Larry Paul Heck is the Rhesa Screven Farmer, Jr., Advanced Computing Concepts Chair, Georgia Research Alliance Eminent Scholar, Co-Executive Director of the Machine Learning Center and Professor at the Georgia Institute of Technology. His career spans many of the sub-disciplines of artificial intelligence, including conversational AI, speech recognition and speaker recognition, natural language processing, web search, online advertising and acoustics. He is best known for his role as a co-founder of the Microsoft Cortana Personal Assistant and his early work in deep learning for speech processing. == Education and career == Larry Heck was born in Havre, Montana. After receiving the Bachelor of Science in electrical engineering at Texas Tech University, he was admitted to graduate school at the Georgia Institute of Technology in 1986. Heck received the MSEE in 1989 and the PhD in 1991 under advisor Prof. James H. McClellan. From 1992 to 1998, he was a senior research engineer at SRI International with the Acoustics and Radar Technology Lab (ARTL) and Speech Technology and Research (STAR) Lab, and in 1998 joined Nuance Communications, serving as vice president of R&D. Funded by the US government's NSA and DARPA from 1995-1998, Heck led the SRI team that was the first to successfully create large-scale deep neural network (DNN) deep learning technology in the field of speech processing. The deep learning technology was used to win the 1998 National Institute of Standards and Technology Speaker Recognition evaluation. The approach trained a 5-layer deep neural network, with the first two layers used as a (learned) feature extractor. To stabilize the training of the DNN, a weight normalization method was used (later rediscovered in 2010 by Xavier, et.al). Heck deployed this DNN in 1999 with Nuance Communications at the Home Shopping Network, representing the first major industrial application of deep learning with over 100K Nuance Verifier voiceprints. From 2005 to 2008, he was vice president of search & advertising quality at Yahoo!. In 2008, Heck and Ron Brachman combined search & advertising quality with Yahoo! Research to form Yahoo! Labs. Beginning in 2009, he was the chief scientist of speech products at Microsoft. In this role, he established the vision, mission and long-range plan and hired the initial team to create Microsoft’s digital-personal-assistant Cortana. Heck was named a Microsoft Distinguished Engineer in 2012 and joined Microsoft Research that same year. In 2014, he joined Google as a principal research scientist, where he founded the deep learning-based conversational AI team "Deep Dialogue". The team works on advanced research for the Google Assistant. In 2017, Heck joined Samsung as SVP and co-head of global AI Research. In 2019, he became head of Bixby (virtual assistant) North America and the CEO of Viv Labs, an independent subsidiary of Samsung. In that same year, Heck led one of the first large scale deployments of Transformer-Based LLMs as part of the Bixby Categories launch at the 2019 Samsung Developer Conference. In 2021, Heck returned to the Georgia Institute of Technology as a Professor. == Awards and honors == Larry Heck was named Fellow of the Institute of Electrical and Electronics Engineers (IEEE) in 2016 for leadership in application of machine learning to spoken and text language processing. Heck was inducted as a Fellow of the National Academy of Inventors (NAI) in 2024. Heck received the 2017 Academy of Distinguished Engineering Alumni Award from the Georgia Institute of Technology. In the same year, he also received the Texas Tech University Whitacre College of Engineering Distinguished Engineer Award. Larry Heck has several best papers including the 2020 IEEE Signal Processing Society (SPS) Best Paper Award: “Using Recurrent Neural Networks for Slot Filling in Spoken Language Understanding” published in the IEEE/ACM Transactions on Audio, Speech, and Language Processing in March 2015, and the 2020 ACM Conference on Information and Knowledge Management (CIKM) Test of Time Award for the paper "Learning Deep Structured Semantic Models for Web Search using Clickthrough Data".
