Variational message passing (VMP) is an approximate inference technique for continuous- or discrete-valued Bayesian networks, with conjugate-exponential parents, developed by John Winn. VMP was developed as a means of generalizing the approximate variational methods used by such techniques as latent Dirichlet allocation, and works by updating an approximate distribution at each node through messages in the node's Markov blanket. == Likelihood lower bound == Given some set of hidden variables H {\displaystyle H} and observed variables V {\displaystyle V} , the goal of approximate inference is to maximize a lower-bound on the probability that a graphical model is in the configuration V {\displaystyle V} . Over some probability distribution Q {\displaystyle Q} (to be defined later), ln P ( V ) = ∑ H Q ( H ) ln P ( H , V ) P ( H | V ) = ∑ H Q ( H ) [ ln P ( H , V ) Q ( H ) − ln P ( H | V ) Q ( H ) ] {\displaystyle \ln P(V)=\sum _{H}Q(H)\ln {\frac {P(H,V)}{P(H|V)}}=\sum _{H}Q(H){\Bigg [}\ln {\frac {P(H,V)}{Q(H)}}-\ln {\frac {P(H|V)}{Q(H)}}{\Bigg ]}} . So, if we define our lower bound to be L ( Q ) = ∑ H Q ( H ) ln P ( H , V ) Q ( H ) {\displaystyle L(Q)=\sum _{H}Q(H)\ln {\frac {P(H,V)}{Q(H)}}} , then the likelihood is simply this bound plus the relative entropy between P {\displaystyle P} and Q {\displaystyle Q} . Because the relative entropy is non-negative, the function L {\displaystyle L} defined above is indeed a lower bound of the log likelihood of our observation V {\displaystyle V} . The distribution Q {\displaystyle Q} will have a simpler character than that of P {\displaystyle P} because marginalizing over P {\displaystyle P} is intractable for all but the simplest of graphical models. In particular, VMP uses a factorized distribution Q ( H ) = ∏ i Q i ( H i ) , {\displaystyle Q(H)=\prod _{i}Q_{i}(H_{i}),} where H i {\displaystyle H_{i}} is a disjoint part of the graphical model. == Determining the update rule == The likelihood estimate needs to be as large as possible; because it's a lower bound, getting closer log P {\displaystyle \log P} improves the approximation of the log likelihood. By substituting in the factorized version of Q {\displaystyle Q} , L ( Q ) {\displaystyle L(Q)} , parameterized over the hidden nodes H i {\displaystyle H_{i}} as above, is simply the negative relative entropy between Q j {\displaystyle Q_{j}} and Q j ∗ {\displaystyle Q_{j}^{}} plus other terms independent of Q j {\displaystyle Q_{j}} if Q j ∗ {\displaystyle Q_{j}^{}} is defined as Q j ∗ ( H j ) = 1 Z e E − j { ln P ( H , V ) } {\displaystyle Q_{j}^{}(H_{j})={\frac {1}{Z}}e^{\mathbb {E} _{-j}\{\ln P(H,V)\}}} , where E − j { ln P ( H , V ) } {\displaystyle \mathbb {E} _{-j}\{\ln P(H,V)\}} is the expectation over all distributions Q i {\displaystyle Q_{i}} except Q j {\displaystyle Q_{j}} . Thus, if we set Q j {\displaystyle Q_{j}} to be Q j ∗ {\displaystyle Q_{j}^{}} , the bound L {\displaystyle L} is maximized. == Messages in variational message passing == Parents send their children the expectation of their sufficient statistic while children send their parents their natural parameter, which also requires messages to be sent from the co-parents of the node. == Relationship to exponential families == Because all nodes in VMP come from exponential families and all parents of nodes are conjugate to their children nodes, the expectation of the sufficient statistic can be computed from the normalization factor. == VMP algorithm == The algorithm begins by computing the expected value of the sufficient statistics for that vector. Then, until the likelihood converges to a stable value (this is usually accomplished by setting a small threshold value and running the algorithm until it increases by less than that threshold value), do the following at each node: Get all messages from parents. Get all messages from children (this might require the children to get messages from the co-parents). Compute the expected value of the nodes sufficient statistics. == Constraints == Because every child must be conjugate to its parent, this has limited the types of distributions that can be used in the model. For example, the parents of a Gaussian distribution must be a Gaussian distribution (corresponding to the Mean) and a gamma distribution (corresponding to the precision, or one over σ {\displaystyle \sigma } in more common parameterizations). Discrete variables can have Dirichlet parents, and Poisson and exponential nodes must have gamma parents. More recently, VMP has been extended to handle models that violate this conditional conjugacy constraint. == Literature == John Winn; Christopher M. Bishop (2005). "Variational Message Passing" (PDF). Journal of Machine Learning Research. 6: 661–694. ISSN 1533-7928. Wikidata Q139488859. Beal, M.J. (2003). Variational Algorithms for Approximate Bayesian Inference (PDF) (PhD). Gatsby Computational Neuroscience Unit, University College London. Archived from the original (PDF) on 2005-04-28. Retrieved 2007-02-15.
