Canonical Representation Research Articles

Simplification of tensor expressions in computer algebra

Computer algebra is widely used in various fields of mathematics, physics and other sciences. The simplification of tensor expressions is an important special case of computer algebra. In this paper, we consider the reduction of tensor polynomials to canonical form, taking into account the properties of symmetry under permutations of indices, the symmetries associated with the renaming of summation indices, and also linear relations between tensors of a general form. We give a definition of the canonical representation for polynomial (multiplicative) expressions of variables with abstract indices, which is the result of averaging of the original expression by the action of some finite group (the signature stabilizer). In practice, the proposed algorithms demonstrate high efficiency for expressions made of Riemann curvature tensors.

Time Scales of Benthic Macrofaunal Response to Pelagic Production Differ Between Major Feeding Groups

Benthic macrofauna, as an element of rich and diverse benthic communities of the shelf seas, play a key role in marine biogeochemical cycles and support a wide range of ecosystem services. To better understand how macrofauna affects mass and energy fluxes within the seabed and between the bed and the pelagic, it is fundamental to characterise their structural and dynamic response to the quantity, quality and timing of food supply. To do so, we have combined long-term time-series of pelagic productivity and macrofaunal abundance with a model of benthic food web to: (1) estimate the characteristic response time scales of major groups of benthic macrofauna to food availability, (2) relate these to carbon fluxes within sediments and across the benthic-pelagic boundary, and (3) explore the mechanisms responsible. The model was designed as a canonical representation of the benthic system, retaining the key pathways that connect benthic macrofauna to pelagic environment, but aggregating variables and groups that were not explicitly observed. Both observations and model simulations revealed pronounced differences between deposit and suspension feeders in their rate of response to phytoplankton blooms: deposit feeders showed a dampened response lagging 26-125 days behind the peak in pelagic production, while suspension feeders responded rapidly, within only 5-7 days. The model parametrisation obtained during calibration relates this to differences in feeding modes, in (trophic) proximity to primary production and in rates of ingestion and losses. Specifically, suspension feeders are predicted to act as a gateway to pelagic productivity, controlling the quantity of organic carbon reaching sediment-dwelling fauna.

A new representation of polygonal curves based on piecewise-linear functions with potential application in shape identification

In this paper, a new representation of polygonal curves is proposed. The motivation behind this proposal is to find a descriptive model with potential application in shape identification. In particular, our work addresses the problem of identifying the shape of a given polygonal curve from a set of different ones. In accordance with the proposed representation, a curve is originally described as a series of consecutive points whose connection by straight lines sketches its characteristic shape. In order to define a proper identification scheme, each series of points is mapped to a one-dimensional piecewise-linear function that assigns to each point its corresponding angle. Depending on the case under study, this assignation is performed by following two possible alternatives: (i) in the case where the shape to be identified exactly corresponds with one of the previously stored ones, a sorted sequence beginning from the minimum value of angle and ending with the maximum one is considered, and (ii) in the case where the shape to be identified is similar to one of the previously stored ones, the assignation is performed by following the sequence of points as they appear in the polygonal curve (without sorting). Under this scheme of representation, by a cyclic comparative process between the computed functions, after several steps it is decided whether the graph of any input piecewise-linear function matches the polygonal curve to be recognized within a certain tolerance. In particular, in the case of the identification of two equal shapes, this proposal exploits the well-known principle of similarity geometry, which allows a polygon to be recognized independently of scale, translation and rotation. In order to validate this representation, a comparative analysis between two different shape identification methods (Fourier descriptors and canonical representation) and the piecewise-linear proposal is performed.

Multispectral camera as spatio-spectrophotometer under uncontrolled illumination.

Multispectral constancy enables the illuminant invariant representation of multi-spectral data. This article proposes an experimental investigation of multispectral constancy through the use of multispectral camera as a spectrophotometer for the reconstruction of surface reflectance. Three images with varying illuminations are captured and the spectra of material surfaces is reconstructed. The acquired images are transformed into canonical representation through the use of diagonal transform and spectral adaptation transform. Experimental results show that use of multispectral constancy is beneficial for both filter-wheel and snapshot multi-spectral cameras. The proposed concept is robust to errors in illuminant estimation and is able to perform well with linear spectral reconstruction method. This work makes us one step closer to the use of multispectral imaging for computer vision.

Rough integrators on Banach manifolds

We introduce a notion of p-rough integrator on any Banach manifolds which plays the role of weak geometric Hölder p-rough paths in the usual Banach space setting, for any p⩾1. The awaited results on rough differential equations driven by such objects are proved, and a canonical representation is given if the manifold is equipped with a connection.

