Rectangular Representation Research Articles

Large N and large representations of Schur line defect correlators

We study the large N and large representation limits of the Schur line defect correlators of the Wilson line operators transforming in the (anti)symmetric, hook and rectangular representations for \U0001d4a9 = 4 U(N) super Yang-Mills theory. By means of the factorization property, the large N correlators of the Wilson line operators in arbitrary representations can be exactly calculated in principle. In the large representation limit they turn out to be expressible in terms of certain infinite series such as Ramanujan’s general theta functions and the q-analogues of multiple zeta values (q-MZVs). Several generating functions for combinatorial objects, including partitions with non-negative cranks and conjugacy classes of general linear groups over finite fields, emerge from the large N correlators. Also we find conjectured properties of the automorphy and the hook-length expansion satisfied by the large N correlators.

Triangular Regions Representations for Matching Images With Viewpoint Changes

This paper proposes triangular region representations based on keypoints detected in images with viewpoint changes. The strongest keypoints in the reference and query images are allocated individually using a previously published contourlet-based approach to determine the keypoints. These selected keypoints serve as vertices of the triangular regions to be transformed into rectangular representations as simple numeric matrices. The suggested representation methods used to form rectangular matrices are full triangle representation (FTR) and a lighter representation called triangle medians and sides representation (TMSR). For the former, the intensity values along the lines connecting the anchor keypoint to the points between the other two triangular vertices form the rows of the representation matrix. These two triangle vertices are located in an allotted window around the anchor keypoint from the nearby keypoints. For the latter, the intensity values on the six triangular lines, medians, and sides formed the resulting rectangular matrix. The proposed representations are validated for image-matching applications using a descriptor-less matching method. Moreover, the performances of these algorithms are compared with those of traditional algorithms. The results confirmed the superiority of the proposed method over these algorithms.

Differential Neuromagnetic Mismatch Responses to Spoken Action and Abstract Verbs in the Motor Cortex

This paper discusses a novel implementable approach to an exact linearization procedure based on the implicit systems techniques. The formal procedure we propose includes a specific “splitting” of the nonlinear state representation in two parts that involve a basic rectangular representation and an auxiliary nonlinear algebraic equation. The proposed linear implicit systems description makes it possible to apply the conventional linear control techniques to an initially given sophisticated nonlinear dynamic model.

The KNTZ trick from arborescent calculus and the structure of the differential expansion

The recently suggested KNTZ trick completed the lasting search for exclusive Racah matrices $\bar S$ and $S$ for all rectangular representations and has a potential to help in the non-rectangular case as well. This was the last lacking insight about the structure of differential expansion of (rectangularly-)colored knot polynomials for twist knots -- and the resulting success is a spectacular achievement of modern knot theory in a classical field of representation theory, which was originally thought to be a tool for knot calculus but instead appeared to be its direct beneficiary. In this paper we explain that the KNTZ ansatz is actually a suggestion to convert the arborescent evolution matrix $\bar S\bar T^2\bar S$ into triangular form ${\cal B}$ and demonstrate how this works and what is the form of the old puzzles and miracles of the differential expansions from this perspective. The main new fully result is the conjecture for the triangular matrix ${\cal B}$ in the case of non-rectangular representation $[3,1]$. This paper does not simplify any calculations, but highlights the remaining problems, which one needs to overcome in order to {\it prove} that things really work. We believe that this discussion is also useful for further steps towards non-rectangular case and the related search of the gauge-invariant arborescent vertices. As an example we formulate a puzzling, still experimentally supported conjecture, that the study of twist knots only is sufficient to describe the shape of the differential expansion for all knots.

Tightening QC Relaxations of AC Optimal Power Flow Problems via Complex Per Unit Normalization

Optimal power flow (OPF) is a key problem in power system operations. OPF problems that use the nonlinear AC power flow equations to accurately model the network physics have inherent challenges associated with non-convexity. To address these challenges, recent research has applied various convex relaxation approaches to OPF problems. The QC relaxation is a promising approach that convexifies the trigonometric and product terms in the OPF problem by enclosing these terms in convex envelopes. The accuracy of the QC relaxation strongly depends on the tightness of these envelopes. This paper presents two improvements to these envelopes. The first improvement leverages a polar representation of the branch admittances in addition to the rectangular representation used previously. The second improvement is based on a coordinate transformation via a complex per unit base power normalization that rotates the power flow equations. The trigonometric envelopes resulting from this rotation can be tighter than the corresponding envelopes in previous QC relaxation formulations. Using an empirical analysis with a variety of test cases, this paper suggests an appropriate value for the angle of the complex base power. Comparing the results with a state-of-the-art QC formulation reveals the advantages of the proposed improvements.

