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"Everglades pygmy sunfish, Elassoma evergladei Elassoma okefenokee Elassoma is a genus of freshwater fish, the only member of subfamily Elassomatinae of the sunfish family Centrarchidae in the order Perciformes. It is sometimes considered to be a family, the Elassomatidae, in a monotypic suborder, the Elassomatoidei of the Perciformes. The type species is E. zonatum, the banded pygmy sunfish. The Elassomatinae are known collectively as pygmy sunfishes, but are considered by some authorities not to be true sunfishes, which are members of family Centrarchidae. Some researchers believe they are related to sticklebacks and pipefishes (order Syngnathiformes) rather than Perciformes, though genetic research strongly implies a close relationship with the Centarchids. http://www.nanfa.org/ac/AC2012vol37no4_Elassoma%20zonatum%20E_okejenokee%20and%20E_evergladei- habitats%20and%20comparative%20observations_Bohlen_Nolte.pdf Currently the Integrated Taxonimic Information System classisfies them as belonging to the family Elassomatidae rather than Centrarchidae. https://www.itis.gov/servlet/SingleRpt/SingleRpt?search_topic=TSN&search;_value=202000#null The pygmy sunfishes grow to a maximum overall length of . They occur mostly in temperate and subtropical swamps, marshes, and other shallow, slow-moving, and heavily vegetated waters, across an area of the American South stretching from the Coastal Plain of North Carolina to central Florida, west along the Gulf Coast to eastern Texas, and north up the Mississippi River Valley to southern Illinois. The bluebarred, Carolina, and spring pygmy sunfishes have small localized populations and are considered Vulnerable. The pygmy sunfishes are too small to be game fish, but are relatively popular as aquarium fish because of the males' iridescent colors and fascinating breeding behaviors. Eggs are laid on or beneath dense vegetation, and the male guards the nest area until the fry hatch and scatter. They adapt well to small aquaria and are relatively adaptable to a range of conditions, but seldom take conventional prepared fish foods, instead requiring small live worms, insects, or crustaceans as food. Etymology The generic name Elassoma derives from the Greek ελάσσων (elasson) meaning smaller and σώμα (soma) meaning body, in reference to the fishes' diminutive size compared to the typical sunfishes. Species The currently recognized species in this genus are: * Elassoma alabamae Mayden, 1993 (spring pygmy sunfish) * Elassoma boehlkei Rohde & R. G. Arndt, 1987 (Carolina pygmy sunfish) * Elassoma evergladei D. S. Jordan, 1884 (Everglades pygmy sunfish) * Elassoma gilberti Snelson, Krabbenhoft & Quattro, 2009 (Gulf Coast pygmy sunfish) * Elassoma okatie Rohde & R. G. Arndt, 1987 (bluebarred pygmy sunfish) * Elassoma okefenokee J. E. Böhlke, 1956 (Okefenokee pygmy sunfish) * Elassoma zonatum D. S. Jordan, 1877 (banded pygmy sunfish) See also * List of fish families References *North American Native Fishes Association (NANFA) Elassoma forum * Duzen, Bill. "The Pygmy Sunfish." The Native Fish Conservancy Website. Elassomatoidei Centrarchidae Fauna of the Southeastern United States "
"Frederick Wallace Smith (born August 11, 1944) is the founder, chairman and CEO of FedEx. The company is headquartered in Smith's hometown of Memphis, Tennessee. Early years Frederick Smith was born in Marks, Mississippi, the son of James Frederick "Fred" Smith, the founder of the Toddle House restaurant chain and the Smith Motor Coach Company (renamed the Dixie Greyhound Lines after The Greyhound Corporation bought a controlling interest in 1931). The elder Smith died when his son was only 4, and the boy was raised by his mother and uncles. Smith was crippled by bone disease as a small boy but regained his health by age 10. He attended elementary school at Presbyterian Day School in Memphis and high school at Memphis University School, and became an amateur pilot as a teen. In 1962, Smith entered Yale University. While attending Yale, he wrote a paper for an economics class, outlining overnight delivery service . Additionally, Smith became a member and eventually the president of the Delta Kappa Epsilon (DKE) fraternity and the Skull and Bones secret society. "Frederick W. Smith." Contemporary