Uncertainty in medical image segmentation tasks, especially inter-rater variability, arising from differences in interpretations and annotations by various experts, presents a significant challenge in achieving consistent and reliable image segmentation. This variability not only reflects the inherent complexity and subjective nature of medical image interpretation but also directly impacts the development and evaluation of automated segmentation algorithms. Accurately modeling and quantifying this variability is essential for enhancing the robustness and clinical applicability of these algorithms. We report the set-up and summarize the benchmark results of the Quantification of Uncertainties in Biomedical Image Quantification Challenge (QUBIQ), which was organized in conjunction with International Conferences on Medical Image Computing and Computer-Assisted Intervention (MICCAI) 2020 and 2021. The challenge focuses on the uncertainty quantification of medical image segmentation which considers the omnipresence of inter-rater variability in imaging datasets. The large collection of images with multi-rater annotations features various modalities such as MRI and CT; various organs such as the brain, prostate, kidney, and pancreas; and different image dimensions 2D-vs-3D. A total of 24 teams submitted different solutions to the problem, combining various baseline models, Bayesian neural networks, and ensemble model techniques. The obtained results indicate the importance of the ensemble models, as well as the need for further research to develop efficient 3D methods for uncertainty quantification methods in 3D segmentation tasks.

We compute Out-of-Time-Order correlators (OTOCs) for conformal field theories (CFTs) subjected to either continuous or discrete periodic drive protocols. This is achieved by an appropriate analytic continuation of the stroboscopic time. After detailing the general structure, we perform explicit calculations in large-$c$ CFTs where we find that OTOCs display an exponential, an oscillatory and a power-law behaviour in the heating phase, the non-heating phase and on the phase boundary, respectively. In contrast to this, for the Ising CFT representing an integrable model, OTOCs never display such exponential growth. This observation hints towards how OTOCs can demarcate between integrable and chaotic CFT models subjected to a periodic drive. We further explore properties of the light-cone which is characterized by the corresponding butterfly velocity as well as the Lyapunov exponent. Interestingly, as a consequence of the spatial inhomogeneity introduced by the drive, the butterfly velocity, in these systems, has an explicit dependence on the initial location of the operators. We chart out the dependence of the Lyapunov exponent and the butterfly velocities on the frequency and amplitude of the drive for both protocols and discuss the fixed point structure which differentiates such driven CFTs from their un-driven counterparts.

We investigate the occurrence of additive and multiplicative structures in random subsets of the natural numbers. Specifically, for a Bernoulli random subset of $\mathbb{N}$ where each integer is included independently with probability $p\in (0,1)$, we prove that almost surely such a set contains finite sumsets (FS-sets) and finite product sets (FP-sets) of every finite length. In addition, we establish a novel connection between Hindman's partition theorem and the central limit theorem, providing a probabilistic perspective on the asymptotic Gaussian behavior of monochromatic finite sums and products. These results can be interpreted as probabilistic analogues of finite-dimensional versions of Hindman's theorem. Applications, implications, and open questions related to infinite FS-sets and FP-sets are discussed.

In this study, we analyze Krylov Complexity in two-dimensional conformal field theories subjected to deformed SL$(2,\mathbb{R})$ Hamiltonians. In the vacuum state, we find that the K-complexity exhibits a universal phase structure. The phase structure involves the K-complexity exhibiting an oscillatory behaviour in the non-heating phase, which contrasts with the exponential growth observed in the heating phase, while it displays polynomial growth at the phase boundary. Furthermore, we extend our analysis to compute the K-complexity of a light operator in excited states, considering both large-c CFT and free field theory. In the free field theory, we find a state-independent phase structure of K-complexity. However, in the large-c CFT, the behavior varies, with the K-Complexity once again displaying exponential growth in the heating phase and polynomial growth at the phase boundary. Notably, the precise exponent governing this growth depends on the heaviness of the state under examination. In the non-heating phase, we observe a transition in K-complexity behavior from oscillatory to exponential growth, akin to findings in [1], as it represents a special case within the non-heating phase.

Here, we present an algebraic and kinematical analysis of non-commutative $\kappa$-Minkowski spaces within Galilean (non-relativistic) and Carrollian (ultra-relativistic) regimes. Utilizing the theory of Wigner-Inönu contractions, we begin with a brief review of how one can apply these contractions to the well-known Poincaré algebra, yielding the corresponding Galilean (both massive and mass-less) and Carrollian algebras as $c \to \infty$ and $c\to 0$, respectively. Subsequently, we methodically apply these contractions to non-commutative $\kappa$-deformed spaces, revealing compelling insights into the interplay among the non-commutative parameters $a^\mu$ (with $|a^\nu|$ being of the order of Planck length scale) and the speed of light $c$ as it approaches both infinity and zero. Our exploration predicts a sort of "branching" of the non-commutative parameters $a^\mu$, leading to the emergence of a novel length scale and time scale in either limit. Furthermore, our investigation extends to the examination of curved momentum spaces and their geodesic distances in appropriate subspaces of the $\kappa$-deformed Newtonian and Carrollian space-times. We finally delve into the study of their deformed dispersion relations, arising from these deformed geodesic distances, providing a comprehensive understanding of the nature of these space-times.

