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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Alaska
|
This is a list of law enforcement agencies in the state of Alaska.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 50 law enforcement agencies employing 1,298 sworn police officers, about 189 for each 100,000 residents.
State agencies
Alaska Department of Commerce, Community, and Economic Development
Alaska Alcohol and Marijuana Control Office
Alaska Department of Corrections
Alaska Department of Environmental Conservation
Alaska Department of Public Safety
Alaska State Crime Lab
Alaska State Fire Marshal's Office
Alaska State Troopers
Alaska Wildlife Troopers
Alaska Court Services Officers
Village Public Safety Officer Program
Alaska State Parks
Alaska State Park Rangers
Alaska Commercial Vehicle Enforcement
Fairbanks International Airport Police and Fire Department
Ted Stevens Anchorage International Airport Police and Fire Department
City agencies
Adak Police Department
Anchorage Police Department
Angoon Police Department
Bethel Police Department
Cordova Police Department
Craig Police Department
Dillingham Police Department
Fairbanks Police Department
Fort Yukon Police Department
Galena Police Department
Haines Police Department
Homer Police Department
Hoonah Police Department
Hooper Bay Police Department
Juneau Police Department
Kake Police Department
Kenai Police Department
Ketchikan Police Department
King Cove Police Department
Klawock Police Department
Kodiak Police Department
Kotzebue Police Department
Metlakatla Police Department
Nome Police Department
North Pole Police Department
Palmer Police Department
Petersburg Police Department
Pilot Station Police Department
Saint Mary's Police Department
Seldovia Police Department
Seward Police Department
Sitka Police Department
Skagway Police Department
Soldotna Police Department
Valdez Police Department
Wasilla Police Department
Whittier Police Department
Wrangell Police Department
Department of Public Safety
Chignik Bay Department of Public Safety
Saint Paul Department of Public Safety
Sand Point Department of Public Safety
Tanana Department of Public Safety
Unalakleet Department of Public Safety
Unalaska Department of Public Safety
Yakutat Department of Public Safety
Other agencies
Alaska Railroad Corporation Police Department
Office of the United States Marshal for the District of Alaska
University & College Agencies
University of Alaska Anchorage Police Department
University of Alaska Fairbanks Police Department
References
Alaska
Law enforcement agencies
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Arizona
|
This is a list of law enforcement agencies in the state of Arizona.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 141 law enforcement agencies employing 14,591 sworn police officers, about 224 for each 100,000 residents.
State agencies
Arizona Counter Terrorism Information Center
Arizona Department of Corrections (ADC)
Arizona Department of Economic Security
Office of Special Investigations
Arizona Division of Emergency Management
Arizona Department of Revenue Criminal Investigations Unit
Arizona Department of Homeland Security
Arizona Department of Transportation (ADOT)
Enforcement and Compliance Division (ECD)
Arizona Department of Public Safety (DPS)
Arizona State Capitol Police
Arizona Highway Patrol (DPS)
Arizona Rangers
Agency Support Division
Criminal Investigations Division (CID)
Technical Services Division (TSD)
Arizona Commercial Vehicle Enforcement
Arizona Game and Fish Department
Arizona State Park Ranger Police
Arizona Office of Inspector General (OIG)
Arizona Counter Assault Team (CAT)
Arizona HEAT Unit (High Speed Unit) (HEAT)
Arizona Motor Unit (High Speed Unit) (Motor)
Arizona Department of Wildlife Rangers (Wildlife)
Regional agencies
Arizona Gang Task Force
East Valley DUI Task Force
Southern Arizona DUI Task Force
Southeast Arizona Task Force
County agencies
Apache County Sheriff's Office
Cochise County Sheriff's Office
Coconino County Sheriff's Office
Gila County Sheriff's Office
Graham County Sheriff's Office
Greenlee County Sheriff's Office
La Paz County Sheriff's Office
Maricopa County Sheriff's Office
Mohave County Sheriff's Office
Navajo County Sheriff's Office
Pima County Sheriff's Department
Pinal County Sheriff's Office
Santa Cruz County Sheriff's Office
Yavapai County Sheriff's Office
Yuma County Sheriff's Office
Municipal agencies
Apache Junction Police Department
Avondale Police Department
Benson Police Department
Bisbee Police Department
Buckeye Police Department
Bullhead City Police Department
Camp Verde Marshal's Office
Casa Grande Police Department
Chandler Police Department
Chino Valley Police Department
Clarkdale Police Department
Clifton Police Department
Coolidge Police Department
Cottonwood Police Department
Douglas Police Department
Eagar Police Department
El Mirage Police Department
Eloy Police Department
Flagstaff Police Department
Florence Police Department
Fredonia Marshal's Office
Gilbert Police Department
Glendale Police Department
Globe Police Department
Goodyear Police Department
Hayden Police Department
Holbrook Police Department
Huachuca City Police Department
Jerome Police Department
Kearny Police Department
Kingman Police Department
Lake Havasu City Police Department
Mammoth Police Department
Marana Police Department
Maricopa Police Department
Mesa Police Department
Miami Police Department
Nogales Police Department
Oro Valley Police Department
Page Police Department
Paradise Valley
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Arkansas
|
This is a list of law enforcement agencies in the state of Arkansas.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 237 law enforcement agencies employing 6,779 sworn police officers, about 236 for each 100,000 residents.
State agencies
Arkansas Department of Public Safety
Arkansas Alcohol Beverage Control Enforcement
Arkansas Department of Community Corrections
Arkansas Department of Correction
Arkansas Forestry Commission
Arkansas Game and Fish Commission
Arkansas Highway Police
Arkansas Law Enforcement Commission on Standards and Training
Arkansas Law Enforcement Training Academy
Arkansas State Capitol Police
Arkansas State Crime Laboratory
Arkansas State Forestry Commission Rangers
Arkansas State Hospital Police
Arkansas State Park Rangers
Arkansas State Parole and Probation
Arkansas State Police
Arkansas Supreme Court Police
Arkansas Tobacco Control Board Enforcement
Arkansas Crime Information Center
Arkansas Insurance Department(Criminal Investigations)
County agencies
Arkansas County Sheriff's Office
Ashley County Sheriff's Office
Baxter County Sheriff's Office
Benton County Sheriff's Office
Boone County Sheriff's Office
Bradley County Sheriff's Office
Calhoun County Sheriff's Office
Carroll County Sheriff's Office
Chicot County Sheriff's Office
Clark County Sheriff's Office
Clay County Sheriff's Office
Cleburne County Sheriff's Office
Cleveland County Sheriff's Office
Columbia County Sheriff's Office
Conway County Sheriff's Office
Craighead County Sheriff's Office
Crawford County Sheriff's Office
Crittenden County Sheriff's Office
Cross County Sheriff's Office
Dallas County Sheriff's Office
Desha County Sheriff's Office
Drew County Sheriff's Office
Faulkner County Sheriff's Office
Franklin County Sheriff's Office
Fulton County Sheriff's Office
Garland County Sheriff's Office
Grant County Sheriff's Office
Greene County Sheriff's Office
Hempstead County Sheriff's Office
Hot Spring County Sheriff's Office
Howard County Sheriff's Office
Independence County Sheriff's Office
Izard County Sheriff's Office
Jackson County Sheriff's Office
Jefferson County Sheriff's Office
Johnson County Sheriff's Office
Lafayette County Sheriff's Office
Lawrence County Sheriff's Office
Lee County Sheriff's Office
Lincoln County Sheriff's Office
Little River County Sheriff's Office
Logan County Sheriff's Office
Lonoke County Sheriff's Office
Madison County Sheriff's Office
Marion County Sheriff's Office
Miller County Sheriff's Office
Mississippi County Sheriff's Office
Monroe County Sheriff's Office
Montgomery County Sheriff's Office
Nevada County Sheriff's Office
Newton County Sheriff's Office
Ouachita County Sheriff's Office
Perry County Sheriff's Office
Phillips County Sheriff's Office
Pike County Sheriff's Office
Poinsett County Sheriff's Office
Polk County Sheriff's Office
Pope County Sheriff's Office
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20California
|
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, California had 509 law enforcement agencies employing 79,431 sworn peace officers, about 217 for each 100,000 residents.
State agencies
California Department of Justice
California Department of Justice Special agents
California Bureau of Investigation
California Bureau of Firearms
California Bureau of Forensic Services
California Bureau of Gambling Control
California Bureau of Medi-Cal Fraud & Elder Abuse
California Highway Patrol
California Department of Corrections and Rehabilitation
State Fugitive Apprehension Teams & Special Service Unit
California Department of Alcoholic Beverage Control
Department of Alcoholic Beverage Control
California Department of Alcoholic Beverage Control Agents Investigators
California Department of Alcoholic Beverage Control Task Forces Co-Operations
California Department of Fish and Wildlife
California State Parks Peace Officer
California Department of Toxic Substances Control
California Department of Motor Vehicles
The department was reduced to the Division of Motor Vehicles Investigations Division
California Department of Insurance Investigations Division
California Franchise Tax Board Criminal Investigations Bureau (CFTB-CIB)
California Lottery Security Law Enforcement Division
California Department of Consumer Affairs
California Department of Consumer Affairs
California Department of Consumer Affairs Division of Investigation
California Department of Health Care Services
California Department of Public Health
California Health and Human Services Agency
California Department of State Hospitals
California Department of State Hospitals Law Enforcement Office of Protective Services (California Department of State Hospitals Police)
California Department of Developmental Services
California Department of Developmental Services Police
County agencies
Alameda County
Alameda County Sheriff's Office
Alameda County Probation Department
Alpine County Sheriff's Office
Amador County Sheriff's Office
Butte County # Law enforcement
Calaveras County Sheriff's Department
Colusa County Sheriff's Office
Contra Costa County Sheriff's Office
Del Norte County Sheriff's Office
El Dorado County Sheriff's Office
Fresno County Sheriff's Department
Glenn County Sheriff's Office
Humboldt County Sheriff's Office
Imperial County Sheriff's Office
Inyo County Sheriff's Office
Kern County
Kern County Sheriff's Office
Kern County Probation Department
Kern County Parks and Recreation: Park Rangers
Kings County Sheriff's Department
Lake County Sheriff's Department
Lassen County Sheriff's Office
Los Angeles County
Los Angeles County Sheriff's Department
Los Angeles County Probation Department
Madera County Sheriff's Department
Marin County Sheriff's Office
Mariposa County Sheriff's Department
Mendocino County Sheriff's Office
Merced County Sheriff's Office
Modoc County Sheriff's Office
Mono County Sheriff's Department
Monterey Cou
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Colorado
|
This is a list of law enforcement agencies in the U.S. state of Colorado.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 246 law enforcement agencies employing 12,069 sworn police officers, about 245 for each 100,000 residents.
State agencies
Colorado Bureau of Investigation
Colorado Department of Corrections
Colorado Department of Natural Resources
Colorado Department of Public Safety
Colorado Division of Youth Services
Colorado State Patrol
Colorado Rangers
County agencies
Adams County Sheriff's Office
Alamosa County Sheriff's Office
Arapahoe County Sheriff's Office
Archuleta County Sheriff's Office
Baca County Sheriff's Office
Bent County Sheriff's Office
Boulder County Sheriff's Office
Broomfield Police Department
Chaffee County Sheriff's Office
Cheyenne County Sheriff's Office
Clear Creek County Sheriff's Office
Conejos County Sheriff's Office
Costilla County Sheriff's Office
Crowley County Sheriff's Office
Custer County Sheriff's Office
Delta County Sheriff's Office
Denver Sheriff Department
Dolores County Sheriff's Office
Douglas County Sheriff's Office
Eagle County Sheriff's Office
El Paso County Coroner's Office
El Paso County Department of Human Services - Fraud & Investigations Unit
El Paso County District Attorney's Office - 4th Judicial District
El Paso County Security Department
El Paso County Sheriff's Office
Elbert County Sheriff's Office
Fremont County Sheriff's Office
Garfield County Sheriff's Office
Gilpin County Sheriff's Office
Grand County Sheriff's Office
Gunnison County Sheriff's Office
Hinsdale County Sheriff's Office
Huerfano County Sheriff's Office
Humane Society of the Pikes Peak Region - El Paso County
Jackson County Sheriff's Office
Jefferson County Sheriff's Office
Kiowa County Sheriff's Office
Kit Carson County Sheriff's Office
La Plata County Sheriff's Office
Lake County Sheriff's Office
Larimer County Sheriff's Office
Las Animas County Sheriff's Office
Lincoln County Sheriff's Office
Logan County Sheriff's Office
Mesa County Sheriff's Office
Mineral County Sheriff's Office
Moffat County Sheriff's Office
Montezuma County Sheriff's Office
Montrose County Sheriff's Office
Morgan County Sheriff's Office
Otero County Sheriff's Office
Ouray County Sheriff's Office
Park County Sheriff's Office
Phillips County Sheriff's Office
Pitkin County Sheriff's Office
Prowers County Sheriff's Office
Pueblo County Sheriff's Office
Rio Blanco County Sheriff's Office
Rio Grande County Sheriff's Office
Routt County Sheriff's Office
Saguache County Sheriff's Office
San Juan County Sheriff's Office
San Miguel County Sheriff's Office
Sedgwick County Sheriff's Office
Summit County Sheriff's Office
Teller County District Attorney's Office - 4th Judicial District
Teller County Sheriff's Office
Washington County Sheriff's Office
Weld County Sheriff's Office
Yuma County Sheriff's Office
Municipality agencies
Alamosa Police Department
Arvada Police Department
Aspe
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Connecticut
|
This is a list of law enforcement agencies in the state of Connecticut.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 143 law enforcement agencies employing 8,281 sworn police officers, about 236 for each 100,000 residents.
State agencies
Connecticut State Capitol Police
Connecticut Department of Correction
Connecticut Department of Motor Vehicles
Commercial Vehicle Safety Division
Connecticut Department of Energy and Environmental Protection
Connecticut State Environmental Conservation Police
Connecticut Department of Mental Health and Addiction Services
Public Safety Division
Connecticut Judicial Branch
Superior Court Operations Division
Judicial Marshal Services
Connecticut Department of Emergency Services and Public Protection
Connecticut State Police
Connecticut State Division of Criminal Justice
Municipal agencies
Ansonia Police Department
Avon Police Department
Beacon Falls Police Department
Berlin Police Department
Bethany Police Department*
Bethel Police Department
Bethlehem Police Department*
Bloomfield Police Department
Branford Police Department
Bridgeport Police Department
Bridgewater Police Department*
Bristol Police Department
Brookfield Police Department
Burlington Police Department*
Canton Police Department
Cheshire Police Department
Chester Police Department*
Clinton Police Department
Colchester Police Department*
Coventry Police Department
Cromwell Police Department
Danbury Police Department
Darien Police Department
Deep River Police Department*
Derby Police Department
East Granby Police Department*
East Haddam Police Department*
East Hampton Police Department
East Hartford Police Department
East Haven Police Department
East Lyme Police Department
East Windsor Police Department
Easton Police Department
Enfield Police Department
Ellington Police Department*
Essex Police Department*
Fairfield Police Department
Farmington Police Department
Glastonbury Police Department
Granby Police Department
Greenwich Police Department
Groton Police Department (City of Groton)
Groton Police Department (Town of Groton)
Groton Long Point Police Department
Guilford Police Department
Hamden Police Department
Hartford Police Department
Hebron Police Department*
Killingly Police Department*
Lebanon Police Department*
Ledyard Police Department
Litchfield Police Department*
Madison Police Department
Manchester Police Department
Marlborough Police Department
Meriden Police Department
Middlebury Police Department
Middletown Police Department
Milford Police Department
Monroe Police Department
Montville Police Department*
Naugatuck Police Department
New Britain Police Department
New Canaan Police Department
New Fairfield Police Department*
New Hartford Police Department*
New Haven Police Department
New London Police Department
New Milford Police Department
Newington Police Department
Newtown Police Department
North Branford Police Department
North Haven Police De
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Delaware
|
This is a list of law enforcement agencies in the state of Delaware.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 49 law enforcement agencies employing 2,131 sworn police officers, about 243 for each 100,000 residents.
State agencies
Delaware Division of Alcohol and Tobacco Enforcement
Delaware Capitol Police
Delaware Department of Correction
Delaware State Probation and Parole
Delaware Department of Justice
Criminal Division
Delaware Department of Natural Resources and Environmental Control
Environmental Protection Officers
Fish & Wildlife Natural Resources Police Officers
Parks & Recreation Natural Resources Police Officers (State Park Rangers)
Delaware Office of Animal Welfare (State Animal Control Officers)
Delaware State Police
Delaware Office of the Fire Marshal
Delaware Justice of the Peace Court Constables
County agencies
New Castle County Police Department
Municipal agencies
Bethany Beach Police Department
Blades Police Department
Bridgeville Police Department
Camden Police Department
Cheswold Police Department
Clayton Police Department
Dagsboro Police Department
Delaware City Police Department
Delmar Police Department
Dewey Beach Police Department
Dover Police Department
Ellendale Police Department
Elsmere Police Department
Felton Police Department
Fenwick Island Police Department
Frederica Police Department
Georgetown Police Department
Greenwood Police Department
Harrington Police Department
Kenton Police Department
Laurel Police Department
Lewes Police Department
Middletown Police Department
Milford Police Department
Millsboro Police Department
Milton Police Department
New Castle City Police Department
Newark Police Department
Newport Police Department
Ocean View Police Department
Rehoboth Beach Police Department
Seaford Police Department
Selbyville Police Department
Smyrna Police Department
South Bethany Police Department
Wilmington Police Department
Wyoming Police Department
University agencies
Delaware State University Police Department
Delaware Technical Community College Public Safety Department
University of Delaware Police Department
Wilmington University Public Safety & Constables
Other agencies
Bayhealth Medical Center Constable
Christiana Care Health System Department of Public Safety Constables
Beebe Healthcare Department of Public Safety
Nemours Alfred I. duPont Hospital for Children Constables
Tower Hill School Constables
Delaware River and Bay Authority Police Department
Ellendale Recovery Response Center
Port of Wilmington Harbor Police Department
Delaware Metropolitan Transit Authority Constables and Public Safety
Department of Veterans Affairs Police, Wilmington Delaware.
References
Delaware
Law enforcement agencies of Delaware
Law enforcement agencies
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20the%20District%20of%20Columbia
|
This is a list of law enforcement agencies in the District of Columbia.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the District has six local law enforcement agencies employing 4,262 sworn police officers, about 722 for each 100,000 residents. This is the highest proportion of police officers to citizens of any state or territory.
Listed by age
The oldest agencies are the:
United States Marshals Service, founded September 24, 1789
United States Park Police, founded in 1791 as park watchmen to guard federal property in DC
United States Mint Police, founded in 1792
United States Capitol Police, founded in 1828
Metropolitan Police Department of the District of Columbia, founded in 1861 (took the place of DC City Watch, founded in 1802)
United States Secret Service, founded July 5, 1865
District of Columbia Protective Services Division, founded by Congress in 1899 under the Watchmen in Municipal Facilities Act
Primary DC law enforcement (local and federal)
District of Columbia Department of Corrections
District of Columbia Housing Authority Police Department of Public Safety (Has city-wide jurisdiction throughout Metropolitan area)
District of Columbia Metropolitan Police Department (local municipal police covering all of DC with approximately 3,800 officers)
District of Columbia Protective Services Division
District of Columbia Public Schools Police - Law Enforcement Division (has city-wide jurisdiction on 119 DCPS owned and leased properties)
District of Columbia Public Library Police
Metro Transit Police Department (has jurisdiction in Metro rail and near Metro bus stops in DC, VA, and MD; 526 officers)
Metropolitan Washington Airports Authority Police (jurisdiction actually falls within specific locations in VA [Reagan National and Dulles airports]; formerly FAA Police)
United States Marshals (deputies at the Superior Court of the District of Columbia fulfill duties similar to those of a sheriff in local court matters, while deputies at the United States District Court for the District of Columbia perform more traditional and federal district court duties)
United States Park Police (national parks federal police for the National Mall, monuments, parkways, and all national park service properties in D.C and surrounding regions; several hundred officers; shares jurisdiction with D.C. Metropolitan Police in addition to exercising federal authority)
Washington National Cathedral Police (officers licensed by the MPD as special police officers)
Washington Humane Society Law Enforcement (charted by Congress in 1870 to enforce the Districts anti-cruelty laws)
Federal police agencies with a uniformed presence in District of Columbia area
The majority of federal law enforcement agencies have some type of jurisdiction and/or headquarter offices in the District of Columbia; however, some are more overt than others.
Amtrak Police Department (quasi-federal, as Amtrak is gov
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Florida
|
This is a list of Law Enforcement Agencies in the state of Florida.
According to the US Bureau of Justice Statistics' 2018 Census of State and Local Law Enforcement Agencies, the state had 373 law enforcement agencies employing 47,177 sworn police officers, about 222 for each 100,000 residents.
Federal Agencies
These are federal agencies that have common operations within the state.
Administrative Office of the United States Courts, Office of Probation and Pretrial Services
Amtrak Police Department
Bureau of Alcohol, Tobacco, Firearms and Explosives
Bureau of Industry and Security, Office of Export Enforcement
Department of the Air Force Police
Department of the Navy Police
Drug Enforcement Administration
Federal Air Marshal Service
Federal Bureau of Investigation
Federal Bureau of Prisons
Federal Protective Service
Internal Revenue Service, Criminal Investigations Division
National Aeronautics and Space Administration, Protective Services
National Oceanic and Atmospheric Administration Office of Law Enforcement
National Park Service
National Nuclear and Security Administration, Office of Secure Transport
United States Air Force Office of Special Investigations
United States Air Force Security Forces
United States Army Military Police
United States Coast Guard
United States Customs and Border Protection
United States Department of Veterans Affairs Police
United States Diplomatic Security Service
United States Environmental Protection Agency Criminal Investigation Division
United States Fish and Wildlife Service Office of Law Enforcement
United States Fish and Wildlife Service Refuge Law Enforcement
United States Food and Drug Administration Office of Criminal Investigations
United States Forest Service Law Enforcement and Investigations
United States Immigration and Customs Enforcement
United States Marine Corps Criminal Investigation Division
United States Marine Corps Military Police
United States Marine Corps Police
United States Marshal Service
United States Naval Criminal Investigative Service
United States Navy Security Forces
United States Postal Inspection Service
United States Secret Service
State Agencies
Florida Department of Agriculture and Consumer Services
Office of Agricultural Law Enforcement
Florida Department of Business and Professional Regulation
Division of Alcoholic Beverages and Tobacco
Bureau of Law Enforcement
Florida Department of Corrections
Office of Inspector General
Bureau of State Investigations/Law Enforcement
Florida Department of Environmental Protection
Division of Law Enforcement
Environmental Crimes Unit
Florida Department of Financial Services
Division of Investigative and Forensic Services
Florida Department of Highway Safety and Motor Vehicles
Division of Florida Highway Patrol
Florida Department of Law Enforcement
Division of Investigations and Forensic Science
Division of Florida Capitol Police
Florida Fish and Wildlife Conservation Commission
Division of Law Enforcement
F
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Georgia
|
This is a list of law enforcement agencies in the U.S. state of Georgia.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 628 law enforcement agencies employing 26,551 sworn police officers, about 274 for each 100,000 residents.
Georgia also hosts the FLETC, which serves as the primary training facility for numerous federal law enforcement agencies.
State agencies
Georgia Department of Juvenile Justice (DJJ)
Georgia Bureau of Investigation (GBI)
Georgia Department of Corrections
Georgia Department of Human Resource and Developmental Disabilities
Georgia Department of Natural Resources
Georgia Department of Public Safety
Executive Security
Georgia Capitol Police
Georgia Motor Carrier Compliance Division
Georgia State Patrol
Georgia Department of Revenue
Georgia Alcohol and Tobacco Division
Office of Special Investigations
Georgia Public Defender Council
Georgia Department of Transportation
Georgia Insurance and Safety Fire Commissioner
Georgia Ports Authority Police Department
Georgia World Congress Center
The Medical Center, Navicent Health Police Department
Stone Mountain Department of Public Safety
County marshal agencies
Cherokee County Marshal's Office
Clayton County Marshal's Office
Columbia County Marshal's Office
DeKalb County Marshal's Office
Fayette County Marshal's Office
Fulton County Marshal's Office
Gwinnett County Marshal's Office
Hall County Marshal's Office
Paulding County Marshal's Office
Richmond County Marshal's Office
Troup County Marshal's Office
Stephens County Marshal's Office
County sheriff agencies
Appling County Sheriff's Office
Atkinson County Sheriff's Office
Bacon County Sheriff's Office
Baker County Sheriff's Office
Baldwin County Sheriff's Office
Banks County Sheriff's Office
Barrow County Sheriff's Office
Bartow County Sheriff's Office
Ben Hill County Sheriff's Office
Berrien County Sheriff's Office
Bibb County Sheriff's Office
Bleckley County Sheriff's Office
Brantley County Sheriff's Office
Brooks County Sheriff's Office
Bryan County Sheriff's Office
Bulloch County Sheriff's Office
Burke County Sheriff's Office
Butts County Sheriff's Office
Calhoun County Sheriff's Office
Camden County Sheriff's Office
Candler County Sheriff's Office
Carroll County Sheriff's Office
Catoosa County Sheriff's Office
Charlton County Sheriff's Office
Chatham County Sheriff's Office
Chattahoochee County Sheriff's Office
Chattooga County Sheriff's Office
Cherokee County Sheriff's Office
Clarke County Sheriff's Office
Clay County Sheriff's Office
Clayton County Sheriff's Office
Clinch County Sheriff's Office
Cobb County Sheriff's Office
Coffee County Sheriff's Office
Colquitt County Sheriff's Office
Columbia County Sheriff's Office
Coweta County Sheriff's Office
Crawford County Sheriff's Office
Crisp County Sheriff's Office
Dade County Sheriff's Office
Dawson County Sheriff's Office
Decatur County Sheriff's Office
DeKalb County Sheriff's Of
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Hawaii
|
This is a list of law enforcement agencies located in Hawaii.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 7 law enforcement agencies employing 3,234 sworn police officers, about 251 for each 100,000 residents.
State agencies
Hawaii Division of Conservation and Resource Enforcement
Hawaii Department of Public Safety
Corrections Division
Narcotics Enforcement Division
Sheriff Division
Hawaii Department of Transportation
Hawaii Harbor Police
The police in the Hawaii State Airports is the Honolulu Police Department,Airport Security along with Hawaii Department of Public Safety's Law Enforcement Division Sheriff Division Hawaii State Sheriff Airport Detail including those airports in the County.
