Dual Basis Research Articles

Tensor reduction of loop integrals

The computational cost associated with reducing tensor integrals to scalar integrals using the Passarino-Veltman method is dominated by the diagonalisation of large systems of equations. These systems of equations are sized according to the number of independent tensor elements that can be constructed using the metric and external momenta. In this article, we present a closed-form solution of this diagonalisation problem in arbitrary tensor integrals. We employ a basis of tensors whose building blocks are the external momentum vectors and a metric tensor transverse to the space of external momenta. The scalar integral coefficients of the basis tensors are obtained by mapping the basis elements to the elements of an orthogonal dual basis. This mapping is succinctly expressed through a formula that resembles the ordering of operators in Wick’s theorem.Finally, we provide examples demonstrating the application of our tensor reduction formula to Feynman diagrams in QCD 2 → 2 scattering processes, specifically up to three loops.

Examples of Riesz Bases of Exponentials for Multi-tiling Domains and Their Duals

A well-studied problem in sampling theory is to find an expansion of a function in terms of a Riesz basis of exponentials for L2(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(\\Omega )$$\\end{document}, where Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} is a bounded, measurable set. For such a basis, we are guaranteed the existence of a unique biorthogonal dual basis that can be used to calculate the expansion coefficients. Much attention has been paid to the existence of Riesz bases of exponentials for various domains; however, the sampling and reconstruction problems in these cases are less understood. Recently, explicit formulas for the corresponding dual Riesz bases were introduced in Frederick and Okoudjou in [Appl Comput Harmon Anal 51:104–117, 2021; Frederick and Mayeli in J Fourier Anal Appl 27(5):1–21, 2021] for a class of multi-tiling domains. In this paper, we further this work by presenting explicit examples of a finite co-measurable union of intervals or multi-rectangles. In the higher-dimensional case, we also discuss how different sampling strategies lead to different Riesz bounds.

Grey Interest Chain Identification and Control Model for Government Investment Engineering Projects Based on Node Identification

In the bidding process of government investment engineering projects, collusion between the government and bidders occurs repeatedly, which seriously affects the quality of engineering projects and the effectiveness of the government investment. Therefore, it is necessary to analyze and discuss the collusion between the government and bidders in government investment engineering projects, so as to provide a healthy and sustainable environment for the government investment engineering bidding market. There are two main types of collusion in engineering bidding: horizontal collusion and vertical collusion, and this paper focuses on the vertical collusion process in the engineering bidding process. A conceptual framework of the grey interest chain based on three stages of benefit creation, benefit distribution, and benefit realization was established, 15 major nodes in the grey interest chain were identified, and a grey interest chain control model was constructed, which further identified and classified the nodes into four levels: key nodes, important nodes, general nodes, and unimportant nodes. Finally, through the application of the model in the case, measures such as establishing a cracking mechanism for grey resource integration, increasing the supervision of grey interest chain, and strengthening post-bid audit are proposed. Measures such as including the preparation of bidding documents into the work assessment system and entrusting consulting units or third parties to prepare bidding documents are proposed to establish a crack mechanism for grey resource integration. In the benefit distribution stage, the penalties for the government and the bidders can be appropriately increased, the responsibilities of enterprises and project leaders can be implemented in the system on a reciprocal basis, and a perfect reputation mechanism information can be established. At the stage of benefit realization, the bidding system should be improved and post-bid audit should be strengthened to increase the difficulty of grey benefit realization. This paper will provide a reference for the prevention and governance of vertical collusion in bidding and tendering.

A dual basis approach to multidimensional scaling

Classical multidimensional scaling (CMDS) is a technique that embeds a set of objects in a Euclidean space given their pairwise Euclidean distances. The main part of CMDS involves double centering a squared distance matrix and using a truncated eigendecomposition to recover the point coordinates. In this paper, motivated by a study in Euclidean distance geometry, we explore a dual basis approach to CMDS. We give an explicit formula for the dual basis vectors and fully characterize the spectrum of an essential matrix in the dual basis framework. We make connections to a related problem in metric nearness.

Rational Cherednik algebras of G(ℓ,p,n) from the Coulomb perspective

We prove a number of results on the structure and representation theory of the rational Cherednik algebra of the imprimitive reflection group G(ℓ,p,n). In particular, we: (1) show a relationship to the Coulomb branch construction of Braverman, Finkelberg, and Nakajima, and 3-dimensional quantum field theory; (2) show that the spherical Cherednik algebra carries the structure of a principal Galois order; (3) construct a graded lift of category O and the larger category of Dunkl-Opdam modules, whose simple modules have the properties of a dual canonical basis and (4) give the first classification of simple Dunkl-Opdam modules for the rational Cherednik algebra of the imprimitive reflection group G(ℓ,p,n).

