Topological Classification Research Articles

Symmetry-preserving quadratic Lindbladian and dissipation driven topological transitions in Gaussian states

The dynamical evolution of an open quantum system can be governed by the Lindblad equation of the density matrix. In this paper, we propose to characterize the density matrix topology by the topological invariant of its modular Hamiltonian. Since the topological classification of such Hamiltonians depends on their symmetry classes, a primary issue we address is determining the requirement for the Lindbladian operators, under which the modular Hamiltonian can preserve its symmetry class during the dynamical evolution. We solve this problem for the fermionic Gaussian state and for the modular Hamiltonian being a quadratic operator of a set of fermionic operators. When these conditions are satisfied, along with a nontrivial topological classification of the symmetry class of the modular Hamiltonian, a topological transition can occur as time evolves. We present two examples of dissipation-driven topological transitions where the modular Hamiltonian lies in the AIII class with U(1) symmetry and the DIII class without U(1) symmetry. By a finite size scaling, we show that this density matrix topology transition occurs at a finite time. We also present the physical signature of this transition.

Symmetry, topology, and geometry: The many faces of the topological magnetoelectric effect

A delicate tension complicates the relationship between the topological magnetoelectric effect (TME) in three-dimensional (3D) Z2 topological insulators (TIs) and time-reversal symmetry (TRS). TRS underlies a particular Z2 topological classification of the electronic ground state of crystalline band insulators and the associated quantization of the magnetoelectric response coefficient calculated using bulk linear-response theory but, according to standard symmetry arguments, simultaneously forbids a nonzero magnetoelectric coefficient in any physical finite-size system. This contrast between theories of magnetoelectric response in formal bulk models and in real finite-sized materials originates from the distinct approaches required to introduce notions of (electronic) polarization and orbital magnetization in these fundamentally different environments. In this work we argue for a modified interpretation of the bulk linear-response calculations in nonmagnetic Z2 TIs that is more plainly consistent with TRS and use this interpretation to discuss the effect's observation—still absent over a decade after its prediction. Our analysis is reinforced by microscopic bulk and thin-film calculations carried out using a simplified but still realistic effective model for the well established V2VI3 [V=(Sb,Bi) and VI=(Se,Te)] family of nonmagnetic Z2 TIs. When a uniform dc magnetic field is included in this model, the anomalous n=0 Landau levels (LLs) play the central role, both in thin films and in bulk. In the former case, only the n=0 LL eigenfunctions can support a dipole moment, which vanishes if there are no magnetic surface dopants and is quantized in the thick-film limit if magnetic dopants at the top and bottom surfaces have opposite orientation. In the latter case, the Hamiltonian projected into the n=0 LL subspace is a one-dimensional Su-Schrieffer-Heeger model with ground-state polarization that is quantized in accordance with the bulk linear-response coefficient calculated for (a lattice regularization of) the starting 3D model. Motivated by analytical results, we conjecture a type of microscopic bulk-boundary correspondence: a bulk insulator with (generalized) TRS supports a magnetoelectric coefficient that is purely itinerant (which is generically related to the geometry of the ground state) if and only if magnetic surface dopants are required for the TME to manifest in finite samples thereof. We conclude that in nonmagnetic Z2 TIs the TME is activated by magnetic surface dopants, that the charge-density response to a uniform dc magnetic field is localized at the surface and specified by the configuration of those dopants, and that the TME is qualitatively less robust against disorder than the integer quantum Hall effect. Published by the American Physical Society 2024

Bulk-boundary and RPS thermodynamics from topology perspective

In this study, we investigate the bulk-boundary and restricted phase space (RPS) thermodynamics of Rissner-Nordström (R-N) AdS and 6-dimensional charged Gauss-Bonnet AdS black holes. Additionally, we examine the topological characteristics of the considered black holes and compare them with the results of extended thermodynamics. We determine that the topological behavior of the bulk-boundary thermodynamics is the same as that of the extended thermodynamics, whereas the RPS thermodynamics exhibits a distinct behavior. Furthermore, we demonstrate that within the RPS formalism, there is only one critical point with a topological charge of +1 . Moreover, in the RPS formalism, the inclusion of higher-derivative curvature terms in the form of Gauss-Bonnet gravity does not alter the topological classification of critical points in charged AdS black holes.

