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transversally

transˈversally, adv. [f. as prec. + -ly2.] In a transversal manner, transversely, athwart. (In quot. 1641, app. = obliquely.)1641 Wilkins Math. Magick i. xviii. (1707) 77 The several Proportions of Swiftness and Distance in an Arrow shot Vertically, or Horizontally, or Transversally. 1762 tr. Buschi... Oxford English Dictionary

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Transversality (mathematics)

Curves that are tangent to a surface at a point (for instance, curves lying on a surface) do not intersect the surface transversally. The space is the tangent space at to the adjoint orbit and so the affine space intersects the orbit of transversally. wikipedia.org

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Transverse Manifolds. Intersection is a Manifold again. Suppose we have two manifolds $A,B$ in $\mathbb{R}^n$. I heard that the intersection $A\cap B$ is again a manifold in $\mathbb{R}^n$ if $A$ and $B$ intersect tra...

This result is a generalization of the pre-image theorem (which you can find on page 21 of Guillemin & Pollack) which states > If $y$ is a regular value of $f: X \to Y$, then the preimage $f^{-1}(y)$ is a submanifold of $X$, with $\dim f^{-1}(y) = \dim X - \dim Y$. Given two manifolds that intersect...

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Acanthogeophilus dentifer

anterior tubercles on a forcipular coxosternum, smooth internal margin of forcipular tarsungulum, presence of basal tubercle on forcipular tarsungulum, and transversally wikipedia.org

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Transversal intersection. In my textbook, it says: "Consider two curves in the plane, one of which is the x-axis, the other being the graph of a function $f(x)$. The two curves intersect transversally at a point x if ...

The tangent space of the x-axis (at a given point) is again the x-axis (because it's horizontal, i.e. derivative is zero). Then the tangent space of the graph of $f(x)$ is whatever it is, but the key is that when it intersects the x-axis, the tangent space at that intersection point is _not_ horizon...

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Pseudomeritastis orphnoxantha

The forewings are light grey with brownish-ferruginous markings, edged and in part transversally strigulated (finely streaked) with dark brown. wikipedia.org

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transversal intersection Hartshorne Lemma V.1.2 p.358 I try to understand an argument on page 358 of Hartshorne, proof of Lemma V.1.2. Consider a surface X, irreducible curves $C_i $on X and a very ample divisor D. Ha...

First of all, he picks $D'$ to be some _nonsingular_ curve in the linear system $|D|$. The intersection $C_i\cap D'$ is ment scheme-theoretically. If the intersection is nonsingular, this means that the local rings of each point in that dimension zero scheme are regular ~~points~~. A regular local r...

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How to check when two manifolds $X_a$ and $Y$ intersect transversally The question is to find the values of $a$ for which $X_a = \\{(x,y,z)\in R^3 | x^2+y^2+z^2 = a\\}$ and $Y=\\{(x,y,z)\in R^3 | x+y^2+2z = 1\\}$ inte...

From calculus 3, the gradient of $f(x,y,z)=x^2+y^2+z^2$ evaluated at any $(x,y,z)$ satisfying $f(x,y,z)=a$ is a vector that is perpendicular to the surface $f(x,y,z)=a$. Similarly for $g(x,y,z)=x+y^2+2z$, its gradient, and the surface $g=1$. Note that $X_a$ and $Y$ are surfaces and, generically, the...

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Tangent sheaf of a (specific) nodal curve Given a nodal (= reduced, connected, projective, having only ordinary double points as singularities) curve $C$ consisting of 5 $\mathbb P^1$ labeled $C_0, D_1, D_2, D_3, D_4$...

I've found the solution in Hartshorne's book Deformation Theory on p. 183. I formulate the solution for an arbitrary nodal curve $C$ consisting of irreducible compontents $C_i \cong \mathbb P^1$. Let $S$ be the set of singular points in $C$. Locally $C$ around each node in $S$ the curve looks like $...

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Transversal Intersections - Some Examples Here is a part of an exercise from Guillemin-Pollack: > Which of the following linear subspaces intersect transversally? > > (d) $R^k\times \\{0\\}$ and $\\{0\\} \times R^l$...

The difference is crucial, when you write $$\mathbb{R}^k\times \\{0\\}\qquad\text{ in }\mathbb{R}^n$$ you really mean $$\\{(x_1,\ldots,x_k,0,\ldots, 0)\in \mathbb{R}^n\\}$$ whereas $$\\{0\\}\times \mathbb{R}^k\qquad\text{ in }\mathbb{R}^n$$ means $$\\{(0,\ldots, 0, x_{n-k+1},\ldots,x_n)\in \mathbb{R...

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Intersection of topological manifolds. A condition for the intersection of two smooth manifolds to be a smooth manifold is that they intersect transversally. Is this only an obstruction because of the smooth structure...

