Loop Group Research Articles

Solitons for the Schrödinger flows on Hermitian symmetric spaces

We use loop group factorizations to construct Darboux transformations, Permutability formulas, Scaling Transformations, and explicit soliton solutions of the [Formula: see text]-d Schrödinger flows on compact Hermitian symmetric spaces.

Loop Groups and QNEC

We construct and study solitonic representations of the conformal net associated to some vacuum Positive Energy Representation (PER) of a loop group LG. For the corresponding solitonic states, we prove the Quantum Null Energy Condition (QNEC) and the Bekenstein Bound. As an intermediate result, we show that a Positive Energy Representation of a loop group LG can be extended to a PER of H^{s}(S^1,G) for s>3/2, where G is any compact, simple and simply connected Lie group. We also show the existence of the exponential map of the semidirect product LG rtimes R, with R a one-parameter subgroup of mathrm{Diff}_+(S^1), and we compute the adjoint action of H^{s+1}(S^1,G) on the stress energy tensor.

Solitons and dynamics for the integrable nonlocal pair-transition-coupled nonlinear Schrödinger equation

In this paper, we investigate the integrable nonlocal pair-transition-coupled nonlinear Schrödinger equation with the help of the loop group method. By constructing the Darboux transformation associated with a variable separation technique, some new soliton solutions such as the doubly-periodic bright/dark/antidark soliton, X-shaped kink, rogue wave-like soliton, mixed type kink and double breather (hump) and different types of novel rational solutions are easily derived. The obtained results are different from the solutions of the local nonlinear equations and nonlocal nonlinear Schrödinger equation. With appropriate choices of free parameters, the wave structures have large changes. Furthermore, the dynamic characteristic of soliton solutions are discussed on the basics of figures.

On Symmetry Properties of Frobenius Manifolds and Related Lie-Algebraic Structures

The aim of this paper is to develop an algebraically feasible approach to solutions of the oriented associativity equations. Our approach was based on a modification of the Adler–Kostant–Symes integrability scheme and applied to the co-adjoint orbits of the diffeomorphism loop group of the circle. A new two-parametric hierarchy of commuting to each other Monge type Hamiltonian vector fields is constructed. This hierarchy, jointly with a specially constructed reciprocal transformation, produces a Frobenius manifold potential function in terms of solutions of these Monge type Hamiltonian systems.

The robust inverse scattering method for focusing Ablowitz–Ladik equation on the non-vanishing background

In this paper, we consider the robust inverse scattering method for the Ablowitz–Ladik (AL) equation on the non-vanishing background, which can be used to deal with arbitrary-order poles on the branch points and spectral singularities in a unified way. The Darboux matrix is constructed with the aid of loop group method and considered within the framework of robust inverse scattering transform. Various soliton solutions are constructed without using the limit technique. These solutions include general soliton, breathers, as well as high order rogue wave solutions.

Factors Affecting the Removal Time of Separated Instruments

IntroductionSeparated endodontic instruments may adversely affect the outcome of endodontic treatment. The combination of ultrasonic techniques and dental operating microscopes appears to be effective in the removal of separated instruments compared with more randomized techniques. This study evaluated the roles of root canal curvature and separated instrument length on the time needed to loosen and retrieve the instrument fragments. MethodsThe retrieval procedures of 128 separated instruments referred to a private endodontic practice for retreatment by general practitioners were evaluated in patients who were monitored for a minimum of 6 months. Preoperative cone-beam computed tomographic images were used to measure separated instrument lengths in relation to the degrees of canal curvatures. Ultrasonic instruments were used in the initial phase to remove the tooth structure and to loosen the fractured instrument. In the second phase, ultrasonic instruments, wire loops, or XP Shapers (FKG Dentaire SA, La Chaux-de-Fonds, Switzerland) were used for fragment removal. The time periods for all procedures were recorded. Statistical analysis was completed applying log-normal regression, structural equation modeling, and linear regression using Stata Version 14.2 software (StataCorp LLC, College Station, TX). ResultsAll separated instruments were successfully retrieved. Using the protocol in this study, 89.8% of the instruments were removed using ultrasonic instruments alone with a mean time of 221 seconds. The instrument removal time was dependent on both the instrument length and the root canal curvature. Additionally, preparation times were proportionately longer with increasing separated instrument lengths when the loop device was required. ConclusionsThe preparation phase appears to have an important role in the retrieval of separated instruments. Preparation times for both non-loop and loop groups demonstrate that length and curvature are independent predictors of the log-transformed time. Generally, procedure times were extended with increasing file lengths and higher degrees of canal curvature.

