Invariant Functions Research Articles

The inverse Frobenius-Perron problem: A survey of solutions to the original problem formulation

<abstract><p>The inverse Frobenius-Perron problem (IFPP) is a collective term for a family of problems that requires the construction of an ergodic dynamical system model with prescribed statistical characteristics. Solutions to this problem draw upon concepts from ergodic theory and are scattered throughout the literature across domains such as physics, engineering, biology and economics. This paper presents a survey of the original formulation of the IFPP, wherein the invariant probability density function of the system state is prescribed. The paper also reviews different strategies for solving this problem and demonstrates several of the techniques using examples. The purpose of this survey is to provide a unified source of information on the original formulation of the IFPP and its solutions, thereby improving accessibility to the associated modeling techniques and promoting their practical application. The paper is concluded by discussing possible avenues for future work.</p></abstract>

On solvability of Riemann problems in Banach hardy classes

This work deals with the rearrangement invariant Banach function space X and Banach Hardy classes generated by this space, which consist of analytic functions inside and outside the unit circle. In these Hardy classes we consider homogeneous and nonhomogeneous Riemann problems with piecewise continuous coefficient. We define new characteristic of the space X related to the power functions in X. Canonical solution is defined depending on the jumps of the argument of the coefficient of the problem. In terms of the above characteristic, we find a condition on the jumps of the argument depending on the Boyd indices of space X, which is sufficient for the solvability of these problems, and, in case of solvability, we construct a general solution. We also give an orthogonality condition for the solvability of nonhomogeneous problem. As X, considering specific spaces, we obtain previously known results.

Conformal hypersurface geometry via a boundary Loewner–Nirenberg–Yamabe problem

We develop an approach to the conformal geometry of embedded hyper- surfaces by treating them as conformal infinities of conformally compact manifolds. This involves the Loewner-Nirenberg-type problem of finding on the interior a metric that is both conformally compact and of constant scalar curvature. Our first main result is an asymptotic solution to all orders. This involves log terms. We show that the coefficient of the first of these is a hypersurface conformal invariant which gener- alises to higher dimensions the Willmore invariant of embedded surfaces. We call this the obstruction density and for even dimensional hypersurfaces this is a fundamental curvature invariant. We make the latter notion precise and show that the obstruction density and the trace-free second fundamental form are, in a suitable sense, the only such invariants. We also show that this obstruction to smoothness is a scalar density analog of the Fefferman-Graham obstruction tensor for Poincare-Einstein metrics; in part this is achieved by exploiting Bernstein-Gel'fand-Gel'fand machinery. The solu- tion to the constant scalar curvature problem provides a smooth hypersurface defining density determined canonically by the embedding up to the order of the obstruction. We give two key applications: the construction of conformal hypersurface invariants and the construction of conformal differential operators. In particular we present an infinite family of conformal powers of the Laplacian determined canonically by the conformal embedding. In general these depend non-trivially on the embedding and, in contrast to Graham-Jennes-Mason-Sparling operators intrinsic to even dimensional hypersurfaces, exist to all orders. These extrinsic conformal Laplacian powers deter- mine an explicit holographic formula for the obstruction density. We also use these to construct conformally invariant hypersurface functionals. On each even dimensional hypersurface they provide a higher dimensional generalisation of the Willmore energy functional. In particular the functional gradient, with respect to variation of embed- ding, has linear leading term and thus provides another generalisation of the Willmore equation. It agrees with the obstruction density equation at leading derivative order.

