Elementary Maps Research Articles

Exposedness of elementary positive maps between matrix algebras

ABSTRACT The positive linear maps Ad s which send matrices x to s ∗ x s play important roles in quantum information theory as well as matrix theory. It was proved by [Marciniak M. Rank properties of exposed positive maps. Linear Multilinear Alg. 2013;61:970–975] that the map Ad s generates an exposed ray of the convex cone of all positive linear maps. In this note, we provide two alternative proofs, using Choi matrices and Woronowicz's method, respectively.

A Novel 3D Topography Stitching Algorithm Based on Reflectance and Multimap

Surface topography is an efficient tool for the understanding of physical phenomena, especially if multiscale roughness analysis is performed. However, the observable scale range in a topography measured with 3D optical profilometers is quite limited. Therefore, all scales linked to a physical phenomenon might not be measured, which impedes the correct analysis of the surface. Stitching of 3D topographies, a technique combining elementary topographic maps into a larger one, can be used to increase the scale range for an objective lens. A high resolution over a large field of measurement topography is then generated. A literature review of 3D topography stitching algorithm highlights the stitching procedure, and detailed explanations on in-plane registration algorithms are provided. However, some existing 3D topography stitching algorithms are not sufficiently accurate for the registration of surface, especially at smaller scales. This paper proposes a new reflectance-based multimap 3D stitching algorithm and three of its variants. These algorithm variants are compared to three existing 3D stitching algorithms (geometric, cross-correlation and global optimization of differences) on four test cases, containing measured elementary topographic maps obtained on four surfaces and with four 3D optical profilometers (two focus variation microscopes and two interferometers). Five qualitative and quantitative criteria and indicators are proposed for the comparison of 3D topography stitching algorithms: visual inspection, run time, memory usage, mean repositioning error and stitching error estimator. Lastly, two quantitative indicators and criteria are new indicators proposed in this article. Overall, the new 3D stitching algorithms based on reflectance and multimaps have a lower mean repositioning error and stitching error estimator compared to other existing algorithms. This highlights the relevance of multimap stitching algorithms in the case of 3D topographies. A new decision-helping tool, the stitching gain lift plot (SGL plot), is described for the selection of the best stitching algorithm for a given test case. The SGL plot especially highlights the higher performance of two of the variants of the novel algorithm compared to the three existing 3D stitching algorithms.

Compositions and tensor products of linear maps between matrix algebras

In this semi-expository paper, we first explain key notions from current quantum information theory and criteria for them in a coherent way. These include separability/entanglement, Schmidt numbers of bi-partite states and block-positivity, together with various kinds of positive maps between matrix algebras like entanglement breaking maps, k-superpositive maps, completely positive maps, k-positive maps. We will begin with concrete examples of elementary positive maps given by x↦s⁎xs, and use Choi matrices and duality to explain all the notions mentioned above. We also show that the Choi matrix can be defined free from coordinates. The above notions of positive maps give rise to mapping cones, whose dual cones are characterized in terms of compositions or tensor products of linear maps. Through the discussion, we exhibit an identity which connects tensor products and compositions of linear maps between matrix algebras through the Choi matrices. Using this identity, we show that the description of the dual cone with tensor products is possible only when the involving cones are mapping cones, and recover various known criteria with ampliation for the notions mentioned above. As another applications of the identity, we construct various mapping cones arising from ampliation and factorization, and provide several equivalent statements to PPT (positive partial transpose) square conjecture in terms of tensor products.

Integrable matrix models in discrete space-time

We introduce a class of integrable dynamical systems of interacting classical matrix-valued fields propagating on a discrete space-time lattice, realized as many-body circuits built from elementary symplectic two-body maps. The models provide an efficient integrable Trotterization of non-relativistic \sigmaσ-models with complex Grassmannian manifolds as target spaces, including, as special cases, the higher-rank analogues of the Landau–Lifshitz field theory on complex projective spaces. As an application, we study transport of Noether charges in canonical local equilibrium states. We find a clear signature of superdiffusive behavior in the Kardar–Parisi–Zhang universality class, irrespectively of the chosen underlying global unitary symmetry group and the quotient structure of the compact phase space, providing a strong indication of superuniversal physics.

Additivity of elementary maps on gamma rings

Let M and M´ be Gamma rings, respectively. We study the additivity of surjective elementary maps of M × M´. We prove that if M contains a non-trivial γ-idempotent satisfying some conditions, then they are additive

