Manuel Baltieri, Martin Biehl, Matteo Capucci
et al.
The internal model principle, originally proposed in the theory of control of linear systems, nowadays represents a more general class of results in control theory and cybernetics. The central claim of these results is that, under suitable assumptions, if a system (a controller) can regulate against a class of external inputs (from the environment), it is because the system contains a model of the system causing these inputs, which can be used to generate signals counteracting them. Similar claims on the role of internal models appear also in cognitive science, especially in modern Bayesian treatments of cognitive agents, often suggesting that a system (a human subject, or some other agent) models its environment to adapt against disturbances and perform goal-directed behaviour. It is however unclear whether the Bayesian internal models discussed in cognitive science bear any formal relation to the internal models invoked in standard treatments of control theory. Here, we first review the internal model principle and present a precise formulation of it using concepts inspired by categorical systems theory. This leads to a formal definition of ``model'' generalising its use in the internal model principle. Although this notion of model is not a priori related to the notion of Bayesian reasoning, we show that it can be seen as a special case of possibilistic Bayesian filtering. This result is based on a recent line of work formalising, using Markov categories, a notion of ``interpretation'', describing when a system can be interpreted as performing Bayesian filtering on an outside world in a consistent way.
We consider the fundamental scenario where a single item is to be sold to one of two agents. Both agents draw their valuation for the item from the same probability distribution. However, only one of them submits a bid to the mechanism. The other agent is profiled, i.e., the mechanism receives a prediction for her valuation, which can be true or false. Our goal is to design mechanisms for selling the item that make as much revenue as possible in cases of a correct or incorrect prediction. As a benchmark for proving our revenue-approximation guarantees, we use the maximum expected revenue that can be obtained by a strategic and an honest bidder. We study two mechanisms. The first one yields optimal revenue when the prediction is guaranteed to be correct and a constant revenue approximation when the prediction is incorrect, assuming that the agent valuations are drawn from a monotone hazard rate (MHR) distribution. The second mechanism ignores the prediction for the second agent and simulates the revenue-optimal mechanism when no bid information for the bidders is available. We prove, again assuming that valuations are drawn from MHR distributions, that this mechanism achieves a constant revenue approximation guarantee compared to our revenue benchmark. The MHR assumption is necessary; we show that there are non-MHR but regular probability distributions for which no constant approximation of our revenue benchmark is possible.
Large language models (LLMs) have demonstrated impressive capabilities in various reasoning tasks, aided by techniques like chain-of-thought prompting that elicits verbalized reasoning. However, LLMs often generate text with obvious mistakes and contradictions, raising doubts about their ability to robustly process and utilize generated rationales. In this work, we investigate reasoning in LLMs through the lens of internal representations, focusing on how these representations are influenced by generated rationales. Our preliminary analysis reveals that while generated rationales improve answer accuracy, inconsistencies emerge between the model's internal representations in middle layers and those in final layers, potentially undermining the reliability of their reasoning processes. To address this, we propose internal consistency as a measure of the model's confidence by examining the agreement of latent predictions decoded from intermediate layers. Extensive empirical studies across different models and datasets demonstrate that internal consistency effectively distinguishes between correct and incorrect reasoning paths. Motivated by this, we propose a new approach to calibrate reasoning by up-weighting reasoning paths with high internal consistency, resulting in a significant boost in reasoning performance. Further analysis uncovers distinct patterns in attention and feed-forward modules across layers, providing insights into the emergence of internal inconsistency. In summary, our results demonstrate the potential of using internal representations for self-evaluation of LLMs. Our code is available at github.com/zhxieml/internal-consistency.
