In reasoning experiments, participants usually evaluate conclusions by clicking on predefined response alternatives or rating scales. However, such response formats may elicit specific cognitive reasoning strategies. To avoid such biases, we employed an open response format, allowing participants to formulate and explain their conclusions freely in complete sentences. Across two experiments, participants generated over 1,300 written responses, which we categorized as certain, uncertain based on counterexamples, or uncertain based on probabilities, using a predefined coding scheme. Participants frequently provided certain responses rather than expressing graded degrees of belief. When uncertainty was expressed, it was more often justified by concrete counterexamples than by probabilistic reasoning. We introduce the term probabilistic masking to describe the phenomenon whereby graded response formats may suppress the consideration of certain inferences or counterexamples, thereby biasing empirical accounts of human inference toward probabilistic interpretations.
William Farr, a renowned nineteenth-century disease theoretician, successfully predicted mortality rates due to cholera in London and Liverpool through the law of elevation, a law inversely correlating mortality from cholera with the elevation of the land. Farr's elevation law and its successful predictions were made within the framework of miasma theory - a theory now considered false, which postulated that diseases, such as cholera, were caught when inhaling toxic odours. Tulodziecki (2021, 2017) points out that Farr's elevation law is a counterexample to selective realism, the position according to which the theoretical elements essentially responsible for the success of a theory are likely to be true. This paper argues that Farr's elevation law was an empirical discovery, essentially independent of the false assumptions of miasma theory, even though the law was considered compatible with these assumptions. Given the empirical nature of Farr's discovery, the success-to-truth inference defended by selective realists does not apply in the first place, and Farr's elevation law is not a counterexample to selective realism.
We present examples where successive ionization potentials of an isolated electronic system are nonincreasing; ergo, the energy is a nonconvex function of particle number. The basic strategy, which is based on minimizing the energy of the N ± 1-electron systems by leveraging solutions to the Thomson problem, seems quite general, giving large families of systems with nonconvex behavior; we show examples for N = 5 and N = 7 charged particles. As with previous counterexamples in the literature, the key ingredient is a qualitative change in configuration upon ionization. We propose two families of molecular structures, F4Br65- and F6Br87-, that are expected to show nonconvex behavior with respect to the number of electrons.
In this paper, we present explicit examples of Denjoy minimal sets that exhibit two- to one-hole transitions at some parameter values 0<ϵ<1 and cantorus to circle transitions as ϵ→1 and collapse to finite sets as ϵ→0. The limit ϵ→0 is an anti-integrable limit in the sense of Aubry. We describe all the transitions in terms of multi-hole Sturmian symbolic systems.
暂无摘要（点击查看详情）
There is a widely accepted material characterization paradigm in the success of synthesis of luminescent metal nanoclusters (NCs) in the aqueous phase: new emission, metal reduction, and ultrasmall particles (size < 3 nm). Herein, we falsified well-known fluorescent histidine (His)-directed Au NCs and a new model of metastable His-Au(I) complexes with emissive His oxidation products has been established. The redox reaction of His and Au(III) yields His oligomers with blue-green fluorescence and reducible Au(I) self-assemblies, which can form ultrasmall particles at electron bombardment. The resultant Au(I) complexes can be further reduced by d-penicillamine (DPA) via forming anisotropic Au nanoparticles with distinct local surface plasmon resonance absorption. The emerging absorption can quench the fluorescence of the His oxidation products through the inner filter effect pathway. A facile dual-model analytical approach is thus proposed to directly detect DPA fluorometrically and colorimetrically without interference from common biothiols, including cysteine and glutathione. Thus, with the help of a smartphone app, a highly sensitive and selective point-of-care testing for DPA direct detection can be realized. Our study warrants the importance of thinking twice about characterization results and supports corrective models for finding new reactions and possible applications.
In the recent years there has been an increased interest in studying regularity properties of the derivatives of semilinear parabolic stochastic evolution equations (SEEs) with respect to their initial values. In particular, in the scientific literature it has been shown for every natural number n ∈ N that if the nonlinear drift coefficient and the nonlinear diffusion coefficient of the considered SEE are n-times continuously Fréchet differentiable, then the solution of the considered SEE is also n-times continuously Fréchet differentiable with respect to its initial value and the corresponding derivative processes satisfy a suitable regularity property in the sense that the n-th derivative process can be extended continuously to n-linear operators on negative Sobolev-type spaces with regularity parameters δ 1 , δ 2 , … , δ n ∈ [ 0 , ∞ ) provided that the condition ∑ i = 1 n δ i < 1 2 is satisfied. The main contribution of this paper is to reveal that this condition can essentially not be relaxed.