Statistical Analysis of the Mixed Fractional Ornstein--Uhlenbeck Process

This paper addresses the problem of estimating the drift parameter of the Ornstein--Uhlenbeck-type process driven by the sum of independent standard and fractional Brownian motions. With the help of some recent results on the canonical representation and spectral structure of mixed processes, the maximum likelihood estimator is shown to be consistent and asymptotically normal in the large-sample limit.

Explicit Description of Some Classes of Non-Bending Surfaces

Here we consider a family of axially symmetric surfaces modeling the shape of thin mechanical shells that are deformable without bending under uniform loading. With the exception of very few surfaces, like the well known right circular cylinder and the sphere, the surfaces of this family have no closed form description in elementary functions. Our main goal is to present their explicit parameterizations including both classes of open and closed families. We distinguish four classes of non-bending surfaces differing by their canonical representations using the normal elliptic integrals and the Jacobian elliptic functions.

Study on a Spinorial Representation of Linear Canonical Transformation

This paper is mainly focused on the description of an approach for establishing a spinorial representation of linear canonical transformations. It can be considered as a continuation of our previous works concerning linear canonical transformations and phase space representation of quantum theory. The said method is based on the development of an adequate parameterization of linear canonical transformations which permits to represent them with special pseudo-orthogonal transformations. Obtaining this pseudo-orthogonal representation makes it possible to establish the spinorial representation exploiting the well-known relation existing between special pseudo-orthogonal and spin groups. The cases of one dimension and general multidimensional theories are both studied. The design of the pseudo-orthogonal transformation associated to a linear canonical transformation is achieved by introducing adequate operators which are linear combinations of reduced momentum and coordinate operators. It is shown that a linear canonical transformation is equivalent to a special pseudo-orthogonal transformation defined in the set formed by these adequate operators. The spinorial representation is then deduced by defining a composite operator which is linear combinations of the tensorial products of the generators of the Clifford algebra with the adequate operators defining the special pseudo-orthogonal representation. It is established that unlike the case of a spinorial representation associated with an ordinary commutative vector space, the main invariant corresponding to the transformation is not the square of the composite operator but a higher degree polynomial function of it.

Разложение граничных представлений на плоскости Лобачевского в сечениях линейных расслоений

Earlier we described canonical (labelled by λ ∈C) and accompanying boundary representations of the group G = SU(1,1) on the Lobachevsky plane D in sections of linear bundles and decomposed canonical representations into irreducible ones. Now we decompose representations acting on distributions concentrated at the boundary of D. In the generic case 2λ ∉N they are diagonalizable, in the exceptional case Jordan blocks appear.

Elicitation of software testing requirements from the selected set of software's requirements in GOREP

Requirements engineering (RE) is employed to elicit, model, and analyse the requirements of software. Software requirements elicitation is the first sub-process of RE; and it is used to identify the requirements of software according to the need of the stakeholders. In literature, we identify that goal oriented requirements elicitation process (GOREP) do not support the identification of testing requirements from the functional requirements (FR) of software in early phase of RE. Therefore, to tackle this research issue, we proposed a method for the identification of the testing requirements from the FR in GOREP. In real life applications, only those requirements are implemented and tested, which are selected by the stakeholders. So in the proposed method we used fuzzy based technique for FR selection on the basis of non-functional requirements (NFR). The 'canonical representation of multiplication operation' (CRMO) associated with L−1 R−1 inverse arithmetic principle and 'graded mean integration representation' (GMIR) based on 'triangular fuzzy numbers' (TFNs) have been applied for the selection of the FR. In our work, an Institute Examination System (IES) is used as a case study to explain the steps of the proposed method.

Intelligent character recognition using fully convolutional neural networks

The recognition of handwritten text is challenging as there are virtually infinite ways a human can write the same message. Deep learning approaches for handwriting analysis have recently demonstrated breakthrough performance using both lexicon-based architectures and recurrent neural networks. This paper presents a fully convolutional network architecture which outputs arbitrary length symbol streams from handwritten text. A preprocessing step normalizes input blocks to a canonical representation which negates the need for costly recurrent symbol alignment correction. When a lexicon is known, we further introduce a probabilistic character error rate to correct errant word blocks. Our multi-state convolutional method is the first to demonstrate state-of-the-art results on both lexicon-based and arbitrary symbol based handwriting recognition benchmarks.

