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Problems in probability theory prove to be one of the most challenging for students. Here, we formulate and discuss four related problems in probability theory that proved difficult for first to fourth-year undergraduate students whose first language was not English. These examples emphasize how crucial it is to understand the conditions and requirements of the problems precisely before starting to solve them. We discuss the solutions to those problems in detail, complement them with numerical estimations, and link the conditions in the problems to the logical statements in Python programming language. We also tested two widely used chatbots (GPT-4o and Claude 3.5 Sonnet) by checking their responses to these problems., Comment: 7 pages

We investigate the dynamical sampling space-time trade-off problem within a graph setting. Specifically, we derive necessary and sufficient conditions for space-time sampling that enable the reconstruction of an initial band-limited signal on a graph. Additionally, we develop and test numerical algorithms for approximating the optimal placement of sensors on the graph to minimize the mean squared error when recovering signals from time-space measurements corrupted by i.i.d.~additive noise. Our numerical experiments demonstrate that our approach outperforms previously proposed algorithms for related problems.

We explore the distribution in space and time of a single-photon excitation shared by a network of dipole-dipole interacting atoms that are also coupled to a common photonic field mode. Time-averaged distributions reveal partial trapping of the excitation near the initially excited atom. This trapping is associated with resonances of the excitation at crossing points of the photon-dressed energy eigenvalues of the network. The predicted photon-induced many-atom trapped excitation (PIMATE) is sensitive to atomic position disorder which broadens the excitation resonances and transforms them to avoided crossings. PIMATE is shown to allow highly effective and accurate sensing of multi-atom networks and their disorder.

The development of high-performance materials for microelectronics, energy storage, and extreme environments depends on our ability to describe and direct property-defining microstructural order. Our present understanding is typically derived from laborious manual analysis of imaging and spectroscopy data, which is difficult to scale, challenging to reproduce, and lacks the ability to reveal latent associations needed for mechanistic models. Here, we demonstrate a multi-modal machine learning (ML) approach to describe order from electron microscopy analysis of the complex oxide La$_{1-x}$Sr$_x$FeO$_3$. We construct a hybrid pipeline based on fully and semi-supervised classification, allowing us to evaluate both the characteristics of each data modality and the value each modality adds to the ensemble. We observe distinct differences in the performance of uni- and multi-modal models, from which we draw general lessons in describing crystal order using computer vision., Comment: 30 pages, 5 figures, 2 tables

In this paper, we study the regularity of the "regular" part of the free boundary for almost minimizers in the parabolic Signorini problem with zero thin obstacle. This work is a continuation of our earlier research on the regularity of almost minimizers. We first establish the Weiss-type monotonicity formula by comparing almost minimizers with parabolically homogeneous replacements and utilizing conformal self-similar coordinates. Subsequently, by deriving the Almgren-type frequency formula and applying the epiperimetric inequality, we obtain the optimal growth near regular free boundary points and achieve the regularity of the regular set., Comment: 46 pages

We study the finite temperature and edge induced effects on the charge and current densities for a massive spinor field localized on a 2D conical space threaded by a magnetic flux. The field operator is constrained on a circular boundary, concentric with the cone apex, by the bag boundary condition and by the condition with the opposite sign in front of the term containing the normal to the edge. In two-dimensional spaces there exist two inequivalent representations of the Clifford algebra and the analysis is presented for both the fields realizing those representations. The circular boundary divides the conical space into two parts, referred as interior (I-) and exterior (E-) regions. The radial current density vanishes. The edge induced contributions in the expectation values of the charge and azimuthal current densities are explicitly separated in the both regions for the general case of the chemical potential. They are periodic functions of the magnetic flux and odd functions under the simultaneous change of the signs of magnetic flux and chemical potential. In the E-region all the spinorial modes are regular and the total charge and current densities are continuous functions of the magnetic flux. In the I-region the corresponding expectation values are discontinuous at half-integer values of the ratio of the magnetic flux to the flux quantum. 2D fermionic models, symmetric under the parity and time-reversal transformations (in the absence of magnetic fields) combine two spinor fields realizing the inequivalent representations of the Clifford algebra. The total charge and current densities in those models are discussed for different combinations of the boundary conditions for separate fields. Applications are discussed for electronic subsystem in graphitic cones described by the 2D Dirac model., Comment: 37 pages, 12 figures

A system of singular integral equations with monotone and concave nonlinearity in the subcritical case is investigated. The specified system and its scalar analog have direct applications in various areas of physics and biology. In particular, scalar and vector equations of this nature are encountered in the dynamic theory of p-adic strings, in the theory of radiative transfer, in the kinetic theory of gases, and in the mathematical theory of the spread of epidemic diseases. A constructive theorem of existence in the space of continuous and bounded vector functions is proved. The integral asymptotic of the constructed solution is studied. An effective iterative process for constructing an approximate solution to this system is proposed. In a certain class of bounded vector functions, the uniqueness of the solution is proved. In the class of bounded vector functions (with non-negative coordinates) with a zero limit at infinity, the absence of an identically zero solution is proved. Examples of the matrix kernel and nonlinearities are given to illustrate the importance of the obtained results. Some of the examples have applications in the above-mentioned branches of natural science.

