scholarly journals EQUIVALENCE OF THE HUNTER-SAXON EQUATION AND THE GENERALIZED HEISENBERG FERROMAGNET EQUATION

2021 ◽  
Vol 2 (336) ◽  
pp. 33-38
Author(s):  
Zh. R. Myrzakulova ◽  
K. R. Yesmakhanova ◽  
Zh. S. Zhubayeva
Keyword(s):  
Integrable Systems ◽  
Heisenberg Ferromagnet ◽  
Lax Representation ◽  
Director Field ◽  
Weakly Nonlinear ◽  
Scattering Transform ◽  
The Family ◽  
The Matrix ◽  
Special Solutions ◽  
Modern Mathematics

Integrable systems play an important role in modern mathematics, theoretical and mathematical physics. The display of integrable equations with exact solutions and some special solutions can provide important guarantees for the analysis of its various properties. The Hunter-Saxton equation belongs to the family of integrable systems. The extensive and interesting mathematical theory, underlying the Hunter-Saxton equation, creates active mathematical and physical research. The Hunter-Saxton equation (HSE) is a high-frequency limit of the famous Camassa-Holm equation. The physical interpretation of HSE is the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal director field. In this paper, we propose a matrix form of the Lax representation for HSE in 𝑠𝑢ሺ𝑛 ൅ 1ሻ/𝑠ሺ𝑢ሺ1ሻ ⊕ 𝑢ሺ𝑛ሻሻ - symmetric space for the case 𝑛 ൌ 2. Lax pairs, introduced in 1968 by Peter Lax, are a tool for finding conserved quantities of integrable evolutionary differential equations. The Lax representation expands the possibilities of the equation we are considering. For example, in this paper, we will use the matrix Lax representation for the HSE to establish the gauge equivalence of this equation with the generalized Heisenberg ferromagnet equation (GHFE). The famous Heisenberg Ferromagnet Equation (HFE) is one of the classical equations integrable through the inverse scattering transform. In this paper, we will consider its generalization. Andalso the connection between the decisions of the HSE and the GHFE will be presented.

scholarly journals TWISTED SYMMETRIES AND INTEGRABLE SYSTEMS

2009 ◽  
Vol 06 (08) ◽  
pp. 1305-1321 ◽  
Author(s):  
GIUSEPPE GAETA ◽  
GIAMPAOLO CICOGNA
Keyword(s):  
Integrable Systems ◽  
Lagrangian Systems ◽  
Symmetry Properties ◽  
Special Solutions

Symmetry properties are at the basis of integrability. In recent years, it appeared that so-called twisted symmetries are as effective as standard symmetries in many respects (integrating ODEs, finding special solutions to PDEs). Here we discuss how twisted symmetries can be used to detect integrability of Lagrangian systems which are not integrable via standard symmetries.

scholarly journals AUTOMORPHISMS OF THE UHF ALGEBRA THAT DO NOT EXTEND TO THE CUNTZ ALGEBRA

2010 ◽  
Vol 89 (3) ◽  
pp. 309-315 ◽  
Author(s):  
ROBERTO CONTI
Keyword(s):  
Sufficient Conditions ◽  
Extension Property ◽  
Cuntz Algebra ◽  
Maximal Abelian Subalgebra ◽  
Abelian Subalgebra ◽  
Matrix Algebras ◽  
The Family ◽  
The Matrix ◽  
Inner Automorphisms ◽  
Necessary And Sufficient

AbstractThe automorphisms of the canonical core UHF subalgebra ℱn of the Cuntz algebra 𝒪n do not necessarily extend to automorphisms of 𝒪n. Simple examples are discussed within the family of infinite tensor products of (inner) automorphisms of the matrix algebras Mn. In that case, necessary and sufficient conditions for the extension property are presented. Also addressed is the problem of extending to 𝒪n the automorphisms of the diagonal 𝒟n, which is a regular maximal abelian subalgebra with Cantor spectrum. In particular, it is shown that there exist product-type automorphisms of 𝒟n that do not extend to (possibly proper) endomorphisms of 𝒪n.

