arXiv subjects
Publications and source records attributed to Sudeshna Roy.
In this paper we construct {\em adic}, {\em saturated} and $\varepsilon$-density functions for a torsion-free module in a graded setup. Then we give some simple criteria for checking the integral dependence of two graded modules $N\subseteq M$ in terms of various well-studied invariants.
Let $C$ be a commutative Noetherian ring containing a field $K$ of characteristic zero. Let $R=C[X_1, \ldots, X_n, Y_1, \ldots, Y_m]$ be a polynomial ring over $C$ with $\mathrm{bideg}~ c=(0,0)$ for all $c \in C$, $\mathrm{bideg}~ X_i=(1,0)$ and $\mathrm{bideg}~ Y_j=(0,1)$ for $i=1, \ldots, n$ and $j=1, \ldots, m$. Let $I$ be a bihomogeneous ideal in $R$. In this article, we study asymptotic behaviour of bigraded pieces of the local cohomology module $H^i_I(R)$. Moreover, under the extra assumption that $C$ is regular, we investigate the asymptotic stability of invariants associated to its bigraded components. Consequently, we obtain certain properties of components of the bigraded local cohomology module $H^i_I(R)$, where $C=K$ is a field and $I$ is a binomial edge ideal.
When granular materials with interstitial liquid bridges are sheared in a split-bottom cell, a localized shear band develops, accompanied by a surface elevation. Cohesion, governed by the surface tension of the interstitial liquid, enhances dilatancy in dense cohesive packings, leading to expansion within the shear band and the emergence of a surface elevation. Surface deformation is observed not only in cohesive systems with high particle density and large liquid surface tension, but also in those with lower values of these parameters. The equivalent Bond number arises as a key control parameter for the surface deformation, shaping both the evolution of the surface profile and the packing density. At higher shear rates, inertial effects dominate dilatancy, resulting in less pronounced surface deformation.
When a granular material composed of elongated grains is sheared in a split-bottom shear cell, a pressure difference develops within the material. This pressure difference depends on the interparticle friction ($\mu$), which affects shear localization and particle alignment. For large $\mu$, alignment is confined to a narrow shear band, leading to localized increases in packing density and pressure. For small $\mu$, particles align over a wider region, leading to a nearly uniform packing density and pressure throughout the material. In contrast, spherical particles, regardless of $\mu$, maintain a uniform packing density and pressure throughout the material. We observe a phenomenological similarity to the Weissenberg effect in non-Newtonian fluids, where normal stress differences induce radial pressure gradients, unlike the uniform pressure in Newtonian fluids.
The extremely weak gravity on small bodies makes landers prone to rebound and uncontrolled drift. To mitigate this, the Hayabusa2 mission employed a particle-filled flexible shell, but the coupled dynamics of shell deformation and internal particle dissipation remain unclear. We develop a computational model representing the flexible shell as a spring-mass network and fully resolve particle collisions, friction, and interactions with granular beds. Results show the flexible shell-granule system dissipates over 90 percent of impact energy, far exceeding rigid shells. Energy loss arises from shell-particle coupling, with the particle filling ratio dominating. Impacts on rigid planes produce large shell deformation, while granular beds limit deformation. Scaling and velocity analyses reveal distinct dissipation regimes. These findings clarify energy transfer mechanisms and inform the design of microgravity impact mitigation devices.
We study a class of double determinantal ideals denoted $I_{mn}^r$, which are generated by minors of size 2, and show that they are equal to the Hibi rings of certain finite distributive lattices. We compute the number of minimal generators of $I_{mn}^r$, as well as the multiplicity, regularity, a-invariant, Hilbert function, and $h$-polynomial of the ring $R/I_{mn}^r$, and we give a new proof of the dimension of $R/I_{mn}^r$. We also characterize when the ring $R/I_{mn}^r$ is Gorenstein, thereby answering a question of Li in the toric case. Finally, we give combinatorial descriptions of the facets of the Stanley-Reisner complex of the initial ideal of $I_{mn}^r$ with respect to a diagonal term order.
The interplay between dilatancy and particle alignment in sheared granular materials composed of non-spherical particles leads to morphological inhomogeneity. Dilatancy, driven by interparticle friction, causes the packing to expand, while particle alignment tends to densify it. We examine the influence of friction, particle aspect ratio, and initial packing conditions on the steady-state particle alignment and packing density. Unlike spherical particles, non-spherical particles with higher AR exhibit either dilatancy or compaction under shear, leading to spontaneous heaping or depression formation. We analyzed the evolution of packing density to identify whether dilatancy or compression prevails within the shear band.
We perform Discrete Element Method simulations of wet granular matter in a split-bottom shear cell. To calculate the capillary forces from the liquid bridges between the grains, we used three different approximations. The simulations of the shear cell showed a linear increase in bulk cohesion with the surface tension of the liquid, consistently for all approximations. However, the macroscopic friction coefficient shows only a weak dependence on surface tension.