Holographic description of Narain CFTs and their code-based ensembles

We provide a precise relation between an ensemble of Narain conformal field theories (CFTs) with central charge c = n, and a sum of (U(1) × U(1))n Chern-Simons theories on different handlebody topologies. We begin by reviewing the general relation of additive codes to Narain CFTs. Then we describe a holographic duality between any given Narain theory and a pure Chern-Simons theory on a handlebody manifold. We proceed to consider an ensemble of Narain theories, defined in terms of an ensemble of codes of length n over ℤk × ℤk for prime k. We show that averaging over this ensemble is holographically dual to a level-k (U(1) × U(1))n Chern-Simons theory, summed over a finite number of inequivalent classes of handlebody topologies. In the limit of large k the ensemble approaches the ensemble of all Narain theories, and its bulk dual becomes equivalent to “U(1)-gravity” — the sum of the pertubative part of the Chern-Simons wavefunction over all possible handlebodies — providing a bulk microscopic definition for this theory. Finally, we reformulate the sum over handlebodies in terms of Hecke operators, paving the way for generalizations.

A New Symmetrical Source-Based DC/AC Converter with Experimental Verification

This research paper introduces a new topology for multilevel inverters, emphasizing the reduction of harmonic distortion and the optimization of the component count. The complexity of an inverter is determined by the number of power switches, which is significantly reduced in the presented topology, as fewer switches require fewer driver circuits. In this proposed topology, a new single-phase generalized multilevel inverter is analyzed with an equal magnitude of voltage supply. A 9-level, 11-level, or 13-level symmetrical inverter with RL load is analyzed in MATLAB/Simulink 2019b and then experimentally validated using the dSPACE-1103 controller. The experimental verification of the load voltage and current with different modulation indices is also presented. The analysis of the proposed topology concludes that the total required number of components is lower than that necessary for the classical inverter topologies, as well as for some new proposed multilevel inverters that are also compared with the proposed topology in terms of gate driver circuits, power switches, and DC sources, which thereby enhances the goodness of the proposed topology. Thus, a comparison of this inverter with the other topologies validates its acceptance.

Neutrosophic Generalized Topological Space

In classical topology, the generalized topological space develops the topological space and supra topological space. In this work, we define and study neutrosophic generalized topological space which is more generalization than the neutrosophic supra topological space, this space may be opening a new horizon and new direction in neutrosophic topology. Also, we study new “neutrosophic open (closed) sets” in this new space such as “neutrosophic generalized open (closed) sets”, “neutrosophic generalized semi open (closed) sets”, “neutrosophic generalized pre open (closed) sets”, “neutrosophic generalized a-open (closed) sets”, and we obtain its basic Properties and study the relations between them. Finally, we introduced “the associated interior and closure operators” in neutrosophic generalized topological space.

Isolated bidirectional DC-DC Converter: A topological review

Bidirectional DC-DC converters (BDCs) are certainly an important power electronic converter for managing bidirectional power flow in various applications. It offers the ability to flow power in both directions, which is useful in systems with renewable energy sources and energy storage. BDCs are becoming increasingly important in various applications, including renewable energy systems, electric vehicles, and energy storage. The development of new BDC topologies and control strategies is expected to further improve their efficiency and performance. Non-Isolated Bidirectional DC-DC Converters (NBDCs) and Isolated Bidirectional DC-DC Converters (IBDCs) are the two main types of BDCs with different isolation properties and trade-offs in terms of size, cost and performance. Topological classification of BDCs further distinguishes various BDC types based on their configurations and voltage-boosting techniques. IBDCs are preferred over NBDCs due to high-power applications, and wide voltage range with higher efficiency; the performance of IBDCs is improving day by day. To provide a framework for comparison in implementing novel configurations or identifying the best converter for a given application, this study is concentrated on each topology and its usual applications. Soft switching converters are also discussed pointed to Resonant Power Converters (RPCs) for the application into Electric Vehicles (EVs). It is worth noting that the choice of BDC depends on the specific requirements of the application, such as power level, voltage range, efficiency, and cost. A detailed comparison, application, advantage and disadvantage table can be useful in comparing the characteristics of different BDCs and selecting the most suitable one.

