Topological Loops Research Articles

On a Well-Conditioned Electric Field Integral Operator for Multiply Connected Geometries

All known integral equation techniques for simulating scattering and radiation from arbitrarily shaped, perfect electrically conducting objects suffer from one or more of the following shortcomings: (i) they give rise to ill-conditioned systems when the frequency is low (ii) and/or when the discretization density is high, (iii) their applicability is limited to the quasi-static regime, (iv) they require a search for global topological loops, (v) they suffer from numerical cancellations in the solution when the frequency is very low. This work presents an equation that does not suffer from any of the above drawbacks when applied to smooth and closed objects. The new formulation is obtained starting from a Helmholtz decomposition of two discretizations of the electric field integral operator obtained by using RWGs and dual bases respectively. The new decomposition does not leverage Loop and Star/Tree basis functions, but projectors that derive from them. Following the decomposition, the two discretizations are combined in a Calderon-like fashion resulting in a new overall equation that is shown to exhibit self-regularizing properties without suffering from the limitations of existing formulations. Numerical results show the usefulness of the proposed method both for closed and open structures.

Theoretical analysis of the role of chromatin interactions in long-range action of enhancers and insulators.

Long-distance regulatory interactions between enhancers and their target genes are commonplace in higher eukaryotes. Interposed boundaries or insulators are able to block these long-distance regulatory interactions. The mechanistic basis for insulator activity and how it relates to enhancer action-at-a-distance remains unclear. Here we explore the idea that topological loops could simultaneously account for regulatory interactions of distal enhancers and the insulating activity of boundary elements. We show that while loop formation is not in itself sufficient to explain action at a distance, incorporating transient nonspecific and moderate attractive interactions between the chromatin fibers strongly enhances long-distance regulatory interactions and is sufficient to generate a euchromatin-like state. Under these same conditions, the subdivision of the loop into two topologically independent loops by insulators inhibits interdomain interactions. The underlying cause of this effect is a suppression of crossings in the contact map at intermediate distances. Thus our model simultaneously accounts for regulatory interactions at a distance and the insulator activity of boundary elements. This unified model of the regulatory roles of chromatin loops makes several testable predictions that could be confronted with in vitro experiments, as well as genomic chromatin conformation capture and fluorescent microscopic approaches.

Topological loops with three-dimensional solvable left translation group

We classify all connected topological loops having a three-dimensional solvable Lie group G as the group topologically generated by their left translations. It is surprising that to the non-nilpotent Lie group G having precisely one one-dimensional normal subgroup there are topological but no differentiable strongly left alternative loops.

The multiplication groups of 2-dimensional topological loops

(Communicated by R. M. Guralnick)Abstract. If the multiplication group MultðLÞ of a connected simply connected 2-dimensionaltopological loop L is a Lie group, then MultðLÞ is an elementary filiform Lie group F of di-mension nþ2 for some nd2, and any such group is the multiplication group of a connectedsimply connected 2-dimensional topological loop L. Moreover, if the group topologicallygenerated by the left translations of L has dimension 3, then L is uniquely determined by areal polynomial of degree n.

Rad54 Oligomers Translocate and Cross-bridge Double-stranded DNA to Stimulate Synapsis

Rad54 is a key component of the eukaryotic recombination machinery. Its presence in DNA strand-exchange reactions in vitro results in a significant stimulation of the overall reaction rate. Using untagged Rad54, we show that this stimulation can be attributed to enhancement of the formation of a key reaction intermediate known as DNA networks. Using a novel, single DNA molecule, dual-optical tweezers approach we show how Rad54 stimulates DNA network formation. We discovered that Rad54 oligomers possess a unique ability to cross-bridge or bind double-stranded DNA molecules positioned in close proximity. Further, Rad54 oligomers rapidly translocate double-stranded DNA while simultaneously inducing topological loops in the DNA at the locus of the oligomer. The combination of the cross-bridging and double-stranded DNA translocation activities of Rad54 stimulates the formation of DNA networks, leading to rapid and efficient DNA strand exchange by Rad51.

A minimal set of dynamic equations in systems modelled with bond graphs

The current paper presents a procedure for converting to a minimal set the dynamic equations of a system modelled with bond graphs starting from a large set of differential-algebraic equations including any type of topological loops and zero-order causal paths (ZCPs). This minimal set is composed only of differential equations expressed in terms of the degrees of freedom of the system. These equations are suitable for use in real-time simulations. Initially, flows corresponding to inertances and displacements associated with compliances are used to establish the dynamic equations and to find the ZCPs of the system. Two different methods are used to solve the ZCPs. With the first method, Lagrange multipliers are introduced by means of new flows and efforts as break variables of causal paths, adding constraint equations. With the second method, break variables are used directly to open the ZCPs. The procedure developed in the present paper solves the problem originated by the use of Lagrange multipliers with the introduction of break variables that imply the presence of new variables and constraint equations. This procedure, before numerical integration of the dynamic equations, converts the equations to a minimal set in order to gain computational efficiency. Several examples are also provided to illustrate the conversion steps.

