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In this paper introduction to the covering spaces is given. We have proved some theorems and problems for the covering spaces. Examples related to the covering spaces are also discussed. [ABSTRACT FROM AUTHOR]

In 1960 algebraic topology was at the centre of the mathematical stage, but Michael Atiyah burst into the field and changed its focus and its language. I describe his work of the following decade and its influence, keeping to the themes of K-theory and generalized cohomology to minimise the overlap with Dan Freed's account of Atiyah's work on index theory, which also appears in this issue. [ABSTRACT FROM AUTHOR]

The precise arrangement of single entity is a crucial objective of nanoscience and holds great promise in various fields such as biology and material science. In this work, we develop a "DNA framework‐mediated geometric renormalization" (DFMGR) strategy to reassemble gold nanoparticles into specific geometric shapes on a 2‐dimensional (2D) fluidic membrane interface. Cholesterol‐modified AuNPs are randomly anchored on the supported lipid bilayer (SLB) via the cholesterol‐lipid interaction. We demonstrate that AuNPs are laterally mobile on SLB and could be further rearranged into a specific geometric shape by DNA framework containing algebraically topological DNA arms. Using scanning electron microscope (SEM) imaging approach, simple geometric shapes, such as points assembled by monomers, line segments assembled by dimers, triangles assembled by trimers are visually presented. Interestingly, we found that the statistic angle (58.77°) and side length (12.21 nm) of triangles obtained from SEM images were both agreed well with the theoretical angle of 60° and side length of 12.58 nm. And the relative error of the angle calculated was as low as 0.33 %. These results indicated that the DFMGR strategy showed precise regulation ability for the AuNPs renormalization. We believe that DNA framework‐mediated geometric renormalization strategy would be a powerful means for regulating ligand‐receptor interactions in biosystems and for nanoparticle assembling in material science. [ABSTRACT FROM AUTHOR]

Shooting methods are an efficient approach to solving nonlinear optimal control problems. As they use local optimization, they exhibit favorable convergence when initialized with a good warm-start but may not converge at all if provided with a poor initial guess. Recent work has focused on providing an initial guess from a learned model trained on samples generated during an offline exploration of the problem space. However, in practice, the solutions contain discontinuities introduced by system dynamics or the environment. Additionally, in many cases, multiple equally suitable, i.e., multimodal, solutions exist to solve a problem. Classic learning approaches smooth across the boundary of these discontinuities and thus generalize poorly. In this work, we apply tools from algebraic topology to extract information on the underlying structure of the solution space. In particular, we introduce a method based on persistent homology to automatically cluster the dataset of precomputed solutions to obtain different candidate initial guesses. We then train a mixture-of-experts within each cluster to predict state and control trajectories to warm-start the optimal control solver and provide a comparison with modality-agnostic learning. We demonstrate our method on a cartpole toy problem and a quadrotor avoiding obstacles, and show that clustering samples based on inherent structure improves the warm-start quality. [ABSTRACT FROM AUTHOR]

Noise modifies the behavior of chaotic systems in both quantitative and qualitative ways. To study these modifications, the present work compares the topological structure of the deterministic Lorenz (1963) attractor with its stochastically perturbed version. The deterministic attractor is well known to be "strange" but it is frozen in time. When driven by multiplicative noise, the Lorenz model's random attractor (LORA) evolves in time. Algebraic topology sheds light on the most striking effects involved in such an evolution. In order to examine the topological structure of the snapshots that approximate LORA, we use branched manifold analysis through homologies—a technique originally introduced to characterize the topological structure of deterministically chaotic flows—which is being extended herein to nonlinear noise-driven systems. The analysis is performed for a fixed realization of the driving noise at different time instants in time. The results suggest that LORA's evolution includes sharp transitions that appear as topological tipping points. [ABSTRACT FROM AUTHOR]

