Preference regression is a statistical technique used by marketers to determine consumers’ preferred core benefits. It usually supplements product positioning techniques like multi dimensional scaling or factor analysis and is used to create ideal vectors on perceptual maps. == Application == Starting with raw data from surveys, researchers apply positioning techniques to determine important dimensions and plot the position of competing products on these dimensions. Next they regress the survey data against the dimensions. The independent variables are the data collected in the survey. The dependent variable is the preference datum. Like all regression methods, the computer fits weights to best predict data. The resultant regression line is referred to as an ideal vector because the slope of the vector is the ratio of the preferences for the two dimensions. If all the data is used in the regression, the program will derive a single equation and hence a single ideal vector. This tends to be a blunt instrument so researchers refine the process with cluster analysis. This creates clusters that reflect market segments. Separate preference regressions are then done on the data within each segment. This provides an ideal vector for each segment. == Alternative methods == Self-stated importance method is an alternative method in which direct survey data is used to determine the weightings rather than statistical imputations. A third method is conjoint analysis in which an additive method is used.
Umple
Umple is a language for both object-oriented programming and modelling with class diagrams and state diagrams. The name Umple is a portmanteau of "UML", "ample" and "Simple", indicating that it is designed to provide ample features to extend programming languages with UML capabilities. == History and philosophy == The design of Umple started in 2008 at the University of Ottawa. Umple was open-sourced and its development was moved to Google Code in early 2011 and to GitHub in 2015. Umple was developed, in part, to address certain problems observed in the modelling community. Most specifically, it was designed to bring modelling and programming into alignment, It was intended to help overcome inhibitions against modelling common in the programmer community. It was also intended to reduce some of the difficulties of model-driven development that arise from the need to use large, expensive or incomplete tools. One design objective is to enable programmers to model in a way they see as natural, by adding modelling constructs to programming languages. == Features and capabilities == Umple can be used to represent in a textual manner many UML modelling entities found in class diagrams and state diagrams. Umple can generate code for these in various programming languages. Currently Umple fully supports Java, C++ and PHP as target programming languages and has functional, but somewhat incomplete support for Ruby. Umple also incorporates various features not related to UML, such as the singleton pattern, keys, immutability, mixins and aspect-oriented code injection. The class diagram notations Umple supports includes classes, interfaces, attributes, associations, generalizations and operations. The code Umple generates for attributes include code in the constructor, 'get' methods and 'set' methods. The generated code differs considerably depending on whether the attribute has properties such as immutability, has a default value, or is part of a key. Umple generates many methods for manipulating, querying and navigating associations. It supports all combinations of UML multiplicity and enforces referential integrity. Umple supports the vast majority of UML state machine notation, including arbitrarily deep nested states, concurrent regions, actions on entry, exit and transition, plus long-lasting activities while in a state. A state machine is treated as an enumerated attribute where the value is controlled by events. Events encoded in the state machine can be methods written by the user, or else generated by the Umple compiler. Events are triggered by calling the method. An event can trigger transitions (subject to guards) in several different state machines. Since a program can be entirely written around one or more state machines, Umple enables automata-based programming. The bodies of methods are written in one of the target programming languages. The same is true for other imperative code such as state machine actions and guards, and code to be injected in an aspect-oriented manner. Such code can be injected before many of the methods in the code Umple generates, for example before or after setting or getting attributes and associations. The