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2. Unimodular matrix - Wikipedia

   en.wikipedia.org/wiki/Unimodular_matrix

   Specifically, if A is TU and b is integral, then linear programs of forms like {,} or {} have integral optima, for any c. Hence if A is totally unimodular and b is integral, every extreme point of the feasible region (e.g. { x ∣ A x ≥ b } {\displaystyle \{x\mid Ax\geq b\}} ) is integral and thus the feasible region is an integral polyhedron.

3. Tubular neighborhood - Wikipedia

   en.wikipedia.org/wiki/Tubular_neighborhood

   Generalizations of smooth manifolds yield generalizations of tubular neighborhoods, such as regular neighborhoods, or spherical fibrations for Poincaré spaces. These generalizations are used to produce analogs to the normal bundle, or rather to the stable normal bundle , which are replacements for the tangent bundle (which does not admit a ...

4. Calculus on Manifolds (book) - Wikipedia

   en.wikipedia.org/wiki/Calculus_on_Manifolds_(book)

   Calculus on Manifolds is a brief monograph on the theory of vector-valued functions of several real variables (f : R n →R m) and differentiable manifolds in Euclidean space. . In addition to extending the concepts of differentiation (including the inverse and implicit function theorems) and Riemann integration (including Fubini's theorem) to functions of several variables, the book treats ...

5. Fundamental class - Wikipedia

   en.wikipedia.org/wiki/Fundamental_class

   April 2023) (Learn how and when to remove this message) In mathematics , the fundamental class is a homology class [ M ] associated to a connected orientable compact manifold of dimension n , which corresponds to the generator of the homology group H n ( M , ∂ M ; Z ) ≅ Z {\displaystyle H_{n}(M,\partial M;\mathbf {Z} )\cong \mathbf {Z} } .

6. Submanifold - Wikipedia

   en.wikipedia.org/wiki/Submanifold

   This image of the open interval (with boundary points identified with the arrow marked ends) is an immersed submanifold. An immersed submanifold of a manifold is the image of an immersion map :; in general this image will not be a submanifold as a subset, and an immersion map need not even be injective (one-to-one) – it can have self-intersections.

7. Does Gatorade Hydrate You? - AOL

   www.aol.com/does-gatorade-hydrate-152500028.html

   Go to any sports game—whether it’s a high school game or a pro one—and you’re bound to see athletes on the sidelines drinking Gatorade. It’s likely a staple at your local gym too. A ...

8. G-structure on a manifold - Wikipedia

   en.wikipedia.org/wiki/G-structure_on_a_manifold

   Let P be a G-structure on a manifold M, and Q a G-structure on a manifold N. Then an isomorphism of the G-structures is a diffeomorphism f : M → N such that the pushforward of linear frames f * : FM → FN restricts to give a mapping of P into Q. (Note that it is sufficient that Q be contained within the image of f *.)

9. Parents Accused of Leaving Toddler in Closet Overnight with ...

   www.aol.com/parents-accused-leaving-toddler...

   Two Indiana parents are in custody after allegedly leaving their 2-year-old daughter in a closet overnight with a space heater turned all the way up.