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Journal articles on the topic 'Z K-Theory'

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Published: 10 December 2022

Last updated: 28 January 2023

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1

Elmrabty, Adnane. "Some Connections Between Bunke-Schick Differential K-theory and Topological $\mathbb{Z}/k\mathbb{Z}$ K-theory." Missouri Journal of Mathematical Sciences 30, no. 1 (May 2018): 32–44. http://dx.doi.org/10.35834/mjms/1534384951.

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2

Elmrabty, Adnane. "Some geometric results on K-theory with $${\mathbb{Z}}/k{\mathbb{Z}}$$-coefficients." São Paulo Journal of Mathematical Sciences 14, no. 2 (July 6, 2020): 562–79. http://dx.doi.org/10.1007/s40863-020-00179-z.

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3

Arlettaz, Dominique. "On the algebraic K-theory of Z." Journal of Pure and Applied Algebra 51, no. 1-2 (March 1988): 53–64. http://dx.doi.org/10.1016/0022-4049(88)90077-1.

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4

Bárcenas, Noé, and Mario Velásquez. "Equivariant $K$-theory of central extensions and twisted equivariant $K$-theory: $SL_{3}\mathbb{Z}$ and $St_{3}{\mathbb{Z}}$." Homology, Homotopy and Applications 18, no. 1 (2016): 49–70. http://dx.doi.org/10.4310/hha.2016.v18.n1.a4.

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5

Boyde, Guy. "p-Hyperbolicity of homotopy groups via K-theory." Mathematische Zeitschrift 301, no. 1 (January 7, 2022): 977–1009. http://dx.doi.org/10.1007/s00209-021-02917-1.

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AbstractWe show that $$S^n \vee S^m$$ S n ∨ S m is $${\mathbb {Z}}/p^r$$ Z / p r -hyperbolic for all primes p and all $$r \in {\mathbb {Z}}^+$$ r ∈ Z + , provided $$n,m \ge 2$$ n , m ≥ 2 , and consequently that various spaces containing $$S^n \vee S^m$$ S n ∨ S m as a p-local retract are $${\mathbb {Z}}/p^r$$ Z / p r -hyperbolic. We then give a K-theory criterion for a suspension $$\Sigma X$$ Σ X to be p-hyperbolic, and use it to deduce that the suspension of a complex Grassmannian $$\Sigma Gr_{k,n}$$ Σ G r k , n is p-hyperbolic for all odd primes p when $$n \ge 3$$ n ≥ 3 and $$0<k<n$$ 0 < k < n . We obtain similar results for some related spaces.

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6

Gutierrez, Julio. "Bivariant K-theory of locally convex Z-graded algebras." Selecciones Matemáticas 9, no. 01 (June 30, 2022): 167–72. http://dx.doi.org/10.17268/sel.mat.2022.01.14.

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In the present work, we describe some results about the K-theory of Z-graded algebras. First, in the context of C* algebras, we begin with the Pimsner-Voiculescu sequence for crossed products and its generalizations. We will see that there are results analog to these in the context of locally convex algebras and we conclude with results for generalized Weyl algebras.

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7

Lee, Jae Min, and Byungdo Park. "A Superbundle Description of Differential K-Theory." Axioms 12, no. 1 (January 12, 2023): 82. http://dx.doi.org/10.3390/axioms12010082.

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We construct a model of differential K-theory using superbundles with a Z/2Z-graded connection and a differential form on the base manifold and prove that our model is isomorphic to the Freed–Lott–Klonoff model of differential K-theory.

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8

Orlitzky, Michael. "Positive and Z-operators on Closed Convex Cones." Electronic Journal of Linear Algebra 34 (February 21, 2018): 444–58. http://dx.doi.org/10.13001/1081-3810.3782.

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Let $K$ be a closed convex cone with dual $\dual{K}$ in a finite-dimensional real Hilbert space. A \emph{positive operator} on $K$ is a linear operator $L$ such that $L\of{K} \subseteq K$. Positive operators generalize the nonnegative matrices and are essential to the Perron-Frobenius theory. It is said that $L$ is a \emph{\textbf{Z}-operator} on $K$ if % \begin{equation*} \ip{L\of{x}}{s} \le 0 \;\text{ for all } \pair{x}{s} \in \cartprod{K}{\dual{K}} \text{ such that } \ip{x}{s} = 0. \end{equation*} % The \textbf{Z}-operators are generalizations of \textbf{Z}-matrices (whose off-diagonal elements are nonpositive) and they arise in dynamical systems, economics, game theory, and elsewhere. In this paper, the positive and \textbf{Z}-operators are connected. This extends the work of Schneider, Vidyasagar, and Tam on proper cones, and reveals some interesting similarities between the two families.

