Geometric Point Of View Research Articles

Aligned and oblique dynamics in recurrent neural networks

The relation between neural activity and behaviorally relevant variables is at the heart of neuroscience research. When strong, this relation is termed a neural representation. There is increasing evidence, however, for partial dissociations between activity in an area and relevant external variables. While many explanations have been proposed, a theoretical framework for the relationship between external and internal variables is lacking. Here, we utilize recurrent neural networks (RNNs) to explore the question of when and how neural dynamics and the network’s output are related from a geometrical point of view. We find that training RNNs can lead to two dynamical regimes: dynamics can either be aligned with the directions that generate output variables, or oblique to them. We show that the choice of readout weight magnitude before training can serve as a control knob between the regimes, similar to recent findings in feedforward networks. These regimes are functionally distinct. Oblique networks are more heterogeneous and suppress noise in their output directions. They are furthermore more robust to perturbations along the output directions. Crucially, the oblique regime is specific to recurrent (but not feedforward) networks, arising from dynamical stability considerations. Finally, we show that tendencies toward the aligned or the oblique regime can be dissociated in neural recordings. Altogether, our results open a new perspective for interpreting neural activity by relating network dynamics and their output.

Aligned and oblique dynamics in recurrent neural networks.

The relation between neural activity and behaviorally relevant variables is at the heart of neuroscience research. When strong, this relation is termed a neural representation. There is increasing evidence, however, for partial dissociations between activity in an area and relevant external variables. While many explanations have been proposed, a theoretical framework for the relationship between external and internal variables is lacking. Here, we utilize recurrent neural networks (RNNs) to explore the question of when and how neural dynamics and the network's output are related from a geometrical point of view. We find that training RNNs can lead to two dynamical regimes: dynamics can either be aligned with the directions that generate output variables, or oblique to them. We show that the choice of readout weight magnitude before training can serve as a control knob between the regimes, similar to recent findings in feedforward networks. These regimes are functionally distinct. Oblique networks are more heterogeneous and suppress noise in their output directions. They are furthermore more robust to perturbations along the output directions. Crucially, the oblique regime is specific to recurrent (but not feedforward) networks, arising from dynamical stability considerations. Finally, we show that tendencies toward the aligned or the oblique regime can be dissociated in neural recordings. Altogether, our results open a new perspective for interpreting neural activity by relating network dynamics and their output.

A geometrical viewpoint on the benign overfitting property of the minimum $$\ell _2$$-norm interpolant estimator and its universality

A geometrical viewpoint on the benign overfitting property of the minimum $$\ell _2$$-norm interpolant estimator and its universality

On the non-flatness nature of noncommutative Minkowski spacetime and the singular behavior of probes

It is more than a century-old concept that the Minkowski spacetime is flat. From the pure geometric point of view, we explicitly address the issue of whether a noncommutative Minkowski spacetime is flat or not. In the framework of the twisted-diffeomorphism approach to noncommutative gravity with canonical type noncommutative (NC) coordinate structure, one important result that we get is that the NC Minkowski spacetime parametrized either with spherical polar coordinates or with parabolic coordinates has nontrivial NC corrections to Riemann curvature tensor, Ricci tensor and curvature scalars. Another crucial result is that the curvature scalars have singular behavior at certain points, and these singularities are not coordinate singularities. The nature of these singularities clearly points towards the idea of high-energetic probes turning into black holes. The absence of any such noncommutative corrections, and thus any such singularities, in the cases of Cartesian coordinates and cylindrical coordinates leads to the conclusion that high-energetic probes do not turn into black holes when the canonical NC structure is considered for these coordinates.

Parametrical Choice of the Optimized Fusion System for a FFHR

Fusion–fission hybrid reactors are concepts of subcritical reactors based on the coupling of fusion and fission devices. In this case, the fusion reactor would work as an external neutron supplier for the fission core of the machine. Such systems could, in principle, operate as multi-purpose machines, such as energy generators, breeders and waste burners. The large availability of fusion and fission technologies makes the choice of devices to couple quite chaotic. In fact, most of the concepts proposed in the literature are based on attempts without real optimization. The purpose of this paper is to propose a parameter which could provide practical information regarding the choice or the design of the fusion system of an FFHR. An engineering approach based on the estimation of the energy efficiency of FFHRs was used. An evaluation of the parameter and some of its possible practical applications are shown. Obtained results indicate that, from a geometrical point of view, compact machines would need lower Q-values to reach high neutron source performance.

