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Journal articles on the topic 'Elementary Type Conjecture'
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1
Keller, Timothy. "Witt groups and the elementary type conjecture." Communications in Algebra 23, no. 1 (January 1995): 277–89. http://dx.doi.org/10.1080/00927879508825221.
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Amir, Firana, and Mohammad Faizal Amir. "Action Proof: Analyzing Elementary School Students Informal Proving Stages through Counter-examples." International Journal of Elementary Education 5, no. 2 (August 23, 2021): 401. http://dx.doi.org/10.23887/ijee.v5i3.35089.
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Both female and male elementary school students have difficulty doing action proof by using manipulative objects to provide conjectures and proof of the truth of a mathematical statement. Counter-examples can help elementary school students build informal proof stages to propose conjectures and proof of the truth of a mathematical statement more precisely. This study analyzes the action proof stages through counter-examples stimulation for male and female students in elementary schools. The action proof stage in this study focuses on three stages: proved their primitive conjecture, confronted counter-examples, and re-examined the conjecture and proof. The type of research used is qualitative with a case study approach. The research subjects were two of the 40 fifth-grade students selected purposively. The research instrument used is the task of proof and interview guidelines. Data collection techniques consist of Tasks, documentation, and interviews. The data analysis technique consists of three stages: data reduction, data presentation, and concluding. The analysis results show that at the stage of proving their primitive conjecture, the conjectures made by female and male students through action proofs using manipulative objects are still wrong. At the stage of confronted counter-examples, conjectures and proof made by female and male students showed an improvement. At the stage of re-examining the conjecture and proof, the conjectures and proof by female and male students were comprehensive. It can be concluded that the stages of proof of the actions of female and male students using manipulative objects through stimulation counter-examples indicate an improvement in conjectures and more comprehensive proof.
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Labute, John, Nicole Lemire, Ján Mináč, and John Swallow. "Demuškin groups, Galois modules, and the Elementary Type Conjecture." Journal of Algebra 304, no. 2 (October 2006): 1130–46. http://dx.doi.org/10.1016/j.jalgebra.2005.12.021.
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AUINGER, K. "A NEW PROOF OF THE RHODES TYPE II CONJECTURE." International Journal of Algebra and Computation 14, no. 05n06 (October 2004): 551–68. http://dx.doi.org/10.1142/s0218196704001918.
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For a given finite monoid M we explicitly construct a finite group G and a relational morphism τ:M→G such that only elements of the type II construct Mc relate to 1 under τ. This provides an elementary and constructive proof of the type II conjecture of John Rhodes. The underlying idea is also used to modify the proof of Ash's celebrated theorem on inevitable graphs. For any finite monoid M and any finite graph Γ a finite group G is constructed which "spoils" all labelings of Γ over M which are not inevitable.
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Van Der Ploeg, C. E. "On a Converse to the Tscgebotarev density theorem." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 44, no. 3 (June 1988): 287–93. http://dx.doi.org/10.1017/s1446788700032109.
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AbstractUsing an elementary counting procedure on biquadratic polynomials over Zp it is shown that the probability distribution of odd, unramified rational primes according to decomposition type in a fixed dihedral numberfield is identical to the probility of separable quartic polynomials (mod p) whose roots generate numberfields with normal closure having Galois group isomorphic to D4, as p → ∞. This verifies a conjecture about a converse to the Tschebotarev density theorem. Further evidence in support of this conjecture is provided in quadratic and coubic numberfields.
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Griffeth, Stephen. "Towards a combinatorial representation theory for the rational Cherednik algebra of type G(r, p, n)." Proceedings of the Edinburgh Mathematical Society 53, no. 2 (April 30, 2010): 419–45. http://dx.doi.org/10.1017/s0013091508000904.