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9

Snaith, Victor. "l-adic and Z/l∞-algebraic and topological K-theory." Proceedings of the Edinburgh Mathematical Society 28, no. 1 (February 1985): 73–90. http://dx.doi.org/10.1017/s0013091500003217.

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Let l be an odd prime and let A be a commutative ring containing 1/l. Let K*(A;Z/lv) denote the mod lv algebraic K-theory of A [3]. As explained in [4] there exists a “Bott element” βv∈K21v–1(l–1)(Z[1/l];Z/lv) and, using the K-theory product we may, following [16, Part IV], formwhich is defined as the direct limit of iterated multiplication by βv. There is a canonical localisation map

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10

Karoubi, Max. "Clifford Modules and Twisted K-Theory." Advances in Applied Clifford Algebras 18, no. 3-4 (May 27, 2008): 765–69. http://dx.doi.org/10.1007/s00006-008-0101-z.

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11

Ara, P., M. A. Moreno, and E. Pardo. "Nonstable K-theory for Graph Algebras." Algebras and Representation Theory 10, no. 2 (November 25, 2006): 157–78. http://dx.doi.org/10.1007/s10468-006-9044-z.

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12

Ananyevskiy, Alexey, Oliver Röndigs, and Paul Arne Østvær. "On very effective hermitian K-theory." Mathematische Zeitschrift 294, no. 3-4 (April 29, 2019): 1021–34. http://dx.doi.org/10.1007/s00209-019-02302-z.

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13

Cerri, Cristina. "Non-Commutative Deformations of C(T2) and K-Theory." International Journal of Mathematics 08, no. 05 (August 1997): 555–71. http://dx.doi.org/10.1142/s0129167x97000287.

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For each α ≥ 0, let Bα be the universal C*-algebra generated by unitary elements uα, vα and a self-adjoint hα such that ||hα|| ≤ α and [Formula: see text]. In this work we prove that the family {Bα}α ∈ [0,∞[ extend the family of soft torus with the same basic properties, i.e., the field of C*-algebras {Bα}α ∈ [0,α0] is continuous and each Bα is a crossed product of a C*-algebra homotopically equivalent to C(S1) by Z. We then show that the K-groups of Bα are isomorphic to Z ⊕ Z. Applying results from the theory of rotation algebras we prove that every positive element (n,m) in K0(Bα) satisfies |m|α ≤ 2πn. It follows that these C*-algebras are not all homotopically equivalent to each other, although they have the same K-groups.

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14

Kashiwabara, Takuji. "Mod p K-theory of Ω∞Σ∞X revisited." Mathematical Proceedings of the Cambridge Philosophical Society 114, no. 2 (September 1993): 219–21. http://dx.doi.org/10.1017/s0305004100071553.

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In this note we present a new proof of a theorem of McClure on K*(Ω∞Σ∞X, Z/p) [11], in the special case when X is a finite complex with K1(X; Z/p) = 0. Although our method does not work in the full generality covered by his work, our argument requires neither a geometric interpretation of complex k-theory nor all the delicate coherence properties of its multiplication. Since BP-theory is not likely to possess such coherence properties [9], the possibility of generalizing his approach to the case of higher Morava K-theory does not seem feasible. On the contrary, the main ingredient of our approach is the rank formula for the Morava K-theory of the Borel construction [5], which works for any K(n); thus our approach is better adapted to the potential generalization [8]. Throughout the paper we assume that p > 2 so that mod p K-theory possesses a commutative multiplication, and denote by K*(−) the mod p K-theory. Since it is simpler to state our results in terms of CX, the combinatorial model for QX, rather than QX itself, we shall do so. This is sufficient, as when X is connected CX is homotopy equivalent to QX, and when not, K*(QX) can be easily recovered from K*(CX) (see e.g. [11]).

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15

Basu, Devraj. "K-Theory with R/Z Coefficients and von Neumann Algebras." K-Theory 36, no. 3-4 (December 2005): 327–43. http://dx.doi.org/10.1007/s10977-006-7110-2.

