Topological Restrictions Research Articles

Self-Assembled Copolymer Adsorption Layer-Induced Block Copolymer Nanostructures in Thin Films.

In polymer thin films, the bottom polymer chains are irreversibly adsorbed onto the substrates creating an ultrathin layer. Although this thin layer (only a few nanometers thick) governs all film properties, an understanding of this adsorbed layer remains elusive, and thus, its effective control has yet to be achieved, particularly in block copolymer (BCP) thin films. Herein, we employ self-assembled copolymer adsorption layers (SCALs), transferred from the air/water interfacial self-assembly of BCPs, as an effective control of the adsorbed layer in BCP thin films. SCALs replace the natural adsorbed layer, irreversibly adsorbing onto the substrates when other BCP is additionally coated on the SCALs. We further show that SCALs guide the thin film nanostructures because they provide topological restrictions and enthalpic/entropic preferences for a BCP self-assembly. The SCAL-induced self-assembly enables unprecedented control of nanostructures, creating novel nanopatterns such as spacing-controlled hole/dot patterns, dotted-line patterns, dash-line patterns, and anisotropic cluster patterns with exceptional controllability.

Unravelling Why and to What Extent the Topology of Similar Ce-Based MOFs Conditions their Photodynamic: Relevance to Photocatalysis and Photonics.

Metal–organic frameworks (MOFs) are emerging materials for luminescent and photochemical applications. Armed with femto to millisecond spectroscopies, and fluorescence microscopy, the photobehaviors of two Ce‐based MOFs are unravelled: Ce‐NU‐1000 and Ce‐CAU‐24‐TBAPy. It is observed that both MOFs show ligand‐to‐cluster charge transfer reactions in ≈100 and ≈70 fs for Ce‐NU‐1000 and Ce‐CAU‐24‐TBAPy, respectively. The formed charge separated states, resulting in electron and hole generation, recombine in different times for each MOF, being longer in Ce‐CAU‐24‐TBAPy: 1.59 and 13.43 µs than in Ce‐NU‐1000: 0.64 and 4.91 µs. The linkers in both MOFs also undergo a very fast intramolecular charge transfer reaction in ≈160 fs. Furthermore, the Ce‐NU‐1000 MOF reveals excimer formation in 50 ps, and lifetime of ≈14 ns. The lack of this interlinkers event in Ce‐CAU‐24‐TBAPy arises from topological restriction and demonstrates the structural differences between the two frameworks. Single‐crystal fluorescence microscopy of Ce‐CAU‐24‐TBAPy shows the presence of a random distribution of defects along the whole crystal, and their impact on the observed photobehavior. These findings reflect the effect of linkers topology and metal clusters orientations on the outcome of electronic excitation of reticular structure, key to their applicability in different fields of science and technology, such as photocatalysis and photonics.

Prescribing the curvature of Riemannian manifolds with boundary

Let M be a compact connected surface with boundary. We prove that the signal condition given by the Gauss–Bonnet theorem is necessary and sufficient for a given smooth function f on \(\partial M\) (resp. on M) to be geodesic curvature of the boundary (resp. the Gauss curvature) of some flat metric on M (resp. metric on M with geodesic boundary). In order to provide analogous results for this problem with \(n\ge 3,\) we prove some topological restrictions which imply, among other things, that any function that is negative somewhere on \(\partial M\) (resp. on M) is a mean curvature of a scalar flat metric on M (resp. scalar curvature of a metric on M and minimal boundary with respect to this metric). As an application of our results, we obtain a classification theorem for manifolds with boundary.

