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Direct sum of groups

In mathematics, a group G is called the direct sum of two subgroups H1 and H2 if each H1 and H2 are normal subgroups of G, the subgroups H1 and H2 have trivial intersection, G = <H1, H2>. More G is called the direct sum of a finite set of subgroups if each Hi is a normal subgroup of G, each Hi has trivial intersection with the subgroup <>, G = <>. If G is the direct sum of subgroups H and K we write G = H + K, if G is the direct sum of a set of subgroups we write G = ∑Hi. Loosely speaking, a direct sum is isomorphic to a weak direct product of subgroups. In abstract algebra, this method of construction can be generalized to direct sums of vector spaces and other structures; this direct sum is commutative up to isomorphism. That is, if G = H + K also G = K + H and thus H + K = K + H, it is associative in the sense that if G = H + K, K = L + M G = H + = H + L + M. A group which can be expressed as a direct sum of non-trivial subgroups is called decomposable, if a group cannot be expressed as such a direct sum it is called indecomposable.

If G = H + K it can be proven that: for all h in H, k in K, we have that h*k = k*h for all g in G, there exists unique h in H, k in K such that g = h*k There is a cancellation of the sum in a quotient. Since hi*hj = hj*hi for all i ≠ j, it follows that multiplication of elements in a direct sum is isomorphic to multiplication of the corresponding elements in the direct product. Given a group G, we say that a subgroup H is a direct summand of G if there exists another subgroup K of G such that G = H + K. In abelian groups, if H is a divisible subgroup of G H is a direct summand of G. If we take G = ∏ i ∈ I H i it is clear that G is the direct product of the subgroups H i 0 × ∏ i ≠ i 0 H i. If H is a divisible subgroup of an abelian group G there exists another subgroup K of G such that G = K + H. If G has a vector space structure G can be written as a direct sum of R and another subspace K that will be isomorphic to the quotient G / K. In the decomposition of a finite group into a direct sum of indecomposable subgroups the embedding of the subgroups is not unique.

For example, in the Klein group V 4 ≅ C 2 × C 2 we have that V 4 = ⟨ ⟩ + ⟨ ⟩, V 4 = ⟨ ⟩ + ⟨ ⟩. However, the Remak-Krull-Schmidt theorem states that given a finite group G = ∑Ai = ∑Bj, where each Ai and each Bj is non-trivial and indecomposable, the two sums have equal terms up to reordering and isomorphism; the Remak-Krull-Schmidt theorem fails for infinite groups. To describe the above properties in the case where G is the direct sum of an infinite set of subgroups, more care is needed. If g is an element of the cartesian product ∏ of a set of groups, let gi be the ith element of g in the product; the external direct sum of a set of groups is the subset of ∏, for each element g of ∑E, gi is the identity e H i for all but a finite number of gi (equivalently


Maglie is a town and comune in the province of Lecce in the Apulia region of south-east Italy. The Maglie area was settled as early as the Bronze Age and the early Iron Age, before, as testified by the presence of archaic dolmens and menhirs, by the Cattìe site, discovered in 1980, featuring 12,000 tools and 800 bone remains. Maglie a countryside casale, developed around the castle built in the 13th century under the Angevine kings of Naples and renewed by Andriolo Lubello, the local baron under king Alfonso I of Aragon. Duomo, it was built in the late 18th century on the site of two previous buildings tracing back to 14th and 16th century. Its bell tower, standing at about 48 metres, is the tallest in the province; the four upper storeys are attributed to Giuseppe Zimbalo. Church of Madonna delle Grazie, in Baroque style; the interior has a 1645 Baroque altar and 17th century canvasses. It features a column, similar to the Lecce's Column of St. Oronzo by Zimbalo, built in 1684–87. Church of Madonna Addolorata Palazzo Baronale Maglie's economy is based on commerce and tourism.

Maglie is served by Maglie-Otranto lines. It is crossed by two state roads, the SS16 Adriatica Lecce-Maglie-Otranto, the SS275 from Santa Maria di Leuca. Aldo Moro, politician

Claude Reeds

Claude Edwin Reeds was an American football player and coach. He played college football at the University of Oklahoma as a fullback from 1910 to 1913. Reeds served as the head football coach at Southwestern Normal School—now Southwestern Oklahoma State University—from 1914 to 1915, at West Texas State Teachers College—now West Texas A&M University—from 1929 to 1930, at Central State Teachers College—now the University of Central Oklahoma—from 1931 to 1940, compiling a career coaching record of 72–41–11, he was inducted into the College Football Hall of Fame as a player in 1961. Claude Reeds at the College Football Hall of Fame Claude Reeds at Find a Grave

Jimmy Burke (baseball)

James Timothy Burke was a Major League Baseball third baseman and manager. He played for the Cleveland Spiders, St. Louis Perfectos, Milwaukee Brewers, Chicago White Stockings, Pittsburgh Pirates, St. Louis Cardinals. Burke was the regular third baseman for the Cardinals from 1903 to 1905, he was named player-manager in the middle of the 1905, season but was replaced by Stanley Robison after amassing a record of 34–56. From 1914 through 1917, Burke was a coach for the Detroit Tigers, he served as manager for the St. Louis Browns from 1918 through 1920. In 1921, he became a coach for a position he held for three seasons. Burke was a coach for the Chicago Cubs from 1926 through 1930, was last a coach with the New York Yankees from 1931 through 1933. Burke made his debut in October of 1898 for the Cleveland Spiders, he was one of many players moved from Cleveland to St. Louis the following season, a move that stocked Cleveland with inferior and inept ball players that resulted in that team producing the worst record in Major league baseball.