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16

Basu, Devraj. "K-Theory with R/Z Coefficients and von Neumann Algebras." K-Theory 38, no. 1 (November 2007): 83. http://dx.doi.org/10.1007/s10977-007-9008-z.

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17

Kaufmann, Ralph M. "Global Stringy Orbifold Cohomology, K-Theory and de Rham Theory." Letters in Mathematical Physics 94, no. 2 (September 22, 2010): 165–95. http://dx.doi.org/10.1007/s11005-010-0427-z.

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18

Neshitov, Alexander. "FRAMED CORRESPONDENCES AND THE MILNOR–WITT -THEORY." Journal of the Institute of Mathematics of Jussieu 17, no. 4 (June 30, 2016): 823–52. http://dx.doi.org/10.1017/s1474748016000190.

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Given a field $k$ of characteristic zero and $n\geqslant 0$, we prove that $H_{0}(\mathbb{Z}F(\unicode[STIX]{x1D6E5}_{k}^{\bullet },\mathbb{G}_{m}^{\wedge n}))=K_{n}^{MW}(k)$, where $\mathbb{Z}F_{\ast }(k)$ is the category of linear framed correspondences of algebraic $k$-varieties, introduced by Garkusha and Panin [The triangulated category of linear framed motives $DM_{fr}^{eff}(k)$, in preparation] (see [Garkusha and Panin, Framed motives of algebraic varieties (after V. Voevodsky), Preprint, 2014, arXiv:1409.4372], § 7 as well), and $K_{\ast }^{MW}(k)$ is the Milnor–Witt $K$-theory of the base field $k$.

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19

Bárcenas, Noé, and Mario Velásquez. "The completion theorem in twisted equivariant K-theory for proper actions." Journal of Homotopy and Related Structures 17, no. 1 (January 31, 2022): 77–104. http://dx.doi.org/10.1007/s40062-021-00299-z.

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20

Bartels, Arthur, Wolfgang Lück, Holger Reich, and Henrik Rüping. "K- and L-theory of group rings over GL n (Z)." Publications mathématiques de l'IHÉS 119, no. 1 (May 25, 2013): 97–125. http://dx.doi.org/10.1007/s10240-013-0055-0.

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21

SRINIVAS, Dr.BRAOU, and Mr. THIRUCHINARPALLI SRINIVAS. "Proof Of Fermat’s Last Theorem By Choosing Two Unknowns in the Integer Solution Are Prime Exponents." Pacific International Journal 3, no. 4 (December 31, 2020): 147–51. http://dx.doi.org/10.55014/pij.v3i4.108.

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In this paper we are revisits well known problem in number theory ‘ proof of Fermat’s last theorem ‘ with different perspective .Also we are presented for n greater than 2, Diophantine equations K(xn+yn)=zn and xn+yn=L zn are satisfied by some positive prime exponents of x,y,z with some sufficient values of K and L. But it is not possible to find positive integers x,y and z, which are satisfies above equations with exactly K=1 and L=1. Clearly it proves Fermat’s last theorem, which states that No positive integers of x, y, z are satisfies the equation xn+yn=zn for n greater than 2.

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22

FORREST, ALAN, and JOHN HUNTON. "The cohomology and $K$-theory of commuting homeomorphisms of the Cantor set." Ergodic Theory and Dynamical Systems 19, no. 3 (June 1999): 611–25. http://dx.doi.org/10.1017/s0143385799130189.

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Given a $\mathbb{Z}^d$ homeomorphic action, $\alpha$, on the Cantor set, $X$, we consider the higher order continuous integer valued dynamical cohomology, $H^*(X,\alpha)$. We also consider the dynamical $K$-theory of the action, the $K$-theory of the crossed product $C^*$-algebra $C(X)\times_{\alpha}\mathbb{Z}^d$. We show that these two invariants are essentially equivalent. We also show that they only take torsion free values. Our work links the two invariants via a third invariant which is based on topological complex $K$-theory evaluated on an associated mapping torus.

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23

Phillips, N. Christopher. "Classifying Algebras for the K-Theory of σ-C*-Algebras." Canadian Journal of Mathematics 41, no. 6 (December 1, 1989): 1021–89. http://dx.doi.org/10.4153/cjm-1989-046-2.

