Weak Isomorphism Research Articles

Special types of bipolar single valued neutrosophic graphs

Neutrosophic theory has many applications in graph theory, bipolar single valued neutrosophic graphs (BSVNGs) is the generalization of fuzzy graphs and intuitionistic fuzzy graphs, SVNGs. In this paper we introduce some types of BSVNGs, such as subdivision BSVNGs, middle BSVNGs, total BSVNGs and bipolar single valued neutrosophic line graphs (BSVNLGs), also investigate the isomorphism, co weak isomorphism and weak isomorphism properties of subdivision BSVNGs, middle BSVNGs,total BSVNGs and BSVNLGs.

Special types of bipolar single valued neutrosophic graphs

Neutrosophic theory has many applications in graph theory, bipolar single valued neutrosophic graphs (BSVNGs) is the generalization of fuzzy graphs and intuitionistic fuzzy graphs, SVNGs. In this paper we introduce some types of BSVNGs, such as subdivision BSVNGs, middle BSVNGs, total BSVNGs and bipolar single valued neutrosophic line graphs (BSVNLGs), also investigate the isomorphism, co weak isomorphism and weak isomorphism properties of subdivision BSVNGs, middle BSVNGs,total BSVNGs and BSVNLGs.

Spaces of Dirichlet series with the complete Pick property

We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form $$k\left( {s,u} \right) = \sum {{a_n}} {n^{ - s - \overline u }}$$ , and characterize when such a space is a complete Pick space. We then discuss what it means for two reproducing kernel Hilbert spaces to be “the same”, and introduce a notion of weak isomorphism. Many of the spaces we consider turn out to be weakly isomorphic as reproducing kernel Hilbert spaces to the Drury–Arveson space H 2 in d variables, where d can be any number in {1, 2,...,∞}, and in particular their multiplier algebras are unitarily equivalent to the multiplier algebra of H 2 . Thus, a family of multiplier algebras of Dirichlet series is exhibited with the property that every complete Pick algebra is a quotient of each member of this family. Finally, we determine precisely when such a space of Dirichlet series is weakly isomorphic as a reproducing kernel Hilbert space to H 2 and when its multiplier algebra is isometrically isomorphic to Mult(H 2 ).

Размер памяти для хранения упорядоченного корневого графа

The paper considers boundaries of memory necessary and sufficient for storage of undirected ordered rooted connected graphs, both numbered and unnumbered. The introduction contains the basic definitions and the problem statement. A graph is rooted if one of its vertices is marked as a root. A graph is ordered if for each of its vertices all the incident edges are ordered (numbered). A graph is numbered if all its vertices are numbered with different integer numbers (from 0 to n -1, where n is the number of vertices). Two undirected ordered graphs G and G` are weakly isomorphic if there exists a one-to-one correspondence between their vertices, for which corresponding vertices has the same degrees (numbers of incident edges) and two edges having corresponding ends and the same numbers in these ends, also have the other ends corresponding. Isomorphism of rooted graphs should also correspond their roots. Isomorphism of numbered graphs should also correspond the vertices with the same numbers. Graphs are considered up to weak isomorphism. It is shown that the memory necessary and sufficient for storage of any graph has the size Q( mlogn ) for numbered graphs, Q( n +( m - n +1) logn ) for unnumbered graphs with the number of vertices n and the number of edges m , and Q( n 2 logn ) for graphs without multiple edges and loops with the number of vertices n . It is also shown that the memory sufficient for storage of an edge sequence of length O ( n ) or a spanning tree, has the O ( nlog ( n D)) or O ( nlog D) size, respectively, where D is the maximum vertex degree.

Morphism of m-Polar Fuzzy Graph

The main purpose of this paper is to introduce the notion of m-polar h-morphism on m-polar fuzzy graphs. The action of m-polar h-morphism on m-polar fuzzy graphs is studied. Some elegant theorems on weak and coweak isomorphism are obtained. Also, some properties of highly irregular, edge regular, and totally edge regular m-polar fuzzy graphs are studied.

