- 1 April 23
- 2 April 24
- 3 April 25
- 4 April 28
Coiled Phone Cord
I have tried this experiment repeatedly. If I pick up the handset of a landline phone cord, make a call, and then hang up, I invariably find the cord has become tangled by several loops. Even if I let the cord hang free, so that it lies untangled, after the next call, it will have multiple loops in it. At most I might change hands once or twice, but that does not in my mind account for it then having six or more loops in it when I hang up. Any material that addresses this? Thanks. μηδείς (talk) 02:43, 23 April 2017 (UTC)
-  gets multiple interesting hits.  has some terminology. Our article tangle (mathematics) isn't much help, though. 22.214.171.124 (talk) 08:05, 23 April 2017 (UTC)
I could have asked this question at any time, but it came to my attention when I saw recent edits to the polygons template.
Triangles can be classified as:
- Equilateral = 3 lines of symmetry, all sides are equal and all angles are 60 degrees.
- Isosceles = 1 line of symmetry, 2 sides are equal and one side is different; 2 angles are equal and one angle (the angle formed by the legs) is different.
- Scalene = no lines of symmetry, no congruent sides and no congruent angles.
To generalize these 3 classifications of triangles into n-gons for n > 3, we can do so as follows:
The generalization of the equilateral triangle is clearly the regular polygon. This is the square for n = 4.
But how about generalizing the isosceles triangle?? An isosceles polygon (a generalization of an isosceles triangle) would be a polygon that is not regular but that has at least one line of symmetry. The non-square rectangle, rhombus, kite, and isosceles trapezoid are all examples of isosceles quadrilaterals.
Likewise, a scalene polygon is a polygon with no lines of symmetry. I don't know whether to categorize a polygon with no lines of symmetry but rotational symmetry (a parallelogram that's not a rectangle or a rhombus) is correctly classified as scalene.
These categories can continue for n-gons for any value n. Do isosceles polygons in general have special properties?? How about scalene polygons in general?? Georgia guy (talk) 12:09, 23 April 2017 (UTC)
- Properties of scalene polygons in general would be properties of all polygons. These would be in the article Polygon. As for all polygons with a line of symmetry, you could look at Isosceles trapezoid or Rhombus, and see if any of the properties there generalize. Loraof (talk) 18:59, 23 April 2017 (UTC)
- The three disjoint sets into which triangles are classified (by some) do not easily extend to other polygons. Classification follows an inclusive structure for other n-gons. I would regard the concept of "scalene polygons" as original research and therefore inappropriate for Wikipedia (but you are welcome to prove me wrong) Dbfirs 11:13, 25 April 2017 (UTC)
- One approach would be to look at the symmetry group of each regular polygon, determine the subgroups, and then find geometric examples of the subgroups. For triangles, symmetry group of the regular (equilateral) triangle has 3 rotations (including 360 degrees) and 3 reflections. It basically has 2 subgroups: 1 reflection, which corresponds to the isoceles triangle, and the identity, which corresponds to the scalene triangle. Square symmetry can be divided up in more interesting ways, so you'll get objects (e.g. rectangles) that don't correspond to triangle subgroups, as well as objects that do (isosceles trapezoid). I expect after working out the details up to octagons or decagons, the general properties of any n-gon will become apparent.--Wikimedes (talk) 15:56, 27 April 2017 (UTC)
derivative of constrained function is sum of unconstrained partials?
Hello, while studying neural networks I came upon what was described by the lecturer as a "math trick" to solve a particular type of optimization problem using gradient descent. Basically, when optimizing a neural net that requires two parameters to be equal, you can replace the partial derivative for each constrained parameter by the sum of the partials with respect to each constrained parameter. So if you have , then the derivative of with respect to is the same as the sum of the partial derivatives of with respect to and . So far I have not been able to find a counter example, but I also do not know how to prove it. If anyone has pointers, clues, or proofs please help! Sorry if its obvious, and thanks! Brusegadi (talk) 01:09, 24 April 2017 (UTC)
- It's a particular case of the multivariable chain rule; it's easier to see what's going on from the more general form . --JBL (talk) 01:52, 24 April 2017 (UTC)
How about a slightly different situation involving total derivative of a function G of constrained independent variables xi with constant sum, for instance 1. Can the partial derivative with respect to xi exist by keeping the other xj constant even if only the sum of xi is constant, not every xi? Is this due to fact that dxi is around zero? Thanks.--126.96.36.199 (talk) 23:49, 24 April 2017 (UTC)
- The partial derivative is a feature of the function, not of the combination of function and constraint. So yes, the partial derivative can exist regardless of what context the function will be used in. You can take the total differential of G, which in the n=2 case is
- Then if you impose and hence hence you get
- Loraof (talk) 16:16, 25 April 2017 (UTC)
- But if n>2, not enough information has been provided—for the last step, we need to know how the offset of is distributed among etc. Loraof (talk) 16:21, 25 April 2017 (UTC)
- But can x2 be held constant when taking the partial derivative in respect to x1 as requested by the definition of the partial derivative? Similar question for the other independent variable x1 to be held constant when taking the partial derivative in respect to x2. Isn't the situation a bit stretched because strictly one independent variable cannot be made constant when the partial derivative is taken in respect to the other indepedent variable in such cases where only the sum of independent variables can be constant? Or it is about quasi-constancy of independent variables which is satisfactory in this situation?--188.8.131.52 (talk) 00:17, 26 April 2017 (UTC)
Face value numbers
When we count, for example, coins or banknotes by their face value rather than actual quantity expressed by natural numbers, are those still natural numbers or some other kind? Thanks.--184.108.40.206 (talk) 16:59, 24 April 2017 (UTC)
- Natural numbers don't include decimals, and most currencies have a sub-denomination (like cents for dollars), which makes it a decimal number. Yen is one that doesn't, so, for that case, I suppose you could use natural numbers for prices, in most cases (with exceptions for buying in quantity, where they might break the price down by tenths of a yen, etc.) StuRat (talk) 17:38, 24 April 2017 (UTC)
Broadcasting on a dynamic grid
We are given a nxn grid, where each node can maintain a single "active" edge with one of its 4 neighbors at each time step. The active edge of each node changes periodically clockwise or anticlockwise, in the sense that if, for example, at time t node v has an active edge with its upper neighbor, and it rotates clockwise, at times t+1, t+2 and t+3 it will maintain an active edge with its right, lower and left neighbors, respectively.
at time t=0 node (i,j) holds a message, and at each time step a node that has already received the message can transfer it through an active edge to one of its neighbors, if that neighbor also maintains this edge as active at this time step.
I am looking for sufficient conditions (initial active edges configurations and rotation direction of all the nodes) such that the message can be transmitted to all nodes in the grid.
- If you just want a sufficient condition: every node rotates clockwise. The node at position (i,j) starts with the right edge active if i+j is even, and with the left edge active if i+j is odd.--2406:E006:2C7:1:1521:7BAE:223B:27D9 (talk) 03:09, 28 April 2017 (UTC)
line of sight from building height
There's a simple formula for calculating your line of sight given a certain height H: .