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In topology, the representable K-theory of a topological space X is defined by the formulas RK0(X) = [X,Z x BU] and RKl(X) = [X, U], where square brackets denote sets of homotopy classes of continuous maps, is the infinite unitary group, and BU is a classifying space for U. (Note that ZxBU is homotopy equivalent to the space of Fredholm operators on a separable infinite-dimensional Hilbert space.) These sets of homotopy classes are made into abelian groups by using the H-group structures on Z x BU and U. In this paper, we give analogous formulas for the representable K-theory for α-C*-algebras defined in [20].

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24

Lee, Dae-Woong, Sunyoung Lee, Yeonjeong Kim, and Jeong-Eun Lim. "On automorphisms of graded quasi-lie algebras." Filomat 34, no. 9 (2020): 3141–50. http://dx.doi.org/10.2298/fil2009141l.

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Let Z be the ring of integers and let K(Z,2n) denote the Eilenberg-MacLane space of type (Z,2n) for n ? 1. In this article, we prove that the graded group Am := Aut(??2mn+1(?K(Z,2n))=torsions) of automorphisms of the graded quasi-Lie algebras ?? 2mn+1(?K(Z,2n)) modulo torsions that preserve the Whitehead products is a finite group for m ? 2 and an infinite group for m ? 3, and that the group Aut(?*(K(Z,2n))=torsions) is non-abelian. We extend and apply those results to techniques in localization (or rationalization) theory.

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25

Park, Byungdo. "Geometric models of twisted differential K-theory I." Journal of Homotopy and Related Structures 13, no. 1 (May 13, 2017): 143–67. http://dx.doi.org/10.1007/s40062-017-0177-z.

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26

Rosenschon, Andreas, and Paul Arne Østvær. "K-Theory of Surfaces at the Prime 2." K-Theory 33, no. 3 (November 2004): 215–50. http://dx.doi.org/10.1007/s10977-004-5928-z.

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27

MATTHEY, MICHEL, and HERVÉ OYONO-OYONO. "ALGEBRAIC K-THEORY IN LOW DEGREE AND THE NOVIKOV ASSEMBLY MAP." Proceedings of the London Mathematical Society 85, no. 1 (March 2002): 43–61. http://dx.doi.org/10.1112/s0024611502013461.

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We prove that the Novikov assembly map for a group $\Gamma$ factorizes, in ‘low homological degree’, through the algebraic K-theory of its integral group ring. In homological degree 2, this answers a question posed by N. Higson and P. Julg. As a direct application, we prove that if $\Gamma$ is torsion-free and satisfies the Baum-Connes conjecture, then the homology group $H_{1}(\Gamma;\,\mathbb{Z})$ injects in $K_{1}(C^{*}_{r}\Gamma)$ and in $K_{1}^{\rm alg}(A)$, for any ring $A$ such that $\mathbb{Z}\Gamma\subseteq A\subseteq C^{*}_{r}\Gamma$. If moreover $B\Gamma$ is of dimension less than or equal to 4, then we show that $H_{2}(\Gamma;\,\mathbb{Z})$ injects in $K_{0}(C^{*}_{r}\Gamma)$ and in $K_{2}^{\rm alg}(A)/\Delta_{2}$, where $A$ is as before, and $\Delta_{2}$ is generated by the Steinberg symbols $\{\gamma,\,\gamma\}$, for $\gamma\in\Gamma$. 2000 Mathematical Subject Classification: primary 19D55, 19Kxx, 58J22; secondary: 19Cxx, 19D45, 43A20, 46L85.

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28

Galashin, Pavel, and Thomas Lam. "Parity duality for the amplituhedron." Compositio Mathematica 156, no. 11 (November 2020): 2207–62. http://dx.doi.org/10.1112/s0010437x20007411.

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The (tree) amplituhedron $\mathcal {A}_{n,k,m}(Z)$ is a certain subset of the Grassmannian introduced by Arkani-Hamed and Trnka in 2013 in order to study scattering amplitudes in $N=4$ supersymmetric Yang–Mills theory. Confirming a conjecture of the first author, we show that when $m$ is even, a collection of affine permutations yields a triangulation of $\mathcal {A}_{n,k,m}(Z)$ for any $Z\in \operatorname {Gr}_{>0}(k+m,n)$ if and only if the collection of their inverses yields a triangulation of $\mathcal {A}_{n,n-m-k,m}(Z)$ for any $Z\in \operatorname {Gr}_{>0}(n-k,n)$. We prove this duality using the twist map of Marsh and Scott. We also show that this map preserves the canonical differential forms associated with the corresponding positroid cells, and hence obtain a parity duality for amplituhedron differential forms.

