A simple proof of attainability for the Sobolev inequality
We give a simple proof of the existence of a minimizer for the Sobolev inequality. Our proof is based on a representation formula via a cut-off fundamental solution.
math.FA↗
arXiv subjects
Megumi Sano
Publications and source records attributed to Megumi Sano.
Weighted Trudinger-Moser inequalities in the subcritical Sobolev spaces and their applications
We study boundedness, optimality and attainability of Trudinger-Moser type maximization problems in the radial and the subcritical homogeneous Sobolev spaces $\dot{W}^{1,p}_{0, \text{rad}}(B_R^N)\,(p<N)$. Our results give a revision of an error in \cite[Theorem C]{HL}. Also, our inequality converges to the original Trudinger-Moser inequality as $p \nearrow N$ including optimal exponent and concentration limit. Finally, we consider an application of our inequality to elliptic problems with exponential nonlinearity.
math.AP↗
Developmental Curiosity and Social Interaction in Virtual Agents
Infants explore their complex physical and social environment in an organized way. To gain insight into what intrinsic motivations may help structure this exploration, we create a virtual infant agent and place it in a developmentally-inspired 3D environment with no external rewards. The environment has a virtual caregiver agent with the capability to interact contingently with the infant agent in ways that resemble play. We test intrinsic reward functions that are similar to motivations that have been proposed to drive exploration in humans: surprise, uncertainty, novelty, and learning progress. These generic reward functions lead the infant agent to explore its environment and discover the contingencies that are embedded into the caregiver agent. The reward functions that are proxies for novelty and uncertainty are the most successful in generating diverse experiences and activating the environment contingencies. We also find that learning a world model in the presence of an attentive caregiver helps the infant agent learn how to predict scenarios with challenging social and physical dynamics. Taken together, our findings provide insight into how curiosity-like intrinsic rewards and contingent social interaction lead to dynamic social behavior and the creation of a robust predictive world model.
cs.LG↗
On eigenvalue problems involving the critical Hardy potential and Sobolev type inequalities with logarithmic weights in two dimensions
We consider the two-dimensional eigenvalue problem for the Laplacian with the Neumann boundary condition involving the critical Hardy potential. We prove the existence of the second eigenfunction and study its asymptotic behavior around the origin. A key tool is the Sobolev type inequality with a logarithmic weight, which is shown in this paper as an application of the weighted nonlinear potential theory.
math.AP↗
On the compactness of the (non)radial Sobolev spaces
In this note, we give the affirmative answer of the question in [18], which is a compactness result of the non-radial Sobolev spaces. As an application, we show the existence of an extremal function of the critical Hardy inequality under spherical average zero. Next, we give an improvement of the compactness results of the radial Sobolev spaces in [8]. In Appendix, we give an alternative proof of Hardy type inequalities under spherical average zero.
math.FA↗
Critical Hardy inequality on the half-space via the harmonic transplantation
We prove a critical Hardy inequality on the half-space by using the harmonic transplantation. Also we give an improvement of the subcritical Hardy inequality on the half-space, which converges to the critical Hardy inequality. Sobolev type inequalities are also discussed.
math.AP↗
ThreeDWorld: A Platform for Interactive Multi-Modal Physical Simulation
We introduce ThreeDWorld (TDW), a platform for interactive multi-modal physical simulation. TDW enables simulation of high-fidelity sensory data and physical interactions between mobile agents and objects in rich 3D environments. Unique properties include: real-time near-photo-realistic image rendering; a library of objects and environments, and routines for their customization; generative procedures for efficiently building classes of new environments; high-fidelity audio rendering; realistic physical interactions for a variety of material types, including cloths, liquid, and deformable objects; customizable agents that embody AI agents; and support for human interactions with VR devices. TDW's API enables multiple agents to interact within a simulation and returns a range of sensor and physics data representing the state of the world. We present initial experiments enabled by TDW in emerging research directions in computer vision, machine learning, and cognitive science, including multi-modal physical scene understanding, physical dynamics predictions, multi-agent interactions, models that learn like a child, and attention studies in humans and neural networks.