Fabric Connect
Fabric Connect, in computer networking usage, is the name used by Extreme Networks to market an extended implementation of the IEEE 802.1aq and IEEE 802.1ah-2008 standards. The Fabric Connect technology was originally developed by the Enterprise Solutions R&D department within Nortel Networks. In 2009, Avaya, Inc acquired Nortel Networks Enterprise Business Solutions; this transaction included the Fabric Connect intellectual property together with all of the Ethernet Switching platforms that supported it. Subsequently, the Fabric Connect technology became part of the Extreme Networks portfolio by virtue of their 2017 purchase of the Avaya Networking business and assets. It was during the Avaya era that this technology was promoted as the lead element of the Virtual Enterprise Network Architecture (VENA). == Technologies == === Fabric Connect === Fabric Connect's provides network-wide, end-to-end, multi-layer virtualization. A network virtualization capability, based on an enhanced implementation of the IEEE 802.1aq Shortest Path Bridging (SPB) standard, Fabric Connect offers the ability to create a simplified network that can dynamically virtualize elements to efficiently provision and utilize resources, thus reducing the strain on the network and personnel. Extreme Networks base the Fabric Connect technology on the SPB standard, including support for RFC 6329, and have integrated IP Routing and IP Multicast support; this unified technology allows for the replacement of multiple conventional protocols such as Spanning Tree, RIP and/or OSPF, ECMP, and PIM. === Fabric Attach === An adjunct to the Fabric Connect technology, Fabric Attach allows network operators to extend network virtualization directly into conventional wiring closets (using existing non-Fabric Ethernet switches) and automate the provisioning of devices to their appropriate virtual network. This is particularly relevant for the mass of unattended network end-point that are now appearing, such as IP Phones, Wireless Access Points, and IP Cameras. Fabric Attach standardized protocols such as 802.1AB LLDP to exchange credentials and obtain provisioning information that allows "Client" Switches to be automatically re-configured on the fly with parameters that let Traffic Flows Map through to Fabric Connect Edge Switches (aka "Backbone Edge Bridge" in SPB definition) functioning as a Fabric Attach "Server" Switch. This method is described by an IETF "Internet Draft", pending further standardization activity. Fabric Attach is typically used to automate Wiring Closet connectivity, but has the potential to be extensible for use in the Data Center, with Virtual Machines being able to dynamically request VLAN/VSN (Virtual Service Network) assignment based upon application requirements. == Hardware products == === Virtual Services Platform 9000 Series === A range of modular chassis-based products, featuring a carrier-grade Linux operation system, and designed for high-performance deployment scenarios that need to scale to multiple terabits of switching capacity and support 10 and 40 gigabit Ethernet connections, and is designed eventually to support 100 gigabit Ethernet. === Virtual Services Platform 8000 Series === A compact form-factor platform delivering high-density 10/40 gigabit Ethernet connectivity, and targeted at mid-market through to mid-size enterprise core switch applications. === Virtual Services Platform 7000 Series === A range of high-end 10 gigabit Ethernet stackable switches that extend fabric-based networking to the data center top-of-rack. They support 40 gigabit Ethernet via the MDA Slot. === Virtual Services Platform 4000 Series === A range of high-end gigabit Ethernet stackable switches that extend Fabric-based networking to branch and metro locations. === Ethernet Routing Switch 5000 Series === A range of high-end gigabit Ethernet stackable switches that provides enterprise-class desktop features, including PoE, and offers 10 Gbit/s uplink connections. Each Switch supports up to 144 Gbit/s of virtual backplane capacity, delivering up to 1.152 Tbit/s for a system of eight, creating a virtual backplane through a stacking configuration. === Ethernet Routing Switch 4000 Series === A range of gigabit Ethernet stackable switches that provide enterprise-class desktop features, including PoE/PoE+, and offer 1/10 Gbit/s uplink connections. Each switch supports up to 48 Gbit/s of virtual backplane capacity, delivering up to 384 Gbit/s for a system of 8, creating a virtual backplane through a stacking configuration. === Ethernet Routing Switch 3500 Series === These entry-level gigabit Ethernet stackable switches provide enterprise-class desktop features, including PoE/PoE+, and 1 Gbit/s uplink connections.
