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  • Biohybrid microswimmer

    Biohybrid microswimmer

    A biohybrid microswimmer also known as biohybrid nanorobot, can be defined as a microswimmer that consist of both biological and artificial constituents, for instance, one or several living microorganisms attached to one or various synthetic parts. In recent years nanoscopic and mesoscopic objects have been designed to collectively move through direct inspiration from nature or by harnessing its existing tools. Small mesoscopic to nanoscopic systems typically operate at low Reynolds numbers (Re ≪ 1), and understanding their motion becomes challenging. For locomotion to occur, the symmetry of the system must be broken. In addition, collective motion requires a coupling mechanism between the entities that make up the collective. To develop mesoscopic to nanoscopic entities capable of swarming behaviour, it has been hypothesised that the entities are characterised by broken symmetry with a well-defined morphology, and are powered with some material capable of harvesting energy. If the harvested energy results in a field surrounding the object, then this field can couple with the field of a neighbouring object and bring some coordination to the collective behaviour. Such robotic swarms have been categorised by an online expert panel as among the 10 great unresolved group challenges in the area of robotics. Although investigation of their underlying mechanism of action is still in its infancy, various systems have been developed that are capable of undergoing controlled and uncontrolled swarming motion by harvesting energy (e.g., light, thermal, etc.). Over the past decade, biohybrid microrobots, in which living mobile microorganisms are physically integrated with untethered artificial structures, have gained growing interest to enable the active locomotion and cargo delivery to a target destination. In addition to the motility, the intrinsic capabilities of sensing and eliciting an appropriate response to artificial and environmental changes make cell-based biohybrid microrobots appealing for transportation of cargo to the inaccessible cavities of the human body for local active delivery of diagnostic and therapeutic agents. == Background == Biohybrid microswimmers can be defined as microswimmers that consist of both biological and artificial constituents, for instance, one or several living microorganisms attached to one or various synthetic parts. The pioneers of this field, ahead of their time, were Montemagno and Bachand with a 1999 work regarding specific attachment strategies of biological molecules to nanofabricated substrates enabling the preparation of hybrid inorganic/organic nanoelectromechanical systems, so called NEMS. They described the production of large amounts of F1-ATPase from the thermophilic bacteria Bacillus PS3 for the preparation of F1-ATPase biomolecular motors immobilized on a nanoarray pattern of gold, copper or nickel produced by electron beam lithography. These proteins were attached to one micron microspheres tagged with a synthetic peptide. Consequently, they accomplished the preparation of a platform with chemically active sites and the development of biohybrid devices capable of converting energy of biomolecular motors into useful work. One of the most fundamental questions in science is what defines life. Collective motion is one of the hallmarks of life. This is commonly observed in nature at various dimensional levels as energized entities gather, in a concerted effort, into motile aggregated patterns. These motile aggregated events can be noticed, among many others, as dynamic swarms; e.g., unicellular organisms such as bacteria, locust swarms, or the flocking behaviour of birds. Ever since Newton established his equations of motion, the mystery of motion on the microscale has emerged frequently in scientific history, as famously demonstrated by a couple of articles that should be discussed briefly. First, an essential concept, popularized by Osborne Reynolds, is that the relative importance of inertia and viscosity for the motion of a fluid depends on certain details of the system under consideration. The Reynolds number Re, named in his honor, quantifies this comparison as a dimensionless ratio of characteristic inertial and viscous forces: R e = ρ u l μ {\displaystyle \mathrm {Re} ={\frac {\rho ul}{\mu }}} Here, ρ represents the density of the fluid; u is a characteristic velocity of the system (for instance, the velocity of a swimming particle); l is a characteristic length scale (e.g., the swimmer size); and μ is the viscosity of the fluid. Taking the suspending fluid to be water, and using experimentally observed values for u, one can determine that inertia is important for macroscopic swimmers like fish (Re = 100), while viscosity dominates the motion of microscale swimmers like bacteria (Re = 10−4). The overwhelming importance of viscosity for swimming at the micrometer scale has profound implications for swimming strategy. This has been discussed memorably by E. M. Purcell, who invited the reader into the world of microorganisms and theoretically studied the conditions of their motion. In the first place, propulsion strategies of large scale swimmers often involve imparting momentum to the surrounding fluid in periodic discrete events, such as vortex shedding, and coasting between these events through inertia. This cannot be effective for microscale swimmers like bacteria: due to the large viscous damping, the inertial coasting time of a micron-sized object is on the order of 1 μs. The coasting distance of a microorganism moving at a typical speed is about 0.1 angstroms (Å). Purcell concluded that only forces that are exerted in the present moment on a microscale body contribute to its propulsion, so a constant energy conversion method is essential. Microorganisms have optimized their metabolism for continuous energy production, while purely artificial microswimmers (microrobots) must obtain energy from the environment, since their on-board-storage-capacity is very limited. As a further consequence of the continuous dissipation of energy, biological and artificial microswimmers do not obey the laws of equilibrium statistical physics, and need to be described by non-equilibrium dynamics. Mathematically, Purcell explored the implications of low Reynolds number by taking the Navier-Stokes equation and eliminating the inertial terms: μ ∇ 2 u − ∇ p = 0 {\displaystyle {\begin{aligned}\mu \nabla ^{2}\mathbf {u} -{\boldsymbol {\nabla }}p&={\boldsymbol {0}}\\\end{aligned}}} where u {\displaystyle \mathbf {u} } is the velocity of the fluid and ∇ p {\displaystyle {\boldsymbol {\nabla }}p} is the gradient of the pressure. As Purcell noted, the resulting equation — the Stokes equation — contains no explicit time dependence. This has some important consequences for how a suspended body (e.g., a bacterium) can swim through periodic mechanical motions or deformations (e.g., of a flagellum). First, the rate of motion is practically irrelevant for the motion of the microswimmer and of the surrounding fluid: changing the rate of motion will change the scale of the velocities of the fluid and of the microswimmer, but it will not change the pattern of fluid flow. Secondly, reversing the direction of mechanical motion will simply reverse all velocities in the system. These properties of the Stokes equation severely restrict the range of feasible swimming strategies. Recent publications of biohybrid microswimmers include the use of sperm cells, contractive muscle cells, and bacteria as biological components, as they can efficiently convert chemical energy into movement, and additionally are capable of performing complicated motion depending on environmental conditions. In this sense, biohybrid microswimmer systems can be described as the combination of different functional components: cargo and carrier. The cargo is an element of interest to be moved (and possibly released) in a customized way. The carrier is the component responsible for the movement of the biohybrid, transporting the desired cargo, which is linked to its surface. The great majority of these systems rely on biological motile propulsion for the transportation of synthetic cargo for targeted drug delivery/ There are also examples of the opposite case: artificial microswimmers with biological cargo systems. Over the past decade, biohybrid microrobots, in which living mobile microorganisms are physically integrated with untethered artificial structures, have gained growing interest to enable the active locomotion and cargo delivery to a target destination. In addition to the motility, the intrinsic capabilities of sensing and eliciting an appropriate response to artificial and environmental changes make cell-based biohybrid microrobots appealing for transportation of cargo to the inaccessible cavities of the human body for local active delivery of diagnostic and therapeutic agents. Active locomotion, targeting and steering of concentrated therape

