Pathwidth Research Articles

Estimating the Light Interception and Photosynthesis of Greenhouse-Cultivated Tomato Crops under Different Canopy Configurations

Understanding the spatial heterogeneity of light and photosynthesis distribution within a canopy is crucial for optimizing plant growth and yield, especially in the context of greenhouse structures. In previous studies, we developed a 3D functional-structural plant model (FSPM) of the Chinese solar greenhouse (CSG) and tomato plants, in which the greenhouse was reconstructed as a 3D mockup and implemented in the virtual scene. This model, which accounts for various environmental factors, allows for precise calculations of radiation, temperature, and photosynthesis at the organ level. This study focuses on elucidating optimal canopy configurations for mechanized planting in greenhouses, building upon the commonly used north–south (N–S) orientation by exploring the east–west (E–W) orientation. Investigating sixteen scenarios with varying furrow distance (1 m, 1.2 m, 1.4 m, 1.6 m) and row spacing (0.3 m, 0.4 m, 0.5 m, 0.6 m), corresponding to 16 treatments of plant spacing, four planting patterns (homogeneous row, double row, staggered row, incremental row) and two orientations were investigated. The results show that in Shenyang city, an E–W orientation with the path width = 0.5 (furrow distance + row distance) = 0.8 m (homogeneous row), and a plant distance of 0.32 m, is the optimal solution for mechanized planting at a density of 39,000 plants/ha. Our findings reveal a nuanced understanding of how altering planting configurations impacts the light environment and photosynthesis rate within solar greenhouses. Looking forward, these insights not only contribute to the field of CSG mechanized planting, but also provide a basis for enhanced CSG planting management. Future research could further explore the broader implications of these optimized configurations in diverse geographic and climatic conditions.

Williams’ path decomposition for self-similar Markov processes in Rd

Williams’ path decomposition for self-similar Markov processes in Rd

Evaluating cyclist ride quality on different bicycle streets

The ride quality is considered essential for the acceptance of bicycles as a mode of transportation. The quality of the ride for a cyclist can be affected by several different variables. In addition, the ride quality on a bicycle path or street can vary from person to person. Hence, this research investigates the ride quality along different bicycle paths, considering the perception of cyclists. Thirteen bicycle paths and streets were selected around the city of Hasselt, Belgium, to measure ride quality. The bicycle paths and streets were chosen to represent different conditions. They were selected based on infrastructure types, scenery along the path, bicycle path width, and unevenness. Twenty volunteer cyclists were recruited and asked to complete a questionnaire and rate the selected factors and ride quality after the completion of the ride. A linear mixed model was used to predict bicycle ride quality. The results suggest a significant correlation between the variables and overall ride quality. The infrastructure type was a critical variable (Estimate = 0.342, p < 0.001), showing that higher infrastructure scores were connected with a better ride experience. Cyclists reported that the scenery along bike lanes had the most negligible impact on the quality of their rides. In addition, a comparison of the ride quality across various types of infrastructure was carried out. Asphalt-paved bicycle pathways provide the highest riding quality for bikers.

Pricing a resettable convertible bond based on decomposition method and PDE models

In this paper, a partial differential equation approach based on the underlying stock price path decomposition is developed to price an American-style resettable convertible bond. The American-style resettable convertible bond is viewed as a mixture of three simple securities, which can be used to replicate the feature of payoffs of the resettable convertible bond completely. The partial differential equations under the Black–Scholes framework are established to price these simple securities. An implicit Euler method is used to discretize the first-order time derivative while a central finite difference method on a piecewise uniform mesh is used to discretize the spatial derivatives. The error estimates are developed by using the maximum principle in two mesh sets both for the time semi-discretization scheme and the spatial discretization scheme, respectively. It is proved that the scheme is first-order convergent for the time variable and second-order convergent for the spatial variable. Numerical experiments support these theoretical results.

