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t \multimap F(t,x) is measurable, for every x \in \mathbb{R}^{n} ,

x \multimap F(t,x) is u.s.c., for almost all (a.a.) t\in J ,

|y| \leq r(t) (1+|x|) , for every (t,x) \in J \times \mathbb{R}^{n} , and every y\in F(t,x) , where r\colon J \rightarrow [0, \infty) is a Lebesgue integrable function.

Let F \colon X \multimap Y be a multivalued mapping and f \colon X \rightarrow Y be a single-valued mapping. We say that f is a selection of F (written f \subset F ) if f(x) \in F(x) , for every x \in X .

We will employ the following definitions and statements.

(Castaing representation, cf. Theorem III.7 in [ 35 ])

\varphi(x)=\overline{\bigl\{ f_{n}(x)\mid n \in\mathbb{N}\bigr\} }= \overline {\bigcup_{n \in\mathbb{N}} f_{n}(x)},

Lemma 3

(, , Lemma7.1 in [ 36 ])

F\colon J\times\mathbb{R}^{n} \multimap\mathbb{R}^{n} -. F(t, q(t)) , q \in C(J, \mathbb{R}^{n}) , -.

If X \subset Y and \varphi\colon X \multimap Y , then a point x \in X is called a fixed point of φ if x \in\varphi (x) . We set \operatorname{Fix} (\varphi) := \{ x \in X \mid x \in\varphi(x) \} .

Definition 1

Let and be subsets of normed linear spaces and \varphi\colon X \multimap Y be a multivalued mapping. If is convex, then is called a , provided is u.s.c. with (nonempty) compact, convex values.

Let and be metric spaces. A multivalued mapping \varphi\colon X \multimap Y is called if its image \varphi(X)= \bigcup\{\varphi(x) \mid x \in X \} is contained in a compact subset of .

The following statement, which we state in the form of proposition, is usually called the Kakutani-Ky Fan-type fixed point theorem.

Proposition 2

(, , Theorem II.8.4 in [ 21 ])

() , \varphi\colon C \multimap C . .

\Phi(t,x)=\bigcap_{\delta>0} \bigcap _{\tiny{\mu(N)=0}} \overline {\operatorname{conv}}\, \varphi\bigl(B \bigl((t,x),\delta\bigr) \backslash N\bigr)

Observe that Φ is a bounded u.s.c. mapping with (nonempty) compact and convex values.

\int_{J} P(t) \,\mathrm{d}t := \biggl\{ \int _{J} p(t) \,\mathrm{d}t \Bigm| p \subset P \mbox{ is a Lebesgue integrable selection of } P \biggr\} .
L u(t) = f(t), \quad a \leq t \leq b, a, b \in\mathbb{R},
u(a)=0, \qquad u(b)=0.

( cf. , e.g. , Theorem3.2.1 in [ 38 ])

Slope of a line is a lower level of Algebra, and once you take statistics you find that a simple linear progression is rarely useful. Understanding what it means, and how to use modern software and tools to calculate it, is far more valuable than knowing rise over run. I’d hire a guy who knew Excel over FOIL anyday.

Here’s the learning outcomes for Intermediate Algebra; the student will be able to: Identify and graph inequalities, absolute value, rational, radical, polynomial, exponential, and logarithmic functions Simplify rational, radical, polynomial, exponential, and logarithmic expressions Solve both analytically and graphically inequalities, absolute value, rational, radical, polynomial, exponential, and logarithmic equations Use technology such as graphing calculators and/or computer algebra system to assist in solving problems involving any of the topics in (A) through (C) Discuss mathematical problems and write solutions in accurate mathematical language and notation interpret mathematical solutions.

Wow. First of all, I received my high school diploma and BE in Mathematics in New York. When I moved to California to teach, it totally blew my mind that most college freshman were taking "College Algebra" as they entered university (Cal State, CU or private). Where I come from if you were going to college straight from high school, you started with Calculus. If not ready for that, you took remedial courses through … Read More

Wow. First of all, I received my high school diploma and BE in Mathematics in New York. When I moved to California to teach, it totally blew my mind that most college freshman were taking “College Algebra” as they entered university (Cal State, CU or private). Where I come from if you were going to college straight from high school, you started with Calculus. If not ready for that, you took remedial courses through junior colleges. It was never expected that they would transfer as general education credit. Statistics could be used if you were not a math major but two semesters of math was required for a BA regardless of major. One had to be Calculus. It actually floors me as a secondary math teacher that students are graduating with no less than 2 years of Algebraic coursework only to repeat it. This from ALL races! It’s NOT A civil rights issue it’s a phobia that hopefully Common Core teaching strategies will help!! Algebraic thinking/problem solving is key to all critical thinking!

