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The zigzag product was introduced by Reingold, Vadhan & Wigderson (2000). When the zig-zag product was first introduced, it was used for the explicit construction of constant degree expanders and extractors. Later on, the zig-zag product was used in computational complexity theory to prove that symmetric logspace and logspace are equal ...
In geometry, a polygonal chain [a] is a connected series of line segments. More formally, a polygonal chain is a curve specified by a sequence of points (,, …,) called its vertices. The curve itself consists of the line segments connecting the consecutive vertices.
A 2-metre carpenter's ruler with centimetre divisions Road sign warning for upcoming zigzag turn. A seismograph showing zigzag lines. The trace of a triangle wave or a sawtooth wave is a zigzag. Pinking shears are designed to cut cloth or paper with a zigzag edge, to lessen fraying. [2] In sewing, a zigzag stitch is a machine stitch in a zigzag ...
The bottom part of the diagram shows some contour lines with a straight line running through the location of the maximum value. The curve at the top represents the values along that straight line. A three-dimensional surface, whose contour graph is below. A two-dimensional contour graph of the three-dimensional surface in the above picture.
The incidence posets of path graphs form examples of fences. A linear extension of a fence is called an alternating permutation; André's problem of counting the number of different linear extensions has been studied since the 19th century. [1] The solutions to this counting problem, the so-called Euler zigzag numbers or up/down numbers, are:
Therefore, a zig-zag product can also be used to construct families of expander graphs. If G is a ( n , d , λ 1 ) -graph and H is an ( m , d , λ 2 ) -graph, then the zig-zag product G H is a ( nm , d 2 , φ ( λ 1 , λ 2 )) -graph where φ has the following properties.
This line has been called the amphoteric line, [2] the metal-nonmetal line, [3] the metalloid line, [4] [5] the semimetal line, [6] or the staircase. [2] [n 1] While it has also been called the Zintl border [8] or the Zintl line [9] [10] these terms instead refer to a vertical line sometimes drawn between groups 13 and 14.
In its simplest form, an arrow is a triangle, chevron, or concave kite, usually affixed to a line segment or rectangle, [1] and in more complex forms a representation of an actual arrow (e.g. U+27B5). The direction indicated by an arrow is the one along the length of the line or rectangle toward the single pointed end.