Laws of Form (hereinafter LoF) is a book by G. Spencer-Brown, written by August 1967 and published in 1969. The book straddles the boundary between mathematics and philosophy. LoF describes three distinct logical systems: The primary arithmetic (described in Chapter 4 of LoF), whose models include Boolean arithmetic; The primary algebra (Chapter 6 of LoF), whose models include the two-element Boolean algebra (hereinafter abbreviated 2), Boolean logic, and the classical propositional calculus; Equations of the second degree (Chapter 11), whose interpretations include finite automata and Alonzo Church's Restricted Recursive Arithmetic (RRA). "Boundary algebra" is a Meguire (2011) term for the union of the primary algebra and the primary arithmetic. Laws of Form sometimes loosely refers to the "primary algebra" as well as to LoF. == Contents == The preface states that the work was first explored in 1959, and Spencer Brown cites Bertrand Russell as being supportive of his endeavour. He also thanks J. C. P. Miller of University College London for helping with the proofreading and offering other guidance. In 1963 Spencer Brown was invited by Harry Frost, staff lecturer in the physical sciences at the department of Extra-Mural Studies of the University of London, to deliver a course on the mathematics of logic. LoF emerged from work in electronic engineering its author did around 1960. Key ideas of the LOF were first outlined in his 1961 manuscript Design with the Nor, which remained unpublished until 2021, and further refined during subsequent lectures on mathematical logic he gave under the auspices of the University of London's extension program. LoF has appeared in several editions. The second series of editions appeared in 1972 with the "Preface to the First American Edition", which emphasised the use of self-referential paradoxes, and the most recent being a 1997 German translation. LoF has never gone out of print. LoF's mystical and declamatory prose and its love of paradox make it a challenging read for all. Spencer-Brown was influenced by Ludwig Wittgenstein and R. D. Laing. LoF also echoes a number of themes from the writings of Charles Sanders Peirce, Bertrand Russell, and Alfred North Whitehead. The work has had curious effects on some classes of its readership; for example, on obscure grounds, it has been claimed that the entire book is written in an operational way, giving instructions to the reader instead of telling them what "is", and that in accordance with G. Spencer-Brown's interest in paradoxes, the only sentence that makes a statement that something is, is the statement which says no such statements are used in this book. Furthermore, the claim asserts that except for this one sentence the book can be seen as an example of E-Prime. What prompted such a claim, is obscure, either in terms of incentive, logical merit, or as a matter of fact, because the book routinely and naturally uses the verb to be throughout, and in all its grammatical forms, as may be seen both in the original and in quotes shown below. == Reception == Ostensibly a work of formal mathematics and philosophy, LoF became something of a cult classic: it was praised by Heinz von Foerster when he reviewed it for the Whole Earth Catalog. Those who agree point to LoF as embodying an enigmatic "mathematics of consciousness", its algebraic symbolism capturing an (perhaps even "the") implicit root of cognition: the ability to "distinguish". LoF argues that primary algebra reveals striking connections among logic, Boolean algebra, and arithmetic, and the philosophy of language and mind. Stafford Beer wrote in a review for Nature in 1969, "When one thinks of all that Russell went through sixty years ago, to write the Principia, and all we his readers underwent in wrestling with those three vast volumes, it is almost sad". Banaschewski (1977) argues that the primary algebra is nothing but new notation for Boolean algebra. Indeed, the two-element Boolean algebra 2 can be seen as the intended interpretation of the primary algebra. Yet the notation of the primary algebra: Fully exploits the duality characterizing not just Boolean algebras but all lattices; Highlights how syntactically distinct statements in logic and 2 can have identical semantics; Dramatically simplifies Boolean algebra calculations, and proofs in sentential and syllogistic logic. Moreover, the syntax of the primary algebra can be extended to formal systems other than 2 and sentential logic, resulting in boundary mathematics. LoF has influenced, among others, Heinz von Foerster, Louis Kauffman, Niklas Luhmann, Humberto Maturana, Francisco Varela and William Bricken. Some of these authors have modified the primary algebra in a variety of interesting ways. LoF claimed that