Airports Division
State of Hawaii Department of the Attorney General
Divisions
Legal services
Division
Hawaii Department of Taxation
Hawaii Department of Human Services
County agencies
Hawai‘i County Police Department - Hawai‘i
Kaua‘i County Police Department - Kaua‘i, Ni‘ihau
Maui County Police Department - Maui, Moloka‘i, Lāna‘ī, Kaho‘olawe
Consolidated City-County Agencies
Honolulu Police Department
Prosecutor Offices
Hawaii Prosecuting Attorney Office, Criminal Investigations Unit
Kauai Prosecuting Attorney Office, Criminal Investigations Unit
Maui Prosecuting Attorney Office, Investigative Services Division
City and County of Honolulu Prosecuting Attorney Office, Criminal Investigations Unit
References
Hawaii
Law enforcement agencies of Hawaii
Law enforcement agencies
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Idaho
|
This is a list of law enforcement agencies in the U.S. state of Idaho.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 117 law enforcement agencies employing 3,146 sworn police officers, about 206 for each 100,000 residents.
State agencies
Idaho Department of Correction
Idaho Department of Juvenile Corrections
Idaho Fish and Game Commission
Idaho Conservation Officers
Idaho State Police
Idaho State Fire Marshal
Idaho Weigh Station Division
Idaho State Brand Inspector
County agencies
Ada County Sheriff's Office
Adams County Sheriff's Office
Bannock County Sheriff's Office
Bear Lake County Sheriff's Office
Benewah County Sheriff's Office
Bingham County Sheriff's Office
Blaine County Sheriff's Office
Boise County Sheriff's Office
Bonner County Sheriff's Office
Bonneville County Sheriff's Office
Boundary County Sheriff's Office
Butte County Sheriff's Office
Camas County Sheriff's Office
Canyon County Sheriff's Office
Caribou County Sheriff's Office
Cassia County Sheriff's Office
Clark County Sheriff's Office
Clearwater County Sheriff's Office
Custer County Sheriff's Office
Elmore County Sheriff's Office
Franklin County Sheriff's Office
Fremont County Sheriff's Office
Gem County Sheriff's Office
Gooding County Sheriff's Office
Idaho County Sheriff's Office
Jefferson County Sheriff's Office
Jerome County Sheriff's Office
Kootenai County Sheriff's Office
Latah County Sheriff's Office
Lemhi County Sheriff's Office
Lewis County Sheriff's Office
Lincoln County Sheriff's Office
Madison County Sheriff's Office
Minidoka County Sheriff's Office
Nez Perce County Sheriff's Office
Oneida County Sheriff's Office
Owyhee County Sheriff's Office
Payette County Sheriff's Office
Power County Sheriff's Office
Shoshone County Sheriff's Office
Teton County Sheriff's Office
Twin Falls County Sheriff's Office
Valley County Sheriff's Office
Washington County Sheriff's Office
City agencies
Aberdeen Police Department
Albion Police Department
American Falls Police Department
Ashton Police Department
Blackfoot Police Department
Boise Police Department
Bonners Ferry Police Department
Buhl Police Department
Caldwell Police Department
Cascade Police Department
Chubbuck Police Department
Coeur d'Alene Police Department
Coeur d'Alene Tribal Police Department
Emmett Police Department
Filer Police Department
Fort Hall Tribal Police Department
Fruitland Police Department
Garden City Police Department
Gooding Police Department
Grangeville Police Department
Hailey Police Department
Heyburn Police Department
Homedale Police Department
Idaho City Police Department
Idaho Falls Police Department
Inkom Police Department
Iona Police Department
Jerome Police Department
Kamiah Marshal's Office
Kellogg Police Department
Ketchum Police Department
Kimberly-Hansen Police Department
Lewiston Police Department
McCall Police Department
Meridian Police
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Illinois
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This is a list of law enforcement agencies in the state of Illinois.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 877 law enforcement agencies employing 41,277 sworn police officers, about 321 for each 100,000 residents.
State agencies
Illinois Attorney General's Office
Department of Investigations
Illinois Attorney General Police
Illinois Commerce Commission Police
Illinois Department of Corrections
Illinois Department of Human Services Police Department
Illinois Department of Natural Resources
Illinois Conservation Police
Illinois Department of Revenue Police
Bureau of Criminal Investigations
Illinois Gaming Board
Illinois Law Enforcement Training and Standards Board
Illinois Secretary of State
Illinois Secretary of State Capitol Police
Illinois Secretary of State Police
Illinois State Fire Marshal Arson Investigation
Illinois State Police
County agencies
Sheriff's departments/offices
Adams County Sheriff's Office
Alexander County Sheriff's Office
Bond County Sheriff's Office
Boone County Sheriff's Office
Brown County Sheriff's Office
Bureau County Sheriff's Office
Calhoun County Sheriff's Office
Carroll County Sheriff's Office
Cass County Sheriff's Office
Champaign County Sheriff's Office
Christian County Sheriff's Office
Clark County Sheriff's Office
Clay County Sheriff's Office
Clinton County Sheriff's Office
Coles County Sheriff's Office
Cook County Sheriff's Office
Crawford County Sheriff's Office
Cumberland County Sheriff's Office
DeKalb County Sheriff's Office
Dewitt County Sheriff's Office
Douglas County Sheriff's Office
DuPage County Sheriff's Office
Edgar County Sheriff's Office
Edwards County Sheriff's Office
Effingham County Sheriff's Office
Fayette County Sheriff's Office
Ford County Sheriff's Office
Franklin County Sheriff's Office
Fulton County Sheriff's Office
Gallatin County Sheriff's Office
Greene County Sheriff's Office
Grundy County Sheriff's Office
Hamilton County Sheriff's Office
Hancock County Sheriff's Office
Hardin County Sheriff's Office
Henderson County Sheriff's Office
Henry County Sheriff's Office
Iroquois County Sheriff's Office
Jackson County Sheriff's Office
Jasper County Sheriff's Office
Jefferson County Sheriff's Office
Jersey County Sheriff's Office
Jo Daviess County Sheriff's Office
Johnson County Sheriff's Office
Kane County Sheriff's Office
Kankakee County Sheriff's Office
Kendall County Sheriff's Office
Knox County Sheriff's Office
LaSalle County Sheriff's Office
Lake County Sheriff's Office
Lawrence County Sheriff's Office
Lee County Sheriff's Office
Livingston County Sheriff's Office
Logan County Sheriff's Office
Macon County Sheriff's Office
Macoupin County Sheriff's Office
Madison County Sheriff's Office
Marion County Sheriff's Office
Marshall County Sheriff's Office
Mason County Sheriff's Office
Massac County Sheriff's Office
Mc Donough Coun
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Indiana
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This is a list of law enforcement agencies in the state of Indiana.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 482 law enforcement agencies employing 13,171 sworn police officers, about 206 for each 100,000 residents.
Since 2012, the Indiana Law Enforcement Training Board (ILETB) has instituted a three-tier system of training for the state's various law enforcement agencies.
Tier I: Full Basic Recruit Academy at one of the state's ILETB certified academies
Tier II: Eight-week program held at the Indiana Law Enforcement Academy (ILEA) in Plainfield
Tier III: Four-week program held at the ILEA in Plainfield
Tier I State Agencies
department
Indiana Department of Natural Resources Law Enforcement Division
Indiana Law Enforcement Academy
Indiana Office of Inspector General
Indiana State Excise Police
Indiana State Fire Marshal's Office
Indiana State Police
Indiana State Police Capitol Police Section
Commercial Vehicle Enforcement Division
Tier II State Agencies
Indiana Department of Correction
Indiana Gaming Commission Office of Enforcement, Investigation, and Control
Tier III State Agencies
Indiana Attorney General's Office Medicaid Fraud Unit
Indiana Secretary of State Securities Enforcement Division
Tier I County Sheriff Agencies
Tier I Municipal Police and Tier II Town Marshal Agencies
Tier I College and University Police Agencies
Anderson University Police Department
Ball State University Police Department
Butler University Police Department
DePauw University Department of Public Safety
Huntington University Police Department
Indiana Purdue Fort Wayne (IPFW) Police Department
Indiana State University Police Department
Indiana University Public Safety
Indiana University Police Department (Bloomington)
Indiana University East Police Department (Richmond)
Indiana University Kokomo Police Department
Indiana University Northwest Police Department (Gary)
Indiana University South Bend Police Department
Indiana University Southeast Police Department (New Albany)
Indiana University–Purdue University Indianapolis Police Department
Indiana Wesleyan University Police Department
Marian University Police Department
Notre Dame University Police Department
Purdue University Police Department
Purdue University Northwest Police Department
Taylor University Police Department
University of Indianapolis Police Department
Valparaiso University Police Department
Vincennes University Police Department
Tier I School Corporation Police Agencies Set Forth Under I.C. 20-26-16
Brownsburg Community School Corporation Police Department
Center Grove Community School Corporation Police Department
Concord Community Schools Police Department
Evansville Vanderburgh Community School Corporation Police Department
Franklin Township Community School Corporation Police Department
Indiana School for the Deaf Police Department
Lebanon Schools Police Department
Metropolitan School District of P
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Iowa
|
This is a list of law enforcement agencies in the state of Iowa.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 392 law enforcement agencies employing 5,830 sworn police officers, about 195 for each 100,000 residents.
Federal Agencies
Office of the United States Marshal for the Northern District of Iowa
Office of the United States Marshal for the Southern District of Iowa
Department of Veterans Affairs
State agencies
Iowa Department of Corrections
Division of Parole
Iowa Department of Natural Resources Enforcement
Iowa Department of Public Safety
Iowa State Division of Criminal Investigation
Iowa State Division of Narcotics Enforcement
Iowa State Fire Marshal
Iowa State Patrol
County agencies
Adair County Sheriff's Office
Adams County Sheriff's Office
Allamakee County Sheriff's Office
Appanoose County Sheriff's Office
Audubon County Sheriff's Office
Benton County Sheriff's Office
Black Hawk County Sheriff's Office
Boone County Sheriff's Office
Buchanan County Sheriff's Office
Buena Vista County Sheriff's Office
Butler County Sheriff's Office
Calhoun County Sheriff's Office
Carroll County Sheriff's Office
Cass County Sheriff's Office
Cedar County Sheriff's Office
Cerro Gordo County Sheriff's Office
Cherokee County Sheriff's Office
Chickasaw County Sheriff's Office
Clarke County Sheriff's Office
Clay County Sheriff's Office
Clayton County Sheriff's Office
Clinton County Sheriff's Office
Crawford County Sheriff's Office
Dallas County Sheriff's Office
Davis County Sheriff's Office
Decatur County Sheriff's Office
Delaware County Sheriff's Office
Des Moines County Sheriff's Office
Dickinson County Sheriff's Office
Dubuque County Sheriff's Office
Emmet County Sheriff's Office
Fayette County Sheriff's Office
Floyd County Sheriff's Office
Franklin County Sheriff's Office
Fremont County Sheriff's Office
Greene County Sheriff's Office
Grundy County Sheriff's Office
Guthrie County Sheriff's Office
Hamilton County Sheriff's Office
Hancock County Sheriff's Office
Hardin County Sheriff's Office
Harrison County Sheriff's Office
Henry County Sheriff's Office
Howard County Sheriff's Office
Humboldt County Sheriff's Office
Ida County Sheriff's Office
Iowa County Sheriff's Office
Jackson County Sheriff's Office
Jasper County Sheriff's Office
Jefferson County Sheriff's Office
Johnson County Sheriff's Office
Jones County Sheriff's Office
Keokuk County Sheriff's Office
Kossuth County Sheriff's Office
Lee County Sheriff's Office
Linn County Sheriff's Office
Louisa County Sheriff's Office
Lucas County Sheriff's Office
Lyon County Sheriff's Office
Madison County Sheriff's Office
Mahaska County Sheriff's Office
Marion County Sheriff's Office
Marshall County Sheriff's Office
Mills County Sheriff's Office
Mitchell County Sheriff's Office
Monona County Sheriff's Office
Monroe County Sheriff's Office
Montgomery County
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Kansas
|
This is a list of law enforcement agencies in the state of Kansas.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 371 law enforcement agencies employing 7,450 sworn police officers, about 266 for each 100,000 residents.
State agencies
Kansas Bureau of Investigation
Kansas Department of Corrections
Kansas Department of Revenue
Kansas State Alcoholic Beverage Control
Kansas Department of Wildlife, Parks and Tourism
Kansas Highway Patrol
Kansas State Fire Marshal
Kansas Lottery Security & Law Enforcement Division (S.L.E.D.)
Kansas Department of Labor
Kansas Attorney General's Office
County agencies
Allen County Sheriff's Office
Anderson County Sheriff's Office
Atchison County Sheriff's Office
Barber County Sheriff's Office
Barton County Sheriff's Office
Bourbon County Sheriff's Office
Brown County Sheriff's Office
Butler County Sheriff's Office
Chase County Sheriff's Office
Chautauqua County Sheriff's Office
Cherokee County Sheriff's Office
Cheyenne County Sheriff's Office
Clark County Sheriff's Office
Clay County Sheriff's Office
Cloud County Sheriff's Office
Coffey County Sheriff's Office
Comanche County Sheriff's Office
Cowley County Sheriff's Office
Crawford County Sheriff's Office
Decatur County Sheriff's Office
Dickinson County Sheriff's Office
Doniphan County Sheriff's Office
Douglas County Sheriff's Office
Edwards County Sheriff's Office
Elk County Sheriff's Office
Ellis County Sheriff's Office
Ellsworth County Sheriff's Office
Finney County Sheriff's Office
Ford County Sheriff's Office
Franklin County Sheriff's Office
Geary County Sheriff's Office
Gove County Sheriff's Office
Graham County Sheriff's Office
Grant County Sheriff's Office
Gray County Sheriff's Office
Greeley County Sheriff's Office
Greenwood County Sheriff's Office
Hamilton County Sheriff's Office
Harper County Sheriff's Office
Harvey County Sheriff's Office
Haskell County Sheriff's Office
Hodgeman County Sheriff's Office
Jackson County Sheriff's Office
Jefferson County Sheriff's Office
Jewell County Sheriff's Office
Johnson County Sheriff's Office
Kearny County Sheriff's Office
Kingman County Sheriff's Office
Kiowa County Sheriff's Office
Labette County Sheriff's Office
Lane County Sheriff's Office
Leavenworth County Sheriff's Office
Lincoln County Sheriff's Office
Linn County Sheriff's Office
Logan County Sheriff's Office
Lyon County Sheriff's Office
Marion County Sheriff's Office
Marshall County Sheriff's Office
McPherson County Sheriff's Office
Meade County Sheriff's Office
Miami County Sheriff's Office
Mitchell County Sheriff's Office
Montgomery County Sheriff's Office
Morris County Sheriff's Office
Morton County Sheriff's Office
Nemaha County Sheriff's Office
Neosho County Sheriff's Office
Ness County Sheriff's Office
Norton County Sheriff's Office
Osage County Sheriff's Office
Osborne County Sheriff's Offic
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Kentucky
|
This is a list of law enforcement agencies in the Commonwealth of Kentucky.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 389 law enforcement agencies employing 7,833 sworn police officers, about 183 for each 100,000 residents.
State agencies
Kentucky Department of Fish and Wildlife Resources
Kentucky State Conservation Officers
Kentucky Department of Parks
Kentucky State Park Rangers
Kentucky Horse Park Police
Kentucky Justice and Public Safety Cabinet
Kentucky Department of Corrections
Kentucky Department of Juvenile Justice
Kentucky State Police
Division of Commercial Vehicle Enforcement
Facilities Security Branch
Kentucky Office of the Attorney General
Kentucky Department of Criminal Investigation
Kentucky Public Protection Cabinet
Kentucky Department of Charitable Gaming Enforcement
Kentucky Department of Alcoholic Beverage Control
County agencies
Adair County Sheriff's Office
Adair County Constable
Adair County Constable District 1
Adair County Constable District 2
Adair County Constable District 3
Adair County Constable District 4
Adair County Constable District 5
Adair County Constable District 6
Adair County Constable District 7
Allen County Sheriff's Office
Allen County Constable
Allen County Constable District 1
Allen County Constable District 2
Allen County Constable District 3
Allen County Constable District 4
Allen County Constable District 5
Anderson County Sheriff's Office
Anderson County Constable
Anderson County Constable District 1
Anderson County Constable District 2
Anderson County Constable District 3
Anderson County Constable District 4
Anderson County Constable District 5
Anderson County Constable District 6
Ballard County Sheriff's Office
Ballard County Constable
Ballard County Constable District 2
Ballard County Constable District 3
Ballard County Constable District 5
Barren County Sheriff's Office
Barren County Constable
Barren County Constable District 1
Barren County Constable District 2
Barren County Constable District 3
Barren County Constable District 4
Barren County Constable District 5
Barren County Constable District 6
Barren County Constable District 7
Bath County Sheriff's Office
Bath County Constable
Bath County Constable District 1
Bath County Constable District 2
Bath County Constable District 3
Bell County Sheriff's Office
Boone County Sheriffs Office
Boone County Constable
Boone County Constable District 1
Boone County Constable District 2
Boone County Constable District 3
Bourbon County Sheriff's Office
Bourbon County Constable
Bourbon County Constable District 1
Bourbon County Constable District 5
Boyd County Sheriff's Office
Boyd County Detention Center
Boyd County Constables Office District 1
Boyle County Sheriff's Office
Bracken County Sheriff's Office
Breathitt County Sheriff's Office
Breckinridge County Sheriff's Office
Bullitt County Sheriff's Office
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Louisiana
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This is a list of law enforcement agencies in Louisiana.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 348 law enforcement agencies employing 18,050 sworn police officers, about 405 for each 100,000 residents. This is the largest ratio of policemen to residents of any state and compares to a national average of 251 to 100,000.
State agencies
Louisiana Department of Children and Family Services
Child Support Enforcement
Louisiana Department of Environmental Quality
Louisiana Department of Health and Hospitals
Louisiana Medical Center Police
Louisiana Department of Justice
Louisiana Department of Public Safety
Louisiana Division of Levee District Police
Atchafalaya Basin Levee District Police Department
East Jefferson Levee District Police Department
Lafourche Basin Levee District Police Department
Lake Borgne Levee District Police Department
Orleans Levee District Police Department
Pontchartrain Levee District Police Department
Tensas Basin Levee District Police Department
Louisiana State Fire Marshal
Louisiana State Police
Louisiana Department of Public Safety & Corrections
Louisiana Department of Corrections - Adult Probation and Parole
Division of Youth Services - Office of Juvenile Justice
Louisiana Department of Revenue & Taxation
Louisiana Alcoholic Beverage Control
Louisiana Department of Transportation and Development
Crescent City Connection Police
Weight Enforcement Police
Louisiana Department of Wildlife & Fisheries - Enforcement Division
Louisiana Livestock Brand Commission Enforcement
Louisiana Office of State Parks - Enforcement Division
Louisiana State Museum Police
Louisiana Military Department Police
New Orleans City Park Police
Parish agencies
Acadia Parish Sheriff's Office
Allen Parish Sheriff's Office
Ascension Parish Sheriff's Office
Assumption Parish Sheriff's Office
Avoyelles Parish Sheriff's Office
Beauregard Parish Sheriff's Office
Bienville Parish Sheriff's Office
Bossier Parish Sheriff's Office
Caddo Parish Constable's Office
Caddo Parish Sheriff's Office
Calcasieu Parish Sheriff's Office
Caldwell Parish Sheriff's Office
Cameron Parish Sheriff's Office
Catahoula Parish Sheriff's Office
Claiborne Parish Sheriff's Office
Concordia Parish Sheriff's Office
Desoto Parish Sheriff's Office
East Baton Rouge Parish Sheriff's Office
East Carroll Parish Sheriff's Office
East Feliciana Parish Sheriff's Office
Evangeline Parish Sheriff's Office
Franklin Parish Sheriff's Office
Grant Parish Sheriff's Office
Iberia Parish Sheriff's Office
Iberville Parish Sheriff's Office
Jackson Parish Sheriff's Office
Jefferson Parish Sheriff's Office
Jefferson Davis Parish Sheriff's Office
Lafayette Parish Sheriff's Office
Lafourche Parish Sheriff's Office
LaSalle Parish Sheriff's Office
Lincoln Parish Sheriff's Office
Livingston Parish Sheriff's Office
Madison Parish Sheriff's Office
Morehouse Parish Sheriff's Office
Natchitoches Parish
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Maine
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This is a list of law enforcement agencies in the state of Maine.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 146 law enforcement agencies employing 2,569 sworn police officers, about 195 for each 100,000 residents.
State agencies
Maine Department of Corrections
Maine Department of Inland Fisheries and Wildlife
Maine Warden Service
Maine Department of Marine Resources
Maine Marine Patrol
Maine Department of Public Safety
Maine Bureau of Capitol Police
Maine Drug Enforcement Agency
Maine State Fire Marshal's Office
Maine State Police
Maine Department of Agriculture, Conservation and Forestry
Maine Forest Service
Maine Department of the Secretary of State
Bureau of Motor Vehicles
Division of Enforcement, Anti-Theft and Regulations
State of Maine Judicial Branch
Office of Judicial Marshals
Office of the Maine Attorney General
Investigation Division
County agencies
Androscoggin County Sheriff's Office
Aroostook County Sheriff's Office
Cumberland County Sheriff's Office
Franklin County Sheriff's Office
Hancock County Sheriff's Office
Kennebec County Sheriff's Office
Knox County Sheriff's Office
Lincoln County Sheriff's Office
Oxford County Sheriff's Office
Penobscot County Sheriff's Office
Piscataquis County Sheriff's Office
Sagadahoc County Sheriff's Office
Somerset County Sheriff's Office
Waldo County Sheriff's Office
Washington County Sheriff's Office
York County Sheriff's Office
Municipal agencies
Androscoggin County
Auburn Police Department
Lewiston Police Department
Lisbon Police Department
Livermore Falls Police Department
Mechanic Falls Police Department
Sabattus Police Department
Aroostook County
Ashland Police Department
Caribou Police Department
Fort Fairfield Police Department
Fort Kent Police Department
Houlton Police Department
Madawaska Police Department
Presque Isle Police Department
Washburn Police Department
Cumberland County
Bridgton Police Department
Brunswick Police Department
Cape Elizabeth Police Department
Cumberland Police Department
Falmouth Police Department
Freeport Police Department
Gorham Police Department
Portland Police Department
Scarborough Police Department
South Portland Police Department
Westbrook Police Department
Windham Police Department
Yarmouth Police Department
Franklin County
Carrabassett Valley Police Department
Farmington Police Department
Jay Police Department
Rangeley Police Department
Wilton Police Department
Hancock County
Bar Harbor Police Department
Bucksport Police Department
Ellsworth Police Department
Gouldsboro Police Department
Mount Desert Police Department
Southwest Harbor Police Department
Winter Harbor Police Department
Kennebec County
Augusta Police Department
Clinton Police Department
Gardiner Police Department
Hallowell Police Department
Monmouth Police Department
Oakland Police Department
Vassalboro Police Department
Waterville P
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Maryland
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This is a list of law enforcement agencies in the state of Maryland.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 142 law enforcement agencies employing 16,013 sworn police officers, about 283 for each 100,000 residents.
State agencies
Maryland Capitol Police
Maryland Department of Health and Mental Hygiene Police
Maryland Department of Labor Police
Maryland Natural Resources Police
Maryland Motor Vehicle Administration Police Department
Maryland Office of the Comptroller
Maryland State Police
Maryland State Fire Marshal
Maryland Transit Administration Police
Maryland Transportation Authority Police
Washington Suburban Sanitary Commission Police Department
Maryland Department of Public Safety and Correctional Services
County agencies
Allegany County Sheriff's Office
Anne Arundel County Police Department
Anne Arundel County Sheriff's Office
Baltimore County Police Department
Baltimore County Sheriff's Office
Calvert County Sheriff's Office
Carroll County Sheriff's Office
Caroline County Sheriff's Department
Cecil County Sheriff's Office
Charles County Sheriff's Office
Dorchester County Sheriff's Office
Frederick County Sheriff's Office
Garrett County Sheriff's Office
Harford County Sheriff's Office
Howard County Police Department
Howard County Sheriff's Office
Kent County Sheriff's Office
Montgomery County Police Department
Montgomery County Sheriff's Office
Prince George's County Police Department
Prince George's County Sheriff's Office
Queen Anne's County Sheriff's Office
St. Mary's County Sheriff's Office
Somerset County Sheriff's Office
Talbot County Sheriff's Office
Washington County Sheriff's Office
Wicomico County Sheriff's Office
Worcester County Sheriff's Office
Municipal agencies
Aberdeen Police Department
Annapolis Police Department
Baltimore Police Department
Baltimore City Schools Police
Baltimore City Sheriff's Office
Baltimore City Environmental Police
Bel Air Police Department
Berlin Police Department
Berwyn Heights Police Department
Bladensburg Police Department
Boonsboro Police Department
Bowie Police Department
Brentwood Police Department
Brunswick Police Department
Cambridge Police Department
Capitol Heights Police Department
Centreville Police Department
Chestertown Police Department
Cheverly Police Department
Chevy Chase Village Police Department
Colmar Manor Police Department
Cottage City Police Department
Crofton Police Department
Crisfield Police Department
Cumberland Police Department
Delmar Police Department
Denton Police Department
District Heights Police Department
Easton Police Department
Edmonston Police Department
Elkton Police Department
Fairmont Heights Police Department
Federalsburg Police Department
Forest Heights Police Department
Frederick City Police Department
Frostburg City Police Department
Fruitland Police Department
Gaithersburg Police Department
Gibson Island Police
Glenarden Police Department
Greenbelt
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Massachusetts
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This is a list of law enforcement agencies in the Commonwealth of Massachusetts.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 357 law enforcement agencies employing 18,342 sworn police officers, about 280 for each 100,000 residents.
Federal Agencies
These are federal agencies that have common operations within the state.
Administrative Office of the United States Courts, Office of Probation and Pretrial Services
Amtrak Police Department
Bureau of Alcohol, Tobacco, Firearms and Explosives
Bureau of Industry and Security, Office of Export Enforcement
Department of the Air Force Police
Department of the Navy Police
Drug Enforcement Administration
Federal Air Marshal Service
Federal Bureau of Investigation
Federal Bureau of Prisons
Federal Protective Service
Internal Revenue Service, Criminal Investigations Division
National Aeronautics and Space Administration, Protective Services
National Oceanic and Atmospheric Administration Office of Law Enforcement
National Park Service
National Nuclear and Security Administration, Office of Secure Transport
United States Air Force Office of Special Investigations
United States Air Force Security Forces
United States Army Military Police
United States Coast Guard
United States Customs and Border Protection
United States Department of Veterans Affairs Police
United States Diplomatic Security Service
United States Environmental Protection Agency Criminal Investigation Division
United States Food and Drug Administration Office of Criminal Investigations
United States Immigration and Customs Enforcement
United States Marine Corps Criminal Investigation Division
United States Marine Corps Military Police
United States Marine Corps Police
United States Marshal Service
United States Naval Criminal Investigative Service
United States Navy Master-at-Arms
United States Postal Inspection Service
State agencies
Office of the Treasurer and Receiver General of Massachusetts
Alcoholic Beverages Control Commission
Enforcement Division
Massachusetts Attorney General's Office
Massachusetts Executive Office of Energy and Environmental Affairs
Massachusetts Environmental Police
Massachusetts Executive Office of Public Safety and Security
Massachusetts State Police
Massachusetts Department of Correction
Massachusetts Court System
Executive Office of the Trial Court
Trial Court Security Department
County agencies
Barnstable County Sheriff's Office
Berkshire County Sheriff's Office
Bristol County Sheriff's Office
Dukes County Sheriff's Office
Essex County Sheriff's Office
Franklin County Sheriff's Office
Hampden County Sheriff's Office
Hampshire County Sheriff's Office
Middlesex County Sheriff's Office
Nantucket County Sheriff's Office
Norfolk County Sheriff's Office
Plymouth County Sheriff's Office
Suffolk County Sheriff's Department
Worcester County Sheriff's Office
Municipal agencies
Abington Police Department
Act
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Michigan
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This is a list of law enforcement agencies in the state of Michigan.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 571 law enforcement agencies employing 19,009 sworn police officers, about 190 for each 100,000 residents.
