We present a new Fermi gas formalism for the ABJ matrix model. This formulation identifies the effect of the fractional M2-brane in the ABJ matrix model as that of a composite Wilson loop operator in the corresponding ABJM matrix model. Using this formalism, we study the phase part of the ABJ partition function numerically and find a simple expression for it. We further compute a few exact values of the partition function at some coupling constants. Fitting these exact values against the expected form of the grand potential, we can determine the grand potential with exact coefficients. The results at various coupling constants enable us to conjecture an explicit form of the grand potential for general coupling constants. The part of the conjectured grand potential from the perturbative sum, worldsheet instantons and bound states is regarded as a natural generalization of that in the ABJM matrix model, though the membrane instanton part contains a new contribution.

We give a Fourier-type formula for computing the orthogonal Weingarten formula. The Weingarten calculus was introduced as a systematic method to compute integrals of polynomials with respect to Haar measure over classical groups. Although a Fourier-type formula was known in the unitary case, the orthogonal counterpart was not known. It relies on the Jack polynomial generalization of both Schur and zonal polynomials. This formula substantially reduces the complexity involved in the computation of Weingarten formulas. We also describe a few more new properties of the Weingarten formula, state a conjecture and give a table of values.

We complete the analysis of all 2018 prime-field microlensing planets identified by the Korea Microlensing Telescope Network (KMTNet) Anomaly Finder. Among the ten previously unpublished events with clear planetary solutions, eight are either unambiguously planetary or are very likely to be planetary in nature: OGLE-2018-BLG-1126, KMT-2018-BLG-2004, OGLE-2018-BLG-1647, OGLE-2018-BLG-1367, OGLE-2018-BLG-1544, OGLE-2018-BLG-0932, OGLE-2018-BLG-1212, and KMT-2018-BLG-2718. Combined with the four previously published new Anomaly Finder events and 12 previously published (or in preparation) planets that were discovered by eye, this makes a total of 24 2018 prime-field planets discovered or recovered by Anomaly Finder. Together with a paper in preparation on 2018 subprime planets, this work lays the basis for the first statistical analysis of the planet mass-ratio function based on planets identified in KMTNet data. By systematically applying the heuristic analysis to each event, we identified the small modification in their formalism that is needed to unify the so-called close-wide and inner-outer degeneracies.

In this paper, we study the relationship between polynomial integrals on the unitary group and the conjugacy class expansion of symmetric functions in Jucys-Murphy elements. Our main result is an explicit formula for the top coefficients in the class expansion of monomial symmetric functions in Jucys-Murphy elements, from which we recover the first order asymptotics of polynomial integrals over U(N ) as N → ∞. Our results on class expansion include an analogue of Macdonald's result for the top connection coefficients of the class algebra, a generalization of Stanley and Olshanski's result on the polynomiality of content statistics on Plancherel-random partitions, and an exact formula for the multiplicity of the class of full cycles in the expansion of a complete symmetric function in Jucys-Murphy elements. The latter leads to a new combinatorial interpretation of the Carlitz-Riordan central factorial numbers.are the Catalan numbers. This estimate has been known to physicists for some time, see e.g.[1] for an application to quantum transport in mesoscopic systems.The first proof of (1.2) is due to Collins [4], and is quite involved. The appearance of the Catalan numbers, which are ubiquitous in enumerative combinatorics [44, Exercise 6.19], motivates the search for combinatorial structure underlying the averages u 11 . . . u nn , u 1π(1) . . . u nπ(n) N which might lead to a conceptually simple proof of (1.2). Along these lines, it was observed in [33] that the evaluation of the basic inner products is intimately related to a certain "inverse problem" in the character theory of the symmetric groups. This connection is explored in the present article. Working on the symmetric group side, we obtain a number of results concerning this "inverse problem" which give a new perspective on the structure of polynomial integrals on the unitary group (leading in particular to a new proof of (1.2)) and are moreover of significant interest in their own right.

Aspergillus nidulans possesses three pmt genes encoding protein O-D-mannosyltransferases (Pmt). Previously, we reported that PmtA, a member of the PMT2 subfamily, is involved in the proper maintenance of fungal morphology and formation of conidia (T. Oka, T. Hamaguchi, Y. Sameshima, M. Goto, and K. Furukawa, Microbiology 150: [1973][1974][1975][1976][1977][1978][1979][1980][1981][1982] 2004). In the present paper, we describe the characterization of the pmtA paralogues pmtB and pmtC. PmtB and PmtC were classified as members of the PMT1 and PMT4 subfamilies, respectively. A pmtB disruptant showed wild-type (wt) colony formation at 30°C but slightly repressed growth at 42°C. Conidiation of the pmtB disruptant was reduced to approximately 50% of that of the wt strain; in addition, hyperbranching of hyphae indicated that PmtB is involved in polarity maintenance. A pmtA and pmtB double disruptant was viable but very slow growing, with morphological characteristics that were cumulative with respect to either single disruptant. Of the three single pmt mutants, the pmtC disruptant showed the highest growth repression; the hyphae were swollen and frequently branched, and the ability to form conidia under normal growth conditions was lost. Recovery from the aberrant hyphal structures occurred in the presence of osmotic stabilizer, implying that PmtC is responsible for the maintenance of cell wall integrity. Osmotic stabilization at 42°C further enabled the pmtC disruptant to form conidiophores and conidia, but they were abnormal and much fewer than those of the wt strain. Apart from the different, abnormal phenotypes, the three pmt disruptants exhibited differences in their sensitivities to antifungal reagents, mannosylation activities, and glycoprotein profiles, indicating that PmtA, PmtB, and PmtC perform unique functions during cell growth.Protein glycosylation, which is a major posttranslational modification, plays essential roles in eukaryotic cells from fungi to mammals (19). N-linked oligosaccharides in glycoproteins that share relatively common structures are structurally classified into high-mannose, complex, and hybrid types (3). Olinked oligosaccharides in glycoproteins are diverse with respect to their sugar components and the mode of sugar linkages among the eukaryotic organisms (8,19). O mannosylation, which is commonly found in the glycoproteins of fungi, has been extensively studied in the budding yeast Saccharomyces cerevisiae (4, 21, 35). The initial reaction of mannose transfer to serine and threonine residues in proteins is catalyzed by protein O-D-mannosyltransferase (Pmt) in the endoplasmic reticulum (ER), where dolichyl phosphate-mannose is required as an immediate sugar donor (4). In the Golgi complex, O mannosylation in S. cerevisiae is linearly elongated by up to five mannose residues by mannosyltransferases (Mnt) that utilize GDP-mannose as the mannosyl donor. At least six Pmt-encoding genes (PMT1 to -6), three ␣-1,2-Mnt-encoding genes (KRE2, KTR1, and KTR3), and three ␣-1,3-Mnt-encoding genes (...