Symmetric Tree Research Articles

The branching number of intermediate growth trees

We introduce an “intermediate branching number” (IBN) which captures the branching of intermediate growth trees, similar in spirit to the well-studied branching number of exponential growth trees. We show that the IBN is the critical threshold for several random processes on trees, and analyze the IBN on some examples of interest. Our main result is an algorithm to find spherically symmetric trees with large IBN inside some permutation wreath products. We demonstrate the usefulness of these trees to the study of intermediate growth groups by using them to get the first tight bounds for the firefighter problem on some intermediate growth groups.

A Likelihood Ratio Test for Hybridization Under the Multispecies Coalescent

Several methods have been developed to carry out a statistical test for hybridization at the species level, including the ABBA-BABA test and HyDe. Here, we propose a new method for detecting hybridization and quantifying the extent of hybridization. Our test computes the likelihood of a species tree that is possibly subject to hybridization using site pattern frequencies from genomic-scale datasets under the multispecies coalescent. To do this, we extend the calculation of the likelihood for site pattern frequency data for the 4-taxon symmetric and asymmetric species trees proposed in Chifman and Kubatko (2015) by incorporating an inheritance parameter, resulting in efficient computation of the likelihood under a scenario of hybridization. We use this likelihood computation to construct a likelihood ratio test that a given species is a hybrid of two parental species. Simulations demonstrate that our test is more powerful than existing tests of hybridization, including HyDe, and that it achieves the desired type I error rate. We apply the method to two empirical data sets, one for which hybridization is believed to have occurred and one for which previous methods have failed to detect hybridization.

SMARTboost Learning for Tabular Data

We introduce SMARTboost (boosting of symmetric smooth additive regression trees), an extension of gradient boosting machines with improved accuracy when the underlying function is smooth or the sample small or noisy. In extensive simulations, we find that the combination of smooth symmetric trees and of carefully designed priors gives SMARTboost a large edge (in comparison with XGBoost and BART) on data generated by the most common parametric models in econometrics, and on a variety of other smooth functions. XGBoost outperforms SMARTboost only when the sample is large, and the underlying function is highly discontinuous. SMARTboost’s performance is illustrated in two applications to global equity returns and realized volatility prediction.

Predictive slope stability early warning model based on CatBoost

A model for predicting slope stability is developed using Categorical Boosting (CatBoost), which incorporates 6 slope features to characterize the state of slope stability. The model is trained using a symmetric tree as the base model, utilizing ordered boosting to replace gradient estimation, which enhances prediction accuracy. Comparative models including Support Vector Machine (SVM), Light Gradient Boosting Machine (LGBM), Random Forest (RF), and Logistic Regression (LR) were introduced. Five performance evaluation metrics are utilized to assess the predictive capabilities of the CatBoost model. Based on CatBoost model, the predicted probability of slope instability is calculated, and the early warning model of slope instability is further established. The results suggest that the CatBoost model demonstrates a 6.25% disparity in accuracy between the training and testing sets, achieving a precision of 100% and an Area Under Curve (AUC) value of 0.95. This indicates a high level of predictive accuracy and robust ordering capabilities, effectively mitigating the problem of overfitting. The slope instability warning model offers reasonable classifications for warning levels, providing valuable insights for both research and practical applications in the prediction of slope stability and instability warning.

An example related to Whitney’s extension problem for L 2,p (R2) when 1 < p < 2

Abstract In this paper, we prove the existence of a bounded linear extension operator T : L 2 , p ( E ) → L 2 , p ( R 2 ) $T:{L}^{2,p}\left(E\right)\to {L}^{2,p}\left({\mathbb{R}}^{2}\right)$ when 1 < p < 2, where E ⊂ R 2 $E\subset {\mathbb{R}}^{2}$ is a certain discrete set with fractal structure. Our proof makes use of a theorem of Fefferman–Klartag (“Linear extension operators for Sobolev spaces on radially symmetric binary trees,” Adv. Nonlinear Stud., vol. 23, no. 1, p. 20220075, 2023) on the existence of linear extension operators for radially symmetric binary trees.

