Sum Of Squares Research Articles

Polynomial Fuzzy Observer-Based Feedback Control for Nonlinear Hyperbolic PDEs Systems.

This article explores the observer-based feedback control problem for a nonlinear hyperbolic partial differential equations (PDEs) system. Initially, the polynomial fuzzy hyperbolic PDEs (PFHPDEs) model is established through the utilization of the fuzzy identification approach, derived from the nonlinear hyperbolic PDEs model. Various types of state estimation and controller design problems for the polynomial fuzzy PDEs system are discussed concerning the state estimation problem. To investigate the relaxed stability problem, Euler's homogeneous theorem, Lyapunov-Krasovskii functional with polynomial matrices (LKFPM), and the sum-of-squares (SOSs) approach are adopted. The exponential stabilization condition is formulated in terms of the spatial-derivative-SOSs (SD-SOSs). Additionally, a segmental algorithm is developed to find the feasible solution for the SD-SOS condition. Finally, a hyperbolic PDEs system and several numerical examples are provided to illustrate the validity and effectiveness of the proposed results.

Prospects for neutron star parameter estimation using gravitational waves from f modes associated with magnetar flares

ABSTRACT Magnetar vibrational modes are theorized to be associated with energetic X-ray flares. Regular searches for gravitational waves from these modes have been performed by Advanced LIGO (Laser Interferometer Gravitational-wave Observatory) and Advanced Virgo, with no detections so far. Presently, search results are given in upper limits on the root sum square of the integrated gravitational-wave strain. However, the increased sensitivity of current detectors and the promise of future detectors invite the consideration of more astrophysically motivated methods. We present a framework for augmenting gravitational-wave searches to measure or place direct limits on magnetar astrophysical properties in various search scenarios using a set of phenomenological and analytical models.

Test for the mean of high-dimensional functional time series

The one-sample test and two-sample test for the mean of high-dimensional functional time series are considered in this study. The proposed tests are built on the dimension-wise max-norm of the sum of squares of diverging projections. The null distribution of the test statistics is investigated using normal approximation, and the asymptotic behavior under the alternative is studied. The approach is robust to the cross-series dependence of unknown forms and magnitude. To approximate the critical values, a blockwise wild bootstrap method for functional time series is employed. Both fully and partially observed data are analyzed in theoretical research and numerical studies. Evidence from simulation studies and an IT stock data case study demonstrates the usefulness of the test in practice. The proposed methods have been implemented in a R package.

Generalized identifiability of sums of squares

Let f be a homogeneous polynomial of even degree d. We study the decompositions f=∑i=1rfi2 where deg⁡fi=d/2. The minimal number of summands r is called the 2-rank of f, so that the polynomials having 2-rank equal to 1 are exactly the squares. Such decompositions are never unique and they are divided into O(r)-orbits, the problem becomes counting how many different O(r)-orbits of decomposition exist. We say that f is O(r)-identifiable if there is a unique O(r)-orbit. We give sufficient conditions for generic and specific O(r)-identifiability. Moreover, we show the generic O(r)-identifiability of ternary forms.

GENERALIZATION OF PELL SEQUENCE AND PELL-LUCAS SEQUENCE, NEW RESULTS AND CIRCULANT MATRICES ASPECTS

Some generalizations of Pell sequence and Pell-Lucas sequence, namely, (k,h)-Pell sequence and (k,h)-Pell-Lucas sequence are considered in this paper. We obtain generating functions, some identities and formulas for the sums of a finite number of terms and consecutive terms, sums of squares of consecutive terms and alternating sums of consecutive terms of these sequences. Then, we obtain the eigenvalues and determinants of particular circulant matrices involving (k,h)-Pell sequence and (k,h)-Pell-Lucas sequence. Finally, we obtain some upper and lower bounds for the spectral norms of these circulant matrices.