Weak Cayley table groups III: PSL(2,q)

ABSTRACTA weak Cayley table isomorphism is a bijection φ:G→H of groups such that φ(xy)∼φ(x)φ(y) for all x,y∈G. Here ∼ denotes conjugacy. When G = H the set of all weak Cayley table isomorphisms φ:G→G forms a group 𝒲(G) that contains the automorphism group Aut(G) and the inverse map I:G→G,x↦x−1. Let 𝒲0(G) = ⟨Aut(G),I⟩≤𝒲(G) and say that G has trivial weak Cayley table group if 𝒲(G) = 𝒲0(G). We show that PSL(2,pn) has trivial weak Cayley table group, where p≥5 is a prime and n≥1.

Formula omitted]-homology of the free group

We analyze the reduced ℓ2-homology groups of a finitely generated nonabelian free group, F. Specifically, the projection map onto the space of harmonic ℓ2-1-chains is explicitly described and a weak isomorphism from (ℓ2(F))rk(F)−1 to the space of harmonic ℓ2-1-chains is given.

A study on vague graphs.

The main purpose of this paper is to introduce the notion of vague h-morphism on vague graphs and regular vague graphs. The action of vague h-morphism on vague strong regular graphs are studied. Some elegant results on weak and co weak isomorphism are derived. Also, mu-complement of highly irregular vague graphs are defined.

Weak containment rigidity for distal actions

We prove that if a measure distal action α of a countable group Γ is weakly contained in a strongly ergodic probability measure preserving action β of Γ, then α is a factor of β. In particular, this applies when α is a compact action.As a consequence, we show that the weak equivalence class of any strongly ergodic action completely remembers the weak isomorphism class of the maximal distal factor arising in the Furstenberg–Zimmer Structure Theorem.

Some properties of vague graphs with application

Theoretical concepts of graphs are highly utilized by computer science applications. Especially in research areas of computer science such as data mining, image segmentation, clustering, image capturing and networking. The vague graphs are more flexible and compatible than fuzzy graphs due to the fact that they have many applications in networks. In this paper, busy vertices, free vertices, strong arc, busy value, isolated vertex, fully free vertex and partial free vertex in vague graphs are defined. Likewise, some interesting properties of isomorphism and weak isomorphism on them are introduced. Finally, we give an application of vague digraph in vulnerability assessment of gas pipeline networks.

Isomorphic properties of neighborly irregular vague graphs

The main purpose of this paper is to introduce weak isomorphism, co-weak isomorphism and isomorphism of neighborly irregular vague graphs. Some results on order, size and degree of nodes in isomorphic neighborly irregular and isomorphic highly irregular vague graphs are discussed. Isomorphism between neighborly irregular and highly irregular vague graphs are proved to be an equivalence relation. Density and balanced irregular vague graphs are introduced. Finally, an application of vague graphs is given.

Weak Cayley Table Groups II: Alternating Groups and Finite Coxeter Groups

A weak Cayley table isomorphism is a bijection φ: G → H of groups such that φ(xy) ∼ φ(x)φ(y) for all x, y ∈ G. Here ∼denotes conjugacy. When G = H the set of all weak Cayley table isomorphisms φ: G → G forms a group 𝒲(G) that contains the automorphism group Aut(G) and the inverse map I: G → G, x → x −1. Let 𝒲0(G) = ⟨Aut(G), I⟩ ≤ 𝒲(G) and say that G has trivial weak Cayley table group if 𝒲(G) = 𝒲0(G). We show that all finite irreducible Coxeter groups (except possibly E 8) have trivial weak Cayley table group, as well as most alternating groups. We also consider some sporadic simple groups.

Are Neural Networks Imitations of Mind?

