Showing papers on "Ring (mathematics) published in 2007"
16 Apr 2007
TL;DR: The security definitions for this primitive are formalized and an efficient and simple construction in the random oracle model is shown that can suit to many applications, such as an anonymous voting on a BBS.
Abstract: The ring signature allows a signer to leak secrets anonymously, without the risk of identity escrow. At the same time, the ring signature provides great flexibility: No group manager, no special setup, and the dynamics of group choice. The ring signature is, however, vulnerable to malicious or irresponsible signers in some applications, because of its anonymity. In this paper, we propose a traceable ring signature scheme. A traceable ring scheme is a ring signature except that it can restrict "excessive" anonymity. The traceable ring signature has a tag that consists of a list of ring members and an issue that refers to, for instance, a social affair or an election. A ring member can make any signed but anonymous opinion regarding the issue, but only once (per tag). If the member submits another signed opinion, possibly pretending to be another person who supports the first opinion, the identity of the member is immediately revealed. If the member submits the same opinion, for instance, voting "yes" regarding the same issue twice, everyone can see that these two are linked. The traceable ring signature can suit to many applications, such as an anonymous voting on a BBS. We formalize the security definitions for this primitive and show an efficient and simple construction in the random oracle model.
188 citations
TL;DR: In this paper, a solution-generating technique was proposed to transform the fat black ring to the thin ring with the same mass and angular momentum by way of the diring solutions.
Abstract: We show that the ${S}^{1}$-rotating black rings can be superposed by the solution-generating technique. We analyze the black diring solution for the simplest case of multiple rings. There exists an equilibrium black diring where the conical singularities are cured by the suitable choice of physical parameters. Also there are infinite numbers of black dirings with the same mass and angular momentum. These dirings can have two different continuous limits of single black rings. Therefore, we can transform the fat black ring to the thin ring with the same mass and angular momentum by way of the diring solutions.
165 citations
16 Apr 2007
TL;DR: The first efficient ring signature scheme without random oracles was described in this article, which is based on bilinear groups and requires 2l + 3 pairings to verify for l members of a ring.
Abstract: We describe the first efficient ring signature scheme secure, without random oracles, based on standard assumptions. Our ring signatures are based in bilinear groups. For l members of a ring our signatures consist of 2l +2 group elements and require 2l +3 pairings to verify. We prove our scheme secure in the strongest security model proposed by Bender, Katz, and Morselli: namely, we show our scheme to be anonymous against full key exposure and unforgeable with respect to insider corruption. A shortcoming of our approach is that all the users' keys must be defined in the same group.
158 citations
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TL;DR: In this paper, a new method for defining a notion of support for objects in any compactly generated triangulated category admitting small coproducts is proposed, based on a construction of local cohomology functors on triangulation categories, with respect to a central ring of operators.
Abstract: We propose a new method for defining a notion of support for objects in any compactly generated triangulated category admitting small coproducts. This approach is based on a construction of local cohomology functors on triangulated categories, with respect to a central ring of operators. Suitably specialized one recovers, for example, the theory for commutative noetherian rings due to Foxby and Neeman, the theory of Avramov and Buchweitz for complete intersection local rings, and varieties for representations of finite groups according to Benson, Carlson, and Rickard. We give explicit examples of objects whose triangulated support and cohomological support differ. In the case of group representations, this allows us to correct and establish a conjecture of Benson.
141 citations
TL;DR: The main properties of the amalgamated duplication of a ring R along an ideal I, denoted by ''R\bowtie I'' were studied in this paper, where every regular fractional ideal of ''R'' is divisorial.
Abstract: After recalling briefly the main properties of the amalgamated duplication of a ring R along an ideal I, denoted by \(R\bowtie I\), see M. D’Anna and M. Fontana, to appear in J. Algebra Appl., we restrict our attention to the study of the properties of \(R\bowtie I\), when I is a multiplicative canonical ideal of R, see W. J. Heinzer, J. A. Huckaba and I. J. Papick, Comm. Algebra. In particular, we study when every regular fractional ideal of \(R\bowtie I\) is divisorial.
