In 1967, Klarner proposed a problem concerning the existence of reflecting $n$-queens configurations. The problem considers the feasibility of placing $n$ mutually non-attacking queens on the reflecting chessboard, an $n\times n$ chessboard with a $1\times n$ "reflecting strip" of squares added along one side of the board. A queen placed on the reflecting chessboard can attack the squares in the same row, column, and diagonal, with the additional feature that its diagonal path can be reflected via the reflecting strip. Klarner noted the equivalence of this problem to a number theory problem proposed by Slater, which asks: for which $n$ is it possible to pair up the integers 1 through $n$ with the integers $n+1$ through $2n$ such that no two of the sums or differences of the $n$ pairs of integers are the same. We prove the existence of reflecting $n$-queens configurations for all sufficiently large $n$, thereby resolving both Slater's and Klarner's questions for all but a finite number of integers.
The purpose of this paper is to introduce the concept of reflecting numbers to the realm of number theory and to classify reflecting numbers of certain types. For us, reflecting numbers are coming from congruent numbers, above congruent numbers, and away from congruent numbers. Explicitly speaking, a reflecting number of type $(k,m)$ is the average of two distinct rational $k$th powers, between which the distance is twice another nonzero rational $m$th power. In particular, reflecting numbers of type $(2,2)$ are all congruent numbers and thus will be called reflecting congruent numbers in this paper. We can show that all prime numbers $p\equiv5\mod8$ are reflecting congruent and in general for any integer $k\ge0$ there are infinitely many square-free reflecting congruent numbers in the residue class of $5$ modulo $8$ with exactly $k+1$ prime divisors. Moreover, we conjecture that all prime congruent numbers $p\equiv1\mod8$ are reflecting congruent. In addition, we show that there are no reflecting numbers of type $(k,m)$ if $\gcd(k,m)\ge3$.
In Molecular Communications via Diffusion (MCvD), messenger molecules are emitted by a transmitter and propagate randomly through the fluidic environment. In biological systems, the environment can be considered a bounded space, surrounded by various structures such as tissues and organs. The propagation of molecules is affected by these structures, which reflect the molecules upon collision. Deriving the channel response of MCvD systems with an absorbing spherical receiver requires solving the 3-D diffusion equation in the presence of reflecting and absorbing boundary conditions, which is extremely challenging. In this paper, the method of images is brought to molecular communication (MC) realm to find a closed-form solution to the channel response of a single-input single-output (SISO) system near an infinite reflecting surface. We showed that a molecular SISO system in a 3-D half-space with an infinite reflecting surface could be approximated as a molecular single-input multiple-output (SIMO) system in a 3-D space, which consists of two symmetrically located, with respect to the reflecting surface, identical absorbing spherical receivers.
In this study, we investigate the use of intelligent reflecting surfaces (IRSs) in multi-operator communication systems for 6G networks, focusing on sustainable and efficient resource management. This research is motivated by two critical challenges: limited coverage provided by mmWave frequencies and high infrastructure costs associated with current technologies. IRSs can help eliminate these issues because they can reflect electromagnetic waves to enhance signal propagation, thereby reducing blockages and extending network coverage. However, deploying a separate IRS for each mobile network operator (MNO) can result in inefficiencies, redundant infrastructure, potential conflicts over placement, and interoperator interference. To address these challenges, in this study, an IRS sharing system is proposed in which multiple MNOs collaborate to use a common IRS infrastructure. This approach not only enhances network flexibility and reduces costs but also minimizes the effect of interoperator interference. Through numerical analysis, we demonstrate that IRS sharing effectively balances performance and fairness among MNOs, outperforming MNO-specific deployment methods in multi-MNO scena
Recently the Event Horizon Telescope observed black holes at event horizon scales for the first time, enabling us to now test the existence of event horizons. Although event horizons have by definition no observable features, one can look for their non-existence. In that case, it is likely that there is some kind of surface, which like any other surface could absorb (and thermally emit) and/or reflect radiation. In this paper, we study the potential observable features of such rotating reflecting surfaces. We construct a general description of reflecting surfaces in arbitrary spacetimes. This is used to define specific models for static and rotating reflecting surfaces, of which we study the corresponding light paths and synthetic images. This is done by numerical integration of the geodesic equation and by the use of the general relativistic radiative transfer code RAPTOR. The reflecting surface creates an infinite set of ring-like features in synthetic images inside the photon ring. There is a central ring in the middle and higher order rings subsequently lie exterior to each other converging to the photon ring. The shape and size of the ring features change only slightly with th
We study static massive scalar field condensations in the regular asymptotically flat reflecting star background. We impose Neumann reflecting surface boundary conditions for the scalar field. We show that the no hair theorem holds in the neutral reflecting star background. For charged reflecting stars, we provide bounds for radii of hairy reflecting stars. Below the lower bound, there is no regular compact reflecting star and a black hole will form. Above the upper bound, the scalar field cannot condense around the reflecting star or no hair theorems exist. And in between the bounds, we obtain scalar configurations supported by Neumann reflecting stars.