AI effect
The AI effect is a phenomenon in which advances in artificial intelligence lead to a redefinition of what is considered intelligence, such that capabilities achieved by AI systems are no longer regarded as examples of "real" intelligence. The concept has been used to describe both a cognitive tendency and a sociotechnical pattern, in which successful AI techniques are reclassified as routine computation or absorbed into other domains. Historian Pamela McCorduck described this as a recurring feature of AI research, noting in her 2004 book Machines Who Think that once a problem is solved, it is no longer considered evidence of intelligence. Researcher Rodney Brooks similarly observed in 2002 that once systems are understood, they are often regarded as "just computation". == Definition == The AI effect refers to a shift in how intelligence is defined as machines acquire new capabilities. Tasks such as playing chess, recognizing speech, or interpreting images were historically considered indicators of intelligence, but after successful automation they are often reclassified as routine computation. McCorduck described this as an "odd paradox", in which successful AI systems are assimilated into other domains, leaving AI researchers to focus on unsolved problems. The phenomenon is often interpreted as an instance of moving the goalposts. A commonly cited formulation is Tesler's theorem, often expressed as "AI is whatever hasn't been done yet". When problems are not fully formalised, they may be described using models involving human computation, such as human-assisted Turing machines. == Historical examples == === Game playing === Early AI systems capable of playing games such as checkers and chess were initially regarded as demonstrations of machine intelligence. As these systems improved and became better understood, their achievements were often reinterpreted as examples of computation rather than intelligence. The victory of IBM's Deep Blue over Garry Kasparov in 1997 is a frequently cited example. Critics argued that the system relied on brute-force methods rather than genuine understanding. === Pattern recognition === Technologies such as optical character recognition and speech recognition were once considered core problems in artificial intelligence. As these systems became reliable and widely deployed, they were increasingly treated as standard engineering solutions. === Integration into applications === Many techniques originally developed within AI research have been incorporated into broader technological systems, including marketing, automation, and software applications. Michael Swaine reported in 2007 that AI advances are often presented as developments in other fields. Marvin Minsky observed that successful AI innovations often evolve into separate disciplines. Nick Bostrom noted in 2006 that widely adopted technologies are often no longer labeled as AI. == Contemporary discussion == The AI effect continues to be discussed in the context of recent advances in machine learning, particularly large language models and other generative AI systems. As these systems have become more widely used, some researchers and commentators have noted that their capabilities are frequently described as statistical or mechanical once understood, rather than as intelligence. A 2016 survey of artificial intelligence also noted that AI systems are increasingly embedded in everyday applications, reinforcing earlier observations that successful AI technologies tend to become normalized and no longer identified as AI. At the same time, the widespread commercial use of artificial intelligence has led to greater visibility of the field, contrasting with earlier periods in which AI techniques were often present but unacknowledged. == Interpretations == === Cognitive bias === Some authors describe the AI effect as a cognitive bias in which expectations of intelligence shift as machines achieve new capabilities. === Sociotechnical perspective === Another interpretation emphasizes how technologies are reclassified over time as they become widespread and commercially successful. === Philosophical debate === Some philosophers argue that reclassification reflects genuine conceptual distinctions rather than bias. == Historical context == During periods such as the AI winter, researchers sometimes avoided the term "artificial intelligence" due to negative perceptions. In the 21st century, however, the term "AI" has become widely used in public discourse and marketing. == Broader implications == The AI effect has been linked to broader questions about human uniqueness and the nature of intelligence. Michael Kearns suggested that people may seek to preserve a special role for humans. Similar patterns have been observed in studies of animal cognition. Herbert A. Simon noted that artificial intelligence can provoke strong emotional reactions.
