Constant Sum Research Articles

Using Transposition to Efficiently Solve Constant Matrix-Vector Multiplication and Sum of Product Problems

In this work, we present an approach to alleviate the potential benefit of adder graph algorithms by solving the transposed form of the problem and then transposing the solution. The key contribution is a systematic way to obtain the transposed realization with a minimum number of cascaded adders subject to the input realization. In this way, wide and low constant matrix multiplication problems, with sum of products as a special case, which are normally exceptionally time consuming to solve using adder graph algorithms, can be solved by first transposing the matrix and then transposing the solution. Examples show that while the relation between the adder depth of the solution to the transposed problem and the original problem is not straightforward, there are many cases where the reduction in adder cost will more than compensate for the potential increase in adder depth and result in implementations with reduced power consumption compared to using sub-expression sharing algorithms, which can both solve the original problem directly in reasonable time and guarantee a minimum adder depth.

Estimating Physical Composition of Municipal Solid Waste in China by Applying Artificial Neural Network Method.

Physical composition of municipal solid waste (PCMSW) is the fundamental parameter in domestic waste management; however, high fidelity, wide coverage, upscaling, and year continuous data sets of PCMSW in China are insufficient. A traceable and predictable methodology for estimating PCMSW in China is established for the first time by analyzing 503 PCMSW data sets of 135 prefecture-level cities in China. A hyperspherical transformation method was used to eliminate the constant sum constraint in statistically analyzing PCMSW data. Moreover, a back-propagation (BP) neural network methodology was applied to establish quantitative models between city-level PCMSW and its socio-economic factors, including city size, per capita gross regional product, geographical location, gas coverage rate, and year. Results show that (1) national-level PCMSW in 2017 was estimated as organic fraction (53.7%), ash and stone (8.3%), paper (16.9%), plastic and rubber (13.6%), textile (2.3%), wood (2.2%), metal (0.6%), glass (1.5%), and others (1.0%); (2) organic fraction, paper, and plastics showed an increasing trend from 1990 to 2017, while ash and stone decreased significantly; (3) organic fractions in East, North, and Central-South China were higher than those in other regions. This enables us to fill the data gap in the practice of municipal solid waste management in China.

Construction and existence of constant block sum PBIB designs

The present article proposes three series of partially balanced incomplete constant block sum designs constructed using regular graphs. Two series develop a three-associate class PBIB designs and one develops a five-associate class PBIB design. Moreover, we were able to show the existence of constant block sum PBIB designs with their parameters for a specified association scheme. Towards the end, we have discussed few possible designs, the blocks of which may be constructed from the association schemes discussed in the present research paper.

Deep learning for computational structural optimization

We investigate a novel computational approach to computational structural optimization based on deep learning. After employing algorithms to solve the stiffness formulation of structures, we used their improvement to optimize the structural computation. A standard illustration of 10 bar-truss was revisited to illustrate the mechanism of neural networks and deep learning. Several benchmark problems of 2D and 3D truss structures were used to verify the reliability of the present approach, and its extension to other engineering structures is straightforward. To enhance computational efficiency, a constant sum technique was proposed to generate data for the input of multi-similar variables. Both displacement and stress enforcements were the constraints of the optimized problem. The optimization data for cross sections with the objective function of total weight were then employed in the context of deep learning. The stochastic gradient descent (SGD) with Nesterov’s accelerated gradient (NAG), root mean square propagation (RMSProp) and adaptive moment estimation (Adam) optimizers were compared in terms of convergence. In addition, this paper devised Chebyshev polynomials for a new approach to activation functions in single-layer neural networks. As expected, its convergence was quicker than the popular learning functions, especially in a short training with a small number of epochs for tested problems. Finally, a split data technique for linear regression was proposed to deal with some sensitive data.

