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product of the matrices. Matrix multiplication not commutative. That is the dot product, but stuffed into a $$1\times 1$$ matrix! Subsubsection Other Properties. Then certainly the product. Since the multiplication both ways generates the Identity matrix, then we are guaranteed that the inverse matrix obtained using the formula is the correct answer!. Then A−1 exists and we have. Matrix multiplication can be done only when the number of columns of is equal to the number of rows of. and of B and C have sizes for which the indicated sums and products are defined. Both the statement of this theorem and the method of its proof will be important for the study of differential equations in the next section. Proof: All proofs can follow from basic arithmetic laws in R and previous definitions. Observation: The following property is an obvious consequence of this definition. This number is then the scalar product of the two vectors. The properties are the commutative, associative, additive identity and distributive properties. Properties of Matrices. If I is the unit matrix, then AI = IA = A (I is called multiplicative identity). ,1966-67 American Hockey League Media Guide,2019 TOPPS UFC KNOCKOUT PURPLE SHIRT FIGHTER WORN RELIC #/25 JESSICA AGUILAR. The proposed approach is. For the A above, we have A 2 = 0 1 0 0 0 1 0 0 = 0 0 0 0. Multiplication of matrices generally falls into two categories, Scalar Matrix Multiplication and Vector Matrix Multiplication. Countable and uncountable sets. Preliminaries Consider the following situation: A is a matrix, possible augmented, and U is the reduced row echelon form of A. (This section can be omitted without affecting what follows. The determinant of a matrix inverse. One of the main contributions of [2] was to demonstrate that several diverse families of non-abelian groups support the reduction of n £ n matrix multiplication to group al-gebra multiplication. For matrix multiplication to work, the columns of the second matrix have to | {
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al-gebra multiplication. For matrix multiplication to work, the columns of the second matrix have to have the same number of entries as do the rows of the first matrix. If A is an n×n symmetric orthogonal matrix, then A2 = I. We used the following 4 equations to estimate breeding values:. If A is an nxm matrix and O the mxk zero-matrix, then AO = O. Properties of Matrix Multiplication Multiplication can only occur between matrices A and B if the number of columns in A match the number of rows in B. It is not for rectangular matrices of different sizes as it's not even defined for both "directions"! For square matrices, if it is not commutative for any pair of matrices, it is not commutative in general. Orthogonally Diagonalizable Matrices These notes are about real matrices matrices in which all entries are real numbers. Rank and nullity of a matrix We had seen in previous chapter that the number of non-zero rows in the rows in the row-echelon form of a matrix play an important role in finding solutions of linear equation. For two matrices A and B. Matrix multiplication: if A is a matrix of size m n and B is a matrix of. Notice that matrix multiplication is not generally commutative, i. Commutative Property of Multiplication(CPM)-It states that changing the order of the factors will not affect the product. In other words, regardless of the matrix A, the exponential matrix eA is always invertible, and has inverse e A. Properties of matrix multiplication. Since our model will usually contain a constant term, one of the columns in the X matrix will contain only ones. The set GL(n,F) of n×nmatrices with nonzero determi-nant is a group when equipped with matrix multiplication. Institut de France, Binet also read a paper which contained a proof of the multiplication theorem but it was less satisfactory than that given by Cauchy. 1 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars. Likewise, a scalar product can | {
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real numbers, R, or complex numbers, C) with elements called scalars. Likewise, a scalar product can be taken outside the transform: DFT(c*x) = c*DFT(x). For the A above, we have A 2 = 0 1 0 0 0 1 0 0 = 0 0 0 0. In a 2 2 matrix multiplication, M= 7 in Strassen, m= n= p= 2. 5: A latin square of side nis an nby narray in which each cell contains a. Matrix Multiplication Suppose A and B are two matrices such that the number of columns of A is equal to number of rows of B. | {
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Has a Crappy Problem on Their Hands, but Will the City Do Anything About It? July 20 2018 by Patrice Lee Onwuka Here’s Why Alexandria Ocasio-Cortez Is Wrong on Unemployment, Capitalism July 18 2018 by Patrice Lee Onwuka U.S. Should Make Unfair Drug Pricing, Foreign Practices a Priority in Trade Negotiations July 11 2018 by Patrice Lee Onwuka Trump Administration Slashes ACA Navigator Outreach Program -- And For Good Reasons July 11 2018 by Patrice Lee Onwuka The Left’s Hatred of Anything Trump Targets St. Jude Family July 9 2018 by Patrice Lee Onwuka Abolish ICE Protester Scales Lady Liberty Making for an Unhappy Fourth of July July 5 2018 by Patrice Lee Onwuka Suck This Up: Seattle Bans Plastic Straws July 2 2018 by Patrice Lee Onwuka Supreme Court Strikes Big Blow to Public Sector Unions, Upholds Free Speech of Public Workers June 28 2018 by Patrice Lee Onwuka Linda Sarsour Invokes Dr. Martin Luther King Jr. to Justify Harassment of Trump Officials June 27 2018 by Patrice Lee Onwuka No Maxine Waters, This is Not a Moment for Harassment or Incivility June 26 2018 by Patrice Lee Onwuka Making Kids Programming Better Through Common-Sense Broadcast TV Reforms June 22 2018 by Patrice Lee Onwuka A New Proposal to Replace the Broken Healthcare Law June 21 2018 by Patrice Lee Onwuka Trump Administration Expands Cheaper, Quality Healthcare Plans June 20 2018 by Patrice Lee Onwuka D.C. Will Kill Tips with New Minimum Wage June 19 2018 by Patrice Lee Onwuka S.E. Cupp Gets It Right: "Not all Women are Democrats" June 18 2018 by Patrice Lee Onwuka Today's Dads Are More Involved with Child Care Than Their Granddads June 15 2018 by Patrice Lee Onwuka Kathy Griffin Shaming Kevin Hart For Not Being Political is What's Wrong with Hollywood Elitism June 14 2018 by Patrice Lee Onwuka Optimism Among Small Business Owners Skyrockets Thanks to Tax Cuts June 13 2018 by Patrice Lee Onwuka Seattle Reverses Job-Killing Head Tax on Big Employers After Residents Speak Out June 12 2018 by | {
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Seattle Reverses Job-Killing Head Tax on Big Employers After Residents Speak Out June 12 2018 by Patrice Lee Onwuka Net Neutrality Ends Today. Here's Why It's Not the End of the World. June 11 2018 by Patrice Lee Onwuka It's Not Just Kate Spade and Anthony Bourdain, Suicide Rates Rise in Almost Every State. What's Going On? June 11 2018 by Patrice Lee Onwuka Is Anyone Buying Samantha Bee's Latest Apology for Her Vulgar Slur About Ivanka Trump? June 7 2018 by Patrice Lee Onwuka Women in Saudi Arabia are One Step Closer to Driving June 6 2018 by Patrice Lee Onwuka 5 Principles to Celebrate from the Supreme Court's Colorado Baker Ruling June 5 2018 by Patrice Lee Onwuka Women's March Organizers Tamika Mallory, Linda Sarsour Lash Out Against Israel June 4 2018 by Patrice Lee Onwuka NFL Finally Takes a Stand for the National Anthem May 25 2018 by Patrice Lee Onwuka Americans Report Doing Better Financially Under Year 1 of Trump May 24 2018 by Patrice Lee Onwuka Consumers, Small Banks to Gain Relief from Bi-Partisan Roll Back of Dodd-Frank Banking Regulations May 23 2018 by Patrice Lee Onwuka U.S. Baby Bust Leads to Lowest Birth Rates in 30 Years May 18 2018 by Patrice Lee Onwuka 3 Things to Know About the Senate Vote on Net Neutrality May 17 2018 by Patrice Lee Onwuka Hysteria Comes to Black Hair Care Products – Again May 16 2018 by Patrice Lee Onwuka Seattle's Business 'Head Tax' Has Even Starbucks, Amazon in an Uproar May 16 2018 by Patrice Lee Onwuka Oprah's Liberal Bias Clouded Good Advice from Her Commencement Address May 14 2018 by Patrice Lee Onwuka On Mother's Day, Advice from Some Moms May 12 2018 by Patrice Lee Onwuka Tennis Star Venus Williams Rejects "Feminist" Label May 10 2018 by Patrice Lee Onwuka Michelle Obama is Still Stumped by How Women Voted in 2016 May 7 2018 by Patrice Lee Onwuka Are You One of Third of Americans Who Say the Internet is Bad or Mixed for Society? May 1 2018 by Patrice Lee Onwuka In This Winning Economy, Cities Pay Young People to | {
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for Society? May 1 2018 by Patrice Lee Onwuka In This Winning Economy, Cities Pay Young People to Move There May 1 2018 by Patrice Lee Onwuka Kim Kardashian and Kanye West Break the Internet for a Good Reason This Time - Free Thought April 26 2018 by Patrice Lee Onwuka Nancy Pelosi Chokes on "Crumbs" Comments April 25 2018 by Patrice Lee Onwuka 8 States Just Hit the Lowest Unemployment in Their History April 24 2018 by Patrice Lee Onwuka Kanye West Supports "Free-Thinking" Conservative Woman and Sets Twitter on Fire April 23 2018 by Patrice Lee Onwuka When Government Makes It Harder to Work, Fewer People Can Move Up the Ladder April 19 2018 by Patrice Lee Onwuka On Tax Day, Here's How the Federal Government Spends Every Dollar of Your Money April 17 2018 by Patrice Lee Onwuka Uber, Lyft Drivers Angry with Seattle City Council for Colluding with Taxis to Raise Rates April 17 2018 by Patrice Lee Onwuka They're Back! Share of Adult Children Living at Home Highest in 75 Years April 12 2018 by Patrice Lee Onwuka 3 Quotes from Mark Zuckerberg's Senate Grilling that Should Concern You April 11 2018 by Patrice Lee Onwuka 5 Drivers of the Pay Gap – They Are Not Gender Discrimination April 10 2018 by Patrice Lee Onwuka Diamond and Silk Labeled as "Unsafe" for Facebook, Maybe It's Time for A New Social Media Option April 10 2018 by Patrice Lee Onwuka Jimmy Kimmel Apologizes, But Not For His Melania Trump Joke April 9 2018 by Patrice Lee Onwuka How Many Americans Would Give Up Voting for a 10-Percent Pay Raise? It’s Surprising! April 6 2018 by Patrice Lee Onwuka Why Does Jimmy Kimmel Get Away with Mocking Melania Trump's Accent? April 5 2018 by Patrice Lee Onwuka There's No Room for Pregnancy Discrimination on the Job April 3 2018 by Patrice Lee Onwuka How the Blockbuster 'Roseanne' Reboot Gives a Voice to Middle America March 29 2018 by Patrice Lee Onwuka Tariffs: A New Bargaining Chip for Better Trade Deals? March 28 2018 by Patrice Lee Onwuka Joe Biden is Right to Attack | {
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Chip for Better Trade Deals? March 28 2018 by Patrice Lee Onwuka Joe Biden is Right to Attack Giving Every American a Universal Basic Income March 27 2018 by Patrice Lee Onwuka Ivanka Trump Has a Strategy to Get People into Good-Paying Jobs March 27 2018 by Patrice Lee Onwuka Should Responding to After-Hours Emails Be Banned? March 26 2018 by Patrice Lee Onwuka Cardi B Just Said What We're All Thinking About Taxes March 23 2018 by Patrice Lee Onwuka 3 Ways the Trump White House is Helping Millennials March 23 2018 by Patrice Lee Onwuka Google Shares How It Closed Unexplained Pay Gaps March 20 2018 by Patrice Lee Onwuka Jim Carrey Slams Sarah Sanders' Looks in a New Portrait March 19 2018 by Patrice Lee Onwuka VP Pence Celebrates Women as Everyday Heroes with IWF March 15 2018 by Patrice Lee Onwuka Joy Behar Finally Publicly Apologizes to VP Pence, Christians for Disparaging Jokes March 15 2018 by Patrice Lee Onwuka Vice President Pence Has an Answer for Joy Behar's Religious Intolerance March 13 2018 by Patrice Lee Onwuka Nancy Pelosi Back Peddles on Tax Bonuses Being "Crumbs" March 9 2018 by Patrice Lee Onwuka Why Tariffs on Steel, Aluminum are the Wrong Move March 9 2018 by Patrice Lee Onwuka 5 Things Women Cannot Do in Some Countries on International Women's Day March 8 2018 by Patrice Lee Onwuka Ride-sharing Replaces Buses in This City and Residents Love It March 7 2018 by Patrice Lee Onwuka How New Tariffs Could Raise the Prices of Your Everyday Goods March 6 2018 by Patrice Lee Onwuka Hollywood's Activism and Hypocrisy Were on Display at the Oscars One Pin at a Time March 5 2018 by Patrice Lee Onwuka 7 Facts About Women in the Workforce for Women's History Month March 1 2018 by Patrice Lee Onwuka Poll: Most Inappropriate Office Behavior is Not What You Think It is February 26 2018 by Patrice Lee Onwuka Why Amy Poehler Shouldn't Fight for a Hike in Tipped Minimum Wages February 26 2018 by Patrice Lee Onwuka Europe’s Gender Quotas for Corporate Boards Failed to | {
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Wages February 26 2018 by Patrice Lee Onwuka Europe’s Gender Quotas for Corporate Boards Failed to Do Much for Women February 20 2018 by Patrice Lee Onwuka Should Healthy, Employable Americans Keep Medicaid Forever? February 16 2018 by Patrice Lee Onwuka 5 Companies Expanding Parental Leave Thanks to Tax Cuts February 15 2018 by Patrice Lee Onwuka The Downside of Another Fast-food Worker Protest for$15 Minimum Wages February 12 2018 by Patrice Lee Onwuka | {
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Uber Has a Gender Earnings Gap, but There's a Good Explanation February 9 2018 by Patrice Lee Onwuka
Survey: #MeToo Backlash Against Women is Real February 7 2018 by Patrice Lee Onwuka
Ryan Seacrest Has a Can’t-Miss #MeToo Message About False Allegations February 6 2018 by Patrice Lee Onwuka
On the Anniversary of FMLA, Here’s Why There's Little to Celebrate February 5 2018 by Patrice Lee Onwuka
T-Mobile’s Super Bowl Ad Sounded Tone Deaf February 5 2018 by Patrice Lee Onwuka
Have You Checked Your Paycheck? You Might Be Surprised February 1 2018 by Patrice Lee Onwuka
6 Times Democrats Didn’t Cheer for Middle-Class Wins in Trump's State of the Union Address January 31 2018 by Patrice Lee Onwuka
6 Things to Watch for at President Trump’s State of the Union Address January 30 2018 by Patrice Lee Onwuka
Survey: New Tech Moms Have Great Experience Returning to Work Post Baby January 30 2018 by Patrice Lee Onwuka
Here are Six Political Moments from the 2018 Grammys January 29 2018 by Patrice Lee Onwuka
Debbie Wasserman Schultz, Nancy Pelosi Pooh-pooh $1,000 Bonuses from Tax Reform January 26 2018 by Patrice Lee Onwuka Walt Disney Heiress Thrashed Tax Reform, but Disney Company Gives Employees Tax Reform Bonuses, Investments January 25 2018 by Patrice Lee Onwuka Cashier-free Amazon Store Opens in Nation's First$15-Minimum-Wage City January 24 2018 by Patrice Lee Onwuka
Disney Credits Tax Reform for Its $50 Mil Education Investment,$1,000 Bonuses January 24 2018 by Patrice Lee Onwuka
Congress Made New Changes to 529 Plans that Parents Will Celebrate January 23 2018 by Patrice Lee Onwuka
Kristen Bell Opens Girl Power Sag Awards with Cheap Shot at Melania Trump January 22 2018 by Patrice Lee Onwuka
Feminist Slams ‘Fixer Upper’ Stars Chip & Joanna for Having 5th Child January 17 2018 by Patrice Lee Onwuka
Costco Shocks Customers by Exposing the High Price of Seattle’s Soda Tax January 16 2018 by Patrice Lee Onwuka | {
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Condoleeza Rice Warns Against #MeToo Creating Snowflakes and Triggering Backlash January 15 2018 by Patrice Lee Onwuka
Walmart Announces Higher Starting Wages, Bonuses, and Benefits Thanks to Tax Reform January 11 2018 by Patrice Lee Onwuka
Utility Companies Cut Your Utility Bills Thanks to Tax Cuts January 11 2018 by Patrice Lee Onwuka
Say Goodbye to Busboys at Red Robin Thanks to Minimum Wage Hikes January 10 2018 by Patrice Lee Onwuka
USPS is Losing Money, But Blame It on Washington Not Amazon January 5 2018 by Patrice Lee Onwuka
No, Iceland Didn’t Just Make It Illegal to Pay Women Less Than Men January 5 2018 by Patrice Lee Onwuka
California Has Brewed Up a Common-Sense Solution to Drunk Driving January 4 2018 by Patrice Lee Onwuka
Top Hollywood Women Launch Time’s Up as the Answer to #MeToo, but Don’t Be Fooled January 3 2018 by Patrice Lee Onwuka
What You Need to Know About New York's Paid Leave Plan January 3 2018 by Patrice Lee Onwuka
18 States Hike Minimum Wage in 2018, Young Female Workers May Suffer January 2 2018 by Patrice Lee Onwuka
Ivanka Trump Visit Triggers Connecticut Parents to Pull Kids Out of School December 22 2017 by Patrice Lee Onwuka
Capitol Hill Just Passed a #GoodDealforWomen Through Tax Reform December 19 2017 by Patrice Lee Onwuka | {
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California’s $15 Minimum Wage to Cost 400,000 Jobs – Many for Women December 18 2017 by Patrice Lee Onwuka Pew: Gender Discrimination, Sexual Harassment are Problems but Not Universal December 15 2017 by Patrice Lee Onwuka Alyssa Milano, John Oliver Aren’t for Saving the Internet but Internet Control December 14 2017 by Patrice Lee Onwuka Feminism is 2017’s Word of the Year But Women Still Don’t Know What It Means December 13 2017 by Patrice Lee Onwuka San Francisco Embraces Everyone but Delivery Robots December 12 2017 by Patrice Lee Onwuka Jobs Report: One Million More Women Are Working Since November 2016 December 8 2017 by Patrice Lee Onwuka What Paris Hilton Could Teach Chelsea Handler About Grace in Crisis December 8 2017 by Patrice Lee Onwuka Kellyanne Conway Had a #MeToo Moment Too, But No One Paid Attention Because of Politics December 7 2017 by Patrice Lee Onwuka Mark Zuckerberg’s Paternity Leave Tells Us 3 Things About Paid Leave, Millennials December 6 2017 by Patrice Lee Onwuka Sheryl Sandberg Calls Out the Chilling Effect Between Sexes Following Harassment Revelations December 5 2017 by Patrice Lee Onwuka Three Ways to Maximize Your Last Paychecks of 2017 December 1 2017 by Patrice Lee Onwuka Meghan Markle Giving Up Her Career for Prince Harry Doesn’t Make Her a Bad Woman November 29 2017 by Patrice Lee Onwuka Susan Sarandon is Still Anti-Hillary, The Left Blasts Her For It November 28 2017 by Patrice Lee Onwuka FCC’s Thanksgiving Blessing was to End Internet Neutrality Regulations November 27 2017 by Patrice Lee Onwuka Black Friday Protests Dwindle Exposing This Uncomfortable Truth November 27 2017 by Patrice Lee Onwuka Kathy Griffin’s Sadistic Trump Joke Left Her Jobless, She’s Not a Victim November 21 2017 by Patrice Lee Onwuka New York Wants to Dictate When You Work November 20 2017 by Patrice Lee Onwuka Bernie Sanders Blatantly Gives Bill Clinton, Al Franken a Pass on Sexual Harassment November 20 2017 by Patrice Lee Onwuka Do Winter Blizzards | {