Analysis of harmonic functions under lower bounds of N-weighted Ricci curvature with ε-range

The behavior of harmonic functions on Riemannian manifolds under lower bounds of the Ricci curvature has been studied from both analytic and geometric viewpoints. For example, some Liouville type theorems are obtained under lower bounds of the Ricci curvature. Recently, those results are generalized under lower bounds of the N-weighted Ricci curvature RicψN with N∈[n,∞]. In this paper, we present a Liouville type theorem for harmonic functions of sublinear growth and a gradient estimate of harmonic functions on weighted Riemannian manifolds under weaker lower bounds of RicψN with N<0. We also prove an Lp-Liouville theorem under lower bounds of RicψN with N∈[n,∞] in a different way from previous studies. Our results are obtained as a consequence of an argument under lower bounds of RicψN with ε-range, which is a unification of constant and variable curvature bounds. Among various methods considered for the analysis of harmonic functions, this paper focuses on methods using the Moser's iteration procedure.

Numerical methods for shape optimal design of fluid–structure interaction problems

We consider the method of mappings for performing shape optimization for unsteady fluid–structure interaction (FSI) problems. In this work, we focus on the numerical implementation. We model the optimization problem such that it takes several theoretical results into account, such as regularity requirements on the transformations and a differential geometrical point of view on the manifold of shapes. Moreover, we discretize the problem such that we can compute exact discrete gradients. This allows for the use of general purpose optimization solvers. We focus on problems derived from an FSI benchmark to validate our numerical implementation. The method is used to optimize parts of the outer boundary and the interface. The numerical simulations build on FEniCS, dolfin-adjoint and IPOPT. Moreover, as an additional theoretical result, we show that for a linear special case the adjoint attains the same structure as the forward problem but reverses the temporal flow of information.

Geometric interpretation of efficient weight vectors

Pairwise comparison matrices (PCMs) are frequently used in different multicriteria decision making problems. A weight vector is said to be efficient if no other weight vector is at least as good in estimating the elements of the PCM, and strictly better in at least one position. Understanding the efficient weight vectors is crucial to determine the appropriate weight calculation technique for a given problem. In this paper we study the set of efficient weight vectors for three and four dimensions (alternatives) from a geometric viewpoint, which is a complementary to the algebraic approach used in the literature. Besides providing well-interpretable demonstrations, we also draw attention to the particular role of weight vectors calculated from spanning trees. Weight vectors corresponding to line graphs are vertices of the (polyhedral, but usually nonconvex) set of efficient weight vectors, while weight vectors corresponding to other spanning trees are also on the boundary.

Remotely Controlled 3D-Engineered Scaffolds for Biomimetic In Vitro Investigations on Brain Cell Cocultures.

Most in vitro studies regarding new anticancer treatments are performed on 2D cultures, despite this approach imposes several limitations in recapitulating the real tumor behavior and in predicting the effects of therapy on both cancer and healthy tissues. Herein, advanced in vitro models based on scaffolds that support the 3D growth of glioma cells, further allowing the cocultures with healthy brain cells, are presented. These scaffolds, doped with superparamagnetic iron oxide nanoparticles and obtained through 2-photon polymerization, can be remotely manipulated thanks to an external magnet, thus obtaining biomimetic 3D organization recapitulating the brain cancer microenvironment. From a geometric point of view, the structure is functional to both cell culture on individual unit scaffolds and to tailored cocultures fostered by magnetic-driven unit assembly, also allowing for cell migration thanks to passages/fenestrations on adjacent structures. Leveraging magnetic dragging, for which a mathematical model is introduced, multiple cocultures are achieved, highlighting the high versatility and the user-friendly character of the proposed platform that can help overcome the current challenges in 3D cocultures handling, and open the way to the construction of increasingly biomimetic artificial systems.

