Linear Operators in Information Retrieval

In this paper, we propose a pseudo-relevance feedback approach based on linear operators: vector space basis change and cross product. The aim of pseudo-relevance feedback methods based on vector space basis change IBM (Ideal Basis Method) is to optimally separate relevant and irrelevant documents. Whereas the aim of pseudo-relevance feedback method based on cross product AI (Absorption of irrelevance) is to effectively exploit irrelevant documents. We show how to combine IBM methods with AI methods. The combination methods IBM+AI are evaluated experimentally on two TREC collections (TREC-7 ad hoc and TREC-8 ad hoc). The experiments show that these methods improve previous works

Rocchio\u27s Model Based on Vector Space Basis Change for Pseudo Relevance Feedback

Rocchio\u27s relevance feedback model is a classic query expansion method and it has been shown to be effective in boosting information retrieval performance. The main problem with this method is that the relevant and the irrelevant documents overlap in the vector space because they often share same terms (at least the terms of the query). With respect to the initial vector space basis (index terms), it is difficult to select terms that separate relevant and irrelevant documents. The Vector Space Basis Change is used to separate relevant and irrelevant documents without any modification on the query term weights. In this paper, first, we study how to incorporate Vector Space Basis Change into the Rocchio\u27s model. Second, we propose Rocchio\u27s models based on Vector Space Basis Change, called VSBCRoc models. Experimental results on a TREC collection show that our proposed models are effective

Equicontinuous local dendrite maps

[EN] Let X be a local dendrite, and f : X → X be a map. Denote by E(X) the set of endpoints of X. We show that if E(X) is countable, then the following are equivalent:(1) f is equicontinuous;(2)  fn (X) = R(f);(3) f|  fn (X) is equicontinuous;(4) f| fn (X) is a pointwise periodic homeomorphism or is topologically conjugate to an irrational rotation of S 1 ;(5) ω(x, f) = Ω(x, f) for all x ∈ X.This result generalizes [17, Theorem 5.2], [24, Theorem 2] and [11, Theorem 2.8].Salem, AH.; Hattab, H.; Rejeiba, T. (2021). Equicontinuous local dendrite maps. Applied General Topology. 22(1):67-77. https://doi.org/10.4995/agt.2021.13446OJS6777221H. Abdelli, ω-limit sets for monotone local dendrite maps. Chaos, Solitons and Fractals, 71 (2015), 66-72. https://doi.org/10.1016/j.chaos.2014.12.003H. Abdelli and H. Marzougui, Invariant sets for monotone local dendrite maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 26, no. 9 (2016), 1650150 (10 pages). https://doi.org/10.1142/S0218127416501509E. Akin, J. Auslander and K. Berg, When is a transitive map chaotic?, in: Convergence in Ergodic Theory and Probability, Walter de Gruyter and Co., Berlin, 1996, pp. 25-40. https://doi.org/10.1515/9783110889383.25G. Askri and I. Naghmouchi, Pointwise recurrence on local dendrites, Qual. Theory Dyn Syst 19, 6 (2020). https://doi.org/10.1007/s12346-020-00347-8F. Balibrea, T. Downarowicz, R. Hric, L. Snoha and V. Spitalsky, Almost totally disconnected minimal systems, Ergodic Th. & Dynam Sys. 29, no. 3 (2009), 737-766. https://doi.org/10.1017/S0143385708000540F. Blanchard, B. Host and A. Maass, Topological complexity, Ergodic Th. & Dynam Sys. 20 (2000), 641-662. https://doi.org/10.1017/S0143385700000341A. M. Blokh, Pointwise-recurrent maps on uniquely arcwise connected locally arcwise connected spaces, Proc. Amer. Math. Soc. 143 (2015), 3985-4000. https://doi.org/10.1090/S0002-9939-2015-12589-0A. M. Blokh, The set of all iterates is nowhere dense in C([0,1],[0,1]), Trans. Amer. Math. Soc. 333, no. 2 (1992), 787-798. https://doi.org/10.1090/S0002-9947-1992-1153009-7W. Boyce, Γ-compact maps on an interval and fixed points, Trans. Amer. Math. Soc. 160 (1971), 87-102. https://doi.org/10.1090/S0002-9947-1971-0280655-1A. M. Bruckner and T. Hu, Equicontinuity of iterates of an interval map, Tamkang J. Math. 21, no. 3 (1990), 287-294.J. Camargo, M. Rincón and C. Uzcátegui, Equicontinuity of maps on dendrites, Chaos, Solitons and Fractals 126 (2019), 1-6. https://doi.org/10.1016/j.chaos.2019.05.033J. Cano, Common fixed points for a class of commuting mappings on an interval, Trans. Amer. Math. Soc. 86, no. 2 (1982), 336-338. https://doi.org/10.1090/S0002-9939-1982-0667301-2R. Gu and Z. Qiao, Equicontinuity of maps on figure-eight space, Southeast Asian Bull. Math. 25 (2001), 413-419. https://doi.org/10.1007/s100120100004A. Haj Salem and H. Hattab, Group action on local dendrites, Topology Appl. 247, no. 15 (2018), 91-99. https://doi.org/10.1016/j.topol.2018.08.002K. Kuratowski, Topology, vol. 2. New York: Academic Press; 1968.J. Mai, Pointwise-recurrent graph maps, Ergodic Th. & Dynam Sys. 25 (2005), 629-637. https://doi.org/10.1017/S0143385704000720J. Mai, The structure of equicontinuous maps, Trans. Amer. Math. Soc. 355, no. 10 (2003), 4125-4136. https://doi.org/10.1090/S0002-9947-03-03339-7C. A. Morales, Equicontinuity on semi-locally connected spaces, Topology Appl. 198 (2016), 101-106. https://doi.org/10.1016/j.topol.2015.11.011S. Nadler, Continuum Theory. Inc., New York: Marcel Dekker; 1992.G. Su and B. Qin, Equicontinuous dendrites flows, Journal of Difference Equations and Applications 25, no. 12 (2019), 1744-1754. https://doi.org/10.1080/10236198.2019.1694012T. Sun, Equicontinuity of σ-maps, Pure and Applied Math. 16, no. 3 (2000), 9-14.T. Sun, Z. Chen, X. Liu and H. G. Xi, Equicontinuity of dendrite maps, Chaos, Solitons and Fractals 69 (2014), 10-13. https://doi.org/10.1016/j.chaos.2014.08.010T. Sun, G. Wang and H. J. Xi, Equicontinuity of maps on a dendrite with finite branch points. Acta Mat. Sin. 33, no. 8 (2017), 1125-1130. https://doi.org/10.1007/s10114-017-6289-xT. Sun, Y. Zhang and X. Zhang, Equicontinuity of a graph map, Bull. Austral Math. Soc. 71 (2005), 61-67. https://doi.org/10.1017/S0004972700038016A. Valaristos, Equicontinuity of iterates of circle maps, Internat. J. Math. and Math. Sci. 21 (1998), 453-458. https://doi.org/10.1155/S016117129800062