Numerical Solutions of High-Order Differential Equations with Polynomial Coefficients Using a Bernstein Polynomial Basis

The paper presents a novel method that allows one to establish numerical solutions of linear and nonlinear ordinary differential equations—with polynomial coefficients—that contain any finite products of the unknown functions and/or their general derivatives. The presented algorithm provides numerical solutions of these differential equations subject to initial or boundary conditions. This algorithm proposes the desired solution in terms of B-polynomials (Bernstein polynomial basis) and then uses the orthonormal relation of B-polynomials with its weighted dual basis with respect to the Jacobi weight function to construct a linear/nonlinear system in the unknown expansion coefficients which can be solved using a suitable solver. The properties of B-polynomials provide greater flexibility in which to impose the initial or boundary conditions at the end points of the interval [0, R] and enable us to obtain exactly and explicitly some of the unknown expansion coefficients in the form of a suggested numerical solution. Consequently, the presented algorithm leads to a linear or nonlinear algebraic system in the unknown expansion coefficients that has a simpler form than that was obtained by the other algorithms. So that, this procedure is a powerful tool that we may utilize to overcome the difficulties associated with boundary and initial value problems with less computational effort than the other techniques. An accepted agreement is obtained between the exact and approximate solutions for the given examples. The error analysis was also studied, and the obtained numerical results clarified the validity of the theoretical results.

India–US Strategic Cooperation through the LEMOA Lens

Since Narendra Modi took office, India and the United States’ strategic ties have become stronger. It took almost 12 years to sign the ‘Logistics Exchange Memorandum of Agreement (LEMOA)’. The LEMOA offers a framework for governing the logistical assistance, supplies and services exchanged between the Indian and US forces on a reciprocal and reimbursable basis. Concerns in India’s strategic elite and political class over the endangerment of India’s strategic autonomy and compromising India’s sovereignty for a long time prevented the execution of this agreement. However, after the BJP government came to power in 2014, it negotiated this agreement with the United States. India has strategically moved closer to the United States by signing this agreement, which will have a lot of positive effects on the development of India’s strategic capabilities and the strengthening of Indian defence forces’ operational capabilities. It will also allow India to expand its strategic influence in the Indo-Pacific region. This study analyses the LEMOA, its implications for India’s strategic autonomy, and its subsequent implications for expanding India’s strategic footprint in the region.

Towards higher-order accurate mass lumping in explicit isogeometric analysis for structural dynamics

We present a mass lumping approach based on an isogeometric Petrov–Galerkin method that preserves higher-order spatial accuracy in explicit dynamics calculations irrespective of the polynomial degree of the spline approximation. To discretize the test function space, our method uses an approximate dual basis, whose functions are smooth, have local support and satisfy approximate bi-orthogonality with respect to a trial space of B-splines. The resulting mass matrix is “close” to the identity matrix. Specifically, a lumped version of this mass matrix preserves all relevant polynomials when utilized in a Galerkin projection. Consequently, the mass matrix can be lumped (via row-sum lumping) without compromising spatial accuracy in explicit dynamics calculations. We also address the imposition of Dirichlet boundary conditions and the preservation of approximate bi-orthogonality under geometric mappings. In addition, we establish a link between the exact dual and approximate dual basis functions via an iterative algorithm that improves the approximate dual basis towards exact bi-orthogonality. We demonstrate the performance of our higher-order accurate mass lumping approach via convergence studies and spectral analyses of discretized beam, plate and shell models.

METHODOLOGICAL FOUNDATIONS OF INNOVATIVE MODERNIZATION OF THE NATIONAL ECONOMY

The methodological tools of innovative modernization of the national economy are investigated.
 It has been determined that for theoretical substantiation of innovative modernization of the national economy, determination of trajectory and clarification of objective tendencies of evolutionary movement, forecasting prospects for formation of innovative system at the level of industry or economy as a whole, the most acceptable methodological platform is evolutionary approach.
 It is stated that the methodology of evolutionary approach is based on biological analogies, evolutionary language of Darwinism, mechanisms and elements of heredity, variability, selection, which determine the evolution of economic institutions. To transfer evolutionary ideas to the study of economic dynamics, a unit of selection is allocated, or an institution that has stability over time, the ability to change, can be transformed.
 It is argued that innovative changes in the national economy are complicated by the dual institutional basis of society and economic entities, which are not always motivated for competitive relations and innovative changes. It is emphasized that innovative modernization of the national economy requires a qualitatively different economic mechanism, which is focused on overcoming the technological lag and structural imbalance and implements the modern trend of economic growth on the basis of high-tech innovative production.
 It is substantiated that the process of modernization requires taking measures to form effective basic institutions of the national economy, efficiency of market mechanisms, mobilization of economic resources and scientific potential for the development of fundamental research and applied developments and implementation of a modern model of the innovation process. 
 It is proved that the methodological synergy of evolutionary and institutional approaches in studying the prospects for innovative modernization of the national economy determines the preservation of links between the essence and form of manifestations of innovations and allows analysing economic institutions in the process of evolution, exploring the influences of evolutionary mechanisms and provides a higher level of theoretical generalizations.

