The 2018 Google data breach was a major data privacy scandal in which the Google+ API exposed the private data of over five hundred thousand users. Google+ managers first noticed harvesting of personal data in March 2018, during a review following the Facebook–Cambridge Analytica data scandal. The bug, despite having been fixed immediately, exposed the private data of approximately 500,000 Google+ users to the public. Google did not reveal the leak to the network's users. In November 2018, another data breach occurred following an update to the Google+ API. Although Google found no evidence of failure, approximately 52.5 million personal profiles were potentially exposed. In August 2019, Google declared a shutdown of Google+ due to low use and technological challenges. == Overview of Google+ == Google+ was launched in June 2011 as an invite-only social network, but was opened for public access later in the year. It was managed by Vic Gundotra. Similar to Facebook, Google+ also included key features Circles, Hangouts and Sparks. Circles let users personalize their social groups by sorting friends into different categories. Once allowed into a Circle, users could regulate information in their individual spaces. Hangouts included video chatting and instant messaging between users. Sparks allowed Google to track users' past searches to find news and content related to their interests. Google+ was linked to other Google services, such as YouTube, Google Drive and Gmail, giving it access to roughly 2 billion user accounts. However, less than 400 million consumers actively used Google+, with 90% of those users using it for less than five seconds. == The breaches == In March 2018, Google developers found a data breach within the Google+ People API in which external apps acquired access to Profile fields that were not marked as public. According to The Wall Street Journal, Google didn’t disclose the breach when it was first discovered in March to avoid regulatory scrutiny and reputational damage. 500,000 Google+ accounts were included in the breach, which allowed 438 external apps unauthorized access to private users' names, emails, addresses, occupations, genders and ages. This information was available between 2015 and 2018. Google found no evidence of any user's personal information being misused, nor that any third-party app developers were aware of the leak. In November 2018, a software update created another data breach within the Google+ API. The bug impacted 52.5 million users, where, similarly to the March breach, unauthorized apps were able to access Google+ profiles, including users' names, email addresses, occupations and ages. Apps could not access financial information, national identification, numbers, or passwords. Blog posts, messages and phone numbers also remained inaccessible if marked as private. Unlike the previous breach, access was only available for six days before Google+ learned of the breach. Once more, Google+ found no evidence of data being misused by third-party developers. == Responses == In October 2018, the Wall Street Journal published an article outlining the initial breach and Google's decision to not disclose it to users. At the time, there was no federal law that required Google to inform their consumers of data breaches. Google+ originally did not disclose the breach out of fears of being compared to Facebook's recent data leak and subsequent loss of consumer confidence. In response to the Wall Street Journal article, Google announced the shutdown of Google+ in August 2019. After the second data leak, the date was moved to April 2019. In response to the data breach, enterprise consumers were notified of the bug's impact and given instructions on how to save, download and delete their data prior to the Google+ shut down. Google's Privacy and Data Protection Office found no misuse of user data. Prior to the Google+ shutdown, Google set a 10-month period in which users could download and migrate their data. After the 10-month period, user content was deleted. On 4 February 2019, consumers were no longer able to create new Google+ profiles. Google shut down Google+ APIs on 7 March 2019 to ensure that developers did not continue to rely on the APIs prior to the Google+ shutdown. Google is the principal entity of its parent company, Alphabet Inc. After the data breach, Alphabet Inc. share prices fell by 1% to $1,157.06 on 9 October 2018 after an earlier drop of $1,135.40 that morning, the lowest price since 5 July 2018. After the publication of The Wall Street Journal article, share prices dropped as low as 2.1% in two days on 10 October 2018. Share prices steadily increased from this point and met the 8 October 2018 share price on 5 February 2019. Google planned to rebuild Google+ as a corporate enterprise network. Google Play will now assess which apps can ask for permission to access the user's SMS data. Only the default app for telephone distribution is able to make requests. Prior to the data breaches, apps were able to request access to all of a consumer's data simultaneously. Now, each app must request permission for each aspect of a consumer's profile.