Newsmakers 1985, Issue Cumulation. Gale Research, 1986. He received his bachelor's degree in economics in 1966. In his college years, he was a friend and DKE fraternity brother of future U.S. president George W. Bush. Smith was also friends with future U.S. Senator and Secretary of State John Kerry; the two shared an enthusiasm for aviation and were flying partners. Marine Corps service After graduation, Smith was commissioned in the U.S. Marine Corps, serving for three years (from 1966 to 1969) as a platoon leader and a forward air controller (FAC), flying in the back seat of the OV-10. He served two tours of duty in Vietnam and was honorably discharged in 1969 with the rank of Captain, having received the Silver Star, the Bronze Star, and two Purple Hearts . His Silver Star citation reads: > The President of the United States of America takes pleasure in presenting > the Silver Star to First Lieutenant Frederick Wallace Smith, United States > Marine Corps, for conspicuous gallantry and intrepidity in action while > serving as Commanding Officer of Company K, 3rd Battalion, 5th Marines, 1st > Marine Division in connection with operations against the enemy in the > Republic of Vietnam. On the morning of 27 May 1968, while conducting a > search and destroy operation, Company K became heavily engaged with a North > Vietnamese Army battalion occupying well-entrenched emplacements on Goi Noi > Island in Quang Nam Province. As Lieutenant Smith led his men in an > aggressive assault upon the enemy positions, the North Vietnamese force > launched a determined counterattack, supported by mortars, on the Marines' > left flank. Unhesitatingly rushing through the intense hostile fire to the > position of heaviest contact, Lieutenant Smith fearlessly removed several > casualties from the hazardous area and, shouting words of encouragement to > his men, directed their fire upon the advancing enemy soldiers, successfully > repulsing the hostile attack. Moving boldly across the fire-swept terrain to > an elevated area, he calmly disregarded repeated North Vietnamese attempts > to direct upon him as he skillfully adjusted artillery fire and air strikes > upon the hostile positions to within fifty meters of his own location and > continued to direct the movement of his unit. Accurately assessing the > confusion that supporting arms was causing among the enemy soldiers, he > raced across the fire-swept terrain to the right flank of his company and > led an enveloping attack on the hostile unit's weakest point, routing the > North Vietnamese unit and inflicting numerous casualties. His aggressive > tactics and calm presence of min [sic] under fire inspired all who observed > him and were instrumental in his unit accounting for the capture of two > hostile soldiers as well as numerous documents and valuable items of > equipment. By his courage, aggressive leadership and unfaltering devotion to > duty at great personal risk, Lieutenant Smith upheld the highest traditions > of the Marine Corps and of the United States Naval Service." Business career In 1970, Smith purchased the controlling interest in an aircraft maintenance company, Ark Aviation Sales, and by 1971 turned its focus to trading used jets. On June 18, 1971, Smith founded Federal Express with his $4 million inheritance (approximately $23 million in 2013 dollars),The Federal Reserve Bank of Minneapolis . Minneapolisfed.org. Retrieved on 2009-02-01. and raised $91 million (approximately $525 million in 2013 dollars) in venture capital. In 1973, the company began offering service to 25 cities, and it began with small packages and documents and a fleet of 14 Falcon 20 (DA-20) jets. His focus was on developing an integrated air-ground system. Smith developed FedEx on the business idea of a shipment version of a bank clearing house where one bank clearing house was located in the middle of the representative banks and all their representatives would be sent to the central location to exchange materials. In the early days of FedEx, Smith had to go to great lengths to keep the company afloat. In one instance, after a crucial business loan was denied, he took the company's last $5,000 to Las Vegas and won $27,000 gambling on blackjack to cover the company's $24,000 fuel bill. It kept FedEx alive for