Umple notation for UML constructs can be embedded in any of its supported target programming languages. When this is done, Umple can be seen as a pre-processor: The Umple compiler expands the UML constructs into code of the target language. Code in a target language can be passed to the Umple compiler directly; if no Umple-specific notation is found, then the target-language code is emitted unchanged by the Umple compiler. Umple, combined with one of its target languages for imperative code, can be seen and used as a complete programming language. Umple plus Java can therefore be seen as an extension of Java. Alternatively, if imperative code and Umple-specific concepts are left out, Umple can be seen as a way of expressing a large subset of UML in a purely textual manner. Code in one of the supported programming languages can be added in the same manner as UML envisions adding action language code. == License == Umple is licensed under an MIT-style license. == Examples == Here is the classic Hello world program written in Umple (extending Java): This example looks just like Java, because Umple extends other programming languages. With the program saved in a file named HelloWorld.ump, it can be compiled from the command line: $ java -jar umple.jar HelloWorld.ump To run it: $ java HelloWorld The following is a fully executable example showing embedded Java methods and declaration of an association. The following example describes a state machine called status, with states Open, Closing, Closed, Opening and HalfOpen, and with various events that cause transitions from one state to another. class GarageDoor { status { Open { buttonOrObstacle -> Closing; } Closing { buttonOrObstacle -> Opening; reachBottom -> Closed; } Closed { buttonOrObstacle -> Opening; } Opening { buttonOrObstacle -> HalfOpen; reachTop -> Open; } HalfOpen { buttonOrObstacle -> Opening; } } } == Umple use in practice == The first version of the Umple compiler was written in Java, Antlr and Jet (Java Emitter Templates), but in a bootstrapping process, the Java code was converted to Umple following a technique called Umplification. The Antlr and Jet were also later converted to native Umple. Umple is therefore now written entirely in itself, in other words it is self-hosted and serves as its own largest test case. Umple and UmpleOnline have been used in the classroom by several instructors to teach UML and modelling. In one study it was found to help speed up the process of teaching UML, and was also found to improve the grades of students. == Tools == Umple is available as a Jar file so it can be run from the command line, and as an Eclipse plugin. There is also an online tool for Umple called UmpleOnline , which allows a developer to create an Umple system by drawing a UML class diagram, editing Umple code or both. Umple models created with UmpleOnline are stored in the cloud. Currently UmpleOnline only supports Umple programs consisting of a single input file. In addition to code, Umple's tools can generate a variety of other types of output, including user interfaces based on the Umple model.
Framework Convention on Artificial Intelligence
The Framework Convention on Artificial Intelligence and Human Rights, Democracy and the Rule of Law (also called Framework Convention on Artificial Intelligence or AI convention) is an international treaty on artificial intelligence. It was adopted under the auspices of the Council of Europe (CoE) and signed on 5 September 2024. The treaty aims to ensure that the development and use of AI technologies align with fundamental human rights, democratic values, and the rule of law, addressing risks such as misinformation, algorithmic discrimination, and threats to public institutions. More than 50 countries, including the EU member states, have endorsed the Framework Convention on Artificial Intelligence. == Background == The development of the Framework Convention on AI emerged in response to growing concerns over the ethical, legal, and societal impacts of artificial intelligence. The Council of Europe, which has historically played a key role in setting human rights standards across Europe, initiated discussions on AI governance in 2020, leading to the drafting of a binding legal framework. The process of creating the Framework Convention began in 2019 with the ad hoc Committee on Artificial Intelligence (CAHAI) assessing the feasibility of the instrument. In 2022, the Committee on Artificial Intelligence (CAI) took over the process, drafting and negotiating the text of the Convention. The treaty is designed to complement existing international human rights instruments, including the European Convention on Human Rights and