Reconfigurable intelligent surface (RIS) has been regarded as a promising technique due to its high array gain and low power. However, the traditional passive RIS suffers from the ``double fading'' effect, which has restricted the performance of passive RIS-aided communications. Fortunately, active RIS can alleviate this problem since it can adjust the phase shift and amplify the received signal simultaneously. Nevertheless, a high beamforming gain often requires a number of reflecting elements, which leads to non-negligible power consumption, especially for the active RIS. Thus, one challenge is how to improve the scalability of the RIS and the energy efficiency. Different from the existing works where all reflecting elements are activated, we propose a novel element on-off mechanism where reflecting elements can be flexibly activated and deactivated. Two different optimization problems for passive RIS and active RIS are formulated by maximizing the total energy efficiency. We develop two different alternating optimization-based iterative algorithms to obtain sub-optimal solutions. Furthermore, we consider special cases involving rate maximization problems for given the same total
We study condensation behaviors of static scalar fields in the regular asymptotically AdS reflecting star spacetime. With analytical methods, we provide upper bounds for the radii of the scalar hairy reflecting stars. Above the bound, there is no scalar hair theorem for the star. Below the bound, we numerically obtain charged scalar hairy reflecting star solutions and in particular, the radii of the hairy stars are discrete, which is similar to known results in other reflecting object backgrounds. For every set of parameters, we search for the largest AdS hairy star radius, study effects of parameters on the largest hairy star radius and also find difference between properties in this AdS reflecting star background and those in the flat reflecting star spacetime. Moreover, we show that scalar fields cannot condense around regular AdS reflecting stars when the star charge is small or the cosmological constant is negative enough.
Reconfigurable intelligent surface (RIS) has emerged as a promising technique for future wireless communication networks. How to reliably transmit information in a RIS-based communication system arouses much interest. This paper proposes a reflecting modulation (RM) scheme for RIS-based communications, where both the reflecting patterns and transmit signals can carry information. Depending on that the transmitter and RIS jointly or independently deliver information, RM is further classified into two categories: jointly mapped RM (JRM) and separately mapped RM (SRM). JRM and SRM are naturally superior to existing schemes, because the transmit signal vectors, reflecting patterns, and bit mapping methods of JRM and SRM are more flexibly designed. To enhance transmission reliability, this paper proposes a discrete optimization-based joint signal mapping, shaping, and reflecting (DJMSR) design for JRM and SRM to minimize the bit error rate (BER) with a given transmit signal candidate set and a given reflecting pattern candidate set. To further improve the performance, this paper optimizes multiple reflecting patterns and their associated transmit signal sets in continuous fields for JRM
In this paper, we introduce an intelligent reflecting surface (IRS) to provide a programmable wireless environment for physical layer security. By adjusting the reflecting coefficients, the IRS can change the attenuation and scattering of the incident electromagnetic wave so that it can propagate in a desired way toward the intended receiver. Specifically, we consider a downlink multiple-input single-output (MISO) broadcast system where the base station (BS) transmits independent data streams to multiple legitimate receivers and keeps them secret from multiple eavesdroppers. By jointly optimizing the beamformers at the BS and reflecting coefficients at the IRS, we formulate a minimum-secrecy-rate maximization problem under various practical constraints on the reflecting coefficients. The constraints capture the scenarios of both continuous and discrete reflecting coefficients of the reflecting elements. Due to the non-convexity of the formulated problem, we propose an efficient algorithm based on the alternating optimization and the path-following algorithm to solve it in an iterative manner. Besides, we show that the proposed algorithm can converge to a local (global) optimum. Fur