Three-Way Alignment Improves Multiple Sequence Alignment of Highly Diverged Sequences

The standard approach for constructing a phylogenetic tree from a set of sequences consists of two key stages. First, a multiple sequence alignment (MSA) of the sequences is computed. The aligned data are then used to reconstruct the phylogenetic tree. The accuracy of the resulting tree heavily relies on the quality of the MSA. The quality of the popularly used progressive sequence alignment depends on a guide tree, which determines the order of aligning sequences. Most MSA methods use pairwise comparisons to generate a distance matrix and reconstruct the guide tree. However, when dealing with highly diverged sequences, constructing a good guide tree is challenging. In this work, we propose an alternative approach using three-way dynamic programming alignment to generate the distance matrix and the guide tree. This three-way alignment incorporates information from additional sequences to compute evolutionary distances more accurately. Using simulated datasets on two symmetric and asymmetric trees, we compared MAFFT with its default guide tree with MAFFT with a guide tree produced using the three-way alignment. We found that (1) the three-way alignment can reconstruct better guide trees than those from the most accurate options of MAFFT, and (2) the better guide tree, on average, leads to more accurate phylogenetic reconstruction. However, the improvement over the L-INS-i option of MAFFT is small, attesting to the excellence of the alignment quality of MAFFT. Surprisingly, the two criteria for choosing the best MSA (phylogenetic accuracy and sum-of-pair score) conflict with each other.

Topology-aware Parallel Joins

We study the design and analysis of parallel join algorithms in a topology-aware computational model. In this model, the network is modeled as a directed graph, where each edge is associated with a cost function that depends on the data transferred between the two endpoints and the link bandwidth. The computation proceeds in synchronous rounds and the cost of each round is measured as the maximum cost over all the edges in the network. Our main result is an asymptotically optimal join algorithm over symmetric tree topologies. The algorithm generalizes prior topology-aware protocols for set intersection and cartesian product to a binary join over an arbitrary input distribution with possible data skew.

Some symmetric sign patterns requiring unique inertia

A sign pattern is a matrix whose entries are from the set $\{+,-,0\}$. A sign pattern requires unique inertia if every matrix in its qualitative class has the same inertia. For symmetric tree sign patterns, several necessary and sufficient conditions to require unique inertia are known. In this paper, sufficient conditions for symmetric tree sign patterns to require unique inertia based on the sign and position of the loops in the underlying graph are given. Further, some sufficient conditions for a symmetric sign pattern to require unique inertia if the underlying graph contains cycles are determined.

A lumped parameter model for evaluating coronary artery blood supply capacity.

The coronary artery constitutes a vital vascular system that sustains cardiac function, with its primary role being the conveyance of indispensable nutrients to the myocardial tissue. When coronary artery disease occurs, it will affect the blood supply of the heart and induce myocardial ischemia. Therefore, it is of great significance to numerically simulate the coronary artery and evaluate its blood supply capacity. In this article, the coronary artery lumped parameter model was derived based on the relationship between circuit system parameters and cardiovascular system parameters, and the blood supply capacity of the coronary artery in healthy and stenosis states was studied. The aortic root pressure calculated by the aortic valve fluid-structure interaction (AV FSI) simulator was employed as the inlet boundary condition. To emulate the physiological phenomenon of sudden pressure drops resulting from an abrupt reduction in blood vessel radius, a head loss model was connected at the coronary artery's entrance. For each coronary artery outlet, the symmetric structured tree model was appended to simulate the terminal impedance of the missing downstream coronary arteries. The particle swarm optimization (PSO) algorithm was used to optimize the blood flow viscous resistance, blood flow inertia, and vascular compliance of the coronary artery model. In the stenosis states, the relative flow and fractional flow reserve (FFR) calculated by numerical simulation corresponded to the published literature data. It was anticipated that the proposed model can be readily adapted for clinical application, serving as a valuable reference for diagnosing and treating patients.

On the distinguishing chromatic number of the Kronecker products of graphs

In this paper, we investigate the distinguishing chromatic number of Kronecker product of paths, cycles, star graphs, symmetric trees, almost symmetric trees, and bisymmetric trees.

