Theoretic Techniques Research Articles

A decentralized control approach in hypergraph distributed optimization decomposition cases

We study the main decomposition approaches (primal, dual and primal–dual) for a distributed optimization problem from a dynamical system perspective where the couplings among the variables of the optimization problem are described by an undirected, unweighted hypergraph. We conduct stability analysis for the respective dynamical systems of the decomposition cases by using non linear decentralized control theoretic techniques and spectral properties of the respective communication matrices, i.e., the incidence and the Laplacian matrices of the hypergraph. Finally, we provide numerical simulations under a specific coalitional setting that demonstrate the superiority of the hypergraph compared to its respective graph analogue, the clique expansion graph, for the given decomposition algorithms in terms of convergence rate and information transmission efficiency.

On a recent extension of a family of biprojective APN functions

APN functions play a big role as primitives in symmetric cryptography as building blocks that yield optimal resistance to differential attacks. In this note, we consider a recent extension, done by Calderini et al. (2023), of a biprojective APN family introduced by Göloğlu (2022) defined on F22m. We show that this generalization yields functions equivalent to Göloğlu's original family if 3∤m. If 3|m we show exactly how many inequivalent APN functions this new family contains. We also show that the family has the minimal image set size for an APN function and determine its Walsh spectrum, hereby settling some open problems. In our proofs, we leverage a group theoretic technique recently developed by Göloğlu and the author in conjunction with a group action on the set of projective polynomials.

Group Theoretic Approach towards the Balaban Index of Catacondensed Benzenoid Systems and Linear Chain of Anthracene

In this work, we present the analytical closed forms of the Balaban index for anthracene and catacondensed benzenoid systems using group theoretic techniques. The Balaban index is a distance-based topological index that provides valuable information about the properties of chemical structures. We emphasize the importance of determining analytical closed forms of the Balaban index for catacondensed benzenoid systems and linear chains of anthracene, as it enables a deeper understanding of these systems and their behavior. Our analysis utilizes the group action of the automorphism group of these chains on the set of vertices, which refer to the points where the chains intersect. In future work, we plan to determine the Balaban index of other polymeric linear chains using group theoretic techniques and extend the applications of this index to other fields, such as materials science and biology. It is clear that the Balaban index remains a valuable tool in theoretical and computational chemistry, and its applications are constantly evolving.

Amplification of the anomalous scaling in the Kazantsev-Kraichnan model with finite-time correlations and spatial parity violation.

By using the field theoretic renormalization group technique together with the operator product expansion, simultaneous influence of the spatial parity violation and finite-time correlations of an electrically conductive turbulent environment on the inertial-range scaling behavior of correlation functions of a passively advected weak magnetic field is investigated within the corresponding generalized Kazantsev-Kraichnan model in the second order of the perturbation theory (in the two-loop approximation). The explicit dependence of the anomalous dimensions of the leading composite operators on the fixed point value of the parameter that controls the presence of finite-time correlations of the turbulent field as well as on the parameter that drives the amount of the spatial parity violation (helicity) in the system is found even in the case with the presence of the large-scale anisotropy. In accordance with the Kolmogorov's local isotropy restoration hypothesis, it is shown that, regardless of the amount of the spatial parity violation, the scaling properties of the model are always driven by the anomalous dimensions of the composite operators near the isotropic shell. The asymptotic (inertial-range) scaling form of all single-time two-point correlation functions of arbitrary order of the passively advected magnetic field is found. The explicit dependence of the corresponding scaling exponents on the helicity parameter as well as on the parameter that controls the finite-time velocity correlations is determined. It is shown that, regardless of the amount of the finite-time correlations of the given Gaussian turbulent environment, the presence of the spatial parity violation always leads to more negative values of the scaling exponents, i.e., to the more pronounced anomalous scaling of the magnetic correlation functions. At the same time, it is shown that the stronger the violation of spatial parity, the larger the anomalous behavior of magnetic correlations.

Fusion category symmetry. Part I. Anomaly in-flow and gapped phases

We study generalized discrete symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. In particular, we describe ’t Hooft anomalies and classify gapped phases stabilized by these symmetries, including new 1+1D topological phases. The algebra of these operators is not a group but rather is described by their fusion ring and crossing relations, captured algebraically as a fusion category. Such data defines a Turaev-Viro/Levin-Wen model in 2+1D, while a 1+1D system with this fusion category acting as a global symmetry defines a boundary condition. This is akin to gauging a discrete global symmetry at the boundary of Dijkgraaf-Witten theory. We describe how to “ungauge” the fusion category symmetry in these boundary conditions and separate the symmetry-preserving phases from the symmetry-breaking ones. For Tambara-Yamagami categories and their generalizations, which are associated with Kramers-Wannier-like self-dualities under orbifolding, we develop gauge theoretic techniques which simplify the analysis. We include some examples of CFTs with fusion category symmetry derived from Kramers-Wannier-like dualities as an appetizer for the Part II companion paper.