Coverings of topological loops

In this paper, we show that in some cases, no proper covering of a locally compact group topologically generated by left translations of a topological loop can occur as the group topologically generated by left translations of a topological loop.

The Mechanical Properties of Human Angiostatin Can Be Modulated by Means of Its Disulfide Bonds: A Single-Molecule Force-Spectroscopy Study

Human angiostatin 1 ± 5 [Ang(1 ± 5)], a 57 kDa proteolitic fragment of human plasminogen, consists of five compact globular modules called Kringle domains with very similar gross architecture and remarkable sequence homology. They are built around a hydrophobic core and exhibit a triple-loop topology defined by three internal disulfide bonds: cys1 ± cys80 (1), cys22 ± cys63 (2), and cys51 ± cys75 (3) (Figure 1a).[1] Here we report on a single-molecule force-spectroscopy study that shows how the redox environment can control the topology and mechanical properties of angiostatin by modifying the extent of pairing of the internal disulfide bonds. Ang(1 ± 5) molecules were deposited on a polystyrene surface; the tip of a scanning force microscope (SFM) was brought into contact with this surface and retracted after reaching a force of 1 nN. This cycle was repeated many hundreds of times, first under phosphate buffer saline (PBS), Figure 1. a) Schematic representation of the Ang(1 ± 5) structure with the triple-loop topology of its five Kringle domains (K1 ±K5), determined in each case by a sequence of three disulfide bonds: cys1 ± cys80 (1), cys22 ± cys63 (2), and cys51 ± cys75 (3). The two asterisks (*) indicate the position of an additional, inter-Kringle disulfide bond. b) Pathway of the sequential reduction of the most exposed (1) and then of the more interior (2) and (3) bonds. This reduction pathway leads to gradually lengthening topological loops (red) that are exposed to the force exerted by the SFM tip at the two ends of the Kringle domain.

多体システムのための高速な順動力学解法の開発 １ 動力学の定式化及び他の漸化式解法との比較検討

The aim of this research is to develop an efficient and stable dynamic analysis approach for multibody systems. In this first report, an efficient extension of Rosenthal's order-n algorithm, which is originally limited for only multibody systems with open-loop topology, to multibody systems containing closed-loops is presented. Relative coordinates are employed for modeling systems. Closed topological loops are handled by cut joint technique. Equations of motion and cut joint constraint equations establish a set of differential-algebraic equation (DAE). Then, Rosenthal's order-n algorithm is extended to solve this DAE. Dynamic analysis of a multibody system is carried out to demonstrate efficiency of the proposed approach. Finally, number of arithmetic operations required by the proposed method and other well-known recursive formulations for analyzing a single closed-loop multibody system are counted in order to examine the efficiency of the proposed method compared to other well-known recursive methods

An Efficient Dynamic Formulation for Multibody Systems

The objective of this article is to present an efficient extension ofRosenthal's order-n algorithm to multibody systems containing closedloops. The equations of motion are created by using relative coordinatesand partial velocity theory. Closed topological loops are handled by cutjoint technique. The set of constraint equations of cut joints isadjoined to the system's equation of motion by using Lagrangemultipliers. This results in the equation of motion as adifferential-algebraic equation (DAE) rather than an ordinarydifferential equation. This DAE is then solved by applying the extendedRosenthal's order-n algorithm proposed in this article. While solvingDAE, violation of the kinematic constraint equations of cut joints iscorrected by coordinate projection method. Some numerical simulationsare carried out to demonstrate efficiency of the proposed method.

Deriving simulation models from bond graphs with algebraic loops.: The extension to multibond graph systems

The first part of this paper makes an introduction to the causal problems raised in some bond graph models when implementing causality. As a consequence, zero-order causal paths (ZCPs) provided of algebraic loops appear. Variables involved in these loops are related themselves by means of algebraic assignments. An algorithm to detect and open all the zero-order causal paths existing in any bond graph model is described. Opening of topological loops will be the employed method. Break variables will be the key to open the loops. The algorithm selects in an optimized way the number of variables necessary to open all the topological loops corresponding to the ZCPs existing in the model. The result is a set of differential-algebraic equations solved using a backward-differential formulae numerical method. How to solve multibond graph systems including zero-order causal paths is the subject of the second part of this paper. The constitutive relationships of multibond resistors, transformers and gyrators give way to a new class of zero-order causal paths whose most important characteristic is that their associated topological loops involve more than one direction. In medium–large models these relationships produce a combinatory explosion of causal paths only treatable via software. The same presented algorithms will be able to analyze and optimally choose the appropriate break variables to open all the zero-order causal paths.