We provide necessary and sufficient conditions to extend the Hopf-Galois algebra structure on an algebra R to a generalized ambiskew ring based on R, in a way such that the added variables for the extension are skew-primitive in an appropriate sense. We show that the associated Hopf algebra is again a generalized ambiskew ring, based on a suitable Hopf algebra H.R/. Several examples are examined, including the Hopf-Galois objects over Uq(sl2). [ABSTRACT FROM AUTHOR]

In many scientific and technological contexts, we have only a poor understanding of the structure and details of appropriate mathematical models. We often, therefore, need to compare different models. With available data we can use formal statistical model selection to compare and contrast the ability of different mathematical models to describe such data. There is, however, a lack of rigorous methods to compare different models a priori. Here, we develop and illustrate two such approaches that allow us to compare model structures in a systematic way by representing models as simplicial complexes. Using well-developed concepts from simplicial algebraic topology, we define a distance between models based on their simplicial representations. Employing persistent homology with a flat filtration provides for alternative representations of the models as persistence intervals, which represent model structure, from which the model distances are also obtained. We then expand on this measure of model distance to study the concept of model equivalence to determine the conceptual similarity of models. We apply our methodology for model comparison to demonstrate an equivalence between a positional-information model and a Turing-pattern model from developmental biology, constituting a novel observation for two classes of models that were previously regarded as unrelated.

We have studied metamorphoses of the angular rainbow pattern generated by classical rainbow scattering of protons by graphene. To analyze the rainbow pattern, a morphological method was developed. It focuses on the shape of the rainbow pattern rather than on the exact position of rainbow lines or the particle count. It comprises elements of the catastrophe theory, which provides a local model of the rainbow pattern and the reduced potential and an index theory of algebraic topology that allows the evolution of the rainbow pattern to be linked with bifurcations of critical points of the reduced potential. The obtained insight is summarized into five principles that allow an experimentalist to sketch a qualitatively correct rainbow pattern in the impact parameter plane and the distribution of the reduced potential critical points, just by observing the evolution of the angular rainbows. The morphological method should be applicable for the analysis of all structurally stable patterns in nature. [ABSTRACT FROM AUTHOR]

Sensitivities for Stochastic Optimization In optimization, sensitivities of optimal values with respect to the data of the problem are of practical as well as theoretical interest. In the paper "Envelope Theorems for Multistage Linear Stochastic Optimization," Terça and Wozabal propose a framework to compute derivatives of optimal values for a certain class of stochastic optimization problems. The results make use of classical envelope theorems and almost sure smoothness properties for optimal values of linear optimization problems, which are derived using tools from real algebraic topology. The authors show that derivatives can be sampled based on solutions obtained from stochastic dual dynamic programming and discuss two numerical case studies, demonstrating that the approach is superior, both in terms of accuracy as well as computationally, to naïve methods of computing derivatives that are based on difference quotients. We propose a method to compute derivatives of multistage linear stochastic optimization problems with respect to parameters that influence the problem's data. Our results are based on classical envelope theorems and can be used in problems directly solved via their deterministic equivalents as well as in stochastic dual dynamic programming for which the derivatives of the optimal value are sampled. We derive smoothness properties for optimal values of linear optimization problems, which we use to show that the computed derivatives are valid almost everywhere under mild assumptions. We discuss two numerical case studies, demonstrating that our approach is superior, both in terms of accuracy and computationally, to naïve methods of computing derivatives that are based on difference quotients. [ABSTRACT FROM AUTHOR]

The higher Leray–Serre spectral sequence associated with a tower of fibrations represents a generalization of the classical Leray–Serre spectral sequence of a fibration. In this work, we present algorithms to compute higher Leray–Serre spectral sequences leveraging the effective homology technique, which allows to perform computations involving chain complexes of infinite type associated with interesting objects in algebraic topology. In order to develop the programs, implemented as a new module for the Computer Algebra system Kenzo, we translated the original construction of the higher Leray–Serre spectral sequence in a simplicial framework and studied some of its fundamental properties. [ABSTRACT FROM AUTHOR]

