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Complex systems, as interwoven miscellaneous interacting entities that emerge and evolve through self-organization in a myriad of spiraling contexts, exhibit subtleties on global scale besides steering the way to understand complexity which has been under evolutionary processes with unfolding cumulative nature wherein order is viewed as the unifying framework. Indicating the striking feature of non-separability in components, a complex system cannot be understood in terms of the individual isolated constituents' properties per se, it can rather be comprehended as a way to multilevel approach systems behavior with systems whose emergent behavior and pattern transcend the characteristics of ubiquitous units composing the system itself. This observation specifies a change of scientific paradigm, presenting that a reductionist perspective does not by any means imply a constructionist view; and in that vein, complex systems science, associated with multiscale problems, is regarded as ascendancy of emergence over reductionism and level of mechanistic insight evolving into complex system. While evolvability being related to the species and humans owing their existence to their ancestors' capability with regards to adapting, emerging and evolving besides the relation between complexity of models, designs, visualization and optimality, a horizon that can take into account the subtleties making their own means of solutions applicable is to be entailed by complexity. Such views attach their germane importance to the future science of complexity which may probably be best regarded as a minimal history congruent with observable variations, namely the most parallelizable or symmetric process which can turn random inputs into regular outputs. Interestingly enough, chaos and nonlinear systems come into this picture as cousins of complexity which with tons of its components are involved in a hectic interaction with one another in a nonlinear fashion amongst the other related systems and fields. Relation, in mathematics, is a way of connecting two or more things, which is to say numbers, sets or other mathematical objects, and it is a relation that describes the way the things are interrelated to facilitate making sense of complex mathematical systems. Accordingly, mathematical modeling and scientific computing are proven principal tools toward the solution of problems arising in complex systems' exploration with sound, stimulating and innovative aspects attributed to data science as a tailored-made discipline to enable making sense out of voluminous (-big) data. Regarding the computation of the complexity of any mathematical model, conducting the analyses over the run time is related to the sort of data determined and employed along with the methods. This enables the possibility of examining the data applied in the study, which is dependent on the capacity of the computer at work. Besides these, varying capacities of the computers have impact on the results; nevertheless, the application of the method on the code step by step must be taken into consideration. In this sense, the definition of complexity evaluated over different data lends a broader applicability range with more realism and convenience since the process is dependent on concrete mathematical foundations. All of these indicate that the methods need to be investigated based on their mathematical foundation together with the methods. In that way, it can become foreseeable what level of complexity will emerge for any data desired to be employed. With relation to fractals, fractal theory and analysis are geared toward assessing the fractal characteristics of data, several methods being at stake to assign fractal dimensions to the datasets, and within that perspective, fractal analysis provides expansion of knowledge regarding the functions and structures of complex systems while acting as a potential means to evaluate the novel areas of research and to capture the roughness of objects, their nonlinearity, randomness, and so on. The idea of fractional-order integration and differentiation as well as the inverse relationship between them lends fractional calculus applications in various fields spanning across science, medicine and engineering, amongst the others. The approach of fractional calculus, within mathematics-informed frameworks employed to enable reliable comprehension into complex processes which encompass an array of temporal and spatial scales notably provides the novel applicable models through fractional-order calculus to optimization methods. Computational science and modeling, notwithstanding, are oriented toward the simulation and investigation of complex systems through the use of computers by making use of domains ranging from mathematics to physics as well as computer science. A computational model consisting of numerous variables that characterize the system under consideration allows the performing of many simulated experiments via computerized means. Furthermore, Artificial Intelligence (AI) techniques whether combined or not with fractal, fractional analysis as well as mathematical models have enabled various applications including the prediction of mechanisms ranging extensively from living organisms to other interactions across incredible spectra besides providing solutions to real-world complex problems both on local and global scale. While enabling model accuracy maximization, AI can also ensure the minimization of functions such as computational burden. Relatedly, level of complexity, often employed in computer science for decision-making and problem-solving processes, aims to evaluate the difficulty of algorithms, and by so doing, it helps to determine the number of required resources and time for task completion. Computational (-algorithmic) complexity, referring to the measure of the amount of computing resources (memory and storage) which a specific algorithm consumes when it is run, essentially signifies the complexity of an algorithm, yielding an approximate sense of the volume of computing resources and seeking to prove the input data with different values and sizes. Computational complexity, with search algorithms and solution landscapes, eventually points toward reductions vis à vis universality to explore varying degrees of problems with different ranges of predictability. Taken together, this line of sophisticated and computer-assisted proof approach can fulfill the requirements of accuracy, interpretability, predictability and reliance on mathematical sciences with the assistance of AI and machine learning being at the plinth of and at the intersection with different domains among many other related points in line with the concurrent technical analyses, computing processes, computational foundations and mathematical modeling. Consequently, as distinctive from the other ones, our special issue series provides a novel direction for stimulating, refreshing and innovative interdisciplinary, multidisciplinary and transdisciplinary understanding and research in model-based, data-driven modes to be able to obtain feasible accurate solutions, designed simulations, optimization processes, among many more. Hence, we address the theoretical reflections on how all these processes are modeled, merging all together the advanced methods, mathematical analyses, computational technologies, quantum means elaborating and exhibiting the implications of applicable approaches in real-world systems and other related domains. [ABSTRACT FROM AUTHOR]

