Search

We study the realization problem of finite groups as the group of homotopy classes of self-homotopy equivalences of finite spaces. Let $G$ be a finite group. Using an infinite family of pairwise non weakly homotopic asymmetric spaces we present a new construction of a finite space whose group of homotopy classes of self-homotopy equivalences is isomorphic to $G$., Comment: 8 pages, 5 figures

Average-reward Markov decision processes (MDPs) provide a foundational framework for sequential decision-making under uncertainty. However, average-reward MDPs have remained largely unexplored in reinforcement learning (RL) settings, with the majority of RL-based efforts having been allocated to episodic and discounted MDPs. In this work, we study a unique structural property of average-reward MDPs and utilize it to introduce Reward-Extended Differential (or RED) reinforcement learning: a novel RL framework that can be used to effectively and efficiently solve various subtasks simultaneously in the average-reward setting. We introduce a family of RED learning algorithms for prediction and control, including proven-convergent algorithms for the tabular case. We then showcase the power of these algorithms by demonstrating how they can be used to learn a policy that optimizes, for the first time, the well-known conditional value-at-risk (CVaR) risk measure in a fully-online manner, without the use of an explicit bi-level optimization scheme or an augmented state-space.

Intelligent vision control systems for surgical robots should adapt to unknown and diverse objects while being robust to system disturbances. Previous methods did not meet these requirements due to mainly relying on pose estimation and feature tracking. We propose a world-model-based deep reinforcement learning framework "Grasp Anything for Surgery" (GAS), that learns a pixel-level visuomotor policy for surgical grasping, enhancing both generality and robustness. In particular, a novel method is proposed to estimate the values and uncertainties of depth pixels for a rigid-link object's inaccurate region based on the empirical prior of the object's size; both depth and mask images of task objects are encoded to a single compact 3-channel image (size: 64x64x3) by dynamically zooming in the mask regions, minimizing the information loss. The learned controller's effectiveness is extensively evaluated in simulation and in a real robot. Our learned visuomotor policy handles: i) unseen objects, including 5 types of target grasping objects and a robot gripper, in unstructured real-world surgery environments, and ii) disturbances in perception and control. Note that we are the first work to achieve a unified surgical control system that grasps diverse surgical objects using different robot grippers on real robots in complex surgery scenes (average success rate: 69%). Our system also demonstrates significant robustness across 6 conditions including background variation, target disturbance, camera pose variation, kinematic control error, image noise, and re-grasping after the gripped target object drops from the gripper. Videos and codes can be found on our project page: https://linhongbin.github.io/gas/.

In this work, we study the effects of random temperature fluctuations on the equation of state of a non-interacting, relativistic fermion gas by means of the replica method. This picture provides a conceptual model for a non-equilibrium system, depicted as an ensemble of subsystems at different temperatures, randomly distributed with respect to a given mean value. We then assume the temperature displays stochastic fluctuations $T = T_0 + \delta T$ with respect to its ensemble average value $T_0$, with zero mean $\overline{\delta T} = 0$ and standard deviation $\overline{\delta T^2} = \Delta$. By means of the replica method, we obtain the average grand canonical potential, leading to the equation of state of the fermion gas expressed in terms of the excess pressure caused by these fluctuations with respect to the ideal gas at uniform temperature. We further extend our results for the ideal Bose gas as well. Our findings reveal an increase in pressure as the system's ensemble average temperature $T_0$ rises, consistently exceeding the pressure observed in an equilibrium state. Finally, we explore the implications for the deconfinement transition in the context of the simple Bag model, where we show that the critical temperature decreases., Comment: 9 pages, 3 figures

We consider the effects of a noisy magnetic field background over the fermion propagator in QED, as an approximation to the spatial inhomogeneities and time-fluctuations that would naturally arise in certain physical scenarios, such as heavy-ion collisions or the quark-gluon plasma in the early stages of the evolution of the Universe. We considered a classical, finite and uniform average magnetic field background $\langle \mathbf{B}(\mathbf{x})\rangle_{\Delta} = \mathbf{B}$, subject to white-noise fluctuations with auto-correlation of magnitude $\Delta_B$. By means of the Schwinger representation of the propagator in the average magnetic field as a reference system, we used the replica formalism to study the effects of the magnetic noise at the mean field level, in terms of a vector order parameter $Q_j = \ii e \Delta_B\langle\langle \bar{\psi}\gamma_j\psi \rangle\rangle_{\Delta}$ whose magnitude represents the ensemble average (over magnetic noise) of the fermion currents. We identified the region where this order parameter acquires a finite value, thus breaking the $U(1)$-symmetry of the model due to the presence of the magnetic noise., Comment: 14 pages, 7 figures. This work is an extension of arXiv:2211.16985. arXiv admin note: text overlap with arXiv:2211.16985

In this research, an optimal algorithm for the Collatz conjecture is presented. Properties such as the convergence of the algorithm and an equation that relates the algorithm to the classical Collatz conjecture are obtained. It is validated that the proposed theory is correct with several examples; as a sector of mathematicians believes that the Collatz conjecture fails in some power of $3$. The algorithm was applied to the first $600$ powers of $3$, where the convergence of the proposed algorithm was verified., Comment: The proof of the main lemma of the document was complemented and more examples were added, the document is made up of 17 pages and 5 figures

In this paper, using an algebraic approach, it is intended to show that the Goldbach's and Twin primes conjectures are true, building, for each $m>2$, an isomorphism between posets. One of the posets is the set of coprimes less than $m$, while the other is endowed with an operation that grants it an abelian group structure. Special features of this operation are demonstrated in the document, which allow characterizing the even numbers, as if they were their fingerprint; furthermore, such an operation locates, in a natural way, the pairs that satisfy the conjecture. Moreover, an algorithm that generates pairs of numbers that satisfy the conjecture is presented, and examples of some of the beautiful symmetries of the orbits of cyclic subgroups are shown, for the proposed abelian group.

DQ-algebroids locally defined on a symplectic manifold form a 2-gerbe. By adapting the method of P. Deligne to the setting of DQ-algebroids we show that this 2-gerbe admits a canonical global section, namely that every symplectic manifold admits a canonical DQ-algebroid quantizing the structure sheaf. The construction relies on methods of non-abelian cohomology and local computations in the Weyl algebra. As a corollary we obtain a classification of symplectic DQ-algebroids., Comment: 36 pages