Heart rate variability (hrv) is a physiological phenomenon of the variation in the length of the time interval between consecutive heartbeats. In many cases it could be an indicator of the development of pathological states. The classical approach to the analysis of hrv includes time domain methods and frequency domain methods. However, attempts are still being made to define new and more effective hrv assessment tools. Persistent homology is a novel data analysis tool developed in the recent decades that is rooted at algebraic topology. The Topological Data Analysis (TDA) approach focuses on examining the shape of the data in terms of connectedness and holes, and has recently proved to be very effective in various fields of research. In this paper we propose the use of persistent homology to the hrv analysis. We recall selected topological descriptors used in the literature and we introduce some new topological descriptors that reflect the specificity of hrv, and we discuss their relation to the standard hrv measures. In particular, we show that this novel approach provides a collection of indices that might be at least as useful as the classical parameters in differentiating between series of beat-to-beat intervals (RR-intervals) in healthy subjects and patients suffering from a stroke episode. [ABSTRACT FROM AUTHOR]

• A novel foundation for topological pattern recognition of point cloud data, through persistent homology of graph approximations. • An extension of persistent homology to point cloud data of metric trees, including theoretical and experimental verification. • The first charting of cell trajectory data sets that explains some of the difficulties this field is confronted with. The rising field of Topological Data Analysis (TDA) provides a new approach to learning from data through persistence diagrams , which are topological signatures that quantify topological properties of data in a comparable manner. For point clouds, these diagrams are often derived from the Vietoris-Rips filtration—based on the metric equipped on the data—which allows one to deduce topological patterns such as components and cycles of the underlying space. In metric trees these diagrams often fail to capture other crucial topological properties, such as the present leaves and multifurcations. Prior methods and results for persistent homology attempting to overcome this issue mainly target Rips graphs, which are often unfavorable in case of non-uniform density across our point cloud. We therefore introduce a new theoretical foundation for learning a wider variety of topological patterns through any given graph. Given particular powerful functions defining persistence diagrams to summarize topological patterns, including the normalized centrality or eccentricity , we prove a new stability result, explicitly bounding the bottleneck distance between the true and empirical diagrams for metric trees. This bound is tight if the metric distortion obtained through the graph and its maximal edge-weight are small. Through a case study of gene expression data, we demonstrate that our newly introduced diagrams provide novel quality measures and insights into cell trajectory inference. [ABSTRACT FROM AUTHOR]

Copyright of Revista EntreLínguas is the property of Revista EntreLinguas and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Let F be a fixed field and let X be a simplicial complex on the vertex set V. The Leray number L(X;F) is the minimal d such that for all i⩾d and S⊂V, the induced complex X[S] satisfies H∼i(X[S];F)=0. Leray numbers play a role in formulating and proving topological Helly‐type theorems. For two complexes X,Y on the same vertex set V, define the relative Leray number LY(X;F) as the minimal d such that H∼i(X[V∖τ];F)=0 for all i⩾d and τ∈Y. In this paper we extend the topological colorful Helly theorem to the relative setting. Our main tool is a spectral sequence for the intersection of complexes indexed by a geometric lattice. [ABSTRACT FROM AUTHOR]

We provide sufficient conditions for a monotone function with a finite set of outcomes to be cyclically monotone. Using these conditions, we show that any monotone function defined on the domain of gross substitutes is cyclically monotone. The result also extends to the domain of generalized gross substitutes and complements. [ABSTRACT FROM AUTHOR]

Copyright of Categories & General Algebraic Structures with Applications is the property of Shahid Beheshti University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Given a Hopf algebra with distinguished bases indexed by combinatorial objects, along with a primitive generating set for this algebra, it is natural to consider how we can combinatorially interpret the primitivity of the elements in this generating set, by expanding these generators in terms of the distinguished bases of this algebra and then applying the comultiplication operation to these expansions and constructing sign-reversing involutions that determine the resultant cancellations that we obtain. In this article, we explore this idea, as applied to the power sum generators of the algebra of symmetric functions. [ABSTRACT FROM AUTHOR]