Link encryption
Link encryption is an approach to communications security that encrypts and decrypts all network traffic at each network routing point (e.g. network switch, or node through which it passes) until arrival at its final destination. This repeated decryption and encryption is necessary to allow the routing information contained in each transmission to be read and employed further to direct the transmission toward its destination, before which it is re-encrypted. This contrasts with end-to-end encryption where internal information, but not the header/routing information, is encrypted by the sender at the point of origin and only decrypted by the intended recipient. Link encryption offers two main advantages: encryption is automatic so there is less opportunity for human error. if the communications link operates continuously and carries an unvarying level of traffic, link encryption defeats traffic analysis. On the other hand, end-to-end encryption ensures only the intended recipient has access to the plaintext. Link encryption can be used with end-to-end systems by superencrypting the messages. Bulk encryption refers to encrypting a large number of circuits at once, after they have been multiplexed.
List of cryptosystems
A cryptosystem is a set of cryptographic algorithms that map ciphertexts and plaintexts to each other. == Private-key cryptosystems == Private-key cryptosystems use the same key for encryption and decryption. Caesar cipher Substitution cipher Enigma machine Data Encryption Standard Twofish Serpent Camellia Salsa20 ChaCha20 Blowfish CAST5 Kuznyechik RC4 3DES Skipjack Safer IDEA Advanced Encryption Standard, also known as AES and Rijndael. == Public-key cryptosystems == Public-key cryptosystems use a public key for encryption and a private key for decryption. Diffie–Hellman key exchange RSA encryption Rabin cryptosystem Schnorr signature ElGamal encryption Elliptic-curve cryptography Lattice-based cryptography McEliece cryptosystem Multivariate cryptography Isogeny-based cryptography
Trace zero cryptography
First proposed by Gerhard Frey in 1998, trace zero cryptography refers to the use of trace zero varieties (TZV) for cryptographic purpose. Trace zero varieties are subgroups of the divisor class group on a low genus hyperelliptic curve defined over a finite field. These groups can be used to establish asymmetric cryptography using the discrete logarithm problem as cryptographic primitive. Trace zero varieties feature a better scalar multiplication performance than elliptic curves. This allows fast arithmetic in these groups, which can speed up the calculations with a factor 3 compared with elliptic curves and hence speed up the cryptosystem. Another advantage is that for groups of cryptographically relevant size, the order of the group can simply be calculated using the characteristic polynomial of the Frobenius endomorphism. This is not the case, for example, in elliptic curve cryptography when the group of points of an elliptic curve over a prime field is used for cryptographic purpose. However, to represent an element of the trace zero variety more bits are needed compared with elements of elliptic or hyperelliptic curves. Another disadvantage is the fact that it is possible to reduce the security of the TZV of 1/6th of the bit length using cover attack. == Mathematical background == A hyperelliptic curve C of genus g over a prime field F q {\displaystyle \mathbb {F} _{q}} where q = pn (p prime) of odd characteristic is defined as C : y 2 + h ( x ) y = f ( x ) , {\displaystyle C:~y^{2}+h(x)y=f(x),} where f monic, deg(f) = 2g + 1 and deg(h) ≤ g. The curve has at least one F q {\displaystyle \mathbb {F} _{q}} -rational Weierstraßpoint. The Jacobian variety J C ( F q n ) {\displaystyle J_{C}(\mathbb {F} _{q^{n}})} of C is for all finite extension F q n {\displaystyle \mathbb {F} _{q^{n}}} isomorphic to the ideal class group Cl ( C / F q n ) {\displaystyle \operatorname {Cl} (C/\mathbb {F} _{q^{n}})} . With the Mumford's representation it is possible to represent the elements of J C ( F q n ) {\displaystyle J_{C}(\mathbb {F} _{q^{n}})} with a pair of polynomials [u, v], where u, v ∈ F q n [ x ] {\displaystyle \mathbb {F} _{q^{n}}[x]} . The Frobenius endomorphism σ is used on an element [u, v] of J C ( F q n ) {\displaystyle J_{C}(\mathbb {F} _{q^{n}})} to raise the power of each coefficient of that element to q: σ([u, v]) = [uq(x), vq(x)]. The characteristic polynomial of this endomorphism has the following form: χ ( T ) = T 2 g + a 1 T 2 g − 1 + ⋯ + a g T g + ⋯ + a 1 q g − 1 T + q g , {\displaystyle \chi (T)=T^{2g}+a_{1}T^{2g-1}+\cdots +a_{g}T^{g}+\cdots +a_{1}q^{g-1}T+q^{g},} where ai in Z {\displaystyle \mathbb {Z} } With the Hasse–Weil theorem it is possible to receive the group order of any extension field F q n {\displaystyle \mathbb {F} _{q^{n}}} by using the complex roots τi of χ(T): | J C ( F q n ) | = ∏ i = 1 2 g ( 1 − τ i n ) {\displaystyle |J_{C}(\mathbb {F} _{q^{n}})|=\prod _{i=1}^{2g}(1-\tau _{i}^{n})} Let D be an element of the J C ( F q n ) {\displaystyle J_{C}(\mathbb {F} _{q^{n}})} of C, then it is possible to define an endomorphism of J C ( F q n ) {\displaystyle J_{C}(\mathbb {F} _{q^{n}})} , the so-called trace of D: Tr ( D ) = ∑ i = 0 n − 1 σ i ( D ) = D + σ ( D ) + ⋯ + σ n − 1 ( D ) {\displaystyle \operatorname {Tr} (D)=\sum _{i=0}^{n-1}\sigma ^{i}(D)=D+\sigma (D)+\cdots +\sigma ^{n-1}(D)} Based on this endomorphism one can reduce the Jacobian variety to a subgroup G with the property, that every element is of trace zero: G = { D ∈ J C ( F q n ) | Tr ( D ) = 0 } , ( 0 neutral element in J C ( F q n ) {\displaystyle G=\{D\in J_{C}(\mathbb {F} _{q^{n}})~|~{\text{Tr}}(D)={\textbf {0}}\},~~~({\textbf {0}}{\text{ neutral element in }}J_{C}(\mathbb {F} _{q^{n}})} G is the kernel of the trace endomorphism and thus G is a group, the so-called trace zero (sub)variety (TZV) of J C ( F q n ) {\displaystyle J_{C}(\mathbb {F} _{q^{n}})} . The intersection of G and J C ( F q ) {\displaystyle J_{C}(\mathbb {F} _{q})} is produced by the n-torsion elements of J C ( F q ) {\displaystyle J_{C}(\mathbb {F} _{q})} . If the greatest common divisor gcd ( n , | J C ( F q ) | ) = 1 {\displaystyle \gcd(n,|J_{C}(\mathbb {F} _{q})|)=1} the intersection is empty and one can compute the group order of G: | G | = | J C ( F q n ) | | J C ( F q ) | = ∏ i = 1 2 g ( 1 − τ i n ) ∏ i = 1 2 g ( 1 − τ i ) {\displaystyle |G|={\dfrac {|J_{C}(\mathbb {F} _{q^{n}})|}{|J_{C}(\mathbb {F} _{q})|}}={\dfrac {\prod _{i=1}^{2g}(1-\tau _{i}^{n})}{\prod _{i=1}^{2g}(1-\tau _{i})}}} The actual group used in cryptographic applications is a subgroup G0 of G of a large prime order l. This group may be G itself. There exist three different cases of cryptographical relevance for TZV: g = 1, n = 3 g = 1, n = 5 g = 2, n = 3 == Arithmetic == The arithmetic used in the TZV group G0 based on the arithmetic for the whole group J C ( F q n ) {\displaystyle J_{C}(\mathbb {F} _{q^{n}})} , But it is possible to use the Frobenius endomorphism σ to speed up the scalar multiplication. This can be archived if G0 is generated by D of order l then σ(D) = sD, for some integers s. For the given cases of TZV s can be computed as follows, where ai come from the characteristic polynomial of the Frobenius endomorphism : For g = 1, n = 3: s = q − 1 1 − a 1 mod ℓ {\displaystyle s={\dfrac {q-1}{1-a_{1}}}{\bmod {\ell }}} For g = 1, n = 5: s = q 2 − q − a 1 2 q + a 1 q + 1 q − 2 a 1 q + a 1 3 − a 1 2 + a 1 − 1 mod ℓ {\displaystyle s={\dfrac {q^{2}-q-a_{1}^{2}q+a_{1}q+1}{q-2a_{1}q+a_{1}^{3}-a_{1}^{2}+a_{1}-1}}{\bmod {\ell }}} For g = 2, n = 3: s = − q 2 − a 2 + a 1 a 1 q − a 2 + 1 mod ℓ {\displaystyle s=-{\dfrac {q^{2}-a_{2}+a_{1}}{a_{1}q-a_{2}+1}}{\bmod {\ell }}} Knowing this, it is possible to replace any scalar multiplication mD (|m| ≤ l/2) with: m 0 D + m 1 σ ( D ) + ⋯ + m n − 1 σ n − 1 ( D ) , where m i = O ( ℓ 1 / ( n − 1 ) ) = O ( q g ) {\displaystyle m_{0}D+m_{1}\sigma (D)+\cdots +m_{n-1}\sigma ^{n-1}(D),~~~~{\text{where }}m_{i}=O(\ell ^{1/(n-1)})=O(q^{g})} With this trick the multiple scalar product can be reduced to about 1/(n − 1)th of doublings necessary for calculating mD, if the implied constants are small enough. == Security == The security of cryptographic systems based on trace zero subvarieties according to the results of the papers comparable to the security of hyper-elliptic curves of low genus g' over F p ′ {\displaystyle \mathbb {F} _{p'}} , where p' ~ (n − 1)(g/g' ) for |G| ~128 bits. For the cases where n = 3, g = 2 and n = 5, g = 1 it is possible to reduce the security for at most 6 bits, where |G| ~ 2256, because one can not be sure that G is contained in a Jacobian of a curve of genus 6. The security of curves of genus 4 for similar fields are far less secure. == Cover attack on a trace zero crypto-system == The attack published in shows, that the DLP in trace zero groups of genus 2 over finite fields of characteristic diverse than 2 or 3 and a field extension of degree 3 can be transformed into a DLP in a class group of degree 0 with genus of at most 6 over the base field. In this new class group the DLP can be attacked with the index calculus methods. This leads to a reduction of the bit length 1/6th.