Manual override
A manual override (MO) or manual analog override (MAO) is a mechanism where control is taken from an automated system and given to the user. For example, a manual override in photography refers to the ability for the human photographer to turn off the automatic aperture sizing, automatic focusing, or any other automated system on the camera. Some manual overrides can be used to veto an automated system's judgment when the system is in error. An example of this is a printer's ink level detection: in one case, a researcher found that when he overrode the system, up to 38% more pages could be printed at good quality by the printer than the automated system would have allowed. Automated systems are becoming increasingly common and integrated into everyday objects such as automobiles and domestic appliances. This development of ubiquitous computing raises general issues of policy and law about the need for manual overrides for matters of great importance such as life-threatening situations and major economic decisions. The loyalty of such autonomous devices then becomes an issue. If they follow rules installed by the manufacturer or required by law and refuse to cede control in some situations then the owners of the devices may feel disempowered, alienated and lacking true ownership. == Major incidents == China Airlines Flight 140 crashed, causing many deaths, due to a misunderstanding about the manual overrides for the autopilot. The Take-Off/Go Around system had been activated to abort a landing. It was programmed to ignore manual controls in this situation but the human pilots tried to continue the landing. The conflicting control signals from the pilots and autopilot then resulted in the aircraft stalling and crashing. The autopilot for this aircraft type was then reprogrammed so that it would never ignore a manual override.
Prescription monitoring program
In the United States, prescription monitoring programs (PMPs) or prescription drug monitoring programs (PDMPs) are state-run programs which collect and distribute data about the prescription and dispensation of federally controlled substances and, depending on state requirements, other potentially abusable prescription drugs. PMPs are meant to help prevent adverse drug-related events such as opioid overdoses, drug diversion, and substance abuse by decreasing the amount and/or frequency of opioid prescribing, and by identifying those patients who are obtaining prescriptions from multiple providers (i.e., "doctor shopping") or those physicians overprescribing opioids. Most US health care workers support the idea of PMPs, which intend to assist physicians, physician assistants, nurse practitioners, dentists and other prescribers, the pharmacists, chemists and support staff of dispensing establishments. The database, whose use is required by State law, typically requires prescribers and pharmacies dispensing controlled substances to register with their respective state PMPs and (for pharmacies and providers who dispense from their offices) to report the dispensation of such prescriptions to an electronic online database. The majority of PMPs are authorized to notify law enforcement agencies or licensing boards or physicians when a prescriber, or patients receiving prescriptions, exceed thresholds established by the state or prescription recipient exceeds thresholds established by the State. All states have implemented PDMPs, although evidence for the effectiveness of these programs is mixed. While prescription of opioids has decreased with PMP use, overdose deaths in many states have actually increased, with those states sharing data with neighboring jurisdictions or requiring reporting of more drugs experiencing highest increases in deaths. This may be because those declined opioid prescriptions turn to street drugs, whose potency and contaminants carry greater overdose risk. == History == Prescription drug monitoring programs, or PDMPs, are an example of one initiative proposed to alleviate effects of the opioid crisis. The programs are designed to restrict prescription drug abuse by limiting a patient's ability to obtain similar prescriptions from multiple providers (i.e. “doctor shopping”) and reducing diversion of controlled substances. This is meant to reduce risk of fatal overdose caused by high doses of opioids or interactions between opioids and benzodiazepenes, and to enable better decision making on the part of healthcare providers who may be unaware of a patient's prescription drug use, history or other prescriptions. PDMPs have been implemented in state legislations since 1939 in California, a time before electronic medical records, though implementation rose alongside increased awareness of overprescribing of opioids and overdose. A later New York state program was struck down by the U.S. Supreme Court in Whalen v. Roe. But, by 2019, 49 states, the District of Columbia, and Guam had enacted PDMP