Axiomatizing centrality measures often requires proving that certain properties do not hold by exhibiting a counterexample (i.e., a graph for which a given centrality measure does not satisfy a specified property). In the context of geometric centralities, constructing such counterexamples requires building a graph with prescribed distance counts, as encoded in its distance-count matrix (DCM). We prove that deciding whether a matrix is the distance-count matrix of an undirected graph is strongly NP-complete. This negative result implies that a brute-force approach to constructing such counterexamples is out of the question. We complement this negative result with some positive findings: while recognizing DCM matrices is strongly NP-hard, the construction of DCM matrices is algorithmically well-behaved under some natural graph operations (which we call DCM-stable): that is, for many important graph operations ⊗, the DCM of G⊗H can be computed efficiently from those of G and H, without having to reconstruct the graphs themselves. This observation shows that, although the inverse problem is intractable in general, distance-count matrices admit a rich and tractable compositional theory on structured graph classes generated by DCM-stable operations.
We resolve a $1,000 Erdős prize problem, complete with formal verification generated by a large language model. In over a dozen papers, beginning in 1976 and spanning two decades, Paul Erdős repeatedly posed one of his "favorite" conjectures: every finite Sidon set can be extended to a finite perfect difference set. We establish that {1, 2, 4, 8, 13} is a counterexample to this conjecture. During the preparation of this paper, we found that although this problem was presumed to be open for half a century, Marshall Hall, Jr. published a different counterexample three decades before Erdős first posed the problem. With a healthy skepticism of this apparent oversight, and out of an abundance of caution, we used ChatGPT to vibe prove both Hall's and our counterexamples in Lean.
Children have been shown to generalize information acquired from social partners better when it is provided to them in a communicative manner. Two distinct learning mechanisms have been proposed to explain how communicative demonstrations assist the acquisition of generic knowledge in infants: by strengthening or speeding up inductive generalization processes (Boosted Induction), or by providing a direct generic statement in a nonverbal form (Nonverbal Generics), which makes additional generalization processes unnecessary. We sought to adjudicate between these accounts by exploring how children treat negative evidence, because disregarding negative evidence is considered to be a hallmark of having acquired generic knowledge. In two studies with 18-month-old infants, we tested the distinct predictions these two accounts offer on how infants should respond to experiencing a counterexample after acquiring information about a single object either from a communicative demonstration or from non-communicative observation. We found that the infants who learned a non-obvious property of an object from simple observation were discouraged from further generalizing this property after having encountered a counterexample. In contrast, the infants who received a communicative demonstration of the same property persisted in trying to elicit it from a novel exemplar even after they had failed to elicit it from a counterexample. These results support the Nonverbal Generics account, and suggest that human infants can learn generic knowledge directly, without induction, from others even before mastering the linguistic skills necessary for the comprehension of generic expressions.
The fragility index (FI) is intended to quantify how many outcome changes would be required to convert a statistically significant two-arm trial result into a nonsignificant one. A reliable statistical metric should produce a result for every valid case it evaluates. This study examined whether a fragility value is always attainable for every statistically significant trial result. FI was analyzed as follows: baseline significance was required (p < 0.05), one-way movement only, and outcome changes were restricted to converting a nonevent to an event in the arm with fewer events, while keeping the arm size fixed. Nonattainability was assessed by determining whether valid 2×2 tables exist for which no finite FI can be obtained under these rules. Evidence is provided through formal counterexamples, complete enumeration of all valid nondegenerate 2 × 2 tables up to total sample size N = 60, and empirical evaluation of published two-arm trials with binary outcomes. Valid baseline-significant 2 × 2 tables exist for which FI is not attainable. A simple counterexample is {3,0,4,11}: baseline two-sided Fisher's exact p = 0.0429, the arm with fewer events is uniquely identified, but that arm has no nonevents available for the required toggle; thus, no legal FI path exists. Enumeration revealed that unattainable cases first appeared at N = 18 and then recurred at every larger sample size through N = 60; by N = 60, a total of 2,390 of 20,774 evaluable baseline-significant tables were unattainable (11.5%). In an empirical dataset of published trials, 2 of 82 baseline-significant evaluable trials (2.4%) were not attainable. The FI is not universally attainable. This is a structural property of the FI algorithm, confirmed by mathematical proof, a complete table enumeration, and published trial data.