Diffeomorphic structure of evolution equations

In this work the problem of the consistent treatment of multidimensional quadratic Hamiltonians is analyzed and developed from the geometric point of view. To this end, two approaches to the treatment for the problem are studied and developed: the pure matrix representation that involves Madelung type transforms (maps), and the evolution type based in a group manifold endowed with the symplectic groups and their coverings. Some of our goals is to introduce important geometric and group theoretical tools in the proposed approaches, such as Fermi normal coordinates in the first and a generalization of the method of nonlinear realizations in the second one. Several interesting results appear and some examples of application of these concepts in different physical scenarios are developed and presented such as the relationship with the Zeldovich approximation for the dynamic of large-scale cosmological structure, the classic case of Gertsner waves or the evolution problem with an inverted Hamiltonian of Caldirola-Kanai type.

The worst approximable rational numbers

We classify and enumerate all rational numbers with approximation constant at least 13 using hyperbolic geometry. Rational numbers correspond to geodesics in the modular torus with both ends in the cusp, and the approximation constant measures how far they stay out of the cusp neighborhood in between. Compared to the original approach, the geometric point of view eliminates the need to discuss the intricate symbolic dynamics of continued fraction representations, and it clarifies the distinction between the two types of worst approximable rationals: (1) There is a plane forest of Markov fractions whose denominators are Markov numbers. They correspond to simple geodesics in the modular torus with both ends in the cusp. (2) For each Markov fraction, there are two infinite sequences of companions, which correspond to non-simple geodesics with both ends in the cusp that do not intersect a pair of disjoint simple geodesics, one with both ends in the cusp and one closed.

Kummer–Witt–Jackson algebras

This paper is concerned with the construction of a small, but non-trivial, example of a polynomial identity algebra, which we call the Jackson algebra, that will be used in sequels to this paper to study non-commutative arithmetic geometry. In this paper this algebra is studied from a ring-theoretic and geometric viewpoint. Among other things it turns out that this algebra is a “non-commutative family” of central simple algebras and thus parameterizes Brauer classes over extensions of the base.

Possible influence of a cosmic repulsion on large-scale jets: Geometric viewpoint

Cosmic repulsion represented by a small positive value of the cosmological constant changes significantly properties of central gravitational fields at large distances, leading to existence of a static (or turnaround) radius where gravitational attraction of a center is just balanced by cosmic repulsion. Analyzing behavior of radial timelike geodesics in the Schwarzschild–de Sitter spacetime near its static radius we show that the particles with specific energy close to unity have tendency to slow down and cluster just below the static radius, forming clumps which, subsequently, start to expand uniformly due to cosmic repulsion. For central masses of (106–1011)M⊙ and current value of the cosmological constant 1.1×10−52m−2, this phenomenon takes place at distances of tens to hundreds of kiloparsecs from the center, being comparable with distances in which huge radio-lobes from some active galaxies were observed.

The Development of the Application of Centrifugal Force in the Geometric Design of Highways for Driving Comfort on the Pangalengan-Banjaran Roads

The Pangalengan-Banjaran road section has a winding geometric condition. The large number of bend geometries resulted in many accidents due to visibility, radius, and widening of the pavement at the corner, inappropriate road slopes caused many accidents to occur. The research method used in this research is the evaluative research method and quantitative method. The growth of suburban highway infrastructure in Indonesia has previously improved suddenly lengthways with economic growth. The geometrical strategy of the highway has previously remained planned by the typical requirements and needs of the purposes of the highway amenities constructed. The submission of physical science ideas in planning the geometrical stage as a meaning in smearing theoretic physical science to the identical-geometric design. A centrifugal strength in the physical science idea is a straightforward value in scheming geometric straight configurations of the road. Theoretical thoughtful of engineering term of scheming geometrical design viewpoint, has remained created on the method of centrifugal strength in the circumstance of effort on vehicles drive in the highway caisson disease. The operation has constantly remained built on variable quantity which remained strong-minded by thoughtful to spread on reason of rational usefully. These measures are lumped through dipping the result of bigger centrifugal strength. Furthermore, the submission is also able to decrease chance-disposed to areas due to the impact of size-centrifugal services on the geometrical plan of the highway.&#x0D; Keywords: centrifugal force, geometric design, highways, driving comfort, Pangalengan-Banjaran roads