The Influence of the Skin Phenomenon on the Impedance of Thin Conductive Layers

This paper analyzes the influence of the skin effect and the proximity effect on the inductance and impedance of thin conductive layers. The motivation for taking up this topic is the initial assessment of the possibility of using conductive layers deposited with the PVD technique on textile materials as strip or planar transmission lines of high-frequency signals (e.g., for transmitting images). This work pursues two goals. The first of them is to develop and test a numerical procedure for calculating the electromagnetic field distribution in this type of issue, based on the fundamental solution method (FSM). The second aim is to examine the impact of the skin phenomenon on the resistance, inductance and impedance of thin conductive paths. The correctness and effectiveness of FSM for the analysis of harmonics of electromagnetic fields in systems containing thin conductive layers were confirmed. Based on the performed simulations, it was found that in the frequency range above 10 MHz, the dependence of resistance and impedance on frequency is a power function with an exponent independent of the path width. Moreover, it was found that for paths with a width at least several times greater than their thickness, the dependence of the phase shift between current and voltage on frequency practically does not depend on the path width.

Two Contributions to Rolling Contact Fatigue Testing Considering Different Diameters of Rail and Wheel Discs

Scaled rolling contact fatigue tests, used to practically simulate the wear of the wheel and rail material under laboratory conditions, are typically classified into two categories. Tests in the first category use twin-disc stands, while the second group of test rigs use two discs of different diameters considering the rail disc as the larger one. The latter setup is closer to the real situation, but problems can occur with high contact pressures and tractions. The focus of this paper is on two main contributions. Firstly, a case study based on finite element analysis is presented, allowing the optimization of the specimen geometry for high contact pressures. Accumulated plastic deformation caused by cycling is responsible for abrupt lateral deformation, which requires the use of an appropriate cyclic plasticity model in the finite element analysis. In the second part of the study, two laser profilers are used to measure the dimensions of the specimen in real time during the rolling contact fatigue test. The proposed technique allows the changes in the specimen dimensions to be characterized during the test itself, and therefore does not require the test to be interrupted. By using real-time values of the specimen’s dimensional contours, it is possible to calculate an instantaneous value of the slip ratio or the contact path width.

Experimental study on pedestrian behavior in right-angled corners: Influence of departure position and passage width

This article presents an experimental investigation into pedestrian behavior while navigating through right-angled corners (defined as angles measuring exactly 90 degrees), emphasizing on the effects of departure position and path width, which have not been thoroughly explored in previous research. The study utilizes controlled experiments and introduces criteria for identifying the initiation and completion of turning maneuvers. Key findings reveal that participants adjust their trajectories based on their initial departure positions, with those farther from inner boundaries displaying more pronounced boundary-approaching behavior. Notably, the starting positions of turning are significantly influenced by the pedestrians’ initial departure positions, while the passage width has only a limited impact on the starting positions for turning. Conversely, the ending positions exhibit a consistent and substantial trend of increase with the widening of the path width, while showing minimal correlation with the initial departure position. Kinematic features, such as speed, acceleration, and angular speed, exhibit distinctive patterns during turning. These insights enhance our understanding of pedestrian behavior in corner navigation, offering implications for facility design and pedestrian dynamic modeling.

Succinct data structure for path graphs

We consider the problem of designing a succinct data structure for path graphs, that generalizes interval graphs, on n vertices while efficiently supporting degree, adjacency, and neighbourhood queries. We provide the following two solutions for this problem:1.an nlog⁡n+o(nlog⁡n)-bit succinct data structure that supports adjacency query in O(log⁡n) time, neighbourhood query in O(dlog⁡n) time and finally, degree query in min⁡{O(log2⁡n),O(dlog⁡n)} time where d is the degree of the queried vertex.2.an O(nlog2⁡n)-bit space-efficient data structure that supports adjacency, neighborhood, and degree queries optimally. Central to our data structures is the usage of the heavy path decomposition, followed by careful bookkeeping using an orthogonal range search data structure using wavelet trees among others, which may be of independent interest for designing succinct data structures for other graph classes.