legislation. In 2021 Missouri, the last State to not use a PMP, adopted legislation to create one. PMPs are constantly being updated to increase speed of data collection, sharing of data across States, and ease of interpretation. This is being done by integrating PDMP reports with other health information technologies such as health information exchanges (HIE), electronic health record (EHR) systems, and/ or pharmacy dispensing software systems. One program that has been implemented in nine states is called the PDMP Electronic Health Records Integration and Interoperability Expansion, also known as PEHRIIE. Another software, marketed by Bamboo Health and integrated with PMPs in 43 states, uses an algorithm to track factors thought to increase risk of diversion, abuse or overdose, and assigns patients a three digit score based on presumed indicators of risk. While some studies have suggested that PDMP-HIT integration and sharing of interstate data brings benefits such as reduced opioid-related inpatient morbidity, others have found no or negative impact on mortality compared to states without PMP data sharing. Patient and media reports suggest need for testing and evaluation of algorithmic software used to score risk, with some patients reporting denial of prescriptions without c explanation or clarity of data. == Goals == Most health care workers support PMPs which intend to assist physicians, physician assistants, nurse practitioners, dentists and other prescribers, the pharmacists, chemists and support staff of dispensing establishments, as well as law-enforcement agencies. The collaboration supports the legitimate medical use of controlled substances while limiting their abuse and diversion. Pharmacies dispensing controlled substances and prescribers typically must register with their respective state PMPs and (for pharmacies and providers who dispense controlled substances from their offices) report the dispensation to an electronic online database. Some pharmacy software can submit these reports automatically to multiple states. == Usage == === List of programs by state === === Software systems === NarxCare is a prescription drug monitoring program (PDMP) run by Bamboo Health. Bamboo Health was formerly known as Appriss. It is widely used across the United States by pharmacies including Rite Aid as well as those at Walmart and Sam’s Club. The NarxCare software allows doctors to view data about a patient, combining data from the prescription registries of various U.S. states to make the registries interoperable nationally. It also uses machine learning to generate an "Overdose Risk Score" that potentially includes EMS and criminal justice data; these scores have been criticized by researchers and patient advocates for the lack of transparency in the process as well as the potential for disparate treatment of women and minority groups. Advertised as an "analytics tool and care management platform", the NarxCare software allows doctors to view data about a patient including how many pharmacies they have visited and the combinations of medication they are prescribed. It combines data from the prescription registries of various U.S. states, making the registries interoperable nationally. It additionally uses machine learning to generate various three-digit "risk scores" and an overall "Overdose Risk Score", collectively referred to as Narx Scores, in a process that potentially includes EMS and criminal justice data as well as court records. == Controversy == Many doctors and researchers support the idea of PDMPs as a tool in combatting the opioid epidemic. Opioid prescribing, opioid diversion and supply, opioid misuse, and opioid-related morbidity and mortality are common elements in data entered into PDMPs. Prescription Monitoring Programs are purported to offer economic benefits for the states who implement them by decreasing overall health care costs, lost productivity, and investigation times. However, there are many studies that conclude the impact of PDMPs is unclear. While use of PMPs has been accompanied by decrease in opioid prescribing, few analyses consider corresponding use of street opioids, extramedical use, or diversion, which might provide a more holistic method for evaluation of PMP intent and efficacy. Evidence for PDMP impact on fatal overdoses is decidedly mixed, with multiple studies finding increased overdose rates in some states, decreases in others, or no clear impact. Interestingly, an increase in heroin overdoses after PDMP implementation has been commonly reported, presumably as denial of prescription opioids sends patients in search of street drugs. Narx Scores have been criticized by researchers and patient advocates for the lack of transparency in the generation process as well as the potential for disparate treatment of women and minority groups. Writing in Duke Law Journal, Jennifer Oliva stated that "black-box algorithms" are used to generate the scores.