Previous work treated the problem of a mass sliding over a rough spherical surface in broad generality, providing both analytic and numerical solutions. This paper examines special cases of 2D motion along a surface meridian when the initial speed is precisely chosen so that the sliding mass nearly stops before speeding up and subsequently leaving the surface. Carrying the solution for these critical cases into the time domain via both an analytical method and numerical integration adds richness that might otherwise be missed. The numerical method raises practical mathematical issues that must be handled carefully to obtain accurate results. Although conceptually simple, this classical mechanics problem is an excellent vehicle for students to gain proficiency with mathematical analysis tools and further their appreciation for how applied mathematics can bring new insight into situations where intuition may fall short. [ABSTRACT FROM AUTHOR]

The paper continues by examining the composite maps Graph HT ht and Graph HT ht to identify period two and period three points of I f i ( I z i ). If Graph HT ht , then Graph HT ht , which implies that Graph HT ht , which is not helpful. Logically, one might simply just extend complex numbers with one more term, Graph HT ht , where Graph HT ht . [Extracted from the article]

A unified study of necessary and sufficient conditions for the existence of intersection points and other important points in mathematical analysis is presented. The results are established in topologically based settings and shown to have equivalent formulations for various kinds of important points. For set-valued maps from a set to a topological space, instead of a convexity structure on this set, we use two general structures. The first structure is based on continuous single-valued maps from simpleces to the aforementioned set in order to extend the ideas of Knaster–Kuratowski–Mazurkiewicz and Fan for convex hulls. The second one is based on single-valued maps from the unit interval of the real line to the considered set and extends the classical connectedness properties. The general existence theorems of the above types are crucial tools in studies of the solution existence in optimization and other areas of applied mathematics. Here we choose minimax problems for applications of our general theorems. Our results are new, or improve or include as special cases many known results. [ABSTRACT FROM AUTHOR]

Special functions are particular mathematical functions which arise in the solutions of various classical problems in physics. The functions are quite useful in mathematical analysis. Mostly, they find applications in applied mathematics, theoretical physics, and engineering. Special functions are usually uniquely described by a generating function, a Rodrigues' formula, and a recurrence relation. In this work, we derive analytical spherical Bessel-like functions corresponding to the infinite degree power series expansion of the functions obtained in the multipole series expansion of the Coulomb repulsion term. The convergence of the truncated power series expansion of the functions with the corresponding derived analytical functions, as well as the comparison of analytical functions of the first and second kinds, is analyzed. Rodrigues' formula and a recurrence relation for obtaining these analytical functions are also presented. It is shown that the higher order even and odd spherical Bessel-like functions of the first kind are derivable from the lowest order functions of the first and second kinds, respectively. [ABSTRACT FROM AUTHOR]

Lutsky in 2008 and 2009, where they investigated the invertibility of parabolic pseudo-differential operators with rapidly growing symbols and obtained estimates of exponential behaviour of solutions of pseudo-differential operators with growing and discontinuous symbols. During several years which followed the defense of the PhD thesis there appeared Vladimir's publications presenting interesting results for quasi-elliptic pseudodifferential operators and on solvability of Cauchy and Goursat problems for parabolic pseudodifferential equations (1971-1979 and later on), on solvability difference-differential equations (1978-1985 and later on), on Fredholmness of elliptic pseudodifferential operators and associated boundary value problems on noncompact manifolds (papers of 1971-1979). In 1993 he defended his doctoral thesis "The method of limit operators in solvability of pseudo-differential equations and convolution equations". [Extracted from the article]