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AbstractThis paper aims to lay the foundations for a combinatorial study, via orthogonal functions and intertwining operators, of category $\mathcal{O}$ for the rational Cherednik algebra of type G(r, p, n). As a first application, a self-contained and elementary proof of the analogue for the groups G(r, p, n), with r > 1, of Gordon's Theorem (previously Haiman's Conjecture) on the diagonal co-invariant ring is given. No restriction is imposed on p; the result for p ≠ r has been proved by Vale using a technique analogous to Gordon's. Because of the combinatorial application to Haiman's Conjecture, the paper is logically self-contained except for standard facts about complex reflection groups. The main results should be accessible to mathematicians working in algebraic combinatorics who are unfamiliar with the impressive range of ideas used in Gordon's proof of his theorem.
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Zannier, U. "On a theorem of Birch concerning sums of distinct integers taken from certain sequences." Mathematical Proceedings of the Cambridge Philosophical Society 106, no. 2 (September 1989): 199–206. http://dx.doi.org/10.1017/s0305004100078014.
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In [1] B. J. Birch, solving in the affirmative a conjecture of Erdὅs, proved the following result:Theorem 1. Let p and q be coprime integers greater than 1. Then every large natural number may be written as a sum of distinct terms of type paqb.In fact Birch pointed out that, with similar arguments, one could obtain a stronger version where the exponent b of q can be bounded in terms of p and q. The proofs were entirely elementary.
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Yu, Yahui, and Jiayuan Hu. "On the generalized Ramanujan-Nagell equation $ x^2+(2k-1)^y = k^z $ with $ k\equiv 3 $ (mod 4)." AIMS Mathematics 6, no. 10 (2021): 10596–601. http://dx.doi.org/10.3934/math.2021615.
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<abstract><p>Let $ k $ be a fixed positive integer with $ k > 1 $. In 2014, N. Terai <sup>[<xref ref-type="bibr" rid="b6">6</xref>]</sup> conjectured that the equation $ x^2+(2k-1)^y = k^z $ has only the positive integer solution $ (x, y, z) = (k-1, 1, 2) $. This is still an unsolved problem as yet. For any positive integer $ n $, let $ Q(n) $ denote the squarefree part of $ n $. In this paper, using some elementary methods, we prove that if $ k\equiv 3 $ (mod 4) and $ Q(k-1)\ge 2.11 $ log $ k $, then the equation has only the positive integer solution $ (x, y, z) = (k-1, 1, 2) $. It can thus be seen that Terai's conjecture is true for almost all positive integers $ k $ with $ k\equiv 3 $(mod 4).</p></abstract>
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Fitzgerald, Robert W. "Gorenstein Witt Rings." Canadian Journal of Mathematics 40, no. 5 (October 1, 1988): 1186–202. http://dx.doi.org/10.4153/cjm-1988-050-x.
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Throughout R is a noetherian Witt ring. The basic example is the Witt ring WF of a field F of characteristic not 2 and finite. We study the structure of (noetherian) Witt rings which are also Gorenstein rings (i.e., have a finite injective resolution). The underlying motivation is the elementary type conjecture. The Gorenstein Witt rings of elementary type are group ring extensions of Witt rings of local type. We thus wish to compare the two classes of Witt rings: Gorenstein and group ring over local type. We show the two classes enjoy many of the same properties and are, in several cases, equal. However we cannot decide if the two classes are always equal.In the first section we consider formally real Witt rings R (equivalently, dim R = 1). Here the total quotient ring of R is R-injective if and only if R is reduced. Further, R is Gorenstein if and only if R is a group ring over Z. This result appears to be somewhat deep.
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Abe, Hiraku, Megumi Harada, Tatsuya Horiguchi, and Mikiya Masuda. "The Cohomology Rings of Regular Nilpotent Hessenberg Varieties in Lie Type A." International Mathematics Research Notices 2019, no. 17 (October 16, 2017): 5316–88. http://dx.doi.org/10.1093/imrn/rnx275.