Burke only played a couple of games for St. Louis, now called the St. Louis Perfectos. In 1901, he split Chicago White Sox of the American League. After being released by Chicago in 1901, he signed with the National League's Pittsburgh Pirates where once again he'd see spot duty. Burke never got anywhere near close to regular playing time until he was traded by Pittsburgh to the St. Louis Cardinals. In 1903, his first season with significant playing time, he rove in 42 runs. However, despite in appearing in over hundred games in both of the next seasons, Burke would find himself playing for the Kansas City Blues in the minor leagues. Burke never again played in the majors, finishing his career with the Fort Wayne Champs of the Central League in 1913. List of Major League Baseball player–managers Career statistics and player information from Baseball-Reference, or Baseball-Reference, or Retrosheet Managerial record Jimmy Burke at Find a Grave

Memory inhibition

In psychology, memory inhibition is the ability not to remember irrelevant information. The scientific concept of memory inhibition should not be confused with everyday uses of the word "inhibition". Scientifically speaking, memory inhibition is a type of cognitive inhibition, the stopping or overriding of a mental process, in whole or in part, with or without intention. Memory inhibition is a critical component of an effective memory system. While some memories are retained for a lifetime, most memories are forgotten. According to evolutionary psychologists, forgetting is adaptive because it facilitates selectivity of rapid, efficient recollection. For example, a person trying to remember where they parked their car would not want to remember every place they have parked. In order to remember something, therefore, it is essential not only to activate the relevant information, but to inhibit irrelevant information. There are many memory phenomena that seem to involve inhibition, although there is debate about the distinction between interference and inhibition.

In the early days of psychology, the concept of inhibition was influential. These psychologists applied the concept of inhibition to early theories of forgetting. Starting in 1894, German scientists Muller and Shumann conducted empirical studies that demonstrated how learning a second list of items interfered with memory of the first list. Based on these experiments, Muller argued. Arguing for a different explanation, Wundt claimed that selective attention was accomplished by the active inhibition of unattended information, that to attend to one of several simultaneous stimuli, the others had to be inhibited. American Psychologist Walter Pillsbury combined Muller and Wundt's arguments, claiming that attention both facilitates information, wanted and inhibits information, unwanted. In the face of behaviorism during the late 1920s through the 1950s, through the early growth of cognitive psychology in the late 1950s and early 1960s, inhibition disappeared as a theory. Instead, classical interference theory dominated memory research until as late as 1960.

By the early 1970s, classical interference theory began to decline due to its reliance on associationism, its inability to explain the facts of interference or how interference applies to everyday life, to newly published reports on proactive and retroactive inhibition. Since the mid-1980s, there has been a renewed interest in understanding the role of inhibition in cognition. Research on a wide variety of psychological processes, including attention, perception and memory, psycholinguistics, cognitive development, learning disabilities, neuropsychology, suggests that resistance to interference is an important part of cognition. More researchers suggest that the hippocampus plays a role in the regulation of disliked and competing memories, fMRI studies have shown hippocampus activity during inhibition processes; the "part-set cuing effect" was discovered by Slamecka, who found that providing a portion of to-be-remembered items as test cues impairs retrieval of the remaining un-cued items compared with performance in a no-cue control condition.

Such an effect is intriguing because cues are expected to aid recall. A prominent figure in retrieval-based inhibition research, Henry L. Roediger III was another one of the first psychologists to propose the idea that retrieving an item reduces the subsequent accessibility of other stored items. Becoming aware of the part-set cueing effect reduces the effect, such that relearning part of a set of learned associations can improve recall of the non-relearned associations. Using inhibition to explain memory processes began with the work of Hasher and Zacks, which focused on the cognitive costs associated with aging and bridging the attention-memory gap. Hasher and Zacks found that older adults show impairments on tasks that require inhibiting irrelevant information in working memory, these impairments may lead to problems in a variety of contexts. Anderson and Spellman's model of retrieval-induced forgetting suggests that when items compete during retrieval, an inhibitory process will serve to suppress those competitors.