the Convention for the Protection of Individuals with regard to Automatic Processing of Personal Data. == Structure and content == The Convention establishes fundamental principles for AI governance, including transparency, accountability, non-discrimination, and human rights protection through eight chapters and 26 articles. Adopted in 2024, this landmark treaty addresses AI governance through seven core principles and detailed implementation mechanisms. It mandates risk and impact assessments to mitigate potential harms and provides safeguards such as the right to challenge AI-driven decisions. It applies to public authorities and private entities acting on their behalf but excludes national security and defense activities. Implementation is overseen by a Conference of the Parties, ensuring compliance and international cooperation. Activities within the AI system lifecycle must adhere to seven fundamental principles, ensuring compliance with human rights, democracy, and the rule of law. The treaty also establishes remedies, procedural rights and safeguards, and risk and impact management requirements to promote accountability, transparency, and responsible AI development. The treaty consists of five chapters. Chapter I contains general provisions. Chapter II states the general obligation to protect human rights and the integrity of democratic processes and respect of the rule of law. The main principles and rights are contained in Chapter III, which consists of Articles 6 to 13. Chapter IV (Articles 14 to 15) sets up the legal remedies. Chapter V states the risk and impact management framework. Chapter VI facilitates the implementation criteria of the treaty. Chapter VII sets the co-operation and oversight mechanisms. Chapter VIII contains various concluding clauses. Article 1 declares the objectives of the treaty, to ensure that activities within the lifecycle of artificial intelligence systems are fully consistent with human rights, democracy and the rule of law. == Entry into force == The treaty will enter into force on the first day of the month following the expiration of a period of three months after the date on which five ratification made by five countries, including three member states of the Council of Europe. == Competing approaches == While the CoE's AI Convention represents a multilateral effort to regulate AI through a human rights-based approach, alternative frameworks have also been proposed. One notable example is the Munich Draft for a Convention on AI, Data and Human Rights, an initiative led by legal scholars and policymakers in Germany. The Munich Draft advocates for stronger safeguards against AI-related risks, emphasizing stricter data protection measures, accountability for AI developers, and explicit prohibitions on high-risk AI applications, such as mass surveillance and autonomous lethal weapons. Unlike the CoE convention, which focuses on balancing innovation with regulation, the Munich Draft takes a more precautionary stance, calling for tighter controls over AI deployment in sensitive domains. Other competing international efforts include the OECD’s AI Principles, the GPAI (Global Partnership on AI), and the European Union's AI Act, each of which offers different regulatory strategies to govern AI at regional and global levels. == Signatories == Signatories include Andorra, Canada, the European Union, Georgia, Iceland, Israel, Japan, Liechtenstein, the Republic of Moldova, Montenegro, Norway, San Marino, Switzerland, Ukraine, the United Kingdom, the United States, and Uruguay. == Endorsement == The treaty was widely endorsed by leading AI policy experts, including Stuart J. Russell, Virginia Dignum, Emma Ruttkamp-Bloem, Pascal Pichonnaz, Maria Helen Murphy, Angella Ndaka, Hannes Werthner, Katja Langenbucher, Gry Hasselbalch, Ricardo Baeza-Yates, Kutoma Wakunuma, Gianclaudio Malgieri, Oreste Pollicino, Nagla Rizk, Giovanni Sartor, Lee Tiedrich, Ingrid Schneider, Eduardo Bertoni, Garry Kasparov, Merve Hikcok, and Marc Rotenberg. The treaty was also endorsed by notable political leaders, including Theodoros Roussopoulos, President of the Parliamentart Assembly in the Council of Europe, and Christopher Holmes, Member of the House of Lords of the United Kingdom, and by the International Bar Association (IBA), and personally by Almudena Arpón de Mendívil, President of the IBA. The Center for AI and Digital Policy (CAIDP) has been carrying out a campaign to promote endorsement of the treaty by urging various countries to sign and ratify the treaty. The CAIDP further urged the countries to make a clear and firm commitment to ensure the full inclusion of the private sector under the treaty’s provisions.
