Iterative Roots Research Articles

Iterative roots of multifunctions

Iterative roots of multifunctions

Fast-SNARF: A Fast Deformer for Articulated Neural Fields.

Neural fields have revolutionized the area of 3D reconstruction and novel view synthesis of rigid scenes. A key challenge in making such methods applicable to articulated objects, such as the human body, is to model the deformation of 3D locations between the rest pose (a canonical space) and the deformed space. We propose a new articulation module for neural fields, Fast-SNARF, which finds accurate correspondences between canonical space and posed space via iterative root finding. Fast-SNARF is a drop-in replacement in functionality to our previous work, SNARF, while significantly improving its computational efficiency. We contribute several algorithmic and implementation improvements over SNARF, yielding a speed-up of 150×. These improvements include voxel-based correspondence search, pre-computing the linear blend skinning function, and an efficient software implementation with CUDA kernels. Fast-SNARF enables efficient and simultaneous optimization of shape and skinning weights given deformed observations without correspondences (e.g. 3D meshes). Because learning of deformation maps is a crucial component in many 3D human avatar methods and since Fast-SNARF provides a computationally efficient solution, we believe that this work represents a significant step towards the practical creation of 3D virtual humans.

Iterative roots of PM functions extended from the characteristic interval

Iterative roots of PM functions extended from the characteristic interval

The non-iterates are dense in the space of continuous self-maps

In this paper we first prove a nonexistence result on iterative roots, which presents several sufficient conditions for identifying self-maps on arbitrary sets that have no iterative roots of any order. Then, using this result, we prove that when X is , or every non-empty open set of the space of continuous self-maps on X endowed with the compact-open topology contains a map that does not have even discontinuous iterative roots of order . This, in particular, proves that the complement of , the set of non-iterates, is dense in for these X.

On Zhang-Yang's open question about iterative roots of PM functions of height 1 (I): Increasing case

An open question was raised by Zhang-Yang in 1983: Does a PM function F with height 1 have an iterative root f of an order n≤N(F)+1 if the ‘characteristic endpoints condition’ is not satisfied? When F is strictly increasing on its characteristic interval K(F), this question was answered in the case n∈{N(F),N(F)+1} and the case n≤N(F)+1 for f increasing on K(F) and for f decreasing on K(F) with H(f)<n−1 respectively. In this paper we work on this question for F increasing on K(F) in some remaining cases. In the case n<N(F) and the case n≤N(F)+1 we discuss existence and nonexistence of f with H(f)=n and H(f)=n−1 respectively.

Iterative roots of two-dimensional mappings

AbstractAs a weak version of embedding flow, the problem of iterative roots is studied extensively in one dimension, especially in monotone case. There are few results in high dimensions because the constructive method dealing with monotone mappings is unavailable. In this paper, by introducing a kind of partial order, we define the monotonicity for two-dimensional mappings and then present some results on the existence of iterative roots for linear mappings, triangle-type mappings, and co-triangle-type mappings, respectively. Our theorems show that even the property of monotonicity for iterative roots of monotone mappings, which is a trivial result in one dimension, does not hold anymore in high dimensions. At the end of this paper, the problem of iterative roots for two well-known planar mappings, that is, Hénon mappings and coupled logistic mappings, are also discussed.

Monster-producing Islands: Prospects for island detourning in contemporary times

Colonialism produces monsters thriving in island ecologies, but without a structural/historical treatment of how island monsters are created, how the agential relation between colonisers and natives accentuates the dynamics of interpellation, the island would reveal nothing of its past, the reusable trace that can be interpellated in the present. In the modern-day Philippines, a former colonial object to three foreign aggressions (Spanish, American and Japanese) that spanned almost half a millennium, such reusable trace has recently entered the international streaming platform Netflix. Trese (2021), a Philippine-made anime, navigates the myth of aswang, one of the dominant features of Filipino folklore, which centres on the image of a female, vampire-like monster. This image traces its iterative root in the lost history of the babaylans (female shamans) amid the creation of folk Christianity and diffracted engagements with the Christian indoctrination of the islands. This article navigates this subject and how Trese, among other actants in this play of figuration, is itself interpellated by traces of historical, geographical, and non-human ecologies that, in turn, reflect the fundamental role of liquidity in monster-creation as a material-semiotic intervention. As the discussions expand on the two most essential concepts in Island Studies today, the archipelagic and the aquapelagic, the article deploys the concept of detourning, a critical rhetorical arc that binds the article’s multi-faceted discussions, connections, and combinations, from human to the non-human, thus completing its assessment of the dynamics of islands being interpellated by traces.

