Fuzzy mathematics is a branch of mathematics that extends classical set theory and logic to model reasoning under uncertainty. Initiated by Lotfi Asker Zadeh in 1965 with the introduction of fuzzy sets, the field has since evolved to include fuzzy set theory, fuzzy logic, and various fuzzy analogues of traditional mathematic structures. Unlike classical mathematics, which usually relies on binary membership (an element either belongs to a set or it does not), fuzzy mathematics allows elements to partially belong to a set, with degrees of membership represented by values in the interval [0, 1]. This framework enables more flexible modeling of imprecise or vague concepts. Fuzzy mathematics has found applications in numerous domains, including control theory, artificial intelligence, decision theory, pattern recognition, and linguistics, where the modeling of gradations and uncertainty is essential. == Definition == A fuzzy subset A of a set X is defined by a function A: X → L, where L is typically the interval [0, 1]. This function is called the membership function of the fuzzy subset and assigns to each element x in X a degree of membership A(x) in the fuzzy set A. In classical set theory, a subset of X can be represented by an indicator function (also known as a characteristic function), which maps elements to either 0 or 1, indicating non-membership or full membership, respectively. Fuzzy subsets generalize this concept by allowing any real value between 0 and 1, thereby enabling partial membership. More generally, the codomain L of the membership function can be replaced with any complete lattice, resulting in the broader framework of L-fuzzy sets. == Fuzzification == The development of fuzzification in mathematics can be broadly divided into three historical stages: Initial, straightforward fuzzifications (1960s–1970s), Expansion of generalization techniques (1980s), Standardization, axiomatization, and L-fuzzification (1990s). Fuzzification generally involves extending classical mathematical concepts from binary (crisp) logic, where membership is determined by characteristic functions, to fuzzy logic, where membership is expressed by values in the interval [0, 1] via membership functions. Let A and B be fuzzy subsets of a set X. The fuzzy versions of set-theoretic operations are commonly defined as: ( A ∩ B ) ( x ) = min ( A ( x ) , B ( x ) ) {\displaystyle (A\cap B)(x)=\min(A(x),B(x))} ( A ∪ B ) ( x ) = max ( A ( x ) , B ( x ) ) {\displaystyle (A\cup B)(x)=\max(A(x),B(x))} for all x ∈ X {\displaystyle x\in X} . These operations can be generalized using t-norms and t-conorms, respectively. For example, the minimum operation can be replaced by multiplication: ( A ∩ B ) ( x ) = A ( x ) ⋅ B ( x ) {\displaystyle (A\cap B)(x)=A(x)\cdot B(x)} Fuzzification of algebraic structures often relies on generalizing the closure property. Let ∗ {\displaystyle } be a binary operation on X, and let A be a fuzzy subset of X. Then A is said to satisfy fuzzy closure if: A ( x ∗ y ) ≥ min ( A ( x ) , A ( y ) ) {\displaystyle A(xy)\geq \min(A(x),A(y))} for all x , y ∈ X {\displaystyle x,y\in X} . If ( G , ∗ ) {\displaystyle (G,)} is a group, then a fuzzy subset A of G is a fuzzy subgroup if: A ( x ∗ y − 1 ) ≥ min ( A ( x ) , A ( y − 1 ) ) {\displaystyle A(xy^{-1})\geq \min(A(x),A(y^{-1}))} for all x , y ∈ G {\displaystyle x,y\in G} . Similar generalizations apply to relational properties. For example, for example, for fuzzification of the transitivity property, a fuzzy relation R {\displaystyle R} on X {\displaystyle X} (i.e., a fuzzy subset of X × X {\displaystyle X\times X} ) is said to be fuzzy transitive if: R ( x , z ) ≥ min ( R ( x , y ) , R ( y , z ) ) {\displaystyle R(x,z)\geq \min(R(x,y),R(y,z))} for all x , y , z ∈ X {\displaystyle x,y,z\in X} . == Fuzzy analogues == Fuzzy subgroupoids and fuzzy subgroups were introduced in 1971 by A. Rosenfeld. Analogues of other mathematical subjects have been translated to fuzzy mathematics, such as fuzzy field theory and fuzzy Galois theory, fuzzy topology, fuzzy geometry, fuzzy orderings, and fuzzy graphs.