Correspondence analysis
Correspondence analysis (CA) is a multivariate statistical technique proposed by Herman Otto Hartley (Hirschfeld) and later developed by Jean-Paul Benzécri. It is conceptually similar to principal component analysis, but applies to categorical rather than continuous data. In a manner similar to principal component analysis, it provides a means of displaying or summarising a set of data in two-dimensional graphical form. Its aim is to display in a biplot any structure hidden in the multivariate setting of the data table. As such it is a technique from the field of multivariate ordination. Since the variant of CA described here can be applied either with a focus on the rows or on the columns it should in fact be called simple (symmetric) correspondence analysis. It is traditionally applied to the contingency table of a pair of nominal variables where each cell contains either a count or a zero value. If more than two categorical variables are to be summarized, a variant called multiple correspondence analysis should be chosen instead. CA may also be applied to binary data given the presence/absence coding represents simplified count data i.e. a 1 describes a positive count and 0 stands for a count of zero. Depending on the scores used CA preserves the chi-square distance between either the rows or the columns of the table. Because CA is a descriptive technique, it can be applied to tables regardless of a significant chi-squared test. Although the χ 2 {\displaystyle \chi ^{2}} statistic used in inferential statistics and the chi-square distance are computationally related they should not be confused since the latter works as a multivariate statistical distance measure in CA while the χ 2 {\displaystyle \chi ^{2}} statistic is in fact a scalar not a metric. == Details == Like principal components analysis, correspondence analysis creates orthogonal components (or axes) and, for each item in a table i.e. for each row, a set of scores (sometimes called factor scores, see Factor analysis). Correspondence analysis is performed on the data table, conceived as matrix C of size m × n where m is the number of rows and n is the number of columns. In the following mathematical description of the method capital letters in italics refer to a matrix while letters in italics refer to vectors. Understanding the following computations requires knowledge of matrix algebra. === Preprocessing === Before proceeding to the central computational step of the algorithm, the values in matrix C have to be transformed. First compute a set of weights for the columns and the rows (sometimes called masses), where row and column weights are given by the row and column vectors, respectively: w m = 1 n C C 1 , w n = 1 n C 1 T C . {\displaystyle w_{m}={\frac {1}{n_{C}}}C\mathbf {1} ,\quad w_{n}={\frac {1}{n_{C}}}\mathbf {1} ^{T}C.} Here n C = ∑ i = 1 n ∑ j = 1 m C i j {\displaystyle n_{C}=\sum _{i=1}^{n}\sum _{j=1}^{m}C_{ij}} is the sum of all cell values in matrix C, or short the sum of C, and 1 {\displaystyle \mathbf {1} } is a column vector of ones with the appropriate dimension. Put in simple words, w m {\displaystyle w_{m}} is just a vector whose elements are the row sums of C divided by the sum of C, and w n {\displaystyle w_{n}} is a vector whose elements are the column sums of C divided by the sum of C. The weights are transformed into diagonal matrices W m = diag ( 1 / w m ) {\displaystyle W_{m}=\operatorname {diag} (1/{\sqrt {w_{m}}})} and W n = diag ( 1 / w n ) {\displaystyle W_{n}=\operatorname {diag} (1/{\sqrt {w_{n}}})} where the diagonal elements of W n {\displaystyle W_{n}} are 1 / w n {\displaystyle 1/{\sqrt {w_{n}}}} and those of W m {\displaystyle W_{m}} are 1 / w m {\displaystyle 1/{\sqrt {w_{m}}}} respectively i.e. the vector elements are the inverses of the square roots of the masses. The off-diagonal elements are all 0. Next, compute matrix P {\displaystyle P} by dividing C {\displaystyle C} by its sum P = 1 n C C . {\displaystyle P={\frac {1}{n_{C}}}C.} In simple words, Matrix P {\displaystyle P} is just the data matrix (contingency table or binary table) transformed into portions i.e. each cell value is just the cell portion of the sum of the whole table. Finally, compute matrix S {\displaystyle S} , sometimes called the matrix of standardized residuals, by matrix multiplication as S = W m ( P − w m w n ) W n {\displaystyle S=W_{m}(P-w_{m}w_{n})W_{n}} Note, the vectors w m {\displaystyle w_{m}} and w n {\displaystyle w_{n}} are combined in an outer product resulting in a matrix of the same dimensions as P {\displaystyle P} . In words the formula reads: matrix outer ( w m , w n ) {\displaystyle \operatorname {outer} (w_{m},w_{n})} is subtracted from matrix P {\displaystyle P} and the resulting matrix is scaled (weighted) by the diagonal matrices W m {\displaystyle W_{m}} and W n {\displaystyle W_{n}} . Multiplying the resulting matrix by the diagonal matrices is equivalent to multiply the i-th row (or column) of it by the i-th element of the diagonal of W m {\displaystyle W_{m}} or W n {\displaystyle W_{n}} , respectively. === Interpretation of preprocessing === The vectors w m {\displaystyle w_{m}} and w n {\displaystyle w_{n}} are the row and column masses or the marginal probabilities for the rows and columns, respectively. Subtracting matrix outer ( w m , w n ) {\displaystyle \operatorname {outer} (w_{m},w_{n})} from matrix P {\displaystyle P} is the matrix algebra version of double centering the data. Multiplying this difference by the diagonal weighting matrices results in a matrix containing weighted deviations from the origin of a vector space. This origin is defined by matrix outer ( w m , w n ) {\displaystyle \operatorname {outer} (w_{m},w_{n})} . In fact matrix outer ( w m , w n ) {\displaystyle \operatorname {outer} (w_{m},w_{n})} is identical with the matrix of expected frequencies in the chi-squared test. Therefore S {\displaystyle S} is computationally related to the independence model used in that test. But since CA is not an inferential method the term independence model is inappropriate here. === Orthogonal components === The table S {\displaystyle S} is then decomposed by a singular value decomposition as S = U Σ V ∗ {\displaystyle S=U\Sigma V^{}\,} where U {\displaystyle U} and V {\displaystyle V} are the left and right singular vectors of S {\displaystyle S} and Σ {\displaystyle \Sigma } is a square diagonal matrix with the singular values σ i {\displaystyle \sigma _{i}} of S {\displaystyle S} on the diagonal. Σ {\displaystyle \Sigma } is of dimension p ≤ ( min ( m , n ) − 1 ) {\displaystyle p\leq (\min(m,n)-1)} hence U {\displaystyle U} is of dimension m×p and V {\displaystyle V} is of n×p. As orthonormal vectors U {\displaystyle U} and V {\displaystyle V} fulfill U ∗ U = V ∗ V = I {\displaystyle U^{}U=V^{}V=I} . In other words, the multivariate information that is contained in C {\displaystyle C} as well as in S {\displaystyle S} is now distributed across two (coordinate) matrices U {\displaystyle U} and V {\displaystyle V} and a diagonal (scaling) matrix Σ {\displaystyle \Sigma } . The vector space defined by them has as number of dimensions p, that is the smaller of the two values, number of rows and number of columns, minus 1. === Inertia === While a principal component analysis may be said to decompose the (co)variance, and hence its measure of success is the amount of (co-)variance covered by the first few PCA axes - measured in eigenvalue -, a CA works with a weighted (co-)variance which is called inertia. The sum of the squared singular values is the total inertia I {\displaystyle \mathrm {I} } of the data table, computed as I = ∑ i = 1 p σ i 2 . {\displaystyle \mathrm {I} =\sum _{i=1}^{p}\sigma _{i}^{2}.} The total inertia I {\displaystyle \mathrm {I} } of the data table can also computed directly from S {\displaystyle S} as I = ∑ i = 1 n ∑ j = 1 m s i j 2 . {\displaystyle \mathrm {I} =\sum _{i=1}^{n}\sum _{j=1}^{m}s_{ij}^{2}.} The amount of inertia covered by the i-th set of singular vectors is ι i {\displaystyle \iota _{i}} , the principal inertia. The higher the portion of inertia covered by the first few singular vectors i.e. the larger the sum of the principal inertiae in comparison to the total inertia, the more successful a CA is. Therefore, all principal inertia values are expressed as portion ϵ i {\displaystyle \epsilon _{i}} of the total inertia ϵ i = σ i 2 / ∑ i = 1 p σ i 2 {\displaystyle \epsilon _{i}=\sigma _{i}^{2}/\sum _{i=1}^{p}\sigma _{i}^{2}} and are presented in the form of a scree plot. In fact a scree plot is just a bar plot of all principal inertia portions ϵ i {\displaystyle \epsilon _{i}} . === Coordinates === To transform the singular vectors to coordinates which preserve the chi-square distances between rows or columns an additional weighting step is necessary. The resulting coordinates are called principal coordinates in CA text books. If principal coordinates are used for
Inverted pendulum