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  • Pachinko allocation

    Pachinko allocation

    In machine learning and natural language processing, the pachinko allocation model (PAM) is a topic model. Topic models are a suite of algorithms to uncover the hidden thematic structure of a collection of documents. The algorithm improves upon earlier topic models such as latent Dirichlet allocation (LDA) by modeling correlations between topics in addition to the word correlations which constitute topics. PAM provides more flexibility and greater expressive power than latent Dirichlet allocation. While first described and implemented in the context of natural language processing, the algorithm may have applications in other fields such as bioinformatics. The model is named for pachinko machines—a game popular in Japan, in which metal balls bounce down around a complex collection of pins until they land in various bins at the bottom. == History == Pachinko allocation was first described by Wei Li and Andrew McCallum in 2006. The idea was extended with hierarchical Pachinko allocation by Li, McCallum, and David Mimno in 2007. In 2007, McCallum and his colleagues proposed a nonparametric Bayesian prior for PAM based on a variant of the hierarchical Dirichlet process (HDP). The algorithm has been implemented in the MALLET software package published by McCallum's group at the University of Massachusetts Amherst. == Model == PAM connects words in V and topics in T with an arbitrary directed acyclic graph (DAG), where topic nodes occupy the interior levels and the leaves are words. The probability of generating a whole corpus is the product of the probabilities for every document: P ( D | α ) = ∏ d P ( d | α ) {\displaystyle P(\mathbf {D} |\alpha )=\prod _{d}P(d|\alpha )}

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  • Best AI Copywriting Tools in 2026

    Best AI Copywriting Tools in 2026

    Looking for the best AI copywriting tool? An AI copywriting tool is software that uses machine learning to help you get more done — it can save you hours every week by automating repetitive work. Most options offer a generous free tier, with paid plans unlocking higher limits, faster processing, and team features. Whether you are a beginner or a pro, the right AI copywriting tool slots into your workflow and pays for itself fast. This guide breaks down the top picks, their pros and cons, and who each one is best for.

