Vinberg's algorithm

In mathematics, Vinberg's algorithm is an algorithm, introduced by Ernest Borisovich Vinberg, for finding a fundamental domain of a hyperbolic reflection group. Conway (1983) used Vinberg's algorithm to describe the automorphism group of the 26-dimensional even unimodular Lorentzian lattice II25,1 in terms of the Leech lattice. == Description of the algorithm == Let Γ < I s o m ( H n ) {\displaystyle \Gamma <\mathrm {Isom} (\mathbb {H} ^{n})} be a hyperbolic reflection group. Choose any point v 0 ∈ H n {\displaystyle v_{0}\in \mathbb {H} ^{n}} ; we shall call it the basic (or initial) point. The fundamental domain P 0 {\displaystyle P_{0}} of its stabilizer Γ v 0 {\displaystyle \Gamma _{v_{0}}} is a polyhedral cone in H n {\displaystyle \mathbb {H} ^{n}} . Let H 1 , . . . , H m {\displaystyle H_{1},...,H_{m}} be the faces of this cone, and let a 1 , . . . , a m {\displaystyle a_{1},...,a_{m}} be outer normal vectors to it. Consider the half-spaces H k − = { x ∈ R n , 1 | ( x , a k ) ≤ 0 } . {\displaystyle H_{k}^{-}=\{x\in \mathbb {R} ^{n,1}|(x,a_{k})\leq 0\}.} There exists a unique fundamental polyhedron P {\displaystyle P} of Γ {\displaystyle \Gamma } contained in P 0 {\displaystyle P_{0}} and containing the point v 0 {\displaystyle v_{0}} . Its faces containing v 0 {\displaystyle v_{0}} are formed by faces H 1 , . . . , H m {\displaystyle H_{1},...,H_{m}} of the cone P 0 {\displaystyle P_{0}} . The other faces H m + 1 , . . . {\displaystyle H_{m+1},...} and the corresponding outward normals a m + 1 , . . . {\displaystyle a_{m+1},...} are constructed by induction. Namely, for H j {\displaystyle H_{j}} we take a mirror such that the root a j {\displaystyle a_{j}} orthogonal to it satisfies the conditions (1) ( v 0 , a j ) < 0 {\displaystyle (v_{0},a_{j})<0} ; (2) ( a i , a j ) ≤ 0 {\displaystyle (a_{i},a_{j})\leq 0} for all i < j {\displaystyle i

Web application firewall

A Web application firewall (WAF) is a specific form of application firewall that filters, monitors, and blocks HTTP traffic to and from a web service. By inspecting HTTP traffic, it can prevent attacks exploiting a Web application's known vulnerabilities, such as SQL injection, cross-site scripting (XSS), file inclusion, and improper system configuration. Financial institutions often utilize WAFs to help in the mitigation of Web application zero-day vulnerabilities, as well as hard-to-patch bugs or weaknesses through custom attack signature strings. == History == Dedicated Web application firewalls entered the market in the late 1990s during a time when web server attacks were becoming more prevalent. Early WAF products, from Kavado and Gilian technologies, tried to solve the increasing amount of attacks on Web applications in the late 1990s. In 2002, the open-source project ModSecurity was formed in order to make WAF technology more accessible. They finalized a core rule set for protecting Web applications, based on OASIS Web Application Security Technical Committee’s (WAS TC) vulnerability work. In 2003, they expanded and standardized rules through the Open Web Application Security Project’s (OWASP) Top 10 List, an annual ranking for Web security vulnerabilities. This list would become the industry standard for Web application security compliance. Since then, the market has continued to grow and evolve, especially focusing on credit card fraud prevention. With the development of the Payment Card Industry Data Security Standard (PCI DSS), a standardization of control over cardholder data, security has become more regulated in this sector. == Description == A Web application firewall is a special type of application firewall that applies specifically to Web applications. It is deployed in front of Web applications and analyzes bi-directional web-based (HTTP) traffic – detecting and blocking anything malicious. The OWASP provides a broad technical definition for a WAF as “a security solution on the Web application level which – from a technical point of view – does not depend on the application itself”. According to the PCI DSS Information Supplement for requirement 6.6, a WAF is defined as “a security policy enforcement point positioned between a Web application and the client endpoint. This functionality can be implemented in software or hardware, running in an appliance device, or in a typical server running a common operating system. It may be a stand-alone device or integrated into other network components.” In other words, a WAF can be a virtual or physical appliance that prevents vulnerabilities in Web applications from being exploited by outside threats. These vulnerabilities may be because the application itself is a legacy type or was insufficiently coded