certain well-known mathematical conjectures of very long standing, such as the four color theorem, Fermat's Last Theorem, and the Goldbach conjecture, are provable using extensions of the primary algebra. Spencer-Brown eventually circulated a purported proof of the four color theorem, but it was met with skepticism. == The form (Chapter 1) == The symbol: Also called the "mark" or "cross", is the essential feature of the Laws of Form. In Spencer-Brown's inimitable and enigmatic fashion, the Mark symbolizes the root of cognition, i.e., the dualistic Mark indicates the capability of differentiating a "this" from "everything else but this". In LoF, a Cross denotes the drawing of a "distinction", and can be thought of as signifying the following, all at once: The act of drawing a boundary around something, thus separating it from everything else; That which becomes distinct from everything by drawing the boundary; Crossing from one side of the boundary to the other. All three ways imply an action on the part of the cognitive entity (e.g., person) making the distinction. As LoF puts it: "The first command: Draw a distinction can well be expressed in such ways as: Let there be a distinction, Find a distinction, See a distinction, Describe a distinction, Define a distinction, Or: Let a distinction be drawn". (LoF, Notes to chapter 2) The counterpoint to the Marked state is the Unmarked state, which is simply nothing, the void, or the un-expressable infinite represented by a blank space. It is simply the absence of a Cross. No distinction has been made and nothing has been crossed. The Marked state and the void are the two primitive values of the Laws of Form. The Cross can be seen as denoting the distinction between two states, one "considered as a symbol" and another not so considered. From this fact arises a curious resonance with some theories of consciousness and language. Paradoxically, the Form is at once Observer and Observed, and is also the creative act of making an observation. LoF (excluding back matter) closes with the words: ...the first distinction, the Mark and the observer are not only interchangeable, but, in the form, identical. C. S. Peirce came to a related insight in the 1890s; see § Related work. == The primary arithmetic (Chapter 4) == The syntax of the primary arithmetic goes as follows. There are just two atomic expressions: The empty Cross ; All or part of the blank page (the "void"). There are two inductive rules: A Cross may be written over any expression; Any two expressions may be concatenated. The semantics of the primary arithmetic are perhaps nothing more than the sole explicit definition in LoF: "Distinction is perfect continence". Let the "unmarked state" be a synonym for the void. Let an empty Cross denote the "marked state". To cross is to move from one value, the unmarked or marked state, to the other. We can now state the "arithmetical" axioms A1 and A2, which ground the primary arithmetic (and hence all of the Laws of Form): "A1. The law of Calling". Calling twice from a state is indistinguishable from calling once. To make a distinction twice has the same effect as making it once. For example, saying "Let there be light" and then saying "Let there be light" again, is the same as saying it once. Formally: = {\displaystyle \ =} "A2. The law of Crossing". After crossing from the unmarked to the marked state, crossing again ("recrossing") starting from the marked state returns one to the unmarked state. Hence recrossing annuls crossing. Formally: = {\displaystyle \ =} In both A1 and A2, the expression to the right of '=' has fewer symbols than the expression to the left of '='. This suggests that every primary arithmetic expression can, by repeated application of A1 and A2, be simplified to one of two states: the marked or the unmarked state. This is indeed the case, and the result is the expression's "simplification". The two fundamental metatheorems of the primary arithmetic state that: Every finite expression has a unique simplification. (T3 in LoF); Starting from an initial marked or unmarked state, "complicating" an expression by a finite number of repeated application of A1 and A2 cannot yield
Gcore
