Diagonal Class Research Articles

Heegner points, stark-Heegner points, and diagonal classes

Heegner points, stark-Heegner points, and diagonal classes

Torsion properties of modified diagonal classes on triple products of modular curves

AbstractConsider three normalized cuspidal eigenforms of weight $2$ and prime level p. Under the assumption that the global root number of the associated triple product L-function is $+1$ , we prove that the complex Abel–Jacobi image of the modified diagonal cycle of Gross–Kudla–Schoen on the triple product of the modular curve $X_0(p)$ is torsion in the corresponding Hecke isotypic component of the Griffiths intermediate Jacobian. The same result holds with the complex Abel–Jacobi map replaced by its étale counterpart. As an application, we deduce torsion properties of Chow–Heegner points associated with modified diagonal cycles on elliptic curves of prime conductor with split multiplicative reduction. The approach also works in the case of composite square-free level.

A Novel Diagonal Class Entropy-Based Multilevel Image Thresholding Using Coral Reef Optimization

In the normal image thresholding methods based on two-dimensional histogram, the edge information of the regions is not maintained because of the local averaging activity used. Moreover, the computation time increases with the increase in the level of thresholds. This paper focusses on retaining more edge information by calculating the image entropy along the diagonal regions of the gray level co-occurrence matrix inspired from the partitioned design structure matrix, which is a novel idea. In addition, the key to our success is the theoretical investigation of a novel diagonal class entropy (DCE) concept that utilizes the minimum area for computation. The benefits of the proposed method are: 1) improved results; 2) efficient to preserve more precise shape of the edges; and 3) the computation time decreases with the increase in the threshold levels. The optimal thresholds are obtained by minimizing the DCE using coral reef optimization (CRO). A first hand fitness function for multilevel image thresholding is derived. The fight for space and the efficient reproduction characteristics of the CRO makes it attractive for this application. Benchmark images from the Berkley segmentation dataset are taken to experiment. Our results are compared with other state-of-the-art thresholding methods. The results obtained are encouraging and may set the path for further investigation in the domain of multilevel thresholding.

Classification of holomorphic endomorphisms of Hopf manifolds

We give a complete classification of holomorphic endomorphisms of Hopf manifolds in dimensions two and three. In particular, we show that these endomorphisms are automorphisms except for the diagonal class, in which case they are quasi-homogeneous.

An improved thermal exchange optimization based GLCM for multi-level image segmentation

The gray-level co-occurrence matrix (GLCM) can obtain the pixel matrix of the image, and selecting multiple thresholds for the matrix can obtain better segmentation results. However, as the number of threshold increases, the computational complexity of the algorithm will also increase. In order to solve this problem, this paper proposes a multi-threshold image segmentation method based on thermal exchange optimization (TEO) algorithm, and take a novel diagonal class entropy (DCE) as the fitness function. We improve TEO algorithm by using two strategic methods of Levy flight (LF) and opposition-based learning (OBL). In order to verify the segmentation ability of the proposed algorithm, color natural images, satellite images and Berkeley images are taken as experimental objects to analyze the segmentation result graph and image segmentation quality evaluation indexes. Experimental results show that the GLCM-ITEO algorithm has good segmentation capability, less CPU time.

Multilevel Color Image Segmentation Based on GLCM and Improved Salp Swarm Algorithm

The grayscale co-occurrence matrix (GLCM) can be adapted to segment the image according to the pixels, but the segmentation effect becomes worse as the number of threshold increases. To solve this problem, we propose an improved salp swarm algorithm (LSSA) to optimize GLCM, with the novel diagonal class entropy (DCE) as the fitness function of the GLCM algorithm. At the same time, in order to increase the optimization ability of traditional SSA algorithm, Levy flight (LF) strategy should be improved. Through experiments on the LSSA algorithm of the color natural images, the satellite images, and the Berkeley images, the segmentation quality of the segmented images is evaluated by peak signal-to-noise ratio, feature similarity, probability rand index, variation of information, global consistency error, and boundary displacement error. The experimental results show that the segmentation ability of the GLCM-LSSA algorithm is superior to other comparison algorithms and has a good segmentation ability.

Anisotropic loop quantum cosmology with self-dual variables

A loop quantization of the diagonal class A Bianchi models starting from the complex-valued self-dual connection variables is presented in this paper. The basic operators in the quantum theory correspond to areas and generalized holonomies of the Ashtekar connection and the reality conditions are implemented via the choice of the inner product on the kinematical Hilbert space. The action of the Hamiltonian constraint operator is given explicitly for the case when the matter content is a massless scalar field (in which case the scalar field can be used as a relational clock), and it is shown that the big-bang and big-crunch singularities are resolved in the sense that singular and non-singular states decouple under the action of the Hamiltonian constraint operator.

Some remarks on modified diagonals

We prove a number of basic vanishing results for modified diagonal classes. We also obtain some sharp results for modified diagonals of curves and abelian varieties, and we prove a conjecture of O’Grady about modified diagonals on double covers.

Finite-dimensionality and cycles on powers of $K3$ surfaces

For a K3 surface S , consider the subring of \mathrm {CH}(S^n) generated by divisor and diagonal classes (with \mathbb Q -coefficients). Voisin conjectures that the restriction of the cycle class map to this ring is injective. We prove that Voisin's conjecture is equivalent to the finite-dimensionality of S in the sense of Kimura-O'Sullivan. As a consequence, we obtain examples of S whose Hilbert schemes satisfy the Beauville-Voisin conjecture.