For any geodesic metric space $X$, we give a complete cohomological characterisation of the hyperbolicity of $X$ in terms of vanishing of its second $\ell^{\infty}$-cohomology. We extend this result to the relative setting of $X$ with a collection of uniformly hyperbolic subgraphs. As an application, we give a cohomological characterisation of acylindrical hyperbolicity., Comment: 26 pages, 4 figures

An \emph{interval $t$-coloring} of a graph $G$ is a proper edge-coloring with colors $1,\dots,t$ such that the colors on the edges incident to every vertex of $G$ are colored by consecutive colors. A graph $G$ is called \emph{interval colorable} if it has an interval $t$-coloring for some positive integer $t$. Let $\mathfrak{N}$ be the set of all interval colorable graphs. For a graph $G\in \mathfrak{N}$, we denote by $w(G)$ and $W(G)$ the minimum and maximum number of colors in an interval coloring of a graph $G$, respectively. In this paper we present some new sharp bounds on $W(G\square H)$ for graphs $G$ and $H$ satisfying various conditions. In particular, we show that if $G,H\in \mathfrak{N}$ and $H$ is an $r$-regular graph, then $W(G\square H)\geq W(G)+W(H)+r$. We also derive a new upper bound on $W(G)$ for interval colorable connected graphs with additional distance conditions. Based on these bounds, we improve known lower and upper bounds on $W(C_{2n_{1}}\square C_{2n_{2}}\square\cdots \square C_{2n_{k}})$ for $k$-dimensional tori $C_{2n_{1}}\square C_{2n_{2}}\square\cdots \square C_{2n_{k}}$ and on $W(K_{2n_{1}}\square K_{2n_{2}}\square\cdots \square K_{2n_{k}})$ for Hamming graphs $K_{2n_{1}}\square K_{2n_{2}}\square\cdots \square K_{2n_{k}}$, and these new bounds coincide with each other for hypercubes. Finally, we give several results on interval colorings of Fibonacci cubes $\Gamma_{n}$., Comment: 23 pages, 6 fidures. arXiv admin note: text overlap with arXiv:2303.11466 by other authors

The present work is the continuation of our paper [1]. It is devoted to the modeling of the interplay of equilibrium and non-equilibrium phase transitions. The modeling of equilibrium phase transition is based on the modified Cahn-Hilliard equation. The non-equilibrium phase transition is modeled by the Second Schl\"ogl reaction system. We consider the advancing front, which combines these both transitions. Different from [1], we consider here the memory effects, i.e. the effects of non-Fickian diffusion. The traveling wave solution is obtained, its dependence on the model parameters is studied in detail. The relative importance of memory effects for different process regimes is estimated., Comment: arXiv admin note: text overlap with arXiv:2002.07597

We present a novel protocol to implement quantum gates between distant atomic qubits connected by an array of neutral atoms playing the role of a quantum bus. The protocol is based on adiabatically transferring the atoms in the array to an antiferromagnetic-like state of Rydberg excitations using chirped laser pulses. Upon exciting and de-exciting the atoms in the array under the blockage of nearest neighbors, depending on the state of the two qubits, the system acquires a conditional geometric $\pi$-phase, while the dynamical phase cancels exactly, even when the atomic positions are disordered but nearly frozen in time, which requires sufficiently low temperatures. For van der Waals interacting atoms, under the optimal parameters of the pulses minimizing the Rydberg-state decay and non-adiabatic errors, the gate infidelity scales with the distance $L$ and the number of atoms $N$ between the qubits as $\sim L^6/N^3$. Hence, increasing the number of atoms in the quantum bus connecting the qubits at a given spatial separation will lead to higher gate fidelity., Comment: 16 pages, 8 figures

We give an extension of a criterion of van der Corput on uniform distribution of sequences. Namely, we prove that a sequence $x_n$ is uniformly distributed modulo 1 if it is weakly monotonic and satisfies the conditions $\Delta^2x_n\to 0,\quad n^2\Delta^2x_n\to \infty $. Our proof is straightforward and uses a Diophantine approximation by rational numbers, while van der Corput's approach is based on some estimates of exponential sums., Comment: 17 pages