Amplitude equations at the critical points of unstable dispersive physical systems

1981 ◽  
Vol 377 (1769) ◽  
pp. 185-219 ◽  
Keyword(s):  
Critical Point ◽  
Pulse Propagation ◽  
Baroclinic Instability ◽  
Soliton Solutions ◽  
Complex Conjugate ◽  
Amplitude Equations ◽  
Weakly Nonlinear ◽  
Physical Systems ◽  
Scattering Transform ◽  
Induced Transparency

The amplitude equations that govern the motion of wavetrains near the critical point of unstable dispersive, weakly nonlinear physical systems are considered on slow time and space scales T m ═ ε m t ; X m ═ ε m x ( m ═ 1, 2,...). Such systems arise when the dispersion relation for the harmonic wavetrain is purely real and complex conjugate roots appear when a control parameter ( μ ) is varied. At the critical point, when the critical wavevector k c is non-zero, a general result for this general class of unstable systems is that the typical amplitude equations are either of the form ( ∂/∂ T 1 + c 1 ∂/∂ X 1 ) (∂/∂ T 1 + c 2 ∂/∂ X 1 ) A ═ ±α A ─ β AB , ( ∂/∂ T 1 + c 2 ∂/∂ X 1 ) B ═ (∂/∂ T 1 + c 1 ∂/∂ X 1 ) | A | 2 , or of the form ( ∂/∂ T 1 + c 1 ∂/∂ X 1 ) (∂/∂ T 1 + c 2 ∂/∂ X 1 ) A ═ ±α A - β A | A | 2 . The equations with the AB -nonlinearity govern for example the two-layer model for baroclinic instability and self-induced transparency (s. i. t.) in ultra-short optical pulse propagation in laser physics. The second equation occurs for the two-layer Kelvin-Helmholtz instability and a problem in the buckling of elastic shells. This second type of equation has been considered in detail by Weissman. The AB -equations are particularly important in that they are integrable by the inverse scattering transform and have a variety of multi-soliton solutions. They are also reducible to the sine-Gordon equation ϕ ξƬ ═ ± sin ϕ when A is real. We prove some general results for this type of instability and discuss briefly their applications to various other examples such as the two-stream instability. Examples in which dissipation is the dominant mechanism of the instability are also briefly considered. In contrast to the dispersive type which operates on the T 1 -time scale, this type operates on the T 2 -scale.

scholarly journals On Optimality of Designs with Three Distinct Eigenvalues

10.37236/2709 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
M. R. Faghihi ◽  
E. Ghorbani ◽  
G. B. Khosrovshahi ◽  
S. Tat
Keyword(s):  
Nonlinear Programming ◽  
Optimal Design ◽  
Design Theory ◽  
Incidence Matrix ◽  
Information Matrix ◽  
Block Designs ◽  
Nonzero Eigenvalue ◽  
Group Divisible ◽  
The Family ◽  
The Matrix

Let ${\cal D}_{v,b,k}$ denote the family of all connected block designs with $v$ treatments and $b$ blocks of size $k$. Let $d\in{\cal D}_{v,b,k}$. The replication of a treatment is the number of times it appears in the blocks of $d$. The matrix $C(d)=R(d)-\frac{1}{k}N(d)N(d)^\top$ is called the information matrix of $d$ where $N(d)$ is the incidence matrix of $d$ and $R(d)$ is a diagonal matrix of the replications. Since $d$ is connected, $C(d)$ has $v-1$ nonzero eigenvalues $\mu_1(d),\ldots,\mu_{v-1}(d)$.Let ${\cal D}$ be the class of all binary designs of ${\cal D}_{v,b,k}$. We prove that if there is a design $d^*\in{\cal D}$ such that (i) $C(d^*)$ has three distinct eigenvalues, (ii) $d^*$ minimizes trace of $C(d)^2$ over $d\in{\cal D}$, (iii) $d^*$ maximizes the smallest nonzero eigenvalue and the product of the nonzero eigenvalues of $C(d)$ over $d\in{\cal D}$, then for all $p>0$, $d^*$ minimizes $\left(\sum_{i=1}^{v-1}\mu_i(d)^{-p}\right)^{1/p}$ over $d\in{\cal D}$. In the context of optimal design theory, this means that if there is a design $d^*\in{\cal D}$ such that its information matrix has three distinct eigenvalues satisfying the condition (ii) above and that $d^*$ is E- and D-optimal in ${\cal D}$, then $d^*$ is $\Phi_p$-optimal in ${\cal D}$ for all $p>0$. As an application, we demonstrate the $\Phi_p$-optimality of certain group divisible designs. Our proof is based on the method of KKT conditions in nonlinear programming.