Extremal statistics for a one-dimensional Brownian motion with a reflective boundary

In this work, we investigate the extreme value statistics of a one-dimensional Brownian motion (with the diffusion constant D) during a time interval 0,t in the presence of a reflective boundary at the origin, when starting from a positive position x0. We first obtain the distribution P(M|x0,t) of the maximum displacement M and its expectation 〈M〉. In the short-time limit, i.e., t≪td where td=x02/D is the diffusion time from the starting position x0 to the reflective boundary at the origin, the particle behaves like a free Brownian motion without any boundaries. In the long-time limit, t≫td, 〈M〉 grows with t as 〈M〉∼t, which is similar to the free Brownian motion, but the prefactor is π/2 times of the free Brownian motion, embodying the effect of the reflective boundary. By solving the propagator and using a path decomposition technique, we then obtain the joint distribution P(M,tm|x0,t) of M and the time tm at which this maximum is achieved, from which the marginal distribution P(tm|x0,t) of tm is also obtained. For t≪td, P(tm|x0,t) looks like a U-shaped attributed to the arcsine law of free Brownian motion. For t equal to or larger than order of magnitude of tm, P(tm|x0,t) deviates from the U-shaped distribution and becomes asymmetric with respect to t/2. Moreover, we compute the expectation 〈tm〉 of tm, and find that 〈tm〉/t is an increasing function of t. In two limiting cases, 〈tm〉/t→1/2 for t≪td and 〈tm〉/t→(1+2G)/4≈0.708 for t≫td, where G≈0.916 is the Catalan’s constant. Finally, we analytically compute the statistics of the last time tℓ the particle crosses the starting position x0 and the occupation time to spent above x0. We find that 〈tℓ〉/t→1/2 in the short-time and long-time limits, and reaches its maximum at an intermediate value of t. The fraction of the occupation time 〈to〉/t is a monotonic function of t, and tends towards 1 in the long-time limit. All the theoretical results are validated by numerical simulations.

Optimization of Laser Direct Structuring Process Parameters for Material Extrusion of Polycarbonate

Laser direct structuring (LDS) is critical in the integration of circuits onto 3D‐shaped plastic parts, such as antennas and radio frequency components. The LDS process encompasses three stages: deposition of 3D parts, laser structuring, and metallization. While laser‐direct structurable parts have been manufactured through plastic injection molding, material extrusion (MEX) is a favored additive manufacturing process for economic low‐volume production and fast prototyping advantages. Although injection‐molded LDS literature is available, 3D‐printed laser‐direct structured components merit further investigation. This study focuses on the MEX of catalyst‐loaded polycarbonate (PC) parts and its LDS process. The parameters from the nanosecond fiber laser, including scan speed, power, and frequency, are thoroughly analyzed to understand the surface property changes and metallization performance of the printed PC parts. The single scan track width, which corresponds to the accuracy of conductive path width and metallization thickness, is employed to elucidate the findings. A process map is built to keep the track width constant aimed at enhancing the uniform metallization of intricate components. Thresholds are established, identifying a minimum track width of 22.1 μm and metallization thickness of 2.5 μm. These delineate clusters of process parameters that yield conductivity levels suitable for various applications.

Voxel-based variable width continuous spiral path planning for 3D printing

Discontinuous printing paths in 3D printing lead to wasted time in fabricating products. This paper proposes a voxel-based continuous spiral path planning algorithm for 3D printing parts with less time consumption. Interestingly, we found our method can also improve the surface qualities and mechanical properties of the printed parts. The proposed path planning strategy in this paper is based on voxels which is different from traditional path planning strategies in literature. Firstly, our proposed algorithm determines the voxel size by considering the printable width of the printer nozzle. It then generates discrete elements by voxelizing the model and creates spirals and line segments by restricting the print direction of the connectable discrete elements. Next, the algorithm employs the Euler circuit to generate continuous sub-paths for the spirals and line segments. It utilizes the depth-first traversal algorithm to traverse and open the continuous sub-paths, resulting in printable continuous paths. To maintain surface accuracy for complex models, contour lines are introduced on the model's exterior and connected internally to ensure global path continuity. Lastly, the algorithm controls the path width in different directions by adjusting the extrusion rate of the printhead, preventing path overlap. Experimental results demonstrate that the proposed algorithm outperforms traditional path planning algorithms and the Connected Fermat spirals algorithm. The proposed strategy effectively mitigates the negative effects of path discontinuity, reduces printing time, and improves printed accuracy, and mechanical properties.

