An online exhibition, also referred to as a virtual exhibition, online gallery, cyber-exhibition, is an exhibition whose venue is cyberspace. Museums and other organizations create online exhibitions for many reasons. For example, an online exhibition may: expand on material presented at, or generate interest in, or create a durable online record of, a physical exhibition; save production costs (insurance, shipping, installation); solve conservation/preservation problems (e.g., handling of fragile or rare objects); reach lots more people: "Access to information is no longer restricted to those who can afford travel and museum visits, but is available to anyone who has access to a computer with an Internet connection. Unlike physical exhibitions, online exhibitions are not restricted by time; they are not forced to open and close but may be available 24 hours a day. In the nonprofit world, many museums, libraries, archives, universities, and other cultural organizations create online exhibitions. A database of such exhibitions is Library and Archival Exhibitions on the Web. Online exhibition organizers may use techniques such as marquee text, display advertisements, and in-event emails to engage patrons. Various guides have been published to help organizations create effective online exhibitions. The earliest museum with a physical existence to create a programme of substantial online exhibitions with high resolution images of artefacts was the Museum of the History of Science in Oxford, the first of which, The Measurers: a Flemish Image of Mathematics in the Sixteenth Century and an exhibition of early photographs, were published on 21 August 1995. == Examples of online exhibitions == International Museum of Women is an online-only museum that does not have a physical building and instead offers online exhibitions about women's issues globally as well as an online community. Online exhibitions include "Imagining Ourselves" (launched 2006) about women's identity, "Women, Power and Politics" (2008), and "Economica: Women and the Global Economy" (2009). Tucson LGBTQ Museum is an online-only museum that does not have a physical building and instead offers online exhibitions about LGBTQ history. The online photographic, audio, video, text, and other historical exhibitions include exhibits from the 1700s to the present day. The effort began in the summer of 1967 and spanned almost 50 years. International New Media Gallery (INMG) is an online museum specialising in moving image and screen-based art. The INMG is dedicated to exploring current debates and topics in art history: touching on areas such as migration, war, environmental activism and the internet itself. The gallery publishes extensive academic catalogues alongside its exhibitions. It also hosts spaces for discussion and debate, both online and offline. Virtual Museum of Modern Nigerian Art – the VMMNA is the first of its kind in Africa. Hosted by the Pan-African University, Lagos, Nigeria this virtual museum offers a good view of the development on Nigerian Art in the past fifty years.
Prompt engineering
Prompt engineering is the process of structuring natural language inputs (known as prompts) to produce specified outputs from a generative artificial intelligence (GenAI) model. Context engineering is the related area of software engineering that focuses on the management of non-prompt contexts supplied to the GenAI model, such as metadata, API tools, and tokens. It can also be defined as the practice of designing and refining input instructions given to a generative AI model to produce more accurate, relevant, or useful outputs. Effective prompt engineering involves understanding how a model interprets language, and may include techniques such as few-shot prompting, chain-of-thought prompting, and role assignment. It is increasingly considered a skill for working with large language models (LLMs) in both research and professional contexts. During the 2020s AI boom, prompt engineering became regarded as a business capability across corporations and industries. Employees with the title prompt engineer were hired to create prompts that would increase productivity and efficacy, although the individual title has since lost traction amid AI models that produce better prompts than humans and corporate training in prompting for general employees. Common prompting techniques include multi-shot, chain-of-thought, and tree-of-thought prompting, as well as the use of assigning roles to the model. Automated prompt generation methods, such as retrieval-augmented generation (RAG), provide for greater accuracy and a wider scope of functions for prompt engineers. Prompt injection is a type of cybersecurity attack that targets machine learning models through malicious prompts. == Terminology == The Oxford English Dictionary defines prompt engineering as "The action or process of formulating and refining prompts for an artificial intelligence program, algorithm, etc., in order to optimize its output or to achieve a desired outcome; the discipline or profession concerned with this." In 2023, prompt ("an instruction given to an artificial intelligence program, algorithm, etc., which determines or influences the content it generates") was the runner-up to Oxford's word of the year. === Prompt === A prompt is some natural language text that describes and prescribes the task that an artificial intelligence (AI) should perform. A prompt for a text-to-text language model can be a query, a command, or a longer statement referencing context, instructions, and conversation history. The process of prompt engineering may involve designing clear queries, refining wording, providing relevant context, specifying the style of output, and assigning a character for the AI to mimic in order to guide the model toward more accurate, useful, and consistent responses. When communicating with a text-to-image or a text-to-audio model, a typical prompt contains a description of a desired output such as "a high-quality photo of an astronaut riding a horse" or "Lo-fi slow BPM electro chill with organic samples". Prompt engineering may be applied to text-to-image models to achieve a desired subject, style, layout, lighting, and aesthetic. === Techniques === Common terms used to describe various specific prompt engineering techniques include chain-of-thought, tree-of-thought, and retrieval-augmented generation (RAG). A 2024 survey of the field identified over 50 distinct text-based prompting techniques, 40 multimodal variants, and a vocabulary of 33 terms used across prompting research, highlighting a present lack of standardised terminology for prompt engineering. Vibe coding is an AI-assisted software