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In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements x1... xs in G such that every x in G can be written in the form x = n1x1 + n2x2 +... + nsxswith integers n1... ns. In this case, we say that the set is a generating set of G or that x1... xs generate G. Every finite abelian group is finitely generated; the finitely generated abelian groups can be classified. The integers, are a finitely generated abelian group; the integers modulo n, are a finite abelian group. Any direct sum of finitely many finitely generated abelian groups is again a finitely generated abelian group; every lattice forms. There are no other examples. In particular, the group of rational numbers is not finitely generated: if x 1, …, x n are rational numbers, pick a natural number k coprime to all the denominators; the group of non-zero rational numbers is not finitely generated. The groups of real numbers under addition and non-zero real numbers under multiplication are not finitely generated.

The fundamental theorem of finitely generated abelian groups can be stated two ways, generalizing the two forms of the fundamental theorem of finite abelian groups. The theorem, in both forms, in turn generalizes to the structure theorem for finitely generated modules over a principal ideal domain, which in turn admits further generalizations; the primary decomposition formulation states that every finitely generated abelian group G is isomorphic to a direct sum of primary cyclic groups and infinite cyclic groups. A primary cyclic group is one; that is, every finitely generated abelian group is isomorphic to a group of the form Z n ⊕ Z q 1 ⊕ ⋯ ⊕ Z q t, where n ≥ 0 is the rank, the numbers q1... qt are powers of prime numbers. In particular, G is finite if and only if n = 0; the values of n, q1... qt are uniquely determined by G, that is, there is one and only one way to represent G as such a decomposition. We can write any finitely generated abelian group G as a direct sum of the form Z n ⊕ Z k 1 ⊕ ⋯ ⊕ Z k u, where k1 divides k2, which divides k3 and so on up to ku.

Again, the rank n and the invariant factors k1... ku are uniquely determined by G. The rank and the sequence of invariant factors determine the group up to isomorphism; these statements are equivalent as a result of the Chinese remainder theorem, which implies that Z j k ≃ Z j ⊕ Z k if and only if j and k are coprime. The history and credit for the fundamental theorem is complicated by the fact that it was proven when group theory was not well-established, thus early forms, while the modern result and proof, are stated for a specific case. An early form of the finite case was proven in, the finite case was proven in, stated in group-theoretic terms in; the finitely presented case is solved by Smith normal form, hence credited to, though the finitely generated case is sometimes instead credited to. Group theorist László Fuchs states: As far as the fundamental theorem on finite abelian groups is concerned, it is not clear how far back in time one needs to go to trace its origin.... It took a long time to formulate and prove the fundamental theorem in its present form...

The fundamental theorem for finite abelian groups was proven by Leopold Kronecker in, using a group-theoretic proof, though without stating it in group-theoretic terms. This generalized an earlier result of Carl Friedrich Gauss from Disquisitiones Arithmeticae, which classified quadratic f

In probability theory and statistics, a complex random vector is a tuple of complex-valued random variables, is a random variable taking values in a vector space over the field of complex numbers. If Z 1, …, Z n are complex-valued random variables the n-tuple is a complex random vector. Complex random variables can always be considered as pairs of real random vectors: their real and imaginary parts; some concepts of real random vectors have a straightforward generalization to complex random vectors. For example, the definition of the mean of a complex random vector. Other concepts are unique to complex random vectors. Applications of complex random vectors are found in digital signal processing. A complex random vector Z = T on the probability space is a function Z: Ω → C n such that the vector T is a real real random vector on where ℜ denotes the real part of z and ℑ denotes the imaginary part of z; the generalization of the cumulative distribution function from real to complex random variables is not obvious because expressions of the form P make no sense.

However expressions of the form P make sense. Therefore, the cumulative distribution function F Z: C n ↦ of a random vector Z = T is defined as where z = T; as in the real case the expectation of a complex random vector is taken component-wise. The covariance matrix K Z Z contains the covariances between all pairs of components; the covariance matrix of an n × 1 random vector is an n × n matrix whose th element is the covariance between the i th and the j th random variables. Unlike in the case of real random variables, the covariance between two random variables involves the complex conjugate of one of the two, thus the covariance matrix is a Hermitian matrix. K Z Z = [ E E ⋯ E

The Ned Brown Forest Preserve, popularly known as Busse Woods, adjoining Elk Grove Village and Schaumburg in Illinois, is a 3,700-acre unit of the Cook County Forest Preserve system. It is named after Edward "Ned" Eagle Brown. A section of the northeast quadrant of the forest preserve is the Busse Forest Nature Preserve, registered as a National Natural Landmark in February 1980. Busse Forest Preserve was named for Cook County Commissioner William Busse in 1949. Busse Woods, the heart of the forest preserve, is a mature Great Lakes hardwood forest. A 440-acre segment of the woods, the Busse Forest Nature Preserve, is listed as a national natural landmark as a surviving fragment of flatwoods, a type of damp-ground forest typical of level patches of ground in the Great Lakes region. Parcels of land with slow rates of precipitation runoff into adjacent wetlands and streams were to develop into flatwoods. A flatwoods forest is characterized by red maple, swamp white oak, black ash trees; the black ash trees of Busse Woods are threatened by the emerald ash borer, reported in Illinois for the first time in 2006.

