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We present a method to engineer various zero-energy localized states on disorder-free hypercube graphs. Previous works have already indicated that disorder is not essential for observing localization phenomena in noninteracting systems, with some prominent examples including the 1D Aubry-Andr\'e model, characterized solely by incommensurate potentials, or 2D incommensurate Moir\'e lattices, which exhibit localization due to the flat band spectrum. Moreover, flat band systems with translational invariance can also possess so-called compact localized states, characterized by exactly zero amplitude outside a finite region of the lattice. Here we show that both compact and non-compact (i.e., Anderson-like) localized states can be observed on disorder-free hypercubes that can be constructed in a standard iterative manner via Cartan products. This construction also makes these localized states robust against perturbations. We illustrate and visualize our results on the example of an $8$D hypercube projected on its Petrie polygon. Our findings can be readily tested on existing photonic experimental setups, where hypercube graphs studied can be emulated by bosonic networks of coupled optical cavities and/or waveguides, thus offering potential advancements in the development of various information and wave manipulation protocols., Comment: 10 pages

The $k$-forest problem asks to find $k$ forests in a graph $G$ maximizing the number of edges in their union. We show how to solve this problem in $O(k^3 \min\{kn, m\} \log^2 n + k \cdot{\rm MAXFLOW}(m, m) \log n)$ time, breaking the $O_k(n^{3/2})$ complexity barrier of previously known approaches. Our algorithm relies on three subroutines: the directed $k$-forest problem with bounded indegree condition, the $k$-pseudoforest problem, and the top clump computation.

We consider two problems for a directed graph $G$, which we show to be closely related. The first one is to find $k$ edge-disjoint forests in $G$ of maximal size such that the indegree of each vertex in these forests is at most $k$. We describe a min-max characterization for this problem and show that it can be solved in $O(k \delta m \log n)$ time, where $(n,m)$ is the size of $G$ and $\delta$ is the difference between $k$ and the edge connectivity of the graph. The second problem is the directed edge-connectivity augmentation problem, which has been extensively studied before: find a smallest set of directed edges whose addition to the graph makes it strongly $k$-connected. We improve the complexity for this problem from $O(k \delta (m+\delta n)\log n)$ [Gabow, STOC 1994] to $O(k \delta m \log n)$, by exploiting our solution for the first problem. A similar approach with the same complexity also works for the undirected version of the problem.

This paper focuses on Majority Dynamics in sparse graphs, in particular, as a tool to study internal cuts. It is known that, in Majority Dynamics on a finite graph, each vertex eventually either comes to a fixed state, or oscillates with period two. The empirical evidence acquired by simulations suggests that for random odd-regular graphs, approximately half of the vertices end up oscillating with high probability. We notice a local symmetry between oscillating and non-oscillating vertices, that potentially can explain why the fraction of the oscillating vertices is concentrated around $\frac{1}{2}$. In our simulations, we observe that the parts of random odd-regular graph under Majority Dynamics with high probability do not contain $\lceil \frac{d}{2} \rceil$-cores at any timestep, and thus, one cannot use Majority Dynamics to prove that internal cuts exist in odd-regular graphs almost surely. However, we suggest a modification of Majority Dynamics, that yields parts with desired cores with high probability.

Recently, the possibility of the generation and interaction of unipolar half-cycle electromagnetic pulses with quantum systems has been the subject of active research. Such pulses can have many different and interesting applications. They are able to excite quantum systems very fast. Based on the numerical solution of Maxwell-Bloch equations, this paper theoretically studies the possibility of guiding and ultrafast controlling population difference gratings by a sequence of half-cycle attosecond pulses in a three-level resonant medium. The parameters of the model medium (transition frequencies and dipole moments of the transitions) are close to those found in the hydrogen atom. We show the possibility of guiding periodic gratings and dynamic microcavities on different resonance transitions of the medium. In this case, the medium considered is an example of a spatiotemporal photonic crystal. The refractive index varies rapidly in space and time. The possibility of the use of such gratings as Bragg mirrors in ultrafast optics is discussed. The results obtained are of interest in two actively developing areas of modern optics - the physics of half-cycle pulses and space-time photonic crystals.