Examining Authorship Dynamics, Publication Patterns, and Affiliations in the Pakistan Journal of Medical and Health Sciences (PJMHS) from 2009-2019

Introduction: The Pakistan Journal of Medical and Health Sciences have been peer-reviewed and is included in the Scopus-Elsevier database. This study represents (PJMHS)'s contribution to research productivity from 2019 to 2009. Methods and Materials: The published data records were obtained from the PJMHS website and the Scopus-Elsevier database for MS Office Excel Sheet tabulation. The frequency of publications, the type of papers, the use of pages, citations, and the initial author's affiliation with their country. Results: In 43 issues, 15401 (4%) authors wrote 3817 items in seven categories. The average number of pages consumed was (2.8%), with 896 (23.4%) citations in publications. The collaboration of three authors provided 1339 (36.6%) papers and received the top spot, while four authors, 3344 (21.7%), contributed 836 (22%). The first authors from 29 nations engaged in publications, with Pakistan leading the way with 3484 (91%), Iran second with 79 (2%), Indonesia third with 77 (2%), and the Kingdom of Saudi Arabia fourth with 65 (1.7%). Practical Implication: The information assess the influence and relevance of the research published in the Pakistan Journal of Medical and Health Sciences. In nut-shell, the published data empowers readers to stay informed about the journal's publication trends, authorship patterns, and the breadth of content available. It enables researchers to identify potential collaborators, explore different document types, and gauge the impact of published articles through citation analysis. Conclusion: Pakistan Journal of Medical and Health Sciences (PJMHS) is a peer-reviewed journal that is published continuously and assists medical and health sciences research communities in gaining their attention in sharing ideas on a reciprocal basis from similar specialties or sub-specialties of common disciplines Keywords: Bibliometric, Pakistan Journal of Medical and Health Sciences, Pakistan, medical education and research, bibliometric, Zameer Baladi.

Optimization of captain’s safety management

The article is devoted to the maritime safety issues in terms of captain’s management activity. Effective and safe management of captains can be achieved by identifying the importance of considering psychophysiological factors in the selection of seamen personnel, including captains, in conjunction with their managerial qualifications and high level of legal culture (knowledge and real willingness to strictly follow legal prescriptions and regulations), and by using a formal safety assessment method in relation to maritime crew management risks.
 The authors made a conclusion that the causes of accidents are rooted in the conflict between two main components: subjective (behavior – managerial decision) and objective (vessel and situation), in fact, the captain's personality (or behavior) complies with the requirements of the situation or technical condition of the vessel.
 Psychophysiological selection aims to identify abilities and qualities that appropriate professions, accompanied by significant neuropsychological strain, the severity of stressful situations, etc. It appeared that reaction disorder play a decisive role in one-third of all accidents, ergo, psychophysiological selection can reduce maritime casualties by 20-25%. In this respect, the application of the formal safety assessment method in combination with psychophysiological selection in the context of optimizing captain's safety management is proposed. The authors considered the professional selection of captains based on stable psychophysiological states and qualification while assessing the risks via formal safety assessment methodology as a dual basis of the maritime safety.
 The functional dependence of maritime safety on the policy of the shipping companies is defined: a) risk assessment, which allows to connect legal norms of safety directly with actions of the captain as much as possible, based on modelling of a situation; b) personnel activity, carried out on the basis of professional and managerial abilities of captains and taking into account the psychophysiological states; c) the level of legal culture (knowledge and strict adherence to safety rules and regulations) of the captain, who operates within the commercial, social and safety sectors, based on the complex of legal rules and regulations governing these sectors.