Plotting algorithms for the Mandelbrot set
There are many programs and algorithms used to plot the Mandelbrot set and other fractals, some of which are described in fractal-generating software. These programs use a variety of algorithms to determine the color of individual pixels efficiently. == Escape time algorithm == The simplest algorithm for generating a representation of the Mandelbrot set is known as the "escape time" algorithm. A repeating calculation is performed for each x, y point in the plot area and based on the behavior of that calculation, a color is chosen for that pixel. === Unoptimized naïve escape time algorithm === In both the unoptimized and optimized escape time algorithms, the x and y locations of each point are used as starting values in a repeating, or iterating calculation (described in detail below). The result of each iteration is used as the starting values for the next. The values are checked during each iteration to see whether they have reached a critical "escape" condition, or "bailout". If that condition is reached, the calculation is stopped, the pixel is drawn, and the next x, y point is examined. For some starting values, escape occurs quickly, after only a small number of iterations. For starting values very close to but not in the set, it may take hundreds or thousands of iterations to escape. For values within the Mandelbrot set, escape will never occur. The programmer or user must choose how many iterations–or how much "depth"–they wish to examine. The higher the maximal number of iterations, the more detail and subtlety emerge in the final image, but the longer time it will take to calculate the fractal image. Escape conditions can be simple or complex. Because no complex number with a real or imaginary part greater than 2 can be part of the set, a common bailout is to escape when either coefficient exceeds 2. A more computationally complex method that detects escapes sooner, is to compute distance from the origin using the Pythagorean theorem, i.e., to determine the absolute value, or modulus, of the complex number. If this value exceeds 2, or equivalently, when the sum of the squares of the real and imaginary parts exceed 4, the point has reached escape. More computationally intensive rendering variations include the Buddhabrot method, which finds escaping points and plots their iterated coordinates. The color of each point represents how quickly the values reached the escape point. Often black is used to show values that fail to escape before the iteration limit, and gradually brighter colors are used for points that escape. This gives a visual representation of how many cycles were required before reaching the escape condition. To render such an image, the region of the complex plane we are considering is subdivided into a certain number of pixels. To color any such pixel, let c {\displaystyle c} be the midpoint of that pixel. We now iterate the critical point 0 under P c {\displaystyle P_{c}} , checking at each step whether the orbit point has modulus larger than 2. When this is the case, we know that c {\displaystyle c} does not belong to the Mandelbrot set, and we color our pixel according to the number of iterations used to find out. Otherwise, we keep iterating up to a fixed number of steps, after which we decide that our parameter is "probably" in the Mandelbrot set, or at least very close to it, and color the pixel black. In pseudocode, this algorithm would look as follows. The algorithm does not use complex numbers and manually simulates complex-number operations using two real numbers, for those who do not have a complex data type. The program may be simplified if the programming language includes complex-data-type operations. for each pixel (Px, Py) on the screen do x0 := scaled x coordinate of pixel (scaled to lie in the Mandelbrot X scale (-2.00, 0.47)) y0 := scaled y coordinate of pixel (scaled to lie in the Mandelbrot Y scale (-1.12, 1.12)) x := 0.0 y := 0.0 iteration := 0 max_iteration := 1000 while (xx + yy ≤ 22 AND iteration < max_iteration) do xtemp := xx - yy + x0 y := 2xy + y0 x := xtemp iteration := iteration + 1 color := palette[iteration] plot(Px, Py, color) Here, relating the pseudocode to c {\displaystyle c} , z {\displaystyle z} and P c {\displaystyle P_{c}} : z = x + i y {\displaystyle z=x+iy\ } z 2 = x 2 + 2 i x y {\displaystyle z^{2}=x^{2}+2ixy} - y 2 {\displaystyle y^{2}\ } c = x 0 + i y 0 {\displaystyle c=x_{0}+iy_{0}\ } and so, as can be seen in the pseudocode in the computation of x and y: x = R e ( z 2 + c ) = x 2 − y 2 + x 0 {\displaystyle x=\mathop {\mathrm {Re} } (z^{2}+c)=x^{2}-y^{2}+x_{0}} and y = I m ( z 2 + c ) = 2 x y + y 0 . {\displaystyle y=\mathop {\mathrm {Im} } (z^{2}+c)=2xy+y_{0}.