Vector Length Research Articles

On semi-orthogonal matrices with row vectors of equal lengths

When does a rectangular matrix with an orthonormal set of column vectors have row vectors of equal lengths? The column spaces of such matrices are multidimensional generalizations of the projection plane used in isometric perspective. We show that in the absence of unexpected linear relations, any rectangular matrix can be row-scaled so that if we were to orthonormalize the column vectors, the row vectors would attain equal lengths in the process. We use Grassmann coordinates to reduce the question into an instance of the famous matrix scaling problem, and with the help of existing theory introduce simple numerical solutions.

Schrödinger connections: from mathematical foundations towards Yano–Schrödinger cosmology

Abstract Schrödinger connections are a special class of affine connections, which despite being metric incompatible, preserve length of vectors under autoparallel transport. In the present paper, we introduce a novel coordinate-free formulation of Schrödinger connections. After recasting their basic properties in the language of differential geometry, we show that Schrödinger connections can be realized through torsion, non-metricity, or both. We then calculate the curvature tensors of Yano–Schrödinger geometry and present the first explicit example of a non-static Einstein manifold with torsion. We generalize the Raychaudhuri and Sachs equations to the Schrödinger geometry. The length-preserving property of these connections enables us to construct a Lagrangian formulation of the Sachs equation. We also obtain an equation for cosmological distances. After this geometric analysis, we build gravitational theories based on Yano–Schrödinger geometry, using both a metric and a metric-affine approach. For the latter, we introduce a novel cosmological hyperfluid that will source the Schrödinger geometry. Finally, we construct simple cosmological models within these theories and compare our results with observational data as well as the ΛCDM model.

A Genetic Algorithm to Search the Space of Ehrhart h*-Vectors

We describe a genetic algorithm to find candidates for h * -vectors satisfying given properties in the space of integers vectors of finite length. We use an implementation of such algorithm to find a 52-dimensional lattice polytope having a non-unimodal h * -vector which is the Cartesian product of two lattice polytopes having unimodal h * -vectors. This counterexample answers negatively to a question by Ferroni and Higashitani.

Rubber-Based Agroforestry Ecosystems Enhance Soil Enzyme Activity but Exacerbate Microbial Nutrient Limitations

Agroforestry ecosystems are an efficient strategy for enhancing soil nutrient conditions and sustainable agricultural development. Soil extracellular enzymes (EEAs) are important drivers of biogeochemical processes. However, changes in EEAs and chemometrics in rubber-based agroforestry systems and their mechanisms of action are still not fully understood. Distribution of EEAs, enzymatic stoichiometry, and microbial nutrient limitation characteristics of rubber plantations under seven planting patterns (RM, rubber monoculture system; AOM, Hevea brasiliensis-Alpinia oxyphylla Miq; PAR, Hevea brasiliensis-Pandanus amaryllifolius Roxb; AKH, Hevea brasiliensis-Alpinia katsumadai Hayata; CAA, Hevea brasiliensis-Coffea Arabica; CCA, Hevea brasiliensis-Cinnamomum cassia (L.) D. Don, and TCA, Hevea brasiliensis-Theobroma Cacao) were analyzed to investigate the metabolic limitations of microorganisms and to identify the primary determinants that restrict nutrient limitation. Compared with rubber monoculture systems, agroforestry ecosystems show increased carbon (C)-acquiring enzyme (EEAC), nitrogen (N)-acquiring enzyme (EEAN), and phosphorus (P)-acquiring enzyme (EEAP) activities. The ecoenzymatic stoichiometry model demonstrated that all seven plantation patterns experienced C and N limitation. Compared to the rubber monoculture system, all agroforestry systems exacerbated the microbial limitations of C and N by reducing the vector angle and increasing vector length. P limitation was not detected in any plantation pattern. In agroforestry systems, progression from herbs to shrubs to trees through intercropping results in a reduction in soil microbial nutrient constraints. This is primarily because of the accumulation of litter and root biomass in tree-based systems, which enhances the soil nutrient content (e.g., soil organic carbon, total nitrogen, total phosphorus, and ammonium nitrogen) and accessibility. Conversely, as soil depth increased, microbial nutrient limitations tended to become more pronounced. Partial least squares path modelling (PLS-PM) indicated that nutrient ratios and soil total nutrient content were the most important factors influencing microbial C limitation (−0.46 and 0.40) and N limitation (−0.30 and −0.42). This study presented novel evidence regarding the constraints and drivers of soil microbial metabolism in rubber agroforestry systems. Considering the constraints of soil nutrients and microbial metabolism, intercropping of rubber trees with arboreal species is recommended over that of herbaceous species to better suit the soil environment of rubber plantation areas on Hainan Island.