We tackle the problem of generating highly realistic and plausible mirror reflections using diffusion-based generative models. We formulate this problem as an image inpainting task, allowing for more user control over the placement of mirrors during the generation process. To enable this, we create SynMirror, a large-scale dataset of diverse synthetic scenes with objects placed in front of mirrors. SynMirror contains around 198k samples rendered from 66k unique 3D objects, along with their associated depth maps, normal maps and instance-wise segmentation masks, to capture relevant geometric properties of the scene. Using this dataset, we propose a novel depth-conditioned inpainting method called MirrorFusion, which generates high-quality, realistic, shape and appearance-aware reflections of real-world objects. MirrorFusion outperforms state-of-the-art methods on SynMirror, as demonstrated by extensive quantitative and qualitative analysis. To the best of our knowledge, we are the first to successfully tackle the challenging problem of generating controlled and faithful mirror reflections of an object in a scene using diffusion-based models. SynMirror and MirrorFusion open up new av
We investigate the gravity system constructed with static scalar fields coupled to asymptotically flat regular reflecting stars. We consider the matter field's backreaction on the reflecting star. We analytically show that there is an upper bound on the radius of the reflecting star. When the star radius is above the bound, the reflecting star cannot support the existence of scalar field hairs. That means large reflecting stars cannot have scalar field hairs.
We investigate anomalous diffusion processes governed by the fractional Langevin equation and confined to a finite or semi-infinite interval by reflecting potential barriers. As the random and damping forces in the fractional Langevin equation fulfill the appropriate fluctuation-dissipation relation, the probability density on a finite interval converges for long times towards the expected uniform distribution prescribed by thermal equilibrium. In contrast, on a semi-infinite interval with a reflecting wall at the origin, the probability density shows pronounced deviations from the Gaussian behavior observed for normal diffusion. If the correlations of the random force are persistent (positive), particles accumulate at the reflecting wall while anti-persistent (negative) correlations lead to a depletion of particles near the wall. We compare and contrast these results with the strong accumulation and depletion effects recently observed for non-thermal fractional Brownian motion with reflecting walls, and we discuss broader implications.
We investigate scalar condensations around noncommutative compact reflecting stars. We find that the neutral noncommutative reflecting star cannot support the existence of scalar field hairs. In the charged noncommutative reflecting star spacetime, we provide upper bounds for star radii. Above the bound, scalar fields cannot exist outside the star. In contrast, when the star radius is below the bound, we show that the scalar field can condense. We also obtain the largest radii of scalar hairy reflecting stars.
Diffusion processes $(\underline{\bf X}_d(t))_{t\geq 0}$ moving inside spheres $S_R^d \subset\mathbb{R}^d$ and reflecting orthogonally on their surfaces $\partial S_R^d$ are considered. The stochastic differential equations governing the reflecting diffusions are presented and their kernels and distributions explicitly derived. Reflection is obtained by means of the inversion with respect to the sphere $S_R^d$. The particular cases of Ornstein-Uhlenbeck process and Brownian motion are examined in detail. The hyperbolic Brownian motion on the Poincarè half-space $\mathbb{H}_d$ is examined in the last part of the paper and its reflecting counterpart within hyperbolic spheres is studied. Finally a section is devoted to reflecting hyperbolic Brownian motion in the Poincarè disc $D$ within spheres concentric with $D$.