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  • Flex (lexical analyzer generator)

    Flex (lexical analyzer generator)

    Flex (fast lexical analyzer generator) is a free and open-source software alternative to lex. It is a computer program that generates lexical analyzers (also known as "scanners" or "lexers"). It is frequently used as the lex implementation together with Berkeley Yacc parser generator on BSD-derived operating systems (as both lex and yacc are part of POSIX), or together with GNU bison (a version of yacc) in BSD ports and in Linux distributions. Unlike Bison, flex is not part of the GNU Project and is not released under the GNU General Public License, although a manual for Flex was produced and published by the Free Software Foundation. == History == Flex was written in C around 1987 by Vern Paxson, with the help of many ideas and much inspiration from Van Jacobson. Original version by Jef Poskanzer. The fast table representation is a partial implementation of a design done by Van Jacobson. The implementation was done by Kevin Gong and Vern Paxson. == Example lexical analyzer == This is an example of a Flex scanner for the instructional programming language PL/0. The tokens recognized are: '+', '-', '', '/', '=', '(', ')', ',', ';', '.', ':=', '<', '<=', '<>', '>', '>='; numbers: 0-9 {0-9}; identifiers: a-zA-Z {a-zA-Z0-9} and keywords: begin, call, const, do, end, if, odd, procedure, then, var, while. == Internals == These programs perform character parsing and tokenizing via the use of a deterministic finite automaton (DFA). A DFA is a theoretical machine accepting regular languages, and is equivalent to read-only right moving Turing machines. The syntax is based on the use of regular expressions. See also nondeterministic finite automaton. == Issues == === Time complexity === A Flex lexical analyzer usually has time complexity O ( n ) {\displaystyle O(n)} in the length of the input. That is, it performs a constant number of operations for each input symbol. This constant is quite low: GCC generates 12 instructions for the DFA match loop. Note that the constant is independent of the length of the token, the length of the regular expression and the size of the DFA. However, using the REJECT macro in a scanner with the potential to match extremely long tokens can cause Flex to generate a scanner with non-linear performance. This feature is optional. In this case, the programmer has explicitly told Flex to "go back and try again" after it has already matched some input. This will cause the DFA to backtrack to find other accept states. The REJECT feature is not enabled by default, and because of its performance implications its use is discouraged in the Flex manual. === Reentrancy === By default the scanner generated by Flex is not reentrant. This can cause serious problems for programs that use the generated scanner from different threads. To overcome this issue there are options that Flex provides in order to achieve reentrancy. A detailed description of these options can be found in the Flex manual. === Usage under non-Unix environments === Normally the generated scanner contains references to the unistd.h header file, which is Unix specific. To avoid generating code that includes unistd.h, %option nounistd should be used. Another issue is the call to isatty (a Unix library function), which can be found in the generated code. The %option never-interactive forces flex to generate code that does not use isatty. === Using flex from other languages === Flex can only generate code for C and C++. To use the scanner code generated by flex from other languages a language binding tool such as SWIG can be used. === Unicode support === Flex is limited to matching 1-byte (8-bit) binary values and therefore does not support Unicode. RE/flex and other alternatives do support Unicode matching. == Flex++ == flex++ is a similar lexical scanner for C++ which is included as part of the flex package. The generated code does not depend on any runtime or external library except for a memory allocator (malloc or a user-supplied alternative) unless the input also depends on it. This can be useful in embedded and similar situations where traditional operating system or C runtime facilities may not be available. The flex++ generated C++ scanner includes the header file FlexLexer.h, which defines the interfaces of the two C++ generated classes.