Kajian Imageability dan Permeability Kawasan Pecinan Kesawan di Kota Medan

Kesawan Chinatown is an urban area in the city of Medan, specifically in the Kesawan District. This area is a Chinatown district that refers to a city area outside of China where the majority of its inhabitants are of Chinese ethnicity. The area offers opportunities for tourists and local residents to learn more about colonial-era buildings. It is also an office, culinary, and other building areas. Some famous locations in this area include Tjong A Fie, the Post Office, the London Sumatra Building, and many other well-known buildings. This research aims to assess the imageability and permeability in Kesawan Chinatown. Imageability is the physical quality of a city that can create a strong image in the minds of observers. Permeability, on the other hand, refers to one's ability to access a specific location in a relatively close and easy manner in terms of time and distance. This research uses a descriptive research method with a qualitative research design (qualitative description) and mental mapping based on the city image theory used, along with sampling techniques using snowball sampling. The results of the research indicate that Kesawan Chinatown has a city image with good visual quality in terms of paths, nodes, edges, districts, and landmarks, and the permeability regarding block accessibility, path width, and path connections is excellent, making Kesawan Chinatown an easily legible area.

To die for: attractiveness, fashion, and health risks

The purpose of this research was to investigate perceptions of risks and benefits associated with popular risky fashion practices (tanning and wearing stilettos) among young adult women. Objectives were to (a) determine if fashion interest predicted benefit perceptions among young women, (b) determine the role of antecedents (fashion interest, risk perceptions, benefit perceptions, consumption emotions) in predicting the behavioral frequency of risky fashion practices, and (c) explore the extent to which tanning and wearing stilettos are similar/different in underlying mechanisms. Undergraduate women were emailed a link to a Qualtrics online survey and a total of 671 provided usable responses; 328 were tanners and 343 wore stilettos. Benefits tapped the attractiveness and fashionability that is expected from engaging in tanning and wearing stilettos, while risks focused on health risks such as melanoma or joint damage associated with tanning or wearing stilettos. Path analyses and decomposition of effects found that fashion interest was positively related to frequency of wearing stilettos, but was not related directly to tanning frequency. Fashion interest was a significant driver of perceived benefits for both risky behaviors. Benefit perceptions increased positive emotion and risk perceptions increased negative emotion; only positive emotion influenced the frequency of risky fashion practices. For both tanning and wearing stilettos, perceived benefits increased risky fashion practice frequency both directly and indirectly through positive emotion. However, perceived risks decreased fashion practice frequency only directly. Thus, perceived benefits outweigh perceived risks among women who engage in these risky fashion practices.

Extremal statistics for a resetting Brownian motion before its first-passage time.

We study the extreme value statistics of a one-dimensional resetting Brownian motion (RBM) till its first passage through the origin starting from the position x_{0} (>0). By deriving the exit probability of RBM in an interval [0,M] from the origin, we obtain the distribution P_{r}(M|x_{0}) of the maximum displacement M and thus gives the expected value 〈M〉 of M as functions of the resetting rate r and x_{0}. We find that 〈M〉 decreases monotonically as r increases, and tends to 2x_{0} as r→∞. In the opposite limit, 〈M〉 diverges logarithmically as r→0. Moreover, we derive the propagator of RBM in the Laplace domain in the presence of both absorbing ends, and then leads to the joint distribution P_{r}(M,t_{m}|x_{0}) of M and the time t_{m} at which this maximum is achieved in the Laplace domain by using a path decomposition technique, from which the expected value 〈t_{m}〉 of t_{m} is obtained explicitly. Interestingly, 〈t_{m}〉 shows a nonmonotonic dependence on r, and attains its minimum at an optimal r^{*}≈2.71691D/x_{0}^{2}, where D is the diffusion coefficient. Finally, we perform extensive simulations to validate our theoretical results.

Safety profile and technical success of narrow window CT-guided percutaneous biopsy with blunt needle approach in the abdomen and pelvis.