Значение плюсквамперфекта: лексика и грамматика

The paper is aimed at the analysis of expressing the meaning ‘long ago’ or the meaning of pluperfect. In the meaning of ‘long ago’ the Russian word davno is the focus of a sentence. Meanwhile, if an event began long ago and is taking place at the time of reference (for instance, at the moment of speech), such context does not require davno to function as the focus. At present, the hypothesis was verified on the spoken corpora material. The key to the analysis is the prosody of the focus. On the whole, the hypothesis appeared to be true. At the same time, it acquired some substantial corrections. If earlier it was presumed that the context of pluperfect arises from the combination of davno and the aspectual forms of a verb, at present, the corpus analysis has revealed the third factor: it is the lexical semantics of the verb. The corpus analysis shows that the verbs of information transfer and the verbs in the negative can recover the connection of the past events with the point of reference. This meaning, in its turn, can cause the loss of the invariable focal function of davno in the context of the general factual in the past.

A new view on risk measures associated with acceptance sets

In this paper, we study properties of certain risk measures associated with acceptance sets. These sets describe regulatory preconditions that have to be fulfilled by financial institutions to pass a given acceptance test. If the financial position of an institution is not acceptable, the decision maker has to raise new capital and invest it into a basket of so called eligible assets to change the current position such that the resulting one corresponds with an element of the acceptance set. Risk measures have been widely studied. The risk measure that is considered here determines the minimal costs of making a financial position acceptable. In the literature, monetary risk measures are often defined as translation invariant functions and, thus, there is an equivalent formulation as Gerstewitz-Functional. The Gerstewitz-Functional is an useful tool for separation and scalarization in multiobjective optimization in the non-convex case. In our paper, we study properties of the sublevel sets, strict sublevel sets and level lines of a risk measure defined on a linear space. Furthermore, we discuss the finiteness of the risk measure and relax the closedness assumptions.

Composition operators and closures of a class of Möbius invariant function spaces in the Bloch space

Composition operators and closures of a class of Möbius invariant function spaces in the Bloch space

Preserved MAIT cell numbers and function in idiopathic CD4 lymphocytopenia.

Mucosal-associated invariant T (MAIT) cells constitute a subset of unconventional, MR1-restricted T-cells involved in antimicrobial responses as well as inflammatory, allergic and autoimmune diseases. Chronic infection and inflammatory disorders as well as immunodeficiencies are often associated with decline and/or dysfunction of MAIT cells. Herein, we investigate the MAIT cells in patients with idiopathic CD4 + lymphocytopenia (ICL), a syndrome characterized by consistently low CD4 T-cell counts (<300 cell/µL) in the absence of HIV infection or other known immunodeficiency, and by susceptibility to certain opportunistic infections. The numbers, phenotype and function of MAIT cells in peripheral blood were preserved in ICL patients compared to healthy controls. Furthermore, administration of IL-7 to ICL patients expanded the CD8 + MAIT cell subset, with maintained responsiveness and effector functions after IL-7 treatment. In conclusion, ICL patients maintain normal levels and function of MAIT cells preserving some antibacterial responses despite the deficiency in CD4 + T cells.

On the discriminant locus of a rank $n-1$ vector bundle on $\mathbb P^{n-1}$

In this paper, we show that if a vector bundle of rank n-1 on projective (n-1) -space is very ample and if its top Chern class is greater than one, then the discriminant locus of the vector bundle, the locus of singular sections of the vector bundle, is an irreducible hypersurface and that the degree of the hypersurface can be expressed as a function of invariants of the vector bundle. As applications, we study the eigendiscriminant locus (the locus of tensors whose eigenschemes are singular) and the discriminant locus of the generic graded matrix (the locus of singular determinantal schemes).

Modulation Transfer Function Measurements Using a Learning Approach From Multiple Diffractive Grids for Optical Cameras