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  • Algorithmic inference

    Algorithmic inference

    Algorithmic inference gathers new developments in the statistical inference methods made feasible by the powerful computing devices widely available to any data analyst. Cornerstones in this field are computational learning theory, granular computing, bioinformatics, and, long ago, structural probability (Fraser 1966). The main focus is on the algorithms which compute statistics rooting the study of a random phenomenon, along with the amount of data they must feed on to produce reliable results. This shifts the interest of mathematicians from the study of the distribution laws to the functional properties of the statistics, and the interest of computer scientists from the algorithms for processing data to the information they process. == The Fisher parametric inference problem == Concerning the identification of the parameters of a distribution law, the mature reader may recall lengthy disputes in the mid 20th century about the interpretation of their variability in terms of fiducial distribution (Fisher 1956), structural probabilities (Fraser 1966), priors/posteriors (Ramsey 1925), and so on. From an epistemology viewpoint, this entailed a companion dispute as to the nature of probability: is it a physical feature of phenomena to be described through random variables or a way of synthesizing data about a phenomenon? Opting for the latter, Fisher defines a fiducial distribution law of parameters of a given random variable that he deduces from a sample of its specifications. With this law he computes, for instance "the probability that μ (mean of a Gaussian variable – omeur note) is less than any assigned value, or the probability that it lies between any assigned values, or, in short, its probability distribution, in the light of the sample observed". == The classic solution == Fisher fought hard to defend the difference and superiority of his notion of parameter distribution in comparison to analogous notions, such as Bayes' posterior distribution, Fraser's constructive probability and Neyman's confidence intervals. For half a century, Neyman's confidence intervals won out for all practical purposes, crediting the phenomenological nature of probability. With this perspective, when you deal with a Gaussian variable, its mean μ is fixed by the physical features of the phenomenon you are observing, where the observations are random operators, hence the observed values are specifications of a random sample. Because of their randomness, you may compute from the sample specific intervals containing the fixed μ with a given probability that you denote confidence. === Example === Let X be a Gaussian variable with parameters μ {\displaystyle \mu } and σ 2 {\displaystyle \sigma ^{2}} and { X 1 , … , X m } {\displaystyle \{X_{1},\ldots ,X_{m}\}} a sample drawn from it. Working with statistics S μ = ∑ i = 1 m X i {\displaystyle S_{\mu }=\sum _{i=1}^{m}X_{i}} and S σ 2 = ∑ i = 1 m ( X i − X ¯ ) 2 , where X ¯ = S μ m {\displaystyle S_{\sigma ^{2}}=\sum _{i=1}^{m}(X_{i}-{\overline {X}})^{2},{\text{ where }}{\overline {X}}={\frac {S_{\mu }}{m}}} is the sample mean, we recognize that T = S μ − m μ S σ 2 m − 1 m = X ¯ − μ S σ 2 / ( m ( m − 1 ) ) {\displaystyle T={\frac {S_{\mu }-m\mu }{\sqrt {S_{\sigma ^{2}}}}}{\sqrt {\frac {m-1}{m}}}={\frac {{\overline {X}}-\mu }{\sqrt {S_{\sigma ^{2}}/(m(m-1))}}}} follows a Student's t distribution (Wilks 1962) with parameter (degrees of freedom) m − 1, so that f T ( t ) = Γ ( m / 2 ) Γ ( ( m − 1 ) / 2 ) 1 π ( m − 1 ) ( 1 + t 2 m − 1 ) m / 2 . {\displaystyle f_{T}(t)={\frac {\Gamma (m/2)}{\Gamma ((m-1)/2)}}{\frac {1}{\sqrt {\pi (m-1)}}}\left(1+{\frac {t^{2}}{m-1}}\right)^{m/2}.} Gauging T between two quantiles and inverting its expression as a function of μ {\displaystyle \mu } you obtain confidence intervals for μ {\displaystyle \mu } . With the sample specification: x = { 7.14 , 6.3 , 3.9 , 6.46 , 0.2 , 2.94 , 4.14 , 4.69 , 6.02 , 1.58 } {\displaystyle \mathbf {x} =\{7.14,6.3,3.9,6.46,0.2,2.94,4.14,4.69,6.02,1.58\}} having size m = 10, you compute the statistics s μ = 43.37 {\displaystyle s_{\mu }=43.37} and s σ 2 = 46.07 {\displaystyle s_{\sigma ^{2}}=46.07} , and obtain a 0.90 confidence interval for μ {\displaystyle \mu } with extremes (3.03, 5.65). == Inferring functions with the help of a computer == From a modeling perspective the entire dispute looks like a chicken-egg dilemma: either fixed data by first and probability distribution of