State agencies
Michigan Department of Corrections
Michigan Department of Natural Resources
Michigan Conservation Officers
Michigan State Police
Regional agencies
Central Michigan Enforcement Team CMAT
Flint Area Narcotics Group FANG
Huron Undercover Narcotics Team HUNT
Livingston and Washtenaw Narcotics Enforcement Team LAWNET
Metropolitan Enforcement Team MET
Straits Area Narcotics Enforcement SANE
State, Sheriffs, Chiefs Enforcement Team SSCENT
Strike Team Investigative Narcotics Group STING
Southwest Enforcement Team SWET
Traverse Narcotics Team TNT
Thumb Narcotics Unit TNU
Tri County Metro Narcotics Tri-County
Upper Peninsula Substance Enforcement Team UPSET
West Michigan Enforcement Team WEMET
County agencies
Alcona County Sheriff's Office
Alger County Sheriff's Office
Allegan County Sheriff's Office
Alpena County Sheriff's Office
Antrim County Sheriff's Office
Arenac County Sheriff's Office
Baraga County Sheriff's Office
Barry County Sheriff's Office
Bay County Sheriff's Office
Benzie County Sheriff's Office
Berrien County Sheriff's Office
Branch County Sheriff's Office
Calhoun County Sheriff's Office
Cass County Sheriff's Office
Charlevoix County Sheriff's Office
Cheboygan County Sheriff's Office
Chippewa County Sheriff's Office
Clare County Sheriff's Office
Clinton County Sheriff's Office
Crawford County Sheriff's Office
Delta County Sheriff's Office
Dickinson County Sheriff's Office
Eaton County Sheriff's Office
Emmet County Sheriff's Office
Genesee County Sheriff's Office
Gladwin County Sheriff's Office
Gogebic County Sheriff's Office
Grand Traverse County Sheriff's Office
Gratiot County Sheriff's Office
Hillsdale County Sheriff's Office
Houghton County Sheriff's Office
Huron County Sheriff's Office
Ingham County Sheriff's Office
Ionia County Sheriff's Office
Iosco County Sheriff's Office
Iron County Sheriff's Office
Isabella County Sheriff's Office
Jackson County Sheriff's Office
Kalamazoo County Sheriff's Office
Kalkaska County Sheriff's Office
Kent County Sheriff's Office
Keweenaw County Sheriff's Office
Lake County Sheriff's Office
Lapeer County Sheriff's Office
Leelanau County Sheriff's Office
Lenawee County Sheriff's Office
Livingston County Sheriff's Office
Luce County Sheriff's Office
Mackinac County Sheriff's Office
Macomb County Sheriff's Office
Manistee County Sheriff's Office
Marquette County Sheriff's Office
Mason County Sheriff's Office
Mecosta County Sheriff's Office
Menominee County Sheriff's Office
Midland County Sheriff's Office
Missaukee County Sheriff's Office
Monroe County Sheriff's Office
Montcalm County Sheriff's Office
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Minnesota
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This is a list of law enforcement agencies in the state of Minnesota.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 448 law enforcement agencies employing 9,667 sworn police officers, about 185 for each 100,000 residents.
State agencies
Minnesota Department of Corrections
Minnesota Department of Natural Resources
Law Enforcement Division
Minnesota Department of Public Safety
Minnesota Bureau of Criminal Apprehension
Minnesota State Fire Marshal
Minnesota Alcohol Gambling Enforcement Division
Minnesota State Patrol
Capitol Security
Minnesota Department of Commerce
Fraud Bureau
Minnesota National Guard
Dept. of Military Affairs Security Police
Military Police
34th Military Police Company
257th Military Police Company
Security Forces
148th Security Forces Squadron
133rd Security Forces Squadron
Phoenix Raven Team
County agencies
Aitkin County Sheriff's Office
Anoka County Sheriff's Office
Becker County Sheriff's Office
Beltrami County Sheriff's Office
Benton County Sheriff's Office
Big Stone County Sheriff's Office
Blue Earth County Sheriff's Office
Brown County Sheriff's Office
Carlton County Sheriff's Office
Carver County Sheriff's Office
Cass County Sheriff's Office
Chippewa County Sheriff's Office
Chisago County Sheriff's Office
Clay County Sheriff's Office
Clearwater County Sheriff's Office
Cook County Sheriff's Office
Cottonwood County Sheriff's Office
Crow Wing County Sheriff's Office
Dakota County Sheriff's Office
Dodge County Sheriff's Office
Douglas County Sheriff's Office
Faribault County Sheriff's Office
Fillmore County Sheriff's Office
Freeborn County Sheriff's Office
Goodhue County Sheriff's Office
Grant County Sheriff's Office
Hennepin County Sheriff's Office
Houston County Sheriff's Office
Hubbard County Sheriff's Office
Isanti County Sheriff's Office
Itasca County Sheriff's Office
Jackson County Sheriff's Office
Kanabec County Sheriff's Office
Kandiyohi County Sheriff's Office
Kittson County Sheriff's Office
Koochiching County Sheriff's Office
Lac Qui Parle County Sheriff's Office
Lake County Sheriff's Office
Lake of the Woods County Sheriff's Office
Le Sueur County Sheriff's Office
Lincoln County Sheriff's Office
Lyon County Sheriff's Office
Mahnomen County Sheriff's Office
Marshall County Sheriff's Office
Martin County Sheriff's Office
McLeod County Sheriff's Office
Meeker County Sheriff's Office
Mille Lacs County Sheriff's Office
Morrison County Sheriff's Office
Mower County Sheriff's Office
Murray County Sheriff's Office
Nicollet County Sheriff's Office
Nobles County Sheriff's Office
Norman County Sheriff's Office
Olmsted County Sheriff's Office
Otter Tail County Sheriff's Office
Pennington County Sheriff's Office
Pine County Sheriff's Office
Pipestone County Sheriff's Office
Polk County Sheriff's Office
Pope County Sheriff's Office
Ramsey County Sheriff's Office
Red Lake County Sheri
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https://en.wikipedia.org/wiki/Ulsan%20Airport
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Ulsan Airport is an airport in Ulsan, South Korea. In 2018, 817,341 passengers used the airport.
Airlines and destinations
Statistics
Facility
Passenger Terminal: 8,651 m2
Runway: Direction= 18/36 ; Length x Width = 2000 m x 45 m
Apron: 3,480 m2 (Four B737s can be parked simultaneously.)
Ground Transportation
Bus
102, 111, 122, 203, 205, 216, 225, 235, 236, 256, 266, 402, 412, 422, 432, 442, 453, 702, 714, 732, 1127, 5005 (to KTX Ulsan Station)
References
External links
Ulsan Airport (in English)
Ulsan's Travelguide
Airports in South Korea
Airports established in 1970
1970 establishments in South Korea
20th-century architecture in South Korea
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Mississippi
|
This is a list of law enforcement agencies in the state of Mississippi.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 342 law enforcement agencies employing 7,707 sworn police officers, about 262 for each 100,000 residents. Per the state constitution, all "civil officers" of the state, including those in the legislative and judicial branches, can exercise the power of arrest, though this is rarely exercised by said officials.
Multi-County agencies
Metro Narcotics Unit (Oxford and Layfette Counties and the University of Mississippi)
State agencies
Mississippi Attorney General's Office
Mississippi Bureau of Investigation
Mississippi Bureau of Narcotics
Mississippi Capitol Police
Mississippi Department of Corrections
Mississippi Department of Revenue
Criminal Investigations Division
Alcoholic Beverage Control
Mississippi Department of Wildlife, Fisheries and Parks
Law Enforcement Division
Mississippi Highway Patrol
Mississippi Department of Transportation Office of Law Enforcement
Mississippi Reservoir Police
Mississippi Gaming Commission
Mississippi Agricultural & Livestock Theft Bureau
Mississippi Department of Human Services | Office of Inspector General | Bureau of Investigation
School district agencies
North Bolivar Consolidated Schools School Safety
Hazlehurst City School District Safety and Security Department
Hattiesburg Public Schools Police Department
Petal School Police Department
Biloxi School District Police Department
Jackson Public Schools Campus Enforcement Department
Ocean Springs School District Police Department
Meridian Public School District Police Department
North Panola School District Bullying/School Safety Department
Pearl River County School District Police Department
Greenville Public School District Campus Safety Department
Leland School District Police Department
George County School District Police Department
Lawrence County School District Police Department
Tishomingo County School District Police Department
Vicksburg-Warren School District Campus Police Department
County agencies
Adams County Sheriff's Office
Alcorn County Sheriff's Office
Amite County Sheriff's Office
Attala County Sheriff's Office
Benton County Sheriff's Office
Bolivar County Sheriff's Office
Calhoun County Sheriff's Office
Carroll County Sheriff's Office
Chickasaw County Sheriff's Office
Choctaw County Sheriff's Office
Claiborne County Sheriff's Office
Clarke County Sheriff's Office
Clay County Sheriff's Office
Coahoma County Sheriff's Office
Copiah County Sheriff's Office
Covington County Sheriff's Office
De Soto County Sheriff's Office
Forrest County Sheriff's Office
Franklin County Sheriff's Office
George County Sheriff's Office
Greene County Sheriff's Office
Grenada County Sheriff's Office
Hancock County Sheriff's Office
Harrison County Sheriff's Department
Hinds County Sheriff's Office
Holmes C
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https://en.wikipedia.org/wiki/Yeosu%20Airport
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Yeosu Airport is an airport in Yeosu, South Korea . In 2018, 590,112 passengers used the airport.
Airlines and destinations
Statistics
Ground transportation
City Bus
No. 31, 32, 33, 34, 35, 96
References
Yeosu Airport (in English)
Airports in South Korea
Yeosu
Airports established in 1972
1972 establishments in South Korea
20th-century architecture in South Korea
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Missouri
|
This is a list of law enforcement agencies in the state of Missouri.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 576 law enforcement agencies employing 14,554 sworn police officers, about 244 for each 100,000 residents.
State agencies
Missouri Department of Conservation
Protection Division
Missouri Department of Corrections
Missouri Department of Natural Resources
Missouri State Park Rangers
Missouri Department of Public Safety
Missouri Gaming Commission
Missouri Homeland Security
Missouri State Capitol Police
Missouri State Emergency Management Agency
Missouri State Fire Marshal Investigation Unit
Missouri State Highway Patrol
Missouri State Water Patrol
Missouri State Marshal
County agencies
Adair County Sheriff's Office
Andrew County Sheriff's Office
Atchison County Sheriff's Office
Audrain County Sheriff's Office
Barry County Sheriff's Office
Barton County Sheriff's Office
Bates County Sheriff's Office
Benton County Sheriff's Office
Bollinger County Sheriff's Office
Boone County Sheriff's Office
Buchanan County Sheriff's Office
Butler County Sheriff's Office
Caldwell County Sheriff's Office
Callaway County Sheriff's Office
Camden County Sheriff's Office
Cape Girardeau County Sheriff's Office
Carroll County Sheriff's Office
Carter County Sheriff's Office
Cass County Sheriff's Office
Cedar County Sheriff's Office
Chariton County Sheriff's Office
Christian County Sheriff's Office
Clark County Sheriff's Office
Clay County Sheriff's Office
Clinton County Sheriff's Office
Cole County Sheriff's Office
Cooper County Sheriff's Department
Crawford County Sheriff's Office
Dade County Sheriff's Office
Dallas County Sheriff's Office
Daviess County Sheriff's Office
Dekalb County Sheriff's Office
Douglas County Sheriff's Office
Dunklin County Sheriff's Office
Franklin County Sheriff's Office
Gasconade County Sheriff's Office
Gentry County Sheriff's Office
Greene County Sheriff's Office
Grundy County Sheriff's Office
Harrison County Sheriff's Office
Henry County Sheriff's Office
Hickory County Sheriff's Office
Holt County Sheriff's Office
Howard County Sheriff's Office
Howell County Sheriff's Office
Iron County Sheriff's Office
Jackson County Department of Corrections
Jackson County Park Rangers (Missouri)
Jackson County Sheriff's Office
Jasper County Sheriff's Office
Jefferson County Sheriff's Office
Johnson County Sheriff's Office
Knox County Sheriff's Office
Laclede County Sheriff's Office
Lafayette County Sheriff's Office
Lawrence County Sheriff's Office
Lewis County Sheriff's Office
Lincoln County Sheriff's Office
Linn County Sheriff's Office
Livingston County Sheriff's Office
Macon County Sheriff's Office
Madison County Sheriff's Office
Maries County Sheriff's Office
Marion County Sheriff's Office
McDonald County Sheriff's Office
Mercer County Sheriff's Office
Miller County Sheriff's Office
Mississippi County Sheriff's Office
Moniteau Count
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Montana
|
This is a list of law enforcement agencies in the state of Montana.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 119 law enforcement agencies employing 3,229 sworn police officers, about 201 for each 100,000 residents.
State agencies
Montana Department of Corrections
Montana Department of Justice
Montana Highway Patrol
Division of Criminal Investigations
Gambling Control Division
Investigation Bureau
Montana Department of Livestock
Brands Enforcement Division
Montana Department of Fish, Wildlife, and Parks
Law Enforcement Bureau
Montana Department of Transportation
Motor Carrier Services
Enforcement Bureau
County agencies
Beaverhead County Sheriff's Office
Big Horn County Sheriff's Office
Blaine County Sheriff's Office
Broadwater County Sheriff's Office
Carbon County Sheriff's Office
Carter County Sheriff's Office
Cascade County Sheriff's Office
Chouteau County Sheriff's Office
Custer County Sheriff's Office
Daniels County Sheriff's Office
Dawson County Sheriff's Office
Fallon County Sheriff's Office
Fergus County Sheriff's Office
Flathead County Sheriff's Office
Gallatin County Sheriff's Office
Garfield County Sheriff's Office
Glacier County Sheriff's Office
Golden Valley County Sheriff's Office
Granite County Sheriff's Office
Hill County Sheriff's Office
Jefferson County Sheriff's Office
Judith Basin County Sheriff's Office
Lake County Sheriff's Office
Lewis and Clark County Sheriff's Office
Liberty County Sheriff's Office
Lincoln County Sheriff's Office
Madison County Sheriff's Office
McCone County Sheriff's Office
Meagher County Sheriff's Office
Mineral County Sheriff's Office
Missoula County Sheriff's Office
Musselshell County Sheriff's Office
Park County Sheriff's Office
Petroleum County Sheriff's Office
Phillips County Sheriff's Office
Pondera County Sheriff's Office
Powder River County Sheriff's Office
Powell County Sheriff's Office
Prairie County Sheriff's Office
Ravalli County Sheriff's Office
Richland County Sheriff's Office
Roosevelt County Sheriff's Office
Rosebud County Sheriff's Office
Sanders County Sheriff's Office
Sheridan County Sheriff's Office
Stillwater County Sheriff's Office
Sweet Grass County Sheriff's Office
Teton County Sheriff's Office
Toole County Sheriff's Office
Treasure County Sheriff's Office
Valley Treasure County Sheriff's Office
Wheatland County Sheriff's Office
Wibaux County Sheriff's Office
Yellowstone County Sheriff's Office
Combined city and county agencies
Anaconda-Deer Lodge County Law Enforcement Department
Butte-Silver Bow Law Enforcement Department
City agencies
Baker Police Department
Belgrade Police Department
Billings Police Department
Boulder Police Department
Bozeman Police Department
Bridger Police Department
Chinook Police Department
Colstrip Police Department
Columbia Falls Police Department
Columbus Police Department
Conrad Police Department
Cut Bank Police Department
Darby Police Department
Deer Lodge Pol
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Nebraska
|
This is a list of law enforcement agencies in the state of Nebraska.
According to the U.S. Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 225 law enforcement agencies employing 3,765 sworn officers, about 211 for each 100,000 residents.
State agencies
Nebraska Brand Committee
Nebraska Department of Agriculture — responsible for enforcement of the Nebraska Pesticide Act
Nebraska Department of Insurance - Insurance Fraud Prevention Division
Nebraska Department of Motor Vehicles, Driver & Vehicle Records Division, Fraud Investigation Section - "performs investigative work relating to vehicle title and registration fraud, driver's license fraud, and odometer fraud. Motor Vehicle Fraud Investigators are certified Nebraska Law Enforcement Officers commissioned as Deputy State Sheriffs."
Nebraska Department of Revenue
Nebraska Emergency Management Agency - Homeland Security
Nebraska Game and Parks Commission - conservation officers enforce Nebraska wildlife laws
Nebraska Law Enforcement Training Center
Nebraska Public Service Commission - Transportation Investigators
Nebraska State Fire Marshal
Nebraska State Patrol
Nebraska State Racing Commission
University of Nebraska at Kearney Police Department
University of Nebraska at Lincoln Police Department
University of Nebraska at Omaha Police Department
County agencies
Adams County Sheriff's Department
Antelope County Sheriff's Office
Arthur County Sheriff's Office
Banner County Sheriff's Office
Blaine County Sheriff's Office
Boone County Sheriff's Office
Box Butte County Sheriff's Office
Boyd County Sheriff's Office
Brown County Sheriff's Office
Buffalo County Sheriff's Office
Burt County Sheriff's Office
Butler County Sheriff's Office
Cass County Sheriff's Office
Cedar County Sheriff's Office
Chase County Sheriff's Office
Cherry County Sheriff's Office
Cheyenne County Sheriff's Office
Clay County Sheriff's Office
Colfax County Sheriff's Office
Cuming County Sheriff's Office
Custer County Sheriff's Office
Dakota County Sheriff's Office
Dawes County Sheriff's Office
Dawson County Sheriff's Office
Deuel County Sheriff's Office
Dixon County Sheriff's Office
Dodge County Sheriff's Office
Douglas County Sheriff's Office
Dundy County Sheriff's Office
Fillmore County Sheriff's Office
Franklin County Sheriff's Office
Frontier County Sheriff's Office
Furnas County Sheriff's Office
Gage County Sheriff's Office
Garden County Sheriff's Office
Garfield County Sheriff's Office
Gosper County Sheriff's Office
Grant County Sheriff's Office
Greeley County Sheriff's Office
Hall County Sheriff's Office
Hamilton County Sheriff's Office
Harlan County Sheriff's Office
Hayes County Sheriff's Office
Hitchcock County Sheriff's Office
Holt County Sheriff's Office
Hooker County Sheriff's Office
Howard County Sheriff's Office
Jefferson County Sheriff's Office
Johnson County Sheriff's Office
Kearney County Sheriff's Office
Keith County Sheriff's Office
Keya Paha County Sheriff'
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20New%20Hampshire
|
This is a list of law enforcement agencies in the state of New Hampshire.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 208 law enforcement agencies employing 2,936 sworn police officers, about 222 for each 100,000 residents.
State agencies
New Hampshire Administrative Office of Courts, Security Department
New Hampshire Liquor Commission
Division of Enforcement
New Hampshire Department of Corrections
New Hampshire Department of Safety
Division of Fire Safety (Office of State Fire Marshal)
Division of State Police
Field Operations Bureau
New Hampshire Marine Patrol
State Office Complex Police
New Hampshire Fish and Game Department
Law Enforcement Division
New Hampshire Office of the Attorney General
Criminal Justice Bureau
New Hampshire Department of Natural and Cultural Resources
Division of Forests and Lands
Forest Protection Bureau
County agencies
Belknap County Sheriff's Office
Carroll County Sheriff's Office
Cheshire County Sheriff's Department
Coös County Sheriff's Office
Grafton County Sheriff's Office
Hillsborough County Sheriff's Office
Merrimack County Sheriff's Office
Rockingham County Sheriff's Office
Strafford County Sheriff's Office
Sullivan County Sheriff's Office
Municipal agencies
Belknap County
Alton Police Department
Barnstead Police Department
Belmont Police Department
Center Harbor Police Department
Gilford Police Department
Gilmanton Police Department
Laconia Police Department
Meredith Police Department
New Hampton Police Department
Sanborton Police Department
Tilton Police Department
Carroll County
Bartlett Police Department
Conway Police Department
Effingham Police Department
Freedom Police Department
Jackson Police Department
Madison Police Department
Moultonborough Police Department
Ossipee Police Department
Sandwich Police Department
Tamworth Police Department
Tuftonboro Police Department
Wakefield Police Department
Wolfeboro Police Department
Cheshire County
Alstead Police Department
Chesterfield Police Department
Dublin Police Department
Fitzwilliam Police Department
Harrisville Police Department
Hinsdale Police Department
Jaffrey Police Department
Keene Police Department
Marlborough Police Department
Marlow Police Department
Nelson Police Department
Rindge Police Department
Roxbury Police Department
Stoddard Police Department
Swanzey Police Department
Troy Police Department
Walpole Police Department
Winchester Police Department
Coös County
Berlin Police Department
Carroll Police Department
Colebrook Police Department
Gorham Police Department
Lancaster Police Department
Northumberland Police Department
Pittsburg Police Department
Randolph Police Department]
Whitefield Police Department
Grafton County
Alexandria Police Department
Ashland Police Department
Bethlehem Police Department
Bridgewater Police Department
Bristol Police Department
Campton Police Department
Canaan Police Department
Enfield Police D
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https://en.wikipedia.org/wiki/1%20%E2%88%92%202%20%2B%203%20%E2%88%92%204%20%2B%20%E2%8B%AF
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In mathematics, 1 − 2 + 3 − 4 + ··· is an infinite series whose terms are the successive positive integers, given alternating signs. Using sigma summation notation the sum of the first m terms of the series can be expressed as
The infinite series diverges, meaning that its sequence of partial sums, , does not tend towards any finite limit. Nonetheless, in the mid-18th century, Leonhard Euler wrote what he admitted to be a paradoxical equation:
A rigorous explanation of this equation would not arrive until much later. Starting in 1890, Ernesto Cesàro, Émile Borel and others investigated well-defined methods to assign generalized sums to divergent series—including new interpretations of Euler's attempts. Many of these summability methods easily assign to a "value" of . Cesàro summation is one of the few methods that do not sum , so the series is an example where a slightly stronger method, such as Abel summation, is required.
The series 1 − 2 + 3 − 4 + ... is closely related to Grandi's series . Euler treated these two as special cases of the more general sequence , where and respectively. This line of research extended his work on the Basel problem and leading towards the functional equations of what are now known as the Dirichlet eta function and the Riemann zeta function.
Divergence
The series' terms do not approach 0; therefore diverges by the term test. Divergence can also be shown directly from the definition: an infinite series converges if and only if the sequence of partial sums converges to limit, in which case that limit is the value of the infinite series. The partial sums of are:
The sequence of partial sums shows that the series does not converge to a particular number: for any proposed limit x, there exists a point beyond which the subsequent partial sums are all outside the interval , so diverges.
The partial sums include every integer exactly once—even 0 if one counts the empty partial sum—and thereby establishes the countability of the set of integers.
Heuristics for summation
Stability and linearity
Since the terms follow a simple pattern, the series can be manipulated by shifting and term-by-term addition to yield a numerical value. If it can make sense to write for some ordinary number s, the following manipulations argue for
So .
Although does not have a sum in the usual sense, the equation can be supported as the most natural answer if such a sum is to be defined. A generalized definition of the "sum" of a divergent series is called a summation method or summability method. There are many different methods and it is desirable that they share some properties of ordinary summation. What the above manipulations actually prove is the following: Given any summability method that is linear and stable and sums the series , the sum it produces is . Furthermore, since
such a method must also sum Grandi's series as
Cauchy product
In 1891, Ernesto Cesàro expressed hope that divergent series would be rigorously
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https://en.wikipedia.org/wiki/John%20Friedlander
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John Friedlander is a Canadian mathematician specializing in analytic number theory. He received his B.Sc. from the University of Toronto in 1965, an M.A. from the University of Waterloo in 1966, and a Ph.D. from Pennsylvania State University in 1972. He was a lecturer at M.I.T. in 1974–76, and has been on the faculty of the University of Toronto since 1977, where he served as Chair during 1987–91. He has also spent several years at the Institute for Advanced Study. In addition to his individual work, he has been notable for his collaborations with other well-known number theorists, including Enrico Bombieri, William Duke, Andrew Granville, and especially Henryk Iwaniec.
In 1997, in joint work with Henryk Iwaniec, Friedlander proved that infinitely many prime numbers can be obtained as the sum of a square and fourth power: . Friedlander and Iwaniec improved Enrico Bombieri's "asymptotic sieve" technique to construct their proof.
Awards and honors
In 1999, Friedlander received the Jeffery–Williams Prize.
In 1988, Friedlander became a fellow of the Royal Society of Canada.
In 2002, CRM-Fields-PIMS prize
In 2012, he became a fellow of the American Mathematical Society.
In 2017, he received the Joseph L. Doob prize, jointly with Henryk Iwaniec, for their book Opera de Cribro.
Selected publications
See also
List of University of Waterloo people
References
External links
John Friedlander's profile in Toronto's Focus on Research.
On Bombieri's asymptotic sieve
Living people
Year of birth missing (living people)
Canadian mathematicians
University of Toronto alumni
Academic staff of the University of Toronto
Number theorists
Fellows of the American Mathematical Society
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20New%20Jersey
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This is a list of law enforcement agencies in the state of New Jersey.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 550 law enforcement agencies employing 33,704 sworn police officers, about 389 for each 100,000 residents.
