Canonical Form Research Articles

Worldvolume fermions as baryons in holographic quantum chromodynamics with instantons

In this study, we investigated worldvolume fermions on the flavor brane in the D0–D4/D8 model, which is holographically equivalent to four-dimensional quantum chromodynamics with instantons or equivalently with a theta angle. The action involving the worldvolume fermions was obtained by the T-duality rules in string theory, and we accordingly derived their effective five-dimensional and canonical four-dimensional forms by using the systematic dimensional reduction and decomposition of the spinor. Subsequently, we used the AdS/CFT dictionary to evaluate the two-point correlation function as the spectral function for the worldvolume fermions and interpreted the fermions as baryons by analyzing their quantum number with the baryon vertex in holography. In this sense, the interacted action involving the worldvolume fermions and gauge field on the flavor brane was finally derived in holography, which describes the various interactions of mesons and baryons with instantons in the large-N limit. Therefore, this study provides a holographic picture to describe baryons and their interactions based on string theory, particularly in the presence of instantons or a theta angle.

The g6 pressure of hot Yang-Mills theory: canonical form of the integrand

We present major progress towards the determination of the last missing piece for the pressure of a Yang-Mills plasma at high temperatures at order g6 in the strong coupling constant. This order is of key importance due to its role in resolving the long-standing infrared problem of finite-temperature field theory within a dimensionally reduced effective field theory setup. By systematically applying linear transformations of integration variables, or momentum shifts, we resolve equivalences between different representations of Feynman sum-integrals on the integrand level, transforming those into a canonical form. At the order g6, this results in reducing a sum of O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}(100000) distinct sum-integrals which are produced from all four-loop vacuum diagrams down to merely 21. Furthermore, we succeed to map 11 of those onto known lower-loop structures. This leaves only 10 genuine 4-loop sum-integrals to be evaluated, thereby bringing the finalization of three decades of theoretical efforts within reach.

Adjoints and canonical forms of tree amplituhedra

We consider semi-algebraic subsets of the Grassmannian of lines in three-space called tree amplituhedra. These arise in the study of scattering amplitudes from particle physics. Our main result states that tree amplituhedra in $\mathrm {Gr}(2,4)$ are positive geometries. The numerator of their canonical form plays the role of the adjoint in Wachspress geometry, and is uniquely determined by explicit interpolation conditions.

Enhanced Oscillation Criteria for Non-Canonical Second-Order Advanced Dynamic Equations on Time Scales

This study aims to establish novel iterative oscillation criteria for second-order half-linear advanced dynamic equations in non-canonical form. The results extend and enhance recently established criteria for this type of equation by various authors and also encompass the classical criteria for related ordinary differential equations. Our methodology involves transforming the non-canonical equation into its corresponding canonical form. The inherent symmetry of these canonical forms plays a pivotal role in deriving our new criteria. By employing techniques from the theory of symmetric differential equations and utilizing symmetric functions, we establish precise conditions for oscillation. Several illustrative examples highlight the accuracy, applicability, and versatility of our results.

The study of the canonical forms of Killing tensor in vacuum with $$\Lambda $$

The study of the canonical forms of Killing tensor in vacuum with $$\Lambda $$

Mixed orthogonality graphs for continuous‐time state space models and orthogonal projections

In this article, we derive (local) orthogonality graphs for the popular continuous‐time state space models, including in particular multivariate continuous‐time ARMA (MCARMA) processes. In these (local) orthogonality graphs, vertices represent the components of the process, directed edges between the vertices indicate causal influences and undirected edges indicate contemporaneous correlations between the component processes. We present sufficient criteria for state space models to satisfy the assumptions of Fasen‐Hartmann and Schenk (2024a) so that the (local) orthogonality graphs are well‐defined and various Markov properties hold. Both directed and undirected edges in these graphs are characterised by orthogonal projections on well‐defined linear spaces. To compute these orthogonal projections, we use the unique controller canonical form of a state space model, which exists under mild assumptions, to recover the input process from the output process. We are then able to derive some alternative representations of the output process and its highest derivative. Finally, we apply these representations to calculate the necessary orthogonal projections, which culminate in the characterisations of the edges in the (local) orthogonality graph. These characterisations are given by the parameters of the controller canonical form and the covariance matrix of the driving Lévy process.

Semantic Effects of Long-Distance Wh-Scrambling in Japanese

This paper investigates the semantic effects of long-distance wh-scrambling in Japanese. Through qualitative analysis, it demonstrates that long-distance scrambling is a semantically vacuous movement in declarative sentences where a wh-phrase undergoes radical reconstruction for interpretation. It is demonstrated that it need not flow into semantic interpretations, in contrast to wh-movement. Long-distance wh-scrambling, on the other hand, has an impact on sentential readings in interrogative sentences since the wh-phrase must draw scope from its nearest question marker, which must be understood through the c-command relationship. The difference is discovered to originate from a processing technique, with each alternative interpretation being more prominent in both the jumbled sentence and the canonical form. This study provides a syntactic-semantic perspective and offers new insights into the semantic implications of scrambling, thus contributing to the existing literature.