Regression for compositions based on a generalization of the Dirichlet distribution

The simplex is the geometrical locus of D-dimensional positive data with constant sum, called compositions. A possible distribution for compositions is the Dirichlet. In Dirichlet models, there are no scale parameters and the D shapes are assumed dependent on auxiliary variables. This peculiar feature makes Dirichlet models difficult to apply and to interpret. Here, we propose a generalization of the Dirichlet, called the simplicial generalized Beta (SGB) distribution. It includes an overall shape parameter, a scale composition and the D Dirichlet shapes. The SGB is flexible enough to accommodate many practical situations. SGB regression models are applied to data from the United Kingdom Time Use Survey. The R-package SGB makes the methods accessible to users.

Automated Market Makers and Decentralized Exchanges: a DeFi Primer

Recent advancements in decentralized finance (DeFi) have resulted in a rapid increase in the use of Automated Market Makers (AMMs) for creating decentralized exchanges (DEXs). In this paper, we organize these developments by treating an AMM as a neoclassical black-box characterized by the conversion of inputs (tokens) to outputs (prices). The conversion is governed by the technology of the AMM summarized by an ‘exchange function’. Various types of AMMs are examined, including: Constant Product Market Makers; Constant Mean Market Makers; Constant Sum Market Makers; Hybrid Function Market Makers; and, Dynamic Automated Market Makers. The paper also looks at the impact of introducing concentrated liquidity in an AMM. Overall, the framework presented here provides an intuitive geometric representation of how an AMM operates, and a clear delineation of the similarities and differences across the various types of AMMs.

Lagrangian for Circuits with Higher-Order Elements

The necessary and sufficient conditions of the validity of Hamilton’s variational principle for circuits consisting of (α,β) elements from Chua’s periodical table are derived. It is shown that the principle holds if and only if all the circuit elements lie on the so-called Σ-diagonal with a constant sum of the indices α and β. In this case, the Lagrangian is the sum of the state functions of the elements of the L or +R types minus the sum of the state functions of the elements of the C or −R types. The equations of motion generated by this Lagrangian are always of even-order. If all the elements are linear, the equations of motion contain only even-order derivatives of the independent variable. Conclusions are illustrated on an example of the synthesis of the Pais–Uhlenbeck oscillator via the elements from Chua’s table.

The Geometry of Elliptical Probability Contours for a Fix using Multiple Lines of Position

Navigation methods, traditional and modern, use lines of position in the plane. Standard Gaussian assumptions about errors leads to a constant sum of squared distances from the lines defining a probability contour. It is well known these contours are a family of ellipses centred on the most probable position and they can be computed using algebraic methods. In this paper we show how the most probable position, the axes and foci of ellipses can be found using geometric methods. This results in a ruler and compasses construction of these points and this gives insight into the way the shape and orientation of the probability contours depend on the angles between the lines of position. We start with the classical case of three lines of position with equal variances, we show how this can be extended to the case where the variances in the errors in the lines of position differ, and we go on to consider the case of four lines of position using a methodology that generalises to an arbitrary number of lines.

On H-supermagic labeling of friendship edge corona product with path and sun edge corona product with path

Let G be a simple, finite, connected and undirected graph with vertex set V and edge set E. Graph G admits an H-covering if every edge in E belongs to a subgraph of G isomorphic to H. Graph G is H− magic if there is a total labeling , such that each subgraph H′ = (V′, E′) of G isomorphic to H and satisfying where m(f) is a constant magic sum. Additionaly, G admits H− supermagic if f(V ) = {1, 2, …, |V(G)|}. The edge corona product between graph G1 and G2 is a graph obtained by taking one copy of G1 and |E(G1)| copies of G2 and then joining two end − vertices of the ith edge of G1 to every vertex in ith copy of G2. This research provides C3 ◊ Pn-supermagic labeling on fn ◊ Pn and P3 ◊ Pn-supermagic labeling on Sn ◊ Pn.