Thermodynamic modeling of the solid-liquid equilibrium of two phenolic compounds in several solvents at temperatures ranging from 283.15 K to 323.15 K

The solubility of two natural phenolic compounds (vanillic acid (VA) and vanillin (V)) in four individual solvents (ethanol, butan-2-ol, ethylene glycol, and ethyl acetate) was measured using the gravimetric method from 283.15 to 323.15 K at atmospheric pressure. The optimization of the complexes between VA–solvents and V–solvents using the B3LYP theoretic technique and the 6-31+G (d, p) basis set indicated that the absolute value of interaction energy was higher, suggesting that the interaction between solute and the solvent was stronger. In addition, two excellent activity coefficient models (NRTL and Wilson) were used to correlate and evaluate the equilibrium solubility results. The highest relative average deviation and root-mean-square deviation values for VA are 4.36% and 3.09 × 10-3, respectively, and 3.35% and 5.52 × 10-3 for V. Furthermore, the excess enthalpy of the solution was determined on the basis of thermodynamic relations and the Wilson model. The findings indicate that the dissolution process was endothermic.
 KEY WORDS: Solid-liquid equilibrium, Vanillin, Vanillic acid, NRTL and Wilson models, Thermodynamic parameters
 Bull. Chem. Soc. Ethiop. 2024, 38(3), 799-810. 
 DOI: https://dx.doi.org/10.4314/bcse.v38i3.19 

Distinct Topological Properties of the Reward Anticipation Network in Preadolescent Children With Binge Eating Disorder Symptoms

Few studies have considered the neural underpinnings of binge eating disorder (BED) in children, despite clinical and subclinical symptom presentation occurring in this age group. Symptom presentation at this age is of clinical relevance, as early onset of binge eating is linked to negative health outcomes. Studies in adults have highlighted dysfunction in the frontostriatal reward system as a potential candidate for binge eating pathophysiology, although the exact nature of such dysfunction is currently unclear. Data from 83 children (mean age 9.9 years, SD= 0.60) with symptoms of BED (57% girls) and 123 control participants (mean age 10.0 years, SD= 0.60) (52% girls) were acquired from the 4.0 baseline release of the Adolescent Brain Cognitive Development Study. Task-based graph theoretic techniques were used to analyze data from anticipation trials of the monetary incentive delay task. Network and nodal properties were compared between groups. The BED-S group showed alterations in topological properties associated with the frontostriatal subnetwork, such as reduced nodal efficiency in the superior frontal gyrus, nucleus accumbens, putamen, and in normal sex-difference patterns of these properties, such as diminished girls-greater-than-boys pattern of betweenness-centrality in nucleus accumbens observed in controls. Distinct network properties and sex-difference patterns in preadolescent children with BED-S suggest dysregulation in the reward system compared to those of matched controls. For the first time, these results quantify this dysregulation in terms of systems-level properties during anticipation of monetary reward and significantly inform the early and sex-related brain markers of BED symptoms. Binge eating disorder is the most common eating disorder. One factor that may contribute to binge eating is dysregulation of the reward system in the brain. This study analyzed brain activity during anticipation of monetary rewards in 83 youth with and 123 children without binge eating disorder symptoms from the Adolescent Brain Cognitive Development Study. The authors found specific alterations in the frontostriatal system, responsible for reward processing, in children with binge eating disorder symptoms, compared to the control group, suggesting dysregulation of the reward system.

Enriched Presheaf Model of Quantum FPC

Selinger gave a superoperator model of a first-order quantum programming language and proved that it is fully definable and hence fully abstract. This paper proposes an extension of the superoperator model to higher-order programs based on modules over superoperators or, equivalently, enriched presheaves over the category of superoperators. The enriched presheaf category can be easily proved to be a model of intuitionistic linear logic with cofree exponential, from which one can cave out a model of classical linear logic by a kind of bi-orthogonality construction. Although the structures of an enriched presheaf category are usually rather complex, a morphism in the classical model can be expressed simply as a matrix of completely positive maps. The model inherits many desirable properties from the superoperator model. A conceptually interesting property is that our model has only a state whose “total probability” is bounded by 1, i.e. ‍does not have a state where true and false each occur with probability 2/3. Another convenient property inherited from the superoperator model is a ωCPO-enrichment. Remarkably, our model has a sufficient structure to interpret arbitrary recursive types by the standard domain theoretic technique. We introduce Quantum FPC , a quantum λ-calculus with recursive types, and prove that our model is a fully abstract model of Quantum FPC.