The right kind of theoretical treatment of direct Coulomb ionization of inner-shell of target atoms including multiple ionization of their outer-shells by using accurate x-ray fluorescence yield data and electron capture by projectile ions from inner-shell electrons of target atoms enables us to fully understand the complex physics issues with the heavy-ion-induced inner-shell ionization phenomenon. Such great success has only been achieved recently [Phys. Rev. A 111 (2025) 042827]. Aftermath, further investigations exhibit such a picture only if the Fermi velocity of the elemental target is accurate, as it takes a significant role in correct evaluation of charge-state distribution of the projectile ions inside the target, which contributes an invaluable share in calculating the electron capture-induced ionization cross section correctly. In this work, we devise a powerful method that enables us to measure the correct and accurate Fermi velocity for almost every elemental metal in the periodic table. As per our present knowledge, this in turn not only improves our understanding of the said complex physics issues one step ahead but also helps move toward further miniaturization of integrated circuits and use the heavy-ion-induced X-ray emission in impurity analysis more reliable and accurate.

This paper investigates the normal modes of a probe scalar field in a five-dimensional AdS-Schwarzschild black hole with the brick wall boundary condition near the horizon. We employ various techniques to compute the spectrum and analyze its properties. Our results reveal a linear dependence of the spectrum on the principal quantum number while demonstrating a non-trivial dependence on the angular momentum quantum number. We compute the Spectral Form Factor (SFF) and find a dip-ramp-plateau structure, with the slope of the ramp approaching unity as the brick wall nears the horizon. We also observe that as the brick wall approaches the horizon, the poles of the retarded Green's function condense on the real line, leading to an emergent thermal behavior in the boundary theory. This work extends previous studies on lower-dimensional black holes to higher dimensions, providing insights into the connection between black hole microstate models and boundary chaos. Our findings contribute to the ongoing discussions on the information paradox and the nature of black hole interiors in the context of AdS/CFT correspondence.

We explore the physics of scrambling in the moving mirror models, in which a two-dimensional CFT is subjected to a time-dependent boundary condition. It is well-known that by choosing an appropriate mirror profile, one can model quantum aspects of black holes in two-dimensions, ranging from Hawking radiation in an eternal black hole (for an "escaping mirror") to the recent realization of Page curve in evaporating black holes (for a "kink mirror"). We explore a class of OTOCs in the presence of such a boundary and explicitly demonstrate the following primary aspects: First, we show that the dynamical CFT data directly affect an OTOC and maximally chaotic scrambling occurs for the escaping mirror for a large-$c$ CFT with identity block dominance. We further show that the exponential growth of OTOC associated with the physics of scrambling yields a power-law growth in the model for evaporating black holes which demonstrates a unitary dynamics in terms of a Page curve. We also demonstrate that, by tuning a parameter, one can naturally interpolate between an exponential growth associated to scrambling and a power-law growth in unitary dynamics. Our work explicitly exhibits the role of higher-point functions in CFT dynamics as well as the distinction between scrambling and Page curve. We also discuss several future possibilities based on this class of models.

We use a Hybrid Deep Neural Network (HDNN) to identify a boosted dark photon jet as a signature of a heavy vector-like fermionic portal matter (PM) connecting the visible and the dark sectors. In this work, the fermionic PM, which mixes only with the Standard Model (SM) third-generation up-type quark, predominantly decays into a top quark and a dark photon pair. The dark photon then $promptly$ decays to a pair of standard model fermions via the gauge kinetic mixing. We have analyzed two different final states, namely, (i) exactly one tagged dark photon and exactly one tagged top quark jet, and (ii) at least two tagged dark photons and at least one tagged top quark jet at the 13 and 14 TeV LHC center of mass energies. Both these final states receive significant contributions from the pair and single production processes of the top partner. The rich event topology of the signal processes, $i.e.$, the presence of a boosted dark photon and top quark jet pair, along with the fact that the invariant mass of the system corresponds to the mass of the top partner, help us to significantly suppress potential SM backgrounds. We have shown that one can set a $2\sigma$ exclusion limit of $\sim 2.3$ TeV on the top partner mass with $\sin\theta_L=0.1$ and assuming $100\%$ branching ratio of the top partner in the final state with exactly one tagged dark photon and exactly one tagged top quark jet at the 14 TeV LHC center of mass energy assuming 300 fb$^{-1}$ of integrated luminosity.