An inverted pendulum is a pendulum that has its center of mass above its pivot point. It is unstable and falls over without additional help. It can be suspended stably in this inverted position by using a control system to monitor the angle of the pole and move the pivot point horizontally back under the center of mass when it starts to fall over, keeping it balanced. The inverted pendulum is a classic problem in dynamics and control theory and is used as a benchmark for testing control strategies. It is often implemented with the pivot point mounted on a cart that can move horizontally under control of an electronic servo system as shown in the photo; this is called a cart and pole apparatus. Most applications limit the pendulum to 1 degree of freedom by affixing the pole to an axis of rotation. Whereas a normal pendulum is stable when hanging downward, an inverted pendulum is inherently unstable, and must be actively balanced in order to remain upright; this can be done either by applying a torque at the pivot point, by moving the pivot point horizontally as part of a feedback system, changing the rate of rotation of a mass mounted on the pendulum on an axis parallel to the pivot axis and thereby generating a net torque on the pendulum, or by oscillating the pivot point vertically. A simple demonstration of moving the pivot point in a feedback system is achieved by balancing an upturned broomstick on the end of one's finger. A second type of inverted pendulum is a tiltmeter for tall structures, which consists of a wire anchored to the bottom of the foundation and attached to a float in a pool of oil at the top of the structure that has devices for measuring movement of the neutral position of the float away from its original position. == Overview == A pendulum with its bob hanging directly below the support pivot is at a stable equilibrium point, where it remains motionless because there is no torque on the pendulum. If displaced from this position, it experiences a restoring torque that returns it toward the equilibrium position. A pendulum with its bob in an inverted position, supported on a rigid rod directly above the pivot, 180° from its stable equilibrium position, is at an unstable equilibrium point. At this point again there is no torque on the pendulum, but the slightest displacement away from this position causes a gravitation torque on the pendulum that accelerates it away from equilibrium, causing it to fall over. In order to stabilize a pendulum in this inverted position, a feedback control system can be used, which monitors the pendulum's angle and moves the position of the pivot point sideways when the pendulum starts to fall over, to keep it balanced. The inverted pendulum is a classic problem in dynamics and control theory and is widely used as a benchmark for testing control algorithms (PID controllers, state-space representation, neural networks, fuzzy control, genetic algorithms, etc.). Variations on this problem include multiple links, allowing the motion of the cart to be commanded while maintaining the pendulum, and balancing the cart-pendulum system on a see-saw. The inverted pendulum is related to rocket or missile guidance, where the center of gravity is located behind the center of drag causing aerodynamic instability. The understanding of a similar problem can be shown by simple robotics in the form of a balancing cart. Balancing an upturned broomstick on the end of one's finger is a simple demonstration, and the problem is solved by self-balancing personal transporters such as the Segway PT, the self-balancing hoverboard and the self-balancing unicycle. Another way that an inverted pendulum may be stabilized, without any feedback or control mechanism, is by oscillating the pivot rapidly up and down. This is called Kapitza's pendulum. If the oscillation is sufficiently strong (in terms of its acceleration and amplitude) then the inverted pendulum can recover from perturbations in a strikingly counterintuitive manner. If the driving point moves in simple harmonic motion, the pendulum's motion is described by the Mathieu equation. == Equations of motion == The equations of motion of inverted pendulums are dependent on what constraints are placed on the motion of the pendulum. Inverted pendulums can be created in various configurations resulting in a number of Equations of Motion describing the behavior of the pendulum. === Stationary pivot point === In a configuration where the pivot point of the pendulum is fixed in space, the equation of motion is similar to that for an uninverted pendulum. The equation of motion below assumes no friction or any other resistance to movement, a rigid massless rod, and the restriction to 2-dimensional movement. θ ¨ − g ℓ sin θ = 0 {\displaystyle {\ddot {\theta }}-{g \over \ell }\sin \theta =0} Where θ ¨ {\displaystyle {\ddot {\theta }}} is the angular acceleration of the pendulum, g {\displaystyle g} is the standard gravity on the surface of the Earth, ℓ {\displaystyle \ell } is the length of the pendulum, and θ {\displaystyle \theta } is the angular displacement measured from the equilibrium position. When θ ¨ {\displaystyle {\ddot {\theta }}} added to both sides, it has the same