scholarly journals Articulated Actions of the Family Health Strategy Teams and their Centres of Support in the State of Piauí, Brazil

10.3823/2275 ◽  
2017 ◽  
Vol 10 ◽  
Author(s):  
Izaias Almeida Belas ◽  
Jorge Henrique Alves da Rocha ◽  
Filipe Melo da Silva ◽  
João Victor Batista Lustosa ◽  
Wendell Soares Carneiro ◽  
...  
Keyword(s):  
Primary Care ◽  
Clinical Care ◽  
Family Health ◽  
The State ◽  
Cross Sectional ◽  
Health Strategy ◽  
The Family ◽  
The Matrix ◽  
Family Health Strategy ◽  
Care Support

Objective: From the perspective of professionals acting in the Family Health Strategy (FHS) in the state of Piauí, Brazil, the aim of this study was to asses the articulated actions of technical-pedagogical and clinical-care support offered by the Family Health Support Centres (FHSC) to the FHS’s professionals. Methodology: This is an analytical census retrospective study, with a cross sectional design developed in a quantitative approach with a descriptive and exploratory nature. The research data was collected through the Program of Improvement in Quality of Access in Primary Care (PIAQ-PC) in Brazil, on its second cycle in 2013, and were analysed by using descriptive statistics. Results: The actions of clinical-care support has been further developed by FHSC, all with frequency greater than 85%. In Piauí, the fields where FHSC has showed to be the nutritional care, rehabilitation and maternal and child care and also non-communicative diseases NCD that showed frequency higher than 85%. Conclusion: The FHSC initiative contributes significantly with their services to the FHS to achieve its goals. However, to make the work of these teams more effective there must be ownership of Primary Care Services by its user and appreciation of it by the managers. The developed actions are being supported and agreed on among the matrix support teams.  Keywords: Primary Health Care; Health  Promotion; Family Health.

scholarly journals Lax Representation and Darboux Solutions of the Classical Painlevé Second Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Irfan Mahmood ◽  
Muhammad Waseem
Keyword(s):  
Linear System ◽  
Darboux Transformation ◽  
Integrable Systems ◽  
Lax Pair ◽  
Lax Representation ◽  
Gauge Transformations ◽  
Additive Form

In this article, we present Darboux solutions of the classical Painlevé second equation. We reexpress the classical Painlevé second Lax pair in new setting introducing gauge transformations to yield its Darboux expression in additive form. The new linear system of that equation carries similar structure as other integrable systems possess in the AKNS scheme. Finally, we generalize the Darboux transformation of the classical Painlevé second equation to the N -th form in terms of Wranskian.

scholarly journals Enter the Matrix: An Examination of Patrick Pound's The Great Exhibition

2021 ◽  
Author(s):  
◽  
Milly Mitchell-Anyon
Keyword(s):  
Computer Algorithms ◽  
Georges Perec ◽  
Historical Knowledge ◽  
Marcel Duchamp ◽  
Algorithmic Approach ◽  
Found Objects ◽  
Vernacular Photography ◽  
The Family ◽  
The Matrix ◽  
Great Exhibition

<p>This thesis considers the practice of New Zealand-born artist, Patrick Pound (b. 1962) through an analysis of his survey show, Patrick Pound: The Great Exhibition, which was staged at the National Gallery of Victoria in Melbourne between 31 March and 30 July 2017. The Great Exhibition demonstrates the complexity and multiplicity of Pound’s practice, exemplifying the interconnectedness of his thinking and his use of an algorithmic approach to collecting, curating and categorisation. His depth of art-historical knowledge plays out as an intricate puzzle. The scope of The Great Exhibition is vast and, while it might appear to mostly involve the arrangement of more than 4,000 vernacular photographs and found objects, alongside 300 items from the NGV’s collection, the methodologies of collecting and curation employed by Pound are multifaceted.  I consider the constancy of Pound’s interrogation of authorship and meaning throughout his practice, which is integrally related to his use of vernacular photographs and found objects within The Great Exhibition. I examine our relationship with vernacular photography and how this is exposed in The Great Exhibition. The practices of artists such as Erik Kessels, Joachim Schmid and Marcel Duchamp provide context here. Chapter Three asks how The Great Exhibition fits within a wider context of exhibitions by artist-as-curators such as Fred Wilson’s Mining the Museum and Edward Steichen’s The Family of Man. This chapter also examines how computer algorithms can be applied as a framework for understanding The Great Exhibition’s curatorial logic. Pound’s complex system of sorting and categorising into matrices and intersections is considered in relation to writer Georges Perec and his understanding of Alan Turing’s conceptualisation of the ‘Automatic’ and ‘Oracle’ machines. My conclusion reflects on what can and cannot be learned from Patrick Pound’s The Great Exhibition.</p>