cs.CV↗
Visual resemblance and communicative context constrain the emergence of graphical conventions
From photorealistic sketches to schematic diagrams, drawing provides a versatile medium for communicating about the visual world. How do images spanning such a broad range of appearances reliably convey meaning? Do viewers understand drawings based solely on their ability to resemble the entities they refer to (i.e., as images), or do they understand drawings based on shared but arbitrary associations with these entities (i.e., as symbols)? In this paper, we provide evidence for a cognitive account of pictorial meaning in which both visual and social information is integrated to support effective visual communication. To evaluate this account, we used a communication task where pairs of participants used drawings to repeatedly communicate the identity of a target object among multiple distractor objects. We manipulated social cues across three experiments and a full internal replication, finding pairs of participants develop referent-specific and interaction-specific strategies for communicating more efficiently over time, going beyond what could be explained by either task practice or a pure resemblance-based account alone. Using a combination of model-based image analyses and crowdsourced sketch annotations, we further determined that drawings did not drift toward arbitrariness, as predicted by a pure convention-based account, but systematically preserved those visual features that were most distinctive of the target object. Taken together, these findings advance theories of pictorial meaning and have implications for how successful graphical conventions emerge via complex interactions between visual perception, communicative experience, and social context.
cs.CV↗
Improvements and generalizations of two Hardy type inequalities and their applications to the Rellich type inequalities
We give improvements and generalizations of both the classical Hardy inequality and the geometric Hardy inequality based on the divergence theorem. Especially, our improved Hardy type inequality derives both two Hardy type inequalities with best constants. Besides, we improve two Rellich type inequalities by using the improved Hardy type inequality.
math.AP↗
Active World Model Learning with Progress Curiosity
World models are self-supervised predictive models of how the world evolves. Humans learn world models by curiously exploring their environment, in the process acquiring compact abstractions of high bandwidth sensory inputs, the ability to plan across long temporal horizons, and an understanding of the behavioral patterns of other agents. In this work, we study how to design such a curiosity-driven Active World Model Learning (AWML) system. To do so, we construct a curious agent building world models while visually exploring a 3D physical environment rich with distillations of representative real-world agents. We propose an AWML system driven by $γ$-Progress: a scalable and effective learning progress-based curiosity signal. We show that $γ$-Progress naturally gives rise to an exploration policy that directs attention to complex but learnable dynamics in a balanced manner, thus overcoming the "white noise problem". As a result, our $γ$-Progress-driven controller achieves significantly higher AWML performance than baseline controllers equipped with state-of-the-art exploration strategies such as Random Network Distillation and Model Disagreement.
cs.LG↗
Minimization problem associated with an improved Hardy-Sobolev type inequality
We consider the existence and the non-existence of a minimizer of the following minimization problems associated with an improved Hardy-Sobolev type inequality introduced by Ioku. $$ I_a := \inf_{u \in W_0^{1,p}(B_R ) \setminus \{ 0\} } \frac{\int_{B_R} |\nabla u |^{p} \,dx}{\left( \int_{B_R} |u|^{p^*(s)} V_a(x) \,dx \right)^{\frac{p}{p^*(s)}}}, \,\,\text{where}\,\, V_a (x) =\frac{1}{|x|^s \left( 1- a \,\left( \frac{|x|}{R} \right)^{\frac{N-p}{p-1}} \right)^β} \ge \frac{1}{|x|^s}. $$ Only for radial functions, the minimization problem $I_a$ is equivalent to it associated with the classical Hardy-Sobolev inequality on $\mathbb{R}^N$ via a transformation. First, we summarize various transformations including that transformation and give a viewpoint of such transformations. As an application of this viewpoint, we derive {\it an infinite dimensional form} of the classical Sobolev inequality in some sense. Next, without the transformation, we investigate the minimization problems $I_a$ on balls $B_R$. In contrast to the classical results for $a=0$, we show the existence of non-radial minimizers for the Hardy-Sobolev critical exponent $p^* (s)=\frac{p (N-s)}{N-p}$ on bounded domains. Finally, we give remarks of a different structure between two nonlinear scalings which are equivalent to the usual scaling only for radial functions under some transformations.
