PHASE-FIELD FRACTURE MODELING FOR INTERLOCKING MICRO-ARCHITECTURED MATERIALS

It is fascinating to see how natural materials like teeth enamel, bone and nacre possess a very high stiffness and strength in spite of the fact that they are composed of minerals mostly. Studies have shown the reason for this aberration as the presence of weaker interfaces with intricate interlocking architectures at microscopic levels in these materials. Inspired by the architecture of these materials, micro-architectured sutures with jig-saw like geometry is being studied in this research study. The main focus of this study is to examine the effects of friction co-efficient and interlocking angles of the jig-saw tabs on pullout strength, fracture toughness and energy absorption. We are using Phase-Field Fracture model to study these effects in ABS (acrylonitrile-butadiene-styrene) material. We will also simulate the fracture of the interlocking tabs when the interlocking angle goes beyond a certain limit

Cores of partitions in rectangles

For a positive integer $t \geq 2$, the $t$-core of a partition plays an important role in modular representation theory and combinatorics. We initiate the study of $t$-cores of partitions contained in an $s \times r$ rectangle. Our main results are as follows. We first give a simple formula for the number of partitions in the rectangle which are themselves $t$-cores and compute its asymptotics for large $r,s$. We then prove that the number of partitions inside the rectangle whose $t$-cores are a fixed partition $\rho$ is given by a product of binomial coefficients. Finally, we use this formula to compute the distribution of the $t$-core of a uniformly random partition inside the rectangle extending our previous work on all partitions of a fixed integer $n$ (Ann. Appl. Prob. 2023). In particular, we show that in the limit as $r,s \to \infty$ maintaining a fixed aspect ratio, we again obtain a Gamma distribution with the same shape parameter $\alpha = (t-1)/2$ and scale parameter $\beta$ that depends on the aspect ratio.Comment: 16 pages, 1 figure, improved exposition, references adde

The size of tt-cores and hook lengths of random cells in random partitions

Fix $t \geq 2$. We first give an asymptotic formula for certain sums of the number of $t$-cores. We then use this result to compute the distribution of the size of the $t$-core of a uniformly random partition of an integer $n$. We show that this converges weakly to a gamma distribution after dividing by $\sqrt{n}$. As a consequence, we find that the size of the $t$-core is of the order of $\sqrt{n}$ in expectation. We then apply this result to show that the probability that $t$ divides the hook length of a uniformly random cell in a uniformly random partition equals $1/t$ in the limit. Finally, we extend this result to all modulo classes of $t$ using abacus representations for cores and quotients.Comment: 28 pages, 3 figures, significant revisions. Several minor errors fixed and results stated in a more concise manner. From v1, Sections 2.4 and 5.2 deleted and Corollary 5.6 is stated as Lemma 5.16 in this version, thanks to a suggestion of D. Grinber

Emergent Supersymmetry at Large NN

We search for infrared fixed points of Gross-Neveu Yukawa models with matrix degrees of freedom in $d=4-\varepsilon$. We consider three models -- a model with $SU(N)$ symmetry in which the scalar and fermionic fields both transform in the adjoint representation, a model with $SO(N)$ symmetry in which the scalar and fermion fields both transform as real symmetric-traceless matrices, and a model with $SO(N)$ symmetry in which the scalar field transforms as a real symmetric-traceless matrix, while the fermion transforms in the adjoint representation. These models differ at finite $N$, but their large-$N$ limits are perturbatively equivalent. The first two models contain a supersymmetric fixed point for all $N$, which is attractive to all classically-marginal deformations for $N$ sufficiently large. The third model possesses a fixed point that, although non-supersymmetric for any finite $N$, possesses emergent supersymmetry when $N$ is sufficiently large. We also find several non-supersymmetric fixed points at finite and large-$N$. Planar diagrams dominate the large-$N$ limit of these fixed points, which suggests the possibility of a stringy holographic dual description.Comment: 80 pages, 19 figure