one more week. In addition to FedEx, Smith is also a minority owner of the Washington Football Team of the National Football League (NFL). His son, Arthur Smith is the offensive coordinator for the Tennessee Titans. This partnership resulted in FedEx sponsorship of the Joe Gibbs NASCAR racing team. Smith also owns or co-owns several other entertainment companies, such as Alcon Entertainment. In 2000, Smith made an appearance as himself in the Tom Hanks movie Cast Away, when Tom's character is welcomed back, which was filmed on location at FedEx's home facilities in Memphis, Tennessee. A DKE Fraternity Brother of George W. Bush while at Yale, after Bush's 2000 election, there was some speculation that Smith might be appointed to the Bush Cabinet as Defense Secretary. While Smith was Bush's first choice for the position, he declined for medical reasons -- Donald Rumsfeld was named instead. Although Smith was friends with both 2004 major candidates, John Kerry and George W. Bush, Smith chose to endorse Bush's re- election in 2004. When Bush decided to replace Rumsfeld, Smith was offered the position again, but he declined in order to spend time with his terminally ill daughter. Smith was a supporter of Senator John McCain's 2008 Presidential bid, and had been named McCain's National Co-Chairman of his campaign committee. Smith was inducted into the Junior Achievement U.S. Business Hall of Fame and also awarded the Golden Plate Award of the American Academy of Achievement in 1998. He was inducted into the SMEI Sales & Marketing Hall of Fame in 2000. His other awards include "CEO of the Year 2004" by Chief Executive Magazine and the 2008 Kellogg Award for Distinguished Leadership, presented by the Kellogg School of Management on May 29, 2008. He was also awarded the 2008 Bower Award for Business Leadership from The Franklin Institute in Philadelphia, Pennsylvania. He is the 2011 recipient of the Tony Jannus Award for distinguished contributions to commercial aviation. While CEO of FedEx in 2008, Frederick W. Smith earned a total compensation of $10,434,589, which included a base salary of $1,430,466, a cash bonus of $2,705,000, stocks granted of $0, and options granted of $5,461,575.2008 CEO Compensation for Frederick W. Smith , Equilar.com In June 2009, Smith expressed interest in purchasing the controlling share (60%) of the St. Louis Rams from Chip Rosenbloom and Lucia Rodriguez. In 2009, Frederick W. Smith earned a total compensation of $7,740,658, which included a base salary of $1,355,028, a cash bonus of $0, stocks granted of $0, options granted of $5,079,191, and other compensation totaling $1,306,439.2009 CEO Compensation for Frederick W. Smith , Equilar.com In March 2014, Fortune Magazine ranked him 26th among the list of "World's 50 Greatest Leaders" The World's 50 Greatest Leaders Personal life Smith has ten children, including photographer Windland Smith Rice and football coach Arthur Smith.https://eu.commercialappeal.com/story/entertainment/movies/2017/03/02/rachel- smith-follows-sisters-footsteps-movie-producer/98587216/ Notes References * *Profile in Fortune Magazine's Innovators Hall of Fame *Article by Smith on how Fedex came to be, includes the story of the paper he wrote while at Yale. *USA Today Q&A; on his love of history *Chief Executive Magazine Names Fred Smith 2004 CEO of the Year External links * *Interview *Frederick W. Smith at Redskins.com 1944 births Living people 20th-century American businesspeople 21st-century American businesspeople American billionaires American chief executives United States Marine Corps personnel of the Vietnam War Businesspeople from Tennessee FedEx Military personnel from Mississippi People from Memphis, Tennessee Recipients of the Silver Star United States Marine Corps officers United States Naval Aviators Yale University alumni People from Quitman County, Mississippi Council on Foreign Relations "
