Negative / Null Result ReportOpen accessMathematics
Ingo Steinwart · 2024 · arXiv
We investigate for which Gaussian processes there do or do not exist reproducing kernel Hilbert spaces (RKHSs) that contain almost all of their paths. In particular, we establish a new result that makes it possible to exclude the existence of such RKHSs in many cases. Moreover, we combine this negative result with some known techniques to establish positive results. Here it turns out that for many classical families of Gaussian processes we can fully characterize for which members of these families there exist RKHSs containing the paths. Similar characterizations are obtained for Gaussian proc
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Andrey Sarychev · 2011 · arXiv
We study controllability of $d$-dimensional defocusing cubic Schroedinger equation under periodic boundary conditions. The control is applied additively, via a source term, which is a linear combination of few complex exponentials (modes) with time-variant coefficients - controls. We manage to prove that controlling at most $2^d$ modes one can achieve controllability of the equation in any finite-dimensional projection of the evolution space $H^{s}(\mathbb{T}^d), \ s>d/2$, as well as approximate controllability in $H^{s}(\mathbb{T}^d)$. We also present negative result regarding exact controlla
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Swagato K. Ray, Rudra P. Sarkar · 2012 · arXiv
Generalizing a result of Roe \cite{Roe} Strichartz proved in \cite{Str} that if a doubly-infinite sequence $\{f_k\}$ of functions on $\R^n$ satisfies $f_{k+1}=Δf_k$ and $|f_{k}(x)|\leq M$ for all $k=0,\pm 1,\pm 2,...$ and $x\in \R^n$, then $Δf_0(x)= -f_0$. Strichartz also showed that the result fails for hyperbolic 3-space. This negative result can be indeed extended to any Riemannian symmetric space of noncompact type. Taking this into account we shall prove that for all Riemannian symmetric spaces of noncompact type the theorem actually holds true when uniform boundedness is modified suitabl
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Fausto Di Biase, Haguma Gratien, Olof Svensson · 2025 · arXiv
In 1906 Fatou proved that bounded holomorphic functions on the unit disc converge a.e. on the boundary along nontangential approach regions. In 1927 Littlewood proved a negative result, i.e., that a.e. convergence fails for certain approach regions: More precisely, it fails for the rotationally invariant families of tangential approach regions that end curvilinearly at the boundary. The fact that tangential approach regions which end sequentially at the boundary may instead be very well conducive to a.e. convergence was understood more recently by W. Rudin (in 1979) and A. Nagel and E.M. Stein
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Antoine Salomon, Jean-Yves Audibert · 2011 · arXiv
This paper studies the deviations of the regret in a stochastic multi-armed bandit problem. When the total number of plays n is known beforehand by the agent, Audibert et al. (2009) exhibit a policy such that with probability at least 1-1/n, the regret of the policy is of order log(n). They have also shown that such a property is not shared by the popular ucb1 policy of Auer et al. (2002). This work first answers an open question: it extends this negative result to any anytime policy. The second contribution of this paper is to design anytime robust policies for specific multi-armed bandit pro
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David Preinerstorfer · 2014 · arXiv
We analytically investigate size and power properties of a popular family of procedures for testing linear restrictions on the coefficient vector in a linear regression model with temporally dependent errors. The tests considered are autocorrelation-corrected F-type tests based on prewhitened nonparametric covariance estimators that possibly incorporate a data-dependent bandwidth parameter, e.g., estimators as considered in Andrews and Monahan (1992), Newey and West (1994), or Rho and Shao (2013). For design matrices that are generic in a measure theoretic sense we prove that these tests eithe
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Dano Kim, Hoseob Seo · 2021 · arXiv
In $L^2$ extension theorems from a singular hypersurface in a complex manifold, important roles are played by certain measures such as the Ohsawa measure which determine when a given function can be extended. We show that the singularity of the Ohsawa measure can be identified in terms of singularity of pairs from algebraic geometry. Using this, we give an analytic proof of the inversion of adjunction in this setting. Then these considerations enable us to compare various positive and negative results on $L^2$ extension from singular hypersurfaces. In particular, we generalize a recent negativ
