| 1. |
- Abdallah, Nancy, et al.
(författare)
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Topology of posets with special partial matchings
- 2019
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Ingår i: Advances in Mathematics. - : Academic Press. - 0001-8708 .- 1090-2082. ; 348, s. 255-276
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Tidskriftsartikel (refereegranskat)abstract
- Special partial matchings (SPMs) are a generalisation of Brentis special matchings. Let a pircon be a poset in which every non-trivial principal order ideal is finite and admits an SPM. Thus pircons generalise Mariettis zircons. We prove that every open interval in a pircon is a PL ball or a PL sphere. It is then demonstrated that Bruhat orders on certain twisted identities and quasiparabolic W-sets constitute pircons. Together, these results extend a result of Can, Cherniaysky, and Twelbeck, prove a conjecture of Hultman, and confirm a claim of Rains and Vazirani.
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| 2. |
- Azimi, Alireza, et al.
(författare)
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Targeting CDK2 overcomes melanoma resistance against BRAF and Hsp90 inhibitors
- 2018
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Ingår i: Molecular Systems Biology. - : EMBO. - 1744-4292. ; 14:3
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Tidskriftsartikel (refereegranskat)abstract
- Novel therapies are undergoing clinical trials, for example, the Hsp90 inhibitor, XL888, in combination with BRAF inhibitors for the treatment of therapy-resistant melanomas. Unfortunately, our data show that this combination elicits a heterogeneous response in a panel of melanoma cell lines including PDX-derived models. We sought to understand the mechanisms underlying the differential responses and suggest a patient stratification strategy. Thermal proteome profiling (TPP) identified the protein targets of XL888 in a pair of sensitive and unresponsive cell lines. Unbiased proteomics and phosphoproteomics analyses identified CDK2 as a driver of resistance to both BRAF and Hsp90 inhibitors and its expression is regulated by the transcription factor MITF upon XL888 treatment. The CDK2 inhibitor, dinaciclib, attenuated resistance to both classes of inhibitors and combinations thereof. Notably, we found that MITF expression correlates with CDK2 upregulation in patients; thus, dinaciclib would warrant consideration for treatment of patients unresponsive to BRAF-MEK and/or Hsp90 inhibitors and/or harboring MITF amplification/overexpression.
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| 3. |
- Hansson, Mikael, 1986-
(författare)
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Combinatorics and topology related to involutions in Coxeter groups
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This dissertation consists of three papers in combinatorial Coxeter group theory.A Coxeter group is a group W generated by a set S, where all relations can be derived from the relations s2 = e for all s ? S, and (ss′)m(s,s′) = e for some pairs of generators s ≠ s′ in S, where e ? W is the identity element and m(s, s′) is an integer satisfying that m(s, s′) = m(s′, s) ≥ 2. Two prominent examples of Coxeter groups are provided by the symmetric group Sn (i.e., the set of permutations of {1, 2, . . . , n}) and finite reflection groups (i.e., finite groups generated by reflections in some real euclidean space). There are also important infinite Coxeter groups, e.g., affine reflection groups.Every Coxeter group can be equipped with various natural partial orders, the most important of which is the Bruhat order. Any subset of a Coxeter group can then be viewed as an induced subposet.In Paper A, we study certain posets of this kind, namely, unions of conjugacy classes of involutions in the symmetric group. We obtain a complete classification of the posets that are pure (i.e., all maximal chains have the same length). In particular, we prove that the set of involutions with exactly one fixed point is pure, which settles a conjecture of Hultman in the affirmative. When the posets are pure, we give their rank functions. We also give a short, new proof of the EL-shellability of the set of fixed-point-free involutions, established by Can, Cherniavsky, and Twelbeck.Paper B also deals with involutions in Coxeter groups. Given an involutive automorphism θ of a Coxeter system (W, S), letℑ(θ) = {w ? W | θ(w) = w−1}be the set of twisted involutions. In particular, ℑ(id) is the set of ordinary involutions in W. It is known that twisted involutions can be represented by words in the alphabet = { | s ? S}, called -expressions. If ss′ has finite order m(s, s′), let a braid move be the replacement of ′ ⋯ by ′ ′ ⋯, both consisting of m(s, s′) letters. We prove a word property for ℑ(θ), for any Coxeter system (W, S) with any θ. More precisely, we provide a minimal set of moves, easily determined from the Coxeter graph of (W, S), that can be added to the braid moves in order to connect all reduced -expressions for any given w ? ℑ(θ). This improves upon a result of Hamaker, Marberg, and Pawlowski, and generalises similar statements valid in certain types due to Hu, Zhang, Wu, and Marberg.In Paper C, we investigate the topology of (the order complexes of) certain posets, called pircons. A special partial matching (SPM) on a poset is a matching of the Hasse diagram satisfying certain extra conditions. An SPM without fixed points is precisely a special matching as defined by Brenti. Let a pircon be a poset in which every non-trivial principal order ideal is finite and admits an SPM. Thus pircons generalise Marietti’s zircons. Our main result is that every open interval in a pircon is a PL ball or a PL sphere.An important subset of ℑ(θ) is the set ?(θ) = {θ(w−1)w | w ? W} of twisted identities. We prove that if θ does not flip any edges with odd labels in the Coxeter graph, then ?(θ), with the order induced by the Bruhat order on W, is a pircon. Hence, its open intervals are PL balls or spheres, which confirms a conjecture of Hultman. It is also demonstrated that Bruhat orders on Rains and Vazirani’s quasiparabolic W-sets (under a boundedness assumption) form pircons. In particular, this applies to all parabolic quotients of Coxeter groups.
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| 4. |
- Hansson, Mikael, 1986-
(författare)
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Generalised Ramsey numbers and Bruhat order on involutions
- 2015
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis consists of two papers within two different areas of combinatorics.Ramsey theory is a classic topic in graph theory, and Paper A deals with two of its most fundamental problems: to compute Ramsey numbers and to characterise critical graphs. More precisely, we study generalised Ramsey numbers for two sets Γ1 and Γ2 of cycles. We determine, in particular, all generalised Ramsey numbers R(Γ1, Γ2) such that Γ1 or Γ2 contains a cycle of length at most 6, or the shortest cycle in each set is even. This generalises previous results of Erdös, Faudree, Rosta, Rousseau, and Schelp. Furthermore, we give a conjecture for the general case. We also characterise many (Γ1, Γ2)-critical graphs. As special cases, we obtain complete characterisations of all (Cn,C3)-critical graphs for n ≥ 5, and all (Cn,C5)-critical graphs for n ≥ 6.In Paper B, we study the combinatorics of certain partially ordered sets. These posets are unions of conjugacy classes of involutions in the symmetric group Sn, with the order induced by the Bruhat order on Sn. We obtain a complete characterisation of the posets that are graded. In particular, we prove that the set of involutions with exactly one fixed point is graded, which settles a conjecture of Hultman in the affirmative. When the posets are graded, we give their rank functions. We also give a short, new proof of the EL-shellability of the set of fixed-point-free involutions, recently proved by Can, Cherniavsky, and Twelbeck.
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