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29

KATSURADA, MASANORI, and TAKUMI NODA. "DIFFERENTIAL ACTIONS ON THE ASYMPTOTIC EXPANSIONS OF NON-HOLOMORPHIC EISENSTEIN SERIES." International Journal of Number Theory 05, no. 06 (September 2009): 1061–88. http://dx.doi.org/10.1142/s1793042109002559.

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Let k be an arbitrary even integer, and Ek(s;z) denote the non-holomorphic Eisenstein series (of weight k attached to SL2(ℤ)), defined by (1.1) below. In the present paper we first establish a complete asymptotic expansion of Ek(s;z) in the descending order of y as y → + ∞ (Theorem 2.1), upon transferring from the previously derived asymptotic expansion of E0(s;z) (due to the first author [16]) to that of Ek(s;z) through successive use of Maass' weight change operators. Theorem 2.1 yields various results on Ek(s;z), including its functional properties (Corollaries 2.1–2.3), its relevant specific values (Corollaries 2.4–2.7), and its asymptotic aspects as z → 0 (Corollary 2.8). We then apply the non-Euclidean Laplacian ΔH,k (of weight k attached to the upper-half plane) to the resulting expansion, in order to justify the eigenfunction equation for Ek(s;z) in (1.6), where the justification can be made uniformly in the whole s-plane (Theorem 2.2).

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30

Fajstrup, Lisbeth. "A Z/2 Descent Theorem for the Algebraic K-Theory of a Semisimple Real Algebra." K-Theory 14, no. 1 (May 1998): 43–77. http://dx.doi.org/10.1023/a:1007701514327.

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31

Hüttemann, Thomas. "The "fundamental theorem" for the higher algebraic $K$-theory of strongly $\mathbb{Z}$-graded rings." Documenta Mathematica 26 (2021): 1557–99. http://dx.doi.org/10.4171/dm/849.

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32

DRENSKY, VESSELIN, and LEONID MAKAR-LIMANOV. "THE CONJECTURE OF NOWICKI ON WEITZENBÖCK DERIVATIONS OF POLYNOMIAL ALGEBRAS." Journal of Algebra and Its Applications 08, no. 01 (February 2009): 41–51. http://dx.doi.org/10.1142/s0219498809003217.

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The Weitzenböck theorem states that if Δ is a linear locally nilpotent derivation of the polynomial algebra K[Z] = K[z1,…,zm] over a field K of characteristic 0, then the algebra of constants of Δ is finitely generated. If m = 2n and the Jordan normal form of Δ consists of 2 × 2 Jordan cells only, we may assume that K[Z] = K[X,Y] and Δ(yi) = xi, Δ(xi) = 0, i = 1,…,n. Nowicki conjectured that the algebra of constants K[X,Y]Δ is generated by x1,…,xn and xiyj – xjyi, 1 ≤ i < j ≤ n. Recently this conjecture was confirmed in the Ph.D. thesis of Khoury with a very computational proof, and also by Derksen whose proof is based on classical results of invariant theory. In this paper we give an elementary proof of the conjecture of Nowicki which does not use any invariant theory. Then we find a very simple system of defining relations of the algebra K[X,Y]Δ which corresponds to the reduced Gröbner basis of the related ideal with respect to a suitable admissible order, and present an explicit basis of K[X,Y]Δ as a vector space.

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33

Golasiński, Marek, and Francisco Gómez Ruiz. "On the algebraic K-theory of $R[X,Y,Z]/(X^2+Y^2+Z^2-1)$." Bulletin of the Belgian Mathematical Society - Simon Stevin 18, no. 5 (November 2011): 849–60. http://dx.doi.org/10.36045/bbms/1323787172.

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34

Magalhães, L. "Some Results in the Connective K-Theory of Lie Groups." Canadian Mathematical Bulletin 31, no. 2 (June 1, 1988): 194–99. http://dx.doi.org/10.4153/cmb-1988-030-9.