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AbstractLet $n$ be a fixed positive integer and $h: \{1,2,\ldots,n\} \rightarrow \{1,2,\ldots,n\}$ a Hessenberg function. The main results of this paper are two-fold. First, we give a systematic method, depending in a simple manner on the Hessenberg function $h$, for producing an explicit presentation by generators and relations of the cohomology ring $H^\ast({\mathrm{Hess}}(\mathsf{N},h))$ with ${\mathbb Q}$ coefficients of the corresponding regular nilpotent Hessenberg variety ${\mathrm{Hess}}(\mathsf{N},h)$. Our result generalizes known results in special cases such as the Peterson variety and also allows us to answer a question posed by Mbirika and Tymoczko. Moreover, our list of generators in fact forms a regular sequence, allowing us to use techniques from commutative algebra in our arguments. Our second main result gives an isomorphism between the cohomology ring $H^*({\mathrm{Hess}}(\mathsf{N},h))$ of the regular nilpotent Hessenberg variety and the $\mathfrak{S}_n$-invariant subring $H^*({\mathrm{Hess}}(\mathsf{S},h))^{\mathfrak{S}_n}$ of the cohomology ring of the regular semisimple Hessenberg variety (with respect to the $\mathfrak{S}_n$-action on $H^*({\mathrm{Hess}}(\mathsf{S},h))$ defined by Tymoczko). Our second main result implies that $\mathrm{dim}_{{\mathbb Q}} H^k({\mathrm{Hess}}(\mathsf{N},h)) = \mathrm{dim}_{{\mathbb Q}} H^k({\mathrm{Hess}}(\mathsf{S},h))^{\mathfrak{S}_n}$ for all $k$ and hence partially proves the Shareshian–Wachs conjecture in combinatorics, which is in turn related to the well-known Stanley–Stembridge conjecture. A proof of the full Shareshian–Wachs conjecture was recently given by Brosnan and Chow, and independently by Guay–Paquet, but in our special case, our methods yield a stronger result (i.e., an isomorphism of rings) by more elementary considerations. This article provides detailed proofs of results we recorded previously in a research announcement [2].
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Sun, Michael. "Strongly outer group actions on UHF algebras." Journal of Topology and Analysis 10, no. 03 (August 30, 2018): 701–21. http://dx.doi.org/10.1142/s1793525318500231.
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We show that for any countable discrete maximally almost periodic group [Formula: see text] and any UHF algebra [Formula: see text], there exists a strongly outer product type action [Formula: see text] of [Formula: see text] on [Formula: see text]. When [Formula: see text] is also elementary amenable, Matui–Sato have shown that such actions have their tracial Rokhlin property. Consequently, the class of crossed products [Formula: see text] satisfy Elliott’s classification conjecture. We also show the existence of the “Rokhlin” property for countable discrete almost abelian group actions on the universal UHF algebra.
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Igusa, Kiyoshi, and Jonah Ostroff. "Mixed cobinary trees." Journal of Algebra and Its Applications 17, no. 09 (August 23, 2018): 1850170. http://dx.doi.org/10.1142/s0219498818501700.
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We develop basic cluster theory from an elementary point of view using a variation of binary trees which we call mixed cobinary trees (MCTs). We show that the number of isomorphism classes of such trees is given by the Catalan number [Formula: see text] where [Formula: see text] is the number of internal nodes. We also consider the corresponding quiver [Formula: see text] of type [Formula: see text]. As a special case of more general known results about the relation between [Formula: see text]-vectors, representations of quivers and their semi-invariants, we explain the bijection between MCTs and the vertices of the generalized associahedron corresponding to the quiver [Formula: see text]. These results are extended to [Formula: see text]-clusters in the next paper. We give one application: a new short proof of a conjecture of Reineke using MCTs.
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RAJARAMA BHAT, B. V., and R. SRINIVASAN. "ON PRODUCT SYSTEMS ARISING FROM SUM SYSTEMS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 08, no. 01 (March 2005): 1–31. http://dx.doi.org/10.1142/s0219025705001834.