Willmore Surfaces in Spheres via Loop Groups IV: On Totally Isotropic Willmore Two-Spheres in S6

In this paper the author derives a geometric characterization of totally isotropic Willmore two-spheres in S6, which also yields to a description of such surfaces in terms of the loop group language. Moreover, applying the loop group method, he also obtains an algorithm to construct totally isotropic Willmore two-spheres in S6. This allows him to derive new examples of geometric interests. He first obtains a new, totally isotropic Willmore two-sphere which is not S-Willmore (i.e., has no dual surface) in S6. This gives a negative answer to an open problem of Ejiri in 1988. In this way he also derives many new totally isotropic, branched Willmore two-spheres which are not S-Willmore in S6.

Comparison of the clinical outcomes of skin bridge loop ileostomy and traditional loop ileostomy in patients with low rectal cancer

To compare the clinical results of patients with low rectal cancer who underwent skin bridge loop ileostomy and traditional loop ileostomy, and provide clinical evidence for choosing a better ostomy method. We retrospectively collected data of 118 patients with rectal cancer who underwent low anterior resection and loop ileostomy. To investigate the patients characteristics, postoperative stoma-related complications and the frequency of exchanged ostomy bags. The differences of these indicators between the two groups of patients who underwent skin bridge loop ileostomy and traditional loop ileostomy were compared. The Visual Analog Scale (VAS) score of the skin bridge loop ileostomy group was lower than that of the traditional ileostomy loop group (P < 0.05). The skin bridge group had a lower Discoloration, Erosion, Tissue overgrowth (DET) score and incidence of mucocutaneous separation than the traditional group at the 1st and 2nd weeks after operation (P < 0.05). The average number of weekly exchanged ostomy bags was significantly less in the skin bridge group than in the traditional group within 4 weeks after surgery (P < 0.05). Our experience demonstrates that the skin bridge loop ileostomy may significantly reduce early postoperative stoma-related complications, the frequency of exchanged ostomy bags and patients’ medical costs after discharge.

A Construction of Representations of Loop Group and Affine Lie Algebra of $\frak {sl}_{n}$

A Construction of Representations of Loop Group and Affine Lie Algebra of $\frak {sl}_{n}$

Out of the Loop: The Value of a Preoperative Loopogram for Colostomy Reversal in Trauma.

Stoma is occasionally fashioned during trauma surgery. A loopogram is routinely conducted in the surgical planning for stoma reversal. This is associated with medical and cost implications. A study was undertaken to evaluate the influence of loopograms on the management of trauma patients at a Johannesburg hospital. A retrospective analysis of records in the stoma database (January 2013 to December 2018) was conducted. The patient demographics, method of injury, stoma-type, loopogram findings and post-operative courses were analysed. 112 records were obtained. 9 (8%) patients were excluded for pending investigations or surgery. 13 (11.6%) patients were excluded for incomplete data. The remaining 90 (80.3%) patients, with a mean age of 32.9 had non-contributory loopograms and underwent a reversal procedure. 43 (47.8%) had a loop colostomy while 47 (52.2%) had undergone a Hartmann's procedure. Mechanism of injury was stab wounds (81.4%L; 61.7%H); gunshot wounds (13.9%L; 29.7%H) and blunt trauma (L5% and 9%H). The post-operative complication rate was 30% for the loop group (2.3% ≥ Clavien-Dindo 3) and 25.5% for the Hartmann's group (4% ≥ Clavien-Dindo 3). The average timing to reversal was 38weeks (range 12-60) in the Hartmann's group and 22weeks (range 12-32) the loop colostomy group. Significant findings are infrequent on loopogram for trauma patients. When these findings are detected, the effect on management is questionable. They are not without complications and have cost and time implications. Loopograms are helpful in selective cases rather than as a routine investigation, particularly in resource-limited settings.