On factorization of the shift semigroup

Let $\mathscr{S}^\mathcal{E}=(S_t^\mathcal{E})_{t\ge 0}$ on $L^2(\mathbb{R}_+,\mathcal{E})$ be the right shift semigroup for a separable Hilbert space $\mathcal{E}$. Let $\mathscr{V}_1 = (V_{1,t})_{t\ge 0}$ and $\mathscr{V}_2 = (V_{2,t})_{t\ge 0}$ be a pair of semigroups of contractions which satisfy $V_{1,t}V_{2,s} = V_{2,s}V_{1,t}$ and $S^\mathcal{E}_t=V_{1,t} V_{2,t}$ for every $s,t \ge 0$. Such a pair is called a factorization of $\mathscr{S}^\mathcal{E}$. The main result of this note completely describes all factorizations of $\mathscr{S}^\mathcal{E}$ when $\mathcal{E}$ is finite dimensional. Using the known fact that $\mathscr{S}^\mathcal{E}$ is unitarily equivalent to a semigroup of multiplication operators on the vector valued Hardy space $H^2_\mathbb{D}(\mathcal{E})$, we employ novel function theoretic methods and classical convex analysis to arrive at the factorization

Efficient Threshold FHE with Application to Real-Time Systems

Threshold Fully Homomorphic Encryption (ThFHE) enables arbitrary computation over encrypted data while keeping the decryption key distributed across multiple parties at all times. ThFHE is a key enabler for threshold cryptography and, more generally, secure distributed computing. Existing ThFHE schemes inherently require highly inefficient parameters and are unsuitable for practical deployment. In this paper, we take the first step towards making ThFHE practically usable by (i) proposing a novel ThFHE scheme with a new analysis resulting in significantly improved parameters; (ii) and providing the first practical ThFHE implementation benchmark based on Torus FHE. • We propose the first practical ThFHE scheme with a polynomial modulus-to-noise ratio that supports practically efficient parameters while retaining provable security based on standard quantum-safe assumptions. We achieve this via a novel Rényi divergence-based security analysis of our proposed threshold decryption mechanism. • We present an optimized software implementation of a Torus-FHE based instantiation of our proposed ThFHE scheme that builds upon the existing Torus FHE library and supports (distributed) decryption on highly resource-constrained ARM-based handheld devices. Along the way, we implement several extensions to the Torus FHE library, including a Torus-based linear integer secret sharing subroutine to support ThFHE key sharing and distributed decryption for any threshold access structure. We illustrate the efficacy of our proposal via an end-to-end use case involving encrypted computations over a real medical database, and distributed decryptions of the computed result on resource-constrained ARM-based handheld devices

DESIGN AND PERFORMANCE VERIFICATION OF NEWLY DEVELOPED DISPOSABLE STATIC DIFFUSION CELL FOR DRUG DIFFUSION/PERMEABILITY STUDIES

Objectives: The present study describes a disposable static diffusion cell for in vitro diffusion studies to achieve better results as compared to well existing Franz diffusion cell (FDC) in terms of the absence of bubbles, variable receptor compartment, ease of handling, and faster results.Materials and Methods: The cell consists of a cup-shaped donor compartment made of semi permeable that could be either cellophane membrane or, animal skin fitted to a rigid frame, which is supported on a plastic plate that contains a hole for the sample withdrawal. The receptor compartment is a separate unit, and it could be any container up to 500ml volume capacity. The most preferred receptor compartment is glass beaker. In the present study, goatskin was used as semi-permeable membrane and verification of its performance was carried out through diffusion studies using gel formulations of one each of the four-selected biopharmaceutical classification system (BCS) class drugs. Metronidazole, diclofenac sodium, fluconazole, and sulfadiazine were used as model drugs for BCS Class I, II, III, and IV, respectively.Results: The newly developed diffusion cell (NDDC) was found to provide faster and more reproducible results as compared to FDC. At the time interval of 24 h, the cell was found to exhibit a higher diffusion of metronidazole, diclofenac sodium, fluconazole, and sulfadiazine by 0.65, 0.65, 0.32, and 0.81 folds, respectively. The faster release obtained with NDDC was attributed to a larger surface area of skin as compared to that in FDC.Conclusion: It was concluded that better reproducibility of results could be achieved with NDDC