Remote sensing images are easily degraded because the imaging process is influenced by the camera itself link (CIL), such as optical systems, electronic systems, and image sensors. The invariant modulation transfer function (IMTF) of a remote sensing camera can represent the degeneration produced by the CIL. Thus, the IMTF compensation can remove the degeneration of the CIL to improve the image quality. However, conventional IMTF measurement methods using a single feature (e.g., edge or pulse) have low measurement accuracy due to the influence of noises, resulting in low image compensation quality. Here, we propose an efficient measurement method of the IMTF in conjunction with a learning approach to compensate for the imaging degeneration. The IMTF from the CIL is measured using multiple diffractive grids, which are generated by two 1-D gratings. A learning approach is proposed to estimate the IMTF from the diffractive grids, where diffractive grids are not directly used as a point spread function (PSF) because grids are easily affected by various noises and result in low reconstruction accuracy of the IMTF. We experimentally confirm that the proposed method is useful for the IMTF estimations and compensation for image quality degeneration of remote sensing cameras under noise conditions. The proposed methods can double the MTF reconstruction accuracy and yield higher image compensation quality compared with traditional methods.

H2Pack

Dense kernel matrices represented in H 2 matrix format typically require less storage and have faster matrix-vector multiplications than when these matrices are represented in the standard dense format. In this article, we present H2Pack, a high-performance, shared-memory library for constructing and operating with H 2 matrix representations for kernel matrices defined by non-oscillatory, translationally invariant kernel functions. Using a hybrid analytic-algebraic compression method called the proxy point method, H2Pack can efficiently construct an H 2 matrix representation with linear computational complexity. Storage and matrix-vector multiplication also have linear complexity. H2Pack also introduces the concept of “partially admissible blocks” for H 2 matrices to make H 2 matrix-vector multiplication mathematically identical to the fast multipole method (FMM) if analytic expansions are used. We optimize H2Pack from both the algorithm and software perspectives. Compared to existing FMM libraries, H2Pack generally has much faster H 2 matrix-vector multiplications, since the proxy point method is more effective at producing block low-rank approximations than the analytic methods used in FMM. As a tradeoff, H 2 matrix construction in H2Pack is typically more expensive than the setup cost in FMM libraries. Thus, H2Pack is ideal for applications that need a large number of matrix-vector multiplications for a given configuration of data points.

DESIGN AND TECHNOLOGICAL PROPOSALS FOR IMPROVING A SINGLE-PHASE TRANSFORMER WITH LAMINATED MAGNETIC CORE

Goal. The purpose of the work is to analyze the features and substantiate the advantages of the spatial radial electromagnetic system of a single-phase transformer with rectangular sections of the rods of a charged magnetic circuit. Methodology. Improvement of single-phase transformers is possible by structural-geometric transformations of active elements. The justification of the advantages of such transformations is achieved by the method of invariant objective functions with dimensionless optimization components and universal relative controlled variables. Results. Replacing the rectangular contours of the variants of a planar magnetic circuit with a hexagonal circuit leads to a decrease in the volume of steel and the loss of three angular zones of changes in the direction of the magnetic flux relative to the texture compared to planar counterparts. The separation of the windings into sections with an arrangement on three rods leads to a decrease in the average length of the coil. Originality. A three-core magnetic core can consist of rod and jugular-angular parallelogram plates of anisotropic and isotropic electrical steel. It is advisable to combine a combination of steels by integral welding of billets with subsequent separation into chevron jugular-rod elements. The method called on has developed mathematical models of the mass, cost, and active power losses of variants of a single-phase electromagnetic system of a transformer with rod and armor planar and spatial radial three-core magnetic cores. Practical significance. It was found that the decrease in the extrema of the mass and cost indicators, as well as the calculated decrease, under the condition that the coefficients of additional losses are identical, the extrema of the losses of the single-phase radial three-rod system of the transformer relative to the planar analogue with a bar stocked magnetic wire from anisotropic steel, respectively 13,0-12,3 %, 15,3-10,1 % and 15,1-18,1 % with copper windings and 10,8-10,2 %, 12,8-13,7 % and 12,6-12,3 % with aluminum windings. The main indicators of traditional single-phase systems with rod and armored magnetic circuits differ insignificantly, which is consistent with early known optimization and calculation data of transformers.