String^{𝑐} structures and modular invariants

In this paper, we study some algebraic topology aspects of String c ^c structures, more precisely, from the perspective of Whitehead tower and the perspective of the loop group of S p i n c ( n ) Spin^c(n) . We also extend the generalized Witten genera constructed for the first time by Chen et al. [J. Differential Geom. 88 (2011), pp. 1–40] to correspond to String c ^c structures of various levels and give vanishing results for them.

An amenability-like property of finite energy path and loop groups

We show that the groups of finite energy loops and paths (that is, those of Sobolev class H 1 ) with values in a compact connected Lie group, as well as their central extensions, satisfy an amenability-like property: they admit a left-invariant mean on the space of bounded functions uniformly continuous with regard to a left-invariant metric. Every strongly continuous unitary representation π of such a group (which we call skew-amenable) has a conjugation-invariant state on B(ℋ π ).

A de Rham model for complex analytic equivariant elliptic cohomology

We construct a cocycle model for complex analytic equivariant elliptic cohomology that refines Grojnowski's theory when the group is connected and Devoto's when the group is finite. We then construct Mathai–Quillen type cocycles for equivariant elliptic Euler and Thom classes, explaining how these are related to positive energy representations of loop groups. Finally, we show that these classes give a unique complex analytic equivariant refinement of Hopkins' “theorem of the cube” construction of the MString-orientation of elliptic cohomology.

Uniformization of semistable bundles on elliptic curves

Let G be a connected reductive complex algebraic group, and E a complex elliptic curve. Let GE denote the connected component of the trivial bundle in the stack of semistable G-bundles on E. We introduce a complex analytic uniformization of GE by adjoint quotients of reductive subgroups of the loop group of G. This can be viewed as a nonabelian version of the classical complex analytic uniformization E≃C⁎/qZ. We similarly construct a complex analytic uniformization of G itself via the exponential map, providing a nonabelian version of the standard isomorphism C⁎≃C/Z, and a complex analytic uniformization of GE generalizing the standard presentation E≃C/(Z⊕Zτ). Finally, we apply these results to the study of sheaves with nilpotent singular support. As an application to Betti geometric Langlands conjecture in genus 1, we define a functor from ShN(GE) (the semistable part of the automorphic category) to IndCohNˇ(LocsysGˇ(E)) (the spectral category).

Loop group and homotopy class as additive group

Loop group and homotopy class as additive group

Boundary $\mathcal{N} = 2$ theory, Floer homologies, affine algebras, and the Verlinde formula

Generalizing our ideas in [arXiv:1006.3313], we explain how topologically-twisted N=2 gauge theory on a four-manifold with boundary, will allow us to furnish purely physical proofs of (i) the Atiyah-Floer conjecture, (ii) Munoz's theorem relating quantum and instanton Floer cohomology, (iii) their monopole counterparts, and (iv) their higher rank generalizations. In the case where the boundary is a Seifert manifold, one can also relate its instanton Floer homology to modules of an affine algebra via a 2d A-model with target the based loop group. As an offshoot, we will be able to demonstrate an action of the affine algebra on the quantum cohomology of the moduli space of flat connections on a Riemann surface, as well as derive the Verlinde formula.

Micro-Rheological Changes of Red Blood Cells in the Presence of an Arterio-Venous Fistula or a Loop-Shaped Venous Graft in the Rat.

