Dyadic vector

Like the FFT, the fast wavelet transform (FWT) is a fast, linear operation that operates on a data vector in which the length is an integer power of two (i.e., a dyadic vector), transforming it into a numerically different vector of the same length. Like the FFT, the FWT is invertible and in fact orthogonal that is, the inverse transform, when viewed as a matrix, is simply the transpose of the transform. Both the FFT and the discrete wavelet transform (DWT) can be regarded as a rotation in function... 

The dimensions for a descriptor are dehned in two groups — Cartesian distance and 2D property — where the minimum, maximum, and resolution of the vector in the hrst dimension of the descriptor can be dehned. The track bars are adapted automatically to changes for example, resolution is calculated and minimum-maximum dependencies are corrected. When the binary checkbox is clicked, only selections are possible that result in dyadic vector length (i.e., the dimension is a factor of 2"). This feature prevents the complicated adjustment of all settings to gain a binary vector that is necessary for transformations. 

Because of the generic nature of Fig. 1, no details of the interconnection between the VPU and the memory are shown. Still, these details are very important for the effective speed of a vector operation when the bandwidth between the memory and the VPU is too small, it is not possible to take full advantage of the VPU because it has to wait for operands and/or has to wait before it can store results. When the ratio of arithmetic-to-load/store operations is not high enough to compensate for such situations, severe performance losses may be incurred. The influence of the number of load/store paths for the dyadic vector operation c = a- -b (a, b, and c vectors) is depicted in Fig. 2. 

Equation (9.15) defines the scalar product (or inner product, dot product vjv7), and (9.16) defines the corresponding dyadic product (or outer product y7yl) of vectors v -and Vj, which are different kinds of mathematical object. 

Use of Dirac notation allows us to recognize at a glance that v) is a column vector, (v is the adjoint row vector, (v v) is the scalar product of these two vectors, and v)(v is a corresponding matrix dyadic, all referring to underlying object v. Further examples of Dirac notation are shown in Sidebar 9.2. 

This equation shows that the stress contribution tensor is essentially a dyadic product of the end-to-end vector r and the statistical force /, which is exerted by the chain on the considered end-point. The angular brackets indicate the averaging with the aid of the mentioned distribution function. Eq. (2.25) can be explained as follows Factor rt in the brackets gives the probability that the mentioned statistical force actually contributes to the stress. This factor gives the projection of the end-to-end vector of the chain on the normal of the considered sectional plane. If a unit area plane is considered, as is usual in stress-analysis, the said projection gives that part of the unit of volume, from which molecules possessing just this projection, actually contribute to the stress on the sectional plane. 

The hypothesis of small deformations means that c/.v. the change in die displacement vector when we go from P tu the neighboring point Q, is very small compared m dr. the position vector of Q relative to P. Consequently, the scalar components of the dyadic Vs arc al) very snlull compared lo unity. The geometrical meaning of ihe dyadic Vs is obtained by separating it into its symmetric part S = j(Vs + sV) and iis antisymmetric part R = - I x (V x si. where 1 is the unity dyadic. The antisymmetric part is interpreted as follows if at some point M the symmetric part vanishes, ilien we have for die neighborhood ul M the relation... 

These vector relations are not sufficient for the complete determination of the symmetric dyadic S. To insure that a solution of the above equations corresponds to a possible displacement veclor S. we must be able to integrate the relation... 

Finally, by expressing the strain dyadic in terms of the displacement vector, we obtain Navicr s form of the equilibrium equations ... 

A few words of explanation are not useless in order to understand this formalism. As a consequence of mixing, the medium is assumed to have a lamellar structure and n is a unit vector which remains normal to the material slices undergoing deformations in the velocity field, n n denotes a dyadic product (the dyadic product of vectors a and b is the tensor a.jbj) and 13 n n denotes the scalar product of the two tensors (the scalar product of tensors i = Tij and W = is the scalar quantity T W = E Z T j wji)- Assume that we start with two miscible fluids A J and B (having for instance different colors). Upon mixing, we obtain a lamellar marbled structure characterized by a striation thickness 6 and a specific "interfacial" area av. If the fluid is incompressible, avS = 1. Then,by application of (7-1)... 

This result requires some additional explanation. The expression (r2l — rr) will occur frequently in the following analysis, summed for both electrons and nuclei. The symbol 1 is the unit dyadic and is represented by a unit matrix. The product rr, which is not to be confused with either a scalar product or a vector product, is also a dyadic. Explicitly the above expression is evaluated in the following manner ... 

The rotational motion of the rigid set of mass points about any axis through its center of mass in the absence of exterior forces is known as the free rotation of the rigid body. The planar moment tensor for this motion, with the position vectors ra referred to an arbitrary basis system, can be compactly written as a dyadic (T denotes transposition) [8,32],... 

Suppose that the variables BJ are to be determined by a least-squares fit of the relations, Eq. 16, to the measured values T exp (vector Yexp). Assume that the measurements Yexp are unbiased ( (Yexp) = Ytrue where E() represents the mean or expectation value) and that the measurement errors and their correlations are described by the positive-definite nxn variance-covariance matrix 0Y which can be written as the dyadic P... 

Consider the centers of the identical spherical particles of radii a to be instantaneously located at the lattice points R . As such, the simplest geometric state exists, in which only one particle is contained within each unit cell. When the latter suspension is sheared, the three basic lattice vectors 1( (1 = 1,2, 3) (or, equivalently, the dyadic L) become functions of time t. Under a homogeneous deformation, the lattice composed of the sphere centers remains spatially periodic, although its instantaneous spatially periodic configuration necessarily changes with time. 

Here, I is the dyadic idemfactor and m = n — n moreover, V = d/dr is the local gradient operator defined only within a unit cell (r e t0). The material functions P, J, and n are the respective dyadic, triadic, and vector fields... 

Here, the vector constants V and 1 are to be determined so as to satisfy the respective pair of dyadic and triadic particle-surface boundary conditions... 

This principle as originally stated by Curie in 1908, is quantities whose tensorial characters differ by an odd number of ranks cannot interact (couple) in an isotropic medium. Consider a flow J, with tensorial rank m. The value of m is zero for a scalar, it is unity for a vector, and it is two for a dyadic. If a conjugate force A) also has a tensorial rank m, than the coefficient Ltj is a scalar, and is consistent with the isotropic character of the system. The coefficients Lij are determined by the isotropic medium they need not vanish, and hence the flow J, and the force A) can interact or couple. If a force A) has a tensorial rank different from m by an even integer k, then Ltj has a tensor at rank k. In this case, Lfj Xj is a tensor product. Since a tensor coefficient Lt] of even rank is also consistent with the isotropic character of the... 

This derivation is valid for a liquid consisting of rigid bodies but it can be generalised to flexible molecules. In this case one can form order tensors by creating dyadics of the eigenvectors of the inertia tensors or some other vector in the molecules. 

We define a tensor G as the external or dyadic product of two vectors a and b from a Euclidean space by the formula... 

The SLLOD equations of motion presented in Eqs. [123] are for the specific case of planar Couette flow. It is interesting to consider how one could write a version of Eqs. [123] for a general flow. One way to do this is introduce a general strain tensor, denoted by Vu. For the case of planar Couette flow, Vu = j iy in dyadic form, where 1 and j denote the unit vector in the x and y directions, respectively. The matrix representation is... 

In general, the notation AB will be used to denote a multiplication of two vectors to yield a dyadic. A review of the properties of dyadics is given in... 

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