Cluster-Cluster Co-Nucleation Induced Defective Polyoxometalate-Based Metal-Organic Frameworks for Efficient Tandem Catalysis.

The construction of defective sites is one of the effective strategies to create high-activity Metal-Organic frameworks (MOFs) catalysts. However, traditional synthesis methods usually suffer from cumbersome synthesis steps and disordered defect structures. Herein, a cluster-cluster co-nucleation (CCCN) strategy is presented that involves the in situ introduction of size-matched functional polyoxometalates (H6P2W18O62, {P2W18}) to intervene the nucleation process of cluster-based MOFs (UiO-66), achieving one-step inducement of exposed defective sites without redundant post-processing. POM-induced UiO-66 ({P2W18}-0.1@UiO-66) exhibits a classical reo topology for well-defined cluster defects. Moreover, the defective sites and the interaction between POM and skeletal cluster nodes are directly observed by Integrated Differential Phase Contrast in Scanning Transmission Electron Microscopy (iDPC-STEM). Owing to the molecular-level proximity between defective sites and POM in the same nano-reaction space, {P2W18}-0.1@UiO-66 exhibits efficient tandem catalysis in the preparation of γ-valerolactone (γ-GVL) from laevulinic acid (LA) by the combination of Lewis and Brønsted acids with 11 times higher performance than defective UiO-66 formed by conventional coordination modulation strategy. The CCCN strategy is applicable to different POM and has the potential to be extended to other cluster-based MOFs, which will pave a new way for the construction of functional MOFs with multi-centered synergistic catalysis.

Topological Deformations of Manifolds by Algebraic Compositions in Polynomial Rings

The interactions between topology and algebraic geometry expose various interesting properties. This paper proposes the deformations of topological n-manifolds over the automorphic polynomial ring maps and associated isomorphic imbedding of locally flat submanifolds within the n-manifolds. The manifold deformations include topologically homeomorphic bending of submanifolds at multiple directions under algebraic operations. This paper introduces the concept of a topological equivalence class of manifolds and the associated equivalent class of polynomials in a real ring. The concepts of algebraic compositions in a real polynomial ring and the resulting topological properties (homeomorphism, isomorphism and deformation) of manifolds under algebraic compositions are introduced. It is shown that a set of ideals in a polynomial ring generates manifolds retaining topological isomorphism under algebraic compositions. The numerical simulations are presented in this paper to illustrate the interplay of topological properties and the respective real algebraic sets generated by polynomials in a ring within affine 3-spaces. It is shown that the coefficients of polynomials generated by a periodic smooth function can induce mirror symmetry in manifolds. The proposed formulations do not consider the simplectic class of manifolds and associated quantizable deformations. However, the proposed formulations preserve the properties of Nash representations of real algebraic manifolds including Nash isomorphism.