This document is an editorial for the Mathematics section of a journal called Symmetry. The editorial highlights the importance of mathematics in various fields and its role as a universal language for understanding natural and artificial systems. It acknowledges the close relationship between mathematics and other sciences like physics and computer science, as well as the existence of mathematical fields that develop independently. The editorial aims to collect high-quality works on different topics in mathematics, including algebra, geometry, applied mathematics, and more, to refine and improve the current level of knowledge. The author declares no conflict of interest. [Extracted from the article]

In this work, we introduce a minimal model for the current pandemic. It incorporates the basic compartments: exposed, and both symptomatic and asymptomatic infected. The dynamical system is formulated by means of fractional operators. The model equilibria are analyzed. The theoretical results indicate that their stability behavior is the same as for the corresponding system formulated via standard derivatives. This suggests that, at least in this case for the model presented here, the memory effects contained in the fractional operators apparently do not seem to play a relevant role. The numerical simulations instead reveal that the order of the fractional derivative has a definite influence on both the equilibrium population levels and the speed at which they are attained. [ABSTRACT FROM AUTHOR]

After graduating in geodesy at the Faculty of Civil Engineering of the Slovak University of Technology in Bratislava in 1954, you started your work at the Geodetic Institute in Bratislava, where you stayed from 1954 until 1962. This time both were from the Faculty of Natural Sciences of Comenius University, the first in mathematical analysis in 1957 and the second in probability & mathematical statistics in 1964. You are an honorary member of the Learned Society of Slovakia, and in 2002 you were awarded the honorary degree Dr.h.c. of the Faculty of Civil Engineering of the Slovak University of Technology. [Extracted from the article]

Let B \xi =U \xi \times \BbbR m for any \xi \in \BbbR m with (\xi,\xi) / \in E. Next, we are going to prove that Downloaded 03/08/21 to 140.234.253.9. In the case \xi (i) \in A co and \xi (l) / \in A co, we invoke again (1.7) to obtain that W sc (\xi (i),\xi (l)) = W (\alpha,\xi (l)) for any \alpha \in A; hence, (6.8) holds in light of (6.6). We can now infer from (5.12) and (5.13) that G coincides with W sc everywhere on the diagonal, and since I W sc\leq I rlx W = I G, Lemma 4.4 implies that G\geq W sc \geq 0 on\BbbR \times \BbbR. The generalized closed interval [\xi,\zeta ] with \xi,\zeta \in \BbbR m is the set\{ \lambda \xi + (1- \lambda)\zeta \in \BbbR m: \lambda \in [0, 1]\}. We associate with any suitable W:\BbbR m \times \BbbR m \rightarrow \BbbR \infty the double-integral functional I W:L p (\Omega; \BbbR m)\rightarrow \BbbR \infty with I W (u):= \int \Omega \int \Omega W (u(x),u(y))dxdy (2.13) for u\in L p (\Omega; \BbbR m). [Extracted from the article]

The paper presents some interesting aspects related to the biography and works of Romanian mathematician Caius Iacob (1912--1992). He was famous for his works in the fields of mathematical analysis, fluid mechanics, classical hydrodynamics and compressible-flow theory. At the age of 19, he graduated from the Mathematics Faculty in Bucharest, and then he went to Paris to continue his studies at the Faculty of Sciences, where he worked on a PhD thesis under the advice of famous French mathematician Henri Villat. On 24 June 1935, Caius Iacob successfully presented to the Sorbonne committee his PhD thesis about "Determination of conjugated harmonic functions with some limit conditions, and their applications in hydrodynamics". Returning to Romania, Caius Iacob had a long and successful career teaching mathematics and mechanics at the universities of Timişoara, Cluj and Bucharest. His most important work is considered the "Mathematical introduction to the mechanics of fluids". This book, providing original ways to work with classical hydrodynamics and compressibleflow theory, was published in Romanian in 1952 and in French in 1959. In 1955, he was elected a Corresponding Member of the Romanian Academy, becoming a titular Member in 1963. He was also President of the Mathematics Section of the Romanian Academy from 1980 until the end of his life, in 1992. In 1991, he initiated the foundation of the "Romanian Academy Institute of Applied Mathematics". In 2001 the institute merged with the "Centre for Mathematical Statistics", which had been created in 1964 by mathematician Gheorghe Mihoc, thus creating the "Gheorghe Mihoc -- Caius Iacob Institute of Mathematical Statistics and Applied Mathematics" of the Romanian Academy. [ABSTRACT FROM AUTHOR]