by design. The WAF addresses these code shortcomings by special configurations of rule-sets, also known as policies. Previously unknown vulnerabilities can be discovered through penetration testing or via a vulnerability scanner. A Web application vulnerability scanner, also known as a web application security scanner, is defined in the SAMATE NIST 500-269 as “an automated program that examines Web applications for potential security vulnerabilities. In addition to searching for Web application-specific vulnerabilities, the tools also look for software coding errors.” Resolving vulnerabilities is commonly referred to as remediation. Corrections to the code can be made in the application, but typically a more prompt response is necessary. In these situations, the application of a custom policy for a unique Web application vulnerability to provide a temporary but immediate fix (known as a virtual patch) may be necessary. WAFs are not an ultimate security solution, rather they are meant to be used in conjunction with other network perimeter security solutions such as network firewalls and intrusion prevention systems to provide a holistic defense strategy. WAFs typically follow a positive security model, a negative security, or a combination of both as mentioned by the SANS Institute. WAFs use a combination of rule-based logic, parsing, and signatures to detect and prevent attacks such as cross-site scripting and SQL injection. In general, features like browser emulation, obfuscation and virtualization, and IP obfuscation are used to attempt to bypass WAFs. The OWASP produces a list of the top ten Web application security flaws. All commercial WAF offerings cover these ten flaws at a minimum. There are non-commercial options as well. As mentioned earlier, the well-known open-source WAF engine called ModSecurity is one of these options. A WAF engine alone is insufficient to provide adequate protection, therefore OWASP along with Trustwave's Spiderlabs help organize and maintain a Core-Rule Set via GitHub to use with the ModSecurity WAF engine. == Deployment options == Although the names for operating mode may differ, WAFs are basically deployed inline in three different ways. According to NSS Labs, deployment options are transparent bridge, transparent reverse proxy, and reverse proxy. "Transparent" refers to the fact that the HTTP traffic is sent straight to the Web application, therefore the WAF is transparent between the client and server. This is in contrast to reverse proxy, where the WAF acts as a proxy, and the client’s traffic is sent directly to the WAF. The WAF then separately sends filtered traffic to Web applications. This can provide additional benefits such as IP masking but may introduce disadvantages such as performance latencies. == JA3 fingerprint == JA3, developed by Salesforce in 2017, is a technique for generating a unique fingerprint for SSL/TLS traffic based on specific fields in the handshake, such as the version, cipher suites, and extensions used by the client. This fingerprint enables the identification and tracking of clients based on the characteristics of their encrypted traffic. In the context of distributed denial of service (DDoS) protection, JA3 fingerprints are used to detect and differentiate malicious traffic, often associated with attack bots, from legitimate traffic, allowing for more precise filtering of potential threats. In September 2023, AWS WAF announced built-in support for JA3, enabling customers to inspect the JA3 fingerprints of incoming requests. JA3 was deprecated in May 2025 in favor of JA4. JA4 is currently patent pending.

Flux (text-to-image model)

Flux (also known as FLUX.1 and FLUX.2) is a text-to-image model developed by Black Forest Labs (BFL), based in Freiburg im Breisgau, Germany. Black Forest Labs was founded by former employees of Stability AI. As with other text-to-image models, Flux generates images from natural language descriptions, called prompts. == History == Black Forest Labs (BFL) was founded in 2024 by Robin Rombach, Andreas Blattmann, and Patrick Esser, former employees of Stability AI. All three founders had previously researched the artificial intelligence image generation at LMU Munich as research assistants under Björn Ommer. They published their research results on image generation in 2022, which resulted in creation of Stable Diffusion. Investors in BFL included venture capital firm Andreessen Horowitz, Brendan Iribe, Michael Ovitz, Garry Tan, and Vladlen Koltun. The company received an initial investment of US$31 million. In August 2024, Flux was integrated into the Grok chatbot developed by xAI and made available as part of premium feature on X (formerly Twitter). Grok