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Yvain Bruned, Paul Laubie · 2025 · arXiv
Rooted trees are essential for describing numerical schemes via the so-called B-series. They have also been used extensively in rough analysis for expanding solutions of singular Stochastic Partial Differential Equations (SPDEs). When one considers scalar-valued equations, the most efficient combinatorial set is multi-indices. In this paper, we investigate the existence of intermediate combinatorial sets that will lie between multi-indices and rooted trees. We provide a negative result stating that there is no combinatorial set encoding elementary differentials in dimension $d\neq 1$, and comp
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Buddhadev Hajra · 2026 · arXiv
We study the descent behaviour of homotopy-theoretic properties of smooth complex affine surfaces under finite surjective morphisms. We first examine the Eilenberg-MacLane property and show, by means of an explicit counterexample, that it does not descend under proper morphisms in general. This negative result motivates the introduction of a weaker notion, the finite homotopy rank-sum property. Our main theorem establishes that this property does descend under proper morphisms between smooth affine surfaces of logarithmic Kodaira dimension at most zero. The proof relies essentially on the rece
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Bradley Stich · 2021 · arXiv
The group synchronization problem is to estimate unknown group elements at the vertices of a graph when given a set of possibly noisy observations of group differences at the edges. We consider the group synchronization problem on finite graphs with size tending to infinity, and we focus on the question of whether the true edge differences can be exactly recovered from the observations (i.e., strong recovery). We prove two main results, one positive and one negative. In the positive direction, we prove that for a sequence of synchronization problems containing the complete digraph along with a
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Austin Conner, Hang Huang, J. M. Landsberg · 2020 · arXiv
We determine the border ranks of tensors that could potentially advance the known upper bound for the exponent $ω$ of matrix multiplication. The Kronecker square of the small $q=2$ Coppersmith-Winograd tensor equals the $3\times 3$ permanent, and could potentially be used to show $ω=2$. We prove the negative result for complexity theory that its border rank is $16$, resolving a longstanding problem. Regarding its $q=4$ skew cousin in $ C^5\otimes C^5\otimes C^5$, which could potentially be used to prove $ω\leq 2.11$, we show the border rank of its Kronecker square is at most $42$, a remarkable
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Mirna Džamonja, Saharon Shelah · 1997 · arXiv
This note gives two results on guessing unbounded subsets of lambda^+. The first is a positive result and applies to the situation of lambda regular, while the second is a negative consistency result which applies to the situation of lambda singular. Both results are connected to an earlier result of the same authors in which they showed that a certain version of clubsuit holds at a successor of singular just in ZFC. The first result here shows that that fact can to a certain extent be extended to the successor of singular. The negative result here gives limitations to the extent to which one
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H. Beirao da Veiga, F. Crispo · 2010 · arXiv
We consider the problem of the strong convergence, as the viscosity goes to zero, of the solutions to the three-dimensional evolutionary Navier-Stokes equations under a Navier slip-type boundary condition to the solution of the Euler equations under the zero flux boundary condition. In spite of the arbitrarily strong convergence results proved in the flat boundary case, see [4], it was shown in reference [5] that the result is false in general, by constructing an explicit family of smooth initial data in the sphere, for which the result fails. Our aim here is to present a more general, simpler
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Michel Duprez, Pierre Lissy · 2017 · arXiv
This paper is devoted to studying the null and approximate controllability of two linear coupled parabolic equations posed on a smooth domain of R^N (N>1) with coupling terms of zero and first orders and one control localized in some arbitrary nonempty open subset of the domain. We prove the null controllability under a new sufficient condition and we also provide the first example of a not approximately controllable system in the case where the support of one of the nontrivial first order coupling terms intersects the control domain.
