Nearly tight bounds for testing function isomorphism

We study the problem of testing isomorphism (equivalence up to relabelling of the variables) of two Boolean functions f, g: {0, 1} n → {0, 1}. Our main focus is on the most studied case, where one of the functions is given (explicitly) and the other function may be queried. We prove that for every k ≤ n, the worst-case query complexity of testing isomorphism to a given k-junta is Ω(k) and O(k log k). Consequently, the query complexity of testing function isomorphism is e Θ(n). Prior to this work, only lower bounds of Ω(log k) queries were known, for limited ranges of k, proved by Fischer et al. (FOCS 2002), Blais and O’Donnell (CCC 2010), and recently by Alon and Blais (RANDOM 2010). The nearly tight O(k log k) upper bound improves on the e O(k 4) upper bound from Fischer et al. (FOCS 2002). Extending the lower bound proof, we also show polynomial query-complexity lower bounds for the problems of testing whether a function can be computed by a circuit of size ≤ s, and testing whether the Fourier degree of a function is ≤ d. This answers questions posed by Diakonikolas et al. (FOCS 2007). We also address two closely related problems – 1. Testing isomorphism to a k-junta with one-sided error: we prove that for any 1 < k < n − 1, the query complexity is Ω(log ` ´ n), which is almost optimal. Thi

Molecular Markers in the Study of Nonmodel Vertebrates: Their Significant Contributions to the Current Knowledge of Tetrapod Glial Cells and Fish Olfactory Neurons

The knowledge of the morphological and functional aspects of mammalian glial cells has greatly increased in the last few decades. Glial cells represent the most diffused cell type in the central nervous system, and they play a critical role in the development and function of the brain. Glial cell dysfunction has recently been shown to contribute to various neurological disorders, such as autism, schizophrenia, pain, and neurodegeneration. For this reason, glia constitutes an interesting area of research because of its clinical, diagnostic, and pharmacological relapses. In this chapter, we present and discuss the cytoarchitecture of glial cells in tetrapods from an evolutive perspective. GFAP and vimentin are main components of the intermediate filaments of glial cells and are used as cytoskeletal molecular markers because of their high degree of conservation in the various vertebrate groups. In the anamniotic tetrapods and their progenitors, Rhipidistia (Dipnoi are the only extant rhipidistian fish), the cytoskeletal markers show a model based exclusively on radial glial cells. In the transition from primitive vertebrates to successively evolved forms, the emergence of a new model has been observed which is believed to support the most complex functional aspects of the nervous system in the vertebrates. In reptiles, radial glial cells are prevalent, but star-shaped astrocytes begin to appear in the midbrain. In endothermic amniotes (birds and mammals), star-shaped astrocytes are predominant. In glial cells, vimentin is indicative of immature cells, while GFAP indicates mature ones. Olfactory receptor neurons undergo continuous turnover, so they are an easy model for neurogenesis studies. Moreover, they are useful in neurotoxicity studies because of the exposed position of their apical pole to the external environment. Among vertebrates, fish represent a valid biological model in this field. In particular, zebrafish, already used in laboratories for embryological, neurobiological, genetic, and pathophysiological studies, is the reference organism in olfactory system research. Smell plays an important role in the reproductive behavior of fish, with direct influences also on the numerical consistency of their populations. Taking into account that a lot of species have considerable economic importance, it is necessary to verify if the model of zebrafish olfactory organ is also directly applicable to other fish. In this chapter, we focus on crypt cells, a morphological type of olfactory cells specific of fish. We describe hypothetical function (probably related with social behavior) and evolutive position of these cells (prior to the appearance of the vomeronasal organ in tetrapods).We also offer the first comparison of the molecular characteristics of these receptors between zebrafish and the guppy. Interestingly, the immunohistochemical expression patterns of known crypt cell markers are not overlapping in the two species