H− Supermagic labeling on domino ʘ Pn and Cn ◊ Sn

A finite simple graph G with vertex set V and edge set E admits an H-covering if every edge in E belongs to a subgraph of G isomorphic to H. We said G to be an H magic if there exists a total labeling , such that for each subgraph H’ = (V’, E’) of G isomorphic to H, ∑v∈V f(v) + ∑e∈E f(e) = m(f) where m(f) is a constant magic sum. Then G is H supermagic if f(V ) = {1, 2, …, |V (G)|}. The edge corona product between graph G1 and G2 is a graph obtained by taking one copy of G1 and |E(G1)| copies of G2 and then joining two end − vertices of the ith edge of G1 to every vertex in ith copy of G2. This research focuses on C4 ʘ Pn− supermagic labeling on Domino ʘ Pn and P2 ◊ Sn supermagic labeling on Cn ◊ Sn.

A Bayesian Network-Based Integrated for Flood Risk Assessment (InFRA)

Floods are natural disasters that should be considered a top priority in disaster management, and various methods have been developed to evaluate the risks. However, each method has different results and may confuse decision-makers in disaster management. In this study, a flood risk assessment method is proposed to integrate various methods to overcome these problems. Using factor analysis and principal component analysis (PCA), the leading indicators that affect flood damage were selected and weighted using three methods: the analytic hierarchy process (AHP), constant sum scale (CSS), and entropy. However, each method has flaws due to inconsistent weights. Therefore, a Bayesian network was used to present the integrated weights that reflect the characteristics of each method. Moreover, a relationship is proposed between the elements and the indicators based on the weights called the Integrated Index for Flood Risk Assessment (InFRA). InFRA and other assessment methods were compared by receiver operating characteristics (ROC)-area under curve (AUC) analysis. As a result, InFRA showed better applicability since InFRA was 0.67 and other methods were less than 0.5.

The null set of the join of paths

For positive integer [Formula: see text], a graph [Formula: see text] is said to be [Formula: see text]-magic if the edges of [Formula: see text] can be labeled with the nonzero elements of Abelian group [Formula: see text], where [Formula: see text] (the set of integers) and [Formula: see text] is the group of integers mod [Formula: see text], so that the sum of the labels of the edges incident to any vertex of [Formula: see text] is the same. When this constant sum is [Formula: see text], we say that [Formula: see text] is a zero-sum [Formula: see text]-magic graph. The set of all [Formula: see text] for which [Formula: see text] is a zero-sum [Formula: see text]-magic graph is the null set of [Formula: see text]. In this paper, we will completely determine the null set of the join of a finite number of paths.

Some practical aspects of predicting texture data in digital soil mapping

Soil texture is the most well-known composition in soil science. When separate components of the texture (sand, silt, and clay) are predicted independently in digital soil mapping (DSM), there is no guarantee that the separate estimates will sum to 100%. Log-ratio transformations before DSM modelling are alternatives to guarantee a constant sum of the estimates. Little is known about the effect of non-summing to 100% and transformations of particle-size fractions (PSFs) when DSM products were used to predict soil functional properties using pedotransfer functions (PTFs). Therefore, this study was conducted to investigate the effect of different soil texture modelling methods on the estimation of available soil water capacity (AWC) and the total amount of irrigation water (TIW) required for wheat production on a 4600 ha area in Khuzestan province, southwestern Iran. Specifically, this study aimed i) to assess the performance of random forest models (RF) to predict untransformed (UT) and transformed PSFs using environmental covariates; ii) to study the effects of three widely used log-ratio transformations including additive, centroid and isometric log-ratio transformations (alr, clr, and ilr respectively) on the estimations of AWC and TIW. A total of 150 soil samples were collected from the surface layers (0–30 cm) based on the conditioned Latin hypercube sampling (cLHS) procedure. Results indicated that, in terms of root mean square error (RMSE), RF provided similar accuracies in predicting PSFs for both untransformed and transformed data. However, transformation resulted in biased estimates. In addition, RF prediction based on untransformed data resulted in more correctly soil texture classes allocation when compared to transformed data. The spatial distribution of the sum of the predicted untransformed fractions indicated only small parts of the area conformed to the 100% sum. Almost the same accuracies for estimates of AWC were obtained when both untransformed and transformed predicted texture components were used as the inputs to PTFs. Data transformation can result in biased estimates of AWC. The findings indicated no significant difference between transformation methods in predicting AWC and TIW. The general patterns of the spatial distribution of the predicted AWCs across the whole area were the same for transformed and untransformed data (except for clr transformation).