Quantum spin chains and classical integrable systems

2018 ◽  
Author(s):  
Anton Zabrodin
Keyword(s):  
Integrable Systems ◽  
Quantum Spin ◽  
Spin Chains ◽  
Algebraic Equations ◽  
Quantum Spin Chains ◽  
Many Body ◽  
Finite Dimensional ◽  
The Family ◽  
Classical Correspondence ◽  
Remarkable Relation

This chapter is a review of the recently established quantum-classical correspondence for integrable systems based on the construction of the master T-operator. For integrable inhomogeneous quantum spin chains with gl(N)-invariant R-matrices in finite-dimensional representations, the master T-operator is a sort of generating function for the family of commuting quantum transfer matrices depending on an infinite number of parameters. Any eigenvalue of the master T-operator is the tau-function of the classical modified KP hierarchy. It is a polynomial in the spectral parameter which is identified with the 0th time of the hierarchy. This implies a remarkable relation between the quantum spin chains and classical many-body integrable systems of particles of the Ruijsenaars-Schneider type. As an outcome, a system of algebraic equations can be obtained for the spectrum of the spin chain Hamiltonians.

The anatomy of an unstable node: a Levantine relict precipitates phylogenomic dissolution of higher-level relationships of the armoured harvestmen (Arachnida: Opiliones: Laniatores)

2019 ◽  
Author(s):  
Shlomi Aharon ◽  
Jesus A. Ballesteros ◽  
Audrey R. Crawford ◽  
Keyton Friske ◽  
Guilherme Gainett ◽  
...  
Keyword(s):  
New Species ◽  
Phylogenetic Signal ◽  
Future Studies ◽  
Phylogenetic Placement ◽  
Unstable Node ◽  
The Family ◽  
Family Level ◽  
The Matrix ◽  
Tree Topologies ◽  
Unstable Position

After tumultuous revisions to the family-level systematics of Laniatores (the armored harvestmen), the basally branching family Phalangodidae presently bears a disjunct and irregular distribution, attributed to the fragmentation of Pangea. One of the curious lineages assigned to Phalangodidae is the monotypic Israeli genus Haasus, the only Laniatores species that occurs in Israel, and whose presence in the Levant has been inferred to result from biogeographic connectivity with Eurasia. Recent surveys of Israeli caves have also yielded a new troglobitic morphospecies of Haasus. Here, we describe this new species as Haasus naasane sp. nov. So as to test the biogeographic affinity of Haasus, we sequenced DNA from both species and RNA from Haasus naasane sp. nov., to assess their phylogenetic placement. Our results showed that the new species is clearly closely related to Haasus judaeus, but Haasus itself is unambiguously nested within the largely Afrotropical family Pyramidopidae. In addition, the Japanese ‘phalangodid’ Proscotolemon sauteri was recovered as nested within the Southeast Asian family Petrobunidae. Phylogenomic placement of Haasus naasane sp. nov. in a 1550-locus matrix indicates that Pyramidopidae has an unstable position in the tree of Laniatores, with alternative partitioning of the matrix recovering high nodal support for mutually exclusive tree topologies. Exploration of phylogenetic signal showed the cause of this instability to be a considerable conflict between partitions, suggesting that the basal phylogeny of Laniatores may not yet be stable to addition of taxa. We transfer Haasus to Pyramidopidae (new familial assignment). Additionally, we transfer Proscotolemon to the family Petrobunidae (new familial assignment). Future studies on basal Laniatores phylogeny should emphasise the investigation of small-bodied and obscure groups that superficially resemble Phalangodidae.

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