Gcore is an edge AI, cloud, network, and security company headquartered in Luxembourg. Founded in 2014, the company provides low-latency services to industries including finance, healthcare, manufacturing, gaming, media and telecommunications internationally. As of March 2024, its global network includes over 180 Points of Presence (PoPs) across six continents. == History == Gcore was founded in 2014 in Luxembourg. The company built its own content delivery network, originally designed for the needs of the gaming industry. In 2016, Gcore's infrastructure expanded to multiple regions that were underserved by hyperscale cloud providers. In 2020, the company formed partnerships with Intel and Equinix. In 2022, Gcore launched the European AI Cloud, providing access to infrastructure for machine learning tasks. In March 2024, Gcore announced the acquisition of a web application and API protection (WAAP) solution from StackPath. In April 2024, Gcore received a commendation in the Industry Innovation category at the NVIDIA Partner Network Awards EMEA for developing the first speech-to-text technology for Luxembourgish, using the LuxemBERT AI model. In May 2024, Philipp Rösler, former vice-chancellor of Germany and federal minister of health joined the Gcore board. In July 2024, Gcore raised $60 million in a Series A funding round, marking the company's first external investment since its founding. In August 2024, Gcore was recognized as a Major Player in the IDC MarketScape report for European public cloud Infrastructure (IaaS) 2024 by IDC, the global market intelligence firm. In May 2025, Feiyu Xu became a member of the Gcore advisory board. == Network infrastructure == According to the company's website, Gcore has network locations in six continents: Europe, North America, Asia, South America, Africa, and Australia with over 14,000 peering partners and a network capacity exceeding 200 Tbps. According to a 2025 review by Geekflare, Gcore's CDN achieved an average global response time of around 30 milliseconds. Gcore offers AI cloud clusters, including a generative AI cluster with Nvidia GPUs in Luxembourg and additional sites in the Netherlands and Wales, as part of its European AI infrastructure. == Products and services == Gcore offers a range of services, including content delivery network (CDN), cloud computing,virtual machines, bare-metal servers, object storage AI infrastructure and inference, Kubernetes, video streaming, DDoS mitigation, web application and API protection (WAAP), Domain Name System (DNS). Gcore provides AI services and GPU cloud infrastructure to support model development, training, fine-tuning, and inference. In January 2025, the company introduced Everywhere Inference, a serverless inference solution that enables AI model deployment. == Controversies == Correctiv and Tageszeitung reported that Gcore supported the distribution of the TV network RT until April 2023, which has been under sanctions by the EU since March 2022. However, Gcore denies these allegations. == Collaborations == In 2024, Gcore and Qareeb Data Centres, a data center provider in the Middle East, launched a collaboration to integrate Gcore's AI, cloud and edge services across data centers in multiple Middle Eastern countries. In June 2025, Gcore joined the SmartSpires initiative, a €3.1 million smart city project co-funded by the Connecting Europe Facility. The three-year programme is coordinated by a public–private consortium including 5SKYE, the Luxembourg Institute of Science and Technology (LIST), Orange Luxembourg, and Gcore. The project aims to transform the Belval campus into a smart city by deploying 5G-enabled smart towers that integrate edge computing, artificial intelligence and IoT services. Within the consortium, Gcore acts as project coordinator and is responsible for the deployment of the edge infrastructure.
AdBlock
AdBlock is an ad-blocking browser extension for Google Chrome, Apple Safari (desktop and mobile), Firefox, Samsung Internet, Microsoft Edge and Opera. AdBlock allows users to prevent page elements, such as advertisements, from being displayed. It is free to download and use, and it includes optional donations to the developers. The AdBlock extension was created on December 8, 2009, which is the day that supports for extensions was added to Google Chrome. It was one of the first Google Chrome extensions that was made. Since 2016, AdBlock has been based on the Adblock Plus source code. In July 2018, AdBlock acquired uBlock, a commercial ad-blocker owned by uBlock LLC and based on uBlock Origin. In April 2021, eyeo GmbH (developer of Adblock Plus) announced its purchase of AdBlock, Inc (formerly BetaFish, Inc). == Crowdfunding == Gundlach launched a crowdfunding campaign on Crowdtilt in August 2013 in order to fund an ad campaign to raise awareness of ad-blocking and to rent a billboard at Times Square. After the one-month campaign, it raised $55,000. == Sales and acceptable ads == AdBlock was sold to an anonymous buyer in 2015 and on October 15, 2015, Gundlach's name was taken down from the site. In the terms of the deal, the original developer Michael Gundlach left operations to Adblock's continuing director, Gabriel Cubbage, and as of October 2, 2015, AdBlock began participating in the Acceptable Ads program. Acceptable Ads identifies "non-annoying" ads, which AdBlock