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At least 1,099 records · Page 61

Experimental Validation of a Command and Control Traffic Detection Model

Network intrusion detection systems (NIDS) are commonly used to detect malware communications, including command-and-control (C2) traffic from botnets. NIDS performance assessments have been studied for decades, but mathematical modeling has rarely been used to explore NIDS performance. This paper details a mathematical model that describes a NIDS performing packet inspection and its detection of malware's C2 traffic. Here, the paper further describes an emulation testbed and a set of cyber experiments that used the testbed to validate the model. These experiments included a commonly used NIDS (Snort) and traffic with contents from a pervasive malware (Emotet). Results are presented for two scenarios: a nominal scenario and a “stressed” scenario in which the NIDS cannot process all incoming packets. Model and experiment results match well, with model estimates mostly falling within 95 % confidence intervals on the experiment means. Model results were produced 70-3000 times faster than the experimental results. Consequently, the model's predictive capability could potentially be used to support decisions about NIDS configuration and effectiveness that require high confidence results, quantification of uncertainty, and exploration of large parameter spaces. Furthermore, the experiments provide an example for how emulation testbeds can be used to validate cyber models that include stochastic variability.

mathematical model↗

Distributed Delay Differential Equation Representations of Cyclic Differential Equations

Compartmental ordinary differential equation (ODE) models are used extensively in mathematical biology. When transit between compartments occurs at a constant rate, the well-known linear chain trick can be used to show that the ODE model is equivalent to an Erlang distributed delay differential equation (DDE). Here, we demonstrate that compartmental models with nonlinear transit rates and possibly delayed arguments are also equivalent to a scalar distributed DDE. To illustrate the utility of these equivalences, we calculate the equilibria of the scalar DDE, and compute the characteristic function---without calculating a determinant. Further, we derive the equivalent scalar DDE for two examples of models in mathematical biology and use the DDE formulation to identify physiological processes that were otherwise hidden by the compartmental structure of the ODE model.

97 MATHEMATICS AND COMPUTING↗

Quantification of Ebola virus replication kinetics in vitro

Mathematical modelling has successfully been used to provide quantitative descriptions of many viral infections, but for the Ebola virus, which requires biosafety level 4 facilities for experimentation, modelling can play a crucial role. Ebola virus modelling efforts have primarily focused on in vivo virus kinetics, e.g., in animal models, to aid the development of antivirals and vaccines. But, thus far, these studies have not yielded a detailed specification of the infection cycle, which could provide a foundational description of the virus kinetics and thus a deeper understanding of their clinical manifestation. Here, we obtain a diverse experimental data set of the Ebola virus infection in vitro, and then make use of Bayesian inference methods to fully identify parameters in a mathematical model of the infection. Our results provide insights into the distribution of time an infected cell spends in the eclipse phase (the period between infection and the start of virus production), as well as the rate at which infectious virions lose infectivity. We suggest how these results can be used in future models to describe co-infection with defective interfering particles, which are an emerging alternative therapeutic.

59 BASIC BIOLOGICAL SCIENCES↗

Practical Understanding of Cancer Model Identifiability in Clinical Applications

Mathematical models are a core component in the foundation of cancer theory and have been developed as clinical tools in precision medicine. Modeling studies for clinical applications often assume an individual’s characteristics can be represented as parameters in a model and are used to explain, predict, and optimize treatment outcomes. However, this approach relies on the identifiability of the underlying mathematical models. In this study, we build on the framework of an observing-system simulation experiment to study the identifiability of several models of cancer growth, focusing on the prognostic parameters of each model. Our results demonstrate that the frequency of data collection, the types of data, such as cancer proxy, and the accuracy of measurements all play crucial roles in determining the identifiability of the model. We also found that highly accurate data can allow for reasonably accurate estimates of some parameters, which may be the key to achieving model identifiability in practice. As more complex models required more data for identification, our results support the idea of using models with a clear mechanism that tracks disease progression in clinical settings. For such a model, the subset of model parameters associated with disease progression naturally minimizes the required data for model identifiability.

