Dimensional Singularities Research Articles

The spectrum of an operator associated with G 2-instantons with 1-dimensional singularities and Hermitian Yang–Mills connections with isolated singularities

abstract: This is the first step in an attempt at a deformation theory for $G_{2}$-instantons with $1$-dimensional conic singularities. Under a set of model data, the linearization yields a Dirac operator $P$ on a certain bundle over $\mathbb{S}^{5}$, called the \textit{link operator}. As a dimension reduction, the link operator also arises from Hermitian Yang--Mills connections with isolated conic singularities on a Calabi--Yau $3$-fold. Using the quaternion structure in the Sasakian geometry of $\mathbb{S}^{5}$, we describe the set of all eigenvalues of $P$, denoted by $\Spec P$. We show that $\Spec P$ consists of finitely many integers induced by certain sheaf cohomologies on $\mathbb{P}^{2}$, and infinitely many real numbers induced by the spectrum of the rough Laplacian on the pullback endomorphism bundle over $\mathbb{S}^{5}$. The multiplicities and the form of an eigensection can be described fairly explicitly. In particular, there is a relation between the spectrum on $\mathbb{S}^{5}$ to certain sheaf cohomologies on~$\mathbb{P}^{2}$. Moreover, on a Calabi--Yau $3$-fold, the index of the linearized operator for admissible singular Hermitian Yang--Mills connections is also calculated, in terms of these sheaf cohomologies. Using the representation theory of $\SU(3)$ and the subgroup $S[U(1)\times U(2)]$, we show an example in which $\Spec P$ and the multiplicities can be completely determined.

Research on movement of dimensional singularities without elastic interaction in gasketed joints

Loss of clamping force due to elastic interaction is unpredictable and nonnegligible with the typical non-linear characteristics of gasketed joints. It is hard to weaken or eliminate elastic interaction by suitable joint dimensions. An analytical model of elastic interaction with dimensional representation is constructed using gasket deformation as a bridge. The influence of multi-dimensional dimensions on clamping force losses, such as bolt spacing, joint width and thickness, was explored by non-linear finite element analysis for a typical sandwich structure with two-bolt gasketed joints. Dimensional singularities without elastic interaction are validated and their approximate range is determined. It is clarified that initial clamping force or joint material can cause dimensional singularity movement. This paper provides a theoretical basis for solving the problem of clamping force losses and inhomogeneities effected by elastic interactions in engineering.

Complexity of direct and iterative solvers on space–time formulations and time-marching schemes for h-refined grids towards singularities

We study computational complexity aspects for Finite Element formulations considering hypercubic space–time full and time-marching discretization schemes for h-refined grids towards singularities. We perform a relatively comprehensive study comparing the computational time via time complexities of direct and iterative solvers. We focus on the space–time formulation with refined computational grids and the corresponding time slabs, namely, computational grids obtained by taking the “cross-sections” of the refined space–time mesh. We estimate the computational complexity of solving systems of linear equations for multidimensional meshes with arbitrary dimensional singularities encountered in space–time formulation and time-marching schemes. The choice between space–time formulation and time-marching schemes depends entirely on the problem’s nature and the properties that need to be addressed. Thus, the paper aims to discuss the computational complexities of both approaches rather than suggest a better formulation. Our considerations concern the computational complexity of sequental execution of the multi-frontal solvers, the iterative solvers, and the static condensation. Numerical experiments with Octave confirm our theoretical findings.

Fujiki-Oka Resolution for Three-dimensional Cyclic Quotient Singularities via Binary Trees

For three dimensional cyclic quotient singularities of type 1r(1,a,r−a−1) (resp. type 1r(1,a,r−a)), the Fujiki-Oka resolution coincides with one of crepant resolutions (resp. an economic resolution). In this paper, we will characterize binary trees which gives the Fujiki-Oka resolution for the above two series of cyclic quotient singularities.

The Jordan property for local fundamental groups

We show the Jordan property for regional fundamental groups of klt singularities of fixed dimension. Furthermore, we prove the existence of effective simultaneous index one covers for $n$-dimensional klt singularities. We give an application to the study of local class groups of klt singularities.