Completion of topological loops

A Hausdorff topological group equipped with the right uniformity admits a group completion iff the inversion mapping preserves Cauchy filters, cf. [1], III. §3, No.5, Theoreme 1. Up until today a general theorem on the completion of topological loops is not available, for partial results see [9], [10]. This is among others due to the fact that topological loops will not necessarily have a compatible right uniformity.

Specificity and Functional Significance of DNA Interaction with the Nuclear Matrix: New Approaches to Clarify the Old Questions

In this chapter the specificity of chromosomal DNA partitioning into topological loops is discussed. Different experimental approaches used for the analysis of the above problem are critically reviewed. This discussion is followed by presentation of a novel approach for mapping the DNA loop anchorage sites that we have developed. This approach, based on the excision of the whole DNA loops by topoisomerase II-mediated DNA cleavage at matrix attachment sites, seems to constitute a unique tool for the analysis of topological organization of chromosomal DNA in living cells. We also discuss experimental results indicating that the DNA-loop anchorage sites form "weak points" in chromosomes that are preferentially sensitive to cleavage with both endogenous and exogenous nucleases. In connection with this discussion, rationales for the supposition that DNA loops constitute basic units of eukaryotic genome organization and evolution are considered. The chapter concludes by suggesting a new model of spatial organization of eukaryotic genome within the cell nucleus that resolves apparent contradictions between different data on the specificity of DNA interaction with the nuclear matrix.

Automorphism groups of differentiable double loops

In this paper, we study local and global topological loops as well as topological double loops having a differentiable structure such that the loop operations are differentiable. The main result states that the group of differentiable automorphisms of a differentiable double loop is compact with respect to the compact-open topology.

Multibody system order n dynamics formulation based on velocity transform method

In the multibody-simulation discipline, equations-of-motion formulations in which the number of calculations per integration step increase only linearly with the number n of bodies are called order it or 0(n) formulations. The development of such formulations is an area of current research because of their capability of yielding simulations that run much faster than conventional ones when n is large. This paper presents a new order n algorithm. It is applicable to systems of rigid or nonrigid bodies. It permits the bodies' interconnection joints to have an arbitrary number of degrees of freedom between 0 and 6. The system can have open-chain, tree, or closed-loop topology. Both absolute accelerations and relative accelerations are established by the technique. Closed topological loops are handled by the concept of cut joints. Constraint forces are calculated only at the cut joints, not at the uncut ones. The derivation of the algorithm uses a velocity transformation to eliminate the appearance of forces due to constraints at the uncut joints. The algorithm entails sequential computational passes—backward and forward—through the system of bodies.

Errata: Transformation Groups in the Theory of Topological Loops

Errata: Transformation Groups in the Theory of Topological Loops

Errata: Transformation groups in the theory of topological loops

Errata: Transformation groups in the theory of topological loops

Transformation Groups in the Theory of Topological Loops

Transformation Groups in the Theory of Topological Loops

Transformation groups in the theory of topological loops

In a recent paper [5] the author has applied the techniques of topological transformation group theory to the study of certain topological loops. The purpose of this note is to show more explicitly the close connection between topological loops and topological transformation groups. It is shown that for every transformation group G which acts reasonably on a space X so that (a) there is a global cross-section a from the coset space GIG, into G for pEX, (b) 5(G/G,) is a strongly transitive collection of homeomorphisms of X, and (c) G, is compact, such a space X may be given a binary operation so that it becomes a topological loop with a left invariant uniformity. Conversely, it is shown that in certain cases a topological loop L allows a transformation group G to act reasonably on L with a cross-section from G/G1 for the identity 1 of L satisfying (a) and (b) above. In certain cases (c) will also be satisfied. The reader may consult K. H. Hofmann [3] for the appropriate terminology in topological loops, with the following exceptions. A topological loop L has a left invariant uniformity (see [5 ]) when there is a uniform structure 'IU on L compatible with its topology so that 'I has a base a of entourages satisfying (x, y) EB if and only if (ax, ay) EB for all BE 6 and a EL. An invariant uniformity is a left invariant uniformity which also satisfies a right invariant condition. If a and b are elements of a loop, the unique solutions x and y to the equations ax =b and ya = b will be designated in this note by a(-I)b and ba(-1), respectively. The parentheses embracing -1 are used as a reminder that a(1)6 is not in general the product of 'a- and b as in a group. (This is at least the third notation appearing in print for these solutions. At the present this seems to be the most satisfactory.) The reference for transformation group topics is Montgomery and Zippin [7]. If G is a transformation group acting on a space X, and if G, is the isotropy or stability group at pEX, then there is a canonical mapping 7r: G/Gp--X defined by r(gG.) =g(p). Whenever ir is a homeomorphism, G is said to act reasonably on X (see [8]). The canonical mapping Xis continuous, onto, and one-one when G acts effectively and transitively [7, p. 43], but not in general open. A cross-section from G/G, into G is a continuous mapping a of the coset

Topological loops with invariant uniformities

Topological loops with invariant uniformities