Other parts of Busse Woods are better-drained and include species more typical of the forests of northern Illinois, such as the basswood, sugar maple, white oak, the latter species being the state tree of Illinois. There are 10.4 miles of paved bicycle trail, the Busse Woods Trail, through the forest preserve: a 7.3-mile loop and two spurs providing pedestrian and bicycle access to the preserve. In contrast to the natural area, the northwest and southwest quadrants of the preserve are dominated by Busse Lake, a 590-acre artificial reservoir that serves as a flood-control catchment for Salt Creek and by the tall skyscrapers of eastern Schaumburg; the Friends of Busse Woods, a non-governmental organization, began operations in 2008. It cooperates with the Forest Preserve District of Cook County to oversee stewardship partnering operations at Busse Woods. Partnership operations include invasive species management, trash removal, native plant reseeding and restocking; the operations are carried out by volunteer stewards.

Cook County Forest Preserves Sauk Trail Woods North Creek Woods Forest Preserve page Friends of Busse Woods Map of Park

The 66th Military Intelligence Brigade is a United States Army brigade, subordinate to United States Army Intelligence and Security Command and based at Wiesbaden Army Airfield, Germany. After years of history as a counter intelligence/intelligence group with headquarters in Munich and geographically dispersed detachments, it became a brigade on 16 October 1986, but was inactivated in July 1995. Reformed again as an intelligence group in 2002, it became a brigade again in 2008; the unit's mission is to provide intelligence support to U. S. Army Europe and U. S. Army Africa. Part of the 66th Military Intelligence Brigade supports near real-time missions for deployed soldiers such as operations in Afghanistan and Iraq. Members of the brigade provide mission support by utilizing databases running on computer clusters and communicate on encrypted networks, such as the NSA-certified TACLANE encrypted network; the 66th MIB includes the 2nd Military Intelligence Battalion. Soldiers of the 66th MIB can be individually attached to other U.

S. Army units in the course of their duties. Members are on duty at U. S. Air Force installations, such as RAF Mildenhall. One brigade soldier was killed in action near a Forward Operating Base in Afghanistan in 2010. Unit members analyze sources in, among other languages and Persian. Soldiers in the brigade ideally hold qualifications in military intelligence and counter-intelligence, depending on their specific roles; some hold military and/or civilian academic degrees. Entrance and intermediate training of military intelligence personnel is provided by the United States Army Intelligence Center at Fort Huachuca, Arizona. 66th Military Intelligence Brigade Headquarters and Headquarters Company 2nd Military Intelligence Battalion 24th Military Intelligence Battalion 323rd Military Intelligence Battalion, Fort Meade, Maryland Soldiers of 66th MI Brigade have been involved in various degrees at the detention facility of Abu Ghraib. The current head of the unit is COL Gregory L. Holden. Description On a silver gray hexagon, one point up, with a 1⁄8 inch oriental blue border 3 inches in height and 2 5⁄8 inches in width overall, an oriental blue hexagon bearing a yellow sphinx superimposed by a silver gray dagger hilted black.

Symbolism Oriental blue and silver gray, representing loyalty and determination, are the colors of the Military Intelligence branch. Yellow/gold symbolizes excellence; the hexagon borders reflect the numerical designation of the unit. The sphinx, a traditional military intelligence symbol, indicates observation and discreet silence; the unsheathed dagger reflects the aggressive and protective requirements and the element of physical danger inherent in the mission of the unit. Background The shoulder sleeve insignia was approved on 27 August 1987 for the 66th Military Intelligence Brigade, it was cancelled on 17 July 2002. The insignia was reinstated effective 18 June 2003 and redesignated as an exception to policy for the 66th Military Intelligence Group, with description and symbolism updated. Description A Gold color metal and enamel device 1 3⁄16 inches in width overall consisting of a hexagon composed of a chequy of Black and White sections, surmounted throughout by a smaller hexagon composed of a chequy of nine sections of Gold and Blue with the center square charged with a Gold sphinx head, facing to the right, all above a Gold scroll inscribed "HONOR VALOR AND SECURITY" in Blue letters.

Symbolism The black and white symbolize enlightenment and knowledge both day and night around the world. The chequy represents the unit's strategic capabilities in counterintelligence; the sphinx is a traditional intelligence symbol and indicates observation and discreet silence. The hexagon within a hexagon "6-6" further distinguishes the numerical designation of the organization. Background The distinctive unit insignia was approved for the 66th Military Intelligence Group on 16 July 1969, it was redesignated for the 66th Military Intelligence Brigade on 8 October 1986. The insignia was redesignated effective 16 October 2002, with the description updated, for the 66th Military Intelligence Group. Constituted 21 June 1944 in the Army of the United States as the 66th Counter Intelligence Corps Detachment. Activated 1 July 1944 at Camp Rucker, Alabama. Inactivated 12 November 1945 at Camp Kilmer, New Jersey. Activated 10 November 1949 in Germany. Allotted 20 September 1951 to the Regular Army. Reorganized and redesignated 20 December 1952 as the 66th Counter Intelligence Corps Group.