Symbol level
In knowledge-based systems, agents choose actions based on the principle of rationality to move closer to a desired goal. The agent is able to make decisions based on knowledge it has about the world (see knowledge level). But for the agent to actually change its state, it must use whatever means it has available. This level of description for the agent's behavior is the symbol level. The term was coined by Allen Newell in 1982. For example, in a computer program, the knowledge level consists of the information contained in its data structures that it uses to perform certain actions. The symbol level consists of the program's algorithms, the data structures themselves, and so on.
Communications system
A communications system is a collection of individual telecommunications networks systems, relay stations, tributary stations, and terminal equipment usually capable of interconnection and interoperation to form an integrated whole. Communication systems allow the transfer of information from one place to another or from one device to another through a specified channel or medium. The components of a communications system serve a common purpose, are technically compatible, use common procedures, respond to controls, and operate in union. In the structure of a communication system, the transmitter first converts the data received from the source into a light signal and transmits it through the medium to the destination of the receiver. The receiver connected at the receiving end converts it to digital data, maintaining certain protocols e.g. FTP, ISP assigned protocols etc. Telecommunications is a method of communication (e.g., for sports broadcasting, mass media, journalism, etc.). Communication is the act of conveying intended meanings from one entity or group to another through the use of mutually understood signs and semiotic rules. == Types == === By media === An optical communication system is any form of communications system that uses light as the transmission medium. Equipment consists of a transmitter, which encodes a message into an optical signal, a communication channel, which carries the signal to its destination, and a receiver, which reproduces the message from the received optical signal. Fiber-optic communication systems transmit information from one place to another by sending light through an optical fiber. The light forms a carrier signal that is modulated to carry information. A radio communication system is composed of several communications subsystems that give exterior communications capabilities. A radio communication system comprises a transmitting conductor in which electrical oscillations or currents are produced and which is arranged to cause such currents or oscillations to be propagated through the free space medium from one point to another remote therefrom and a receiving conductor at such distant point adapted to be excited by the oscillations or currents propagated from the transmitter. Power-line communication systems operate by impressing a modulated carrier signal on power wires. Different types of power-line communications use different frequency bands, depending on the signal transmission characteristics of the power wiring used. Since the power wiring system was originally intended for transmission of AC power, the power wire circuits have only a limited ability to carry higher frequencies. The propagation problem is a limiting factor for each type of power line communications. === By technology === A duplex communication system is a system composed of two connected parties or devices which can communicate with one another in both directions. The term duplex is used when describing communication between two parties or devices. Duplex systems are employed in nearly all communications networks, either to allow for a communication "two-way street" between two connected parties or to provide a "reverse path" for the monitoring and remote adjustment of equipment in the field. An antenna is basically a small length of a conductor that is used to radiate or receive electromagnetic waves. It acts as a conversion device. At the transmitting end it converts high frequency current into electromagnetic waves. At the receiving end it transforms electromagnetic waves into electrical signals that is fed into the input of the receiver. several types of antenna are used in communication. Examples of communications subsystems include the Defense Communications System (DCS). === Examples: by technology === Telephone Mobile phone Tablet computer Television Telegraph Edison Telegraph TV cable Computer === By application area === The term transmission system is used in the telecommunications industry to emphasize the intermediate media, protocols, and equipment in the circuit, rather than particular end-user applications. A tactical communications system is a communications system that (a) is used within, or in direct support of tactical forces (b) is designed to meet the requirements of changing tactical situations and varying environmental conditions, (c) provides securable communications, such as voice, data, and video, among mobile users to facilitate command and control within, and in support of, tactical forces, and (d) usually requires extremely short installation times, usually on the order of hours, in order to meet the requirements of frequent relocation. An Emergency communication system is any system (typically computer based) that is organized for the primary purpose of supporting the two way communication of emergency messages between both individuals and groups of individuals. These systems are commonly designed to integrate the cross-communication of messages between are variety of communication technologies. An Automatic call distributor (ACD) is a communication system that automatically queues, assigns and connects callers to handlers. This is used often in customer service (such as for product or service complaints), ordering by telephone (such as in a ticket office), or coordination services (such as in air traffic control). A Voice Communication Control System (VCCS) is essentially an ACD with characteristics that make it more adapted to use in critical situations (no waiting for dial tone, or lengthy recorded announcements, radio and telephone lines equally easily connected to, individual lines immediately accessible etc..) == Key components == =