"In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof: * Biquaternions when the coefficients are complex numbers. * Split-biquaternions when the coefficients are split-complex numbers. * Dual quaternions when the coefficients are dual numbers. This article is about the ordinary biquaternions named by William Rowan Hamilton in 1844 (see Proceedings of the Royal Irish Academy 1844 & 1850 page 388Proceedings of the Royal Irish Academy November 1844 (NA) and 1850 page 388 from Google Books ). Some of the more prominent proponents of these biquaternions include Alexander Macfarlane, Arthur W. Conway, Ludwik Silberstein, and Cornelius Lanczos. As developed below, the unit quasi-sphere of the biquaternions provides a representation of the Lorentz group, which is the foundation of special relativity. The algebra of biquaternions can be considered as a tensor product (taken over the reals) where is the field of complex numbers and is the division algebra of (real) quaternions. In other words, the biquaternions are just the complexification of the quaternions. Viewed as a complex algebra, the biquaternions are isomorphic to the algebra of complex matrices . They are also isomorphic to several Clifford algebras including ,D. J. H. Garling (2011) Clifford Algebras: An Introduction, Cambridge University Press. the Pauli algebra ,Francis and Kosowsky (2005) The construction of spinors in geometric algebra. Annals of Physics, 317, 384—409. Article link and the even part of the spacetime algebra. Definition Let be the basis for the (real) quaternions , and let be complex numbers, then :q = u mathbf 1 + v mathbf i + w mathbf j + x mathbf k is a biquaternion.William Rowan Hamilton (1853) Lectures on Quaternions, Article 669. This historical mathematical text is available on- line courtesy of Cornell University To distinguish square roots of minus one in the biquaternions, HamiltonHamilton (1899) Elements of Quaternions, 2nd edition, page 289 and Arthur W. Conway used the convention of representing the square root of minus one in the scalar field ℂ by h to avoid confusion with the in the quaternion group. Commutativity of the scalar field with the quaternion group is assumed: : h mathbf i = mathbf i h, h mathbf j = mathbf j h, h mathbf k = mathbf k h . Hamilton introduced the terms bivector, biconjugate, bitensor, and biversor to extend notions used with real quaternions . Hamilton's primary exposition on biquaternions came in 1853 in his Lectures on Quaternions. The editions of Elements of Quaternions, in 1866 by William Edwin Hamilton (son of Rowan), and in 1899, 1901 by Charles Jasper Joly, reduced the biquaternion coverage in favor of the real quaternions. Considered with the operations of component-wise addition, and multiplication according to the quaternion group, this collection forms a 4-dimensional algebra over the complex numbers ℂ. The algebra of biquaternions is associative, but not commutative. A biquaternion is either a unit or a zero divisor. The algebra of biquaternions forms a composition algebra and can be constructed from bicomplex numbers. See ' below. Place in ring theory =Linear representation= Note the matrix product :begin{pmatrix}h & 00 & -hend{pmatrix}begin{pmatrix}0 & 1-1 & 0end{pmatrix} = begin{pmatrix}0 & hh & 0end{pmatrix}. Because h is the imaginary unit, each of these three arrays has a square equal to the negative of the identity matrix. When this matrix product is interpreted as i j = k, then one obtains a subgroup of matrices that is isomorphic to the quaternion group. Consequently, :begin{pmatrix}u+hv & w+hx-w+hx & u-hvend{pmatrix} represents biquaternion q = u 1 + v i + w j + x k. Given any 2 × 2 complex matrix, there are complex values u, v, w, and x to put it in this form so that the matrix ring M(2,C) is isomorphicLeonard Dickson (1914) Linear Algebras, §13 "Equivalence of the complex quaternion and matric algebras", page 13, via HathiTrust to the biquaternion ring. =Subalgebras= Considering the biquaternion algebra over the scalar field of real numbers , the set :{mathbf 1, h, mathbf i, hmathbf i, mathbf j, hmathbf j, mathbf k, hmathbf k } forms a basis so the algebra has eight real dimensions. The squares of the elements , and are all positive one, for example, . The subalgebra given by :lbrace x + y(hmathbf i) : x, y in mathbb R rbrace is ring isomorphic to the plane of split-complex numbers, which has an algebraic structure built upon the unit hyperbola. The elements and also determine such subalgebras. Furthermore, :lbrace x + ymathbf j : x,y in mathbb C rbrace is a subalgebra isomorphic to the tessarines. A third subalgebra called coquaternions is generated by and . It is seen that , and that