math.AP↗
Explicit optimal constants of two critical Rellich inequalities for radially symmetric functions
We consider two critical Rellich inequalities with singularities at both the origin and the boundary in the higher order critical radial Sobolev spaces $W_{0, {\rm rad}}^{k, p}$, where $1< p = \frac{N}{k}$. We give the explicit values of the optimal constants of two critical Rellich inequalities for radially symmetric functions in $W_{0, {\rm rad}}^{k, \frac{N}{k}}$. Furthermore the (non-)attainability of the optimal constants are also discussed.
math.AP↗
Biological Blueprints for Next Generation AI Systems
Diverse subfields of neuroscience have enriched artificial intelligence for many decades. With recent advances in machine learning and artificial neural networks, many neuroscientists are partnering with AI researchers and machine learning experts to analyze data and construct models. This paper attempts to demonstrate the value of such collaborations by providing examples of how insights derived from neuroscience research are helping to develop new machine learning algorithms and artificial neural network architectures. We survey the relevant neuroscience necessary to appreciate these insights and then describe how we can translate our current understanding of the relevant neurobiology into algorithmic techniques and architectural designs. Finally, we characterize some of the major challenges facing current AI technology and suggest avenues for overcoming these challenges that draw upon research in developmental and comparative cognitive neuroscience.
q-bio.NC↗
Two limits on Hardy and Sobolev inequalities
It is known that classical Hardy and Sobolev inequalities hold when the exponent $p$ and the dimension $N$ satisfy $p < N < \infty$. In this note, we consider two limits of Hardy and Sobolev inequalities as $p \nearrow N$ and $N \nearrow \infty$ in some sense.
math.FA↗
Remarks on a limiting case of Hardy type inequalities
The classical Hardy inequality holds in Sobolev spaces $W_0^{1,p}$ when $1\le p< N$. In the limiting case where $p=N$, it is known that by adding a logarithmic function to the Hardy potential, some inequality which is called the critical Hardy inequality holds in $W_0^{1,N}$. In this note, in order to give an explanation of appearance of the logarithmic function at the potential, we derive the logarithmic function from the classical Hardy inequality with the best constant via some limiting procedure as $p \nearrow N$. And we show that our limiting procedure is also available for the classical Rellich inequality in second order Sobolev spaces $W_0^{2,p}$ with $p \in (1, \frac{N}{2})$ and the Poincaré inequality.
math.AP↗
Extremal functions of generalized critical Hardy inequalities
In this paper, we show the existence and non-existence of minimizers of the following minimization problems which include an open problem mentioned by Horiuchi and Kumlin in 2012: \begin{align*} G_a := \inf_{u \in W_0^{1,N}(Ω) \setminus \{ 0\} } \dfrac{\int_Ω |\nabla u |^{N} \,dx}{\left( \int_Ω |u|^{q} f_{a,β}(x) dx \right)^{\frac{N}{q}}}, \,\text{where} \,\,f_{a, β}(x):=|x|^{-N}\left( \log \frac{aR}{|x|} \right)^{-β}. \end{align*} First, we give an answer to the open problem when $Ω=B_R(0)$. Next, we investigate the minimization problems on general bounded domains. In this case, the results depend on the shape of the domain $Ω$. Finally, symmetry breaking property of the minimizers is proved for sufficiently large $β$.
math.AP↗
Weighted Hardy's inequality in a limiting case and the perturbed Kolmogorov equation
In this paper, we show a weighted Hardy inequality in a limiting case for functions in weighted Sobolev spaces with respect to an invariant measure. We also prove that the constant in the left-hand side of the inequality is optimal. As applications, we establish the existence and nonexistence of positive exponentially bounded weak solutions to a parabolic problem involving the Ornstein-Uhlenbeck operator perturbed by a critical singular potential in two dimensional case, according to the size of the coefficient of the critical potential. These results can be considered as counterparts in the limiting case of results which established in \cite{GGR(AA)} \cite{Hauer-Rhandi} in the non-critical cases, and are also considered as extensions of a result in \cite{Cabre-Martel} to the Kolmogorov operator case perturbed by a critical singular potential.
math.AP↗