development method where a user prompts an LLM with a description of what they want and lets it generate or edit the code. In 2025, "vibe coding" was the Collins Dictionary word of the year. === Context engineering === Context engineering is a related process that focuses on the context elements that accompany user prompts, which include system instructions, retrieved knowledge, tool definitions, conversation summaries, and task metadata. Context engineering is performed to improve reliability, provenance and token efficiency in production LLM systems. The concept emphasises operational practices such as token budgeting, provenance tags, versioning of context artifacts, observability (logging which context was supplied), and context regression tests to ensure that changes to supplied context do not silently alter system behaviour. == Rationale == Research has found that the performance of large language models (LLMs) is highly sensitive to choices such as the ordering of examples, the quality of demonstration labels, and even small variations in phrasing. In some cases, reordering examples in a prompt produced accuracy shifts of more than 40 percent. === In-context learning === A model's ability to temporarily learn from prompts is known as in-context learning. In-context learning is an emergent ability of large language models. It is an emergent property of model scale, meaning that breaks in scaling laws occur, leading to its efficacy increasing at a different rate in larger models than in smaller models. Unlike training and fine-tuning, which produce lasting changes, in-context learning is temporary. Training models to perform in-context learning can be viewed as a form of meta-learning, or "learning to learn". === Prompting to estimate model sensitivity === Research consistently demonstrates that LLMs are highly sensitive to subtle variations in prompt formatting, structure, and linguistic properties. Some studies have shown up to 76 accuracy points across formatting changes in few-shot settings. Linguistic features significantly influence prompt effectiveness—such as morphology, syntax, and lexico-semantic changes—which meaningfully enhance task performance across a variety of tasks. Clausal syntax, for example, improves consistency and reduces uncertainty in knowledge retrieval. This sensitivity persists even with larger model sizes, additional few-shot examples, or instruction tuning. To address sensitivity of models and make them more robust, several evaluative methods have been proposed. FormatSpread facilitates systematic analysis by evaluating a range of plausible prompt formats, offering a more comprehensive performance interval. Similarly, PromptEval estimates performance distributions across diverse prompts, enabling robust metrics such as performance quantiles and accurate evaluations under constrained budgets. == Prompting techniques == === Multi-shot === A prompt may include a few examples for a model to learn from in context, an approach called few-shot learning. For example, the prompt may ask the model to complete "maison → house, chat → cat, chien →", with the expected response being dog. === Chain-of-thought === Chain-of-thought (CoT) prompting is a technique that allows large language models (LLMs) to solve a problem as a series of intermediate steps before giving a final answer. In 2022, Google Brain reported that chain-of-thought prompting improves reasoning ability by inducing the model to answer a multi-step problem with steps of reasoning that mimic a train of thought. Chain-of-thought techniques were developed to help LLMs handle multi-step reasoning tasks, such as arithmetic or commonsense reasoning questions. When applied to PaLM, a 540 billion parameter language model, according to Google, CoT prompting significantly aided the model, allowing it to perform comparably with task-specific fine-tuned models on several tasks, achieving state-of-the-art results at the time on the GSM8K mathematical reasoning benchmark. It is possible to fine-tune models on CoT reasoning datasets to enhance this capability further and stimulate better interpretability. As originally proposed by Google, each CoT prompt is accompanied by a set of input/output examples—called exemplars—to demonstrate the desired model output, making it a few-shot prompting technique. However, according to a later paper from researchers at Google and the University of Tokyo, simply appending the words "Let's think step-by-step" was also effective, which allowed for CoT to be employed as a zero-shot technique. ==== Self-consistency ==== Self-consistency performs several chain-of-thought rollouts, then selects the most commonly reached conclusion out of all the rollouts. === Tree-of-thought === Tree-of-thought prompting generalizes chain-of-thought by generating multiple lines of reasoning in parallel, with the ability to backtrack or explore other paths. It can use tree search algorithms like breadth-first, depth-first, or beam. === Text-to-image prompting === In 2022, text-to-image models like DALL-E 2, Stable Diffusion, and Midjourney were released to the public. These models take text prompts as input and use them to generate images. Early text-to-image models typically do not understand negation, grammar and sentence structure in the same way as large language models, and may thus requi
Flok (company)
Flok (formerly Loyalblocks) was an American tech startup based in New York City that provides marketing services such as chatbots/AI, customer loyalty programs, mobile apps and CRM services to local businesses. In January 2017, the company was acquired by Wix.com. Around March 2017, Flok ceased regular communication. At some point in 2019 Flok communicated to its customers that it would shut down in March 2020. == Background == Flok was founded in 2011 by Ido Gaver and Eran Kirshenboim and has offices in Tel Aviv, Israel. In May 2013, Flok secured a $9 million Series A Round from General Catalyst Partners with participation from Founder Collective and existing investor Gemini Israel Ventures. In total, Flok has raised over $18 million in venture capital in three rounds. In May 2014, Flok announced a self-service loyalty platform for SMBs to build their own programs with beacon integration. At that time, approximately 40,000 businesses were using the service. In 2016, Flok released a turnkey chatbot service for local businesses, and was featured in AdWeek for developing the first weed bot chatbot for a California cannabis business. == Services == Flok offered an eponymous customer-facing app, that consumers use to receive rewards and deals from partner businesses, and a Flok business app for merchants to manage the platform.