T-norm
In mathematics, a t-norm (also T-norm or, unabbreviated, triangular norm) is a kind of binary operation used in the framework of probabilistic metric spaces and in multi-valued logic, specifically in fuzzy logic. A t-norm generalizes intersection in a lattice and conjunction in logic. The name triangular norm refers to the fact that in the framework of probabilistic metric spaces t-norms are used to generalize the triangle inequality of ordinary metric spaces. == Definition == A t-norm is a function T: [0, 1] × [0, 1] → [0, 1] that satisfies the following properties: Commutativity: T(a, b) = T(b, a) Monotonicity: T(a, b) ≤ T(c, d) if a ≤ c and b ≤ d Associativity: T(a, T(b, c)) = T(T(a, b), c) The number 1 acts as identity element: T(a, 1) = a Since a t-norm is a binary algebraic operation on the interval [0, 1], infix algebraic notation is also common, with the t-norm usually denoted by ∗ {\displaystyle } . The defining conditions of the t-norm are exactly those of a partially ordered abelian monoid on the real unit interval [0, 1]. (Cf. ordered group.) The monoidal operation of any partially ordered abelian monoid L is therefore by some authors called a triangular norm on L. === Classification of t-norms === A t-norm is called continuous if it is continuous as a function, in the usual interval topology on [0, 1]2. (Similarly for left- and right-continuity.) A t-norm is called strict if it is continuous and strictly monotone. A t-norm is called nilpotent if it is continuous and each x in the open interval (0, 1) is nilpotent, that is, there is a natural number n such that x ∗ {\displaystyle } ... ∗ {\displaystyle } x (n times) equals 0. A t-norm ∗ {\displaystyle } is called Archimedean if it has the Archimedean property, that is, if for each x, y in the open interval (0, 1) there is a natural number n such that x ∗ {\displaystyle } ... ∗ {\displaystyle } x (n times) is less than or equal to y. The usual partial ordering of t-norms is pointwise, that is, T1 ≤ T2 if T1(a, b) ≤ T2(a, b) for all a, b in [0, 1]. As functions, pointwise larger t-norms are sometimes called stronger than those pointwise smaller. In the semantics of t-norm fuzzy logics, however, the larger a t-norm, the weaker (in terms of logical strength) conjunction it represents. == Prominent examples == Minimum t-norm ⊤ m i n ( a , b ) = min { a , b } , {\displaystyle \top _{\mathrm {min} }(a,b)=\min\{a,b\},} also called the Gödel t-norm, as it is the standard semantics for conjunction in Gödel fuzzy logic. Besides that, it occurs in most t-norm based fuzzy logics as the standard semantics for weak conjunction. It is the pointwise largest t-norm (see the properties of t-norms below). Product t-norm ⊤ p r o d ( a , b ) = a ⋅ b {\displaystyle \top _{\mathrm {prod} }(a,b)=a\cdot b} (the ordinary product of real numbers). Besides other uses, the product t-norm is the standard semantics for strong conjunction in product fuzzy logic. It is a strict Archimedean t-norm. Łukasiewicz t-norm ⊤ L u k ( a , b ) = max { 0 , a + b − 1 } . {\displaystyle \top _{\mathrm {Luk} }(a,b)=\max\{0,a+b-1\}.} The name comes from the fact that the t-norm is the standard semantics for strong conjunction in Łukasiewicz fuzzy logic. It is a nilpotent Archimedean t-norm, pointwise smaller than the product t-norm. Drastic t-norm ⊤ D ( a , b ) = { b if a = 1 a if b = 1 0 otherwise. {\displaystyle \top _{\mathrm {D} }(a,b)={\begin{cases}b&{\mbox{if }}a=1\\a&{\mbox{if }}b=1\\0&{\mbox{otherwise.}}\end{cases}}} The name reflects the fact that the drastic t-norm is the pointwise smallest t-norm (see the properties of t-norms below). It is a right-continuous Archimedean t-norm. Nilpotent minimum ⊤ n M ( a , b ) = { min ( a , b ) if a + b > 1 0 otherwise {\displaystyle \top _{\mathrm {nM} }(a,b)={\begin{cases}\min(a,b)&{\mbox{if }}a+b>1\\0&{\mbox{otherwise}}\end{cases}}} is a standard example of a t-norm that is left-continuous, but not continuous. Despite its name, the nilpotent minimum is not a nilpotent t-norm. Hamacher product ⊤ H 0 ( a , b ) = { 0 if a = b = 0 a b a + b − a b otherwise {\displaystyle \top _{\mathrm {H} _{0}}(a,b)={\begin{cases}0&{\mbox{if }}a=b=0\\{\frac {ab}{a+b-ab}}&{\mbox{otherwise}}\end{cases}}} is a strict Archimedean t-norm, and an important representative of the parametric classes of Hamacher t-norms and Schweizer–Sklar t-norms. == Properties of t-norms == The drastic t-norm is the pointwise smallest t-norm and the minimum is the pointwise largest t-norm: ⊤ D ( a , b ) ≤ ⊤ ( a , b ) ≤ ⊤ m i n ( a , b ) , {\displaystyle \top _{\mathrm {D} }(a,b)\leq \top (a,b)\leq \mathrm {\top _{min}} (a,b),} for any t-norm ⊤ {\displaystyle \top } and all a, b in [0, 1]. In particular, we have that: ⊤ D ( a , b ) ≤ ⊤ L u k ( a , b ) ≤ ⊤ p r o d ( a , b ) ≤ ⊤ m i n ( a , b ) , {\displaystyle \top _{\mathrm {D} }(a,b)\leq \top _{\mathrm {Luk} }(a,b)\leq \top _{\mathrm {prod} }(a,b)\leq \mathrm {\top _{min}} (a,b),} for all a, b in [0, 1]. For every t-norm T, the number 0 acts as null element: T(a, 0) = 0 for all a in [0, 1]. A t-norm T has zero divisors if and only if it has nilpotent elements; each nilpotent element of T is also a zero divisor of T. The set of all nilpotent elements is an interval [0, a] or [0, a), for some a in [0, 1]. === Properties of continuous t-norms === Although real functions of two variables can be continuous in each variable without being