One-dimensional Affine Research Articles

Swelling Effect on the Yielding, Elasticity, and Fracture of Double-Network Hydrogels with an Inhomogeneous First Network

Typical double-network (DN) gels are formed by free radical polymerization, and the constituent network strands have a wide molecular weight distribution. It is important to clarify what determines the specific mechanical behaviors of DN gels with a strongly inhomogeneous first network, including the criteria for the occurrence of yielding and strain hardening during tensile deformation, as well as their impact on the fracture toughness. In this work, we first studied the influence of swelling ratio λs of the first network synthesized from radical polymerization on the yielding and strain hardening of DN and triple-network (TN) gels by tensile tests. We observed scaling relations for the yielding stretch ratio λy ∝ λs–1 and engineering yielding stress σy ∝ λs–2, like those found for DN hydrogels with a homogeneous first-network structure. Furthermore, we found a universal relationship λyσy ∝ λs–3 that is independent of the cross-linking structure of the first network. We found that these scaling relations agree well with the theoretical predictions derived from a simple one-dimensional affine model for the fracture of a homogeneous network, suggesting the affine swelling effect on the fracture of network strands. Next, we showed that swelling induces more tension in the inhomogeneous network than in the homogeneous network, suggesting non-affine swelling behavior at the network strand scale for the inhomogeneous network. We discussed the molecular origin of the discrepancy between the affine swelling effect on the fracture of network strands and the non-affine swelling behavior for strand tension. Finally, we investigated the influence of swelling ratio λs on the local yielding behavior ahead of the crack tip and the fracture toughness of the DN and TN hydrogels using pure shear tests. We found that crack-tip yielding zone size hy ∝ λs1 and fracture energy Γ ∝ λs–2. Furthermore, we found that the relationship Γhy∝λs−3 is also independent of cross-linking structure and obeys a universal relationship with the true yielding stress λyσy ∝ λs–3 determined from the tensile tests, which is in excellent agreement with nonlinear elastic fracture mechanics theory. This work provides important insight into the specific mechanical behaviors of the DN gels with an inhomogeneous first network.

Rapid antenna optimization with restricted sensitivity updates by automated dominant direction identification

Meticulous tuning of geometry parameters turns pivotal in improving performance of antenna systems. It is more and more often realized using formal optimization methods, which is demonstrably the most efficient way of handling multiple design variables, objectives, and constraints. Although in some cases a need for launching global search arises, a typical design scenario only requires local optimization, especially when a decent initial design can be rendered using engineering experience or parametric studies. At the same time, antenna optimization is typically conducted using full-wave electromagnetic (EM) simulations, which entails considerable computational expenses. In this paper, we introduce a novel procedure for expedited antenna tuning. Its fundamental mechanism is to restrict the antenna response sensitivity updates to the selected dominant directions within the parameter space, determined based on the problem-specific knowledge, in particular, the estimated changes of antenna characteristics when moving across one-dimensional affine subspaces spanned by these directions. Thus, the said dominant directions affect the most the responses of the antenna structure under design as assessed using the introduced metrics. The decision making process concerning the number of directions to be used relies on quantification of the aggregated system response variability metrics. The proposed approach is demonstrated by means of several antenna structures and benchmarked against conventional trust-region algorithm, but also its accelerated versions. The results indicate considerable (up to over 60%) speedup over the reference procedure without noticeable quality degradation.

New families of flag-transitive linear spaces

In this paper, we construct new families of flag-transitive linear spaces with q2n points and q2 points on each line that admit a one-dimensional affine automorphism group. We achieve this by building a natural connection with permutation polynomials of Fq2 of a particular form and following the scheme of Pauley and Bamberg in (2008) [14].