Understanding implicit and explicit consumer desires for protein bars, powders, and beverages

AbstractThe objective of this study was to assess the implicit and explicit qualities of protein products that consumers find attractive. To do so, an Adaptive Choice‐Based Conjoint (ACBC) survey and a Constant Sum (CS) survey were conducted. Participants (n = 457) were the same for both studies and results were compared on both aggregate and individual levels. Additionally, ACBC utility scores were used to calculate “Share of Preference” scores so that each attribute level could be directly compared to CS. Attribute importance scores were relatively similar between methods, with the exception of the platform, sweetener claim, and sweetener type, which were different (p < 0.05) between surveys. Overall, both techniques agreed that the ideal protein product was a chocolate‐flavored protein bar with 20–29 g of primarily whey protein, all‐natural, and naturally sweetened by stevia. Additionally, both methods agreed on relative scaling of levels within each attribute. ACBC resulted in smaller standard deviations for shares of preference ratings than the CS survey. Differences in results between the surveys suggest that protein product consumers have differences in implicit and explicit preferences. Exploration of where these preferences disagree may provide insight on product development initiatives for select consumers.Practical applicationsThis study found that there are subgroups of protein product consumers who express different ideals and desires. Furthermore, these preferences exist both explicitly and implicitly within individual consumers, and differences in those preferences may influence subsequent segmentation of consumer groups. Understanding the benefits and shortcomings of these different survey methods, and reviewing the sentiments each method captures, may help manufacturers and product developers reach new consumer groups and satisfy existing consumers to a greater extent.

From telluric helix to telluric remix

The first attempt to represent the Periodic system graphically was the Telluric Helix (Vis Tellurique) presented in 1862 by Alexandre-Emile Béguyer de Chancourtois, in which the sequence of elements was wound round a cylinder. This has hardly been attempted since, because the intervals between periodic returns vary in length from 2 to 32 elements, but Charles Janet presented a model wound round four nested cylinders. The rows in Janet’s table are defined by a constant sum of the first two quantum numbers, n and l, so that they end with the s-block, headed by hydrogen and helium. By combining Janet’s table, Edward Mazurs’ version, in which each row represents an electron shell and Valery Tsimmerman’s use of a half square for each element, I have produced a representation that can be printed out and wound round to make a cylinder with manageable dimensions. In the unwound version, I have placed the s-block in the middle, to emphasise its pivotal nature, since it both ends each (n + ℓ) row and contributes electrons to the valence of elements in the next (n + ℓ) row; it thus does not necessarily belong either on the left or the right side of a table. The downward arrows that link subshells within each (n + ℓ) series graphically illustrate the Janet Effect (or Madelung Rule). To acknowledge my debt to Chancourtois, Janet, Mazurs and Tsimmerman, I call my design the ‘Telluric Remix’.

Criterion of globally complete chaos synchronization for diverse three-node VCSEL networks with coupling delays

For diverse three node vertical-cavity surface-emitting laser (VCSEL) networks with uniform time-delayed coupling, we propose a new criterion of the globally complete chaos synchronization (GCCS) among all node lasers by using the mater-stability function (MSF). It is found that GCCS can be achieved for an arbitrarily given three-node VCSEL networks when two points determined by constant row sum and two transversal eigenvalues fall into the region of stability where maximum Lyapunov exponent (MLE) is negative and the MLE as a function of constant row sum and the eigenvalue associated to perturbations within the synchronization manifold is positive.

Structural transition in interdependent networks with regular interconnections.