In this paper, we investigate the internal (finite) sum of submonoids of rank-$1$ torsion-free abelian groups. These submonoids, when not groups, are isomorphic to nontrivial submonoids of the nonnegative cone of $\mathbb Q$, known as Puiseux monoids, and have been actively studied during the last few years. Here we study how the atomicity and arithmetic of Puiseux monoids behave under their internal (finite) sum inside the abelian group $\mathbb Q$. We study the factorization properties of such internal sums, giving priority to Cohn's notion of atomicity and the classical bounded and finite factorization properties introduced and studied in 1990 by Anderson, Anderson, and Zafrullah in the setting of integral domains, and then generalized by Halter-Koch to commutative monoids. We pay special attention to how each of the considered properties behaves under the internal sum of a Puiseux monoid with a finitely generated Puiseux monoid. Throughout the paper, we also discuss examples showing that our primary results do not hold for submonoids of torsion-free abelian groups with rank larger than $1$.
One of the most fundamental facts in topos theory is the internal parameterization of subtoposes: the bijective correspondence between subtoposes and Lawvere-Tierney topologies. In this paper, we introduce a new but elementary concept, "a local state classifier," and give an analogous internal parameterization of hyperconnected quotients (i.e., hyperconnected geometric morphisms from a topos). As a corollary, we obtain a solution to the Boolean case of the first problem of Lawvere's open problems.
We introduce categories $\M$ and $§$ internal in the tricategory $\Bicat_3$ of bicategories, pseudofunctors, pseudonatural transformations and modifications, for matrices and spans in a 1-strict tricategory $V$. Their horizontal tricategories are the tricategories of matrices and spans in $V$. Both the internal and the enriched constructions are tricategorifications of the corresponding constructions in 1-categories. Following \cite{FGK} we introduce monads and their vertical morphisms in categories internal in tricategories. We prove an equivalent condition for when the internal categories for matrices $\M$ and spans $§$ in a 1-strict tricategory $V$ are equivalent, and deduce that in that case their corresponding categories of (strict) monads and vertical monad morphisms are equivalent, too. We prove that the latter categories are isomorphic to those of categories enriched and discretely internal in $V$, respectively. As a byproduct of our tricategorical constructions we recover some results from \cite{Fem}. Truncating to 1-categories we recover results from \cite{CFP} and \cite{Ehr} on the equivalence of enriched and discretely internal 1-categories.
In this study, internal categories in the category of the crossed modules are characterized and it has been shown that there is a natural equivalence between the category of the crossed modules over crossed modules, i.e. crossed squares, and the category of the internal categories within the category of crossed modules. Finally, we obtain examples of crossed squares using this equivalence.
Tourism is a worldwide practice with international tourism revenues increasing from US\$495 billion in 2000 to US\$1340 billion in 2017. Its relevance to the economy of many countries is obvious. Even though the World Airline Network (WAN) is global and has a peculiar construction, the International Tourism Network (ITN) is very similar to a random network and barely global in its reach. To understand the impact of global distances on local flows, we map the flow of tourists around the world onto a complex network and study its topological and dynamical balance. We find that although the WAN serves as infrastructural support for the ITN, the flow of tourism does not correlate strongly with the extent of flight connections worldwide. Instead, unidirectional flows appear locally forming communities that shed light on global travelling behaviour inasmuch as there is only a 15% probability of finding bidirectional tourism between a pair of countries. We conjecture that this is a consequence of one-way cyclic tourism by analyzing the triangles that are formed by the network of flows in the ITN. Finally, we find that most tourists travel to neighbouring countries and mainly cover larger distances when there is a direct flight, irrespective of the time it takes.
In this paper we construct a new solution which represents Pollard-like three-dimensional nonlinear geophysical internal water waves. The Pollard-like solution includes the effects of the rotation of Earth and describes the internal water wave which exists at all latitudes across Earth and propagates above the thermocline. The solution is provided in Lagrangian coordinates. In the process we derive the appropriate dispersion relation for the internal water waves in a stable stratification and discuss the particles paths. An analysis of the dispersion relation for the constructed model identifies one mode of the internal water waves.
A generalization of the recently discovered cooperative internal conversion process is investigated theoretically. In the cooperative internal conversion process by proton exchange investigated the coupling of bound-free electron and proton transitions due to the dipole term of their Coulomb interaction permits cooperation of two nuclei leading to proton exchange and an electron emission. General expression of the cross section of the process obtained in the one particle spherical nuclear shell model is presented. As a numerical example the cooperative internal conversion process by proton exchange in $Al$ is dealt with. As a further generalization, cooperative internal conversion process by heavy charged particle exchange and as an example of it the cooperative internal conversion process by triton exchange is discussed. The process is also connected to the field of nuclear waste disposal.