the square of this element is . These elements generate the dihedral group of the square. The linear subspace with basis thus is closed under multiplication, and forms the coquaternion algebra. In the context of quantum mechanics and spinor algebra, the biquaternions , and (or their negatives), viewed in the representation, are called Pauli matrices. Algebraic properties The biquaternions have two conjugations: * the biconjugate or biscalar minus bivector is q^* = w - xmathbf i - ymathbf j - zmathbf k ! , and * the complex conjugation of biquaternion coefficients q^{star} = w^{star} + x^{star}mathbf i + y^{star}mathbf j + z^{star}mathbf k where z^{star} = a - bh when z = a + bh,quad a,b in mathbb R,quad h^2 = -mathbf 1. Note that (pq)^* = q^* p^*, quad (pq)^{star} = p^{star} q^{star} , quad (q^*)^{star} = (q^{star})^*. Clearly, if q q^* = 0 then is a zero divisor. Otherwise lbrace q q^* rbrace^{-mathbf 1} is defined over the complex numbers. Further, q q^* = q^* q is easily verified. This allows an inverse to be defined by * q^{-mathbf 1} = q^* lbrace q q^* rbrace^{-mathbf 1}, if qq^* eq 0. =Relation to Lorentz transformations= Consider now the linear subspace See equation 94.16, page 305. The following algebra compares to Lanczos, except he uses ~ to signify quaternion conjugation and * for complex conjugation :M = lbrace q : q^* = q^{star} rbrace = lbrace t + x(hmathbf i) + y(h mathbf j) + z(h mathbf k) : t, x, y, z in mathbb R rbrace . is not a subalgebra since it is not closed under products; for example (hmathbf i)(hmathbf j) = h^2 mathbf{ij} = -mathbf k otin M.. Indeed, cannot form an algebra if it is not even a magma. Proposition: If is in , then q q^* = t^2 - x^2 - y^2 - z^2. Proof: From the definitions, :begin{align}q q^* &= (t+xhmathbf i+yhmathbf j+zhmathbf k)(t-xhmathbf i-yhmathbf j-zhmathbf k) &= t^2 - x^2(hmathbf i)^2 - y^2(hmathbf j)^2 - z^2(hmathbf k)^2 = t^2 - x^2 - y^2 - z^2.end{align} Definition: Let biquaternion satisfy g g^* = mathbf 1. Then the Lorentz transformation associated with is given by :T(q) = g^* q g^{star}. Proposition: If is in , then is also in . Proof: (g^* q g^{star})^* = (g^{star})^* q^* g = (g^*)^{star} q^{star} g = (g^* q g^{star})^{star}. Proposition: quad T(q) (T(q))^* = q q^* Proof: Note first that means that the sum of the squares of its four complex components is one. Then the sum of the squares of the complex conjugates of these components is also one. Therefore, g^{star} (g^{star})^* = mathbf 1. Now :(g^* q g^{star})(g^* q g^{star})^* = g^* q g^{star} (g^{star})^* q^* g = g^* q q^* g = q q^*. Associated terminology As the biquaternions have been a fixture of linear algebra since the beginnings of mathematical physics, there is an array of concepts that are illustrated or represented by biquaternion algebra. The transformation group G = lbrace g : g g^* = 1 rbrace has two parts, G cap H and G cap M. The first part is characterized by g = g^{star} ; then the Lorentz transformation corresponding to is given by T(q) = g^{-1} q g since g^* = g^{-1}. Such a transformation is a rotation by quaternion multiplication, and the collection of them is O(3) cong G cap H . But this subgroup of is not a normal subgroup, so no quotient group can be formed. To view G cap M it is necessary to show some subalgebra structure in the biquaternions. Let represent an element of the sphere of square roots of minus one in the real quaternion subalgebra . Then and the plane of biquaternions given by D_r = lbrace z = x + yhr : x, y in mathbb R rbrace is a commutative subalgebra isomorphic to the plane of split-complex numbers. Just as the ordinary complex plane has a unit circle, D_r has a unit hyperbola given by :exp(ahr) = cosh(a) + hr sinh(a),quad a in R. Just as the unit circle turns by multiplication through one of its elements, so the hyperbola turns because exp(ahr) exp(bhr) = exp((a+b)hr). Hence these algebraic operators on the hyperbola are called hyperbolic versors. The unit circle in and unit hyperbola in are examples of one-parameter groups. For every square root of minus one in , there is a one-parameter group in the biquaternions given by G cap D_r. The space of biquaternions has a natural topology through the Euclidean metric on -space. With respect to this topology, is a topological group. Moreover, it has