Новый подход к индоевропейской классификации: фонетико-семантическая реконструкция базисных лексических единиц для промежуточных праязыковых состояний (A New Approach to the Indo-European Classification: Phonetic-Semantic Reconstruction of Basic Lexical Units for Intermediate Original Language States)

Russian Abstract: В настоящем препринте предлагается реконструкция индоевропейской филогении на основе тринадцати 110-словных списков для праязыков ИЕ-подгрупп (протогерманский, протославянский и т. д.) или древних языков соответствующих подгрупп (хеттский, древнегреческий и т. д.). Мы применяем более или менее формальные методы сбора лингвистических данных и последующей обработки (семантическая реконструкция, деривационная дрейфовая элиминация, оптимизация гомоплазий), недавно предложенные или разработанные для настоящего исследования. Мы используем последовательную филогенетическую процедуру и получаем консенсусное дерево, основанное на нескольких алгоритмах (байесовский анализ, максимальная экономность, присоединение соседей, отсутствие применения топологических ограничений). Полученная топология дерева и датировки полностью совместимы с традиционными представлениями экспертов. Нашей главной находкой является мультифуркация внутренней ИЕ клады на четыре ветви ок. 3357-2162 гг. до н.э.: (1) греко-армянскую, (2) албанскую, (3) итало-германо-кельтскую, (4) балто-славяно-индо-иранскую. Предлагаемый сценарий расхождения примиряет друг с другом различные мнения о внутреннем ИЕ ветвлении, высказывавшиеся ранее индоевропеистами. English Abstract: This preprint suggests the reconstruction of Indo-European phylogeny based on thirteen 110-word lists for the proto-languages ​​of the IE subgroups (Proto-German, Proto-Slavonic, etc.) or the ancient languages ​​of the corresponding subgroups (Hittite, Ancient Greek, etc.). We use more or less formal methods for the collection of linguistic data and subsequent processing (semantic reconstruction, derivational drift elimination, optimization of homoplasias) recently proposed or developed for this study. We use a consistent phylogenetic procedure and obtain a consensus tree based on several algorithms (Bayesian analysis, maximum economy, joining of neighbors, lack of application of topological restrictions). The obtained tree topology and dating are fully compatible with the traditional views of experts. Our main finding is the multifurcation of the internal clade AI into four branches approx. 3357-2162 BC: (1) Greek-Armenian, (2) Albanian, (3) Italo-Germanic-Celtic, (4) Balto-Slavic-Indo-Iranian. The proposed scenario of discrepancy reconciles with each other the different opinions on the internal AI branching expressed earlier by the Indo-Europeists.

Space-time singularities and the theory of retracts

In previous publications, we have started investigating some possible applications to the retraction theory in gravitational physics and showed that it can be very useful in providing proofs and explaining the topological bases. In the current work, we investigate the role of the retraction theory in the mathematical study of space-time singularities and suggest a topological restriction on the formation of singularities. Although we present a toy model using a 5D cosmological metric as an example, the current study applies for any collapsing system under the influence of gravity such as a massive star or the whole universe. After showing that the cosmological space-time M can be retracted into lower dimensional circles , we prove the existence of a homotopy between this retraction and the identity map which defines a deformation retract on M. Since a circle does not deformation retract onto a point but it does retract to a point, the defined deformation retract stops any circle Si from retracting into a point which means no singularity is formed. The paper underlines the role of algebraic topology in the mathematical study of space-time singularities and represents a new application of the retraction theory in mathematical physics.

Conformal vector fields and Yamabe solitons

In this paper, we use less topological restrictions and more geometric and analytic conditions to obtain some sufficient conditions on Yamabe solitons such that their metrics are Yamabe metrics, that is, metrics of constant scalar curvature. More precisely, we use properties of conformal vector fields to find several sufficient conditions on the soliton vector fields of Yamabe solitons under which their metrics are Yamabe metrics.

Polymer Dispersity Control by Organocatalyzed Living Radical Polymerization

AbstractMolecular weight distribution of polymers, termed dispersity (Đ), is a fundamental parameter for determining polymer material properties. This paper reports a novel approach for controlling Đ by exploiting a temperature‐selective radical generation in organocatalyzed living radical polymerization. The polymers with tailored Đ were synthesized in a batch system without the assistance of an external pump. A unique aspect of this approach is that Đ was tuneable from 1.11 to 1.50 in any segment in diblock, triblock, and multiblock copolymers and in any form of star and brush polymer without segmental or topological restriction. This approach is amenable to various monomers and free from metals and thus attractive for applications. The approach also generated polymer brushes on surfaces with tailored Đ. An interesting finding was that the polymer brushes exhibited unique interaction with external molecules, depending on the Đ value.