Modern logistics processes and systems can feature extremely complicated dynamics. Agent Based Modeling is emerging as a powerful modeling tool for design, analysis and control of such logistics systems. However, the complexity of the model itself can be overwhelming and mathematical meta-modeling tools are needed that aggregate information and enable fast and accurate decision making and control system design. Here we present Koopman Mode Analysis (KMA) as such a tool. KMA uncovers exponentially growing, decaying or oscillating collective patterns in dynamical data. We apply the methodology to two problems, both of which exhibit a bifurcation in dynamical behavior, but feature very different dynamics: Medical Treatment Facility (MTF) logistics and ship fueling (SF) logistics. The MTF problem features a transition between efficient operation at low casualty rates and inefficient operation beyond a critical casualty rate, while the SF problem features a transition between short mission life at low initial fuel levels and sustained mission beyond a critical initial fuel level. Both bifurcations are detected by analyzing the spectrum of the associated Koopman operator. Mathematical analysis is provided justifying the use of the Dynamic Mode Decomposition algorithm in punctuated linear decay dynamics that is featured in the SF problem. [ABSTRACT FROM AUTHOR]

Periodic and soliton solutions are derived for the (1+1)-dimensional van der Waals gas system in the viscosity-capillarity regularization form. The system is handled via the e -ϕ(ξ) -expansion method. The obtained solutions have been articulated by the hyperbolic, trigonometric, exponential and rational functions with arbitrary constants. Mathematical analysis and numerical graphs are provided for some solitons, periodic and kink solitary wave solutions to visualize the dynamics of equations. Obtained results reveal that the method is very influential and effective tool for solving nonlinear partial differential equations in applied mathematics. [ABSTRACT FROM AUTHOR]

Abstract For an anisotropic porous medium equation with L 1 regular lower order terms we prove the local boundedness and continuity of weak solutions in terms of the linear Riesz potential of the right-hand side of equation. [ABSTRACT FROM AUTHOR]

Abstract We show that the kernel of the linearization of the blow-up problem at the regular part of the interface that separates segregated BECs is one-dimensional, generated by translations in the normal direction to the interface. This useful non-degeneracy property was previously known only in one and two dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we present an analytical, numerical, and experimental description of the period addition phenomenon in a dynamical system arising from a boost converter controlled by ZAD strategy. The ideal model is presented, and compared with a model that includes parasitic resistances. We show the presence of chaos in the two systems, which is controlled by TDAS. In particular, numerical simulation shows the chaos control zone is greater in the system with internal resistances. [ABSTRACT FROM AUTHOR]

In analysis of multi-component complex systems, such as neural systems, identifying groups of units that share similar functionality will aid understanding of the underlying structures of the system. To find such a grouping, it is useful to evaluate to what extent the units of the system are separable. Separability or inseparability can be evaluated by quantifying how much information would be lost if the system were partitioned into subsystems, and the interactions between the subsystems were hypothetically removed. A system of two independent subsystems are completely separable without any loss of information while a system of strongly interacted subsystems cannot be separated without a large loss of information. Among all the possible partitions of a system, the partition that minimizes the loss of information, called the Minimum Information Partition (MIP), can be considered as the optimal partition for characterizing the underlying structures of the system. Although the MIP would reveal novel characteristics of the neural system, an exhaustive search for the MIP is numerically intractable due to the combinatorial explosion of possible partitions. Here, we propose a computationally efficient search to precisely identify the MIP among all possible partitions by exploiting the submodularity of the measure of information loss, when the measure of information loss is submodular. Submodularity is a mathematical property of set functions which is analogous to convexity in continuous functions. Mutual information is one such submodular information loss function, and is a natural choice for measuring the degree of statistical dependence between paired sets of random variables. By using mutual information as a loss function, we show that the search for MIP can be performed in a practical order of computational time for a reasonably large system (N = 100 ∼ 1000). We also demonstrate that MIP search allows for the detection of underlying global structures in a network of nonlinear oscillators. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new concept of completely linear degeneracy for quasilinear hyperbolic systems in several space variables, and then get an interesting property for multidimensional hyperbolic conservation laws satisfying our new definition. For applications, we give some examples arising from mathematics and physics at last. [ABSTRACT FROM AUTHOR]