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Kevin Barreto, Jiwon Kang, Sang-hyun Kim et al. · 2026 · arXiv
Answering a question of Erdős and Graham, we show that the double exponential growth condition $\limsup_{n\to\infty}a_n^{1/φ^n}=\infty$ for a strictly increasing sequence of positive integers $\{a_n\}_{n=1}^\infty$ is sufficient for the series $\sum_{n=1}^\infty 1/(a_n a_{n+1})$ to have an irrational sum; here $φ$ denotes the golden ratio. We also provide a positive generalization to $\sum_{n=1}^\infty 1/(a_n^{w_0}\cdots a_{n+d-1}^{w_{d-1}})$, and a negative result showing that some of its instances are essentially optimal. The original problem was autonomously solved by the AI agent \emph{Ale
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Itai Benjamini, Alexander Shamov, Barak Weiss · 2026 · arXiv
We present an open problem about non-colliding freely moving hard disks in the Euclidean plane, together with related positive and negative partial results. The open problem is stated in a non-degenerate form: velocities are required to be pairwise distinct and their speeds are required to be uniformly bounded away from infinity. The positive deterministic result gives a bounded, injective, non-colliding velocity assignment for the integer lattice; after a common velocity shift, the speeds are also bounded away from zero. The negative result shows that no bounded continuous vector field on the
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Han Huang · 2016 · arXiv
It is natural to ask whether the center of mass of a convex body $K\subset \mathbb{R}^n$ lies in its John ellipsoid $B_K$, i.e., in the maximal volume ellipsoid contained in $K$. This question is relevant to the efficiency of many algorithms for convex bodies. In this paper, we obtain an unexpected negative result. There exists a convex body $K\subset \mathbb{R}^n$ such that its center of mass does not lie in the John ellipsoid $B_K$ inflated $(1-C\sqrt{\frac{\log(n)} {n}})n$ times about the center of $B_K$. Moreover, there exists a polytope $P \subset \mathbb{R}^n$ with $O(n^2)$ facets whose
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Petra Laketa, Stanislav Nagy · 2022 · arXiv
The halfspace depth of a $d$-dimensional point $x$ with respect to a finite (or probability) Borel measure $μ$ in $\mathbb{R}^d$ is defined as the infimum of the $μ$-masses of all closed halfspaces containing $x$. A natural question is whether the halfspace depth, as a function of $x \in \mathbb{R}^d$, determines the measure $μ$ completely. In general, it turns out that this is not the case, and it is possible for two different measures to have the same halfspace depth function everywhere in $\mathbb{R}^d$. In this paper we show that despite this negative result, one can still obtain a substan
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Yossi Arjevani, Ohad Shamir · 2016 · arXiv
Finite-sum optimization problems are ubiquitous in machine learning, and are commonly solved using first-order methods which rely on gradient computations. Recently, there has been growing interest in \emph{second-order} methods, which rely on both gradients and Hessians. In principle, second-order methods can require much fewer iterations than first-order methods, and hold the promise for more efficient algorithms. Although computing and manipulating Hessians is prohibitive for high-dimensional problems in general, the Hessians of individual functions in finite-sum problems can often be effic
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Tobias Fissler, Yannick Hoga · 2024 · arXiv
This paper lays out a principled approach to compare copula forecasts via strictly consistent scores. We first establish the negative result that, in general, copulas fail to be elicitable, implying that copula predictions cannot sensibly be compared on their own. A notable exception is on Fréchet classes, that is, when the marginal distribution structure is given and fixed, in which case we give suitable scores for the copula forecast comparison. As a remedy for the general non-elicitability of copulas, we establish novel multi-objective scores for copula forecast along with marginal forecast
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Francisco Javier Gallego, Miguel Gonzalez, Bangere P. Purnaprajna · 2015 · arXiv
In this paper, we show that if $X$ is a smooth variety of general type of dimension $m \geq 2$, for which its canonical map induces a double cover onto $Y$, where $Y$ is a projective bundle over $\mathbf P^1$, or onto a projective space or onto a quadric hypersurface, embedded by a complete linear series, then the general deformation of the canonical morphism of $X$ again is canonical and again induces a double cover. The second part of the article deals with the existence or non existence of canonical double structures on rational varieties. The negative result in this article has consequence
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Amadeo Irigoyen · 2006 · arXiv
In the theory of approximation there are some problems on approximation of compacts in functional spaces by nonlinear families : first we deal with the polynomial case, and then we consider the analytic case. We demonstrate a negative result in which we claim that an analytic familie of functions with $N$ parameters can not approach the compact $Λ_l(I^s)$ closer than of order $(N\log N)^{\frac{l}{s}}$, when $N$ increases. As applied to an inverse problem in Sturm-Liouville theory, this assertion provides an answer to a question about the best possible reconstruction of the negative potential $