Polymer Dispersity Control by Organocatalyzed Living Radical Polymerization.

Molecular weight distribution of polymers, termed dispersity (Đ), is a fundamental parameter for determining polymer material properties. This paper reports a novel approach for controlling Đ by exploiting a temperature-selective radical generation in organocatalyzed living radical polymerization. The polymers with tailored Đ were synthesized in a batch system without the assistance of an external pump. A unique aspect of this approach is that Đ was tuneable from 1.11 to 1.50 in any segment in diblock, triblock, and multiblock copolymers and in any form of star and brush polymer without segmental or topological restriction. This approach is amenable to various monomers and free from metals and thus attractive for applications. The approach also generated polymer brushes on surfaces with tailored Đ. An interesting finding was that the polymer brushes exhibited unique interaction with external molecules, depending on the Đ value.

Semiglobal Nonmemoryless Attitude Controls on the Special Orthogonal Group

This paper presents tracking strategies for the attitude dynamics of a rigid body that are global on the configuration space SO(3) and semiglobal over the phase space SO(3)×ℝ3. It is well known that global attractivity is prohibited for continuous attitude control systems on the special orthogonal group. Such topological restriction has been dealt with either by constructing smooth attitude control systems that exclude a set of zero measure in the region of attraction or by introducing discontinuities in the control input. This paper proposes nonmemoryless attitude control systems that are continuous in time, where the region of attraction guaranteeing exponential convergence completely covers the special orthogonal group. This provides a new framework to address the topological restriction in attitude controls. The efficacy of the proposed methods is illustrated by numerical simulations and an experiment.

An Atypical Mechanism of Split Intein Molecular Recognition and Folding.

Split inteins associate to trigger protein splicing in trans, a post-translational modification in which protein sequences fused to the intein pair are ligated together in a traceless manner. Recently, a family of naturally split inteins has been identified that is split at a noncanonical location in the primary sequence. These atypically split inteins show considerable promise in protein engineering applications; however, the mechanism by which they associate is unclear and must be different from that of previously characterized canonically split inteins due to unique topological restrictions. Here, we use a consensus design strategy to generate an atypical split intein pair (Cat) that has greatly improved activity and is amenable to detailed biochemical and biophysical analysis. Guided by the solution structure of Cat, we show that the association of the fragments involves a disorder-to-order structural transition driven by hydrophobic interactions. This molecular recognition mechanism satisfies the topological constraints of the intein fold and, importantly, ensures that premature chemistry does not occur prior to fragment complementation. Our data lead a common blueprint for split intein complementation in which localized structural rearrangements are used to drive folding and regulate protein-splicing activity.

Robustness and universality of surface states in Dirac materials

Ballistically propagating topologically protected states harbor exotic transport phenomena of wide interest. Here we describe a nontopological mechanism that produces such states at the surfaces of generic Dirac materials, giving rise to propagating surface modes with energies near the bulk band crossing. The robustness of surface states originates from the unique properties of Dirac-Bloch wavefunctions which exhibit strong coupling to generic boundaries. Surface states, described by Jackiw-Rebbi-type bound states, feature a number of interesting properties. Mode dispersion is gate tunable, exhibiting a wide variety of different regimes, including nondispersing flat bands and linear crossings within the bulk bandgap. The ballistic wavelike character of these states resembles the properties of topologically protected states; however, it requires neither topological restrictions nor additional crystal symmetries. The Dirac surface states are weakly sensitive to surface disorder and can dominate edge transport at the energies near the Dirac point.