MPEG-4 AVC stream-based saliency detection. Application to robust watermarking

By bridging uncompressed-domain saliency detection and MPEG-4 AVC compression principles, the present paper advances a methodological framework for extracting the saliency maps directly from the stream syntax elements. In this respect, inside each GOP, the intensity, color, orientation and motion elementary saliency maps are related to the energy of the luma coefficients, to the energy of chroma coefficients, to the gradient of the prediction modes and to the amplitude of the motion vectors, respectively. The three spatial saliency maps are pooled according to an average formula, while the static-temporal fusion is achieved by six different formulas. The experiments consider both ground-truth and applicative evaluations. The ground-truth benchmarking investigates the relation between the predicted MPEG-4 AVC saliency map and the actual human saliency, captured by eye-tracking devices. It is based on two corpora (representing density fixation maps and saccade locations), two objective criteria (related to the closeness between the predicted and the real saliency maps and to the difference between the behavior of the predicted saliency map in fixation and random locations), two objective measures (KLD – the Kullback Leibler Divergence and AUC – the Area Under the ROC Curve) and 5 state-of-the-art saliency models (3 acting in spatial domain and 2 acting in compressed domain). The applicative validation is carried out by integrating the MPEG-4 AVC saliency map into a robust watermarking application. As an overall conclusion, the paper demonstrates that although the MPEG-4 AVC standard does not explicitly relies on any visual saliency principle, its stream syntax elements preserve this property. Four main benefits for the MPEG-4 AVC based saliency extraction are thus brought to light: (1) it outperforms (or, at least, is as good as) state-of-the-art uncompressed domain methods, (2) it allows significant gains to be obtained in watermarking transparency (for prescribed data payload and robustness), (3) it is less sensitive to the randomness in the processed visual content, and (4) it has a linear computational complexity. For instance, the ground truth results exhibit absolute relative gains between 60% and 164% in KLD, between 17% and 21% in AUC, and relative gains in KLD sensitivity between 1.18 and 6.12 and in AUC sensitivity between 1.06 and 33.7; the applicative validation brings to light transparency gains up to 10 dB in PSNR.

Additivity of $$\alpha $$ α -elementary maps on triangular algebras

Let \(\mathfrak {T}\) be a triangular algebra and \(\mathfrak {S}\) an arbitrary ring. We study the additivity of surjective maps \(M:\mathfrak {T}\rightarrow \mathfrak {S}\) and \(M^{*}:\mathfrak {S}\rightarrow \mathfrak {T}\) preserving a family of sums of triple products on these rings. In this paper, we give sufficient conditions on \(\mathfrak {T}\) such that both M and \(M^{*}\) are additives. In particular, if \(\mathfrak {T}\) is a standard subalgebra of a nest algebra, then M and \(M^{*}\) are additives.

Additivity of Jordan elementary maps on nest subalgebras of factor von Neumann algebras

In this paper, we will show that if the ordered pair (F, F*) is a Jordan elementary map of A × B. Then both F and F* are additive.

HEVC saliency map computation

This paper investigates whether the information related to the human visual saliency is still preserved at the level of the HEVC compressed stream syntax elements. In this respect, a new saliency model, matched to the peculiarities of this emerging standard is defined. It consists of four elementary maps, describing the four main saliency features: intensity, color, orientation and motion. These maps are defined based on the energies of the luma and chroma coefficients, on the variations of the intra prediction modes and on the energy of motion vectors, respectively. They are fusioned according to 48 static and static-dynamic pooling formulas. The results are compared to three state-of-the-art uncompressed (pixel) domain as well as to the MPEG-4 AVC compressed domain saliency maps. It is brought to light that the HEVC saliency model outperforms (with singular exceptions) the state-of-the-art uncompressed domain and is as good as MPEG-4 AVC saliency model. We can thus state that, as its MPEG-4 AVC ancestor, although not designed based upon visual saliency principles, the HEVC compression standard preserves this human visual property at the level of its syntax elements.

Jordan Elementary Maps on Alternative Rings

In this article we study the Jordan elementary maps on alternative rings. We prove that if ℛ is a 2 and 3-torsion free prime alternative ring containing a nontrivial idempotent and ℛ′ is an arbitrary alternative ring, then every surjective Jordan elementary map (M, M*) of ℛ × ℛ′ is additive.

A Family of Nonparametric Density Estimation Algorithms

AbstractA new methodology for density estimation is proposed. The methodology, which builds on the one developed by Tabak and Vanden‐Eijnden, normalizes the data points through the composition of simple maps. The parameters of each map are determined through the maximization of a local quadratic approximation to the log‐likelihood. Various candidates for the elementary maps of each step are proposed; criteria for choosing one includes robustness, computational simplicity, and good behavior in high‐dimensional settings. A good choice is that of localized radial expansions, which depend on a single parameter: all the complexity of arbitrary, possibly convoluted probability densities can be built through the composition of such simple maps. © 2012 Wiley Periodicals, Inc.

“A Few of the Brightest, Cleanest Mexican Children”: School Segregation as a Form of Mundane Racism in Oxnard, California, 1900–1940

In this article, David G. García, Tara J. Yosso, and Frank P. Barajas examine the early twentieth-century origins of a dual schooling system that facilitated the reproduction of a cheap labor force and the marginalization of Mexicans in Oxnard, California. In their analysis of the 1930s Oxnard Elementary School District board minutes, alongside newspapers, maps, scholarly accounts, and oral history interviews, they argue that school segregation privileged Whites and discriminated against Mexicans as a form of mundane racism. The authors build on previous scholarship documenting the pervasiveness of racism in U.S. society to define mundane racism as the systematic subordination of Mexicans that occurred as a commonplace, ordinary way of conducting business within and beyond schools. Their findings complicate narratives that emphasize complete segregation in “Mexican schools,” while acknowledging the resistance of parents and the resilience of their children.