A characterization of exceptional pseudocyclic association schemes by multidimensional intersection numbers

Recent classification of 3/2-transitive permutation groups leaves us with three infinite families of groups which are neither 2-transitive, nor Frobenius, nor one-dimensional affine. The groups of the first two families correspond to special actions of PSL(2, q) and PΓL(2, q), whereas those of the third family are the affine solvable subgroups of AGL(2, q) found by D. Passman in 1967. The association schemes of the groups in each of these families are known to be pseudocyclic. It is proved that apart from three particular cases, each of these exceptional pseudocyclic schemes is characterized up to isomorphism by the tensor of its 3-dimensional intersection numbers.

Jump-induced mixed-mode oscillations through piecewise-affine maps

Mixed-mode oscillations (MMOs) are complex oscillatory patterns in which large-amplitude oscillations (LAOs) of relaxation type alternate with small-amplitude oscillations (SAOs). MMOs are found in singularly perturbed systems of ordinary differential equations of slow-fast type, and are typically related to the presence of so-called folded singularities and the corresponding canard trajectories in such systems. Here, we introduce a canonical family of three-dimensional slow-fast systems that exhibit MMOs which are induced by relaxation-type dynamics, and which are hence based on a “jump mechanism”, rather than on a more standard canard mechanism. In particular, we establish a correspondence between that family and a class of associated one-dimensional piecewise affine maps (PAMs) which exhibit MMOs with the same signature. Finally, we give a preliminary classification of admissible mixed-mode signatures, verifying results of (Rajpathak, Pillai, and Bandyopahdyay (2012) [29]) in the process, and we illustrate our findings with numerical examples.

On 2-closures of rank 3 groups

A permutation group G on Ω is called a rank 3 group if it has precisely three orbits in its induced action on Ω × Ω . The largest permutation group on Ω having the same orbits as G on Ω × Ω is called the 2 - closure of G . A description of 2 -closures of rank 3 groups is given. As a special case, it is proved that the 2 -closure of a primitive one-dimensional affine rank 3 group of sufficiently large degree is also affine and one-dimensional.

Impartial achievement games on convex geometries

We study a game where two players take turns selecting points of a convex geometry until the convex closure of the jointly selected points contains all the points of a given winning set. The winner of the game is the last player able to move. We develop a structure theory for these games and use it to determine the nim number for several classes of convex geometries, including one-dimensional affine geometries, vertex geometries of trees, and games with a winning set consisting of extreme points.

A classification of one-dimensional affine rank three graphs

The rank three subgroups of a one-dimensional affine group over a finite field were classified in 1978 by Foulser and Kallaher. Although one can use their results for a classification of corresponding rank three graphs, the author did not find such a classification in a literature. The goal of this note is to present such a classification. It turned out that graph classification is much simpler than the group one. More precisely, it is shown that the graphs in the title are either the Paley graphs or one of the graphs constructed by Van Lint and Schrijver or by Peisert. Our approach is based on elementary group theory and does not use the classification of rank three affine groups.

Comment intégrer les données massives dans la pratique du pneumologue : le syndrome d’apnées du sommeil comme cas d’usage: How to integrate big data into the practice of the pulmonologist: sleep apnea syndrome as a use case

The rank three subgroups of a one-dimensional affine group over a finite field were classified in 1978 by Foulser and Kallaher. Although one can use their results for a classification of corresponding rank three graphs, the author did not find such a classification in a literature. The goal of this note is to present such a classification. It turned out that graph classification is much simpler than the group one. More precisely, it is shown that the graphs in the title are either the Paley graphs or one of the graphs constructed by Van Lint and Schrijver or by Peisert. Our approach is based on elementary group theory and does not use the classification of rank three affine groups.

Generalised Paley Graphs with a Product Structure

A graph is Cartesian decomposable if it is isomorphic to a Cartesian product of strictly smaller graphs, each of which has more than one vertex and admits no such decomposition. These smaller graphs are called the Cartesian-prime factors of the Cartesian decomposition, and were shown, by Sabidussi and Vizing independently, to be uniquely determined up to isomorphism. We characterise by their parameters those generalised Paley graphs which are Cartesian decomposable, and we prove that for such graphs, the Cartesian-prime factors are themselves smaller generalised Paley graphs. This generalises a result of Lim and the second author which deals with the case where all the Cartesian-prime factors are complete graphs. These results contribute to the determination, by parameters, of generalised Paley graphs with automorphism groups larger than the one-dimensional affine subgroups used to define them. They also shed light on the structure of primitive cyclotomic association schemes.

