PETRI NETS MODELING FOR SDN TOPOLOGIES

The article examines some new algorithms and focuses mainly on suggesting new working topologies for software-defined controllers in order to ensure SDN security and to prevent the occurrence of a potential central point of failure (SPOF) by overcoming the centralization problem. This is a positive feature of the SDN structure, but could also be a threat, caused by the use of several controllers in different working topologies. This article focuses on exactly one of the suggested topologies, which features and models based on the Petri Nets system. The usual topology of a single controller is compared to verify the advantages and privileges of the proposed serial topology over the existing one. The paper tries to obtain a formula from the modeling of the serial topology and its advantages over the usual topology and that formula will be used to measure the level of security or the defense capacity of the network defined by the software against cyber attacks; in particular, denial of service attacks / distributed denial of service attacks / DDoS.

Spectral gap characterization of full type III factors

We give a spectral gap characterization of fullness for type {\mathrm{III}} factors which is the analog of a theorem of Connes in the tracial case. Using this criterion, we generalize a theorem of Jones by proving that if M is a full factor and {\sigma:G\rightarrow\mathrm{Aut}(M)} is an outer action of a discrete group G whose image in {\mathrm{Out}(M)} is discrete, then the crossed product von Neumann algebra {M\rtimes_{\sigma}G} is also a full factor. We apply this result to prove the following conjecture of Tomatsu–Ueda: the continuous core of a type {\mathrm{III}_{1}} factor M is full if and only if M is full and its τ invariant is the usual topology on {\mathbb{R}}.

Well-definedness and observational equivalence for inductive–coinductive programs

We define notions of well-definedness and observational equivalence for programs of mixed inductive and coinductive types. These notions are defined by means of tests formulas which combine structural congruence for inductive types and modal logic for coinductive types. Tests also correspond to certain evaluation contexts. We define a program to be well-defined if it is strongly normalizing under all tests, and two programs are observationally equivalent if they satisfy the same tests. We show that observational equivalence is sufficiently coarse to ensure that least and greatest fixed point types are initial algebras and final coalgebras, respectively. This yields inductive and coinductive proof principles for reasoning about program behaviour. On the other hand, we argue that observational equivalence does not identify too many terms, by showing that tests induce a topology that, on streams, coincides with usual topology induced by the prefix metric. As one would expect, observational equivalence is, in general, undecidable, but in order to develop some practically useful heuristics we provide coinductive techniques for establishing observational normalization and observational equivalence, along with up-to techniques for enhancing these methods.

Strongly ergodic equivalence relations: spectral gap and type III invariants

We obtain a spectral gap characterization of strongly ergodic equivalence relations on standard measure spaces. We use our spectral gap criterion to prove that a large class of skew-product equivalence relations arising from measurable $1$-cocycles with values in locally compact abelian groups are strongly ergodic. By analogy with the work of Connes on full factors, we introduce the Sd and $\unicode[STIX]{x1D70F}$ invariants for type $\text{III}$ strongly ergodic equivalence relations. As a corollary to our main results, we show that for any type $\text{III}_{1}$ ergodic equivalence relation ${\mathcal{R}}$, the Maharam extension $\text{c}({\mathcal{R}})$ is strongly ergodic if and only if ${\mathcal{R}}$ is strongly ergodic and the invariant $\unicode[STIX]{x1D70F}({\mathcal{R}})$ is the usual topology on $\mathbb{R}$. We also obtain a structure theorem for almost periodic strongly ergodic equivalence relations analogous to Connes’ structure theorem for almost periodic full factors. Finally, we prove that for arbitrary strongly ergodic free actions of bi-exact groups (e.g. hyperbolic groups), the Sd and $\unicode[STIX]{x1D70F}$ invariants of the orbit equivalence relation and of the associated group measure space von Neumann factor coincide.

SUBSPACES OF THE FREE TOPOLOGICAL VECTOR SPACE ON THE UNIT INTERVAL

For a Tychonoff space $X$, let $\mathbb{V}(X)$ be the free topological vector space over $X$, $A(X)$ the free abelian topological group over $X$ and $\mathbb{I}$ the unit interval with its usual topology. It is proved here that if $X$ is a subspace of $\mathbb{I}$, then the following are equivalent: $\mathbb{V}(X)$ can be embedded in $\mathbb{V}(\mathbb{I})$ as a topological vector subspace; $A(X)$ can be embedded in $A(\mathbb{I})$ as a topological subgroup; $X$ is locally compact.

Spaces of polynomial knots in low degree

We show that all knots up to six crossings can be represented by polynomial knots of degree at most [Formula: see text], among which except for [Formula: see text] and [Formula: see text] all are in their minimal degree representation. We provide concrete polynomial representation of all these knots. Durfee and O’Shea had asked a question: Is there any [Formula: see text]-crossing knot in degree [Formula: see text]? In this paper we try to partially answer this question. For an integer [Formula: see text], we define a set [Formula: see text] to be the set of all polynomial knots given by [Formula: see text] such that [Formula: see text] and [Formula: see text]. This set can be identified with a subset of [Formula: see text] and thus it is equipped with the natural topology which comes from the usual topology [Formula: see text]. In this paper we determine a lower bound on the number of path components of [Formula: see text] for [Formula: see text]. We define a path equivalence for polynomial knots in the space [Formula: see text] and show that it is stronger than the topological equivalence.

Operator Valued Measures as Multipliers of L1(I,X) with order Convolution

Let I = (0;∞) with the usual topology and product as max multiplication. Then I becomes a locally compact topo- logical semigroup. Let X be a Banach Space. Let L1(I;X) be the Banach space of X-valued measurable functions f such that ,we define It turns out that ƒ ∗ g ∈ L1(I;X) and L1(I;X) becomes an L1(I)-Banach module. A bounded linear operator T on L1(I;X) is called a multiplier of L1(I;X) if T(f ∗ g) = f ∗ Tg for all f ∈ L1(I) and g ∈ L1(I;X). We characterize the multipliers of L1(I;X) in terms of operator valued measures with point-wise finite variation and give an easy proof of some results of Tewari[12].

Some Properties of the Sorgenfrey Line and the Sorgenfrey Plane

Summary We first provide a modified version of the proof in [3] that the Sorgenfrey line is T1. Here, we prove that it is in fact T2, a stronger result. Next, we prove that all subspaces of ℝ1 (that is the real line with the usual topology) are Lindel¨of. We utilize this result in the proof that the Sorgenfrey line is Lindel¨of, which is based on the proof found in [8]. Next, we construct the Sorgenfrey plane, as the product topology of the Sorgenfrey line and itself. We prove that the Sorgenfrey plane is not Lindel¨of, and therefore the product space of two Lindel¨of spaces need not be Lindel¨of. Further, we note that the Sorgenfrey line is regular, following from [3]:59. Next, we observe that the Sorgenfrey line is normal since it is both regular and Lindel¨of. Finally, we prove that the Sorgenfrey plane is not normal, and hence the product of two normal spaces need not be normal. The proof that the Sorgenfrey plane is not normal and many of the lemmas leading up to this result are modelled after the proof in [3], that the Niemytzki plane is not normal. Information was also gathered from [15].