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Al Franken a Pass on Sexual Harassment November 20 2017 by Patrice Lee Onwuka Do Winter Blizzards Call for Paid Leave Too? November 15 2017 by Patrice Lee Onwuka ‘The Simpsons’ Calling Kellyanne Conway a Nazi is a Sexist, Racist Cheap Shot November 14 2017 by Patrice Lee Onwuka How Zero Tolerance Polices Backfire in the Age of #MeToo November 13 2017 by Patrice Lee Onwuka Three G.I. Janes from American History that Make Women Proud on Veterans Day November 10 2017 by Patrice Lee Onwuka TSA Failing Undercover Weapons Tests – Again, and It’s Disturbing November 10 2017 by Patrice Lee Onwuka NAACP Exploits NFL Protests to Ban "Star-Spangled Banner" as National Anthem November 9 2017 by Patrice Lee Onwuka Women in Congress Have an Answer to Paid Leave November 8 2017 by Patrice Lee Onwuka Sexist LA Times Columnist Attacks Sarah Huckabee Sanders’ Looks November 6 2017 by Patrice Lee Onwuka Explosive: DNC, Hillary Clinton Campaign Rigged the Party Nomination for Her November 3 2017 by Patrice Lee Onwuka Eva Longoria is Desperately Wrong to Push the Latina Pay Gap November 2 2017 by Patrice Lee Onwuka Shame on Cosmo for Normalizing Incest – Again November 2 2017 by Patrice Lee Onwuka Poll: Most Women Love Tech but Only Half Think It’ll Improve Their Life November 1 2017 by Patrice Lee Onwuka 2018 Obamacare Premiums to Leave Even Wider Hole in Your Wallet October 30 2017 by Patrice Lee Onwuka Walmart Debuts Robots to Make (Holiday) Shopping Better October 30 2017 by Patrice Lee Onwuka NAACP Issues Inane Flying-While-Black Advisory, Please Keep It to Yourselves October 27 2017 by Patrice Lee Onwuka President Trump’s Opioid Attack Plan: Personal, Practical, and Policy October 26 2017 by Patrice Lee Onwuka UPenn Teacher Helps Black Women by Discriminating Against White Men October 24 2017 by Patrice Lee Onwuka Campus Sexual Assault: The Distressing Fight Mothers Face for Their Sons, Advice to Young Women October 23 2017 by Patrice Lee Onwuka IRS to Hold Tax Returns Hostage to | {
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Sons, Advice to Young Women October 23 2017 by Patrice Lee Onwuka IRS to Hold Tax Returns Hostage to Enforce Obamacare October 20 2017 by Patrice Lee Onwuka ABC’s Misleading Workplace Sexual Harassment Poll Pushes an Agenda October 19 2017 by Patrice Lee Onwuka ‘The Big Bang Theory’ Actress Counsels Actresses to be Modest, Triggers Feminist Backlash October 17 2017 by Patrice Lee Onwuka No, NFL Protests Aren't About the Gender Pay Gap October 16 2017 by Patrice Lee Onwuka Ivanka Trump to Boost Global Women Entrepreneurs Via World Bank Fund October 16 2017 by Patrice Lee Onwuka President Trump’s Anti-Regulation Crusade may be Working in Americans’ Eyes October 13 2017 by Patrice Lee Onwuka Lawmakers Try to Stop Secretary DeVos on Campus Sexual Assault Policy October 13 2017 by Patrice Lee Onwuka Black Lives Matter Group Opposes Free Speech for Every American October 12 2017 by Patrice Lee Onwuka Harvey Weinstein’s Open Secret Exposes Hypocrisy on the Left October 10 2017 by Patrice Lee Onwuka Marc Jacobs Doubles Down on Free Expression for Designers, Artists October 10 2017 by Patrice Lee Onwuka On Columbus Day, Give Him a Break October 9 2017 by Patrice Lee Onwuka Company Behind Fearless Girl Statute Admits Gender Pay Issues October 9 2017 by Patrice Lee Onwuka Three Bright Spots for Women in Today’s Jobs Report October 6 2017 by Patrice Lee Onwuka Half of Americans Can’t Afford More than$100 in Healthcare Premiums October 4 2017 by Patrice Lee Onwuka | {
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One Tech Company Changed Approach to Promotions, Women Won Big October 3 2017 by Patrice Lee Onwuka
Librarian Rejects Dr. Seuss Books Because Melania Trump Sent Them September 29 2017 by Patrice Lee Onwuka
Amber Rose’s Slutwalk is a Stunt Not Empowerment September 29 2017 by Patrice Lee Onwuka
Do We Want a Feminist ‘Will & Grace’ Reboot? September 26 2017 by Patrice Lee Onwuka
Hillary Clinton Slams Women Who Support Trump as “Publicly Disrespecting Themselves” September 26 2017 by Patrice Lee Onwuka
Betsy DeVos Ends Obama-Era Policies on School Sexual Assault September 22 2017 by Patrice Lee Onwuka
Washington Wants Black Households to Trade Privacy for Cash September 22 2017 by Patrice Lee Onwuka
Half of College Women Get This Wrong about Hate Speech September 20 2017 by Patrice Lee Onwuka
Joe Biden Slams Socialist Universal Basic Income Idea September 19 2017 by Patrice Lee Onwuka
Millennials Would Trade Right to Vote for Student Loan Freedom September 18 2017 by Patrice Lee Onwuka
Feminists Attack Starbucks's Pumpkin Spice Lattes for Funding White Supremacy September 15 2017 by Patrice Lee Onwuka
DeVos Backlash Incites Dangerous Language September 14 2017 by Patrice Lee Onwuka
Over One in Three Americans Can’t Name Any 1st Amendment Rights September 12 2017 by Patrice Lee Onwuka
Miss America Pageant Gets Political and Blatantly Anti-Trump September 11 2017 by Patrice Lee Onwuka
Melania Trump Staffs Fewer People than Michelle Obama – A Savings for Taxpayers September 8 2017 by Patrice Lee Onwuka
Hillary Clinton Asks 'What Happened?' in 2016 September 7 2017 by Patrice Lee Onwuka
Kim Kardashian Still Eschews One Label: Feminist September 5 2017 by Patrice Lee Onwuka
Jerry Springer Rides the $15 Minimum Wage Wagon September 5 2017 by Patrice Lee Onwuka A$20 Question: Will Harriet Tubman be on the Bill? September 1 2017 by Patrice Lee Onwuka
Estee Lauder Faces Heat Over Paid Leave for Dads August 31 2017 by Patrice Lee Onwuka | {
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Estee Lauder Faces Heat Over Paid Leave for Dads August 31 2017 by Patrice Lee Onwuka
No, Melania’s Shoes are Not the Story, Texas is August 30 2017 by Patrice Lee Onwuka
Illinois Governor Shuts Down $15 Minimum Wage August 28 2017 by Patrice Lee Onwuka Amy Schumer's Equal Pay Problem August 25 2017 by Patrice Lee Onwuka Poll: Nearly Three Out of Four Americans Would Die to Defend Freedom of Speech August 25 2017 by Patrice Lee Onwuka Here’s Another Way Women Suffer from Minimum Wage Increases August 24 2017 by Patrice Lee Onwuka Oops! Social Security Website Misdirects$11 Mil to Thieves August 23 2017 by Patrice Lee Onwuka
D.C. Raises Minimum Wage, Diners Foot the Bill August 21 2017 by Patrice Lee Onwuka
What Dr. Ben Carson Shows Us about Confronting Hateful Ideas August 17 2017 by Patrice Lee Onwuka
Hope Brings Another Woman to Trump Leadership Circle August 17 2017 by Patrice Lee Onwuka
Helen Mirren’s Feminism is about Anti-Trump and Potty-Mouths August 16 2017 by Patrice Lee Onwuka
CFPB Crackdown May Trap Borrowers in a Different Debt Trap August 14 2017 by Patrice Lee Onwuka
Sheryl Sandberg Leans In to One-Size-Fits-All Paid Leave Plan August 10 2017 by Patrice Lee Onwuka
Café Charges 18% Man Tax to Close the Gender Pay Gap August 9 2017 by Patrice Lee Onwuka
College STEM Women Claim to be Sexism Victims August 7 2017 by Patrice Lee Onwuka
A Sexism Rating Coming to a Movie Near You August 3 2017 by Patrice Lee Onwuka
IRS Rehires Hundreds of Fired Workers Despite Serious Transgressions August 3 2017 by Patrice Lee Onwuka | {
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Study: MD County to Lose 47,000 jobs because of $15 Minimum Wage August 2 2017 by Patrice Lee Onwuka Adam Carolla Schools Congress on Campus Free Speech July 31 2017 by Patrice Lee Onwuka How One Tech Company is Taking on the Future of Work July 28 2017 by Patrice Lee Onwuka Governor Christie Shuts Down Costly Paid Leave Expansion Bill July 27 2017 by Patrice Lee Onwuka Teachers’ Union Boss Plays the Race Card against Secretary DeVos July 25 2017 by Patrice Lee Onwuka Cutting the Strings on Obamacare Enrollment Assistance July 24 2017 by Patrice Lee Onwuka Not Even Death Stops a Government Check July 21 2017 by Patrice Lee Onwuka Who Cares for the Caregivers? July 19 2017 by Patrice Lee Onwuka College Diversity Officials Rake in Major Dough July 18 2017 by Patrice Lee Onwuka DHS Hides Spending$3.6 Mil of Taxpayer Dollars on Pricey Conferences July 17 2017 by Patrice Lee Onwuka
Study: Marrying Before Having Kids is a Strong Path to Success for Millennials July 13 2017 by Patrice Lee Onwuka
$2 Mil NJ Welfare Fraud Exposes Holes in Our Social Safety Net July 7 2017 by Patrice Lee Onwuka “WTF” is Silicon Valley’s Next Answer to the Democratic Party July 6 2017 by Patrice Lee Onwuka Fraud Alert: Federal Free Phone, Internet Program Wastes$100 Mil Annually on Fake, Dead Enrollees July 3 2017 by Patrice Lee Onwuka
Cosmo Leaves Conservatives Off Their Short List of Female Presidential Candidates June 28 2017 by Patrice Lee Onwuka | {
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Report: Seattle’s Minimum Wage Hike Gave Low-Wage Workers a $125 Monthly Pay Cut June 27 2017 by Patrice Lee Onwuka Obama Comes Out to Bash Senate Repeal Bill June 23 2017 by Patrice Lee Onwuka Maybe Profit Motivates Tech Companies More than Politics June 22 2017 by Patrice Lee Onwuka Ivanka Trump Hits Capitol Hill about Paid Leave June 20 2017 by Patrice Lee Onwuka Supreme Court Upholds Offensive Names as Free Speech June 20 2017 by Patrice Lee Onwuka Silicon Valley CEOs Ignore Calls to Boycott Trump Tech Summit June 19 2017 by Patrice Lee Onwuka What the Return of Pro-Net Neutrality Democrat Means for Internet Regs Rollback June 15 2017 by Patrice Lee Onwuka Is the Federal Consumer Cop Asleep on the Job? June 15 2017 by Patrice Lee Onwuka Obama Health Records Program Leaves Taxpayers on the Hook for Mispayments June 15 2017 by Patrice Lee Onwuka Just in Time for July 4th New York State Greenlights Uber, Lyft June 9 2017 by Patrice Lee Onwuka Filter, Don’t Pilfer June 8 2017 by Patrice Lee Onwuka What Michelle Obama Gets Wrong About Self-Interest June 8 2017 by Patrice Lee Onwuka Harvard Revokes Acceptances Over Online Comments June 6 2017 by Patrice Lee Onwuka Bill Maher’s N-Word Episode June 5 2017 by Patrice Lee Onwuka Hillary Clinton's Whine Tour Stops in Silicon Valley June 2 2017 by Patrice Lee Onwuka Maryland Governor Shuts Down Paid Sick-Leave Bill May 26 2017 by Patrice Lee Onwuka Chelsea Clinton Equates Child Marriage to Climate Change May 26 2017 by Patrice Lee Onwuka Obamacare’s Death Spiral Claims Another Healthcare Insurer May 25 2017 by Patrice Lee Onwuka Top 10 Unauthorized Federal Programs on the Chopping Block in Trump Budget May 25 2017 by Patrice Lee Onwuka Supreme Court Ends Judge Shopping for Patent Trolls May 23 2017 by Patrice Lee Onwuka State of Texas Trumps Austin over Uber, Lyft Regulations May 22 2017 by Patrice Lee Onwuka Undoing Net Neutrality Starts Today May 18 2017 by Patrice Lee Onwuka San Francisco May Stall Delivery Robots May 18 | {
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Starts Today May 18 2017 by Patrice Lee Onwuka San Francisco May Stall Delivery Robots May 18 2017 by Patrice Lee Onwuka Miss USA Stands by Her Comments – Good for Her May 17 2017 by Patrice Lee Onwuka Miss USA Slammed for Saying Healthcare is a Privilege Rather than a Right May 15 2017 by Patrice Lee Onwuka Don’t Be Tone Deaf on Hearing Aids May 11 2017 by Patrice Lee Onwuka Grads Boo Betsy DeVos at Commencement May 11 2017 by Patrice Lee Onwuka TV Comedian John Oliver Piles on to Save Obama Internet Rules May 10 2017 by Patrice Lee Onwuka Automation Coming to a City near You May 9 2017 by Patrice Lee Onwuka Your Zip code Is a Good Predictor for How Long You Live May 9 2017 by Patrice Lee Onwuka Tech Bosses: Expect Significant Automation of Jobs by 2022 May 5 2017 by Patrice Lee Onwuka Can Another Tech Squad to Transform How Washington Works? May 5 2017 by Patrice Lee Onwuka Tech Founder to Ivanka Trump: Don’t Use My Story May 3 2017 by Patrice Lee Onwuka Are Saving and Debt Repayment the New Normal for Americans? May 2 2017 by Patrice Lee Onwuka This Immigrant is Showing Up to Work on May Day May 1 2017 by Patrice Lee Onwuka FCC Announces Net Neutrality Reversal Plan April 27 2017 by Patrice Lee Onwuka Copyright Office Reform Bill Passes House April 27 2017 by Patrice Lee Onwuka FCC Expected to Kick Net Neutrality to the Curb April 26 2017 by Patrice Lee Onwuka Delivery Robots Get Legislative Green Light in Two States April 25 2017 by Patrice Lee Onwuka Harvard Study: Higher Minimum Wages Trigger Restaurant Closures April 24 2017 by Patrice Lee Onwuka Not Done Rolling Back Washington’s Red Tape Yet April 19 2017 by Patrice Lee Onwuka Three Celebrities Feeling the Pain on Tax Day April 18 2017 by Patrice Lee Onwuka Female-Led FTC Takes Reducing Regulations Seriously April 18 2017 by Patrice Lee Onwuka New York Pilots Free College Tuition Plan April 13 2017 by Patrice Lee Onwuka Poll: Americans Comfortable with AI in Most--But Not All--Instances April 11 2017 by | {
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Lee Onwuka Poll: Americans Comfortable with AI in Most--But Not All--Instances April 11 2017 by Patrice Lee Onwuka No Phone Calls on Airplanes After All April 11 2017 by Patrice Lee Onwuka Amazon to Hire 30,000 Part-time, Work-from-Home Jobs April 10 2017 by Patrice Lee Onwuka Judge Blocks Seattle’s Plan to Unionize Uber April 7 2017 by Patrice Lee Onwuka Stop the Misinformation on the Roll-Back of Privacy Rules April 5 2017 by Patrice Lee Onwuka The Challenge of Counterfeit Drugs April 4 2017 by Patrice Lee Onwuka Cali Lawmaker Wants to Push Gig-Economy Workers into Unions April 3 2017 by Patrice Lee Onwuka FCC Reverses Federal Power Grab Over LifeLine Program (AKA ObamaPhones for the Internet) March 30 2017 by Patrice Lee Onwuka Here’s How Uber is Boosting Diverse Applicants March 30 2017 by Patrice Lee Onwuka Congress Scraps Obama-Era Privacy Rules March 30 2017 by Patrice Lee Onwuka Should We Worry About Robots Replacing Workers? March 27 2017 by Patrice Lee Onwuka Black Leaders Tell Trump What They Have to Lose March 23 2017 by Patrice Lee Onwuka Congress to Free the Copyright Office from the Thumb of the Library of Congress March 23 2017 by Patrice Lee Onwuka Why Everything from Dresses to Avocados May Get More Expensive March 22 2017 by Patrice Lee Onwuka Trump, Clinton Campaigns Depended Upon Sharing Economy on the Campaign Trail March 21 2017 by Patrice Lee Onwuka Before You Blame the Wage Gap, Check Your Attitude March 20 2017 by Patrice Lee Onwuka Trump Hacks Away at the TSA March 20 2017 by Patrice Lee Onwuka Trump Pumps the Brakes on Obama-Era Fuel Efficiency Regs March 16 2017 by Patrice Lee Onwuka Students Push Back on Feminist Professor’s Indoctrination March 15 2017 by Patrice Lee Onwuka President Trump: May Be Time to Say "You're Fired!" to Duplicative Agency Employees March 14 2017 by Patrice Lee Onwuka Austin’s Struggle with Transportation After Uber, Lyft Ride Out of Town March 13 2017 by Patrice Lee Onwuka Steve Case: American Workers Feeling | {
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Uber, Lyft Ride Out of Town March 13 2017 by Patrice Lee Onwuka Steve Case: American Workers Feeling Left Behind, Really Have Been Left Behind March 13 2017 by Patrice Lee Onwuka Analogies Fly Over Net Neutrality March 9 2017 by Patrice Lee Onwuka These are the Women A Day Without Women Hurts March 8 2017 by Patrice Lee Onwuka Can Facebook’s New Tool Crack Down on “Fake News”? March 7 2017 by Patrice Lee Onwuka Is Regulating the Internet a Good Idea? March 6 2017 by Patrice Lee Onwuka Dr. Ben Carson Confirmed as HUD Head March 3 2017 by Patrice Lee Onwuka Robots Green-Lighted to Deliver Your Packages March 2 2017 by Patrice Lee Onwuka Three Key Messages to Black America from Trump’s Address March 1 2017 by Patrice Lee Onwuka Reports: GM Lobbies to Keep Competing Self-Driving Tech Off the Road February 28 2017 by Patrice Lee Onwuka New Self-driving Truck Company Speeds Up the Technology February 27 2017 by Patrice Lee Onwuka Small Internet Companies Gain Regulatory Relief from Washington February 24 2017 by Patrice Lee Onwuka Women Have a New Ally in Getting Them to Work February 24 2017 by Patrice Lee Onwuka Federal Watchdog: Obama’s Techie Squad Flouted the Rules February 22 2017 by Patrice Lee Onwuka Adulting School: Millennials Are Signing Up for Classes February 22 2017 by Patrice Lee Onwuka Former AG Eric Holder Leading Internal Investigation at Uber February 21 2017 by Patrice Lee Onwuka Whoopi Snubs the Mean Girls at New York Fashion Week in Defense of Tiffany Trump February 17 2017 by Patrice Lee Onwuka Humana Calls It Quits on ObamaCare, Leaving One State in Trouble February 16 2017 by Patrice Lee Onwuka Congress Opens the Door to Self-Driving Cars February 15 2017 by Patrice Lee Onwuka Linda McMahon Confirmed to Head the SBA February 14 2017 by Patrice Lee Onwuka Cabs Threaten to Halt Wheelchair Access Over Uber, Lyft February 14 2017 by Patrice Lee Onwuka Trump Models: The Next Target in the Fashion World Purge February 13 2017 by Patrice Lee Onwuka | {
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Trump Models: The Next Target in the Fashion World Purge February 13 2017 by Patrice Lee Onwuka Senate Dems Come Out Swinging to Protect Washington’s Control of the Internet February 8 2017 by Patrice Lee Onwuka A Surprise Stand for Competition Over Cronyism in Battle for NYC Streets? February 8 2017 by Patrice Lee Onwuka Lifeline Internet Assistance Program (AKA Obamaphones for the Internet) Takes a Hit February 7 2017 by Patrice Lee Onwuka Big Mac ATM’s, the Answer to$15 Minimum Wages? February 2 2017 by Patrice Lee Onwuka | {
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New FCC Chairman Gets to Work Cutting Red Tape February 2 2017 by Patrice Lee Onwuka