State agencies
New Jersey Department of Corrections
New Jersey Department of Environmental Protection
Division of Fish and Wildlife
Bureau of Law Enforcement (State Conservation Officers)
Division of Parks and Forestry
New Jersey Forest Fire Service
New Jersey DEP Marine Law Enforcement Unit
New Jersey Department of Human Services Police
New Jersey Office of the Attorney General
Department of Law and Public Safety
Division of Alcoholic Beverage Control
Division of Consumer Affairs
Office of Consumer Protection
Enforcement Bureau
Office of Weights and Measures
NJ Division of Criminal Justice
Division of Gaming Enforcement
New Jersey Department of the Treasury
Division of Taxation
Office of Criminal Investigations
New Jersey Juvenile Justice Commission
New Jersey Office of Homeland Security and Preparedness
New Jersey Election Law Enforcement Commission
New Jersey State Park Police
New Jersey State Parole Board
Division of Parole (State Parole Officers)
New Jersey State Police
New Jersey State Detective Agency
New Jersey Transit Police Department
County Prosecutor's Office
Atlantic County Prosecutor's Office
Bergen County Prosecutor's Office
Burlington County Prosecutor's Office
Camden County Prosecutor's Office
Cape May County Prosecutor's Office
Cumberland County Prosecutor's Office
Essex County Prosecutor's Office
Gloucester County Prosecutor's Office
Hudson County Prosecutor's Office
Hunterdon County Prosecutor's Office
Mercer County Prosecutor's Office
Middlesex County Prosecutor's Office
Monmouth County Prosecutor's Office
Morris County Prosecutor's Office []
Ocean County Prosecutor's Office
Passaic County Prosecutor's Office
Salem County Prosecutor's Office
Somerset County Prosecutor's Office
Sussex County Prosecutor's Office
Union County Prosecutor's Office
Warren County Prosecutor's Office
County agencies
Atlantic County Sheriff's Office
Bergen County Sheriff's Office
Burlington County Sheriff's Department
Camden County Sheriff's Office
Camden County Police Department
Cape May County Sheriff's Office
Cumberland County Sheriff's Department
Essex County Sheriff's Office
Gloucester County Sheriff's Office
Hudson County Sheriff's Office
Hunterdon County Sheriff's Office
Mercer County Sheriff's Office
Middlesex County Sheriff's Department
Monmouth County Sheriff's Office
Morris County Sheriff's Office
Morris County Park Police
Ocean County Sheriff's Department
Passaic County Sheriff's Department
Salem County Sheriff's Office
Somerset County Sheriff's Office
Sussex County Sheriff's Office
Union County Sheriff's Office
Union County Police Department
Warren Co
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20New%20Mexico
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This is a list of law enforcement agencies in the state of New Mexico.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 146 law enforcement agencies employing 5,010 sworn police officers, about 252 for each 100,000 residents. This is almost exactly the national average of policemen to residents in the United States.
State agencies
New Mexico Department of Public Safety
New Mexico State Police
New Mexico Mounted Patrol
New Mexico Attorney General's Office
New Mexico Board of Pharmacy
New Mexico Corrections Department
New Mexico Department of Game and Fish
New Mexico Energy, Minerals and Natural Resources Department
New Mexico State Forestry Division
Special Agents
New Mexico State Parks Division
Park Rangers
New Mexico Livestock Inspector Board
New Mexico Taxation and Revenue Department
Office of the Inspector General
Tax Fraud Investigations Division
County agencies
Bernalillo County Sheriff's Department
Catron County Sheriff's Department
Chaves County Sheriff’s Office
Cibola County Sheriff's Office
Colfax County Sheriff's Office
Curry County Sheriff's Office
De Baca County Sheriff's Office
Doña Ana County Sheriff's Office
Eddy County Sheriff's Office
Grant County Sheriff's Department
Guadalupe County Sheriff's Office
Harding County Sheriff's Department
Hidalgo County Sheriff's Department
Lea County Sheriff's Department
Lincoln County Sheriff's Office
Los Alamos County Police Department
Luna County Sheriff's Office
McKinley County Sheriff's Office
Mora County Sheriff's Office
Otero County Sheriff's Department
Quay County Sheriff's Office
Rio Arriba County Sheriff's Office
Roosevelt County Sheriff's Office
San Juan County Sheriff's Office
San Miguel County Sheriff's Office
Sandoval County Sheriff's Office
Santa Fe County Sheriff's Office
Sierra County Sheriff's Office
Socorro County Sheriff's Department
Taos County Sheriff's Department
Torrance County Sheriff's Department
Union County Sheriff's Office
Valencia County Sheriff's Office
City agencies
Alamogordo Police Department
Albuquerque Police Department
Angel Fire Police Department
Anthony Police Department
Artesia Police Department
Aztec Police Department
Bayard Police Department
Belen Police Department
Bernalillo Police Department
Bloomfield Police Department
Bosque Farms Police Department
Buena Vista Police Department
Capitan Police Department
Carlsbad Police Department
Carrizozo Police Department
Cerrillos Police Department
Chama Police Department
Clayton Police Department
Cloudcroft Police Department
Clovis Police Department
Corrales Police Department
Cuba Police Department
Deming Police Department
Edgewood Police Department
Española Police Department
Estancia Police Department
Farmington Police Department
Gallup Police Department
Grants Police Department
Hatch Police Department
Hobbs Police Department
Hurley Police Department
Jemez Springs Police Department
Lamy Police Department
Las
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20North%20Carolina
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This is a list of law enforcement agencies in the state of North Carolina.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 504 law enforcement agencies employing 23,442 sworn police officers, about 254 for each 100,000 residents. As of June 2022, this is the latest data available, as the report has not been conducted since 2008.
State agencies
Black Mountain Neuro-Medical Treatment Center Police - Black Mountain, North Carolina (NC Department of Health & Human Services)
Broughton Hospital Police - Morganton, North Carolina (NC Department of Health & Human Services)
Cherry Hospital Police – Goldsboro, North Carolina (NC Department of Health & Human Services)
Longleaf Neuro-Medical Treatment Center Police - Wilson, North Carolina (NC Department of Health & Human Services)
North Carolina Alcohol Law Enforcement
North Carolina Arboretum Campus Police - Asheville, North Carolina
North Carolina Department of Agriculture State Fairgrounds Police - Raleigh, North Carolina
North Carolina Department of Insurance Criminal Investigations Division
North Carolina Division of Marine Fisheries Marine Patrol
North Carolina Division of Motor Vehicles License and Theft Bureau
North Carolina Division of Parks Law Enforcement Rangers
North Carolina General Assembly Police
North Carolina State Bureau of Investigation
North Carolina State Capitol Police
North Carolina State Highway Patrol
North Carolina Division of Adult Corrections (Probation/Parole)
North Carolina Museum of Art Special Police – Raleigh, North Carolina (Duties are now part of NC State Capitol Police)
North Carolina State Ports Authority Police - Morehead City/Wilmington
North Carolina Wildlife Resources Commission Law Enforcement
North Carolina Forest Service, Criminal Investigations Unit
North Carolina Department of Revenue, Criminal Investigations Unit, Unauthorized Substances, Motor Fuels Investigations
North Carolina Secretary of State's Office, Securities, Trademarks, Notary
North Carolina Department of Correction
North Carolina Supreme Court Police
O’Berry Neuro-Medical Treatment Center Police - Goldsboro, North Carolina (NC Department of Health & Human Services)
County agencies
Sheriffs have been required in each county of North Carolina since the North Carolina Constitution of 1776. Article VII, Section 2 of the 1971 Constitution of North Carolina gives the authority and qualifications for a sheriff in each county:
"In each county a Sheriff shall be elected by the qualified voters thereof at the same time and places as members of the General Assembly are elected and shall hold his office for a period of four years, subject to removal for cause as provided by law. No person is eligible to serve as Sheriff if that person has been convicted of a felony against this State, the United States, or another state, whether or not that person has been restored to the rights of citizenship in the man
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20North%20Dakota
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This is a list of law enforcement agencies in the state of North Dakota.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 114 law enforcement agencies employing 1,324 sworn officers, about 206 for each 100,000 residents.
State agencies
North Dakota Bureau of Criminal Investigation
North Dakota Department of Corrections and Rehabilitation
North Dakota Game and Fish Department
North Dakota Highway Patrol
North Dakota State Fire Marshal
North Dakota Parks and Recreation Department
County agencies
Adams County Sheriff's Office
Barnes County Sheriff's Office
Benson County Sheriff's Office
Billings County Sheriff's Office
Bottineau County Sheriff's Office
Bowman County Sheriff's Office
Burke County Sheriff's Office
Burleigh County Sheriff's Office
Cass County Sheriff's Office
Cavalier County Sheriff's Office
Dickey County Sheriff's Office
Divide County Sheriff's Office
Dunn County Sheriff's Office
Eddy County Sheriff's Office
Emmons County Sheriff's Office
Foster County Sheriff's Office
Golden Valley County Sheriff's Office
Grand Forks County Sheriff's Office
Grant County Sheriff's Office
Griggs County Sheriff's Office
Hettinger County Sheriff's Office
Kidder County Sheriff's Office
Lamoure County Sheriff's Office
Logan County Sheriff's Office
McHenry County Sheriff's Office
McIntosh County Sheriff's Office
McKenzie County Sheriff's Office
McLean County Sheriff's Office
Mercer County Sheriff's Office
Morton County Sheriff's Office
Mountrail County Sheriff's Office
Nelson County Sheriff's Office
Oliver County Sheriff's Office
Pembina County Sheriff's Office
Pierce County Sheriff's Office
Ramsey County Sheriff's Office
Ransom County Sheriff's Office
Renville County Sheriff's Office
Richland County Sheriff's Office
Rolette County Sheriff's Office
Sargent County Sheriff's Office
Sheridan County Sheriff's Office
Sioux County Sheriff's Office
Slope County Sheriff's Office
Stark County Sheriff's Office
Steele County Sheriff's Office
Stutsman County Sheriff's Office
Towner County Sheriff's Office
Traill County Sheriff's Office
Walsh County Sheriff's Office
Ward County Sheriff's Office
Wells County Sheriff's Office
Williams County Sheriff's Office
City agencies
Ashley Police Department
Belfield Police Department
Beulah Police Department
Bismarck Police Department
Bowman Police Department
Burlington Police Department
Cando Police Department
Carrington Police Department
Casselton Police Department
Cavalier Police Department
Center Police Department
Cooperstown Police Department
Crosby Police Department
Devils Lake Police Department
Dickinson Police Department
Elgin Police Department
Ellendale Police Department
Emerado Police Department
Fargo Police Department
Fessenden Police Department
Grafton Police Department
Grand Forks Police Department
Harvey Police Department
Hazen Police Department
Hillsboro Police Department
Jamestown Police Department
Kenmare Pol
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Ohio
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This is a list of law enforcement agencies in the state of Ohio.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 831 law enforcement agencies employing 25,992 sworn police officers, about 225 for each 100,000 residents.
State agencies
Ohio Attorney General
Ohio Bureau of Criminal Investigation
Ohio State Auditor
Ohio Board of Pharmacy
Ohio Casino Control Commission
Ohio Department of Commerce
Ohio State Fire Marshal
Ohio Department of Developmental Disabilities
Cambridge Developmental Center
Columbus Developmental Center
Gallipolis Developmental Center
Mount Vernon Developmental Center
Northwest Ohio Developmental Center
Southwest Ohio Developmental Center
Tiffin Developmental Center
Warrensville Developmental Center
Ohio Department of Mental Health and Addiction Services
Appalachian Behavioral Healthcare
Heartland Behavioral Healthcare
Northcoast Behavioral Healthcare
Northwest Ohio Psychiatric Hospital
Summit Behavioral Healthcare
Twin Valley Behavioral Healthcare
Ohio Department of Natural Resources
Division of Parks and Watercraft
Division of Wildlife
Office of Law Enforcement
Ohio Department of Public Safety
Ohio State Highway Patrol
Investigative Unit
Highway Patrol Police
Ohio Department of Rehabilitation and Corrections
Ohio Department of Taxation
Ohio Department of Veterans Services
Veterans Home Police Department
Ohio House of Representatives
Ohio Senate
County agencies
Ohio has 88 counties, each with its own elected county sheriff.
Adams County Sheriff's Office
Allen County Sheriff's Office
Ashland County Sheriff's Office
Ashtabula County Sheriff's Office
Athens County Sheriff's Office
Auglaize County Sheriff's Office
Belmont County Sheriff's Office
Brown County Sheriff's Office
Butler County Sheriff's Office
Carroll County Sheriff's Office
Champaign County Sheriff's Office
Clark County Sheriff's Office
Clermont County Sheriff's Office
Clinton County Sheriff's Office
Columbiana County Sheriff's Office
Coshocton County Sheriff's Office
Crawford County Sheriff's Office
Cuyahoga County Sheriff's Office
Darke County Sheriff's Office
Defiance County Sheriffs Office
Delaware County Sheriff's Office
Erie County Sheriff's Office
Fairfield County Sheriff's Office
Fayette County Sheriff's Office
Franklin County Sheriff's Office
Fulton County Sheriff's Office
Gallia County Sheriff's Office
Geauga County Sheriff's Office
Greene County Sheriff's Office
Guernsey County Sheriff's Office
Hamilton County Sheriff's Office
Hancock County Sheriff's Office
Hardin County Sheriff's Office
Harrison County Sheriff's Office
Henry County Sheriff's Office
Highland County Sheriff's Office
Hocking County Sheriff's Office
Holmes County Sheriff's Office
Huron County Sheriff's Office
Jackson County Sheriff's Office
Jefferson County Sheriff's Office
Knox County Sheriff's Office
Lake County Sheri
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Oklahoma
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This is a list of law enforcement agencies in the state of Oklahoma.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 483 law enforcement agencies employing 8,639 sworn police officers, about 237 for each 100,000 residents.
State agencies
Attorney General of Oklahoma
Grand River Dam Authority Police Department
Oklahoma Alcoholic Beverage Laws Enforcement Commission
Oklahoma Bureau of Narcotics and Dangerous Drugs Control
Oklahoma Office of the Chief Medical Examiner
Oklahoma Council on Law Enforcement Education and Training
Oklahoma Criminal Justice Resource Center
Oklahoma Department of Corrections
Oklahoma Department of Emergency Management
Oklahoma Department of Environmental Quality
Criminal Investigation Unit
Oklahoma Department of Public Safety
Oklahoma Highway Patrol
Oklahoma Office of Homeland Security
Oklahoma Department of Tourism and Recreation
Oklahoma State Park Rangers
Oklahoma Department of Wildlife Conservation
Oklahoma District Attorneys Council
Oklahoma Department of Insurance
Anti-Fraud Unit
Oklahoma Office of Juvenile Affairs
Oklahoma Pardon and Parole Board
Oklahoma State Board of Pharmacy
Oklahoma State Bureau of Investigation
Oklahoma State Fire Marshal
Oklahoma Law Enforcement Retirement System
Oklahoma Police Pension and Retirement System
Oklahoma Department of Agriculture Investigative Services Unit
Oklahoma Department of Human Services Office of Inspector General
County sheriffs
Adair County Sheriff's Office
Alfalfa County Sheriff's Office
Atoka County Sheriff's Office
Beaver County Sheriff's Office
Beckham County Sheriff's Office
Blaine County Sheriff's Office
Bryan County Sheriff's Office
Caddo County Sheriff's Office
Canadian County Sheriff's Office
Carter County Sheriff's Office
Cherokee County Sheriff's Office
Choctaw County Sheriff's Office
Cimarron County Sheriff's Office
Cleveland County Sheriff's Office
Coal County Sheriff's Office
Comanche County Sheriff's Office
Cotton County Sheriff's Office
Craig County Sheriff's Office
Creek County Sheriff's Office
Custer County Sheriff's Office
Delaware County Sheriff's Office
Dewey County Sheriff's Office
Ellis County Sheriff's Office
Garfield County Sheriff's Office
Garvin County Sheriff's Office
Grady County Sheriff's Office
Grant County Sheriff's Office
Greer County Sheriff's Office
Harmon County Sheriff's Office
Harper County Sheriff's Office
Haskell County Sheriff's Office
Hughes County Sheriff's Office
Jackson County Sheriff's Office
Jefferson County Sheriff's Office
Johnston County Sheriff's Office
Kay County Sheriff's Office
Kingfisher County Sheriff's Office
Kiowa County Sheriff's Office
Latimer County Sheriff's Office
Leflore County Sheriff's Office
Lincoln County Sheriff's Office
Logan County Sheriff's Office
Love County Sheriff's Office
Major County Sheriff's Office
Marshall County Sheriff's Office
Mayes County Sheriff's Office
McClain County Sheriff's Office
McCurtain County Sh
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Oregon
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This is a list of law enforcement agencies in the U.S. state of Oregon.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 174 law enforcement agencies employing 6,695 sworn police officers, about 177 for each 100,000 residents.
State agencies
Oregon Board of Parole and Post-Prison Supervision
Oregon Department of Corrections
Oregon Department of Justice
Oregon Department of Transportation Motor Carrier Enforcement
Oregon Liquor and Cannabis Commission
Oregon State Police
Oregon State Fire Marshal
Oregon Youth Authority
County agencies
Baker County Sheriff's Office
Benton County Sheriff's Office
Clackamas County Sheriff's Office
Clatsop County Sheriff's Office
Columbia County Sheriff's Office
Coos County Sheriff's Office
Crook County Sheriff's Office
Curry County Sheriff's Office
Deschutes County Sheriff's Office
Douglas County Sheriff's Office
Gilliam County Sheriff's Office
Grant County Sheriff's Office
Harney County Sheriff's Office
Hood River County Sheriff's Office
Jackson County Sheriff's Office
Jefferson County Sheriff's Office
Josephine County Sheriff's Office
Klamath County Sheriff's Office
Lake County Sheriff's Office
Lane County Sheriff's Office
Lincoln County Sheriff's Office
Linn County Sheriff's Office
Malheur County Sheriff's Office
Marion County Sheriff's Office
Morrow County Sheriff's Office
Multnomah County Sheriff's Office
Polk County Sheriff's Office
Sherman County Sheriff's Office
Tillamook County Sheriff's Office
Umatilla County Sheriff's Office
Union County Sheriff's Office
Wallowa County Sheriff's Office
Wasco County Sheriff's Office
Washington County Sheriff's Office
Wheeler County Sheriff's Office
Yamhill County Sheriff's Office
City agencies
Adair Village Police Department
Albany Police Department
Amity Police Department
Ashland Police Department
Astoria Police Department
Athena Police Department
Aumsville Police Department
Baker City Police Department
Bandon Police Department
Beaverton Police Department
Bend Police Department
Black Butte Ranch Police Department
Boardman Police Department
Brookings Police Department
Burns Police Department
Butte Falls Police Department
Canby Police Department
Cannon Beach Police Department
Carlton Police Department
Central Point Police Department
Clatskanie Police Department
Coburg Police Department
Columbia City Police Department
Condon Police Department
Coos Bay Police Department
Coquille Police Department
Cornelius Police Department
Corvallis Police Department
Cottage Grove Police Department
Dallas Police Department (Oregon)
Eagle Point Police Department
Enterprise Police Department
Eugene Police Department
Fairview Police Department
Florence Police Department
Forest Grove Police Department
Gearhart Police Department
Gervais Police Department
Gladstone Police Department
Gold Beach Police Department
Grants Pass Department
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Rhode%20Island
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According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 48 law enforcement agencies employing 2,828 sworn police officers, about 268 for each 100,000 residents.
State agencies
Rhode Island Department of Corrections
Rhode Island Department of Environmental Management
Division of Law Enforcement
Rhode Island Department of Public Safety
Rhode Island State Police
Rhode Island Capitol Police
Division of Sheriffs
Office of the State Fire Marshal
Rhode Island Office of the Attorney General
Bureau of Criminal Identification and Investigation
Municipal agencies
Barrington Police Department
Bristol Police Department
Burrillive Police Department
Central Falls Police Department
Charlestown Police Department
Coventry Police Department
Cranston Police Department
Cumberland Police Department
East Greenwich Police Department
East Providence Police Department
Foster Police Department
Glocester Police Department
Hopkinton Police Department
Jamestown Police Department
Johnston Police Department
Lincoln Police Department
Little Compton Police Department
Middletown Police Department
Narragansett Police Department
Newport Police Department
New Shoreham Police Department
North Kingstown Police Department
North Providence Police Department
North Smithfield Police Department
Pawtucket Police Department
Portsmouth Police Department
Providence Police Department
Richmond Police Department
Scituate Police Department
Smithfield Police Department
South Kingstown Police Department
Tiverton Police Department
Warren Police Department
Warwick Police Department
West Greenwich Police Department
West Warwick Police Department
Westerly Police Department
Woonsocket Police Department
College and university agencies
Brown University Department of Public Safety
Bryant University Department of Public Safety
Community College of Rhode Island Police Department
Providence College Office of Public Safety
Rhode Island College Campus Police Department
University of Rhode Island Police Department
Other agencies
Narragansett Indian Tribe Environmental Police Department
Narragansett Tribal Police Department
Rhode Island Airport Police
References
Rhode Island
Law enforcement agencies of Rhode Island
Law enforcement agencies
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20South%20Carolina
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This is a list of law enforcement agencies in the state of South Carolina.
According to the US Bureau of Justice Statistics' 2022 'Census of State and Local Law Enforcement Agencies,'' the state had 272 law enforcement agencies employing 11,674 sworn police officers, about 259 for each 100,000 residents.
State agencies
South Carolina Department of Corrections (SCDC)
Office of Inspector General(OIG)
South Carolina Department of Natural Resources, Law Enforcement Division (SCDNR)
South Carolina Department of Public Safety (SCDPS)
South Carolina Highway Patrol (SCHP)
South Carolina State Transport Police Division (SCSTP)
South Carolina Bureau of Protective Services (BPS)
South Carolina Law Enforcement Division (SLED)
South Carolina State Constable's Office
South Carolina State Ports Authority Port Police
South Carolina Department of Probation, Parole, and Pardon Services (SCDPPPS)
South Carolina Department of Mental Health Public Safety (SCDMHPS)
South Carolina State Forestry Commission Law Enforcement
South Carolina Department of Juvenile Justice
County agencies
City and town agencies
College agencies
Allen University Police Department
Benedict College Police Department
Bob Jones University Police Department
Clemson University Police Department
Coastal Carolina University Department of Public Safety
Denmark Technical College Department of Public Safety
Erskine College Police Department
Francis Marion University Police Department
Furman University Police Department
Greenville Technical College Police Department
Lander University Police Department
Limestone University Police Department
Medical University of South Carolina Department of Public Safety
Midlands Technical College Campus Police
Orangeburg-Calhoun Technical College Police Department
Presbyterian College Campus Police Department
South Carolina State University Police Department
Spartanburg Community College Police Department
Spartanburg Methodist College Campus Safety Department
The Citadel Department of Public Safety
Tri County Technical College Campus Safety
Trident Technical College Department of Public Safety
University of South Carolina-Aiken Police Department
University of South Carolina-Beaufort Department of Public Safety
University of South Carolina Police Department
University of South Carolina Upstate Department of Public Safety
Winthrop University Police Department
Wofford College Campus Safety
Other agencies
Charleston County Aviation Authority Police Department
Columbia Metro Housing Authority Police Department
Columbia Metropolitan Airport Police Department
Dorn VA Medical Center Police Department
Florence Regional Airport Department of Public Safety
Joint Base Charleston Security Forces
Lexington Medical Center Police Department
Marine Corps Air Station Beaufort Provost Marshal
Former agencies
Bethune Police Department
References
South Carolina
Law enforcement agencies of South Carolina
Law enforcement agencies
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20South%20Dakota
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This is a list of law enforcement agencies in the state of South Dakota.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 155 law enforcement agencies employing 1,636 sworn police officers, about 203 for each 100,000 residents.
State agencies
South Dakota Division of Criminal Investigation
South Dakota Department of Corrections
South Dakota Department of Public Safety
South Dakota Department of Public Safety Safety & Enforcement
South Dakota Highway Patrol
South Dakota Office of Highway Safety
South Dakota Department of Homeland Security
South Dakota Game, Fish, & Parks Conservation Officers
South Dakota Commission on Gaming
South Dakota Motor Carrier
The Huron Police Department is the South Dakota's State Fair Police Department.
Federal Agencies within South Dakota
Bureau of Alcohol, Tobacco, Firearms and Explosives Field Divisions
Bureau of Alcohol, Tobacco, Firearms and Explosives St. Paul Field Division/South Dakota Field Offices
Bureau of Indian Affairs
Bureau of Indian Affairs Regional Offices
Bureau of Indian Affairs Office of Justice Services
Bureau of Indian Affairs Office of Justice Services Districts
Bureau of Indian Affairs Office of Justice Services Drug Division Enforcement
Drug Enforcement Administration
Drug Enforcement Administration Field Offices
Federal Bureau of Investigation
National Park Service
United States Marshal's Service
County agencies
Aurora County Sheriff's Office
Beadle County Sheriff's Office
Bennett County Sheriff's Office
Bon Homme County Sheriff's Office
Brookings County Sheriff's Office
Brown County Sheriff's Office
Brule County Sheriff's Office
Buffalo County Sheriff's Office
Butte County Sheriff's Office
Campbell County Sheriff's Office
Charles Mix County Sheriff's Office
Clark County Sheriff's Office
Clay County Sheriff's Office
Codington County Sheriff's Office
Corson County Sheriff's Office
Custer County Sheriff's Office
Davison County Sheriff's Office
Day County Sheriff's Office
Deuel County Sheriff's Office
Dewey County Sheriff's Office
Douglas County Sheriff's Office
Edmunds County Sheriff's Office
Fall River County Sheriff's Office
Faulk County Sheriff's Office
Grant County Sheriff's Office
Gregory County Sheriff's Office
Haakon County Sheriff's Office
Hamlin County Sheriff's Office
Hand County Sheriff's Office
Hanson County Sheriff's Office
Harding County Sheriff's Office
Hughes County Sheriff's Office
Hutchinson County Sheriff's Office
Hyde County Sheriff's Office
Jackson County Sheriff's Office
Jerauld County Sheriff's Office
Jones County Sheriff's Office
Kingsbury County Sheriff's Office
Lake County Sheriff's Office
Lawrence County Sheriff's Office
Lincoln County Sheriff's Office
Lyman County Sheriff's Office
Marshall County Sheriff's Office
McCook County Sheriff's Office
McPherson County Sheriff's Office
Meade County Sheriff's Office
Mellette County Sheriff's Office
Miner County Sheriff's
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Tennessee
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This is a list of law enforcement agencies in the state of Tennessee.
According to the United States Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 375 law enforcement agencies employing 15,976 sworn police officers, about 256 for each 100,000 residents.