We study a control problem where the state equation is a nonlinear partial differential equation of the calculus of variation in a bounded domain, perturbed by noise. We allow the control to act on the boundary and set boundary conditions which result in a stochastic differential equation for the trace of the solution on the boundary. This work provides necessary and sufficient conditions of optimality in the form of a maximum principle. We also provide a result of existence for the optimal control in the case where the control acts linearly. [ABSTRACT FROM AUTHOR]

Bayesian network classifiers (BNCs) have demonstrated competitive classification performance in a variety of real-world applications. A highly scalable BNC with high expressivity is extremely desirable. This paper proposes Redundant Dependence Elimination (RDE) for improving the classification performance and expressivity of k-dependence Bayesian classifier (KDB). To demonstrate the unique characteristics of each case, RDE identifies redundant conditional dependencies and then substitute/remove them. The learned personalized k-dependence Bayesian Classifier (PKDB) can achieve high-confidence conditional probabilities, and graphically interpret the dependency relationships between attributes. Two thyroid cancer datasets and four other cancer datasets from the UCI machine learning repository are selected for our experimental study. The experimental results prove the effectiveness of the proposed algorithm in terms of zero-one loss, bias, variance and AUC. [ABSTRACT FROM AUTHOR]

We study the distribution of strategies in a large game that models how agents choose among different double auction markets. We classify the possible mean field Nash equilibria, which include potentially segregated states where an agent population can split into subpopulations adopting different strategies. As the game is aggregative, the actual equilibrium strategy distributions remain undetermined, however. We therefore compare with the results of a reinforcement learning dynamics inspired by Experience-Weighted Attraction (EWA) learning, which at long times leads to Nash equilibria in the appropriate limits of large intensity of choice, low noise (long agent memory) and perfect imputation of missing scores (fictitious play). The learning dynamics breaks the indeterminacy of the Nash equilibria. Non-trivially, depending on how the relevant limits are taken, more than one type of equilibrium can be selected. These include the standard homogeneous mixed and heterogeneous pure states, but also heterogeneous mixed states where different agents play different strategies that are not all pure. The analysis of the reinforcement learning involves Fokker-Planck modeling combined with large deviation methods. The theoretical results are confirmed by multi-agent simulations. [ABSTRACT FROM AUTHOR]

I Readership: i Statisticians and data scientists interested in learning the fundamentals of wavelets methodology; graduate and advanced undergraduate students in statistics and data science; mathematicians and students of mathematics interested in applications and data analysis; and engineers and engineering students interested in image analysis. This does not diminish the undeniable value of the book for presenting, in a compact and comprehensive way, many of the fundamental and modern aspects of wavelet theory that were not a part of the previous takes on the subject. The statistical flavour to the presented material is own mostly to treating the wavelet decomposition problem in the framework of functional regression. [Extracted from the article]

We prove a maximality theorem for one-parameter dynamical systems that include W*-, C*- and multiplier one-parameter dynamical systems. Our main result is new even for one-parameter actions on commutative multiplier algebras including the algebra C b ( R ) of bounded continuous functions on R acted upon by translations. The methods we develop and use in our characterization of maximality include harmonic analysis, topological vector spaces and operator algebra techniques. [ABSTRACT FROM AUTHOR]