later switched to its own text-to-image model Aurora in December 2024. On 18 November 2024, Mistral AI announced that its Le Chat chatbot had integrated Flux Pro as its image generation model. On 21 November 2024, BFL announced the release of Flux.1 Tools, a suite of editing tools designed to be used on top of existing Flux models. The tools consisting of Flux.1 Fill for inpainting and outpainting, Flux.1 Depth for control based on extracted depth map of input images and prompts, Flux.1 Canny for control based on extracted canny edges of input images and prompts, and Flux.1 Redux for mixing existing input images and prompts. Each tools are available in both Pro and Dev models. In January 2025, BFL announced a partnership with Nvidia for inclusion of Flux models as foundation models for Nvidia's Blackwell microarchitecture. The company also announced the release of Flux Pro Finetuning API, designed for customisation and fine-tuning of Flux-generated images and a partnership with German media company Hubert Burda Media for usage of Flux Pro as part of content creation. On 29 May 2025, BFL announced Flux.1 Kontext, a suite of models that enable in-context image generation and editing, allowing users to prompt with both text and images. Alongside this, BFL Playground, an interface for testing Flux models was released. On 31 July 2025, BFL announced Flux.1 Krea Dev, a model developed in collaboration with Krea AI that trained to achieve better performance, more varied aesthetics, and better realism compared to existing text-to-image models. In September 2025, Adobe Inc. announced that Photoshop (beta) users can use Flux.1 Kontext Pro as a model for its generative fill tool. BFL collaborated with Meta on Vibes, a video-generation app. On 25 November 2025, BFL announced the release of Flux.2 model series, consisting of Pro, Flex, Dev, and Apache 2.0-licensed Klein (meaning Little or Small in German language) models along with Flux.2 variational autoencoder which also released as open-source software under Apache 2.0 licence. This series claimed improvements for image reference, photorealism, typography, and prompt understanding. == Models == Flux is a series of text-to-image models. The models are based on rectified flow transformer blocks scaled to 12 billion parameters. Flux.1 models were released under different licences with Schnell (meaning Fast or Quick in German language) released as open-source software under Apache License, Dev released as source-available software under a non-commercial licence (users can obtain a self-serving commercial licence for Dev from BFL), and Pro released as proprietary software and only available as API that can be licensed by third-party users. Users retained the ownership of resulting output regardless of models used. An improved flagship model, Flux 1.1 Pro was released on 2 October 2024. Two additional modes were added on 6 November, Ultra which can generate image at four times higher resolution and up to 4 megapixel without affecting generation speed and Raw which can generate hyper-realistic image in the style of candid photography. Flux.1 Kontext is a series with in-context image generation and editing capabilities. It is available in Max, Pro, and Dev models. Max is the highest quality model and can be used to iteratively modify an existing image by using prompt while Pro is optimized to balance quality and speed of generation. Dev is an open-weight model released under non-commercial license, same as Flux.1 Dev. Flux.2 models are based on latent flow matching architecture with Mistral AI's Mistral-3 model (24 billion parameters) for its vision-language model. As with Flux.1, Flux.2 models were also released under different licences with Klein released as open-source software under Apache License, Dev released as source-available software under a non-commercial licence (users can obtain a self-serving commercial licence from BFL), and both Flex and Pro released as proprietary software and only available as API. The models can be used either online or locally by using generative AI user interfaces such as ComfyUI, Recraft Studio and Stable Diffusion WebUI Forge (a fork of Automatic1111 WebUI). Related to Flux is a text-to-video model by Black Forest Labs, under development as of February 2026. == Reception == According to a test performed by Ars Technica, the outputs generated by Flux.1 Dev and Flux.1 Pro are comparable with DALL-E 3 in terms of prompt fidelity, with the photorealism closely matched Midjourney 6 and generated human hands with more consistency over previous models such as Stable Diffusion XL. Flux has been criticised for its very realistic generated images. According to media reports, depictions ranged