59 BASIC BIOLOGICAL SCIENCES↗

MFANS 2024 - Formally Proving Characteristics of Cyber-Physical Systems

Cyber-physical systems (CPS) are engineered systems that rely on the smooth integration of computational algorithms and physical elements. This integration presents new challenges for verifying that systems will behave as expected. The goal of this presentation is to present current challenges and potential solutions for the formal verification of cyber-physical systems. For cyber systems, formal methods refer to systematically rigorous mathematical techniques employed in the specification, development, analysis, and verification of both software and hardware systems. Recent advancements in computer science have yielded sophisticated tools specifically designed to address challenges associated with formal methods in complex systems. These tools leverage various foundational concepts such as logic, formal languages, program semantics, type systems, type theory, and automata theory. A notable achievement in the application of formal methods is the seL4 microkernel, claimed to be the first general-purpose operating-system kernel to be verified. Its proof implies the absence of bugs and guarantees that the kernel meets specifications. For physical systems, dynamic and control theory has a history of using rigorous analytic techniques to prove functional correctness. Lyapunov, optimal, classical, modern, and robust control theories all provide rigorous mathematical methods both to analyze system performance and to design controller that can be guaranteed to meet certain objectives. Recent computational techniques like level set theory and reachability analysis provide assertions that a system's state will avoid unsafe regions. Even though success has been independently achieved for cyber systems and physical systems, the integration of such systems creates new challenges. In particular, there is an obvious discrepancy between finite-state machines and infinite-state systems, resulting in different approaches for modeling and analyzing these system. While it is possible to simulate hybrid systems, this provides only a demonstration of a performance and not proof. For hybrid systems, current formal methods and system analysis approaches typically require a workarounds to work on hybrid systems like CPS. This paper will outline the state of the art and limits of current practice for formally verifying CPS and will identify possible research directions that require attention.

97 MATHEMATICS AND COMPUTING↗

What is the gradient of a scalar function defined on a subspace of square matrices?

We illustrate a technique to calculate the gradient of scalar functions that are defined on any arbitrary matrix subspace. It generalizes our earlier work titled “What is the gradient of a scalar function of a symmetric matrix ?”(Indian Journal of Pure and Applied Mathematics (2022), https://doi.org/10.1007/s13226-022-00313-x), in which we considered the special case of the subspace of symmetric matrices. Extant methods to calculate the gradient in such cases have an inherent flaw which leads to spurious results that populate several publications, as well as respected textbooks and handbooks on matrix calculus. Here, we examine these sources and results in a rigorous and concrete mathematical setting of a finite-dimensional inner-product space and discover the inherent flaw and also a remedy. We demonstrate two ways to calculate the derivative/gradient and second derivative for scalar functions of matrices defined over an arbitrary matrix subspace; the first method is by considering any (differentiable) extension to the space of square matrices and projection of its gradient onto the given subspace. The second method utilizes an ordered basis and computes each component of the gradient through evaluation of the directional derivative. All the ideas presented are illustrated by non-trivial examples, namely, considering the subspace of 3 x 3 circulant and Toeplitz matrices and presenting the results of gradient-descent with both the spurious and correct gradients. Moreover, our bibliography makes it clear that a rigorous approach to matrix calculus is not common in practice, and our presentation of matrix calculus in the language of inner-product spaces will be significant and meaningful for applied mathematicians, engineers and researchers working in inter-disciplinary fields to avoid the conceptual pitfalls that exist.

97 MATHEMATICS AND COMPUTING↗

Mocking the u-plane integral

Abstract The u -plane integral is the contribution of the Coulomb branch to correlation functions of $${\mathcal {N}}=2$$ N = 2 gauge theory on a compact four-manifold. We consider the u -plane integral for correlators of point and surface observables of topologically twisted theories with gauge group $$\mathrm{SU}(2)$$ SU ( 2 ) , for an arbitrary four-manifold with $$(b_1,b_2^+)=(0,1)$$ ( b 1 , b 2 + ) = ( 0 , 1 ) . The u -plane contribution equals the full correlator in the absence of Seiberg–Witten contributions at strong coupling, and coincides with the mathematically defined Donaldson invariants in such cases. We demonstrate that the u -plane correlators are efficiently determined using mock modular forms for point observables, and Appell–Lerch sums for surface observables. We use these results to discuss the asymptotic behavior of correlators as function of the number of observables. Our findings suggest that the vev of exponentiated point and surface observables is an entire function of the fugacities.