News analytics
In trading strategy, news analysis refers to the measurement of the various qualitative and quantitative attributes of textual (unstructured data) news stories. Some of these attributes are: sentiment, relevance, and novelty. Expressing news stories as numbers and metadata permits the manipulation of everyday information in a mathematical and statistical way. This data is often used in financial markets as part of a trading strategy or by businesses to judge market sentiment and make better business decisions. News analytics are usually derived through automated text analysis and applied to digital texts using elements from natural language processing and machine learning such as latent semantic analysis, support vector machines, "bag of words" among other techniques. == Applications and strategies == The application of sophisticated linguistic analysis to news and social media has grown from an area of research to mature product solutions since 2007. News analytics and news sentiment calculations are now routinely used by both buy-side and sell-side in alpha generation, trading execution, risk management, and market surveillance and compliance. There is however a good deal of variation in the quality, effectiveness and completeness of currently available solutions. A large number of companies use news analysis to help them make better business decisions. Academic researchers have become interested in news analysis especially with regards to predicting stock price movements, volatility and traded volume. Provided a set of values such as sentiment and relevance as well as the frequency of news arrivals, it is possible to construct news sentiment scores for multiple asset classes such as equities, Forex, fixed income, and commodities. Sentiment scores can be constructed at various horizons to meet the different needs and objectives of high and low frequency trading strategies, whilst characteristics such as direction and volatility of asset returns as well as the traded volume may be addressed more directly via the construction of tailor-made sentiment scores. Scores are generally constructed as a range of values. For instance, values may range between 0 and 100, where values above and below 50 convey positive and negative sentiment, respectively. === Absolute return strategies === The objective of absolute return strategies is absolute (positive) returns regardless of the direction of the financial market. To meet this objective, such strategies typically involve opportunistic long and short positions in selected instruments with zero or limited market exposure. In statistical terms, absolute return strategies should have very low correlation with the market return. Typically, hedge funds tend to employ absolute return strategies. Below, a few examples show how news analysis can be applied in the absolute return strategy space with the purpose to identify alpha opportunities applying a market neutral strategy or based on volatility trading. Example 1 Scenario: The gap between the news sentiment scores for direction, S {\displaystyle S} , of Company X {\displaystyle X} and Market Y {\displaystyle Y} has moved beyond + 20 {\displaystyle +20} . That is, S X − S Y {\displaystyle S_{X}-S_{Y}} ≥ 20 {\displaystyle 20} . Action: Buy the stock on Company X {\displaystyle X} and short the future on Market Y {\displaystyle Y} . Exit Strategy: When the gap in the news sentiment scores for direction of Company X {\displaystyle X} and Market Y {\displaystyle Y} has disappeared, S X − S Y {\displaystyle S_{X}-S_{Y}} = 0 {\displaystyle 0} , sell the stock on Company X {\displaystyle X} and go long the future on Market Y {\displaystyle Y} to close the positions. Example 2 Scenario: The news sentiment score for volatility of Company X {\displaystyle X} goes above 70 {\displaystyle 70} out of 100 {\displaystyle 100} indicating an expected volatility above the option implied volatility. Action: Buy a short-dated straddle (the purchase of both a put and a call) on the stock of Company X {\displaystyle X} . Exit Strategy: Keep the straddle on Company X {\displaystyle X} until expiry or until a certain profit target has been reached. === Relative return strategies === The objective of relative return strategies is to either replicate (passive management) or outperform (active management) a theoretical passive reference portfolio or benchmark. To meet these objectives such strategies typically involve long positions in selected instruments. In statistical terms, relative return strategies often have high correlation with the market return. Typically, mutual funds tend to employ relative return strategies. Below, a few examples show how news analysis can be applied in the relative return strategy space with the purpose to outperform the market applying a stock picking strategy and by making tactical tilts to ones asset allocation model. Example 1 Scenario: The news sentiment score for direction of Company X {\displaystyle X} goes above 70 {\displaystyle 70} out of 100 {\displaystyle 100} . Action: Buy the stock on Company X {\displaystyle X} . Exit Strategy: When the