To assess success and safety of CT-guided procedures with narrow window access for biopsy. Three hundred ninety-six consecutive patients undergoing abdominal or pelvic CT-guided biopsy or fiducial placement between 01/2015 and 12/2018 were included (183 women, mean age 63 ± 14 years). Procedures were classified into "wide window" (width of the needle path between structures > 15 mm) and "narrow window" (≤ 15 mm) based on intraprocedural images. Clinical information, complications, technical and clinical success, and outcomes were collected. The blunt needle approach is preferred by our interventional radiology team for narrow window access. There were 323 (81.5%) wide window procedures and 73 (18.5%) narrow window procedures with blunt needle approach. The median depth for the narrow window group was greater (97 mm, interquartile range (IQR) 82-113 mm) compared to the wide window group (84 mm, IQR 60-106 mm); p = 0.0017. Technical success was reached in 100% (73/73) of the narrow window and 99.7% (322/323) of the wide window procedures. There was no difference in clinical success rate between the two groups (narrow: 86.4%, 57/66; wide: 89.5%, 265/296; p = 0.46). There was no difference in immediate complication rate (narrow: 1.3%, 1/73; wide: 1.2%, 4/323; p = 0.73) or delayed complication rate (narrow: 1.3%, 1/73; wide: 0.6%, 1/323; p = 0.50). Narrow window (< 15 mm) access biopsy and fiducial placement with blunt needle approach under CT guidance is safe and successful. CT-guided biopsy and fiducial placement can be performed through narrow window access of less than 15 mm utilizing the blunt-tip technique. • A narrow window for CT-guided abdominal and pelvic biopsies and fiducial placements was considered when width of the needle path between vital structures was ≤ 15 mm. • Seventy-three biopsies and fiducial placements performed through a narrow window with blunt needle approach had a similar rate of technical and clinical success and complications compared to 323 procedures performed through a wide window approach, with traditional approach (> 15 mm). • This study confirmed the safety of the CT-guided percutaneous procedures through < 15 mm window with blunt-tip technique.

Bat activity is related to habitat structure and time since prescribed fire in managed pine barrens in New England.

Several insectivorous bat species are found in New England, yet research on them is still scarce. Current research shows the ecological importance of bats due to their control of insect populations, but they are endangered by habitat loss and White Nose Syndrome, among other threats. Pine barrens are an uncommon ecosystem found in New England that supports other rare taxa and could be important for these bat species. With hand-held audio recorders, we surveyed for bats in Montague Plains Wildlife Management Area in Massachusetts and Concord Pine Barrens in New Hampshire in June 2022. Our study objectives were to (1) describe the most common bat species and (2) compare bat activity across different habitat types at two managed pine barrens in New England. In particular, we examined bat activity related to habitat type (scrub oak, mature pitch pine, treated pitch pine, hardwood forest), habitat structure (i.e., canopy closure), time since prescribed fire, and path width. We analyzed our data through generalized linear modeling and logistic regression. Overall, we were able to measure the presence of five out of the nine total species found in the area, including the endangered Myotis lucifugus (little brown bat). We recorded 293 bat calls, with the majority of calls from big brown bats (71%). We found significant differences (p<0.05) in bat activity in relation to time since prescribed fire and habitat structure. The index of bat activity was greatest in pitch pine and hardwood forests and lowest in scrub oak and treated pitch pine habitats. With preliminary data, we also found that silver-haired bat presence was influenced by habitat type, with more detections at survey points in hardwood forests. These findings demonstrate the importance of pine barrens as an ecosystem that supports bats in New England. According to the activity of bats in our study, closed canopy and mature pitch pine habitats may be prioritized in conservation efforts at managed barrens for bat species. Further research is recommended to better understand the relationship between prescribed fires, which are common in managed barrens, and bat activity.

Characterisation of print path deviation in material extrusion

Material extrusion, also known as fused filament fabrication, is one of the most popular additive manufacturing techniques due to its lower cost and ease of handling. However, parts produced by material extrusion have relatively poor mechanical performance, dimensional accuracy and thermal performance as compared to parts produced by subtractive manufacturing due to high void content. Previous studies have suggested print path deviation as the cause of these voids, but no attempt has yet been made to characterise these deviations. In this study, we propose a method to assess print path deviation for material extrusion that may reduce the formation of voids in printed parts. Geometric features including straight paths, various angled corners and curves of varying radii are printed at different print speeds using an open-source printer and then imaged under a microscopic. The deviation in print path centroid and width is classified as being a combination of systematic and stochastic deviations. Systematic deviation is determined by the difference between the mean of the actual print path and the ideal print path sent to the printer by the user. Stochastic deviation represents the randomness across print samples and is given by the root mean square deviation. The relationship between stochastic deviation between any two points along the print path is determined by a correlation coefficient. The results show that both print speed and different geometric print features affect the amount of deviation in the print path. In the case of correlation of the stochastic deviation along print paths, geometric features are found to have a much greater effect than print speed. The proposed method provides a low cost and highly transferrable technique to characterise the print path deviation within material extrusion parts with respect to varying printing parameters. An accurate understanding of local print deviations within a part plays a major role in the analysis and topology optimisation of 3D printed parts and gives the ability to assess the print quality and identify the root cause of print deviations, thus reducing voids and improving mechanical performance, dimensional accuracy and thermal properties of the printed part.