continuous on [0, 1]2, this is not the case with t-norms: a t-norm T is continuous if and only if it is continuous in one variable, i.e., if and only if the functions fy(x) = T(x, y) are continuous for each y in [0, 1]. Analogous theorems hold for left- and right-continuity of a t-norm. A continuous t-norm is Archimedean if and only if 0 and 1 are its only idempotents. A continuous Archimedean t-norm is strict if 0 is its only nilpotent element; otherwise it is nilpotent. By definition, moreover, a continuous Archimedean t-norm T is nilpotent if and only if each x < 1 is a nilpotent element of T. Thus with a continuous Archimedean t-norm T, either all or none of the elements of (0, 1) are nilpotent. If it is the case that all elements in (0, 1) are nilpotent, then the t-norm is isomorphic to the Łukasiewicz t-norm; i.e., there is a strictly increasing function f such that ⊤ ( x , y ) = f − 1 ( ⊤ L u k ( f ( x ) , f ( y ) ) ) . {\displaystyle \top (x,y)=f^{-1}(\top _{\mathrm {Luk} }(f(x),f(y))).} If on the other hand it is the case that there are no nilpotent elements of T, the t-norm is isomorphic to the product t-norm. In other words, all nilpotent t-norms are isomorphic, the Łukasiewicz t-norm being their prototypical representative; and all strict t-norms are isomorphic, with the product t-norm as their prototypical example. The Łukasiewicz t-norm is itself isomorphic to the product t-norm undercut at 0.25, i.e., to the function p(x, y) = max(0.25, x ⋅ y) on [0.25, 1]2. For each continuous t-norm, the set of its idempotents is a closed subset of [0, 1]. Its complement—the set of all elements that are not idempotent—is therefore a union of countably many non-overlapping open intervals. The restriction of the t-norm to any of these intervals (including its endpoints) is Archimedean, and thus isomorphic either to the Łukasiewicz t-norm or the product t-norm. For such x, y that do not fall into the same open interval of non-idempotents, the t-norm evaluates to the minimum of x and y. These conditions actually give a characterization of continuous t-norms, called the Mostert–Shields theorem, since every continuous t-norm can in this way be decomposed, and the described construction always yields a continuous t-norm. The theorem can also be formulated as follows: A t-norm is continuous if and only if it is isomorphic to an ordinal sum of the minimum, Łukasiewicz, and product t-norm. A similar characterization theorem for non-continuous t-norms is not known (not even for left-continuous ones), only some non-exhaustive methods for the construction of t-norms have been found. == Residuum == For any left-continuous t-norm ⊤ {\displaystyle \top } , there is a unique binary operation ⇒ {\displaystyle \Rightarrow } on [0, 1] such that ⊤ ( z , x ) ≤ y {\displaystyle \top (z,x)\leq y} if and only if z ≤ ( x ⇒ y ) {\displaystyle z\leq (x\Rightarrow y)} for all x, y, z in [0, 1]. This operation is called the residuum of the t-norm. In prefix notation, the residuum of a t-norm ⊤ {\displaystyle \top } is often denoted by ⊤ → {\displaystyle {\vec {\top }}} or by the letter R. The interval [0, 1] equipped with a t-norm and its residuum forms a residuated lattice. The relation between a t-norm T and its residuum R is an instance of adjunction (specifically, a Galois connection): the residuum forms a right adjoint R(x, –) to the functor T(–, x) for each x in the lattice [0, 1] taken as a poset category. In the standard semantics of t-norm based fuzzy logics, where conjunction is interpreted by a t-norm, the residuum plays the role of implication (often
Stable Diffusion
Stable Diffusion is a deep learning, text-to-image model released in 2022 based on diffusion techniques. The generative artificial intelligence technology is the premier product of Stability AI and is considered to be a part of the ongoing AI boom. It is primarily used to generate detailed images conditioned on text descriptions, though it can also be applied to other tasks such as inpainting, outpainting, and generating image-to-image translations guided by a text prompt. Its development involved researchers from the CompVis Group at LMU Munich and Runway with a computational donation from Stability and training data from non-profit organizations. Stable Diffusion is a latent diffusion model, a kind of deep generative artificial neural network. Its code and model weights have been released publicly, and an optimized version can run on most consumer hardware equipped with a modest GPU with as little as 2.4 GB VRAM. This marked a departure from previous proprietary text-to-image models such as DALL-E and Midjourney which were accessible only via cloud services. == Development == Stable Diffusion originated from a project called Latent Diffusion, developed in Germany by researchers at LMU Munich in Munich and Heidelberg University. Four of the original 5 authors (Robin Rombach, Andreas Blattmann, Patrick Esser and Dominik Lorenz) later joined Stability AI and released subsequent versions of Stable Diffusion. The technical license for the model was released by the CompVis group at LMU Munich. Development was led by Patrick Esser of Runway and Robin Rombach of CompVis, who were among the researchers who had earlier invented the latent diffusion model architecture used by Stable Diffusion. Stability AI also credited EleutherAI and LAION (a German nonprofit which