We introduce the notion of query substitution, that is, generating a new query to replace a user's original search query. Our technique uses modifications based on typical substitutions web searchers make to their queries. In this way the new query is strongly related to the original query, containing terms closely related to all of the original terms. This contrasts with query expansion through pseudo-relevance feedback, which is costly and can lead to query drift. This also contrasts with query relaxation through boolean or TFIDF retrieval, which reduces the specificity of the query. We define a scale for evaluating query substitution, and show that our method performs well at generating new queries related to the original queries. We build a model for selecting between candidates, by using a number of features relating the query-candidate pair, and by fitting the model to human judgments of relevance of query suggestions. This further improves the quality of the candidates generated. Experiments show that our techniques significantly increase coverage and effectiveness in the setting of sponsored search.
We present the numerical solution of the full gap equation in a weak coupling constant g. It is found that the standard approximations to derive the gap equation to the leading order of coupling constant are essential for a secure numerical evaluation of the logarithmic singularity with a small coupling constant. The approximate integral gap equation with a very small g should be inverted to a soft integral equation to smooth the logarithmic singularity near the Fermi surface. The full gap equation is solved for a rather large coupling constant g ≥ 2.0. The approximate and soft integral gap equations are solved for small g values. When their solutions are extrapolated to larger g values, they coincide the full gap equation solution near the Fermi surface. Furthermore, the analytical solution matches the numerical one up to the order one O(1). Our results confirm the previous estimates that the gap energy is of the order tens to 100 MeV for the chemical potential µ ≤ 1000 MeV. They also support the validity of leading approximations applied to the full gap equation to derive the soft integral gap equation and its analytical solution near the Fermi surface.
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