Wilson loops and free energies in 3D $\mathcal{N}=4$ SYM: exact results, exponential asymptotics, and duality

We show that $U(N)$ $3d$ $\mathcal{N}=4$ supersymmetric gauge theories on $S^{3}$ with $N_{f}$ massive fundamental hypermultiplets and with a Fayet-Iliopoulos (FI) term are solvable in terms of generalized Selberg integrals. Finite $N$ expressions for the partition function and Wilson loop in arbitrary representations are given. We obtain explicit analytical expressions for Wilson loops with symmetric, antisymmetric, rectangular and hook representations, in terms of Gamma functions of complex argument. The free energy for orthogonal and symplectic gauge group is also given. The asymptotic expansion of the free energy is also presented, including a discussion of the appearance of exponentially small contributions. Duality checks of the analytical expressions for the partition functions are also carried out explicitly.

On exclusive Racah matrices [formula omitted] for rectangular representations

We elaborate on the recent observation that evolution for twist knots simplifies when described in terms of triangular evolution matrixB, not just its eigenvalues Λ, and provide a universal formula for B, applicable to arbitrary rectangular representation R=[rs]. This expression is in terms of skew characters and it remains literally the same for the 4-graded rectangularly-colored hyperpolynomials, if characters are substituted by Macdonald polynomials. Due to additional factorization property of the differential-expansion coefficients for the double-braid knots, explicit knowledge of twist-family evolution leads to a nearly explicit answer for Racah matrix S¯ in arbitrary rectangular representation R. We also relate matrix evolution to existence of a peculiar rotation U of Racah matrix, which diagonalizes the Z-factors in the differential expansion – what can be a key to further generalization to non-rectangular representations R.

Separated variables and wave functions for rational gl(N) spin chains in the companion twist frame

We propose a basis for rational gl(N) spin chains in an arbitrary rectangular representation (SA) that factorises the Bethe vectors into products of Slater determinants in Baxter Q-functions. This basis is constructed by repeated action of fused transfer matrices on a suitable reference state. We prove that it diagonalises the so-called B-operator; hence, the operatorial roots of the latter are the separated variables. The spectrum of the separated variables is also explicitly computed, and it turns out to be labeled by Gelfand-Tsetlin patterns. Our approach utilises a special choice of the spin chain twist which substantially simplifies derivations.

The Rectangular Representation of the Double Affine Hecke Algebra via Elliptic Schur–Weyl Duality

Abstract Given a module $M$ for the algebra ${\mathcal{D}}_{\mathtt{q}}(G)$ of quantum differential operators on $G$, and a positive integer $n$, we may equip the space $F_n^G(M)$ of invariant tensors in $V^{\otimes n}\otimes M$, with an action of the double affine Hecke algebra of type $A_{n-1}$. Here $G= SL_N$ or $GL_N$, and $V$ is the $N$-dimensional defining representation of $G$. In this paper, we take $M$ to be the basic ${\mathcal{D}}_{\mathtt{q}}(G)$-module, that is, the quantized coordinate algebra $M= {\mathcal{O}}_{\mathtt{q}}(G)$. We describe a weight basis for $F_n^G({\mathcal{O}}_{\mathtt{q}}(G))$ combinatorially in terms of walks in the type $A$ weight lattice, and standard periodic tableaux, and subsequently identify $F_n^G({\mathcal{O}}_{\mathtt{q}}(G))$ with the irreducible “rectangular representation” of height $N$ of the double affine Hecke algebra.

Performance of Understanding Students’ Construction In The Naming Fraction of The Three Representation

The purpose of this study is to describe students’ thinking processes in constructing the names of the fraction in circular, rectangular and number line representations by using APOS Theory (Action, Process, Object, and Scheme). For that purpose, the researcher used qualitative research approach. The subject of this research is the students of class XI in Malang. The process of collecting data begins with giving the questions to the subject to be completed. Then the students are interviewed to reinforce the answers that have given. From the performance of the subject and results of the interview, the data are analyzed using indicators of APOS Theory is action, process, object, and scheme. The results of this study indicate that for the thinking process students with circle representation S1 reached the stage of the scheme, while for S2 and S3 only reached the stage of the process. With the rectangular representation, S1 and S2 reach the stage of the scheme, but different with S3, S3 that reached the stage of the process. Moreover, the thinking process by using the number line representation S1, S2 and S3 can reach the stage of the scheme. To construct the name of the fraction with a circle and rectangular representation the subject is using the whole-part concept and area model of the fraction. As for the representation that uses the number line the subject using the concept of measurement and associates it with the length model in constructing the name of the fraction.

