Related papers: Depth and Stanley depth of multigraded modules
A new sub-grid-scale model is developed for studying influences of the Hall term on macroscopic aspects of magnetohydrodynamic turbulence. Although the Hall term makes numerical simulations extremely expensive by exciting high-wave-number…
Accurate stereo depth estimation plays a critical role in various 3D tasks in both indoor and outdoor environments. Recently, learning-based multi-view stereo methods have demonstrated competitive performance with a limited number of views.…
We investigate scalar restriction, scalar extension, and scalar coextension functors for graded modules, including their interplay with coarsening functors, graded tensor products, and graded Hom functors. This leads to several…
We study vertex algebras and their modules associated with possibly degenerate even lattices, using an approach somewhat different from others. Several known results are recovered and a number of new results are obtained. We also study…
The aim of this paper is to study the Stanley depth of symbolic powers of a squarefree monomial ideal. We prove that for every squarefree monomial ideal $I$ and every pair of integers $k, s\geq 1$, the inequalities ${\rm sdepth}…
Submodular functions are discrete functions that model laws of diminishing returns and enjoy numerous algorithmic applications. They have been used in many areas, including combinatorial optimization, machine learning, and economics. In…
Stein variational gradient descent (SVGD) is a deterministic sampling algorithm that iteratively transports a set of particles to approximate given distributions, based on an efficient gradient-based update that guarantees to optimally…
We obtain the exact values for depth and projective dimension and lower bounds for Stanley depth of the quotient rings of the edge ideals associated with all cubic circulant graphs.
Scaling properties of the quantum Hall metal-insulator transition are severely affected by finite size effects in small systems. Surprisingly, despite the narrow spatial range where probability structure functions exhibit multifractal…
We study the relationship between depth and regularity of a homogeneous ideal I and those of (I,f) and I:f, where f is a linear form or a monomial. Our results has several interesting consequences on depth and regularity of edge ideals of…
The observation of scaling in processes in which a weakly interacting probe delivers large momentum ${\bf q}$ to a many-body system reflects the dominance of incoherent scattering off target constituents. While a suitably defined scaling…
Let $R$ be a Cohen-Macaulay local ring, and let $I\subset R$ be an ideal with minimal reduction $J$. In this paper we attach to the pair $I$, $J$ a non-standard bigraded module $\Sigma^{I,J}$. The study of the bigraded Hilbert function of…
The aim of this work is to show how we can decompose a module (if decomposable) into an indecomposable module with the help of the minimization process.
A persistence module with $m$ discrete parameters is a diagram of vector spaces indexed by the poset $\mathbb{N}^m$. If we are only interested in the large scale behavior of such a diagram, then we can consider two diagrams equivalent if…
The subfactor approach to modular invariants gives insight into the fusion rule structure of the modular invariants.
We study modular forms of some congruence subgroups. In this paper, we treat the cases level is 2-power, 3-power or 5. Structures of graded rings and many identities of infinite sum or infinite product are given. Theory of rational (1/3,…
Existing analyses of optimization in deep learning are either continuous, focusing on (variants of) gradient flow, or discrete, directly treating (variants of) gradient descent. Gradient flow is amenable to theoretical analysis, but is…
A transverse multipole expansion is derived, including the longitudinal components necessarily present in regions of varying magnetic field profile. It can be used for exact numerical orbit following through the fringe field regions of…
Efforts to understand the generalization mystery in deep learning have led to the belief that gradient-based optimization induces a form of implicit regularization, a bias towards models of low "complexity." We study the implicit…
The London penetration depth $\lambda$ is the basic length scale for electromagnetic behavior in a superconductor. Precise measurements of $\lambda$ as a function of temperature, field, and impurity scattering have been instrumental in…