While learning statistics represents a less challenging experience, it should never replace mathematics, which is a valid discipline by itself. Intermediate algebra provides a set of skills that every college students should be able to cope with. The argument that such knowledge students "will never use, either in college or in life" is invalid. Numerous other components of education do not call for an immediate application but they are still inevitable components of a … Read More

An IndieFlix Original Documentary

The filmmakers' goal is to have a global conversation and raise awareness around anxiety. Angst features candid interviews with kids and young adults who suffer, or have suffered, from anxiety and what they've learned about it. The film includes discussions with mental health experts about the causes of anxiety and its sociological effects, as well as help, resources and tools.

The Angst Weekly

The Mighty: 47 ‘Little Signs’ You’re Recovering From Anxiety


The Mighty’s 4 Things That Help Ease Anxiety


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Build Up Your Community

Is anxiety affecting you and the people living in your community? Are you a teacher who sees many students dealing with stress or anxiety? Or a parent with children experiencing anxiety? Help break the stigma around anxiety and get the discussion started today.

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Why isn't this online?

Angst is not available online at this time because we are currently screening the film at schools and community centers across the country. We believe it is important to follow up this film with an in-person discussion. At some point in the near future, we hope to make the film available online as well.

Find a Screening

Experts, Educators Public Figures

"Angst is inspirational; the courageous effort made by teens in the film to face their fears and become competent in coping is heart-warming and breath-taking."

"I welcomed the opportunity to be a part of to further the dialogue around mental health and to help people understand the impact anxiety has on our mental state and encourage people, especially kids, to ask for help."

"I also find that anxiety is a debilitating disease with our students and wanted to share your vision and hope with them. "

Students, Teens Parents

"So glad my dad got to see this movie now he knows I'm not making it up."

"I wish this film had existed when I was growing up it would've saved me and my family a lot of heartache."

"I never knew that what I was experiencing was severe anxiety I'm so grateful to know that there is treatment."

"It's powerful, spot-on, genuine, educational and hopeful. Well done!!"

Angst Gallery

We know anxiety can cause you, your loved ones, and your friends a lot of stress. Here are some of the resources we believe can help. This list has been examined by experts in the mental health field and will continue to grow. Please note these are just recommendations, and not "one size fits all".

Hockey historians say Fred Sasakamoose became the NHL’s first indigenous player with treaty status when he skated for the Chicago Blackhawks in 1953-54 (Photo/Courtesy Hockey Hall of Fame).

Maracle is starting to get his due, thanks to Schmidt-Adeney’s doggedness. She reached out to the Rangers and obtained two official jerseys, complete with Maracle’s last name and Number 14 on the back.

One of the sweaters will be presented at a ceremony next month to North Dumfries Mayor Sue Foxton on behalf of the township. Former New York Islanders broadcaster Jiggs McDonald , an Ayr native, and Walter Gretzky , father of Wayne Gretzky , the NHL’s all-time leading scorer, plan to attend the event, Schmidt-Adeney said.

North Dumfries Mayor Sue Foxton New York Islanders Jiggs McDonald Walter Gretzky Wayne Gretzky

Plans are being formulated to present the second Rangers jersey to a representative of Six Nations of Grand River , hopefully to coincide with National Aboriginal Heritage Day on the June 21.

Six Nations of Grand River National Aboriginal Heritage Day

In addition, the Hockey Hall of Fame in Toronto asked Schmidt-Adeney for her research on Maracle.

Hockey Hall of Fame

She noted in her March 21, 2018 article that the Hall had biographies for Maracle and Sasakamoose in its data base, but only had Sasakamoose described as indigenous.

So who was Henry Elmer “Buddy” Maracle?

He was a 5-foot-11 left wing whose professional career began in 1926-27 with the Springfield Indians of the old Canadian-AmericanHockey League . He spent four season with the Indians before he was traded to the Rangers in 1930-31. His hockey exploits garnered racist headlines like “Indian Puck Star” and “Redskin Icer.”

Canadian-AmericanHockey League

He ended his professional playing career in 1936-37 after skating for the Indians, Philadelphia Arrows , New Haven Eagles , and Bronx Tigers of the old Canadian-American Hockey League , and the Tulsa Oilers of the American Hockey Association. He briefly served the Oilers’ player/head coach during the 1936-37 season, according to .

Philadelphia Arrows New Haven Eagles Bronx Tigers Canadian-American Hockey League Tulsa Oilers American Hockey Association. .

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