3D geometrical modelling of the non-cylindrical Vélodrome Miocene fold in the southwestern Alps

Fold geometries and kinematics within foreland basins is a major issue for understanding the late evolution of thrust fronts. In the foreland of the southwestern Alps, the Vélodrome complex fold involves the whole Tertiary series which have recorded the evolution of the alpine front. From a geometrical and kinematic point of view, the Vélodrome is classically described as a recumbent Mio-Pliocene syncline with a strongly curved axis; interpreted either as a classic growth-fold, a post-deposit fold, or a result of coeval salt activity. The debate reveals a lack of consensus on the internal structure of the Vélodrome and the potential role of salt motion. By a detailed field analysis coupled with an implicit 3D geometrical modelling approach, we test the different hypotheses and provide a new 3D illustrated interpretation of the Vélodrome series. The Vélodrome consists in a complex non-cylindrical structure composed by several folds with different orientations together with inter- and intra-formations unconformities. Folding is partly syn-depositional and began earlier in the south, near Esclangon village, than in the north. In the north, deformation started with the marly and sandy molasse (m2, middle Burdigalian), attested by the transition from the conformable conglomeratic marine molasse (m1–2, Aquitanian and early Burdigalian) to the unconformable m2 to the Valensole formation. Deformation began earlier in the south, during the deposition of the m1–2, as shown by internal unconformities. We discuss the ingredients controlling the deformation in the Vélodrome, regional tectonics (mainly since middle Burdigalian) versus salt tectonics. This study further brings new constraints on the timing and pattern of deformation of the southwestern orogenic front of the Alps. It also highlights the power of the 3D geometrical modelling approach for testing different hypotheses and better understanding 3D complex structures.

High-order kinematics of uniform flexures

In the last decades, many efforts have been made to analyze and model small and large deflections of flexures, considering complex load cases and different solution techniques. However, few investigations focused on the kinematic aspects related to the deflection analysis of the flexible elements, and limited the study to the second-order kinematics. In this paper, an analytical formulation based on the instantaneous geometric invariants is developed to give deep kinematic insight, up to the fourth order, into the motion generated by the deflections. The problem is addressed from a geometrical point of view, defining the fundamental geometric entities that characterize the motion, that are inflection circle, cubic of stationary curvature and its derivative, Ball’s point, and Burmester’s points. Thanks to these special points on the plane, straight and circular paths can be approximated to the third and to the fourth order, respectively. The proposed formulation defines the geometric characteristics of flexures with different curvatures. An application regarding the definition of pseudo-rigid body models is discussed. Finite element simulations are performed to validate the results.

From the classical Frenet–Serret apparatus to the curvature and torsion of quantum-mechanical evolutions. Part II. Nonstationary Hamiltonians