from an image of Donald Trump posing with guns to disturbing scenes, which triggered discussions about ethical implications of Flux models. After the release of the model, social media platform X was flooded with Flux-generated images. Black Forest Labs has not provided exact details of the data used to train the model. Ars Technica suspected that Flux is based on a large, unauthorised collection of images scraped from the internet, a controversial practice with potential legal consequences. According to a test performed by Japanese technology news website Gigazine for Flux.1 Kontext, the model series has a good understanding of the English language and can easily transfer style of the image from photorealistic into anime-style according to prompts given by the user; however, its ability to understand Japanese is quite poor. == Availability == In addition to the official BFL Playground on its website, the Flux models are also widely available through various third-party platforms for creative and professional use. These include repositories on platforms like Hugging Face and Replicate. == Further readings == FLUX.1 Kontext: Flow Matching for In-Context Image Generation and Editing in Latent Space (29 May 2025) FLUX.2: Analyzing and Enhancing the Latent Space of FLUX – Representation Comparison (25 November 2025)

Type-1 OWA operators

Type-1 OWA operators are a set of aggregation operators that generalise the Yager's OWA (ordered weighted averaging) operators in the interest of aggregating fuzzy sets rather than crisp values in soft decision making and data mining. These operators provide a mathematical technique for directly aggregating uncertain information with uncertain weights via OWA mechanism in soft decision making and data mining, where these uncertain objects are modelled by fuzzy sets. The two definitions for type-1 OWA operators are based on Zadeh's Extension Principle and α {\displaystyle \alpha } -cuts of fuzzy sets. The two definitions lead to equivalent results. == Definitions == === Definition 1 === Let F ( X ) {\displaystyle F(X)} be the set of fuzzy sets with domain of discourse X {\displaystyle X} , a type-1 OWA operator is defined as follows: Given n linguistic weights { W i } i = 1 n {\displaystyle \left\{{W^{i}}\right\}_{i=1}^{n}} in the form of fuzzy sets defined on the domain of discourse U = [ 0 , 1 ] {\displaystyle U=[0,1]} , a type-1 OWA operator is a mapping, Φ {\displaystyle \Phi } , Φ : F ( X ) × ⋯ × F ( X ) ⟶ F ( X ) {\displaystyle \Phi \colon F(X)\times \cdots \times F(X)\longrightarrow F(X)} ( A 1 , ⋯ , A n ) ↦ Y {\displaystyle (A^{1},\cdots ,A^{n})\mapsto Y} such that μ Y ( y ) = sup ∑ k = 1 n w ¯ i a σ ( i ) = y ( μ W 1 ( w 1 ) ∧ ⋯ ∧ μ W n ( w n ) ∧ μ A 1 ( a 1 ) ∧ ⋯ ∧ μ A n ( a n ) ) {\displaystyle \mu _{Y}(y)=\displaystyle \sup _{\displaystyle \sum _{k=1}^{n}{\bar {w}}_{i}a_{\sigma (i)}=y}\left({\begin{array}{{1}l}\mu _{W^{1}}(w_{1})\wedge \cdots \wedge \mu _{W^{n}}(w_{n})\wedge \mu _{A^{1}}(a_{1})\wedge \cdots \wedge \mu _{A^{n}}(a_{n})\end{array}}\right)} where w ¯ i = w i ∑ i = 1 n w i {\displaystyle {\bar {w}}_{i}={\frac {w_{i}}{\sum _{i=1}^{n}{w_{i}}}}} , and σ : { 1 , ⋯ , n } ⟶ { 1 , ⋯ , n } {\displaystyle \sigma \colon \{1,\cdots ,n\}\longrightarrow \{1,\cdots ,n\}} is a permutation function such that a σ ( i ) ≥ a σ ( i + 1 ) , ∀ i = 1 , ⋯ , n − 1 {\displaystyle a_{\sigma (i)}\geq a_{\sigma (i+1)},\ \forall i=1,\cdots ,n-1} , i.e., a σ ( i ) {\displaystyle a_{\sigma (i)}} is the i {\displaystyle i} th highest element in the set { a 1 , ⋯ , a n } {\displaystyle \left\{{a_{1},\cdots ,a_{n}}\right\}} . === Definition 2 === Using the alpha-cuts of fuzzy sets: Given the n linguistic weights { W i } i = 1 n {\displaystyle \left\{{W^{i}}\right\}_{i=1}^{n}} in the form of fuzzy sets defined on the domain of discourse U = [ 0 , 1 ] {\displaystyle U=[0,\;\;1]} , then for each α ∈ [ 0 , 1 ] {\displaystyle \alpha \in [0,\;1]} , an α {\displaystyle \alpha } -level type-1 OWA operator with α {\displaystyle \alpha } -level sets { W α i } i = 1 n {\displaystyle \left\{{W_{\alpha }^{i}}\right\}_{i=1}^{n}} to aggregate the α {\displaystyle \alpha } -cuts of fuzzy sets { A i } i = 1 n {\displaystyle \left\{{A^{i}}\right\}_{i=1}^{n}} is: Φ α ( A α 1 , … , A α n ) = { ∑ i = 1 n w i a σ ( i ) ∑ i = 1 n w i | w i ∈ W α i , a i ∈ A α i , i = 1 , … , n } {\displaystyle \Phi _{\alpha }\left({A_{\alpha }^{1},\ldots ,A_{\alpha }^{n}}\right)=\left\{{{\frac {\sum \limits _{i=1}^{n}{w_{i}a_{\sigma (i)}}}{\sum \limits _{i=1}^{n}{w_{i}}}}\left|{w_{i}\in W_{\alpha }^{i},\;a_{i}}\right.