A novel model for representing a plane target and finding the worst-case coverage in wireless sensor network based on Clifford algebra

Wireless ad hoc sensor networks have recently emerged as a premier research topic. They have great long-term economic potential and ability to transform our lives and pose many new system building challenges. Sensor networks also pose a number of new conceptual and optimization problems. Most of researches in wireless sensor networks are focused in obtaining better target coverage in order to reduce energy and cost of the network. The problem of planar target analysis is one of the crucial problems that should be considered while studying coverage problem of sensor networks. By combining computational geometry and graph theoretic techniques, specifically the Voronoi diagram and graph search algorithms, this paper introduces a novel sensor network coverage model that deals with plane target problem based on Clifford algebra which is a powerful tool that is coordinate free. Also, the calculations of the node coverage rate for the plane target in the sensor network using Clifford algebra are presented. Then, the maximum clearance path (worst-case coverage) of the sensor network for a plane target is proposed. The optimality and reliability of the proposed algorithm have been proved using simulation. Also, a comparison between the breach weight of the point target and the plane target is provided.

Existence of horizontal immersions in fat distributions

Contact structures, as well as their holomorphic and quaternionic counterparts, are the primary examples of strongly bracket generating (or fat) distributions. In this paper, we associate a numerical invariant to corank-[Formula: see text] fat distribution on manifolds, referred to as the degree of the distribution. The real distribution underlying a holomorphic contact structure is of degree [Formula: see text]. Using Gromov’s sheaf theoretic and analytic techniques of [Formula: see text]-principle, we prove the existence of horizontal immersions of an arbitrary manifold into degree [Formula: see text] fat distributions and the quaternionic contact structures. We also study isocontact immersions, i.e. immersions that pull back a fat distribution to a given contact structure.

DISTRIBUTION OF FROBENIUS ELEMENTS IN FAMILIES OF GALOIS EXTENSIONS

AbstractGiven a Galois extension $L/K$ of number fields, we describe fine distribution properties of Frobenius elements via invariants from representations of finite Galois groups and ramification theory. We exhibit explicit families of extensions in which we evaluate these invariants and deduce a detailed understanding and a precise description of the possible asymmetries. We establish a general bound on the generic fluctuations of the error term in the Chebotarev density theorem, which under GRH is sharper than the Murty–Murty–Saradha and Bellaïche refinements of the Lagarias–Odlyzko and Serre bounds, and which we believe is best possible (assuming simplicity, it is of the quality of Montgomery’s conjecture on primes in arithmetic progressions). Under GRH and a hypothesis on the multiplicities of zeros up to a certain height, we show that in certain families, these fluctuations are dominated by a constant lower order term. As an application of our ideas, we refine and generalize results of K. Murty and of Bellaïche, and we answer a question of Ng. In particular, in the case where $L/{\mathbb {Q}}$ is Galois and supersolvable, we prove a strong form of a conjecture of K. Murty on the unramified prime ideal of least norm in a given Frobenius set. The tools we use include the Rubinstein–Sarnak machinery based on limiting distributions and a blend of algebraic, analytic, representation theoretic, probabilistic and combinatorial techniques.

Shock waves, periodic, topological kink and singular soliton solutions of a new generalized two dimensional nonlinear wave equation of engineering physics with applications in signal processing, electromagnetism and complex media

This article investigates a new generalized two-dimensional nonlinear wave equation of engineering physics with various applications in the fields of sciences and engineering. In this study, shock wave and solitary wave solutions were secured via the sine–Gordon technique. Moreover, various new group invariants along-side exact classical results of the equation were achieved through the utilization of Lie group theoretic techniques. Some of the solutions are gained with regards to Weierstrass functions, complex soliton, topological kink soliton as well as singular soliton. Besides, several algebraic and other solitary-wave-type solutions are obtained. Wave dynamics of the solutions are plotted to give more physical meanings to the obtained results and have a better knowledge of what the nonlinear wave equation represents in terms of physical phenomena. Application of the secured results in engineering (signal processing), physics (electromagnetism) and complex media are presented.

Finite-temperature equilibrium density profiles of integrable systems in confining potentials.