Topology of Born–Infeld-AdS black hole phase transitions: Bulk and CFT sides

The thermodynamic criticality of the AdS black holes serves as an important structure during the thermal phase transition. This paper discusses about the critical points and their topology during thermal phase transitions of the Born–Infeld AdS black holes. We make such investigations using two different topological approaches, namely, using Duan’s topological current ϕ-mapping theory, and the off-shell free energy. Within Duan’s formalism, we observe that for a given value of the Born–Infeld parameter b, there exists an associated electric charge parameter Q, which is highly sensitive to the topological phase transitions. This way we examine the connections of the first-order phase transition and the topological nature of the critical points. We find that the topological nature has a possible breakdown in certain parametric ranges. In effect, we determine the unconventional and the conventional phase critical points as the creation (topologically vortex) and annihilation (topologically anti-vortex) points (pairs). As the second approach, we call the off-shell free energy to determine the topological classes: of which one corresponds to the AdS-Schwarzschild black hole phases, while the other provides a possible topological phase transition. Here we also reveal a novel phase transition between two unstable phases, namely, the unstable small black hole and the intermediate black holes. For certain parametric values of the Born–Infeld parameter and the pressure, we also study the different topological descriptions that inevitably correspond to the AdS-Reissner–Nordström black hole phases. As a consistency check of the critical points during the topological phase transitions, we study the vortex/anti-vortex annihilation thermodynamics from local as well as global thermodynamic viewpoints. At the end, a special emphasis has been put on phase transition from the conformal field theory perspective revealing that the Born–Infeld-AdS black hole and its dual CFT share identical topology. Consequently, any topological transition that occurs in the bulk corresponds to a parallel transition in the dual CFT.

Abstract PO3-18-04: MBRC-101: a novel antibody-drug conjugate (ADC) targeting the membrane-associated tyrosine kinase receptor EphA5 in breast cancer

Abstract Objectives: ADCs have emerged as a major advance in contemporary medical oncology. The receptor tyrosine kinase (RTK) EphA5 possesses the classical receptor kinase topology with an extracellular-binding domain, a single-pass transmembrane domain, and a cytoplasmic kinase domain. We have previously shown that EphA5 is highly expressed in non-small cell lung cancer and contributes to DNA-damage repair and radiation therapy resistance. Others have reported EphA5 expression in gastric, ovarian, and pancreatic cancers. Here, we investigated the expression of EphA5 in breast cancer (BC) and assessed EphA5 as a potential cancer-specific target using a novel ADC designated MBRC-101. Methods: MBRC-101 is composed of a humanized anti-EphA5 IgG1 kappa monoclonal antibody conjugated to the small molecule microtubule-disrupting agent monomethyl auristatin E (MMAE) via a protease cleavable valine-citrulline (vc) ThioBridge linker (ThioBridge™-Glu-[Val-Cit-PAB-MMAE] PEG [24u]). MBRC-101 averages a drug-to-antibody ratio (DAR)=4. The specificity profile of the anti-EphA5 monoclonal antibody was characterized using a Membrane Proteome Array (MPA). Receptor-mediated antibody internalization was determined by real-time live-cell analysis; cell killing assays used both live-cell analysis of cell confluence and the CellTiter-Glo® luminescent cell viability assay. Expression of EphA5 in archival human breast tumor tissue sections was evaluated by immunohistochemistry (IHC) using a commercial polyclonal antibody against human EphA5. In vivo efficacy studies used patient-derived xenograft (PDX) models of BC. Results: IHC demonstrated robust and selective EphA5 expression in 87% (20 of 23) of triple negative breast cancer (TNBC) and 88% (23 of 26) of hormone receptor-positive (HR+) patient tumors tested. EphA5 expression was not detected in adjacent, non-malignant breast tissue. Antibody-specificity profile testing showed that the anti-EphA5 antibody bound exclusively to EphA5 and did not cross-react with other members of the Eph receptor/ephrin ligand family. Binding of the anti-EphA5 antibody to EphA5 on the cell surface triggered rapid internalization and processing of the EphA5/antibody complex through the endosomal-lysosomal system. Consistently, MBRC-101 was cytotoxic to EphA5-expressing cells and its activity was concentration-dependent and commensurate to the levels of EphA5 on the cell surface. In PDX murine models of TNBC, once weekly administrations of intravenous (IV) MBRC-101 showed dose-dependent, robust, and reproducible anti-tumor activity in vivo. Partial tumor responses were observed at doses of 2.5 mg/Kg IV and complete and nearly complete tumor responses starting at 5 mg/Kg IV. Doses up to 10 mg/Kg IV were well-tolerated in tumor-bearing mice with no observable weight loss. Pre-clinical safety confirmed MBRC-101 is also well tolerated in rats and in cynomolgus monkeys. Conclusions: These results support EphA5 as a new and promising membrane-associated molecular target for the design and development of ADC-based therapies against hormone-positive breast cancer and TNBC and support the conduct of a Phase 1/1b study of MBRC-101 in patients with breast cancer. Citation Format: Fernanda Staquicini, Fenny Tang, Vanessa de Oliveira, Daniela Staquicini, Yongjian Wu, Kirstin Barnhart, J Kellogg Parsons, Wadih Arap, Isan Chen, Renata Pasqualini. MBRC-101: a novel antibody-drug conjugate (ADC) targeting the membrane-associated tyrosine kinase receptor EphA5 in breast cancer [abstract]. In: Proceedings of the 2023 San Antonio Breast Cancer Symposium; 2023 Dec 5-9; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2024;84(9 Suppl):Abstract nr PO3-18-04.