their properties as a consequence, or fixed properties by first and probability distribution of the observed data as a corollary. The classic solution has one benefit and one drawback. The former was appreciated particularly back when people still did computations with sheet and pencil. Per se, the task of computing a Neyman confidence interval for the fixed parameter θ is hard: you do not know θ, but you look for disposing around it an interval with a possibly very low probability of failing. The analytical solution is allowed for a very limited number of theoretical cases. Vice versa a large variety of instances may be quickly solved in an approximate way via the central limit theorem in terms of confidence interval around a Gaussian distribution – that's the benefit. The drawback is that the central limit theorem is applicable when the sample size is sufficiently large. Therefore, it is less and less applicable with the sample involved in modern inference instances. The fault is not in the sample size on its own part. Rather, this size is not sufficiently large because of the complexity of the inference problem. With the availability of large computing facilities, scientists refocused from isolated parameters inference to complex functions inference, i.e. re sets of highly nested parameters identifying functions. In these cases we speak about learning of functions (in terms for instance of regression, neuro-fuzzy system or computational learning) on the basis of highly informative samples. A first effect of having a complex structure linking data is the reduction of the number of sample degrees of freedom, i.e. the burning of a part of sample points, so that the effective sample size to be considered in the central limit theorem is too small. Focusing on the sample size ensuring a limited learning error with a given confidence level, the consequence is that the lower bound on this size grows with complexity indices such as VC dimension or detail of a class to which the function we want to learn belongs. === Example === A sample of 1,000 independent bits is enough to ensure an absolute error of at most 0.081 on the estimation of the parameter p of the underlying Bernoulli variable with a confidence of at least 0.99. The same size cannot guarantee a threshold less than 0.088 with the same confidence 0.99 when the error is identified with the probability that a 20-year-old man living in New York does not fit the ranges of height, weight and waistline observed on 1,000 Big Apple inhabitants. The accuracy shortage occurs because both the VC dimension and the detail of the class of parallelepipeds, among which the one observed from the 1,000 inhabitants' ranges falls, are equal to 6. == The general inversion problem solving the Fisher question == With insufficiently large samples, the approach: fixed sample – random properties suggests inference procedures in three steps: === Definition === For a random variable and a sample drawn from it a compatible distribution is a distribution having the same sampling mechanism M X = ( Z , g θ ) {\displaystyle {\mathcal {M}}_{X}=(Z,g_{\boldsymbol {\theta }})} of X with a value θ {\displaystyle {\boldsymbol {\theta }}} of the random parameter Θ {\displaystyle \mathbf {\Theta } } derived from a master equation rooted on a well-behaved statistic s. === Example === You may find the distribution law of the Pareto parameters A and K as an implementation example of the population bootstrap method as in the figure on the left. Implementing the twisting argument method, you get the distribution law F M ( μ ) {\displaystyle F_{M}(\mu )} of the mean M of a Gaussian variable X on the basis of the statistic s M = ∑ i = 1 m x i {\textstyle s_{M}=\sum _{i=1}^{m}x_{i}} when Σ 2 {\displaystyle \Sigma ^{2}} is known to be equal to σ 2 {\displaystyle \sigma ^{2}} (Apolloni, Malchiodi & Gaito 2006). Its expression is: F M ( μ ) = Φ ( m μ − s M σ m ) , {\displaystyle F_{M}(\mu )=\Phi {\left({\frac {m\mu -s_{M}}{\sigma {\sqrt {m}}}}\right)},} shown in the figure on the right, where Φ {\displaystyle \Phi } is the cumulative distribution function of a standard normal distribution. Computing a confidence interval for M given its distribution function is straightforward: we need only find two quantiles (for instance δ / 2 {\displaystyle \delta /2} and 1 − δ / 2 {\displaystyle 1-\delta /2} quantiles in case we are interested in a confidence interval of level δ symmetric in the tail's probabilities) as indicated on the left in the diagram showing the behavior of