Mathematics↗

Bifurcation Analysis Reveals Solution Structures of Phase Field Models

The phase field method is playing an increasingly important role in understanding and predicting morphological evolution in materials and biological systems. Here, in this study, we develop a new analytical approach based on the bifurcation analysis to explore the mathematical solution structure of phase field models. Revealing such solution structures not only is of great mathematical interest but also may provide guidance to experimentally or computationally uncover new morphological evolution phenomena in materials undergoing electronic and structural phase transitions. To elucidate the idea, we apply this analytical approach to three representative phase field equations: the Allen-Cahn equation, the Cahn-Hilliard equation, and the Allen-Cahn-Ohta-Kawasaki system. The solution structures of these three phase field equations are also verified numerically by the homotopy continuation method.

97 MATHEMATICS AND COMPUTING↗

PyOECP: A flexible open-source software library for estimating and modeling the complex permittivity based on the open-ended coaxial probe (OECP) technique

Here, we present PyOECP, a Python-based flexible open-source software for estimating and modeling the complex permittivity obtained from the open-ended coaxial probe (OECP) technique. The transformation of the measured reflection coefficient to complex permittivity is performed based on three different methods. The software library contains the dielectric spectra of common reference liquids, which can be used to transform the reflection coefficient into the dielectric spectra. Several Python routines that are commonly employed (e.g., SciPy and NumPy) in the field of science and engineering are required only so that the users can alter the software structure depending on their needs. The modeling algorithm exploits the Markov Chain Monte Carlo method for the data regression. The discrete relaxation models can be built by a proper combination of well-known relaxation models. In addition to these models, electrode polarization, a typical measurement artifact for interpreting dielectric spectra, can be incorporated into the modeling algorithm. A continuous relaxation model, which solves the Fredholm integral equation of the first kind (a mathematically ill-posed problem), is also included. This open-source software enables users to freely adjust the physical parameters to obtain physical insight into their materials under test and will be consistently updated for more accurate measurement and interpretation of dielectric spectra in an automated manner. This work describes the theoretical and mathematical background of the software, lays out the workflow, and validates the software functionality based on both synthetic and empirical data included in the software.

97 MATHEMATICS AND COMPUTING↗

Analytical solution for partial heating on the exterior of the pipe with application to measuring fluid flow rate

An analytical model using Green’s functions for partial external heating of a pipe is developed, which results in an exact mathematical solution for the radial and axial temperature distribution in the pipe wall. Partial heating consists of a constant heat flux function imposed over a small section of the exterior of a pipe and for a limited time duration. The solution comprises steady-state and transient parts, and an algebraic identity is used to decrease the number of summation terms in the slowly-converging steady- state part. Intrinsic verification principles are used to verify the solution. As an example application, this transient solution is applied toward the development of a simple, noninvasive method for in-field measurement of the flow rate in pipes. Furthermore, to simulate this application, a pulse of energy is imposed to the wall of the pipe, and the developed mathematical solution is used to find the flow rate of the fluid inside the pipe. An optimal experiment is designed to find the best measurement location and time.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗

Emergence of Orchestrated and Dynamic Metabolism of Saccharomyces cerevisiae

Microbial metabolism is a fundamental cellular process that involves many biochemical events and is distinguished by its emergent properties. While the molecular details of individual reactions have been increasingly elucidated, it is not well understood how these reactions are quantitatively orchestrated to produce collective cellular behaviors. Here we developed a coarse-grained, systems, and dynamic mathematical framework, which integrates metabolic reactions with signal transduction and gene regulation to dissect the emergent metabolic traits of Saccharomyces cerevisiae. Our framework mechanistically captures a set of characteristic cellular behaviors, including the Crabtree effect, diauxic shift, diauxic lag time, and differential growth under nutrient-altered environments. It also allows modular expansion for zooming in on specific pathways for detailed metabolic profiles. Furthermore, this study provides a systems mathematical framework for yeast metabolic behaviors, providing insights into yeast physiology and metabolic engineering.