Association of Changes in Vector Length with Changes in Right Ventricular Magnetic Resonance Imaging Indices among Patients on Maintenance Hemodialysis

Association of Changes in Vector Length with Changes in Right Ventricular Magnetic Resonance Imaging Indices among Patients on Maintenance Hemodialysis

Some remarks on spin-orbits of unit vectors

For n∈N and a commutative ring R with 2∈R×, the group SLn(R) acts on the set Umn(R) of unimodular vectors of length n and Spin2n(R) acts on the set of unit vectors U2n−1(R). We give an example of a ring for which the comparison map Umn(R)/SLn(R)→U2n−1(R)/Spin2n(R) fails to be bijective.

Multivariable generalizations of bivariate means via invariance

AbstractFor a given p-variable mean $$M :I^p \rightarrow I$$ M : I p → I (I is a subinterval of $${\mathbb {R}}$$ R ), following (Horwitz in J Math Anal Appl 270(2):499–518, 2002) and (Lawson and Lim in Colloq Math 113(2):191–221, 2008), we can define (under certain assumptions) its $$(p+1)$$ ( p + 1 ) -variable $$\beta $$ β -invariant extension as the unique solution $$K :I^{p+1} \rightarrow I$$ K : I p + 1 → I of the functional equation $$\begin{aligned}&K\big (M(x_2,\dots ,x_{p+1}),M(x_1,x_3,\dots ,x_{p+1}),\dots ,M(x_1,\dots ,x_p)\big )\\&\quad =K(x_1,\dots ,x_{p+1}), \text { for all }x_1,\dots ,x_{p+1} \in I \end{aligned}$$ K ( M ( x 2 , ⋯ , x p + 1 ) , M ( x 1 , x 3 , ⋯ , x p + 1 ) , ⋯ , M ( x 1 , ⋯ , x p ) ) = K ( x 1 , ⋯ , x p + 1 ) , for all x 1 , ⋯ , x p + 1 ∈ I in the family of means. Applying this procedure iteratively we can obtain a mean which is defined for vectors of arbitrary lengths starting from the bivariate one. The aim of this paper is to study the properties of such extensions.

Power quality disturbances categorization using Identity Feature Vector and Extreme Learning Machine

Power quality disturbances are variations or anomalies in the voltage, current, or frequency of electrical power that can affect the proper operation of electrical equipment. These disturbances are usually classified into different categories based on their attributes and effects. This article presents an intelligent technique based on an Identity Feature Vector and an Extreme Learning Machine (ELM). This study first derives a constant length vector for each disturbance signal. A wavelet transform is applied to derive attributes from the input disturbance signal, and the identity vector is formed using the approximation coefficients. After the required normalization procedures, the normalized identity vector is classified using an ELM. To assess the productivity of the suggested approach, 12 types of disturbances, single and combined, are generated, and the system's efficiency is studied. The results indicate that ten out of 12 combinations, including Harmonic, Sag, and Flicker, were detected with 100% accuracy. Additionally, the combination "Harmonic + Swell" exhibited the lowest accuracy, identified with 98% accuracy. The total average accuracy of this method is 99.75%. The outcomes demonstrate the highly favorable performance of this approach. This study evaluated the analyzed algorithm under noisy conditions with three different noise levels: 30 dB, 40 dB, and 50 dB, respectively. The average prediction accuracy for these three noise levels is 99.16%, 99.25%, and 98.91%. The outcomes demonstrate that the evaluated algorithm accurately detects power quality disturbances across various noisy conditions.

Enzymatic Stoichiometry Reveals the Metabolic Limitations of Soil Microbes under Nitrogen and Phosphorus Addition in Chinese Fir Plantations.