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  • Barney Pell

    Barney Pell

    Barney Pell (born March 18, 1968) is an American entrepreneur, angel investor and computer scientist. He was co-founder and CEO of Powerset, a pioneering natural language search startup, search strategist and architect for Microsoft's Bing search engine, a pioneer in the field of general game playing in artificial intelligence, and the architect of the first intelligent agent to fly onboard and control a spacecraft. He was co-founder, Vice Chairman and Chief Strategy Officer of Moon Express; co-founder and chairman of LocoMobi; and Associate Founder of Singularity University. == Career == === Education === Pell received his Bachelor of Science degree in symbolic systems from Stanford University in 1989, where he graduated Phi Beta Kappa and was a National Merit Scholar. Pell earned a PhD in computer science from Cambridge University in 1993, supervised by Stephen Pulman, where he was a Marshall Scholar. === Research === Pell's research is focused on basic problems in the study of intelligence, computer game playing, machine learning, natural language processing, autonomous robotics, and web search. Barney Pell has published over 30 technical papers on topics related to information retrieval, knowledge management, machine learning, artificial intelligence, and scheduling systems. In computer game playing and machine learning, he was a pioneer in the field of General Game Playing, and created programs to generate the rules of chess-like games and programs to play individual games directly from the rules without human assistance. He also did early work on machine learning in the game of Go and on an architecture for pragmatic reasoning for bidding in the game of Bridge. In natural language processing, he was a scientist in the Artificial Intelligence Center at SRI International, where he worked on the Core Language Engine. Barney Pell was the Technical Area Manager of the Collaborative and Assistant Systems area within the Computational Sciences Division (now the Intelligent Systems Division) at NASA Ames Research Center, where he oversaw a staff of 80 scientists working on information retrieval, search, knowledge management, machine learning, semantic technology, human centered systems, collaboration technology, adaptive user interfaces, human robot interaction, and other areas of artificial intelligence. From 1993 to 1998, Barney Pell worked as a Principal Investigator and Senior Computer Scientist at NASA Ames, where he conducted advanced research and development of autonomous control software for NASA's deep space missions. He was the Architect for the Deep Space One Remote Agent Experiment and the Project Lead for the Executive component of the Remote Agent Experiment, the first intelligent agent to fly onboard and control a spacecraft. === Business === Pell is an entrepreneur who has founded or co-founded several business ventures, including Powerset, Moon Express, and LocoMobi. He was the founder and CEO of Powerset, a San Francisco startup company that built a search engine based on natural language processing technology originally developed at XEROX PARC. On May 11, 2008, the company unveiled a tool for searching a fixed subset of Wikipedia using conversational phrases rather than keywords. On July 1, 2008, Microsoft signed an agreement to acquire Powerset for an estimated $100 million. Powerset became a part of Microsoft's search engine, Bing. From 2008 until August 2011, Pell served as Partner, Search Strategist, and Evangelist for Microsoft's search engine, Bing and as Head of Bing's Local and Mobile Search teams. Prior to joining Powerset, Pell was an Entrepreneur-in-Residence at Mayfield Fund, a venture capital firm in Silicon Valley. Pell is also a founder of Moon Express, Inc., a U.S. company awarded a $10M commercial lunar contract by NASA and a competitor in the Google Lunar X PRIZE. Pell was also co-founder and chairman of LocoMobi, Inc., a U.S. company developing mobile, software and hardware technology solutions for the parking industry. LocoMobi was winner of the Tie50 Award in 2014. Pell is also an associate founder of Singularity University and a Machine Learning Fellow at the Creative Destruction Lab at the Rotman School of Management From 1998 to 2000, Pell served as chief strategist and vice president of business development at StockMaster.com (acquired by Red Herring in March, 2000). From 2000 to 2002, Pell was Chief Strategist and Vice President of Business Development for Whizbang Labs. Pell has been an angel investor and advisor to numerous startup companies, including Pulse.io (acquired by Google), Aardvark (acquired by Google), Appjet (acquired by Google), Jibe Mobile (acquired by Google), Movity (acquired by Trulia), QuestBridge, BrandYourself, CrowdFlower (acquired by Appen), and LinkedIn. === Views and predictions === Pell has expressed views and predictions regarding technological advancements in coming years. He believes that humans will soon have "brain-machine interfaces that will let people interact with each other as if they had 'hangouts' in their mind." Pell predicts these interfaces to become available within 20 to 30 years. Pell also predicts advancements in bodily augmentation, such as "even-better-than-human prosthetics and high-quality tissue engineering within 10 years." Pell believes that with advancements in space exploration technology the moon will soon be a commercially viable resource for material such as platinum and water. == Awards and recognition == In 1986, Pell was awarded a National Merit Scholarship. In 1989, Pell was awarded a Marshall Scholarship. In 1989, Pell was elected Phi Beta Kappa. In 1997, Pell was part of the team award a NASA Software of the Year Award for the Deep Space 1 Remote Agent.

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  • The Best Free AI Bug Finder for Beginners

    The Best Free AI Bug Finder for Beginners

    Shopping for the best AI bug finder? An AI bug finder is software that uses machine learning to help you get more done — it keeps getting smarter as the underlying models improve. Pricing, accuracy, and the size of the model behind the tool are the three factors that most affect daily usefulness. Whether you are a beginner or a pro, the right AI bug finder slots into your workflow and pays for itself fast. We tested the leading options and ranked them by quality, value, and ease of use.

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  • The Best Free AI Art Generator for Beginners

    The Best Free AI Art Generator for Beginners

    Trying to pick the best AI art generator? An AI art generator is software that uses machine learning to help you get more done — it scales effortlessly from a single task to thousands. The best picks balance beginner-friendly simplicity with the depth power users need, and they ship updates often. Whether you are a beginner or a pro, the right AI art generator slots into your workflow and pays for itself fast. This guide breaks down the top picks, their pros and cons, and who each one is best for.