shows by default. The intent is to allow non-invasive advertising, to either maintain support for websites that rely on advertising as a main source of revenue or for websites that have an agreement with the program. == Filters == AdBlock uses EasyList, the same filter syntax as Adblock Plus for Firefox, and natively supports the use of a number of filter lists. == Partnership with Amnesty International == On March 12, 2016, in support of World Day Against Cyber Censorship, and in partnership with Amnesty International, instead of blocking ads, AdBlock replaced ads with banners linked to articles on Amnesty's website, written by prominent free speech advocates such as Edward Snowden, to raise awareness of government-imposed online censorship and digital privacy issues around the world. The campaign was met with both praise and criticism, with AdBlock's CEO, Gabriel Cubbage, defending the decision in an essay on AdBlock's website, saying "We’re showing you Amnesty banners, just for today, because we believe users should be part of the conversation about online privacy. Tomorrow, those spaces will be vacant again. But take a moment to consider that in an increasingly information-driven world, when your right to digital privacy is threatened, so is your right to free expression." Meanwhile, Simon Sharwood of The Register characterized Cubbage's position as "'You should control your computer except when we feel political', says AdBlock CEO". == AdBlock for Firefox == On September 13, 2014, the AdBlock team released a version for Firefox users, ported from the code for Google Chrome, released under the same free software license as the original Adblock. The extension was removed on April 2, 2015, by an administrator on Mozilla Add-ons. On December 7, 2015, the official AdBlock site's knowledge base article stated that with version 44 or higher of Firefox desktop and Firefox Mobile, AdBlock will not be supported. The last version of Adblock for those platforms will work on older versions of Firefox. AdBlock was released again on Mozilla Add-ons on November 17, 2016. On April 1, 2012, Adblock developer Michael Gundlach tweaked the code to display LOLcats instead of simply blocking ads. Initially developed as a short-lived April Fools joke, the response was so positive that CatBlock was continued to be offered as an optional add-on supported by a monthly subscription. On October 23, 2014, the developer decided to end official support for CatBlock, and made it open-source, under GPLv3 licensing, as the original extension.
Comparison of video editing software
This is a comparison of non-linear video editing software applications. See also a more complete list of video editing software. == General information == This table gives basic general information about the different editors: === Active === === Discontinued / Inactive === ==== Definition ==== professional: used for full length Hollywood movies; professional (small): mainly used for paid commercials, short films or podcasts/YouTube channels; prosumer: Mainly targeting private use, anything that can do more than just trimming a film; basic: trimming a film; == System requirements == This table lists the operating systems that different editors can run on without emulation, as well as other system requirements. Note that minimum system requirements are listed; some features (like High Definition support) may be unavailable with these specifications. "Unix" includes the similar Linux, BSD and Unix-like operating systems. == High definition/High resolution import == The table below indicates the ability of each program to import various High Definition video or High resolution video formats for editing. == Feature set == == Output options == Please note that recording to Blu-ray does not imply 1080@50p/60p . Most only support up to 1080i 25/30 frames per second recording. Also not all formats can be output.
Audio-visual speech recognition
Audio visual speech recognition (AVSR) is a technique that uses image processing capabilities in lip reading to aid speech recognition systems in recognizing indeterministic phones or giving preponderance among near probability decisions. Each system of lip reading and speech recognition works separately, then their results are mixed at the stage of feature fusion. As the name suggests, it has two parts. First one is the audio part and second one is the visual part. In audio part we use features like log mel spectrogram, mfcc etc. from the raw audio samples and we build a model to get feature vector out of it . For visual part generally we use some variant of convolutional neural network to compress the image to a feature vector after that we concatenate these two vectors (audio and visual ) and try to predict the target object.