sign as the angular acceleration term: θ ¨ = g ℓ sin θ {\displaystyle {\ddot {\theta }}={g \over \ell }\sin \theta } Thus, the inverted pendulum accelerates away from the vertical unstable equilibrium in the direction initially displaced, and the acceleration is inversely proportional to the length. Tall pendulums fall more slowly than short ones. Derivation using torque and moment of inertia: The pendulum is assumed to consist of a point mass, of mass m {\displaystyle m} , affixed to the end of a massless rigid rod, of length ℓ {\displaystyle \ell } , attached to a pivot point at the end opposite the point mass. The net torque of the system must equal the moment of inertia times the angular acceleration: τ n e t = I θ ¨ {\displaystyle {\boldsymbol {\tau }}_{\mathrm {net} }=I{\ddot {\theta }}} The torque due to gravity providing the net torque: τ n e t = m g ℓ sin θ {\displaystyle {\boldsymbol {\tau }}_{\mathrm {net} }=mg\ell \sin \theta \,\!} Where θ {\displaystyle \theta \ } is the angle measured from the inverted equilibrium position. The resulting equation: I θ ¨ = m g ℓ sin θ {\displaystyle I{\ddot {\theta }}=mg\ell \sin \theta \,\!} The moment of inertia for a point mass: I = m R 2 {\displaystyle I=mR^{2}} In the case of the inverted pendulum the radius is the length of the rod, ℓ {\displaystyle \ell } . Substituting in I = m ℓ 2 {\displaystyle I=m\ell ^{2}} m ℓ 2 θ ¨ = m g ℓ sin θ {\displaystyle m\ell ^{2}{\ddot {\theta }}=mg\ell \sin \theta \,\!} Mass and ℓ 2 {\displaystyle \ell ^{2}} is divided from each side resulting in: θ ¨ = g ℓ sin θ {\displaystyle {\ddot {\theta }}={g \over \ell }\sin \theta } === Inverted pendulum on a cart === An inverted pendulum on a cart consists of a mass m {\displaystyle m} at the top of a pole of length ℓ {\displaystyle \ell } pivoted on a horizontally moving base as shown in the adjacent image. The cart is restricted to linear motion and is subject to forces resulting in or hindering motion. === Essentials of stabilization === The essentials of stabilizing the inverted pendulum can be summarized qualitatively in three steps. 1. If the tilt angle θ {\displaystyle \theta } is to the right, the cart must accelerate to the right and vice versa. 2. The position of the cart x {\displaystyle x} relative to track center is stabilized by slightly modulating the null angle (the angle error that the control system tries to null) by the position of the cart, that is, null angle = θ + k x {\displaystyle =\theta +kx} where k {\displaystyle k} is small. This makes the pole want to lean slightly toward track center and stabilize at track center where the tilt angle is exactly vertical. Any offset in the tilt sensor or track slope that would otherwise cause instability translates into a stable position offset. A further added offset gives position control. 3. A normal pendulum subject to a moving pivot point such as a load lifted by a crane, has a peaked response at the pendulum radian frequency of ω p = g / ℓ {\displaystyle \omega _{p}={\sqrt {g/\ell }}} . To prevent uncontrolled swinging, the frequency spectrum of the pivot motion should be suppressed near ω p {\displaystyle \omega _{p}} . The inverted pendulum requires the same suppression filter to achieve stability. As a consequence of the null angle modulation strategy, the position feedback is positive, that is, a sudden command to move right produces an initial cart motion to the left followed by a move right to rebalance the pendulum. The interaction of the pendulum instability and the positive position feedback instability to produce a stable system is a feature that makes the mathematical analysis an interesting and challenging problem. === From Lagrange's equations === The equations of motion c
GamePigeon
GamePigeon is a mobile app for iOS devices, developed by Vitalii Zlotskii and released on September 13, 2016. The game takes advantage of the iOS 10 update, which expanded how users could interact with Apple's Messages app. GamePigeon is only available through the Messages app, which allows players to start and respond to different party games in conversations. == Release == The app was first released on September 13, 2016, coinciding with the launch of iOS 10. The app was released for free, although it includes in-app purchases to unlock additional items, such as cosmetic skins, avatar items, new game modes, and an option to remove ads. == Games in the app == The following is a list of games that users can play within GamePigeon: Sources: Poker was one of the games included in GamePigeon at launch, although it has since been removed and is no longer listed on the game's App Store description. == Reception == GamePigeon has enjoyed commercial success, with VentureBeat noting that