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AbstractIn this paper we give a description of:(1) the Hopf algebra structure of k*(G; L) when G is a compact, connected Lie group and L is a ring of type Q(P) so that H*(G; L) is torsion free;(2) the algebra structure of k*(G2; L) for L = Z2 or Z.

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35

Kononov, Ya, A. Okounkov, and A. Osinenko. "The 2-Leg Vertex in K-theoretic DT Theory." Communications in Mathematical Physics 382, no. 3 (March 2021): 1579–99. http://dx.doi.org/10.1007/s00220-021-03936-z.

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36

Duflot, Jeanne. "Simplicial groups that are models for algebraic K-theory." manuscripta mathematica 113, no. 4 (April 1, 2004): 423–70. http://dx.doi.org/10.1007/s00229-003-0431-z.

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37

Melo, Severino T., and Cíntia C. Silva. "K-Theory of pseudodifferential operators with semi-periodic symbols." K-Theory 37, no. 3 (March 2006): 235–48. http://dx.doi.org/10.1007/s10977-006-0018-z.

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38

Baum, Paul F., Piotr M. Hajac, Rainer Matthes, and Wojciech Szymański. "The K-Theory of Heegaard-Type Quantum 3-Spheres." K-Theory 37, no. 1-2 (February 2006): 211. http://dx.doi.org/10.1007/s10977-006-0026-z.

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39

Kambor, J., J. Missimer, and D. Wyler. "K→2π and K→3π decays in next-to-leading order chiral perturbation theory." Physics Letters B 261, no. 4 (June 1991): 496–503. http://dx.doi.org/10.1016/0370-2693(91)90463-z.

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40

HØYNES, SIRI-MALÉN. "Toeplitz flows and their ordered K-theory." Ergodic Theory and Dynamical Systems 36, no. 6 (February 11, 2015): 1892–921. http://dx.doi.org/10.1017/etds.2014.144.

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To a Toeplitz flow $(X,T)$ we associate an ordered $K^{0}$-group, denoted $K^{0}(X,T)$, which is order isomorphic to the $K^{0}$-group of the associated (non-commutative) $C^{\ast }$-crossed product $C(X)\rtimes _{T}\mathbb{Z}$. However, $K^{0}(X,T)$ can be defined in purely dynamical terms, and it turns out to be a complete invariant for (strong) orbit equivalence. We characterize the $K^{0}$-groups that arise from Toeplitz flows $(X,T)$ as exactly those simple dimension groups $(G,G^{+})$ that contain a non-cyclic subgroup $H$ of rank one that intersects $G^{+}$ non-trivially. Furthermore, the Bratteli diagram realization of $(G,G^{+})$ can be chosen to have the ERS property, i.e. the incidence matrices of the Bratteli diagram have equal row sums. We also prove that for any Choquet simplex $K$ there exists an uncountable family of pairwise non-orbit equivalent Toeplitz flows $(X,T)$ such that the set of $T$-invariant probability measures $M(X,T)$ is affinely homeomorphic to $K$, where the entropy $h(T)$ may be prescribed beforehand. Furthermore, the analogous result is true if we substitute strong orbit equivalence for orbit equivalence, but in that case we can actually prescribe both the entropy and the maximal equicontinuous factor of $(X,T)$. Finally, we present some interesting concrete examples of dimension groups associated to Toeplitz flows.

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41

Barlak, Selçuk. "$K$-theory for the crossed products by certain actions of $\mathbb Z^2$." Journal of Noncommutative Geometry 10, no. 4 (2016): 1559–87. http://dx.doi.org/10.4171/jncg/266.

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42

Phillips, N. Christopher, and Larry B. Schweitzer. "Representable $K$-theory of smooth crossed products by ${\bf R}$ and ${\bf Z}$." Transactions of the American Mathematical Society 344, no. 1 (January 1, 1994): 173–201. http://dx.doi.org/10.1090/s0002-9947-1994-1219733-4.

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43

Antonini, Paolo, Sara Azzali, and Georges Skandalis. "Bivariant K-theory with R/Z-coefficients and rho classes of unitary representations." Journal of Functional Analysis 270, no. 1 (January 2016): 447–81. http://dx.doi.org/10.1016/j.jfa.2015.06.017.

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44

Kragh, Thomas. "Orientations on 2-vector Bundles and Determinant Gerbes." MATHEMATICA SCANDINAVICA 113, no. 1 (September 1, 2013): 63. http://dx.doi.org/10.7146/math.scand.a-15482.

