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Hadi Zare

Publications and source records attributed to Hadi Zare.

32 records · Page 2Linked to original sources
arXiv2017

IEDC: An Integrated Approach for Overlapping and Non-overlapping Community Detection

Community detection is a task of fundamental importance in social network analysis that can be used in a variety of knowledge-based domains. While there exist many works on community detection based on connectivity structures, they suffer from either considering the overlapping or non-overlapping communities. In this work, we propose a novel approach for general community detection through an integrated framework to extract the overlapping and non-overlapping community structures without assuming prior structural connectivity on networks. Our general framework is based on a primary node based criterion which consists of the internal association degree along with the external association degree. The evaluation of the proposed method is investigated through the extensive simulation experiments and several benchmark real network datasets. The experimental results show that the proposed method outperforms the earlier state-of-the-art algorithms based on the well-known evaluation criteria.

cs.SI↗
arXiv2016

Splitting Madsen-Tillmann spectra I. Twisted transfer maps

We record various properties of twisted Becker-Gottlieb transfer maps and study their multiplicative properties analogous to Becker-Gottlieb transfer. We show these twisted transfer maps factorise through Becker-Schultz-Mann-Miller-Miller transfer; some of these might be well known. We apply this to show that $BSO(2n+1)_+$ splits off $MTO(2n)$, which after localisation away from $2$, refines to a homotopy equivalence $MTO(2n)\simeq BO(2n)_+$ as well as $MTO(2n+1)\simeq *$ for all $n\geqslant0$. This reduces the study of $MTO(n)$ to the $2$-localised case. At the prime $2$ our splitting allows to identify some algebraically independent classes in mod $2$ cohomology of $Ω^\infty MTO(2n)$. We also show that $BG_+$ splits off $MTK$ for some pairs $(G,K)$ at appropriate set of primes $p$, and investigate the consequences for characteristic classes, including algebraic independence and non-divisibility of some universally defined characteristic classes, generalizing results of Ebert and Randal-Williams.

math.AT↗
arXiv2016

Spherical classes in some finite loop spaces of spheres

Working at the prime $2$, Curtis conjecture predicts that, in positive dimensions, spherical classes in $H_*QS^0$ only arise from Hopf invariant one and Kervaire invariant one elements. Eccles conjecture states that, in positive dimensional, for a path connected space $X$, a class in $H_nQX$ is spherical if its either stably spherical or it arises from a stable map $S^n\to X$ which is detected by a primary operation in its mapping cone. (i) We use Hopf invariant one result to verify Eccles conjecture on some finite loop spaces of spheres, namely, for $X=S^k$ , we completely determine spherical classes in $H_*(Ω^dS^{k+d};\mathbb{Z}/2)$ for specific values of $d$, $k$ showing that a spherical classes in these cases only do arise from Hopf invariant one elements. (ii) We completely determine spherical classes in homology of single, double, and triple loop spaces of spheres, namely $ΩS^{n+1}$, $Ω^2 S^{n+2}$, and $Ω^3S^{n+3}$ with $n>0$. These computations, verify Eccles conjecture on the finite loop spaces that we have considered. We also record some observations on the relation between two conjectures. The latter conjecture for $X=S^k$, with $k>0$, provides some evidence for the former to be true.

math.AT↗
arXiv2016

Relevant based structure learning for feature selection

Feature selection is an important task in many problems occurring in pattern recognition, bioinformatics, machine learning and data mining applications. The feature selection approach enables us to reduce the computation burden and the falling accuracy effect of dealing with huge number of features in typical learning problems. There is a variety of techniques for feature selection in supervised learning problems based on different selection metrics. In this paper, we propose a novel unified framework for feature selection built on the graphical models and information theoretic tools. The proposed approach exploits the structure learning among features to select more relevant and less redundant features to the predictive modeling problem according to a primary novel likelihood based criterion. In line with the selection of the optimal subset of features through the proposed method, it provides us the Bayesian network classifier without the additional cost of model training on the selected subset of features. The optimal properties of our method are established through empirical studies and computational complexity analysis. Furthermore the proposed approach is evaluated on a bunch of benchmark datasets based on the well-known classification algorithms. Extensive experiments confirm the significant improvement of the proposed approach compared to the earlier works.

cs.LG↗
arXiv2016

On the Bott periodicity, $\mathcal{A}$-annihilated classes in $H_*QX$, and the stable symmetric hit problem

We provide a characterisation of $\mathcal{A}$-annihilated generators in the homology ring $H_*(QX;\mathbb{Z}/2)$ and $H_*(Q(X_+);\mathbb{Z}/2)$ when $X$ is some path connected space. We also introduce a method to construct such classes. We comment on the application of this result to illustrate how to use the infinite loop space structure on $\mathbb{Z}\times BO$, provided by the Bott periodicity can be used to obtain some information on the (stable) symmetric hit problem of Wood and Janfada. Our methods seem to allow much straightforward calculations. The numerical conditions of our Theorem 3 look very similar to the `spikes' considered by Wood \cite{Wood-Ioa} and Janfada-Wood \cite{JanfadaWood} as well as Janfada \cite{Janfada-P(3)}.