GamePigeon was ranked number-one in the "Top Free" category of the iMessage App Store, six months after its release. Critically, GamePigeon has been generally well received, being highlighted by online media publications early on shortly after the iOS 10 launch. It has since been included on many "best iMessage apps" lists. Based on over 162,000 ratings, the game holds a 4.0 out of 5 rating on the App Store. Julian Chokkattu of Digital Trends wrote "GamePigeon should be like the pre-installed versions of Solitaire and Minesweeper that used to come with older iterations of Windows." On its launch day, Boy Genius Report included it on a list of "10 of the best iMessage apps, games and stickers for iOS 10 on launch day." The Daily Dot wrote, "GamePigeon is easily the best current gaming option within iMessages." 8-ball and cup pong have been particularly well received by media outlets. The Daily Dot had specific praise for the app's billiards game: "8-Ball controls shockingly smoothly with your fingers, and there’s nothing quite like destroying a dear friend in poker." During his 2020 U.S. presidential campaign, Cory Booker was cited as playing the game with his family. In 2017, CNBC cited one teenager who expressed that GamePigeon was one of just a few reasons that those in her age range use the iMessage app. The game has received particular positive reception for allowing introverted individuals to exercise a form social activity; similarly, the game was highlighted as a way to maintain social distancing guidelines during the COVID-19 pandemic. As an April Fools' Day joke in 2020, The Chronicle, a Duke University newspaper, published that Duke's athletic program adopted GamePigeon's Cup Pong as an official varsity sport.
Synaptic weight
In neuroscience and computer science, synaptic weight refers to the strength or amplitude of a connection between two nodes, corresponding in biology to the amount of influence the firing of one neuron has on another. The term is typically used in artificial and biological neural network research. == Computation == In a computational neural network, a vector or set of inputs x {\displaystyle {\textbf {x}}} and outputs y {\displaystyle {\textbf {y}}} , or pre- and post-synaptic neurons respectively, are interconnected with synaptic weights represented by the matrix w {\displaystyle w} , where for a linear neuron y j = ∑ i w i j x i or y = w x {\displaystyle y_{j}=\sum _{i}w_{ij}x_{i}~~{\textrm {or}}~~{\textbf {y}}=w{\textbf {x}}} . where the rows of the synaptic matrix represent the vector of synaptic weights for the output indexed by j {\displaystyle j} . The synaptic weight is changed by using a learning rule, the most basic of which is Hebb's rule, which is usually stated in biological terms as Neurons that fire together, wire together. Computationally, this means that if a large signal from one of the input neurons results in a large signal from one of the output neurons, then the synaptic weight between those two neurons will increase. The rule is unstable, however, and is typically modified using such variations as Oja's rule, radial basis functions or the backpropagation algorithm. == Biology == For biological networks, the effect of synaptic weights is not as simple as for linear neurons or Hebbian learning. However, biophysical models such as BCM theory have seen some success in mathematically describing these networks. In the mammalian central nervous system, signal transmission is carried out by interconnected networks of nerve cells, or neurons. For the basic pyramidal neuron, the input signal is carried by the axon, which releases neurotransmitter chemicals into the synapse which is picked up by the dendrites of the next neuron, which can then generate an action potential which is analogous to the output signal in the computational case. The synaptic weight in this process is determined by several variable factors: How well the input signal propagates through the axon (see myelination), The amount of neurotransmitter released into the synapse and the amount that can be absorbed in the following cell (determined by the number of AMPA and NMDA receptors on the cell membrane and the amount of intracellular calcium and other ions), The number of such connections made by the axon to the dendrites, How well the signal propagates and integrates in the postsynaptic cell. The changes in synaptic weight that occur is known as synaptic plasticity, and the process behind long-term changes (long-term potentiation and depression) is still poorly understood. Hebb's original learning rule was originally applied to biological systems, but has had to undergo many modifications as a number of theoretical and experimental problems came to light.