Cleverpath AION Business Rules Expert
Cleverpath AION Business Rules Expert (formerly Platinum AIONDS, and before that Trinzic AIONDS, and originally Aion) is an expert system and Business rules engine owned by Computer Associates by 2000. == History == The product was created around 1986 as "Aion" by the Aion company. In its initial release Aion was multi-platform and continues to be deliverable to the PC, Unixs, and Mainframe computer's. In addition it ties in seamlessly with a variety of databases including Oracle, Microsoft SQL Server, and ODBC. Aion was founded by Harry Reinstein, Larry Cohn, Garry Hallee, Scott Grinis, and others. From Scott Grinis's bio: Scott founded Aion, a company that developed expert systems and whose advanced inference engine and object technology were used by financial services and insurance firms to develop risk-scoring and underwriting applications. Harry Reinstein was quoted as saying: “Our biggest competitor was not AICorp, it was COBOL” Trinzic owned AION by 1993. A reference in a 1993 announcement indicates that Trinzic's formation was the result of a merger (paraphased): Trinzic set three development initiatives shortly after its formation from the merger of Aion Corp. and AICorp. The other initiatives -- adding SQL extensions to Aion/DS and evaluating the unbundling of some of that product's object-oriented programming capabilities -- are still active. Writing in 1993 Judith Hodges and Deborah Melewski give the date for the merger: Two rival artificial intelligence software vendors -- AICorp, Inc. and Aion Corp. -- merged in September 1992 to form Trinzic Corp. As part of the merger, redundant jobs were eliminated (20% of the combined work force), leaving a total work force of 245 employees worldwide. The new firm also boasted a combined installed base of more than 1,200 sites representing more than 10,000 software licenses. Although in the merger, technically AICorp bought Aion, as AICorp was a public company and Aion was still private, the reality was that Aion's leadership and technology subsumed AICorp's. Jim Gagnard, the CEO of Aion, became CEO of Trinzic and AICorp's flagship product, KBMS, was discontinued, while the Aion Development System continued to be enhanced and KBMS customers were assisted in converting to AIONDS, under the continued technical leadership of Garry Hallee and Scott Grinis. On August 1, 1994 Trinzic released version 6.4 of AIONDS saying, in part: Trinzic Corp., Palo Alto, Calif., has unveiled The Aion Development System (AionDS) Version 6.4, an upgrade to the company's development environment for building business process automation applications. Version 6.4 provides a visual development environment for Microsoft Windows or OS/2 PM applications using business rules. Trinzic was acquired by PLATINUM Technologies in 1995 which retained at least some of Trinzic's acquisitions Platinum Technologies was acquired by Computer Associates in 1999. CA changed the system's name to CA Aion Business Rules Expert" on or before 2009. It is currently (June 2011) at Release 11 on a wide range of supported platforms. == Applications using Aion == Aion has been used in a variety of industries including Energy, Insurance, Military, Aviation, and Banking. At one point an Aion expert system application written by Covia, LLC existed to do airport gate assignment. Colossus, a computer program, developed by Computer Sciences Corporation is the insurance industry’s leading expert system for assisting adjusters in the evaluation of bodily injury claims (aka "pain and suffering"). Colossus helps adjusters reduce variance in payouts on similar bodily injury claims through objective use of industry standard rules.
Trustworthy AI
Trustworthy AI refers to artificial intelligence systems that are designed to have transparent reasoning, are explainable (XAI), accountable, robust, fair and honest, respectful of data privacy, and steerable or alignable with human goals. == Terminology == Recent work in AI ethics distinguishes trustworthiness and trustability as two different conditions relevant to trustworthy AI. Trustworthiness is concerned with whether an AI system or the institutions deploying it merit trust by being reliable, fair, and accountable. Trustability, on the other hand, is the prior question of whether a given entity is even the kind of thing to which interpersonal trust can coherently apply as opposed to mere instrumental reliance. Some philosophers argue that current AI systems are best understood as tools that are not genuine targets of interpersonal trust. They argue that trust should be directed toward