We study the equilibrium density profile of particles in two one-dimensional classical integrable models, namely hard rods and the hyperbolic Calogero model, placed in confining potentials. For both of these models the interparticle repulsion is strong enough to prevent particle trajectories from intersecting. We use field theoretic techniques to compute the density profile and their scaling with system size and temperature, and we compare them with results from Monte Carlo simulations. In both cases we find good agreement between the field theory and simulations. We also consider the case of the Toda model in which interparticle repulsion is weak and particle trajectories can cross. In this case, we find that a field theoretic description is ill-suited and instead, in certain parameter regimes, we present an approximate Hessian theory to understand the density profile. Our work provides an analytical approach toward understanding the equilibrium properties for interacting integrable systems in confining traps.

Group theoretic transformation techanique for MHD viscous fluid with heat transfer over a non-linear porous surface

Using group theoretic transformation technique, class of similarity solution of heat transfer for MHD boundary layer flow of viscous fluid over a non linear porous surface. This similarity equation which is highly nonlinear ordinary differential equations which is solved numerically for particular fluids so called power-law fluids. The results obtain are found in good agreement with those available in literature.

Flood Risk Mitigation and Valve Control in Stormwater Systems: State-Space Modeling, Control Algorithms, and Case Studies

The increasing access to inexpensive sensors, computing power, and more accurate forecasting of storm events provides unique opportunities to shift flood management practices from static approaches to an optimization-based real-time control (RTC) of urban drainage systems. Recent studies have addressed a plethora of strategies for flood control in stormwater reservoirs; however, advanced control theoretic techniques are not yet fully investigated and applied to these systems. In addition, there is an absence of a coupled integrated control model for systems composed of watersheds, reservoirs, and channels for flood mitigation. To this end, we developed a novel nonlinear state-space model of hydrologic and hydrodynamic processes in watersheds, reservoirs, and one-dimensional channels. The model was tested under different types of reservoir control strategies based on real-time measurements (reactive control) and predictions of the future behavior of the system (predictive control) using rainfall forecastings. We applied the modeling approach in a system composed of a single watershed, reservoir, and channel connected in series for the observed rainfall data in San Antonio. Results indicate that for flood mitigation, the predictive control strategy outperforms the reactive controls not only when applied for synthetic design storm events, but also for a continuous simulation. Moreover, the predictive control strategy requires smaller valve operations while still guaranteeing efficient hydrological performance. From the results, we recommend the use of the nonlinear model predictive control strategy to control stormwater systems because of the ability to handle different objective functions, which can be altered according to rainfall forecasting and shift the reservoir operation from flood-based control to strategies focused on increasing detention times, depending on the forecasting.

Graph Theoretic Technique to the Global Stability Analysis of the Discrete Age Structured SVEIR Model with Application to Ebola Virus Disease

Discrete age structured models provide a better approach of discussing the spread of infectious diseases with infectivity, mortality and recovery being age dependent. Measures such as vaccination are carried out in discrete time or applied to individuals in certain age-groups. In the past Ebola Virus disease (EVD) outbreaks, infections in children under 5 years was associated with high mortality rates. In this work, a general discrete SVEIR epidemic model with age structure is formulated. The study establishes the existence of the endemic equilibrium and further shows that it is globally stable using the graph theoretic approach on the method of Lyapunov functions. The model is then applied to EVD dynamics to study the impact of vaccination on the age structured population. The numerical simulation of the discrete age case scenario demonstrates that vaccination of children under 5 years of age against EVD has a great impact in reducing their susceptibility because of the active immune system as compared to the older population who have a poor response to vaccine immunity. For the older population, vaccination is not very eective in reducing their susceptibility.

Semihypergroup-Based Graph for Modeling International Spread of COVID-n in Social Systems

Graph theoretic techniques have been widely applied to model many types of links in social systems. Also, algebraic hypercompositional structure theory has demonstrated its systematic application in some problems. Influenced by these mathematical notions, a novel semihypergroup-based graph (SBG) of G=H,E is constructed through the fundamental relation γn on H, where semihypergroup H is appointed as the set of vertices and E is addressed as the set of edges on SBG. Indeed, two arbitrary vertices x and y are adjacent if xγny. The connectivity of graph G is characterized by xγ*y, whereby the connected components SBG of G would be exactly the elements of the fundamental group H/γ*. Based on SBG, some fundamental characteristics of the graph such as complete, regular, Eulerian, isomorphism, and Cartesian products are discussed along with illustrative examples to clarify the relevance between semihypergroup H and its corresponding graph. Furthermore, the notions of geometric space, block, polygonal, and connected components are introduced in terms of the developed SBG. To formulate the links among individuals/countries in the wake of the COVID (coronavirus disease) pandemic, a theoretical SBG methodology is presented to analyze and simplify such social systems. Finally, the developed SBG is used to model the trend diffusion of the viral disease COVID-n in social systems (i.e., countries and individuals).