Dynamically emergent gravity from hidden local Lorentz symmetry

Gravity can be regarded as a consequence of local Lorentz (LL) symmetry, which is essential in defining a spinor field in curved spacetime. The gravitational action may admit a zero-field limit of the metric and vierbein at a certain ultraviolet cutoff scale such that the action becomes a linear realization of the LL symmetry. Consequently, only three types of term are allowed in the four-dimensional gravitational action at the cutoff scale: a cosmological constant, a linear term of the LL field strength, and spinor kinetic terms, whose coefficients are in general arbitrary functions of LL and diffeomorphism invariants. In particular, all the kinetic terms are prohibited except for spinor fields, and hence the other fields are auxiliary. Their kinetic terms, including those of the LL gauge field and the vierbein, are induced by spinor loops simultaneously with the LL gauge field mass. The LL symmetry is necessarily broken spontaneously and hence is nothing but a hidden local symmetry, from which gravity is emergent.

PINE: Universal Deep Embedding for Graph Nodes via Partial Permutation Invariant Set Functions.

Graph node embedding aims at learning a vector representation for all nodes given a graph. It is a central problem in many machine learning tasks (e.g., node classification, recommendation, community detection). The key problem in graph node embedding lies in how to define the dependence to neighbors. Existing approaches specify (either explicitly or implicitly) certain dependencies on neighbors, which may lead to loss of subtle but important structural information within the graph and other dependencies among neighbors. This intrigues us to ask the question: can we design a model to give the adaptive flexibility of dependencies to each node's neighborhood. In this paper, we propose a novel graph node embedding method (named PINE) via a novel notion of partial permutation invariant set function, to capture any possible dependence. Our method 1) can learn an arbitrary form of the representation function from the neighborhood, without losing any potential dependence structures, and 2) is applicable to both homogeneous and heterogeneous graph embedding, the latter of which is challenged by the diversity of node types. Furthermore, we provide theoretical guarantee for the representation capability of our method for general homogeneous and heterogeneous graphs. Empirical evaluation results on benchmark data sets show that our proposed PINE method outperforms the state-of-the-art approaches on producing node vectors for various learning tasks of both homogeneous and heterogeneous graphs.

Dynamic Evaluation of Traffic Noise through Standard and Multifractal Models

Traffic microsimulation models use the movement of individual driver-vehicle-units (DVUs) and their interactions, which allows a detailed estimation of the traffic noise using Common Noise Assessment Methods (CNOSSOS). The Dynamic Traffic Noise Assessment (DTNA) methodology is applied to real traffic situations, then compared to on-field noise levels from measurement campaigns. This makes it possible to determine the influence of certain local traffic factors on the evaluation of noise. The pattern of distribution of vehicles along the avenue is related to the logic of traffic light control. The analysis of the inter-cycles noise variability during the simulation and measurement time shows no influence from local factors on the prediction of the dynamic traffic noise assessment tool based on CNOSSOS. A multifractal approach of acoustic waves propagation and the source behaviors in the traffic area are implemented. The novelty of the approach also comes from the multifractal model’s freedom which allows the simulation, through the fractality degree, of various behaviors of the acoustic waves. The mathematical backbone of the model is developed on Cayley–Klein-type absolute geometries, implying harmonic mappings between the usual space and the Lobacevsky plane in a Poincaré metric. The isomorphism of two groups of SL(2R) type showcases joint invariant functions that allow associations of pulsations–velocities manifolds type.

Long-Range Orientational Organization of Dipolar and Steric Liquids.

Long-range orientational correlations in liquids have received recent renewed interest, in particular for the neat water case. These long-range orientational correlations, exceeding several tens of nanometers, originate from the presence of the strong permanent water dipolar moment. However, the exact dependence with the dipolar moment and the role of other local forces like steric hindrance has never been addressed. In this work, we experimentally measure long-range correlations for a set of liquids differing by their molecular weight and dipolar moment, in order to reveal the origin of their long-range organization. Hence, we show that the dipolar moment of a solvent molecule is not the unique feature determining the orientational correlation. Steric hindrance significantly helps to structure the liquids as well. In order to quantify these long-range correlations, we also derive theoretically the polarization resolved second harmonic scattering intensity as a function of the rotational invariants describing the dipolar and octupolar interaction.