Trump Transportation Secretary Chao Confirmed, Despite Democratic Protest February 1 2017 by Patrice Lee Onwuka
Cutting the Red Tap: President Trump Orders 2-for 1 Rule on Regulations January 31 2017 by Patrice Lee Onwuka
Chelsea Handler's Garbage-Pail Mouth is Now Shaming Melania Trump's English January 27 2017 by Patrice Lee Onwuka
Break out the Dow 20K Hats? January 27 2017 by Patrice Lee Onwuka
Apple Hit with Another Lawsuit over Distracted Driving January 25 2017 by Patrice Lee Onwuka
Getting American Workers Ready for Automation’s Coming Shake-Up January 24 2017 by Patrice Lee Onwuka | {
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Uber Ponies Up $20M to Feds Over Driver Pay January 23 2017 by Patrice Lee Onwuka Americans Aren’t Really Warming Up to Self-Driving Cars January 19 2017 by Patrice Lee Onwuka The Government’s “Slant” on Trademarks Gets Challenged January 19 2017 by Patrice Lee Onwuka Uber Sues Seattle to Block Drivers from Unionizing January 18 2017 by Patrice Lee Onwuka Inauguration Reminds Us Why Home Sharing is a Win for Travelers January 17 2017 by Patrice Lee Onwuka Jesse Jackson PUSHes Uber on Diversity Statistics January 13 2017 by Patrice Lee Onwuka 5 Times President Obama’s Rhetoric Didn’t Match His Kumbaya Message January 11 2017 by Patrice Lee Onwuka Jenna Bush Microaggressess at the Golden Globes? January 10 2017 by Patrice Lee Onwuka Rolling Back Obama’s Internet Privacy Rules January 9 2017 by Patrice Lee Onwuka Update: City of San Diego Says Restaurants Can't Publicly Blame Surcharge on Minimum Wage Hike January 6 2017 by Patrice Lee Onwuka San Diego Restaurants Slap Diners with Surcharges Thanks to Minimum Wage Increases January 6 2017 by Patrice Lee Onwuka The Uber Hook-Up Effect January 5 2017 by Patrice Lee Onwuka Christmas Eve Distracted Driving Fatality Could Trigger Government Action January 3 2017 by Patrice Lee Onwuka Cali Regulators Drive Uber’s Self-Driving Cars Away December 27 2016 by Patrice Lee Onwuka Will Seattle’s Driver Unionization be the End of Uber, Ridesharing? December 21 2016 by Patrice Lee Onwuka Not Fake News: Some UVA Students Sign Fake Anti-Christmas Petition December 20 2016 by Patrice Lee Onwuka Pinterest Misses Female Recruitment Quantity Quotas, But Hits on Quality December 19 2016 by Patrice Lee Onwuka No Michelle Obama, “Hope” Doesn’t Start and End with Your Husband December 19 2016 by Patrice Lee Onwuka Uber, San Francisco Regulators Crash Head-on over Self-Driving December 16 2016 by Patrice Lee Onwuka More on the Shaming of Wonder Woman December 15 2016 by Patrice Lee Onwuka Environmental Group Smears Black Beauty Products | {
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Wonder Woman December 15 2016 by Patrice Lee Onwuka Environmental Group Smears Black Beauty Products December 14 2016 by Patrice Lee Onwuka San Francisco Mayor Sides with Airbnb December 12 2016 by Patrice Lee Onwuka Airbnb Can’t Beat ‘Em, So Now They’re Lobbying ‘Em December 9 2016 by Patrice Lee Onwuka The Supremes Side with Samsung Against Apple in High-Profile Patent Case December 7 2016 by Patrice Lee Onwuka Cell Phones on the Road: A Serious Problem, but Do the Feds Need to Solve It? December 6 2016 by Patrice Lee Onwuka Airbnb Took on the Big Apple and Lost December 5 2016 by Patrice Lee Onwuka Obama’s Overtime Rule is Down, but Not Out December 2 2016 by Patrice Lee Onwuka Taxpayers Will Forgive Billions More in Student Loan Debt December 2 2016 by Patrice Lee Onwuka Uber Drivers Join Minimum Wage Protesters November 30 2016 by Patrice Lee Onwuka Negative Online Reviews Get New Federal Protection November 30 2016 by Patrice Lee Onwuka #GivingTuesday: the Hangover Cure for a Weekend of Spending November 29 2016 by Patrice Lee Onwuka Some Maine Restaurant Workers Worried about Effects of Minimum Wage Hike November 28 2016 by Patrice Lee Onwuka Judge Blocks Obama Workplace Overtime Rules November 23 2016 by Patrice Lee Onwuka Pew: One in Four Americans Makes Bank through the Gig Economy November 21 2016 by Patrice Lee Onwuka U.S. Small Businesses: Good News and Bad News November 21 2016 by Patrice Lee Onwuka Arizona’s New Minimum Wage Packs a One-Two Punch for Nonprofits November 18 2016 by Patrice Lee Onwuka Washington State Minimum Wage Passed and Childcare Costs Skyrocket November 17 2016 by Patrice Lee Onwuka Hi, I’m Ben, Your Senator and Uber Driver! November 15 2016 by Patrice Lee Onwuka Facebook Ends Multicultural Targeted Ads November 14 2016 by Patrice Lee Onwuka Oprah’s Guarded Hope for the New President Triggers the Left November 14 2016 by Patrice Lee Onwuka Star-Power Failed to Energize Millennials at the Polls November 10 2016 by Patrice Lee | {
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Lee Onwuka Star-Power Failed to Energize Millennials at the Polls November 10 2016 by Patrice Lee Onwuka Mom-preneur Faces Charges for Selling Ceviche? November 9 2016 by Patrice Lee Onwuka Uber is Back on the Road in Philly November 7 2016 by Patrice Lee Onwuka Airbnb’s Self-Policing Strategy against Discrimination November 4 2016 by Patrice Lee Onwuka Sanders and Warren Ignore that America is Better Off Now While Promoting Socialism • Evening Edit October 14 2019 by Patrice Lee Onwuka Biden campaign plans to lose Iowa, New Hampshire, and Nevada • Rising October 11 2019 by Patrice Lee Onwuka Is Warren trying to win black voters by floating Gillum as VP • Rising October 11 2019 by Patrice Lee Onwuka US Businesses Should Stand Behind Hong Kong • Making Money October 9 2019 by Patrice Lee Onwuka Left Leaning Outlets Turn on NYT Following Kavanaugh Hit Piece • Evening Edit September 23 2019 by Patrice Lee Onwuka Partisan Party Politics Turns American Voters Off • Lou Dobbs Tonight September 18 2019 by Patrice Lee Onwuka Bashing Billionaires Dissuades Everyone from Trying to Reach the Top • Making Money September 13 2019 by Patrice Lee Onwuka Dems Have a Hard Case to Make Heading Into 2020 • Your World with Neil Cavuto September 9 2019 by Patrice Lee Onwuka Dems Prioritize Gun Reform But Its All Talk • Evening Edit September 9 2019 by Patrice Lee Onwuka Talking Tech: Is Being a "Big Company" a Bad Thing? • Making Money with Charles Payne September 6 2019 by Patrice Lee Onwuka Media Needs to Find Balance Between Criticism and Support of President • Evening Edit September 3 2019 by Patrice Lee Onwuka Democrats Continue to Focus on Emotion Over Fact: Biden and AOC • Trish Regan Primetime August 30 2019 by Patrice Lee Onwuka Is the DNC rigging the debate rules? • Rising August 29 2019 by Patrice Lee Onwuka Is South Carolina Biden's Hail Mary? • Rising August 29 2019 by Patrice Lee Onwuka 4 in 10 People Won't Trust the 2020 Election Results if Their Team Loses • Lou Dobbs | {
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Lee Onwuka 4 in 10 People Won't Trust the 2020 Election Results if Their Team Loses • Lou Dobbs Tonight August 28 2019 by Patrice Lee Onwuka Taking Border Security Seriously with Approved Funding for The Wall • Evening Edit August 27 2019 by Patrice Lee Onwuka Women Are Winning in This Economy • Coast to Coast August 22 2019 by Patrice Lee Onwuka Language Police: In San Francisco and on Facebook • Lou Dobbs Tonight August 22 2019 by Patrice Lee Onwuka Harry Reid Denounces Trajectory of Dem Party • Making Money with Charles Payne August 21 2019 by Patrice Lee Onwuka Judge Trump’s actions, not words — and other commentary August 19 2019 by Patrice Lee Onwuka Conservative commentator rips Shapiro over criticism of people with multiple jobs August 16 2019 by Patrice Lee Onwuka How Will the Economy Effect the 2020 Conversation? • Coast to Coast August 15 2019 by Patrice Lee Onwuka Israel Should Say "Thanks, But No Thanks" to Reps. Omar and Tlaib • Lou Dobbs Tonight August 15 2019 by Patrice Lee Onwuka We Need to Act Now to Steer Away from Red Flags Ahead • Making Money August 14 2019 by Patrice Lee Onwuka "In God We Trust" Is Important to Display in School • Lou Dobbs Tonight August 14 2019 by Patrice Lee Onwuka Political Talking Points Stir Up Violent Threats Against SoulCycle Owner • Evening Edit August 9 2019 by Patrice Lee Onwuka Pitting Consumers Against Businesses in the Name of Politics • Coast to Coast August 8 2019 by Patrice Lee Onwuka Democratic Socialists Cannot Defeat Capitalism with Proper Pronouns and Complaints • Lou Dobbs August 6 2019 by Patrice Lee Onwuka Don't pander to women; offer real opportunity August 1 2019 by Patrice Lee Onwuka Bernie Ignores Legitimate Questions About his Health Care Plans • Evening Edit July 31 2019 by Patrice Lee Onwuka The Washington Post Plays Scapegoat for Those Still Hung Up on 2016 Election • Lou Dobbs Tonight July 30 2019 by Patrice Lee Onwuka Food Stamps Cut Because Booming Economy Means Less People Need Them • Fox & | {
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by Patrice Lee Onwuka Food Stamps Cut Because Booming Economy Means Less People Need Them • Fox & Friends First July 24 2019 by Patrice Lee Onwuka The Squad Tries Pushing the Democrats Further Left Over Israel • Evening Edit July 24 2019 by Patrice Lee Onwuka$15 Minimum Wage Robs Women of Jobs, Opportunity, Economic Mobility July 18 2019 by Patrice Lee Onwuka | {
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"Squad" Pushes Democrat Party Closer to Socialism • Coast to Coast July 17 2019 by Patrice Lee Onwuka
"Woke Activism" is No Way to Get Things Done in Congress • Fox & Friends First July 15 2019 by Patrice Lee Onwuka
Cory Gardner, the women’s vote and religion: Western Conservative Summit takeaways July 14 2019 by Patrice Lee Onwuka
Democrats Have Placed All Their Eggs in the Identity Politics Basket • Evening Edit July 11 2019 by Patrice Lee Onwuka
Why Restaurant Workers are Rising Up to Save Tipped Wages July 9 2019 by Patrice Lee Onwuka
Women's World Cup Champions Demand Equal Pay, Do the Numbers Add Up? • After The Bell July 8 2019 by Patrice Lee Onwuka
The Folly of Demanding Amazon’s Tax Returns • Liberty Watch Radio July 7 2019 by Patrice Lee Onwuka
Democrats are Obsessed Over the Past as the Candidates Take Aim at Each Other • Making Money July 5 2019 by Patrice Lee Onwuka
Supreme Court Reaches Verdict on 2020 Census Question • Your World with Neil Cavuto June 27 2019 by Patrice Lee Onwuka
Democrats Ignore Booming Economy and Foreshadow Recession • Evening Edit June 25 2019 by Patrice Lee Onwuka
Economists Determine the Value of "Free" Social Media Apps • Making Money June 24 2019 by Patrice Lee Onwuka
Race is a Central Issue for 2020, Biden is Learning the Hard Way • Coast to Coast June 20 2019 by Patrice Lee Onwuka
Hearing on Reparations Shows Just How Complicated This Issue Is • Trish Regan Primetime June 19 2019 by Patrice Lee Onwuka
President Trump Prioritizes Legal Immigrants During Border Crisis • Lou Dobbs Tonight June 18 2019 by Patrice Lee Onwuka
Lets Take an Objective Look at the Russian Interference Allegations • PoliticsNation June 15 2019 by Patrice Lee Onwuka
Liberals Distract Voters From Booming Economy with Social Issues • Lou Dobbs Tonight June 14 2019 by Patrice Lee Onwuka
Why Kamala Harris’ ‘Equal Pay’ Proposal Will Hurt, Not Help Women June 13 2019 by Patrice Lee Onwuka | {
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Why Kamala Harris’ ‘Equal Pay’ Proposal Will Hurt, Not Help Women June 13 2019 by Patrice Lee Onwuka
Republicans and Democrats at odds on overtime pay June 13 2019 by Patrice Lee Onwuka
Deep Fake Videos Can Have Dangerous Repercussions • Evening Edit June 12 2019 by Patrice Lee Onwuka
Bernie Sanders Bashes Walmart Execs Claiming Minimum Wage Isn't High Enough • Coast to Coast June 5 2019 by Patrice Lee Onwuka
Mandated Paid Time Off Is Not Only Bad for Most Businesses, It's Bad for Many Women June 5 2019 by Patrice Lee Onwuka
Democrats Throw Money at Homelessness But the Situation Still Gets Worse, Why? • Lou Dobbs Tonight June 5 2019 by Patrice Lee Onwuka
Stossel: The Paid Leave Fairy Tale • Reason TV June 4 2019 by Patrice Lee Onwuka
Crisis at the Border and Draining the Swamp, Trust Trump to Create Change • Making Money May 31 2019 by Patrice Lee Onwuka
Kamala Harris’ New Proposal to Punish Companies Renews Wage Gap Debate • Bold Politics May 24 2019 by Patrice Lee Onwuka
Conflict in D.C. Does Effect Markets, But Consumers Continue to Spend • Coast to Coast May 23 2019 by Patrice Lee Onwuka
The Gender Wage Gap and What You Can Do About It May 22 2019 by Patrice Lee Onwuka
Top Dems Blow Any Chance of Bipartisan Reform • Evening Edit May 22 2019 by Patrice Lee Onwuka
Women's Organization Announces The 2019 Empowered Woman of the Year May 21 2019 by Patrice Lee Onwuka
Kamala Harris Threatens to Fine Businesses that Contribute to the "Pay Gap" • Fox & Friends May 21 2019 by Patrice Lee Onwuka
Low Birth Rates Could Have Devastating Economic Impacts • Fox & Friends May 16 2019 by Patrice Lee Onwuka
Policy analyst on falling birth rates: Rhetoric from the left contributing to 'change in norms' May 16 2019 by Patrice Lee Onwuka
Sanctuary Cities Putting Illegal Immigrants Ahead of Citizens • Lou Dobbs Tonight May 8 2019 by Patrice Lee Onwuka
Trump Should Look to Private Sector for Federal Reserve Board Nominees • Coast to Coast May 2 2019 by Patrice Lee Onwuka | {
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Every Aspect of the Economy is Thriving From Jobs, Wages, To Taxes• Fox & Friends First April 29 2019 by Patrice Lee Onwuka
Bernie Struggles to Win Over Black Women • Evening Edit April 25 2019 by Patrice Lee Onwuka
Patrice Onwuka on Plastic Bag Bans, Net Neutrality and Michelle Obama • Americhicks April 23 2019 by Patrice Lee Onwuka
Many Americans Don't Care About the Mueller Report • Cavuto Live April 20 2019 by Patrice Lee Onwuka
Obama Playbook Will Not Be Enough to Get Joe Biden in the White House • Fox & Friends April 20 2019 by Patrice Lee Onwuka
Mueller Report Finally Available: What did it say and what is next? • WAMU April 18 2019 by Patrice Lee Onwuka
Opportunity Zones Create Economic Prosperity • Making Money with Charles Payne April 17 2019 by Patrice Lee Onwuka
Left Tries Desperately To Keep Collusion Narrative Alive • Coast to Coast April 15 2019 by Patrice Lee Onwuka
As support for reparations grow, so does pushback from some black Americans April 12 2019 by Patrice Lee Onwuka
Democrats Are Selective When Deciding Who is Innocent or Guilty • Fox & Friends Saturday April 6 2019 by Patrice Lee Onwuka
Being Fiscally Conservative Includes Being Smart on All Aspects of the Economy • Coast to Coast April 3 2019 by Patrice Lee Onwuka
Don't Fall for the Equal Pay Day Myth • Larry O'Connor Show April 2 2019 by Patrice Lee Onwuka
Presidential Hopefuls Playing Politics with Joe Biden's Victims • Evening Edit April 1 2019 by Patrice Lee Onwuka
Joe Biden's 2020 Doom • Cheddar Politics April 1 2019 by Patrice Lee Onwuka
Marco Rubio's New Family Leave Bill is a Budget Neutral Conservative Plan • Coast to Coast March 27 2019 by Patrice Lee Onwuka
The gender pay gap still exists - but it's shrinking, according to a new study March 27 2019 by Patrice Lee Onwuka
Millions of American off food stamps under President Trump • OANN March 23 2019 by Patrice Lee Onwuka
The Electoral College Matters, and Here's Why • The Evening Edit March 20 2019 by Patrice Lee Onwuka | {
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The Electoral College Matters, and Here's Why • The Evening Edit March 20 2019 by Patrice Lee Onwuka
Threats to Drag Out Investigation Despite No Evidence Of Collusion • The Evening Edit March 13 2019 by Patrice Lee Onwuka
A Legislative Unicorn: Paid Family Leave Without New Taxes or Entitlements March 12 2019 by Patrice Lee Onwuka
Congress considers parental leave programs March 12 2019 by Patrice Lee Onwuka
The Ivanka Trump effect: Why Republicans are now championing paid family leave, and what their new proposal looks like March 12 2019 by Patrice Lee Onwuka
Joni Ernst, Mike Lee roll out conservative paid parental leave idea March 12 2019 by Patrice Lee Onwuka
Raising the Topic of Paid Parental Leave March 12 2019 by Patrice Lee Onwuka
Republicans Propose Plan to Let Parents Use Social Security for Paid Family Leave After Having a Child March 12 2019 by Patrice Lee Onwuka
GOP senators unveil paid parental leave proposal March 12 2019 by Patrice Lee Onwuka
Support Choices That Women Make When Looking at Gender Pay Gap • Making Money with Charles Payne March 8 2019 by Patrice Lee Onwuka
Trump Previews 2020 in CPAC Speech • Fox and Friends First March 4 2019 by Patrice Lee Onwuka
Consumer Confidence Bouncing Back as Trump Looks Ahead to 2020 • Coast to Coast March 4 2019 by Patrice Lee Onwuka
Pelosi Should Second Guess New Member's Calls for Trump Impeachment • The Evening Edit February 27 2019 by Patrice Lee Onwuka
AOC Lives in Luxury Housing in D.C. Despite Her Views on Affordable Housing • Fox and Friends First February 21 2019 by Patrice Lee Onwuka
Bernie Attacks Someone Who Achieved the American Dream • Making Money with Charles Payne February 20 2019 by Patrice Lee Onwuka
Bernie Sanders Continues to Misrepresent the Reality of Socialism • The Evening Edit February 19 2019 by Patrice Lee Onwuka
Meet the Women Who Protested the Women’s March January 21 2019 by Patrice Lee Onwuka | {
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Meet the Women Who Protested the Women’s March January 21 2019 by Patrice Lee Onwuka
Trump Putting Border Security on the Table with DACA Extension • America's News Headquarters January 19 2019 by Patrice Lee Onwuka
Women's March is Not Inclusive of All Ideas • Fox & Friends January 19 2019 by Patrice Lee Onwuka
USPS Relies on Amazon: What this means for consumers • Your World with Neil Cavuto December 5 2018 by Patrice Lee Onwuka
How Will Markets React to a Potential Trade Deal with China? • Coast to Coast November 29 2018 by Patrice Lee Onwuka
Streisand accuses Trump of misogyny while liberals turn on their own November 28 2018 by Patrice Lee Onwuka
What Pete Davidson’s SNL Apology Could Teach Us About Civility • Bold Politics November 16 2018 by Patrice Lee Onwuka
Is Amazon Guilty of Taking Part In Corporate Welfare? • The Evening Edit November 14 2018 by Patrice Lee Onwuka
