Term
| Write the sum and product formulas for the Riemann zeta function. |
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Definition
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| Information about what is coded in the Riemann zeta function? |
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Definition
| information regarding the distribution of primes |
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Term
| What was Gauss's attitude towards non-Euclidean geometry? |
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Definition
| He verified that it was consistent, but kept it under wraps so as not to have to deal with the philosophers. |
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Term
| What surfaces are Elliptic functions typically associated with? |
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Definition
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Term
| State the Gauss-Bonnet Theorem. |
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Definition
| Let k be Gaussian curvature and g be the genus, then integral kdS=2pi(2-2g) |
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Term
| What is integral kdS for the Torus? |
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Definition
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Term
| State Euler's formula for the genus of a surface via a map on the surface |
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Definition
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Term
| Who brought complex numbers out of the shadows and established them as the "real" deal? How? |
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Definition
| It was Gauss with the placing of points on the plane in the year 1790!! (CLUE reference) |
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Term
| Describe the Riemann surface of sqrt(2). |
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Definition
1850s
This is a concrete non-singular Riemann surface. The two horizontal axes represent the real and complex parts of z and the vertical axis represents the real parts of sqrt(z) |
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Term
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Definition
The curve along which a small object moves, under the influence of friction, when pulled on a horizontal plane by a piece of thread and a puller that moves at a right angle to the initial line between the object and the puller at an infinitesimal speed.
1670 |
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Term
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Definition
Constant-negative Gaussian curvature surface of revolution generated by a tractrix about its asymptote.
1868 |
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Term
| What do the pseudosphere and sphere have in common? |
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Definition
| They have the same surface area and volume. They both have constant curvature. Sphere is positive and pseudo is negative... |
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Term
| Angle sums of triangles in the different geometry. |
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Definition
Euclid = 180
Hyperbolic < 180
Spherical >180 |
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Term
| Parallel lines in geometries. |
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Definition
| Know them (I can't draw them...) |
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Term
| What were the effects of the invention of Lobachevski geometry in mathematics and philosophy? |
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Definition
1871
Killed philosopher's Absolute Truth (synthetic a priori)
Math now based on axioms. |
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Term
| Compare the effects of Hamilton's Quaternions on algebra and Lobachevski's geometry on geometry. |
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Definition
| Both broke the mould and substantially furthered their fields. |
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Term
| What is the most important property of elliptic functions? |
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Definition
| Elliptic functions are doubly periodic |
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Term
| How did elliptic integrals come about? |
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Definition
| They were originated by Euler from Fagano's study of the lemniscate. |
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