Multipliers and invariants of endomorphisms of projective space in dimension greater than 1

There is a natural conjugation action on the set of endomorphism of $¶^N$ of fixed degree $d \geq 2$. The quotient by this action forms the moduli of degree $d$ endomorphisms of $¶^N$, denoted $\mathcal{M}_d^N$. We construct invariant functions on this moduli space coming from to set of multiplier matrices of the periodic points. The basic properties of these functions are demonstrated such as that they are in the ring of regular functions of $\mathcal{M}_d^N$, methods of computing them, as well as the existence of relations. The main part of the article examines to what extend these invariant functions determine the conjugacy class in the moduli space. Several different types of isospectral families are constructed and a generalization of McMullen's theorem on the multiplier mapping of dimension 1 is proposed. Finally, this generalization is shown to hold when restricted to several specific families in $\mathcal{M}_d^N$.

Bayesian estimation of the expected queue length of a system M/M/1 with certain and uncertain priors

One of characteristics of the traffic intensity in M/M/1 queue is the expected queue length. In this article, the problem of Bayes, E-Bayes, and robust Bayes estimation of the expected length in a M/M/1 queue system under a scale invariant squared error loss function is considered. We find the Bayes, E-Bayes, and robust Bayes estimators of the expected length. Finally, a simulation study is conducted for comparison of the proposed Bayes estimators.

Turbulence anisotropy with higher-order moments in flow through passive grid under rigid boundary influence

The investigation presents the estimate of the degree of deviation from the isotropic turbulence in terms of Reynolds stress tensor for grid generated turbulence under the influence of bottom boundary. The turbulence triangle, Eigen values, and the invariant functions are presented at near and far field regions of the grids with different solidity ratio. In addition, the work also deals with the analysis based on third-order moments of the velocity fluctuations and the ratio of momentum flux to the turbulent kinetic energy in the frequency domain. The Reynolds stress anisotropy exposes that the anisotropic invariant maps possess a closed looping trend in the near field region and an open looping trend in the far-field region of the grid. Further, to describe the physical behaviour of the velocity time-series of random fluctuating components in the stream-wise directions, the probability distribution function are estimated and interpreted.

Invariants of the gas dynamics inside the mushroom clouds

The material conservation laws are derived for the axisymmetric flows of the inviscid barotropic gas inside the mushroom clouds. The invariant functions Ψ(μ) and ΨG(μ) of an independent variable μ are constructed for any pure poloidal compressible gas flow.

Image Stitching for Chest Digital Radiography Using the SIFT and SURF Feature Extraction by RANSAC Algorithm

Image stitching is one of the branches of computer vision. It combines two or more images for a scene to acquire a high-resolution panoramic image. An invariant local function often uses to stitch two images together. Since the flat plate of a digital radiography (DR) system does not cover all parts of the body, the whole bone structure image cannot seize in a single scan. To solved this problem, image stitching is broadly utilized by medical systems to stitch DR images, which can be helpful for scoliosis or lower extremity deformities in the diagnosis, and pre-operative planning are of great importance. In this paper, the stitching and retrieval of medical images planned. To conquer the background noise in medical images, and improve the recovery of quality and stitching rapidity of medical images, a random sample consensus (RANSAC) algorithm is useful to stitching the images of Chest digital radiography by scale-invariant feature transform (SIFT) and speeded-up robust features (SURF) feature extraction. Down-sampling utilizes to lessen the size of the images and reduction the measure of calculation. In the interim, the phase correlation is engaged to discover the overlapping region. After feature matching and perspective transformation, the stitched image is gotten dependent on the homography. At last, experimentation has finished showing the presentation.