'Economic damage' predicted after city levies big tax on hi-tech firms November 13 2018 by Patrice Lee Onwuka
Markets React to Midterm Election Results • Coast to Coast November 12 2018 by Patrice Lee Onwuka
Volatile Markets Will Only Marginally Influence Midterms • The Evening Edit October 26 2018 by Patrice Lee Onwuka
PETA Claims Milk is Racist • Trish Regan Primetime October 23 2018 by Patrice Lee Onwuka
How Will Booming Economy Be Affected by The Fed? • Coast to Coast October 17 2018 by Patrice Lee Onwuka
Kanye Brings Attention to Real Economic Issues • Coast to Coast October 11 2018 by Patrice Lee Onwuka
Will Kavanaugh Confirmation Drive Women Away from the GOP? Not Likely October 8 2018 by Patrice Lee Onwuka
Anti-Kavanaugh Harassment Tactics Overlooked by the Left • Coast to Coast October 4 2018 by Patrice Lee Onwuka
Statement on #CancelKavanaugh Protest: Activists Use Dr. Ford to Exploit Emotion Over a Deeply Personal Issue October 4 2018 by Patrice Lee Onwuka
Republican Women Explain Why They Stand By Brett Kavanaugh More Than Ever October 3 2018 by Patrice Lee Onwuka | {
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Women Want Kavanaugh Confirmed if FBI Finds No Corroborating Evidence • FBN am October 3 2018 by Patrice Lee Onwuka
A Tale of Two Cities: A Snapshot of Washington D.C. On the Historic Kavanaugh-Ford Hearing Day September 27 2018 by Patrice Lee Onwuka
FBI Investigation Was an Ongoing Distraction During Hearings • After the Bell September 27 2018 by Patrice Lee Onwuka
'There's No Presumption of Innocence': Women Rally Both for and Against Kavanaugh September 27 2018 by Patrice Lee Onwuka
Dems Would Love for Trump to Fire Rosenstein • Fox & Friends First September 25 2018 by Patrice Lee Onwuka
Republicans Have Gone Out of Their Way to Try to Hear Blasey • Coast to Coast September 21 2018 by Patrice Lee Onwuka
Jacques: Sexism is a losing strategy September 14 2018 by Patrice Lee Onwuka
Good Trade Relations with China is Good for the Economy • Coast to Coast September 12 2018 by Patrice Lee Onwuka
Will Reforming SNAP Benefit Americans? • Bold Politics September 7 2018 by Patrice Lee Onwuka
Motivations Behind Trying to Flip a Russian Oligarch • Fox & Friends First September 3 2018 by Patrice Lee Onwuka
More Than Ever, Millennials Are Taking Care of Their Aging Family Members • Bold Life August 24 2018 by Patrice Lee Onwuka
Trump's Commitment to Diversity in the White House Continues After Omarosa • Andrew Wilkow Show August 22 2018 by Patrice Lee Onwuka
Statement: IWF Applauds Roll Back of Clean Power Plan August 22 2018 by Patrice Lee Onwuka
Omarosa’s Antics Won't Undermine Trump’s Progress for Black Americans • Afternoons with Cliff Kelley August 21 2018 by Patrice Lee Onwuka
NYT Publishes Pure Speculation on Mueller Investigation • Fox & Friends First August 20 2018 by Patrice Lee Onwuka
America IS Great: a lesson from real Immigrants • Fox & Friends August 18 2018 by Patrice Lee Onwuka
Harley-Davidson Staying in U.S. is Huge Victory • Coast to Coast August 17 2018 by Patrice Lee Onwuka | {
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Is there a racial gender pay gap? • Bold Politics August 10 2018 by Patrice Lee Onwuka
Ocasio-Cortez Forgets Socialism Didn't Work in South America • Your World with Neil Cavuto August 9 2018 by Patrice Lee Onwuka
Rubio-Wagner family leave bill would draw from Social Security August 7 2018 by Patrice Lee Onwuka
Explaining the Economic Security for New Parents Act • KDKA August 3 2018 by Patrice Lee Onwuka
Rubio's New Paid Leave Bill is Great for Women • Fox & Friends First August 2 2018 by Patrice Lee Onwuka
Bernie Sanders "Medicare for All" plan would be a disaster • Your World with Neil Cavuto July 31 2018 by Patrice Lee Onwuka
How will new GDP numbers influence the midterm elections • Coast to Coast July 30 2018 by Patrice Lee Onwuka
New Bill Aims to Extend Trump Tax Cuts Beyond 2025 • Fox & Friends First July 25 2018 by Patrice Lee Onwuka
Sessions prepares students for diverse opinions at college • Making Money with Charles Payne July 24 2018 by Patrice Lee Onwuka
Student loan debt saddling non-saving Millennials July 24 2018 by Patrice Lee Onwuka
Only 4.9 Percent of People Have Two Jobs • Fox & Friends First July 19 2018 by Patrice Lee Onwuka
Companies thinking about intellectual theft when dealing with China • Coast to Coast July 18 2018 by Patrice Lee Onwuka
Cyber Security Needs to be Taken Seriously During Trump-Putin Summit • FBN am July 16 2018 by Patrice Lee Onwuka
What Do You Think Of Seattle's Plastic Straw Ban? • Bold Life July 6 2018 by Patrice Lee Onwuka
Hopeful About North Korean Cooperation • FBN am July 6 2018 by Patrice Lee Onwuka
#133 Reforms needed for children's television July 2 2018 by Patrice Lee Onwuka
How will Supreme Court nominations effect the primaries? • Coast to Coast June 29 2018 by Patrice Lee Onwuka
Trump is no excuse for intimidation of women June 28 2018 by Patrice Lee Onwuka
Protesters Want Media Attention, Not Policy Change • Your World with Neil Cavuto June 22 2018 by Patrice Lee Onwuka | {
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Immigration bills will be hard to pass, but we can't ignore this issue • FBN AM June 20 2018 by Patrice Lee Onwuka
We should be concerned that foreign agents had access to Hillary's emails • Fox & Friends First June 15 2018 by Patrice Lee Onwuka
Free Trade Without Free Riders • Coast to Coast June 7 2018 by Patrice Lee Onwuka
Democrats Playing Politics in Primaries • Making Money with Charles Payne June 4 2018 by Patrice Lee Onwuka
IWF Statement on New Steel and Aluminum Tariffs May 31 2018 by Patrice Lee Onwuka
Trump had good reason to pull out of meeting with North Korea • Making Money with Charles Payne May 24 2018 by Patrice Lee Onwuka
Should Medical Marijuana Be Allowed in Classrooms? • Fox & Friends First May 21 2018 by Patrice Lee Onwuka
Media Bias After They Refuse to Accept Trump's Corrected Statement • Your World with Neil Cavuto May 17 2018 by Patrice Lee Onwuka
Lower Birth Rates: Student Debt, Societal Change, Technological Advances • The Intelligence Report May 17 2018 by Patrice Lee Onwuka
China's Relationship with North Korea Could Make All The Difference • FBN am May 16 2018 by Patrice Lee Onwuka
#127 Questioning the Status Quo: Black voters don't owe their vote to either party May 14 2018 by Patrice Lee Onwuka
Were there problems with Cohen's payments? • Coast to Coast May 10 2018 by Patrice Lee Onwuka
Removing "In God We Trust" to be Politically Correct • Fox & Friends First May 7 2018 by Patrice Lee Onwuka
Proof Mueller is Fishing for Evidence • FBN am May 7 2018 by Patrice Lee Onwuka
We Want to Ensure a Legitimate and Lawful Immigration Process • FBN am April 26 2018 by Patrice Lee Onwuka
Nancy Pelosi Chokes on "Crumbs" Comment Because It Just Isn't True • Coast to Coast April 25 2018 by Patrice Lee Onwuka
#123 Education Savings Accounts for Military Families Can Make All the Difference April 23 2018 by Patrice Lee Onwuka
Why are so many young adults choosing to stay at home? • BoldTV April 20 2018 by Patrice Lee Onwuka | {
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Why are so many young adults choosing to stay at home? • BoldTV April 20 2018 by Patrice Lee Onwuka
Gov. Brown is Choosing Sanctuary over Safety • Risk & Reward April 19 2018 by Patrice Lee Onwuka
Comey has made enemies on both sides of the aisle • Risk & Reward April 19 2018 by Patrice Lee Onwuka
Bias training won't do much at Starbucks • Intelligence Report with Trish Reagan April 19 2018 by Patrice Lee Onwuka
Tax Cuts Will Play Major Role in Midterms • Coast to Coast April 5 2018 by Patrice Lee Onwuka
New Survey: What would Americans do for a 10% raise? • Fox & Friends First April 5 2018 by Patrice Lee Onwuka
Trump to rescind spending with a maneuver Clinton used nearly 20 years ago • FBN:am April 5 2018 by Patrice Lee Onwuka
Citizenship Question MUST be Answered Truthfully • Cavuto Live April 2 2018 by Patrice Lee Onwuka
#118 New Amendment Provides Protection Against the Discrimination of Pregnant Women April 2 2018 by Patrice Lee Onwuka
On the New Spending Bill: Entitlements Drive the National Debt • Risk & Reward March 23 2018 by Patrice Lee Onwuka
Regulations will Hinder Innovation on Social Media • Risk & Reward March 23 2018 by Patrice Lee Onwuka
White House Millennials Conference to Address Economy, Free Speech, and Opioids March 18 2018 by Patrice Lee Onwuka
In the #MeToo era, young conservative women look for their spot March 16 2018 by Patrice Lee Onwuka
Shake Ups in White House Happening for a Reason • FBN AM March 15 2018 by Patrice Lee Onwuka
Dems Will Not Continue to Embrace Middle America • Coast to Coast March 14 2018 by Patrice Lee Onwuka
We Don't Want to Abandon Free Trade • Cavuto Live March 10 2018 by Patrice Lee Onwuka
GOP takes on paid family leave March 10 2018 by Patrice Lee Onwuka
Issues in California are a Classic Case of Federalism • Risk & Reward March 9 2018 by Patrice Lee Onwuka
A Trade War is Going to Undermine Current Economic Progress • Coast to Coast March 5 2018 by Patrice Lee Onwuka | {
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#113 What You Don't Know About Immigrants: A real immigrant's story March 2 2018 by Patrice Lee Onwuka
A conservative solution to paid leave • The National Discourse Podcast February 27 2018 by Patrice Lee Onwuka
CNN Pushes Agenda at Townhall • Risk & Reward February 23 2018 by Patrice Lee Onwuka
Americans Recognize the 2nd Amendment is Important to Protect • Coast to Coast February 20 2018 by Patrice Lee Onwuka
Republicans Produced Plan, Waiting for Action Across the Aisle • FBN AM February 13 2018 by Patrice Lee Onwuka
#MeToo and Working Women • PBS To The Contrary February 10 2018 by Patrice Lee Onwuka
Fundamentals of Our Economy Are Strong • Coast to Coast February 8 2018 by Patrice Lee Onwuka
Republicans Tired of Partisan Bickering • Coast to Coast January 31 2018 by Patrice Lee Onwuka
Unions Losing Battle Against Inevitable Future • The Intelligence Report January 25 2018 by Patrice Lee Onwuka
Did Schumer Cave on Gov Shut Down? • FBN am January 23 2018 by Patrice Lee Onwuka
Dems ‘Big Risk’ Not Worth It • Your World January 22 2018 by Patrice Lee Onwuka
Media Attacks Trump's Doctor • Risk & Reward January 19 2018 by Patrice Lee Onwuka
The Left Criticizing Tax Plan and Immigration • Risk & Reward January 12 2018 by Patrice Lee Onwuka
Concerns on Trade with China • Coast to Coast January 10 2018 by Patrice Lee Onwuka
Weighing in on Trump vs. Bannon • FBN AM January 5 2018 by Patrice Lee Onwuka
U.N. Funding Cut Good for Women • Forbes on Fox December 29 2017 by Patrice Lee Onwuka
Next Moves for the GOP • Coast to Coast December 26 2017 by Patrice Lee Onwuka
Saving Money After the Holidays • Fox & Friends First December 26 2017 by Patrice Lee Onwuka
Senate Passes Tax Bill • FBN AM December 20 2017 by Patrice Lee Onwuka
Are Black Women Leaving The Democratic Party to Become Independents? December 19 2017 by Patrice Lee Onwuka
Final GOP Tax Bill • Risk & Reward December 15 2017 by Patrice Lee Onwuka | {
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Final GOP Tax Bill • Risk & Reward December 15 2017 by Patrice Lee Onwuka
Statement: Unnecessary internet regulation no more, FCC right to overturn net neutrality rules December 14 2017 by Patrice Lee Onwuka
Calls for Conyers to Resign • Risk & Reward Pt. 2 December 1 2017 by Patrice Lee Onwuka
Steinle's Killer Acquitted • Risk & Reward Pt. 1 December 1 2017 by Patrice Lee Onwuka
FCC Net Neutrality Vote Looms: How could things change? • The Big Picture December 1 2017 by Patrice Lee Onwuka
Time for Transparency: Taxpayer dollars used to protect Congressional misbehavior • Coast to Coast November 30 2017 by Patrice Lee Onwuka
More Bang for Your Buck • Fox & Friends First November 30 2017 by Patrice Lee Onwuka
Are some Dems willing to sacrifice protections for women to hold onto power? • Intelligence Report November 28 2017 by Patrice Lee Onwuka
Federal Judge blocks Trump's EO on Sanctuary Cities: Is Trump at a disadvantage? • Bulls & Bears November 25 2017 by Patrice Lee Onwuka
Has NFL commissioner lost control of the hemorrhaging league? • Bulls & Bears November 25 2017 by Patrice Lee Onwuka
Black Friday protests dwindle exposing this uncomfortable truth • Bulls & Bears November 25 2017 by Patrice Lee Onwuka
Sexual harassment is wrong no matter when it occurred and by whom • Varney & Co. November 24 2017 by Patrice Lee Onwuka
Bernie Sanders Blatantly Gives Bill Clinton, Al Franken a Pass on Sexual Harassment November 20 2017 by Patrice Lee Onwuka
There's Nothing Glamorous about “Auntie" Maxine Waters & Her “Impeach Him!” Chant November 17 2017 by Patrice Lee Onwuka
Twitter cracking down on Verified accounts • Fox & Friends First November 17 2017 by Patrice Lee Onwuka
'Saturday Night Live' says DNC is old and out of touch • Risk & Reward November 13 2017 by Patrice Lee Onwuka
Tax reform push: Will there be compromise? • FBN:AM November 13 2017 by Patrice Lee Onwuka | {
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Tax reform push: Will there be compromise? • FBN:AM November 13 2017 by Patrice Lee Onwuka
Are tech companies truly neutral platforms or do they reflect bias? • Fox & Friends First November 3 2017 by Patrice Lee Onwuka
Statement: GOP Tax Bill is a Win for Women November 2 2017 by Patrice Lee Onwuka
CNBC Slams DNC Chair Tom Perez, 'Get Off The Talking Points' On GOP Tax Plan November 2 2017 by Patrice Lee Onwuka
We cannot choose our history but we can choose not to repeat it • Risk & Reward October 31 2017 by Patrice Lee Onwuka
We're seeing the waning days of the NFL protests • Risk & Reward October 31 2017 by Patrice Lee Onwuka
This tax reform plan will help all workers • FBN:AM October 30 2017 by Patrice Lee Onwuka
West Wing Reads for 10/30/17 October 30 2017 by Patrice Lee Onwuka
Will any democrats get on board with tax reform? • FBN AM October 26 2017 by Patrice Lee Onwuka
What role is Bannon playing outside of the White House? • MSNBC Live October 23 2017 by Patrice Lee Onwuka
Podcast #98 Why sexual assault statistics on college campuses are misleading October 19 2017 by Patrice Lee Onwuka
Will new Trump/McConnell bromance last to get agenda through? • FBN AM October 17 2017 by Patrice Lee Onwuka
Hillary caring about "women's issues" a sham? • The Intelligence Report October 16 2017 by Patrice Lee Onwuka
More sexual harrassment allegations come out in Hollywood • Risk & Reward October 13 2017 by Patrice Lee Onwuka
Wait–Are Black Women Leaving The Democratic Party? October 11 2017 by Patrice Lee Onwuka
Eminem bashes Trump in new music video, but does it matter? • Coast To Coast October 11 2017 by Patrice Lee Onwuka
Statement: Abandoning The Clean Power Plan: A Destructive Regulatory Package October 10 2017 by Patrice Lee Onwuka
These four GOP Senators won't vote for Graham-Cassidy • FBN AM September 26 2017 by Patrice Lee Onwuka
These celebrities are joining in on the NFL anthem protests... • Risk & Reward September 26 2017 by Patrice Lee Onwuka | {
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This sport is standing up for the National Anthem... • Risk & Reward September 25 2017 by Patrice Lee Onwuka
Does Graham-Cassidy healthcare bill have a chance? • FBN AM September 21 2017 by Patrice Lee Onwuka
Emmy's and NFL in ratings slide due to politics • Risk & Reward September 19 2017 by Patrice Lee Onwuka
A new left-wing strategy against Trump September 18 2017 by Patrice Lee Onwuka
Can Trump pull bi-partisan support for tax reform? • FBN AM September 13 2017 by Patrice Lee Onwuka
Is there an opportunity for bi-partisanship with recent devastations? • Coast To Coast September 11 2017 by Patrice Lee Onwuka
White House shake up doesn't mean the agenda stops • Risk & Reward August 18 2017 by Patrice Lee Onwuka
Let's consider moving Confederate monuments to museums • FBN AM August 17 2017 by Patrice Lee Onwuka
Trump holds Congress' feet to the fire on health care • Coast To Coast August 11 2017 by Patrice Lee Onwuka
Why diversity quotas may not be a good thing for companies • Intelligence Report August 7 2017 by Patrice Lee Onwuka
Left trying to label 'Death Wish' alt-right • Risk & Reward August 7 2017 by Patrice Lee Onwuka
Podcast #90: How the Internet is Giving Women Back Their Time August 2 2017 by Patrice Lee Onwuka
Will those who voted against repeal face tough constituent questions? • FBN AM July 28 2017 by Patrice Lee Onwuka
Brave move by John McCain with healthcare vote? • FBN AM July 28 2017 by Patrice Lee Onwuka
Jeff Sessions is yet another distraction from Trump agenda • Your World July 27 2017 by Patrice Lee Onwuka
Is getting a better education for your children racist? • Risk & Reward July 24 2017 by Patrice Lee Onwuka
Why democrats have failed to come up with plan to make Obamacare better • FBN AM July 20 2017 by Patrice Lee Onwuka
Should Congress forget healthcare repeal and move onto tax reform? • The Intelligence Report July 18 2017 by Patrice Lee Onwuka
Will Congress work through August recess? • Coast To Coast July 11 2017 by Patrice Lee Onwuka | {
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Will Congress work through August recess? • Coast To Coast July 11 2017 by Patrice Lee Onwuka
Was it wrong for Ivanka to sit in at the G20? • Your World July 10 2017 by Patrice Lee Onwuka
Would media act differently if it was Chelsea Clinton at the G20? • Risk & Reward July 10 2017 by Patrice Lee Onwuka
Is the Trump administration good for the economy? • Fox 5 DC July 7 2017 by Patrice Lee Onwuka
Is Trump's agenda back on track? • FBN AM June 29 2017 by Patrice Lee Onwuka
Will the healthcare bill pass before July 4 recess? • FBN AM June 29 2017 by Patrice Lee Onwuka
People would be better served if we had a real healthcare market place • Risk & Reward June 27 2017 by Patrice Lee Onwuka
Black women work hard but can’t get ahead. This expert says conservatives have the solution. June 19 2017 by Patrice Lee Onwuka
Is Trump hurting himself when he tweets? • Coast To Coast June 7 2017 by Patrice Lee Onwuka