127 citations
09 Jul 2007
TL;DR: A variation of the ring signature scheme is offered, where the signer is guaranteed anonymity even if the common reference string is maliciously generated, and an additional feature of this scheme is that it has perfect anonymity.
Abstract: Ring signatures, introduced by Rivest, Shamir and Tauman, enable a user to sign a message anonymously on behalf of a "ring". A ring is a group of users, which includes the signer. We propose a ring signature scheme that has size O(√N) where N is the number of users in the ring. An additional feature of our scheme is that it has perfect anonymity.
Our ring signature like most other schemes uses the common reference string model. We offer a variation of our scheme, where the signer is guaranteed anonymity even if the common reference string is maliciously generated.
123 citations
TL;DR: In this article, the authors derived a quantum group gp,q that is Kazhdan-Lusztig dual to the W-algebra Wp,q of the logarithmic (p, q) conformal field theory model.
Abstract: We derive and study a quantum group gp,q that is Kazhdan-Lusztig dual to the W-algebra Wp,q of the logarithmic (p,q) conformal field theory model. The algebra Wp,q is generated by two currents W+(z) and W−(z) of dimension (2p−1)(2q−1) and the energy-momentum tensor T(z). The two currents generate a vertex-operator ideal R with the property that the quotient Wp,q∕R is the vertex-operator algebra of the (p,q) Virasoro minimal model. The number (2pq) of irreducible gp,q representations is the same as the number of irreducible Wp,q representations on which R acts nontrivially. We find the center of gp,q and show that the modular group representation on it is equivalent to the modular group representation on the Wp,q characters and “pseudocharacters.” The factorization of the gp,q ribbon element leads to a factorization of the modular group representation on the center. We also find the gp,q Grothendieck ring, which is presumably the “logarithmic” fusion of the (p,q) model.
116 citations
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TL;DR: In this paper, a theory of mutations of maximal rigid objects of a finite quiver without oriented cycles was developed, analogous to the mutations of clusters in Fomin and Zelevinsky's theory of cluster algebras.
Abstract: Let Q be a finite quiver without oriented cycles, and let $\Lambda$ be the associated preprojective algebra. To each terminal representation M of Q (these are certain preinjective representations), we attach a natural subcategory $C_M$ of $mod(\Lambda)$. We show that $C_M$ is a Frobenius category,and that its stable category is a Calabi-Yau category of dimension 2. Then we develop a theory of mutations of maximal rigid objects of $C_M$, analogous to the mutations of clusters in Fomin and Zelevinsky's theory of cluster algebras. We show that $C_M$ yields a categorification of a cluster algebra $A(C_M)$, which is not acyclic in general. We give a realization of $A(C_M)$ as a subalgebra of the graded dual of the enveloping algebra $U(
)$, where $
$ is a maximal nilpotent subalgebra of the symmetric Kac-Moody Lie algebra $\g$ associated to the quiver Q. Let $S^*$ be the dual of Lusztig's semicanonical basis $S$ of $U(
)$. We show that all cluster monomials of $A(C_M)$ belong to $S^*$, and that $S^* \cap A(C_M)$ is a basis of $A(C_M)$. Next, we prove that $A(C_M)$ is naturally isomorphic to the coordinate ring of the finite-dimensional unipotent subgroup $N(w)$ of the Kac-Moody group $G$ attached to $\g$. Here w = w(M) is the adaptable element of the Weyl group of $\g$ which we associate to each terminal representation M of Q. Moreover, we show that the cluster algebra obtained from $A(C_M)$ by formally inverting the generators of the coefficient ring is isomorphic to the coordinate ring of the unipotent cell $N^w := N \cap (B_-wB_-)$ of G. We obtain a corresponding dual semicanonical basis of this coorindate ring.
116 citations
TL;DR: In this article, the concept of a comultiplication $R$-module was introduced and some related results were obtained for a ring with an identity (not necessary commutative).
Abstract: Let $R$ be a ring with an identity (not necessary commutative) and let $M$ be a left $R$-module. In this paper we will introduce the concept of a comultiplication $R$-module and we will obtain some related results.
90 citations
TL;DR: In this paper, it was shown that the Galois group of a local field can be computed using the associated étale (φ, Γ)-module over the Robba ring.