In this article, we investigate homogeneous versions of certain nonlinear Ramsey-theoretic results, with three significant applications. As the first application, we prove that for every finite coloring of $\mathbb{Z}^+$, there exist an infinite set $A$ and an arbitrarily large finite set $B$ such that $A \cup (A+B) \cup A \cdot B$ is monochromatic. This result resolves the finitary version of a question posed by Kra, Moreira, Richter, and Robertson regarding the partition regularity of $(A+B) \cup A \cdot B$ for infinite sets $A, B$ (see (Question 8.4, J. Amer. Math. Soc., 37 (2024))), which is closely related to a question of Erd\H{o}s. As the second application, we make progress on a nonlinear extension of the partition regularity of Pythagorean triples. Specifically, we demonstrate that the equation $x^2 + y^2 = z^2 + P(u_1, \dots, u_n)$ is $2$-regular for certain appropriately chosen polynomials $P$ of any desired degree. Finally, as the third application, we establish a nonlinear variant of Rado's conjecture concerning the degree of regularity. We prove that for every $m, n \in \mathbb{Z}^+$, there exists an$m$-degree homogeneous equation that is $n$-regular but not $(n+1)$-regular. The case $m = 1$ corresponds to Rado's conjecture, originally proven by Alexeev and Tsimerman (J. Combin. Theory Ser. A, 117 (2010), and later independently by Golowich (Electron. J. Combin. 21 (2014)).

We construct an infinite class of eigenmodes with integer eigenvalues for the Vacuum Modular Hamiltonian of a single interval $N$ in 2d CFT and study some of its interesting properties, which includes its action on OPE blocks as well as its bulk duals. Our analysis suggests that these eigenmodes, like the OPE blocks have a natural description on the so called kinematic space of CFT$_2$ and in particular realize the Virasoro algebra of the theory on this kinematic space. Taken together, our results hints at the possibility of an effective description of the CFT$_2$ in the kinematic space language.

This contributory article begins with our fond and sincere reminiscences about our beloved Prof. A.P. Balachandran. In the main part, we discuss our recent formulation of quantum mechanics on (1+1)D noncommutative space-time using Hilbert-Schmidt operators. As an application, we demonstrate how geometrical phase in a system of time-dependent forced harmonic oscillator living in the Moyal space-time can emerge.

We study the homological algebra of edge ideals of Erd\"{o}s-R\'enyi random graphs. These random graphs are generated by deleting edges of a complete graph on $n$ vertices independently of each other with probability $1-p$. We focus on some aspects of these random edge ideals - linear resolution, unmixedness and algebraic invariants like the Castelnuovo-Mumford regularity, projective dimension and depth. We first show a double phase transition for existence of linear presentation and resolution and determine the critical windows as well. As a consequence, we obtain that except for a very specific choice of parameters (i.e., $n,p := p(n)$), with high probability, a random edge ideal has linear presentation if and only if it has linear resolution. This shows certain conjectures hold true for large random graphs with high probability even though the conjectures were shown to fail for determinstic graphs. Next, we study asymptotic behaviour of some algebraic invariants - the Castelnuovo-Mumford regularity, projective dimension and depth - of such random edge ideals in the sparse regime (i.e., $p = \frac{\lambda}{n}, \lambda \in (0,\infty)$). These invariants are studied using local weak convergence (or Benjamini-Schramm convergence) and relating them to invariants on Galton-Watson trees. We also show that when $p \to 0$ or $p \to 1$ fast enough, then with high probability the edge ideals are unmixed and for most other choices of $p$, these ideals are not unmixed with high probability. This is further progress towards the conjecture that random monomial ideals are unlikely to have Cohen-Macaulay property (see De Loera et al. 2019a,2019b) in the setting when the number of variables goes to infinity but the degree is fixed.

Sanskrit, one of humanity's most ancient languages, has a vast collection of books and manuscripts on diverse topics that have been accumulated over millennia. However, its digital content (audio and text), which is vital for the training of AI systems, is profoundly limited. Furthermore, its intricate linguistics make it hard to develop robust NLP tools for wider accessibility. Given these constraints, we have developed an automatic speech recognition model for Sanskrit by employing transfer learning mechanism on OpenAI's Whisper model. After carefully optimising the hyper-parameters, we obtained promising results with our transfer-learned model achieving a word error rate of 15.42% on Vaksancayah dataset. An online demo of our model is made available for the use of public and to evaluate its performance firsthand thereby paving the way for improved accessibility and technological support for Sanskrit learning in the modern era.