Increasing nitrogen (N) deposition alters the availability of soil nutrients and is likely to intensify phosphorus (P) limitations, especially in P-limited tropical and subtropical forests. Soil microorganisms play vital roles in carbon (C) and nutrient cycling, but it is unclear whether and how much N and P imbalances affect the soil's microbial metabolism and mechanisms of nutrient limitations. In this study, a 3-year field experiment of N and P addition (control (CK), 100 kg N ha-1 yr-1 (N), 50 kg P ha-1 yr-1 (P), and NP) was set up to analyze the extracellular enzyme activities and stoichiometry characteristics of the top mineral soils in Chinese fir plantations with different stand ages (7, 20, and 33 years old). The results showed that the enzyme activities associated with the acquisition of C (β-1,4-glucosidase (BG) and β-d-cellobiohydrolase (CBH)) and P (acid phosphatases (APs)) in the N treatment were significantly higher than those in the CK treatment. Moreover, vector analysis revealed that both the vector's length and angle increased in stands of all ages, which indicated that N addition aggravated microbial C and P limitations. The P and NP treatments both significantly decreased the activity of AP and the enzymes' N:P ratio, thereby alleviating microbial P limitations, as revealed by the reduction in the vector's angle. Stand age was found to promote all enzymatic activities but had no obvious effects on the limitation of microbial metabolism with or without added nutrients in the soils under Chinese fir. Available N, Olsen-P, and pH were the main drivers of microbial metabolic limitations related to C nutrients. These results provide useful data for understanding the change in soil microbial activity in response to environmental changes, and suggest that P fertilization should be considered for management to improve productivity and C sequestration in Chinese fir plantation in the context of increased deposition of N.

Effects of 40 Years of Fertilizer Application on the Characteristics of Soil Enzymatic Stoichiometry in Black Soil

Microorganisms produce extracellular enzymes to meet elemental requirements and cope with stoichiometric imbalances of resources. To gain insights into the cycling of C, N, and P, the activities of the C∶N∶P acquisition enzymes have been extensively investigated. To detect the effects of long-term fertilization practices on soil nutrient balance and characteristics of soil enzymatic stoichiometry in black soil, four different fertilization treatments were selected： no fertilization （CK）, nitrogen fertilizer （N）, phosphorus fertilizer （P）, and combination of nitrogen and phosphorus fertilizers （NP）. Soil samples were collected in both April 2021 and April 2022 to determine soil enzyme activities and their stoichiometric characteristics. The results showed that soil acid phosphatase and β-D-glucosidase activities were significantly higher in the N and NP treatments than in CK by 68%-158% and 26%-222%, respectively. Soil β-N-acetylaminoglucosidase activities were significantly higher in the P and NP treatments, with the highest around 75.48 nmol·（g·h）-1 and 106.81 nmol·（g·h）-1, respectively. Two-way ANOVA analysis showed that N and P inputs had a great impact on soil enzyme activities. Redundancy analysis showed that the main factors controlling enzyme activities were soil pH, microbial biomass phosphorus, and soil available P content. It was found that N inputs significantly increased enzyme vector length, which was ranged from 1.32 to 1.52, and the enzyme vector angles were all larger than 45°, suggesting C and P co-limited in the black soils. These findings suggest that 40 years of fertilization have had a great impact on soil enzymes and the related resource use strategy, which provides great implications for assessing soil nutrients balance and soil sustainability.

Long-term agricultural cultivation decreases microbial nutrient limitation in coastal saline soils

Soil enzyme activities are pivotal for diverse biochemical processes and are sensitive to land use changes. They can indicate soil microbial nutrient limitations. Nonetheless, the mechanism governing the response of soil microbial nutrient limitation to land use alterations in coastal regions remains elusive. We assessed soil nutrients, microbial biomass, and extracellular enzyme activities across various land use types—natural (wasteland and woodland) and agricultural (farmland and orchard)—in the Hangzhou Bay area, China. All four land use types experience co-limitation by carbon (C) and phosphorus (P). However, the extent of microbial resource limitations varies among them. Long-term agricultural practices diminish microbial C and P limitations in farmland and orchard soils compared to natural soils, as evidenced by lower ecoenzymatic C:N ratios and vector lengths, alongside higher microbial carbon use efficiency (CUE). Soil nutrient stoichiometric ratios and CUE are primary factors influencing microbial C and P limitations. Thus, fostering appropriate land use and management practices proves imperative to regulate soil nutrient cycles and foster the sustainable management of coastal areas.

