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Recent advancements in hyperspectral imaging have significantly increased the acquired data volume, creating a need for more efficient compression methods for handling the growing storage and transmission demands. These challenges are particularly critical for onboard satellite systems, where power and computational resources are limited, and real-time processing is essential. In this article, we present a novel FPGA-based hardware acceleration of a near-lossless compression technique for hyperspectral images by leveraging a division-free quadrature-based square rooting method. In this regard, the two division operations inherent in the original approach were replaced with pre-computed reciprocals, multiplications, and a geometric series expansion. Optimized for real-time applications, the synthesis results show that our approach achieves a high throughput of 1611.77 Mega Samples per second (MSps) and a low power requirement of 0.886 Watts on the economical Cyclone V FPGA. This results in an efficiency of 1819.15 MSps/Watt, which, to the best of our knowledge, surpasses recent state-of-the-art hardware implementations in the context of near-lossless compression of hyperspectral images. [ABSTRACT FROM AUTHOR]

Efficient computation of expected values is paramount in scenario analysis and decision-making, especially for problems involving large finite state machines with complex state dependencies and transitions. Traditional approaches relying on linear equation systems often fall short under such demanding conditions. Addressing this critical need, the proposed method integrates finite state machines to model transitions and matrices to manage state properties, delivering a breakthrough in efficiency. The matrix transition model, enhanced with an optimized memory management technique, achieves an average computation time of just 0.179 seconds--over 22 times faster than the strict 4-second time limit set by the problem setter--and consumes only 5.63 MB of memory, a mere 0.36% of the 1536 MB limit. These results underscore the solution's exceptional capability to not only meet but vastly exceed stringent performance requirements, redefining expectations for large-scale finite state machine calculations. [ABSTRACT FROM AUTHOR]

AbstractWe observe the distribution of odd and even elements in the gap set of a numerical semigroup. For embedding dimension 2 or 3 and depending on the number of odd minimal generators of a numerical semigroup S, we compute O(G(S))−E(G(S)), the difference between the number of odd and even elements in the gap set. We also obtain the results on O(G(S))−E(G(S)) when the minimal generators of S are one of the followings: some compound sequences, a geometric sequence, or an arithmetic sequence. [ABSTRACT FROM AUTHOR]

An equal temperament is a musical tuning system that approximates just intervals within an octave in order to suppress some beat acoustic frequencies known to be dissonant to the human ear. Division of an octave into n segments always results in n frequencies that form a geometric sequence with common ratio rn = 21/n. This ratio is a golden ratio for any choice of integer n ≥ 2. In western music, the widely used golden ratio of r12 = 21/12 ≃ 1.059463 has heretofore gone unappreciated because of the unfortunate widespread perception that the only golden ratio is the Euclidean φ = (1 +√5)/2 ≃ 1.618034 that results from the section of a line segment into only two noncongruent parts. Confusion has also been compounded by lack of distinction between golden ratios of lengths versus those of discrete physical values (such as frequencies in music). In this work, we clarify these issues, and we extend the n = 2 Euclidean golden section to n ≥ 3 partitions. [ABSTRACT FROM AUTHOR]

Total leaf area per shoot (AT) can reflect the photosynthetic capacity of a shoot. A prior study hypothesized that AT is proportional to the product of the sum of the individual leaf widths per shoot (LKS) and the maximum individual leaf length per shoot (WKS), referred to as the Montgomery–Koyama–Smith equation (MKSE). However, empirical evidence does not support such a proportional relationship hypothesis, as AT was found to allometrically scale with LKSWKS, i.e., A T ∝ L K S W K S α , where α ≠ 1 , referred to as the power law equation (PLE). Given that there is variation in the total number of leaves per shoot (n), little is known about whether the leaf area distribution has an explicit mathematical link with the sorted leaf area sequence per shoot, and it is unknown whether the mathematical link can affect the prediction accuracy of the MKSE and PLE. In the present study, the leaves of 500 shoots of a dwarf bamboo (Sasaella kongosanensis 'Aureostriatus') were scanned, and the leaf area, length, and width values were obtained by digitizing the leaf images. We selected the shoots with n ranging from 3 to 10, which accounted for 76.6% of the totally sampled shoots (388 out of 500 shoots). We used the formula for the sum of the first j terms (j ranging from 1 to n) of a geometric series (GS), with the mean of the quotients of any adjacent two terms (denoted as q ¯ A ) per shoot as the common ratio of the GS, to fit the cumulative leaf area observations. Mean absolute percentage error (MAPE) was used to measure the goodness of fit of the GS. We found that there were 367 out of 388 shoots (94.6%) where 1 < q ¯ A < 1.618 and MAPE < 15%, and these 367 shoots were defined as valid samples. The GS hypothesis for leaf area distribution was supported by the result that the MAPE values for most valid samples (349 out of 367, i.e., 95.1%) were smaller than 5%. Here, we provide a theoretical basis using the GS hypothesis to demonstrate the validity of the MKSE and PLE. The MAPE values for the two equations to predict AT were smaller than 5%. This work demonstrates that the leaf area sequence per shoot follows a GS and provides a useful tool for the calculation of total leaf area per shoot, which is helpful to assess the photosynthetic capacity of plants. [ABSTRACT FROM AUTHOR]

