Who Sits at the Bridal Table at the Reception?
i've got been to weddings the place in basic terms the instantaneous wedding ceremony occasion became into at an prolonged table. i've got additionally been to a pair the place the marriage occasion's companion became into invited to sit down down with him/her. at the instant i've got observed (my very own daughters did this) there is not any "actual table" yet what is referred to as a sweetheart table in basic terms for the bride and groom. the marriage occasion sits (with their spouses or significant others, babies) at tables like all different travelers. for my section i like this, because it saves room for the dance floor
1. Is it safe to give a dog table food?
I think it is okay as long as it's in moderation and do not do it at the dinner table, as it will obviously keep him begging when you are eating. Be sure to tell your dad to not feed him chocalate. That can kill the dog
2. Prove/Explain the centroids equations on the table?
It looks like the figure is showing the area under a curve, and breaking that area up into discrete areas by taking slices out of it. On the left, the slices are taken vertically. In the middle, they are taken horizontally, and on the right, they are taken radially. Let's start with the vertical slice. Centered around some point P(x,y) on the curve, we can take a vertical slice of the area, with an infinitesimally small width dx. How tall is that slice? It goes all the way up from the x axis to the curve, so it's y tall. The slice is an infinitesimally skinny rectangle, so we know its area is dA = y*dx. You can do that at every point on the curve, and sum up all those areas (in other words, integrate), to find the total area under the curve. But if you keep track of the centroid of each slice, you can figure out the centroid of the whole area. So where is the centroid of the vertical slice? Well, since it's a rectangle, the centroid should be in the middle of the base, and the middle of the height. In other words, the horizontal position of the centroid is equal to the average location of the left and right sides, and the vertical position of the centroid is equal to the average location of the top and bottom. Since the slice of width dx is centered horizontally on the point P(x,y), the middle of the base has position x. Since the height is y, and the bottom of the rectangle is located at 0, that means that Yel = (y0)/2 = y/2. Okay, now let's imagine that instead of dividing the area up into vertical slices, we take horizontal slices. Centered on a point P(x,y), we can take a horizontal slice of the area, with infinitessimally small height dy. This time, the left side of the slice is at the point P(x,y), and the right side is at the line x = a, so the total width of the slice is a - x. Now, we've got a infinitessimally short rectangle, so the area is dA = (a - x)*dy. Again, we could integrate the differential areas to find the total area, and we should come up with the same value as with the vertical slices. As for the centroid, again, the position of the centroid of a rectangle is in the middle of the base, and in the middle of the height. This time, our base starts on the left at x, and ends on the right at a, so the average is Xel = (x a)/2. Since the height is centered on the point P(x,y), the centroid is located at Yel = y. So, that takes care of the rectangular slices. Now let's look at the radial slices, which are similar, but a little bit more complicated. This time, instead of having an infinitessimally small width or height, our slice has an infinitessimally small angle, d, and it runs from the origin to the point P(r,). So, the slice is a sector of a circle of radius r. What is its area, dA? You know the area of a circle is r^2, and the internal angle of the circle is 2. The area of a sector divided by the area of the full circle is equal to the sector angle divided by the circle angle, so dA / r^2 = d / 2 dA = r^2 * d / 2 dA = r^2 * d / 2 Now, the centroid of that sector is also a little tricky. Since the sector is symmetric about the line between the origin and the point P(r,), we know that the centroid lies on the line of symmetry. So it has an angular coordinate of . Where is the centroid located radially? For that, I like to consider the sector as a triangle. Instead of this: ....../ ..../.... ../........ /-__...__- ......-- it has a flat base, like this: ....../ ..../.... ../........ /_______ When the angle gets very small, the curved edge gets very close to flat. Since d is infinitesimal, this is a pretty good approximation. Then, you might remember that the centroid of a triangle is located at 1/3 of its height measured perpendicularly from any side. In this case, our triangle (i.e. our sector) has a height r, so the centroid is located 2/3 * r along the radial line (1/3 * r from the base). .......__________ ....../ ............. ..../.... ....2r/3....r ../...... .__.....| /_______ _____ Now, all that's left is to convert the centroid from polar to cartesian coordinates. We know it's at (2r/3, ). Using trigonometry, that's equivalent to Xel = 2r/3 * cos , and Yel = 2r/3 * sin . I think that's everything. Message me if you need anything else. I hope that helps you out!
3. Constructing the character table of a group
An easy (read: that can be implemented in a computer) way to compute the character table of a finite group is Burnside's algorithm, which translates the problem of computing the character table to the problem of finding eigenvalues/eigenvectors of matrices
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