Generalizing multi-branching radial symmetric flow structures

This study revisits the design of tree-shaped flow networks in disc-shaped bodies, focusing on thermal management applications. The key objectives are to achieve equal point distribution along the disk periphery and minimize flow resistance at each bifurcation point. The literature highlights the challenge of balancing fluid and thermal performance simultaneously. To address this, the authors propose a multi-branching approach and hydraulic diameter setting, allowing flexibility in network growth.The optimization process involves minimizing flow resistance as the objective function, employing a constrained minimization technique with an interior point algorithm. By solving this optimization problem, the geometric features of the network are determined. Notably, the research uncovers a memory effect in the geometry and overcomes computational time limitations. The Svelteness, defined using the concept of configuration area, aids in distinguishing between different design possibilities and identifying the evolutionary direction.Increasing the complexity of the network while minimizing the total branch length leads to the discovery of the optimal architecture configuration. These findings provide valuable insights for optimizing symmetric tree designs, improving thermal and hydraulic performance.

Cutoff on Trees is Rare

We study the simple random walk on trees and give estimates on the mixing and relaxation times. Relying on a seminal result by Basu, Hermon and Peres characterizing cutoff on trees, we give geometric criteria that are easy to verify and allow to determine whether the cutoff phenomenon occurs. We provide a general characterization of families of trees with cutoff, and show how our criteria can be used to prove the absence of cutoff for several classes of trees, including spherically symmetric trees, Galton–Watson trees of a fixed height, and sequences of random trees converging to the Brownian continuum random tree.

Symmetric Key generation And Tree Construction in Cryptosystem based on Pythagorean and Reciprocal Pythagorean Triples

Key generation and Secure are critical to the security of a Cryptosystem. Key generation and key exchange is the most challenging part of cryptography. In this paper, a scheme for symmetric key generation based on Pythagorean and Reciprocal Pythagorean triple has been presented. The proposed scheme incorporates a Key Distribution Centre (KDC) for user authentication and the secure exchange of secret information to generate keys. The KDC operation involves a request from a user for initiation. The KDC authenticates authenticate and secures the exchange of secret information to generate keys. The KDC authenticates the initiator. If the authentication is successful, KDC generates and sends an encrypted timestamp to both the initiator and responder. The proposed system is based on a novel mechanism to determine Pythagorean and Reciprocal Pythagorean triples to generate keys.

Linear extension operators for Sobolev spaces on radially symmetric binary trees

Abstract Let 1 < p < ∞ 1\lt p\lt \infty and suppose that we are given a function f f defined on the leaves of a weighted tree. We would like to extend f f to a function F F defined on the entire tree, so as to minimize the weighted W 1 , p {W}^{1,p} -Sobolev norm of the extension. An easy situation is when p = 2 p=2 , where the harmonic extension operator provides such a function F F . In this note, we record our analysis of the particular case of a radially symmetric binary tree, which is a complete, finite, binary tree with weights that depend only on the distance from the root. Neither the averaging operator nor the harmonic extension operator work here in general. Nevertheless, we prove the existence of a linear extension operator whose norm is bounded by a constant depending solely on p p . This operator is a variant of the standard harmonic extension operator, and in fact, it is harmonic extension with respect to a certain Markov kernel determined by p p and by the weights.

A multiscale framework for defining homeostasis in distal vascular trees: applications to the pulmonary circulation.

Pulmonary arteries constitute a low-pressure network of vessels, often characterized as a bifurcating tree with heterogeneous vessel mechanics. Understanding the vascular complexity and establishing homeostasis is important to study diseases such as pulmonary arterial hypertension (PAH). The onset and early progression of PAH can be traced to changes in the morphometry and structure of the distal vasculature. Coupling hemodynamics with vessel wall growth and remodeling (G&R) is crucial for understanding pathology at distal vasculature. Accordingly, the goal of this study is to provide a multiscale modeling framework that embeds the essential features of arterial wall constituents coupled with the hemodynamics within an arterial network characterized by an extension of Murray's law. This framework will be used to establish the homeostatic baseline characteristics of a pulmonary arterial tree, including important parameters such as vessel radius, wall thickness and shear stress. To define the vascular homeostasis and hemodynamics in the tree, we consider two timescales: a cardiac cycle and a longer period of vascular adaptations. An iterative homeostatic optimization, which integrates a metabolic cost function minimization, the stress equilibrium, and hemodynamics, is performed at the slow timescale. In the fast timescale, the pulsatile blood flow dynamics is described by a Womersley's deformable wall analytical solution. Illustrative examples for symmetric and asymmetric trees are presented that provide baseline characteristics for the normal pulmonary arterial vasculature. The results are compared with diverse literature data on morphometry, structure, and mechanics of pulmonary arteries. The developed framework demonstrates a potential for advanced parametric studies and future G&R and hemodynamics modeling of PAH.

