arXiv subjects
Publications and source records attributed to Jie Wu.
We prove two homotopy decomposition theorems for the loops on co-H-spaces, including a generalization of the Hilton-Milnor Theorem. These are applied to problems arising in algebra, representation theory, toric topology, and the study of quasi-symmetric functions.
In this paper homotopical methods for the description of subgroups determined by ideals in group rings are introduced. It is shown that in certain cases the subgroups determined by symmetric product of ideals in group rings can be described with the help of homotopy groups of spheres.
Let $E$ be an elliptic curve over $\F_p$ without complex multiplication, and for each prime $p$ of good reduction, let $n_E(p) = | E(\F_p) |$. Let $Q_{E,b}(x)$ be the number of primes $p \leq x$ such that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$, and $π_{E, b}^{\rm pseu}(x)$ be the number of {\it compositive} $n_E(p)$ such that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$ (also called elliptic curve pseudoprimes). Motivated by cryptography applications, we address in this paper the problem of finding upper bounds for $Q_{E,b}(x)$ and $π_{E, b}^{\rm pseu}(x)$, generalising some of the literature for the classical pseudoprimes \cite{Erdos56, Pomerance81} to this new setting.
We consider some questions related to the signs of Hecke eigenvalues or Fourier coefficients of classical modular forms. One problem is to determine to what extent those signs, for suitable sets of primes, determine uniquely the modular form, and we give both individual and statistical results. The second problem, which has been considered by a number of authors, is to determine the size, in terms of the conductor and weight, of the first sign-change of Hecke eigenvalues. Here we improve significantly the recent estimate of Iwaniec, Kohnen and Sengupta.
The nitrogen-vacancy defect center (NV center) is a promising candidate for quantum information processing due to the possibility of coherent manipulation of individual spins in the absence of the cryogenic requirement. We report a room-temperature implementation of the Deutsch-Jozsa algorithm by encoding both a qubit and an auxiliary state in the electron spin of a single NV center. By thus exploiting the specific S=1 character of the spin system, we demonstrate how even scarce quantum resources can be used for test-bed experiments on the way towards a large-scale quantum computing architecture.
Let $X$ be a co-$H$-space of $(p-1)$-cell complex with all cells in even dimensions. Then the loop space $ΩX$ admits a retract $\bar A^{\min}(X)$ that is the evaluation of the functor $\bar A^{\min}$ on $X$. In this paper, we determine the homology $H_*(\bar A^{\min}(X))$ and give the $\EHP$ sequence for the spaces $\bar A^{\min}(X)$.
In this paper, we determine the homotopy groups π_4(ΣK(A,1)) and π_5(ΣK(A,1)) for abelian groups A by using different facts and methods from group theory and homotopy theory: derived functors, the Carlsson simplicial construction, the Baues-Goerss spectral sequence, homotopy decompositions and the methods of algebraic K-theory. As the applications, we also determine π_i(ΣK(G,1)) with i=4,5 for some non-abelian groups G=Σ_3 and SL(Z), and π_4(ΣK(A_4,1)) for the 4-th alternating group A_4.
We introduce the (general) homotopy groups of spheres as link invariants for Brunnian-type links through the investigations on the intersection subgroup of the normal closures of the meridians of strongly nonsplittable links. The homotopy groups measure the difference between the intersection subgroup and symmetric commutator subgroup of the normal closures of the meridians and give the invariants of the links obtained in this way. Moreover the higher homotopy-group invariants can produce some links that could not be detected by the Milnor invariants. Furthermore all homotopy groups of spheres can be obtained from the geometric Massey products on links.
Natural linear and coalgebra transformations of tensor algebras are studied. The representations of certain combinatorial groups are given. These representations are connected to natural transformations of tensor algebras and to the groups of the homotopy classes of maps from the James construction to loop spaces. Applications to homotopy theory appear in a sequel.
Combinatorial groups together with the groups of natural coalgebra transformations of tensor algebras are linked to the groups of homotopy classes of maps from the James construction to a loop space. This connection gives rise to applications to homotopy theory. The Hopf invariants of the Whitehead products are studied and a rate of exponent growth for the strong version of the Barratt Conjecture is given.
This article is an exposition of certain connections between the braid groups, classical homotopy groups of the 2-sphere, as well as Lie algebras attached to the descending central series of pure braid groups arising as Vassiliev invariants of pure braids. Natural related questions are posed at the end of this article.
Let F_n = denote the free group with generators {x_1,...,x_n}. Nielsen and Magnus described generators for the kernel of the canonical epimorphism from the automorphism group of F_n to the general linear group over the integers. In particular among them are the automorphisms chi_{k,i} which conjugate the generator x_k by the generator x_i leaving the x_j fixed for j not k. A computation of the cohomology ring as well as the Lie algebra obtained from the descending central series of the group generated by chi_{k,i} for i<k is given here. Partial results are obtained for the group generated by all chi_{k,i}.
We give the best possible lower bounds in order of magnitude for the number of positive and negative Hecke eigenvalues. This improves upon a recent work of Kohnen, Lau & Shparlinski. Also, we study an analogous problem for short intervals.
Let $E$ be an elliptic curve over $\Q$ without complex multiplication, and which is not isogenous to a curve with non-trivial rational torsion. For each prime $p$ of good reduction, let $|E(\F_p)|$ be the order of the group of points of the reduced curve over $\F_p$. We prove in this paper that, under the GRH, there are at least $2.778 C_E^{\rm twin} x / (\log{x})^2$ primes $p$ such that $|E(\F_p)|$ has at most 8 prime factors, counted with multiplicity. This improves previous results of Steuding & Weng and Murty & Miri. This is also the first result where the dependence on the conjectural constant $C_E^{\rm twin}$ appearing in the twin prime conjecture for elliptic curves (also known as Koblitz's conjecture) is made explicit. This is achieved by sieving a slightly different sequence than the one used by previous authors. By sieving the same sequence and using Selberg's linear sieve, we can also improve the constant appearing in the upper bound for the number of primes $p$ such that $|E(\F_p)|$ is prime. Finally, we remark that our results still hold under an hypothesis weaker than the GRH.
We sharpen some estimates of Rankin on power sums of Hecke eigenvalues, by using Kim & Shahidi's recent results on higher order symmetric powers. As an application, we improve Kohnen, Lau & Shparlinski's lower bound for the number of Hecke eigenvalues of same signs.
In this paper, we study the limiting behavior of the Brown-York mass and Hawking mass along nearly round surfaces at infinity of an asymptotically flat manifold. Nearly round surfaces can be defined in an intrinsic way. Our results show that the ADM mass of an asymptotically flat 3-manifold can be approximated by some geometric invariants of a family of nearly round surfaces which approach to infinity of the manifold.
For every even integer N, denote by D_{1,2}(N) the number of representations of N as a sum of a prime and an integer having at most two prime factors. In this paper, we give a new lower bound for D_{1,2}(N).
In this paper, we prove a weaker form of a conjecture of Montgomery-Vaughan on extreme values of automorphic L-functions at 1.