math.AT↗
arXiv2016

On the relation between Dyer-Lashof algebra and the hit problems

The aim of this note is to use geometric methods to study the hit problem of Peterson for $H_*\mathbb{R} P^{\times k}$ as well as the symmetric hit problem of Janfada and Wood for $H_*BO(k)$. We continue by exploring the applications of the results of \cite{Zare-symmetric} on the $\mathcal{A}$-annihilated generators of $H_*QX$ to obtain a family of generic `new' examples of $\mathcal{A}$-annihilated in $H_*(\mathbb{Z}\times BO)$ and $H_*BO$, i.e. the case of stable symmetric hit problem, where an essential step is provided by the infinite loop space structure on $BO$ implied by the Bott periodicity. Applying a length filtration allows to state our results in the case of symmetric hit problem for $H_*BO(k)$. Using the Becker-Gottlieb transfer associated to $\mathbb{R} P^{\times k}=BO(1)^{\times k}\to BO(k)$ we are able to restate our results for the classic hit problem of $H_*\mathbb{R} P^{\times k}$. We use the phrase `stable hit problem' to the study of hit problem for $H_*\mathbb{R} P^{\times k}$ for all $k>0$ at once, which allows to use multiplicative structures on which we seem to have taken some new steps after \cite{Ault-Singer}. Our new examples depend on specific numerical conditions of which we have provided an algorithm to construct in \cite{Zare-symmetric}. The methodological outcome is that such conditions also have to taken into account while dealing with counting arguments. The numerical conditions we obtain seem to have not appeared in the literature in this context, although they may include previous ones as examples. Therefore, our work provides an infinite family of new examples and consequently raises the lower bounds obtained previously.

math.AT↗
arXiv2015

Generalised geometric weak conjecture on spherical classes and non-factorisation of Kervaire invariant one elements

This paper is on the Curtis conjecture. We show that the image of the Hurewicz homomorhism $h:π_*Q_0S^0\to H_*(Q_0S^0;\mathbb{Z})$, when restricted to product of positive dimensional elements, is determined by $\mathbb{Z}\{h(η^2),h(ν^2),h(σ^2)\}$. Localised at $p=2$, this proves a geometric version of a result of Hung and Peterson for the Lannes-Zarati homomorphism. We apply this to show that, for $p=2$ and $G=O(1)$ or any prime $p$ and $G$ any compact Lie group with Lie algebra $\mathfrak{g}$ so that $\dim\mathfrak{g}>0$, the composition $${_pπ_*}QΣ^{n\dim\mathfrak{g}}BG_+^{\wedge n}\to {_pπ_*}Q_0S^0\stackrel{h}{\to}H_*(Q_0S^0;\mathbb{Z}/p)$$ where $Σ^{n\dim\mathfrak{g}}BG_+^{\wedge n}\to S^0$ is the $n$-fold transfer, is trivial if $n>2$. Moreover, we show that for $n=2$, the image of the above composition vanishes on all elements of Adams filtration at least $1$, i.e. those elements of ${_2π_*^s}Σ^{n\dim\mathfrak{g}}BG_+^{\wedge n}$ represented by a permanent cycle $\mathrm{Ext}_{A_p}^{s,t}(\widetilde{H}^*Σ^{n\dim\mathfrak{g}}BG_+^{\wedge n},\mathbb{Z}/p)$ with $s>0$, map trivially under the above composition. The case of $n>2$ of the above observation proves and generalises a geometric variant of the weak conjecture on spherical classes due to Hung, later on verified by Hung and Nam. We also show that, for a compact Lie group $G$, Curtis conjecture holds if we restrict to the image of the $n$-fold transfer $Σ^{n\dim\mathfrak{g}}BG_+^{\wedge n}\to S^0$ with $n>1$. Finally, we show that the Kervaire invariant one elements $θ_j\in{_2π_{2^{j+1}-2}^s}$ with $j>3$ do not factorise through the $n$-fold transfer $Σ^{n\dim\mathfrak{g}}BG_+^{\wedge n}\to S^0$ with $n>1$ for $G=O(1)$ or any compact Lie group as above.

math.AT↗
arXiv2015

Splitting Madsen-Tillmann spectra II. The Steinberg idempotents and Whitehead conjecture