Preferences with changing ambiguity aversion

We provide two extensions of Gilboa and Schmeidler (J Math Econ 18:141–153, 1989)’s maxmin expected utility decision rule to accommodate a decision maker’s changing ambiguity attitudes. The two rules are, respectively, a weighted maxmin rule and a variant constraint rule. The former evaluates an act by a weighted average of its worst and best possible expected utilities over a set of priors, with the weights depending on the act. The latter evaluates an act by its worst expected utility over a neighborhood of a set of approximating priors, with the size of the neighborhood depending on the act. Canonical representations of the two rules are provided for classes of preference relations that exhibit, respectively, ambiguity aversion a la Schmeidler (Econometrica 57:571–587, 1989) and ambiguity aversion a la Ghirardato and Marinacci (J Econ Theory 102:251–289, 2002). In the second part of this paper, we study wealth effect under ambiguity. We propose axioms on absolute and relative ambiguity aversion and derive three representations for the ambiguity averse preference relations exhibiting decreasing (increasing) absolute ambiguity aversion. In particular, decreasing absolute ambiguity aversion implies that as the baseline utility of an act increases, a weighted maxmin decision maker puts less weight on the worst case, and a variant constraint decision maker considers a smaller neighborhood of approximating priors.

Naming polyhedra by general face-spirals – Theory and applications to fullerenes and other polyhedral molecules

We present a general face-spiral algorithm for cubic polyhedral graphs (including fullerenes and fulleroids), and extend it to the full class of all polyhedral graphs by way of the leapfrog transform. This yields compact canonical representations of polyhedra with a simple and intuitive geometrical interpretation, well suited for use by both computers and humans. Based on the algorithm, we suggest a unique, unambiguous, and simple notation for canonical naming of polyhedral graphs, up to automorphism, from which the graph is easily reconstructed. From this, we propose a practical nomenclature for all polyhedral molecules, and an especially compact form for the special class of fullerenes. A unique numbering of vertices is obtained as a byproduct of the spiral algorithm. This is required to denote modifications of the parent cage in IUPAC naming schemes. Similarly, the symmetry group of the molecule can be found together with the canonical general spiral at negligible cost. The algorithm is fully compatible with the classical spiral algorithm developed by Manolopoulos for fullerenes, i.e., classical spirals are accepted as input, and spiralable graphs lead to identical output. We prove that the algorithm is correct and complete.The worst case runtime complexity is for general N-vertex polyhedral graphs, with J the sum of all jump lengths. When the number of faces of any particular size is bounded by a constant, such as the case for fullerenes, this reduces to . We have calculated canonical general spirals for all 2,157,751,423 fullerene isomers from C20 to C200, as well as for all fullerene graphs that require jumps up to C400. Further, we have calculated canonical general spirals for large fullerenes with few or no classical spirals: all the Goldberg-Coxeter transforms up to C50,000 of the the non-spiralable chiral T-C380, D3-C384, D3-C440, and D3-C672 fullerenes, and for assorted fullerenes with no pentagon spiral starts. We verify exhaustively that the algorithm is linear for all the 2.7 × 1012 fullerene isomers up to C400, and show that this holds also for 11,413 large GC-transform fullerenes up to C50,000. On the used hardware, each single general spiral took about N × 200ns to produce for a CN fullerene, and the canonical general spiral was found in N × 22μs–32μs. Hence, we claim the algorithm to be efficient even for very large polyhedra.The algorithm is implemented in our program package Fullerene. In addition, the source code for a reference implementation of our proposed nomenclature for polyhedral molecules can be downloaded from http://erda.ku.dk/vgrid/Polyhedra/spirals/.

Remote state preparation using Einstein–Podolsky–Rosen representations

Because Einstein–Podolsky–Rosen states are well-defined on the Weyl algebra, their canonical cyclic representations are suitable for demonstrating the process of remote state preparation. Local projection-valued measurements and the corresponding recovery unitary operator are formulated in this representation, whereupon generalized equatorial states are prepared remotely and faithfully.

Convergent Temperature Representations in Artificial and Biological Neural Networks

Artificial Neural Networks (ANN) are increasingly successful in solving tasks long considered hallmarks of cognition in Biological Neural Networks (BNN), such as visual discrimination, playing Go and navigation. While the design of ANNs has been inspired by discoveries in BNNs, it is controversial whether both network types utilize the same fundamental principles and hence if ANNs can serve as models of animal cognition. However, if representations and algorithms are shared between BNNs and ANNs, then the analysis of processing in ANNs could lead to fundamental insights into their biological counterparts. Here, we generated and trained a deep convolutional neural network to solve a heat gradient navigation task using the behavioral repertoire of larval zebrafish. We found that these behavioral constraints led to striking similarities in temperature processing and representation in this ANN with biological circuits and neural dynamics underlying heat avoidance in larval zebrafish. This includes stimulus representation in ON and OFF types as well as ANN units showing adapting and sustained responses. Importantly, ANN performance critically relied on units representing temperature in a fish-like manner while other nodes were dispensable for network function. We next used the accessibility of the ANN to uncover new features of the zebrafish BNN. We 25 identified a novel neuronal response type in the zebrafish brain that was predicted by the ANN but escaped detection in previous brain wide calcium imaging experiments. Finally, our approach generalized since training the same ANN constrained by the C. elegans motor repertoire resulted in distinct neural representations that match closely features observed in the worm. Together, these results emphasize convergence of ANNs and BNNs on canonical representations and that man made ANNs form a powerful tool to understand their biological counterparts.