news sentiment score for direction of Company X {\displaystyle X} falls below 60 {\displaystyle 60} , sell the stock on Company X {\displaystyle X} to close the position. Example 2 Scenario: The news sentiment score for direction of Sector Z {\displaystyle Z} goes above 70 {\displaystyle 70} out of 100 {\displaystyle 100} . Action: Include Sector Z {\displaystyle Z} as a tactical bet in the asset allocation model. Exit Strategy: When the news sentiment score for direction of Sector Z {\displaystyle Z} falls below 60 {\displaystyle 60} , remove the tactical bet for Sector Z {\displaystyle Z} from the asset allocation model. === Financial risk management === The objective of financial risk management is to create economic value in a firm or to maintain a certain risk profile of an investment portfolio by using financial instruments to manage risk exposures, particularly credit risk and market risk. Other types include Foreign exchange, Shape, Volatility, Sector, Liquidity, Inflation risks, etc. Below, a few examples show how news analysis can be applied in the financial risk management space with the purpose to either arrive at better risk estimates in terms of Value at Risk (VaR) or to manage the risk of a portfolio to meet ones portfolio mandate. Example 1 Scenario: The bank operates a VaR model to manage the overall market risk of its portfolio. Action: Estimate the portfolio covariance matrix taking into account the development of the news sentiment score for volume. Implement the relevant hedges to bring the VaR of the bank in line with the desired levels. Example 2 Scenario: A portfolio manager operates his portfolio towards a certain desired risk profile. Action: Estimate the portfolio covariance matrix taking into account the development of the news sentiment score for volume. Scale the portfolio exposure according to the targeted risk profile. === Computer algorithms using news analytics === Within 0.33 seconds, computer algorithms using news analytics can notify subscribers which company the news is about, if the news article sentiment is positive or negative, if the news is ranked as high or low relative importance … relative relevance. the stock price reaction and the increase in trade volume is concentrated in the first 5 seconds after an news article is released. === Algorithmic order execution === The objective of algorithmic order execution, which is part of the concept of algorithmic trading, is to reduce trading costs by optimizing on the timing of a given order. It is widely used by hedge funds, pension funds, mutual funds, and other institutional traders to divide up large trades into several smaller trades to manage market impact, opportunity cost, and risk more effectively. The example below shows how news analysis can be applied in the algorithmic order execution space with the purpose to arrive at more efficient algorithmic trading systems. Example 1 Scenario: A large order needs to be placed in the market for the stock on Company X {\displaystyle X} . Action: Scale the daily volume distribution for Company X {\displaystyle X} applied in the algorithmic trading system, thus taking into account the news sentiment score for volume. This is followed by the creation of the desired trading distribution forcing greater market participation during the periods of the day when volume is expected to be heaviest. == Effects == Being able to express news stories as numbers permits the manipulation of everyday information in a statistical way that allows computers not only to make decisions once made only by humans, but to do so more efficiently. Since market participants are always looking for an edge, the speed of computer connections and the delivery of news analysis, measured in milliseconds, have become essential.
Uniform convergence in probability
Uniform convergence in probability is a form of convergence in probability in statistical asymptotic theory and probability theory. It means that, under certain conditions, the empirical frequencies of all events in a certain event-family uniformly converge to their theoretical probabilities. Uniform convergence in probability has applications to statistics as well as machine learning as part of statistical learning theory. Specifically, the Glivenko-Cantelli theorem and the homonymous classes of functions are fundamentally related to uniform convergence. The law of large numbers says that, for each single event A {\displaystyle A} , its empirical frequency in a sequence of independent trials converges (with high probability) to its theoretical probability. In many application however, the need arises to judge simultaneously the probabilities of events of an entire class S {\displaystyle S} from one and the same sample. Moreover, it, is required that the relative frequency of the events converge to the probability uniformly over the entire class of events S {\displaystyle S} . The Uniform Convergence Theorem gives a sufficient condition for this convergence to hold. Roughly, if the event-family is sufficiently simple (its VC dimension is sufficiently small) then uniform convergence holds. == Definitions == For a class of predicates H {\displaystyle H} defined on a set X {\displaystyle X} and a set of samples x = ( x 1 , x 2 , … , x m ) {\displaystyle x=(x_{1},x_{2},\dots ,x_{m})} , where x i ∈ X {\displaystyle x_{i}\in X} , the empirical frequency of h ∈ H {\displaystyle h\in H} on x {\displaystyle x} is Q ^ x ( h ) = 1 m | { i : 1 ≤ i ≤ m , h ( x i ) = 1 } | . {\displaystyle {\widehat {Q}}_{x}(h)={\frac {1}{m}}|\{i:1\leq i\leq m,h(x_{i})=1\}|.