Showing papers on "Ring (mathematics) published in 2013"

The K-Book: An Introduction to Algebraic K-theory

Abstract: Projective modules and vector bundles The Grothendieck group $K_0$ $K_1$ and $K_2$ of a ring Definitions of higher $K$-theory The fundamental theorems of higher $K$-theory The higher $K$-theory of fields Nomenclature Bibliography Index

A universal characterization of higher algebraic K-theory

Abstract: In this paper we establish a universal characterization of higher algebraic K–theory in the setting of small stable ∞–categories. Specifically, we prove that connective algebraic K–theory is the universal additive invariant, ie the universal functor with values in spectra which inverts Morita equivalences, preserves filtered colimits and satisfies Waldhausen’s additivity theorem. Similarly, we prove that nonconnective algebraic K–theory is the universal localizing invariant, ie the universal functor that moreover satisfies the Thomason–Trobaugh–Neeman Localization Theorem. To prove these results, we construct and study two stable ∞–categories of “noncommutative motives”; one associated to additivity and another to localization. In these stable ∞–categories, Waldhausen’s S∙–construction corresponds to the suspension functor and connective and nonconnective algebraic K–theory spectra become corepresentable by the noncommutative motive of the sphere spectrum. In particular, the algebraic K–theory of every scheme, stack and ring spectrum can be recovered from these categories of noncommutative motives. In the case of a connective ring spectrum R, we prove moreover that its negative K–groups are isomorphic to the negative K–groups of the ordinary ring π0R. In order to work with these categories of noncommutative motives, we establish comparison theorems between the category of spectral categories localized at the Morita equivalences and the category of small idempotent-complete stable ∞–categories. We also explain in detail the comparison between the ∞–categorical version of Waldhausen K–theory and the classical definition. As an application of our theory, we obtain a complete classification of the natural transformations from higher algebraic K–theory to topological Hochschild homology (THH) and topological cyclic homology (TC). Notably, we obtain an elegant conceptual description of the cyclotomic trace map.

Off-diagonal Bethe ansatz and exact solution of a topological spin ring

Abstract: A general method is proposed for constructing the Bethe ansatz equations of integrable models without U(1) symmetry. As an example, the exact spectrum of the XXZ spin ring with a Mobius-like topological boundary condition is derived by constructing a modified T-Q relation based on the functional connection between the eigenvalues of the transfer matrix and the quantum determinant of the monodromy matrix. With the exact solution, the elementary excitations of the topological XX spin ring are discussed in detail. It is found that the excitation spectrum indeed shows a nontrivial topological nature.

Baxter's Relations and Spectra of Quantum Integrable Models

Abstract: Generalized Baxter's relations on the transfer-matrices (also known as Baxter's TQ relations) are constructed and proved for an arbitrary untwisted quantum affine algebra. Moreover, we interpret them as relations in the Grothendieck ring of the category O introduced by Jimbo and the second author in arXiv:1104.1891 involving infinite-dimensional representations constructed in arXiv:1104.1891, which we call here "prefundamental". We define the transfer-matrices associated to the prefundamental representations and prove that their eigenvalues on any finite-dimensional representation are polynomials up to a universal factor. These polynomials are the analogues of the celebrated Baxter polynomials. Combining these two results, we express the spectra of the transfer-matrices in the general quantum integrable systems associated to an arbitrary untwisted quantum affine algebra in terms of our generalized Baxter polynomials. This proves a conjecture of Reshetikhin and the first author formulated in 1998 (arXiv:math/9810055). We also obtain generalized Bethe Ansatz equations for all untwisted quantum affine algebras.

The Ring of Algebraic Functions on Persistence Bar Codes

Abstract: We study the ring of algebraic functions on the space of persistence barcodes, with applications to pattern recognition.

Equations of tropical varieties

Abstract: We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as $\mathbb{T} = (\mathbb{R}\cup \{-\infty\}, \mathrm{max}, +)$ by realizing them as solution sets to explicit systems of tropical equations that are uniquely determined by idempotent module theory. We then define a tropicalization functor that sends closed subschemes of a toric variety over a ring R with non-archimedean valuation to closed subschemes of the corresponding tropical toric variety. Upon passing to the set of $\mathbb{T}$-points this reduces to Kajiwara-Payne's extended tropicalization, and in the case of a projective hypersurface we show that the scheme structure determines the multiplicities attached to the top-dimensional cells. By varying the valuation, these tropicalizations form algebraic families of $\mathbb{T}$-schemes parameterized by a moduli space of valuations on R that we construct. For projective subschemes, the Hilbert polynomial is preserved by tropicalization, regardless of the valuation. We conclude with some examples and a discussion of tropical bases in the scheme-theoretic setting.