Artificial neural networks are often understood as a good way to imitate mind through the web structure of neurons in brain, but the very high complexity of human brain prevents to consider neural networks as good models for human mind;anyway neural networks are good devices for computation in parallel. The difference between feed-forward and feedback neural networks is introduced; the Hopfield network and the multi-layers Perceptron are discussed. In a very weak isomorphism (not similitude) between brain and neural networks, an artificial form of short term memory and of acknowledgement, in Elman neural networks, is proposed.

On the tradeoff between compositionality and exactness in weak bisimilarity for integrated-time Markovian process calculi

Integrated-time Markovian process calculi rely on actions whose durations are quantified by exponentially distributed random variables. The Markovian bisimulation equivalences defined so far for these calculi treat exponentially timed internal actions like all the other actions, because each such action has a nonzero duration and hence can be observed if it is executed between a pair of exponentially timed noninternal actions. However, no difference may be noted, at stationary state, between a sequence of exponentially timed internal actions and a single exponentially timed internal action, if their expected durations and execution probabilities coincide, a fact exploited in Hillston's weak isomorphism. We show that Milner's approach can be adapted on the basis of this fact, so to derive a weak bisimulation equivalence for integrated-time Markovian process calculi, up to a tradeoff between compositionality and exactness inherent to the Markovian setting. The resulting weak Markovian bisimulation equivalence induces a pseudo-aggregation that is exact at stationary state for all the considered processes, but turns out to be a congruence only over sequential processes. To achieve compositionality over concurrent processes, we need to enhance the abstraction capability of the equivalence in the presence of interleaved computations. However, the corresponding pseudo-aggregation turns out to be exact at stationary state only for a subset of concurrent processes. In addition to this tradeoff, we present, for the first equivalence, a sound and complete axiomatization over sequential processes, which is instrumental to characterize pseudo-aggregations, and a polynomial-time equivalence-checking algorithm, which can be exploited for the compositional minimization of concurrent processes.

Improved Enumeration of Simple Topological Graphs

A simple topological graph T = (V(T), E(T)) is a drawing of a graph in the plane where every two edges have at most one common point (an endpoint or a crossing) and no three edges pass through a single crossing. Topological graphs G and H are isomorphic if H can be obtained from G by a homeomorphism of the sphere, and weakly isomorphic if G and H have the same set of pairs of crossing edges. We generalize results of Pach and Toth and the author's previous results on counting different drawings of a graph under both notions of isomorphism. We prove that for every graph G with n vertices, m edges and no isolated vertices the number of weak isomorphism classes of simple topological graphs that realize G is at most 2^O(n^2 log(m/n)), and at most 2^O(mn^{1/2} log n) if m < n^{3/2}. As a consequence we obtain a new upper bound 2^O(n^{3/2} log n) on the number of intersection graphs of n pseudosegments. We improve the upper bound on the number of weak isomorphism classes of simple complete topological graphs with n vertices to 2^{n^2 alpha(n)^O(1)}, using an upper bound on the size of a set of permutations with bounded VC-dimension recently proved by Cibulka and the author. We show that the number of isomorphism classes of simple topological graphs that realize G is at most 2^{m^2+O(mn)} and at least 2^Omega(m^2) for graphs with m > (6+epsilon)n.