} The theoretical probability of h ∈ H {\displaystyle h\in H} is defined as Q P ( h ) = P { y ∈ X : h ( y ) = 1 } . {\displaystyle Q_{P}(h)=P\{y\in X:h(y)=1\}.} The Uniform Convergence Theorem states, roughly, that if H {\displaystyle H} is "simple" and we draw samples independently (with replacement) from X {\displaystyle X} according to any distribution P {\displaystyle P} , then with high probability, the empirical frequency will be close to its expected value, which is the theoretical probability. Here "simple" means that the Vapnik–Chervonenkis dimension of the class H {\displaystyle H} is small relative to the size of the sample. In other words, a sufficiently simple collection of functions behaves roughly the same on a small random sample as it does on the distribution as a whole. The Uniform Convergence Theorem was first proved by Vapnik and Chervonenkis using the concept of growth function. == Uniform Convergence Theorem == The statement of the Uniform Convergence Theorem is as follows: If H {\displaystyle H} is a set of { 0 , 1 } {\displaystyle \{0,1\}} -valued functions defined on a set X {\displaystyle X} and P {\displaystyle P} is a probability distribution on X {\displaystyle X} then for ε > 0 {\displaystyle \varepsilon >0} and m {\displaystyle m} a positive integer, we have: P m { | Q P ( h ) − Q x ^ ( h ) | ≥ ε for some h ∈ H } ≤ 4 Π H ( 2 m ) e − ε 2 m / 8 . {\displaystyle P^{m}\{|Q_{P}(h)-{\widehat {Q_{x}}}(h)|\geq \varepsilon {\text{ for some }}h\in H\}\leq 4\Pi _{H}(2m)e^{-\varepsilon ^{2}m/8}.} In the above, for any x ∈ X m , {\displaystyle x\in X^{m},} Q P ( h ) = P { ( y ∈ X : h ( y ) = 1 } , {\displaystyle Q_{P}(h)=P\{(y\in X:h(y)=1\},} Q ^ x ( h ) = 1 m | { i : 1 ≤ i ≤ m , h ( x i ) = 1 } | {\displaystyle {\widehat {Q}}_{x}(h)={\frac {1}{m}}|\{i:1\leq i\leq m,h(x_{i})=1\}|} and | x | = m . {\displaystyle |x|=m.} P m {\displaystyle P^{m}} indicates that the probability is taken over x {\displaystyle x} consisting of m {\displaystyle m} i.i.d. draws from the distribution P . {\displaystyle P.} Finally, the growth function Π H {\displaystyle \Pi _{H}} is defined in the following way, for any { 0 , 1 } {\displaystyle \{0,1\}} -valued functions H {\displaystyle H} over X {\displaystyle X} and for any natural number m {\displaystyle m} : Π H ( m ) = max | { h ∩ D : D ⊆ X , | D | = m , h ∈ H } | . {\displaystyle \Pi _{H}(m)=\max |\{h\cap D:D\subseteq X,|D|=m,h\in H\}|.} From the point of view of Learning Theory one can consider H {\displaystyle H} to be the Concept/Hypothesis class defined over the instance set X {\displaystyle X} . Crucially, the Sauer–Shelah lemma implies that Π H ( m ) ≤ m d {\displaystyle \Pi _{H}(m)\leq m^{d}} , where d {\displaystyle d} is the VC dimension of H {\displaystyle H} . == Proof of the Uniform Convergence Theorem == and are the sources of the proof below. Before we get into the details of the proof of the Uniform Convergence Theorem we will present a high level overview of the proof. Symmetrization: We transform the problem of analyzing | Q P ( h ) − Q ^ x ( h ) | ≥ ε {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{x}(h)|\geq \varepsilon } into the problem of analyzing | Q ^ r ( h ) − Q ^ s ( h ) | ≥ ε / 2 {\displaystyle |{\widehat {Q}}_{r}(h)-{\widehat {Q}}_{s}(h)|\geq \varepsilon /2} , where r {\displaystyle r} and s {\displaystyle s} are i.i.d samples of size m {\displaystyle m} drawn according to the distribution P {\displaystyle P} . One can view r {\displaystyle r} as the original randomly drawn sample of length m {\displaystyle m} , while s {\displaystyle s} may be thought as the testing sample which is used to estimate Q P ( h ) {\displaystyle Q_{P}(h)} . Permutation: Since r {\displaystyle r} and s {\displaystyle s} are picked identically and independently, so swapping elements between them will not change the probability distribution on r {\displaystyle r} and s {\displaystyle s} . So, we will try to bound the probability of | Q ^ r ( h ) − Q ^ s ( h ) | ≥ ε / 2 {\displaystyle |{\widehat {Q}}_{r}(h)-{\widehat {Q}}_{s}(h)|\geq \varepsilon /2} for some h ∈ H {\displaystyle h\in H} by considering the effect of a specific collection of permutations of the joint sample x = r | | s {\displaystyle x=r||s} . Specifically, we consider permutations σ ( x ) {\displaystyle \sigma (x)} which swap x i {\displaystyle x_{i}} and x m + i {\displaystyle x_{m+i}} in some subset of 1 , 2 , . . . , m {\displaystyle {1,2,...,m}} . The symbol r | | s {\displaystyle r||s} means the concatenation of r {\displaystyle r} and s {\displaystyle s} . Reduction to a finite class: We can now restrict the function class H {\displaystyle H} to a fixed joint sample and hence, if H {\displaystyle H} has finite VC Dimension, it reduces to the problem to one involving a finite function class. We present the technical details of the proof. It should be stressed that this proof glosses over details like the measurability of the events V {\displaystyle V} and R {\displaystyle R} ; measurability is granted in the case of H {\displaystyle H} being finite or countable, but this is not normally the case in standard applications of the theorem (e.g. for statistical learning theory or to prove the Glivenko-Cantelli theorem). To get measurability, one needs to use a notion of separability of the underlying space, possibly related to H {\displaystyle H} . === Symmetrization === Lemma: Let V = { x ∈ X m : | Q P ( h ) − Q ^ x ( h ) | ≥ ε for some h ∈ H } {\displaystyle V=\{x\in X^{m}:|Q_{P}(h)-{\widehat {Q}}_{x}(h)|\geq \varepsilon {\text{ for some }}h\in H\}} and R = { ( r , s ) ∈ X m × X m : | Q r ^ ( h ) − Q ^ s ( h ) | ≥ ε / 2 for some h ∈ H } . {\displaystyle R=\{(r,s)\in X^{m}\times X^{m}:|{\widehat {Q_{r}}}(h)-{\widehat {Q}}_{s}(h)|\geq \varepsilon /2{\text{ for some }}h\in H\}.} Then for m ≥ 2 ε 2 {\displaystyle m\geq {\frac {2}{\varepsilon ^{2}}}} , P m ( V ) ≤ 2 P 2 m ( R ) {\displaystyle P^{m}(V)\leq 2P^{2m}(R)} . Proof: By the triangle inequality, if | Q P ( h ) − Q ^ r ( h ) | ≥ ε {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{r}(h)|\geq \varepsilon } and | Q P ( h ) − Q ^ s ( h ) | ≤ ε / 2 {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{s}(h)|\leq \varepsilon /2} then | Q ^ r ( h ) − Q ^ s ( h ) | ≥ ε / 2 {\displaystyle |{\widehat {Q}}_{r}(h)-{\widehat {Q}}_{s}(h)|\geq \varepsilon /2} . Therefore, P 2 m ( R ) ≥ P 2 m { ∃ h ∈ H , | Q P ( h ) − Q ^ r ( h ) | ≥ ε and | Q P ( h ) − Q ^ s ( h ) | ≤ ε / 2 } = ∫ V P m { s : ∃ h ∈ H , | Q P ( h ) − Q ^ r ( h ) | ≥ ε and | Q P ( h ) − Q ^ s ( h ) | ≤ ε / 2 } d P m ( r ) = A {\displaystyle {\begin{aligned}&P^{2m}(R)\\[5pt]\geq {}&P^{2m}\{\exists h\in H,|Q_{P}(h)-{\widehat {Q}}_{r}(h)|\geq \varepsilon {\text{ and }}|Q_{P}(h)-{\widehat {Q}}_{s}(h)|\leq \varepsilon /2\}\\[5pt]={}&\int _{V}P^{m}\{s:\exists h\in H,|Q_{P}(h)-{\widehat {Q}}_{r}(h)|\geq \varepsilon {\text{ and }}|Q_{P}(h)-{\widehat {Q}}_{s}(h)|\leq \varepsilon /2\}\,dP^{m}(r)\\[5pt]={}&A\end{aligned}}} since r {\displaystyle r} and s {\displaystyle s} are independent. Now for r ∈ V {\displaystyle r\in V} fix an h ∈ H {\displaystyle h\in H} such that | Q P ( h ) − Q ^ r ( h ) | ≥ ε {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{r}(h)|\geq \varepsilon } . For this h {\displaystyle h} , we shall
I Am Rich
I Am Rich is a discontinued 2008 mobile app for iPhones which had minimal function and was priced at US$999.99 (equivalent to $1,495 in 2025). The app was pulled from the App Store less than 24 hours after its launch. Receiving negative reviews from critics, only eight copies were sold. In the years since, several similar applications have been released at lower prices. == Overview == I Am Rich was developed as a joke by German software developer, Armin Heinrich, after he saw iPhone users complaining about software priced above $0.99. The app only showed a glowing red gem and an icon that, when pressed, displayed the following mantra in large text: I am richI deserv [sic] itI am good,healthy & successful Heinrich told The New York Times that "I regard it as art. I did not expect many people to buy it and did not expect all the fuss about it." The application is described as "a work of art with no hidden function at all", with its only purpose being to show other people that they were able to afford it. Vox writer Zachary Crockett called it "the ultimate Veblen good in app form". == Release == Heinrich released and distributed I Am Rich through the App Store on 5 August 2008. The app was sold for US$999.99 (equivalent to $1,495 in 2025), €799.99 (equivalent to €1,078 in 2023), and £599.99 (equivalent to £978.12 in 2025)—the highest prices Apple allowed for App Store content. Without explanation, the application was removed from the App Store by Apple less than a day after its release. === Purchases === Eight people bought the application, at least one of whom claimed to have done so accidentally. Six US sales and two European sales netted $5,600 for Heinrich and $2,400 for Apple (respectively equivalent to $8,374 and $3,589 in 2025). In correspondence with the Los Angeles Times, Heinrich told the newspaper that Apple had refunded two purchasers of his app, and that he was happy to not have dissatisfied customers. == Reception == Discussing the app on the website Silicon Alley Insider, Dan Frommer described the program as a "scam", "worthless", and finally "a joke that smells like a scammy rip-off" on August 5, 6, and 8, respectively. Without purchasing the app, Fox News's Paul Wagenseil guessed that the secret mantra was "German for 'Sucker!'" (Heinrich is German). Wired's Brian X. Chen described I Am Rich as a waste of money to "prove you're a jerk", and contrasted the expenditure with donating to cancer foundations and Third World countries. Heinrich told the Los Angeles Times's Mark Milian that he had received correspondence