Extremal Area of Polygons, sliding along a circle

We determine all critical configurations for the Area function on polygons with vertices on a circle or an ellipse. For isolated critical points we compute their Morse index, resp index of the gradient vector field. We relate the computation at an isolated degenerate point to an eigenvalue question about combinations. In the even dimensional case non-isolated singularities occur as ‘zigzag trains’.

On the Briançon-Skoda theorem for analytic local rings with singularities

In this note, we study validity of the original Briançon-Skoda theorem for analytic local rings. By introducing a new concept of multiplier ideal sheaves associated to plurisubharmonic functions on any (reduced) complex space of pure dimension, we establish the original Briançon-Skoda theorem for normal weakly rational singularities (not essentially of finite type over C and Cohen-Macaulay). Moreover, we also characterize two dimensional rational singularities by the original Briançon-Skoda theorem.

Covers of rational double points in mixed characteristic

We further the classification of rational surface singularities. Suppose $(S, \mathfrak{n}, \mathcal{k})$ is a strictly Henselian regular local ring of mixed characteristic $(0, p > 5)$. We classify functions $f$ for which $S/(f)$ has an isolated rational singularity at the maximal ideal $\mathfrak{n}$. The classification of such functions are used to show that if $(R, \mathfrak{m}, \mathcal{k})$ is an excellent, strictly Henselian, Gorenstein rational singularity of dimension $2$ and mixed characteristic $(0, p > 5)$, then there exists a split finite cover of $\mbox{Spec}(R)$ by a regular scheme. We give an application of our result to the study of $2$-dimensional BCM-regular singularities in mixed characteristic.

Dark Energy, Dark Matter, and the Multiverse

Our third spatial dimension (3-D) is currently in the process of building the fourth spatial dimension. Dark Energy is being used to create higher dimensions of space from the first to the nth, and it is also responsible for the formation and the rotation of the fourth dimension from the point singularities of the vacuum space of the third dimension. To start the creation up to the nth dimension, n! zero dimensional singularities arise simultaneously from the n = 0 level of the void. Once the creation of an n dimensional ellipsoidal matter Universe has been completed, the outward Dark Energy force ends, and then the inward force of gravity takes over to crush the n dimensional Universe into the center of the nth dimensional singularity of the void. The system is now in a non-equilibrium state because the entropy of the system has attained its lowest possible value. To bring the system back to equilibrium there will arise out of the nth dimensional void an antimatter Universe which will reverse the process to deposit all the Dark Energy back into the original levels of the void so that the net change in entropy of the system becomes zero. Dark Matter is the matter that exists in the fourth spatial dimension as 4-D matter spewed there through the Black Hole singularities of our third dimension. Ellipsoidal matter and antimatter Multiverses rotating in opposite directions are a natural byproduct of this new theory of Physics which also solves certain particle Physics problems of the Standard Model.

An Implementation of the Lê–Teissier Method for Computing Local Euler Obstructions

Local Euler obstruction of a hypersurface with possibly positive dimensional singularities, a constructible function on the hypersurface, is considered in the context of symbolic computation. It is shown that the method due to Le and Teissier (Ann Math 114:457–491, 1981) that requires genericity conditions for computing local Euler obstructions can be realized as an algorithm in a deterministic way. The key ideas of the proposed algorithm are the use of comprehensive Grobner systems and of parametric local cohomology systems.

Examples of four dimensional cusp singularities

Examples of four dimensional cusp singularities

Towards a link theoretic characterization of smoothness

A theorem of Mumford states that, on complex surfaces, any normal isolated singularity whose link is diffeomorphic to a sphere is actually a smooth point. While this property fails in higher dimensions, McLean asks whether the contact structure that the link inherits from its embedding in the variety may suffice to characterize smooth points among normal isolated singularities. He proves that this is the case in dimension 3. In this paper, we use techniques from birational geometry to extend McLean's result to a large class of higher dimensional singularities. We also introduce a more refined invariant of the link using CR geometry, and conjecture that this invariant is strong enough to characterize smoothness in full generality.