In this study, topology optimized, patient specific osteosynthesis plates (TOPOS-implants) are evaluated for the mandibular reconstruction using fibula segments. These shape optimized implants are compared to a standard treatment with miniplates (thickness: 1.0 mm, titanium grade 4) in biomechanical testing using human cadaveric specimen. Mandible and fibula of 21 body donors were used. Geometrical models were created based on automated segmentation of CT-scans of all specimens. All reconstructions, including cutting guides for osteotomy as well as TOPOS-implants, were planned using a custom-made software tool. The TOPOS-implants were produced by electron beam melting (thickness: 1.0 mm, titanium grade 5). The fibula-reconstructed mandibles were tested in static and dynamic testing in a multi-axial test system, which can adapt to the donor anatomy and apply side-specific loads. Static testing was used to confirm mechanical similarity between the reconstruction groups. Force-controlled dynamic testing was performed with a sinusoidal loading between 60 and 240 N (reconstructed side: 30% reduction to consider resected muscles) at 5 Hz for up to 5 · 105 cycles. There was a significant difference between the groups for dynamic testing: All TOPOS-implants stayed intact during all cycles, while miniplate failure occurred after 26.4% of the planned loading (1.32 · 105 ± 1.46 · 105 cycles). Bone fracture occurred in both groups (miniplates: n = 3, TOPOS-implants: n = 2). A correlation between bone failure and cortical bone thickness in mandible angle as well as the number of bicortical screws used was demonstrated. For both groups no screw failure was detected. In conclusion, the topology optimized, patient specific implants showed superior fatigue properties compared to miniplates in mandibular reconstruction. Additionally, the patient specific shape comes with intrinsic guiding properties to support the reconstruction process during surgery. This demonstrates that the combination of additive manufacturing and topology optimization can be beneficial for future maxillofacial surgery. [ABSTRACT FROM AUTHOR]

In this paper, we demonstrate that algebraic topology can be used to perform 2D+t object detection. After the construction of a topological complex for a 2D+t image sequence, we build a nested sequence of cell complexes on which relative persistent homology is computed. The relative homology adds to "absolute" homology the computation of classes related to the first and last frames of the sequence. By identifying 2D chains with large life spans, the most persistent classes are extracted. This allows for the identification of the interesting parts in a sequence and for the detection of the movement of objects despite continuous deformations in the image domain. The results obtained on a synthetic image and on two real biomedical images with moving vesicles recorded by a quantitative phase time‐lapse technique show the potential of this method. Comparing the method with a newly developed tracking tool proves that the strength of this method is its independence from prior parameters. [ABSTRACT FROM AUTHOR]

We use derived localization of the bar and nerve constructions to provide simple proofs of a number of results in algebraic topology, both known and new. This includes a recent generalization of Adams's cobar-construction to the non-simply connected case, and a new algebraic model for the homotopy theory of connected topological spaces as an ∞ -category of discrete monoids. [ABSTRACT FROM AUTHOR]

This research presents a numerical method to design a two-dimensional auxetic microstructure with negative Poisson's ratios. The method is established by combining the finite element method (FEM) with two different optimization procedures called bi-directional evolutionary structural optimization (BESO) and solid isotropic material with penalization (SIMP), respectively. The results show that by choosing any of the two methods of optimization (FEM and BESO or FEM and SIMP), each with a different objective function, it is possible to create microstructures that show negative Poisson's ratio. In addition, there is a significant resemblance between the results of both optimization methodologies. The final part of this study is to verify the outcome of optimization, both numerically and experimentally using three-dimensional (3D) printing techniques. [ABSTRACT FROM AUTHOR]

Copyright of Revista EntreLínguas is the property of Revista EntreLinguas and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Journal of College of Education is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