assembled the dataset on which Stable Diffusion was trained) as supporters of the project. == Technology == === Architecture === Diffusion models, introduced in 2015, are trained with the objective of removing successive applications of Gaussian noise on training images, which can be thought of as a sequence of denoising autoencoders. The name diffusion is from the thermodynamic diffusion, since they were first developed with inspiration from thermodynamics. Models in Stable Diffusion series before SD 3 all used a variant of diffusion models, called latent diffusion model (LDM), developed in 2021 by the CompVis (Computer Vision & Learning) group at LMU Munich. Stable Diffusion consists of 3 parts: the variational autoencoder (VAE), U-Net, and an optional text encoder. The VAE encoder compresses the image from pixel space to a smaller dimensional latent space, capturing a more fundamental semantic meaning of the image. Gaussian noise is iteratively applied to the compressed latent representation during forward diffusion. The U-Net block, composed of a ResNet backbone, denoises the output from forward diffusion backwards to obtain a latent representation. Finally, the VAE decoder generates the final image by converting the representation back into pixel space. The denoising step can be flexibly conditioned on a string of text, an image, or another modality. The encoded conditioning data is exposed to denoising U-Nets via a cross-attention mechanism. For conditioning on text, the fixed, pretrained CLIP ViT-L/14 text encoder is used to transform text prompts to an embedding space. Researchers point to increased computational efficiency for training and generation as an advantage of LDMs. With 860 million parameters in the U-Net and 123 million in the text encoder, Stable Diffusion is considered relatively lightweight by 2022 standards, and unlike other diffusion models, it can run on consumer GPUs, and even CPU-only if using the OpenVINO version of Stable Diffusion. ==== SD XL ==== The XL version uses the same LDM architecture as previous versions, except larger: larger UNet backbone, larger cross-attention context, two text encoders instead of one, and trained on multiple aspect ratios (not just the square aspect ratio like previous versions). The SD XL Refiner, released at the same time, has the same architecture as SD XL, but it was trained for adding fine details to preexisting images via text-conditional img2img. ==== SD 3.0 ==== The 3.0 version completely changes the backbone. Not a UNet, but a Rectified Flow Transformer, which implements the rectified flow method with a Transformer. The Transformer architecture used for SD 3.0 has three "tracks", for original text encoding, transformed text encoding, and image encoding (in latent space). The transformed text encoding and image encoding are mixed during each transformer block. The architecture is named "multimodal diffusion transformer (MMDiT), where the "multimodal" means that it mixes text and image encodings inside its operations. This differs from previous versions of DiT, where the text encoding affects the image encoding, but not vice versa. === Training data === Stable Diffusion was trained on pairs of images and captions taken from LAION-5B, a publicly available dataset derived from Common Crawl data scraped from the web, where 5 billion image-text pairs were classified based on language and filtered into separate datasets by resolution, a predicted likelihood of containing a watermark, and predicted "aesthetic" score (e.g. subjective visual quality). The dataset was created by LAION, a German non-profit which receives funding from Stability AI. The Stable Diffusion model was trained on three subsets of LAION-5B: laion2B-en, laion-high-resolution, and laion-aesthetics v2 5+. A third-party analysis of the model's training data identified that out of a smaller subset of 12 million images taken from the original wider dataset used, approximately 47% of the sample size of images came from 100 different domains, with Pinterest taking up 8.5% of the subset, followed by websites such as WordPress, Blogspot, Flickr, DeviantArt and Wikimedia Commons. An investigation by Bayerischer Rundfunk showed that LAION's datasets, hosted on Hugging Face, contain large amounts of private and sensitive data. === Training procedures === The model was initially trained on the laion2B-en and laion-high-resolution subsets, with the last few rounds of training done on LAION-Aesthetics v2 5+, a subset of 600 million captioned images which the LAION-Aesthetics Predictor V2 predicted that humans would, on average, give a score of at least 5 out of 10 when asked to rate how much they liked them. The LAION-Aesthetics v2 5+ subset also excluded low-resolution images and images which LAION-5B-WatermarkDetection identified as carrying a watermark with greater than 80% probability. Final rounds of training additionally dropped 10% of text conditioning to improve Classifier-Free Diffusion Guidance. The model was trained using 256 Nvidia A100 GPUs on Amazon Web Services for a total of 150,000 GPU-hours, at a cost of $600,000. === Limitations === Stable Diffusion has issues with degradation and inaccuracies in certain scenarios. Initial releases of the model were trained on a dataset that consists of 512×512 resolution images, meaning that the quality of generated images noticeably