An epistemic relative necessity account is proposed to treat nonmodal deductive reasoning as modal reasoning. It assumes that an epistemically valid conclusion from the factual premises is pragmatically necessary relative to the premises. Three studies on modal Modus Ponens problems (with the form: given the premises of p and if p then q, individuals were asked to judge whether the conclusion is "necessarily q," "q," or "possibly q") revealed (1) Participants generally defaulted to interpreting arbitrary conditionals "if p then q" as "if p then must q." (2) Modal MP problems without retrievable counterexamples to conditionals tended to elicit inferences "necessarily q" rather than "q." (3) The influence of level of relevance in conditionals (arbitrary vs. causal conditionals) on modal inferences was modulated by whether causal conditionals had retrievable counterexamples: Causal conditionals with retrievable counterexamples elicited more "possibly q" inferences (belief bias responses) and fewer "necessarily q" inferences than arbitrary and causal conditionals without retrievable counterexamples. The overall response pattern favors only the epistemic relative necessity account, indicating that a mentally valid nonmodal deductive inference can be transformed into a modal inference including the modal word "necessary" in the conclusion. Our research bridges linguistic and psychological research on epistemic necessity.
Gihawi et al. (mBio 14:e01607-23, 2023, https://doi.org/10.1128/mbio.01607-23) argued that the analysis of tumor-associated microbiome data by Poore et al. (Nature 579:567-574, 2020, https://doi.org/10.1038/s41586-020-2095-1) is invalid because features that were originally very sparse (genera with mostly zero read counts) became associated with the phenotype following batch correction. Here, we examine whether such an observation should necessarily indicate issues with processing or machine learning pipelines. We show counterexamples using the centered log ratio (CLR) transformation, which is often used for analysis of compositional microbiome data. The CLR transformation has similarities to voom-SNM, the batch-correction method brought into question by Gihawi et al., and yet is a sample-wise operation that cannot, in itself, "leak" information or invalidate downstream analyses. We show that because the CLR transformation divides each value by the geometric mean of its sample, common imputation strategies for missing or zero values result in transformed features that are associated with the geometric mean. Through analyses of both synthetic and vaginal microbiome data sets, we demonstrate that when the geometric mean is associated with a phenotype, sparse and CLR-transformed features will also become associated with it. We re-analyze features highlighted by Gihawi et al. and demonstrate that the phenomenon of sparse features becoming phenotype-associated can also be observed after a CLR transformation, which serves as a counterexample to the claim that such an observation necessarily means information leakage. While we do not intend to address other concerns regarding tumor microbiome analyses, validate Poore et al.'s results, or evaluate batch-correction pipelines, we conclude that because phenotype-associated features that were initially sparse can be created by a sample-wise transformation that cannot artifactually inflate machine learning performance, their detection is not independently sufficient to demonstrate information leakage in machine learning pipelines. Microbiome data are multivariate, and as such, a value of 0 carries a different meaning for each sample. Many transformations, including CLR and other batch-correction methods, are likewise multivariate, and, as these issues demonstrate, each individual feature should be interpreted with caution. Gihawi et al. claim that finding that a transformation turned highly sparse (mostly zero) features into features that are associated with a phenotype is sufficient to conclude that there is information leakage and to invalidate an analysis. This claim has critical implications for both the debate regarding The Cancer Genome Atlas (TCGA) cancer microbiome analysis and for interpretation and evaluation of analyses in the microbiome field at large. We show by counterexamples and by reanalysis that such transformations can be valid.
Nonlinear entanglement witnesses constructed from multiple linear entanglement witnesses and multiple copies of quantum states have recently been proposed as a powerful tool for entanglement detection. In this work, we show, via an explicit counterexample, that the fineness of linear witnesses generally fails to transfer to their tensor-product nonlinear counterparts. For the canonical family of nonlinear witnesses in the form of (αI-L)⊗(βI-T), we rigorously prove that the optimal nonlinear witness is uniquely attained with weakly optimal parameters of α=λmax(L) and β=λmax(T). Meanwhile, we analytically demonstrate that the self-tensor products of two representative linear witnesses fail to detect any entangled state. The question of whether a nonlinear entanglement witness capable of detecting entanglement can be constructed by tensoring a linear witness with itself remains open.