analytic structure making it a six- parameter Lie group. Consider the subspace of bivectors A = lbrace q : q^* = -q rbrace . Then the exponential map exp:A to G takes the real vectors to G cap H and the -vectors to G cap M. When equipped with the commutator, forms the Lie algebra of . Thus this study of a six-dimensional space serves to introduce the general concepts of Lie theory. When viewed in the matrix representation, is called the special linear group SL(2,C) in . Many of the concepts of special relativity are illustrated through the biquaternion structures laid out. The subspace corresponds to Minkowski space, with the four coordinates giving the time and space locations of events in a resting frame of reference. Any hyperbolic versor corresponds to a velocity in direction of speed where is the velocity of light. The inertial frame of reference of this velocity can be made the resting frame by applying the Lorentz boost given by since then g^{star} = exp(-0.5ahr) = g^* so that T(exp(ahr)) = 1 . Naturally the hyperboloid G cap M, which represents the range of velocities for sub-luminal motion, is of physical interest. There has been considerable work associating this "velocity space" with the hyperboloid model of hyperbolic geometry. In special relativity, the hyperbolic angle parameter of a hyperbolic versor is called rapidity. Thus we see the biquaternion group provides a group representation for the Lorentz group. After the introduction of spinor theory, particularly in the hands of Wolfgang Pauli and Élie Cartan, the biquaternion representation of the Lorentz group was superseded. The new methods were founded on basis vectors in the set :lbrace q : q q^* = 0 rbrace = lbrace w + xmathbf i + ymathbf j + zmathbf k : w^2 + x^2 + y^2 + z^2 = 0 rbrace which is called the complex light cone. The above representation of the Lorentz group coincides with what physicists refer to as four-vectors. Beyond four-vectors, the standard model of particle physics also includes other Lorentz representations, known as scalars, and the -representation associated with e.g. the electromagnetic field tensor. Furthermore, particle physics makes use of the representations (or projective representations of the Lorentz group) known as left- and right- handed Weyl spinors, Majorana spinors, and Dirac spinors. It is known that each of these seven representations can be constructed as invariant subspaces within the biquaternions. As a composition algebra Although W.R. Hamilton introduced biquaternions in the 19th century, its delineation of its mathematical structure as a special type of algebra over a field was accomplished in the 20th century: the biquaternions may be generated out of the bicomplex numbers in the same way that Adrian Albert generated the real quaternions out of complex numbers in the so-called Cayley–Dickson construction. In this construction, a bicomplex number (w,z) has conjugate (w,z)* = (w, – z). The biquaternion is then a pair of bicomplex numbers (a,b), where the product with a second biquaternion (c, d) is :(a,b)(c,d) = (a c - d^* b, d a + b c^* ). If a = (u, v), b = (w,z), then the biconjugate (a, b)^* = (a^*, -b). When (a,b)* is written as a 4-vector of ordinary complex numbers, :(u, v, w, z)^* = (u, -v, -w, -z). The biquaternions form an example of a quaternion algebra, and it has norm :N(u,v,w,z) = u^2 + v^2 + w^2 + z^2 . Two biquaternions p and q satisfy N(p q) = N(p) N(q) indicating that N is a quadratic form admitting composition, so that the biquaternions form a composition algebra. See also *Biquaternion algebra *Hypercomplex number *MacFarlane's use *Quotient ring Notes References * Arthur Buchheim (1885) "A Memoir on biquaternions", American Journal of Mathematics 7(4):293 to 326 from Jstor early content. *. * *Hamilton (1866) Elements of Quaternions University of Dublin Press. Edited by William Edwin Hamilton, son of the deceased author. *Hamilton (1899) Elements of Quaternions volume I, (1901) volume II. Edited by Charles Jasper Joly; published by Longmans, Green & Co.. *Kravchenko, Vladislav (2003), Applied Quaternionic Analysis, Heldermann Verlag . *. *. *. *. *. *. *. *. Composition algebras Quaternions Ring theory Special relativity Articles containing proofs William Rowan Hamilton de:Biquaternion#Hamilton Biquaternion "