We show that, at the prime $p=2$, the spectrum $Σ^{-n}D(n)$ splits off the Madsen-Tillmann spectrum $MTO(n)=BO(n)^{-γ_n}$ which is compatible with the classic splitting of $M(n)$ off $BO(n)_+$. For $n=2$, together with our previous splitting result on Madsen-Tillmann spectra, this shows that $MTO(2)$ is homotopy equivalent to $BSO(3)_+\veeΣ^{-2}D(2)$.

math.AT↗
arXiv2015

On the Hurewicz homomorphism on the extensions of ideals in $π_*^s$ and spherical classes in $H_*Q_0S^0$

This is about Curtis conjecture on the image of the Hurewicz map $h:{_2π_*}Q_0S^0\to H_*(Q_0S^0;\Z/2)$. First, we show that if $f\in{_2π_*^s}$ is of Adams filtration at least $3$ with $h(f)\neq 0$ then $f$ is not a decomposable element in ${_2π_*^s}$. Moreover, it is shown if $k$ is the least positive integer that $f$ is represented by a cycle in $\mathrm{Ext}^{k,k+n}_A(\Z/2,\Z/2)$, then (i) if $e_*h(f)\neq 0$ then $n\geqslant 2^k-1$; (ii) if $e_*h(f)=0$ then $n\geqslant 2^k-2^t$ for some $t>1$. Second, for $S\subseteq{_2π_{*>0}^s}$ we show that: (i) if the conjecture holds on $S$, then it holds on $\la S\ra$; (ii) if $h(S)=0$ then $h$ acts trivially on any extension of $S$ obtained by applying homotopy operations arising from ${_2π_*}D_rS^n$ with $n>0$. We also provide partial results on the extensions of $\la S\ra$ by taking (possible) Toda brackets of its elements. We also discuss how the $EHP$-sequence information maybe applied to eliminate classes from being spherical.

math.AT↗
arXiv2011

On spherical classes in $H_*QX$

We give an upper bound on the set of spherical classes in $H_*QX$ when $X = P,S^1$. This is related to the Curtis conjecture on spherical classes in $H_*Q_0S^0$. The results also provide some control over the bordism classes on of immersions when $X$ is a Thom complex.

math.AT↗
arXiv2010

On the Bott periodicity, $J$-homomorphisms, transfer maps and $H_*Q_0S^{-n}$

The Curtis conjecture predicts that the only spherical classes in $H_*(Q_0S^0;\Z/2)$ are the Hopf invariant one and the Kervaire invariant one elements. We consider Sullivan's decomposition $$Q_0S^0=J\times\cok J$$ where $J$ is the fibre of $ψ^q-1$ ($q=3$ at the prime $2$) and observe that the Curtis conjecture holds when we restrict to $J$. We then use the Bott periodicity and the $J$-homomorphism $SO\to Q_0S^0$ to define some generators in $H_*(Q_0S^0;\Z/p)$, when $p$ is any prime, and determine the type of subalgebras that they generate. For $p=2$ we determine spherical classes in $H_*(Ω^k_0J;\Z/2)$. We determine truncated subalgebras inside $H_*(Q_0S^{-k};\Z/2)$. Applying the machinery of the Eilenberg-Moore spectral sequence we define classes that are not in the image of by the $J$-homomorphism. We shall make some partial observations on the algebraic structure of $H_*(Ω^k_0\cok J;\Z/2)$. Finally, we shall make some comments on the problem in the case equivariant $J$-homomorphisms.

math.AT↗
arXiv2009

The Hurewicz image of the $η_i$ family, a polynomial subalgebra of $H_*Ω_0^{2^{i+1}-8+k}S^{2^i-2}$

We consider the problem of calculating the Hurewicz image of Mahowald's family $η_i\in{_2π_{2^i}^S}$. This allows us to identify specific spherical classes in $H_*Ω_0^{2^{i+1}-8+k}S^{2^i-2}$ for $0\leqslant k\leqslant 6$. We then identify the type of the subalgebras that these classes give rise to, and calculate the $A$-module and $R$-module structure of these subalgebras. We shall the discuss the relation of these calculations to the Curtis conjecture on spherical classes in $H_*Q_0S^0$, and relations with spherical classes in $H_*Q_0S^{-n}$.

math.AT↗
arXiv2009

On the form of potential spherical classes in $H_*Q_0S^0$

This note is about spherical classes in $H_*Q_0S^0$. A conjecture, due to Ed. Curtis, predicts that only Hopf invariant one and Kervaire invariant one elements will give rise to spherical classes in $H_*Q_0S^0$. Yet, there has been no proof of this conjecture around. Assuming that this conjecture fails, there must exist some other spherical classes in $H_*Q_0S^0$. This note determines the form of these potential spherical classes, and sets the target for someone who wishes to prove the conjecture, in the sense that correctness of the Curtis conjecture will be the same as failure of any classes predicted in this paper being spherical.

math.AT↗