Response of soil microbial homeostasis to soil ecological stoichiometric balance in a World Natural Heritage area

Soil microbial biomass stoichiometry homeostasis is essential for microorganism survival and ecosystem stability. Despite its importance, research on soil microbial homeostasis in Natural World Heritage Sites (NWHS) is lacking. This study analyzed ecological stoichiometry and microbial homeostasis in surface (0–20 cm) and subsurface (20–40 cm) soils across various vegetation types in Fanjing Mountain, an NWHS in China. The objective was to explore microbial homeostasis in relation to soil ecological stoichiometry and identify key influencing factors. Results indicated that microbial biomass stoichiometry in surface soil is higher than in subsurface soil for 5 vegetation types, mirroring nutrient stoichiometry but contrasting enzyme stoichiometry. Vector length (VL) suggests higher C limitation in subsurface soil for all vegetation types, while vector angle (VA) shows P limitation in surface soil of certain types and N limitation in subsurface soil across all types. Random forest analysis revealed that the microbial carbon-nitrogen ratio (MB C/N) was mainly contributed by C/N (14.11 %), SOC (12.31 %), pH (10.52 %), NH4+-N, SWC, NO3--N in the surface soil, and NO3--N (15.74 %), altitude, SWC, SOC, C/N in the subsurface soil, whereas for the microbial carbon-phosphorus ratio (MB C/P), altitude (12.16 %), SWC (9.86 %), and AP were the main contributing factors in the surface soil, and in the subsurface soil, altitude (10.49 %), C/P, SWC, SOC, and TP. This study provides insights into ecological stoichiometry, homeostasis, and nutrient limitations in Fanjing Mountain, aiding vegetation nutrient balance management in NWHS.

Generating m-Ary Gray Codes and Related Algorithms

In this work, we systematize several implementations of the Gray code over an alphabet with m≥2 elements, which we present in C code so that they can be used directly after copying from the text. We consider two variants—reflected and modular (or shifted) m-ary Gray codes. For both variants, we present the ranking and unranking functions, as well as algorithms for generating only a part of the code, more precisely the codewords between two given vectors. Finally, we give algorithms that generate a maximal set of non-proportional vectors of length n over the given alphabet in a Gray code.

The pluralization palette: unveiling semantic clusters in English nominal pluralization through distributional semantics.

The online version contains supplementary material available at 10.1007/s11525-024-09428-9.

Stockpiling turf alters microbial carbon and nitrogen use efficiency on the Tibetan Plateau

Microbial carbon use efficiency (CUE) and nitrogen use efficiency (NUE) are crucial parameters reflecting soil C and N sequestration. Concerns about how artificial activities disturb alpine meadow ecosystem are increasing, but the knowledge of variances in microbial CUE and NUE in response to turf storage remains scarce. Here, we conducted a turf storage experiment on the Tibetan Plateau with two common storage methods, laying turfs method (LT) and stacking turfs method (ST). Plant litter, aboveground and belowground biomass declined considerably in the LT and ST than those in natural meadow. Soil pH and available phosphorus were significantly lower, but soil organic carbon, total nitrogen, dissolved organic carbon, and available nitrogen were substantially higher in stored turfs (both ST and LT) than in natural meadow. These results led to a differentiation in nutrient status among treatments. Vetor model indicated a stronger C limitation (vector length > 0.61) in ST than that in the LT and a shift from N to P limitation (vector angle >55°) in all stored turfs. Microbial CUE was prominently higher in the LT than those in the ST, signifying that microbes allocated more exogenous C to self-growth in the LT. Microbial NUE declined considerably in stored turfs, indicating a great proportion of N used for catabolic process instead of anabolic process. Microbial CUE and NUE were tightly linked to nutrient content and availability, enzymatic stoichiometry, microbial traits and plant biomass. Our results suggest that variations in microbial CUE and NUE were indirectly regulated by soil physicochemical properties via mediating nutrient imbalance and enzymatic stoichiometry in stored turfs.