Stability Analysis of Yield and Its Components in Snap Melon (Cucumis melo var momordica)

Snap melon, scientifically known as Cucumis melo L. var. momordica (Roxb.), is a crop native to India and belonging to Cucurbitaceae family. This study was carried out to assess the nature and magnitude of genotype-environment interaction and stability among some parameters in snap melon genotypes. Twenty five local cultivars along with three standard checks were evaluated in randomized block design with three replications at three environments during summer 2023 at Navsari. The Eberhart and Russell model of stability analysis was employed which has been proved to be a reliable model. The study revealed that the mean sum of squares due to genotypes when tested against pooled deviation were highly significant for all the traits studied. Environment ´ Genotype (linear) showed significant differences for all characteristics except fruit weight and yield per vine, indicating that prediction could be possible by considering individual genotypes for these traits. A significant non-linear component (pooled deviation) was observed for fruit weight and fruit yield per vine, suggesting that genotypes varied considerably in terms of stable performance for these traits. The G ´ E interaction was only significant for flesh thickness and yield per vine, with all other traits being non-significant. The genotypes T 14 (HUB 18) and T 17 (HUB 21) were relatively stable and can be further used as parents to develop a more robust stable genotype. We identified genotypes which could pave way introducing high performing snap melon varieties.

Estimation of heterosis on CGMS based pigeonpea

Thirty-five hybrids generated from seven cytoplasmic male sterile lines and five fertility restorer lines were tested for determining heterosis. All studied traits except pod length, number of seeds per pod, and leaf area showed significant variations between the parents indicating a sufficient variability amongthe parents (lines and testers). The mean sum of squares of the hybrids suggested high heterosis for studied traits except for test weight and number of branches per plant. The findings showed noteworthy and affirmative relative heterosis for eaj hybrid. Top three hybrids viz. CMS GT 1615 A × GTR 105 (28.34% and 38.97%), CMS GT 1615 A × GTR 18 (24.36% and 34.66%) and CMS GT 1615 × GTR 23 (22.23% and 32.35%) identified for seed yield and its contributing traits on the basis of standard heterosis against standard checks GTH 1 and GT 101, respectively. These can be exploited commercially for heterosis breeding in pigeonpea.

Cones between the cones of positive semidefinite forms and sums of squares

Abstract For n, d ∈ ℕ, the cone 𝓟 n+1,2d of positive semidefinite real forms in n + 1 variables of degree 2d contains the subcone Σ n+1,2d of those representable as finite sums of squares of real forms. Hilbert [11] proved that these cones coincide exactly in the Hilbert cases (n + 1, 2d) with n + 1 = 2 or 2d = 2 or (n + 1, 2d) = (3, 4). In this paper, we induce a filtration of intermediate cones between Σ n+1,2d and 𝓟 n+1,2d via the Gram matrix approach in [4] on a filtration of irreducible projective varieties V k−n ⊊ … ⊊ Vn ⊊ … ⊊ V 0 containing the Veronese variety. Here, k is the dimension of the vector space of real forms in n + 1 variables of degree d. By showing that V 0, …, V n (and V n+1 when n = 2) are varieties of minimal degree, we demonstrate that the corresponding intermediate cones coincide with Σ n+1,2d . We moreover prove that, in the non-Hilbert cases of (n + 1)-ary quartics for n ≥ 3 and (n + 1)-ary sextics for n ≥ 2, all the remaining cone inclusions are strict.

Numbers of Exchangeable Hydrogens from LC-MS Data of Heavy Water Metabolically Labeled Samples.

Labeling with deuterium oxide (D2O) has emerged as one of the preferred approaches for measuring the synthesis of individual proteins in vivo. In these experiments, the synthesis rates of proteins are determined by modeling mass shifts in peptides during the labeling period. This modeling depends on a theoretical maximum enrichment determined by the number of labeling sites (NEH) of each amino acid in the peptide sequence. Currently, NEH is determined from one set of published values. However, it has been demonstrated that NEH can differ between species and potentially tissues. The goal of this work was to determine the number of NEH for each amino acid within a given experiment to capture the conditions unique to that experiment. We used four methods to compute the NEH values. To test these approaches, we used two publicly available data sets. In a de novo approach, we compute NEH values and the label enrichment from the abundances of three mass isotopomers. The other three methods use the complete isotope profiles and body water enrichment in deuterium as an input parameter. They determine the NEH values by (1) minimizing the residual sum of squares, (2) from the mole percent excess of labeling, and (3) the time course profile of the depletion of the relative isotope abundance of monoisotope. In the test samples, the method using residual sum of squares performed the best. The methods are implemented in a tool for determining the NEH for each amino acid within a given experiment to use in the determination of protein synthesis rates using D2O.