Multipoint evaluation is the computational task of evaluating a polynomial given as a list of coefficients at a given set of inputs. Besides being a natural and fundamental question in computer algebra on its own, fast algorithms for this problem are also closely related to fast algorithms for other natural algebraic questions such as polynomial factorization and modular composition. And while nearly linear time algorithms have been known for the univariate instance of multipoint evaluation for close to five decades due to a work of Borodin and Moenck [7], fast algorithms for the multivariate version have been much harder to come by. In a significant improvement to the state-of-the-art for this problem, Umans [25] and Kedlaya & Umans [16] gave nearly linear time algorithms for this problem over field of small characteristic and over all finite fields, respectively, provided that the number of variables n is at most do(1) where the degree of the input polynomial in every variable is less than d. They also stated the question of designing fast algorithms for the large variable case (i.e., n x do(1)) as an open problem. In this work, we show that there is a deterministic algorithm for multivariate multipoint evaluation over a field Fq of characteristic p, which evaluates an n-variate polynomial of degree less than d in each variable on N inputs in time provided that p is at most do(1), and q is at most (exp(exp(exp(· · · (exp(d))))), where the height of this tower of exponentials is fixed. When the number of variables is large (e.g., n x do(1)), this is the first nearly linear time algorithm for this problem over any (large enough) field. Our algorithm is based on elementary algebraic ideas, and this algebraic structure naturally leads to the following two independently interesting applications: --We show that there is an algebraic data structure for univariate polynomial evaluation with nearly linear space complexity and sublinear time complexity over finite fields of small characteristic and quasipolynomially bounded size. This provides a counterexample to a conjecture of Miltersen [21] who conjectured that over small finite fields, any algebraic data structure for polynomial evaluation using polynomial space must have linear query complexity. We also show that over finite fields of small characteristic and quasipolynomially bounded size, Vandermonde matrices are not rigid enough to yield size-depth tradeoffs for linear circuits via the current quantitative bounds in Valiant's program [26]. More precisely, for every fixed prime p, we show that for every constant e > 0, and large enough n, the rank of any n × n Vandermonde matrix V over the field Fpa can be reduced to (n/exp(O(poly(e)log0.53 n))) by changing at most nT(e) entries in every row of V, provided a = poly(logn). Prior to this work, similar upper bounds on rigidity were known only for special Vandermonde matrices. For instance, the Discrete Fourier Transform matrices and Vandermonde matrices with generators in a geometric progression [9]. [ABSTRACT FROM AUTHOR]

In this paper we give a Voronovskaya type theorem for the operators introduced by U. Abel, which are defined as the geometric series of Bernstein - Durrmeyer operators. [ABSTRACT FROM AUTHOR]

The article explores the theory of body motion in confined Stokesian fluids, specifically analyzing the convergence of a reflection method for determining velocity fields. It delves into the scaling analysis of approximate expressions for forces and torques on bodies in bounded fluids, emphasizing conditions for convergence, particularly in close proximity to walls. The study sheds light on the dynamics of particles near walls, including sedimentation, translation, and rotation, offering detailed mathematical formulations and derivations to enhance understanding of hydrodynamic interactions in microfluidics and related fields. [Extracted from the article]

The rise in machine-enabled password attacks and the cost per record lost in an average case of a data breach necessitate the need for a more robust technique for combating password attacks. Organizations of different sizes and global reputation have been victims of cyber-attacks. The problem of cyber-attacks has attracted several research responses from researchers with some attending results. This article presents the Dual Combat Technique-based Cyber-Systems protection against password attack. The proposed system utilizes a-three-tier model for detection, notification, and combat. The dual combat technique involves the System Protection Model (SPM) and the User Protection Model (UPM). While the SPM implemented a time delay algorithm powered by a geometric progression model, the UPM uses a dual handshake method for data communication between the user and the server. In the first instance, the UPM sends data to the Cyber-system server through an HTTP Request over an SMS gateway to virtualize a user's account upon a trigger by the attack detection model. In the second instance, the deactivation of the virtualization operation uses the authentication of the user's email and phone number. The result of the work presents a system that introduces a time-delay after a number of login attempts defined by a certain threshold value, and a user response action for account virtualization. The application testing presented a success rate of 90.16% on the number of times the request response was induced over an eightday period of testing and 9.84% failed attempts. [ABSTRACT FROM AUTHOR]

An additive submonoid of the nonnegative cone of the real line is called a positive monoid. Positive monoids consisting of rational numbers (also known as Puiseux monoids) have been the subject of several recent papers. Moreover, those generated by a geometric sequence have also received a great deal of recent attention. Our purpose is to survey many of the recent advances regarding positive monoids, and we provide numerous examples to illustrate the complexity of their atomic and arithmetic structures. [ABSTRACT FROM AUTHOR]

