Tangent Hyperplane Research Articles

DISCO-DJ I: a differentiable Einstein-Boltzmann solver for cosmology

We present the Einstein-Boltzmann module of the Disco-Dj (DIfferentiable Simulations for COsmology — Done with J ax) software package. This module implements a fully differentiable solver for the linearised cosmological Einstein-Boltzmann equations in the Jax framework, and allows computing Jacobian matrices of all solver output with respect to all input parameters using automatic differentiation. This implies that along with the solution for a given set of parameters, the tangent hyperplane in parameter space is known as well, which is a key ingredient for cosmological inference and forecasting problems as well as for many other applications. We discuss our implementation and demonstrate that our solver agrees at the per-mille level with the existing non-differentiable solvers Camb and Class, including massive neutrinos and a dark energy fluid with parameterised equation of state. We illustrate the dependence of various summary statistics in large-scale structure cosmology on model parameters using the differentiable solver, and finally demonstrate how it can be easily used for Fisher forecasting, with a forecast for Euclid as an example. Since the implementation is significantly shorter and more modular than existing solvers, we believe it will be more straightforward to extend our solver to include additional physics, such as additional dark energy and dark matter models, modified gravity, or other non-standard physics in the future.

Оснащенное гиперполосное распределение аффинного пространства

A rigged hyperstrip distribution is a special class of hyperstrips. The study of hyperstrips and their generalizations in spaces with various fun­damental groups is of great interest due to numerous applications in mathematics and physics. A special place is occupied by regular hyper­strips, for which the characteristic planes of families of principal tangent hyperplanes do not contain directions tangent to the base surface of the hyperstrip. In this work, we use E. Cartan’s method of external differen­tial forms and the group-theoretic method of G. F. Laptev. In affine space, a hyperstrip distribution is considered, which at each point of the base surface is equipped with a tangent plane and a conjugate tangent line. The specification of the studied hyperstrip distribution in an affine space with respect to a 1st order reference and an existence theorem are given. The fields of affine normals of the 1st kind for Blaschke and Transon are constructed and the conditions for their coincidence are found. The definition of normal affine connection and normal centroaffine connection on the studied framed hyperstrip distribution is given.

On concave functions over lotteries

This note discusses functions over lotteries that are concave and continuous, but are not necessarily superdifferentiable. Earlier work claims that concave continuous utility for lotteries that satisfy best-outcome independence can be written as the minimum of affine functions. We give a counter-example that cannot be written as the minimum of affine functions, because there is no tangent hyperplane that dominates the functions at the boundary. We then review the fact that concavity and upper semi-continuity are equivalent to a representation as the infimum of affine functions, and show that these assumptions imply continuity for functions on finite-dimensional lotteries. Therefore, in finite-dimensional simplices, concavity and continuity are equivalent to the “infimum” representation. The “minimum” representation is equivalent to the existence of local utilities (supporting affine functions) at every lottery, a property that is equivalent to superdifferentiability.

Transient Stability Assessment of Renewable Dominated Power Systems by the Method of Hyperplanes

With continuous increase of penetration of renewable energies and power electronic equipment, power grid stability incidents are emerging frequently worldwide. In this paper, the transient stability assessment of renewable dominated power systems with multiple PLL-based voltage source converters (VSCs) is studied by calculating critical clearing time (CCT). A tangent hyperplane method is developed, which approximates the boundary of basin of attraction near unstable equilibrium point by linear system theory. The detailed algorithm is presented, accompanying with four typical case studies, including a single machine infinite system, two parallel VSCs, modified IEEE 9- bus, and modified 39- bus systems. Broad EMT simulations are performed to verify the accuracy and computational effectiveness of the algorithm. In all these tests, we find that the hyperplane method can give accurate estimations of CCTs. Hence it is expected to be an efficient method. In addition, we find that generally the CCT in renewable dominated power systems is around tens of milliseconds, which is much shorter than that in traditional power systems around hundreds of milliseconds. Therefore, it might be imperative to adopt faster controls and protections for renewable dominated power systems.