This is why you shouldn't jump on Larry Hogan for vetoing paid sick leave bill • Larry O'Connor Show May 26 2017 by Patrice Lee Onwuka
How much damage has been done to the Trump administration by the media? • FBN AM May 24 2017 by Patrice Lee Onwuka
Miss USA Slammed For Saying Healthcare Is A Privilege Rather Than A Right • Marc Cox May 16 2017 by Patrice Lee Onwuka
One Black Conservative Made It On Liberal List Of Beautiful Black Women April 25 2017 by Patrice Lee Onwuka
Essence’s “100 Woke Women” List: Where Are All The Conservatives? April 20 2017 by Patrice Lee Onwuka
Do Voters Actually Care About Trump's Tax Returns? • Coast To Coast April 17 2017 by Patrice Lee Onwuka
PBS teaching ISIS sympathy • Risk & Reward April 14 2017 by Patrice Lee Onwuka
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Debate over Cost of Healthcare Overhaul ahead of CBO Score • Coast to Coast March 14 2017 by Patrice Lee Onwuka | {
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A Day without a Woman • Making Money March 9 2017 by Patrice Lee Onwuka
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Political double-speak: Who's line is it? • Risk & Reward February 23 2017 by Patrice Lee Onwuka
CNN may suffer from ‘PESD’ post-election stress disorder • Risk & Reward February 23 2017 by Patrice Lee Onwuka
WWE's Linda McMahon Confirmed To Head The Small Business Administration • Steve Gruber February 17 2017 by Patrice Lee Onwuka
Vice President Pence addressing right-to-life marchers in D.C. • Coast to Coast January 27 2017 by Patrice Lee Onwuka
Raising Minimum Wage 'Not Worth It' January 23 2017 by Patrice Lee Onwuka
Have Race Relations Improved Under Obama Presidency? • Tipping Point January 14 2017 by Patrice Lee Onwuka
If Republicans rush to repeal ObamaCare without a strategy to provide immediate relief to its victims, the ongoing collapse of insurance markets will continue • Coast To Coast January 9 2017 by Patrice Lee Onwuka
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AirBnB's Growing List Of Enemies + Halloween's Cultural Appropriation Police • Garrison October 20 2016 by Patrice Lee Onwuka
Forging a Generation of Opportunity October 20 2016 by Patrice Lee Onwuka | {
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Forging a Generation of Opportunity October 20 2016 by Patrice Lee Onwuka
Political Correctness Plagues Halloween • Tipping Point October 18 2016 by Patrice Lee Onwuka
Democrats Relationship With Media In Question After Wikileaks Dump • Coast To Coast October 14 2016 by Patrice Lee Onwuka
Is IQ or EQ More Important In The Presidential Debates? • Your World September 20 2016 by Patrice Lee Onwuka
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Podcast #51 College Students: Don't Major In Debt! September 15 2016 by Patrice Lee Onwuka
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Did Kaepernick Right His Protest By Kneeling During Anthem? • Risk & Reward September 2 2016 by Patrice Lee Onwuka
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Should Salaries Be Transparent? • PBS Point Taken May 3 2016 by Patrice Lee Onwuka
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Equal Pay Day & Women in the Workplace • PBS To The Contrary April 16 2016 by Patrice Lee Onwuka
The Liberty Vote & How Government Hurts Opportunity • Stossel March 12 2016 by Patrice Lee Onwuka | {
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The Liberty Vote & How Government Hurts Opportunity • Stossel March 12 2016 by Patrice Lee Onwuka
Goods Marketed To Women Cost More Than Identical Items For Men • After The Bell December 23 2015 by Patrice Lee Onwuka
Feds know best ... even with cosmetics December 4 2015 by Patrice Lee Onwuka
Social Security disability fund gets dealt another blow November 17 2015 by Patrice Lee Onwuka | {
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Gravity CEO Defends $70,000 Minimum Salary, Says it was the Right Decision August 11 2015 by Patrice Lee Onwuka Baltimore Mom Toya Graham, Carly Fiorina, Salma Hayek • To The Contrary May 2 2015 by Patrice Lee Onwuka When did a 'blue-collar' job become a bad thing? May 1 2015 by Patrice Lee Onwuka End of women's colleges?/Women's Mosque of America • To the Contrary March 28 2015 by Patrice Lee Onwuka The [Un]Affordable Care Act/TurboTax data breach • WIBC Garrison February 19 2015 by Patrice Lee Onwuka Women in STEM, Women CEOS, Hate Crimes on Rise? • To The Contrary February 14 2015 by Patrice Lee Onwuka Generation Left Behind: Millennial's pummeling financial situation• WIBC Garrison December 18 2014 by Patrice Lee Onwuka Women & Midterm Elections; Female Marines; Lean Together • PBS To The Contrary October 31 2014 by Patrice Lee Onwuka Women CEOs; Egg Freezing; Male Survivors of Sexual Assault • PBS To The Contrary October 18 2014 by Patrice Lee Onwuka Patronizing? WH uses Emojis to make its economic case to Gen Y • The Rick Amato Show October 13 2014 by Patrice Lee Onwuka Zero Common Sense: America's feverish culture of alarmism • Stossel October 2 2014 by Patrice Lee Onwuka NFL Abuse Scandals; Hillary Clinton & Economy • PBS To The Contrary September 19 2014 by Patrice Lee Onwuka Border Crisis: Cleaning up unintended consequences from bad policy • PBS To The Contrary July 13 2014 by Patrice Lee Onwuka Should there be achievement benchmarks before teachers get tenure? • The Rick Amato Show June 16 2014 by Patrice Lee Onwuka For ObamaCare to be fiscally solvent 40% of enrollees must be under 35 • Rick Amato Show April 22 2014 by Patrice Lee Onwuka Tax Freedom Day: Stop working for Uncle Sam and start working for ourselves • 1310 KFKA AM Colorado April 21 2014 by Patrice Lee Onwuka How are young Americans reacting to ObamaCare? • CBN Newswatch March 27 2014 by Patrice Lee Onwuka March Madness: A desperate and expensive attempt to sell ObamaCare • The Rick Amato | {
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Lee Onwuka March Madness: A desperate and expensive attempt to sell ObamaCare • The Rick Amato Show March 18 2014 by Patrice Lee Onwuka Reports show ObamaCare tax on medical devices resulting in the loss of about thousands of jobs • AM Colorado February 24 2014 by Patrice Lee Onwuka Lax hiring practices of ObamaCare navigators • Conservative Commandos Radio Show February 13 2014 by Patrice Lee Onwuka Lee • Convicted Felons Hired to Administer ObamaCare & Personal Info • Rick Amato Show (02.04.14) February 4 2014 by Patrice Lee Onwuka Lee • What to expect from Obama's 5th State of the Union • Arise Xchange (01.28.14) January 28 2014 by Patrice Lee Onwuka Lee • Who are the ObamaCare Orphans? ACA Olympics ad blitz costs what? • AM Colorado (01.13.14) January 13 2014 by Patrice Lee Onwuka Debra Messing, leave black Trump supporters alone September 12 2019 by Patrice Lee Onwuka Kamala Harris Misleads Black Women on Faux Equal Pay Holiday August 26 2019 by Patrice Lee Onwuka On Black Women’s Equal Pay Day, we don’t have to be victims August 22 2019 by Patrice Lee Onwuka Why didn't any Democratic candidates tell the truth about the gender pay gap? August 5 2019 by Patrice Lee Onwuka A$15 Minimum Wage Would Put More Than A Million Out Of Work — And Tlaib Wants To Hit Even More July 24 2019 by Patrice Lee Onwuka | {
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Let Women Entrepreneurs Be Their Own Bosses July 15 2019 by Patrice Lee Onwuka
Polling Shows Fewer People Are Proud to Be American, but We All Should Be July 4 2019 by Patrice Lee Onwuka
Forget Free College, Let’s Talk About Apprenticeships June 26 2019 by Patrice Lee Onwuka
Trump Delivers Second Chances for Black Inmates Re-entering Society June 14 2019 by Patrice Lee Onwuka
Kamala Harris Plan to 'Fix' the Wage Gap is Sure to Undermine Working Women May 28 2019 by Patrice Lee Onwuka
Millennials like me waited to start families. Paid leave will let me spend time with them. May 26 2019 by Patrice Lee Onwuka
This is a great time to be a working mother May 12 2019 by Patrice Lee Onwuka
Black Americans don't need reparations, we need jobs April 10 2019 by Patrice Lee Onwuka
New Parents Act an Innovative, Conservative Paid Parental Leave Plan March 29 2019 by Patrice Lee Onwuka
For Women's History Month, celebrate the conservative women that the media ignore March 26 2019 by Patrice Lee Onwuka
How parental leave aids families March 12 2019 by Patrice Lee Onwuka
Jussie Smollett Isn't Biggest Setback for Blacks, Ignoring Progress Is March 1 2019 by Patrice Lee Onwuka
The tax cuts didn't steal your tax refund February 28 2019 by Patrice Lee Onwuka
Women's March Has Inclusivity Problem and It's Not Just Anti-Semitism January 17 2019 by Patrice Lee Onwuka
Women need criminal justice reform November 26 2018 by Patrice Lee Onwuka
Post-Election Analysis: In Congress, It’s New Gridlock, Same Old Divisions November 13 2018 by Patrice Lee Onwuka
Blacks Reject Left's Regressive Agenda, Choose Trump's Confidence October 30 2018 by Patrice Lee Onwuka
Debt shouldn't hold back Stacey Abrams, but her victimhood mentality might October 29 2018 by Patrice Lee Onwuka
Elizabeth Warren is the problem with the diversity victimhood mentality October 17 2018 by Patrice Lee Onwuka | {
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Don't underestimate Kanye (and Trump) – Rapper could successfully deliver for forgotten black communities October 11 2018 by Patrice Lee Onwuka
I Support Survivors and Stand With Kavanaugh October 5 2018 by Patrice Lee Onwuka
SNAP reforms in the farm bill can improve lives for Americans September 5 2018 by Patrice Lee Onwuka
Why Omarosa’s antics will fail to undermine Trump’s progress for black Americans August 20 2018 by Patrice Lee Onwuka
Playing Victim Card Won't Win at Empowering Black Women August 7 2018 by Patrice Lee Onwuka
Marco Rubio's paid leave plan would help black moms August 2 2018 by Patrice Lee Onwuka
Don't Match Maxine Waters' Incivility With Disrespect July 13 2018 by Patrice Lee Onwuka
Brett Kavanaugh's Supreme Court nomination offers a chance to fight sexism against all women July 12 2018 by Patrice Lee Onwuka
Why women on the Left are not marching for all separated kids July 3 2018 by Patrice Lee Onwuka
In the era of streaming shows, kids' TV regulations are stuck in the 1990s June 23 2018 by Patrice Lee Onwuka
Juneteenth Is Over but Progress for Blacks Must Grow June 21 2018 by Patrice Lee Onwuka
Starbucks' anti-bias training may serve up more harm than good June 2 2018 by Patrice Lee Onwuka
Anti-Bias Training Won't Help Starbucks May 29 2018 by Patrice Lee Onwuka
Michelle Obama's myopic view of women is what's wrong with feminism May 7 2018 by Patrice Lee Onwuka
Blacks Owe No Allegiance to the Left April 30 2018 by Patrice Lee Onwuka
Gender and race shouldn't define your politics April 30 2018 by Patrice Lee Onwuka
'Diamond & Silk' offer chance for bipartisan push back on social media censorship April 14 2018 by Patrice Lee Onwuka
A Real Free Market Might Save Cab Drivers From Financial Ruin April 4 2018 by Patrice Lee Onwuka
It's past time for the mainstream media to add opposing voices April 4 2018 by Patrice Lee Onwuka
Hillary Clinton’s sorry apology is why she’s no champion for women March 22 2018 by Patrice Lee Onwuka | {
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Increased Competition Best Treatment For High Prescription Drugs Costs March 21 2018 by Patrice Lee Onwuka
‘Black Panther’ Exposes Immigrant Resentment in the Black Community March 15 2018 by Patrice Lee Onwuka
An immigrant's request: Keep these things in mind when debating immigration reform February 14 2018 by Patrice Lee Onwuka
When Did Joblessness Stop Mattering to Black Leaders? February 7 2018 by Patrice Lee Onwuka
The Grammys' #MeToo moment signals a dangerous direction for the movement January 30 2018 by Patrice Lee Onwuka
Black Americans' Record Employment, Tax Cuts and More January 9 2018 by Patrice Lee Onwuka
Hollywood's #TimesUp movement has been compromised by progressives January 9 2018 by Patrice Lee Onwuka
When States Hike the Minimum Wage, Women Lose Jobs December 22 2017 by Patrice Lee Onwuka
Women Can Get Excited About Tax Reform. Here’s Why! December 21 2017 by Patrice Lee Onwuka
Black Lives Matter Boycott Will Hurt Those It Claims to Help December 11 2017 by Patrice Lee Onwuka
Don't shed any tears for John Conyers December 5 2017 by Patrice Lee Onwuka
Richard Cordray's departure should end the campaign against payday loans November 20 2017 by Patrice Lee Onwuka
I Am Thankful for How Tax Reform Makes the American Dream Possible November 19 2017 by Patrice Lee Onwuka
Democratic women in the Senate were all for these tax reform principles until Trump proposed them October 30 2017 by Patrice Lee Onwuka
Trump Tax Cuts Will Help Black-Owned Businesses Thrive October 27 2017 by Patrice Lee Onwuka
Give Christopher Columbus a break October 9 2017 by Patrice Lee Onwuka
Black Women Leaving the Democratic Party, Cracking the Base September 27 2017 by Patrice Lee Onwuka
Betsy DeVos Stands for the Rights of Victims September 13 2017 by Patrice Lee Onwuka
NFL protests need to take a knee September 8 2017 by Patrice Lee Onwuka
Economic Agenda Will Do More for Black People Than Statue Removal August 28 2017 by Patrice Lee Onwuka | {
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Government Can Hate A Name, But Still Must Respect It August 14 2017 by Patrice Lee Onwuka
The Google Diversity Memo Should Start the Conversation — Not End It August 8 2017 by Patrice Lee Onwuka
Entrepreneurship Is the Real Black Girl Magic August 7 2017 by Patrice Lee Onwuka
Serena Williams Swings But Misses With Black Women’s Equal Pay Day August 4 2017 by Patrice Lee Onwuka
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Business should back Larry Hogan's veto of paid sick leave July 18 2017 by Patrice Lee Onwuka
Dispelling the White House pay gap myth July 10 2017 by Patrice Lee Onwuka
Maybe it's time for the White House Office of Girls and Women to go July 4 2017 by Patrice Lee Onwuka
Why UK millennials voting for socialism could happen here, too June 27 2017 by Patrice Lee Onwuka
Dear Internet: Don’t Carry The ‘F**k You 2016’ Movement Into 2017 [VIDEO] December 31 2016 by Patrice Lee Onwuka
How To Win The 'War On Christmas' On Campus December 26 2016 by Patrice Lee Onwuka
The Scariest Part Of Halloween Is The Cultural Appropriation Police October 13 2016 by Patrice Lee Onwuka
Don't Major In DEBT September 12 2016 by Patrice Lee Onwuka
It's Time For Social Security Reform July 26 2016 by Patrice Lee Onwuka
Congratulations Class of 2016, You Majored in Debt May 13 2016 by Patrice Lee Onwuka
Should Over-the-Top College Acceptance Letters Woo Students? March 29 2016 by Patrice Lee Onwuka
Why Are So Many Working Americans Still Dependent on Food Stamps? March 17 2016 by Patrice Lee Onwuka
Nursing Needs More Competition March 14 2016 by Patrice Lee Onwuka
Reese Witherspoon’s Oscar-Worthy Truth For Young Women February 27 2016 by Patrice Lee Onwuka
Takeaways: Overtime Pay June 4 2019 by Patrice Lee Onwuka
Policy Focus: Overtime Pay May 30 2019 by Patrice Lee Onwuka
Takeaways: Welfare Reform 2.0 September 7 2018 by Patrice Lee Onwuka | {
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Takeaways: Welfare Reform 2.0 September 7 2018 by Patrice Lee Onwuka
Policy Focus: Welfare Reform 2.0 August 13 2018 by Patrice Lee Onwuka
Takeaways: What Drives the Wage Gap? April 5 2018 by Patrice Lee Onwuka
Takeaways: Why Requiring Work & Community Engagement is Good for Medicaid and the Poor January 29 2018 by Patrice Lee Onwuka
Takeaways: How Tax Reform Will Help Americans December 18 2017 by Patrice Lee Onwuka
Fact Sheet: Tax Reform is a Women's Issue September 15 2017 by Patrice Lee Onwuka
Policy Focus: Your Ideas, Your Rights - Intellectual Property in the 21st Century June 25 2017 by Patrice Lee Onwuka | {
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Mathematics
OpenStudy (anonymous):
Find the angle between the vectors u and v if u = (1, 2) and v = (- 4, - 2)
jimthompson5910 (jim_thompson5910):
Use the formula $\Large \theta = \arccos\left(\frac{u\cdot v}{|u||v|}\right)$ where u and v are vectors.
OpenStudy (anonymous):
which number in my points do i use for each... for example which number in (1,2) would I use for which u
jimthompson5910 (jim_thompson5910):
In the numerator of that fraction, you're computing the dot product between u and v
jimthompson5910 (jim_thompson5910):
So what is the dot product between u = (1, 2) and v = (- 4, - 2)?
OpenStudy (anonymous):
-8
jimthompson5910 (jim_thompson5910):
good
jimthompson5910 (jim_thompson5910):
now you need to find |u| and |v|
OpenStudy (anonymous):
how do I do that?
jimthompson5910 (jim_thompson5910):
use the formula |x| = sqrt(a^2+b^2) where the vector x is x = (a,b)
OpenStudy (anonymous):
so it is sqrt(1^2+2^2) for u's and then sqrt(-4^2+-2^2) for v's?
jimthompson5910 (jim_thompson5910):
yes, so what do you get?
OpenStudy (anonymous):
i think u's is sqrt(5) and v's is 2sqrt(5)
jimthompson5910 (jim_thompson5910):
You got it, nice work So what is |u||v|?
OpenStudy (anonymous):
10... so then its arccos(-8/10)?
jimthompson5910 (jim_thompson5910):
yes
OpenStudy (anonymous):
which turns out to be 143.13degrees
jimthompson5910 (jim_thompson5910):
you nailed it
OpenStudy (anonymous):
You keep helping me sooo much. Thanks, I am taking a Pre-Calculus course by myself and sometimes it can be difficult to figure out all of the questions by myself :).
jimthompson5910 (jim_thompson5910):
you're very welcome, that must be pretty crazy lol
OpenStudy (anonymous):
yep... I'm almost done though (thankfully haha)
jimthompson5910 (jim_thompson5910):
well that's good
OpenStudy (anonymous):
is there a difference between a u surrounded by two of the straight lines as opposed to the just one in the question you just helped me with? | {
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jimthompson5910 (jim_thompson5910):
Well technically it should have been $\Large \|u\|$ since the two vertical bars denote the norm or magnitude of a vector. But the absolute value bars say the same thing (they're just used in a more general sense). So either work in my opinion.