the human and institutional arrangements that govern the systems' design, deployment, and oversight. This stance supports interpreting "trustworthy AI" as trustworthy governance and use of AI rather than trust in the artifacts themselves. Transparency in AI involves making the processes and decisions of such systems understandable to users and stakeholders. Accountability ensures that there are protocols for addressing adverse outcomes or biases that may arise, with designated responsibilities for oversight and remediation. Robustness and security aim to ensure that AI systems perform reliably under various conditions and are safeguarded against malicious attacks. Harmlessness can be achieved by refusal training: training the models to avoid problematic requests, and by adding filters to detect and prevent discussion on biased, unethical, or dangerous outputs. There is research on how to train AI so that it aligns with human goals. == Techniques and ITU standardization == Trustworthy AI creation is a goal of AI governance and policymaking. To achieve transparency and data privacy, several privacy-enhancing technologies (PETs) can be used. These include: Homomorphic encryption for computing with encrypted data without ever decrypting it. Federated learning and secure multi-party computation (MPC) for distributing the model training without sharing information between the learning centers and computing servers. Differential privacy for exposing statistical data while guaranteeing that no private information is exposed. Zero-knowledge proof - providing proven validity for statements without disclosing any extra information. A work programme for achieving Trustworthy AI was set up by the International Telecommunication Union, an agency of the United Nations, initiated under its AI for Good programme. Its origin lies with the ITU-WHO Focus Group on Artificial Intelligence for Health, where a strong need for both privacy and analytics created demand for a standard in these technologies. In 2020, AI for Good moved online, and the TrustworthyAI seminar series was established to initiate discussions on these topics. This eventually led to standardization activities. === Multi-party computation === Secure multi-party computation (MPC) is being standardized under "Question 5" (the incubator) of ITU-T Study Group 17. === Homomorphic encryption === Homomorphic encryption allows for computing on encrypted data, where the outcomes or result is still encrypted and unknown to those performing the computation, but can be deciphered by the original encryptor. It is often developed with the goal of enabling use in jurisdictions different from the data creation (under, for instance, GDPR). ITU has been collaborating since the early stage of the HomomorphicEncryption.org standardization meetings, which has developed a standard on homomorphic encryption. The fifth homomorphic encryption meeting was hosted at ITU HQ in Geneva. === Federated learning === Zero-sum masks as used by federated learning for privacy preservation are used extensively in the multimedia standards of ITU-T Study Group 16 (VCEG) such as JPEG, MP3, H.264, and H.265 (commonly known as MPEG). === Zero-knowledge proof === Previous pre-standardization work on the topic of zero-knowledge proof has been conducted in the ITU-T Focus Group on Digital Ledger Technologies. === Differential privacy === The application of differential privacy in the preservation of privacy was examined at several of the "Day 0" machine learning workshops at AI for Good Global Summits. == Mozilla "Rebel Alliance" == In January 2026, the Mozilla Foundation and its subsidiaries announced a strategic shift to deploy their entire $1.4 billion reserve into building what foundation president Mark Surman termed a "rebel alliance" for trustworthy AI. Framed by Surman as a mission-driven alternative to the market dominance of OpenAI and Anthropic, the initiative seeks to establish an open-source AI stack by 2028. The alliance includes several startups funded via Mozilla Ventures, specifically focusing on decentralized governance and transparency: Trail: A firm developing AI compliance frameworks for regulated industries. Transformer Lab: A developer of open-source tools for AI model management. Oumi: A platform for training and deploying open-source models. The "rebel alliance" terminology is a historical reference to Mozilla's efforts in 1998 to challenge Microsoft's browser monopoly. While the $1.4 billion in funding is significant, it has been contrasted with the tens of billions in capital raised by proprietary competitors like OpenAI.