Monotonic functions in Bianchi models: why they exist and how to find them

All rigorous and detailed dynamical results in Bianchi cosmology rest upon the existence of a hierarchical structure of conserved quantities and monotonic functions. In this paper we uncover the underlying general mechanism and derive this hierarchical structure from the scale-automorphism group for an illustrative example, vacuum and diagonal class A perfect fluid models. First, kinematically, the scale-automorphism group leads to a reduced dynamical system that consists of a hierarchy of scale-automorphism invariant sets. Second, we show that, dynamically, the scale-automorphism group results in scale-automorphism invariant monotone functions and conserved quantities that restrict the flow of the reduced dynamical system.

Acquisition/Processing—A world of reality—Designing land 3D programs for signal, noise, and prestack migration—Part 2

This article is the continuation of a two-part tutorial. The first part (October, 2004) reviewed methods of estimating signal and understanding noise in a given project area, discussed the concepts of trace density and statistical diversity, and addressed concepts in prestack migration. This part begins with a brief discussion of the merits of various model types, which will lead to a discussion of robustness during implementation. The tutorial concludes by suggesting a data simulation method to evaluate the characteristics of a survey design. Much discussion about survey design has focused around the potential advantages of different survey geometries or model types (orthogonals, staggered orthogonals, bricks, diagonals, etc.). When compared using fixed source and receiver densities, Cooper and Herrera (2002) found very little difference amongst many of these variations. By focusing on limited statistics (for example nearest contributing offset in each bin), differences can be presented that favor some designs over others. However, when other statistics are presented (for example, the gap from nearest to second nearest offsets) the preferences often reverse. I have attempted to study the various common models using more generalized statistics as well as data simulations. The model types were divided into two classes: orthogonal designs (rigid orthogonal, offset orthogonal and staggered orthogonal) and diagonal designs (double brick, triple brick, skewed diagonals, rotated diagonals). I concluded that there were some clear benefits within each category. The staggered orthogonal design showed less bin-to-bin erratic behavior (refer to the offset homogeneity histograms at the top of Figure 14 and the azimuth homogeneity histograms at the top of Figure 15). Similarly, the 26.56° skewed diagonal design exhibited somewhat better statistics than related brick patterns, but the differences between the best of the orthogonal class (the staggered orthogonal) and the best of the diagonal class (the 26.56° skewed) were very subtle. …

Characteristic numbers of rational curves with cusp or prescribed triple contact

This note pursues the techniques of [Graber-Kock-Pandharipande] to give concise solutions to the characteristic number problem of rational curves in $\boldsymbol P^2$ or $\boldsymbol P^1\times\boldsymbol P^1$ with a cusp or a prescribed triple contact. The classes of such loci are computed in terms of modified psi classes, diagonal classes, and certain codimension-2 boundary classes. Via topological recursions the generating functions for the numbers can then be expressed in terms of the usual characteristic number potentials.

Removal of additive white noise using an adaptive Wiener filter with edge retention

In this paper, an adaptive Wiener filter for removal of additive white noise is proposed. Images are partitioned into a set of blocks of pixels, divided into five subsets of blocks according to their edge contents and directions, namely, shade, horizontal, vertical, and two diagonal classes. Each subset of blocks is used to define a covariance matrix, from which a Wiener filter is derived. For classification of blocks in the presence of noise, five eigenvectors calculated from the five covariance matrices are used. For a block classified into the appropriate class, five inner products are calculated between five eigenvectors and the block in the presence of noise. After classification, by switching the Wiener filter according to the input block, edge-preserving image filtering is useful. Experimental results are included to verify the usefulness of the proposed method. © 1999 Scripta Technica, Syst Comp Jpn, 30(1): 81–89, 1999

Splitting the Square of a Schur Function into its Symmetric and Antisymmetric Parts

We propose a new combinatorial description of the product of two Schur functions. In the particular case of the square of a Schur function S I, it allows to discriminate in a very natural way between the symmetric and antisymmetric parts of the square. In other words, it describes at the same time the expansion on the basis of Schur functions of the plethysms S 2(S I) and Λ2(S I). More generally our combinatorial interpretation of the multiplicities c IJ K = SISJ, SKleads to interesting q-analogues c IJ K (q) of these multiplicities. The combinatorial objects that we use are domino tableaux, namely tableaux made up of 1 × 2 rectangular boxes filled with integers weakly increasing along the rows and strictly increasing along the columns. Standard domino tableaux have already been considered by many authors [33], [6], [34], [8], [1], but, to the best of our knowledge, the expression of the Littlewood-Richardson coefficients in terms of Yamanouchi domino tableaux is new, as well as the bijection described in Section 7, and the notion of the diagonal class of a domino tableau, defined in Section 8. This construction leads to the definition of a new family of symmetric functions (H-functions), whose relevant properties are summarized in Section 9.

Diagonalization, uniformity, and fixed-point theorems

We derive new fixed-point theorems for subrecursive classes, together with a theorem on the uniformity of certain reductions, from a general formulation of the technique of delayed diagonalization. This formulation extends the main theorem of U. Schöning (Theoret. Comput. Sci. 18 (1982), 95–103) to cases which involve infinitely many diagonal classes Ck, and which allow each Ck to contain uncountably many members. The main technical work ties the familiar concept of a witness function directly to the often-studied Cantor-set topology on languages, and provides a “delay construction” which refines those due to Schöning, S. Breidtbart, and D. Schmidt. Our “a.e.” fixed-point theorems do not require that the “programming system” for the subrecursive class in question be well-behaved; we compare them to results which do. The other theorem is similar to the “uniform boundedness theorem” of classical analysis, and extends work of J. Grollmann and A. Selman.