In a recent work, arxiv:2412.20407, we have mentioned two Egyptian kings, Shepseskaf and Userkaf, of the IV and V Dynasties. Here we continue discussing their burial places, that is their pyramids, in a new framework based on the study of orientation of pyramid substructures (entrances, corridors, burial chambers). Shepseskaf and Userkaf opted for the same architectonic solutions, exhibiting a continuity in funerary architecture. After showing the substructure orientations of the pyramids of the IV and V Dynasties, we will stress that the substructure in the Shepseskaf's pyramid, the Mastabat Fara'un, is oriented as in the Fourth Dynasty architecture of pyramids and that it has a layout which is closer to that used by the following Fifth Dynasty. In our discussion, we will start with the first ruler of the Fourth Dynasty, Snefru, who introduced a new form of external and internal layout of the burial complexes of the king, through an evolution based on four attempts, the Seila, Meidum, Bent and Red pyramids. In the final model, the Snefru's Red pyramid, the substructure has a north entrance and a burial chamber east-west oriented. This orientation persisted in the pyramids of the F
Gromov introduced the notion of a pyramid as a generalization of a metric measure space, based on the idea of the concentration of measure phenomenon. In this paper, we introduce the concept of a direct sum of pyramids, which naturally appears as a limit of a sequence of metric measure spaces whose measures concentrate on finitely or countably many regions, with the distances between these regions diverging to infinity. As one of our main results, we prove that any pyramid admits a unique direct sum decomposition. Moreover, as an application, we establish the method for checking whether a given pyramid is an extended metric measure space.
Pyramids introduced by Gromov are generalized objects of metric spaces with Borel probability measures. We study non-trivial pyramids, where non-trivial means that they are not represented as metric measure spaces. In this paper, we establish general theory of invariants of pyramids and construct several invariants. Using them, we distinguish concrete pyramids. Furthermore, we study a space consisting of non-trivial pyramids and prove that the space have infinite dimension.
Understanding the relationship between population dynamics and disease-specific mortality is central to evidence-based health policy. This study introduces two novel metrics, PoPDivergence and PoPStat, one to quantify the difference between population pyramids and the other to assess the strength and nature of their association with the mortality of a given disease. PoPDivergence, based on Kullback-Leibler divergence, measures deviations between a countrys population pyramid and a reference pyramid. PoPStat is the correlation between these deviations and the log form of disease-specific mortality rates. The reference population is selected by a brute-force optimization that maximizes this correlation. Utilizing mortality data from the Global Burden of Disease 2021 and population statistics from the United Nations, we applied these metrics to 371 diseases across 204 countries. Results reveal that PoPStat outperforms traditional indicators such as median age, GDP per capita, and Human Development Index in explaining the mortality of most diseases. Noncommunicable diseases (NCDs) like neurological disorders and cancers, communicable diseases (CDs) like neglected tropical diseases, and
In this experimental work we study billiard trajectories in triangular pyramids and try to establish conditions that guarantee the existence (or absence) of 4-cycles (there can be not more, than three of them). We formulate conjectures and prove some statements. For example, if a pyramid has two orthogonal faces, then it has not more than two 4-cycles. Also we study 4-cycles of the "physical" billiard in pyramids, i.e. in the presence of gravity. Here we present our observations for a generic case.
Purpose: A geometric simulation of a possible two-plane detector was developed to test the abilities of the detector to generate high-resolution images of the Great Pyramid using muon tomography. Methods and Materials: Trajectory range, angular resolution, and acceptance of the detector were calculated with a simulation. Trajectories and the corresponding sinogram space covered were simulated first with one detector in one location, and then two moving detectors on adjacent sides of the pyramid. The resolution at the center slice of the pyramid was calculated using the angular resolution of the detector. Results: The simulation returned trajectory range encompassing the pyramid and peak angular resolution of .0004sr. Sinogram space covered by one position was inadequate, however two moving detectors on adjacent sides of the pyramid cover a significant portion. Resolution at the center of the pyramid is roughly 3m. Conclusions: The simulation provides a way to calculate the detector positions needed to cover an adequate amount of sinogram space for high-resolution cosmic-ray tomographic reconstruction of the Great Pyramids. Key Words: high-resolution muon tomography, one-sided tomog
For natural numbers $k<n$ we study the graphs $T_{n,k}:=K_{k}\lor\overline{K_{n-k}}$. For $k=1$, $T_{n,1}$ is the star $S_{n-1}$. For $k>1$ we refer to $T_{n,k}$ as a \emph{graph of pyramids}. We prove that the graphs of pyramids are determined by their spectrum, and that a star $S_{n}$ is determined by its spectrum iff $n$ is prime. We also show that the graphs $T_{n,k}$ are completely positive iff $k\le2$.