59 BASIC BIOLOGICAL SCIENCES↗

Diffractive optical computing in free space

Abstract Structured optical materials create new computing paradigms using photons, with transformative impact on various fields, including machine learning, computer vision, imaging, telecommunications, and sensing. This Perspective sheds light on the potential of free-space optical systems based on engineered surfaces for advancing optical computing. Manipulating light in unprecedented ways, emerging structured surfaces enable all-optical implementation of various mathematical functions and machine learning tasks. Diffractive networks, in particular, bring deep-learning principles into the design and operation of free-space optical systems to create new functionalities. Metasurfaces consisting of deeply subwavelength units are achieving exotic optical responses that provide independent control over different properties of light and can bring major advances in computational throughput and data-transfer bandwidth of free-space optical processors. Unlike integrated photonics-based optoelectronic systems that demand preprocessed inputs, free-space optical processors have direct access to all the optical degrees of freedom that carry information about an input scene/object without needing digital recovery or preprocessing of information. To realize the full potential of free-space optical computing architectures, diffractive surfaces and metasurfaces need to advance symbiotically and co-evolve in their designs, 3D fabrication/integration, cascadability, and computing accuracy to serve the needs of next-generation machine vision, computational imaging, mathematical computing, and telecommunication technologies.

36 MATERIALS SCIENCE↗

Advantages of imperfect dice rolls over coin flips for random number generation

With an eye toward neural-inspired probabilistic computation, recent work has examined the development of true random number generators via stochastic devices. Typically, these devices are operated in a two-state regime to produce a sequence of binary outcomes (i.e., coin flips). However, there is no guarantee that stochastic devices will infallibly produce fair outputs and small deviations from a uniform distribution may have unwanted complications in applications. Using mathematical analysis, we contend that opting instead for a multi-state device (i.e., a dice roll) has benefits in these unfair paradigms. To demonstrate these benefits, we apply this framework to the analysis of a tunnel diode operated in a stochastic regime. In particular, interpreting the binary stochastic output of the tunnel diode as a multi-state die roll output also sees advantages in remaining closer to uniform. Overall, our approach provides a compelling argument for mathematical driven co-design and development of novel probabilistic computing devices and hardware.

applied mathematics↗

A probabilistic graphical model foundation for enabling predictive digital twins at scale

A unifying mathematical formulation is needed to move from one-off digital twins built through custom implementations to robust digital twin implementations at scale. This work proposes a probabilistic graphical model as a formal mathematical representation of a digital twin and its associated physical asset. We create an abstraction of the asset–twin system as a set of coupled dynamical systems, evolving over time through their respective state spaces and interacting via observed data and control inputs. The formal definition of this coupled system as a probabilistic graphical model enables us to draw upon well-established theory and methods from Bayesian statistics, dynamical systems and control theory. The declarative and general nature of the proposed digital twin model make it rigorous yet flexible, enabling its application at scale in a diverse range of application areas. Here, we demonstrate how the model is instantiated to enable a structural digital twin of an unmanned aerial vehicle (UAV). The digital twin is calibrated using experimental data from a physical UAV asset. Its use in dynamic decision-making is then illustrated in a synthetic example where the UAV undergoes an in-flight damage event and the digital twin is dynamically updated using sensor data. The graphical model foundation ensures that the digital twin calibration and updating process is principled, unified and able to scale to an entire fleet of digital twins.

42 ENGINEERING↗

Spatial modeling algorithms for reactions and transport in biological cells

Biological cells rely on precise spatiotemporal coordination of biochemical reactions to control their functions. Such cell signaling networks have been a common focus for mathematical models, but they remain challenging to simulate, particularly in realistic cell geometries. Here we present Spatial Modeling Algorithms for Reactions and Transport (SMART), a software package that takes in high-level user specifications about cell signaling networks and then assembles and solves the associated mathematical systems. SMART uses state-of-the-art finite element analysis, via the FEniCS Project software, to efficiently and accurately resolve cell signaling events over discretized cellular and subcellular geometries. We demonstrate its application to several different biological systems, including yes-associated protein (YAP)/PDZ-binding motif (TAZ) mechanotransduction, calcium signaling in neurons and cardiomyocytes, and ATP generation in mitochondria. Throughout, we utilize experimentally derived realistic cellular geometries represented by well-conditioned tetrahedral meshes. These scenarios demonstrate the applicability, flexibility, accuracy and efficiency of SMART across a range of temporal and spatial scales.