The generalized entropy ergodic theorem for Markov chains indexed by a spherically symmetric tree

In information theory, the entropy ergodic theorem is a fundamental one well-known to all. In this article, we will study the generalized entropy ergodic theorem for Markov chains indexed by a spherically symmetric tree. First, we give the definition of Markov chains indexed by a spherically symmetric tree. Meanwhile, the generalized strong law of large numbers for Markov chains indexed by a spherically symmetric tree is proved. Finally, we obtain the generalized entropy ergodic theorem for Markov chains indexed by a spherically symmetric tree.

The Origins of the Buckeye Nickname

The Origins of the Buckeye Nickname Raymond D. Irwin (bio) Others have called Ohioans—and Ohioans have called themselves—"buckeyes" for nearly two centuries, but the origins of that appellation have been the subject of limited thoughtful debate, unfounded assertions, and uncritically accepted assumptions. Three related issues frequently appear in the literature regarding the buckeye nickname: the length of time the title has been in use, where it comes from, and how it became affixed to human beings. Although many have asserted that the moniker has been commonly applied to persons since the 1780s, that it was derived only tangentially from the tree of the same name, and that it owes its existence to a quaint tale involving friendly indigenous persons, the evidence suggests otherwise. The real story involves the changing image of the buckeye tree, and thus, the altered status and broad acceptability of the word "buckeye," which only occurred in the early nineteenth century. One of the most common explanations for Ohioans' bearing the nickname "buckeye" comes from antiquarian and physician Samuel Prescott Hildreth in his 1852 description of the opening of the first judicial body in the Northwest Territory on September 2, 1788, in Marietta. There, the leader of the ceremonial procession for the local court of common pleas was the community's sheriff, Col. Ebenezer Sproat, "a man six feet and four inches high, and large in proportion," who carried "a drawn sword in his right hand, and wand of office in the left." His "imposing appearance" elicited "the admiration of the friendly savages, a number of whom were loitering about the new city." The "tall, commanding" Sproat, having attracted rapt attention, earned from natives "the name of Hetuck, or Big [End Page 9] Buckeye." "From this," Hildreth confidently asserted, "originated the name of Buckeye, now applied to the natives of Ohio, as the phrase was familiar to all the early settlers of Marietta."1 In relatively short order, Hildreth's account became a standard origin story for the buckeye sobriquet. The 1857 edition of The American Biographical Dictionary repeated the tale in the conclusion of its entry for Sproat,2 and a decade and a half later, the hetuck narrative appeared nearly verbatim in the editorial notes for the Historical and Philosophical Society of Ohio edition of the journal and letters of early Marietta settler Col. John May.3 The 1881 History of Washington County speculated that the byname "buckeye" came from one of three possibilities, all related to the founding of Marietta: a woodcutting contest among those who had landed at the mouth of the Muskingum and who had felled several buckeye trees; from the flowers of the tree that would have been in bloom in the spring of 1788; or from the Sproat story. The authors cited unnamed pioneers' explanations that "agree with, rather than contradict, Dr. Hildreth's statement," which put Sproat and the indigenous observers at the center of the narrative.4 Four years later Harper's Magazine repeated both the woodchopping and "hetuck" stories, adding the detail that Delaware tribesmen compared Sproat "with the stately, symmetrical tree which grows by the Western water-courses."5 As the nineteenth century came to a close, Hildreth's narrative flourished, bolstered by often inexplicable and usually uncited details. In his centennial history of the Marietta settlement, William H. Venable offered a brief version, noting that Sproat's nickname "was extended to other white men, and finally to Ohioans in general," but that "it is certain that the term Buckeye became attached to Ohio and its citizens at a very early period in the history of the [End Page 10] Northwestern Territory."6 In his centennial address before the Ohio Archaeological and Historical Society, William Farrar repeated the Hildreth story, declaring that the use of "buckeye" to describe Colonel Sproat "was certainly its first known application to an individual in the sense now used," but Farrar immediately conceded that "there is no evidence that the name continued to be so used and applied from that time forward, or that it became a fixed and accepted sobriquet of the State and people until more than a half a century afterwards."7 Others were less critical...