\in A_{\alpha }^{i},\;i=1,\ldots ,n}\right\}} where W α i = { w | μ W i ( w ) ≥ α } , A α i = { x | μ A i ( x ) ≥ α } {\displaystyle W_{\alpha }^{i}=\{w|\mu _{W_{i}}(w)\geq \alpha \},A_{\alpha }^{i}=\{x|\mu _{A_{i}}(x)\geq \alpha \}} , and σ : { 1 , ⋯ , n } → { 1 , ⋯ , n } {\displaystyle \sigma :\{\;1,\cdots ,n\;\}\to \{\;1,\cdots ,n\;\}} is a permutation function such that a σ ( i ) ≥ a σ ( i + 1 ) , ∀ i = 1 , ⋯ , n − 1 {\displaystyle a_{\sigma (i)}\geq a_{\sigma (i+1)},\;\forall \;i=1,\cdots ,n-1} , i.e., a σ ( i ) {\displaystyle a_{\sigma (i)}} is the i {\displaystyle i} th largest element in the set { a 1 , ⋯ , a n } {\displaystyle \left\{{a_{1},\cdots ,a_{n}}\right\}} . == Representation theorem of Type-1 OWA operators == Given the n linguistic weights { W i } i = 1 n {\displaystyle \left\{{W^{i}}\right\}_{i=1}^{n}} in the form of fuzzy sets defined on the domain of discourse U = [ 0 , 1 ] {\displaystyle U=[0,\;\;1]} , and the fuzzy sets A 1 , ⋯ , A n {\displaystyle A^{1},\cdots ,A^{n}} , then we have that Y = G {\displaystyle Y=G} where Y {\displaystyle Y} is the aggregation result obtained by Definition 1, and G {\displaystyle G} is the result obtained by in Definition 2. == Programming problems for Type-1 OWA operators == According to the Representation Theorem of Type-1 OWA Operators, a general type-1 OWA operator can be decomposed into a series of α {\displaystyle \alpha } -level type-1 OWA operators. In practice, this series of α {\displaystyle \alpha } -level type-1 OWA operators is used to construct the resulting aggregation fuzzy set. So we only need to compute the left end-points and right end-points of the intervals Φ α ( A α 1 , ⋯ , A α n ) {\displaystyle \Phi _{\alpha }\left({A_{\alpha }^{1},\cdots ,A_{\alpha }^{n}}\right)} . Then, the resulting aggregation fuzzy set is constructed with the membership function as follows: μ G ( x ) = ⋁ α : x ∈ Φ α ( A α 1 , ⋯ , A α n ) α ⁡ α {\displaystyle \mu _{G}(x)=\operatorname {\bigvee } \limits _{\alpha :x\in \Phi _{\alpha }\left({A_{\alpha }^{1},\cdots ,A_{\alpha }^{n}}\right)_{\alpha }}\alpha } For the left end-points, we need to solve the following programming problem: Φ α ( A α 1 , ⋯ , A α n ) − = min W α − i ≤ w i ≤ W α + i A α − i ≤ a i ≤ A α + i ⁡ ∑ i = 1 n w i a σ ( i ) / ∑ i = 1 n w i {\displaystyle \Phi _{\alpha }\left({A_{\alpha }^{1},\cdots ,A_{\alpha }^{n}}\right)_{-}=\operatorname {\min } \limits _{\begin{array}{l}W_{\alpha -}^{i}\leq w_{i}\leq W_{\alpha +}^{i}A_{\alpha -}^{i}\leq a_{i}\leq A_{\alpha +}^{i}\end{array}}\sum \limits _{i=1}^{n}{w_{i}a_{\sigma (i)}/\sum \limits _{i=1}^{n}{w_{i}}}} while for the right end-points, we need to solve the following programming problem: Φ α ( A α 1 , ⋯ , A α n ) + = max W α − i ≤ w i ≤ W α + i A α − i ≤ a i ≤ A α + i ⁡ ∑ i = 1 n w i a σ ( i ) / ∑ i = 1 n w i {\displaystyle \Phi _{\alpha }\left({A_{\alpha }^{1},\cdots ,A_{\alpha }^{n}}\right)_{+}=\operatorname {\max } \limits _{\begin{array}{l}W_{\alpha -}^{i}\leq w_{i}\leq W_{\alpha +}^{i}A_{\alpha -}^{i}\leq a_{i}\leq A_{\alpha +}^{i}\end{array}}\sum \limits _{i=1}^{n}{w_{i}a_{\sigma (i)}/\sum \limits _{i=1}^{n}{w_{i}}}} A fast method has been presented to solve two programming problem so that the type-1 OWA aggregation operation can be performed efficiently, for details, please see the paper. == Alpha-level approach to Type-1 OWA operation == Three-step process: Step 1—To set up the α {\displaystyle \alpha } - level resolution in [0, 1]. Step 2—For each α ∈ [ 0 , 1 ] {\displaystyle \alpha \in [0,1]} , Step 2.1—To calculate ρ α + i 0 ∗ {\displaystyle \rho _{\alpha +}^{i_{0}^{\ast }}} Let i 0 = 1 {\displaystyle i_{0}=1} ; If ρ α + i 0 ≥ A α + σ ( i 0 ) {\displaystyle \rho _{\alpha +}^{i_{0}}\geq A_{\alpha +}^{\sigma (i_{0})}} , stop, ρ α + i 0 {\displaystyle \rho _{\alpha +}^{i_{0}}} is the solution; otherwise go to Step 2.1-3. i 0 ← i 0 + 1 {\displaystyle i_{0}\leftarrow i_{0}+1} , go to Step 2.1-2. Step 2.2 To calculate ρ α − i 0 ∗ {\displaystyle \rho _{\alpha -}^{i_{0}^{\ast }}} Let i 0 = 1 {\displaystyle i_{0}=1} ; If ρ α − i 0 ≥ A α − σ ( i 0 ) {\displaystyle \rho _{\alpha -}^{i_{0}}\geq A_{\alpha -}^{\sigma (i_{0})}} , stop, ρ α − i 0 {\displaystyle \rho _{\alpha -}^{i_{0}}} is the solution; otherwise go to Step 2.2-3. i 0 ← i 0 + 1 {\displaystyle i_{0}\leftarrow i_{0}+1} , go to step Step 2.2-2. Step 3—To construct the aggregation resulting fuzzy set G {\displaystyle G} based on all the available intervals [ ρ α − i 0 ∗ , ρ α + i 0 ∗ ] {\displaystyle \left[{\rho _{\alpha -}^{i_{0}^{\ast }},\;\rho _{\alpha +}^{i_{0}^{\ast }}}\right]} : μ G ( x ) = ⋁ α : x ∈ [ ρ α − i 0 ∗ , ρ α + i 0 ∗ ] ⁡ α {\displaystyle \mu _{G}(x)=\operatorname {\bigvee } \limits _{\alpha :x\in \left[{\rho _{\alpha -}^{i_{0}^{\ast }},\;\rho _{\alpha +}^{i_{0}^{\ast }}}\right]}\alpha } == Some Examples == The type-1 OWA operator with the weights shown in the top figure