Factorization of differential expansion for non-rectangular representations

Factorization of the differential expansion (DE) coefficients for colored HOMFLY-PT polynomials of antiparallel double braids, originally discovered for rectangular representations [Formula: see text], in the case of rectangular representations [Formula: see text], is extended to the first non-rectangular representations [Formula: see text] and [Formula: see text]. This increases chances that such factorization will take place for generic [Formula: see text], thus fixing the shape of the DE. We illustrate the power of the method by conjecturing the DE-induced expression for double-braid polynomials for all [Formula: see text]. In variance with the rectangular case, the knowledge for double braids is not fully sufficient to deduce the exclusive Racah matrix [Formula: see text] — the entries in the sectors with nontrivial multiplicities sum up and remain unseparated. Still, a considerable piece of the matrix is extracted directly and its other elements can be found by solving the unitarity constraints.

Integration processes during frequency graph comprehension: Performance and eye movements while processing tree maps versus pie charts

SummaryFrequency graph types differ in the way how data are translated into visual representations. We compared 2 visualization methods, a traditional circular representation (pie chart) and a rectangular representation (constant column width tree map), which were hypothesized to differ regarding the cognitive ease of visual comparison processes. Performance was evaluated in tasks involving proportion and comparison judgments under both highly controlled and more realistic circumstances. The results showed performance benefits (in terms of reduced response times or error rates) for rectangular representations. Additional eye movement analyses revealed that this benefit was mainly due to a facilitation of scanning the graph for relevant information. The results suggest that facilitating comparison processes by representing the critical variable in less complex visual dimensions (i.e., straight length with constant orientation instead of surface area or curved length) eventually enhances the efficiency of integration processes during graph comprehension.

Generalized hypergeometric series for Racah matrices in rectangular representations

One of the spectacular results in mathematical physics is the expression of Racah matrices for symmetric representations of the quantum group [Formula: see text] through the Askey–Wilson polynomials, associated with the [Formula: see text]-hypergeometric functions [Formula: see text]. Recently it was shown that this is in fact the general property of symmetric representations, valid for arbitrary [Formula: see text] — at least for exclusive Racah matrices [Formula: see text]. The natural question then is what substitutes the conventional [Formula: see text]-hypergeometric polynomials when representations are more general? New advances in the theory of matrices [Formula: see text], provided by the study of differential expansions of knot polynomials, suggest that these are multiple sums over Young sub-diagrams of the one which describes the original representation of [Formula: see text]. A less trivial fact is that the entries of the sum are not just the factorized combinations of quantum dimensions, as in the ordinary hypergeometric series, but involve non-factorized quantities, like the skew characters and their further generalizations — as well as associated additional summations with the Littlewood–Richardson weights.

HOMFLY for twist knots and exclusive Racah matrices in representation [333

Next step is reported in the program of Racah matrices extraction from the differential expansion of HOMFLY polynomials for twist knots: from the double-column rectangular representations R=[rr] to a triple-column and triple-hook R=[333]. The main new phenomenon is the deviation of the particular coefficient f[332][21] from the corresponding skew dimension, what opens a way to further generalizations.

An improved ORNAM representation of grey images

An efficient image representation can save space and facilitate the manipulation of the acquired images. In order to further enhance the reconstructed image quality and reduce the number of the homogeneous blocks of the overlapping rectangular non-symmetry and anti-packing model (ORNAM) representation, in this paper, we propose an improved overlapping rectangular non-symmetry and anti-packing model representation (IORNAM) of grey images. Compared with most of the up-to-date and the state-of-the-art hierarchical representation methods, the new IORNAM representation is characterised by two properties: 1) it adopts a ratio parameter of the length and the width of a homogenous block to improve the reconstructed image quality; 2) it uses a new expansion method to anti-pack the subpatterns of grey images to further decrease the number of homogenous blocks, which is important for improving the compression ratios of image representation and reducing the complexities of many image manipulation algorithms. The experimental results presented in this paper demonstrate that: 1) the new IORNAM representation is able to achieve high representation efficiency for grey images; 2) the new IORNAM representation outperforms most of the up-to-date and the state-of-the-art hierarchical representation methods of grey images.

Rectangular superpolynomials for the figure-eight knot 41

We rewrite the recently proposed differential expansion formula for HOMFLY polynomials of the knot $4_1$ in arbitrary rectangular representation $R=[r^s]$ as a sum over all Young sub-diagrams $\lambda$ of $R$ with extraordinary simple coefficients $D_{\lambda^{tr}}(r)\cdot D_\lambda(s)$ in front of the $Z$-factors. Somewhat miraculously, these coefficients are made from quantum dimensions of symmetric representations of the groups $SL(r)$ and $SL(s)$ and restrict summation to diagrams with no more than $s$ rows and $r$ columns. They possess a natural $\beta$-deformation to Macdonald dimensions and produces positive Laurent polynomials, which can be considered as plausible candidates for the role of the rectangular superpolynomials. Both polynomiality and positivity are non-evident properties of arising expressions, still they are true. This extends the previous suggestions for symmetric and antisymmetric representations (when $s=1$ or $r=1$ respectively) to arbitrary rectangular representations. As usual for differential expansion, there are additional gradings. In the only example, available for comparison -- that of the trefoil knot $3_1$, to which our results for $4_1$ are straightforwardly extended, -- one of them reproduces the "fourth grading" for hyperpolynomials. Factorization properties are nicely preserved even in the 5-graded case.