Leveraging sUAS for Infrastructure Network Exploration and Failure Isolation

Large-scale infrastructures are prone to simultaneous faults when struck by a natural or man-made event. The current operating procedure followed by many utilities needs improvement, both in terms of monitoring performance and time to repair. Motivated by the recent technological progress on small Unmanned Aerial Systems (sUAS), we propose a practical framework to integrate the monitoring capabilities of sUAS into standard utility repair operations. A key aspect of our framework is the use of monitoring locations for sUAS-based inspection of failures within a certain spatial zone (called a localization set). This set is defined based on the alerts from fixed sensors or customer calls. The positioning of monitoring locations is subject to several factors such as sUAS platform, network topology, and airspace restrictions. We formulate the problem of minimizing the maximum time to respond to all failures by routing repair vehicles to various localization sets and exploring these sets using sUAS. The formulation admits a natural decomposition into two sub-problems: the sUAS Network Exploration Problem (SNEP); and the Repair Vehicle Routing Problem (RVRP). Standard solvers can be used to solve the RVRP in a scalable manner; however, solving the SNEP for each localization set can be computationally challenging. To address this limitation, we propose a set cover based heuristic to approximately solve the SNEP. We implement the overall framework on a benchmark network.

Ensemble equivalence for dense graphs

In this paper we consider a random graph on which topological restrictions are imposed, such as constraints on the total number of edges, wedges, and triangles. We work in the dense regime, in which the number of edges per vertex scales proportionally to the number of vertices $n$. Our goal is to compare the micro-canonical ensemble (in which the constraints are satisfied for every realisation of the graph) with the canonical ensemble (in which the constraints are satisfied on average), both subject to maximal entropy. We compute the relative entropy of the two ensembles in the limit as $n$ grows large, where two ensembles are said to be \emph{equivalent} in the dense regime if this relative entropy divided by $n^2$ tends to zero. Our main result, whose proof relies on large deviation theory for graphons, is that breaking of ensemble equivalence occurs when the constraints are \emph{frustrated}. Examples are provided for three different choices of constraints.

On the geometry and topology of initial data sets with horizons

We study the relationship between initial data sets with horizons and the existence of metrics of positive scalar curvature. We define a Cauchy Domain of Outer Communications (CDOC) to be an asymptotically flat initial set $(M, g, K)$ such that the boundary $\partial M$ of $M$ is a collection of Marginally Outer (or Inner) Trapped Surfaces (MOTSs and/or MITSs) and such that $M\setminus \partial M$ contains no MOTSs or MITSs. This definition is meant to capture, on the level of the initial data sets, the well known notion of the domain of outer communications (DOC) as the region of spacetime outside of all the black holes (and white holes). Our main theorem establishes that in dimensions $3\leq n \leq 7$, a CDOC which satisfies the dominant energy condition and has a strictly stable boundary has a positive scalar curvature metric which smoothly compactifies the asymptotically flat end and is a Riemannian product metric near the boundary where the cross sectional metric is conformal to a small perturbation of the initial metric on the boundary $\partial M$ induced by $g$. This result may be viewed as a generalization of Galloway and Schoen's higher dimensional black hole topology theorem \cite{GS06} to the exterior of the horizon. We also show how this result leads to a number of topological restrictions on the CDOC, which allows one to also view this as an extension of the initial data topological censorship theorem, established in \cite{EGP13} in dimension $n=3$, to higher dimensions.

On locally conformal symplectic manifolds of the first kind

We present some examples of locally conformal symplectic structures of the first kind on compact nilmanifolds which do not admit Vaisman metrics. One of these examples does not admit locally conformal Kähler metrics and all the structures come from left-invariant locally conformal symplectic structures on the corresponding nilpotent Lie groups. Under certain topological restrictions related with the compactness of the canonical foliation, we prove a structure theorem for locally conformal symplectic manifolds of the first kind. In the non compact case, we show that they are the product of a real line with a compact contact manifold and, in the compact case, we obtain that they are mapping tori of compact contact manifolds by strict contactomorphisms. Motivated by the aforementioned examples, we also study left-invariant locally conformal symplectic structures on Lie groups. In particular, we obtain a complete description of these structures (with non-zero Lee form) on connected simply connected nilpotent Lie groups in terms of locally conformal symplectic extensions and symplectic double extensions of symplectic nilpotent Lie groups. In order to obtain this description, we study locally conformal symplectic structures of the first kind on Lie algebras.

