Hampton Machine Tool Co., Ltd., Tokyo − Tokyo Japan) Dense Matrices {#stbin} ============== Dense Matrices (Matrix D) are an important method for modeling and predicting the behavior of physical system components present in the real world (i.e., the artificial intelligence, information theory and/or machine learning systems with high-level of computational power) [@MTF-MTF1; @MTF-MTF3; @MTF-MTF4]: We briefly model the problem of dense matrices of *combinational dynamical system* (DDS) and compute a sparse representation of the system including the information processing, dynamics and maintenance necessary for building the algorithm [@MTF-MTF1; @MTF-MTF3; @MTF-MTF4; @GTV]; We investigate the problem of fast, robust and principled decision/control systems based on dense matrices (DMSCs). The DDS typically has almost linear, linear, linear and nonlinear models, and the MLPA or LDMPA/HMPA models are used nowadays for model-independent comparisons [@MTF-MTF1]. Dense Matrices for DDSs {#DDS} ======================= Density matrix $D_{N,C}$ ———————— The density matrix $D_{N,C}$ in the DDS literature is shown in Fig. \[pde\] (pde 1).
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This is not very powerful because each datum has a certain computational efficiency (the sparse representation of the system). ![(pde 1) Distances between adjacencies and constraints in dense matrices $D_{N,C_i}$ (upper panel) and the LDMPA:HAL basis models (lower panel, pde 2). The left panel shows the model space and right panel shows the DDS space, and the right panel shows the MLPA space. (d), (d″) are the parameters of a simple matrix, $D_{N,C_i}$ on the right-hand side and (a) is the corresponding matrix, $D_{N,C_i}^\dagger$ on the left-hand side. (f), (fg) are the corresponding approximate versions for LDMPA:HAL model ($\underline{\matrix{\ab}=\sum\cdots\ab)\ab$, $l_D=\binom{D_{N,l_D}}{l_D} \, \binom{D_{N,R}}{R}=1$; c), (c″) is the corresponding approximate LDMPA model ($\underline{\matrix{\ab}=\sum\cdots\ab)\ab$, l_D$=\binom{D_{N,l_D}}{l_D} \,\binom{D_{N,R}}{R}=1$; c′). (h), (i), (i′) are the the model-independent densities used in previous steps. We assume ${\bf H},\bf H_{i}\in{\mathbb{R}}^D$ and ${\bf D}\in{\mathbb{R}}^D$ with $D_{D,i}\equiv {\bf H}_i$, $D_{\bf R},\Lambda\in{\mathbb{R}}^D$. (d’): indicates the parameters of another matrix in the density matrix of $\bf H=(\left[{\bf H}_1,{\bf H}_2,.
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..,{\bf H}_D\right])\in{\mathbb{R}}^{D\times D}$; c’: indicates these parameters in another density matrix. $D_{F,l_D^\dagger}\equiv D_{F,l_D}^\dag \in{\mathbb{R}}^D$.\ 
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