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  • Convolution

    Convolution

    In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions f {\displaystyle f} and g {\displaystyle g} that produces a third function f ∗ g {\displaystyle fg} , as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The term convolution refers to both the resulting function and to the process of computing it. The integral is evaluated for all values of shift, producing the convolution function. The choice of which function is reflected and shifted before the integral does not change the integral result (see commutativity). Graphically, it expresses how the 'shape' of one function is modified by the other. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution f ∗ g {\displaystyle fg} differs from cross-correlation f ⋆ g {\displaystyle f\star g} only in that either f ( x ) {\displaystyle f(x)} or g ( x ) {\displaystyle g(x)} is reflected about the y-axis in convolution; thus it is a cross-correlation of g ( − x ) {\displaystyle g(-x)} and f ( x ) {\displaystyle f(x)} , or f ( − x ) {\displaystyle f(-x)} and g ( x ) {\displaystyle g(x)} . For complex-valued functions, the cross-correlation operator is the adjoint of the convolution operator. Convolution has applications that include probability, statistics, acoustics, spectroscopy, signal processing and image processing, computer vision and human vision, geophysics, engineering, physics, and differential equations. The convolution can be defined for functions on Euclidean space and other groups (as algebraic structures). For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution. (See row 18 at DTFT § Properties.) A discrete convolution can be defined for functions on the set of integers. Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, and in the design and implementation of finite impulse response filters in signal processing. Computing the inverse of the convolution operation is known as deconvolution. == Definition == The convolution of f {\displaystyle f} and g {\displaystyle g} is written f ∗ g {\displaystyle fg} , denoting the operator with the symbol ∗ {\displaystyle } . It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. As such, it is a particular kind of integral transform: ( f ∗ g ) ( t ) := ∫ − ∞ ∞ f ( τ ) g ( t − τ ) d τ . {\displaystyle (fg)(t):=\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau .} An equivalent definition is (see commutativity): ( f ∗ g ) ( t ) := ∫ − ∞ ∞ f ( t − τ ) g ( τ ) d τ . {\displaystyle (fg)(t):=\int _{-\infty }^{\infty }f(t-\tau )g(\tau )\,d\tau .} While the symbol t {\displaystyle t} is used above, it need not represent the time domain. At each t {\displaystyle t} , the convolution formula can be described as the area under the function f ( τ ) {\displaystyle f(\tau )} weighted by the function g ( − τ ) {\displaystyle g(-\tau )} shifted by the amount t {\displaystyle t} . As t {\displaystyle t} changes, the weighting function g ( t − τ ) {\displaystyle g(t-\tau )} emphasizes different parts of the input function f ( τ ) {\displaystyle f(\tau )} ; If t {\displaystyle t} is a positive value, then g ( t − τ ) {\displaystyle g(t-\tau )} is equal to g ( − τ ) {\displaystyle g(-\tau )} that slides or is shifted along the τ {\displaystyle \tau } -axis toward the right (toward + ∞ {\displaystyle +\infty } ) by the amount of t {\displaystyle t} , while if t {\displaystyle t} is a negative value, then g ( t − τ ) {\displaystyle g(t-\tau )} is equal to g ( − τ ) {\displaystyle g(-\tau )} that slides or is shifted toward the left (toward − ∞ {\displaystyle -\infty } ) by the amount of | t | {\displaystyle |t|} . For functions f {\displaystyle f} , g {\displaystyle g} supported on only [ 0 , ∞ ) {\displaystyle [0,\infty )} (i.e., zero for negative arguments), the integration limits can be truncated, resulting in: ( f ∗ g ) ( t ) = ∫ 0 t f ( τ ) g ( t − τ ) d τ for f , g : [ 0 , ∞ ) → R . {\displaystyle (fg)(t)=\int _{0}^{t}f(\tau )g(t-\tau )\,d\tau \quad \ {\text{for }}f,g:[0,\infty )\to \mathbb {R} .