Elastic wave attenuation in a functionally graded viscoelastic couple stress plate, sandwiched between two elastic half-spaces

Taking the viscoelasticity and size-dependent effect into account, the reflection and transmission of elastic waves through functionally graded viscoelastic plate sandwiched in two elastic half-spaces are investigated. The extended analytical integration Legendre polynomial method and Kelvin-Voigt viscoelastic model in the context of the modified couple stress theory are utilized to perform the derivation. The global matrix method is used to confirm the validity of the extended analytical integration Legendre polynomial method. Compared with the global matrix method, the extended analytical integration Legendre polynomial method provides an efficient numerical calculation method without layering and iterative root seeking. The influences of the couple stress, viscoelasticity and inhomogeneity of medium on elastic wave reflection and transmission are discussed. In addition, the attenuation caused by the viscoelasticity is analyzed. It is found that the couple stress effect weakens the influence of the viscoelasticity, while the viscoelasticity strengthens the couple stress effect. The couple stress effect decreases the attenuation, while the inhomogeneity increases the attenuation.

A rapidly convergent method for solving third-order polynomials

We present a rapidly convergent method for solving cubic polynomial equations with real coefficients. The method is based on a power series expansion of a simplified form of Cardano’s formula using Newton’s generalized binomial theorem. Unlike Cardano’s formula and semi-analytical iterative root finders, the method is free from round-off error amplification when the polynomial coefficients differ by several orders of magnitude or when they do not differ much from each other, but are all large or small by many orders of magnitude. Validation of the method is assessed by casting a cubic equation of state as a polynomial in terms of the compressibility factor and the reduced molar volume for propylene at temperature and pressure conditions where Cardano’s formula and iterative root finders fail.

Non-Iterative Simulation Methods for Virtual Analog Modelling

The simulation of nonlinear components is central to virtual analog simulation. In audio effects, circuit elements often include devices such as diodes and transistors, mostly operating in the strongly nonlinear regime. Mathematical models are of the form of systems of nonlinear ordinary differential equations (ODEs), and traditional integrators, such as the trapezoid and midpoint methods, can be employed as solvers. These methods are fully implicit, and require the solution of a nonlinear algebraic system at each time step, introducing further complications regarding the existence and uniqueness of the solution, as well as the choice of halting conditions for the iterative root finder. On the other hand, fast explicit methods such as Forward Euler, or explicit Runge-Kutta methods, are prone to unstable behaviour at standard audio sample rates, even at moderate amplitudes. For these reasons, in this work a family of linearly-implicit schemes is presented. These schemes take the form of a perturbation expansion, making the construction of higher-order schemes possible. Compared with classic implicit designs, the proposed methods have the advantage of efficiency, since the update is computed in a single iteration, through the solution of a linear system of equations. Furthermore, the existence and uniqueness of the update are proven by simple inspection of the update matrix. Compared to classic explicit designs, the proposed schemes display stable behaviour at standard audio sample rates. In the case of a single scalar ODE, sufficient conditions for numerical stability can be derived, imposing constraints on the choice of the sampling rate. Several theoretical results are provided, as well as numerical examples for typical stiff equations used in virtual analog modelling.

Phonological evidence for morpho-syntactic structure in Athapaskan

Abstract Athapaskan verbal morphology appears to violate the Mirror Principle in multiple ways and, thus, the ordering of affixes in these languages has resisted a straightforward analysis. We adopt a new morphological tool of Iterative Root Prefixation, which allows for a more direct mapping from syntax to morphology in languages of this profile. Apparent violations of affix ordering that remain, namely the puzzling placement of the transitive and causative morphemes, are argued to be explained by overriding phonological constraints.

Topological conjugacy for multifunctions with finitely many set-valued points and an application to iterative roots

Topological conjugacy plays an important role in simplifying the systems and equations in the research of dynamical systems and functional equations. In this paper, we deal with increasing upper semi-continuous multifunctions with finitely many set-valued points. Some sufficient and necessary conditions are given for the topological conjugate relation between those multifunctions, and the methods of construction for all conjugacies are also presented. Moreover, as an application, our results are applied to study the problem of iterative roots for the multifunctions with a unique set-valued point.