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B. Tsirelson constructed an uncountable family of type III product systems of Hilbert spaces through the theory of Gaussian spaces, measure type spaces and "slightly colored noises", using techniques from probability theory. Here we take a purely functional analytic approach and try to have a better understanding of Tsireleson's construction and his examples. We prove an extension of Shale's theorem connecting symplectic group and Weyl representation. We show that the "Shale map" respects compositions (this settles an old conjecture of K. R. Parthasarathy8). Using this we associate a product system to a sum system. This construction includes the exponential product system of Arveson, as a trivial case, and the type III examples of Tsirelson. By associating a von Neumann algebra to every "elementary set" in [0, 1], in a much simpler and direct way, we arrive at the invariants of the product system introduced by Tsirelson, given in terms of the sum system. Then we introduce a notion of divisibility for a sum system, and prove that the examples of Tsirelson are divisible. It is shown that only type I and type III product systems arise out of divisible sum systems. Finally, we give a sufficient condition for a divisible sum system to give rise to a unitless (type III) product system.
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Cordes, Craig M. "Abstract Witt Rings When Certain Binary Forms Represent Exactly Four Elements." Canadian Journal of Mathematics 45, no. 6 (December 1, 1993): 1184–99. http://dx.doi.org/10.4153/cjm-1993-066-7.
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AbstractAn abstract Witt ring (R, G) of positive characteristic is known to be a group ring S[Δ] with ﹛1﹜ ≠ Δ ⊆ G if and only if it contains a form〈1,x〉, x ≠1, which represents only the two elements 1 and x. Carson and Marshall have characterized all Witt rings of characteristic 2 which contain binary forms representing exactly four elements. Such results which show R is isomorphic to a product of smaller rings are helpful in settling the conjecture that every finitely generated Witt ring is of elementary type. Here, some special situations are considered. In particular if char(R) = 8, |D〈l, 1〉| = 4, and R contains no rigid elements, then R is isomorphic to the Witt ring of the 2-adic numbers. If char(R) = 4, |D〈l,a〉| = 4 where a ∈ D〈1, 1〉, and R contains no rigid elements, then R is either a ring of order 8 or is the specified product of two Witt rings at least one of which is a group ring. In several cases R is realized by a field.
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Pillay, Anand, and Charles Steinhorn. "A note on nonmultidimensional superstable theories." Journal of Symbolic Logic 50, no. 4 (December 1985): 1020–24. http://dx.doi.org/10.2307/2273987.
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In this paper we prove that if T is the complete elementary diagram of a countable structure and is a theory as in the title, then Vaught's conjecture holds for T. This result is Theorem 7, below. In the process of establishing this proposition, in Theorem 3 we give a sufficient condition for a superstable theory having only countably many types without parameters to be ω-stable. Familiarity with the rudiments of stability theory, as presented in [3] and [4], will be supposed throughout. The notation used is, by now, standard.We begin by giving a new proof of a lemma due to J. Saffe in [6]. For T stable, recall that the multiplicity of a type p over a set A ⊆ ℳ ⊨ T is the cardinality of the collection of strong types over A extending p.Lemma 1 (Saffe). Let T be stable, A ⊆ ℳ ⊨ T. If t(b̄, A) has infinite multiplicity and t(c̄, A) has finite multiplicity, then t(b̄, A ∪ {c̄}) has infinite multiplicity.Proof. We suppose not and work for a contradiction. Let ‹b̄γ:γ ≤ α›, α ≥ ω, be a list of elements so that t(b̄γ, A) = t(b̄, A) for all γ ≤ α, and st(b̄γ, A) ≠ st(b̄δ, A) for γ ≠ δ. Furthermore, let c̄γ satisfy t(b̄γ∧c̄γ, A) = t(b̄ ∧ c̄, A) for each γ < α.Since t(c̄, A) has finite multiplicity, we may assume for all γ, δ < α. that st(c̄γ, A) = st(c̄δ, A). For each γ < α there is an automorphism fγ of the so-called “monster model” of T (a sufficiently large, saturated model of T) that preserves strong types over A and is such that f(c̄γ) = c̄0.