is used to aggregate the fuzzy sets (solide lines) in the bottom figure, and the dashed line is the aggregation result. == Special cases == Any OWA operators, like maximum, minimum, mean operators; Join operators of (type-1) fuzzy sets, i.e., fuzzy maximum operators; Meet operators of (type-1) fuzzy sets, i.e., fuzzy minimum operators; Join-like operators of (type-1) fuzzy sets; Meet-like operators of (type-1) fuzzy sets. == Generalizations == Type-2 OWA operators have been suggested to aggregate the type-2 fuzzy sets for soft decision making. == Applications == Type-1 OWA operators have been applied to different domains for soft decision making. Improved efficiency of computing approach ; Type reduction of type-2 fuzzy sets ; Group decision making ; Credit risk evaluation ; Information fusion ; Linguistic expressions and symbolic translation ; Sentiment analysis ; Ro

SpreeAI

SpreeAI (stylized as SPREEAI) is an American fashion technology company headquartered in Incline Village, Nevada that develops artificial intelligence software for the apparel and retail industries, including photorealistic virtual try-on, AI-powered sizing recommendations, and digital model generation. Founded in 2022 by John Imah and Bob Davidson, the company achieved unicorn status in 2025 following a Series B round led by Davidson Group that valued the company at approximately US$1.5 billion. TechCrunch identified SpreeAI as one of the more than 100 new tech unicorns minted in 2025. Its board of directors includes supermodel Naomi Campbell and hospitality executive Larry Ruvo. == History == SpreeAI was founded in 2022 by John Imah and Bob Davidson with a focus on artificial intelligence applications in fashion retail. By 2024, the company had raised approximately US$60 million in venture funding. In May 2025, SpreeAI announced a Series B round led by Davidson Group; reporting at the time placed the company's valuation at approximately US$1.5 billion, making it one of a small number of fashion-technology companies to reach unicorn status. In January 2026, TechCrunch listed SpreeAI among the more than 100 new tech unicorns minted in 2025. == Technology == SpreeAI develops a suite of artificial intelligence tools for the apparel industry. Its consumer-facing platform allows shoppers to upload a single photograph or select a digital model and then visualize clothing items on that figure with photorealistic rendering, while a complementary sizing engine generates fit recommendations intended to reduce returns. The platform is designed for integration with online retailers so that shoppers can preview garments before purchase. The company has stated that its models were developed in part through research collaborations with the Massachusetts Institute of Technology and Carnegie Mellon University. == Leadership and board == John Imah, a Nigerian-American technology executive who previously held roles at Samsung, Twitch, Meta Platforms, and Snap Inc., is co-founder and chief executive officer. Co-founder Bob Davidson, through Davidson Group, led the company's Series B financing. The company's board of directors includes supermodel Naomi Campbell, who joined in 2024, and Las Vegas hospitality executive Larry Ruvo. == Partnerships == SpreeAI has formed partnerships across both academia and the fashion industry. Council of Fashion Designers of America (CFDA). In 2025, SpreeAI entered a partnership with the CFDA to support American designers and brands with AI-driven tools; the CFDA described SpreeAI as "a fashion technology leader delivering innovative solutions to help designers and brands thrive." Massachusetts Institute of Technology and Carnegie Mellon University. The company has cited ongoing research and talent collaborations with both institutions. Sergio Hudson and Kai Collective. In 2025, SpreeAI made what WWD described as its Met Gala debut through a custom collaboration with designer Sergio Hudson and Nigerian-British label Kai Collective; the collaboration paired Hudson's couture with SpreeAI's virtual try-on platform. == Recognition == In 2025, TechCrunch named SpreeAI among the new tech unicorns of the year. In 2025, SpreeAI was named an honoree in Inc.'s Best in Business awards, and CEO John Imah was included on Inc.'s list of 40 business leaders who "propelled their organizations to success." In 2025, Imah was named to the Observer's AI Power Index, a list of 100 leaders shaping the future of artificial intelligence. In 2025, Imah was included in AfroTech's Future 50, recognizing Black innovators in technology. SpreeAI and Imah have been the subject of profile coverage in The Washington Post, Rolling Stone UK, WWD, Vogue UA, L'Officiel Arabia, GQ South Africa, and Inc..