On non-monotonicity height of piecewise monotone functions

Non-monotonicity height is an important index to describe the complexity of dynamics for piecewise monotone functions. Although it is used extensively in the theory of iterative roots, its calculation is still difficult especially in the infinite case. In this paper, by introducing the concept of spanning interval, we first present a sufficient condition for piecewise monotone functions to have height infinity and then an algorithm for finding the spanning intervals is given. We further investigate the density of all piecewise monotone functions with infinite and finite height, respectively, and the results indicate the instability of height. At the end of this paper, the variance of height under composition, especially for functions of height 1 and infinity, are also discussed.

Square iterative roots of generalized weighted quasi-arithmetic mean-type mappings

We determine all pairs $$(A_{f,g},A_{g,h})$$ of generalized weighted quasi-arithmetic means being square iterative roots of $$(A_{F,G},A_{G,H})$$ , that is, the equation \begin{equation*} ( A_{f,g},A_{g,h}) \circ ( A_{f,g},A_{g,h}) =(A_{F,G},A_{G,H}), \end{equation*} is solved under three times differentiability of the functions $$f, g, h, F, G, H$$ . As an application, some special cases are presented.

Topological Classifications for a Class of 2-dimensional Quadratic Mappings and an Application to Iterative Roots

Usually, it is difficult to find a topological relation among high dimensional mappings because of the complexity of existence conditions for conjugacies. In this paper, we investigate the conjugation for a class of 2-dimensional quadratic mappings via the theory of polynomial algebra. Two kinds of conjugate classifications are given. Finally, an application to the problem of iterative roots is also presented.

Tooth flank approximation with root point iteration – potentials and limits in gear metrology

Gear production demands high-precision metrology, for which a holistic evaluation approach of the geometric data is proposed to overcome current restrictions. The holistic approximation with integrated partitioning and iterative root point calculation can cope even with modified flanks and is validated for gear parameter estimation with systematic deviations <0.2 µm. Apart from low signal-to-noise-ratio cases, where the approximation suffers from the multidimensionality of the optimization, the accuracy of standard evaluation procedures is achieved. Furthermore, holistic approximation is able to perform the required mathematical separation of the integral geometric elements of a tooth flank automatically when determining unknown gear parameters.

Iterative roots of exclusive multifunctions

In this paper, we investigate iterative roots of strictly monotone upper semi-continuous multifunctions having finitely many jumps. Known results are concerning roots of order 2 for multifunctions of exact one jump. For the general investigation, we introduce a concept ‘intensity’ to formulate the growth of jumps under iteration and find a class of strictly monotone and upper semi-continuous multifunctions of intensity 1, called exclusive multifunctions, each of which has an absorbing interval. Then we use the absorbing interval to construct iterative roots of order n for those exclusive multifunctions.

Genetics of iterative roots for PM functions

It is known that the time-one mapping of a flow defines a discrete dynamical system with the same dynamical behaviors as the flow, but conversely one wants to know whether a flow embedded by a homeomorphism preserves the dynamical behaviors of the homeomorphism. In this paper we consider iterative roots, a weak version of embedded flows, for the preservation. We refer an iterative root to be genetic if it is topologically conjugate to its parent function. We prove that none of PM functions with height being $ &gt;1 $ has a genetic root and none of iterative roots of height being $ &gt;1 $ is genetic even if the height of its parent function is equal to 1. This shows that most functions do not have a genetic iterative root. Further, we obtain a necessary and sufficient conditions under which a PM function $ f $ has a genetic iterative root in the case that $ f $ and the iterative root are both of height 1.

Non-monotonicity and smoothness of iterative roots of one-dimensional mapping

The problem of iterative roots is a weak version of embedding flows,which is a basis of dynamic interpolation.However, even for one-dimensional mappings,it is still difficult to discuss the non-monotonicity and smoothness of their iterative roots.In this paper,some new results are introduced,especially for the existence and construction of continuous iterative roots of strictly piecewise monotone and continuous mappings,and for smoothness of iterative roots about local and global cases.Finally,as a special class of strictly piecewise monotone and smooth mappings,polynomials are discussed and the conditions of existence with calculation methods of their iterative roots are given.

Iterative roots of continuous functions and Hyers–Ulam stability

In this paper, we prove that continuous non-PM functions with non-monotonicity height equal to 1 need not be strictly monotone on its range, unlike PM functions. An existence theorem is obtained for the iterative roots of such functions. We also discuss the Hyers–Ulam stability for the functional equation of the iterative root problem.