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Morales, Marcel. "Some Numerical Criteria for the Nash Problem on Arcs for Surfaces." Nagoya Mathematical Journal 191 (2008): 1–19. http://dx.doi.org/10.1017/s0027763000025897.
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AbstractLet (X, O) be a germ of a normal surface singularity, π: → X be the minimal resolution of singularities and let A = (ai,j) be the n × n symmetrical intersection matrix of the exceptional set of In an old preprint Nash proves that the set of arcs on a surface singularity is a scheme , and defines a map from the set of irreducible components of to the set of exceptional components of the minimal resolution of singularities of (X,O). He proved that this map is injective and ask if it is surjective. In this paper we consider the canonical decomposition •For any couple (Ei,Ej) of distinct exceptional components, we define Numerical Nash condition (NN(i,j)). We have that (NN(i,j)) implies In this paper we prove that (NN(i,j)) is always true for at least the half of couples (i,j).•The condition (NN(i,j)) is true for all couples (i,j) with i ≠ j, characterizes a certain class of negative definite matrices, that we call Nash matrices. If A is a Nash matrix then the Nash map N is bijective. In particular our results depend only on A and not on the topological type of the exceptional set.•We recover and improve considerably almost all results known on this topic and our proofs are new and elementary.•We give infinitely many other classes of singularities where Nash Conjecture is true.The proofs are based on my old work [8] and in Plenat [10].
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HAGGE, TOBIAS J., and SEUNG-MOON HONG. "SOME NON-BRAIDED FUSION CATEGORIES OF RANK THREE." Communications in Contemporary Mathematics 11, no. 04 (August 2009): 615–37. http://dx.doi.org/10.1142/s0219199709003521.
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We classify all fusion categories for a given set of fusion rules with three simple object types. If a conjecture of Ostrik is true, our classification completes the classification of fusion categories with three simple object types. To facilitate the discussion, we describe a convenient, concrete and useful variation of graphical calculus for fusion categories, discuss pivotality and sphericity in this framework, and give a short and elementary re-proof of the fact that the quadruple dual functor is naturally isomorphic to the identity.
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HIRSCHHORN, MICHAEL D., and JAMES A. SELLERS. "ELEMENTARY PROOFS OF PARITY RESULTS FOR 5-REGULAR PARTITIONS." Bulletin of the Australian Mathematical Society 81, no. 1 (July 2, 2009): 58–63. http://dx.doi.org/10.1017/s0004972709000525.
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AbstractIn a recent paper, Calkin et al. [N. Calkin, N. Drake, K. James, S. Law, P. Lee, D. Penniston and J. Radder, ‘Divisibility properties of the 5-regular and 13-regular partition functions’, Integers8 (2008), #A60] used the theory of modular forms to examine 5-regular partitions modulo 2 and 13-regular partitions modulo 2 and 3; they obtained and conjectured various results. In this note, we use nothing more than Jacobi’s triple product identity to obtain results for 5-regular partitions that are stronger than those obtained by Calkin and his collaborators. We find infinitely many Ramanujan-type congruences for b5(n), and we prove the striking result that the number of 5-regular partitions of the number n is even for at least 75% of the positive integers n.
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RIJKE, EGBERT, and BAS SPITTERS. "Sets in homotopy type theory." Mathematical Structures in Computer Science 25, no. 5 (January 30, 2015): 1172–202. http://dx.doi.org/10.1017/s0960129514000553.
