
Strengthening mechanisms of materials Basic description Strengthening_mechanisms_of_materials > Basic description There are five main strengthening mechanisms for metals, each is a method to prevent dislocation motion and propagation, or make it energetically unfavorable for the dislocation to move. For a material that has been strengthened, by some processing method, the amount of force required to start irreversible (plastic) deformation is greater than it was for the original material. In amorphous materials such as polymers, amorphous ceramics (glass), and amorphous metals, the lack of long range order leads to yielding via mechanisms such as brittle fracture, crazing, and shear band formation.
Attack patterns Structure Attack_patterns > Structure An example of this in an Integer Overflow attack would be that the integer-based input field is not checking size of the value of the incoming data to ensure that the target variable is capable of managing the incoming value. This is the vulnerability that the associated exploit will take advantage of in order to carry out the attack. ParticipantsThe Participants are one or more entities that are required for this attack to succeed.
Virtual leak M Glossary_of_engineering:_M–Z > M Taylor's series are named after Brook Taylor, who introduced them in 1715. If zero is the point where the derivatives are considered, a Taylor series is also called a Maclaurin series, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century. Magnetic fieldA magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents,: ch1 and magnetic materials.
MicroRNA Targets MicroRNA > Targets Other experiments show that a single miRNA species may repress the production of hundreds of proteins, but that this repression often is relatively mild (much less than 2-fold). The first human disease discovered to be associated with deregulation of miRNAs was chronic lymphocytic leukemia. Other B cell malignancies followed.
Spectral shape analysis Laplace Spectral_shape_analysis > Laplace {\displaystyle \Delta \varphi _{i}+\lambda _{i}\varphi _{i}=0.} The solutions are the eigenfunctions φ i {\displaystyle \varphi _{i}} (modes) and corresponding eigenvalues λ i {\displaystyle \lambda _{i}} , representing a diverging sequence of positive real numbers. The first eigenvalue is zero for closed domains or when using the Neumann boundary condition.
Attractive (force) Static Attractive_(force) > Combining forces > Equilibrium > Static Static equilibrium was understood well before the invention of classical mechanics. Objects that are at rest have zero net force acting on them.The simplest case of static equilibrium occurs when two forces are equal in magnitude but opposite in direction. For example, an object on a level surface is pulled (attracted) downward toward the center of the Earth by the force of gravity. At the same time, a force is applied by the surface that resists the downward force with equal upward force (called a normal force).
QCD theory Symmetry groups QCD_theory > Theory > Symmetry groups There are two different types of SU(3) symmetry: there is the symmetry that acts on the different colors of quarks, and this is an exact gauge symmetry mediated by the gluons, and there is also a flavor symmetry that rotates different flavors of quarks to each other, or flavor SU(3). Flavor SU(3) is an approximate symmetry of the vacuum of QCD, and is not a fundamental symmetry at all. It is an accidental consequence of the small mass of the three lightest quarks.
Proteostasis Pharmacologic intervention in proteostasis Proteostasis > Diseases of proteostasis > Pharmacologic intervention in proteostasis Amicus sells a regulatory agency approved pharmacologic chaperone for Fabry disease–a lysosomal storage disease. The principle behind proteostasis regulators is different.
Butterfly loop Disadvantages Butterfly_loop > Disadvantages Difficult to tie around a solid ring or similar object, as when a rethreaded figure eight is needed. Improper tying can result in similar looking but inferior "false butterfly" knot. Works best with softer ropes.
Euler acceleration Mathematical description Euler_acceleration > Mathematical description The direction and magnitude of the Euler acceleration is given, in the rotating reference frame, by: a E u l e r = − d ω d t × r , {\displaystyle \mathbf {a} _{\mathrm {Euler} }=-{\frac {d{\boldsymbol {\omega }}}{dt}}\times \mathbf {r} ,} where ω is the angular velocity of rotation of the reference frame and r is the vector position of the point in the reference frame. The Euler force on an object of mass m in the rotating reference frame is then F E u l e r = m a E u l e r = − m d ω d t × r . {\displaystyle \mathbf {F} _{\mathrm {Euler} }=m\mathbf {a} _{\mathrm {Euler} }=-m{\frac {d{\boldsymbol {\omega }}}{dt}}\times \mathbf {r} .}
Graph homology Example Graph_homology > Higher dimensional homologies > Example But now C2 contains the two-dimensional cycle A-B, so ∂ 2 {\displaystyle \partial _{2}} has a non-trivial kernel. This cycle generates the second homology group, corresponding to the fact that there is a single two-dimensional hole: We can proceed and add a 3-cell - a solid 3-dimensional object (called C) bounded by A and B. Define C3 as the free abelian group generated by {C}, and the boundary operator ∂ 3: C 3 → C 2 {\displaystyle \partial _{3}:C_{3}\to C_{2}} . We can orient C such that ∂ 3 ( C ) = A − B {\displaystyle \partial _{3}(C)=A-B} ; note that the boundary of C is a cycle in C2. Now the second homology group is: corresponding to the fact that there are no two-dimensional holes (C "fills the hole" between A and B).

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