Гиперполосное распределение, оснащенное полем сопряженных плоскостей

In this paper, we study a special class of hyperbands, i. e., a framed hyperband distribution. The study of hyperbands and their generalizations in spaces with different fundamental groups is of great interest in connec­tion with numerous applications in mathematics and physics. A special place is occupied by regular hyperstrips, for which the characteristic planes of families of principal tangent hyperplanes do not contain direc­tions tangent to the basal surface of the hyperstrip. In this paper, we use the method of external differential forms of E. Cartan and the group-theoretic method of G. F. Laptev. We consider a regular hyperband distribution of an affine space equipped with a field of conjugate planes with respect to an asymptotic bundle of tensors of the basic surface. The definition of the studied hy­perband distribution in the affine space with respect to the 1st order frame is given and the existence theorem is proved. A sequence of fundamental geometric objects of the 1st and 2nd order of a hyperband distribution equip­ped with a field of conjugate planes is constructed. Fields of qua­sitensors are constructed that define the fields of normals of the first kind of the distribution of the characteristics of the hyperband distribution. In a differential neighborhood of the 2nd order, the fields of Transon normals of the 1st and 2nd kind are constructed. The conditions for the coincidence of the Transon normal and the Blaschke normal are found.

A Convex Hull-Based Machine Learning Algorithm for Multipartite Entanglement Classification

Quantum entanglement becomes more complicated and capricious when more than two parties are involved. There have been methods for classifying some inequivalent multipartite entanglements, such as GHZ states and W states. In this paper, based on the fact that the set of all W states is convex, we approximate the convex hull by some critical points from the inside and propose a method of classification via the tangent hyperplane. To accelerate the calculation, we bring ensemble learning of machine learning into the algorithm, thus improving the accuracy of the classification.

C1–regularity for degenerate diffusion equations

We prove that any solution of a degenerate elliptic PDE is of class C1 provided the inverse of the equation's degeneracy law satisfies an integrability criterium, viz. σ−1∈L1(1λdλ). The proof is based upon the construction of a sequence of converging tangent hyperplanes that approximate u(x), near x0, by an error of order o(|x−x0|). Explicit control of such hyperplanes is carried over through the construction, yielding universal estimates upon the C1–regularity of solutions. Among the main new ingredients required in the proof, we develop an alternative recursive algorithm for renormalization of approximating solutions. This new method is based on a technique tailored to prevent the sequence of degeneracy laws constructed through the process from being, itself, degenerate.

ESTIMATION OF THE TOLERANCE AREA FOR CORRECTION PARAMETERS IN INDUCTION ACCELERATION SYSTEMS

he issues of formalization and numerical solution of the problems of calculating tolerances for the parameters of corrective elements for a linear induction acceleration system, which are directly related to the performance of a real object, are considered. The key moments of the dynamics of the transverse motion of particles for a specific structure of a linear induction system of acceleration are studied for given values of energy for each of the periods of the resonator. The presence of parasitic electric and magnetic fields, which arise as a result of particle displacement relative to the accelerator axis and change the transverse components of the pulses, is taken into account. The original difference model of the induction system has been transformed into a linear form. To formulate the problem statements for calculating tolerances, the scatter vectors of phase coordinates and tolerances for the correction parameters are introduced. In order to apply the methods of practical stability, the set of tolerances for the parameters of the corrective elements is given in the form of an ellipsoid. Provided that the initial displacements of the transverse coordinates relative to the axis of the accelerator are known constant values, the structured tolerance region was estimated under known linear restrictions on the spread of the phase coordinate vectors. Due to the developed algorithms of practical stability, the original problem of calculating tolerances is reduced to the problem of finding the maximum of a linear form on an ellipsoid. Some important types of restrictions on deviations of phase coordinates concerning the estimation of tolerances on the parameters of the first correction element and the number of particles are investigated. For the case of non-linear dynamic constraints on the spread of the phase coordinate vector, it is proposed to approximate the convex closed set by tangent hyperplanes. From the standpoint of practical stability, the problem of estimating tolerances for the case of given restrictions on the spread of the quality criterion is considered. With the help of practical directional stability algorithms, it is proposed to estimate the maximum tolerance ranges for parameters in terms of volume in the presence of dynamic restrictions on the spread of phase coordinates or a quality criterion.