\ } To get colorful images of the set, the assignment of a color to each value of the number of executed iterations can be made using one of a variety of functions (linear, exponential, etc.). One practical way, without slowing down calculations, is to use the number of executed iterations as an entry to a palette initialized at startup. If the color table has, for instance, 500 entries, then the color selection is n mod 500, where n is the number of iterations. === Optimized escape time algorithms === The code in the previous section uses an unoptimized inner while loop for clarity. In the unoptimized version, one must perform five multiplications per iteration. To reduce the number of multiplications the following code for the inner while loop may be used instead: x2:= 0 y2:= 0 w:= 0 while (x2 + y2 ≤ 4 and iteration < max_iteration) do x:= x2 - y2 + x0 y:= w - x2 - y2 + y0 x2:= x x y2:= y y w:= (x + y) (x + y) iteration:= iteration + 1 The above code works via some algebraic simplification of the complex multiplication: ( i y + x ) 2 = − y 2 + 2 i y x + x 2 = x 2 − y 2 + 2 i y x {\displaystyle {\begin{aligned}(iy+x)^{2}&=-y^{2}+2iyx+x^{2}\\&=x^{2}-y^{2}+2iyx\end{aligned}}} Using the above identity, the number of multiplications can be reduced to three instead of five. The above inner while loop can be further optimized by expanding w to w = x 2 + 2 x y + y 2 {\displaystyle w=x^{2}+2xy+y^{2}} Substituting w into y = w − x 2 − y 2 + y 0 {\displaystyle y=w-x^{2}-y^{2}+y_{0}} yields y = 2 x y + y 0 {\displaystyle y=2xy+y_{0}} and hence calculating w is no longer needed. The further optimized pseudocode for the above is: x:= 0 y:= 0 x2:= 0 y2:= 0 while (x2 + y2 ≤ 4 and iteration < max_iteration) do x2:= x x y2:= y y y:= 2 x y + y0 x:= x2 - y2 + x0 iteration:= iteration + 1 Note that in the above pseudocode, 2 x y {\displaystyle 2xy} seems to increase the number of multiplications by 1, but since 2 is the multiplier the code can be optimized via ( x + x ) y {\displaystyle (x+x)y} . == Coloring algorithms == In addition to plotting the set, a variety of algorithms have been developed to efficiently color the set in an aesthetically pleasing way show structures of the data (scientific visualisation) === Histogram coloring === A more complex coloring method involves using a histogram which pairs each pixel with said pixel's maximum iteration count before escape/bailout. This method will equally distribute colors to the same overall area, and, importantly, is independent of the maximum number of iterations chosen. This algorithm has four passes. The first pass involves calculating the iteration counts associated with each pixel (but without any pixels being plotted). These are stored in an array IterationCounts[x][y], where x and y are the x and y coordinates of said pixel on the screen respectively. The first step of the second pass is to create an array NumIterationsPerPixel[n], where the array size n is the maximum iteration count. Next, one must iterate over the array of pixel-iteration count pairs IterationCounts[x][y], and retrieve each pixel's saved iteration count, i, via e.g. i = IterationCounts[x][y]. After each pixel's iteration count i is retrieved, it is necessary to index the NumIterationsPerPixel array at i and increment the indexed value (which is initially zero) -- e.g. NumIterationsPerPixel[i] = NumIterationsPerPixel[i] + 1. for (x = 0; x < width; x++) do for (y = 0; y < height; y++) do i:= IterationCounts[x][y] NumIterationsPerPixel[i]++ The third pass iterates through the NumIterationsPerPixel array and adds up all the stored values, saving them in total. The array index represents the number of pixels that reached that iteration count before bailout. total: = 0 for (i = 0; i < max_iterations; i++) do total += NumIterationsPerPixel[i] After this, the fourth pass begins and all the values in the IterationCounts array are indexed, and, for each iteration count i, associated with each pixel, the count is added to a global sum of all the iteration counts from 1 to i in the NumIterationsPerPixel array . This value is then normalized by dividing the sum by the total value computed earlier. hue[][]:= 0.0 for (x = 0; x < width; x++) do for (y = 0; y < height; y++) do iteration:= Iteration