In nuclear collisions, nuclear bremsstrahlung can cause nuclear Coulomb excitation via photon exchange in the projectile as well as the target nuclei. Such a process originating in nuclear timescales (zeptoseconds) can also influence the atomic phenomenon, which can be observed if it is delayed at least by a few attoseconds as atomic timescales $\ge$ an attosecond. We have found that this may happen due to a mechanism called the Eisenbud-Wigner-Smith (EWS)time delay process. We have estimated EWS time delays in atomic collisions utilizing the non-relativistic version of random phase approximation with exchange as well as Hartree-Fock methods. We present three representative collision systems through which one can experimentally observe the phenomena in attosecond timescales even though they originate from nuclear bremsstrahlung radiation occurring in zeptoseconds. Thus the present work represents an investigation of parallels between two neighboring areas of physics: atomic and nuclear physics. Furthermore the present work suggests the possibilities for atomic physics research near the Coulomb barrier energies, where the nuclear bremsstrahlung can be used as a zeptosecond x-ray source.

We show that a dynamical transition from a non-heating to a heating phase of a periodic $SL(2,\mathbb{R})$ driven two-dimensional conformal field theory (CFT) with a large central charge is perceived as a first order transition by a bulk brane embedded in the dual AdS. We construct the dual bulk metric corresponding to a driven CFT for both the heating and the non-heating phases. These metrics are different AdS$_{2}$ slices of the pure AdS$_{3}$ metric. We embed a brane in the obtained dual AdS space and provide an explicit computation of its free energy both in the probe limit and for an end-of-world (EOW) brane taking into account its backreaction. Our analysis indicates a finite discontinuity in the first derivative of the brane free energy as one moves from the non-heating to the heating phase (by tuning the drive amplitude and/or frequency of the driven CFT) thus demonstrating the presence of the bulk first order transition. Interestingly, no such transition is perceived by the bulk in the absence of the brane. We also provide explicit computations of two-point, four-point out-of-time correlators (OTOC) using the bulk picture. Our analysis shows that the structure of these correlators in different phases match their counterparts computed in the driven CFT. We analyze the effect of multiple EOW branes in the bulk and discuss possible extensions of our work for richer geometries and branes.

We consider the inertial Kuramoto model of $N$ globally coupled oscillators characterized by both their phase and angular velocity, in which there is a time delay in the interaction between the oscillators. Besides the academic interest, we show that the model can be related to a network of phase-locked loops widely used in electronic circuits for generating a stable frequency at multiples of an input frequency. We study the model for a generic choice of the natural frequency distribution of the oscillators, to elucidate how a synchronized phase bifurcates from an incoherent phase as the coupling constant between the oscillators is tuned. We show that in contrast to the case with no delay, here the system in the stationary state may exhibit either a subcritical or a supercritical bifurcation between a synchronized and an incoherent phase, which is dictated by the value of the delay present in the interaction and the precise value of inertia of the oscillators. Our theoretical analysis, performed in the limit $N \to \infty$, is based on an unstable manifold expansion in the vicinity of the bifurcation, which we apply to the kinetic equation satisfied by the single-oscillator distribution function. We check our results by performing direct numerical integration of the dynamics for large $N$, and highlight the subtleties arising from having a finite number of oscillators.

In this paper, we propose necessary upper bounds for all prime divisors of friends of 10. This is precisely a generalization of a recent paper \cite{SS} of the authors where they proposed upper bounds for the second, third, and fourth smallest prime divisors of friends of 10.

Focussing on theories for which the higher derivative terms are considered as small corrections in the Lagrangian to Einstein's two-derivative theory of general relativity (GR), we prove the classical version of the covariant entropy bound (also known as the Bousso bound) in arbitrary diffeomorphism invariant gravitational theories. Even if the higher derivative corrections are treated perturbatively, we provide instances of specific configurations for which they can potentially violate the Bousso bound. To tackle this obstruction, we propose a modification in the Bousso bound that incorporates the offending contributions from the higher derivative corrections. We argue that the modified Bousso bound that we propose holds to all orders in the higher curvature corrections. Our proposed modifications are equivalent to replacing the Bekenstein-Hawking area term by Wald's definition (with dynamical corrections as suggested by Wall) for the black hole entropy. Hence, the modifications are physically well motivated by results from the laws of black hole mechanics in higher derivative theories.

A theoretical methodology for exploring the conventional Bardeen-Cooper-Schrieffer (BCS) pairing instability for superconductivity from a correlated normal phase for all possible degrees of many-body correlation, has been developed. The Gutzwiller projection scheme with a correlation parameter was made use of in generating the BCS pairing state. A variational scheme was thereafter implemented, leading to a self-consistent equation for superconducting gap function. This equation shows explicit dependence of the gap function on the many body correlation parameter. This `pairing-gap' and the corresponding self-consistent gap equation in zero correlation limit, becomes identical in nature with those of the pure (1-well) BCS formalism, as expected and the Coulomb correlation affects the pairing significantly with the strength of correlation. The detailed consequences are being presented here.