} For the multi-dimensional formulation of convolution, see domain of definition (below). === Notation === A common engineering notational convention is: f ( t ) ∗ g ( t ) := ∫ − ∞ ∞ f ( τ ) g ( t − τ ) d τ ⏟ ( f ∗ g ) ( t ) , {\displaystyle f(t)g(t)\mathrel {:=} \underbrace {\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau } _{(fg)(t)},} which has to be interpreted carefully to avoid confusion. For instance, f ( t ) ∗ g ( t − t 0 ) {\displaystyle f(t)g(t-t_{0})} is equivalent to ( f ∗ g ) ( t − t 0 ) {\displaystyle (fg)(t-t_{0})} , but f ( t − t 0 ) ∗ g ( t − t 0 ) {\displaystyle f(t-t_{0})g(t-t_{0})} is in fact equivalent to ( f ∗ g ) ( t − 2 t 0 ) {\displaystyle (fg)(t-2t_{0})} . === Relations with other transforms === Given two functions f ( t ) {\displaystyle f(t)} and g ( t ) {\displaystyle g(t)} with bilateral Laplace transforms (two-sided Laplace transform) F ( s ) = ∫ − ∞ ∞ e − s u f ( u ) d u {\displaystyle F(s)=\int _{-\infty }^{\infty }e^{-su}\ f(u)\ {\text{d}}u} and G ( s ) = ∫ − ∞ ∞ e − s v g ( v ) d v {\displaystyle G(s)=\int _{-\infty }^{\infty }e^{-sv}\ g(v)\ {\text{d}}v} respectively, the convolution operation ( f ∗ g ) ( t ) {\displaystyle (fg)(t)} can be defined as the inverse Laplace transform of the product of F ( s ) {\displaystyle F(s)} and G ( s ) {\displaystyle G(s)} . More precisely, F ( s ) ⋅ G ( s ) = ∫ − ∞ ∞ e − s u f ( u ) d u ⋅ ∫ − ∞ ∞ e − s v g ( v ) d v = ∫ − ∞ ∞ ∫ − ∞ ∞ e − s ( u + v ) f ( u ) g ( v ) d u d v {\displaystyle {\begin{aligned}F(s)\cdot G(s)&=\int _{-\infty }^{\infty }e^{-su}\ f(u)\ {\text{d}}u\cdot \int _{-\infty }^{\infty }e^{-sv}\ g(v)\ {\text{d}}v\\&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-s(u+v)}\ f(u)\ g(v)\ {\text{d}}u\ {\text{d}}v\end{aligned}}} Let t = u + v {\displaystyle t=u+v} , then F ( s ) ⋅ G ( s ) = ∫ − ∞ ∞ ∫ − ∞ ∞ e − s t f ( u ) g ( t − u ) d u d t = ∫ − ∞ ∞ e − s t ∫ − ∞ ∞ f ( u ) g ( t − u ) d u ⏟ ( f ∗ g ) ( t ) d t = ∫ − ∞ ∞ e − s t ( f ∗ g ) ( t ) d t . {\displaystyle {\begin{aligned}F(s)\cdot G(s)&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-st}\ f(u)\ g(t-u)\ {\text{d}}u\ {\text{d}}t\\&=\int _{-\infty }^{\infty }e^{-st}\underbrace {\int _{-\infty }^{\infty }f(u)\ g(t-u)\ {\text{d}}u} _{(fg)(t)}\ {\text{d}}t\\&=\int _{-\infty }^{\infty }e^{-st}(fg)(t)\ {\text{d}}t.\end{aligned}}} Note that F ( s ) ⋅ G ( s ) {\displaystyle F(s)\cdot G(s)} is the bilateral Laplace transform of ( f ∗ g ) ( t ) {\displaystyle (fg)(t)} . A similar derivation can be done using the unilateral Laplace transform (one-sided Laplace transform). The convolution operation also describes the output (in terms of the input) of an important class of operations known as linear time-invariant (LTI). See LTI system theory for a derivation of convolution as the result of LTI constraints. In terms of the Fourier transforms of the input and output of an LTI operation, no new frequency components are created. The existing ones are only modified (amplitude and/or phase). In other words, the output transform is the pointwise product of the input transform with a third transform (known as a transfer function). See Convolution theorem for a derivation of that property of convolution. Conversely, convolution can be derived as the inverse Fourier transform of the pointwise product of two Fourier transforms. == Visual explanation == == Historical developments == One of the earliest uses of the convolution integral appeared in D'Alembert's derivation of Taylor's theorem in Recherches sur différents points importants du système du monde, published in 1754. Also, an expression of the type: ∫ f ( u ) ⋅ g ( x − u ) d u {\displaystyle \int f(u)\cdot g(x-u)\,du} is used by Sylvestre François Lacroix on page 505 of his book entitled Treatise on differences and series, which is the last of 3 volumes of the encyclopedic series: Traité du calcul différentiel et du calcul intégral, Chez Courcier, Paris, 1797–1800. Soon thereafter, convolution operations appear in the works of Pierre Simon Laplace, Jean-Baptiste Joseph Fourier, Siméon Denis Poisson, and others. The term itself did not come into wide use until the 1950s or 1960s. 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