Розрахунок оптимальних параметрів коректувальних елементів в індукційних системах прискорення

The formulations of a number of optimization problems for a linear induction acceleration system with respect to the adjustment parameters are considered. The dynamics of the transverse motion of electrons in the horizontal plane is investigated in the presence of given energy values for each resonator period: the particles at the initial moment of time are somewhat displaced relative to the accelerator axis (we neglect the displacements of the ends of the solenoids and the centers of the accelerating gaps relative to the accelerator axis). A connection is established between the initial and final coordinates and the components of the momentum. The presence of parasitic electric and magnetic fields arising as a result of the displacement of particles relative to the axis of the accelerator, which change the transverse components of the pulses, is taken into account. For the mathematical formulation of problems, in order to apply algorithms of practical stability, the original difference model of the induction system was converted to a linear form. By introducing into consideration the vector of parameters, the scatter of phase coordinates, and tolerances on the parameters, the problem of calculating the tolerances for given linear constraints on the scatter of phase coordinates for the corresponding inhomogeneous system is formulated. For the case of nonlinear dynamic constraints on the spread of the vector of phase coordinates, it is proposed to approximate a convex closed set by tangent hyperplanes. Numerical estimation of the range of tolerances for the parameters of correcting elements is reduced to the problems of practical stability of discrete parametric systems. In this case, the region of the initial conditions on the state vector, the tolerances on the parameters, are given structurally in the form of an ellipsoid, which makes it possible to numerically solve the original problem as an extremal one. From the standpoint of practical stability in the corresponding space of functions, the problem of assessing the range of tolerances for the parameters of correcting elements in the presence of specified restrictions on the spread of the quality criterion is considered. Attention is focused on an important class of problems of numerical modelling of a linear induction acceleration system − problems of practical stability. Numerical estimation of the region of initial displacements of the transverse coordinates of the linear induction acceleration system in the given structures in the presence of linear constraints on the vector of phase coordinates in dynamics is carried out. Key words: modeling, induction system of acceleration, solenoid, parameters, elements of correction, optimization, stability

Locally $C^{1,1}$ convex extensions of $1$-jets

Let E be an arbitrary subset of \mathbb{R}^n , and let f\colon E\to\mathbb{R} , G\colon E\to\mathbb{R}^n be given functions. We provide necessary and sufficient conditions for the existence of a convex function F\in C^{1,1}_{\textrm{loc}}(\mathbb{R}^n) such that F=f and \nabla F=G on E . We give a useful explicit formula for such an extension F , and a variant of our main result for the class C^{1, \omega}_{\textrm{loc}} , where \omega is a modulus of continuity. We also present two applications of these results, concerning how to find C^{1,1}_{\textrm{loc}} convex hypersurfaces with prescribed tangent hyperplanes on a given subset of \mathbb{R}^n , and some explicit formulas for (not necessarily convex) C^{1,1}_{\textrm{loc}} extensions of 1 -jets.

Locally polynomially integrable surfaces and finite stationary phase expansions

Let $M$ be a strictly convex smooth connected hypersurface in $\mathbb R^n$ and $\widehat{M}$ its convex hull. We say that $M$ is locally polynomially integrable if the $(n-1)-$ dimensional volumes of the sections of $\widehat M$ by hyperplanes, sufficiently close to the tangent hyperplanes to $M,$ depend polynomially on the distance of the hyperplanes to the origin. It is conjectured that only quadrics in odd dimensional spaces possess such a property. The main result of this article partially confirms the conjecture. The study of integrable domains and surfaces is motivated by a conjecture of V.I. Arnold about algebraically integrable domains. The result and the proof are related to study oscillating integrals for which the asymptotic stationary phase expansions consist of finite number of terms.

Differentiable points of Sierpinski-like sponges

For the Sierpinski-like sponge in Rn, we obtain the Hausdorff dimension of the set of differentiable points such that there exists an (n−1)-rectifiable subsurface which has a tangent hyperplane at the point, using the techniques such as multi dynamical systems, geometric types of slices and restricted Frobenius problem. Furthermore, we can use the dimension of the set of differentiable points, as a bilipschitz invariant, to classify Sierpinski-like sponges in terms of bilipschitz equivalence.