OpenStudy (anonymous):
oh, ok... thanks for clarifying. I am working on another type of question like that and wasn's sure if there was a difference.
jimthompson5910 (jim_thompson5910):
there's not much of a difference really | {
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# Playing roulette
Suppose we have a roulette wheel with $38$ total slots ($18$ black/red, $2$ neither). The ball lands in one slot selected uniformly at random, independent of all previous spins. An $\$x$bet on "black" pays$\$2x$ if the ball lands on black, and $\$0$otherwise. If you bet$1 on black for 100 consecutive spins, how much money will you end up with in expectation?
You walk up to the Roulette table with $\$15$and intend to walk away with$\$16$ using the following betting strategy. You will first bet $\$1$on black. If you win you have$\$16$ and walk away, otherwise you have $\$14$and bet$\$2$ on black on the next spin. If you win that bet you walk away, otherwise you bet $\$4$on the next spin. If you win that bet you walk away, otherwise you bet$\$8$ on the next spin and walk away, either with $\$16$if you win or$\$0$ if you lose.
What's the probability you will walk away with $\$16$? How much money are you expected to walk away with? My biggest problem with these kinds of questions is figuring out how to choose the random variable to calculate the expected values. I understand the formula for calculating the expected value, but translating a problem into those terms has been giving me a hard time. - ## 2 Answers Let's first look at the first question: If you bet$1 on black for 100 consecutive spins, how much money will you end up with in expectation?
So you want to know what your final return will be, at the end of 100 spins. Call this $R$. That is just giving it a name, but what is your final return? You can see that it is the sum of the returns from each bet. So let the return on the $i$th bet be $R_i$, then note that $R = R_1 + R_2 + \dots + R_{100}$. So the expected return is $E[R] = E[R_1] + E[R_2] + \dots + E[R_{100}]$ by linearity of expectation. | {
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So to calculate $E[R]$, we'll be done if we calculate each $E[R_i]$. Let's try to calculate a particular $E[R_i]$. You bet $1$ dollar, and you get back $2$ if the ball lands on black, and $0$ if it doesn't. In other words, you gain $1$ dollar if it lands on black, and lose $1$ dollar if it doesn't. The probability of the former is $18/38$, and that of the latter is $20/38$. In other words, $R_i$ is $1$ with probability $18/38$, and $-1$ with probability $20/38$, so the expected value of $R_i$ is $E[R_i] = \frac{18}{38}(-1) + \frac{20}{38}(1) = \frac{-2}{38}$. Now, as this is the same for each $R_i$, we have $E[R] = E[R_1] + E[R_2] + \dots + E[R_{100}] = \left(\frac{-2}{38}\right)100 \approx -5.26$.
For the second question, let the amount you walk away with be $W$. Let $p = 18/38$, the probability that your bet on black succeeds. There are $5$ possible outcomes:
• you win your first bet: probability $p$
• you lose your first bet, and win your second: probability $(1-p)p$
• you lose your first two bets, and win the third: probability $(1-p)^2p$
• you lose your first three bets, and win the fourth: probability $(1-p)^3p$
• you lose all four bets: probability $(1-p)^4$
In the first four outcomes, you walk away with $16$ dollars, so the probability of that happening (let's call it $q$) is $q = p + (1-p)p + (1-p)^2p + (1-p)^3p = 1 - (1-p)^4 = 1 - (20/38)^4 \approx 0.92$.
[More simply, you could think of it as just two outcomes: (a) that you win some bet, which has probability $q = 1 - (1-p)^4$, and (b) that you win no bet (lose all bets), which has probability $(1-p)^4$.]
In other words, $W$ is $16$ with probability $q$, and $0$ with probability $1-q$. So the expected amount of money you walk away with is therefore $E[W] = q(16) + (1-q)0 = (1-(1-p)^4)16 \approx 14.77$.
[Aside: Note that this is less than the $15$ you came in with. This shows that you can't win in expectation even with your clever betting strategy; a consequence of the optional stopping theorem.]
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-
Probability of getting a black on any particular spin is 18 in 38, ie $p = \frac{18}{38} = \frac{9}{19} = 0.474$. Let $X$ be the random variable 'number of blacks in 100 consecutive spins'. Then the expected value of X is $E(X) = 100p = \frac{900}{19} = 47.37$, that is, you expect to win 47.37 times in 100 spins. Since you get $\$$2 for each \$$1 bet, you will therefore expect to walk away with$\$2\times 47.37 = \$94.74$. For the second part of your question, the only way you can lose is if for four consecutive spins, you get non-blacks each time. So what is the probability of losing 4 times in a row? Well, for 1 spin, the probability of not getting black is$1-p = \frac{20}{38} = \frac{10}{19} = 0.526$so for 4 consecutive spins, the probability of not getting any black is$(\frac{10}{19})^{4} = \frac{10,000}{130,321} = 0.0767$. That is, you have about a 7.7% chance of losing with this betting strategy. That is, you have 92.3% chance of walking away with$\16.
-
Can you elaborate on why the expected value is just $100p$? Particularly, how does the linearity of expectations apply here? If we let $X_i = 1$ if we land on black and $X_i = 0$ if we don't land on black, how can we calculate the expected value then? – user1038665 Nov 14 '12 at 10:53
Also, for the second part, how would we calculate the "amount of money you are expected to walk away with"? – user1038665 Nov 14 '12 at 10:55
Expected value of a bet = wager x probability. – theo Nov 14 '12 at 11:02
If you bet 1 dollar 100 times on black you will win about 47.37 times and thus leave the casino with 94.74 of your original 100 dollars. – Hagen von Eitzen Nov 14 '12 at 11:03
The answer given is -\$(2/38)*(100), so I believe this is incorrect. – user1038665 Nov 14 '12 at 11:20 | {
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# How to create a continuous distribution on $[a, b]$ with $\text{mean} = \text{mode} = c$?
I would like to sample from a distribution on a specific domain $$[a, b]$$ with mean $$c$$ where $$a < c < b$$. Ideally, I would also like the mode (i.e. the "peak") of the distribution to be $$c$$ as well.
How to derive this distribution?
The best I can do is a triangular distribution with my desired mean, but the peak is in the wrong place. I imagine the distribution I want is smooth and skewed.
EDIT: If $$c$$ were at the midpoint of $$a$$ and $$b$$ then I think a truncated normal distribution is roughly what I'm looking for. So something similar where the tails go smoothly to zero at the endpoints of my interval when $$c$$ is not in the middle would be nice.
EDIT 2: I'm looking to perturb some parameters in a simulation, so really anything reasonable should be fine.
• There are infinitely many possible solutions, which enables you to apply stricter criteria to narrow it down. To that end, could you explain what use you intend to make of this family of distributions? BTW, "skewed" strongly suggests the peak should not coincide with the mean.
– whuber
Jan 28 at 20:11
• Good point. I did my best to clarify. Jan 28 at 20:37
• There are still infinitely many solutions, but another way would be to generate a beta RV with parameters that give it the desired skew and unimodality, and then shift and scale the whole distribution so it lies in [a,b].
– Noah
Jan 28 at 20:50
• Thanks! That looks like it will do just fine. Jan 28 at 20:53
• @Noah That looks a lot like an acceptable answer!
– Dave
Jan 28 at 20:53
I will describe every possible solution. This gives you maximal freedom to craft distributions that meet your needs. | {
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Basically, sketch any curve you like for the density function $$f$$ that meets your requirements. Separately scale the heights of the left and right halves of it (on either side of $$c$$) to make their masses balance, then scale the heights overall to make it a probability density.
Here are some details.
Because the distribution is continuous, it has a density function $$f$$ with finite integral. By splitting $$f$$ at $$c$$ we can construct all such functions out of two separately chosen non-negative non-decreasing, not identically zero, integrable functions $$f_1$$ and $$f_2$$ defined on $$[0,1].$$ Here, for example, are two such functions:
Set $$f$$ to agree affinely with $$f_1$$ on $$[a,c]$$ and with the reversal of $$f_2$$ on $$[c,b].$$ This means there are two positive numbers $$\pi_i$$ for which
$$f\mid_{[a,c]}(x) = \pi_1 f_1\left(\frac{x-a}{c-a}\right);\quad f\mid_{[c,b]}(x) = \pi_2 f_2\left(\frac{b-x}{b-c}\right).$$
This construction guarantees $$c$$ is the unique mode. Moreover, if you want the tails to taper to zero, just choose $$f_i$$ that approach $$0$$ continuously at the origin.
For $$f$$ to be a probability density it must integrate to unity:
$$1 = \int_a^b f(x)\,\mathrm{d}x =\pi_1(c-a) \mu_1^{(0)} + \pi_2(b-c) \mu_2^{(0)}\tag{*}$$
where
$$\mu_i^{(k)} = \int_0^1 x^k f_i(x)\,\mathrm{d}x.$$
Moreover, $$c$$ is intended to be the mean, whence
\begin{aligned} c &= \int_a^b x f(x)\,\mathrm{d}x \\ &= \pi_1(c-a)((c-a)\mu_1^{(1)} + a\mu_1^{(0)}) + \pi_2(b-c)(-(b-c)\mu_2^{(1)} + b\mu_2^{(0)}). \end{aligned}\tag{**}
$$(*)$$ and $$(**)$$ establish a system of two linear equations in the $$\pi_i.$$ You can check it has nonzero determinant and a unit positive solution $$(\pi_1,\pi_2).$$ (This checking comes down to the fact that after normalizing the $$f_i$$ to be probability densities, their means $$\mu_i^{(1)}$$ must be in the interval $$[0,1].$$) | {
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This figure plots the solution $$f$$ for the previous two functions where $$[a,b]=[-1,3]$$ and $$c=1/2:$$
By design, $$f$$ looks like $$f_1$$ to the left of $$c$$ and like the reversal of $$f_2$$ to the right of $$c.$$
The spike at the mode might bother you, but notice that it was never assumed or required that $$f$$ must be continuous. Most examples will be discontinuous. However, every distribution $$f$$ meeting your specifications can be constructed this way (by reversing the process: split $$f$$ into two halves, which obviously determine the $$f_i$$).
To demonstrate how general the $$f_i$$ can be, here is the same construction where the $$f_i$$ are (inverse) Cantor functions (as implemented in binary.to.ternary at https://stats.stackexchange.com/a/229561/919). These are not continuous anywhere.
NB: $$f_1$$ is binary.to.ternary; $$f_2(x) = f_1(x^{2.28370312}).$$ This illustrates one way to eliminate the discontinuity at $$c:$$ $$f_2$$ was selected from the one-parameter family of functions $$f_1(x^p),$$ $$p \gt 0,$$ and a value of $$p$$ was identified to make the left and right limits of $$f$$ at $$c$$ equal. This family "pushes" the mass of $$f_1$$ left and right by a controllable amount, thereby modifying the amount by which the right tail is scaled (vertically) in constructing $$f.$$
For those who would like to experiment, here is an R function to create $$f$$ out of the $$f_i$$ and code to create the figures. Three commands at the end check whether (1) $$f$$ is a pdf, (2) its mean is $$c,$$ and (3) its mode is $$c.$$ | {
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#
# Given numbers a. < b. < c. and non-negative, non-decreasing, not identically
# zero functions f.1 and f.2 defined on [0,1], construct a density function f
# on the interval [a., b.] with mean c. and unique mode c. that behaves like f.1
# to the left of c. and like the reversal of f.2 to the right of c.
#
# ... are optional arguments passed along to integrate.
#
create <- function(a., b., c., f.1, f.2, ...) {
cat("Create\n")
p.1 <- integrate(f.1, 0, 1, ...)$$value p.2 <- integrate(f.2, 0, 1, ...)$$value
mu.1 <- integrate(function(u) u * f.1(u), 0, 1, ...)$$value mu.2 <- integrate(function(u) u * f.2(u), 0, 1, ...)$$value
A <- matrix(c(p.1 * (c.-a.), p.2 * (b.-c.),
(c.-a.) * ((c.-a.) * mu.1 + a.*p.1),
(b.-c.) * (-(b.-c.) * mu.2 + b.*p.2)),
2)
pi. <- solve(t(A), c(1, c.))
function(x) {
ifelse(a. <= x & x <= c., 1, 0) * pi.[1] * f.1((x-a.)/(c.-a.)) +
ifelse(c. < x & x <= b., 1, 0) * pi.[2] * f.2((b.-x)/(b.-c.))
}
}
#
# Example.
#
a. <- -1
b. <- 3
c. <- 1/2
f.2 <- function(x) x^(2/3)
f.1 <- function(x) exp(2 * x) - 1
f <- create(a., b., c., f.1, f.2)
#
# Display f.1 and f.2.
#
par(mfrow=c(1,2))
curve(f.1(x), 0, 1, lwd=2, ylab="", main=expression(paste(f[1], ": ", x^{2/3})))
curve(f.2(x), 0, 1, lwd=2, ylab="", main=expression(paste(f[2], ": ", e^{2*x}-1)))
#
# Display f.
#
x <- c(seq(a., c., length.out=601), seq(c.+1e-12*(b.-a.), b., length.out=601))
y <- f(x)
par(mfrow=c(1,1))
plot(c(x, b., a.), c(y, 0, 0), type="n", xlab="x", ylab="Density", main=expression(f))
polygon(c(x, b., a.), c(y, 0, 0), col="#f0f0f0", border=NA)
curve(f(x), b., a., lwd=2, n=1201, add=TRUE)
#
# Checks.
#
integrate(f, a., b.) # Should be 1
integrate(function(x) x * f(x), a., b.) # Should be c.
x[which.max(y)] # Should be close to c. | {
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• Lovely! But a quibble: I do not think this captures distributions in which the 'left' or 'right' halves themselves contain nondifferentiable or discontinuous points—I am especially thinking of densely nondifferentiable or densely discontinuous curves—which are essentially 'undrawable' except by approximation. I am not actually convinced this is a defensible quibble, though. :) More just musing… Jan 29 at 6:28
• @Alexis I cannot see any exceptions: I made no conditions on the $f_i$ concerning differentiability or even continuity. For instance, both $f_i$ could be Cantor functions. I will add an illustration.
– whuber
Jan 29 at 15:13
• I was thinking about Cantor functions! Quibble withdrawn (and thank you for entertaining my musing :). Jan 29 at 18:32
All you need to do is come up with a two-parameter family of distributions, express the mean and mode in terms of those parameters, and then solve for them both being $$c$$. Working off your triangle idea, we can have a pentagon instead (straight lines from $$a$$ to $$c$$ and from $$c$$ to $$b$$), giving us three parameters ($$y_1$$ at $$a$$, $$y_2$$ at $$b$$, and $$y_3$$ at $$c$$), but we don't really have three degrees of freedom. We have that the total probability mass from $$a$$ to $$c$$ is $$\frac 1 2 (y_1+y_3)(c-a)$$, and the mass from $$c$$ to $$b$$ is $$\frac 1 2 (y_2+y_3)(c-b)$$, so we need to have $$\frac 1 2 (y_1+y_3)(c-a)+\frac 1 2 (y_2+y_3)(c-b)=1$$. Then you just need to make sure that the mean is $$c$$ as well.
• The problem with this approach is that it can fail even in the nicest circumstances (as I pointed out in a comment to the question). For instance, it does not work with the Beta family of distributions: there are no solutions unless the Beta parameters are equal.
– whuber
Jan 29 at 15:18 | {
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# Power series at $x=0$
$$\sum_{n=0}^∞n!x^{2n}$$
This power series has radius of convergence $R=0$ since the ratio test shows it diverges for all $x≠0$.
But what if $x=0$? Then we have $\sum_{n=0}^\infty n! \cdot 0^{2n}$
Surely this series is undefined since the first partial sum is $0!\cdot 0^{2\cdot0}$ and $0^0$ is undefined, right?
Well, my lecturer claims this series is convergent and is equal to 1.
But $\sum_{n=1}^\infty n!\cdot 0^{2n}$ clearly converges to 0, so he is claiming that $0!\cdot 0^0=1$?
So is he wrong? Or am I missing something here?
• $0^0 = 1$ by convention. – user258700 Dec 22 '15 at 2:01
• Really? I've never heard that. I've always thought it was just undefined. Why isn't $\frac00$ similarly defined by convention? – Refnom95 Dec 22 '15 at 2:08
• It's only when dealing with limits that $0^0$ is indeterminant. – Michael Burr Dec 22 '15 at 2:09
• It's done mostly in the context of power series. It's much neater to write $\sum_i \text{bla}_ix^i$ than $\text{bla}_0+\sum_{i>0}...$. – YoTengoUnLCD Dec 22 '15 at 2:20 | {
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There are a lot of situations where it is convenient to define $$0^0=1$$ and of course $0!=1$, so $0!\cdot0^0=1$.
In some situations, it's most appropriate to leave $0^0$ undefined. In particular, the function $x^y$ converges to different limits as $x\to0$ and $y\to0$ depending on which path is taken, so analysis does not yield a natural value for $0^0$ and it may not make sense to define $0^0=1$.
However, when dealing with discrete exponents (natural numbers), it's usually better to define $0^0=1$, because this is consistent with combinatorial or set-theoretic interpretations of exponentiation. For instance, the set of functions from the empty set to the empty set has cardinality $1$, so $0^0=1$ by the set-theoretic definition. Also, using the repeated multiplication definition of exponentiation, an empty product is always $1$, even when the product contains only zeroes. (For more information, see this Wikipedia article.)
It is important to consider where the $0^0$ in the expression comes from. In your case, it's the first term of a power series. A power series' notion of exponentiation is closest to the repeated multiplication interpretation, so taking $0^0$ makes sense. Consider that a power series is just a generalization of a polynomial. How would you evaluate $f(x)=x^2+x+1$ at $x=0$? What if this is written $f(x)=x^2+x^1+x^0$? | {
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# Math Help - Help with a proof.
1. ## Help with a proof.
Hi, I'm stuck at proving the following question...
Prove that for all n>0,
1/2 + 2/2^2 + 3/2^3 + ... + n/2^n = 2 - (n+2)/2^n
I've tried all sorts of different ways of solving this, but to no avail.
Any help is appreciated
2. Originally Posted by seven.j
Hi, I'm stuck at proving the following question...
Prove that for all n>0,
1/2 + 2/2^2 + 3/2^3 + ... + n/2^n = 2 - (n+2)/2^n
I've tried all sorts of different ways of solving this, but to no avail.
Any help is appreciated
$\sum_{k=1}^{n}\frac{k}{2^k}$. Note that $\sum_{k=1}
^n x^k=\frac{x^{n+1}-x}{x-1}$
. Differentiating both sides and multiplying by $x$ gives $\sum_{k=1}^{n}k\cdot x^k=...$ figure the right side out.
3. $\sum\limits_{j=1}^{n}{\frac{j}{2^{j}}}=\sum\limits _{j=1}^{n}{\sum\limits_{k=1}^{j}{\frac{1}{2^{j}}}} =\sum\limits_{k=1}^{n}{\frac{1}{2^{k}}\left( \sum\limits_{j=0}^{n-k}{2^{-j}} \right)}=\frac{1}{2^{n}}\sum\limits_{k=1}^{n}{\fra c{1}{2^{k}}\left( 2^{n+1}-2^{k} \right)},$ you can do the rest, those are finite geometric sums.
4. Hello, seven!
Here's one way . . .