Supersymmetry of the Robinson-Trautman solution

The Robinson-Trautman solution in the Einstein-Maxwell-Λ system admits a shear-free and twist-free null geodesic congruence with a nonvanishing expansion. Restricting to the case where the Maxwell field is aligned, i.e., the spacetime is algebraically special, we provide an exhaustive classification of supersymmetric Robinson-Trautman spacetimes in the four-dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 gauged supergravity. The differential constraints that arise from the integrability conditions of the Killing spinor equation enable us to systematically reconstruct the metric. We derive the explicit form of the Killing spinor either by directly integrating the Killing spinor equation or by casting the solution into the canonical form of supersymmetric solutions given in [13]. In any case, the supersymmetric Robinson-Trautman solution generically exhibits a naked singularity.

Six-derivative Yang-Mills couplings in heterotic string theory

In this work, we present a comprehensive analysis of the structure of six-derivative bosonic couplings in heterotic string theory. First, we determine the maximal covariant and Yang-Mills gauge invariant basis, which consists of 801 independent coupling constants. By imposing T-duality constraints on the circular reduction of these terms, we obtain 468 relations between the coupling constants at the six-derivative order and the known couplings at lower derivative orders. Through the use of field redefinitions, we are able to eliminate the remaining 333 coupling constants. Remarkably, we find that the Yang-Mills field strength only appears through the trace of two field strengths or their derivatives. Finally, we perform further field redefinition to rewrite the remaining couplings in a canonical form characterized by 85 independent couplings.

On the kinetics of internal gravity waves beyond the hydrostatic regime

We present a new derivation of the kinetic equation for weak, non-hydrostatic internal gravity wave turbulence. The equation is equivalent to the one obtained by Caillol & Zeitlin (Dyn. Atmos. Oceans, vol. 32, issue 2, 2000, pp. 81–112), but it takes a canonical form. We show that it conserves the energy without involving the resonance condition in frequency, and look for the isotropic part of the steady, scale-invariant solutions. We provide a parametrization of the resonant manifold of non-hydrostatic internal gravity wave triadic interactions. This allows us to simplify the collision integral, and to evaluate the transfer coefficients of all triadic interactions. In the hydrostatic limit, our equation is equivalent to the Hamiltonian description of Lvov & Tabak (Phys. Rev. Lett., vol. 87, issue 16, 2001, 168501).

Two-loop five-point two-mass planar integrals and double Lagrangian insertions in a Wilson loop

We consider the complete set of planar two-loop five-point Feynman integrals with two off-shell external legs. These integrals are relevant, for instance, for the calculation of the second-order QCD corrections to the production of two heavy vector bosons in association with a jet or a photon at a hadron collider. We construct pure bases for these integrals and reconstruct their analytic differential equations in canonical form through numerical sampling over finite fields. The newly identified symbol alphabet, one of the most complex to date, provides valuable data for bootstrap methods. We then apply our results to initiate the study of double Lagrangian insertions in a four-cusp Wilson loop in planar maximally supersymmetric Yang-Mills theory, computing it through two loops. We observe that it is finite, conformally invariant in four dimensions, and of uniform transcendentality. Furthermore, we provide numerical evidence for its positivity within the amplituhedron region through two loops.

Minimal rank weighted weak Drazin inverses

The concept of a minimal rank weak Drazin inverse for square matrices is extended to rectangular matrices. Precisely, a minimal rank weighted weak Drazin inverse is introduced and its properties are investigated. Some known generalized inverses such as the weighted Drazin inverse, the weighted core-EP inverse, and the weighted $p$-WGI are particular cases of a minimal rank weighted weak Drazin inverse. Thus, a wider class of generalized inverses is proposed. General representation forms of a minimal rank weighted weak Drazin inverse are presented as well as its canonical form. Applying the minimal rank weighted weak Drazin inverse, corresponding systems of linear matrix equations are solved and their solutions are expressed. As consequences of our results, new properties of minimal rank weak Drazin inverse are obtained.

A five-dimensional vacuum-defect wormhole space-time

In this study, we present a novel extension to the Klinkhamer vacuum-defect model by incorporating a fifth spatial dimension. This modification results in the formulation of a comprehensive five-dimensional wormhole space-time. Crucially, this extension preserves its classification as a vacuum solution to the field equations, thereby automatically satisfying all energy conditions. We demonstrate that this five-dimensional vacuum-defect space-time can be expressed in a pure canonical form, and thus, locally reduces to five-dimensional Minkowski space. Finally, we explore the geodesic equations in the vicinity of this five-dimensional vacuum-defect wormhole, offering insights into its intriguing properties.

Weighted index-MP matrices

Solving certain systems of matrix equations, we generalize notions of index-MP, MP-index and MP-index-MP matrices defined for square matrices and we introduce novel types of rectangular matrices, the so-called weighted index-MP, weighted MP-index and weighted MP-index-MP matrices. Many characterizations and expressions for new kinds of matrices are developed. The canonical forms of weighted index-MP, weighted MP-index and weighted MP-index-MP matrices are presented too. We apply the weighted index-MP, weighted MP-index and weighted MP-index-MP matrices to solve systems of linear matrix equations and represent their solutions.

