## What is/are Reduced Space?

Reduced Space - Rather than estimating the full quadratic discriminant rule, we propose to perform simultaneous variable selection and linear dimension reduction on the original data, with the subsequent application of quadratic discriminant analysis on the reduced space.^{[1]}The method for full conformational analysis includes Cartesian coordinates sampling through quantum chemical molecular dynamics simulations, reduction of dimensionality by principal component analysis, determination of probability distributions in a reduced space of Cartesian coordinates and search for all the strict extrema points in probability distribution functions.

^{[2]}Adequate prediction results were achieved in a reduced space of characteristics.

^{[3]}The second method uses the popular IsoMap nonlinear dimensionality reduction algorithm, and assigns an outlier score by comparing the geodesic distances with the distances in the reduced space.

^{[4]}Then QPSO is applied to fetch the optimal features set over the reduced space that contains the best vectors only.

^{[5]}In this work, we design a custom deep autoencoder network to discover a coordinate transformation into a reduced space where the dynamics may be sparsely represented.

^{[6]}Afterwards, we derive efficiently computable and rigorous a posteriori error bounds for various quantities of interest which is a tool to generate the reduced space.

^{[7]}The proposed method is based on relaxations of algorithms using McCormick relaxations in a reduced space (Mitsos et al.

^{[8]}Our formulation is devised in a non-intrusive manner with an equation-free evolution of dynamics in a reduced space with the latter being obtained through a proper orthogonal decomposition.

^{[9]}The reduced space group symmetry of the crystal allows for several channels for spin-operator bilinears to couple to the electric field.

^{[10]}In order to retrieve accuracy, we first project every such coarse solution into the reduced space, and then further improve them via a rectification technique.

^{[11]}The reduced space harmonic balance method is demonstrated to find the maximum vibration responses of rotor systems.

^{[12]}Successful collaboration and interaction between component systems in (increasingly) reduced spaces is contingent to precise and accurate perception, interpretation and management of various spatial representations of self and teaming systems.

^{[13]}Conclusions A marked retardation of displacement was demonstrated for all IRL within the immature fovea of preterms with decreased areal and volume magnification and reduced space available for synaptic communication coupled to the degree of immaturity.

^{[14]}We found that the SCCA provides reasonable correlation scores in the reduced space, enables interpretability, and also provides the best classification performance among the three methods.

^{[15]}Biological samples are a complex and heterogeneous matrix where different macromolecules with different physicochemical parameters cohabit in reduced spaces.

^{[16]}The main drawback is that machine tools generally offer reduced spaces for the actuator location close to the cutting point and, hence, a reduced volume design is required for this external device.

^{[17]}Furthermore a set of hyper-surface functions associated with each POD basis function is constructed for representing the fluid dynamics over the reduced space.

^{[18]}In this research, a heuristic optimisation paradigm Reduced Space Searching Algorithm, combined with soft-computing-based modelling and data analysis techniques, is employed to solve the problem.

^{[19]}Existing methods rely on either the probabilistic distribution of training data or the discriminant power of the reduced space, disregarding the impact of repeatability and uncertainty in features.

^{[20]}In aquatic systems, increasing density of aquatic vegetation creates more habitat structural complexity for fishes, but only up to a certain threshold, beyond which fish abundance and diversity are restricted by reduced space.

^{[21]}In the second approach, we derive an inf-sup stable velocity-pressure reduced-order model by enriching the velocity reduced space with supremizers computed on a velocity reference space.

^{[22]}However, because these monitoring methods use Hotelling’s T 2 statistics in the reduced space, a normality assumption underlies the construction of these tools.

^{[23]}An enterprise that has implemented virtualization can consolidate multiple servers into fewer host servers and get the benefits of reduced space, power, and administrative requirements.

^{[24]}Besides, an explicit nonlinear mapping relationship from data to the reduced space can be obtained as the regression of a nonlinear function.

^{[25]}A Rayleigh-Ritz approach, based on a suitable number of modal shapes of the uniform shear-torsional beam models, allows to discretize the analysis and to evaluate the response in the generalized reduced space.

^{[26]}Results The characteristics of the dimensionality reduction methods are observed on the nine groups of three different tissues in the reduced space with dimensionality of two, three, and four.

^{[27]}By making use of symmetries and suitable first integrals of the system, we explicitly define a global 2-form for which the gauge transformed nonholonomic bracket gives rise to a new bracket on the reduced space codifying the nonholonomic dynamics and carrying an almost symplectic foliation (determined by the common level sets of the first integrals).

^{[28]}Biological samples are a complex and heterogeneous matrix where different macromolecules with different physicochemical parameters cohabit in reduced spaces.