Dynamics and control of a plant-herbivore model incorporating Allee's effect

This research focuses on the interaction between the grape borer and grapevine using a discrete-time plant-herbivore model with Allee's effect. We specifically investigate a model that incorporates a strong predator functional response to better understand the system's qualitative behavior at positive equilibrium points. In the present study, we explore the topological classifications at fixed points, stability analysis, Neimark-Sacker, Transcritical bifurcation and State feedback control in the two-dimensional discrete-time plant-herbivore model. It is proved that for all involved parameters ς1,ϱ1,γ1 and ϒ1, discrete-time plant-herbivore model has boundary and interior fixed points: c1=(0,0), c2=(ς1−1ϱ1,0) and c3=(ϒ1(1−γ1)2γ1−1,γ1(2ς1+ϱ1ϒ1−2)−ϱ1ϒ1+1−ς12γ1−1) respectively. Then by linear stability theory, local dynamics with different topological classifications are investigated at fixed points: c1=(0,0), c2=(ς1−1ϱ1,0) and c3=(ϒ1(1−γ1)2γ1−1,γ1(2ς1+ϱ1ϒ1−2)−ϱ1ϒ1+1−ς12γ1−1). Our investigation uncovers that the boundary equilibrium c2=(ς1−1ϱ1,0) experiences a transcritical bifurcation, whereas the unique positive steady-state c3=(ϒ1(1−γ1)2γ1−1,γ1(2ς1+ϱ1ϒ1−2)−ϱ1ϒ1+1−ς12γ1−1) of the discrete-time plant-herbivore model undergoes a Neimark-Sacker bifurcation. To address the periodic fluctuations in grapevine population density and other unpredictable behaviors observed in the model, we propose implementing state feedback chaos control. To support our theoretical findings, we provide comprehensive numerical simulations, phase portraits, dynamics diagrams, and a graph of the maximum Lyapunov exponent. These visual representations enhance the clarity of our research outcomes and further validate the effectiveness of the chaos control approach.

Enhancing the degrees of freedom of topology optimization via variable-porosity metal foams: Design of heat conduction paths in a volume-to-point problem

Multi-material topology optimization determines the optimal distribution of different materials within a design domain in order to achieve specific performance goals. This work takes advantage of such technique to enhance the thermal performance of a benchmark heat conduction problem. The aim is to investigate the impact of enhancing the degrees of freedom introducing different materials, i.e., variable-porosity metal foams in addition to the high-conductivity solid, in the design of heat conduction paths under a constant weight constraint. The study leverages a well-established case study – volume-to-point problem – in the field of thermal management, ensuring a consistent basis for comparison. It consists of a square domain with a uniform heat source and a Dirichlet condition at a point on the boundary. Interpolation functions – ordered solid isotropic material with penalization (SIMP) type – allow the properties – in this case thermal conductivity – of materials to be correctly assigned. The distinction between materials is made by means of different thresholds on the projection function. By varying the foam porosity, we investigate the trade-offs between weight and heat dissipation efficiency. The objective is to find the ideal combination of materials that maximizes heat transfer – minimizing the average domain temperature – while conforming to weight constraints. Findings reveal that multi-material topology optimization – when applied to heat conduction problems – can outperform other design approaches, such as the growth-based algorithm, the evolved constructal tree, and the classical topology optimization (with one solid material), thereby paving the ground to innovative heat sink designs in weight-constrained environments.