Pridgen v University of Calgary
Pridgen v University of Calgary was freedom of speech case which took place in Alberta, Canada, in 2010. The case deals with two university students, Keith and Steven Pridgen, who were found guilty and punished by the University of Calgary in 2008, on grounds of "non-academic misconduct". The University of Calgary defines "non-academic misconduct" as:(a) conduct which causes injury to a person and/or damage to University property and/or the property of any member of the University community; (b) unauthorized removal and/or unauthorized possession of University property; and (c) conduct which seriously disrupts the lawful educational and related activities of other students and/or University staff.The Court of the Queen's Bench of Alberta found the University of Calgary to be wrong in prosecuting ten students, including the Pridgen brothers, in regards to comments made about a professor on Facebook. The key ruling in this case was that the universities are not exempt from, and that these students were in fact protected under, section 2(b) of the Charter of Rights and Freedoms. This case is notable as it highlights the jurisdiction of the Charter in terms of both new media technologies and university institutions in Canada. == Background == Keith and Steven Pridgen were undergraduate students at the University of Calgary in 2008. The twin brothers shared a Law and Society class being taught by Aruna Mitra. Professor Mitra was teaching this class for the first time in her career, and many of the students were very critical of her knowledge of the course. A Facebook page entitled “I NO Longer Fear Hell, I Took a Course with Aruna Mitra” was created, and many students began posting comments. In particular, Steven Pridgen's comment on November 13, 2007, read: “Somehow I think she just got lazy and gave everybody a 65....that's what I got. Does anybody know how to apply to have it remarked?” Many students had similar concerns to Pridgen's and after having their work re-marked, a number of them did in fact receive higher grades. Keith Pridgen also commented on August 26, 2008: “Hey fellow LWSO. Homees.. So I am quite sure Mitra is NO LONGER TEACHING ANY COURSES WITH THE U OF C !!!!! Remember when she told us she was a long-term professor? Well, Actually she was only sessional and picked up our class at the last moment because another prof wasn't able to do it ...lucky us. Well, anyways I think we should all congratulate ourselves for leaving a Mitra-free legacy for future students!” On September 4, 2008, Aruna Mitra complained about the Facebook page to the Interim Dean of the Faculty of Communication and Culture at the University of Calgary. Dean Tettey called a meeting for the ten students who posted material about Mitra on the Facebook page. The meeting took place on September 18, 2008, and included four professors from the department as well as the Dean. At this meeting, all ten students, including the Pridgen brothers, were found guilty of non-academic misconduct. On November 20, 2008, the Appellant's received a letter from Dean Tettey advising them that their comments “clearly caused unwarranted professional and personal injury to Prof. Mitra and clearly meets the criteria for non-academic misconduct as outlined in the University of Calgary Calendar”. Keith Pridgen was put on probation for 24 months, and both brothers were required to write a letter of apology to Prof. Mitra and refrain from posting or circulating defamatory material regarding any faculty members of the University of Calgary. The Pridgen brothers appealed the decision to the University of Calgary Review Committee and later to the Board of Governors of the University of Calgary however neither of these attempts succeeded in having the decision overturned. == Opinion of the Court == Eight main issues to be determined were laid out by the Honourable Madam Justice J. Strekaf: (a) Does the Charter apply to the disciplinary proceedings taken by the Respondent; (b) If, so were the Applicants' Charter rights infringed; (c) Were the actions taken by the University ultra vires the jurisdiction of the Province of Alberta; (d) Did the Board of Governors err in refusing to hear the Applicants appeals; (e) Were the Applicants' denied a fair hearing; (f) Did the Review Committee provide adequate reasons for its decisions; (g) Did the Review Committee err in concluding that the activities of the Applicants constituted non-academic misconduct; and (h) What, if any, remedy should be granted to the Applicants. The Court determined from previous cases that "a non-government entity may still be subject to the Charter of Rights and freedoms when implementing a specific government policy or program". Justice Strekaf distinguished that the University was acting as agent of the provincial government in providing accessible post-secondary education services to students in Alberta pursuant to the provisions of the PSL Act. Justice Strekaf felt there was sufficient evidence to show that universities in Alberta have some level of reliance on government funds and therefore they are not a "Charter free zone". Justice Strekaf concluded that comments made by Keith and Steven Pridgen, regarding Professor Mitra, on Facebook did not constitute academic misconduct and the Pridgen brothers' right to freedom of expression, under section 2(b) of the Charter, was infringed by the University of Calgary Review Committee.