degrades when user specifications deviate from its "expected" 512×512 resolution; the version 2.0 update of the Stable Diffusion model later introduced the ability to natively generate images at 768×768 resolution. Another challenge is in generating human limbs due to poor data quality of limbs in the LAION database. The model is insufficiently trained to replicate human limbs and faces due to the lack of representative features in the database, and prompting the model to generate images of such type can confound the model. In addition to human limbs, Stable Diffusion is unable to generate legible ambigrams and some other forms of text and typography. Stable Diffusion XL (SDXL) version 1.0, released in July 2023, introduced native 1024x1024 resolution and improved generation for limbs and text. Accessibility for individual developers can also be a problem. In order to customize the model for new use cases that are not included in the dataset, such as generating anime characters ("waifu diffusion"), new data and further training are required. Fine-tuned adaptations of Stable Diffusion created through additional retraining have been used for a variety of different use-cases, from medical imaging to algorithmically generated music. However, this fine-tuning process is sensitive to the quality of new data; low resolution images or different resolutions from the original data can not only fail to learn the new task but degrade the overall performance of the model. Even when the model is additionally trained on high quality images, it is difficult for individuals to run models in consumer electronics. For example, the training process for waifu-diffusion requires a minimum 30 GB of VRAM, which exceeds the usual resource provided in such consumer GPUs as Nvidia's GeForce 30 series, w
Computer-assisted proof
A computer-assisted proof is a mathematical proof that has been at least partially generated by computer. Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of a mathematical theorem. The idea is to use a computer program to perform lengthy computations, and to provide a proof that the result of these computations implies the given theorem. In 1976, the four color theorem was the first major theorem to be verified using a computer program. Attempts have also been made in the area of artificial intelligence research to create smaller, explicit, new proofs of mathematical theorems from the bottom up using automated reasoning techniques such as heuristic search. Such automated theorem provers have proved a number of new results and found new proofs for known theorems. Additionally, interactive proof assistants allow mathematicians to develop human-readable proofs which are nonetheless formally verified for correctness. Since these proofs are generally human-surveyable (albeit with difficulty, as with the proof of the Robbins conjecture) they do not share the controversial implications of computer-aided proofs-by-exhaustion. == Methods == One method for using computers in mathematical proofs is by means of so-called validated numerics or rigorous numerics. This means computing numerically yet with mathematical rigour. One uses set-valued arithmetic and inclusion principle in order to ensure that the set-valued output of a numerical program encloses the solution of the original mathematical problem. This is done by controlling, enclosing and propagating round-off and truncation errors using for example interval arithmetic. More precisely, one reduces the computation to a sequence of elementary operations, say ( + , − , × , / ) {\displaystyle (+,-,\times ,/)} . In a computer, the result of each elementary operation is rounded off by the computer precision. However, one can construct an interval provided by upper and lower bounds on the result of an elementary operation. Then one proceeds by replacing numbers with intervals and performing elementary operations between such intervals of representable numbers. == Philosophical objections == Computer-assisted proofs are the subject of some controversy in the mathematical world, with Thomas Tymoczko first to articulate objections. Those who adhere to Tymoczko's arguments believe that lengthy computer-assisted proofs are not, in some sense, 'real' mathematical proofs because they involve so many logical steps that they are not practically verifiable by human beings, and that mathematicians are effectively being asked to replace logical deduction from assumed axioms with trust in an empirical computational process, which is potentially affected by errors in the computer program, as well as defects in the runtime environment and hardware. Other mathematicians believe that lengthy computer-assisted proofs should be regarded as calculations, rather than proofs: the proof algorithm itself should be proved valid, so that its use can then be regarded as a mere "verification". Arguments that computer-assisted proofs are subject to errors in their source programs, compilers, and hardware can be resolved by providing a formal proof of correctness for the computer program (an approach which was successfully applied to the four color theorem in 2005) as well as replicating the result using different programming languages, different compilers, and different computer hardware. Another possible way of verifying computer-aided proofs is to generate their reasoning steps in a machine