Weak equivalence and non-classifiability of measure preserving actions

AbstractAbért and Weiss have shown that the Bernoulli shift ${s}_{\Gamma } $ of a countably infinite group $\Gamma $ is weakly contained in any free measure preserving action $\boldsymbol{a}$ of $\Gamma $. Proving a conjecture of Ioana, we establish a strong version of this result by showing that ${\boldsymbol{s}}_{\Gamma } \times \boldsymbol{a}$ is weakly equivalent to $\boldsymbol{a}$. Using random Bernoulli shifts introduced by Abért, Glasner, and Virag, we generalize this to non-free actions, replacing ${\boldsymbol{s}}_{\Gamma } $ with a random Bernoulli shift associated to an invariant random subgroup, and replacing the product action with a relatively independent joining. The result for free actions is used along with the theory of Borel reducibility and Hjorth’s theory of turbulence to show that, on the weak equivalence class of a free measure preserving action, the equivalence relations of isomorphism, weak isomorphism, and unitary equivalence are not classifiable by countable structures. This in particular shows that there are no free weakly rigid actions, that is, actions whose weak equivalence class and isomorphism class coincide, answering negatively a question of Abért and Elek. We also answer a question of Kechris regarding two ergodic theoretic properties of residually finite groups. A countably infinite residually finite group $\Gamma $ is said to have property ${\text{EMD} }^{\ast } $ if the action ${\boldsymbol{p}}_{\Gamma } $ of $\Gamma $ on its profinite completion weakly contains all ergodic measure preserving actions of $\Gamma $, and $\Gamma $ is said to have property $\text{MD} $ if $\boldsymbol{\iota} \times {\boldsymbol{p}}_{\Gamma } $ weakly contains all measure preserving actions of $\Gamma $, where $\boldsymbol{\iota} $ denotes the identity action on a standard non-atomic probability space. Kechris has shown that ${\text{EMD} }^{\ast } $ implies $\text{MD} $ and asked if the two properties are actually equivalent. We provide a positive answer to this question by studying the relationship between convexity and weak containment in the space of measure preserving actions.

Isomorphism Properties on Strong Fuzzy Hypergraphs

In this paper, the concept of hypergraph was extended to fuzzy hypergraph and paper discusses the connectedness of isomorphic fuzzy hypergraph. Image of strong arc under isomorphism and co-weak isomorphism are studied. Some properties of strong fuzzy hypergraphs are discussed. The image of strong fuzzy hypergraph under isomorphism, coweak isomorphism and weak isomorphism is also studied.

On the hardness of game equivalence under local isomorphism

We introduce a type of isomorphism among strategic games that we call local isomorphism . Local isomorphisms is a weaker version of the notions of strong and weak game isomorphism introduced in [J. Gabarro, A. Garcia and M. Serna, Theor. Comput. Sci. 412 (2011) 6675–6695]. In a local isomorphism it is required to preserve, for any player, the player’s preferences on the sets of strategy profiles that differ only in the action selected by this player. We show that the game isomorphism problem for local isomorphism is equivalent to the same problem for strong or weak isomorphism for strategic games given in: general, extensive and formula general form. As a consequence of the results in [J. Gabarro, A. Garcia and M. Serna, Theor. Comput. Sci. 412 (2011) 6675–6695] this implies that local isomorphism problem for strategic games is equivalent to (a) the circuit isomorphism problem for games given in general form, (b) the boolean formula isomorphism problem for formula games in general form, and (c) the graph isomorphism problem for games given in explicit form.

The complexity of game isomorphism

We address the question of whether two multiplayer strategic games are equivalent and the computational complexity of deciding such a property. We introduce two notions of isomorphisms, strong and weak. Each one of those isomorphisms preserves a different structure of the game. Strong isomorphisms are defined to preserve the utility functions and Nash equilibria. Weak isomorphisms preserve only the player’s preference relations and thus pure Nash equilibria. We show that the computational complexity of the game isomorphism problem depends on the level of succinctness of the description of the input games but it is independent of which of the two types of isomorphisms is considered. Utilities in games can be given succinctly by Turing machines, boolean circuits or boolean formulas, or explicitly by tables. Actions can be given both explicitly or succinctly. When the games are given in general form, we assume an explicit description of actions and a succinct description of utilities. We show that the game isomorphism problem for general form games is equivalent to the circuit isomorphism when utilities are described by TMs and to the boolean formula isomorphism problem when utilities are described by formulas. When the game is given in explicit form, we show that the game isomorphism problem is equivalent to the graph isomorphism problem.

Weak isomorphisms between Bernoulli shifts

In this note, we prove that if G is a countable group that contains a nonabelian free subgroup then every pair of nontrivial Bernoulli shifts over G are weakly isomorphic.