Elonis v. United States
Elonis v. United States, 575 U.S. 723 (2015), was a United States Supreme Court case concerning whether conviction of threatening another person over interstate lines (under 18 U.S.C. § 875(c)) requires proof of subjective intent to threaten or whether it is enough to show that a "reasonable person" would regard the statement as threatening. In controversy were the purported threats of violent rap lyrics written by Anthony Douglas Elonis and posted to Facebook under a pseudonym. The ACLU filed an amicus brief in support of the petitioner. It was the first time the Court has heard a case considering true threats and the limits of speech on social media. == Background == In May 2010, Elonis was in the process of divorce and made a number of public Facebook posts. Prior to his postings, he had lost his job at an amusement park. He "posted the script of a sketch" by The Whitest Kids U' Know, which originally referenced saying "I want to kill the President of the United States" and replaced the president with his wife: Elonis ended the post with this statement: "Art is about pushing limits. I'm willing to go to jail for my constitutional rights. Are you?" A week later, Elonis posted about local law enforcement and a kindergarten class, which caught the attention of the Federal Bureau of Investigation. Then, he wrote a post on Facebook about one of the agents who visited him: He concluded: == Arrest and Conviction == These actions led to Elonis's arrest on December 8, 2010. He was indicted by a grand jury on five counts of threats to his estranged ex-wife, park employees and visitors, local law enforcement, an FBI agent, and a kindergarten class that had been relayed through interstate communication. At the district court, Elonis moved to dismiss the indictment for failing to allege that he had intended to threaten anyone, claiming his Facebook post was not were not intended as a threat. He argued that, as an aspiring rap artist, his posts were intended to be a form of artistic expression to help him cope with his recent loses. According to him, he did not mean anything said in his posts in a literal sense. His motion was denied. He requested a jury instruction that "the government must prove that he intended to communicate a true threat", which was also denied. He was convicted on the last four of the five counts, and was sentenced to 44 months in prison and three years on supervised release. He appealed unsuccessfully to the Third Circuit, renewing his challenge to the jury instructions. He then appealed to the U.S. Supreme Court based on lack of any attempt to show intent to threaten and on First Amendment rights. == Decision == On June 1, 2015, the U.S. Supreme Court reversed Elonis's conviction in an 8–1 decision. Chief Justice John Roberts wrote for a seven-justice majority, Samuel Alito authored an opinion concurring in part and dissenting in part, and Clarence Thomas authored a dissenting opinion. The finding of the circuit court was reversed and the matter remanded. === Majority opinion === The majority opinion, written by Roberts, did not rule on First Amendment matters or on the question of whether recklessness was sufficient mens rea to show intent. It ruled that mens rea was required to prove the commission of a crime under §875(c). Importantly, the mens rea issue had been preserved for review, since Elonis had raised that objection at every stage of the previous proceedings. The government contended that the presence of the words "intent to extort" in §875(b) and §875(d) implied that the absence in §875(c) was constructive. The court disagreed, holding that the absence of the language in §875(c) was because the section was intended to have a broader scope than threats relating to extortion. The opinion drew on many Supreme Court cases holding that in criminal law, mens rea was required though it had not been mentioned explicitly in statute. Consequently, the Supreme Court ruled in favor of Elonis. === Alito's concurrence === Justice Samuel Alito, concurring in part and dissenting in part, opined that while agreeing that mens rea was required and specifically that showing negligence was not sufficient, the court should have ruled on the question of recklessness. He further opined that recklessness was sufficient to show a crime under that provision on the basis that going further would amount to amending the statute, rather than interpreting it. Since Elonis explicitly argued that recklessness was not sufficient, Alito said: I would therefore remand for the Third