Actually, the intersection of two topological manifolds in a Euclidean space can be almost anything. Here's an example to show how bad things can get. Let $C$ be any closed subset of $\mathbb R^n$ whatsoever, and let $f\colon \mathbb R^n \to \mathbb R$ be the function $$ f(x) = \operatorname{dist}(x...

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Warp-like pattern in a closed curve Given a closed curve in 2D space that intersects itself (transversally, and there's no point in which three paths or more meet), is it possible to look at it as a Celtic knot so whe...

A (reasonably well-behaved) self-intersecting closed curve divides the plane into a number of components that can be coloured alternately black and white in a checkerboard manner. Declare a segment of the curve to go over or under a crossing depending on whether black is to the left or the right of ...

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Invertibility of 2 by 2 continuous maps This should be a simple question, but I've no idea about a strict proof. Consider a $2\times2$ map $(x,y)\mapsto(f(x,y),g(x,y))$, where $f:\mathbb{R}^2\to\mathbb{R}$ and $g:\ma...

Let $h(x) = \begin{cases} x^2 & x \ge 0 \\\ -x^2 & x < 0 \end{cases}$. Let $q(x) = \begin{cases} \sin \frac{1}{x} & x \ne 0 \\\ 0 & x = 0 \end{cases}$. Then I believe that $$ k(x) = (xq(x) + x) + h(x) $$ is strictly positive to the right of $0$, and strictly negative to the left of $0$, but is not i...

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Property of non-transversal manifold In Guillemin and Pollack they ask: > Suppose that X and Z do not intersect transversally in $Y$. May $X \cap Z$ still be a manifold? If so, must its codimensions be $\text{codim}\...

Take a cylinder of radius $1$, and a sphere of radius $1$. Put the sphere inside. Now put it outside, and touch the cylinder with it.

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Intersection of smooth manifolds I am wondering under which conditions the intersection of two smooth manifolds, say $X$ and $Y$, is a smooth manifold. I know that if they intersect transversally then the intersection...

Yes, it sure can. Consider $X=\\{y=0\\}$ and $Y=\\{y=x^2\\}\subset\Bbb R^2$.

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transversally

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transversally

Transversality (mathematics)

Transverse Manifolds. Intersection is a Manifold again. Suppose we have two manifolds $A,B$ in $\mathbb{R}^n$. I heard that the intersection $A\cap B$ is again a manifold in $\mathbb{R}^n$ if $A$ and $B$ intersect tra...

Acanthogeophilus dentifer

Transversal intersection. In my textbook, it says: "Consider two curves in the plane, one of which is the x-axis, the other being the graph of a function $f(x)$. The two curves intersect transversally at a point x if ...

Pseudomeritastis orphnoxantha

transversal intersection Hartshorne Lemma V.1.2 p.358 I try to understand an argument on page 358 of Hartshorne, proof of Lemma V.1.2. Consider a surface X, irreducible curves $C_i $on X and a very ample divisor D. Ha...

How to check when two manifolds $X_a$ and $Y$ intersect transversally The question is to find the values of $a$ for which $X_a = \\{(x,y,z)\in R^3 | x^2+y^2+z^2 = a\\}$ and $Y=\\{(x,y,z)\in R^3 | x+y^2+2z = 1\\}$ inte...

Tangent sheaf of a (specific) nodal curve Given a nodal (= reduced, connected, projective, having only ordinary double points as singularities) curve $C$ consisting of 5 $\mathbb P^1$ labeled $C_0, D_1, D_2, D_3, D_4$...

Transversal Intersections - Some Examples Here is a part of an exercise from Guillemin-Pollack: > Which of the following linear subspaces intersect transversally? > > (d) $R^k\times \\{0\\}$ and $\\{0\\} \times R^l$...

Intersection of topological manifolds. A condition for the intersection of two smooth manifolds to be a smooth manifold is that they intersect transversally. Is this only an obstruction because of the smooth structure...

Warp-like pattern in a closed curve Given a closed curve in 2D space that intersects itself (transversally, and there's no point in which three paths or more meet), is it possible to look at it as a Celtic knot so whe...

Invertibility of 2 by 2 continuous maps This should be a simple question, but I've no idea about a strict proof. Consider a $2\times2$ map $(x,y)\mapsto(f(x,y),g(x,y))$, where $f:\mathbb{R}^2\to\mathbb{R}$ and $g:\ma...

Property of non-transversal manifold In Guillemin and Pollack they ask: > Suppose that X and Z do not intersect transversally in $Y$. May $X \cap Z$ still be a manifold? If so, must its codimensions be $\text{codim}\...

Intersection of smooth manifolds I am wondering under which conditions the intersection of two smooth manifolds, say $X$ and $Y$, is a smooth manifold. I know that if they intersect transversally then the intersection...