On some planar Baumslag–Solitar actions

Let \mathrm{BS}(1,n)= \langle a,b : a b a^{-1}= b^{n}\rangle be the solvable Baumslag–Solitar group for n \geq 2 . We study representations of \mathrm{BS}(1, n) on the plane by orientation-preserving homeomorphisms, assuming that a acts as a linear map and b as a map with bounded displacement. We find that the possibilities for a faithful action depend greatly on the Jordan canonical form of the map h defined by the action of a . In case h is diagonalizable over \mathbb{R} , we shall give examples or prove rigidity theorems depending on the eigenvalues. We also show some rigidity in the cases where h is elliptic or parabolic. Then we give applications to the actions of \mathrm{BS}(1, n) by homeomorphisms of the torus.

Rogue waves and nonzero background solutions for the Gross–Pitaevskii equation with a parabolic potential

In this paper an integrable Gross–Pitaevskii equation with a parabolic potential and a gain term is investigated. Its solutions with a nonzero background are derived. These solutions are constructed by using biliearization reduction approach and connections between the nonlinear Schrödinger equation and the Gross–Pitaevskii equation. The solutions are presented in double-Wronskian form and are classified in terms of canonical forms of a certain matrix. Various breathers and rogue waves are analyzed and illustrated.

Control point modifications that preserve the Pythagorean–hodograph nature of planar quintic curves

Although planar Pythagorean–hodograph (PH) curves are compatible with the standard Bernstein–Bézier representations, freely modifying the control points will compromise their PH nature. The present study focuses on identifying control point displacements that ensure a given planar PH curve remains a PH curve. In particular, for planar quintic PH curves r(t), t∈[0,1] it is shown that finitely-many simultaneous displacements of two control points yield modified quintic PH curves, identified as the solutions of quadratic and cubic equations. As a more practical approach, modification of PH quintics in canonical form with r(0)=0 and r(1)=1 by the displacement of a single interior control point is considered, with the remaining interior control points being used to minimize a measure of deviation from the original PH quintic. As illustrated by several examples, this approach provides an efficient and intuitive means of effecting reasonable shape modifications within the space of planar quintic PH curves.

O(9, 25) symmetry of heterotic string theory at orders α′, α′2

In a recent study, we have observed that by imposing a truncated T-duality transformation on the circular reduction of the bosonic couplings in the heterotic theory at four- and six-derivative orders, we can calculate these couplings in a particular YM gauge where the YM potential vanishes but its field strength remains non-zero. Importantly, the coupling constants are independent of the gauge choice, so these results are valid across different YM gauge choices.In this work, we explore the cosmological reduction of these couplings when the YM gauge fields belong to the Cartan subalgebra of SO(32) or E8 × E8. We demonstrate that after applying appropriate one-dimensional field redefinitions and total derivative terms, the couplings can be expressed in a proposed O(9, 25)-invariant canonical form, which is the extension of the canonical O(9, 9)-invariant form for just the NS-NS fields proposed by Hohm and Zwiebach. This O(9, 25)-invariant expression is in terms of the trace of the first time derivative of the generalized metric, which encompasses both the YM field and the NS-NS fields.

Double Satake Diagrams and Canonical Forms in Compact Symmetric Triads

In this paper, we first introduce the notion of double Satake diagrams for compact symmetric triads. In terms of this notion, we give an alternative proof for the classification theorem for compact symmetric triads, which was originally given by Toshihiko Matsuki. Secondly, we introduce the notion of canonical forms for compact symmetric triads, and prove the existence of canonical forms for compact simple symmetric triads. We also give some properties for canonical forms.

VEXAS syndrome as a mimicker of ANCA-associated vasculitis.

Differentiating VEXAS syndrome from cases of canonical forms of primary vasculitis remains a significant clinical challenge, particularly for ANCA-associated vasculitis (AAV). We reviewed the clinical features of VEXAS as an AAV mimicker, while adding three new cases to the existing literature. We identified three cases of VEXAS with an AAV phenotype in our institution. We performed a comprehensive literature search of available similar cases and summarized and compared the findings. Inclusion criterion was a positive UBA1 mutation analysis. Patient 1 was referred for evaluation of eosinophilic granulomatosis with polyangiitis (GPA), but had no active respiratory symptoms, despite CT imaging showing widespread ground-glass opacities. Patient 2 had no history of sinus disease, despite being referred under the diagnostic construct of limited GPA. Patient 3 developed a novel inflammatory syndrome suspected to represent GPA. Six other cases were identified upon literature review. In all the cases, the most common findings were pulmonary infiltrates (67%), skin involvement (55%) and ocular manifestations (44%). Additionally, 44% of cases had renal involvement, with half of them displaying kidney lesions resembling the typical AAV pattern. VEXAS can mimic different phenotypes of AAV and should be considered in atypical AAV presentations, especially when refractory to multiple treatments. Further studies are needed to explore the immunologic basis for an AAV phenotype within the spectrum of VEXAS.