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Prove that coordinate is cyclic in Lagrangian then it is also cyclic in Hamiltonian?
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∙ 14y ago
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∙ 14y ago
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== ==
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Prove that a group of order 5 must be cyclic?
There's a theorem to the effect that every group of prime order is cyclic. Since 5 is prime, the assertion in the question follows from the said theorem.
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Sides bc and ad of a quadrilateral abcd are parallel a circle meets at the side ab at b and e the side CD at c and f prove that the quadrilateral aefd is cyclic?
AEFD is a cyclic quad. Let angle EBC be x. Then EFC is 180-x, and EFD is x. We also know that BAD is 180- x ( cointerior angles). Hence BAD and EFD are supplementary. Use same technique, let angle BCF be y. Then you will get FDA to be supplementary to FEA. therefore aefd is a cyclic quad.
How do we prove that a finite group G of order p prime is cyclic using Lagrange?
Lagrange theorem states that the order of any subgroup of a group G must divide order of the group G. If order p of the group G is prime the only divisors are 1 and p, therefore the only subgroups of G are {e} and G itself. Take any a not equal e. Then the set of all integer powers of a is by definition a cyclic subgroup of G, but the only subgroup of G with more then 1 element is G itself, therefore G is cyclic. QED.
Prove that a group of order three is abelian?
By LaGrange's Thm., the order of an element of a group must divide the order of the group. Since 3 is prime, up to isomorphism, the only group of order three is {1,x,x^2} where x^3=1. Note that this is a finite cyclic group. Since all cyclic groups are abelian, because they can be modeled by addition mod an integer, the group of order 3 is abelian.
Using coordinate geometry how can you prove that the midpoints of the sides of a rhombus determine a rectangle?
Step 1: Identify the coordinates of the vertices of the rhombus. Step 2: Calculate the coordinates of the midpoints of the sides. x-coordinate of midpoint = average of x-coordinates of the two end points, and similarly the y- coordinate. Step 3: Calculate lengths of sides of the quadrilateral formed (using Pythagoras) Step 4: Use step 3 results to show opposite sides are equal. Step 5: Calculate gradient (slope) of any two adjacent sides, if defined. Step 6: The two gradients multiply to -1 which shows that they are perpendicular. 4 and 6 prove that the quadrilateral is a rectangle. If a side of the quadrilateral is vertical, its gradient (step 5) is not defined, but then the adjacent side will be horizontal. And so the two sides are perpendicular.
What did Avogadro's number prove?
no prove....