Polystability in positive characteristic and degree lower bounds for invariant rings

We develop a representation theoretic technique for detecting closed orbits that is applicable in all characteristics. Our technique is based on Kempf’s theory of optimal subgroups and we make some improvements and simplify the theory from a computational perspective. We exhibit our technique in many examples and in particular, give an algorithm to decide if a symmetric polynomial in n -variables has a closed \operatorname{SL}_n -orbit. As an important application, we prove exponential lower bounds on the maximal degree of a system of generators of invariant rings for two actions that are important from the perspective of Geometric Complexity Theory (GCT). The first is the action of \operatorname{SL}(V) on S^3(V)^{\oplus 3} , the space of 3 -tuples of cubic forms, and the second is the action of \operatorname{SL}(V) \times \operatorname{SL}(W) \times \operatorname{SL}(Z) on the tensor space (V \otimes W \otimes Z)^{\oplus 5} . In both these cases, we prove an exponential lower degree bound for a system of invariants that generate the invariant ring or that define the null cone.

Reducing the question burden of patient reported outcome measures using Bayesian networks

Patient Reported Outcome Measures (PROMs) are questionnaires completed by patients about aspects of their health status. They are a vital part of learning health systems as they are the primary source of information about important outcomes that are best assessed by patients such as pain, disability, anxiety and depression. The volume of questions can easily become burdensome. Previous techniques reduced this burden by dynamically selecting questions from question item banks which are specifically built for different latent constructs being measured. These techniques analyzed the information function between each question in the item bank and the measured construct based on item response theory then used this information function to dynamically select questions by computerized adaptive testing. Here we extend those ideas by using Bayesian Networks (BNs) to enable Computerized Adaptive Testing (CAT) for efficient and accurate question selection on widely-used existing PROMs. BNs offer more comprehensive probabilistic models of the connections between different PROM questions, allowing the use of information theoretic techniques to select the most informative questions. We tested our methods using five clinical PROM datasets, demonstrating that answering a small subset of questions selected with CAT has similar predictions and error to answering all questions in the PROM BN. Our results show that answering 30% − 75% questions selected with CAT had an average area under the receiver operating characteristic curve (AUC) of 0.92 (min: 0.8 – max: 0.98) for predicting the measured constructs. BNs outperformed alternative CAT approaches with a 5% (min: 0.01% – max: 9%) average increase in the accuracy of predicting the responses to unanswered question items.

A fast, high-order scheme for evaluating volume potentials on complex 2D geometries via area-to-line integral conversion and domain mappings

While potential theoretic techniques have received significant interest and found broad success in the solution of linear partial differential equations (PDEs) in mathematical physics, limited adoption is reported in the case of nonlinear and/or inhomogeneous problems (i.e. with distributed volumetric sources) owing to outstanding challenges in producing a particular solution on complex domains while simultaneously respecting the competing ideals of allowing complete geometric flexibility, enabling source adaptivity, and achieving optimal computational complexity. This article presents a new high-order accurate algorithm for finding a particular solution to the PDE by means of a convolution of the volumetric source function with the Green's function in complex geometries. Utilizing volumetric domain decomposition, the integral is computed over a union of regular boxes (lending the scheme compatibility with adaptive box codes) and triangular regions (which may be potentially curved near boundaries). Singular and near-singular quadrature is handled by converting integrals on volumetric regions to line integrals bounding a reference volume cell using cell mappings and elements of the Poincaré lemma, followed by leveraging existing one-dimensional near-singular and singular quadratures appropriate to the singular nature of the kernel. The scheme achieves compatibility with fast multipole methods (FMMs) and thereby optimal asymptotic complexity by coupling global rules for target-independent quadrature of smooth functions to local target-dependent singular quadrature corrections, and it relies on orthogonal polynomial systems on each cell for well-conditioned, high-order and efficient (with respect to number of required volume function evaluations) approximation of arbitrary volumetric sources. Our domain discretization scheme is naturally compatible with standard meshing software such as Gmsh, which are employed to discretize a narrow region surrounding the domain boundaries. We present 8th-order accurate results, demonstrate the success of the method with examples showing up to 12-digit accuracy on complex geometries, and, for static geometries, our numerical examples show well over 99% of evaluation time of the particular solution is spent in the FMM step.