readable form, and then use a proof checker program to demonstrate their correctness. Since validating a given proof is much easier than finding a proof, the checker program is simpler than the original assistant program, and it is correspondingly easier to gain confidence into its correctness. However, this approach of using a computer program to prove the output of another program correct does not appeal to computer proof skeptics, who see it as adding another layer of complexity without addressing the perceived need for human understanding. Another argument against computer-aided proofs is that they lack mathematical elegance—that they provide no insights or new and useful concepts. In fact, this is an argument that could be advanced against any lengthy proof by exhaustion. An additional philosophical issue raised by computer-aided proofs is whether they make mathematics into a quasi-empirical science, where the scientific method becomes more important than the application of pure reason in the area of abstract mathematical concepts. This directly relates to the argument within mathematics as to whether mathematics is based on ideas, or "merely" an exercise in formal symbol manipulation. It also raises the question whether, if according to the Platonist view, all possible mathematical objects in some sense "already exist", whether computer-aided mathematics is an observational science like astronomy, rather than an experimental one like physics or chemistry. This controversy within mathematics is occurring at the same time as questions are being asked in the physics community about whether twenty-first century theoretical physics is becoming too mathematical, and leaving behind its experimental roots. The emerging field of experimental mathematics is confronting this debate head-on by focusing on numerical experiments as its main tool for mathematical exploration. == Theorems proved with the help of computer programs == Inclusion in this list does not imply that a formal computer-checked proof exists, but rather, that a computer program has been involved in some way. See the main articles for details.
Trazzler
Trazzler is a travel destination app that specializes in unique and local destinations. The initial concept was developed by Adam Rugel and Biz Stone in 2006 at Twitter's original offices under the name "71 miles". More than 10,000 writers and photographers have contributed and more than $350,000 in freelance contracts have been issued as a result of Trazzeler's weekly writing and photography contests. Investors in the company include SV Angel, AOL Founder Steve Case, and the Twitter founders, Evan Williams, Jack Dorsey, and Biz Stone. The company's partners are the City of Chicago, Hawaii Tourism Authority, Fairmont Hotels & Resorts, Salon.com, and Air New Zealand. Trazzler is designed for use on the iOS, Android, and Facebook.
Tip and cue
Tip and cue, sometimes referred to as tip and que, tipping and cueing, or tipping and queing, is a method for satellite imagery and reconnaissance satellites to automatically coordinate tracking of objects across different satellites in real or near real-time. This technique ensures continuous tracking of targets as they move across different regions by handing them off between satellites, sharing satellite imagery and collateral across discrete satellites. The coordination between various satellites and their complementary sensors allows for more accurate and efficient data collection. This system is particularly useful in scenarios requiring real-time monitoring and rapid response; the method significantly improves situational awareness and operational effectiveness. Tip and cue techniques involve integrating various sensor systems, each playing a specific role in the tracking process. As a target moves, it is handed off from one satellite to another, ensuring continuous monitoring. This coordination optimizes data collection and analysis, enhancing overall tracking accuracy. The real-time information gathered by these satellites is critical for decision-making in various applications, including defense and surveillance. By leveraging multiple satellites and their sensors, it provides broader coverage and more reliable tracking, and the continuous handoff between satellites ensures there are no gaps in monitoring, essential for high-stakes applications. The real-time data provided by this system allows for timely and informed decisions, improving response times and outcomes. Tip and cue methodologies are a part of geospatial intelligence, or GEOINT. Robert Cardillo, a former director of the National Geospatial-Intelligence Agency, highlighted the importance of tip and cue methods to their data collection efforts in 2015. == Historical Development == The concept of tip and cue in satellite monitoring has its origins in early military applications designed to enhance missile detection and tracking systems. During the Cold War, advancements in infrared sensing technologies laid the groundwork for more sophisticated tip and cue techniques. The integration of different sensor types, such as radar and optical sensors, in the 1990s expanded the capabilities of tip and cue systems beyond military applications. These advancements have made tip and cue techniques essential for various civilian uses, including disaster monitoring and environmental surveillance. Significant