State agencies
Tennessee Alcoholic Beverage Commission
Tennessee Bureau of Investigation
Tennessee Department of Correction
Tennessee Department of Revenue
Tennessee Department of Safety and Homeland Security
Tennessee Highway Patrol
Tennessee Wildlife Resources Agency
Tennessee Governor's Task Force on Marijuana Eradication
County agencies
Anderson County Sheriff's Office
Bedford County Sheriff's Office
Benton County Sheriff's Office
Bledsoe County Sheriff's Office
Blount County Sheriff's Office
Bradley County Sheriff's Office
Campbell County Sheriff's Office
Cannon County Sheriff's Office
Carroll County Sheriff's Office
Carter County Sheriff's Office
Cheatham County Sheriff's Office
Chester County Sheriff's Office
Claiborne County Sheriff's Office
Clay County Sheriff's Office
Cocke County Sheriff's Office
Coffee County Sheriff's Office
Crockett County Sheriff's Office
Cumberland County Sheriff's Office
Davidson County Sheriff's Office
Decatur County Sheriff's Office
DeKalb County Sheriff's Office
Dickson County Sheriff's Office
Dyer County Sheriff's Office
Fayette County Sheriff's Office
Fentress County Sheriff's Office
Franklin County Sheriff's Office
Gibson County Sheriff's Office
Giles County Sheriff's Office
Grainger County Sheriff's Office
Greene County Sheriff's Office
Grundy County Sheriff's Office
Hamblen County Sheriff's Office
Hamilton County Sheriff's Office
Hancock County Sheriff's Office
Hardeman County Sheriff's Office
Hardin County Sheriff's Office
Hawkins County Sheriff's Office
Haywood County Sheriff's Office
Henderson County Sheriff's Office
Henry County Sheriff's Office
Hickman County Sheriff's Office
Houston County Sheriff's Office
Humphreys County Sheriff's Office
Jackson County Sheriff's Office
Jefferson County Sheriff's Office
Johnson County Sheriff's Office
Knox County Sheriff's Office
Lake County Sheriff's Office
Lauderdale County Sheriff's Office
Lawrence County Sheriff's Office
Lewis County Sheriff's Office
Lincoln County Sheriff's Office
Loudon County Sheriff's Office
Macon County Sheriff's Office
Madison County Sheriff's Office
Marion County Sheriff's Office
Marshall County Sheriff's Office
Maury County Sheriff's Office
McMinn County Sheriff's Office
McNairy County Sheriff's Office
Meigs County Sheriff's Office
Monroe County Sheriff's Office
Montgomery County Sheriff's Office
Moore County Sheriff's Office
Morgan County Sheriff's Office
Obion County Sheriff's Office
Overton County Sheriff's Office
Perry County Sheriff's Office
Pickett County Sheriff's Office
Polk County Sheriff's Office
Putnam County Sheriff's Office
Rhea County Sheriff's Office
Roane County Sheriff's Offi
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Texas
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This is a list of law enforcement agencies in the U.S. state of Texas.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 1,913 law enforcement agencies, the most of any state. These agencies employed 59,219 sworn police officers, about 244 for each 100,000 residents.
Federal agencies
There are over 150 federal law enforcement offices in Texas. including those for the Federal Bureau of Prisons, Bureau of Alcohol, Tobacco, Firearms and Explosives; Customs and Border Protection; Drug Enforcement Administration; Federal Bureau of Investigation; Immigration and Customs Enforcement; United States Secret Service; Naval Criminal Investigative Service, and U.S. Marshals. According to the Bureau of Justice Statistics, a federal law enforcement agency is "an organizational unit, or subunit, of the federal government with the principle (sic) functions of prevention, detection, and investigation of crime and the apprehension of alleged offenders."
State agencies
Texas Alcoholic Beverage Commission
Texas Attorney General
Texas Comptroller - Criminal Investigation Division (State Police)
Texas Commission on Law Enforcement (regulatory agency)
Texas Department of Criminal Justice
Texas Department of Insurance
State Fire Marshal's Office
Fraud Unit
Texas Department of Public Safety
Texas Highway Patrol
Texas Ranger Division
Texas Parks and Wildlife Department
Texas Racing Commission - State Police
Texas Juvenile Justice Department – Office of the Inspector General
Texas and Southwestern Cattle Raisers Association Special Rangers
Texas State Board of Dental Examiners - Investigations Division
County sheriff agencies
Anderson County Sheriff's Office
Andrews County Sheriff's Office
Angelina County Sheriff's Office
Aransas County Sheriff's Office
Archer County Sheriff's Office
Armstrong County Sheriff's Office (Texas)
Atascosa County Sheriff's Office
Austin County Sheriff's Office
Bailey County Sheriff's Office
Bandera County Sheriff's Office
Bastrop County Sheriff's Office
Baylor County Sheriff's Office
Bee County Sheriff's Office
Bell County Sheriff's Office
Bexar County Sheriff's Office
Blanco County Sheriff's Office
Borden County Sheriff's Office
Bosque County Sheriff's Office
Bowie County Sheriff's Office
Brazoria County Sheriff's Office
Brazos County Sheriff's Office (Texas)
Brewster County Sheriff's Office
Briscoe County Sheriff's Office
Brooks County Sheriff's Office
Brown County Sheriff's Office
Burleson County Sheriff's Office
Burnet County Sheriff's Office
Caldwell County Sheriff's Office
Calhoun County Sheriff's Office
Callahan County Sheriff's Office
Cameron County Sheriff's Office
Camp County Sheriff's Office
Carson County Sheriff's Office
Cass County Sheriff's Office
Castro County Sheriff's Office
Chambers County Sheriff's Office
Cherokee County Sheriff's Office
Childress County Sheriff's Office
Clay County Sheriff's Office
Cochra
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Utah
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This is a list of law enforcement agencies in the state of Utah.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 136 law enforcement agencies employing 4,782 sworn police officers, about 175 for each 100,000 residents.
State agencies
Utah Department of Corrections
Utah Adult Probation & Parole
Utah Department of Human Services
Division of Juvenile Justice Services
Utah Department of Natural Resources
Utah Division of State Parks and Recreation
Park Rangers
Utah Division of Wildlife Resources
Conservation Officers
Utah Attorney General and Assistant Attorneys General
Utah Department of Public Safety
Liquor Enforcement Section
Utah State Fire Marshal
Utah Highway Patrol
Utah State Bureau of Investigation
Utah Department of Transportation - Motor Carrier Safety - Port of Entry
Utah Division of Insurance Fraud
Utah State Tax Commission - Motor Vehicle Enforcement Division (MVED)
Utah State Hospital - Campus Police
County agencies
Beaver County Sheriff's Office
Box Elder County Sheriff's Office
Cache County Sheriff's Office
Carbon County Sheriff's Office
Daggett County Sheriff's Office
Davis County Sheriff's Office
Duchesne County Sheriff's Office
Emery County Sheriff's Office
Garfield County Sheriff's Office
Grand County Sheriff's Office
Iron County Sheriff's Office
Juab County Sheriff's Office
Kane County Sheriff's Office
Millard County Sheriff's Office
Morgan County Sheriff's Office
Piute County Sheriff's Office
Rich County Sheriff's Office
Salt Lake County Sheriff's Office
San Juan County Sheriff's Office
Sanpete County Sheriff's Office
Sevier County Sheriff's Office
Summit County Sheriff's Office
Tooele County Sheriff's Office
Uintah County Sheriff's Office
Unified Police Department of Greater Salt Lake
Utah County Sheriff's Office
Wasatch County Sheriff's Office
Washington County Sheriff's Office
Wayne County Sheriff's Office
Weber County Sheriff's Office
City agencies
Alta Marshal's Office
Aurora Police Department
American Fork Police Department
Blanding Police Department
Bluffdale Police Department
Bountiful Police Department
Brigham City Police Department
Cedar City Police Department
Centerville Police Department
Clearfield Police Department
Clinton Police Department
Cottonwood Heights Police Department
Draper Police Department
East Carbon Police Department
Enoch Police Department
Ephraim City Police Department
Escalante Police Department
Farmington Police Department
Grantsville Police Department
Gunnison Police Department
Harrisville Police Department
Heber City Police Department
Helper Police Department
Herriman Police Department
Hurricane Police Department
Hyrum Police Department
Kamas Police Department
Kanab Police Department
Kaysville Police Department
La Verkin Police Department
Layton Police Department
Lehi Police Department
Lindon Police Department
Logan Police Departm
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Vermont
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This is a list of law enforcement agencies in the state of Vermont.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 69 law enforcement agencies employing 1,103 sworn police officers, about 178 for each 100,000 residents.
State agencies
Vermont Capitol Police
Vermont Department of Public Safety
Vermont State Police
Vermont Fish and Wildlife Department
Division of Warden Service
Vermont Department of Corrections
Vermont Department of Liquor and Lottery
Division of Liquor Control
Vermont Secretary of State
Office of Professional Regulation
Investigative Unit
Vermont Department of Motor Vehicles
Enforcement and Safety Division
Office of the Vermont Attorney General
County agencies
Addison County Sheriff's Office
Bennington County Sheriff's Department
Caledonia County Sheriff's Office
Chittenden County Sheriff's Department
Essex County Sheriff's Office
Franklin County Sheriff's Department
Grand Isle County Sheriff's Office
Lamoille County Sheriff's Department
Orange County Sheriff's Department
Orleans County Sheriff's Office
Rutland County Sheriff's Office
Washington County Sheriff's Office
Windham County Sheriff's Department
Windsor County Sheriff's Department
Municipal agencies
Barre Police Department (City of Barre)
Barre Police Department (Town of Barre)
Bellows Falls Police Department
Bennington Police Department
Berlin Police Department
Bradford Police Department
Brandon Police Department
Brattleboro Police Department
Brighton Police Department
Bristol Police Department
Burlington Police Department
Canaan Police Department
Castleton Police Department
Chester Police Department
Colchester Police Department
Dover Police Department
Essex Police Department
Fairlee Police Department
Fair Haven Police Department
Hardwick Police Department
Hartford Police Department
Hinesburg Police Department
Killington Police Department
Ludlow Police Department
Lyndonville Police Department
Manchester Police Department (Town of Manchester)
Middlebury Police Department
Milton Police Department
Morristown Police Department
Montpelier Police Department
Mount Tabor Police Department
Newport Police Department (City of Newport)
Northfield Police Department
Norwich Police Department
Pittsford Police Department
Randolph Police Department
Richmond Police Department
Royalton Police Department
Rutland Police Department (City of Rutland)
Rutland Police Department (Town of Rutland)
Shelburne Police Department
South Burlington Police Department
Springfield Police Department
St. Albans Police Department (City of St. Albans)
St. Johnsbury Police Department
Stowe Police Department
Swanton Police Department (Village of Swanton)
Thetford Police Department
Vergennes Police Department
Weathersfield Police Department
Williston Police Department
Wilmington Police Department
Windsor Police Department
Winhall Police Department
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Virginia
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This is a list of law enforcement agencies in the state of Virginia.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 340 law enforcement agencies employing 22,848 sworn police officers, about 293 for each 100,000 residents.
State agencies
Virginia Alcoholic Beverage Control Authority
Virginia Department of Agriculture and Consumer Services - Charitable Gaming
Virginia Department of Corrections
Virginia Department of Fire Programs - State Fire Marshal's Office
Virginia Department of Forestry
Virginia Department of Juvenile Justice
Virginia Department of Motor Vehicles Law Enforcement Division
Virginia Department of Wildlife Resources Conservation Police
General Assembly - Virginia Division of Capitol Police
Virginia Marine Resources Commission - Virginia Marine Police
Virginia Office of State Inspector General
Virginia Port Authority Police
Virginia State Corporation Commission - Bailiffs
Virginia State Lottery Security Division
Virginia State Parks Police
Virginia State Police
Special District agencies
Chesapeake Bay Bridge-Tunnel District and Commission Police Department
County agencies
Accomack County Sheriff's Office
Albemarle County Police Department
Albemarle County Sheriff's Office
Allegheny County Sheriff's Office
Amelia County Sheriff's Office
Amherst County Sheriff's Office
Appomattox County Sheriff's Office
Arlington County Police Department
Arlington County Sheriff's Office
Augusta County Sheriff's Office
Bath County Sheriff's Office
Bedford County Sheriff's Office
Bland County Sheriff's Office
Botetourt County Sheriff's Office
Brunswick County Sheriff's Office
Buchanan County Sheriff's Office
Buckingham County Sheriff's Office
Campbell County Sheriff's Office
Caroline County Sheriff's Office
Carroll County Sheriff's Office
Charles City County Sheriff's Office
Charlotte County Sheriff's Office
Chesterfield County Police Department
Chesterfield County Sheriff's Office
Clarke County Sheriff's Office
Craig County Sheriff's Office
Culpeper County Sheriff's Office
Cumberland County Sheriff's Office
Dickenson County Sheriff's Office
Dinwiddie County Sheriff's Office
Essex County Sheriff's Office
Fairfax County Police Department
Fairfax County Sheriff's Office
Fauquier County Sheriff's Office
Floyd County Sheriff's Office
Fluvanna County Sheriff's Office
Franklin County Sheriff's Office
Frederick County Sheriff's Office
Giles County Sheriff's Office
Gloucester County Sheriff's Office
Goochland County Sheriff's Office
Grayson County Sheriff's Office
Greene County Sheriff's Office
Greensville County Sheriff's Office
Halifax County Sheriff's Office
Hanover County Sheriff's Office
Henrico County Police Department
Henrico County Sheriff's Office
Henry County Sheriff's Office
Highland County Sheriff's Office
Isle of Wight County Sheriff's Office
James City County Police Department
King and Queen Co
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Washington%20%28state%29
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This is a list of law enforcement agencies in the US state of Washington.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 260 law enforcement agencies employing 11,411 sworn police officers, about 174 for each 100,000 residents. The state has the lowest ratio of police officers to residents of any state, compared to a national average of 251 per 100,000 residents.
State agencies
Washington State Attorney General's Office
Washington State Department of Corrections
Washington State Department of Fish and Wildlife
Washington State Department of Labor and Industries
Washington State Department of Natural Resources
Washington State Gambling Commission
Washington State Liquor and Cannabis Board
Washington State Parks
Washington State Patrol
County agencies
Adams County Sheriff's Office
Asotin County Sheriff's Office
Benton County Sheriff's Office
Chelan County Sheriff's Office
Clallam County Sheriff's Office
Clark County Sheriff's Office
Columbia County Sheriff's Office
Cowlitz County Sheriff's Office
Douglas County Sheriff's Office
Ferry County Sheriff's Office
Franklin County Sheriff's Office
Garfield County Sheriff's Office
Grant County Sheriff's Office
Grays Harbor County Sheriff's Office
Island County Sheriff's Office
Jefferson County Sheriff's Office
King County Sheriff's Office
Kitsap County Sheriff's Office
Kittitas County Sheriff's Office
Klickitat County Sheriff's Office
Lewis County Sheriff's Office
Lincoln County Sheriff's Office
Mason County Sheriff's Office
Okanogan County Sheriff's Office
Pacific County Sheriff's Office
Pend Oreille County Sheriff's Office
Pierce County Sheriff's Department
San Juan County Sheriff's Office
Skagit County Sheriff's Office
Skamania County Sheriff's Office
Snohomish County Sheriff's Office
Spokane County Sheriff's Office
Stevens County Sheriff's Office
Thurston County Sheriff's Office
Wahkiakum County Sheriff's Office
Walla Walla County Sheriff's Office
Whatcom County Sheriff's Office
Whitman County Sheriff's Office
Yakima County Sheriff's Office
Local agencies
Aberdeen Police Department
Airway Heights Police Department
Algona Police Department
Anacortes Police Department
Arlington Police Department
Auburn Police Department
Bainbridge Island Police Department
Battle Ground Police Department
Bellevue Police Department
Bellingham Police Department
Bingen-White Salmon Police Department
Black Diamond Police Department
Blaine Police Department
Bonney Lake Police Department
Bothell police Department
Bremerton Police Department
Brewster Police Department
Brier Police Department
Buckley Police Department
Burlington Police Department
Camas Police Department
Castle Rock Police Department
Centralia Police Department
Chehalis Police Department
Chelan Police Department
Cheney Police Department
Chewelah Police Department
Cle Elum Roslyn South Cle Elum Police Department
C
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20West%20Virginia
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This is a list of law enforcement agencies in the state of West Virginia.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 233 law enforcement agencies employing 3,382 sworn police officers, about 186 for each 100,000 residents.
State agencies
Hatfield-McCoy Recreation Area Park Rangers
West Virginia Alcohol Beverage Control Administration
West Virginia Division of Forestry Special Operations and Enforcement Division
West Virginia Division of Natural Resources Law Enforcement Section (Natural Resources Police)
West Virginia Division of Protective Services
West Virginia Public Service Commission Motor Carrier Enforcement
West Virginia State Fire Marshal's Office
West Virginia State Police
West Virginia Military Authority - Camp Dawson Security
State Correctional agencies
West Virginia Division of Corrections and Rehabilitation
West Virginia Division of Juvenile Services (defunct)
West Virginia Regional Jail Authority (defunct)
County agencies
Barbour County Sheriff's Office
Berkeley County Sheriff's Office
Boone County Sheriff's Office
Braxton County Sheriff's Office
Brooke County Sheriff's Office
Cabell County Sheriff's Office
Calhoun County Sheriff's Office
Clay County Sheriff's Office
Doddridge County Sheriff's Office
Fayette County Sheriff's Office
Gilmer County Sheriff's Office
Grant County Sheriff's Office
Greenbrier County Sheriff's Office
Hampshire County Sheriff's Office
Hancock County Sheriff's Office
Hardy County Sheriff's Department
Harrison County Sheriff's Department
Jackson County Sheriff's Department
Jefferson County Sheriff's Department
Kanawha County Sheriff's Department
Lewis County Sheriff's Department
Lincoln County Sheriff's Department
Logan County Sheriff's Department
Marion County Sheriff's Department
Marshall County Sheriff's Department
Mason County Sheriff's Department
McDowell County Sheriff's Department
Mercer County Sheriff's Department
Mineral County Sheriff's Department
Mingo County Sheriff's Department
Monongalia County Sheriff's Department
Monroe County Sheriff's Department
Morgan County Sheriff's Department
Nicholas County Sheriff's Department
Ohio County Sheriff's Department
Pendleton County Sheriff's Department
Pleasants County Sheriff's Department
Pocahontas County Sheriff's Department
Preston County Sheriff's Department
Putnam County Sheriff's Department
Raleigh County Sheriff's Department
Randolph County Sheriff's Department
Ritchie County Sheriff's Department
Roane County Sheriff's Department
Summers County Sheriff's Department
Taylor County Sheriff's Department
Tucker County Sheriff's Department
Tyler County Sheriff's Department
Upshur County Sheriff's Department
Wayne County Sheriff's Department
Webster County Sheriff's Department
Wetzel County Sheriff's Office
Wirt County Sheriff's Office
Wood County Sheriff's Office
Wyoming County Sheriff's Off
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Wisconsin
|
This is a list of law enforcement agencies in the state of Wisconsin.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 529 law enforcement agencies employing 13,730 sworn police officers, about 186 for each 100,000 residents.
State agencies
Wisconsin Capitol Police
Wisconsin Department of Corrections
Wisconsin Department of Justice
Wisconsin Department of Natural Resources
Division of Forestry
Foresters, Forester-Rangers
Division of Enforcement and Science; Bureau of Law Enforcement
Conservation Wardens, Deputy Conservation Wardens
Wisconsin Department of Transportation
Wisconsin State Patrol
Troopers, Inspectors
Wisconsin State Fair Park Police Department
County agencies
Adams County Sheriff's Office
Ashland County Sheriff's Office
Barron County Sheriff's Office
Bayfield County Sheriff's Office
Brown County Sheriff's Office
Buffalo County Sheriff's Office
Burnett County Sheriff's Office
Calumet County Sheriff's Office
Chippewa County Sheriff's Office
Clark County Sheriff's Office
Columbia County Sheriff's Office
Crawford County Sheriff's Office
Dane County Sheriff's Office
Dodge County Sheriff's Office
Door County Sheriff's Office
Douglas County Sheriff's Office
Dunn County Sheriff's Office
Eau Claire County Sheriff's Office
Florence County Sheriff's Office
Fond du Lac County Sheriff's Office
Forest County Sheriff's Office
Grant County Sheriff's Office
Green County Sheriff's Office
Green Lake County Sheriff's Office
Iowa County Sheriff's Office
Iron County Sheriff's Office
Jackson County Sheriff's Office
Jefferson County Sheriff's Office
Juneau County Sheriff's Office
Kenosha County Sheriff's Office
Kewaunee County Sheriff's Office
La Crosse County Sheriff's Office
Lafayette County Sheriff's Office
Langlade County Sheriff's Office
Lincoln County Sheriff's Office
Manitowoc County Sheriff's Office
Marathon County Sheriff's Office
Marinette County Sheriff's Office
Marquette County Sheriff's Office
Menominee County Sheriff's Office
Milwaukee County Sheriff's Office
Monroe County Sheriff's Office
Oconto County Sheriff's Office
Oneida County Sheriff's Office
Outagamie County Sheriff's Office
Ozaukee County Sheriff's Office
Pepin County Sheriff's Office
Pierce County Sheriff's Office
Polk County Sheriff's Office
Portage County Sheriff's Office
Price County Sheriff's Office
Racine County Sheriff's Office
Richland County Sheriff's Office
Rock County Sheriff's Office
Rusk County Sheriff's Office
Saint Croix County Sheriff's Office
Sauk County Sheriff's Office
Sawyer County Sheriff's Office
Shawano County Sheriff's Office
Sheboygan County Sheriff's Office
Taylor County Sheriff's Office
Trempealeau County Sheriff's Office
Vernon County Sheriff's Office
Vilas County Sheriff's Office
Walworth County Sheriff's Office
Washburn County Sheriff's Office
Washington County Sheriff's Office
Waukesha County Sh
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https://en.wikipedia.org/wiki/Riemann%E2%80%93Hilbert%20problem
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In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations in the complex plane. Several existence theorems for Riemann–Hilbert problems have been produced by Mark Krein, Israel Gohberg and others (see the book by Clancey and Gohberg (1981)).
The Riemann problem
Suppose that is a closed simple contour in the complex plane dividing the plane into two parts denoted by (the inside) and (the outside), determined by the index of the contour with respect to a point. The classical problem, considered in Riemann's PhD dissertation (see ), was that of finding a function
analytic inside such that the boundary values of M+ along satisfy the equation
for all , where a, b, and c are given real-valued functions .
By the Riemann mapping theorem, it suffices to consider the case when is the unit circle . In this case, one may seek M+(z) along with its Schwarz reflection:
On the unit circle Σ, one has , and so
Hence the problem reduces to finding a pair of functions M+(z) and M−(z) analytic, respectively, on the inside and the outside of the unit disc, so that on the unit circle
and, moreover, so that the condition at infinity holds:
The Hilbert problem
Hilbert's generalization was to consider the problem of attempting to find M+ and M− analytic, respectively, on the inside and outside of the curve Σ, such that on one has
where α, β, and c are arbitrary given complex-valued functions (no longer just complex conjugates).
Riemann–Hilbert problems
In the Riemann problem as well as Hilbert's generalization, the contour was simple. A full Riemann–Hilbert problem allows that the contour may be composed of a union of several oriented smooth curves, with no intersections. The + and − sides of the "contour" may then be determined according to the index of a point with respect to . The Riemann–Hilbert problem is to find a pair of functions, M+ and M− analytic, respectively, on the + and − side of , subject to the equation
for all z ∈ Σ.
Generalization: Matrix factorization problems
Given an oriented "contour" Σ (technically: an oriented union of smooth curves without points of infinite self-intersection in the complex plane), a Riemann–Hilbert factorization problem is the following.
Given a matrix function V defined on the contour Σ, to find a holomorphic matrix function M defined on the complement of Σ, such that two conditions be satisfied:
If M+ and M− denote the non-tangential limits of M as we approach Σ, then M+ = M−V, at all points of non-intersection in Σ.
As z tends to infinity along any direction outside Σ, M tends to the identity matrix.
In the simplest case V is smooth and integrable. In more complicated cases it could have singularities. The limits M+ and M− could be classical and continuous or they could be taken in the L2 sense.
At end-points or intersection points of the contour Σ the jump condition is not defined; constraints on
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https://en.wikipedia.org/wiki/List%20of%20law%20enforcement%20agencies%20in%20Wyoming
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This is a list of law enforcement agencies in the state of Wyoming.
According to the US Bureau of Justice Statistics' 2008 Census of State and Local Law Enforcement Agencies, the state had 90 law enforcement agencies employing 1,691 sworn police officers, about 317 for each 100,000 residents.
State agencies
Wyoming Department of Transportation#Child agency
511 Info Wyoming DOT 511 Info Road
Wyoming DOT Aeronautics
Wyoming Highway Patrol
WyDOT Tax Fuel
Wyoming Department of Corrections
Wyoming State Attorney General's Office
Division of Criminal Investigation
Wyoming Game and Fish Department
Game Wardens
Wyoming Division of State Parks and Historic Sites
Park Rangers
Wyoming Department of Fire Prevention and Electrical Safety (State Fire Marshal)
Fire Investigation
Wyoming Livestock Board
Law Enforcement
Wyoming Gaming Commission (Employs two sworn investigators)
Wyoming State Board of Outfitters and Professional Guides (Employs one sworn investigator)
Wyoming Supreme Court (Employs one sworn bailiff)
County agencies
Albany County Sheriff's Office
Big Horn County Sheriff's Office
Campbell County Sheriff's Office
Carbon County Sheriff's Office
Converse County Sheriff's Office
Crook County Sheriff's Office
Fremont County Sheriff's Office
Goshen County Sheriff's Office
Hot Springs County Sheriff's Office
Johnson County Sheriff's Office
Laramie County Sheriff's Office
Lincoln County Sheriff's Office
Natrona County Sheriff's Office
Niobrara County Sheriff's Office
Park County Sheriff's Office
Platte County Sheriff's Office
Sheridan County Sheriff's Office
Sublette County Sheriff's Office
Sweetwater County Sheriff's Office
Teton County Sheriff's Office
Uinta County Sheriff's Office
Washakie County Sheriff's Office
Weston County Sheriff's Office
Municipal agencies
Afton Police Department
Bairoil Police Department
Basin Police Department
Buffalo Police Department
Casper Police Department
Cheyenne Police Department
Cody Police Department
Cokeville Police Department
Diamondville Police Department
Douglas Police Department
Encampment Police Department
Evanston Police Department
Evansville Police Department
Frannie Police Department
Gillette Police Department
Glenrock Police Department
Green River Police Department
Greybull Police Department
Guernsey Police Department
Hanna Marshal's Office
Hulett Police Department
Jackson Police Department
Kemmerer Police Department
Lander Police Department
Laramie Police Department
Lingle Police Department
Lovell Police Department
Lusk Police Department
Lyman Police Department
Medicine Bow Marshal's Office
Midwest Police Department
Mills Police Department
Moorcroft Police Department
Mountain View Police Department
Newcastle Police Department
Pine Bluffs Police Department
Powell Police Department
Rawlins Police Department
Riverton Police Department
Rock Springs Police Department
Saratoga Police Department
Sheridan Police Departmen
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https://en.wikipedia.org/wiki/List%20of%20Middlesbrough%20F.C.%20records%20and%20statistics
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This article contains the honours, records and statistics of Middlesbrough Football Club. This article lists all of the major honours won by Middlesbrough since their foundation. This list also lists the major playing honours including top goalscorer and most appearances. The Club records including record transfer fees are shown below as are international player honours.