"Smaller and Smaller" is a woodcut by the Dutch artist M.C. Escher in which the lizards designed in a rosette tiling approach the centre by geometric series. This paper proposes an easy method to generate drawings similar to "Smaller and Smaller". To this end, the geometrical structure of rosette tilings is first considered from the viewpoint of the symmetry group in detail. This gives a simple way to construct rosette tilings. Then, a one-to-one mapping between kite-shaped and square regions is presented to embed a pre-designed template into kite-shaped tiles of rosette tilings. Next, the algorithms for rendering the rosette tilings are discussed. Finally, some detailed implements of producing rosette drawings are specified. The presented examples show that by using the existing rich wallpaper templates, the method proposed in the paper can generate a variety of Escher-like rosette drawings. [ABSTRACT FROM AUTHOR]

Butterflies and plant species are widely used as model systems in biodiversity and conservation research. However, little is known about spatial patterns of butterfly diversity linked to plant communities across coastal dune ecotones. We explored the relation between butterfly and vegetation diversity in a Mediterranean protected dune system in Central Italy. We sampled butterfly and plant communities on shifting and fixed dunes and investigated their distribution patterns by diversity indexes and Species Abundance Diversity models (SADs). We also computed species beta-diversity and analyzed the relation between butterfly communities and vegetation by correlation analyses. Cross-taxon congruence in species richness, composition and abundance was assessed using Mantel correlation tests. Both butterfly and plant diversity were higher on the fixed dunes than on the shifting dunes. In both communities, SADs were adequately described by the geometric series model, which is typical of harsh environments with simple communities dominated by few species. Among both butterflies and plants, we identified species found only on either the shifting or the fixed dunes. Correlation between plant and butterfly diversity was higher for entomophilous plant species than for the more abundant anemophilous ones. Our results indicate that both plant and butterfly communities respond to the sea-inland environmental gradient, forming distinct communities with specialized taxa. Our findings also provide evidence of the importance of butterfly-plant relations for preserving biological diversity and ecosystem functioning in Mediterranean coastal areas. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two types of hyper-dual numbers with components including Pell and Pell-Lucas numbers. This novel approach facilitates our understanding of hyper-dual numbers and properties of Pell and Pell-Lucas numbers. We also investigate fundamental properties and identities associated with Pell and Pell-Lucas numbers, such as the Binet-like formulas, Vajda-like, Catalan-like, Cassini-like, and d'Ocagne-like identities. Furthermore, we also define hyper-dual vectors by using Pell and Pell-Lucas vectors and discuse hyper-dual angles. This extensionis not only dependent on our understanding of dual numbers, it also highlights the interconnectedness between integer sequences and geometric concepts. [ABSTRACT FROM AUTHOR]

The current paper focuses on the assessment of radial mesh influence on the description of the transient event obtained by an axisymmetric model. The objective is to reduce computational effort while accurately calculating hydraulic transients in smooth–turbulent pressurized pipes. The analyzed pipe system has a reservoir–pipe–valve configuration with an inner diameter of 0.02 m and a total length of 14.96 m, with the initial discharge being equal to 120 × 10−3 L/s (Re = 7638). An extensive study is carried out with 80 geometric sequence meshes by varying the total number of cylinders, the geometric common ratio, and the pipe axial discretization. The benefit of increasing the geometric common ratio is highlighted. A detailed comparison between two meshes is presented, in which the best mesh (i.e., the one with the lowest computational effort) has a three-fold higher value of the geometric common ratio. The two meshes show small differences for the instantaneous valve closure, limited to a time interval immediately after the arrival of the pressure surge and only during the first pressure wave. The dynamic characterization of the transient phenomenon demonstrates the in-depth consistency between the model results and the hydraulic transients' phenomenon in terms of the piezometric head, the wall shear stress, and the mean velocity time-history, in comparison to the results obtained with the shear stress, lateral velocity, and axial velocity profiles. [ABSTRACT FROM AUTHOR]

The moment distribution method, originally introduced by Hardy Cross in his influential publication "Analysis of Continuous Frames by Distributing Fixed‐End Moments" (published in Transactions of the ASCE, 1932;96(1793):1–10), is widely recognized as a fundamental aspect of both structural engineering and global education. This iterative technique aims to improve solution accuracy at each step of distribution and has stood the test of time. In our study, we delve into the mathematical foundation of this method and make a crucial discovery: the terms generated during iterations follow a geometric series pattern. By leveraging this property, we propose a streamlined formula for efficiently calculating the sum of all the terms produced in each step of distribution iteration. Through comparative analysis with alternative methods, we highlight the method's superior precision and robustness in structural analysis. Our investigation primarily focuses on demonstrating the geometric series nature of the terms generated through iterative moment distribution, providing specific theorems as references. Ultimately, our objective is to present concise, closed‐form mathematical solutions that showcase the practical effectiveness of this method in various applications within the field of structural engineering. [ABSTRACT FROM AUTHOR]

The study aims to develop vertical radar reflectivity profiles based on lightning density data from the Geostationary Lightning Mapper (GLM) on the GOES-16 satellite in the subtropical region of Brazil. The primary objective is to improve the assimilation of lightning data in numerical weather prediction models. The methodology involves the analysis of polarimetric radar data from Chapecó-SC and Jaraguari-MS, spanning from January 2019 to December 2023, and their correlation with lightning data from the GLM. Radar reflectivity profiles were created for different lightning density classes, categorized into six classes based on geometric progression. Results show a significant relationship between lightning activity and radar reflectivity, with distinct profiles for convective and stratiform events. These findings demonstrate the potential of using GLM data to enhance short-term weather forecasting, particularly for severe weather events. The study concludes that the integration of GLM data into weather models can lead to more accurate predictions of intense precipitation events, contributing to better preparedness and response strategies. [ABSTRACT FROM AUTHOR]