In this paper, a unique and more distinctive encryption algorithm is proposed. This is based on the complexity of highly nonlinear S box in Flesnelet domain. The nonlinear pattern is transformed further to enhance the confusion in the dummy data using Fresnelet technique. The security level of the encrypted image boosts using the algebra of Galois field in Fresnelet domain. At first level, the Fresnelet transform is used to propagate the given information with desired wavelength at specified distance. It decomposes given secret data into four complex subbands. These complex sub-bands are separated into two components of real subband data and imaginary subband data. At second level, the net subband data, produced at the first level, is deteriorated to non-linear diffused pattern using the unique S-box defined on the Galois field . In the diffusion process, the permuted image is substituted via dynamic algebraic S-box substitution. We prove through various analysis techniques that the proposed scheme enhances the cipher security level, extensively. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the notions of supra soft locally closed sets [1] and supra soft a-locally closed sets [2] by using the notions of supra pre-open soft sets [13]. Especially, we introduce the notions of supra soft P-locally closed sets, supra soft P*-locally closed sets and supra soft P**-locally closed sets in supra soft topological spaces. Also, we discuss their relationships with other supra open soft sets in detail, supported by examples and counterexamples. These examples illustrating the notions used in the paper are included. So we can see that all these concepts are independent from each other or does implies the other. Also, we introduce three different notions of generalized supra soft continuity, namely supra SPLC-continuous functions, supra SP* LC-continuous functions and supra SP** LC-continuous functions. Furthermore, we investigated some relations of these functions with other types of soft functions. [ABSTRACT FROM AUTHOR]

Finding dense spanning trees (DST) in unweighted graphs is a variation of the well studied minimum spanning tree problem (MST). We utilize established mathematical properties of extremal structures with the minimum sum of distances between vertices to formulate some general conditions on the sum of vertex degrees. We analyze the performance of various combinations of these degree sum conditions in finding dense spanning subtrees and apply our approach to practical examples. After briefly describing our algorithm we also show how it can be used on variations of DST, motivated by variations of MST. Our work provide some insights on the role of various degree sums in forming dense spanning trees and hopefully lay the foundation for finding fast algorithms or heuristics for related problems. [ABSTRACT FROM AUTHOR]

This paper proposes a new measure for recommendation through integrating Triangle and Jaccard similarities. The Triangle similarity considers both the length and the angle of rating vectors between them, while the Jaccard similarity considers non co-rating users. We compare the new similarity measure with eight state-of-the-art ones on four popular datasets under the leave-one-out scenario. Results show that the new measure outperforms all the counterparts in terms of the mean absolute error and the root mean square error. [ABSTRACT FROM AUTHOR]

The estimation of a finite population distribution function under a dual frame context is considered when auxiliary population information is available. Several procedures are defined and compared to various methods adapted from the literature. The asymptotic distribution of the proposed estimators is established, a brief simulation is implemented and an application to real data is included. [ABSTRACT FROM AUTHOR]

In this paper, L D U factorization of nonsingular sign regular and almost strictly sign regular matrices is analyzed. Besides, nonsingular almost strictly sign regular matrices are characterized through the Q R factorization. Illustrative numerical examples are included. [ABSTRACT FROM AUTHOR]

We consider the problem of inferring the local transition functions of discrete dynamical systems from observed behavior. Our focus is on synchronous systems whose local transition functions are threshold functions. We assume that the topology of the system is known and that the goal is to infer a threshold value for each node so that the system produces the observed behavior. We show that some of these inference problems are efficiently solvable while others are NP -complete, even when the underlying graph of the dynamical system is a simple path. We identify a fixed parameter tractable problem in this context. We also consider constrained versions of threshold inference problems where the input includes a set of equality or inequality constraints (which specify pairs of nodes which must have the same threshold value or different threshold values). We present algorithmic and complexity results for several constrained threshold inference problems. [ABSTRACT FROM AUTHOR]

Resilience, the ability to withstand disruptions and recover quickly, must be considered during system design because any disruption of the system may cause considerable loss, including economic and societal. This work develops analytic maximum flow-based resilience models for series and parallel systems using Zobel’s resilience measure. The two analytic models can be used to evaluate quantitatively and compare the resilience of the systems with the corresponding performance structures. For systems with identical components, the resilience of the parallel system increases with increasing number of components, while the resilience remains constant in the series system. A Monte Carlo-based simulation method is also provided to verify the correctness of our analytic resilience models and to analyze the resilience of networked systems based on that of components. A road network example is used to illustrate the analysis process, and the resilience comparison among networks with different topologies but the same components indicates that a system with redundant performance is usually more resilient than one without redundant performance. However, not all redundant capacities of components can improve the system resilience, the effectiveness of the capacity redundancy depends on where the redundant capacity is located. [ABSTRACT FROM AUTHOR]