Intrinsic dimension
In mathematics, the intrinsic dimension of a subset can be thought of as the minimal number of variables needed to represent the subset. The concept has widespread applications in geometry, dynamical systems, signal processing, statistics, and other fields. Due to its widespread applications and vague conceptualization, there are many different ways to define it rigorously. Consequently, the same set might have different intrinsic dimensions according to different definitions. The intrinsic dimension can be used as a lower bound of what dimension it is possible to compress a data set into through dimension reduction, but it can also be used as a measure of the complexity of the data set or signal. For a data set or signal of N variables, its intrinsic dimension M satisfies 0 ≤ M ≤ N, although estimators may yield higher values. == Exact dimension == === Differential === In differential geometry, given a differentiable manifold N and a submanifold M, the intrinsic dimension of M is its dimension. Suppose N has n dimensions and M has m dimensions, then that means around any point in M, there exists a local coordinate system ( x 1 , … , x m , x m + 1 , … , x n ) {\displaystyle (x_{1},\dots ,x_{m},x_{m+1},\dots ,x_{n})} of N, such that the manifold M is simply the subset of N defined by x m + 1 = 0 , … , x n = 0 {\displaystyle x_{m+1}=0,\dots ,x_{n}=0} . === Metric === Given a mere metric space, we can still define its intrinsic dimension. The most general case is the Hausdorff dimension, though for metric spaces occurring in practice, the box-counting dimension and the packing dimension often are identical to the Hausdorff dimension. Let X , d {\textstyle X,d} be a metric space and A ⊂ X {\textstyle A\subset X} be totally bounded. Define the covering number N ( A , ε ) = min { k : A ⊂ ⋃ i = 1 k B ( x i , ε ) } . {\displaystyle N(A,\varepsilon )=\min \left\{k:A\subset \bigcup _{i=1}^{k}B\left(x_{i},\varepsilon \right)\right\}.} The metric entropy is H ( A , ε ) = log N ( A , ε ) {\textstyle H(A,\varepsilon )=\log N(A,\varepsilon )} (any log base). The upper and lower metric entropy dimensions are dim ¯ E A = lim sup ε ↓ 0 H ( A , ε ) log ( 1 / ε ) , dim _ E A = lim inf ε ↓ 0 H ( A , ε ) log ( 1 / ε ) . {\displaystyle {\overline {\dim }}_{E}A=\limsup _{\varepsilon \downarrow 0}{\frac {H(A,\varepsilon )}{\log(1/\varepsilon )}},\quad {\underline {\dim }}_{E}A=\liminf _{\varepsilon \downarrow 0}{\frac {H(A,\varepsilon )}{\log(1/\varepsilon )}}.} If they are equal, then dim E A {\textstyle \operatorname {dim} _{E}A} is that common value, called the metric entropy dimension. The entropy dimensions are usually used in information theory, and especially coding theory, since entropy is involved in its definition. === Topological === If X {\displaystyle X} is merely a topological space, then we can still define its intrinsic dimension, using the topological dimension or Lebesgue covering dimension. An open cover of a topological space X is a family of open sets Uα such that their union is the whole space, ∪ α {\displaystyle \cup _{\alpha }} Uα = X. The order or ply of an open cover A {\displaystyle {\mathfrak {A}}} = {Uα} is the smallest number m (if it exists) for which each point of the space belongs to at most m open sets in the cover: in other words Uα1 ∩ ⋅⋅⋅ ∩ Uαm+1 = ∅ {\displaystyle \emptyset } for α1, ..., αm+1 distinct. A refinement of an open cover A {\displaystyle {\mathfrak {A}}} = {Uα} is another open cover B {\displaystyle {\mathfrak {B}}} = {Vβ}, such that each Vβ is contained in some Uα. The covering dimension of a topological space X is defined to be the minimum value of n such that every finite open cover A {\displaystyle {\mathfrak {A}}} of X has an open refinement B {\displaystyle {\mathfrak {B}}} with order n + 1. The refinement B {\displaystyle {\mathfrak {B}}} can always be chosen to be finite. Thus, if n is finite, Vβ1 ∩ ⋅⋅⋅ ∩ Vβn+2 = ∅ {\displaystyle \emptyset } for β1, ..., βn+2 distinct. If no such minimal n exists, the space is said to have infinite covering dimension. == Introductory example == Let f ( x 1 , x 2 ) {\textstyle f(x_{1},x_{2})} be a two-variable function (or signal) which is of the form f ( x 1 , x 2 ) = g ( x 1 ) {\textstyle f(x_{1},x_{2})=g(x_{1})} for some one-variable function g which is not constant. This means that f varies, in accordance to g, with the first variable or along the first coordinate. On the other hand, f is constant with respect to the second variable or along