Abstract: Given a p-adic representation of the Galois group of a local field, we show that its Galois cohomology can be computed using the associated étale (φ,Γ)-module over the Robba ring; this is a variant of a result of Herr. We then establish analogues, for not necessarily étale (φ,Γ)-modules over the Robba ring, of the Euler-Poincaré characteristic formula and Tate local duality for p-adic representations. These results are expected to intervene in the duality theory for Selmer groups associated to de Rham representations.
85 citations
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TL;DR: In this paper, the authors studied the first cohomology of a group G with coefficients in l^2(G) and obtained strong results about the existence of free subgroups and the subgroup structure provided the group has a positive first l 2 Betti number.
Abstract: In this article we study cocycles of discrete countable groups with values in l^2(G) and the ring of affiliated operators UG. We clarify properties of the first cohomology of a group G with coefficients in l^2(G) and answer several questions from [CTV]. Moreover, we obtain strong results about the existence of free subgroups and the subgroup structure, provided the group has a positive first l^2-Betti number. We give numerous applications and examples of groups which satisfy our assumptions.
TL;DR: In this article, a new exactly solvable potential which consists of the modified Kratzer potential plus a new ring-shaped potential βctg2θ/r2 was proposed, which is based on the Nikiforov-Uvarov method.
Abstract: We propose a new exactly solvable potential which consists of the modified Kratzer potential plus a new ring-shaped potential βctg2θ/r2. The exact solutions of the bound states of the Schrodinger equation for this potential are presented analytically by using the Nikiforov–Uvarov method, which is based on solving the second-order linear differential equation by reducing to a generalized equation of hypergeometric type. The wavefunctions of the radial and angular parts are taken on the form of the generalized Laguerre polynomials and the total energy of the system is different from the modified Kratzer potential because of the contribution of the angular part.
TL;DR: The K-theoretic Farrell-Jones Conjecture for hyperbolic groups with (twisted) coefficients in any associative ring with unit was proved in this article.
Abstract: We prove the K-theoretic Farrell-Jones Conjecture for hyperbolic groups with (twisted) coefficients in any associative ring with unit.
TL;DR: In this article, it was shown that the dual of a pointed semisimple category with respect to a module category is a Grothendieck ring and the associator of the dual is an associator.
Abstract: A finite tensor category is called pointed if all its simple objects are invertible. We find necessary and sufficient conditions for two pointed semisimple categories to be dual to each other with respect to a module category. Whenever the dual of a pointed semisimple category with respect to a module category is pointed, we give explicit formulas for the Grothendieck ring and for the associator of the dual. This leads to the definition of categorical Morita equivalence on the set of all finite groups and on the set of all pairs (G, ω), where G is a finite group and ω ∊ H 3(G, k ×). A group-theoretical and cohomological interpretation of this relation is given. A series of concrete examples of pairs of groups that are categorically Morita equivalent but have nonisomorphic Grothendieck rings are given. In particular, the representation categories of the Drinfeld doubles of the groups in each example are equivalent as braided tensor categories and hence these groups define the same modular data.
TL;DR: Using deep properties of the Kazhdan-Lusztig basis of the one-parameter Hecke algebra, the authors showed that it has a natural cellular structure in the sense of Graham and Lehrer.
Abstract: Let $\mathcal{H}$
be the one-parameter Hecke algebra associated to a finite Weyl group W, defined over a ground ring in which “bad” primes for W are invertible. Using deep properties of the Kazhdan–Lusztig basis of $\mathcal{H}$
and Lusztig’s a-function, we show that $\mathcal{H}$
has a natural cellular structure in the sense of Graham and Lehrer. Thus, we obtain a general theory of “Specht modules” for Hecke algebras of finite type. Previously, a general cellular structure was only known to exist in types A
n
and B
n
.
17 Dec 2007
TL;DR: It is shown that the minimum number of robots that can explore a ring of size n is O(log n) and that ρ(n) = Ω( log n) for arbitrarily large n.