Two-dimensional Shannon type expansions via one-dimensional affine and wavelet lattice actions

It is rather unexpected, but true, that it is possible to construct reproducing formulae and orthonormal bases of $L^2 (\mathbb{R}^2)$ just by applying the standard one dimensional wavelet action of translations and dilations to the first variable $x_1$ of the generating function $\psi(x_1,x_2)$, $\psi \in L^2 (\mathbb{R}^2)$, i.e., by making use of building blocks $$\psi_{u,s}(x_1,x_2)=s^{-1/2}\psi\left(\frac{x_1-u}{s},x_2\right), \text{where } u\in \mathbb{R}, s>0,$$ in the case of reproducing formulae, and $$\psi_{k,m}(x_1,x_2)=2^{-k/2} \psi\left(\frac{x_1-2^k m}{2^k},x_2 \right), \text{where } k,m\in \mathbb{Z},$$ in the case of orthonormal bases. It is possible to compensate the fact, that the second variable $x_2$ is not acted upon, by a careful selection of the generating function $\psi$. Shannon wavelet tiling of the time-frequency plane $\mathbb{R}^2$, a standard illustration of orthogonality and completeness phenomena corresponding to the Shannon wavelet, $$ \chi_{(2^km,2^k(m+1)]}(x) \chi_{2^{-k}I}(\xi),\, k,m\in \mathbb{Z}, \,I=- (1/2,1]\cup (1/2,1], $$ with $x$ representing time and $\xi$ frequency, is substituted by a phase space tiling of $\mathbb{R}^4$ with unbounded, hyperboloid type blocks of the form $$ \chi_{(2^km,2^k(m+1)]}(x_1)\sum_{n,l}\chi_{2^{-k}I_{D(n,l)}}(\xi_1) \chi_{(n,n+1]}(x_2)\chi_{(l,l+1]}(\xi_2),\, k,m\in \mathbb{Z} $$ where $I_r=2^{-r}I$, $ r\ge 1$, and $D:\mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{N}$ is a bijection, an additional parameter of the generating function, needed for the lift from $L^2(\mathbb{R})$ to $L^2(\mathbb{R}^2)$. Variables $x_1,x_2$ are coordinates of position and variables $\xi_1,\xi_2$ of momentum.

Option Pricing in a One-Dimensional Affine Term Structure Model via Spectral Representations

Under a mild condition on the branching mechanism, we provide an eigenvalue expansion for the pricing semigroup in a one-dimensional positive affine term structure model. This representation, which...

Optimal input design for minimum-variance estimation of parameters in nonlinear state-space models

The paper presents a methodology for optimal input design (OID) for minimum-variance estimation of parameters in nonlinear state-space models. To allow analytical solutions, in Keesman (2015) asequential OID approach, based on Pontryagin’s minimum principle, was proposed for low-dimensional non-linear systems that are affine in their input. In this study, a simultaneous OID approach for a set of two parameters in one-dimensional non-linear systems affine in their input is presented. For the two-parameter case still analytically tractable solutions are found, unlike cases with three or more parameters.

Optimization With Affine Homogeneous Quadratic Integral Inequality Constraints

We introduce a new technique to optimize a linear cost function subject to a one-dimensional affine homogeneous quadratic integral inequality, i.e., the requirement that a homogeneous quadratic integral functional, affine in the optimization variables, is non-negative over a space of functions defined by homogeneous boundary conditions. Such problems arise in stability analysis, input-to-state/output analysis, and control of many systems governed by partial differential equations (PDEs), in particular fluid dynamical systems. First, we derive outer approximations for the feasible set of a homogeneous quadratic integral inequality in terms of linear matrix inequalities (LMIs), and show that under mild assumptions a convergent, non-decreasing sequence of lower bounds for the optimal cost can be computed with a sequence of semidefinite programs (SDPs). Second, we obtain inner approximations in terms of LMIs and sum-of-squares constraints, so upper bounds for the optimal cost and strictly feasible points for the integral inequality can also be computed with SDPs. To aid the formulation and solution of our SDP relaxations, we implement our techniques in QUINOPT, an open-source add-on to YALMIP. We demonstrate our techniques by solving problems arising from the stability analysis of PDEs.