the second coordinate. It is only necessary to know the value of one, namely the first, variable in order to determine the value of f. Hence, it is a two-variable function but its intrinsic dimension is one. A slightly more complicated example is f ( x 1 , x 2 ) = g ( x 1 + x 2 ) {\textstyle f(x_{1},x_{2})=g(x_{1}+x_{2})} . f is still intrinsic one-dimensional, which can be seen by making a variable transformation y 1 = x 1 + x 2 {\textstyle y_{1}=x_{1}+x_{2}} and y 2 = x 1 − x 2 {\textstyle y_{2}=x_{1}-x_{2}} which gives f ( y 1 + y 2 2 , y 1 − y 2 2 ) = g ( y 1 ) {\textstyle f\left({\frac {y_{1}+y_{2}}{2}},{\frac {y_{1}-y_{2}}{2}}\right)=g\left(y_{1}\right)} . Since the variation in f can be described by the single variable y1 its intrinsic dimension is one. For the case that f is constant, its intrinsic dimension is zero since no variable is needed to describe variation. For the general case, when the intrinsic dimension of the two-variable function f is neither zero or one, it is two. In the literature, functions which are of intrinsic dimension zero, one, or two are sometimes referred to as i0D, i1D or i2D, respectively. == Signal processing == In signal processing of multidimensional signals, the intrinsic dimension of the signal describes how many variables are needed to generate a good approximation of the signal. For an N-variable function f, the set of variables can be represented as an N-dimensional vector x: f = f ( x ) where x = ( x 1 , … , x N ) {\textstyle f=f\left(\mathbf {x} \right){\text{ where }}\mathbf {x} =\left(x_{1},\dots ,x_{N}\right)} . If for some M-variable function g and M × N matrix A it is the case that for all x; f ( x ) = g ( A x ) , {\textstyle f(\mathbf {x} )=g(\mathbf {Ax} ),} M is the smallest number for which the above relation between f and g can be found, then the intrinsic dimension of f is M. The intrinsic dimension is a characterization of f, it is not an unambiguous characterization of g nor of A. That is, if the above relation is satisfied for some f, g, and A, it must also be satisfied for the same f and g′ and A′ given by g ′ ( y ) = g ( B y ) {\textstyle g'\left(\mathbf {y} \right)=g\left(\mathbf {By} \right)} and A ′ = B − 1 A {\textstyle \mathbf {A'} =\mathbf {B} ^{-1}\mathbf {A} } where B is a non-singular M × M matrix, since f ( x ) = g ′ ( A ′ x ) = g ( B A ′ x ) = g ( A x ) {\textstyle f\left(\mathbf {x} \right)=g'\left(\mathbf {A'x} \right)=g\left(\mathbf {BA'x} \right)=g\left(\mathbf {Ax} \right)} . == The Fourier transform of signals of low intrinsic dimension == An N variable function which has intrinsic dimension M < N has a characteristic Fourier transform. Intuitively, since this type of function is constant along one or several dimensions its Fourier transform must appear like an impulse (the Fourier transform of a constant) along the same dimension in the frequency domain. === A simple example === Let f be a two-variable function which is i1D. This means that there exists a normalized vector n ∈ R 2 {\textstyle \mathbf {n} \in \mathbb {R} ^{2}} and a one-variable function g such that f ( x ) = g ( n T x ) {\textstyle f(\mathbf {x} )=g(\mathbf {n} ^{\operatorname {T} }\mathbf {x} )} for all x ∈ R 2 {\textstyle \mathbf {x} \in \mathbb {R} ^{2}} . If F is the Fourier transform of f (both are two-variable functions) it must be the case that F ( u ) = G ( n T u ) ⋅ δ ( m T u ) {\textstyle F\left(\mathbf {u} \right)=G\left(\mathbf {n} ^{\mathrm {T} }\mathbf {u} \right)\cdot \delta \left(\mathbf {m} ^{\mathrm {T} }\mathbf {u} \right)} . Here G is the Fourier transform of g (both are one-variable functions), δ is the Dirac impulse function and m is a normalized vector in R 2 {\textstyle \mathbb {R} ^{2}} perpendicular to n. This means that F vanishes everywhere except on a line which passes through the origin of the frequency domain and is parallel to m. Along this line F varies according to G. === The general case === Let f be an N-variable function which has intrinsic dimension M, that is, there exists an M-variable function g and M × N matrix A such that f ( x ) = g ( A x ) ∀ x {\textstyle f(\mathbf {x} )=g(\mathbf {Ax} )\quad \forall \mathbf {x} } . Its Fourier transform F can then be described as follows: F vanishes everywhere except for a subspace of dimension M The subspace M is spanned by the rows of the matrix A In the subspace, F varies according to G the Fourier transform of g == Generalizations == The type of intrinsic dimension described above assume
Resilience (mathematics)