Spectral shape analysis

Spectral shape analysis relies on the spectrum (eigenvalues and/or eigenfunctions) of the Laplace–Beltrami operator to compare and analyze geometric shapes. Since the spectrum of the Laplace–Beltrami operator is invariant under isometries, it is well suited for the analysis or retrieval of non-rigid shapes, i.e. bendable objects such as humans, animals, plants, etc. == Laplace == The Laplace–Beltrami operator is involved in many important differential equations, such as the heat equation and the wave equation. It can be defined on a Riemannian manifold as the divergence of the gradient of a real-valued function f: Δ f := div ⁡ grad ⁡ f . {\displaystyle \Delta f:=\operatorname {div} \operatorname {grad} f.} Its spectral components can be computed by solving the Helmholtz equation (or Laplacian eigenvalue problem): Δ φ i + λ i φ i = 0. {\displaystyle \Delta \varphi _{i}+\lambda _{i}\varphi _{i}=0.} The solutions are the eigenfunctions φ i {\displaystyle \varphi _{i}} (modes) and corresponding eigenvalues λ i {\displaystyle \lambda _{i}} , representing a diverging sequence of positive real numbers. The first eigenvalue is zero for closed domains or when using the Neumann boundary condition. For some shapes, the spectrum can be computed analytically (e.g. rectangle, flat torus, cylinder, disk or sphere). For the sphere, for example, the eigenfunctions are the spherical harmonics. The most important properties of the eigenvalues and eigenfunctions are that they are isometry invariants. In other words, if the shape is not stretched (e.g. a sheet of paper bent into the third dimension), the spectral values will not change. Bendable objects, like animals, plants and humans, can move into different body postures with only minimal stretching at the joints. The resulting shapes are called near-isometric and can be compared using spectral shape analysis. == Discretizations == Geometric shapes are often represented as 2D curved surfaces, 2D surface meshes (usually triangle meshes) or 3D solid objects (e.g. using voxels or tetrahedra meshes). The Helmholtz equation can be solved for all these cases. If a boundary exists, e.g. a square, or the volume of any 3D geometric shape, boundary conditions need to be specified. Several discretizations of the Laplace operator exist (see Discrete Laplace operator) for the different types of geometry representations. Many of these operators do not approximate well the underlying continuous operator. == Spectral shape descriptors == === ShapeDNA and its variants === The ShapeDNA is one of the first spectral shape descriptors. It is the normalized beginning sequence of the eigenvalues of the Laplace–Beltrami operator. Its main advantages are the simple representation (a vector of numbers) and comparison, scale invariance, and in spite of its simplicity a very good performance for shape retrieval of non-rigid shapes. Competitors of shapeDNA include singular values of Geodesic Distance Matrix (SD-GDM) and Reduced BiHarmonic Distance Matrix (R-BiHDM). However, the eigenvalues are global descriptors, therefore the shapeDNA and other global spectral descriptors cannot be used for local or partial shape analysis. === Global point signature (GPS) === The global point signature at a point x {\displaystyle x} is a vector of scaled eigenfunctions of the Laplace–Beltrami operator computed at x {\displaystyle x} (i.e. the spectral embedding of the shape). The GPS is a global feature in the sense that it cannot be used for partial shape matching. === Heat kernel signature (HKS) === The heat kernel signature makes use of the eigen-decomposition of the heat kernel: h t ( x , y ) = ∑ i = 0 ∞ exp ⁡ ( − λ i t ) φ i ( x ) φ i ( y ) . {\displaystyle h_{t}(x,y)=\sum _{i=0}^{\infty }\exp(-\lambda _{i}t)\varphi _{i}(x)\varphi _{i}(y).} For each point on the surface the diagonal of the heat kernel h t ( x , x ) {\displaystyle h_{t}(x,x)} is sampled at specific time values t j {\displaystyle t_{j}} and yields a local signature that can also be used for partial matching or symmetry detection. === Wave kernel signature (WKS) === The WKS follows a similar idea to the HKS, replacing the heat equation with the Schrödinger wave equation. === Improved wave kernel signature (IWKS) === The IWKS improves the WKS for non-rigid shape retrieval by introducing a new scaling function to the eigenvalues and aggregating a new curvature term. === Spectral graph wavelet signature (SGWS) === SGWS is a local descriptor that is not only isometric invariant, but also compact, easy to compute and combines the advantages of both band-pass and low-pass filters. An important facet of SGWS is the ability to combine the advantages of WKS and HKS into a single signature, while allowing a multiresolution representation of shapes. == Spectral Matching == The spectral decomposition of the graph Laplacian associated with complex shapes (see Discrete Laplace operator) provides eigenfunctions (modes) which are invariant to isometries. Each vertex on the shape could be uniquely represented with a combinations of the eigenmodal values at each point, sometimes called spectral coordinates: s ( x ) = ( φ 1 ( x ) , φ 2 ( x ) , … , φ N ( x ) ) for vertex x . {\displaystyle s(x)=(\varphi _{1}(x),\varphi _{2}(x),\ldots ,\varphi _{N}(x)){\text{ for vertex }}x.