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In a paper from 2009, a half magnetic monopole was discovered by Ausoni, Dundas, and Rognes. This describes an obstruction to the existence of a continuous map $K(ku) \to B(ku^*)$ with determinant like properties. This magnetic monopole is in fact an obstruction to the existence of a map from $K(ku)$ to $K(\mathsf{Z},3)$, which is a retract of the natural map $K(\mathsf{Z},3) \to K(ku)$; and any sensible definition of determinant like should produce such a retract. In this paper we describe this obstruction precisely using monoidal categories. By a result from 2011 by Baas, Dundas, Richter and Rognes $K(ku)$ classifies 2-vector bundles. We thus define the notion of oriented 2-vector bundles, which removes the obstruction by the magnetic monopole. We use this to define an oriented K-theory of 2-vector bundles with a lift of the natural map from $K(\mathsf{Z},3)$. It is then possible to define a retraction of this map and since $K(\mathsf{Z},3)$ classifies complex gerbes we call this a determinant gerbe map.

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45

Li, Yangcheng. "A Diophantine equation about polygonal numbers." Notes on Number Theory and Discrete Mathematics 27, no. 3 (September 2021): 113–18. http://dx.doi.org/10.7546/nntdm.2021.27.3.113-118.

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It is well known that the number P_k(x)=\frac{x((k-2)(x-1)+2)}{2} is called the x-th k-gonal number, where x\geq1,k\geq3. Many Diophantine equations about polygonal numbers have been studied. By the theory of Pell equation, we show that if G(k-2)(A(p-2)a^2+2Cab+B(q-2)b^2) is a positive integer but not a perfect square, (2A(p-2)\alpha-(p-4)A + 2C\beta+2D)a + (2B(q-2)\beta-(q-4)B+2C\alpha+2E)b>0, 2G(k-2)\gamma-(k-4)G+2H>0 and the Diophantine equation \[AP_p(x)+BP_q(y)+Cxy+Dx+Ey+F=GP_k(z)+Hz\] has a nonnegative integer solution (\alpha,\beta,\gamma), then it has infinitely many positive integer solutions of the form (at + \alpha,bt + \beta,z), where p, q, k \geq 3 and p,q,k,a,b,t,A,B,G\in\mathbb{Z^+}, C,D,E,F,H\in\mathbb{Z}.

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46

Cheng, Hai-Yang. "K → πππ decays in large Nc chiral perturbation theory." Physics Letters B 238, no. 2-4 (April 1990): 399–405. http://dx.doi.org/10.1016/0370-2693(90)91755-z.

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Luo, Zhenguo, Liping Luo, and Yunhui Zeng. "Positive Periodic Solution for the Generalized Neutral Differential Equation with Multiple Delays and Impulse." Journal of Applied Mathematics 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/592513.

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By using a fixed point theorem of strict-set-contraction, which is different from Gaines and Mawhin's continuation theorem and abstract continuation theory fork-set contraction, we established some new criteria for the existence of positive periodic solution of the following generalized neutral delay functional differential equation with impulse:x'(t)=x(t)[a(t)-f(t,x(t),x(t-τ1(t,x(t))),…,x(t-τn(t,x(t))),x'(t-γ1(t,x(t))),…,x'(t-γm(t,x(t))))], t≠tk, k∈Z+; x(tk+)=x(tk-)+θk(x(tk)), k∈Z+. As applications of our results, we also give some applications to several Lotka-Volterra models and new results are obtained.

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48

Pearson, Paul Thomas. "The mod 2 Hopf ring for connective Morava K-theory." Journal of Homotopy and Related Structures 11, no. 3 (June 2, 2015): 469–91. http://dx.doi.org/10.1007/s40062-015-0113-z.

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Putnam, Ian F. "Some classifiable groupoid $$C^{*}$$ C ∗ -algebras with prescribed K-theory." Mathematische Annalen 370, no. 3-4 (September 22, 2017): 1361–87. http://dx.doi.org/10.1007/s00208-017-1598-z.

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Zheng, Yonghui, and Yuefei Zhu. "Conditional factorization based on lattice theory for 〈K,1〉-integers." Journal of Electronics (China) 25, no. 2 (March 2008): 254–57. http://dx.doi.org/10.1007/s11767-006-0144-z.

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