We present the implementation of two advanced capillary bridge approximations within the Discrete Element Method (DEM) framework of the open-source code MercuryDPM. While MercuryDPM already includes a simplified version of the Willett approximation, our work involves implementing both the classical Willett approximation and the recently published Bagheri approximation in MercuryDPM. Through detailed descriptions and illustrative simulations using a two-particle collision model, we demonstrate the enhanced accuracy and capabilities of these approximations in capturing the complex dynamics of wet granular matter.
Let $R=\oplus_{m\geq 0}R_m$ be a standard graded equidimensional ring over a field $R_0$, and $I\subseteq J$ be two non-nilpotent graded ideals in $R$. Then we give a set of numerical characterizations of the integral dependence of $I$ and $J$ in terms of certain multiplicities. A novelty of this approach is that it does not involve localization and only requires checking computable and well-studied invariants. In particular, we show the following: let $S=R[y]$, $\mathsf{I} = IS$ and $\mathsf{J} = JS$ and $\bf d$ be the maximum of the generating degrees of both $I$ and $J$. Let $c>{\bf d}$ be any given integer. Then $$\overline{I} = \overline{J}\iff e\big(S[\mathsf{I}t]_{\Delta_{(c,1)}}\big) = e\big(S[\mathsf{J}t]_{\Delta_{(c,1)}}\big),$$ where $e\big(S[\mathsf{I}t]_{\Delta_{(c,1)}}\big)$ denotes the Hilbert-Samuel multiplicity of the standard graded domain $S[\mathsf{I}t]_{\Delta_{(c,1)}} = \oplus_{n\geq 0}(\mathsf{I}^n)_{cn}t^n$. Further, if $I$ is of finite colength in $R$ then $e\big(S[\mathsf{I}t]_{\Delta_{(c,1)}}\big) = c^de(R) - e(I,R)$. If $R$ is also a domain, then other numerical criteria are the following: \begin{align*} \overline{I} = \overline{J} & \iff \varepsilon(I)=\varepsilon(J)\;\;\mbox{and}\;\; e_i(R[It]) = e_i(R[Jt])\;\;\mbox{for all}\;\; 0\leq i <\dim(R/I), \end{align*} where $\varepsilon(I)$ denotes the epsilon multiplicity of $I$, and $e_i(R[It])$'s are the mixed multiplicities of the Rees algebra $R[It]$. The relation between $e_i(S[\mathsf{I}t])$ and the polar multiplicities of $\mathsf{I}_{\geq {\bf d}}$ provides another criterion in terms of polar multiplicities of $\mathsf{I}_{\geq {\bf d}}$. The first two characterizations generalize Rees's classical result for ideals of finite colengths. Apart from several well-established results, the proofs of these results use the theory of density functions, which was developed in arXiv:2311.17679.
The quality of the powder spread in additive manufacturing devices depends sensitively on the particles' shapes. Here, we study powder spreading for mixtures of spherical and irregularly shaped particles in Polyamide 12 powders. Using DEM simulations, including heat transfer, we find that spherical particles exhibit better flowability. Thus, the particles are deposited far ahead of the spreading blade. In contrast, a large fraction of non-spherical particles hinders the flow. Therefore, the cold particles are deposited near the front of the spreading blade. This results in a temperature drop of the deposited particles near the substrate, which cannot be seen with spherical particles. The particles of both shapes are homogeneously distributed in the deposited powder layer.
When granular materials of shape-anisotropic grains are sheared in a split-bottom shear cell, a localized shear band is formed with a depression at its center. This effect is closely related to the alignment of the particles with aspect ratio (AR), which, in turn, influences the local packing density, the stress distribution, and the system's overall bulk rheology. Particles with large AR tend to align with the shear direction, which increases the packing density in the shear band and affects rheological properties like stress, macroscopic friction coefficient, and effective viscosity. A scaling law correlates particle AR to macroscopic friction and effective viscosity, revealing shear-thinning behavior in bulk and near the surface.
The thermal and mechanical behaviors of powders are important for various additive manufacturing technologies. For powder bed fusion, capturing the temperature profile and the packing structure of the powders prior to melting is challenging due to both the various pathways of heat transfer and the complicated properties of powder system. Furthermore, these two effects can be coupled due to the temperature dependence of particle properties. This study addresses this challenge using a discrete element model that simulates non-spherical particles with thermal properties in powder spreading. Thermal conduction and radiation are introduced to a multi-sphere particle formulation for capturing the heat transfer among irregular-shaped powders, which have temperature-dependent elastic properties. The model is utilized to simulate the spreading of pre-heated PA12 powder through a hot substrate representing the part under manufacturing. Differences in the temperature profiles were found in the spreading cases with different particle shapes, spreading speed, and temperature dependence of the elastic moduli. The temperature of particles below the spreading blade is found to be dependent on the kinematics of the heap of particles in front, which eventually is influenced by the temperature-dependent properties of the particles.
