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Learning in artificial neural networks usually relies on continuous, externally driven weight updates, in which parameters are modified at every step in response to incoming data, error signals or reward feedback. In this setting, routine and informative inputs contribute similarly to parameter adjustment. We introduce a learning approach in which parameter updates are governed by internally generated events arising from the network own representational dynamics. During ongoing activity, synaptic interactions are accumulated as latent traces encoding recent coactivation patterns, without immediately modifying the underlying parameters. In parallel, an internal predictive process estimates the evolving latent state, while a scalar measure of discrepancy between predicted and observed states is continuously computed. When discrepancy exceeds an adaptive threshold derived from recent error statistics, a learning event is triggered, inducing a retrospective update selectively integrating past activity into the current configuration. We performed simulations using a minimal neural network exposed to structured sequential inputs with transient perturbations. We found that learning occurs

Krueger showed that PFA implies that for all regular $Θ\ge \aleph_2$, there are stationarily many $[H(Θ)]^{\aleph_1}$ that are internally club but not internally approachable. From countably many Mahlo cardinals, we force a model in which, for all positive $n<ω$ and $Θ\ge \aleph_{n+1}$, there is a stationary subset of $[H(Θ)]^{\aleph_n}$ consisting of sets that are internally club but not internally approachable. The theorem is obtained using a new variant of Mitchell forcing. This answers questions of Krueger.

Models of astrophysical convection, such as mixing length theory, typically assume that the heat transport is independent of microphysical diffusivities. Such 'diffusion-free' behaviour is, however, not observed in numerical simulations employing standard fixed-flux or fixed-temperature boundary conditions, except possibly in extreme parameter regimes that are computationally expensive to achieve. Recent numerical and experimental work has suggested that internally heated and cooled convection can exhibit diffusion-free scalings in more numerically accessible regimes. Here, we present direct numerical simulations of 2.5D Cartesian rotating thermal convection driven by an internal heating and cooling function. The use of distributed heating and cooling functions alleviates sharp thermal boundary layers that would otherwise be present, allowing the flows to be simulated with modest computational resources. We show that for high Rossby numbers this set-up recovers mixing length theory scalings for the heat transport. The velocity amplitudes, in contrast, are observed to display diffusion-limited scalings. By comparing against boundary driven rotating convection, we show that internall

A well-known result by Kant [Algorithmica, 1996] implies that n-vertex outerplane graphs admit embedding-preserving planar straight-line grid drawings where the internal faces are convex polygons in $O(n^2)$ area. In this paper, we present an algorithm to compute such drawings in $O(n^{1.5})$ area. We also consider outerplanar drawings in which the internal faces are required to be strictly-convex polygons. In this setting, we consider outerplanar graphs whose weak dual is a path and give a drawing algorithm that achieves $Θ(nk^2)$ area, where $k$ is the maximum size of an internal facial cycle.

This work investigates heat transport in rotating internally heated convection, for a horizontally periodic fluid between parallel plates under no-slip and isothermal boundary conditions. The main results are the proof of bounds on the mean temperature, $\overline{\langle T \rangle }$, and the heat flux out of the bottom boundary, $\mathcal{F}_B$ at infinite Prandtl numbers where the Prandtl number is the nondimensional ratio of viscous to thermal diffusion. The lower bounds are functions of a Rayleigh number quantifying the ratio of internal heating to diffusion and the Ekman number, $E$, which quantifies the ratio of viscous diffusion to rotation. We utilise two different estimates on the vertical velocity, $w$, one pointwise in the domain (Yan 2004, J. Math. Phys., vol. 45(7), pp. 2718-2743) and the other an integral estimate over the domain (Constantin et al . 1999, Phys. D: Non. Phen., vol. 125, pp. 275-284), resulting in bounds valid for different regions of buoyancy-to-rotation dominated convection. Furthermore, we demonstrate that similar to rotating Rayleigh-Bénard convection, for small $E$, the critical Rayleigh number for the onset of convection asymptotically scales as

A sheaf of modules on a site is said to be internally projective if sheaf hom with the module preserves epimorphism. In this note, we give an example showing that internally projective sheaves of abelian groups are not in general stable under base change to a slice. This shows that internal projectivity is weaker than projectivity in the internal logic of the topos, as expressed for example in terms of Shulman's stack semantics. The sheaf of groups that we use as a counterexample comes from recent work by Clausen and Scholze on light condensed sets.

Instruction-following is crucial for building AI agents with large language models (LLMs), as these models must adhere strictly to user-provided constraints and guidelines. However, LLMs often fail to follow even simple and clear instructions. To improve instruction-following behavior and prevent undesirable outputs, a deeper understanding of how LLMs' internal states relate to these outcomes is required. In this work, we investigate whether LLMs encode information in their representations that correlate with instruction-following success - a property we term knowing internally. Our analysis identifies a direction in the input embedding space, termed the instruction-following dimension, that predicts whether a response will comply with a given instruction. We find that this dimension generalizes well across unseen tasks but not across unseen instruction types. We demonstrate that modifying representations along this dimension improves instruction-following success rates compared to random changes, without compromising response quality. Further investigation reveals that this dimension is more closely related to the phrasing of prompts rather than the inherent difficulty of the task

The Ozawa's Intersubjectivity Theorem (OIT) proved within quantum measurement theory supports the new postulate of relational quantum mechanics (RQM), the postulate on internally consistent descriptions. We remark that this postulate was proposed only recently to resolve the problem of intersubjectivity of information in RQM. In contrast to RQM for which OIT is a supporting theoretical statement, QBism is challenged by OIT.