This work presents the first characterization of the internal structure of overpressured steady superfast magnetosonic relativistic jets in connection with their dominant type of energy. To this aim, relativistic magnetohydrodynamic simulations of different jet models threaded by a helical magnetic field have been analyzed covering a wide region in the magnetosonic Mach number - specific internal energy plane. The merit of this plane is that models dominated by different types of energy (internal energy: hot jets; rest-mass energy: kinetically dominated jets; magnetic energy: Poynting-flux dominated jets) occupy well separated regions. The analyzed models also cover a wide range of magnetizations. Models dominated by the internal energy (i.e., hot models, or Poynting-flux dominated jets with magnetizations larger than but close to 1) have a rich internal structure characterized by a series of recollimation shocks and present the largest variations in the flow Lorentz factor (and internal energy density). Conversely, in kinetically dominated models there is not much internal nor magnetic energy to be converted into kinetic one and the jets are featureless, with small variations in the flow Lorentz factor. The presence of a significant toroidal magnetic field threading the jet produces large gradients in the transversal profile of the internal energy density. Poynting-flux dominated models with high magnetization (>10) are prone to be unstable against magnetic pinch modes, which sets limits to the expected magnetization in parsec-scale AGN jets {and/or constrains their magnetic field configuration}.
The possibility of quantum interference of a composite object with many internal degrees of freedom is studied, such that the internal degrees play a role of an internal environment. In particular, if the internal degrees have a capacity for an irreversible record of which-path information, then the internal-environment induced decoherence prevents external experimentalists from observing interference. Interference can be observed only if the interfering object is sufficiently isolated from the external environment, so that the object cannot record which-path information. Extrapolation to a hypothetical interference experiment with a conscious object implies that being a Schrodinger cat would be like being an ordinary cat living in a box without any information about the world external to the box.
Hybrid entities give rise to international tax problems and opportunities. Different countries tax systems treat hybrid entities in fundamentally different ways, allocating income to different parties. The tax consequences of this divergence of approach result in complex and unintended outcomes. Referring to the OECD Report on the taxation of Partnerships, this article looks at whether the treatment of trans-Tasman limited Partnerships under the Australian and New Zealand Convention results in double taxation or double non-taxation. It concludes that hybrid entity double taxation is, mostly, resolved through the operation of the Convention.
Let A(t) denote the cluster produced by internal diffusion limited aggregation (internal DLA) with t particles in dimension d > 2. We show that A(t) is approximately spherical, up to an O(\sqrt{\log t}) error.
Matthieu Mercier, Nicolas Garnier, Thierry Dauxois
We apply the Hilbert transform to the physics of internal waves in two-dimensional fluids. Using this demodulation technique, we can discriminate internal waves propagating in different directions: this is very helpful in answering several fundamental questions in the context of internal waves. We focus more precisely in this paper on phenomena associated with dissipation, diffraction and reflection of internal waves.
Ingmar Kanitscheider, Kostas Skenderis, Marika Taylor
We construct general 2-charge D1-D5 horizon-free non-singular solutions of IIB supergravity on T^4 and K3 describing fuzzballs with excitations in the internal manifold; these excitations are characterized by arbitrary curves. The solutions are obtained via dualities from F1-P solutions of heterotic and type IIB on T^4 for the K3 and T^4 cases, respectively. We compute the holographic data encoded in these solutions, and show that the internal excitations are captured by vevs of chiral primaries associated with the middle cohomology of T^4 or K3. We argue that each geometry is dual to a specific superposition of R ground states determined in terms of the Fourier coefficients of the curves defining the supergravity solution. We compute vevs of chiral primaries associated with the middle cohomology and show that they indeed acquire vevs in the superpositions corresponding to fuzzballs with internal excitations, in accordance with the holographic results. We also address the question of whether the fuzzball program can be implemented consistently within supergravity.