Brain chains as topological signatures for Alzheimer’s disease

AbstractTopology is providing new insights for neuroscience. For instance, graphs, simplicial complexes, directed graphs, flag complexes, persistent homology and convex covers have been used to study functional brain networks, synaptic connectivity, and hippocampal place cell codes. We propose a topological framework to study the evolution of Alzheimer’s disease, the most common neurodegenerative disease. The modeling of this disease starts with the representation of the brain connectivity as a graph and the seeding of a toxic protein in a specific region represented by a vertex. Over time, the accumulation of toxic proteins at vertices and their propagation along edges are modeled by a dynamical system on this graph. These dynamics provide an order on the edges of the graph according to the damage created by high concentrations of proteins. This sequence of edges defines a filtration of the graph. We consider different filtrations given by different disease seeding locations. To study these filtrations we propose a new combinatorial and topological method. A filtration defines a maximal chain in the partially ordered set of spanning subgraphs ordered by inclusion. To identify similar graphs, and define a topological signature, we quotient this poset by graph homotopy equivalence, which gives maximal chains in a smaller poset. We provide an algorithm to compute this direct quotient without computing all subgraphs and then propose bounds on the total number of graphs up to homotopy equivalence. To compare the maximal chains generated by this method, we extend Kendall’s $$d_K$$ d K metric for permutations to more general graded posets and establish bounds for this metric. We then demonstrate the utility of this framework on actual brain graphs by studying the dynamics of tau proteins on the structural connectome. We show that the proposed topological brain chain equivalence classes distinguish different simulated subtypes of Alzheimer’s disease.

Modeling 0.3 THz Coaxial Single-Mode Phase Shifter Designs in Liquid Crystals with Constitutive Loss Quantifications

This work proposes and examines the feasibility of next-generation 0.3 THz phase shifters realized with liquid crystals (LCs) as tunable dielectrics coaxially filled in the transmission line. The classic coaxial transmission line topology is robust to electromagnetic interference and environmental noise, but is susceptible to higher-order modes from microwave to millimeter-wave towards terahertz (THz) wavelength ranges, which impedes the low-insertion-loss phase-shifting functionality. This work thus focuses primarily on the suppression of the risky higher-order modes, particularly the first emerging TE11 mode impacting the dielectric loss and metal losses in diverse manners. Based on impedance matching baselines at diverse tuning states of LCs, this work analytically derives and models two design geometries; i.e., design 1 for the coaxial geometry matched at the isotopically referenced state of LC for 50 Ω, and design 2 for geometry matched at the saturated bias of LC with the maximally achievable permittivity. The Figure-of-Merit for design 1 and design 2 reports as 35.15°/dB and 34.73°/dB per unit length, respectively. We also propose a constitutive power analysis method for understanding the loss consumed by constitutive materials. Notably, for the 0.3 THz design, the isotropic LC state results in an LC dielectric loss of 63.5% of the total input power (assuming 100%), which becomes the primary constraint on achieving low-loss THz operations. The substantial difference in the LC dielectric loss between the isotropic LC state and saturated bias state for the 0.3 THz design (35.76% variation) as compared to that of our past 60 GHz design (13.5% variation) indicates that the LC dielectric loss’s escalating role is further enhanced with the rise in frequency, which is more pronounced than the conductor losses. Overall, the results from analytical and finite-element optimization in this work shape the direction and feasibility of the unconventional THz coaxial phase shifting technology with LCs, actioned as continuously tunable dielectrics.