The magnetic resonance of iron garnets is at the heart of rich physics and provides a crucial technology used inside microwave components. A barrier still stands in the way of integrated cryogenic devices with iron garnets, because their epitaxy requires paramagnetic substrates that cause large microwave losses at low temperatures. Finding alternative combinations of substrates and magnetic garnets is thus essential to enabling cryogenic integrated magnonics. With this motivation, we develop a new approach to the synthesis of epitaxial garnet films, based on the co-sputtering of binary oxides, enabling a precise and continuous tuning of the iron garnet composition. By substituting a controlled proportion of the iron with additional yttrium, we obtain Y$_{3}$(Y$_{x}$Fe$_{5-x}$)O$_{12}$ films of high crystalline quality that can be lattice-matched to yttrium scandium gallium garnet (YSGG), a diamagnetic substrate suitable for low-temperature applications. Ferromagnetic resonance performed at low temperatures confirms the elimination of substrate paramagnetism losses on YSGG. We also show that the Y$_{3}$(Y$_{x}$Fe$_{5-x}$)O$_{12}$ system can be matched to other substrates, such as gadolinium yttrium scandium gallium garnet and gadolinium scandium gallium garnet. Additionally, we demonstrate that Bi-substituted films of (Bi$_{0.8}$Y$_{2.2}$)Fe$_{5}$O$_{12}$ also offer ideal lattice matching to YSGG for low-temperature magnetic resonance. The present approach of controlled substitutions in iron garnets to obtain ideal lattice matching with diamagnetic substrates opens a very promising pathway to magnonic devices operating in low-temperature environments., Comment: Main: 18 pages, 6 figures; Supplement: 9 pages, 6 figures

We consider plane symmetric gravitational fields within the framework of General Relativity in (D+1)-dimensional spacetime. Two classes of vacuum solutions correspond to higher-dimensional generalizations of the Rindler and Taub spacetimes. The general solutions are presented for a positive and negative cosmological constant as the only source of the gravity. Matching conditions on a planar boundary between two regions with distinct plane symmetric metric tensors are discussed. An example is considered with Rindler and Taub geometries in neighboring half-spaces. As another example, we discuss a finite thickness cosmological constant slab embedded into the Minkowski, Rindler and Taub spacetimes. The corresponding surface energy-momentum tensor is found required for matching the exterior and interior geometries., Comment: 12 pages

We propose and analyze an approach for fast syndrome measurements in an array of rubidium and cesium atomic qubits. The scheme works by implementing an inter-species $\textsf{CNOT}_k$ gate, entangling one cesium ancilla qubit with $k\geq 1$ rubidium qubits which are then used for state measurement. Utilizing Rydberg states with different inter- and intra-species interaction strengths, the proposal provides a syndrome measurement fidelity of $\mathcal{F}>0.9999$ in less than 5 $\mu$s of integration time., Comment: Minor edits to text and additional references added

The Spin Physics Detector collaboration proposes to install a universal detector in the second interaction point of the NICA collider under construction (JINR, Dubna) to study the spin structure of the proton and deuteron and other spin-related phenomena using a unique possibility to operate with polarized proton and deuteron beams at a collision energy up to 27 GeV and a luminosity up to $10^{32}$ cm$^{-2}$ s$^{-1}$. As the main goal, the experiment aims to provide access to the gluon TMD PDFs in the proton and deuteron, as well as the gluon transversity distribution and tensor PDFs in the deuteron, via the measurement of specific single and double spin asymmetries using different complementary probes such as charmonia, open charm, and prompt photon production processes. Other polarized and unpolarized physics is possible, especially at the first stage of NICA operation with reduced luminosity and collision energy of the proton and ion beams. This document is dedicated exclusively to technical issues of the SPD setup construction.

A \emph{$2$-partition of a graph $G$} is a function $f:V(G)\rightarrow \{0,1\}$. A $2$-partition $f$ of a graph $G$ is a \emph{locally-balanced with an open neighborhood} if for every $v\in V(G)$, $$\left\vert \vert \{u\in N_{G}(v)\colon\,f(u)=0\}\vert - \vert \{u\in N_{G}(v)\colon\,f(u)=1\}\vert \right\vert\leq 1.$$ A $2$-partition $f^{\prime}$ of a graph $G$ is a \emph{locally-balanced with a closed neighborhood} if for every $v\in V(G)$, $$\left\vert \vert \{u\in N_{G}[v]\colon\,f^{\prime}(u)=0\}\vert - \vert \{u\in N_{G}[v]\colon\,f^{\prime}(u)=1\}\vert \right\vert\leq 1.$$ In this paper we prove that the problem of the existence of locally-balanced $2$-partition with an open (closed) neighborhood is $NP$-complete for some restricted classes of graphs. In particular, we show that the problem of deciding if a given graph has a locally-balanced $2$-partition with an open neighborhood is $NP$-complete for biregular bipartite graphs and even bipartite graphs with maximum degree $4$, and the problem of deciding if a given graph has a locally-balanced $2$-partition with a closed neighborhood is $NP$-complete even for subcubic bipartite graphs and odd graphs with maximum degree $3$. Last results prove a conjecture of Balikyan and Kamalian., Comment: 19 pages, 4 figures