^{[29]}We provide a data‐driven approach to generating nonlinear reduced spaces for deformation dynamics.

^{[30]}We present a direct manipulation technique that allows material scientists to interactively highlight relevant parameterized simulation instances located in dimensionally reduced spaces, enabling a user-defined understanding of a continuous parameter space.

^{[31]}This paper combines the nonnegativity and graph constraints to find the original geometrical information intrinsic to high dimensional data, keeping it in a dimensionality reduced space.

^{[32]}Specifically, we combine symplectic reduction by stages with a dual pair associated with the reduction by rotations to show that the -reduced space with non-zero angular impulse is a coadjoint orbit.

^{[33]}For the iterative diagonalization, pairs of Hermitian and anti-Hermitian trial vectors are used which facilitate, on the one hand, an efficient solution of the pair-structured generalized eigenvalue problem in the reduced space, and on the other hand, make the full multiconfigurational random phase approximation as efficient as the corresponding Tamm-Dancoff approximation.

^{[34]}This is especially applicable for urban areas and bottlenecks, since the compact design of the high-temperature superconductor cables (HTS cables) and the reduced space requirement leads to a significant reduction in the civil engineering expenditure and to a reduction in installation costs.

^{[35]}This model defines a Poisson flow endowed with invariants defining a S M 2 × S M 2 reduced space.

^{[36]}The adjoint of the tangent linear model is replaced by the adjoint of linear reduced model and the minimization problem is then solved in the reduced space at very low computational cost.

^{[37]}These methods transform the time series from the original space into a reduced space, where they can be managed more efficiently.

^{[38]}Our algorithm consists of two steps: in the first step we decompose the output label space into lower dimensions using simple matrix factorization method; subsequently we employ feature selection process in the decoupled reduced space.

^{[39]}We see that the presented algorithm provides a significant improvement in tightness and decrease in computational time, especially in the case studies using the reduced space formulation presented in (Bongartz and Mitsos in J Glob Optim 69:761–796, 2017).

^{[40]}Model reduction be used to approach the solution of the problem in a reduced space dimension.

^{[41]}As a regression model of the RB method for unsteady flows, the neural network approximates the map between the time/parameter value and the projection coefficients of the high-fidelity solution onto the reduced space.

^{[42]}In the current signal reconstruction, the introduced PLL topology has the advantage of reduced space harmonics.

^{[43]}The proposed configuration aids in capturing more data in a reduced space as well as tunable frequency operation.

^{[44]}The intermediate reduced spaces obtained by left or right reduction are shown to be coadjoint orbits of the special Euclidean group $SE(4)$.

^{[45]}We show that the reduced space for this Hamiltonian action for 3 vortices is generically a 2-sphere, and proceed to describe the reduced dynamics under simple hypotheses on the type of Hamiltonian interaction.

^{[46]}This relies upon applying Seiberg-Witten theory to the resolution of symplectic $4$-orbifolds occurring as the reduced spaces of $M$.

^{[47]}The d-MTMDs are also convenient to install owing to the reduced space requirements, being placed at various floors.

^{[48]}These reduced spaces are chosen based on the independent mode displaced harmonic oscillator model.

^{[49]}Case Report: Male patient, 13 years old, Angle Class I in the mixed dentition stage, with the tooth 21 impacted and reduced space for its eruption due to the migration of adjacent teeth.

^{[50]}

## Dimensionally Reduced Space

We present a direct manipulation technique that allows material scientists to interactively highlight relevant parameterized simulation instances located in dimensionally reduced spaces, enabling a user-defined understanding of a continuous parameter space.^{[1]}Overall, a dimensionally reduced space does not significantly affect the prediction performance of the SDP models.

^{[2]}

## reduced space harmonic

The reduced space harmonic balance method is demonstrated to find the maximum vibration responses of rotor systems.^{[1]}In the current signal reconstruction, the introduced PLL topology has the advantage of reduced space harmonics.

^{[2]}Therefore, this paper focuses on modeling and analysis of a novel star-delta winding configuration using unconventional odd slot numbers for fractional-slot wound machines towards maximum torque density and reduced space harmonic content.

^{[3]}

## reduced space requirement

This is especially applicable for urban areas and bottlenecks, since the compact design of the high-temperature superconductor cables (HTS cables) and the reduced space requirement leads to a significant reduction in the civil engineering expenditure and to a reduction in installation costs.^{[1]}The d-MTMDs are also convenient to install owing to the reduced space requirements, being placed at various floors.

^{[2]}Experimental results confirm that the PLS method is faster than Nauty, and has reduced space requirements.

^{[3]}