The coherent interaction of extremely short light pulses with a resonant medium can result in formation of population difference gratings. Such gratings have been created by pulses that are pi/2 or smaller. This paper demonstrates that a microcavity with Bragg-like mirrors can be formed by colliding two single-cycle attosecond self-induced transparency pulses in the center of a two-level medium. The parameters of this structure can be quickly adjusted by increasing the number of collisions, which showcases the ability to control the dynamic properties of the medium on a sub-cycle time scale by using attosecond pulses., Comment: 4 pages

In this technical report, we present three novel datasets of Kotlin code: KStack, KStack-clean, and KExercises. We also describe the results of fine-tuning CodeLlama and DeepSeek models on this data. Additionally, we present a version of the HumanEval benchmark rewritten by human experts into Kotlin - both the solutions and the tests. Our results demonstrate that small, high-quality datasets (KStack-clean and KExercises) can significantly improve model performance on code generation tasks, achieving up to a 16-point increase in pass rate on the HumanEval benchmark. Lastly, we discuss potential future work in the field of improving language modeling for Kotlin, including the use of static analysis tools in the learning process and the introduction of more intricate and realistic benchmarks.

Rapidly changing the refractive index of a medium in space and time (space-time photonic crystal, STPC) has been a challenging task. Such a rapid change can be achieved by carrier-wave Rabi flopping. We show that it can be realized when a train of half-cycle light pulses collide in a simple three-level atomic medium. We found the formation of Bragg microcavities and demonstrated their ultrafast control in this case. The proposed approach is a convenient way to control medium properties on the sub-cycle time scale. It is a demonstration of the formation of STPCs and their ultrafast control in a simple atomic medium, which has been a difficult task until now.

Non-Hermitian systems have attracted much interest in recent decades, driven partly by the existence of exotic spectral singularities, known as exceptional points (EPs), where the dimensionality of the system evolution operator is reduced. Among various intriguing applications, the discovery of EPs has suggested the potential for implementing a symmetric mode switch, when encircling them in a system parameter space. However, subsequent theoretical and experimental works have revealed that {\it dynamical} encirclement of EPs invariably results in asymmetric mode conversion; namely, the mode switching depends only on the winding direction but not on the initial state. This chirality arises from the failure of adiabaticity due to the complex spectrum of non-Hermitian systems. Although the chirality revealed has undoubtedly made a significant impact in the field, a realization of the originally sought symmetric adiabatic passage in non-Hermitian systems with EPs has since been elusive. In this work, we bridge this gap and theoretically demonstrate that adiabaticity, and therefore a symmetric state transfer, is achievable when dynamically winding around an EP. This becomes feasible by specifically choosing a trajectory in the system parameter space along which the corresponding evolution operator attains a real spectrum. Our findings, thus, offer a promise for advancing various wave manipulation protocols in both quantum and classical domains., Comment: 6 pages, 4 figures

In a fixed spectral range, single- and half-cycle electromagnetic pulses have the shortest duration. Half-cycle pulses are promising tools for ultrafast control of quantum systems. Previously, the possibility of using a sequence of single- and half-cycle attosecond pulses to generate and ultrafast control light-induced population difference gratings has been demonstrated. However, such studies have been carried out using different approximations. For example, when the medium is modelled in the two-level approximation. In this paper, based on the numerical solution of the system of Maxwell-Bloch equations, it is shown that it is possible to generate and control population gratings in a three-level medium without using the approximations used in previous studies. It is shown that taking into account the additional level of the medium does not lead to a violation of the effect of generating such gratings. This extends the applicability of previous results.

The generalization of the area theorem is derived for the case of a pulse circulating inside a ring laser cavity. In contrast to the standard area theorem, which is valid for a single pass of a traveling pulse through a resonant medium, the obtained generalized area theorem takes into account the medium-assisted nonlinear self-action effects through the medium excitation left by the pulse at the previous round-trip in the cavity. The generalized area theorem was then applied to the theoretical description of the dynamics of a single-section ring-cavity laser and the steady solutions for the pulse area and for the medium parameters were found both in the limit of a lumped model and for a spatially-extended system. The derived area theorem can be used for the convenient analytical description of different coherent photonic devices, like coherently mode-locked lasers or pulse compressors, as well as for the analysis of the photon echo formation in cavity-based setups.

We propose a simple quantum system, namely, a nested quantum-well structure, which is able to generate a train of half-cycle pulses of a few-fs duration, when driven by a static electric field. We theoretically investigate the emission of such a structure and its dependence on the parameters of the quantum wells. It is shown that the production of a regular output pulse train with tunable properties and the pulse repetition frequencies of tens of THz is possible in certain parameter ranges. We expect the suggested structure can be used as an ultra-compact source of subcycle pulses in the optical range.