In the present paper, we study a set that can be treated as a generalised set of subsums for a geometric series. This object was discovered independently in various mathematical aspects. For instance, it is closely related to various systems of representation of real numbers. The main object of this paper was particularly studied by R. Kenyon, who brought up a question about the Lebesgue measure of the set and conjectured that it is positive. Further, Z. Nitecki confirmed the hypothesis by using nontrivial topological techniques. However, the aforementioned result is quite limited, as this particular case should satisfy a rigid condition of homogeneity. Despite the limited progress, the problem remained understudied in a general framework. The study of topological, metric, and fractal properties of the set of subsums for a numerical series is a separate research direction in mathematics. On the other hand, the topic is related to another modern mathematical problem, namely, deepening of the Jessen-Wintner theorem for infinite Bernoulli convolutions and their generalisations. The essence of the problem is to reveal the necessary and sufficient conditions for the probability distribution of a random subsum of a geometric series to be absolutely continuous or singular. The Jessen-Wintner theorem guarantees that the distribution is pure (pure discrete, pure singular, or pure absolutely continuous). Meanwhile, the L'evy theorem gives us the necessary and sufficient condition for the distribution to be discrete. Since the set of subsums for an absolutely convergent series coincides with the set of possible outcomes of the corresponding probability distribution, under certain conditions, it allows us to apply various probability techniques for its further investigation. In particular, some techniques help us to prove that the above sets have a positive Lebesgue measure and allow to deepen the Jessen-Wintner theorem under certain conditions. [ABSTRACT FROM AUTHOR]

In this short note I present an elementary proof of irrationality for the number e, the base of the natural logarithm. It is simpler than other known proofs as it does not use comparisons with geometric series, nor Beukers' integrals, and it does not assume that e is a rational number from the beginning. [ABSTRACT FROM AUTHOR]

Integrating vendors' and buyers' production-inventory systems can significantly reduce the total costs incurred by both parties. In this paper, we study the synchronised multi-vendor multiple-buyer integrated production-inventory system with a production flow where the lot sizes follow a geometric sequence. We model this new problem with the objective to minimise the total cost incurred by the vendors and the buyers and determine the optimal production shipment policy. The computational experiments realised on real case study show the efficiency of the developed model for optimising production shipment and synchronisation of vendor-buyer inventories. [ABSTRACT FROM AUTHOR]

Recently, some new sequence spaces ℓ p (A α) (0 < p < ∞) , c 0 (A α) , c (A α) , and ℓ ∞ (A α) have been studied by Yaying et al. (Forum Math., 2024, https://doi.org/10.1515/forum-2023-0138) as matrix domains of A α = (a n , v α) , where a m , v α = { v α ρ (α) (m) , v ∣ m , 0 , v ∤ m , and ρ (α) (m) : = sum of the α th power of the positive divisors of m ∈ N . They obtained their duals, matrix transformations and associated compact matrix operators for these matrix classes. This article deals with some geometric properties of these sequence spaces. [ABSTRACT FROM AUTHOR]

Flow field regulation plays an important role in the optimization of column flotation units with agitation function. This paper aims to discuss the regulating effect of baffle design on the flow field in stirred column flotation units by numerical simulation. A series of geometric models of changing baffle structure and layout were established, and detailed flow field simulation analysis was carried out by using COMSOL software. The results show that the internal flow field of the traditional standard cylindrical tank will cause the solid particles to adhere to the wall movement and the bubbles to the middle area, resulting in the uneven distribution of particles and bubbles, and the internal flow field is not conducive to the mineralization process and the stability of the foam layer. This limitation can be significantly improved by introducing baffles. The results show different baffle shapes and layouts are as follows: wide and narrow on top, narrow on the bottom and wide on top, and only arranged at the bottom of the column body, can increase the average turbulent kinetic energy in the collision zone by 22%, 46% and 43%, and the average turbulent dissipation rate by 18%, 43% and 36%, respectively, while maintaining the turbulence intensity in the transport zone. Among them, the baffle plate with wide top and narrow bottom has the best effect. Arranging baffles with a length shorter than 2/3 of the height of the cylinder at the upper part inside the cylinder can also enhance the turbulence intensity in the collision zone, but at the same time, it also increases the turbulence intensity in the transport zone. Moreover, as the length of the baffle decreases, the enhancement effect becomes stronger. Since this method significantly affects the transport zone, it is more suitable for columnar mixing devices. The study also reveals that by adjusting the proportion of wide and narrow baffles, the center of the turbulent region can be laterally shifted to the short baffles, and the flow field distribution can be adjusted. Meanwhile, with the increase in the number of long baffles, the intensity of the flow field in the device will gradually decrease. By moving the center of the turbulent region to the side inlet, the particle dispersion can be accelerated, but in general, the particles will accumulate toward the region with lower turbulence intensity. These findings have important guiding significance for the design and optimization of the flow field of the stirred column flotation unit, which is helpful in improving the agitation uniformity and flotation efficiency. [ABSTRACT FROM AUTHOR]