In this paper, we present a geometric perspective on how to quantify the bending and the twisting of quantum curves traced by state vectors evolving under nonstationary Hamiltonians. Specifically, relying on the existing geometric viewpoint for stationary Hamiltonians, we discuss the generalization of our theoretical construct to time-dependent quantum-mechanical scenarios where both time-varying curvature and torsion coefficients play a key role. Specifically, we present a quantum version of the Frenet–Serret apparatus for a quantum trajectory in projective Hilbert space traced out by a parallel-transported pure quantum state evolving unitarily under a time-dependent Hamiltonian specifying the Schrödinger evolution equation. The time-varying curvature coefficient is specified by the magnitude squared of the covariant derivative of the tangent vector [Formula: see text] to the state vector [Formula: see text] and measures the bending of the quantum curve. The time-varying torsion coefficient, instead, is given by the magnitude squared of the projection of the covariant derivative of the tangent vector [Formula: see text] to the state vector [Formula: see text], orthogonal to [Formula: see text] and [Formula: see text] and, in addition, measures the twisting of the quantum curve. We find that the time-varying setting exhibits a richer structure from a statistical standpoint. For instance, unlike the time-independent configuration, we find that the notion of generalized variance enters nontrivially in the definition of the torsion of a curve traced out by a quantum state evolving under a nonstationary Hamiltonian. To physically illustrate the significance of our construct, we apply it to an exactly soluble time-dependent two-state Rabi problem specified by a sinusoidal oscillating time-dependent potential. In this context, we show that the analytical expressions for the curvature and torsion coefficients are completely described by only two real three-dimensional vectors, the Bloch vector that specifies the quantum system and the externally applied time-varying magnetic field. Although we show that the torsion is identically zero for an arbitrary time-dependent single-qubit Hamiltonian evolution, we study the temporal behavior of the curvature coefficient in different dynamical scenarios, including off-resonance and on-resonance regimes and, in addition, strong and weak driving configurations. While our formalism applies to pure quantum states in arbitrary dimensions, the analytic derivation of associated curvatures and orbit simulations can become quite involved as the dimension increases. Thus, finally we briefly comment on the possibility of applying our geometric formalism to higher-dimensional qudit systems that evolve unitarily under a general nonstationary Hamiltonian.

Towards 2D Borane Chemistry in Hexagonal Cyclic Compounds.

A comprehensive comparison between known benzene mono-substituted compounds R-Ph and the corresponding isoelectronic unknown R-cyclohexaborane(12) molecules is carried out from a geometric and electronic structure point of view, with R={H, BH2, CH3, NH2, OH, F ; AlH2, SiH3, PH2, SH, Cl ; NO2, OCH3}. We suggest new chemical names for the 2D borane compounds and analyze the geometric and electronic structure carbon vs. boron comparatives by means of HOMO-LUMO gaps, bonding schemes, electron density topological properties and predicted NMR chemical shifts. The predictions on the properties in planar hexagonal cyclic boranes may help in the design of synthesis procedures for these yet-unkown compounds.

Elastic energy trapping in Kresling origami

In this study, elastic energy trapping strategy is exploited to investigate the energy absorption capability of origami structures. The twofold novelty of this paper can be summarized as (i) comprehensive analysis of a single Kresling module from a geometrical point of view to illustrate its energy trapping capability in the pathway of compression; (ii) introducing translational and torsional ACKS (achiral coupled Kresling structures); demonstrating their potential to be utilized as a platform for designing energy absorption systems that can trap energy applied by compressional and torsional loads. The outcomes pave the way toward utilizing origami structures in reusable energy-absorbing systems.

Geometrical Interpretation and Design of Multilayer Perceptrons.

The multilayer perceptron (MLP) neural network is interpreted from the geometrical viewpoint in this work, that is, an MLP partition an input feature space into multiple nonoverlapping subspaces using a set of hyperplanes, where the great majority of samples in a subspace belongs to one object class. Based on this high-level idea, we propose a three-layer feedforward MLP (FF-MLP) architecture for its implementation. In the first layer, the input feature space is split into multiple subspaces by a set of partitioning hyperplanes and rectified linear unit (ReLU) activation, which is implemented by the classical two-class linear discriminant analysis (LDA). In the second layer, each neuron activates one of the subspaces formed by the partitioning hyperplanes with specially designed weights. In the third layer, all subspaces of the same class are connected to an output node that represents the object class. The proposed design determines all MLP parameters in a feedforward one-pass fashion analytically without backpropagation. Experiments are conducted to compare the performance of the traditional backpropagation-based MLP (BP-MLP) and the new FF-MLP. It is observed that the FF-MLP outperforms the BP-MLP in terms of design time, training time, and classification performance in several benchmarking datasets. Our source code is available at https://colab.research.google.com/drive/1Gz0L8AnT4ijrUchrhEXXsnaacrFdenn?usp = sharing.