Circuit to determine if Elonis’s failure (indeed, refusal) to argue for recklessness prevents reversal of his conviction. The Third Circuit should also have the opportunity to consider whether the conviction could be upheld on harmless error grounds. Alito also addressed the First Amendment question, elided by the majority opinion. He held that "lyrics in songs that are performed for an audience or sold in recorded form are unlikely to be interpreted as a real threat to a real person. ... Statements on social media that are pointedly directed at their victims, by contrast, are much more likely to be taken seriously." === Thomas's dissent === Justice Clarence Thomas, dissenting, wrote against discarding the "general intent" standard without replacing it with a clearer standard. Thomas argued that "there is no historical practice requiring more than general intent when a statute regulates speech." Thomas cited Rosen v. United States, arguing that general intent was sufficient in this case. However, the majority opinion offers refutation in that Rosen turned on ignorance of the law: knowledge as to whether material was legally obscene, not on whether it was intended to be obscene. Thomas also supported the government's claim that the presence of "intent to extort" language in the adjacent §875(b) and did not address the majority's reasoning on that language. Thomas used precedent, notably from the states and 18th-century England based on other but similar and, arguably, influencing legislation to support his "general intent" claim. Thomas also drew a parallel with general intent in tort. While he sought to address the First Amendment issues, he never strayed far from "general intent". == Aftermath == On remand, the Third Circuit reaffirmed the conviction "concluding beyond a reasonable doubt that Elonis would have been convicted if the jury had been properly instructed" and therefore was harmless error. In 2022, Elonis was once again arrested and indicted on three counts of cyberstalking involving three people. It was discovered that between 2018 and 2021, Elonis had sent numerous threatening messages over email, text, voice mail, and social media platforms like Twitter to a former prosecutor of the Eastern District of Pennsylvania, his ex-girlfriend, and ex-wife. On August 5, after a five-day trial, Elonis was found guilty on all three counts, and on March 23, 2023, he was sentenced by U.S. District Court Judge Edward G. Smith of Easton, Pennsylvania to twelve years and seven months in prison.
Digital Darkroom
Digital Darkroom was a graphics program for editing gray-scale photos, published by Silicon Beach Software for the Macintosh in 1987. It was programmed by Ed Bomke and Don Cone. Digital Darkroom was the first Macintosh program to incorporate a plug-in architecture. Silicon Beach and Ed Bomke are credited with having coined the term "plug-in". Another innovation of Digital Darkroom was the Magic Wand tool, which also appeared later in Photoshop. When Silicon Beach Software was acquired by Aldus Corporation, Digital Darkroom continued to be published by the Aldus Consumer Division, but was never updated to include color. The trademark "Digital Darkroom" was acquired by MicroFrontier in 1997 and used for a completely new image-editing program that does work with color. The software was acquired by Digimage Arts in 2002 and was sold for both Windows and Mac systems.
Fifth Estate
The Fifth Estate is a socio-cultural reference to groupings of outlier viewpoints in contemporary society, and is most associated with bloggers, journalists publishing in non-mainstream media outlets, and online social networks. The "Fifth" Estate extends the sequence of the three classical estates (clergy (first), nobility (second), commoners (third)) and the preceding Fourth Estate, essentially the common press. The use of "fifth estate" dates to the 1960s counterculture, and in particular the influential Fifth Estate, an underground newspaper first published in Detroit in 1965. Web-based technologies have enhanced the scope and power of the Fifth Estate far beyond the modest and boutique conditions of its beginnings. Nimmo and Combs asserted in 1992 that political pundits constitute a Fifth Estate. Media researcher Stephen D. Cooper argued in 2006 that bloggers are the Fifth Estate. In 2009, William Dutton argued that the Fifth Estate is not just the blogging community, nor an extension of the media, but "networked individuals" enabled by the Internet, e.g. social media, in ways that can hold the other estates accountable.