Middlesbrough are an English professional association football club based in Middlesbrough, in the Tees Valley, who currently play in the EFL Championship. The club was founded in 1876 and have played at their current home ground, the Riverside Stadium, since 1995. Middlesbrough were founding members of the Premier League in 1992. They have won one major trophy in their history: the 2004 Football League Cup.
Honours
Domestic
League
Football League Second Division / Football League Division One
Champions 1926–27, 1928–29, 1973–74, 1994–95; runners up 1901–02, 1991–92, 1997–98, 2015–16
Football League Third Division
Runners up 1966–67, 1986–87
Northern League
Champions 1893–94, 1894–95, 1896–97; runners up 1890–91, 1891–92, 1897–98
Cup
League Cup
Winners 2003–04; runners up 1996–97, 1997–98
FA Cup
Runners up 1996–97
FA Amateur Cup
Winners 1894–95, 1897–98
Zenith Data Systems Cup
Runners up 1990
International
UEFA Cup
Runners up 2005–06
Anglo-Scottish Cup
Winners 1975
Kirin Cup
Winners 1980
Player records
Appearances
Youngest first-team player – 16 years and 72 days
Nathan Wood (vs Notts County 14 August 2018)
Oldest first-team player – 40 years and 68 days
Dimitrios Konstantopoulos (vs Newport County 5 February 2019)
Most consecutive appearances – 305
David Armstrong, between March 1973 and August 1980
Most appearances
As of 22 May 2008. Competitive matches only, appearances as substitutes in brackets.
{| class="wikitable sortable" style="text-align: center;"
!width=5%|#
!width=20%|Name
!width=12%|Years
!width=12%|League
!width=12%|FA Cup
!width=14%|League Cup
!width=12%|Other
!width=13%|Total
|-
|1||style="text-align:left;"| ||1902–1923||563 (0)||39 (0)||0 (0)||0 (0)||602 (0)
|-
|2||style="text-align:left;"|
||1960–1973||457 (5)||40 (0)||26 (0)||4 (0)||527 (5)
|-
|3||style="text-align:left;"|
||2004–2023||395 (20)||37 (0)||26 (4)||15 (2)||473 (26)
|-
|4||style="text-align:left;"|
||1971–1982||409 ||33 ||31 ||15 ||488
|-
|5||style="text-align:left;"|
||1971–1983||401 ||34 ||33 ||13 ||481
|-
|6||style="text-align:left;"|
||1925–1939||418 ||35 ||0 (0)||0 (0)||453
|-
|7||style="text-align:left;"|
||1910–1930||421 ||28 ||0 (0)||0 (0)||449
|-
|8||style="text-align:left;"|
||1997–2008||367 ||32 ||26 ||21 ||446
|-
|9||style="text-align:left;"|
||1971–1981||359 ||29 ||28 ||15 ||431
|-
|10=||style="text-align:left;"|
||1982–1992||348 ||23 ||29 ||24 ||424
|-
|10=||style="text-align:left;"|
||1983–1995||339 ||25 ||32 ||28 ||424
|}
Goalscorers
Most goals in a season – 63
George Camsell (Second Division, 1926–1927)
Most League goals in a season – 59
George Camsel
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https://en.wikipedia.org/wiki/Bing%20metrization%20theorem
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In topology, the Bing metrization theorem, named after R. H. Bing, characterizes when a topological space is metrizable.
Formal statement
The theorem states that a topological space is metrizable if and only if it is regular and T0 and has a σ-discrete basis. A family of sets is called σ-discrete when it is a union of countably many discrete collections, where a family of subsets of a space is called discrete, when every point of has a neighborhood that intersects at most one member of
History
The theorem was proven by Bing in 1951 and was an independent discovery with the Nagata–Smirnov metrization theorem that was proved independently by both Nagata (1950) and Smirnov (1951). Both theorems are often merged in the Bing-Nagata-Smirnov metrization theorem. It is a common tool to prove other metrization theorems, e.g. the Moore metrization theorem – a collectionwise normal, Moore space is metrizable – is a direct consequence.
Comparison with other metrization theorems
Unlike the Urysohn's metrization theorem which provides a sufficient condition for metrization, this theorem provides both a necessary and sufficient condition for a topological space to be metrizable.
See also
References
"General Topology", Ryszard Engelking, Heldermann Verlag Berlin, 1989.
Theorems in topology
de:Satz von Bing-Nagata-Smirnow
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https://en.wikipedia.org/wiki/Collectionwise%20normal%20space
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In mathematics, a topological space is called collectionwise normal if for every discrete family Fi (i ∈ I) of closed subsets of there exists a pairwise disjoint family of open sets Ui (i ∈ I), such that Fi ⊆ Ui. Here a family of subsets of is called discrete when every point of has a neighbourhood that intersects at most one of the sets from .
An equivalent definition of collectionwise normal demands that the above Ui (i ∈ I) themselves form a discrete family, which is stronger than pairwise disjoint.
Some authors assume that is also a T1 space as part of the definition, but no such assumption is made here.
The property is intermediate in strength between paracompactness and normality, and occurs in metrization theorems.
Properties
A collectionwise normal space is collectionwise Hausdorff.
A collectionwise normal space is normal.
A Hausdorff paracompact space is collectionwise normal. In particular, every metrizable space is collectionwise normal.Note: The Hausdorff condition is necessary here, since for example an infinite set with the cofinite topology is compact, hence paracompact, and T1, but is not even normal.
Every normal countably compact space (hence every normal compact space) is collectionwise normal.Proof: Use the fact that in a countably compact space any discrete family of nonempty subsets is finite.
An Fσ-set in a collectionwise normal space is also collectionwise normal in the subspace topology. In particular, this holds for closed subsets.
The states that a collectionwise normal Moore space is metrizable.
Hereditarily collectionwise normal space
A topological space X is called hereditarily collectionwise normal if every subspace of X with the subspace topology is collectionwise normal.
In the same way that hereditarily normal spaces can be characterized in terms of separated sets, there is an equivalent characterization for hereditarily collectionwise normal spaces. A family of subsets of X is called a separated family if for every i, we have , with cl denoting the closure operator in X, in other words if the family of is discrete in its union. The following conditions are equivalent:
X is hereditarily collectionwise normal.
Every open subspace of X is collectionwise normal.
For every separated family of subsets of X, there exists a pairwise disjoint family of open sets , such that .
Examples of hereditarily collectionwise normal spaces
Every linearly ordered topological space (LOTS)
Every generalized ordered space (GO-space)
Every metrizable space. This follows from the fact that metrizable spaces are collectionwise normal and being metrizable is a hereditary property.
Every monotonically normal space
Notes
References
Properties of topological spaces
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https://en.wikipedia.org/wiki/BJPS
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BJPS may refer to:
Brebeuf Jesuit Preparatory School
British Journal for the Philosophy of Science
British Journal of Political Science
Brazilian Journal of Probability and Statistics
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https://en.wikipedia.org/wiki/Viacheslav%20Belavkin
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Viacheslav Pavlovich Belavkin (; 20 May 1946 – 27 November 2012) was a Russian-British professor in applied mathematics at the University of Nottingham. An active researcher, he was one of the pioneers of quantum probability. His research spanned areas such as quantum filtering, quantum information and quantum chaos.
Biography
He was born in Lviv, and graduated from Moscow State University in 1970 where his teachers include Evgeny Lifshitz, Victor Pavlovich Maslov, Andrey Kolmogorov and Ruslan L. Stratonovich. In the 1980s Belavkin held visiting professorship in the Dublin Institute for Advanced Studies, and the Volterra Centre in Rome before taking up an appointment at the University of Nottingham in 1992. He was promoted to a Chair in Mathematical Physics in 1996. He and Ruslan L. Stratonovich were awarded the State Prize of the Russian Federation (formerly the Lenin Prize) for outstanding achievements in science and technology, in part due to his work on the measurement problem. He is survived by his wife Nadezda Belavkin and son Roman Belavkin.
References
External links
1946 births
2012 deaths
Academics of the University of Nottingham
Recipients of the Lenin Prize
State Prize of the Russian Federation laureates
Soviet mathematicians
Moscow State University alumni
Probability theorists
Soviet physicists
20th-century British physicists
Academics of the Dublin Institute for Advanced Studies
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https://en.wikipedia.org/wiki/Kalmanson%20combinatorial%20conditions
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In mathematics, the Kalmanson combinatorial conditions are a set of conditions on the distance matrix used in determining the solvability of the traveling salesman problem. These conditions apply to a special kind of cost matrix, the Kalmanson matrix, and are named after Kenneth Kalmanson.
References
.
.
.
.
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Combinatorics
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https://en.wikipedia.org/wiki/Virginia%20Ragsdale
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Virginia Ragsdale (December 13, 1870 – June 4, 1945) was a teacher and mathematician specializing in algebraic curves. She is most known as the creator of the Ragsdale conjecture.
Early life
Ragsdale was born on a farm in Jamestown, North Carolina the third child of John Sinclair Ragsdale and Emily Jane Idol. John was an officer in the Civil War, a teacher in the Flint Hill School, and later a state legislator.
Virginia Ragsdale descended from Godfrey Ragsdale, a settler of the new Jamestown colony. Jamestown was raided by a native-American tribe in 1644 led by the uncle of Pocahontas, during which Godfrey and his wife were killed, but their infant son, Godfrey, Jr., survived. Ragsdale was then descended from the infant.
Virginia documented her early years in a paper titled "Our Early Home and Childhood", writing:
Study
As a junior, Ragsdale entered Salem Academy, and graduated in 1887 as valedictorian with an extra diploma in piano. Ragsdale attended Guilford College in Greensboro, North Carolina, where she earned her B.S. in 1892. She was active in student life, establishing a Y.M.C.A. on campus, expanding collegiate athletics, and contributing to the formation the Guilford's Alumni Association.
Ragsdale was awarded the first scholarship from Bryn Mawr College for the top scholar Guilford College. She studied physics at Bryn Mawr College, obtaining an A.B. degree in 1896. She was elected European fellow for the class of 1896, but waited a year before traveling, working as an assistant demonstrator in physics and mathematics graduate student at Bryn Mawr.
Together with two of her colleagues (including Emilie Martin), she spent 1897-98 abroad at the University of Göttingen, attending lectures of Felix Klein and David Hilbert. After her return to the United States, she taught in Baltimore for three years until a second scholarship, by the Baltimore Association for the Promotion of University Education of Women, permitted her to return to Bryn Mawr college to complete her Ph.D. under the direction of Charlotte Scott.
Her dissertation, "On the Arrangement of the Real Branches of Plane Algebraic Curves," was published in 1906 by the American Journal of Mathematics. Her dissertation addressed the 16th of Hilbert's problems, for which Ragsdale formulated a conjecture that provided an upper bound on the number of topological circles of a certain type. Her result is called the Ragsdale conjecture; it was an open problem for 90 years until counterexamples were derived by Oleg Viro (1979) and Ilya Itenberg (1994).
Career
After completing her degree, Ragsdale taught in New York City and Dr. Sach's School for Girls until 1905. She was head of the Baldwin School in Bryn Mawr from 1906 to 1911, and a reader for Charlotte Scott from 1908 to 1910. Ragsdale returned to North Carolina in 1911 to accept a mathematics position at Woman's College in Greensboro (now known as the University of North Carolina at Greensboro). She remained there for almost
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https://en.wikipedia.org/wiki/Christian%20August%20Hausen
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Christian August Hausen (1693–1743) was a German mathematician who is known for his research on electricity.
Biography
Hausen studied mathematics at the University of Wittenberg and received his master's degree in 1712. He became an extraordinary professor of mathematics at the University of Leipzig at the age of 21 and later (1726) became an ordinary professor.
Hausen also researched electrical phenomena, using a triboelectric generator. In the introduction to his book on this subject, Novi profectus in historia electricitatis, published posthumously, Hausen states that he started these experiments shortly before his death. Hausen's generator was similar to earlier generators, such as that of Francis Hauksbee. It consisted of a glass globe rotated by a cord and a large wheel. An assistant rubbed the globe with his hand to produce static electricity. Hausen's book describes his generator and sets forth a theory of electricity in which electrification is a consequence of the production of vortices in a universal electrical fluid.
References
External links
1693 births
1743 deaths
Scientists from Dresden
18th-century German mathematicians
18th-century German physicists
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https://en.wikipedia.org/wiki/Laser%20power%20scaling
|
Power scaling of a laser is increasing its output power without changing the geometry, shape, or principle of operation. Power scalability is considered an important advantage in a laser design. This means it can increase power without changing outside features.
Usually, power scaling requires a more powerful pump source, stronger cooling, and an increase in size. It may also require reduction of the background loss in the laser resonator and, in particular, in the gain medium.
MOPA
The most popular way of achieving power scalability is the "MOPA" (Master Oscillator Power Amplifier) approach. The master oscillator produces a highly coherent beam, and an optical amplifier is used to increase the power of the beam while preserving its main properties. The master oscillator has no need to be powerful, and has no need to operate at high efficiency because the efficiency is determined mainly by the power amplifier. The combination of several laser amplifiers seeded by a common master oscillator is essential concept of the
High Power Laser Energy Research Facility.
Inherently scalable designs
Disk lasers
One type of solid-state laser designed for good power scaling is the disk laser (or "active mirror"). Such lasers are believed to be scalable to a power of several kilowatts
from a single active element in continuous-wave operation.
Amplified spontaneous emission, overheating and round-trip loss seem to be the most important processes that limit the power of disk lasers. For future power scaling, the reduction of the
round-trip loss and/or combining of several active elements is required.
Fiber lasers
Fiber lasers are another type of solid-state laser with good power scaling. The power scaling of fiber lasers is limited by Raman scattering and Brillouin scattering, and by the fact that such lasers cannot be very long. The limited length of the double-clad fibers limits the usable power of the multi-mode pump, because the pump is not absorbed efficiently in the fiber's active core. Optimization of the shape of the cladding can extend the limit of power scaling.
Fiber disk lasers
The limit of power scaling of fiber lasers can be extended with lateral delivery of the pump. This is realized in so-called fiber disk lasers.
The pump in such a laser is delivered from side of a disk, made of coiled fiber with doped core.
Several such disks (with a coolant between them) can be combined into a stack.
Problem of heat sink
The power scaling is limited by the ability to dissipate the heat. Usually, the thermal conductivity of materials designed for efficient laser action, is small compared to that of materials optimal for the heat transfer (metals, diamonds).
For the efficient drain of heat from a compact device, the active medium should be a narrow slab; in order to give advantage to the amplification of light at wanted direction over the ASE, the energy and head would be withdrawn in orthogonal directions, as it is shown in figure. At low background lo
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https://en.wikipedia.org/wiki/Spherical%20design
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A spherical design, part of combinatorial design theory in mathematics, is a finite set of N points on the d-dimensional unit d-sphere Sd such that the average value of any polynomial f of degree t or less on the set equals the average value of f on the whole sphere (that is, the integral of f over Sd divided by the area or measure of Sd). Such a set is often called a spherical t-design to indicate the value of t, which is a fundamental parameter. The concept of a spherical design is due to Delsarte, Goethals, and Seidel, although these objects were understood as particular examples of cubature formulas earlier.
Spherical designs can be of value in approximation theory, in statistics for experimental design, in combinatorics, and in geometry. The main problem is to find examples, given d and t, that are not too large; however, such examples may be hard to come by.
Spherical t-designs have also recently been appropriated in quantum mechanics in the form of quantum t-designs with various applications to quantum information theory and quantum computing.
Existence of spherical designs
The existence and structure of spherical designs on the circle were studied in depth by Hong. Shortly thereafter, Seymour and Zaslavsky proved that such designs exist of all sufficiently large sizes; that is, given positive integers d and t, there is a number N(d,t) such that for every N ≥ N(d,t) there exists a spherical t-design of N points in dimension d. However, their proof gave no idea of how big N(d,t) is.
Mimura constructively found conditions in terms of the number of points and the dimension which characterize exactly when spherical 2-designs exist. Maximally sized collections of equiangular lines (up to identification of lines as antipodal points on the sphere) are examples of minimal sized spherical 5-designs. There are many sporadic small spherical designs; many of them are related to finite group actions on the sphere.
In 2013, Bondarenko, Radchenko, and Viazovska obtained the asymptotic upper bound
for all positive integers d and t. This asymptotically matches the lower bound given originally by Delsarte, Goethals, and Seidel. The value of Cd is currently unknown, while exact values of are known in relatively few cases.
See also
Thomson problem
External links
Spherical t-designs for different values of N and t can be found precomputed at Neil Sloane's website.
Notes
References
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Algebra
Design of experiments
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https://en.wikipedia.org/wiki/Probability%20matching
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Probability matching is a decision strategy in which predictions of class membership are proportional to the class base rates. Thus, if in the training set positive examples are observed 60% of the time, and negative examples are observed 40% of the time, then the observer using a probability-matching strategy will predict (for unlabeled examples) a class label of "positive" on 60% of instances, and a class label of "negative" on 40% of instances.
The optimal Bayesian decision strategy (to maximize the number of correct predictions, see ) in such a case is to always predict "positive" (i.e., predict the majority category in the absence of other information), which has 60% chance of winning rather than matching which has 52% of winning (where p is the probability of positive realization, the result of matching would be , here ). The probability-matching strategy is of psychological interest because it is frequently employed by human subjects in decision and classification studies (where it may be related to Thompson sampling).
The only case when probability matching will yield same results as Bayesian decision strategy mentioned above is when all class base rates are the same. So, if in the training set positive examples are observed 50% of the time, then the Bayesian strategy would yield 50% accuracy (1 × .5), just as probability matching (.5 ×.5 + .5 × .5).
References
Shanks, D. R., Tunney, R. J., & McCarthy, J. D. (2002). A re‐examination of probability matching and rational choice. Journal of Behavioral Decision Making, 15(3), 233-250.
Statistical classification
Machine learning
Decision-making
Cognitive science
Cognitive biases
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https://en.wikipedia.org/wiki/Base%20rate
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In probability and statistics, the base rate (also known as prior probabilities) is the class of probabilities unconditional on "featural evidence" (likelihoods).
It is the proportion of individuals in a population who have a certain characteristic or trait. For example, if 1% of the population were medical professionals, and remaining 99% were not medical professionals, then the base rate of medical professionals is 1%. The method for integrating base rates and featural evidence is given by Bayes' rule.
In the sciences, including medicine, the base rate is critical for comparison. In medicine a treatment's effectiveness is clear when the base rate is available. For example, if the control group, using no treatment at all, had their own base rate of 1/20 recoveries within 1 day and a treatment had a 1/100 base rate of recovery within 1 day, we see that the treatment actively decreases the recovery.
The base rate is an important concept in statistical inference, particularly in Bayesian statistics. In Bayesian analysis, the base rate is combined with the observed data to update our belief about the probability of the characteristic or trait of interest. The updated probability is known as the posterior probability and is denoted as P(A|B), where B represents the observed data. For example, suppose we are interested in estimating the prevalence of a disease in a population. The base rate would be the proportion of individuals in the population who have the disease. If we observe a positive test result for a particular individual, we can use Bayesian analysis to update our belief about the probability that the individual has the disease. The updated probability would be a combination of the base rate and the likelihood of the test result given the disease status.
The base rate is also important in decision-making, particularly in situations where the cost of false positives and false negatives are different. For example, in medical testing, a false negative (failing to diagnose a disease) could be much more costly than a false positive (incorrectly diagnosing a disease). In such cases, the base rate can help inform decisions about the appropriate threshold for a positive test result.
Base rate fallacy
Many psychological studies have examined a phenomenon called base-rate neglect or base rate fallacy, in which category base rates are not integrated with presented evidence in a normative manner, although not all evidence is consistent regarding how common this fallacy is. Mathematician Keith Devlin illustrates the risks as a hypothetical type of cancer that afflicts 1% of all people. Suppose a doctor then says there is a test for said cancer that is approximately 80% reliable, and that the test provides a positive result for 100% of people who have cancer, but it also results in a 'false positive' for 20% of people - who do not have cancer. Testing positive may therefore lead people to believe that it is 80% likely that they have cancer. Devlin
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https://en.wikipedia.org/wiki/Mark%20L.%20Perkins
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Mark L. Perkins is the president of InnerSight. He served as president of Towson University from July 2001 to April 2002.
Education
Perkins earned a doctorate in psychometrics and statistics from the University of Georgia in 1976. He received his master's in psychometrics and research design from the same institution in 1974. Perkins earned a bachelor's degree from St. Andrews Presbyterian College in 1972.
Towson University
Perkins briefly served a controversial tenure as president of Towson University from July 2001 to April 2002.
He resigned after three members of the University System of Maryland Board of Regents, including the chairman, told him in a meeting that he would be fired if he did not step down, according to a four-page letter he posted on Towson's website.
In the letter, Perkins stated the spending included improvements for coping with handicap accessibility as well as "family health issues" and for making the home a suitable place to entertain prospective donors. The regents were aware these improvements were necessary prior to selecting Perkins.
The university received approval from the regents to buy a six-bedroom mansion in northern Baltimore for $850,000. The university subsequently spent $860,000 on renovations, but $360,000 more had been allocated to complete renovations and provide furnishings for the public spaces of the university home as well. Perkins claimed Towson officials were unaware of flaws in the home when the university bought it. Workers subsequently found deteriorating wall coverings, unabated lead paint and asbestos.
Perkins said in his letter that he was not involved in many of the spending decisions on the home, including $279,000 for an elevator and $25,000 for a multimedia system. However, Perkins wrote, he took "full responsibility" for controversies including the spending on those renovations, which dominated discussions with the Board of Regents ahead of his resignation.
There was also an inauguration ceremony held using funds raised specifically for the event, which reportedly cost $56,000. In his letter, Perkins said he had agreed to the event because "it was designed to celebrate not me or my arrival, but instead, the 135 years of commitment to learning on the Towson campus."
The Baltimore Sun reported that similar concerns about excessive spending by Perkins were raised at the University of Wisconsin–Green Bay, where he was president before coming to Towson. Perkins had pushed for $630,000 in improvements to the president's house on that campus to make it acceptable for parties, but he rarely entertained guests of the university in the house, The Sun reported.
Perkins also sparked controversy when the university spent $25,000 on a university medal symbolizing the president's office. All funds used for the inauguration and the medallion were raised for those purposes by a committee charged with organizing the inaugural events.
References
External links
Presidential Biographies - Towson
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https://en.wikipedia.org/wiki/Charles%20Leedham-Green
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Charles R. Leedham-Green is a retired professor of mathematics at Queen Mary, University of London, known for his work in group theory. He completed his DPhil at the University of Oxford.
His parents were John Charles Leedham-Green (1902–1984), a surgeon and general practitioner in Southwold, and Gertrude Mary Somerville Caldwell.
Work
With Leonard Soicher, Leedham-Green designed the product replacement algorithm; an algorithm within computational group theory that generates random elements of groups by taking a random walk through the group. This algorithm has been implemented in both GAP and MAGMA.
He is responsible for a great body of work in group theory. In recent times, this has involved research in computational group theory and pro-p groups.
The 300th edition of the Journal of Algebra was dedicated to him for his 65th birthday.
On the occasion of his retirement in 2006, the Mathematics Research Centre at Queen Mary held a conference in celebration of his mathematical achievements.
Selected publications
Charles R. Leedham-Green, Leonard H. Soicher: Collection from the Left and Other Strategies. J. Symb. Comput. 9(5/6): 665–675 (1990)
Charles R. Leedham-Green, Cheryl E. Praeger, Leonard H. Soicher: Computing with Group Homomorphisms. J. Symb. Comput. 12(4/5): 527–532 (1991)
Derek F. Holt, C. R. Leedham-Green, E. A. O'Brien and Sarah Rees: Testing Matrix Groups for Primitivity. Journal of Algebra, Volume 184, Issue 3, 15 September 1996, Pages 795–817
Derek F. Holt, C. R. Leedham-Green, E. A. O'Brien and Sarah Rees: Computing Matrix Group Decompositions with Respect to a Normal Subgroup. Journal of Algebra, Volume 184, Issue 3, 15 September 1996, Pages 818–838.
C. R. Leedham-Green and E. A. O'Brien: Tensor Products are Projective Geometries. Journal of Algebra, Volume 189, Issue 2, 15 March 1997, Pages 514–528
Robert Beals, Charles R. Leedham-Green, Alice C. Niemeyer, Cheryl E. Praeger, Ákos Seress: Permutations With Restricted Cycle Structure And An Algorithmic Application. Combinatorics, Probability & Computing 11(5): (2002)
C. R. Leedham-Green and E. A. O'Brien: Recognising tensor-induced matrix groups. Journal of Algebra, Volume 253, Issue 1, 1 July 2002, Pages 14–30
Nigel Boston and Charles Leedham-Green: Explicit computation of Galois p-groups unramified at p. Journal of Algebra, Volume 256, Issue 2, 15 October 2002, Pages 402–413.
Charles Leedham-Green and Sue McKay: The Structure of Groups of Prime Power Order (2002)
John J. Cannon, Bettina Eick, Charles R. Leedham-Green: Special polycyclic generating sequences for finite soluble groups. J. Symb. Comput. 38(5): 1445–1460 (2004)
Robert Beals, Charles R. Leedham-Green, Alice C. Niemeyer, Cheryl E. Praeger and Ákos Seress: Constructive recognition of finite alternating and symmetric groups acting as matrix groups on their natural permutation modules. Journal of Algebra, Volume 292, Issue 1, 1 October 2005, Pages 4–46
S.P. Glasby, C.R. Leedham-Green and E.A. O'Brien: Writing projecti
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https://en.wikipedia.org/wiki/Abel%27s%20summation%20formula
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In mathematics, Abel's summation formula, introduced by Niels Henrik Abel, is intensively used in analytic number theory and the study of special functions to compute series.
Formula
Let be a sequence of real or complex numbers. Define the partial sum function by
for any real number . Fix real numbers , and let be a continuously differentiable function on . Then:
The formula is derived by applying integration by parts for a Riemann–Stieltjes integral to the functions and .
Variations
Taking the left endpoint to be gives the formula
If the sequence is indexed starting at , then we may formally define . The previous formula becomes
A common way to apply Abel's summation formula is to take the limit of one of these formulas as . The resulting formulas are
These equations hold whenever both limits on the right-hand side exist and are finite.
A particularly useful case is the sequence for all . In this case, . For this sequence, Abel's summation formula simplifies to
Similarly, for the sequence and for all , the formula becomes
Upon taking the limit as , we find
assuming that both terms on the right-hand side exist and are finite.
Abel's summation formula can be generalized to the case where is only assumed to be continuous if the integral is interpreted as a Riemann–Stieltjes integral:
By taking to be the partial sum function associated to some sequence, this leads to the summation by parts formula.
Examples
Harmonic numbers
If for and then and the formula yields
The left-hand side is the harmonic number .