Biofuel Blends for Desalination Units: Comparison and Assessments

Although desalinations with renewables were introduced some time ago, conventional desalination units are still applied. Conventional desalinations account for 90% of desalinations worldwide. Yet, they have two significant issues: a high demand for energy and a high level of environmental contaminants. Such issues are studied and remedies are suggested in the current study. Varieties of biofuel blends in dual and ternary bases are investigated experimentally for indirect desalination. Results showed that ternary blends can introduce lower desalination potentials than fossil fuels by about 4–7%. The best ternary blends for the indirect desalination process are iBE, followed by niB, and finally EM. The EGT of iBE is greater than niB and EM by about 1.1 and 1.2%, respectively. Both n-butanol/iso-butanol–gasoline dual blends introduced an almost similar desalination potential as the ternary blends (e.g., lower desalination by about 4.4 and 4.7%). Nevertheless, bio-ethanol/bio-methanol–gasoline dual blends introduced greater desalination potentials than the fossil fuel by 3.2 and 3%, respectively. Regarding environmental issues, both ternary and dual blends introduced lower CO and UHC emissions than fossil fuels in varying degrees. M presented the lowest CO by about 30%, followed by EM by about 21%, and lastly E by about 20%, compared to G. However, the lowest UHC is presented by EM followed by nB and niB with rates of 18, 16.2, and 13.5%. Results also showed that the engine speed has a considerable effect on the desalination process and environment; low engine speed is recommended in the case of applying ternary blends, as well as dual n-butanol/iso-butanol–gasoline blends. Alternatively, in the case of applying bio-ethanol/bio-methanol–gasoline dual blends, moderate engine speed is preferable.

Geometric properties of the Kazhdan–Lusztig Schubert basis

We study classes determined by the Kazhdan-Lusztig basis of the Hecke algebra in the $K$-theory and hyperbolic cohomology theory of flag varieties. We first show that, in $K$-theory, the two different choices of Kazhdan-Lusztig bases produce dual bases, one of which can be interpreted as characteristic classes of the intersection homology mixed Hodge modules. In equivariant hyperbolic cohomology, we show that if the Schubert variety is smooth, then the class it determines coincides with the class of the Kazhdan-Lusztig basis; this was known as the Smoothness Conjecture. For Grassmannians, we prove that the classes of the Kazhdan-Lusztig basis coincide with the classes determined by Zelevinsky's small resolutions. These properties of the so-called KL-Schubert basis show that it is the closest existing analogue to the Schubert basis for hyperbolic cohomology; the latter is a very useful testbed for more general elliptic cohomologies.

Loop-by-loop differential equations for dual (elliptic) Feynman integrals

We present a loop-by-loop method for computing the differential equations of Feynman integrals using the recently developed dual form formalism. We give explicit prescriptions for the loop-by-loop fibration of multi-loop dual forms. Then, we test our formalism on a simple, but non-trivial, example: the two-loop three-mass elliptic sunrise family of integrals. We obtain an ε-form differential equation within the correct function space in a sequence of relatively simple algebraic steps. In particular, none of these steps relies on the analysis of q-series. Then, we discuss interesting properties satisfied by our dual basis as well as its simple relation to the known ε-form basis of Feynman integrands. The underlying K3-geometry of the three-loop four-mass sunrise integral is also discussed. Finally, we speculate on how to construct a “good” loop-by-loop basis at three-loop.

K-positivity of dual canonical basis elements from 1324- and 2143-avoiding Kazhdan–Lusztig immanants

In this note, we show that certain dual canonical basis elements of ℂ[SL m ] are positive when evaluated on k-positive matrices, matrices whose minors of size k×k and smaller are positive. Skandera showed that all dual canonical basis elements of ℂ[SL m ] can be written in terms of Kazhdan–Lusztig immanants, which were introduced by Rhoades and Skandera. We focus on the basis elements which are expressed in terms of Kazhdan–Lusztig immanants indexed by 1324- and 2143-avoiding permutations. This extends previous work of the authors on Kazhdan–Lusztig immanants and uses similar tools, namely Lewis Carroll’s identity (also known as the Desnanot-Jacobi identity).

MULTILAYER ORTHOGONAL STRUCTURES BASED ON MAXIMUM LENGTH SEQUENCES

The work is devoted to the study of the properties of maximum length sequences, the most well-known contemporary representatives of noise-like signals. The unique properties of these combinations, associated primarily with recurrent dependencies within linear sections, make it possible to use special mathematical processing methods based on the dual basis of the Galoisfield. This mathematical apparatus of linear algebra provides the determination of the selected M-sequence current phase according to any of its fragments with preset length. As indicated hereto, such processing algorithms are applicable not only for single M-sequences, but also for multilayer orthogonal constructions created on their basis. These constructions, in turn, can be successfully used in future for the code division technology implementation. The new methods of processing such structures offered in the work should contribute to the development of transmitted signal monitoring algorithms, which will enable tracking its distortions and phase failures. Thus, the results of this work can be used in code division multiple access systems.