FedSHE: privacy preserving and efficient federated learning with adaptive segmented CKKS homomorphic encryption

Unprotected gradient exchange in federated learning (FL) systems may lead to gradient leakage-related attacks. CKKS is a promising approximate homomorphic encryption scheme to protect gradients, owing to its unique capability of performing operations directly on ciphertexts. However, configuring CKKS security parameters involves a trade-off between correctness, efficiency, and security. An evaluation gap exists regarding how these parameters impact computational performance. Additionally, the maximum vector length that CKKS can once encrypt, recommended by Homomorphic Encryption Standardization, is 16384, hampers its widespread adoption in FL when encrypting layers with numerous neurons. To protect gradients’ privacy in FL systems while maintaining practical performance, we comprehensively analyze the influence of security parameters such as polynomial modulus degree and coefficient modulus on homomorphic operations. Derived from our evaluation findings, we provide a method for selecting the optimal multiplication depth while meeting operational requirements. Then, we introduce an adaptive segmented encryption method tailored for CKKS, circumventing its encryption length constraint and enhancing its processing ability to encrypt neural network models. Finally, we present FedSHE, a privacy-preserving and efficient Federated learning scheme with adaptive Segmented CKKS Homomorphic Encryption. FedSHE is implemented on top of the federated averaging (FedAvg) algorithm and is available at https://github.com/yooopan/FedSHE. Our evaluation results affirm the correctness and effectiveness of our proposed method, demonstrating that FedSHE outperforms existing homomorphic encryption-based federated learning research efforts in terms of model accuracy, computational efficiency, communication cost, and security level.

Phylo2Vec: a vector representation for binary trees

Abstract Binary phylogenetic trees inferred from biological data are central to understanding the shared history among evolutionary units. However, inferring the placement of latent nodes in a tree is computationally expensive. State-of-the-art methods rely on carefully designed heuristics for tree search, using different data structures for easy manipulation (e.g., classes in object-oriented programming languages) and readable representation of trees (e.g., Newick-format strings). Here, we present Phylo2Vec, a parsimonious encoding for phylogenetic trees that serves as a unified approach for both manipulating and representing phylogenetic trees. Phylo2Vec maps any binary tree with n leaves to a unique integer vector of length n − 1. The advantages of Phylo2Vec are fourfold: i) fast tree sampling, (ii) compressed tree representation compared to a Newick string, iii) quick and unambiguous verification if two binary trees are identical topologically, and iv) systematic ability to traverse tree space in very large or small jumps. As a proof of concept, we use Phylo2Vec for maximum likelihood inference on five real-world datasets and show that a simple hill-climbing-based optimisation scheme can efficiently traverse the vastness of tree space from a random to an optimal tree.

Enzyme stoichiometry reveals microbial nitrogen limitation in stony soils

Resource limitation for soil microorganisms is the crucial factor in nutrient cycling and vegetation development, which are especially important in arid climate. Given that rock fragments strongly impact hydrologic and geochemical processes in arid areas, we hypothesized that microbial resource (C and N) limitation will increase along the rock fragment content (RFC) gradient. We conducted a field experiment in Minjiang river arid valleys with four RFC content (0 %, 25 %, 50 %, and 75 %, V V−1) and four vegetation types (Artemisia vestita, Bauhinia brachycarpa, Sophora davidii, and the soil without plants). Activities of C (β-1,4-glucosidase, BG), N (β-1,4-N-acetyl-glucosaminidase, NAG; L-leucine aminopeptidase, LAP), and P (acid phosphatase, ACP) acquiring enzymes were investigated to assess the limitations by C, N or P. In unplanted soil, the C acquiring enzyme activity decreased by 43 %, but N acquiring enzyme activity increased by 72 % in 75 % RFC than those in rock-free soils (0 % RFC). Increasing RFC reduced C:N and C:P enzymatic ratios, as well as vector length and vector angle (< 45°). Plants increased the activities of C and N acquiring enzymes in soils, as well as C:P and N:P enzyme activities, as well as vector length (by 5.6 %–25 %), but decreased vector angle (by 13 %–21 %). Enzyme stoichiometry was dependent on biotic and abiotic factors, such as soil water content, soil C:N, and total content of phospholipid fatty acids, reflecting microbial biomass content. Increased RFC shifted enzymatic stoichiometry toward lower C but stronger N limitation for microorganisms. Vegetation increased microbial C and N limitation, and impacted the enzymatic activities and stoichiometry depending on shrub functional groups. Consequently, the direct effects of vegetation, nutrient availability and microbial biomass content, as well as indirect effects of soil properties collectively increased microbial resource limitations along the RFC gradient.