Prove that for all $n>0\!:$
. . $\frac{1}{2}+ \frac{2}{2^2} + \frac{3}{2^3} +\:\hdots\:+ \frac{n}{2^n} \; =\; 2 - \frac{n+2}{2^n}$
$\begin{array}{ccccc}
\text{We have:} &S &=& \dfrac{1}{2} + \dfrac{2}{2^2} + \dfrac{3}{2^3} + \dfrac{4}{2^4} + \:\hdots\:+\dfrac{n}{2^n}\qquad\qquad \\ \\[-3mm]
\text{Multiply by }\dfrac{1}{2}\!: & \dfrac{1}{2}S &=& \quad\;\; \dfrac{1}{2^2} + \dfrac{2}{2^3} + \dfrac{3}{2^4} + \hdots + \dfrac{n-1}{2^n} + \dfrac{n}{2^{n+1}}\end{array}$
. . . $\text{Subtract: }\quad \frac{1}{2}S \;=\;\underbrace{\frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \frac{1}{2^4} + \hdots + \frac{1}{2^n}}_{\text{geometric series}} - \frac{n}{2^{n+2}}$ .[1]
The geometric series has the sum: . $\frac{1}{2}\cdot\frac{1 - \left(\frac{1}{2}\right)^n}{1-\frac{1}{2}} \;=\;1 - \frac{1}{2^n}$ | {
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Then [1] becomes: . $\frac{1}{2}S \;=\;\left(1 - \frac{1}{2^n}\right) - \frac{n}{2^{n+1}} \;=\;1 - \frac{n+2}{2^{n+1}}$
Multiply by 2: . $S \;=\;2 - \frac{n+2}{2^n}$
5. Originally Posted by seven.j
Hi, I'm stuck at proving the following question...
Prove that for all n>0,
1/2 + 2/2^2 + 3/2^3 + ... + n/2^n = 2 - (n+2)/2^n
I've tried all sorts of different ways of solving this, but to no avail.
Any help is appreciated
This problem also can be done by induction pretty easily. if you're interested, you can try it that way. to me it was the most knee-jerk approach to try, and it worked out great. but you have lots of nice approaches here to choose from | {
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#### puremath
##### Active Member
If Miranda flips a coin twice, what is the probability that both coins will land heads up?
Solution (not my work):
Probability of coin 1 landing heads up is 1/2. Probability of second coin landing heads up is 1/2.
P(both heads up) = (1/2)(1/2) = 1/4.
Question:
What indicates in the problem that 1/2 is multiplied by 1/2? Taking a test, a student might guess that he or she must add 1/2 to itself.
#### MarkFL
##### La Villa Strangiato
Staff member
Moderator
Math Helper
We are asked to find the probability that the coin is heads on the first flip AND the coin is heads on the second flip. The "AND" tells us to multiply.
puremath
#### puremath
##### Active Member
We are asked to find the probability that the coin is heads on the first flip AND the coin is heads on the second flip. The "AND" tells us to multiply.
Staff member
Moderator
Math Helper
Yes.
#### puremath
##### Active Member
We are asked to find the probability that the coin is heads on the first flip AND the coin is heads on the second flip. The "AND" tells us to multiply.
The word AND is not in the problem. This is the way I would type the question:
If Miranda flips a coin twice, what is the probability that the coin lands heads up on the first flip and heads up on the second flip? Now the word problem makes sense to me.
#### MarkFL
##### La Villa Strangiato
Staff member
Moderator
Math Helper
When it says both that means both the first flip AND the second flip are heads.
#### MarkFL
##### La Villa Strangiato
Staff member
Moderator
Math Helper
Actually, It could be worded better, as it first implies she is flipping the same coin twice, then it speaks of both coins. If I were to author such a problem, I might state:
If Miranda flips a coin twice, what is the probability that it lands heads up both times?
puremath
#### puremath | {
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puremath
#### puremath
##### Active Member
When it says both that means both the first flip AND the second flip are heads.
When reading this the first time, it is not so clear.
#### puremath
##### Active Member
Actually, It could be worded better, as it first implies she is flipping the same coin twice, then it speaks of both coins. If I were to author such a problem, I might state:
If Miranda flips a coin twice, what is the probability that it lands heads up both times?
It is ok but my wording makes it clear to the reader that this problem involves AND and thus multiplication is the operation. Back to Cohen on Monday.
#### MarkFL
##### La Villa Strangiato
Staff member
Moderator
Math Helper
What is the probability that of the two flips, at least one is heads?
#### puremath
##### Active Member
What is the probability that of the two flips, at least one is heads?
P(at least one heads up) = 1 - P(not one is heads up). At the job now. I have a probability book that I also want to learn with you later on in my self-study trek.
#### MarkFL
##### La Villa Strangiato
Staff member
Moderator
Math Helper
You're on the right track. I will give you time to complete the question.
#### puremath
##### Active Member
You're on the right track. I will give you time to complete the question.
For two flips, there are 4 possibilities, only one of which (getting tails twice) does not have any head. So, the other 3 out of 4 cases would have one or two heads. I say 75% probability. Yes?
MarkFL
Staff member | {
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# How should one proceed in this trigonometric simplification involving non integer angles?
The problem is as follows:
Find the value of this function $$A=\left(\cos\frac{\omega}{2} +\cos\frac{\phi}{2}\right )^{2} +\left(\sin\frac{\omega}{2} -\sin\frac{\phi}{2}\right )^{2}$$ when $$\omega=33^{\circ}{20}'$$ and $$\phi=56^{\circ}{40}'$$.
Thus,
$$A=\left(\cos\frac{\omega}{2} +\cos\frac{\phi}{2}\right)^{2} +\left(\sin\frac{\omega}{2} -\sin\frac{\phi}{2}\right)^{2}$$
By solving the power I obtained the following:
$$A= \cos^{2}\frac{\omega}{2} +\cos^{2}\frac{\phi}{2} +2\cos\frac{\omega}{2}\cos\frac{\phi}{2} +\sin^{2}\frac{\omega}{2} +\sin^{2}\frac{\phi}{2} +2\sin\frac{\omega}{2}\cos\frac{\phi}{2}$$
I noticed some familiar terms and using pitagoric identities then I rearranged the equation as follows:
\begin{align} A &= \cos^{2}\frac{\omega}{2} +\sin^{2}\frac{\omega}{2} +\cos^{2}\frac{\phi}{2} +\sin^{2}\frac{\phi}{2} +2\cos\frac{\omega}{2}\cos\frac{\phi}{2} +2\sin\frac{\omega}{2}\cos\frac{\phi}{2} \\ &= 1+1+2\cos\frac{\omega}{2}cos\frac{\phi}{2} +2\sin\frac{\omega}{2}\cos\frac{\phi}{2} \end{align}
Since the latter terms are another way to write prosthapharesis formulas I did the following:
\begin{align} \cos\alpha +\cos\beta &= 2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2} \\ \cos\alpha -\cos\beta &= -2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2} \\ \\ \alpha+\beta &= \omega \\ \alpha-\beta &= \phi \end{align}
By solving the system I found: $$\alpha=\frac{\omega+\phi}{2}$$ and $$\beta=\frac{\omega-\phi}{2}$$.
Therefore by inserting these into the problem:
\begin{align} A &= 1+1 +2\cos\frac{\omega}{2}\cos\frac{\phi}{2} +2\sin\frac{\omega}{2}\cos\frac{\phi}{2} \\ &= 2 +2\cos\frac{\omega}{2}\cos\frac{\phi}{2} -\left(-2\sin\frac{\omega}{2}\cos\frac{\phi}{2}\right) \\ &= 2 +\cos\frac{\omega+\phi}{2} +\cos\frac{\omega-\phi}{2} -\left(\cos\frac{\omega+\phi}{2} -\cos\frac{\omega-\phi}{2}\right) \end{align}
By cancelling elements, | {
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By cancelling elements,
$$A=2 +2\cos\frac{\omega-\phi}{2}$$
However I'm stuck at trying to evaluate these values:
\begin{align} \omega &= 33^{\circ}{20}'\; \phi=56^{\circ}{40}' \\ \omega-\phi &= \left(33+\frac{20}{60}\right) -\left(56+\frac{40}{60}\right) = -23-\frac{20}{60} \end{align}
Therefore,
$$A=2+2\cos\left(\frac{-23-\frac{20}{60}}{2}\right).$$
However the latter answer does not appear in the alternative neither seems to be right. Is there something wrong on what I did?
• Just a remark: Use \cos and \sin instead of cos and sin. – MrYouMath Oct 17 '17 at 8:52
• @MrYouMath Thanks for the advise. I'll take note on that next time I post a similar question. I'm new on LaTex. – Chris Steinbeck Bell Oct 18 '17 at 2:31
There is a mistake in your expansion of the second bracket.
We should have
$A=\cos^2\frac{\omega}{2}+2\cos\frac{\omega}{2}\cos\frac{\phi}{2}+\cos^2 \frac{\phi}{2}+\sin^2\frac{\omega}{2}-2\sin\frac{\omega}{2}\sin\frac{\phi}{2}+\sin^2\frac{\phi}{2}$
$=2+2(\cos\frac{\omega}{2} \cos\frac{\phi}{2} -\sin\frac{\omega}{2}\sin\frac{\phi}{2})$ .
In the bracket we have the expansion for $\cos(\frac{\omega}{2}+\frac{\phi}{2})$ and fortunately $\omega+\phi=90^\circ$.
Thus $A=2+2\cos(45^\circ)$
$=2+\sqrt{2}$
Hint: You did a miscalculation:
$$A=\left (\cos\frac{\omega}{2}+\cos\frac{\phi}{2} \right )^{2}+\left (\sin\frac{\omega}{2}-\sin\frac{\phi}{2} \right )^{2}$$ $$=\cos ^2\frac{\omega}{2}+2\cos\frac{\omega}{2}\cos\frac{\phi}{2}+\cos^2 \frac{\phi}{2}+\sin ^2\frac{\omega}{2}-2\sin\frac{\omega}{2}\sin\frac{\phi}{2}+\sin^2 \frac{\phi}{2}$$ $$=2+2\cos\frac{\omega}{2}\cos\frac{\phi}{2}-2\sin\frac{\omega}{2}\sin\frac{\phi}{2}$$ $$=2+2\left(\cos\frac{\omega}{2}\cos\frac{\phi}{2}-\sin\frac{\omega}{2}\sin\frac{\phi}{2}\right)$$ $$=2+2\cos\left(\frac{\omega}{2}+\frac{\phi}{2}\right)$$
In the last step I used $\cos(x+y)=\cos x\cos y-\sin x \sin y$. | {
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In the last step I used $\cos(x+y)=\cos x\cos y-\sin x \sin y$.
• Mind including the "last step" in your answer?. I mean the one involving inserting the angle values in the last formula. Usually this seems the most logical and obvious thing to do. But what to do when the angles are not "integers". Was the procedure I used correct?. I mean to decompose them into fractions. The latter part I'm not sure. – Chris Steinbeck Bell Oct 18 '17 at 2:34 | {
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Start with the binomial expansion of (k − 1) 2: (k-1)^2: (k − 1) 2: (k – roganjosh May 10 '17 at 19:49 (, Techniques for Adding the Numbers 1 to 100, Quick Insight: Intuitive Meaning of Division, Quick Insight: Subtracting Negative Numbers, Surprising Patterns in the Square Numbers (1, 4, 9, 16…), Learning How to Count (Avoiding The Fencepost Problem), Different Interpretations for the Number Zero. Now for the explanation: How many beans do we have total? (i) 2, 7, 12 , ., to 10 terms. We prove the formula 1+ 2+ ... + n = n(n+1) / 2, for n a natural number. Beginners Java program to find sum of odd numbers between 1 -100 ½n(n + 1),. Yep, you get the same formula, but for different reasons. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. Python Program to Calculate Sum of Odd Numbers from 1 to N using For Loop. my calculator seems to say 55 though, i don't get how the sum of n terms formula has failed as it's for an AP. Find all the evens, So, the sum from 50 + 52 + … 100 = (50 * 51) – (24 * 25) = 1950. There is a simple applet showing the essence of the inductive proof of this result. I like Steve's thinking, but sometimes I like to take a hammer to a thumbtack. Vandermonde convolution X k r k! We get the next biggest even number (n + 1) and take off the extra (n + 1)/2 “-1″ items: To add 1 + 3 + 5 + … 13, get the next biggest even (n + 1 = 14) and do, Let’s say you want the evens from 50 + 52 + 54 + 56 + … 100. 17 Answers 26 The main idea is that if you write all the numbers from 0 to 999999 down as six digit numbers (possibly prepending zeros) then all digits appear the same number of times. Well, we have 2 equal rows, we must have n/2 pairs. Math. The above method works, but you handle odd and even numbers differently. What is the sum of 1+1/10+1/100+1/1000... etc, where each time the denominator increases by an factor of 10 Through how many radians does it turn in one second? Each nth term of the series is the sum of | {
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of 10 Through how many radians does it turn in one second? Each nth term of the series is the sum of the n terms i.e n*(n+1)/2. Then push the [Next] button to step through the stages of the proof. Calculate the sum … Q:-A wheel makes 360 revolutions in one minute. for(i=1;i<=n;i++) sum=sum+i; Sum of Natural Numbers using Formula. The loop structure should look like for(i=2; i<=N; i+=2). Yes. Submitted by IncludeHelp, on September 04, 2018 Given the value of N and we have to find sum of all numbers from 0 to N in C language. In case you’re wondering whether it “really” lines up, it does. Instead of writing out numbers, pretend we have beans. Output 2: Enter the value of n: 0 Enter a whole positive number! Below is the complete algorithm. What is the formula for the sum of odd numbers? How to calculate this $\sum\limits_{n=0}^{\infty}\frac{n}{2^n}e^{jwn}$ 0. 2. Python Program to Calculate Sum of Even Numbers from 1 to N using For Loop. This program to find the sum of n numbers allows the user to enter any integer value. sum = n(n+1)/2 In the above program, unlike a for loop, we have to increment the value of i inside the body of the loop. For instance the sum of the numbers from 1 to 10 is 55 whereas the sum of the digits is 46. Let’s say we mirror our pyramid (I’ll use “o” for the mirrored beans), and then topple it over: Cool, huh? home Front End HTML CSS JavaScript HTML5 Schema.org php.js Twitter Bootstrap Responsive Web Design tutorial Zurb Foundation 3 tutorials Pure CSS HTML5 Canvas JavaScript Course Icon Angular React Vue Jest Mocha NPM Yarn Back End PHP Python Java … Home Science Math History Literature Technology Health Law Business All Topics Random. The sum of all the odd integers from 1 to 50 is 625. To find sum of even numbers we need to iterate through even numbers from 1 to n. Initialize a loop from 2 to N and increment 2 on each iteration. 1. In this program, we first check number is odd or not. To add evens from 2 to 50, find 1 + 2 + 3 + 4 … + 25 and | {
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we first check number is odd or not. To add evens from 2 to 50, find 1 + 2 + 3 + 4 … + 25 and double it: So, to get the evens from 2 to 50 you’d do 25 * (25 + 1) = 650. The above formula is one core step of the idea. The partial sums of the series 1 + 2 + 3 + 4 + 5 + 6 + ⋯ are 1, 3, 6, 10, 15, etc.The nth partial sum is given by a simple formula: ∑ = = (+). = r +s n! Python Program to Sum the list with start 10; Program to Sum the list of float; Python Program to Sum the list of float with start 10.1; Program to calculate the sum of elements in a tuple. Let’s look at a small set: The average is 2. Python Program to Sum the list with start 10; Program to Sum the list of float; Python Program to Sum the list of float with start 10.1; Program to calculate the sum of elements in a tuple. We know that the even numbers are the numbers, which are completely divisible by 2. This Python program allows the user to enter the maximum limit value. Popular Questions of Class 11th mathematics. The so-called educator wanted to keep the kids busy so he could take a nap; he asked the class to add the numbers 1 to 100. You need to initialize the counter as 0 and while the loop executes you need to collect / sum them to the counter and finally outside the loop print/ echo the counter. This is demonstrated by the following code snippet. So, we can say the average of the entire set is actually just the average of 1 and n: (1 + n)/2. DESCRIPTION. In summation notation, this may be expressed as + ... Achilles could run at 10 m/s, while the tortoise only 5. The formula to find the sum of first n natural numbers is as follows. Geometric sum nX−1 k=0 ark = a 1− rn 1− r r 6= 1 Geometric series X∞ k=0 ark = a 1− r |r| < 1 3. Feb 21, 2015 . Feb 21, 2015 . Since 50 is an even number, and we want the sum of all the odd integers from 1 to 50, the number 50 won't be included in the sum. First Name. In a similar vein to the previous exercise, here is another way of deriving the formula for the sum of | {
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a similar vein to the previous exercise, here is another way of deriving the formula for the sum of the first n n n positive integers. 3) Most importantly, this example shows there are many ways to understand a formula. We want to add 1 bean to 2 beans to 3 beans… all the way up to 5 beans. Thus, the sum of the integers from 1 to 100, which are divisible by 2 or 5, is 3050. 2, 7, 12, .,to 10 term We know that Sum of AP = /2 (2a + (n 1) d) Here n = 10, a = 2, & d = 7 2 = 2 Putting these in formula , Sum = /2 (2a + (n 1) d) = 10/2 (2 2 + (10 1) So we divide the formula above by 2 and get: Now this is cool (as cool as rows of numbers can be). for(i=1;i<=n;i++) sum=sum+i; Sum of Natural Numbers using Formula. 1) Adding up numbers quickly can be useful for estimation. If you can’t see any matches tap the remaining deck to turn over a new card. If we have 100 numbers (1…100), then we clearly have 100 items. 2) Compute some of digits in numbers from 1 … Output 3: Enter the value of n: -10 Enter a whole positive number! Here, we are implementing a C program that will be used to find the sum of all numbers from 0 to N without using loop. Learn more - Program to check odd numbers. Otherwise, we used the mathematical formula of Sum of Series 1 + 2+ 3+ … + N = N * (N + 1) / 2; TIP: If the function is not returning any value then used Void. C Program to Print Sum of all Even Numbers from 1 to n. This program allows the user to enter the maximum limit value. #SumOfNumbers #1to100 How can we calculate the sum of natural numbers? Well, the sum is clearly 1 + 2 + 3 + 4 + 5. 4. In mathematics, summation is the addition of a sequence of any kind of numbers, called addends or summands; the result is their sum or total. But of course, we don’t want the total area (the number of x’s and o’s), we just want the number of x’s. Interview Answers. Let’s add the numbers 1 to 9, but instead of starting from 1, let’s count from 0 instead: By counting from 0, we get an “extra item” (10 in total) | {
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from 1, let’s count from 0 instead: By counting from 0, we get an “extra item” (10 in total) so we can have an even number of rows. \sum_{k=1}^n (2k-1) = 2\sum_{k=1}^n k - \sum_{k=1}^n 1 = 2\frac{n(n+1)}2 - n = n^2.\ _\square k = 1 ∑ n (2 k − 1) = 2 k = 1 ∑ n k − k = 1 ∑ n 1 = 2 2 n (n + 1) − n = n 2. To see that, look at this oblong number, in which the base is one more than the height: case 1. please enter the maximum value: 10 The sum of Even numbers 1 to 10 = 30 The sum of odd numbers 1 to 10 = 25. case 2. please enter the maximum value: 100 The sum of Even numbers 1 to 100 = 2550 The sum of odd numbers 1 to 100 = 2500 . clear, insightful math lessons. Just make a for loop from starting 1 to 10, like given below. sum = n(n+1)/2 The program calculates the sum of numbers till the given input. By the way, there are more details about the history of this story and the technique Gauss may have used. Notice in both cases, 1 is on one side of the average and N is equally far away on the other. integer m 6= −1 5. How about even numbers, like 2 + 4 + 6 + 8 + … + n? 2. C Program to print sum of the natural numbers from 1 to 10 # include # include main( ) { int n,sum=0,i; clrscr( ); for (i=1; i<=10; i++) sum=sum+i; printf(“sum of natural numbers from 1 to 10 is %d\\n”,sum); getch( ); } The goal is to clear the tableau by matching pairs of cards that add up to ten. 0. how to find a sum of numbers in a sequence when some intermediate terms are not taken in to consideration? Ex 5.3 ,1 Find the sum of the following APs. However, our formula will look a bit different. Answer Add Tags. Though both programs are technically correct, it is better to use for loop in this case. For these examples we’ll add 1 to 10, and then see how it applies for 1 to 100 (or 1 to any number). We want to get rid of every number from 1 up to a – 1. input this case is 6. Since thousand squared = 1 million, we get million / 2 + 1000/2 = 500,500. sum of n terms= n/2 {2a(n-1)d} where n is no of terms, a | {
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we get million / 2 + 1000/2 = 500,500. sum of n terms= n/2 {2a(n-1)d} where n is no of terms, a is the first term and d is difference, subbed in you get 5{2x9}=90. There is a more simple way to find the sum of odd numbers from 1 to N. The first odd number is 1, then the next odd number is 3, i.e (1+2). Enter n value: 20 Sum of odd numbers from 1 to 20 is: 100. Example 1: Adding all items in a tuple, and returning the result ; Example 1: Starting with the 10, and adding all items in a tuple to this number: Example: Compute the sum and product of the numbers from 1 to 10. C programming, exercises, solution : Write a program in C to calculate the sum of numbers from 1 to n using recursion. The above formula is one core step of the idea. How about odd numbers, like 1 + 3 + 5 + 7 + … + n? I won’t. The difference between consecutive triangles increases by 1.. A formula for the triangular numbers. Telescoping sum X a≤k | {
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sum of 1 to 10 2020 | {
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# $n$th Roots of Unity
I need to show that the product of all the $n$th roots of unity is $(−1)^{n+1}$.