A Moment-Sum-of-Squares Hierarchy for Robust Polynomial Matrix Inequality Optimization with Sum-of-Squares Convexity

We study a class of polynomial optimization problems with a robust polynomial matrix inequality (PMI) constraint where the uncertainty set itself is also defined by a PMI. These can be viewed as matrix generalizations of semi-infinite polynomial programs because they involve actually infinitely many PMI constraints in general. Under certain sum-of-squares (SOS)-convexity assumptions, we construct a hierarchy of increasingly tight moment-SOS relaxations for solving such problems. Most of the nice features of the moment-SOS hierarchy for the usual polynomial optimization are extended to this more complicated setting. In particular, asymptotic convergence of the hierarchy is guaranteed, and finite convergence can be certified if some flat extension condition holds true. To extract global minimizers, we provide a linear algebra procedure for recovering a finitely atomic matrix-valued measure from truncated matrix-valued moments. As an application, we are able to solve the problem of minimizing the smallest eigenvalue of a polynomial matrix subject to a PMI constraint. If SOS convexity is replaced by convexity, we can still approximate the optimal value as closely as desired by solving a sequence of semidefinite programs and certify global optimality in case that certain flat extension conditions hold true. Finally, an extension to the nonconvexity setting is provided under a rank 1 condition. To obtain the above-mentioned results, techniques from real algebraic geometry, matrix-valued measure theory, and convex optimization are employed. Funding: This work was supported by the National Key Research and Development Program of China [Grant 2023YFA1009401] and the National Natural Science Foundation of China [Grants 11571350 and 12201618].

Hierarchy relaxations for robust equilibrium constrained polynomial problems and applications to electric vehicle charging scheduling

In this paper, we consider a polynomial problem with equilibrium constraints in which the constraint functions and the equilibrium constraints involve data uncertainties. Employing a robust optimization approach, we examine the uncertain equilibrium constrained polynomial optimization problem by establishing lower bound approximations and asymptotic convergences of bounded degree diagonally dominant sum-of-squares (DSOS), scaled diagonally dominant sum-of-squares (SDSOS) and sum-of-squares (SOS) polynomial relaxations for the robust equilibrium constrained polynomial optimization problem. We also provide numerical examples to illustrate how the optimal value of a robust equilibrium constrained problem can be calculated by solving associated relaxation problems. Furthermore, an application to electric vehicle charging scheduling problems under uncertain discharging supplies shows that for the lower relaxation degrees, the DSOS, SDSOS and SOS relaxations obtain reasonable charging costs and for the higher relaxation degrees, the SDSOS relaxation scheme has the best performance, making it desirable for practical applications.

Stability and genetic parameters for cassava yield attributes in the tropical humid region of Brazil

The performance differences in cassava genotypes arising from genotype vs. environment interactions (G × E) often lead to responses that are significantly lower than expected for selection. The objective of this study was to evaluate different stability methods, both parametric and non-parametric, such as additive main-effects and multiplicative interaction (AMMI), main effect of genotypes plus G × E (GGE), and weighted average of absolute scores (WAASB), in order to quantify the G × E in multi-environmental trials. A total of 12 genotypes were assessed across 12 environments using a completely randomized block design, with three replicates for traits such as fresh root yield (FRY) and dry matter content in the roots (DMC). The data were subjected to analysis of variance and the Scott Knott test (p < 0.05). The sum of squares (SQ) of genotypes, environment, and G × E effects were equally distributed for FRY, whereas for DMC, these effects accounted for 64.1%, 21.9%, and 13.8% of the SQ, respectively, indicating a lower environmental effect on this characteristic. Using the AMMI, GGE, and WAASB methods, genotypes with high agronomic performance and stability for FRY (BR11-34–41 and BR11-34–69) (> 32 t ha−1) and DMC (BRS Novo Horizonte, BR12-107–002, and BR11-24–156) (> 37%) were identified. The broad-sense heritability (h2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${h}^{2}$$\\end{document}) for FRY and DMC was estimated to be 0.45 and 0.75, respectively. Approximately 72% of the methods identified BRS Novo Horizonte as the genotype with the highest stability and performance for DMC, while 47% identified genotypes BR11-34–41 and BR11-34–69 for FRY and intermediate DMC. Genotype BR11-24–156 exhibited high static stability according to 50% of the methods. Significant correlations were observed between stability and agronomic performance across the different methods, enabling the formation of groups based on stability concepts. Additionally, it was found that two mega-environments existed for FRY, whereas DMC displayed a single mega-environment with similar patterns, indicating an absence of G × E. We identified superior genotypes that could be promoted to national performance trials to develop stable cultivars with better yield attributes in cassava.