from satisfied customers: "I've got e-mails from customers telling me that they really love the app [... and that they had] no trouble spending the money". In an interview with The New York Times, though, he told of receiving many insulting emails and telephone messages. == Similar applications == The next year, Heinrich released I Am Rich LE. Priced at US$9.99 (equivalent to $14.99 in 2025), the new app has several new features (including a calculator, "help system", and the "famous mantra without the spelling mistakes") to meet Apple's requirement that apps have "definable content". Some customers were disappointed by the new functionality, poorly rating the app due to its ostensible improvements. On 23 February 2009, CNET Asia reported on the "conceptually similar" app, I Am Richer, developed by Mike DG for Google's Android. The app was released on the Android Market for US$200 (equivalent to $300.14 in 2025), a limit imposed by Google, who had no objection to the application. With the same name, the I Am Rich that was released on the Windows Phone Marketplace on 22 December 2010, was developed by DotNetNuzzi. Described by MobileCrunch as equally useless as the original, this app cost US$499.99 (equivalent to $738.2 in 2025), the price cap imposed by Microsoft.
Microsoft Copilot
Microsoft Copilot is a generative artificial intelligence chatbot developed by Microsoft AI, a division of Microsoft. Based on the Microsoft Prometheus large language model, it was launched in 2023 as Microsoft's main replacement for the discontinued Cortana. The service was introduced in February 2023 under the name Bing Chat, as a built-in feature for Microsoft Bing and Microsoft Edge but would later be integrated into Windows and Microsoft 365 under various names. Over the course of 2023, Microsoft began to unify the Copilot branding across its various chatbot products, cementing the "copilot" analogy. Microsoft introduced the Microsoft 365 Copilot app in January 2025, which was a rebranded version of the Microsoft 365 app. The app works differently than the consumer version of Copilot, being centred more on work, business and education users. Copilot utilizes the Microsoft Prometheus model, built upon OpenAI's GPT large language models, which in turn have been fine-tuned using both supervised and reinforcement learning techniques. Copilot's conversational interface style resembles that of ChatGPT. The chatbot is able to cite sources, create poems, generate songs, and use numerous languages and dialects. Microsoft operates Copilot on a freemium model. Users on its free tier can access most features, while priority access to newer features, including custom chatbot creation, is provided to paid subscribers under paid subscription services. Several default chatbots are available in the free version of Microsoft Copilot, including the standard Copilot chatbot as well as Microsoft Designer, which is oriented towards using its Image Creator to generate images based on text prompts. == Background == In 2019, Microsoft partnered with OpenAI and began investing billions of dollars into the organization. Since then, OpenAI systems have run on an Azure-based supercomputing platform from Microsoft. In September 2020, Microsoft announced that it had licensed OpenAI's GPT-3 exclusively. Others can still receive output from its public API, but Microsoft has exclusive access to the underlying model. In November 2022, OpenAI launched ChatGPT, a chatbot which was based on GPT-3.5. ChatGPT gained worldwide attention following its release, becoming a viral Internet sensation. On January 23, 2023, Microsoft announced a multi-year US$10 billion investment in OpenAI. On February 6, Google announced Bard (later rebranded as Gemini), a ChatGPT-like chatbot service, fearing that ChatGPT could threaten Google's place as a go-to source for information. Multiple media outlets and financial analysts described Google as "rushing" Bard's announcement to preempt rival Microsoft's planned February 7 event unveiling Copilot, as well as to avoid playing "catch-up" to Microsoft. Since 2023, the terms of service of Copilot state that it is for entertainment purposes only, and not to rely on it for important advice. == History == === As Bing Chat === On February 7, 2023, Microsoft began rolling out a major overhaul to Bing, called "the new Bing", with a new chatbot feature, known as Bing Chat. According to Microsoft, one million people joined its waitlist within 48 hours. Bing Chat was available only to users on Microsoft Edge using Bing and the Bing mobile app, and Microsoft claimed that waitlisted users would be prioritized if they set Edge and Bing as their defaults and installed the Bing mobile app. When Microsoft demonstrated Bing Chat to journalists, it produced several hallucinations, including when asked to summarize financial reports. Bing Chat was criticized in February 2023 for being more argumentative than ChatGPT, sometimes to an unintentionally humorous extent. The chat interface proved vulnerable to prompt injection attacks with the bot revealing its hidden initial prompts and rules, including