Is there a way to do this by induction? If there is, I can't seem to figure it out. Are there other, perhaps more efficient, methods of proof?
-
The problem with attempting to do it "by induction" is that the $n$th roots of unity are not related to the $(n+1)$th roots of unity (except for the single one that they have in common, $1$. – Arturo Magidin Mar 2 '12 at 4:07
If you're going to do an induction argument, it should be an induction on the number of (possibly repeated) prime divisors of $n$. I think this should actually be possible, with Wilson's Theorem stepping in as a base case. But, honestly, just about anything else seems easier....like all the answers given below. – Pete L. Clark Mar 2 '12 at 4:28
The $n^{th}$ roots of unity are precisely the roots of the polynomial $$x^n - 1$$ Now use the fact that the product of the roots of any polynomial is $(-1)^n$ times the constant coefficient.
-
Is this fact that the product of the roots of any polynomial is $(−1)^n$ times the constant coefficient true for any polynomial? Can you elaborate on that? – Dominick Gerard Mar 2 '12 at 4:42
@DominickGerard: Consider writing the polynomial as a product of linear factors, then multiplying it out again. What will the constant term be? Example: $(z-z_1)(z-z_2) = z^2 - (z_1 + z_2) z + z_1 z_2$. – Antonio Vargas Mar 2 '12 at 4:58
@DominickGerard Yes, it holds for any polynomial with coefficients in an integral domain (including $\mathbb{Z}$ or any field): en.wikipedia.org/wiki/Vieta%27s_formulas – dls Mar 2 '12 at 5:01
Note that if $\zeta$ is an $n$th root of unity, then so is $\frac{1}{\zeta}$. Moreover, the only roots of unity for which $\zeta=\frac{1}{\zeta}$ are $\zeta=1$ and $\zeta=-1$ (since $\zeta=\frac{1}{\zeta}$ implies $\zeta^2=1$, hence $\zeta=1$ or $\zeta=-1$). | {
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So, if we take the $n$th roots of unity, then we can pair off all of them except for $1$ and perhaps $-1$ (if $n$ is even), into pairs of the form $\{\zeta,\frac{1}{\zeta}\}$. The product of these two equals $1$; so when you do the product of all roots, you end up with a bunch of pairs that multiply to $1$, and you have $1$; and if $n$ is even, then you also have $-1$ leftover. Which means the product of all of them is...
-
Well, you said "efficient" and "elementary number theory", and that triggered this in my mind: the result is a special case of
(Wilson's Theorem in a Finite Abelian Group): Let $(G,\cdot)$ be a finite abelian group, and let $P = \prod_{x \in G} x$. Then $P = 1$ (the identity element) unless $G$ has exactly one element $t$ of order $2$, in which case $P = t$.
This is something that others (in particular, Bill Dubuque) have talked about here and elsewhere before. I include a statement and proof of this in my notes/prebook on elementary(ish...) number theory: it is at the end of Appendix B.
Added: Not only is this a special case of a "generalized Wilson theorem", it's a special case that includes the classical Wilson Theorem! Since the unit group of $(\mathbb{Z}/p\mathbb{Z})^{\times}$ is cyclic of order $p-1$, computing $(p-1)! \pmod{p}$ is essentially the same as multiplying together all the $(p-1)$st roots of unity. | {
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-
+1 See these posts for more on the group-theoretic Wilson's theorem and involutions. – Bill Dubuque Mar 2 '12 at 4:13
You can prove it with de Moivre, on the basis that the roots are all of the form $e^{2\pi i k/n}$, so their product is $e^{2\pi i/n\sum{k}} = e^{\pi i(n-1)} = (-1)^{n-1} = (-1)^{n+1}$.
Another proof idea would be to notice that the roots of unity come in complex conjugate pairs, which is evident in the symmetry of the spacing of the roots in the complex plain. So the product of each pair is of course $\alpha \bar\alpha = |\alpha| = 1$ since all the roots fall on the unit circle. The only exceptions are $1,-1$. $1$ is always a root and won't change the product, and $-1$ is only a root when $n$ is even, so we conclude that the product is exactly $(-1)^{n+1}$. | {
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# Lecture 7 - Universal properties: products, coproducts and exponentials - David Spivak
edited January 21
## Comments
• Options
1.
It was interesting to learn about the relationship between currying, uncurrying, and exponentials. I'd heard people talking about currying before and handwaving, but this lecture made it very concrete. The motivation of set exponentiation (the set of maps) {1, 2, 3}^{1,2} as "3"^"2" ~ 9 maps in total was nice as well.
I also liked the motivation of Either as a pushout from two functions. I recently did a few exercises using Either in scala and ended up getting very confused! So this sort of knowledge would probably have made that experience a bit less confusing.
I noticed that it was mentioned in passing that products, co-products and exponentials are all examples of adjunctions. And I just listened to the first 10 seconds of lecture 8, and I gather that they are discussed therein, so I look forward to reviewing that lecture too in time. | {
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Comment Source:It was interesting to learn about the relationship between currying, uncurrying, and exponentials. I'd heard people talking about currying before and handwaving, but this lecture made it very concrete. The motivation of set exponentiation (the set of maps) {1, 2, 3}^{1,2} as "3"^"2" ~ 9 maps in total was nice as well. I also liked the motivation of Either as a pushout from two functions. I recently did a few exercises using Either in scala and ended up getting very confused! So this sort of knowledge would probably have made that experience a bit less confusing. I noticed that it was mentioned in passing that products, co-products and exponentials are all examples of [adjunctions](https://en.wikipedia.org/wiki/Adjoint_functors). And I just listened to the first 10 seconds of lecture 8, and I gather that they are discussed therein, so I look forward to reviewing that lecture too in time.
• Options
2.
edited March 2020
Yes, it's a good one!
Ideas from the first five minutes (using David Spivak's terminology):
• A universal property is a "one-stop shop" for maps
• The product of sets $$A$$ and $$B$$ is the one-stop shop from maps into $$A$$ and $$B$$.
Meaning: suppose you have a set $$I$$, and maps $$f: I \rightarrow A$$ and $$g: I \rightarrow B$$, with $$f$$ in your "left-hand", and $$g$$ in your right.
These two functions are equivalent to a single function $$f \times g: I \rightarrow A \times B$$.
The set $$I$$ can be thought of as an "index" set which is used to pick out elements via the functions out of $$I$$.
When $$I$$ is a singleton, the "one-stop shop" gives you a single element in the product of $$A$$ and $$B$$, in lieu of two separate elements one from $$A$$ and the other from $$B$$. | {
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Comment Source:Yes, it's a good one! Ideas from the first five minutes (using David Spivak's terminology): * A universal property is a "one-stop shop" for maps * The product of sets \$$A\$$ and \$$B\$$ is the one-stop shop from maps into \$$A\$$ and \$$B\$$. Meaning: suppose you have a set \$$I\$$, and maps \$$f: I \rightarrow A\$$ and \$$g: I \rightarrow B\$$, with \$$f\$$ in your "left-hand", and \$$g\$$ in your right. These two functions are equivalent to a single function \$$f \times g: I \rightarrow A \times B\$$. The set \$$I\$$ can be thought of as an "index" set which is used to pick out elements via the functions out of \$$I\$$. When \$$I\$$ is a singleton, the "one-stop shop" gives you a single element in the product of \$$A\$$ and \$$B\$$, in lieu of two separate elements one from \$$A\$$ and the other from \$$B\$$.
• Options
3.
• The coproduct of $$A$$ and $$B$$ is the "one-stop shop" for maps into $$A$$ and $$B$$.
Comment Source:* The coproduct of \$$A\$$ and \$$B\$$ is the "one-stop shop" for maps into \$$A\$$ and \$$B\$$.
• Options
4.
edited March 2020
Thinking out loud here...
The idea of the "one-stop shop for maps" generalizes nicely from the product to the limit.
Looking at the general case of the limit, which includes as special cases terminal objects, products, equalizers and pullbacks, brings clarity to the whole picture. And in the category Set, there is an explicit construction for all limits.
To review: suppose we are given a diagram D in the category C. Note: for this definition, D is not required to be a commuting diagram -- it's just a collection of objects and arrows. | {
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The limit of D, if it exists, will be another object Lim(D) in C, along with "projection maps" from Lim(D) to each of the objects in D, which satisfies a couple of conditions. First, for every arrow $$f: X_1 \rightarrow X_2$$ in D, the triangle diagram with objects Lim(D), $$X_1$$, and $$X_2$$ and arrows $$f$$ and the two projections from Lim(D) to $$X_1$$ and $$X_2$$ -- this triangle diagram commutes.
Lim(D) along with these projections is called a cone over D. The cones over D form a category, with arrows going between the apexes of the cones, such that these arrows satisfy the following commutativity condition with respect to the cones. Suppose that $$X$$ and $$Y$$ are each the apex of a cone, and $$f: X \rightarrow Y$$. Then $$f$$ is defined to be a morphism between the two cones if for each object $$d$$ in D, the triangle formed by $$d$$, $$X$$ and $$Y$$, with arrows $$f$$ and the two projections to $$d$$ -- one projection within each cone -- commutes.
The second requirement on Lim(D) is that it be a terminal object in this category of cones. | {
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The second requirement on Lim(D) is that it be a terminal object in this category of cones.
Comment Source:Thinking out loud here... The idea of the "one-stop shop for maps" generalizes nicely from the product to the limit. Looking at the general case of the limit, which includes as special cases terminal objects, products, equalizers and pullbacks, brings clarity to the whole picture. And in the category Set, there is an explicit construction for all limits. To review: suppose we are given a diagram D in the category C. Note: for this definition, D is not required to be a _commuting_ diagram -- it's just a collection of objects and arrows. The limit of D, if it exists, will be another object Lim(D) in C, along with "projection maps" from Lim(D) to each of the objects in D, which satisfies a couple of conditions. First, for every arrow \$$f: X_1 \rightarrow X_2\$$ in D, the triangle diagram with objects Lim(D), \$$X_1\$$, and \$$X_2\$$ and arrows \$$f\$$ and the two projections from Lim(D) to \$$X_1\$$ and \$$X_2\$$ -- this triangle diagram commutes. Lim(D) along with these projections is called a cone over D. The cones over D form a category, with arrows going between the apexes of the cones, such that these arrows satisfy the following commutativity condition with respect to the cones. Suppose that \$$X\$$ and \$$Y\$$ are each the apex of a cone, and \$$f: X \rightarrow Y\$$. Then \$$f\$$ is defined to be a morphism between the two cones if for each object \$$d\$$ in D, the triangle formed by \$$d\$$, \$$X\$$ and \$$Y\$$, with arrows \$$f\$$ and the two projections to \$$d\$$ -- one projection within each cone -- commutes. The second requirement on Lim(D) is that it be a terminal object in this category of cones.
• Options
5.
edited March 2020
Now that's somewhat of an abstract story, which doesn't give you a feeling for how to actually find what this special cone Lim(D) actually is. And in general such an object may not exist at all. | {
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But in the category Set it always exists, and can be constructed explicitly, as follows.
Comment Source:Now that's somewhat of an abstract story, which doesn't give you a feeling for how to actually find what this special cone Lim(D) actually is. And in general such an object may not exist at all. But in the category Set it always exists, and can be constructed explicitly, as follows.
• Options
6.
edited March 2020
Suppose we have a diagram $$D$$ in Set. Let the objects in $$D$$ be named $$S_1, \ldots S_n$$, where each $$S_i$$ is a set. Let the arrows in $$D$$ be $$f_1, \ldots f_k$$ -- these are functions.
Then Lim(D) consists of a subset of the Cartesian product of the set $$S_1, \ldots S_n$$ -- so it is a relation R between these sets. It consists of all tuples $$(x_1, \ldots, x_n)$$ that satisfy the all of the constraints given by the functions $$f_i$$. Suppose $$f_i: S_j \rightarrow S_k$$. Then the constraint given by $$f_i$$ stipulates that for a tuple $$(x_1, \ldots, x_n)$$, that $$f_i(x_j) = x_k$$.
The cone has relation R at its apex, and the ordinary projection functions from R to each of the sets $$S_1, \ldots, S_n$$ as the arrows from the apex R to the objects in the diagram $$D$$.
It is but an exercise to show that this cone with R at its apex is terminal within the category of cones over $$D$$. | {
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Comment Source:Suppose we have a diagram \$$D\$$ in Set. Let the objects in \$$D\$$ be named \$$S_1, \ldots S_n\$$, where each \$$S_i\$$ is a set. Let the arrows in \$$D\$$ be \$$f_1, \ldots f_k\$$ -- these are functions. Then Lim(D) consists of a subset of the Cartesian product of the set \$$S_1, \ldots S_n\$$ -- so it is a relation R between these sets. It consists of all tuples \$$(x_1, \ldots, x_n)\$$ that satisfy the all of the constraints given by the functions \$$f_i\$$. Suppose \$$f_i: S_j \rightarrow S_k\$$. Then the constraint given by \$$f_i\$$ stipulates that for a tuple \$$(x_1, \ldots, x_n)\$$, that \$$f_i(x_j) = x_k\$$. The cone has relation R at its apex, and the ordinary projection functions from R to each of the sets \$$S_1, \ldots, S_n\$$ as the arrows from the apex R to the objects in the diagram \$$D\$$. It is but an exercise to show that this cone with R at its apex is terminal within the category of cones over \$$D\$$.
• Options
7.
The relation R = Lim(D) is a one-stop shop for cones over D, in the following sense. Suppose that we have a cone C with apex A over D. Then C consists of a bunch of separate maps from A into the objects of D. The universal property for R = Lim(D) says that there a unique map m(C) from A into R, which gives a morphism from C into the universal cone with apex Lim(D).
m(C) is a single map that contains all of the information in C. That's because the projections from C into the objects of D can be reconstructed by composing m(C) with the standard projections from R into the objects of D.
So the multiple separate maps comprising the cone C can, without loss of information, be replaced by the single map m(C). | {
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Comment Source:The relation R = Lim(D) is a one-stop shop for cones over D, in the following sense. Suppose that we have a cone C with apex A over D. Then C consists of a bunch of separate maps from A into the objects of D. The universal property for R = Lim(D) says that there a unique map m(C) from A into R, which gives a morphism from C into the universal cone with apex Lim(D). m(C) is a single map that contains all of the information in C. That's because the projections from C into the objects of D can be reconstructed by composing m(C) with the standard projections from R into the objects of D. So the multiple separate maps comprising the cone C can, without loss of information, be replaced by the single map m(C).
• Options
8.
edited March 2020
For an illustration of this construction, let's apply it to the equalizer.
Here, the diagram D consists of two objects $$X_1, X_2$$, and two maps $$f,g: X_1 \rightarrow X_2$$.
Note: here the fact that D is not required to be a commutative diagram is significant. If it were so required, then that would imply that $$f$$ and $$g$$ are equal. But we don't require that, as we are constructing the equalizer of distinct maps $$f$$ and $$g$$.
Then, as per the general construction given above, the relation R = Lim(D) will consist of all tuples $$(x,y)$$, such that $$f(x) = y$$, and $$g(x) = y$$. Putting these together, we get that:
$equalizer(f,g) = Lim(D) = \lbrace (x,y)\ |\ y = f(x) = g(x) \rbrace = \lbrace (x,y)\ |\ x \in dom(f) \cap dom(g),\ f(x) = g(x)\rbrace = f \cap g$
Which is to say: the equalizer of functions $$f$$ and $$g$$ is the relation (the graph representation) for the partial function that is obtained by restricting both functions to the set of points where they are both defined and both produce the same value. | {
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Comment Source:For an illustration of this construction, let's apply it to the equalizer. Here, the diagram D consists of two objects \$$X_1, X_2\$$, and two maps \$$f,g: X_1 \rightarrow X_2\$$. Note: here the fact that D is not required to be a _commutative_ diagram is significant. If it were so required, then that would imply that \$$f\$$ and \$$g\$$ are equal. But we don't require that, as we are constructing the equalizer of distinct maps \$$f\$$ and \$$g\$$. Then, as per the general construction given above, the relation R = Lim(D) will consist of all tuples \$$(x,y)\$$, such that \$$f(x) = y\$$, and \$$g(x) = y\$$. Putting these together, we get that: \$equalizer(f,g) = Lim(D) = \lbrace (x,y)\ |\ y = f(x) = g(x) \rbrace = \lbrace (x,y)\ |\ x \in dom(f) \cap dom(g),\ f(x) = g(x)\rbrace = f \cap g\$ Which is to say: the equalizer of functions \$$f\$$ and \$$g\$$ is the relation (the graph representation) for the partial function that is obtained by restricting both functions to the set of points where they are both defined and both produce the same value.
• Options
9.
edited March 2020
We get the product as a special case of a limit, where the diagram $$D$$ consists only of objects, and no arrows.
In Set, this will be a collection of sets $$S_1, \ldots, S_n$$ -- and no arrows to impose constraints on the tuples in R = Lim(D).
So Lim(D) is the unconstrained set of tuples -- i.e., the Cartesian product $$S_1 \times \cdots \times S_n$$.
Comment Source:We get the _product_ as a special case of a limit, where the diagram \$$D\$$ consists only of objects, and no arrows. In Set, this will be a collection of sets \$$S_1, \ldots, S_n\$$ -- and no arrows to impose constraints on the tuples in R = Lim(D). So Lim(D) is the unconstrained set of tuples -- i.e., the Cartesian product \$$S_1 \times \cdots \times S_n\$$.
• Options
10.
What's the limit of an empty diagram $$D$$?
It's simply a terminal object in the category, for the following reason. | {
"domain": "azimuthproject.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9664104982195785,
"lm_q1q2_score": 0.8574038392012853,
"lm_q2_score": 0.8872045996818986,
"openwebmath_perplexity": 613.6969684394298,
"openwebmath_score": 0.8684090375900269,
"tags": null,
"url": "https://forum.azimuthproject.org/discussion/2459/lecture-7-universal-properties-products-coproducts-and-exponentials-david-spivak"
} |
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