A Mixed-level Factorial Experiment to Elucidate the Impact of Inorganic Fertilizers on Crop Yield

Lack of nutrients in the soil affects soil fertility which can hinder the growth and yield of any crop. Therefore, soil amendment is important to improve soil nutrient either by organic or inorganic technique. In this paper, we considered different inorganic fertilizers to determine the main optimal effect and their interaction on the yield of crop by adoption of mixed-level factorial experiment. Three inorganic fertilizers namely Nitrogen (N), Phosphorous (P) and Potassium (K) were considered each at different levels using tomato crop for investigation. We set up a multiple linear model to match with the design. We estimated the model parameters with the error component and sums of squares. A 2×3×4 design structure for factorial combinations—of factor N at two levels, factor P at three levels, and factor K at four levels using a randomized complete block design with three replications—was developed. Normality of the data was tested and factors visualization for both main and interaction effects were illustrated by the use of graphs. The result revealed that Nitrogen (N) and Potassium (K) concentration are significant while Phosphorous (P) concentration is not, the effect of two-factor interactions for NP and PK are not significant and NK is significant, which signified that the yields do not depend on the levels irrespective of the factor. Based on the results, we concluded that phosphorus is the best inorganic fertilizer compared with Nitrogen and Potassium in terms of soil nutrient for the yield of crops.

First-order reasoning and efficient semi-algebraic proofs

Semi-algebraic proof systems such as sum-of-squares (SoS) have attracted a lot of attention due to their relation to approximation algorithms: constant degree semi-algebraic proofs lead to conjecturally optimal polynomial-time approximation algorithms for important NP-hard optimization problems. Motivated by the need to allow a more streamlined and uniform framework for working with SoS proofs than the restrictive propositional level, we initiate a systematic first-order logical investigation into the kinds of reasoning possible in algebraic and semi-algebraic proof systems. Specifically, we develop first-order theories that capture in a precise manner constant degree algebraic and semi-algebraic proof systems: every statement of a certain form that is provable in our theories translates into a family of constant degree polynomial calculus or SoS refutations, respectively; and using a reflection principle, the converse also holds.This places algebraic and semi-algebraic proof systems in the established framework of bounded arithmetic, while providing theories corresponding to systems that vary quite substantially from the usual propositional-logic ones.We give examples of how our semi-algebraic theory proves statements such as the pigeonhole principle, we provide a separation between algebraic and semi-algebraic theories, and we describe initial attempts to go beyond these theories by introducing extensions that use the inequality symbol, identifying along the way which extensions lead outside the scope of constant degree SoS. Moreover, we prove new results for propositional proofs, and specifically extend Berkholz's dynamic-by-static simulation of polynomial calculus (PC) by SoS to PC with the radical rule.

Testing independence based on Spearman’s footrule in high dimensions

. Spearman’s footrule is a widely used tool to discover the relationship between two continuous random variables, mainly because of its robustness, natural interpretation, and simplicity of calculation. In this article, we propose a new test statistic based on the sum of squares of pairwise Spearman’s footrule, to test the mutual independence of a high dimensional random vector. The large sample properties of the proposed test method are established. Simulation results show that the new test has higher power than existing test methods under some circumstances. An example with real life data is discussed to illustrate the usefulness and effectiveness of the proposed test statistic.

Strictly positive polynomials in the boundary of the SOS cone

We study the boundary of the cone of real polynomials that can be decomposed as a sum of squares (SOS) of real polynomials. This cone is included in the cone of nonnegative polynomials and both cones share a part of their boundary, which corresponds to polynomials that vanish at at least one point. We focus on the part of the boundary which is not shared, corresponding to strictly positive polynomials.For the cases of polynomials of degree 6 in 3 variables and degree 4 in 4 variables, this boundary has been completely characterized by G. Blekherman. For the cases of a polynomial f in more variables or of higher degree, results by G. Blekherman, R. Sinn and M. Velasco and other authors based on a conjecture by Ottaviani and Paoletti give bounds for the maximum number of linearly independent polynomials that can appear in an SOS decomposition of f, or equivalently the maximum rank of the matrices in the Gram spectrahedron of f. We show that the same bounds can be obtained from the Eisenbud-Green-Harris conjecture. Combining theoretical results and computational techniques, we compute examples that allow us to prove the optimality of the bounds for all degrees and number of variables. Additionally, we give examples for the following problems: examples in the boundary of the cone that are the sum of less than n squares and have common complex roots, and examples of polynomials in the boundary with SOS length larger than the expected one from the dimension.