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Laurent Bienvenu, Santiago Figueira, Benoit Monin et al. · 2014 · arXiv
Suppose that we are given an infinite binary sequence which is random for a Bernoulli measure of parameter $p$. By the law of large numbers, the frequency of zeros in the sequence tends to~$p$, and thus we can get better and better approximations of $p$ as we read the sequence. We study in this paper a similar question, but from the viewpoint of inductive inference. We suppose now that $p$ is a computable real, but one asks for more: as we are reading more and more bits of our random sequence, we have to eventually guess the exact parameter $p$ (in the form of a Turing code). Can one do such a
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Abbas Moameni, Brendan Pass · 2015 · arXiv
We study solutions to the multi-marginal Monge-Kantorovich problem which are concentrated on several graphs over the first marginal. We first present two general conditions on the cost function which ensure, respectively, that any solution must concentrate on either finitely many or countably many graphs. We show that local differential conditions on the cost, known to imply local $d$-rectifiability of the solution, are sufficient to imply a local version of the first of our conditions. We exhibit two examples of cost functions satisfying our conditions, including the Coulomb cost from density
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Tony Se · 2015 · arXiv
Let R be a commutative Noetherian ring, I and J ideals of R and M a finitely generated R-module. Let F be a covariant R-linear functor from the category of finitely generated R-modules to itself. We first show that if F is coherent, then the sets of associated primes of F(M/I^n M) and F(I^n-1 M / I^n M) and the J-depths of F(M/I^n M) and F(I^n-1 M / I^n M) become independent of n for large n. Next, we consider several examples in which F is a rather familiar functor, but is not coherent or not even finitely generated in general. In these cases, the set of associated primes of F(M/I^n M) still
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Samuel Pawel, Rachel Heyard, Charlotte Micheloud et al. · 2023 · arXiv
In several large-scale replication projects, statistically non-significant results in both the original and the replication study have been interpreted as a "replication success". Here we discuss the logical problems with this approach: Non-significance in both studies does not ensure that the studies provide evidence for the absence of an effect and "replication success" can virtually always be achieved if the sample sizes are small enough. In addition, the relevant error rates are not controlled. We show how methods, such as equivalence testing and Bayes factors, can be used to adequately qu
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Dawid Kucharski · 2026 · arXiv
The Brody distribution, originally a phenomenological interpolation between Poisson and Wigner level-spacing statistics in quantum chaos, is calibrated here as a quantitative measure of short-range exclusion in 2D spatial point processes. Two results form the core. First, the 2D complete-spatial-randomness baseline is recalibrated to $β=0.96\pm0.15$, correcting the inappropriate 1D Poisson reference. Second, an empirical $β$--$r_{\text{excl}}$ calibration is validated against the effective hard-core radius with Spearman $ρ=0.988$. The framework is demonstrated on 58 manufactured surfaces (10 m
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Jens Rauch, Jens Hüsers, Birgit Babitsch et al. · 2018 · arXiv
Frequent users of emergency departments (ED) pose a significant challenge to hospital emergency services. Despite a wealth of studies in this field, it is hardly understood, what medical conditions lead to frequent attendance. We examine (1) what ambulatory care sensitive conditions (ACSC) are linked to frequent use, (2) how frequent users can be clustered into subgroups with respect to their diagnoses, acuity and admittance, and (3) whether frequent use is related to higher acuity or admission rate. We identified several ACSC that highly increase the risk for heavy ED use, extracted four majo
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Antonin Novak, Andrzej Gnatowski, Premysl Sucha · 2025 · arXiv
This paper addresses the problem of batching laboratory samples in hospital laboratories where samples of different priorities are received continuously with uncertain transportation times. The focus is on optimizing the control strategy for loading a centrifuge to minimize patient turnaround time (TAT). While focusing on samples of patients in life-threatening situations (i.e., vital samples), we propose several online and offline methods, including a stochastic mixed-integer quadratic programming model integrated within a discrete-event system simulation. This paper aims to enhance patient c
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H. Sherry Zhang, Dianne Cook, Nicolas Langrené et al. · 2024 · arXiv
The projection pursuit (PP) guided tour optimizes a criterion function, known as the PP index, to gradually reveal projections of interest from high-dimensional data through animation. Optimization of some PP indexes can be non-trivial, if they are non-smooth functions, or when the optimum has a small "squint angle", detectable only from close proximity. Here, measures for calculating the smoothness and squintability properties of the PP index are defined. These are used to investigate the performance of a recently introduced swarm-based algorithm, Jellyfish Search Optimizer (JSO), for optimiz
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