Quantum Grothendieck rings and derived Hall algebras

Abstract: We obtain a presentation of the t-deformed Grothendieck ring of a quantum loop algebra of Dynkin type A, D, E. Specializing t at the the square root of the cardinality of a finite field F, we obtain an isomorphism with the derived Hall algebra of the derived category of a quiver Q of the same Dynkin type. Along the way, we study for each choice of orientation Q a tensor subcategory whose t-deformed Grothendieck ring is isomorphic to the positive part of a quantum enveloping algebra of the same Dynkin type, where the classes of simple objects correspond to Lusztig's dual canonical basis.

Square Roots of –1 in Real Clifford Algebras

Abstract: It is well known that Clifford (geometric) algebra offers a geometric interpretation for square roots of –1in the form of blades that square to minus 1. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube. Systematic research has been done [33] on the biquaternion roots of –1, abandoning the restriction to blades. Biquaternions are isomorphic to the Clifford (geometric) algebra Cl 3,0 of ℝ3. Further research on general algebras Cl p,q has explicitly derived the geometric roots of –1for p + q≤4 [20]. The current research abandons this dimension limit and uses the Clifford algebra to matrix algebra isomorphisms in order to algebraically characterize the continuous manifolds of square roots of –1f ound in the different types of Clifford algebras, depending on the type of associated ring (ℝ,ℍ,ℝ2,ℍ2, or ℂ). At the end of the chapter explicit computer generated tables of representative square roots of –1 are given for all Clifford algebras with n = 5,7, and s = 3 (mod 4) with the associated ring ℂ. This includes, e.g., Cl 0,5 important in Clifford analysis, and Cl 4,1 which in applications is at the foundation of conformal geometric algebra. All these roots of –1 are immediately useful in the construction of new types of geometric Clifford–Fourier transformations.

On weakly 2-absorbing ideals of commutative rings

Abstract: Let R be a commutative ring with identity 1 6 0. Various generalizations of prime ideals have been studied. For example, a proper ideal I of R isweakly prime if a;b2 R with 06 ab2 I, then either a2 I or b2 I. Also a proper ideal I of R is said to be 2-absorbing if whenever a;b;c 2 R and abc 2 I, then either ab 2 I or ac 2 I or bc 2 I. In this paper, we introduce the concept of a weakly 2-absorbing ideal. A proper ideal I of R is called a weakly 2-absorbing ideal of R if whenever a;b;c2 R and 0 6 abc 2 I, then either ab 2 I or ac 2 I or bc 2 I. For example, every proper ideal of a quasi-local ring (R;M) with M 3 =f0g is a weakly 2-absorbing ideal of R. We show that a weakly 2-absorbing ideal I of R with I 3 6 0 is a 2-absorbing ideal of R. We show that every proper ideal of a commutative ring R is a weakly 2-absorbing ideal if and only if either R is a quasi-local ring with maximal ideal M such that M 3 =f0g or R is ring- isomorphic to R1 F where R1 is a quasi-local ring with maximal ideal M such that M 2 =f0g and F is a eld or R is ring-isomorphic to F1 F2 F3 for some elds F1;F2;F3.

Algebraic (trapdoor) one-way functions and their applications

Abstract: In this paper we introduce the notion of Algebraic (Trapdoor) One Way Functions, which, roughly speaking, captures and formalizes many of the properties of number-theoretic one-way functions. Informally, a (trapdoor) one way function F: X#8594;Y is said to be algebraic if X and Y are (finite) abelian cyclic groups, the function is homomorphic i.e. F(x)·F(y)=F(x ·y), and is ring-homomorphic, meaning that it is possible to compute linear operations 'in the exponent' over some ring (which may be different from ℤp where p is the order of the underlying group X) without knowing the bases. Moreover, algebraic OWFs must be flexibly one-way in the sense that given y=F(x), it must be infeasible to compute (x′, d) such that F(x′)=yd (for d≠0). Interestingly, algebraic one way functions can be constructed from a variety of standard number theoretic assumptions, such as RSA, Factoring and CDH over bilinear groups. As a second contribution of this paper, we show several applications where algebraic (trapdoor) OWFs turn out to be useful. These include publicly verifiable secure outsourcing of polynomials, linearly homomorphic signatures and batch execution of Sigma protocols.