A promising approach to attain long-distance coherent spin propagation is accessing topological spin-polarized edge states in graphene. Achieving this without external magnetic fields necessitates engineering graphene band structure, obtainable through proximity effects in van der Waals heterostructures. In particular, proximity-induced staggered potentials and spin-orbit coupling are expected to form a topological bulk gap in graphene with gapless helical edge states that are robust against disorder. In this work, we detect the spin-polarized helical edge transport in graphene at zero external magnetic field, allowed by the proximity of an interlayer antiferromagnet, CrPS$_4$. We show the coexistence of the quantum spin Hall (QSH) states and magnetism in graphene, where the induced spin-orbit and exchange couplings also give rise to a large anomalous Hall (AH) effect. The detection of the QSH states at zero external magnetic field, together with the AH signal that persists up to room temperature, opens the route for practical applications of magnetic graphene in quantum spintronic circuitries.

We propose and analyze an efficient algorithm for solving the joint sparse recovery problem using a new regularization-based method, named orthogonally weighted $\ell_{2,1}$ ($\mathit{ow}\ell_{2,1}$), which is specifically designed to take into account the rank of the solution matrix. This method has applications in feature extraction, matrix column selection, and dictionary learning, and it is distinct from commonly used $\ell_{2,1}$ regularization and other existing regularization-based approaches because it can exploit the full rank of the row-sparse solution matrix, a key feature in many applications. We provide a proof of the method's rank-awareness, establish the existence of solutions to the proposed optimization problem, and develop an efficient algorithm for solving it, whose convergence is analyzed. We also present numerical experiments to illustrate the theory and demonstrate the effectiveness of our method on real-life problems.

We present a method for the unsupervised segmentation of electron microscopy images, which are powerful descriptors of materials and chemical systems. Images are oversegmented into overlapping chips, and similarity graphs are generated from embeddings extracted from a domain$\unicode{x2010}$pretrained convolutional neural network (CNN). The Louvain method for community detection is then applied to perform segmentation. The graph representation provides an intuitive way of presenting the relationship between chips and communities. We demonstrate our method to track irradiation$\unicode{x2010}$induced amorphous fronts in thin films used for catalysis and electronics. This method has potential for "on$\unicode{x2010}$the$\unicode{x2010}$fly" segmentation to guide emerging automated electron microscopes., Comment: 7 pages, 3 figures. Accepted to Machine Learning and the Physical Sciences Workshop, NeurIPS 2023

The relativistic Langevin equation poses a number of technical and conceptual problems related to its derivation and underlying physical assumptions. Recently, a method has been proposed in [A. Petrosyan and A. Zaccone, J. Phys. A: Math. Theor. 55 015001 (2022)] to derive the relativistic Langevin equation from a first-principles particle-bath Lagrangian. As a result of the particle-bath coupling, a new ``restoring force'' term appeared, which breaks translation symmetry. Here we revisit this problem aiming at deriving a fully translation-invariant relativistic Langevin equation. We successfully do this by adopting the renormalization potential protocol originally suggested by Caldeira and Leggett. The relativistic renormalization potential is derived here and shown to reduce to Caldeira and Leggett's form in the non-relativistic limit. The introduction of this renormalization potential successfully removes the restoring force and a fully translation-invariant relativistic Langevin equation is derived for the first time. The physically necessary character of the renormalization potential is discussed in analogy with non-relativistic systems, where it emerges due to the renormalization of the tagged particle dynamics due to its interaction with the bath oscillators (a phenomenon akin to level-repulsion or avoided-crossing in condensed matter). We discuss the properties that the corresponding non-Markovian friction kernel has to satisfy, with implications ranging from transport models of the quark-gluon plasma, to relativistic viscous hydrodynamic simulations, and to electrons in graphene.

We exploit controlled breaking of time-reversal symmetry to realize coherent routing of quantum information in spin networks. The key component of our scheme is a spin triangle whose chirality is determined by the quantum state of a control qubit which thus defines the propagation direction, or a superposition thereof, of the quantum information. We then consider a particular realization of a coherent router using Rydberg atoms. Our results can facilitate scalable quantum information processing and communication in large arrays of Rydberg atoms., Comment: 12 pages, 8 figures