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Quantum collision model provides a promising tool for investigating system-bath dynamics. Most of the studies on quantum collision models work in the resonant regime. In quantum dynamics, the off-resonant interaction often brings in exciting ffects. It is thereby attractive to investigate quantum collision models in the off-resonant regime. On the other hand, a bath with a negative absolute temperature is anticipated to be instrumental in developing thermal devices. The design of an effective bath with negative absolute temperature coupled to a qubit is significant for developing such thermal devices. We establish an effective negative-absolute-temperature bath coupled to a qubit with a quantum collision model in a far-off-resonant regime. We conduct a detailed and systematic investigation on the off-resonant collision model. There is an additional constraint on the collision duration resulting from the far-off resonant collision. The dynamics of the collision model in the far-off-resonant regime are different from the one beyond the far-off-resonant regime. Numerical simulations confirm the validity of the proposed approach., Comment: 7 pages, 3 figures

Answering a question asked by Hsia and Tucker in their paper on the finiteness of greatest common divisors of iterates of polynomials, we prove that if $f, g \in \mathbb{C}(X)$ are compositionally independent rational functions and $c \in \mathbb{C}(X)$, then there are at most finitely many $\lambda\in\mathbb{C}$ with the property that there is an $n$ such that $f^n(\lambda) = g^n(\lambda) = c(\lambda)$, except for a few families of $f, g \in Aut(\mathbb{P}^1_\mathbb{C})$ which gives counterexamples.

Entropy production is a fundamental concept that plays a crucial role in the second law of thermodynamics and the measure of irreversibility. It imposes rigorous constraints on the kinds of transformations allowed in thermodynamic processes. Using an optical setup, here we experimentally demonstrate the division of entropy production of an open quantum system into a population-related component and a coherence-related component, validating previous theoretical predictions. The coherence-related component represents a genuine quantum contribution with no classical counterpart. By adjusting bath temperatures and initial coherences of the system, we first derive the total entropy production due to both populations and coherences, then remove all the coherences of the system to solely obtain the population-related contribution. The difference between these two results permits to isolate the coherence-related term. Based on this division, our experiment ultimately proves that irreversibility at the quantum level can be reduced through properly harnessing the two contributions to entropy production., Comment: 7 pages, 3 figures

Motor imagery EEG classification plays a crucial role in non-invasive Brain-Computer Interface (BCI) research. However, the classification is affected by the non-stationarity and individual variations of EEG signals. Simply pooling EEG data with different statistical distributions to train a classification model can severely degrade the generalization performance. To address this issue, the existing methods primarily focus on domain adaptation, which requires access to the target data during training. This is unrealistic in many EEG application scenarios. In this paper, we propose a novel multi-source domain generalization framework called EEG-DG, which leverages multiple source domains with different statistical distributions to build generalizable models on unseen target EEG data. We optimize both the marginal and conditional distributions to ensure the stability of the joint distribution across source domains and extend it to a multi-source domain generalization framework to achieve domain-invariant feature representation, thereby alleviating calibration efforts. Systematic experiments on a simulative dataset and BCI competition datasets IV-2a and IV-2b demonstrate the superiority of our proposed EEG-DG over state-of-the-art methods. Specifically, EEG-DG achieves an average classification accuracy/kappa value of 81.79%/0.7572 and 87.12%/0.7424 on datasets IV-2a and IV-2b, respectively, which even outperforms some domain adaptation methods. Our code is available at https://github.com/XC-ZhongHIT/EEG-DG for free download and evaluation.

Let $K$ be a number field and let $f : (\mathbb{P}^1)^n \to (\mathbb{P}^1)^n$ be a dominant endomorphism defined over $K$. We show that if $V$ is an $f$-invariant subvariety (that is, $f(V)=V$) then there is a positive integer $s_0$ such that $ (f^{-s-1}(V)\setminus f^{-s}(V))(K) = \emptyset$ for every integer $s \geq s_0$, answering the Preimages Question of Matsuzawa, Meng, Shibata, and Zhang in the case of $(\mathbb{P}^1)^n$.