We present counterexamples to the lore that symmetries that cannot be gauged or made on site are necessarily anomalous. Specifically, we construct unitary, internal symmetries of two-dimensional lattice models that cannot be consistently coupled to background or dynamical gauge fields or disentangled to a tensor product of on-site operators. These symmetries are nevertheless anomaly-free in the sense that they admit symmetric, gapped Hamiltonians with unique, invertible ground states. We show that symmetries of this kind are characterized by an index [ω]∈H^{2}(G,Q_{+}), where Q_{+} is the multiplicative group of positive rational numbers labeling one-dimensional quantum cellular automata.
Olfactory dysfunction is an early, prodromal feature of Parkinson's disease (PD), often preceding motor symptoms by years. This has fueled the hypothesis that PD pathology may originate in the olfactory bulb (OB), where early Lewy-type α-synucleinopathy is frequently observed and from which pathology could propagate to neural systems. In this context, Arshamian et al. (2022) hypothesized that isolated congenital anosmia (ICA), a rare lifelong absence of smell typically associated with bilateral OB aplasia, might confer immunity to PD. They argued that identifying a single individual with both ICA and PD would constitute a "black swan", falsifying the claim that an intact OB is necessary for PD initiation. We describe such a case: an individual with lifelong anosmia, MRI-based evidence of bilateral OB aplasia, and a clinically established diagnosis of PD. This counterexample challenges strict OB-necessity models and suggests PD can arise despite apparent congenital absence of an intact OB.
Generalized Feller theory provides an important analog to Feller theory beyond locally compact metric state spaces. This is very useful for solutions of certain stochastic partial differential equations, Markovian lifts of fractional processes, or infinite-dimensional affine and polynomial processes which appear prominently in the theory of signature stochastic differential equations. We extend several folklore results related to generalized Feller processes, in particular on their construction and path properties, and provide the often quite sophisticated proofs in full detail. We also introduce the new concept of extended Feller processes and compare them with classical and generalized ones as well as with Doob's h-transform. A key example relates generalized Feller semigroups of algebra homomorphisms via the method of characteristics to transport equations and continuous semiflows on weighted spaces, i.e., a remarkably generic way to treat differential equations on weighted spaces. We also provide a counterexample, which shows that no condition of the basic definition of generalized Feller semigroups can be dropped.
In dense neutrino environments, the mean field of flavor coherence can develop instabilities. A necessary condition is that the flavor lepton number changes sign as a function of energy and/or angle. Whether such a crossing is also sufficient has been a longstanding question. We construct an explicit counterexample: a spectral crossing without accompanying flavor instability, with an even number of crossings being key. This failure is physically understood as Cherenkov-like emission of flavor waves. If flipped-lepton-number neutrinos never dominate among those kinematically allowed to decay, the waves cannot grow.
Quantum oscillations in magnetization or resistivity are a defining feature of metals in a magnetic field. The phenomenon is generally not expected in insulators without a Fermi surface. Its observation in Kondo and other correlated insulators provided counterexamples and remains poorly understood. Here we report the observation of resistivity oscillations in a gate-controlled excitonic insulator realized in Coulomb-coupled electron-hole double layers. When the electron or hole cyclotron energy is tuned to exceed the exciton binding energy, recurring transitions arise between the excitonic insulator and layer-decoupled quantum Hall states. Compressibility measurements show an oscillatory exciton binding energy as a function of the magnetic field and electron-hole pair density. Coulomb drag measurements further reveal the signature of finite-angular-momentum excitonic correlations. These findings are qualitatively captured by mean-field calculations. Our study establishes a highly tunable platform based on electron-hole double layers for studying quantum oscillations in correlated insulators.
Population-level distributions of fluorescence or molecule counts are often taken to reflect the behaviours of individual cells within that population. In this conceptual article, I argue that counting subpopulations can be a misleading proxy for identifying the number of behavioural modes accessible to individual cells within a system. I show that definitions of behavioural modes based on deterministic modelling can fail when fluctuations in a system's state-or noise-become significant. In such cases, peaks in the probability distribution-emerging from stochastic descriptions-are often interpreted as substitutes for deterministically defined stable modes. However, I demonstrate that this interpretation can break down: it is possible to construct counterexamples in which two subpopulations arise from a system that supports only a single mode of behaviour, driven by non-equilibrium transient dynamics. Better understanding the role of noise in transient biological randomness may allow for the discovery of novel mechanisms of regulation that are not apparent in steady-state behaviours.