For instance, retrieval of one meaning for a word will tend to inhibit the dominant meaning of that word. In 1995, Anderson and Spellman conducted a three-phase study using their retrieval-induced forgetting model to demonstrate unlearning as inhibition. Study phase: Participants study a list of category-exemplar pairings where some exemplars semantically similar in that they belong to another category besides the one they are explicitly paired with. Retrieval-practice phase: Participants are cued to practice remembering some of the exemplars given the category cue. Test phase: Given each category as a cue, the participant tries to recall the exemplar. Anderson and Spellman observed that items that shared a semantic relationship with practiced information was less recallable. Using the example from above, recall of items related to practiced information, including tomato and strawberry was lower than recall for cracker though strawberry is part of a different pair; this finding suggests that associative competition by explicit category cue is not the only factor in retrieval difficulty.

They theorized that the brain suppresses, or inhibit

John A. Elkington

John A. Elkington is an American real estate developer best known for the redevelopment of Beale Street, the entertainment district in Memphis, known in the United States as a popular tourist destination. Elkington earned a B. A. in psychology from Vanderbilt University before graduating from the University of Memphis Cecil C. Humphreys School of Law, he began his business career in 1975 by launching a company that specialized in property management for condominiums. In 1979, with business partner Steve Keltner, Elkington built the first private unit development in Memphis; the success of this venture was followed by many other residential developments and by 1985, Elkington & Keltner Group was ranked #79 on Inc. magazine's "Inc. 5000," an annual list of the fastest-growing private companies in the U. S. For 10 years, Elkington & Keltner was named one of the "Top 100 Builders" in America by Professional Builder magazine. In the 1990s, Elkington formed a new company focused on urban redevelopment, with projects as varied as Nashville's historic Union Station, the Scimitar Building in downtown Memphis, Lenox School in midtown Memphis.

Lenox School, built in 1912, had been closed for years before its conversion into condominiums that retained the integrity of the original building. In 2002, John Elkington was the recipient of the A. W. Willis Preservation Award from the Memphis Heritage Foundation, a non-profit historic preservation society. John Elkington's involvement with the redevelopment of historic Beale Street in downtown Memphis began in 1982. After decades of decline and an ill-conceived urban renewal program that left the vibrant area desolate and the street lined with empty, crumbling buildings, John Elkington agreed to lead the development and management of the massive project, he had three goals for Beale Street's restoration: preserve the music, bring commerce back, maintain a diversity of owners and patrons. He recruited B. B. King to Memphis in 1991 to open the original B. B. King's Blues Club, led development of the W. C. Handy Performing Arts Park on Beale Street, named for the musician and composer regarded as the "Father of the Blues."

By the mid-1990s, Beale Street was drawing thousands of visitors each year. In 2013, Beale Street was voted "Most Iconic American Street" by readers in a USA Today "10Best" poll. Once, when asked why Beale Street was so successful, Elkington replied, "We're genuine. We're real. It's allowing your tenants to be as creative as possible... You want every experience to be unique. We're living in an experience economy, we are. People are looking for things. We're looking for things that have longevity." The success of Beale Street earned John Elkington many honors, including being named by Memphis Magazine in 2011 as one of "The Memphis 35" – people whose influence over the past 35 years "was critical to the growth and evolution of our city." Three years the Memphis Flyer tapped him as one of "25 Who Shaped Memphis: 1989-2014." The Brass Note Walk of Fame along the sidewalks of Beale Street offers "a tangible embodiment of the many talented people who had put Memphis music and Beale Street on the world map," and includes a note honoring the contributions of John Elkington.

Government entities recognized his years of work resurrecting Beale Street: He was honored by the U. S. Congress in 2008, by the Tennessee General Assembly in 2010. Through the development of Beale Street, Elkington became active in expanding music tourism in Memphis. In 1992, he served on the board of the Memphis in May International Festival, a month-long series of events that includes a three-day music festival and a world championship barbecue contest, he was appointed to the Tennessee Film, Entertainment & Music Commission in 1996, was honored with the Blue Note Award from the Blues Foundation. In 2005, he joined the Executive Committee of the Memphis Visitors Bureau board; as a real estate professional, Elkington earned his CCIM designation in 1992, was honored by the MAAR in 2012 as Retail Broker of the Year. Elkington has held positions on several local and state industry boards, including Chairman of the Tennessee Board for Licensing Contractors. Under his leadership, the board first implemented a statewide qualification test for contractors.

Elkington has served as Chairman of the Memphis and Shelby County Land Use Control Board, which has the responsibility of overseeing and guiding the growth of Memphis and Shelby County through zoning regulations. He has served as Vice Chairman of the Downtown Memphis Commission, Chairman of the Downtown Parking Authority, as a board member of the Memphis Park Commission, as Secretary of the Downtown Development Board, he is a member of the Germantown Parks and Recreation Board in suburban Memphis. A founding board member of the National Civil Rights Museum in Memphis, John Elkington played a key role in the redevelopment of the historic Orpheum Theatre, the downtown relocation of the University of Memphis Cecil C. Humphreys School of Law. John Elkington is a longtime member of the Rotary Club of Memphis, was honored in 2015 as the Rotarian of the Year and as a Paul Harris Fellow, he was elected chapter president for 2019-2020