Physics-informed neural networks
In machine learning, physics-informed neural networks (PINNs), also referred to as theory-trained neural networks (TTNs), are a type of universal function approximator that can embed the knowledge of any physical laws that govern a given data-set in the learning process, and can be described by partial differential equations (PDEs). Low data availability for some biological and engineering problems limit the robustness of conventional machine learning models used for these applications. The prior knowledge of general physical laws acts in the training of neural networks (NNs) as a regularization agent that limits the space of admissible solutions, increasing the generalizability of the function approximation. This way, embedding this prior information into a neural network results in enhancing the information content of the available data, facilitating the learning algorithm to capture the right solution and to generalize well even with a low amount of training examples. Because they process continuous spatial and time coordinates and output continuous PDE solutions, they can be categorized as neural fields. == Function approximation == Most of the physical laws that govern the dynamics of a system can be described by partial differential equations. For example, the Navier–Stokes equations are a set of partial differential equations derived from the conservation laws (i.e., conservation of mass, momentum, and energy) that govern fluid mechanics. The solution of the Navier–Stokes equations with appropriate initial and boundary conditions allows the quantification of flow dynamics in a precisely defined geometry. However, these equations cannot be solved exactly and therefore numerical methods must be used (such as finite differences, finite elements and finite volumes). In this setting, these governing equations must be solved while accounting for prior assumptions, linearization, and adequate time and space discretization. Recently, solving the governing partial differential equations of physical phenomena using deep learning has emerged as a new field of scientific machine learning (SciML), leveraging the universal approximation theorem and high expressivity of neural networks. In general, deep neural networks could approximate any high-dimensional function given that sufficient training data are supplied. However, such networks do not consider the physical characteristics underlying the problem, and the level of approximation accuracy provided by them is still heavily dependent on careful specifications of the problem geometry as well as the initial and boundary conditions. Without this preliminary information, the solution is not unique and may lose physical correctness. To remedy this, Physics-Informed Neural Networks (PINNs) leverage governing physical equations in neural network training. Namely, PINNs are designed to be trained to satisfy the given training data as well as the imposed governing equations. In this fashion, a neural network can be guided with training datasets that do not necessarily need to be large or complete. An accurate solution of partial differential equations can potentially be found without knowing the boundary conditions. Therefore, with some knowledge about the physical characteristics of the problem and some form of training data (even sparse and incomplete), PINNs may be used for finding an optimal solution with high fidelity. PINNs can be applied to a wide range of problems in computational science, and are a pioneering technology leading to the development of new classes of numerical solvers for PDEs. PINNs can be thought of as a mesh-free alternative to traditional approaches (e.g., CFD for fluid dynamics), and new data-driven approaches for model inversion and system identification. Notably, a trained PINN network can be used to predict values on simulation grids of different resolutions without needing to be retrained. Additionally, the derivatives used in the partial differential equations can be computed using automatic differentiation (AD), which is assessed to be superior to numerical or symbolic differentiation. == Modeling and computation == A general nonlinear partial differential equation can be written as: u t + N [ u ; λ ] = 0 , x ∈ Ω , t ∈ [ 0 , T ] {\displaystyle u_{t}+{\mathcal {N}}[u;\lambda ]=0,\quad x\in \Omega ,\quad t\in [0,T]} where u ( t , x ) {\displaystyle u(t,x)} denotes the solution, N [ ⋅ ; λ ] {\displaystyle {\mathcal {N}}[\cdot ;\lambda ]} is a nonlinear operator parameterized by λ {\displaystyle \lambda } , and Ω {\displaystyle \Omega } is a subset of R D {\displaystyle \mathbb {R} ^{D}} . This general form of governing equations summarizes a wide range of problems in mathematical physics, such as conservative laws, diffusion process, advection-diffusion systems, and kinetic equations. Given noisy measurements of a generic dynamic system described by the equation above, PINNs can be designed to solve two classes of problems: data-driven solutions of partial differential equations data-driven discovery of partial differential equations === Data-driven solution of partial differential equations === The data-driven solution of PDE computes the hidden state u ( t , x ) {\displaystyle u(t,x)} of the system given boundary data and/or measurements z {\displaystyle