For balloon-borne telescopes, the primary mirror is the most important optical element, but designing a primary mirror with an excellent overall performance is a challenge. To comprehensively consider the contradictory objectives of the root mean square (RMS) surface error under gravity in the X and Z directions, the mass and fundamental frequency of the primary mirror, a parametric primary mirror design using the compromise programming method based on topology optimization is proposed. The parametric design of the compromise programming method based on topology optimization is used to find the optimal solution for X-direction RMS (RMSx), Z-direction RMS (RMSz), mass, and fundamental frequency. Compared with the initial primary mirror structure designed according to traditional experience, the overall performance is improved. Results show that the respective mass of the primary mirror, the RMSx and the RMSz decreased by 8.5%, 14.3% and 10.5% compared to those before optimization. Comprehensive consideration can prove the effectiveness of parametric design based on the topology optimization of the primary mirror. This method provides a reference for the design of other primary mirrors for balloon-borne telescope and space cameras. [ABSTRACT FROM AUTHOR]

The story of the friendship and collaboration between Vladimir Abramovich Rokhlin and the Nizhny Novgorod mathematician Dmitry Andreevich Gudkov during the last period of Rokhlin's mathematical biography, when he worked in the topology of real algebraic varieties, in which he obtained remarkable results. The paper is based on the correspondence of 1971–1982 preserved in Gudkov's archive containing 15 letters by V. A.Rokhlin and 8 letters by D. A. Gudkov. [ABSTRACT FROM AUTHOR]

Most robotic applications demand lightweight and high-speed manipulators for considerably reducing the consumed power and achieving high production rates. The two ways for seeking such high-speed arms are; applying advanced control algorithms and/or performing an extensive optimization of the arm structure itself. Therefore, the topology optimization technique is proposed here for obtaining an optimal robot arm design from both structure and control viewpoints. Results of some researches, that have been previously accomplished by size and shape optimization, were encouraging enough to extend and propose this optimization approach. The method of moving asymptotes (MMA) as an optimization algorithm, the finite element analysis (FEA) by ANSYS, and the time-optimal control method are integrated to gain an optimum design capable of attaining the minimum traveling time. The proposed methodology focuses on performing different comparisons between the proposed optimum topological designs and their initial designs for different robot arms' sizes and materials. It also distinguishes between the proposed optimum design and the previously achieved one by size optimization under the same operational conditions. Therefore, the significance of the proposed technique is emphasized. It shows that the traveling time is reduced by 44.8%, while the previous work only achieved 23.5%. In addition, the mass is reduced to nearly half of its initial value, taking into account the air damping as the real case in all terrestrial applications. [ABSTRACT FROM AUTHOR]

Both spectral sequences and persistent homology are tools in algebraic topology defined from filtrations of objects (e.g. topological spaces or simplicial complexes) indexed over the set Z of integer numbers. A recent work has shown the details of the relation between both concepts. Moreover, generalizations of both concepts have been proposed which originate from a different choice of the set of indices of the filtration, producing the new notions of multipersistence and spectral system. In this paper, we show that these notions are also related, generalizing results valid in the case of filtrations over Z. By using this relation and some previous programs for computing spectral systems, we have developed a new module for the Kenzo system computing multipersistence. We also present a birth-death descriptor and a new invariant providing information on multifiltrations. This new invariant, in some cases, is able to provide more information than the rank invariant. We show some applications of our algorithms to spaces of infinite type via the effective homology technique, where the performance has also been improved by means of discrete vector fields. [ABSTRACT FROM AUTHOR]

Flat metalenses have attracted attention due to an increasing demand for compact electromagnetic devices. For such applications, broadband metalenses are highly desirable; however, conventional metalenses show relatively narrow band operation. Here, we propose a design method of free-form metalenses using topology optimization to operate with enhanced bandwidths. In contrast with preceding reports of topology optimization methods for metalenses, we developed a topology optimization method based on the time domain formulation to deal with broadband frequencies simultaneously. For this purpose, a group delay of optical pulses in the time domain, which is equivalent to the broadband phase matching condition in the frequency domain, is employed in the objective function. A level set based topology optimization method is applied to obtain a clear optimal configuration. To demonstrate the effectiveness of the proposed method, we provide design examples of metalens unit cells at millimeter frequency. We confirm that optimized unit cells of metalenses show superior performance compared to the conventional unit cells for both transmittance efficiency and phase error in broadband wavelength. [ABSTRACT FROM AUTHOR]