Linearization by means of Linear Implicit Rectangular Descriptions

This paper discusses a novel implementable approach to an exact linearization procedure based on the implicit systems techniques. The formal procedure we propose includes a specific “splitting” of the nonlinear state representation in two parts that involve a basic rectangular representation and an auxiliary nonlinear algebraic equation. The proposed linear implicit systems description makes it possible to apply the conventional linear control techniques to an initially given sophisticated nonlinear dynamic model.

On the descriptor variable observation of rectangular implicit representations, in the presence of column minimal indices blocks

Recently, it has been shown that implicit rectangular descriptions can be successfully used for modelling and controlling broad classes of linear systems, including systems with internal switches (i.e., variable structure linear systems where the variation is driven by switching signals). This technique consists in finding first the degree-of-freedom, characterizing the internal variable structure, and then making it unobservable by means of an ideal proportional and derivative descriptor variable feedback. When the proportional and derivative feedback is approximated by a suitable proper controller, then the degree-of-freedom is only attenuated in an epsilon order (i.e., the degree of approximation). In this article, we propose two different ways for observing the descriptor variable for implicit rectangular systems, in the presence of column minimal indices blocks. The first one concerns a descriptor variable observer based on fault detection; an apparent failure signal characterizes the variation of structure, which observation is required to support the synthesis of a standard state observer (this approach is constrained to minimum phase systems, with respect to the output—degree-of-freedom transfer). The second method concerns a descriptor variable observer based on precise finite-time adaptive structure detection.

On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid

Golumbic, Lipshteyn and Stern [12] proved that every graph can be represented as the edge intersection graph of paths on a grid (EPG graph), i.e., one can associate with each vertex of the graph a nontrivial path on a rectangular grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. For a nonnegative integer k, Bk-EPG graphs are defined as EPG graphs admitting a model in which each path has at most k bends. Circular-arc graphs are intersection graphs of open arcs of a circle. It is easy to see that every circular-arc graph is a B4-EPG graph, by embedding the circle into a rectangle of the grid. In this paper, we prove that circular-arc graphs are B3-EPG, and that there exist circular-arc graphs which are not B2-EPG. If we restrict ourselves to rectangular representations (i.e., the union of the paths used in the model is contained in the boundary of a rectangle of the grid), we obtain EPR (edge intersection of paths in a rectangle) representations. We may define Bk-EPR graphs, k≥0, the same way as Bk-EPG graphs. Circular-arc graphs are clearly B4-EPR graphs and we will show that there exist circular-arc graphs that are not B3-EPR graphs. We also show that normal circular-arc graphs are B2-EPR graphs and that there exist normal circular-arc graphs that are not B1-EPR graphs. Finally, we characterize B1-EPR graphs by a family of minimal forbidden induced subgraphs, and show that they form a subclass of normal Helly circular-arc graphs.

Factorization of differential expansion for antiparallel double-braid knots

Continuing the quest for exclusive Racah matrices, which are needed for evaluation of colored arborescent-knot polynomials in Chern-Simons theory, we suggest to extract them from a new kind of a double-evolution -- that of the antiparallel double-braids, which is a simple two-parametric family of two-bridge knots, generalizing the one-parametric family of twist knots. In the case of rectangular representations $R=[r^s]$ we found an evidence that the corresponding differential expansion miraculously factorizes and can be obtained from that for the twist knots. This reduces the problem of rectangular exclusive Racah to constructing the answers for just a few twist knots. We develop a recent conjecture on the structure of differential expansion for the simplest members of this family (the trefoil and the figure-eight knot) and provide the exhaustive answer for the first unknown case of $R=[33]$. The answer includes HOMFLY of arbitrary twist and double-braid knots and Racah matrices $\bar S$ and $S$ -- what allows to calculate $[33]$-colored polynomials for arbitrary arborescent (double-fat) knots. For generic rectangular representations described in detail are only the contributions of the single- and two-floor pyramids, the way to proceed is explicitly illustrated by the examples of $R=[44]$ and $R=[55]$. This solves the difficult part of the problem, but the last tedious step towards explicit formulas for generic exclusive rectangular Racah matrices still remains to be made.