Social commerce
Social commerce is a subset of electronic commerce that involves social media and online media that supports social interaction, and user contributions to assist online buying and selling of products and services. More succinctly, social commerce is the use of social network(s), and user-generated content in the context of e-commerce transactions. The term social commerce was introduced by Yahoo! in November 2005 which describes a set of online collaborative shopping tools such as shared pick lists, user ratings and other user-generated content of online product information and advice. The concept of social commerce was developed by David Beisel to denote user-generated advertorial content on e-commerce sites, and by Steve Rubel to include collaborative e-commerce tools that enable shoppers "to get advice from trusted individuals, find goods and services and then purchase them". The social networks that spread this advice have been found to increase the customer's trust in one retailer over another. Social commerce may assist companies in achieving the following purposes: Firstly, social commerce helps companies engage customers with their brands according to the customers' social behaviors. Secondly, it provides an incentive for customers to return to their website. Thirdly, it provides customers with a platform to talk about their brand on their website. Fourthly, it provides all the information customers need to research, compare, and ultimately choose you over your competitor, thus purchasing from you and not others. In these days, the range of social commerce has been expanded to include social media tools and content used in the context of e-commerce, especially in the fashion industry. Examples of social commerce include customer ratings and reviews, user recommendations and referrals, social shopping tools (sharing the act of shopping online), forums and communities, social media optimization, social applications and social advertising. Technologies such as augmented reality have also been integrated with social commerce, allowing shoppers to visualize apparel items on themselves and solicit feedback through social media tools. Some academics have sought to distinguish "social commerce" from "social shopping", with the former being referred to as collaborative networks of online vendors; the latter, the collaborative activity of online shoppers. == Timeline == 2005: The term "social commerce" was first introduced on Yahoo! in 2005. 2021: The Global Web Index associated one's use of social media to his/her eagerness to buy. Social media with its entertaining and inspirational content can increase a product's profitability. This explains why Instagram expanded its Checkout feature to similar content like IG Stories, IGTV, and Reels. == Elements == The attraction and effectiveness of Social Commerce can be understood in terms of Robert Cialdini's Principles of InfluenceInfluence: Science and Practice": Reciprocity – When a company gives a person something for free, that person will feel the need to return the favor, whether by buying again or giving good recommendations for the company. Community – When people find an individual or a group that shares the same values, likes, beliefs, etc., they find community. People are more committed to a community that they feel accepted within. When this commitment happens, they tend to follow the same trends as a group and when one member introduces a new idea or product, it is accepted more readily based on the previous trust that has been established. It would be beneficial for companies to develop partnerships with social media sites to engage social communities with their products. Social proof – To receive positive feedback, a company needs to be willing to accept social feedback and to show proof that other people are buying, and like, the same things that I like. This can be seen in a lot of online companies such as eBay and Amazon, that allow public feedback of products and when a purchase is made, they immediately generate a list showing purchases that other people have made in relation to my recent purchase. It is beneficial to encourage open recommendation and feedback. This creates trust for you as a seller. 