A Re-Ranking Method Based on Irrelevant Documents in Ad-Hoc Retrieval

In this paper, we propose a novel approach for document re-ranking, which relies on the concept of negative feedback represented by irrelevant documents. In a previous paper, a pseudo-relevance feedback method is introduced using an absorbing document ~d which best fits the user\u27s need. The document ~d is orthogonal to the majority of irrelevant documents. In this paper, this document is used to re-rank the initial set of ranked documents in Ad-hoc retrieval. The evaluation carried out on a standard document collection shows the effectiveness of the proposed approach

The fundamental group of the orbit space

Let G be a subgroup of the group Homeo(X) of homeomorphisms of a topological space X. Let G¯$\bar G$ be the closure of G in Homeo(X). The class of an orbit O of G is the union of all orbits having the same closure as O. We denote by X/G˜X/\widetildeG the space of classes of orbits called the orbit class space. In this paper, we study the fundamental group of the spaces X/G, X/G¯$X/\bar G$ and X/G˜$X/\widetildeG

A model of quotient spaces

Let R be an open equivalence relation on a topological space E. We define on E a new equivalence relation ̃ℜ̅ by x̃ ̃ℜ̅y if the closure of the R-trajectory of x is equal to the closure of the R-trajectory of y. The quotient space E/̃ ̃ℜ̅ is called the trajectory class space. In this paper, we show that the space E/̃ ̃ℜ̅ is a simple model of the quotient space E/R. This model can provide a finite model. Some applications to orbit spaces of groups of homeomorphisms and leaf spaces are given