We propose a balanced $ 5 $ 5 -moduli set $ P = \{{{2^q}, {2^q} - 1,{2^{q + 1}} - 1,{2^{q + 1}} + 1,{2^{q + 2}} - 1} \} $ P = { 2 q , 2 q − 1 , 2 q + 1 − 1 , 2 q + 1 + 1 , 2 q + 2 − 1 } , for odd $ q \ge 3 $ q ≥ 3. Our reverse conversion scheme is based on a multi-stage new Chinese remainder theorem, where some of the difficult-to-derive multiplicative inverse coefficients are expressed as geometric series. Only carry-save and ripple-carry adders are used as components of the corresponding circuits. Analytical gate-level evaluations are performed on the proposed and reference converters, whose results are confirmed by the synthesis tool. A comparison of the converters for moduli sets with equal dynamic range shows that $ P $ P is similar to previous 5-moduli set $ \{{{2^q}, {2^q} - 1,{2^q} + 1,{2^{q + 1}} - 1,{2^{q - 1}} - 1} \} $ { 2 q , 2 q − 1 , 2 q + 1 , 2 q + 1 − 1 , 2 q − 1 − 1 } and shows slight advantage in delay and energy consumption. [ABSTRACT FROM AUTHOR]

Zero inflation, zero deflation, overdispersion, and underdispersion are commonly encountered in count time series. To better describe these characteristics of counts, this article introduces a zero‐modified geometric first‐order integer‐valued autoregressive (INAR(1)) model based on the generalized negative binomial thinning operator, which contains dependent zero‐inflated geometric counting series. The new model contains the NGINAR(1) model, ZMGINAR(1) model, and GNBINAR(1) model with geometric marginals as special cases. Some statistical properties are studied, and estimates of the model parameters are derived by the Yule–Walker, conditional least squares, and maximum likelihood methods. Asymptotic properties and numerical results of the estimators are also studied. In addition, some test and forecasting problems are addressed. Three real‐data examples are given to show the flexibility and practicability of the new model. [ABSTRACT FROM AUTHOR]

We compute some differentials of Sinha's spectral sequence for cohomology of the space of long knots modulo immersions in codimension one, mainly over a field of characteristic 2 or 3. This spectral sequence is closely related to Vassiliev's spectral sequence for the space of long knots in codimension |$\geq2$|⁠. We prove that the d 2-differential of an element is non-zero in characteristic 2, which has already essentially been proved by Salvatore, and the d 3-differential of another element is non-zero in characteristic 3. While the geometric meaning of the sequence is unclear in codimension one, these results have some applications to non-formality of operads. The result in characteristic 3 implies planar non-formality of the standard map |$C_\ast(E_1)\to C_\ast(E_2)$| in characteristic 3, where |$C_\ast(E_k)$| denotes the chain little k -disks operad. We also reprove the result of Salvatore which states that |$C_\ast(E_2)$| is not formal as a planar operad in characteristic 2 using the result in characteristic 2. For the computation, we transfer the structure on configuration spaces behind the spectral sequence onto Thom spaces over fat diagonals through a duality between configuration spaces and fat diagonals. This procedure enables us to describe the differentials by relatively simple maps to Thom spaces. We also show that the d 2-differential of the generator of bidegree |$(-4,2)$| is zero in characteristic |$\not=2$|⁠. This computation illustrates how one can manage the three-term relation using the description. Although the computations in this paper are concentrated to codimension one, our method also works for codimension |$\geq2$| and we prepare most of the basic notions and lemmas for general codimension. [ABSTRACT FROM AUTHOR]

The worldwide pesticide marketplace was approximate 85 billion dollars in 2019, growing at a geometric progression rate of 4.2% from 2015, and by 2023 it is projected to grow at a rate of 11.5% to approximately 130.7 billion dollars. Pesticides' uniqueness is defined by the distinctiveness of their chemistry and their relationship with the environment. Pesticide mobility and bioavailability in water bodies are determined by their desorption and absorption processes from soil particles. Waterbodies are severely affected with the build-up of these poisonous impurities with its imbalance in pH and accumulation of heavy metals, which adversely distress the aquatic ecology in the waterbodies. Pesticides are not only damaging to the ecology of the waterbodies; it is also consequentially harmful to humans. It causes gastrointestinal disorders, cardiac and respiratory issues, and even affects the biotic molecules and organs subsequently leading to acute and/or chronic illnesses. This article reviews 200 scientific literatures for existence of pesticides in drinking water and the various remediation technologies available for the treatment to render the water potable. Upon review of more than 20 technologies, suggestions have been made for the best probable technologies for water with presence of pesticide. [ABSTRACT FROM AUTHOR]