Due to their intriguing optical properties, including stable and chiral excitons, two-dimensional transition metal dichalcogenides (2D-TMDs) hold the promise of applications in nanophotonics. Chemical vapor deposition (CVD) techniques offer a platform to fabricate and design nanostructures with diverse geometries. However, the more exotic the grown nanogeometry, the less is known about its optical response. WS2 nanostructures with geometries ranging from monolayers to hollow pyramids have been created. The hollow pyramids exhibit a strongly reduced photoluminescence with respect to horizontally layered tungsten disulphide, facilitating the study of their clear Raman signal in more detail. Excited resonantly, the hollow pyramids exhibit a great number of higher-order phononic resonances. In contrast to monolayers, the spectral features of the optical response of the pyramids are position dependent. Differences in peak intensity, peak ratio and spectral peak positions reveal local variations in the atomic arrangement of the hollow pyramids crater and sides. The position-dependent optical response of hollow WS2 pyramids is characterized and attributed to growth-induced nanogeometry. T
What can we learn about a scene by watching it for months or years? A video recorded over a long timespan will depict interesting phenomena at multiple timescales, but identifying and viewing them presents a challenge. The video is too long to watch in full, and some occurrences are too slow to experience in real-time, such as glacial retreat. Timelapse videography is a common approach to summarizing long videos and visualizing slow timescales. However, a timelapse is limited to a single chosen temporal frequency, and often appears flickery due to aliasing and temporal discontinuities between frames. In this paper, we propose Video Temporal Pyramids, a technique that addresses these limitations and expands the possibilities for visualizing the passage of time. Inspired by spatial image pyramids from computer vision, we developed an algorithm that builds video pyramids in the temporal domain. Each level of a Video Temporal Pyramid visualizes a different timescale; for instance, videos from the monthly timescale are usually good for visualizing seasonal changes, while videos from the one-minute timescale are best for visualizing sunrise or the movement of clouds across the sky. To he
For some symmetric pyramids of $R^3$ , we find Galois obstruction for their Dehn invariant to be zero, i.e. for the pyramids to be scissor equivalent to a cube. These conditions are that some associated Kummer extensions of number fields must be abelian. This work can be viewed as a complement to [5]. Indeed, in [5], a classification of all rational tetrahedra is given, while we concentrate our attention on much more symmetric pyramids and prove that the only ones which are scissor equivalent to a cube are the rational ones.
Despite the practicality of quantile regression (QR), simultaneous estimation of multiple QR curves continues to be challenging. We address this problem by proposing a Bayesian nonparametric framework that generalizes the quantile pyramid by replacing each scalar variate in the quantile pyramid with a stochastic process on a covariate space. We propose a novel approach to show the existence of a quantile pyramid for all quantiles. The process of dependent quantile pyramids allows for non-linear QR and automatically ensures non-crossing of QR curves on the covariate space. Simulation studies document the performance and robustness of our approach. An application to cyclone intensity data is presented.
Although the existence of robust inverted biomass pyramids seem paradoxical, they have been observed in planktonic communities, and more recently, in pristine coral reefs. Understanding the underlying mechanisms which produce inverted biomass pyramids provides new ecological insights, and for coral reefs, may help mitigate or restore damaged reefs. We present three classes of predator-prey models which elucidate mechanisms that generate robust inverted biomass pyramids. The first class of models exploits well-mixing of predators and prey, the second class has a refuge (with explicit size) for the prey to hide, and the third class incorporates the immigration of prey. Our models indicate that inverted biomass pyramids occur when the prey growth rate, prey carrying capacity, biomass conversion efficiency, the predator life span, or the immigration rate of prey fish is sufficiently large. In the second class, we discuss three hypotheses on how refuge size can impact the amount of prey available to predators. By explicitly incorporating a refuge size, these can more realistically model predator-prey interactions than refgue models with implicit refuge size.
We consider pyramids made of one-dimensional pieces of fixed integer length a and which may have pairwise overlaps of integer length from 1 to a. We prove that the number of pyramids of size m, i.e. consisting of m pieces, equals (am-1,m-1) for each a >= 2. This generalises a well known result for a = 2. A bijective correspondence between so-called right (or left) pyramids and a-ary trees is pointed out, and it is shown that asymptotically the average width of pyramids is proportional to the square root of the size.
In the same way as tree rings give us useful information about the climate many decades ago (or even centuries ago in the case of big trees), population pyramids allow us to know birth or death rates several decades earlier. Naturally, they can fulfill such promises only if they have been recorded accurately. That is why we start this study by listing a number of pitfalls that may occur in the process of taking censuses. In this paper our main goal is to show how it is possible to "read" population pyramids. Sudden changes in birth rate give rise to the clearest signatures. This indicator reveals that major tragedies like famines, wars, or epidemics are generally accompanied by a fall in birth rates. Thanks to this observation, population pyramids can be used to identify and gauge the blows suffered by populations and nations. As an illustration of how this works, we compare the population pyramids of North and South Korea. This allows us, among other things, to gauge the scale of the food shortage that North Korea experienced at the end of the 1990s.