In mathematical modeling, resilience refers to the ability of a dynamical system to recover from perturbations and return to its original stable steady state. It is a measure of the stability and robustness of a system in the face of changes or disturbances. If a system is not resilient enough, it is more susceptible to perturbations and can more easily undergo a critical transition. A common analogy used to explain the concept of resilience of an equilibrium is one of a ball in a valley. A resilient steady state corresponds to a ball in a deep valley, so any push or perturbation will very quickly lead the ball to return to the resting point where it started. On the other hand, a less resilient steady state corresponds to a ball in a shallow valley, so the ball will take a much longer time to return to the equilibrium after a perturbation. The concept of resilience is particularly useful in systems that exhibit tipping points, whose study has a long history that can be traced back to catastrophe theory. While this theory was initially overhyped and fell out of favor, its mathematical foundation remains strong and is now recognized as relevant to many different systems. == History == In 1973, Canadian ecologist C. S. Holling proposed a definition of resilience in the context of ecological systems. According to Holling, resilience is "a measure of the persistence of systems and of their ability to absorb change and disturbance and still maintain the same relationships between populations or state variables". Holling distinguished two types of resilience: engineering resilience and ecological resilience. Engineering resilience refers to the ability of a system to return to its original state after a disturbance, such as a bridge that can be repaired after an earthquake. Ecological resilience, on the other hand, refers to the ability of a system to maintain its identity and function despite a disturbance, such as a forest that can regenerate after a wildfire while maintaining its biodiversity and ecosystem services. With time, the once well-defined and unambiguous concept of resilience has experienced a gradual erosion of its clarity, becoming more vague and closer to an umbrella term than a specific concrete measure. == Definition == Mathematically, resilience can be approximated by the inverse of the return time to an equilibrium given by resilience ≡ − Re ( λ 1 ( A ) ) {\displaystyle {\text{resilience}}\equiv -{\text{Re}}(\lambda _{1}({\textbf {A}}))} where λ 1 {\textstyle \lambda _{1}} is the maximum eigenvalue of matrix A {\textstyle {\textbf {A}}} . The largest this value is, the faster a system returns to the original stable steady state, or in other words, the faster the perturbations decay. == Applications and examples == In ecology, resilience might refer to the ability of the ecosystem to recover from disturbances such as fires, droughts, or the introduction of invasive species. A resilient ecosystem would be one that is able to adapt to these changes and continue functioning, while a less resilient ecosystem might experience irreversible damage or collapse. The exact definition of resilience has remained vague for practical matters, which has led to a slow and proper application of its insights for management of ecosystems. In epidemiology, resilience may refer to the ability of a healthy community to recover from the introduction of infected individuals. That is, a resilient system is more likely to remain at the disease-free equilibrium after the invasion of a new infection. Some stable systems exhibit critical slowing down where, as they approach a basic reproduction number of 1, their resilience decreases, hence taking a longer time to return to the disease-free steady state. Resilience is an important concept in the study of complex systems, where there are many interacting components that can affect each other in unpredictable ways. Mathematical models can be used to explore the resilience of such systems and to identify strategies for improving their resilience in the face of environmental or other changes. For example, when modelling networks it is often important to be able to quantify network resilience, or network robustness, to the loss of nodes. Scale-free networks are particularly resilient since most of their nodes have few links. This means that if some nodes are randomly removed, it is more likely that the nodes with fewer connections are taken out, thus preserving the key properties of the network.