Decoding Successive Computational Stages of Saliency Processing

An important requirement for vision is to identify interesting and relevant regions of the environment for further processing. Some models assume that salient locations from a visual scene are encoded in a dedicated spatial saliency map [1, 2]. Then, a winner-take-all (WTA) mechanism [1, 2] is often believed to threshold the graded saliency representation and identify the most salient position in the visual field. Here we aimed to assess whether neural representations of graded saliency and the subsequent WTA mechanism can be dissociated. We presented images of natural scenes while subjects were in a scanner performing a demanding fixation task, and thus their attention was directed away. Signals in early visual cortex and posterior intraparietal sulcus (IPS) correlated with graded saliency as defined by a computational saliency model. Multivariate pattern classification [3, 4] revealed that the most salient position in the visual field was encoded in anterior IPS and frontal eye fields (FEF), thus reflecting a potential WTA stage. Our results thus confirm that graded saliency and WTA-thresholded saliency are encoded in distinct neural structures. This could provide the neural representation required for rapid and automatic orientation toward salient events in natural environments.

The Additivity of Multiplicative Maps on Rings

In this article, we first present a unified technique in the discussion of the additivity of multiplicative maps on rings with idempotents. We next apply the obtained result to discuss the additivity of surjective elementary maps in rings with idempotents.

Modelling Spatio-Temporal Saliency to Predict Gaze Direction for Short Videos

This paper presents a spatio-temporal saliency model that predicts eye movement during video free viewing. This model is inspired by the biology of the first steps of the human visual system. The model extracts two signals from video stream corresponding to the two main outputs of the retina: parvocellular and magnocellular. Then, both signals are split into elementary feature maps by cortical-like filters. These feature maps are used to form two saliency maps: a static and a dynamic one. These maps are then fused into a spatio-temporal saliency map. The model is evaluated by comparing the salient areas of each frame predicted by the spatio-temporal saliency map to the eye positions of different subjects during a free video viewing experiment with a large database (17000 frames). In parallel, the static and the dynamic pathways are analyzed to understand what is more or less salient and for what type of videos our model is a good or a poor predictor of eye movement.

On fiber products of elementary maps

The following characterization of fully closed maps is proved: a quotient map between regular spaces is fully closed if and only if it coincides with the fiber product of elementary maps between regular spaces.

AUTOMORPHISMS OF A WEYL-TYPE ALGEBRA I

Every non-associative algebra L corresponds to its symmetric semi-Lie algebra <TEX>$L_{[,]}$</TEX> with respect to its commutator. It is an interesting problem whether the equality <TEX>$Aut{non}(L)=Aut_{semi-Lie}(L)$</TEX> holds or not [2], [13]. We find the non-associative algebra automorphism groups <TEX>$Aut_{non}\; \frac\;{(WN_{0,0,1}_{[0,1,r_1...,r_p])}$</TEX> and <TEX>$Aut_{non-Lie}\; \frac\;{(WN_{0,0,1}_{[0,1,r_1...,r_p])}$</TEX> where every automorphism of the automorphism groups is the composition of elementary maps [3], [4], [7], [8], [9], [10], [11]. The results of the paper show that the F-algebra automorphism groups of a polynomial ring and its Laurent extension make easy to find the automorphism groups of the algebras in the paper.

Elementary maps on nest algebras

Let A 1 , A 2 be algebras and let M : A 1 → A 2 , M ∗ : A 2 → A 1 be maps. An elementary map of A 1 × A 2 is an ordered pair ( M , M ∗ ) such that { M ( A M ∗ ( B ) C ) = M ( A ) B M ( C ) , M ∗ ( B M ( A ) D ) = M ∗ ( B ) A M ∗ ( D ) for all A , C ∈ A 1 , B , D ∈ A 2 . In this paper, the general form of surjective elementary maps on standard subalgebras of nest algebras is described. In particular, such maps are automatically additive.

Elementary Birational Maps between Mori Toric Fiber 3-Spaces

In the present paper, we classify elementary equivariant birational maps (links) between Mori toric fiber 3-spaces. These links are naturally divided into several classes, depending on the dimension of the bases. For classes containing finitely many links, we present a complete list, in other cases we provide a local description (see the statements in Sec. 4). Almost all the proofs are of combinatorial nature, which is why we present proofs of the results from Sec. 3 only.

Additivity of Jordan elementary maps on nest algebras

Let A be a standard subalgebra of a nest algebra on a Hilbert space of dimension greater than one and B an arbitrary algebra. A Jordan elementary map of A × B is a pair ( M, M*) where M : A → B and M ∗ : B → A are two maps satisfying M ( AM ∗ ( B ) A ) = M ( A ) BM ( A ) , M ∗ ( BM ( A ) B ) = M ∗ ( B ) AM ∗ ( B ) for A ∈ A , B ∈ B . In this note, it is proved that for a special class of surjective Jordan elementary maps of A × B , every member in it is automatically additive. Also, we construct a counterexample which shows that this result is not necessarily true for all surjective Jordan elementary maps.