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In this paper, images containing mathematical ideas that were generated by different image-generating AI models (DALL-E 2, Leonardo.Ai, and Midjourney) are comparatively interpreted from a mathematics aesthetical perspective. Our exploratory study aimed to scrutinize how AI image models visualize abstract mathematical ideas and whether this imagination holds any potential for use in the context of learning mathematics. Two different mathematics-containing prompts, representing different aspects of mathematics, were used as input. Our results show that based on our examples, currently, Midjourney generates the most detailed outcome that can be used as an initial point for discussing mathematical ideas with students in the classroom. [ABSTRACT FROM AUTHOR]

The article is devoted to the problem of approximation of classes of periodic functions by rectangular linear means of Fourier series. Asymptotic equalities are found for upper bounds of deviations in the uniform metric of rectangular Fejér means on classes of periodic functions of several variables generated by sequences that tend to zero at the rate of geometric progression. In one-dimensional cases, these classes consist of Poisson integrals, namely functions that can be regularly extended in the fixed strip of a complex plane. [ABSTRACT FROM AUTHOR]

The given text discusses the concept of representative points in a graph and how they can be used to define a function. It explains that by choosing three rational points, one inside each region of the graph, a function can be created. The text aims to show that this function is injective, meaning that it is one-to-one. It provides a diagram and references a related problem about the possibility of having uncountably many disjoint figure 8s in the plane. The text concludes by mentioning that the result discussed is due to R. L. Moore. [Extracted from the article]

Finding the Frobenius number and the genus of any numerical semigroup S is a well-known open problem. Similarly, it has been studied how to express the Frobenius number and the genus of a quotient of a numerical semigroup. In this paper, by enumerating the Hilbert series of each type of numerical semigroup, we show an expression for the genus of a quotient of numerical semigroups generated by one of the following series: arithmetic progression, geometric series, and Pythagorean triple. [ABSTRACT FROM AUTHOR]

Data assimilation (DA) plays a pivotal role in diverse applications, ranging from weather forecasting to trajectory planning for autonomous vehicles. A prime example is the widely used ensemble Kalman filter (EnKF), which relies on the Kalman filter's linear update equation to correct each of the ensemble forecast member's state with incoming observations. Recent advancements have witnessed the emergence of deep learning approaches in this domain, primarily within a supervised learning framework. However, the adaptability of such models to untrained scenarios remains a challenge. In this study, we introduce a new DA strategy that utilizes reinforcement learning (RL) to apply state corrections using full or partial observations of the state variables. Our investigation focuses on demonstrating this approach to the chaotic Lorenz 63 and 96 systems, where the agent's objective is to maximize the geometric series with terms that are proportional to the negative root‐mean‐squared error (RMSE) between the observations and corresponding forecast states. Consequently, the agent develops a correction strategy, enhancing model forecasts based on available observations. Our strategy employs a stochastic action policy, enabling a Monte Carlo‐based DA framework that relies on randomly sampling the policy to generate an ensemble of assimilated realizations. Numerical results demonstrate that the developed RL algorithm performs favorably when compared to the EnKF. Additionally, we illustrate the agent's capability to assimilate non‐Gaussian observations, addressing one of the limitations of the EnKF. Plain Language Summary: Reliable forecasts of the state of chaotic systems, such as environmental flows, require combining observational data and dynamical model outputs through a process called data assimilation. The ensemble Kalman filter (EnKF) is the most commonly adopted algorithm for this task, however, is subject to some limitations when applied to nonlinear/non‐Gaussian systems. Recently, there has been interest in using deep learning (DL), particularly within a supervised learning setup, for DA. However, making DL models work well in new situations that differ from those experienced during training is challenging. In this work, we propose a new DA approach that leverages reinforcement learning (RL). RL helps the system make corrections to its predictions based on incoming observations, even when the model has not been trained for those specific scenarios. Compared to the EnKF framework, RL offers a novel algorithm for nonlinear corrections of the forecasts. Numerical results show that the proposed RL algorithm outperforms the EnKF and demonstrates the RL agent's ability at addressing some shortcomings of the EnKF. Key Points: Deep reinforcement learning (RL) is introduced for data assimilation in application to the Lorenz 63 and 96RL generalizes to new situations unseen during training through actively learning from the data and system dynamicsThe RL agent allows for nonlinear correction of the forecast using the observationsThe performance of the proposed RL algorithm generally surpasses that of the standard ensemble Kalman filter (EnKF) [ABSTRACT FROM AUTHOR]

Calculations about compound interest serve as the basis of most financial decisions; therefore, it is imperative to explore what mathematical knowledge people need to correctly calculate simple and compound interest. The aim of this study is to analyze the relationship between college students' competence in calculating simple and compound interest and their understanding of the arithmetic and geometric progressions. It is also pointed out whether the results vary according to gender. Population proportion tests are carried out, and gender proportion differences are considered for inferential analysis. The dichotomous Probit model was used for correlation analysis. Results demonstrate that 59.8% of students know how to formulate a whole-number succession, and only 30.9% in the case of fractional numbers. Less than 50% of students can calculate compound interest, but 76.7% can calculate simple interest. There is no significant difference between men and women. The results show a positive relationship between male students' competence in calculating compound interest with the possibility to correctly formulate a geometric succession. Findings can be useful for mathematics teaching strategy design and its applications in finance contexts with the purpose of training students to be better at finance decision making. [ABSTRACT FROM AUTHOR]