55% of buyers turn to social media when they're looking for information. Authority – Many people need proof that a product is of good quality. This proof can be based on the recommendations of others who have bought the same product. If there are many user reviews about a product, then a consumer will be more willing to trust their own decision to buy this item. Liking – People trust based on the recommendations of others. If there are a lot of "likes" of a particular product, then the consumer will feel more confident and justified in making this purchase. Scarcity – As part of supply and demand, a greater value is assigned to products that are regarded as either being in high demand or are seen as being in a shortage. Therefore, if a person is convinced that they are purchasing something that is unique, special, or not easy to acquire, they will have more of a willingness to make a purchase. If there is trust established from the seller, they will want to buy these items immediately. This can be seen in the cases of Zara and Apple Inc. who create demand for their products by convincing the public that there is a possibility of missing out on being able to purchase them. == Types == === Onsite === Onsite social commerce refers to retailers including social sharing and other social functionality on their website. Some notable examples include Zazzle which enables users to share their purchases, Macy's which allows users to create a poll to find the right product, and Fab.com which shows a live feed of what other shoppers are buying. Onsite user reviews are also considered a part of social commerce. This approach has been successful in improving customer engagement, conversion and word-of-mouth branding according to several industry sources. === Offsite === Offsite social commerce includes activities that happen outside of the retailers' website. This may include posting products on social networks such as Facebook, X, and TikTok. It may also include advertising on shopping forums such as SlickDeals, Red Flag Deals, and LatestDeals.co.uk. == Measurements == Social commerce can be measured by any of the principle ways to measure social media. Return on Investment: measures the effect or action of social media on sales. Reputation: indices measure the influence of social media investment in terms of changes to online reputation – made up of the volume and valence of social media mentions. Reach: metrics use traditional media advertising metrics to measure the exposure rates and levels of an audience with social media. == Business applications == This category is based on individuals' shopping, selling, recommending behaviors. Social network-driven sales (Soldsie) – Facebook commerce and Twitter commerce belong to this part. Sales take place on established social network sites. Peer-to-peer sales platforms (eBay, Etsy, Amazon) – In these websites, users can directly communicate and sell products to other users. Group buying (Groupon, LivingSocial) – Users can buy products or services at a lower price when enough users agree to make this purchase. Peer recommendations and reviews (Amazon, Yelp, Bazaarvoice) – Users can see recommendations and reviews from other users. User-curated shopping (The Fancy, Lyst) – Users create and share lists of products and services for others to shop from. Participatory commerce (Betabrand, Threadless, Kickstarter) – Users can get involved in the production process. Social shopping (Squadded) – Allowing e-commerce to provide their users live chat sessions and shared shopping lists so they can communicate with their friends or other shoppers for advice. == Business examples == Here are some notable business examples of Social Commerce: Betabrand: an online brand using participatory design to release new, community-created ideas every week. Cafepress: an online retailer of stock and user-customized on demand products. Etsy: an e-commerce website focused on handmade or vintage items and supplies, as well as unique factory-manufactured items under Etsy's new guidelines. Eventbrite: an online ticketing service that allows event organizers to plan, set up ticket sales and promote events (event management) and publish them across Facebook, Twitter and other social-networking tools directly from the site's interface. Groupon: a deal-of-the-day website that features discounted gift certificates usable at local or national companies. Houzz: a web site and online community about architecture, interior design and decorating, landscape design and home improvement. LivingSocial: an online marketplace that allows clients to buy and share things to do in their city. Lockerz: an international social commerce website based in Seattle, Washington. OpenSky: is a r