Billiard characterization of spheres

In this paper we study the higher dimensional convex billiards satisfying the so-called Gutkin property. A convex hypersurface S satisfies this property if any chord [p, q] which forms angle $$\delta $$ with the tangent hyperplane at p has the same angle $$\delta $$ with the tangent hyperplane at q. Our main result is that the only convex hypersurface with this property in $$\mathbf {R}^d, d\ge 3$$ is a round sphere. This extends previous results on Gutkin billiards obtained in Bialy (Nonlinearity 31(5):2281–2293, 2018).

Cones generated by random points on half-spheres and convex hulls of Poisson point processes

Let $$U_1,U_2,\ldots $$ be random points sampled uniformly and independently from the d-dimensional upper half-sphere. We show that, as $$n\rightarrow \infty $$ , the f-vector of the $$(d+1)$$ -dimensional convex cone $$C_n$$ generated by $$U_1,\ldots ,U_n$$ weakly converges to a certain limiting random vector, without any normalization. We also show convergence of all moments of the f-vector of $$C_n$$ and identify the limiting constants for the expectations. We prove that the expected Grassmann angles of $$C_n$$ can be expressed through the expected f-vector. This yields convergence of expected Grassmann angles and conic intrinsic volumes and answers thereby a question of Barany et al. (Random Struct Algorithms 50(1):3–22, 2017. https://doi.org/10.1002/rsa.20644 ). Our approach is based on the observation that the random cone $$C_n$$ weakly converges, after a suitable rescaling, to a random cone whose intersection with the tangent hyperplane of the half-sphere at its north pole is the convex hull of the Poisson point process with power-law intensity function proportional to $$\Vert x\Vert ^{-(d+\gamma )}$$ , where $$\gamma =1$$ . We compute the expected number of facets, the expected intrinsic volumes and the expected T-functional of this random convex hull for arbitrary $$\gamma >0$$ .

Reliability sensitivity analysis using axis orthogonal importance Latin hypercube sampling method

In this article, a combined use of Latin hypercube sampling and axis orthogonal importance sampling, as an efficient and applicable tool for reliability analysis with limited number of samples, is explored for sensitivity estimation of the failure probability with respect to the distribution parameters of basic random variables, which is equivalently solved by reliability sensitivity analysis of a series of hyperplanes through each sampling point parallel to the tangent hyperplane of limit state surface around the design point. The analytical expressions of these hyperplanes are given, and the formulas for reliability sensitivity estimators and variances with the samples are derived according to the first-order reliability theory and difference method when non-normal random variables are involved and not involved, respectively. A procedure is established for the reliability sensitivity analysis with two versions: (1) axis orthogonal Latin hypercube importance sampling and (2) axis orthogonal quasi-random importance sampling with the Halton sequence. Four numerical examples are presented. The results are discussed and demonstrate that the proposed procedure is more efficient than the one based on the Latin hypercube sampling and the direct Monte Carlo technique with an acceptable accuracy in sensitivity estimation of the failure probability.

Bipartite Sets of Spheres and Casey-Type Theorems

A bipartite set of spheres in $${\mathbb {R}}^n$$ is a set of colored spheres, where two colors are used, no sphere is contained in the closed ball bounded by another sphere in the set, and spheres of different colors are disjoint. For any two spheres in a bipartite set, the common-tangent distance between them is defined as the distance between two tangent points in a common tangent hyperplane to them, where an external common tangent hyperplane is taken if the two spheres are of the same color; otherwise, a common internal tangent hyperplane is taken. By this common-tangent distance, a bipartite set becomes a semi-metric space. It turns out that bipartite sets of spheres form an interesting family of semi-metric spaces. Casey’s theorem (a generalization of Ptolemy’s theorem) gives a condition for a bipartite set of four circles in $${\mathbb {R}}^2$$ to have a circle that is suitably tangent to all circles in the bipartite set. Ptolemy’s theorem is generalized to the n-dimensional situation via Cayley–Menger determinants. Among other results, we present a Casey-type theorem for a bipartite set of $$n+2$$ spheres in n dimensions as a generalization of the n-dimensional version of Ptolemy’s theorem, and we extend this further to a bipartite set with an arbitrary number of spheres in $${\mathbb {R}}^n$$ .