Enzyme Activity Stoichiometry Suggests That Fertilization, Especially Nitrogen Fertilization, Alleviates Nutrient Limitation of Soil Microorganisms in Moso Bamboo Forests

Rational application of N fertilizer is essential for maintaining the long-term productivity of Moso bamboo forests. Microbial activity is a crucial indicator of soil quality. Changes in soil nutrient resources due to N addition can lead to microbial nutrient limitations, thereby impeding the maintenance of soil quality. Currently, there is limited research on the effects of N application on microbial nutrient limitations in Moso bamboo forest soils. To examine the changes in extracellular enzyme activity and microbial nutrient limitations in Moso bamboo forest soils following N application, we conducted an N application experiment in northern Guizhou. The findings revealed that the N3 treatment (726 kg·N·hm−2·yr−1) significantly reduced β-glucosidase (BG) activity by 27.61% compared to the control group (no fertilization). The N1 (242 kg·N·hm−2·yr−1), N2 (484 kg·N·hm−2·yr−1), and N3 treatments notably increased the activities of leucine aminopeptidase (LAP) and N-acetyl-β-D-glucosidase (NAG) by 11.45% to 15.79%. Acid phosphatase (ACP) activity remained unaffected by fertilization. N application treatments significantly decreased the C:Ne and C:Pe ratios, while the N:Pe ratio was less influenced by N fertilizer application. Scatter plots and vector characteristics of enzyme activity stoichiometry suggested that microorganisms in the study area were limited by C and N, and N fertilizer application reduced the vector length and increased the vector angle, indicating that N application alleviated the C and N limitation of microorganisms in Moso bamboo forests. Redundancy Analysis (RDA) demonstrated that microbial biomass phosphorus (MBP) was the most critical factor affecting extracellular enzyme activity and stoichiometry. Furthermore, Random Forest Regression analysis identified MBP and the N:Pm ratio as the most significant factors influencing microbial C and N limitation, respectively. The study demonstrated that N application modulates the microbial nutrient acquisition strategy by altering soil nutrient resources in Moso bamboo forests. Formulating fertilizer application strategies based on microbial nutrient requirements is more beneficial for maintaining soil quality and sustainably managing Moso bamboo forests. Additionally, our study offers a theoretical reference for understanding carbon cycling in bamboo forest ecosystems in the context of substantial N inputs.

Re-Orthogonalized/Affine GMRES and Orthogonalized Maximal Projection Algorithm for Solving Linear Systems

GMRES is one of the most powerful and popular methods to solve linear systems in the Krylov subspace; we examine it from two viewpoints: to maximize the decreasing length of the residual vector, and to maintain the orthogonality of the consecutive residual vector. A stabilization factor, η, to measure the deviation from the orthogonality of the residual vector is inserted into GMRES to preserve the orthogonality automatically. The re-orthogonalized GMRES (ROGMRES) method guarantees the absolute convergence; even the orthogonality is lost gradually in the GMRES iteration. When η&lt;1/2, the residuals’ lengths of GMRES and GMRES(m) no longer decrease; hence, η&lt;1/2 can be adopted as a stopping criterion to terminate the iterations. We prove η=1 for the ROGMRES method; it automatically keeps the orthogonality, and maintains the maximality for reducing the length of the residual vector. We improve GMRES by seeking the descent vector to minimize the residual in a larger space of the affine Krylov subspace. The resulting orthogonalized maximal projection algorithm (OMPA) is identified as having good performance. We further derive the iterative formulas by extending the GMRES method to the affine Krylov subspace; these equations are slightly different from the equations derived by Saad and Schultz (1986). The affine GMRES method is combined with the orthogonalization technique to generate a powerful affine GMRES (A-GMRES) method with high performance.