It is known that the capacity of the intelligent reflecting surface (IRS) aided cellular network can be effectively improved by reflecting the incident signals from the transmitter in a low-cost passive reflecting way. In this paper, we study the adoption of an IRS for downlink multi-user communication from a multi-antenna base station (BS). Nevertheless, in the actual network operation, the IRS operator can be selfish or have its own objectives due to competing/limited resources as well as deployment/maintenance cost. Therefore, in this paper, we develop a Stackelbeg game model to analyze the interaction between the BS and the IRS operator. Specifically, different from the existing studies on IRS that merely focus on tuning the reflection coefficient of all the reflection elements, we consider the reflection resource (elements) management, which can be realized via trigger module selection under our proposed IRS architecture that all the reflection elements are partially controlled by independent switches of controller. A Stackelberg game-based alternating direction method of multipliers (ADMM) is proposed to jointly optimize the transmit beamforming at the BS and the passive beam
We consider random flights in $\mathbb{R}^d$ reflecting on the surface of a sphere $\mathbb{S}^{d-1}_R,$ with center at the origin and with radius $R,$ where reflection is performed by means of circular inversion. Random flights studied in this paper are motions where the orientation of the deviations are uniformly distributed on the unit-radius sphere $\mathbb{S}^{d-1}_1$. We obtain the explicit probability distributions of the position of the moving particle when the number of changes of direction is fixed and equal to $n\geq 1$. We show that these distributions involve functions which are solutions of the Euler-Darboux-Poisson equation. The unconditional probability distributions of the reflecting random flights are obtained by suitably randomizing $n$ by means of a fractional-type Poisson process. Random flights reflecting on hyperplanes according to the optical reflection form are considered and the related distributional properties derived.
We investigate the use of an intelligent reflecting surface (IRS) in a millimeter-wave (mmWave) vehicular communication network. An intelligent reflecting surface consists of passive elements, which can reflect the incoming signals with adjustable phase shifts. By properly tuning the phase shifts we can improve link performance. This is known as phase optimization or passive beamforming. We consider the problem of rate maximization in the uplink, which utilizes an IRS. However, using an IRS brings more challenges in terms of channel estimation. We propose two schemes to reduce the channel estimation overhead associated with utilizing an IRS. One method uses the grouping of reflecting elements and the other one performs passive beamforming based on the position of the device. Numerical results show IRS can bring significant improvements to existing communication. Furthermore, to get a practical insight into vehicular communications aided by an IRS, we use a commercial ray-tracing tool to evaluate the performance.
The IEEE Signal Processing Society is proud to announce the eighth edition of the Signal Processing Cup: an exciting challenge to control a wireless propagation environment using an intelligent reflecting surface. An intelligent reflecting surface is a two-dimensional array of metamaterial whose interaction with electromagnetic waves can be controlled, e.g., by tuning the impedance variations over the surface. These surfaces might be used in the sixth generation (6G) mobile technology to direct wireless signals from a transmitter towards a receiver, to raise the communication performance. The goal of the challenge is to characterize the behavior of an intelligent reflecting surface based on received signals from an over-the-air signaling phase and develop a control algorithm to configure the surface to aid wireless communications.
This work develops new numerical methods for the solution of the tomography problem in domains with reflecting obstacles. We compare the solution's performance for Lambertian reflection, for classical tomography with unbroken rays and for specular reflection. Our numerical method using Lambertian reflection improves the solution's accuracy by an order of magnitude compared to classical tomography with unbroken rays and for tomography in the presence of a specularly reflecting obstacle the numerical method improves the solution's accuracy approximately by a factor of three times. We present efficient new algorithms for the solution's software implementation and analyze the solution's performance and effectiveness. The new method from this work for reducing the number of equations in tomography linear systems is applicable to improving the performance of a wide class of algebraic tomographic reconstruction methods.