Efficient Source Separation Enhancement Based on Advanced Multi-dimensional Transform Technique

This paper is concerned with blind separation of digital images from mixtures. It suggests the implementation of a blind separation technique on the ridgelet transform (RT) of the mixed images, instead of executing the separation on the mixtures in the time or a trigonometric transform domain. Ridgelet transform is a new orientational multi-resolution transform, and it is widely appropriate for characterizing the signals with dimensional singularities. Finite ridgelet transform is a discrete implementation of the ridgelet transform, which has numerical accuracy as the uninterrupted RT and has soft calculation intricacy. In contrary to the time domain, the RT finds more applications in image separation, because it appears sleek and edge sides of images have sparsity. In addition, the RT involves additional orientation information. The separated images are obtained using independent component analysis. The simulation results reveal that image separation in the RT domain is better, when compared to separation in the time or trigonometric transform domains.

A Computational Approach for Image Fusion using Medical Images based on Ripplet Transform

Image fusion is the process of combine different images from one or multiple imaging to raise the imaging quality. The concept is to improve the image content by combing images. Image fusion is important for improving the image quality and clinical application of medical image. Computed Tomography best suited for bone rift and reduction in providing information of the tissues, at the same time Magnetic Resolution Imaging gives soft tissue information and lacks in boundary information. In this paper Ripplet transform is resolving 2 dimensional singularities and show image edges more efficiently. At beginning the medical images are reconstruct by discrete ripplet transform. Fusion rules are applied to low frequency and high frequency subband. Apply inverse discrete ripplet transform to fused coefficient of low frequency and high frequency, for getting fused image. The performance of proposed system is calculated by quantitative approach such as spatial frequency, entropy, mutual information, peak signal to noise ratio, and root mean square error.

Noncommutative mirror symmetry for punctured surfaces

In 2013, Abouzaid, Auroux, Efimov, Katzarkov and Orlov showed that the wrapped Fukaya categories of punctured spheres and finite unbranched covers of punctured spheres are derived equivalent to the categories of singularities of a superpotential on certain crepant resolutions of toric 3 dimensional singularities. We generalize this result to other punctured Riemann surfaces and reformulate it in terms of certain noncommutative algebras coming from dimer models. In particular, given any consistent dimer model we can look at a subcategory of noncommutative matrix factorizations and show that this category is A ∞ \mathtt {A}_\infty -isomorphic to a subcategory of the wrapped Fukaya category of a punctured Riemann surface. The connection between the dimer model and the punctured Riemann surface then has a nice interpretation in terms of a duality on dimer models.

On the Griffiths numbers for higher dimensional singularities

We show that Yau’s conjecture on the inequalities for (n-1)-th Griffiths number and (n-1)-th Hironaka number does not hold for isolated rigid Gorenstein singularities of dimension greater than 2. But his conjecture on the inequality for (n-1)-th Griffiths number is true for irregular singularities.

Three dimensional canonical singularities in codimension two in positive characteristic

We investigate local structure of a three dimensional variety X defined over an algebraically closed field k of characteristic p>0 with at most canonical singularities. Under the assumption that p⩾3 and a general hyperplane cut of X has at most rational singularities, we show that local structure of X in codimension two is well understood in the level of local equations. Consequently, we find that i) any singularity of such a variety X in codimension two is compound Du Val, ii) it has a crepant resolution, iii) it is analytically a product of a rational double point and a nonsingular curve when p⩾3 with two exceptions in p=3, and iv) R1π⁎OX˜=R1π⁎KX˜=0 holds outside some finite points of X for any resolution of singularities π:X˜→X.

Generating toric noncommutative crepant resolutions

We present an algorithm that finds all toric noncommutative crepant resolutions of a given toric 3-dimensional Gorenstein singularity. The algorithm embeds the quivers of these algebras inside a real 3-dimensional torus such that the relations are homotopy relations. One can project these embedded quivers down to a 2-dimensional torus to obtain the corresponding dimer models. We discuss some examples and use the algorithm to show that all toric noncommutative crepant resolutions of a finite quotient of the conifold singularity can be obtained by mutating one basic dimer model. We also discuss how this algorithm might be extended to higher dimensional singularities.

Maximal ideal cycles over normal surface singularities of Brieskorn type

For normal two dimensional hypersurface singularities of Brieskorn type, concrete descriptions are given to both the fundamental cycle and the maximal ideal cycle on a star-shaped good resolution space. It is determined when these two cycles coincide.

Discussion of Experimental Data for 3D Crack Propagation on the Basis of Three Dimensional Singularities

Discussion of Experimental Data for 3D Crack Propagation on the Basis of Three Dimensional Singularities