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The saturation property for Littlewood--Richardson coefficients was established by Knutson and Tao in 1999. In 2004, Kirillov conjectured that the saturation property extends to Schubert coefficients. We disprove this conjecture in a strong form, by showing that it fails for a large family of instances. We also refute the saturation property for Schubert coefficients under bit scaling and discuss computational complexity implications.
In the context of the Dirac equation with square-summable potential, we study the Jost solutions and prove that the maximal function associated with the argument of the transmission coefficient is unbounded. We also show that the strong version of the nonlinear Carleson conjecture fails for Dirac equations and Krein systems.
Large Language Model (LLM) services such as ChatGPT, DALLE, and Cursor have quickly become essential for society, businesses, and individuals, empowering applications such as chatbots, image generation, and code assistance. The complexity of LLM systems makes them prone to failures and affects their reliability and availability, yet their failure patterns are not fully understood, making it an emerging problem. However, there are limited datasets and studies in this area, particularly lacking an open-access tool for analyzing LLM service failures based on incident reports. Addressing these problems, in this work we propose FAILS, the first open-sourced framework for incident reports collection and analysis on different LLM services and providers. FAILS provides comprehensive data collection, analysis, and visualization capabilities, including:(1) It can automatically collect, clean, and update incident data through its data scraper and processing components;(2) It provides 17 types of failure analysis, allowing users to explore temporal trends of incidents, analyze service reliability metrics, such as Mean Time to Recovery (MTTR) and Mean Time Between Failures (MTBF);(3) It leverag
In this report we flesh out a sketch by Krachun and Kazanin to prove that for a certain family of Reed-Solomon codes, proximity gaps fail at radii that are $O(1/\log n)$ below the capacity rate of the code, where $n$ is the length of the code.
The violation of Bell type inequalities in quantum systems manifests that quantum states cannot be described by classical probability distributions. Yet, Bohmian mechanics is a realistic, non-local theory of classical particle trajectories that is claimed to be indistinguishable by observations from more traditional approaches to quantum mechanics. We set up a spatial version of the GHZ system with qubits realised as positional observables that demonstrates that the Bohmian theory fails to match predictions of textbook quantum mechanics (and most likely experients) unless enlarged by a microscopic theory of collapse of the wave function after observation. For this discrepancy to occur it is essential that positions at different times do not commute.
Contrary to common belief, we show that gradient ascent-based unconstrained optimization methods frequently fail to perform machine unlearning, a phenomenon we attribute to the inherent statistical dependence between the forget and retain data sets. This dependence, which can manifest itself even as simple correlations, undermines the misconception that these sets can be independently manipulated during unlearning. We provide empirical and theoretical evidence showing these methods often fail precisely due to this overlooked relationship. For random forget sets, this dependence means that degrading forget set metrics (which, for a retrained model, should mirror test set metrics) inevitably harms overall test performance. Going beyond random sets, we consider logistic regression as an instructive example where a critical failure mode emerges: inter-set dependence causes gradient descent-ascent iterations to progressively diverge from the ideal retrained model. Strikingly, these methods can converge to solutions that are not only far from the retrained ideal but are potentially even further from it than the original model itself, rendering the unlearning process actively detrimental.
We prove that the Harnack inequality fails for nonlocal kinetic equations. Such equations arise as linearized models for the Boltzmann equation without cutoff and are of hypoelliptic type. We provide a counterexample for the simplest equation in this theory, the fractional Kolmogorov equation. Our result reflects a purely nonlocal phenomenon since the Harnack inequality holds true for local kinetic equations like the Kolmogorov equation.
The Cremona groups are the groups of all birational equivalences of rational varieties and, equivalently, the automorphism groups of the rational function fields. In this note, we explain that homological stability fails for them in both possible ways and comment on their stable homology.
We point out an error in the paper "Linear Time Encoding of LDPC Codes" (by Jin Lu and José M. F. Moura, IEEE Trans). The paper claims to present a linear time encoding algorithm for every LDPC code. We present a family of counterexamples, and point out where the analysis fails. The algorithm in the aforementioned paper fails to encode our counterexample, let alone in linear time.
Given p between 1 and infinity, but not 2, we show that the T1 theorem for the Hilbert transform fails for L^{p}, despite holding for p equal to 2
We show that the Frobenius--stable version of the Grauert--Riemenschneider vanishing theorem fails for threefolds in any positive characteristic, and for terminal 3-folds in characteristic $p \in \{2, 3, 5\}$. To prove this, we introduce the notion of $\mathbb{F}_p$-rationality for singularities in positive characteristic and we show that klt singularities in dimension at most 4 are $\mathbb{F}_p$-rational. We apply this to prove a Frobenius--stable version of the Kawamata--Viehweg vanishing theorem on $K$-trivial 3-folds.
It is known that the first two-variable Links--Gould quantum link invariant $LG\equiv LG^{2,1}$ is more powerful than the HOMFLYPT and Kauffman polynomials, in that it distinguishes all prime knots (including reflections) of up to 10 crossings. Here we report investigations which greatly expand the set of evaluations of $LG$ for prime knots. Through them, we show that the invariant is complete, modulo mutation, for all prime knots (including reflections) of up to 11 crossings, but fails to distinguish some nonmutant pairs of 12-crossing prime knots. As a byproduct, we classify the mutants within the prime knots of 11 and 12 crossings. In parallel, we learn that $LG$ distinguishes the chirality of all chiral prime knots of at most 12 crossings. We then demonstrate that every mutation-insensitive link invariant fails to distinguish the chirality of a number of 14-crossing prime knots. This provides 14-crossing examples of chiral prime knots whose chirality is undistinguished by $LG$.