59 BASIC BIOLOGICAL SCIENCES↗

Bayesian Monte Carlo Evaluation of Imperfect (n, 233 U) Data and Model

Conventional nuclear data evaluation methods using generalized linear least squares make the following assumptions: prior and posterior probability distribution functions (PDFs) of all model parameters and data are normal (Gaussian); the linear approximation is sufficiently accurate to minimize the cost function (even for nonlinear models); the model (e.g., of neutron cross section) and experimental data (including covariance data) are without defect and prior PDFs of parameters and measured data are known perfectly. Neglect of covariance between model parameters and measured data in conventional evaluations contributes to imperfections. These assumptions are inherent to the generalized linear least squares minimization method commonly used for resolved resonance region neutron cross section evaluations but are often not justified due to the presence of non-normal PDFs, nonlinear models (e.g., R-matrix formalism), and inherent imperfections in data and models (e.g., imperfect covariance data). Here, these assumptions are removed in a mathematical framework of Bayes’ theorem, which is implemented using the Metropolis-Hastings Monte Carlo method. Most importantly, new parameters are introduced to parameterize discrepancies between the theoretical model and measured data to quantify judgement about discrepancies or imperfections in a reproducible manner. An evaluation of 233U in the eV region using the ENDF-B/VIII.0 library and transmission data (Guber et al.) is presented, and posterior parameters are compared to those obtained by conventional evaluation methods. This example illustrates the effects of removing the most harmful assumption: that of model-data perfection.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Chaotic Dynamics Driven by Particle-Core Interactions

High-intensity beams in modern linacs are frequently encircled by diffuse halos, which drive sustained particle losses and result in gradual degradation of accelerating structures. In large part, the growth of halos is facilitated by internal space-charge forces within the beams, and detailed characterization of this process constitutes an active area of ongoing research. A partial understanding of dynamics that ensue within space-charge dominated beams is presented by the particle-core interaction paradigm – a mathematical model wherein single particle dynamics, subject to the collective potential of the core, are treated as a proxy for the broader behavior of the beam. In this work, we investigate the conditions for the onset of large-scale chaos within the framework of this model, and demonstrate that the propensity towards stochastic evolution is strongly dependent upon the charge distribution of the beam. In particular, we show that while particle motion within a uniformly charged beam is dominantly regular, rapid deterministic chaos readily arises within space-charge dominated Gaussian beams. Importantly, we find that for sufficiently high values of the beam’s space charge and beam pulsation amplitude, enhanced chaotic mixing between the core and the halo can lead to an enhanced radial diffusion of charged particles. We explain our results from analytic grounds by demonstrating that chaotic motion is driven by the intersection of two principal resonances of the system, and derive the relevant overlap conditions. Additionally, our analysis illuminates a close connection between the mathematical formulation of the particle-core interaction model and the Andoyer family of integrable Hamiltonians

43 PARTICLE ACCELERATORS↗

A multi-dimensional Child–Langmuir law for any diode geometry

While prior theoretical studies of multi-dimensional space-charge limited current (SCLC) assumed emission from a small patch on infinite electrodes, none have considered emission from an entire finite electrode. In this paper, we apply variational calculus (VC) and conformal mapping, which have previously been used to derive analytic solutions for SCLC density (SCLCD) for nonplanar one-dimensional geometries, to obtain mathematical relationships for any multi-dimensional macroscopic diode with finite cathode and anode. We first derive a universal mathematical relationship between space-charge limited potential and vacuum potential for any diode and apply this technique to determine SCLCD for an eccentric spherical diode. We then apply VC and the Schwartz–Christoffel transformation to derive an exact equation for SCLCD in a general two-dimensional planar geometry with emission from a finite emitter. Particle-in-cell simulations using VSim agreed within 4%–13% for a range of ratios of emitter width to gap distance using the thinnest electrodes practical for the memory constraints of our hardware, with the difference partially attributed to the theory's assumption of infinitesimally thin electrodes. After generalizing this approach to determine SCLCD for any orthogonal diode as a function of only the vacuum capacitance and vacuum potential, we derive an analytical formulation of the three-dimensional Child–Langmuir law for finite parallel rectangular and disk geometries. These results demonstrate the utility for calculating SCLCD for any diode geometry using vacuum capacitance and vacuum potential, which are readily obtainable for many diode geometries, to guide experiment and simulation development.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