Abstract: We consider the problem of exploring an anonymous unoriented ring by a team of k identical, oblivious, asynchronous mobile robots that can view the environment but cannot communicate. This weak scenario is standard when the spatial universe in which the robots operate is the two-dimentional plane, but (with one exception) has not been investigated before. We indeed show that, although the lack of these capabilities renders the problems considerably more difficult, ring exploration is still possible.
We show that the minimum number ρ(n) of robots that can explore a ring of size n is O(log n) and that ρ(n) = Ω(log n) for arbitrarily large n. On one hand we give an algorithm that explores the ring starting from any initial configuration, provided that n and k are co-prime, and we show that there always exist such k in O(log n). On the other hand we show that Ω(log n) agents are necessary for arbitrarily large n. Notice that, when k and n are not co-prime, the problem is sometimes unsolvable (i.e., there are initial configurations for which the exploration cannot be done). This is the case, e.g., when k divides n.
TL;DR: In this article, the boundary Harnack inequality for p harmonic functions in Lipschitz and star-like ring domains was shown to hold up to the boundary of the boundary.
Abstract: In this paper we prove new results for p harmonic functions, p ≠ 2 , 1 p ∞ , in Lipschitz and starlike Lipschitz ring domains. In particular we prove the boundary Harnack inequality, Theorem 1, for the ratio of two positive p harmonic functions vanishing on a portion of the boundary of a Lipschitz domain, with constants only depending on p , n and the Lipschitz constant of the domain. For p capacitary functions, in starlike Lipschitz ring domains, we prove an even stronger result, Theorem 2, showing that the ratio is Holder continuous up to the boundary. Moreover, for p capacitary functions in starlike Lipschitz ring domains we prove, Theorems 3 and 4, appropriate extensions to p ≠ 2 , 1 p ∞ , of famous results of Dahlberg [12] and Jerison and Kenig [25] on the Poisson kernel associated to the Laplace operator (i.e. p = 2 ).
TL;DR: In this paper, the authors studied the cohomology of geometric quotients of algebraic groups on smooth projective varieties X and defined sets of stable points X^s and semistable points X^{ss, often explicitly computable via the methods of reductive geometric invariant theory.
Abstract: We study linear actions of algebraic groups on smooth projective varieties X. A guiding goal for us is to understand the cohomology of "quotients" under such actions, by generalizing (from reductive to non-reductive group actions) existing methods involving Mumford's geometric invariant theory (GIT). We concentrate on actions of unipotent groups H, and define sets of stable points X^s and semistable points X^{ss}, often explicitly computable via the methods of reductive GIT, which reduce to the standard definitions due to Mumford in the case of reductive actions. We compare these with definitions in the literature. Results include (1) a geometric criterion determining whether or not a ring of invariants is finitely generated, (2) the existence of a geometric quotient of X^s, and (3) the existence of a canonical "enveloping quotient" variety of X^{ss}, denoted X//H, which (4) has a projective completion given by a reductive GIT quotient and (5) is itself projective and isomorphic to Proj(k[X]^H) when k[X]^H is finitely generated.
TL;DR: In this paper, a characterization of the equivariant quantum cohomology ring of the Grassmannian is presented, and determinantal formulae for equivariants of quantum Schubert classes are given for factorial Schur functions.
Abstract: We find presentations by generators and relations for the equivariant quantum cohomology of the Grassmannian. For these presentations, we also find determinantal formulae for the equivariant quantum Schubert classes. To prove this, we use the theory of factorial Schur functions and a characterization of the equivariant quantum cohomology ring.
TL;DR: In this article, a mathematical model is built on analysis of the reason for the ring enlargement during rolling and the model defines the relationship between the ring outer diameter growth rate and feed speed.
Abstract: Feed speed needs to be controlled during cold ring rolling but first it must be planned. However, up to now, it has been set mainly by trials and experience, as the existing method is inefficient. This paper aims to find a convenient method to plan feed speed. In this study, the ring outer diameter growth rate was regarded as the basic quantity to determine the feed speed. A mathematical model is built on analysis of the reason for the ring enlargement during rolling. The model defines the relationship between the ring outer diameter growth rate and feed speed. Using the model, the feed speed was planned versus the thickness of the rolling ring. The extrema of the feed speed and the ring outer diameter growth rate were also determined. The method of planning the feed speed based on the ring outer diameter growth rate is efficient and economical, and it facilitates process control for cold ring rolling.