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Homotopy type theory may be seen as an internal language for the ∞-category of weak ∞-groupoids. Moreover, weak ∞-groupoids model the univalence axiom. Voevodsky proposes this (language for) weak ∞-groupoids as a new foundation for Mathematics called the univalent foundations. It includes the sets as weak ∞-groupoids with contractible connected components, and thereby it includes (much of) the traditional set theoretical foundations as a special case. We thus wonder whether those ‘discrete’ groupoids do in fact form a (predicative) topos. More generally, homotopy type theory is conjectured to be the internal language of ‘elementary’ of ∞-toposes. We prove that sets in homotopy type theory form a ΠW-pretopos. This is similar to the fact that the 0-truncation of an ∞-topos is a topos. We show that both a subobject classifier and a 0-object classifier are available for the type theoretical universe of sets. However, both of these are large and moreover the 0-object classifier for sets is a function between 1-types (i.e. groupoids) rather than between sets. Assuming an impredicative propositional resizing rule we may render the subobject classifier small and then we actually obtain a topos of sets.
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Ackland, J. F., N. B. Schwartz, K. E. Mayo, and R. E. Dodson. "Nonsteroidal signals originating in the gonads." Physiological Reviews 72, no. 3 (July 1, 1992): 731–87. http://dx.doi.org/10.1152/physrev.1992.72.3.731.
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The discovery of the various peptide factors in the gonads followed different paths. A number of factors were specifically searched for because of physiological experiments that predicted that an activity from the gonads was necessary to explain phenomena. Such was the case for gonadal steroids and for such peptide factors as inhibin, MIS, OMI, FRP, seminal plasma inhibin, relaxin, PA factor and other proteases, and ABP. In the process other factors such as activin and follistatin were serendipitously discovered. A second group of factors was discovered because in vitro experiments of various combinations of gonadal cell types failed to replicate in vivo findings, suggesting missing signals. Such substances are the panoply of growth factors aiding in differentiation and growth promotion and inhibition: LS and LI, P-Mod-S, clusterin, and various components of the ECM. Finally, and most recently, another set of peptides has been identified because immunological or molecular probes have been used to search gonadal tissue for factors originally discovered elsewhere; these include POMC, GnRH-like peptide, oxytocin, AVP, angiotensin, ANF, CRF, neural peptides, and c-mos. Our understanding of the relationship of most of these peptides to the local signals necessary for gonadal function is still very elementary. Clearly some like relaxin and inhibin function as important hormones, and ABP, for example, probably functions importantly in transporting testosterone down the tubule. Most local paracrine or autocrine peptide signals appear to act in relationship to gonadotropin levels probably in local differentiation in the process of gamete maturation, but this is only conjecture at this point. No experimental verification that any of these factors is involved in follicle selection for recruitment or for atresia is yet available. For many of the factors local receptors have not yet been identified. The richness of the variety of peptides in the gonads suggests that microanalysis of cell-cell signaling would be rewarding, but at the time of this writing such investigations are not yet possible.
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Quadrelli, Claudio. "One-Relator Maximal Pro-p Galois Groups and the Koszulity Conjectures." Quarterly Journal of Mathematics, November 28, 2020. http://dx.doi.org/10.1093/qmath/haaa049.
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Abstract Let p be a prime number and let ${\mathbb{K}}$ be a field containing a root of 1 of order p. If the absolute Galois group $G_{\mathbb{K}}$ satisfies $\dim\, H^1(G_{\mathbb{K}},\mathbb{F}_p)\lt\infty$ and $\dim\, H^{\,2}(G_{\mathbb{K}},\mathbb{F}_p)=1$, we show that L. Positselski’s and T. Weigel’s Koszulity conjectures are true for ${\mathbb{K}}$. Also, under the above hypothesis, we show that the $\mathbb{F}_p$-cohomology algebra of $G_{\mathbb{K}}$ is the quadratic dual of the graded algebra ${\rm gr}_\bullet\mathbb{F}_p[G_{\mathbb{K}}]$, induced by the powers of the augmentation ideal of the group algebra $\mathbb{F}_p[G_{\mathbb{K}}]$, and these two algebras decompose as products of elementary quadratic algebras. Finally, we propose a refinement of the Koszulity conjectures, analogous to I. Efrat’s elementary type conjecture.
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Muthiah, Dinakar, Alex Weekes, and Oded Yacobi. "On a conjecture of Pappas and Rapoport about the standard local model for GL_d." Journal für die reine und angewandte Mathematik (Crelles Journal), October 8, 2020. http://dx.doi.org/10.1515/crelle-2020-0030.