We propose a scheme to realize the single-photon transport affected by the temperature. The scheme is composed by a waveguide-atom interacting structure linked to a thermal bath. The single-photon reflection coefficient can be tuned by adjusting the temperature of the thermal bath. This provides a thermal control of the single-photon transport. Moreover, the temperature of the thermal bath can be estimated by measuring the single-photon transport. It is feasible that the estimation on the temperature is sensitive to slight changes of low temperature. This implies an avenue for implementing the optical thermometer with the ability to accurately measure the sample temperature in the low-temperature region., Comment: 7 pages, 4 figures

We improve known estimates for the number of points of bounded height in semigroup orbits of polarized dynamical systems. In particular, we give exact asymptotics for generic semigroups acting on the projective line. The main new ingredient is the Wiener-Ikehara Tauberian theorem, which we use to count functions in semigroups of bounded degree.

Extending the work of Bell, Matsuzawa and Satriano, we consider a finite set of polynomials $S$ over a number field $K$ and give a necessary and sufficient condition for the existence of a $N \in \mathbb{N}_{> 0}$ and a finite set $Z \subset \mathbb{P}^1_K \times \mathbb{P}^1_K$ such that for any $(a,b) \in (\mathbb{P}^1_K \times \mathbb{P}^1_K) \setminus Z$ we have the cancellation result: if $k>N$ and $\phi_1,\ldots ,\phi_k$ are maps in $S$ such that $\phi_{k} \circ \dots \circ \phi_1 (a) = \phi_k \circ \dots \circ \phi_1(b)$, then in fact $\phi_N \circ \dots \circ \phi_1(a) = \phi_N \circ \dots \circ \phi_1(b)$. Moreover, the conditions we give for this cancellation result to hold can be checked by a finite number of computations from the given set of polynomials.

This note provides a succinct survey of the existing literature concerning the H\"older regularity for the gradient of weak solutions of PDEs of the form $$\sum_{i=1}^{2n} X_i A_i(\nabla_0 u)=0 \text{ and } \partial_t u= \sum_{i=1}^{2n} X_i A_i(\nabla_0 u)$$ modeled on the $p$-Laplacian in a domain $\Omega$ in the Heisenberg group $\mathbb H^n$, with $1\le p <\infty$, and of its parabolic counterpart. We present some open problems and outline some of the difficulties they present., Comment: The new version has less typos and updated references

We investigate the heat conduction between two one-dimension waveguides intermediated by a Laser-driving atom. The Laser provides the optical control on the heat conduction. The tunable asymmetric conduction of the heat against the temperature gradient is realized. Assisted by the modulated Laser, the heat conduction from either waveguide to the other waveguide can be suppressed. Meanwhile, the conduction towards the direction opposite to the suppressed one is gained. The heat currents can be significantly amplified by the energy flow of the Laser. Moveover, the scheme can act like a heat engine.

Let $L$ be a field of characteristic zero, let $h:\mathbb{P}^1\to \mathbb{P}^1$ be a rational map defined over $L$, and let $c\in \mathbb{P}^1(L)$. We show that there exists a finitely generated subfield $K$ of $L$ over which both $c$ and $h$ are defined along with an infinite set of inequivalent non-archimedean completions $K_{\mathfrak{p}}$ for which there exists a positive integer $a=a(\mathfrak{p})$ with the property that for $i\in \{0,\ldots ,a-1\}$ there exists a power series $g_i(t)\in K_{\mathfrak{p}}[[t]]$ that converges on the closed unit disc of $K_{\mathfrak{p}}$ such that $h^{an+i}(c)=g_i(n)$ for all sufficiently large $n$. As a consequence we show that the dynamical Mordell-Lang conjecture holds for split self-maps $(h,g)$ of $\mathbb{P}^1 \times X$ with $g$ \'etale., Comment: 12 pages

We consider a deformation of the two-dimensional free scalar field theory by raising the Laplacian to a positive real power. It turns out that the resulting non-local generalized free action is invariant under two commuting actions of the global conformal symmetry algebra, although it is no longer invariant under the full Witt algebra. Furthermore, there is an adelic version of this family of conformal field theories, parameterized by the choice of a number field, together with a Hecke character. Tate's thesis gives the Green's functions of these theories, and ensures that these Green's functions satisfy an adelic product formula. In particular, the local $L$-factors contribute to the prefactors of these Green's functions. Quadratic reciprocity turns out to be a consequence of an adelic version of a holomorphic factorization property of this family of theories on a quadratic extension of $\mathbb{Q}$. We explain that at the Archimedean place, the desired holomorphic factorization follows from the global conformal symmetry., Comment: 19 pages