Unipolar and quasi-unipolar half-cycle pulses having nonzero electric pulse area are a limit of pulse shortening in a given spectral range. In spite of the fact that existence of such pulses was considered by Jackson (1962), V.L. Ginzburg (1960-s), Bullough and Ahmad (1971) as well as Bessonov (1981), the possibility of their existence and propagation in space remained questionable for many years. Only within the past decades both the possibility of unipolar pulse existence and their propagation dynamics were shown and analyzed in detail theoretically and experimentally. So far such pulses became a subject of active research due to their potential in the ultra-fast optics and study of new regimes of light-matter interactions with subcycle resolution. Here, we show the possibility of the effective ultrafast control of the level populations in quantum well nanostructures by the half-cycle unipolar attosecond light pulses in comparison to the single-cycle ones. It is shown that the population dynamics can be determined by the electric pulse area divided to its characteristic "scale" determined by the quantum well width. The selective excitation of quantum states and the feasibility of the population inversion by the subcycle unipolar pulses is demonstrated.

We theoretically demonstrate the possibility to tune the temporal waveform of optical unipolar pulses upon their coherent interaction with a multi-level resonant medium. This is achieved through the coherent control of the response of a multi-level resonant medium by means of half-cycle unipolar pulses. We show that despite the ultrabroad spectrum of half-cycle pulses one can efficiently steer the induced medium polarization through the proper choice of the parameters of the excitation pulses. As the result, producing of unipolar optical pulses of varying profile, like rectangular or triangular ones, from an extended layer of a multi-layer medium was obtained when excited by a pair of half-cycle pulses. Besides, we find out that the response of a multi-level medium for the amplitude of the driving pulses below a certain threshold can be quantitatively well approximated by the two-level model.

The induced polarization oscillations in a one-dimensional rectangular quantum well are modeled by a numerical solution of the time-dependent Schroedinger equation. The finite-difference discretization over time is realized in the framework of the Crank-Nicolson algorithm, whereas over the spatial coordinate it is combined with the exterior complex-scaling technique. A formation of the harmonic oscillations of the dipole moment by an incident short unipolar pulse is shown. It is obtained that the frequency of oscillations is solely defined by the energy of the main resonant transition. Moreover, if two such short unipolar pulses are delayed by a half-period of the oscillation, then these oscillations can be abruptly induced and stopped. Thus, the so-called stopped polarization pulse is obtained. It is shown that both the amplitude and the duration of the incident unipolar pulse, contributing to the so-called electric pulse area, define the impact of the incident pulse on the quantum system., Comment: 11 pages; submitted to Micro and Nanostructures

We define $A_{\infty}$-structures -- algebras, coalgebras, modules, and comodules -- in an arbitrary monoidal DG category or bicategory by rewriting their definitions in terms of unbounded twisted complexes. We develop new notions of strong homotopy unitality and bimodule homotopy unitality to work in this level of generality. For a strong homotopy unital $A_{\infty}$-algebra we construct Free-Forgetful homotopy adjunction, its Kleisli category, and its derived category of modules. Analogous constructions for $A_{\infty}$-coalgebras require bicomodule homotopy counitality. We define homotopy adjunction for $A_{\infty}$-algebra and $A_{\infty}$-coalgebra and show such pair to be derived module-comodule equivalent. As an application, we obtain the notions of an $A_{\infty}$-monad and of an enhanced exact monad. We also show that for any adjoint triple $(L,F,R)$ of functors between enhanced triangulated categories the adjunction monad $RF$ and the adjunction comonad $LF$ are derived module-comodule equivalent., Comment: 97 pages, v1

In order to substantiate claims of quantum computational advantage, it is crucial to develop efficient methods for validating the experimental data. We propose a test of the correct functioning of a boson sampler with single-photon inputs that is based on how photons distribute among partitions of the output modes. Our method is versatile and encompasses previous validation tests based on bunching phenomena, marginal distributions, and even some suppression laws. We show via theoretical arguments and numerical simulations that binned-mode photon number distributions can be used in practical scenarios to efficiently distinguish ideal boson samplers from those affected by realistic imperfections, especially partial distinguishability of the photons., Comment: 30 pages, 14 figures