Introduction: In case of kidney failure, hemodialysis is the primary kidney replacement technique. Several vascular access methods used for the therapy, one of which is the arterio-venous fistula (AVF). In the AVF, the blood flow is altered, which can elevate the mechanical stress on the red blood cells (RBCs). This can affect the RBC hemorheological properties, and it can further cause systemic changes. To lower the turbulence and shear stress, we performed a loop-shaped arterio-arterial venous interposition graft (loop-shaped graft) to compare its effect to the conventional AVF.Materials and Methods: Thirty male Wistar were used (permission registration Nr.: 25/2016/UDCAW). The animals were randomly divided into sham-operated, AVF, and loop groups (n = 10/each). The superficial inferior epigastric vein (SIEV) was used to create the AVF and the loop-shaped graft. Blood samples were taken before/after the surgery and at the 1st, 3rd, and 5th postoperative weeks. We measured hemorhelogical, hematological, and blood gas parameters. The microcirculation of the hind limbs was also monitored using Laser Doppler fluxmetry.Results: Hematocrit, RBC count, and hemoglobin decreased by the 1st postoperative week. The erythrocyte aggregation values significantly increased in the fistula group by the 5th week (6.43 ± 2.31 vs. 13.60; p < 0.0001; vs. before operation). At the postoperative 1st week in the loop group, the values showed a significant decrease in RBC deformability. During the maturation period, dominantly at the 5th week, all values were normalized. The operated hind limb’s skin microcirculation significantly increased in the sham and loop group by the 1st week (39 ± 10.57 vs. 73.93 ± 1.97 BFU, p < 0.01). This increase wasn’t observed in the fistula group probably due to a steal-effect.Conclusion: Unlike in the loop group, in the presence of the fistula, several rheological parameters have changed. The loop-shaped graft had only minimal impact on micro-rheological parameters.

Iterated Laurent series over rings and the Contou-Carrère symbol

This article contains a survey of a new algebro-geometric approach for working with iterated algebraic loop groups associated with iterated Laurent series over arbitrary commutative rings and its applications to the study of the higher-dimensional Contou-Carrère symbol. In addition to the survey, the article also contains new results related to this symbol. The higher-dimensional Contou-Carrère symbol arises naturally when one considers deformation of a flag of algebraic subvarieties of an algebraic variety. The non-triviality of the problem is due to the fact that, in the case , for the group of invertible elements of the algebra of -iterated Laurent series over a ring, no representation is known in the form of an ind-flat scheme over this ring. Therefore, essentially new algebro-geometric constructions, notions, and methods are required. As an application of the new methods used, a description of continuous homomorphisms between algebras of iterated Laurent series over a ring is given, and an invertibility criterion for such endomorphisms is found. It is shown that the higher- dimensional Contou-Carrère symbol, restricted to algebras over the field of rational numbers, is given by a natural explicit formula, and this symbol extends uniquely to all rings. An explicit formula is also given for the higher-dimensional Contou-Carrère symbol in the case of all rings. The connection with higher-dimensional class field theory is described. As a new result, it is shown that the higher-dimensional Contou-Carrère symbol has a universal property. Namely, if one fixes a torsion-free ring and considers a flat group scheme over this ring such that any two points of the scheme are contained in an affine open subset, then after restricting to algebras over the fixed ring, all morphisms from the -iterated algebraic loop group of the Milnor -group of degree to the above group scheme factor through the higher-dimensional Contou-Carrère symbol. Bibliography: 67 titles.

A Matrix Approach for Analyzing Signal Flow Graph

Mason’s gain formula can grow factorially because of growth in the enumeration of paths in a directed graph. Each of the (n − 2)! permutation of the intermediate vertices includes a path between input and output nodes. This paper presents a novel method for analyzing the loop gain of a signal flow graph based on the transform matrix approach. This approach only requires matrix determinant operations to determine the transfer function with complexity O(n3) in the worst case, therefore rendering it more efficient than Mason’s gain formula. We derive the transfer function of the signal flow graph to the ratio of different cofactor matrices of the augmented matrix. By using the cofactor expansion, we then obtain a correspondence between the topological operation of deleting a vertex from a signal flow graph and the algebraic operation of eliminating a variable from the set of equations. A set of loops sharing the same backward edges, referred to as a loop group, is used to simplify the loop enumeration. Two examples of feedback networks demonstrate the intuitive approach to obtain the transfer function for both numerical and computer-aided symbolic analysis, which yields the same results as Mason’s gain formula. The transfer matrix offers an excellent physical insight, because it enables visualization of the signal flow.

A weak homotopy equivalence type result related to Kirchberg algebras

We obtain a weak homotopy equivalence type result between two topological groups associated with a Kirchberg algebra: the unitary group of the continuous asymptotic centralizer and the loop group of the automorphism group of the stabilization. This result plays a crucial role in our subsequent work on the classification of poly- \mathbb Z group actions on Kirchberg algebras. As a special case, we show that the K -groups of the continuous asymptotic centralizer are isomorphic to the KK -groups of the Kirchberg algebra.