This article investigates the computational aspects of the $\varepsilon$-multiplicity. Primarily, we show that the $\varepsilon$-multiplicity of a homogeneous ideal $I$ in a two-dimensional standard graded domain of finite type over an algebraically closed field of arbitrary characteristic, is always a rational number. In this situation, we produce a formula for the $\varepsilon$-multiplicity of $I$ in terms of certain mixed multiplicities associated to $I$. In any dimension, under the assumptions that the saturated Rees algebra of $I$ is finitely generated, we give a different expression of the $\varepsilon$-multiplicity in terms of mixed multiplicities by using the Veronese degree. This enabled us to make various explicit computations of $\varepsilon$-multiplicities. We further write a Macaulay2 algorithm to compute $\varepsilon$-multiplicity (under the Noetherian hypotheses) even when the base ring is not necessarily standard graded.
We study the question of finite generation of saturated multi-Rees algebras and investigate the asymptotic behaviour of related length functions. In the setup of excellent local domains, we show that the saturated multi-Rees algebra of a finite collection of ideals is finitely generated when the analytic spread is not maximal and the associated length function eventually agrees with a polynomial. Similar results are obtained when we restrict to two-dimensional local UFDs with no restrictions on the analytic spread. We further prove that the saturated multi-Rees algebra of finitely many monomial ideals in a polynomial ring modulo an irreducible monomial ideal, is always finitely generated. In this case, the corresponding length function is shown to exhibit piecewise quasi-polynomial behaviour. We also produce multi-ideal versions of a theorem of Amao.
A density function for an algebraic invariant is a measurable function on $\mathbb{R}$ which measures the invariant on an $\mathbb{R}$-scale. This function carries a lot more information related to the invariant without seeking extra data. It has turned out to be a useful tool, which was introduced by the third author, to study the characteristic $p$ invariant, namely Hilbert-Kunz multiplicity of a homogeneous ${\bf m}$-primary ideal. Here we construct density functions $f_{A,\{I_n\}}$ for a Noetherian filtration $\{I_n\}_{n\in\mathbb{N}}$ of homogeneous ideals and $f_{A,\{\widetilde{I^n}\}}$ for a filtration given by the saturated powers of a homogeneous ideal $I$ in a standard graded domain $A$. As a consequence, we get a density function $f_{\varepsilon(I)}$ for the epsilon multiplicity $\varepsilon(I)$ of a homogeneous ideal $I$ in $A$. We further show that the function $f_{A,\{I_n\}}$ is continuous everywhere except possibly at one point, and $f_{A,\{\widetilde{I^n}\}}$ is a continuous function everywhere and is continuously differentiable except possibly at one point. As a corollary the epsilon density function $f_{\varepsilon(I)}$ is a compactly supported continuous function on $\mathbb{R}$ except at one point, such that $\int_{\mathbb{R}_{\geq 0}} f_{\varepsilon(I)} = \varepsilon(I)$. All the three functions $f_{A,\{I^n\}}$, $f_{A,\{\widetilde{I^n}\}}$ and $f_{\varepsilon(I)}$ remain invariant under passage to the integral closure of $I$. As a corollary of this theory, we observe that the `rescaled' Hilbert-Samuel multiplicities of the diagonal subalgebras form a continuous family.
We introduce Spiral, a third-order integration algorithm for the rotational motion of extended bodies. It requires only one force calculation per time step, does not require quaternion normalization at each time step, and can be formulated for both leapfrog and synchronous integration schemes, making it compatible with many particle simulation codes. The stability and precision of Spiral exceed those of state-of-the-art algorithms currently used in popular DEM codes such as Yade, MercuryDPM, LIGGGHTS, PFC, and more, at only slightly higher computational cost. Also, beyond DEM, we see potential applications in all numerical simulations that involve the 3D rotation of extended bodies.
Granular materials show inhomogeneous flows characterized by strain localization. When strain is localized in a sheared granular material, rigid regions of a nearly undeformed state are separated by shear bands, where the material yields and flows. The characteristics of the shear bands are determined by the geometry of the system, the micromechanical material properties, and the kinematics at the particle level. For a split-bottom shear cell, recent experimental work has shown that mixtures of hard, frictional and soft, nearly frictionless particles exhibit wider shear zones than samples with only one of the two components. To explain this finding, we investigate the shear zone properties and the stress response of granular mixtures using discrete element simulations. We show that both interparticle friction and elastic modulus determine the shear-band properties of granular mixtures of various mixing ratios, but their stress response depends strongly on the interparticle friction. Our study provides a fundamental understanding of the micromechanics of shear band formation in granular mixtures.