A result of Crew implies that for an etale Galois p-cover of smooth proper varieties over a field of characteristic p, the alternating sum of the p-ranks of the cohomology groups behave like Euler characteristics in characteristic 0. That is, the sum for the top curve equals the sum for the bottom curve times the degree of the cover. We exhibit an example to show that this relation fails in general if the p-rank is replaced by the part of some fixed slope (other than 0).
Borsuk's conjecture states that any bounded set in R^n can be partitioned into n+1 sets of smaller diameter. It is known to be false for all n bigger or equal to 323. Here we show that Borsuk's conjecture fails in dimensions 321 and 322. (This result has been independently discovered by Hinrichs and Richter.)
I examine a recently proposed failed-supernova scenario for the fading of the yellow supergiant event M31-2014-DS1, and find that it requires unlikely fine-tuned parameters to work, if at all. In the failed-supernova scenario, most of the yellow supergiant collapsed to form a black hole. Due to the energy carried by neutrinos from the cooling, collapsing core, gravity decreases, leading to the ejection of a small fraction of the outer envelope, some of which remains bound. The fallback accreted gas possesses large angular-momentum fluctuations due to the pre-collapse envelope convection. The fallback material forms intermittent accretion disks around the black hole that launch jets (or disk wind), which unbind most of the bound material. The failed-supernova scenario for M31-2014-DS1 requires that only <1% of the bound material be accreted by the black hole, but the jets do not shut down the backflow for over 10 years. I find this fine-tuned requirement unlikely. I also find that, due to the rapid radiative cooling of the outflow interaction zone with the outer gas, the expected radiation is about an order of magnitude or more above the observed value. These, as well as earlier
A central challenge for understanding unconventional superconductivity in most strongly correlated electronic materials is their complicated band structure and presence of competing orders. In contrast, quasi-two-dimensional organic spin liquids are single-band systems with superconductivity arising near the bandwidth-tuned Mott metal-insulator transition in the absence of other orders. Here, we study chemically substituted $κ$-organics in which superconducting fluctuations emerge in the phase coexistence region between the Mott insulator and the Fermi liquid. Using magnetotransport and ac susceptibility measurements, we find that global superconductivity fails to set in as temperature $T\rightarrow 0$. Our results indicate instead the presence of superconducting domains embedded in the metallic percolating cluster that undergo a magnetic field-tuned quantum superconductor-to-metal phase transition. Surprisingly, albeit consistent with the percolation picture, universal conductance fluctuations are seen at high fields in macroscopic samples. The observed interplay of the intrinsic inhomogeneity and quantum phase fluctuations provides a new insight into failed superconductivity, a p
Solana is an emerging blockchain platform, recognized for its high throughput and low transaction costs, positioning it as a preferred infrastructure for Decentralized Finance (DeFi), Non-Fungible Tokens (NFTs), and other Web 3.0 applications. In the Solana ecosystem, transaction initiators submit various instructions to interact with a diverse range of Solana smart contracts, among which are decentralized exchanges (DEXs) that utilize automated market makers (AMMs), allowing users to trade cryptocurrencies directly on the blockchain without the need for intermediaries. Despite the high throughput and low transaction costs of Solana, the advantages have exposed Solana to bot spamming for financial exploitation, resulting in the prevalence of failed transactions and network congestion. Prior work on Solana has mainly focused on the evaluation of the performance of the Solana blockchain, particularly scalability and transaction throughput, as well as on the improvement of smart contract security, leaving a gap in understanding the characteristics and implications of failed transactions on Solana. To address this gap, we conducted a large-scale empirical study of failed transactions o
Although the quality of human-robot interactions has improved with the advent of LLMs, there are still various factors that cause systems to be sub-optimal when compared to human-human interactions. The nature and criticality of failures are often dependent on the context of the interaction and so cannot be generalized across the wide range of scenarios and experiments which have been implemented in HRI research. In this work we propose the use of a technique overlooked in the field of HRI, ethnographic vignettes, to clearly highlight these failures, particularly those that are rarely documented. We describe the methodology behind the process of writing vignettes and create our own based on our personal experiences with failures in HRI systems. We emphasize the strength of vignettes as the ability to communicate failures from a multi-disciplinary perspective, promote transparency about the capabilities of robots, and document unexpected behaviours which would otherwise be omitted from research reports. We encourage the use of vignettes to augment existing interaction evaluation methods.
In this paper we begin the study of well-failed graphs, that is, graphs in which every maximal failed zero forcing set is a maximum failed zero forcing set, or equivalently, in which every minimal fort is a minimum fort. We characterize trees that are well-failed. Along the way, we prove that the set of vertices in a graph that are not in any minimal fort is identical to the set of vertices that are in no minimal zero forcing set, which allows us to characterize vertices in a tree that are in no minimal fort.
Why do banks fail? We create a panel covering most commercial banks from 1863 through 2024 to study the history of failing banks in the United States. Failing banks are characterized by rising asset losses, deteriorating solvency, and an increasing reliance on expensive noncore funding. These commonalities imply that bank failures are highly predictable using simple accounting metrics from publicly available financial statements. Failures with runs were common before deposit insurance, but these failures are strongly related to weak fundamentals, casting doubt on the importance of non-fundamental runs. Furthermore, low recovery rates on failed banks' assets suggest that most failed banks were fundamentally insolvent, barring strong assumptions about the value destruction of receiverships. Altogether, our evidence suggests that the primary cause of bank failures and banking crises is almost always and everywhere a deterioration of bank fundamentals.