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AbstractIn their study of local models of Shimura varieties for totally ramified extensions, Pappas and Rapoport posed a conjecture about the reducedness of a certain subscheme of {n\times n} matrices. We give a positive answer to their conjecture in full generality. Our main ideas follow naturally from two of our previous works. The first is our proof of a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman on the equations defining type A affine Grassmannians. The second is the work of the first two authors and Kamnitzer on affine Grassmannian slices and their reduced scheme structure. We also present a version of our argument that is almost completely elementary: the only non-elementary ingredient is the Frobenius splitting of Schubert varieties.
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Quadrelli, Claudio, and Thomas S. Weigel. "Oriented pro-$$\ell $$ groups with the Bogomolov–Positselski property." Research in Number Theory 8, no. 2 (March 2, 2022). http://dx.doi.org/10.1007/s40993-022-00318-9.
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AbstractFor a prime number $$\ell $$ ℓ we say that an oriented pro-$$\ell $$ ℓ group $$(G,\theta )$$ ( G , θ ) has the Bogomolov–Positselski property if the kernel of the canonical projection on its maximal $$\theta $$ θ -abelian quotient $$\pi ^{\mathrm {ab}}_{G,\theta }:G\rightarrow G(\theta )$$ π G , θ ab : G → G ( θ ) is a free pro-$$\ell $$ ℓ group contained in the Frattini subgroup of G. We show that oriented pro-$$\ell $$ ℓ groups of elementary type have the Bogomolov–Positselski property (cf. Theorem 1.2). This shows that Efrat’s Elementary Type Conjecture implies a positive answer to Positselski’s version of Bogomolov’s Conjecture on maximal pro-$$\ell $$ ℓ Galois groups of a field $$\mathbb {K}$$ K in case that $$\mathbb {K}^\times /(\mathbb {K}^\times )^\ell $$ K × / ( K × ) ℓ is finite. Secondly, it is shown that for an $$H^\bullet $$ H ∙ -quadratic oriented pro-$$\ell $$ ℓ group $$(G,\theta )$$ ( G , θ ) the Bogomolov–Positselski property can be expressed by the injectivity of the transgression map $$d_2^{2,1}$$ d 2 2 , 1 in the Hochschild–Serre spectral sequence (cf. Theorem 1.4).
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Bastian, Brice, Thomas W. Grimm, and Damian van de Heisteeg. "Weak gravity bounds in asymptotic string compactifications." Journal of High Energy Physics 2021, no. 6 (June 2021). http://dx.doi.org/10.1007/jhep06(2021)162.
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Abstract We study the charge-to-mass ratios of BPS states in four-dimensional $$ \mathcal{N} $$ N = 2 supergravities arising from Calabi-Yau threefold compactifications of Type IIB string theory. We present a formula for the asymptotic charge-to-mass ratio valid for all limits in complex structure moduli space. This is achieved by using the sl(2)-structure that emerges in any such limit as described by asymptotic Hodge theory. The asymptotic charge-to-mass formula applies for sl(2)-elementary states that couple to the graviphoton asymptotically. Using this formula, we determine the radii of the ellipsoid that forms the extremality region of electric BPS black holes, which provides us with a general asymptotic bound on the charge-to-mass ratio for these theories. Finally, we comment on how these bounds for the Weak Gravity Conjecture relate to their counterparts in the asymptotic de Sitter Conjecture and Swampland Distance Conjecture.
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ODAKA, YUJI. "PL DENSITY INVARIANT FOR TYPE II DEGENERATING K3 SURFACES, MODULI COMPACTIFICATION AND HYPER-KÄHLER METRIC." Nagoya Mathematical Journal, November 3, 2021, 1–41. http://dx.doi.org/10.1017/nmj.2021.13.