Although it is very easy to calculate the 1st moment and 2nd moment values of the geometric distribution with the methods available in existing books and other articles, it is quite difficult to calculate moment values larger than the 3rd order. Because in order to find these moment values, many higher order derivatives of the geometric series and convergence properties of the series are needed. The aim of this article is to find new formulas for characteristic function of the geometric random variable (with parameter p) in terms of the Apostol–Bernoulli polynomials and numbers, and the Stirling numbers. This characteristic function characterizes the geometric distribution. Using the Euler's identity, we give relations among this characteristic function, the Apostol–Bernoulli polynomials and numbers, and also trigonometric functions including cosine and sine. A relations between the characteristic function and the moment generating function is also given. By using these relations, we derive new moments formulas in terms of the Apostol–Bernoulli polynomials and numbers. Moreover, we give some applications of our new formulas. [ABSTRACT FROM AUTHOR]

We establish lower estimates for the characteristic determinant of a problem with multipoint conditions in the selected variable with simple interpolation nodes and periodicity conditions with respect to the remaining variables for an Euler-type partial differential equation. We consider particular cases in which either the interpolation nodes form a geometric progression or the real parts of the roots of characteristic equation behave as power functions at infinity. [ABSTRACT FROM AUTHOR]

We present an investigation of the infinite sequences of numbers formed by calculating the pairwise averages of three given numbers. The problem has an interesting geometric interpretation related to the sequence of triangles with equal perimeters which tend to an equilateral triangle. Investigative activities of the problem are carried out in several stages and involve various technological tools and various fields of mathematics. The activities can be useful for the instruction of pre-service and experienced mathematics teachers. [ABSTRACT FROM AUTHOR]

The interest in special complex functions and their wide-ranging implementations in geometric function theory (GFT) has developed tremendously. Recently, subordination theory has been instrumentally employed for special functions to explore their geometric properties. In this effort, by using a convolutional structure, we combine the geometric series, logarithm, and Hurwitz–Lerch zeta functions to formulate a new special function, namely, the logarithm-Hurwitz–Lerch zeta function (LHL-Z function). This investigation then contributes to the study of the LHL-Z function in terms of the geometric theory of holomorphic functions, based on the differential subordination methodology, to discuss and determine the univalence and convexity conditions of the LHL-Z function. Moreover, there are other subordination and superordination connections that may be visually represented using geometric methods. Functions often exhibit symmetry when subjected to conformal mappings. The investigation of the symmetries of these mappings may provide a clearer understanding of how subordination and superordination with the Hurwitz–Lerch zeta function behave under different transformations. [ABSTRACT FROM AUTHOR]

Ensuring the alignment of perspectives between unmanned ground vehicles (UGVs) and Building Information Modeling (BIM) is crucial for the precise retrieval and analysis of BIM-stored information during inspection tasks. However, accumulative localization errors often result in deviations between the viewpoints of UGV cameras and their corresponding representations in BIM at specific waypoints. Therefore, this study introduces a sequential rectification method to correct the UGV's pose within the BIM environment to ensure seamless alignment of perspectives. By leveraging visual features and geometric strategies in sequence, this method correlates the UGV-captured point cloud data with the BIM, thereby deriving accurate and robust pose rectification. Experimental validation in a featureless indoor environment demonstrated that this method reduced the angle and distance error of reference lines in two-dimensional (2D) views to approximately 2° and 7 pixels, respectively, and the root mean square error (RMSE) of three-dimensional (3D) lines to approximately 17 cm. The validation also demonstrated that the proposed method was particularly robust in correcting the pose and improving the alignment of perspectives between BIM and the UGV, even in cases of significant misalignment. Hence, this study improves the reliability of decisions made by UGVs for indoor inspection when cross-referencing with BIM data. Practical Applications: UGVs have been leveraged to automate inspection tasks, thereby reducing human participation. Generally, these UGVs are programmed to perform analyses by comparing the as-built condition with the as-designed BIM. However, for these tasks to accurately reference the data stored in BIM, it is crucial to have a seamless alignment between the UGV's perspectives and the as-designed BIM. Unfortunately, due to the accumulation of localization errors, discrepancies often arise between the perspectives in BIM and UGV cameras at waypoints. Such misalignments can compromise the reliability of decisions made for assigned tasks compared with BIM, such as verifying measurements of elements or confirming equipment installation during the project handover phase. Hence, this study, having demonstrated superior performance and robustness, could provide an effective solution to this alignment challenge. By ensuring seamlessly aligned perspectives, this method improves the reliability of UGV inspection assessments when accessing corresponding data stored in BIM. [ABSTRACT FROM AUTHOR]

In this paper, we consider a geometric circulant matrix with geometric sequence i.e. a geometric circulant matrix whose first row is $ (g,gq,gq^{2},\ldots,gq^{n-1}) $ (g , gq , g q 2 , ... , g q n − 1) , where g is a nonzero complex number and q is a nonzero real number. The determinant, the Euclidean norm and bounds for the spectral norm of such matrix are obtained. Also, in the case when such matrix is singular, we obtain the Moore-Penrose inverse of such matrix and show that the group inverse of such matrix does not exist (in most cases). The obtained results are illustrated by examples. [ABSTRACT FROM AUTHOR]