Toward a Topological Classification of Nonadiabaticity in Chemical Reactions

Toward a Topological Classification of Nonadiabaticity in Chemical Reactions

Topological classifications of a piecewise linear Liénard system with three zones

Llibre et al. in [J. Nonlinear Sci., 2019] studied the number of limit cycles of piecewise linear Liénard system with one anti-saddle at the central zone in some parameter regions. However, the exact number and hyperbolicity of limit cycles were still unclear as in these cases of parameter. The goal of this paper is to give the exact number of limit cycles and the hyperbolicity of limit cycles in these parameter cases, where the maximum number of limit cycles is two. Moreover, all corresponding global phase portraits in the Poincaré disc are obtained.

Rényi topology of charged-flat black hole: Hawking-Page and Van-der-Waals phase transitions

In this paper, we extend the proposed setup in Wei et al. (2022); Yerra et al. (2022) for finding the topological charges associated with the Hawking-Page and Van-der-Waals transition points as well as equilibrium phases to catch the nonextensive nature of the black hole entropy. Rigorously speaking we incorporate the Rényi statistics formalism in off-shell Bragg-Williams free energy landscape, to examine topologically the Hawking-Page phase transition related to the uncharged/charged asymptotically flat black hole in grand canonical and the Van-der-Waals transition in the canonical ensemble, where a vortex/anti-vortex structure is found. For this purpose, we introduce three mappings, the ψ- and ξ-mapping, for phase transitions classification and the η-mapping for equilibrium phases classification. We found that Hawking-Page and Van-der-Waals phase transitions belong to different topological classes and exhibit an interplay of total charge values hinting to a possible new correspondence.Our topological study provides further substantiation for a possible conjecture positing a correspondence between the thermodynamic characteristics of black holes in asymptotically flat spacetime using Rényi statistics, and those in asymptotically Anti-de-Sitter spacetime employing Gibbs-Boltzmann statistics.

Design of metadevices based on isotropic materials for eigenfrequency recovery in lightened structures

This work presents a novel approach for the design of metadevices conceived to recover a prescribed set of the natural eigenfrequencies of a structure lightened by machining arbitrary shape holes, using classical topology optimization (TO) techniques. This TO metadevice design (TOMD) is performed by minimizing the error of an eigenfrequency recovery task, using the finite element method (FEM) as an analysis tool, ensuring manufacturability via the solid isotropic material with penalization (SIMP) approach with Heaviside filtering to reduce intermediate-density areas. The spatial variation of mechanical properties derives from the macroscopically distinguishable dual material distribution obtained from the TOMD problem. The methodology was first tested on lightened thin steel plates and then extended to a real-world 3D mechanical component with additional mechanical performance testing. In all cases, the optimized metadevice yielded lower error in recovering eigenfrequencies compared to the lightened component without the metadevice. Furthermore, the metadevice generally achieved further mass reduction, demonstrating its potential for lightweight design while restoring its original eigenfrequencies.

Mathematical topology and geometry-based classification of tauopathies

Neurodegenerative diseases, like Alzheimer’s, are associated with the presence of neurofibrillary lesions formed by tau protein filaments in the cerebral cortex. While it is known that different morphologies of tau filaments characterize different neurodegenerative diseases, there are few metrics of global and local structure complexity that enable to quantify their structural diversity rigorously. In this manuscript, we employ for the first time mathematical topology and geometry to classify neurodegenerative diseases by using cryo-electron microscopy structures of tau filaments that are available in the Protein Data Bank. By employing mathematical topology metrics (Gauss linking integral, writhe and second Vassiliev measure) we achieve a consistent, but more refined classification of tauopathies, than what was previously observed through visual inspection. Our results reveal a hierarchy of classification from global to local topology and geometry characteristics. In particular, we find that tauopathies can be classified with respect to the handedness of their global conformations and the handedness of the relative orientations of their repeats. Progressive supranuclear palsy is identified as an outlier, with a more complex structure than the rest, reflected by a small, but observable knotoid structure (a diagrammatic structure representing non-trivial topology). This topological characteristic can be attributed to a pattern in the beginning of the R3 repeat that is present in all tauopathies but at different extent. Moreover, by comparing single filament to paired filament structures within tauopathies we find a consistent change in the side-chain orientations with respect to the alpha carbon atoms at the area of interaction.