On the automorphisms of [formula omitted] designs

Thirty years ago, a six-person team classified the pairs (D,G) where D is a 2−(v,k,1) design and G is a flag-transitive automorphism group of D, with the exception of those in which G is a one-dimensional affine group. Since then the effort has been to classify those D, designs which are block-transitive but not flag-transitive. This paper contributes to the program for determining the pairs 2−(v,k,1) in which (D,G) has a block-transitive group G of automorphisms. It is clear that if one wishes to study the structure of a finite group acting on a D design then describing the socle is an important first step. Here we prove that if G is a block-transitive group of automorphisms of 2−(v,k,1) which has D as its socle then T is also transitive on the blocks of T=2F4(q).

Subgroups of the Cremona groups: Bass’ problem

A general theorem on the purity of invariant field extensions is proved. Using it, a criterion of rational triangulability of connected solvable affine algebraic subgroups of the Cremona groups is obtained. This criterion is applied for proving the existence of rationally nontriangulable subgroups of the above form and for proving their stable rational triangulability. The latter property answers in the affirmative Bass’ Triangulability Problem in the stable range. A general construction of all rationally triangulable connected solvable affine algebraic subgroups of the Cremona groups is obtained. As an application, a classification of all rationally triangulable connected one-dimensional unipotent affine algebraic subgroups of the Cremona groups up to conjugacy is given.

Quotient-complete arc-transitive graphs

We examine the class of G-arc-transitive graphs with at least one nontrivial complete normal quotient, such that all other normal quotients are either complete or empty graphs. Such graphs arise naturally as basic graphs under normal quotient analysis on families of arc-transitive graphs. We construct all such graphs Γ, together with the corresponding group G≤Aut(Γ), which have 3 or more nontrivial complete G-normal quotients. In particular, we prove that these graphs have order the square of a prime power q, and arise from finite transitive linear groups H acting on a vector space of order q. The only disconnected graphs in this class are isomorphic to q copies of the complete graph on q vertices, and we classify all the connected graphs that arise from the transitive linear groups H where H is not a subgroup of a one-dimensional affine group ΓL(1,q). For the case where H≤ΓL(1,q) we construct some infinite families of examples. In most cases we determine which of the graphs constructed have diameter two.

Rationality of trace and norm L-functions

For a given l-adic sheaf F on a commutative algebraic group over a finite field k and an integer r we define the r-th local norm L-function of F at a point t in G(k) and prove its rationality. This function gives information on the sum of the local Frobenius traces of F over the points of G(k_r) (where k_r is the extension of degree r of k) with norm t. For G the one-dimensional affine line or the torus, these sums can in turn be used to estimate the number of rational points on curves or the absolute value of exponential sums which are invariant under a large group of translations or homotheties.

Dynamical systems and numerical analysis: the study of measures generated by uncountable I.F.S.

Measures generated by Iterated Function Systems composed of uncountably many one-dimensional affine maps are studied. We present numerical techniques as well as rigorous results that establish whether these measures are absolutely or singular continuous.

Star products: a group-theoretical point of view

Adopting a purely group-theoretical point of view, we consider the star product of functions which is associated, in a natural way, with a square integrable (in general, projective) representation of a locally compact group. Next, we show that for this (implicitly defined) star product, explicit formulas can be provided. Two significant examples are studied in detail: the group of translations on phase space and the one-dimensional affine group. The study of the first example leads to the Groenewold–Moyal star product. In the second example, the link with wavelet analysis is clarified.