Whittaker Categories, Properly Stratified Categories and Fock Space Categorification for Lie Superalgebras

We study various categories of Whittaker modules over a type I Lie superalgebra realized as cokernel categories that fit into the framework of properly stratified categories. These categories are the target of the Backelin functor $$\Gamma _\zeta $$ . We show that these categories can be described, up to equivalence, as Serre quotients of the BGG category $$\mathcal O$$ and of certain singular categories of Harish-Chandra $$({\mathfrak {g}},{\mathfrak {g}}_{{\bar{0}}})$$ -bimodules. We also show that $$\Gamma _\zeta $$ is a realization of the Serre quotient functor. We further investigate a q-symmetrized Fock space over a quantum group of type A and prove that, for general linear Lie superalgebras our Whittaker categories, the functor $$\Gamma _\zeta $$ and various realizations of Serre quotients and Serre quotient functors categorify this q-symmetrized Fock space and its q-symmetrizer. In this picture, the canonical and dual canonical bases in this q-symmetrized Fock space correspond to tilting and simple objects in these Whittaker categories, respectively.

Solving Fractional Optimal Control Problems Involving Caputo–Fabrizio Derivative Using Hermite Spline Functions

In the current research, we develop a collocation method based on the biorthogonal Hermite cubic spline functions to solve a class of fractional optimal control problems using Caputo–Fabrizio derivative operator. We design dual bases for Hermite cubic spline functions for the first time in this work. So, we present two direct and efficient algorithms to solve a class of fractional optimal control problems. Two new operational matrices of Caputo–Fabrizio fractional derivative are derived using Hermite cubic spline functions. By employing these matrices, we reduce these problems to systems of algebraic equations. Illustrative examples are examined to demonstrate the important features of the new algorithm.

“Death to the Slavs!” The Italian-Yugoslav relations on mutual minorities and the impact of the 1961 Trieste Riots (1954-1964)

After the 1954 Memorandum of Understanding of London, Italy and Yugoslavia settled their border dispute by partitioning what was formerly the Free Territory of Trieste. Furthermore, they also agreed to extend to each other?s national minorities living in the two zones of the former Free Territory the protection measures established by the Special Statute, an annex to the Memorandum. Neither of the two countries fully complied with the Special Statute but wanted it to be implemented in the Zone administered by the other side. Italy wanted to negotiate all further implementations and new concessions on the basis of reciprocity, the main rationale for the protection measures stipulated in the Special Statute. In contrast, Yugoslavia wanted Italy to agree to the unilateral implementation of the Special Statute in Trieste and to other concessions. This led to a stalemate in the negotiations and consequently to the poor enforcement of the Special Statute, which caused rising tensions on the local level, even though, in the meantime, the overall diplomatic relations between the two countries continued to improve. Yugoslavia?s increasing requests for unilateral implementation of the Special Statute and the Yugoslav-funded Slovene organisations in Italy resulted in a series of major anti-Yugoslav and anti-Slovene demonstrations in Trieste. After these riots and the ensuing debate on the minority issue, the need to rely on reciprocity also became evident to the Yugoslav diplomacy. Yugoslavia, therefore, dropped its old policy and started to improve the treatment of its Italian minority and agreed to negotiate based on the principle of reciprocity. This led to quick benefits for both minorities, and a new path that led Italy and Yugoslavia to sign new agreements on their mutual minorities, going even beyond the Special Statute.

Second-Order Nédélec Curl-Conforming Hexahedral Element for Computational Electromagnetics

We follow a systematic approach to obtain mixed-order curl-conforming basis functions for the hexahedron that are compatible with basis functions for tetrahedra and triangular prisms previously published. The approach is mathematically sound since we obtain the functions as the dual basis with respect to properly discretized Nédélec degrees of freedom. Well-conditioned bases without the need for added orthogonalization procedures are obtained. We provide simple closed-form expressions for second-order basis functions in a reference hexahedron in terms of integer coefficients and monomials. The expressions are ready to use as long as the appropriate geometric mappings are made. We apply the Method of Manufactured Solutions (MMS) to a finite-element double curl vector wave formulation for verification purposes; specifically, we conduct a study of the non-symmetrical structure of the corresponding tensor product finite-element space. We also solve generalized eigenvalue problems for well-known cavities. We provide the open-source code for generating the coefficients, evaluating the basis functions, and computing the finite-element matrices involved in some of the numerical solutions shown in the article.