Bee-inspired insights: Unleashing the potential of artificial bee colony optimized hybrid neural networks for enhanced groundwater level time series prediction.

Analysis of the change in groundwater used as a drinking and irrigation water source is of critical importance in terms of monitoring aquifers, planning water resources, energy production, combating climate change, and agricultural production. Therefore, it is necessary to model groundwater level (GWL) fluctuations to monitor and predict groundwater storage. Artificial intelligence-based models in water resource management have become prevalent due to their proven success in hydrological studies. This study proposed a hybrid model that combines the artificial neural network (ANN) and the artificial bee colony optimization (ABC) algorithm, along with the ensemble empirical mode decomposition (EEMD) and the local mean decomposition (LMD) techniques, to model groundwater levels in Erzurum province, Türkiye. GWL estimation results were evaluated with mean square error (MSE), coefficient of determination (R2), and residual sum of squares (RSS) and visually with violin, scatter, and time series plot. The study results indicated that the EEMD-ABC-ANN hybrid model was superior to other models in estimating GWL, with R2 values ranging from 0.91 to 0.99 and MSE values ranging from 0.004 to 0.07. It has also been revealed that promising GWL predictions can be made with previous GWL data.

Computing sparse Fourier sum of squares on finite abelian groups in quasi-linear time

The problem of verifying the nonnegativity of a function on a finite abelian group is a long-standing challenging problem. The basic representation theory of finite groups indicates that a function f on a finite abelian group G can be written as a linear combination of characters of irreducible representations of G by f(x)=∑χ∈Gˆfˆ(χ)χ(x), where Gˆ is the dual group of G consisting of all characters of G and fˆ(χ) is the Fourier coefficient of f at χ∈Gˆ. In this paper, we show that by performing the fast (inverse) Fourier transform, we are able to compute a sparse Fourier sum of squares (FSOS) certificate of f on a finite abelian group G with complexity that is quasi-linear in the order of G and polynomial in the FSOS sparsity of f. Moreover, for a nonnegative function f on a finite abelian group G and a subset S⊆Gˆ, we give a lower bound of the constant M such that f+M admits an FSOS supported on S. We demonstrate the efficiency of the proposed algorithm by numerical experiments on various abelian groups of orders up to 107. As applications, we also solve some combinatorial optimization problems and the sum of Hermitian squares (SOHS) problem by sparse FSOS.

Spectrahedral shadows and completely positive maps on real closed fields

In this article we develop new methods for exhibiting convex semialgebraic sets that are not spectrahedral shadows. We characterize when the set of nonnegative polynomials with a given support is a spectrahedral shadow in terms of sums of squares. As an application we prove that the cone of copositive matrices of size n\geq5 is not a spectrahedral shadow, answering a question of Scheiderer. Our arguments are based on the model-theoretic observation that any formula defining a spectrahedral shadow must be preserved by every unital \mathbb{R} -linear completely positive map R\to R on a real closed field extension R of \mathbb{R} .

Colour difference detection algorithm for warp‐knitted fabric based on image colour appearance model

AbstractDetecting colour differences in warp‐knitting fabric is essential to ensure its quality. Machine vision algorithms are commonly used for automatic colour difference detection. However, the current algorithm directly extracts RGB (red, green, blue) colour stimulus values from the image without considering the image's colour attributes, leading to a significant error in colour difference detection. To solve this issue, a colour appearance model is introduced into the field of colour difference detection for warp‐knitted fabric. Initially, a colour appearance model based on CAM16 is established, and the critical parameters are calculated to obtain the colour appearance attribute of the fabric. The colour difference calculation formula is then constructed in the uniform colour space of CAM16‐UCS. A template selection algorithm based on principal component analysis was designed to select warp‐knitted cloth images with standard colours. Subsequently, a cloth colour difference detection algorithm was developed using the cloth colour profile model. The performance of the colour difference formula based on the colour profile model was evaluated using the PF/3 method. To compare the CIELab, CMC(2:1), and CIEDE2000 colour difference formulas, standardised residual sum of squares was used. The results indicated that the colour difference formula based on the colour‐appearance model is about 5.32% different from the visual colour difference perceived by the human eye. However, it can perform as well as the CMC(2:1) colour difference formula, which is widely used in the textile industry.