In this paper, we describe minimal string attractors of pseudopalindromic prefixes of standard complementarysymmetric Rote sequences. Such a class of Rote sequences forms a subclass of binary generalized pseudostandard sequences, i.e., of sequences obtained when iterating palindromic and antipalindromic closures. When iterating only palindromic closure, palindromic prefixes of standard Sturmian sequences are obtained and their string attractors are of size two. However, already when iterating only antipalindromic closure, antipalindromic prefixes of binary pseudostandard sequences are obtained and we prove that the minimal string attractors are of size three in this case. We conjecture that the pseudopalindromic prefixes of any binary generalized pseudostandard sequence have a minimal string attractor of size at most four. [ABSTRACT FROM AUTHOR]

Liquid hydrogen (LH2), a key clean energy carrier, requires precise thermal management, especially for long-term storage and long-distance transport. Accurate prediction of thermal stratification, self-pressurization, and Boil-Off Gas (BOG) generation is considered pivotal for optimizing low-temperature cryogenic storage systems. Most of the literature uses the two-zone model, where the vapor-liquid interface is treated as a simple saturation condition. The influence of interfacial thermal behavior on the calculation of thermal stratification, self-pressurization, and Boil-Off Gas (BOG) generation has not been sufficiently analyzed. To overcome such limitations, in this study, the liquid-vapor interface is modeled by establishing a separate governing equation and optimal geometric progression grid discretization instead of treating it as a simple condition. The Diffusion-based Interface Model (DIM) of the present study transcends traditional methods by eliminating the typical saturation condition at the vapor-liquid interface, providing a deeper insight into the thermal dynamics of the interface. The DIM's findings include a daily BOG rate of 0.14%, aligning with literature, and temperature deviations of 0.008% in liquid and 0.13% in vapor near the interface, indicating a greater vapor phase influence on interfacial thermal behavior. Furthermore, a 0.35% deviation is observed in vapor pressure towards the simulation's end. The DIM can be a complementary tool for engineering works. It enables the design of safe storage of liquid hydrogen. • Novel Diffusion-based Interface Model (DIM) for LH2 storage. • Analyze the interfacial thermal behavior of LH2 storage tank. • DIM improved prediction accuracy of boil-off gas (BOG) generation. [ABSTRACT FROM AUTHOR]

A finite zero-sum game defined on a subset of the unit square is considered. The game is a generalized progressive discrete silent duel, in which the kernel is skew-symmetric, and the players, referred to as duelists, have identical linear accuracy functions featured with an accuracy proportionality factor. As the duel starts, time moments of possible shooting become denser by a geometric progression. Apart from the duel beginning and end time moments, every following time moment is the partial sum of the respective geometric series. Due to the skew-symmetry, both the duelists have the same optimal strategies and the game optimal value is 0. If the accuracy factor is not less than 1, the duelist’s optimal strategy is the middle of the duel time span. If the factor is less than 1, the duel solution is not always a pure strategy saddle point. In a boundary case, when the accuracy factor is equal to the inverse numerator of the ratio that is the time moment preceding the duel end moment, the duel has four pure strategy saddle points which are of the mentioned time moments. For a trivial game, where the duelist possesses just one moment of possible shooting between the duel beginning and end moments, and the accuracy factor is 1, any pure strategy situation, not containing the duel beginning moment, is optimal. [ABSTRACT FROM AUTHOR]

Modern giraffids are nowadays represented by few species; however, this family was quite diverse during the Neogene. Several lineages radiated, showing high morphological disparity in metapodial shapes. During the last few years, not many scholars have focused on this subject. Furthermore, modern statistical approaches have been totally neglected. Herein, we present the first attempt to fill this gap of knowledge adopting a plethora of phylogenetic comparative approaches alongside multivariate statistics. Namely, we performed a series of traditional and geometric morphometric analyses for a total of 207 metapodial specimens of extant and fossil giraffid species. Our results recognized four different clusters, grouped in different morphospaces. Moreover, two of these clusters are clearly outliers in respect to an overall homogeneous distribution. The two outlying clusters include short and robust metapodials observed in Sivatherium, and long and slender metapodials observed in Giraffa and Bohlinia, respectively. Similarities between the modern Okapia and the extinct genus Decennatherium were also detected. This study explores the relations between the morphological adaptations and the ecological roles of different members of the Giraffidae. It also provides insights into how different environmental factors might have influenced the evolution of this group and the adaptation of their members to diverse environments. Furthermore, locomotory adaptations of the most hyper-specialized Giraffidae groups are briefly discussed and investigated. [ABSTRACT FROM AUTHOR]

Even though Zaremba's conjecture remains open, Bourgain and Kontorovich solved the problem for a full density subset. Nevertheless, there are only a handful of explicit sequences known to satisfy the strong version of the conjecture, all of which were obtained using essentially the same algorithm. In this note, we provide a refined algorithm using the folding lemma for continued fractions, which both generalizes and improves on the old one. As a result, we uncover new examples that fulfill the strong version of Zaremba's conjecture. [ABSTRACT FROM AUTHOR]