Sine-Gordon kinks on a domain wall ring

Abstract: We construct a stable domain wall ring with lump beads on it in a baby Skyrme model with a potential consisting of two terms linear and quadratic in fields.

The tautological ring of the moduli space of curves

Abstract: The tautological ring of the moduli space of curves Mg is a subring R(Mg) of the Chow ring A(Mg). The tautological ring can also be defined for other moduli spaces of curves, such as the moduli space of curves of compact typeMg or the moduli space of Deligne-Mumford stable pointed curves Mg,n. We conjecture and prove various results about the structure of the tautological ring. In particular, we give two proofs of the Faber-Zagier relations, a large family of relations between the kappa classes in R(Mg) that contains all known relations. The first proof (joint work with R. Pandharipande) uses the virtual geometry of the moduli space of stable quotients developed by Marian, Oprea, and Pandharipande. The second proof (joint work with R. Pandharipande and D. Zvonkine) uses Witten’s class on the moduli space of 3-spin curves and the classification of semisimple cohomological field theories by Givental and Teleman. The second proof has the disadvantage that it only proves the image of the Faber-Zagier relations in cohomology, but the advantage that it also proves an extension of the relations toMg,n that was conjectured by the author. These relations on Mg,n and their restrictions to smaller moduli spaces of curves seem to describe all known relations in the tautological ring. We also prove several combinatorial results about the structure of the Gorenstein quotient rings of R(Mg) and R(Mg). This includes several new families of relations that are similar to the Faber-Zagier relations, as well as joint work with F. Janda giving formulas for ranks of restricted socle pairings in R(Mg). The appendix presents data obtained by computer calculations of the tautological relations on Mg,n and their restrictions toMg,n andM g,n for small values of g and n. The data suggests several new locations in which the tautological ring might not be a Gorenstein ring.

Identifying the Matrix Ring: Algorithms for Quaternion Algebras and Quadratic Forms

Abstract: We discuss the relationship between quaternion algebras and quadratic forms with a focus on computational aspects. Our basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2 ×2-matrix ring M2(R) and, if so, to compute such an embedding. We discuss many variants of this problem, including algorithmic recognition of quaternion algebras among algebras of rank 4, computation of the Hilbert symbol, and computation of maximal orders.

Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras

Abstract: Let $J$ be a set of pairs consisting of good modules over an affine quantum algebra and invertible elements. The distribution of poles of the normalized R-matrices yields Khovanov-Lauda-Rouquier algebras $R^J$. We define a functor $F$ from the category $S_J$ of finite-dimensional graded $R^J$-modules to the category of finite-dimensional integrable $U_q(g)$-modules. The functor $F$ sends convolution products of $R^J$-modules to tensor products of $U_q(g)$-modules. It is exact if $R^J$ is of finite type A,D,E. When $J$ is the vector representation of $A^{(1)}_{n-1}$, we recover the affine Schur-Weyl duality. Focusing on this case, we obtain an abelian rigid graded tensor category $T_J$ by localizing the category $S_J$. The functor $F$ factors through $T_J$. Moreover, the Grothendieck ring of the category $C_J$, the image of $F$, is isomorphic to the Grothendieck ring of $T_J$ at $q=1$.

Generalized inverses modulo ℋ in semigroups and rings

Abstract: The definition of the inverse along an element was very recently introduced, and it contains known generalized inverses such as the group, Drazin and Moore–Penrose inverses. In this article, we first prove a simple existence criterion for this inverse in a semigroup, and then relate the existence of such an inverse in a ring to the ring units.

Finiteness of cominuscule quantum $K$ -theory

Abstract: The product of two Schubert classes in the quantum K-theory ring of a homogeneous space X = G/P is a formal power series with coefficients in the Grothendieck ring of algebraic vector bundles on X. We show that if X is cominuscule, then this power series has only finitely many non-zero terms. The proof is based on a geometric study of boundary Gromov-Witten varieties in the Kontsevich moduli space, consisting of stable maps to X that take the marked points to general Schubert varieties and whose domains are reducible curves of genus zero. We show that all such varieties have rational singularities, and that boundary Gromov-Witten varieties defined by two Schubert varieties are either empty or unirational. We also prove a relative Kleiman-Bertini theorem for rational singularities, which is of independent interest. A key result is that when X is cominuscule, all boundary Gromov-Witten varieties defined by three single points in X are rationally connected.