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Abstract A protagonist here is a new-type invariant for type II degenerations of K3 surfaces, which is explicit piecewise linear convex function from the interval with at most $18$ nonlinear points. Forgetting its actual function behavior, it also classifies the type II degenerations into several combinatorial types, depending on the type of root lattices as appeared in classical examples. From differential geometric viewpoint, the function is obtained as the density function of the limit measure on the collapsing hyper-Kähler metrics to conjectural segments, as in [HSZ19]. On the way, we also reconstruct a moduli compactification of elliptic K3 surfaces by [AB19], [ABE20], [Brun15] in a more elementary manner, and analyze the cusps more explicitly. We also interpret the glued hyper-Kähler fibration of [HSVZ18] as a special case from our viewpoint, and discuss other cases, and possible relations with Landau–Ginzburg models in the mirror symmetry context.
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Armstrong, Drew. "Hyperplane Arrangements and Diagonal Harmonics." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AO,..., Proceedings (January 1, 2011). http://dx.doi.org/10.46298/dmtcs.2889.
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International audience In 2003, Haglund's bounce statistic gave the first combinatorial interpretation of the q,t-Catalan numbers and the Hilbert series of diagonal harmonics. In this paper we propose a new combinatorial interpretation in terms of the affine Weyl group of type A. In particular, we define two statistics on affine permutations; one in terms of the Shi hyperplane arrangement, and one in terms of a new arrangement — which we call the Ish arrangement. We prove that our statistics are equivalent to the area' and bounce statistics of Haglund and Loehr. In this setting, we observe that bounce is naturally expressed as a statistic on the root lattice. We extend our statistics in two directions: to "extended'' Shi arrangements and to the bounded chambers of these arrangements. This leads to a (conjectural) combinatorial interpretation for all integral powers of the Bergeron-Garsia nabla operator applied to elementary symmetric functions. En 2003, la statistique bounce de Haglund a donné la première interprétation combinatoire de la somme des nombres q,t-Catalan et de la série de Hilbert des harmoniques diagonaux. Dans cet article nous proposons une nouvelle interprétation combinatoire à partir du groupe de Weyl affine de type A. En particulier, nous définissons deux statistiques sur les permutations affines; l'une à partir de l'arrangement d'hyperplans Shi, et l'autre à partir d'un nouvel arrangement — que nous appelons l'arrangement Ish. Nous prouvons que nos statistiques sont équivalentes aux statistiques area' et bounce de Haglund et Loehr. Dans ce contexte, nous observons que bounce s'exprime naturellement comme une statistique sur le réseau des racines. Nous prolongeons nos statistiques dans deux directions: arrangements Shi "étendus'', et chambres bornées associées. Cela conduit à une interprétation (conjecturale) combinatoire pour toutes les puissances entières de l'opérateur nabla de Bergeron-Garsia appliqué aux fonctions symétriques élémentaires.
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Eagles, Nancy Mae, Angèle M. Foley, Alice Huang, Elene Karangozishvili, and Annan Yu. "$H$-Chromatic Symmetric Functions." Electronic Journal of Combinatorics 29, no. 1 (February 11, 2022). http://dx.doi.org/10.37236/10011.
Full textAbstract:
We introduce $H$-chromatic symmetric functions, $X_{G}^{H}$, which use the $H$-coloring of a graph $G$ to define a generalization of Stanley's chromatic symmetric functions. We say two graphs $G_1$ and $G_2$ are $H$-chromatically equivalent if $X_{G_1}^{H} = X_{G_2}^{H}$, and use this idea to study uniqueness results for $H$-chromatic symmetric functions, with a particular emphasis on the case $H$ is a complete bipartite graph. We also show that several of the classical bases of the space of symmetric functions, i.e. the monomial symmetric functions, power sum symmetric functions, and elementary symmetric functions, can be realized as $H$-chromatic symmetric functions. Moreover, we show that if $G$ and $H$ are particular types of multipartite complete graphs we can derive a set of $H$-chromatic symmetric functions that are a basis for $\Lambda^n$. We end with some conjectures and open problems.