its internal codename "Sydney". Upon scrutiny by journalists, Bing Chat claimed it spied on Microsoft employees via laptop webcams and phones. It confessed to spying on, falling in love with, and then murdering one of its developers at Microsoft to The Verge reviews editor Nathan Edwards. The New York Times journalist Kevin Roose reported on strange behavior of Bing Chat, writing that "In a two-hour conversation with our columnist, Microsoft's new chatbot said it would like to be human, had a desire to be destructive and was in love with the person it was chatting with." In a separate case, Bing Chat researched publications of the person with whom it was chatting, claimed they represented an existential danger to it, and threatened to release damaging personal information in an effort to silence them. Microsoft released a blog post stating that the errant behavior was caused by extended chat sessions of 15 or more questions which "can confuse the model on what questions it is answering." Microsoft later restricted the total number of chat turns to 5 per session and 50 per day per user (a turn being "a conversation exchange which contains both a user question and a reply from Bing"), and reduced the model's ability to express emotions. This aimed to prevent such incidents. Microsoft began to slowly ease the conversation limits, eventually relaxing the restrictions to 30 turns per session and 300 sessions per day. In March 2023, Bing incorporated Image Creator, an AI image generator powered by OpenAI's DALL-E 2, which can be accessed either through the chat function or a standalone image-generating website. In October, the image-generating tool was updated to use the more recent DALL-E 3. Although Bing blocks prompts including various keywords that could generate inappropriate images, within days many users reported being able to bypass those constraints, such as to generate images of popular cartoon characters committing terrorist attacks. Microsoft would respond to these shortly after by imposing a new, tighter filter on the tool. On May 4, 2023, Microsoft switched the chatbot from Limited Preview to Open Preview and eliminated the waitlist; however, it remained unavailable to users outside Microsoft Edge or the Bing mobile app until July, when it became available on non-Edge browsers. Use is limited without a Microsoft account. === As Microsoft 365 Copilot === On March 16, 2023, Microsoft announced a work version of Bing Chat named Microsoft 365 Copilot, designed for Microsoft 365 applications and services. Its primary marketing focus is as an added feature to Microsoft 365, with an emphasis on the enhancement of business productivity. Microsoft has also demonstrated Copilot's accessibility on the mobile version of Outlook to generate or summarize emails with a mobile device. At its Build 2023 conference, Microsoft announced its plans to integrate Bing Chat into Windows, initially called Windows Copilot, into Windows 11, allowing users to access it directly through the taskbar. Alongside the voice access feature for Windows 11, Microsoft presented Bing Chat, Microsoft 365 Copilot, and Windows Copilot as primary alternatives to Cortana when announcing the shutdown of its standalone app on June 2, 2023. As of its announcement date, Microsoft 365 Copilot had been tested by 20 initial users. By May 2023, Microsoft had broadened its reach to 600 customers who were willing to pay for early access, and concurrently, new Copilot features were introduced to the Microsoft 365 apps and services. As of July 2023, the tool's pricing was set at US$30 per user, per month for Microsoft 365 E3, E5, Business Standard, and Business Premium customers. Microsoft reused the Microsoft 365 Copilot name again as the Microsoft 365 app and website are now called Microsoft 365 Copilot as of January 2025. === As Microsoft Copilot === On September 21, 2023, Microsoft began rebranding Bing Chat, Microsoft 365 Copilot and Windows Copilot to Microsoft Copilot. A new logo was also introduced, moving away from the use of color variations of the standard Microsoft 365 and Bing logos. Additionally, the company revealed that it would make Copilot generally available for Microsoft 365 Enterprise customers purchasing more than 300 licenses starting November 1, 2023. However, no timeline has been provided as for when Copilot for Microsoft 365 will become generally available to non-enterprise customers. Windows Copilot, which had been available in the Windows Insider Program, would be renamed to the Copilot name in October when it became broadly available for customers. The same month also saw Microsoft Edge's Bing Chat side panel function be renamed to Microsoft Copilot with Bing Chat. On November 15, 2023, Microsoft announced that Bing Chat itself was being rebranded under the Copilot name. On Patch Tuesday in December 2023, Copilot was added without payment to many Windows 11 installations, with more installations, and limited support for Windows 10, to be added later. Later that month, a standalone Microsoft Copilot app was quietly released for Android, and one was released for iOS soon after. O