An Estimation and Evaluation of Network Availability in Link State Routing Networks

The internet is taking component in a developing feature in every non-public and professional activity. The real-time, delay sensitive and mission-essential purposes, community availability requirement is beforehand for internet carrier providers (ICPs). The loop-loose criterion (LLC) approach has been extensively deployed through numerous ICPs for handling the best network component failure state of affairs in fantastic internet through. The achievement of LLC lies in its inherent simplicity; however, this comes at the rate of letting certain failure. To reap complete failure safety with LLC without incurring significant extra, a singular link protection scheme, hybrid hyperlink protocol (HLP), to reap failure routing. In contrast with in advance schemes, HLP guarantees tall network in a greater surroundings pleasant way. HLP is carried out in stages. Initial level substances a surroundings pleasant LLC primarily based totally on (MNP-e). The complexity of the set of rules is decrease than that of Dijkstra’s set of rules and might gift similar to network availability with LCC (Loop-loose criterion). Moment level substances backup direction safety based on MNP-e, the area totally a minimum type of need to be protected, to fulfill the network requirement. We don't forget those algorithms in a massive spread of associated, real and actual, and the effects display that HLP can achieve lofty network without introducing apparent.

Polynomial invariants of singular knots and links

We generalize the notion of the quandle polynomial to the case of singquandles. We show that the singquandle polynomial is an invariant of finite singquandles. We also construct a singular link invariant from the singquandle polynomial and show that this new singular link invariant generalizes the singquandle counting invariant. In particular, using the new polynomial invariant, we can distinguish singular links with the same singquandle counting invariant.

Movie moves for singular link cobordisms in 4-dimensional space

Two singular links are cobordant if one can be obtained from the other by singular link isotopy together with a combination of births or deaths of simple unknotted curves, and saddle point transformations. A movie description of a singular link cobordism in 4-space is a sequence of singular link diagrams obtained from a projection of the cobordism into 3-space by taking 2-dimensional cross-sections perpendicular to a fixed direction. We present a set of movie moves that are sufficient to connect any two movies of isotopic singular link cobordisms.

AN INVARIANT FOR SINGULAR KNOTS

In this paper we introduce a Jones-type invariant for singular knots, using a Markov trace on the Yokonuma–Hecke algebras Y d,n(u) and the theory of singular braids. The Yokonuma–Hecke algebras have a natural topological interpretation in the context of framed knots. Yet, we show that there is a homomorphism of the singular braid monoid SBn into the algebra Y d,n(u). Surprisingly, the trace does not normalize directly to yield a singular link invariant, so a condition must be imposed on the trace variables. Assuming this condition, the invariant satisfies a skein relation involving singular crossings, which arises from a quadratic relation in the algebra Y d,n(u).

Linking Number of Singular Links and the Seifert Matrix

We extend the notion of linking number of an ordinary link of two components to that of a singular link with transverse intersections, in which case the linking number is a half-integer. We then apply it to simplify the construction of the Seifert matrix, and therefore the Alexander polynomial, in a natural way.

On the Design of Three Links Robot for Drawing Epicycloid and Hypocycloid Curves

An epicycloid or hypocycloid mechanism is capable of drawing an exact epicycloid or hypocycloid curve. Similar mechanism designs can be found abundantly in industrial machines or educational equipment. Currently, the major type of epicycloid or hypocycloid configurations is planetary gear trains, which contain a binary link that has one fixed and one moving pivot, and a singular link adjacent to the moving pivot. The main feature of the configurations is that any point on the singular link may describe an epicycloid or hypocycloid curve when the binary link is rotated. Presently, the major types of configurations of epicycloid (hypocycloid) mechanisms have one degree of freedom. However, at present, as far as the authors are concerned, there appears to be no approach in designing epicycloid (hypocycloid) mechanisms with two degrees of freedom. Thus, the main aim of this paper is to develop a new design method in designing new configurations of epicycloid (hypocycloid) mechanisms. This paper analyses the characteristics of the topological structures of existing planetary gear train type epicycloid (hypocycloid) mechanisms with one degree of freedom. The equation of motion and kinematical model of the mechanism was derived and appropriate design constraints and criteria were implemented. Subsequently, using the design constraints and criteria, this work designs a new and simple epicycloid (hypocycloid) mechanism that is a three-links robot and has two degrees of freedom. We can easily control the angular velocities of the binary and singular links to satisfy the criterion to draw an epicycloid (hypocycloid) curve. Additionally, an epicycloid (hypocycloid) path of a point on the three links robot is simulated by computer drawing to prove the feasibility of proposed theory. Finally, a prototype of three links robot for drawing an epicycloid (hypocycloid) path is done well. We know the methods of design and manufacture of the proposed epicycloid or hypocycloid mechanism in linkage is easily done.

n-QUASI-ISOTOPY I: QUESTIONS OF NILPOTENCE

It is well-known that no knot can be cancelled in a connected sum with another knot, whereas every link can be cancelled up to link homotopy in a (componentwise) connected sum with another link. In this paper we address the question whether the noncancellation property of knots holds for (piecewise-linear) links up to some stronger analogue of link homotopy, which still does not distinguish between sufficiently close C0-approximations of a topological link. We introduce a sequence of such increasingly stronger equivalence relations under the name of k-quasi-isotopy, k∈ℕ; all of them are weaker than isotopy (in the sense of Milnor). We prove that every link can be cancelled up to peripheral structure preserving isomorphism of any quotient of the fundamental group, functorially invariant under k-quasi-isotopy; functoriality means that the isomorphism between the quotients for links related by any allowable crossing change fits in the commutative diagram with the fundamental group of the complement to the intermediate singular link. The proof invokes Baer's theorem on the join of subnormal locally nilpotent subgroups. On the other hand, the integral generalized ( lk ≠ 0) Sato–Levine invariant [Formula: see text] is invariant under 1-quasi-isotopy, but is not determined by any quotient of the fundamental group (endowed with the peripheral structure), functorially invariant under 1-quasi-isotopy — in contrast to Waldhausen's theorem.As a byproduct, we use [Formula: see text] to determine the image of the Kirk–Koschorke invariant [Formula: see text] of fibered link maps.