Integrated circuit identification and dependability verification using ring oscillator based physical unclonable function and age detection circuitry

Abstract: One feature pertains to an integrated circuit (IC) that includes a first plurality of ring oscillators configured to implement, in part, a physically unclonable function (PUF). The IC further includes a second plurality of ring oscillators configured to implement, in part, an age sensor circuit, and also a ring oscillator selection circuit that is coupled to the first plurality of ring oscillators and the second plurality of ring oscillators. The ring oscillator selection circuit is adapted to select at least two ring oscillator outputs from at least one of the first plurality of ring oscillators and/or the second plurality of ring oscillators. Notably, the ring oscillator selection circuit is commonly shared by the PUF and the age sensor circuit. Also, the IC may further include an output function circuit adapted to receive and compare the two ring oscillator outputs and generate an output signal.

SKEW CYCLIC CODES OVER Fp + vFp

Abstract: 【In this paper, we study a special class of linear codes, called skew cyclic codes, over the ring $R=F_p+vF_p$ , where $p$ is a prime number and $v^2=v$ . We investigate the structural properties of skew polynomial ring $R[x,{\theta} ] $ and the set $R[x,{\theta} ] /(x^n-1)$ . Our results show that these codes are equivalent to either cyclic codes or quasi-cyclic codes. Based on this fact, we give the enumeration of distinct skew cyclic codes over R.】

Pushing forward matrix factorizations

Abstract: We describe the pushforward of a matrix factorization along a ring morphism in terms of an idempotent defined using relative Atiyah classes, and we use this construction to study the convolution of kernels defining integral functors between categories of matrix factorizations. We give an elementary proof of a formula for the Chern character of the convolution generalizing the Hirzebruch–Riemann–Roch formula of Polishchuk and Vaintrob.

Duality and Tilting for Commutative DG Rings

Abstract: We consider commutative DG rings (better known as nonpositive strongly commutative associative unital DG algebras). For such a DG ring $A$ we define the notions of perfect, tilting, dualizing, Cohen-Macaulay and rigid DG $A$-modules. Geometrically perfect DG modules are defined by a local condition on $\operatorname{Spec} \bar{A}$, where $\bar{A} := \operatorname{Spec} \, \operatorname{H}^0(A)$. Algebraically perfect DG modules are those that can be obtained from $A$ by finitely many shifts, direct summands and cones. Tilting DG modules are those that have inverses w.r.t. the derived tensor product; their isomorphism classes form the derived Picard group $\operatorname{DPic}(A)$. Dualizing DG modules are a generalization of Grothendieck's original definition (and here $A$ has to be cohomologically pseudo-noetherian). Cohen-Macaulay DG modules are the duals (w.r.t. a given dualizing DG module) of finite $\bar{A}$-modules. Rigid DG $A$-modules, relative to a commutative base ring $K$, are defined using the squaring operation, and this is a generalization of Van den Bergh's original definition. The techniques we use are the standard ones of derived categories, with a few improvements. We introduce a new method for studying DG $A$-modules: Cech resolutions of DG $A$-modules corresponding to open coverings of $\operatorname{Spec} \bar{A}$. Here are some of the new results obtained in this paper:... [truncated] The functorial properties of Cohen-Macaulay DG modules that we establish here are needed for our work on rigid dualizing complexes over commutative rings, schemes and Deligne-Mumford stacks. We pose several conjectures regarding existence and uniqueness of rigid DG modules over commutative DG rings.

On convergence analysis of space homeomorphisms

Abstract: Various theorems on convergence of general spatial homeomorphisms are proved and, on this basis, theorems on convergence and compactness for classes of the so-called ring Q-homeomorphisms are obtained. In particular, it is established that a family of all ring Q-homeomorphisms f in ℝ n fixing two points is compact provided that the function Q is of finite mean oscillation. The corresponding applications have been given to mappings in the Sobolev classes W loc 1, for the case p > n − 1.

N=1 dynamics with T_N theory

Abstract: We study the dynamics of N=1 supersymmetric systems consisting of the strongly-coupled superconformal theory T_N, SU(N) gauge groups, and fundamental chiral multiplets. We demonstrate that such systems exhibit familiar phenomena such as deformation of the vacuum moduli space, appearance of the dynamical superpotential, and Coulomb branches with N=1 Seiberg-Witten curves. The analysis requires a rather detailed knowledge of the chiral ring of the T_N theory, which will also be discussed at length.