Position (geometry)

Summary

In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point P in space in relation to an arbitrary reference origin O. Usually denoted x, r, or s, it corresponds to the straight line segment from O to P. In other words, it is the displacement or translation that maps the origin to P: :\mathbf{r}=\overrightarrow{OP} The term position vector is used mostly in the fields of differential geometry, mechanics and occasionally vector calculus. Frequently this is used in two-dimensional or three-dimensional space, but can be easily generalized to Euclidean spaces and affine spaces of any dimension. Relative position Directed line segment The relative position of a point Q with respect to point P is the Euclidean vector resulting from the subtraction of the two absolute position vectors (each with respect to the origin): :\Delta \mathbf{r}=\mathbf{s} - \mathbf{r}=\ove

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https://en.wikipedia.org/wiki/Position_(geometry)

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This thesis investigates the growth of the natural cocycle introduced by Klingler for the Steinberg representation. When possible, we extend the framework of simple algebraic groups over a local field to arbitrary Euclidean buildings. In rank one, the growth of the cocycle is determined to be sublinear. In higher rank, the complexity of the problem leads us to study of the geometry of buildings of dimension two, where we describe in details the relative position of three points.

EPFL2016

Effect of inclined ribs on the Aero-Thermal performance of a trailing edge cavity with crossing jets

Tony Arts, Emanuele Facchinetti

The present contribution is devoted to the aero-thermal experimental study of a trapezoidal cross-section model, reproducing a trailing edge cooling cavity with one rib-roughened wall and slots along two opposite walls. Four rib geometries are compared to investigate the effect of the relative position of ribs and slots, as well as the tapering of the turbulators. The Reynolds number, defined at the inlet of the test section, is set at 67500 for all the experiments. Insight into the flow field for the different geometries is gained by means of surface streamline flow visualization. Highly resolved heat transfer distributions over the channel walls, including the rib surface, are obtained using steady state liquid crystal thermography. The results demonstrate how the overall thermal performance of the cooling scheme can be enhanced adjusting the rib position with respect to the slots. Moreover, the beneficial effect of tapering the ribs is shown.

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Loops in AdS: from the spectral representation to position space

Din Carmi

We compute a family of scalar loop diagrams in AdS. We use the spectral representation to derive various bulk vertex/propagator identities, and these identities enable to reduce certain loop bubble diagrams to lower loop diagrams, and often to tree- level exchange or contact diagrams. An important example is the computation of the finite coupling 4-point function of the large-N conformal O(N ) model on AdS(3). Remarkably, the re-summation of bubble diagrams is equal to a certain contact diagram: the D1,1,32,32zz function. Another example is a scalar with phi (4) or phi (3) coupling in AdS(3): we compute various 4-point (and higher point) loop bubble diagrams with alternating integer and half- integer scaling dimensions in terms of a finite sum of contact diagrams and tree-level exchange diagrams. The 4-point function with external scaling dimensions differences obeying (12) = 0 and (34) = 1 enjoys significant simplicity which enables us to compute in quite generality. For integer or half-integer scaling dimensions, we show that the M -loop bubble diagram can be written in terms of Lerch transcendent functions of the cross- ratios z and z. Finally, we compute 2-point bulk bubble diagrams with endpoints in the bulk, and the result can be written in terms of Lerch transcendent functions of the AdS chordal distance. We show that the similarity of the latter two computations is not a coincidence, but arises from a vertex identity between the bulk 2-point function and the double-discontinuity of the boundary 4-point function.

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