We present a new approach combining top down fabrication and bottom up overgrowth to create diamond photonic nanostructures in form of single-crystalline diamond nanopyramids. Our approach relies on diamond nanopillars, that are overgrown with single-crystalline diamond to form pyramidal structures oriented along crystal facets. To characterize the photonic properties of the pyramids, color centers are created in a controlled way using ion implantation and annealing. We find very high collection efficiency from color centers close to the pyramid apex. We further show excellent smoothness and sharpness of our diamond pyramids with measured tip radii on the order of 10 nm. Our results offer interesting prospects for nanoscale quantum sensing using diamond color centers, where our diamond pyramids could be used as scanning probes for nanoscale imaging. There, our approach would offer signifikant advantages compared to the cone-shaped scanning probes which define the current state of the art.
In this research paper, authors propose multimodal brain image registration using discrete wavelet transform(DWT) followed by Gaussian pyramids. The reference and target images are decomposed into their LL, LH, HL and LL DWT coefficients and then are processed for image registration using Gaussian pyramids. The image registration is also done using Gaussian pyramids only and wavelets transforms only for comparison. The quality of registration is measured by comparing the maximum MI values used by the three methods and also by comparing their correlation coefficients. Our proposed technique proves to show better results when compared with the other two methods.
We demonstrate gold coated polymer surface enhanced Raman scattering (SERS) substrates with a pair of complementary structures--positive and inverted pyramids array structures fabricated by multiple-step molding and replication process. The uniform SERS enhancement factors over the entire device surfaces were measured as 7.2 X 10^4 for positive pyramids substrate while 1.6 X 10^6 for inverted pyramids substrate with Rhodamine 6G as the target analyte. Based on the optical reflection measurement and FDTD simulation result, the enhancement factor difference is attributed to plasmon resonance matching and to SERS "hot spots ". With this simple, fast and versatile complementary molding process, we can produce polymer SERS substrates with extremely low cost, high throughput and high repeatability.
Ever since Flinders Petrie undertook a theodolite survey on the Giza plateau in 1881 and drew attention to the extraordinary degree of precision with which the three colossal pyramids are oriented upon the four cardinal directions, there have been a great many suggestions as to how this was achieved and why it was of importance. Surprisingly, given the many astronomical hypotheses and speculations that have been offered in the intervening 130 years, there have been remarkably few attempts to reaffirm or improve on the basic survey data concerning the primary orientations. This paper presents the results of a week-long Total Station survey undertaken by the authors during December 2006 whose principal aim was to clarify the basic data concerning the orientation of each side of the three large pyramids and to determine, as accurately as possible, the orientations of as many as possible of the associated structures. The principal difference between this and all previous surveys is that it focuses upon measurements of sequences of points along multiple straight and relatively well preserved structural segments, with best-fit techniques being used to provide the best estimate of their o
Standing on the shoulders of giants, we resolve the quantum pyramids conjecture, confirming the globally information-optimal measurement for an ensemble of equiangular equiprobable pure states, as conjectured by Englert and Řeháček (arXiv:0905.0510). We do so by proving the remaining entropy inequalities of Holevo and Utkin (arXiv:2506.06700), which certify optimality for obtuse and flat pyramids. For obtuse pyramids, our key contribution is a rigorous proof that local minimizers of the corresponding entropy inequality cannot have three distinct coordinate values. We show that eliminating this family can be reduced to a neat algebraic reciprocal inequality relating branches of the Lambert $W$ function, which may be of independent interest. For flat pyramids, we prove a tight $\ell^p$ inequality for zero-sum vectors that was recently conjectured, proved analytically in dimension $d=3$, and computationally verified for $d\leq 200$ by Holevo and Utkin (arXiv:2603.24017). We prove this bound for all $d\geq 2$ via a technique in symmetric inequalities known as the equal variables method.
Image pyramids are commonly used in modern computer vision tasks to obtain multi-scale features for precise understanding of images. However, image pyramids process multiple resolutions of images using the same large-scale model, which requires significant computational cost. To overcome this issue, we propose a novel network architecture known as the Parameter-Inverted Image Pyramid Networks (PIIP). Our core idea is to use models with different parameter sizes to process different resolution levels of the image pyramid, thereby balancing computational efficiency and performance. Specifically, the input to PIIP is a set of multi-scale images, where higher resolution images are processed by smaller networks. We further propose a feature interaction mechanism to allow features of different resolutions to complement each other and effectively integrate information from different spatial scales. Extensive experiments demonstrate that the PIIP achieves superior performance in tasks such as object detection, segmentation, and image classification, compared to traditional image pyramid methods and single-branch networks, while reducing computational cost. Notably, when applying our method