In this paper, we prove the uniqueness of ground states to the following fractional nonlinear elliptic equation with harmonic potential,

\[ (-\Delta)^s u+ \left(\omega+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\ \mathbb{R}^n, \]

$n \geq 1$

$0< s<1$

$\omega >-\lambda _{1,s}$

$2< p< {2n}/{(n-2s)^+}$

$\lambda _{1,s}>0$

$(-\Delta )^s + |x|^2$

$(-\Delta )^s$

$\mathcal {F}((-\Delta )^{s}u)(\xi )=|\xi |^{2s} \mathcal {F}(u)(\xi )$

$\xi \in \mathbb {R}^n$

$\mathcal {F}$

One famous debate in contemporary epistemology considers whether there is always one unique, epistemically rational way to respond to a given body of evidence. Generally speaking, answering “yes” to this question makes one a proponent of the Uniqueness thesis, while those who answer “no” are called “permissivists”. Another influential recent debate concerns whether non-truth-related factors can be the basis of epistemic justification, knowledge, or rational belief. Traditional theories answer “no”, and are therefore considered “purists”. However, more recently many theorists have argued to the contrary, claiming that impurist factors, such as practical stakes, can sometimes encroach or even override truth-related considerations. This paper bridges the two debates by presenting and defending what I call “Impurist Permissivism”. I support Impurist Permissivism by showing how it can resist Roger White's famous Argument from Arbitrariness (2005).

As it is presently employed, grounding permits grounding many things from one ground. In this paper, I show why this is a mistake by pushing for a uniqueness principle on grounding. After arguing in favor of this principle, I say something about it and kinds of grounding, discuss a similar principle, and consider its import on a formal feature of grounding, ontology, and ontological simplicity.

In this paper I identify a family of explanatory demands facing permissivists, those who deny the uniqueness thesis, according to which every body of evidence rationally permits exactly one doxastic attitude for a person to have in light of that evidence. Call a pair of a body of evidence and a proposition a permissive case just in case there is more than one attitude that is permitted for someone who has that body of evidence to take to that proposition. Uniquers claim that there are no permissive cases, and permissivists deny this. The uniqueness thesis is a strong claim, and is vulnerable to counterexamples because it is a universal generalization, and indeed, many permissivists argue for their view by identifying putative counterexamples to the uniqueness thesis. However, in virtue of advancing these putative counterexamples, permissivists incur explanatory demands. If some, but not all, cases are permissive cases, then permissivists owe us an explanation of why some cases are permissive and others are not. Likewise, for each permissive case, permissivists must explain why some attitudes towards the proposition are permitted and others are not. Permissivists draw arbitrary lines between permissive and impermissive cases, and between permitted and impermitted attitudes, giving rise to distinctions which need explaining.

I shall argue that permissivists cannot discharge these explanatory burdens in a satisfying way. After carefully presenting these explanatory demands in section 2, I consider how permissivists might answer them in section 3. I argue that the only permissivists who are able to successfully answer the explanatory demands are extreme permissivists like subjective Bayesians. Most philosophers, however, will find this epistemological outlook implausible because it contains no substantive constraints on rational belief. I also show that previous attempts by permissivists to explain why there are permissive cases merely relocate, rather than answer, the explanatory demands I have identified. The final section argues that permissivists cannot soften up the explanatory demands, either. I show that permissivist cannot appeal to the vagueness of the notion of a permissive case, or of an attitude's being permitted, in order to make the line between the permitted and the impermitted attitudes appear less arbitrary. Nor, I argue, can the permissivist hold that our ignorance of which cases are permissive, and how permissive they are, lessens the explanatory burdens they face.

Unfortunately, permissivists have been most interested in attempting to identify a counterexample to the uniqueness thesis, and have not been concerned with the explanatory demands which they incur in so doing. If my arguments are successful, then, permissivism is a much less plausible view than it is currently given credit for, and, at the very least, permissivists have a lot of work to do in further articulating and generalizing their view so as to provide an adequate explanation of the scope of epistemic permission.

In this paper, we consider the chemotaxis-Navier-Stokes model with realistic boundary conditions matching the experiments of Hillesdon, Kessler et al. in a two-dimensional periodic strip domain. For the lower boundary, we impose the usual homogeneous Neumann-Neumann-Dirichlet boundary condition. While, for the upper boundary, since it is open to the atmosphere, we consider three kinds of different mixed non-homogeneous boundary conditions, that is, (i) Neumann-Dirichlet-Navier slip boundary condition; (ii) Zero flux-Dirichlet-Navier slip boundary condition; (iii) Zero flux-Robin-Navier slip boundary condition. For boundary conditions (i) and (iii), the existence and uniqueness of global classical solutions for any initial data and any large chemotactic sensitivity coefficient is established, and for boundary condition (ii), the existence and uniqueness of global classical solutions for any initial data and small chemotactic sensitivity coefficient is proved.

We call attention to an important, but overlooked finding in research reported by Longoni, Bonezzi and Morewedge (2019). Longoni et al. claim that people always prefer a human to an artificially intelligent (AI) medical provider. We show that this was only the case when the historical performance of the human and AI providers was equal.  When the AI is known to outperform the human, their data showed a clear preference for the automated provider. We provide additional statistical analyses of their data to support this claim.

We clarify two points made in our commentary (Pezzo & Beckstead, 2020, this issue) on a recent paper by Longoni, Bonezzi, and Morewedge (2019). In both Experiments 1 and 4 from their paper, it is not possible to determine whether accuracy can compensate for algorithm aversion. Experiments 3A-C, however, do show a strong effect of accuracy such that AI that is superior to a human provider is embraced by patients. Many papers, including Longoni et al. tend to minimize the role of this compensatory process, apparently because it seems obvious to the authors (Longoni, Bonezzi, Morewedge, 2020, this issue). Such minimization, however, can lead to (mis)citations in which research that clearly demonstrates a compensatory role of AI accuracy is cited as non-compensatory.

In this work, we study a high order derivative in time problem. First, we show that there exists a sequence of elements of the spectrum which tends to infinity and therefore, it is ill posed. Then, we prove the uniqueness of solutions for this problem by adapting the logarithmic arguments to this situation. Finally, the results are applied to the backward in time problem for the generalized linear Burgers’ fluid, a couple of heat conduction problems and a viscoelastic model.

We present a uniqueness result for Gibbs point processes with interactions that come from a non-negative pair potential; in particular, we provide an explicit uniqueness region in terms of activity z and inverse temperature $\beta$ . The technique used relies on applying to the continuous setting the classical Dobrushin criterion. We also present a comparison to the two other uniqueness methods of cluster expansion and disagreement percolation, which can also be applied for this type of interaction.

Epistemic uniqueness is the view that there is at most one rational doxastic response to a given batch of evidence E, while epistemic permissiveness is the denial of epistemic uniqueness. As several authors have noted, one of the attractions of epistemic permissiveness is that it allows us to believe that more than one doxastic response may be rationally faultless, and so permits us to respect the epistemic credentials of all parties to a dispute. I argue that permissiveness is unable to deliver on this claimed benefit in many philosophical disputes.

Previous research has demonstrated that unique names increased in Japan, which shows a rise in uniqueness-seeking and individualism. To increase the validity of the prior findings, it is important to confirm the robustness of their results. Therefore, this study examined another indicator of historical changes in names in Japan. Specifically, I investigated whether the rates of common names decreased in Japan between 2004 and 2018. The dataset used in the previous study was analyzed. The results consistently showed that the rates of common names decreased for both boys and girls for the period. These results were consistent with the previous research, which further increases the validity of the finding that Japanese culture became more individualistic.