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Web pages also have a z-axis: an imaginary line that runs from the surface of your screen, towards your face (or whatever else you like to have in front of the screen). Cheap Authentic Manolo Blahnik Satin Bow Flats w/ Tags Newest Asy2iV
values affect where positioned elements sit on that axis; positive values move them higher up the stack, and negative values move them lower down the stack. By default, positioned elements all have a z-index of auto , which is effectively 0.

To change the stacking order, try adding the following declaration to your p:nth-of-type(1) rule:

You should now see the finished example, with the green paragraph on top:

Note that z-index only accepts unitless index values; you can't specify that you want one element to be 23 pixels up the Z-axis — it doesn't work like that. Higher values will go above lower values, and it is up to you what values you use. Using 2 and 3 would give the same effect as 300 and 40000.

Note : You can see the example at this point live at ( see source code ).

Let's now look at fixed positioning. This works in exactly the same way as absolutepositioning, with one key difference: whereas absolute positioning fixes an element in place relative to the Saint Laurent Platform Espadrille Wedges w/ Tags Sale Comfortable tcZTmWmg
element or its nearest positioned ancestor, fixed positioning fixes an element in place relative to the browser viewport itself. This means that you can create useful UI items that are fixed in place, like persisting navigation menus.

Let's put together a simple example to show what we mean. First of all, delete the existing p:nth-of-type(1) and .positioned rules from your CSS.

Now, update the body rule to remove the position: relative; declaration and add a fixed height, like so:

Now we're going to give the element position: fixed; , and get it to sit at the top center of the viewport. Add the following rule to your CSS:

The top: 0; is required to make it stick to the top of the screen; we then give the heading the same width as the content column and use the faithful old margin: 0 auto; trick to center it. We then give it a white background and some padding, so the content won't be visible underneath it.

If you save and refresh now, you'll see a fun little effect whereby the heading stays fixed, and the content appears to scroll up and disappear underneath it. But we could improve this more — at the moment some of the content starts off underneath the heading. This is because the positioned heading no longer appears in the document flow, so the rest of the content moves up to the top. We need to move it all down a bit; we can do this by setting some top margin on the first paragraph. Add this now:

You should now see the finished example:

Although this product differs from the tensor product, the kernel of Alt is precisely the ideal I (again, assuming that K has characteristic 0), and there is a canonical isomorphism

Suppose that V has finite dimension n , and that a basis e 1 , ..., e n of V is given. then any alternating tensor t ∈ A r ( V ) ⊂ T r ( V ) can be written in Clearance Discount Salvatore Ferragamo Varina Patent Leather Flats Largest Supplier Online Original For Sale Best Sale Fashion Style Cheap Online zptLf
as

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in its indices.

The exterior product of two alternating tensors t and s of ranks r and p is given by

The components of this tensor are precisely the skew part of the components of the tensor product s t , denoted by square brackets on the indices:

The interior product may also be described in index notation as follows. Let t = t i 0 i 1 i r 1 {\displaystyle t=t^{i_{0}i_{1}\cdots i_{r-1}}} be an antisymmetric tensor of rank r . Then, for α V , i α t is an alternating tensor of rank r − 1 , given by

where n is the dimension of V .

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, the exterior product provides an abstract algebraic manner for describing the determinant and the minors of a matrix . For instance, it is well known that the magnitude of the determinant of a square matrix is equal to the volume of the parallelotope whose sides are the columns of the matrix. This suggests that the determinant can be defined in terms of the exterior product of the column vectors. Likewise, the k × k minors of a matrix can be defined by looking at the exterior products of column vectors chosen k at a time. These ideas can be extended not just to matrices but to Opening Ceremony Leather LowTop Sneakers Clearance Footlocker Finishline b0w7PG
as well: the magnitude of the determinant of a linear transformation is the factor by which it scales the volume of any given reference parallelotope. So the determinant of a linear transformation can be defined in terms of what the transformation does to the top exterior power. The action of a transformation on the lesser exterior powers gives a The Row Casablanca Thong Slide Sandals w/ Tags Sale Footlocker Finishline g3qLF
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