TL;DR: In this article, a ring-rolling design for a large-scale ring product of Ti-6Al-4V alloy was investigated by a calculation method and FEM analysis.
TL;DR: Despite the finite particle size, an excellent agreement is found between the measured eigenvalues and those numerically predicted by Oseen theory applied to the two-dimensional (2D) ring geometry.
Abstract: We use a continuous acquisition, high-speed camera with integrated centroid tracking to simultaneously measure the positions of a ring of micron-sized particles held in holographic optical tweezers. Hydrodynamic coupling between the particles gives a set of eigenmodes, each one independently relaxing with a characteristic decay rate (eigenvalue) that can be measured using our positional data. Despite the finite particle size, we find an excellent agreement between the measured eigenvalues and those numerically predicted by Oseen theory applied to the two-dimensional (2D) ring geometry. Particle motions are also analyzed in terms of the alternative eigenmode set obtained by wrapping onto the ring the eigenmodes of a 1D periodic chain. We identify the modes for which the periodic chain is a good approximation to the ring and those for which it is not.
Book•
02 Oct 2007
TL;DR: Noether and Artin this paper introduced the notion of abstract group theory and defined the abstract definition of a ring in abstract algebra, and showed that the abstract notion of group theory can be expressed in terms of abstract fields.
Abstract: Preface.-Chapter 1: Classical Algebra.-Early roots.-The Greeks.-Al-Khwarizmi.-Cubic and quartic equations.-The cubic and complex numbers.-Algebraic notation: Viete and Descartes.-The theory of equations and the Fundamental Theorem of Algebra.-Symbolical algebra.-References.-Chapter 2: Group Theory.-Sources of group theory.-Development of 'specialized' theories of groups.-Emergence of abstraction in group theory.-Consolidation of the abstract group concept dawn of abstract group theory. Divergence of developments in group theory.-References.-Chapter 3: Ring Theory.-Noncommutative ring theory.-Commutative ring theory.-The abstract definition of a ring.-Emmy Noether and Emil Artin.-Epilogue.-References.-Chapter 4: Field Theory.-Galois theory.-Algebraic number theory.-Algebraic geometry.-Symbolical algebra.-The abstract definition of a field.-Hensel's p-adic numbers.-Steinitz.-A glance ahead.-References.-Chapter 5: Linear Algebra.-Linear equations.-Determinants Matrices and linear transformations.-Linear independence, basis, and dimension.-Vector spaces.-References.-Chapter 6: Emmy Noether and the Advent of Abstract Algebra.-Invariant theory.-Commutative algebra.-Noncommutative algebra and representation theory.-Applications of noncommutative to commutative algebra.-Noether's legacy.-References.-Chapter 7: A course in abstract algebra inspired by history.-Problem I: Why is (-1)(-1) = 1? .-Problem II: What are the integer solutions of x2 + 2 = y3 ? .-Problem III: Can we trisect a 600 angle using only straightedge and compass?.-Problem IV: Can we solve x5 - 6x + 3 = 0? .-Problem V: 'Papa, can you multiply triples?' .-General remarks on the course.-References.-Chapter 8: Biographies of Selected Mathematicians.-Cayley.-Invariants.-Groups.-Matrices. Geometry.-Conclusion.-References.-Dedekind.-Algebraic numbers.-Real numbers.-Natural numbers.-Other works.Conclusion.-References.-Galois.-Mathematics.-Politics.-The duel.-Testament.-Conclusion.-References.-Gauss.-Number theory.-Differential geometry, probability, statistics.-The diary.-Conclusion.-References.-Hamilton.-Optics.-Dynamics.-Complex numbers.-Foundations of algebra.-Quaternions.-Conclusion.-References.-Noether.-Early years.-University studies.-Gottingen.-Noether as a teacher.-Bryn Mawr.-Conclusion.-References.-Index.-Acknowledgments
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Abstract: We study linear actions of algebraic groups on smooth projective varieties X. A guiding goal for us is to understand the cohomology of "quotients" under such actions, by generalizing (from reductive to non-reductive group actions) existing methods involving Mumford's geometric invariant theory (GIT). We concentrate on actions of unipotent groups H, and define sets of stable points X^s and semistable points X^{ss}, often explicitly computable via the methods of reductive GIT, which reduce to the standard definitions due to Mumford in the case of reductive actions. We compare these with definitions in the literature. Results include (1) a geometric criterion determining whether or not a ring of invariants is finitely generated, (2) the existence of a geometric quotient of X^s, and (3) the existence of a canonical "enveloping quotient" variety of X^{ss}, denoted X//H, which (4) has a projective completion given by a reductive GIT quotient and (5) is itself projective and isomorphic to Proj(k[X]^H) when k[X]^H is finitely generated.