Modelling proton tunneling in hydrogen bonds through path integral method

Proton transfer via tunneling in hydrogen bonds is a phenomenon that occurs in many chemical compounds and biological systems. It explains several molecular properties, such as the interconversion of tautomeric forms of nucleic acids-base pairs and the conformational dynamics of proteins. Because of the complexity of these systems, it is often challenging to formulate a mathematical model capable of explaining the mechanism of proton tunneling. This study proposes an approach based on path decomposition expansion, which allows the modelling of the proton transfer reaction in multidimensional space, simplifying the investigation of complex phenomena such as the domino proton transfer or competing proton transfer mechanisms.

Extreme value statistics and Arcsine laws of Brownian motion in the presence of a permeable barrier

The Arcsine laws of Brownian motion are a collection of results describing three different statistical quantities of one-dimensional Brownian motion: the time at which the process reaches its maximum position, the total time the process spends in the positive half-space and the time at which the process crosses the origin for the last time. Remarkably the cumulative probabilities of these three observables all follow the same distribution, the Arcsine distribution. But in real systems, space is often heterogeneous, and these laws are likely to hold no longer. In this paper we explore such a scenario and study how the presence of a spatial heterogeneity alters these Arcsine laws. Specifically we consider the case of a thin permeable barrier, which is often employed to represent diffusion impeding heterogeneities in physical and biological systems such as multilayer electrodes, electrical gap junctions, cell membranes and fragmentation in the landscape for dispersing animals. Using the Feynman–Kac formalism and path decomposition techniques we are able to find the exact time-dependence of the probability distribution of the three statistical quantities of interest. We show that a permeable barrier has a large impact on these distributions at short times, but this impact is less influential as time becomes long. In particular, the presence of a barrier means that the three distributions are no longer identical with symmetry about their means being broken. We also study a closely related statistical quantity, namely, the distribution of the maximum displacement of a Brownian particle and show that it deviates significantly from the usual half-Gaussian form.

Multi-Active Multi-Passive (MAMP)-IRS Aided Wireless Communication: A Multi-Hop Beam Routing Design

Prior studies on intelligent reflecting surface (IRS) have mostly considered wireless communication systems aided by a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">single passive</i> IRS, which, however, has limited control over wireless propagation environment and suffers severe product-distance path-loss. To address these issues, we propose in this paper a new <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">multi-active multi-passive (MAMP)</i> -IRS aided wireless communication system, where a number of active and passive IRSs are deployed to assist the communication between a base station (BS) and a remote user in complex environment, by establishing a multi-hop reflection path across active and passive IRSs. In particular, the active IRSs enable to opportunistically amplify the reflected signal along the multi-reflection link, thus effectively compensating for the severe product-distance path-loss. For the new MAMP-IRS aided system, an optimization problem is formulated to maximize the achievable rate of a typical user by designing the active-and-passive IRS routing path as well as the joint beamforming of the BS and selected active/passive IRSs. To draw useful insights into the optimal design, we first consider a special case of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">single-active multi-passive (SAMP)</i> -IRS aided system. For this case, we propose an efficient algorithm to obtain its optimal solution by first optimizing the joint beamforming given any SAMP-IRS routing path, and then optimizing the routing path by using a new path decomposition method and graph theory. Moreover, we show that the active IRS should be selected to establish the beam routing path when its amplification power and/or number of active reflecting elements are sufficiently large. Next, for the general MAMP-IRS aided system, we show that its challenging beam routing optimization problem can be efficiently solved by a new two-phase approach. Its key idea is to first optimize the inner passive-IRS beam routing between each two active IRSs for effective channel power gain maximization, followed by an outer active-IRS beam routing optimization for rate maximization. Last, numerical results are provided to validate our analytical results and demonstrate the effectiveness of the proposed MAMP-IRS beam routing scheme as compared to various benchmark schemes.