z} , and fixed model parameters λ {\displaystyle \lambda } . We solve: u t + N [ u ] = 0 , x ∈ Ω , t ∈ [ 0 , T ] {\displaystyle u_{t}+{\mathcal {N}}[u]=0,\quad x\in \Omega ,\quad t\in [0,T]} . by defining the residual f ( t , x ) {\displaystyle f(t,x)} as: f := u t + N [ u ] {\displaystyle f:=u_{t}+{\mathcal {N}}[u]} , and approximating u ( t , x ) {\displaystyle u(t,x)} by a deep neural network. This network can be differentiated using automatic differentiation. The parameters of u ( t , x ) {\displaystyle u(t,x)} and f ( t , x ) {\displaystyle f(t,x)} can be then learned by minimizing the following loss function L tot {\displaystyle L_{\text{tot}}} : L tot = L u + L f {\displaystyle L_{\text{tot}}=L_{u}+L_{f}} where: L u = ‖ u − z ‖ Γ {\displaystyle L_{u}=\Vert u-z\Vert _{\Gamma }} is the error between the PINN u ( t , x ) {\displaystyle u(t,x)} and the set of boundary conditions and measured data on the set of points Γ {\displaystyle \Gamma } where the boundary conditions and data are defined. L f = ‖ f ‖ Γ {\displaystyle L_{f}=\Vert f\Vert _{\Gamma }} is the mean-squared error of the residual function. This second term encourages the PINN to learn the structural information expressed by the PDE during the training process. This approach has been used to yield computationally efficient physics-informed surrogate models with applications in the forecasting of physical processes, model predictive control, multi-physics and multi-scale modeling, and simulation. It has been shown to converge to the solution of the PDE. === Data-driven discovery of partial differential equations === Given noisy and incomplete measurements z {\displaystyle z} of the state of the system, the data-driven discovery of PDEs results in computing the unknown state u ( t , x ) {\displaystyle u(t,x)} and learning model parameters λ {\displaystyle \lambda } that best describe the observed data: u t + N [ u ; λ ] = 0 , x ∈ Ω , t ∈ [ 0 , T ] {\displaystyle u_{t}+{\mathcal {N}}[u;\lambda ]=0,\quad x\in \Omega ,\quad t\in [0,T]} By defining f ( t , x ) {\displaystyle f(t,x)} as: f := u t + N [ u ; λ ] = 0 {\displaystyle f:=u_{t}+{\mathcal {N}}[u;\lambda ]=0} , and approximating u ( t , x ) {\displaystyle u(t,x)} by a deep neural network, f ( t , x ) {\displaystyle f(t,x)} results in a PINN. This network can be derived using automatic differentiation. The parameters of u ( t , x ) {\displaystyle u(t,x)} and f ( t , x ) {\displaystyle f(t,x)} , together with the parameter λ {\displaystyle \lambda } of the differential operator can be then learned by minimizing the following loss function L tot {\displaystyle L_{\text{tot}}} : L tot = L u + L f {\displaystyle L_{\text{tot}}=L_{u}+L_{f}} where: L u = ‖ u − z ‖ Γ {\displaystyle L_{u}=\Vert u-z\Vert _{\Gamma }} , with u {\displaystyle u} and z {\displaystyle z} state solutions and measurements at sparse location Γ {\displaystyle \Gamma } , respectively. L f = ‖ f ‖ Γ {\displaystyle L_{f}=\Vert f\Vert _{\Gamma }} is the residual function. This second term requires the structured information represented by the partial differential equations to be satisfied in the training process. This strategy allows for discovering dynamic models described by nonlinear PDEs assembling computationally efficient and fully differentiable surrogate models that may find application in predictive forecasting, control, and data assimilation. == Extensions and applications == === For piece-wise function approximation === PINNs are unable to approximate PDEs that have strong non-linearity or sharp gradients (such as those that commonly occur in practical fluid flow problems). Piecewise approximation has been an old practic
Fully probabilistic design
Decision making (DM) can be seen as a purposeful choice of action sequences. It also covers control, a purposeful choice of input sequences. As a rule, it runs under randomness, uncertainty and incomplete knowledge. A range of prescriptive theories have been proposed how to make optimal decisions under these conditions. They optimise sequence of decision rules, mappings of the available knowledge on possible actions. This sequence is called strategy or policy. Among various theories, Bayesian DM is broadly accepted axiomatically based theory that solves the design of optimal decision strategy. It describes random, uncertain or incompletely known quantities as random variables, i.e. by their joint probability expressing belief in their possible values. The strategy that minimises expected loss (or equivalently maximises expected reward) expressing decision-maker's goals is then taken as the optimal strategy. While the probabilistic description of beliefs is uniquely and deductively driven by rules for joint probabilities, the composition and decomposition of the loss function have no such universally applicable formal machinery. Fully probabilistic design (of decision strategies or control, FPD) removes the mentioned drawback and expresses also the DM goals of by the "ideal" probability, which assigns high (small) values to desired (undesired) behaviours of the closed DM loop formed by the influenced world part and by the used strategy. FPD has axiomatic basis and has Bayesian DM as its restricted subpart. FPD has a range of theoretical consequences , and, importantly, has been successfully used to quite diverse application domains.