We study a class of reinforcement learning problems where the reward signals for policy learning are generated by an internal reward model that is dependent on and jointly optimized with the policy. This interdependence between the policy and the reward model leads to an unstable learning process because reward signals from an immature reward model are noisy and impede policy learning, and conversely, an under-optimized policy impedes reward estimation learning. We call this learning setting $\textit{Internally Rewarded Reinforcement Learning}$ (IRRL) as the reward is not provided directly by the environment but $\textit{internally}$ by a reward model. In this paper, we formally formulate IRRL and present a class of problems that belong to IRRL. We theoretically derive and empirically analyze the effect of the reward function in IRRL and based on these analyses propose the clipped linear reward function. Experimental results show that the proposed reward function can consistently stabilize the training process by reducing the impact of reward noise, which leads to faster convergence and higher performance compared with baselines in diverse tasks.

In stars and planets natural processes heat convective flows in the bulk of a convective region rather than at hard boundaries. By characterizing how convective dynamics are determined by the strength of an internal heating source we can gain insight into the processes driving astrophysical convection. Internally heated convection has been studied extensively in incompressible fluids, but the effects of stratification and compressibility have not been examined in detail. In this work, we study fully compressible convection driven by a spatially uniform heating source in 2D and 3D Cartesian, hydrodynamic simulations. We use a fixed temperature upper boundary condition which results in a system that is internally heated in the bulk and cooled at the top. We find that the flow speed, as measured by the Mach number, and turbulence, as measured by the Reynolds number, can be independently controlled by separately varying the characteristic temperature gradient from internal heating and the diffusivities. 2D simulations at a fixed Mach number (flow speed) demonstrate consistent power at low wavenumber as diffusivities are decreased. We observe convection where the velocity distribution i

We describe what it means for an algebra to be internally d-Calabi-Yau with respect to an idempotent. This definition abstracts properties of endomorphism algebras of (d-1)-cluster-tilting objects in certain stably (d-1)-Calabi-Yau Frobenius categories, as observed by Keller-Reiten. We show that an internally d-Calabi-Yau algebra satisfying mild additional assumptions can be realised as the endomorphism algebra of a (d-1)-cluster-tilting object in a Frobenius category. Moreover, if the algebra satisfies a stronger 'bimodule' internally d-Calabi-Yau condition, this Frobenius category is stably (d-1)-Calabi-Yau. We pay special attention to frozen Jacobian algebras; in particular, we define a candidate bimodule resolution for such an algebra, and show that if this complex is indeed a resolution, then the frozen Jacobian algebra is bimodule internally 3-Calabi-Yau with respect to its frozen idempotent. These results suggest a new method for constructing Frobenius categories modelling cluster algebras with frozen variables, by first constructing a suitable candidate for the endomorphism algebra of a cluster-tilting object in such a category, analogous to Amiot's construction in the coef

In 1977 Stanley proved that the $h$-vector of a matroid is an $\mathcal{O}$-sequence and conjectured that it is a pure $\mathcal{O}$-sequence. In the subsequent years the validity of this conjecture has been shown for a variety of classes of matroids, though the general case is still open. In this paper we use Las Vergnas' internal order to introduce a new class of matroids which we call internally perfect. We prove that these matroids satisfy Stanley's Conjecture and compare them to other classes of matroids for which the conjecture is known to hold. We also prove that, up to a certain restriction on deletions, every minor of an internally perfect ordered matroid is internally perfect.

Let $M$ be a $3$-connected binary matroid; $M$ is internally $4$-connected if one side of every $3$-separation is a triangle or a triad, and $M$ is $(4,4,S)$-connected if one side of every $3$-separation is a triangle, a triad, or a $4$-element fan. Assume $M$ is internally $4$-connected and that neither $M$ nor its dual is a cubic Möbius or planar ladder or a certain coextension thereof. Let $N$ be an internally $4$-connected proper minor of $M$. Our aim is to show that $M$ has a proper internally $4$-connected minor with an $N$-minor that can be obtained from $M$ either by removing at most four elements, or by removing elements in an easily described way from a special substructure of $M$. When this aim cannot be met, the earlier papers in this series showed that, up to duality, $M$ has a good bowtie, that is, a pair, $\{x_1,x_2,x_3\}$ and $\{x_4,x_5,x_6\}$, of disjoint triangles and a cocircuit, $\{x_2,x_3,x_4,x_5\}$, where $M\backslash x_3$ has an $N$-minor and is \ffsc. We also showed that, when $M$ has a good bowtie, either $M\backslash x_3,x_6$ has an $N$-minor and $M\backslash x_6$ is $(4,4,S)$-connected; or $M\backslash x_3/x_2$ has an $N$-minor and is \ffsc. In this paper

Let M and N be internally 4-connected binary matroids such that M has a proper N-minor, and |E(N)| is at least seven. As part of our project to develop a splitter theorem for internally 4-connected binary matroids, we prove the following result: if M\e has no N-minor whenever e is in a triangle of M, and M/e has no N-minor whenever e is in a triad of M, then M has a minor, M', such that M' is internally 4-connected with an N-minor, and 0 < |E(M)|-|E(M')| < 3.