progress was made with the advent of high-speed data processing and communication technologies in the early 2000s, further refining the method. Advanced algorithms and data fusion techniques have been introduced to better integrate information from multiple sensors. Machine learning technologies now play a crucial role in improving detection and prediction capabilities, allowing for more adaptive and efficient tracking. Richmond and Brennan of Lockheed Martin, presenting to the annual technical conference of the Maui Space Surveillance Complex (formerly the Air Force Maui Optical Station (AMOS)), discussed the algorithms needed for 'tip and cue', to facilitate "multi-phenomenology data fusion." The Space Surveillance Telescope (SST) at Naval Communication Station Harold E. Holt in Australia, operated by the United States Space Force and designed by the Massachusetts Institute of Technology Lincoln Laboratory, was reported by the Defense Advanced Research Projects Agency (DARPA) to be a leader in creating and improving tip and cue techniques, from a large library of orbital object data. == Technical overview == Tip and cue systems utilize a network of at least two satellites equipped with complementary sensor technologies to track moving objects in real-time. The method involves detecting a target with a primary sensor, such as an infrared or photographic sensor, which then cues secondary sensors on the same or other satellites for more detailed monitoring. This handoff process between discrete systems ensures continuous tracking as the target moves across different areas, leveraging each systems strengths. Data collected by these systems and sensors are rapidly processed and shared among the network, enhancing situational awareness. This coordination optimizes resource usage and improves the accuracy of tracking moving objects over large areas. The primary sensors detect initial targets based on specific signatures, such as heat or movement, and then cue secondary sensors to gather more precise data. This ensures that each sensor operates within its optimal range, maintaining high tracking accuracy and reliability. The integration of various sensor types, including optical, radar, and infrared, allows the system to function effectively under different conditions and environments. Real-time data processing and communication between satellites and ground stations are crucial for timely and accurate target tracking. Satellites using tip and cue processes may use either passive or active scanning methodoloigies. These systems may also leverage both orbital and ground-based ELINT (electronic signals intelligence). == Known use cases == Tip and cue systems have been extensively utilized in military applications, particularly for missile detection and defense. These systems enable early detection of missile launches using infrared sensors, which then cue other sensors to track the missile's trajectory more accurately. In environmental monitoring, tip and cue techniques help track natural disasters such as wildfires and hurricanes by coordinating various satellite sensors for comprehensive data collection and analysis. Surveillance and reconnaissance operations also benefit from tip and cue systems, which provide continuous and precise tracking of moving objects, enhancing situational awareness. Additionally, these systems are used in maritime surveillance to monitor ship movements and detect illegal activities such as smuggling and piracy. Tip and cue systems are used in disaster management. For instance, during wildfires, infrared sensors can detect heat signatures, prompting other sensors to gather detailed imagery and data on fire spread and intensity. This coordinated approach allows for real-time monitoring and rapid response, crucial for mitigating damage and saving lives. Similarly, in hurricane tracking, satellites equipped with various sensors can monitor storm development and progression, providing timely information for emergency management agencies. The integration of multiple sensor types ensures accurate and comprehensive coverage of these dynamic and fast-changing events. In maritime surveillance, or maritime domain awareness (MDA), tip and cue systems enhance the detection and monitoring of vessel movements, contributing to maritime security. By coordinating satellite sensors, these systems can track ships over vast ocean areas, identifying potential threats or illegal activities such as smuggling, piracy, and illegal fishing. The ability to maintain continuous surveillance and share data in real-time with maritime authorities improves response times and enforcement capabilities. This application of tip and cue systems not only aids in law enforcement but also supports environmental conservation efforts by monitoring protected marine areas. Automatic Identification System (AIS) is one of the most important sources of data for the MDA agencies. AIS is used in order for ships to know each other's whereabouts, they transmit a signal from ship to ship and to shore. Lately, the system has been developed into satellite system, so called satellite AIS, which makes the system more effective. All ocean-going vessels above 300 tons, are supposed to use and transmit via AIS according to the International Maritime Organization. The satellite constellations help facilitate this with tip and cue methodologies.