Networks are often made up of several layers that exhibit diverse degrees of interdependencies. An interdependent network consists of a set of graphs G that are interconnected through a weighted interconnection matrix B, where the weight of each intergraph link is a non-negative real number p. Various dynamical processes, such as synchronization, cascading failures in power grids, and diffusion processes, are described by the Laplacian matrix Q characterizing the whole system. For the case in which the multilayer graph is a multiplex, where the number of nodes in each layer is the same and the interconnection matrix B=pI, I being the identity matrix, it has been shown that there exists a structural transition at some critical coupling p^{*}. This transition is such that dynamical processes are separated into two regimes: if p>p^{*}, the network acts as a whole; whereas when p<p^{*}, the network operates as if the graphs encoding the layers were isolated. In this paper, we extend and generalize the structural transition threshold p^{*} to a regular interconnection matrix B (constant row and column sum). Specifically, we provide upper and lower bounds for the transition threshold p^{*} in interdependent networks with a regular interconnection matrix B and derive the exact transition threshold for special scenarios using the formalism of quotient graphs. Additionally, we discuss the physical meaning of the transition threshold p^{*} in terms of the minimum cut and show, through a counterexample, that the structural transition does not always exist. Our results are one step forward on the characterization of more realistic multilayer networks and might be relevant for systems that deviate from the topological constraints imposed by multiplex networks.

Circumstantial evidence of non-pollen palynomorph palaeoecology: a 5,500 year NPP record from forest hollow sediments compared to pollen and macrofossil inferred palaeoenvironments

The addition of non-pollen palynomorphs (NPPs) to pollen analytical studies has improved the interpretational frame, especially concerning the local regime in anthropogenically disturbed environments. Using advanced ordination techniques this paper explores the variation of NPP-assemblages and the indicative value of individual NPP-types by comparison to independent classical proxies. Sediment samples from a forest hollow at Tarup Lund, Denmark, covering the last 5,500 years, were prepared for NPP-analysis using a non-aggressive procedure. Correlations between non-pollen palynomorphs and sediment, pollen and macrofossil inferred environmental explanatory variables were studied based on a constant counting sum of NPPs. This approach identified main drivers influencing NPP assemblage composition and indicated that a change in these drivers occurred at the onset of the Pre-Roman Iron Age. Twelve known and 12 specific, but so far unknown, NPPs correlating with the environmental variables were identified and their possible indicator value presented. The mild preparation procedure resulted in a very large dataset allowing for sub-assemblages to be explored separately. This approach indicated the potential for identifying further environmental indicators among these groups.

WAR: What Else Is It Good For? A Comparison of Maximum Difference Scaling, Adaptive Choice-based Conjoint, Constant Sum Scaling and Wallet Allocation Rule Utilities

Purpose – This research compares the results of Maximum Difference Scaling (MaxDiff), Adaptive Choice-Based Conjoint (ACBC), and constant sum scaling with the results of a Wallet Allocation Rule (WAR) approach for identifying the features of smartphones deemed most valuable by consumers. Design/methodology/approach – The authors examine the responses of 554 recent purchasers and 598 intended purchasers of smartphones about the features of smartphones that they have considered/will consider in their purchase decisions and the relative value they assign to each feature. MaxDiff, ACBC, constant sum scaling and WAR are used for quantifying the importance of these features and correlation analysis is used to compare the results of the four methods. Findings – The authors find that constant sum scaling and WAR provide nearly identical results at both the aggregate and individual levels, while MaxDiff and ACBC produce similar results to WAR at the aggregate level only. Originality/value – This research validates the feasibility of a simple method for estimating the utility of product features by demonstrating the similarity of results to more established and/or complex methods. The availability of a simpler method creates value by allowing more firms to engage in this kind of co-creative research while reducing costs for customers to participate.

Distance matrices and quadratic embedding of graphs

A connected graph is said to be of Q E class if it admits a quadratic embedding in a Hilbert space, or equivalently, if the distance matrix is conditionally negative definite. Several criteria for a graph to be of Q E class are derived from the point of view of graph operations. For a quantitative criterion the Q E constant is introduced and concrete examples are shown with explicit calculation. If the distance matrix admits a constant row sum, the Q E constant coincides with the second largest eigenvalue of the distance matrix. The Q E constants are determined for all graphs on n vertices with n ≤ 5 , among which two are not of Q E class.