Representation of Riemann's zeta function
Fix a complex number . If for and then and the formula becomes
If , then the limit as exists and yields the formula
where is the Riemann zeta function.
This may be used to derive Dirichlet's theorem that has a simple pole with residue 1 at .
Reciprocal of Riemann zeta function
The technique of the previous example may also be applied to other Dirichlet series. If is the Möbius function and , then is Mertens function and
This formula holds for .
See also
Summation by parts
Integration by parts
References
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Number theory
Summability methods
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https://en.wikipedia.org/wiki/Kuratowski%27s%20free%20set%20theorem
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Kuratowski's free set theorem, named after Kazimierz Kuratowski, is a result of set theory, an area of mathematics. It is a result which has been largely forgotten for almost 50 years, but has been applied recently in solving several lattice theory problems, such as the congruence lattice problem.
Denote by the set of all finite subsets of a set . Likewise, for a positive integer , denote by the set of all -elements subsets of . For a mapping , we say that a subset of is free (with respect to ), if for any -element subset of and any , . Kuratowski published in 1951 the following result, which characterizes the infinite cardinals of the form .
The theorem states the following. Let be a positive integer and let be a set. Then the cardinality of is greater than or equal to if and only if for every mapping from to ,
there exists an -element free subset of with respect to .
For , Kuratowski's free set theorem is superseded by Hajnal's set mapping theorem.
References
P. Erdős, A. Hajnal, A. Máté, R. Rado: Combinatorial Set Theory: Partition Relations for Cardinals, North-Holland, 1984, pp. 282–285.
C. Kuratowski, Sur une caractérisation des alephs, Fund. Math. 38 (1951), 14–17.
John C. Simms (1991) "Sierpiński's theorem", Simon Stevin 65: 69–163.
Set theory
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https://en.wikipedia.org/wiki/Refinement%20monoid
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In mathematics, a refinement monoid is a commutative monoid M such that for any elements a0, a1, b0, b1 of M such that a0+a1=b0+b1, there are elements c00, c01, c10, c11 of M such that a0=c00+c01, a1=c10+c11, b0=c00+c10, and b1=c01+c11.
A commutative monoid M is said to be conical if x+y=0 implies that x=y=0, for any elements x,y of M.
Basic examples
A join-semilattice with zero is a refinement monoid if and only if it is distributive.
Any abelian group is a refinement monoid.
The positive cone G+ of a partially ordered abelian group G is a refinement monoid if and only if G is an interpolation group, the latter meaning that for any elements a0, a1, b0, b1 of G such that ai ≤ bj for all i, j<2, there exists an element x of G such that ai ≤ x ≤ bj for all i, j<2. This holds, for example, in case G is lattice-ordered.
The isomorphism type of a Boolean algebra B is the class of all Boolean algebras isomorphic to B. (If we want this to be a set, restrict to Boolean algebras of set-theoretical rank below the one of B.)
The class of isomorphism types of Boolean algebras, endowed with the addition defined by
(for any Boolean algebras X and Y, where denotes the isomorphism type of X), is a conical refinement monoid.
Vaught measures on Boolean algebras
For a Boolean algebra A and a commutative monoid M, a map μ : A → M is a measure, if μ(a)=0 if and only if a=0, and μ(a ∨ b)=μ(a)+μ(b) whenever a and b are disjoint (that is, a ∧ b=0), for any a, b in A. We say in addition that μ is a Vaught measure (after Robert Lawson Vaught), or V-measure, if for all c in A and all x,y in M such that μ(c)=x+y, there are disjoint a, b in A such that c=a ∨ b, μ(a)=x, and μ(b)=y.
An element e in a commutative monoid M is measurable (with respect to M), if there are a Boolean algebra A and a V-measure μ : A → M such that μ(1)=e---we say that μ measures e. We say that M is measurable, if any element of M is measurable (with respect to M). Of course, every measurable monoid is a conical refinement monoid.
Hans Dobbertin proved in 1983 that any conical refinement monoid with at most ℵ1 elements is measurable. He also proved that any element in an at most countable conical refinement monoid is measured by a unique (up to isomorphism) V-measure on a unique at most countable Boolean algebra.
He raised there the problem whether any conical refinement monoid is measurable. This was answered in the negative by Friedrich Wehrung in 1998. The counterexamples can have any cardinality greater than or equal to ℵ2.
Nonstable K-theory of von Neumann regular rings
For a ring (with unit) R, denote by FP(R) the class of finitely generated projective right R-modules. Equivalently, the objects of FP(R) are the direct summands of all modules of the form Rn, with n a positive integer, viewed as a right module over itself. Denote by the isomorphism type of an object X in FP(R). Then the set V(R) of all isomorphism types of members of FP(R), endowed with the addition defined by , i
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https://en.wikipedia.org/wiki/Ken%20Ono
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Ken Ono (born March 20, 1968) is an American mathematician who specializes in number theory, especially in integer partitions, modular forms, umbral moonshine, the Riemann Hypothesis and the fields of interest to Srinivasa Ramanujan. He is the STEM Advisor to the Provost and the Marvin Rosenblum Professor of Mathematics at the University of Virginia.
Early life and education
Ono was born on March 20, 1968, in Philadelphia, Pennsylvania. He is the son of mathematician Takashi Ono, who emigrated from Japan to the United States after World War II. His older brother, immunologist and university president Santa J. Ono, was born while Takashi Ono was in Canada working at the University of British Columbia, but by the time Ken Ono was born the family had returned to the US for a position at the University of Pennsylvania. In the 1980s, Ono attended Towson High School, but he dropped out. He later enrolled at the University of Chicago without a high school diploma. There he raced bicycles, and he was a member of the Pepsi–Miyata Cycling Team.
He received his BA from the University of Chicago in 1989, where he was a member of the Psi Upsilon fraternity. He earned his PhD in 1993 at UCLA where his advisor was Basil Gordon. Initially he planned to study medicine, but later switched to mathematics. He attributes his interest in mathematics to his father.
Career
Ono worked as an instructor at Woodbury University from 1991 to 1993, as a visiting assistant professor at the University of Georgia from 1993 to 1994, and as a visiting assistant professor at the University of Illinois at Urbana-Champaign from 1994 to 1995. He was a member of the Institute for Advanced Study from 1995 to 1997.
Ono worked at Pennsylvania State University from 1997 to 2000 as an assistant professor and then as the Louis A. Martarano Professor of Mathematics. He moved to the University of Wisconsin-Madison as an associate professor in 1999, and later became the Solle P. and Margaret Manasse Professor of Letters and Science from 2004 to 2011 and as the Hilldale Professor of Mathematics from 2008 to 2011. He was the Candler Professor of Mathematics at Emory University from 2010 to 2019. In 2019, Ono became the Thomas Jefferson Professor of Mathematics at the University of Virginia, and in Fall 2021 he was named the Marvin Rosenblum Professor of Mathematics and the chairman of the Department of Mathematics. He ended his term as chairman in Fall 2022 to become the STEM Advisor to the Provost at the University of Virginia.
Ono was the Vice President of the American Mathematical Society from 2018 to 2021. He is serving as the section chair for mathematics at the American Association for the Advancement of Science from 2020 to 2023.
Research
Integer partitions
In 2000, Ono derived a theory of Ramanujan congruences for the partition function with all prime moduli greater than 3. His paper was published in the Annals of Mathematics.
In a joint work with Jan Bruinier, Ono discovered a f
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https://en.wikipedia.org/wiki/Fraction%20of%20variance%20unexplained
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In statistics, the fraction of variance unexplained (FVU) in the context of a regression task is the fraction of variance of the regressand (dependent variable) Y which cannot be explained, i.e., which is not correctly predicted, by the explanatory variables X.
Formal definition
Suppose we are given a regression function yielding for each an estimate where is the vector of the ith observations on all the explanatory variables. We define the fraction of variance unexplained (FVU) as:
where R2 is the coefficient of determination and VARerr and VARtot are the variance of the residuals and the sample variance of the dependent variable. SSerr (the sum of squared predictions errors, equivalently the residual sum of squares), SStot (the total sum of squares), and SSreg (the sum of squares of the regression, equivalently the explained sum of squares) are given by
Alternatively, the fraction of variance unexplained can be defined as follows:
where MSE(f) is the mean squared error of the regression function ƒ.
Explanation
It is useful to consider the second definition to understand FVU. When trying to predict Y, the most naive regression function that we can think of is the constant function predicting the mean of Y, i.e., . It follows that the MSE of this function equals the variance of Y; that is, SSerr = SStot, and SSreg = 0. In this case, no variation in Y can be accounted for, and the FVU then has its maximum value of 1.
More generally, the FVU will be 1 if the explanatory variables X tell us nothing about Y in the sense that the predicted values of Y do not covary with Y. But as prediction gets better and the MSE can be reduced, the FVU goes down. In the case of perfect prediction where for all i, the MSE is 0, SSerr = 0, SSreg = SStot, and the FVU is 0.
See also
Coefficient of determination
Correlation
Explained sum of squares
Lack-of-fit sum of squares
Linear regression
Regression analysis
Mean absolute scaled error
References
Parametric statistics
Statistical ratios
Least squares
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https://en.wikipedia.org/wiki/Congruence%20lattice%20problem
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In mathematics, the congruence lattice problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of some other lattice. The problem was posed by Robert P. Dilworth, and for many years it was one of the most famous and long-standing open problems in lattice theory; it had a deep impact on the development of lattice theory itself. The conjecture that every distributive lattice is a congruence lattice is true for all distributive lattices with at most ℵ1 compact elements, but F. Wehrung provided a counterexample for distributive lattices with ℵ2 compact elements using a construction based on Kuratowski's free set theorem.
Preliminaries
We denote by Con A the congruence lattice of an algebra A, that is, the lattice of all congruences of A under inclusion.
The following is a universal-algebraic triviality. It says that for a congruence, being finitely generated is a lattice-theoretical property.
Lemma.
A congruence of an algebra A is finitely generated if and only if it is a compact element of Con A.
As every congruence of an algebra is the join of the finitely generated congruences below it (e.g., every submodule of a module is the union of all its finitely generated submodules), we obtain the following result, first published by Birkhoff and Frink in 1948.
Theorem (Birkhoff and Frink 1948).
The congruence lattice Con A of any algebra A is an algebraic lattice.
While congruences of lattices lose something in comparison to groups, modules, rings (they cannot be identified with subsets of the universe), they also have a property unique among all the other structures encountered yet.
Theorem (Funayama and Nakayama 1942).
The congruence lattice of any lattice is distributive.
This says that α ∧ (β ∨ γ) = (α ∧ β) ∨ (α ∧ γ), for any congruences α, β, and γ of a given lattice. The analogue of this result fails, for instance, for modules, as , as a rule, for submodules A, B, C of a given module.
Soon after this result, Dilworth proved the following result. He did not publish the result but it appears as an exercise credited to him in Birkhoff 1948. The first published proof is in Grätzer and Schmidt 1962.
Theorem (Dilworth ≈1940, Grätzer and Schmidt 1962).
Every finite distributive lattice is isomorphic to the congruence lattice of some finite lattice.
It is important to observe that the solution lattice found in Grätzer and Schmidt's proof is sectionally complemented, that is, it has a least element (true for any finite lattice) and for all elements a ≤ b there exists an element x with a ∨ x = b and a ∧ x = 0. It is also in that paper that CLP is first stated in published form, although it seems that the earliest attempts at CLP were made by Dilworth himself. Congruence lattices of finite lattices have been given an enormous amount of attention, for which a reference is Grätzer's 2005 monograph.
The congruence lattice problem (CLP):
Is every distributive algebraic lattice isomorphic to the congruence latti
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https://en.wikipedia.org/wiki/Umbilical%20point
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In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are equal, and every tangent vector is a principal direction. The name "umbilic" comes from the Latin umbilicus (navel).
Umbilic points generally occur as isolated points in the elliptical region of the surface; that is, where the Gaussian curvature is positive.
The sphere is the only surface with non-zero curvature where every point is umbilic. A flat umbilic is an umbilic with zero Gaussian curvature. The monkey saddle is an example of a surface with a flat umbilic and on the plane every point is a flat umbilic. A torus can have no umbilics, but every closed surface of nonzero Euler characteristic, embedded smoothly into Euclidean space, has at least one umbilic. An unproven conjecture of Constantin Carathéodory states that every smooth topological sphere in Euclidean space has at least two umbilics.
The three main types of umbilic points are elliptical umbilics, parabolic umbilics and hyperbolic umbilics. Elliptical umbilics have the three ridge lines passing through the umbilic and hyperbolic umbilics have just one. Parabolic umbilics are a transitional case with two ridges one of which is singular. Other configurations are possible for transitional cases. These cases correspond to the D4−, D5 and D4+ elementary catastrophes of René Thom's catastrophe theory.
Umbilics can also be characterised by the pattern of the principal direction vector field around the umbilic which typically form one of three configurations: star, lemon, and lemonstar (or monstar). The index of the vector field is either −½ (star) or ½ (lemon, monstar). Elliptical and parabolic umbilics always have the star pattern, whilst hyperbolic umbilics can be star, lemon, or monstar. This classification was first due to Darboux and the names come from Hannay.
For surfaces with genus 0 with isolated umbilics, e.g. an ellipsoid, the index of the principal direction vector field must be 2 by the Poincaré–Hopf theorem. Generic genus 0 surfaces have at least four umbilics of index ½. An ellipsoid of revolution has two non-generic umbilics each of which has index 1.
Classification of umbilics
Cubic forms
The classification of umbilics is closely linked to the classification of real cubic forms . A cubic form will have a number of root lines such that the cubic form is zero for all real . There are a number of possibilities including:
Three distinct lines: an elliptical cubic form, standard model .
Three lines, two of which are coincident: a parabolic cubic form, standard model .
A single real line: a hyperbolic cubic form, standard model .
Three coincident lines, standard model .
The equivalence classes of such cubics under uniform scaling form a three-dimensional real projective space and the subset of parabolic forms define a surface –
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https://en.wikipedia.org/wiki/Z%2A%20theorem
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In mathematics, George Glauberman's Z* theorem is stated as follows:
Z* theorem: Let G be a finite group, with O(G) being its maximal normal subgroup of odd order. If T is a Sylow 2-subgroup of G containing an involution not conjugate in G to any other element of T, then the involution lies in Z*(G), which is the inverse image in G of the center of G/O(G).
This generalizes the Brauer–Suzuki theorem (and the proof uses the Brauer–Suzuki theorem to deal with some small cases).
Details
The original paper gave several criteria for an element to lie outside Its theorem 4 states:
For an element t in T, it is necessary and sufficient for t to lie outside Z*(G) that there is some g in G and abelian subgroup U of T satisfying the following properties:
g normalizes both U and the centralizer CT(U), that is g is contained in N = NG(U) ∩ NG(CT(U))
t is contained in U and tg ≠ gt
U is generated by the N-conjugates of t
the exponent of U is equal to the order of t
Moreover g may be chosen to have prime power order if t is in the center of T, and g may be chosen in T otherwise.
A simple corollary is that an element t in T is not in Z*(G) if and only if there is some s ≠ t such that s and t commute and s and t are G-conjugate.
A generalization to odd primes was recorded in : if t is an element of prime order p and the commutator [t, g] has order coprime to p for all g, then t is central modulo the p′-core. This was also generalized to odd primes and to compact Lie groups in , which also contains several useful results in the finite case.
have also studied an extension of the Z* theorem to pairs of groups (G, H) with H a normal subgroup of G.
Works cited
gives a detailed proof of the Brauer–Suzuki theorem.
Theorems about finite groups
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https://en.wikipedia.org/wiki/Maximal%20semilattice%20quotient
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In abstract algebra, a branch of mathematics, a maximal semilattice quotient is a commutative monoid derived from another commutative monoid by making certain elements equivalent to each other.
Every commutative monoid can be endowed with its algebraic preordering ≤ . By definition, x≤ y holds, if there exists z such that x+z=y. Further, for x, y in M, let hold, if there exists a positive integer n such that x≤ ny, and let hold, if and . The binary relation is a monoid congruence of M, and the quotient monoid is the maximal semilattice quotient of M.
This terminology can be explained by the fact that the canonical projection p from M onto is universal among all monoid homomorphisms from M to a (∨,0)-semilattice, that is, for any (∨,0)-semilattice S and any monoid homomorphism f: M→ S, there exists a unique (∨,0)-homomorphism such that f=gp.
If M is a refinement monoid, then is a distributive semilattice.
References
A.H. Clifford and G.B. Preston, The Algebraic Theory of Semigroups. Vol. I. Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I. 1961. xv+224 p.
Lattice theory
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https://en.wikipedia.org/wiki/Principal%20direction
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A principal direction can refer to one of the following:
Principal directions (geometry) - In differential geometry, one of the directions of principal curvature.
Principal directions - a term used in gear nomenclature.
In stress analysis, a set of axes where the normal stress vector is maximized. See Stress (mechanics)
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https://en.wikipedia.org/wiki/ZJ%20theorem
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In mathematics, George Glauberman's ZJ theorem states that if a finite group G is p-constrained and p-stable and has a normal p-subgroup for some odd prime p, then O(G)Z(J(S)) is a normal subgroup of G, for any Sylow p-subgroup S.
Notation and definitions
J(S) is the Thompson subgroup of a p-group S: the subgroup generated by the abelian subgroups of maximal order.
Z(H) means the center of a group H.
O is the maximal normal subgroup of G of order coprime to p, the -core
Op is the maximal normal p-subgroup of G, the p-core.
O,p(G) is the maximal normal p-nilpotent subgroup of G, the ,p-core, part of the upper p-series.
For an odd prime p, a group G with Op(G) ≠ 1 is said to be p-stable if whenever P is a p-subgroup of G such that PO(G) is normal in G, and [P,x,x] = 1, then the image of x in NG(P)/CG(P) is contained in a normal p-subgroup of NG(P)/CG(P).
For an odd prime p, a group G with Op(G) ≠ 1 is said to be p-constrained if the centralizer CG(P) is contained in O,p(G) whenever P is a Sylow p-subgroup of O,p(G).
References
Theorems about finite groups
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https://en.wikipedia.org/wiki/Dupin%20indicatrix
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In differential geometry, the Dupin indicatrix is a method for characterising the local shape of a surface. Draw a plane parallel to the tangent plane and a small distance away from it. Consider the intersection of the surface with this plane. The shape of the intersection is related to the Gaussian curvature. The Dupin indicatrix is the result of the limiting process as the plane approaches the tangent plane. The indicatrix was introduced by Charles Dupin.
For elliptical points where the Gaussian curvature is positive the intersection will either be empty or form a closed curve. In the limit this curve will form an ellipse aligned with the principal directions.
For hyperbolic points, where the Gaussian curvature is negative, the intersection will form a hyperbola. Two different hyperbolas will be formed on either side of the tangent plane. These hyperbolas share the same axis and asymptotes. The directions of the asymptotes are the same as the asymptotic directions.
See also
Euler's theorem (differential geometry)
References
Full 1909 text (now out of copyright)
Differential geometry of surfaces
Surfaces
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https://en.wikipedia.org/wiki/Banach%27s%20matchbox%20problem
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Banach's match problem is a classic problem in probability attributed to Stefan Banach. Feller says that the problem was inspired by a humorous reference to Banach's smoking habit in a speech honouring him by Hugo Steinhaus, but that it was not Banach who set the problem or provided an answer.
Suppose a mathematician carries two matchboxes at all times: one in his left pocket and one in his right. Each time he needs a match, he is equally likely to take it from either pocket. Suppose he reaches into his pocket and discovers for the first time that the box picked is empty. If it is assumed that each of the matchboxes originally contained matches, what is the probability that there are exactly matches in the other box?
Solution
Without loss of generality consider the case where the matchbox in his right pocket has an unlimited number of matches and let be the number of matches removed from this one before the left one is found to be empty. When the left pocket is found to be empty, the man has chosen that pocket times. Then is the number of successes before failures in Bernoulli trials with , which has the negative binomial distribution and thus
.
Returning to the original problem, we see that the probability that the left pocket is found to be empty first is which equals because both are equally likely. We see that the number of matches remaining in the other pocket is
.
The expectation of the distribution is approximately . (This is shown using Stirling's approximation.) So starting with boxes with matches, the expected number of matches in the second box is .
See also
List of things named after Stefan Banach
References
External links
Java applet
Applied probability
Probability problems
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https://en.wikipedia.org/wiki/Sterility%20assurance%20level
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In microbiology, sterility assurance level (SAL) is the probability that a single unit that has been subjected to sterilization nevertheless remains nonsterile.
It is never possible to prove that all organisms have been destroyed, as the likelihood of survival of an individual microorganism is never zero. So SAL is used to express the probability of the survival. For example, medical device manufacturers design their sterilization processes for an extremely low SAL, such as 10−6, which is a 1 in 1,000,000 chance of a non-sterile unit. SAL also describes the killing efficacy of a sterilization process. A very effective sterilization process has a very low SAL.
Terminology
Mathematically, SALs are probabilities, often very small but (by definition) always lying between zero and one. So when they are expressed in scientific notation their exponents are negative, as for instance, "The SAL of this process is 10−6". But the term SAL is sometimes also used to refer to a sterilization's efficacy. This usage (technically the multiplicative inverse) results in positive exponents, as in "The SAL of this process is 106". To avoid ambiguity from these inverse usages, some authors use the term log reduction (e.g., "This process gives a six-log reduction").
SALs can also be used to describe the microbial population that was destroyed by the sterilization process, though this is not the same as the probabilistic definition. What is often called a "log reduction" (technically a reduction by one order of magnitude) represents a 90% reduction in microbial population. Thus a process that achieves a "6-log reduction" (10−6) will theoretically reduce an initial population of one million organisms to very close to zero. The difference in meaning between this and the probabilistic sense can be seen from an example: if careful assays before and after indicate that a procedure has inactivated 90% of the biological agents in some unit, then the procedure can be correctly reported to have achieved a 1-log reduction, even though the probability that the unit is sterile is not 90% but 0.
Because of all these ambiguities, contexts in which it is critical to prevent any confusion—such as in the setting of standards—require that SAL terminology be defined carefully and explicitly.
SALs describing the "Probability of a Non-Sterile Unit" are expressed more specifically in some literature.
References
Microbiology terms
Sterilization (microbiology)
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https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein
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Bose–Einstein may refer to:
Bose–Einstein condensate
Bose–Einstein condensation (network theory)
Bose–Einstein correlations
Bose–Einstein statistics
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https://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac
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Fermi–Dirac may refer to:
Fermi–Dirac statistics or Fermi–Dirac distribution
Fermi–Dirac integral (disambiguation)
Complete Fermi–Dirac integral
Incomplete Fermi–Dirac integral
See also
Fermi (disambiguation)
Dirac (disambiguation)
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https://en.wikipedia.org/wiki/Depth%20%28ring%20theory%29
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In commutative and homological algebra, depth is an important invariant of rings and modules. Although depth can be defined more generally, the most common case considered is the case of modules over a commutative Noetherian local ring. In this case, the depth of a module is related with its projective dimension by the Auslander–Buchsbaum formula. A more elementary property of depth is the inequality
where denotes the Krull dimension of the module . Depth is used to define classes of rings and modules with good properties, for example, Cohen-Macaulay rings and modules, for which equality holds.
Definition
Let be a commutative ring, an ideal of and a finitely generated -module with the property that is properly contained in . (That is, some elements of are not in .) Then the -depth of , also commonly called the grade of , is defined as
By definition, the depth of a local ring with a maximal ideal is its -depth as a module over itself. If is a Cohen-Macaulay local ring, then depth of is equal to the dimension of .
By a theorem of David Rees, the depth can also be characterized using the notion of a regular sequence.
Theorem (Rees)
Suppose that is a commutative Noetherian local ring with the maximal ideal and is a finitely generated -module. Then all maximal regular sequences for , where each belongs to , have the same length equal to the -depth of .
Depth and projective dimension
The projective dimension and the depth of a module over a commutative Noetherian local ring are complementary to each other. This is the content of the Auslander–Buchsbaum formula, which is not only of fundamental theoretical importance, but also provides an effective way to compute the depth of a module.
Suppose that is a commutative Noetherian local ring with the maximal ideal and is a finitely generated -module. If the projective dimension of is finite, then the Auslander–Buchsbaum formula states
Depth zero rings
A commutative Noetherian local ring has depth zero if and only if its maximal ideal is an associated prime, or, equivalently, when there is a nonzero element of such that (that is, annihilates ). This means, essentially, that the closed point is an embedded component.
For example, the ring (where is a field), which represents a line () with an embedded double point at the origin, has depth zero at the origin, but dimension one: this gives an example of a ring which is not Cohen–Macaulay.
References
Winfried Bruns; Jürgen Herzog, Cohen–Macaulay rings. Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993. xii+403 pp.
Module theory
Commutative algebra
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https://en.wikipedia.org/wiki/Robert%20Chote
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Sir Robert William Chote (born 24 January 1968) is a British economist and chair of the UK Statistics Authority. He was previously chairman of the Office of Budget Responsibility from 2010 to 2020.
Education
Chote completed his secondary education at St Mary's College in Bitterne Park, Southampton. In 1989, he graduated in economics from Queens' College, Cambridge (where he was president of the Cambridge University Social Democrats and, after the merger of the SDP with the Liberals, chair of the Cambridge University Social and Liberal Democrats). He then studied journalism at City University, London, and international public policy at the School of Advanced International Studies at Johns Hopkins University in the United States.
Career
Chote began his career as a reporter and columnist at The Independent and was named Young Financial Journalist of the Year in 1993 when working for the Independent on Sunday by the Wincott Foundation. He then moved to the Financial Times to become Economics Editor in 1995.
From 1999 to 2002, he served as an adviser to the senior management of the International Monetary Fund in Washington, DC, where he worked under Stanley Fischer and Anne Krueger. Chote was appointed director of the Institute for Fiscal Studies in October 2002. He has also served as a member of the Statistics Advisory Committee of the Office for National Statistics.
In September 2010, he was appointed chairman of the Office of Budget Responsibility, succeeding Sir Alan Budd. This appointment was subject to Parliamentary approval, which was received. He started as chairman on 4 October 2010. As of 2015, Chote was paid a salary of between £150,000 and £154,999 by the department, making him one of the 328 most highly paid people in the British public sector at that time. He served two five-year terms and stepped down in 2020.
In March 2021, he began as the inaugural chair of the Northern Ireland Fiscal Council, which was established to provide independent scrutiny of the NI public finances.
On 1 June 2022, Chote was appointed chairman of the UK Statistics Authority. Chote also has a position as a Senior Advisor at Francis Maude Associates, a consultancy set up by Francis Maude.
Honours and awards
Chote was knighted in the 2021 New Year Honours for services to fiscal policy and the economy.
Personal life
Since 1997, Chote has been married to Dame Sharon White, chairman of the John Lewis Partnership and previously the chief executive of Ofcom. The couple has two children.
He is the son of Olympic athlete Morville Chote.