Characteristic classes of affine varieties and Plücker formulas for affine morphisms

An enumerative problem on a variety V is usually solved by reduction to intersection theory in the cohomology of a compactification of V . However, if the problem is invariant under a “nice” group action on V (so that V is spherical), then many authors suggested a better home for intersection theory: the direct limit of the cohomology rings of all equivariant compactifications of V . We call this limit the affine cohomology of V and construct affine characteristic classes of subvarieties of a complex torus, taking values in the affine cohomology of the torus. This allows us to make the first steps in computing affine Thom polynomials . Classical Thom polynomials count how many fibers of a generic proper map of a smooth variety have a prescribed collection of singularities and our affine version addresses the same question for generic polynomial maps of affine algebraic varieites. This notion is also motivated by developing an intersection-theoretic approach to tropical correspondence theorems: they can be reduced to the computation of affine Thom polynomials, because the fundamental class of a variety in the affine cohomology is encoded by the tropical fan of this variety. The first concrete answer that we obtain is the affine version of what were, historically speaking, the first three Thom polynomials – the Plücker formulas for the degree and the number of cusps and nodes of a projectively dual curve. This, in particular, characterizes toric varieties whose projective dual is a hypersurface, computes the tropical fan of the variety of double tangent hyperplanes to a toric variety, and describes the Newton polytope of the hypersurface of non-Morse polynomials of a given degree. We also make a conjecture on the general form of affine Thom polynomials; a key ingredient is the n -ary fan, generalizing the secondary polytope.

Selection of corporate spare parts inventory for Brazilian refineries: a smoothed data envelopment analysis frontier function using calculus of variations

This paper intends to give a small contribution to data envelopment analysis (DEA) development, among the huge research carried out in the field, by developing a model for materials selection in a corporately managed inventory that includes high-value MRO spare parts. The goal of the model is to reduce the asset value in the stocks without compromising the reliability of the operating units for Brazilian refineries. We implemented a standard DEA model and defined a smoothed DEA frontier using the efficient DMU from the model. This is achieved through the calculus of variations, generating a non-linear polynomial frontier that is continuously differentiable and contains all DEA efficient DMUs, providing a unique tangent hyperplane for each target on the efficient frontier. The main reason for smoothing the DEA frontier is to create an agile assessment of the radial efficiency of new materials without running the entire DEA mathematical programme.

Constrained Posterior Cramér-Rao Bound for Discrete-Time Systems

This article presents a Cramér-Rao lower bound for the discrete-time filtering problem under linear state constraints. A simple recursive algorithm is presented that extends the computation of the Cramér-Rao lower bound found in previous literature by one additional step in which the full-rank Fisher Information matrix is projected onto the tangent hyperplane of the constraint set. This makes it possible to compute the constrained Cramér-Rao lower bound for the discrete-time filtering problem without reparametrization of the original problem to remove redundancies in the state vector, which improves insights into the problem. It is shown that in case of a positive-definite Fisher Information Matrix the presented constrained Cramér-Rao bound is lower than the unconstrained Cramér-Rao bound. The bound is evaluated on an example.

A Tradeoff-Based Interactive Multi-Objective Optimization Method Driven by Evolutionary Algorithms

Multi-objective optimization problems involve two or more conflicting objectives, and they have a set of Pareto optimal solutions instead of a single optimal solution. In order to support the decision maker (DM) to find his/her most preferred solution, we propose an interactive multi-objective optimization method based on the DM’s preferences in the form of indifference tradeoffs. The method combines evolutionary algorithms with the gradient-based interactive step tradeoff (GRIST) method. An evolutionary algorithm is used to generate an approximate Pareto optimal solution at each iteration. The DM is asked to provide indifference tradeoffs whose projection onto the tangent hyperplane of the Pareto front provides a tradeoff direction. An approach for approximating the normal vector of the tangent hyperplane is proposed which is used to calculate the projection. A water quality management problem is used to demonstrate the interaction process of the interactive method. In addition, three benchmark problems are used to test the accuracy of the normal vector approximation approach and compare the proposed method with GRIST.