Anomalies and entanglement renormalization

We study 't Hooft anomalies of discrete groups in the framework of (1+1)-dimensional multiscale entanglement renormalization ansatz states on the lattice. Using matrix product operators, general topological restrictions on conformal data are derived. An ansatz class allowing for optimization of MERA with an anomalous symmetry is introduced. We utilize this class to numerically study a family of Hamiltonians with a symmetric critical line. Conformal data is obtained for all irreducible projective representations of each anomalous symmetry twist, corresponding to definite topological sectors. It is numerically demonstrated that this line is a protected gapless phase. Finally, we implement a duality transformation between a pair of critical lines using our subclass of MERA.

Graph Realizations Constrained by Skeleton Graphs

In 2008 Amanatidis, Green and Mihail introduced the Joint Degree Matrix (JDM) model to capture the fundamental difference in assortativity of networks in nature studied by the physical and life sciences and social networks studied in the social sciences. In 2014 Czabarka proposed a direct generalization of the JDM model, the Partition Adjacency Matrix (PAM) model. In the PAM model the vertices have specified degrees, and the vertex set itself is partitioned into classes. For each pair of vertex classes the number of edges between the classes in a graph realization is prescribed. In this paper we apply the new skeleton graph model to describe the same information as the PAM model. Our model is more convenient for handling problems with low number of partition classes or with special topological restrictions among the classes. We investigate two particular cases in detail: (i) when there are only two vertex classes and (ii) when the skeleton graph contains at most one cycle.

Local Large Deviations: A McMillian Theorem for Typed Random Graph Processes

For a finite typed graph on $n$ nodes and with type law $\mu,$ we define the so-called spectral potential $\rho_{\lambda}(\,\cdot,\,\mu),$ of the graph.From the $\rho_{\lambda}(\,\cdot,\,\mu)$ we obtain Kullback action or the deviation function, $\mathcal{H}_{\lambda}(\pi\,\|\,\nu),$ with respect to an empirical pair measure, $\pi,$ as the Legendre dual. For the finite typed random graph conditioned to have an empirical link measure $\pi$ and empirical type measure $\mu$, we prove a Local large deviation principle (LLDP), with rate function $\mathcal{H}_{\lambda}(\pi\,\|\,\nu)$ and speed $n.$ We deduce from this LLDP, a full conditional large deviation principle and a weak variant of the classical McMillian Theorem for the typed random graphs. Given the typical empirical link measure, $\lambda\mu\otimes\mu,$ the number of typed random graphs is approximately equal $e^{n\|\lambda\mu\otimes\mu\|H\big(\lambda\mu\otimes\mu/\|\lambda\mu\otimes\mu\|\big)}.$ Note that we do not require any topological restrictions on the space of finite graphs for these LLDPs.

Mean curvature flow with free boundary in embedded cylinders or cones and uniqueness results for minimal hypersurfaces

In this paper we study the mean curvature flow of embedded disks with free boundary on an embedded cylinder or generalised cone of revolution, called the support hypersurface. We determine regions of the interior of the support hypersurface such that initial data is driven to a curvature singularity in finite time or exists for all time and converges to a minimal disk. We further classify the type of the singularity. We additionally present applications of these results to the uniqueness problem for minimal hypersurfaces with free boundary on such support hypersurfaces; the results obtained this way do not require a-priori any symmetry or topological restrictions.

Entanglements and Crystallization of Concentrated Polymer Solutions: Molecular Dynamics Simulations

We carried out molecular dynamics simulations to study the crystallization of long polymers in a concentrated solution made by an explicit solvent of short chains. The weight-averaged entanglement length in the concentrated solution is found to exhibit power-law behavior with respect to the polymer volume fraction. The crystalline stem length obtained by quenching the solutions below melting temperature displays a linear relation with the entanglement length in the homogeneous solution above the crystallization temperature. Chain folding numbers are found to be very close to those in pure melts at different initial temperatures, and they tend to moderately decrease with increasing concentration of the long polymers. While our results are directly obtained from a coarse-grained model, they seem to suggest that the topological restriction of entanglements is a universal property to control the thickness selection during polymer crystallization.