An accurate valvular heart disorders detection model based on a new dual symmetric tree pattern using stethoscope sounds

Background and purposeValvular heart disease (VHD) is an important cause of morbidity and mortality. Echocardiography is the reference standard for VHD diagnosis but is not universally accessible. Manual cardiac auscultation is inadequate for screening VHD. Many machine learning models using heart sounds acquired with an electronic stethoscope may improve the accuracy of VHD diagnosis. We aimed to develop an accurate sound classification model for VHD diagnosis. Materials and methodsA new large stethoscope sound dataset containing 10,366 heart sounds divided into ten categories (nine VHD and one healthy) were prospectively collected. We developed a handcrafted learning model that comprised multilevel feature extraction based on a dual symmetric tree pattern (DSTP) and multilevel discrete wavelet transform (DWT), feature selection, and classification. The multilevel DWT was used to create subbands to extract features at both high and low levels. Then, iterative neighborhood component analysis was used to select the most discriminative 512 features from among the extracted features in the generated feature vector. In the classification phase, a support vector machine (SVM) was used with 10-fold cross-validation (CV) and leave-one-subject-out (LOSO) CV. ResultsOur proposed DSTP-based model attained 99.58% and 99.84% classification accuracies using SVM classifier with 10-fold CV and LOSO CV, respectively. ConclusionsThe presented DSTP-based classification model attained excellent multiclass classification performance on a large prospective heart sound dataset at a low computational cost.

Symmetric Binary Tree Based Co-occurrence Texture Pattern Mining for Fine-grained Plant Leaf Image Retrieval

Leaf image patterns have been actively researched for plant species recognition. However, as a very challenging fine-grained pattern identification issue, cultivar recognition in which the leaf image patterns usually have very subtle difference among cultivars has not yet received considerable attention in computer vision and pattern recognition community. In this paper, a novel symmetric geometric configuration, named Symmetric Binary Tree (SBT) which has multiple symmetric branch pairs and can change in size, is designed to mine the multiple scale co-occurrence texture patterns. The resulting SBT descriptors encode both shape and texture features which make them more informative than the existing individual descriptors and co-occurrence features. A novel feature fusion scheme, named K-NN Based Handcrafted and Deep Features Fusion (KNN-HDFF) that encodes the neighbouring information of distance measure, is proposed for further boosting the retrieval performance. Extensive experiments conducted on the challenging soybean cultivar leaf image dataset and peanut cultivar leaf image dataset consistently indicate the superiority of the proposed method over the state-of-the-art methods on fine-grained leaf image retrieval. We also conduct extensive experiments of feature fusions using the proposed KNN-HDFF on the benchmark datasets and the experimental results prove its potential for improving the performance of cultivar identification which also indicates that fusing handcrafted and deep features may be the direction to address the challenging fine-grained image recognition problem.

A novel approach to quantify ventilation heterogeneity in occluded bronchial tree based on lung admittance

Obstructions in airways result in significant alterations in ventilation distribution and consequently reduce the ventilation to perfusion ratio, affecting gas exchange. This study presents a lumped parameter-based model to quantify the spatial ventilation distribution using constructal theory. An extension of the existing theory is made for the conductive bronchial tree and is represented in matrix frame incorporated with airway admittances. The proposed lung admittance model has a greater advantage over the existing methodologies based on lung impedance, as it can be applicable for both fully and partially blocked regions. We proved the well-posedness of the problem, and the generated matrix is highly sparse in nature. A modified block decomposition method is implemented for symmetric and asymmetric trees of various obstructions 0:20:100% to reduce the memory size. The asymmetry is considered in every left branch of the bronchial tree recursively, following the mathematical relations: Li, 2j=ΓLi, 2j+1 and Di, 2j=ΓDi, 2j+1, where L and D are the length, diameter of the jth branch at ith generation, respectively, for Γ∈0.9:0.01:1.0. It is observed that relative flow rate (Qi,jQi,jhealthy) decreases exponentially with the generation index. In tidal breathing, the regional ventilation pattern is found to vary spatially instead of spatio-temporally. The comparison of our result with the clinical data is found to be accurate when 40% or more obstruction is considered in the proximal region (observed in asthma). Moreover, this predicts an increment of lung impedance by 6%, which can be used for further improvement of clinical observations.