} Spectral matching consists of establishing the point correspondences by pairing vertices on different shapes that have the most similar spectral coordinates. Early work focused on sparse correspondences for stereoscopy. Computational efficiency now enables dense correspondences on full meshes, for instance between cortical surfaces. Spectral matching could also be used for complex non-rigid image registration, which is notably difficult when images have very large deformations. Such image registration methods based on spectral eigenmodal values indeed capture global shape characteristics, and contrast with conventional non-rigid image registration methods which are often based on local shape characteristics (e.g., image gradients).

Model Context Protocol

The Model Context Protocol (MCP) is an open standard and open-source framework introduced by Anthropic in November 2024 to standardize the way artificial intelligence (AI) systems like large language models (LLMs) integrate and share data with external tools, systems, and data sources. MCP provides a standardized interface for reading files, executing functions, and handling contextual prompts. Following its announcement, the protocol was adopted by major AI providers, including OpenAI and Google DeepMind. == Background == MCP was announced by Anthropic in November 2024 as an open standard for connecting AI assistants to data systems such as content repositories, business management tools, and development environments. The protocol was created at Anthropic by engineers David Soria Parra and Justin Spahr-Summers. It aims to address the challenge of information silos and legacy systems. Before MCP, developers often had to build custom connectors for each data source or tool, resulting in what Anthropic described as an "N×M" data integration problem. Earlier stop-gap approaches—such as OpenAI's 2023 "function-calling" API and the ChatGPT plug-in framework—solved similar problems but required vendor-specific connectors. MCP re-uses the message-flow ideas of the Language Server Protocol (LSP) and is transported over JSON-RPC 2.0. In December 2025, Anthropic donated the MCP to the Agentic AI Foundation (AAIF), a directed fund under the Linux Foundation, co-founded by Anthropic, Block and OpenAI, with support from other companies. == Features == The protocol was released with software development kits (SDKs) in programming languages including Python, TypeScript, C# and Java. Anthropic maintains an open-source repository of reference MCP server implementations and SDKs. MCP defines a standardized framework for integrating AI systems with external data sources and tools. It includes specifications for data ingestion and transformation, contextual metadata tagging, and AI interoperability across different platforms. The protocol also supports bidirectional connections between data sources and AI tools. MCP enables applications such as querying structured databases with plain language in the field of natural language data access. The protocol is used in AI-assisted software development tools. Integrated development environments (IDEs), coding platforms such as Replit, and code intelligence tools like Sourcegraph have adopted MCP to grant AI coding assistants real-time access to project context. MCP Apps is an official extension to the Model Context Protocol built on mcp-ui. While the base MCP specification is restricted to text and structured data, MCP Apps standardizes the delivery of interactive user interfaces—such as dashboards, forms, and data visualizations—from MCP servers to host applications like Claude and ChatGPT. == Adoption == In March 2025, OpenAI officially adopted the MCP, after having integrated the standard across its products, including the ChatGPT desktop app. In September 2025, OpenAI added support for MCP to ChatGPT apps. This allows for third-party access inside ChatGPT. MCP can be integrated with Microsoft Semantic Kernel, and Azure OpenAI. MCP servers can be deployed to Cloudflare. In April 2026, the AAIF held the MCP Dev Summit North America in New York City, drawing approximately 1,200 attendees. == Reception == The Verge reported that MCP addresses a growing demand for AI agents that are contextually aware and capable of pulling from diverse sources. In April 2025, security researchers released an analysis that concluded there are multiple outstanding security issues with MCP, including prompt injection, tool permissions that allow for combining tools to exfiltrate data, and lookalike tools that can silently replace trusted ones. MCP has been likened to OpenAPI, a similar specification that aims to describe APIs.