Visual recall memory contains highly detailed and precise object and spatial information

While there has been much research investigating visual recognition memory, little work has examined the specific content of visual free recall memory, despite evidence that these may be two neurally distinct processes. Here, we quantify the capacity and resolution of visual free recall for complex real-world scene images using a drawing task. Participants (N=30) studied 30 real-world scenes (10s each) and after an 11-min digit span distractor task, drew as many images as possible from memory in as much detail as possible. To serve as benchmarks, separate participants made 1) drawings from the scene category names ("lower bound"), reflecting canonical representations of a scene, and 2) drawings created while looking at the image ("upper bound"), reflecting the maximum information that could be drawn. We leveraged online crowd-sourced experiments on Amazon Mechanical Turk to objectively score the content of these 1,782 drawings. These ratings revealed an impressive detail contained in peoples' memories. First, memory drawings were easily matched to their corresponding image, and were nearly as diagnostic as those drawn directly from the image. Second, memory drawings contained 73.9% of the objects in the drawings made directly from the image, and on average, participants recalled 151.3 objects across the experiment with very few false alarms (1.83 objects across the experiment). Third, the spatial arrangement of objects in memory drawings was highly accurate, and almost identical to the original image. Further, computer vision graph-based visual saliency maps significantly predicted which objects would be remembered by participants. Collectively these results suggest that visual recall memory contains diagnostic, detailed, and precise representations of real-world scenes. Meeting abstract presented at VSS 2018

On the Class of Infinitely Divisible Exponential Distribution

The class of infinitely divisible exponential distribution is determined by property of Levy measure from its canonical representation of characteristic functions. Levy measure of the exponential distribution is obtained belonging to a class of completely monotones and measurable function. The class of this Levy measure is governed from exponential distribution similarly to the class of Thorin.

The quest for minimal quotients for probabilistic and Markov automata

One of the prevailing ideas in applied concurrency theory and verification is the concept of automata minimization with respect to strong or weak bisimilarity. The minimal automata can be seen as canonical representations of the behaviour modulo the bisimilarity considered. Together with congruence results wrt. process algebraic operators, this can be exploited to alleviate the notorious state space explosion problem. In this paper, we aim at identifying minimal automata and canonical representations for concurrent probabilistic models. We present minimality and canonicity results for probabilistic and Markov automata modulo strong and weak probabilistic bisimilarity, together with the corresponding minimization algorithms. We also consider weak distribution bisimilarity, originally proposed for Markov automata. For this relation, the quest for minimality does not have a unique answer, since fanout minimality clashes with state and transition minimality. We present an SMT approach to enumerate fanout-minimal models.

G-marked moduli spaces

The aim of this paper is to investigate the closed subschemes of moduli spaces corresponding to projective varieties which admit an effective action by a given finite group [Formula: see text]. To achieve this, we introduce the moduli functor [Formula: see text] of [Formula: see text]-marked Gorenstein canonical models with Hilbert polynomial [Formula: see text], and prove the existence of [Formula: see text], the coarse moduli scheme for [Formula: see text]. Then we show that [Formula: see text] has a proper and finite morphism onto [Formula: see text] so that its image [Formula: see text] is a closed subscheme. In the end we obtain the canonical representation type decomposition [Formula: see text] of [Formula: see text] and use [Formula: see text] to study the structure of [Formula: see text].

Identification of substructures in complex networks using formal concept analysis

Purpose In recent years, the increasing complexity of the hyper-connected world demands new approaches for social network analysis. The main challenges are to find new computational methods that allow the representation, characterization and analysis of these social networks. Nowadays, formal concept analysis (FCA) is considered an alternative to identifying conceptual structures in a social network. In this FCA-based work, this paper aims to show the potential of building computational models based on implications to represent and analyze two-mode networks. Design/methodology/approach This study proposes an approach to find three important substructures in social networks such as conservative access patterns, minimum behavior patterns and canonical access patterns. The present study approach considered as a case study a database containing the access logs of a cable internet service provider. Findings The result allows us to uncover access patterns, conservative access patterns and minimum access behavior patterns. Furthermore, through the use of implications sets, the relationships between event-type elements (websites) in two-mode networks are analyzed. This paper discusses, in a generic form, the adopted procedures that can be extended to other social networks. Originality/value A new approach is proposed for the identification of conservative behavior in two-mode networks. The proper implications needed to handle minimum behavior pattern in two-mode networks is also proposed to be analyzed. The one-item conclusion implications are easy to understand and can be more relevant to anyone looking for one particular website access pattern. Finally, a method for a canonical behavior representation in two-mode networks using a canonical set of implications (steam base), which present a minimal set of implications without loss of information, is proposed.