TL;DR: Jollenbeck and Berglund as mentioned in this paper showed that a polynomial ring is Golod if and only if the product on Tor ∗ P (A, k ) is trivial.
TL;DR: In this article, it was shown that if R is a homomorphic image of a Gorenstein local ring, then all the local cohomology (below the dimension) of such a ring maps to zero in a finite extension of the ring.
TL;DR: In this paper, the ring of algebraic cobordisms of a Pfister quadric was computed for a non-cellular variety and the results showed that the restriction homomorphism in this case is injective for any field extension E/F.
Abstract: In this article we compute the ring of algebraic cobordisms of a Pfister quadric. This is a rare example of a non-cellular variety where such a computation is known. We consider the algebraic cobordisms Ω* of Levine and Morel, as well as the MGL 2*, * of Voevodsky. The methods of computation in these two cases are quite different. However, the results do agree (which supports the expectation that the two theories actually coincide). We show that the restriction homomorphism in our case is injective for any field extension E/F.
TL;DR: In this article, the generating functions which count the BPS operators in the chiral ring of a N = 2$ quiver gauge theory that lives on N D3-branes probing an ALE singularity are presented.
Abstract: This note is presenting the generating functions which count the BPS operators in the chiral ring of a $\cal N=2$ quiver gauge theory that lives on N D3-branes probing an ALE singularity. The difficulty in this computation arises from the fact that this quiver gauge theory has a moduli space of vacua that splits into many branches — the Higgs, the Coulomb, and mixed branches. As a result, there can be operators which explore those different branches and the counting gets complicated by having to deal with such operators while avoiding over or under counting. The solution to this problem turns out to be very elegant and is presented in this note. Some surprises with “surgery” of generating functions arises.
TL;DR: A formal security model for revocable ring signature is provided and an efficient construction which is proven secure under the security model is proposed.
Abstract: Group signature allows the anonymity of a real signer in a group to be revoked by a trusted party called group manager. It also gives the group manager the absolute power of controlling the formation of the group. Ring signature, on the other hand, does not allow anyone to revoke the signer anonymity, while allowing the real signer to form a group (also known as a ring) arbitrarily without being controlled by any other party. In this paper, we propose a new variant for ring signature, called Revocable Ring Signature. The signature allows a real signer to form a ring arbitrarily while allowing a set of authorities to revoke the anonymity of the real signer. This new variant inherits the desirable properties from both group signature and ring signature in such a way that the real signer will be responsible for what it has signed as the anonymity is revocable by authorities while the real signer still has the freedom on ring formation. We provide a formal security model for revocable ring signature and propose an efficient construction which is proven secure under our security model.
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TL;DR: In this article, a modern proof of the Clifford-Munn-Ponizovski result is provided based on a lemma of J. A. Green, which allows us to circumvent the theory of 0-simple semigroups.
Abstract: Work of Clifford, Munn and Ponizovski{\u\i} parameterized the irreducible representations of a finite semigroup in terms of the irreducible representations of its maximal subgroups. Explicit constructions of the irreducible representations were later obtained independently by Rhodes and Zalcstein and by Lallement and Petrich. All of these approaches make use of Rees's theorem characterizing 0-simple semigroups up to isomorphism. Here we provide a short modern proof of the Clifford-Munn-Ponizovski{\u\i} result based on a lemma of J. A. Green, which allows us to circumvent the theory of 0-simple semigroups. A novelty of this approach is that it works over any base ring.