References
External links
Bio at the Institute for Fiscal Studies
Living people
Alumni of Queens' College, Cambridge
Alumni of City, University of London
Johns Hopkins University alumni
People educated at St Mary's College, Southampton
British male journalists
1968 births
Knights Bachelor
20th-century British economists
21st-century British economists
20th-century British journalists
20th-century British male writers
The Indep
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https://en.wikipedia.org/wiki/Fra%C5%88kov%C3%A1%E2%80%93Helly%20selection%20theorem
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In mathematics, the Fraňková–Helly selection theorem is a generalisation of Helly's selection theorem for functions of bounded variation to the case of regulated functions. It was proved in 1991 by the Czech mathematician Dana Fraňková.
Background
Let X be a separable Hilbert space, and let BV([0, T]; X) denote the normed vector space of all functions f : [0, T] → X with finite total variation over the interval [0, T], equipped with the total variation norm. It is well known that BV([0, T]; X) satisfies the compactness theorem known as Helly's selection theorem: given any sequence of functions (fn)n∈N in BV([0, T]; X) that is uniformly bounded in the total variation norm, there exists a subsequence
and a limit function f ∈ BV([0, T]; X) such that fn(k)(t) converges weakly in X to f(t) for every t ∈ [0, T]. That is, for every continuous linear functional λ ∈ X*,
Consider now the Banach space Reg([0, T]; X) of all regulated functions f : [0, T] → X, equipped with the supremum norm. Helly's theorem does not hold for the space Reg([0, T]; X): a counterexample is given by the sequence
One may ask, however, if a weaker selection theorem is true, and the Fraňková–Helly selection theorem is such a result.
Statement of the Fraňková–Helly selection theorem
As before, let X be a separable Hilbert space and let Reg([0, T]; X) denote the space of regulated functions f : [0, T] → X, equipped with the supremum norm. Let (fn)n∈N be a sequence in Reg([0, T]; X) satisfying the following condition: for every ε > 0, there exists some Lε > 0 so that each fn may be approximated by a un ∈ BV([0, T]; X) satisfying
and
where |-| denotes the norm in X and Var(u) denotes the variation of u, which is defined to be the supremum
over all partitions
of [0, T]. Then there exists a subsequence
and a limit function f ∈ Reg([0, T]; X) such that fn(k)(t) converges weakly in X to f(t) for every t ∈ [0, T]. That is, for every continuous linear functional λ ∈ X*,
References
Theorems in analysis
Compactness theorems
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https://en.wikipedia.org/wiki/Regulated%20function
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In mathematics, a regulated function, or ruled function, is a certain kind of well-behaved function of a single real variable. Regulated functions arise as a class of integrable functions, and have several equivalent characterisations. Regulated functions were introduced by Nicolas Bourbaki in 1949, in their book "Livre IV: Fonctions d'une variable réelle".
Definition
Let X be a Banach space with norm || - ||X. A function f : [0, T] → X is said to be a regulated function if one (and hence both) of the following two equivalent conditions holds true:
for every t in the interval [0, T], both the left and right limits f(t−) and f(t+) exist in X (apart from, obviously, f(0−) and f(T+));
there exists a sequence of step functions φn : [0, T] → X converging uniformly to f (i.e. with respect to the supremum norm || - ||∞).
It requires a little work to show that these two conditions are equivalent. However, it is relatively easy to see that the second condition may be re-stated in the following equivalent ways:
for every δ > 0, there is some step function φδ : [0, T] → X such that
f lies in the closure of the space Step([0, T]; X) of all step functions from [0, T] into X (taking closure with respect to the supremum norm in the space B([0, T]; X) of all bounded functions from [0, T] into X).
Properties of regulated functions
Let Reg([0, T]; X) denote the set of all regulated functions f : [0, T] → X.
Sums and scalar multiples of regulated functions are again regulated functions. In other words, Reg([0, T]; X) is a vector space over the same field K as the space X; typically, K will be the real or complex numbers. If X is equipped with an operation of multiplication, then products of regulated functions are again regulated functions. In other words, if X is a K-algebra, then so is Reg([0, T]; X).
The supremum norm is a norm on Reg([0, T]; X), and Reg([0, T]; X) is a topological vector space with respect to the topology induced by the supremum norm.
As noted above, Reg([0, T]; X) is the closure in B([0, T]; X) of Step([0, T]; X) with respect to the supremum norm.
If X is a Banach space, then Reg([0, T]; X) is also a Banach space with respect to the supremum norm.
Reg([0, T]; R) forms an infinite-dimensional real Banach algebra: finite linear combinations and products of regulated functions are again regulated functions.
Since a continuous function defined on a compact space (such as [0, T]) is automatically uniformly continuous, every continuous function f : [0, T] → X is also regulated. In fact, with respect to the supremum norm, the space C0([0, T]; X) of continuous functions is a closed linear subspace of Reg([0, T]; X).
If X is a Banach space, then the space BV([0, T]; X) of functions of bounded variation forms a dense linear subspace of Reg([0, T]; X):
If X is a Banach space, then a function f : [0, T] → X is regulated if and only if it is of bounded φ-variation for some φ:
If X is a separable Hilbert space, then Reg([0, T]; X
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https://en.wikipedia.org/wiki/Divergent%20geometric%20series
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In mathematics, an infinite geometric series of the form
is divergent if and only if | r | ≥ 1. Methods for summation of divergent series are sometimes useful, and usually evaluate divergent geometric series to a sum that agrees with the formula for the convergent case
This is true of any summation method that possesses the properties of regularity, linearity, and stability.
Examples
In increasing order of difficulty to sum:
1 − 1 + 1 − 1 + · · ·, whose common ratio is −1
1 − 2 + 4 − 8 + · · ·, whose common ratio is −2
1 + 2 + 4 + 8 + · · ·, whose common ratio is 2
1 + 1 + 1 + 1 + · · ·, whose common ratio is 1.
Motivation for study
It is useful to figure out which summation methods produce the geometric series formula for which common ratios. One application for this information is the so-called Borel-Okada principle: If a regular summation method sums Σzn to 1/(1 - z) for all z in a subset S of the complex plane, given certain restrictions on S, then the method also gives the analytic continuation of any other function on the intersection of S with the Mittag-Leffler star for f.
Summability by region
Open unit disk
Ordinary summation succeeds only for common ratios |z| < 1.
Closed unit disk
Cesàro summation
Abel summation
Larger disks
Euler summation
Half-plane
The series is Borel summable for every z with real part < 1. Any such series is also summable by the generalized Euler method (E, a) for appropriate a.
Shadowed plane
Certain moment constant methods besides Borel summation can sum the geometric series on the entire Mittag-Leffler star of the function 1/(1 − z), that is, for all z except the ray z ≥ 1.
Everywhere
Notes
References
Divergent series
Geometric series
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https://en.wikipedia.org/wiki/Euler%20summation
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In the mathematics of convergent and divergent series, Euler summation is a summation method. That is, it is a method for assigning a value to a series, different from the conventional method of taking limits of partial sums. Given a series Σan, if its Euler transform converges to a sum, then that sum is called the Euler sum of the original series. As well as being used to define values for divergent series, Euler summation can be used to speed the convergence of series.
Euler summation can be generalized into a family of methods denoted (E, q), where q ≥ 0. The (E, 1) sum is the ordinary Euler sum. All of these methods are strictly weaker than Borel summation; for q > 0 they are incomparable with Abel summation.
Definition
For some value y we may define the Euler sum (if it converges for that value of y) corresponding to a particular formal summation as:
If all the formal sums actually converge, the Euler sum will equal the left hand side. However, using Euler summation can accelerate the convergence (this is especially useful for alternating series); sometimes it can also give a useful meaning to divergent sums.
To justify the approach notice that for interchanged sum, Euler's summation reduces to the initial series, because
This method itself cannot be improved by iterated application, as
Examples
Using y = 1 for the formal sum we get if Pk is a polynomial of degree k. Note that the inner sum would be zero for , so in this case Euler summation reduces an infinite series to a finite sum.
The particular choice provides an explicit representation of the Bernoulli numbers, since (the Riemann zeta function). Indeed, the formal sum in this case diverges since k is positive, but applying Euler summation to the zeta function (or rather, to the related Dirichlet eta function) yields (cf. Globally convergent series) which is of closed form.
With an appropriate choice of y (i.e. equal to or close to −) this series converges to .
See also
Binomial transform
Borel summation
Cesàro summation
Lambert summation
Perron's formula
Abelian and Tauberian theorems
Abel–Plana formula
Abel's summation formula
Van Wijngaarden transformation
Euler–Boole summation
References
Mathematical series
Summability methods
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https://en.wikipedia.org/wiki/Tien-Yien%20Li
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Tien-Yien Li (李天岩) (June 28, 1945 – June 25, 2020) was a University Distinguished Professor of Mathematics and University Distinguished Professor Emeritus at Michigan State University. There, he spent 42 years and supervised 26 Ph.D. dissertations.
Early life and education
Li was born on June 28, 1945, in Sha County, Fujian Province, China. At age three, he was brought to Taiwan by his parents. He earned his B.S. in Mathematics at the National Tsinghua University in 1968. Li received his doctorate in 1974 from University of Maryland under the guidance of James Yorke.
Academic career
Li joined the faculty of the Department of Mathematics at Michigan State University in 1976 and was promoted to the rank of full professor in 1983. He retired as a University Distinguished Professor Emeritus in 2018 after spending 42 years at the university. Li and his supervisor James Yorke published a paper in 1975 entitled Period three implies chaos, in which the mathematical term chaos was coined. He also proved Ulam's conjecture in the field of computation of invariant measures of chaotic dynamical systems. Working with Kellogg and Yorke, Li's ideas and the use of numerical methods in computing Brouwer's fixed point, part of the field of modern Homotopy Continuation methods.
Awards and honors
Guggenheim Fellow, 1995
Distinguished Faculty Award, College of Natural Science, Michigan State University, 1996.
Distinguished Faculty Award, Michigan State University, 1996.
J.S.Frame Teaching Award, 1996.
University Distinguished Professor, Michigan State University, 1998.
Distinguished Alumni, College of Sciences, Tsing Hua University, Taiwan, 2002.
Outstanding Academic Advisor Award, College of Natural Science, Michigan State University, 2006.
National Tsinghua University's Outstanding Alumni Award, Taiwan, 2012.
References
External links
T. Y. Li, and J. A. Yorke, Period Three Implies Chaos, American Mathematical Monthly 82, 985 (1975)
Celebration of Life Dr. Tien-Yien Li (1945-2020)
Dr. Tien-Yien Li
1945 births
2020 deaths
American educators
American people of Chinese descent
20th-century Taiwanese mathematicians
21st-century American mathematicians
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https://en.wikipedia.org/wiki/Predictive%20value%20of%20tests
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Predictive value of tests is the probability of a target condition given by the result of a test, often in regard to medical tests.
In cases where binary classification can be applied to the test results, such yes versus no, test target (such as a substance, symptom or sign) being present versus absent, or either a positive or negative test), then each of the two outcomes has a separate predictive value. For example, for positive or negative test, the predictive values are termed positive predictive value or negative predictive value, respectively.
In cases where the test result is of a continuous value, the predictive value generally changes continuously along with the value. For example, for a pregnancy test that displays the urine concentration of hCG, the predictive value increases with increasing hCG value.
A conversion of continuous values into binary values can be performed, such as designating a pregnancy test as "positive" above a certain cutoff value, but this confers a loss of information and generally results in less accurate predictive values.
See also
Positive predictive value
Negative predictive value
References
Medical tests
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https://en.wikipedia.org/wiki/Trefftz%20method
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In mathematics, the Trefftz method is a method for the numerical solution of partial differential equations named after the German mathematician Erich Trefftz(de) (1888–1937). It falls within the class of finite element methods.
Introduction
The hybrid Trefftz finite-element method has been considerably advanced since its introduction about 30 years ago. The conventional method of finite element analysis involves converting the differential equation that governs the problem into a variational functional from which element nodal properties – known as field variables – can be found. This can be solved by substituting in approximate solutions to the differential equation and generating the finite element stiffness matrix which is combined with all the elements in the continuum to obtain the global stiffness matrix. Application of the relevant boundary conditions to this global matrix, and the subsequent solution of the field variables rounds off the mathematical process, following which numerical computations can be used to solve real life engineering problems.
An important aspect of solving the functional requires us to find solutions that satisfy the given boundary conditions and satisfy inter-element continuity since we define independently the properties over each element domain.
The hybrid Trefftz method differs from the conventional finite element method in the assumed displacement fields and the formulation of the variational functional. In contrast to the conventional method (based on the Rayleigh-Ritz mathematical technique) the Trefftz method (based on the Trefftz mathematical technique) assumes the displacement field is composed of two independent components; the intra-element displacement field which satisfies the governing differential equation and is used to approximate the variation of potential within the element domain, and the conforming frame field which specifically satisfies the inter-element continuity condition, defined on the boundary of the element. The frame field here is the same as that used in the conventional finite element method but defined strictly on the boundary of the element – hence the use of the term "hybrid" in the method's nomenclature. The variational functional must thus include additional terms to account for boundary conditions, since the assumed solution field only satisfies the governing differential equation.
Advantages over conventional finite element method
The main advantages of the hybrid Trefftz method over the conventional method are:
the formulation calls for integration along the element boundaries only which allows for curve-sided or polynomial shapes to be used for the element boundary,
presents expansion bases for elements that do not satisfy inter-element continuity through the variational functional, and
this method allows for the development of crack singular or perforated elements through the use of localized solution functions as the trial functions.
Applications
Since
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https://en.wikipedia.org/wiki/Female%20condom
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An internal condom (also known as a femidom or female condom) is a barrier device that is used during sexual intercourse as a barrier contraceptive to reduce the probability of pregnancy or a sexually transmitted infection (STI). Meant as an alternative to the condom, it was invented by Danish MD Lasse Hessel and designed to be worn internally by the woman during vaginal sex to prevent exposure to semen or other body fluids. His invention was launched in Europe in 1990 and approved by the FDA for sale in the US in 1993. Its protection against STIs is inferior to that of male condoms. Internal condoms can be used by the receptive partner during anal sex.
Description
The female condom is a thin, soft, loose-fitting sheath with a flexible ring/frame or ring/foam disc at the closed end. They typically come in various sizes. For most vaginas, a moderately sized condom is adequate; women who have recently given birth should try a large size first. The inner ring or foam disc at the closed end of the sheath is used to insert the condom inside the vagina and to hold it in place during intercourse. The rolled outer ring or poly frame at the open end of the sheath remains outside the vagina and covers part of the external genitalia.
The female condom was developed in the late 20th century (male condoms have been used for centuries). A primary motive for its creation is the well-documented refusal of some men to use a condom because of loss of sensation and the resulting impact on the hardness of the man's erection, and secondarily by its implication that the male could transmit an STI.
Versions and materials
The FC1 female condom was first made from polyurethane. The second generation female condom is called the FC2 and is made from synthetic nitrile (this material change was announced in September 2005, and full transition of the product line to FC2 was done by October 2009). The newer nitrile condoms are less likely to make potentially distracting crinkling noises. FC2 was developed to take the place of FC1, providing the same safety and efficacy during use, but at a significantly lower cost. FC2 is manufactured by The Female Health Company. The World Health Organization (WHO) has cleared FC2 for purchase by U.N. agencies and the United Nations Population Fund has incorporated the female condom into national programming. They are sold under many brand names, including Reality, Femidom, Dominique, Femy, Myfemy, Protectiv and Care.
A recent version of the female condom is made from natural latex, the same material used in male condoms. This condom does not make the noises some experience with plastic condoms and fits snugly against the female anatomy. This type of female condom is manufactured by HLL Lifecare Ltd., India and IXu LLC of USA. It is sold under the brand name VA w.o.w Condom Feminine. One more clinical trial is required before it can be considered for Food and Drug Administration (FDA) approval in the United States.
The global health no
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https://en.wikipedia.org/wiki/Moulton%20plane
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In incidence geometry, the Moulton plane is an example of an affine plane in which Desargues's theorem does not hold. It is named after the American astronomer Forest Ray Moulton. The points of the Moulton plane are simply the points in the real plane R2 and the lines are the regular lines as well with the exception that for lines with a negative slope, the slope doubles when they pass the y-axis.
Formal definition
The Moulton plane is an incidence structure , where denotes the set of points, the set of lines and the incidence relation "lies on":
is just a formal symbol for an element . It is used to describe vertical lines, which you may think of as lines with an infinitely large slope.
The incidence relation is defined as follows:
For and we have
Application
The Moulton plane is an affine plane in which Desargues' theorem does not hold. The associated projective plane is consequently non-desarguesian as well. This means that there are projective planes not isomorphic to for any (skew) field F. Here is the projective plane determined by a 3-dimensional vector space over the (skew) field F.
Notes
References
Richard S. Millman, George D. Parker: Geometry: A Metric Approach with Models. Springer 1991, , pp. 97-104
Incidence geometry
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https://en.wikipedia.org/wiki/Tibor%20%C5%A0al%C3%A1t
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Tibor Šalát ( – ) was a Slovak mathematician, professor of mathematics, and Doctor of Mathematics who specialized in number theory and real analysis. He was the author and co-author of undergraduate and graduate textbooks in mathematics, mostly in Slovak. And most of his scholarly papers have been published in various scientific journals.
Life
Originally from Žitava by the southern region of Slovakia, he studied at the Faculty of Natural Sciences of Charles University in Prague, where in 1952 he defended a dissertation entitled Príspevok k teorii súčtov a nekonečných radov s reálnými členami and supervised by and Vojtěch Jarník.
In 1952 he went to work at the Faculty of Natural Sciences of Comenius University in Bratislava, where he became an assistant professor in 1962. He was appointed to a full professorship position in 1972. And in 1974, he earned a Ph.D. in Mathematics from the same institution.
He specialized in Cantor's expansions, uniform distribution, statistical convergence, summation methods and theory of numbers.
He wrote several undergraduate and graduate textbooks.
Academic papers
References
P. Kostyrko, O. Strauch: Professor Tibor Šalát (1926-2005), Tatra Mt. Math. Publ. 31 (2005), 1-16
Slovak mathematicians
Number theorists
Charles University alumni
Academic staff of Comenius University
1926 births
2005 deaths
Czechoslovak mathematicians
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https://en.wikipedia.org/wiki/Takashi%20Amano%20%28footballer%29
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is a Japanese football player who currently plays for the J3 League team Nagano Parceiro.
Career statistics
Updated to 23 February 2017.
J-League Firsts
Appearance: April 14, 2007. Yokohama F Marinos 5 vs 0 Ōita Trinita, Nissan Stadium
Honours
Yokohama F. Marinos
Emperor's Cup: 2013
References
External links
Profile at Yokohama F. Marinos
Profile at Nagano Parceiro
1986 births
Living people
Association football people from Kanagawa Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Yokohama F. Marinos players
JEF United Chiba players
AC Nagano Parceiro players
Men's association football defenders
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https://en.wikipedia.org/wiki/Rename%20%28relational%20algebra%29
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In relational algebra, a rename is a unary operation written as where:
is a relation
and are attribute names
is an attribute of
The result is identical to except that the attribute in all tuples is renamed to . For an example, consider the following invocation of on an relation and the result of that invocation:
Formally, the semantics of the rename operator is defined as follows:
where is defined as the tuple , with the attribute renamed to , so that:
References
Relational algebra
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https://en.wikipedia.org/wiki/Solder%20form
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In mathematics, more precisely in differential geometry, a soldering (or sometimes solder form) of a fiber bundle to a smooth manifold is a manner of attaching the fibers to the manifold in such a way that they can be regarded as tangent. Intuitively, soldering expresses in abstract terms the idea that a manifold may have a point of contact with a certain model Klein geometry at each point. In extrinsic differential geometry, the soldering is simply expressed by the tangency of the model space to the manifold. In intrinsic geometry, other techniques are needed to express it. Soldering was introduced in this general form by Charles Ehresmann in 1950.
Soldering of a fibre bundle
Let M be a smooth manifold, and G a Lie group, and let E be a smooth fibre bundle over M with structure group G. Suppose that G acts transitively on the typical fibre F of E, and that dim F = dim M. A soldering of E to M consists of the following data:
A distinguished section o : M → E.
A linear isomorphism of vector bundles θ : TM → o*VE from the tangent bundle of M to the pullback of the vertical bundle of E along the distinguished section.
In particular, this latter condition can be interpreted as saying that θ determines a linear isomorphism
from the tangent space of M at x to the (vertical) tangent space of the fibre at the point determined by the distinguished section. The form θ is called the solder form for the soldering.
Special cases
By convention, whenever the choice of soldering is unique or canonically determined, the solder form is called the canonical form, or the tautological form.
Affine bundles and vector bundles
Suppose that E is an affine vector bundle (a vector bundle without a choice of zero section). Then a soldering on E specifies first a distinguished section: that is, a choice of zero section o, so that E may be identified as a vector bundle. The solder form is then a linear isomorphism
However, for a vector bundle there is a canonical isomorphism between the vertical space at the origin and the fibre VoE ≈ E. Making this identification, the solder form is specified by a linear isomorphism
In other words, a soldering on an affine bundle E is a choice of isomorphism of E with the tangent bundle of M.
Often one speaks of a solder form on a vector bundle, where it is understood a priori that the distinguished section of the soldering is the zero section of the bundle. In this case, the structure group of the vector bundle is often implicitly enlarged by the semidirect product of GL(n) with the typical fibre of E (which is a representation of GL(n)).
Examples
As a special case, for instance, the tangent bundle itself carries a canonical solder form, namely the identity.
If M has a Riemannian metric (or pseudo-Riemannian metric), then the covariant metric tensor gives an isomorphism from the tangent bundle to the cotangent bundle, which is a solder form.
In Hamiltonian mechanics, the solder form is known as the tautological one
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https://en.wikipedia.org/wiki/Covariance%20function
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In probability theory and statistics, the covariance function describes how much two random variables change together (their covariance) with varying spatial or temporal separation. For a random field or stochastic process Z(x) on a domain D, a covariance function C(x, y) gives the covariance of the values of the random field at the two locations x and y:
The same C(x, y) is called the autocovariance function in two instances: in time series (to denote exactly the same concept except that x and y refer to locations in time rather than in space), and in multivariate random fields (to refer to the covariance of a variable with itself, as opposed to the cross covariance between two different variables at different locations, Cov(Z(x1), Y(x2))).
Admissibility
For locations x1, x2, …, xN ∈ D the variance of every linear combination
can be computed as
A function is a valid covariance function if and only if this variance is non-negative for all possible choices of N and weights w1, …, wN. A function with this property is called positive semidefinite.
Simplifications with stationarity
In case of a weakly stationary random field, where
for any lag h, the covariance function can be represented by a one-parameter function
which is called a covariogram and also a covariance function. Implicitly the C(xi, xj) can be computed from Cs(h) by:
The positive definiteness of this single-argument version of the covariance function can be checked by Bochner's theorem.
Parametric families of covariance functions
For a given variance , a simple stationary parametric covariance function is the "exponential covariance function"
where V is a scaling parameter (correlation length), and d = d(x,y) is the distance between two points. Sample paths of a Gaussian process with the exponential covariance function are not smooth. The "squared exponential" (or "Gaussian") covariance function:
is a stationary covariance function with smooth sample paths.
The Matérn covariance function and rational quadratic covariance function are two parametric families of stationary covariance functions. The Matérn family includes the exponential and squared exponential covariance functions as special cases.
See also
Autocorrelation function
Correlation function
Covariance matrix
Kriging
Positive-definite kernel
Random field
Stochastic process
Variogram
References
Geostatistics
Spatial analysis
Covariance and correlation
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https://en.wikipedia.org/wiki/No-wandering-domain%20theorem
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In mathematics, the no-wandering-domain theorem is a result on dynamical systems, proven by Dennis Sullivan in 1985.
The theorem states that a rational map f : Ĉ → Ĉ with deg(f) ≥ 2 does not have a wandering domain, where Ĉ denotes the Riemann sphere. More precisely, for every component U in the Fatou set of f, the sequence
will eventually become periodic. Here, f n denotes the n-fold iteration of f, that is,
The theorem does not hold for arbitrary maps; for example, the transcendental map has wandering domains. However, the result can be generalized to many situations where the functions naturally belong to a finite-dimensional parameter space, most notably to transcendental entire and meromorphic functions with a finite number of singular values.
References
Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993,
Dennis Sullivan, Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains, Annals of Mathematics 122 (1985), no. 3, 401–18.
S. Zakeri, Sullivan's proof of Fatou's no wandering domain conjecture
Ergodic theory
Limit sets
Theorems in dynamical systems
Complex dynamics
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https://en.wikipedia.org/wiki/Power%20transform
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In statistics, a power transform is a family of functions applied to create a monotonic transformation of data using power functions. It is a data transformation technique used to stabilize variance, make the data more normal distribution-like, improve the validity of measures of association (such as the Pearson correlation between variables), and for other data stabilization procedures.
Power transforms are used in multiple fields, including multi-resolution and wavelet analysis, statistical data analysis, medical research, modeling of physical processes, geochemical data analysis, epidemiology and many other clinical, environmental and social research areas.
Definition
The power transformation is defined as a continuously varying function, with respect to the power parameter λ, in a piece-wise function form that makes it continuous at the point of singularity (λ = 0). For data vectors (y1,..., yn) in which each yi > 0, the power transform is
where
is the geometric mean of the observations y1, ..., yn. The case for is the limit as approaches 0. To see this, note that - using Taylor series. Then , and everything but becomes negligible for sufficiently small.
The inclusion of the (λ − 1)th power of the geometric mean in the denominator simplifies the scientific interpretation of any equation involving , because the units of measurement do not change as λ changes.
Box and Cox (1964) introduced the geometric mean into this transformation by first including the Jacobian of rescaled power transformation
with the likelihood. This Jacobian is as follows:
This allows the normal log likelihood at its maximum to be written as follows:
From here, absorbing into the expression for produces an expression that establishes that minimizing the sum of squares of residuals from is equivalent to maximizing the sum of the normal log likelihood of deviations from and the log of the Jacobian of the transformation.
The value at Y = 1 for any λ is 0, and the derivative with respect to Y there is 1 for any λ. Sometimes Y is a version of some other variable scaled to give Y = 1 at some sort of average value.
The transformation is a power transformation, but done in such a way as to make it continuous with the parameter λ at λ = 0. It has proved popular in regression analysis, including econometrics.
Box and Cox also proposed a more general form of the transformation that incorporates a shift parameter.
which holds if yi + α > 0 for all i. If τ(Y, λ, α) follows a truncated normal distribution, then Y is said to follow a Box–Cox distribution.
Bickel and Doksum eliminated the need to use a truncated distribution by extending the range of the transformation to all y, as follows:
where sgn(.) is the sign function. This change in definition has little practical import as long as is less than , which it usually is.
Bickel and Doksum also proved that the parameter estimates are consistent and asymptotically normal under appropriate regula
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