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Yichao Zhang

Publications and source records attributed to Yichao Zhang.

At least 91 records · Page 5Linked to original sources
arXiv2015

Cohomology of $SL_2$ and related structures

Let $SL_2$ be the rank one simple algebraic group defined over an algebraically closed field $k$ of characteristic $p>0$. The paper presents a new method for computing the dimension of the cohomology spaces $\text{H}^n(SL_2,V(m))$ for Weyl $SL_2$-modules $V(m)$. We provide a closed formula for $\text{dim}\text{H}^n(SL_2,V(m))$ when $n\le 2p-3$ and show that this dimension is bounded by the $(n+1)$-th Fibonacci number. This formula is then used to compute $\text{dim}\text{H}^n(SL_2, V(m))$ for $n=1, 2,$ or $3$. For $n>2p-3$, an exponential bound, only depending on $n$, is obtained for $\text{dim}\text{H}^n(SL_2,V(m))$. Analogous results are also established for the extension spaces $\text{Ext}^n_{SL_2}(V(m_2),V(m_1))$ between Weyl modules $V(m_1)$ and $V(m_2)$. In particular, we determine the degree three extensions for all Weyl modules of $SL_2$. As a byproduct, our results and techniques give explicit upper bounds for the dimensions of the cohomology of the Specht modules of symmetric groups, the cohomology of simple modules of $SL_2$, and the finite group of Lie type $SL_2(p^s)$.

math.RT↗
arXiv2015

Eigenform Product Identities For Hilbert Modular Forms

We prove that amongst all real quadratic fields and all spaces of Hilbert modular forms of full level and of weight $2$ or greater, the product of two Hecke eigenforms is not a Hecke eigenform except for finitely many real quadratic fields and finitely many weights. We show that for $\mathbb Q(\sqrt 5)$ there are exactly two such identities.

math.NT↗
arXiv2015

Divisibility properties for weakly holomorphic modular forms with sign vectors

In this paper, we prove some divisibility results for the Fourier coefficients of reduced modular forms of sign vectors. More precisely, we generalize a divisibility result of Siegel on constant terms when the weight is non-positive, which is related to the weight of Borcherds lifts when the weight is zero. By considering Hecke operators for the spaces of weakly holomorphic modular forms with sign vectors, and obtain divisibility results in an "orthogonal" direction on reduced modular forms.

math.NT↗
arXiv2014

Zagier duality and integrality for Fourier coefficients for weakly holomorphic modular forms

In this note, we generalize the isomorphisms to the case when the discriminant form is not necessarily induced from real quadratic fields. In particular, this general setting includes all the subspaces with epsilon-conditions, only two spacial cases of which were treated before. With this established, we shall prove the Zagier duality for canonical bases. Finally, we raise a question on the integrality of the Fourier coefficients of these bases elements, or equivalently we concern the existence of a Miller-like basis for vector valued modular forms.

math.NT↗
arXiv2014

Emergence of Cooperation in Non-scale-free Networks

Evolutionary game theory is one of the key paradigms behind many scientific disciplines from science to engineering. Previous studies proposed a strategy updating mechanism, which successfully demonstrated that the scale-free network can provide a framework for the emergence of cooperation. Instead, individuals in random graphs and small-world networks do not favor cooperation under this updating rule. However, a recent empirical result shows the heterogeneous networks do not promote cooperation when humans play a Prisoner's Dilemma. In this paper, we propose a strategy updating rule with payoff memory. We observe that the random graphs and small-world networks can provide even better frameworks for cooperation than the scale-free networks in this scenario. Our observations suggest that the degree heterogeneity may be neither a sufficient condition nor a necessary condition for the widespread cooperation in complex networks. Also, the topological structures are not sufficed to determine the level of cooperation in complex networks.

physics.soc-ph↗
arXiv2014

Fence-sitters Protect Cooperation in Complex Networks

Evolutionary game theory is one of the key paradigms behind many scientific disciplines from science to engineering. In complex networks, because of the difficulty of formulating the replicator dynamics, most of previous studies are confined to a numerical level. In this paper, we introduce a vectorial formulation to derive three classes of individuals' payoff analytically. The three classes are pure cooperators, pure defectors, and fence-sitters. Here, fence-sitters are the individuals who change their strategies at least once in the strategy evolutionary process. As a general approach, our vectorial formalization can be applied to all the two-strategies games. To clarify the function of the fence-sitters, we define a parameter, payoff memory, as the number of rounds that the individuals' payoffs are aggregated. We observe that the payoff memory can control the fence-sitters' effects and the level of cooperation efficiently. Our results indicate that the fence-sitters' role is nontrivial in the complex topologies, which protects cooperation in an indirect way. Our results may provide a better understanding of the composition of cooperators in a circumstance where the temptation to defect is larger.

physics.soc-ph↗
arXiv2014

An Isomorphism between Scalar-Valued Modular Forms and Modular Forms for Weil Representations

In this note, we consider discriminant forms that are given by the norm form of real quadratic fields and their induced Weil representations. We prove that there exists an isomorphism between the space of vector-valued modular forms for the Weil representations that are invariant under the action of the automorphism group and the space of scalar-valued modular forms that satisfy some epsilon-condition, with which we translate Borcherds's theorem of obstructions to scalar-valued modular forms. In the end, we consider an example in the case of level 12.

math.NT↗
arXiv2013

Fulde-Ferrell-Larkin-Ovchinnikov Phases in Two-dimensional Spin-Orbit Coupled Degenerate Fermi Gases

The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase, a superconducting state with non-zero total momentum Cooper pairs in a strong magnetic field, was predicted more than 50 years ago and now becomes an important concept in many branches of physics. However, no unambiguous experimental evidences for the existence of FFLO phases have been observed yet. Recently, spin-imbalanced ultracold degenerate Fermi gases have emerged as a new powerful platform for the observation of FFLO phases due to their high experimental controllability and the lack of disorder. However, in three dimensional degenerate Fermi gases where the FFLO phases can be described within the simple mean field theory, the parameter region for the FFLO phases is too small to be observed in experiments. Recently, we showed that the Rashba type spin-orbit coupling and in-plane Zeeman field in three dimensional degenerate Fermi gases provide a more efficient way to create the FFLO phase because (1) the parameter region for the FFLO phase is greatly enlarged in the phase diagram and (2) the FFLO phase is stabilized due to the enhanced energy difference between FFLO phase and conventional Bardeen-Cooper-Schrieffer (BCS) phase. In this work we investigate the FFLO phase in two dimensional spin-orbit coupled degenerate Fermi gases using the mean field theory, with special concerns on the physical origin of the FFLO state. The basic properties of the FFLO phase are discussed and the phase diagram is obtained at zero temperature. The symmetry of different quantum phases is examined, which provide important basis for the experimental observation of FFLO phases using the time-of-flight imaging.

cond-mat.quant-gas↗
arXiv2013

Weakly holomorphic modular forms and rank two hyperbolic Kac-Moody algebras

In this paper, we compute basis elements of certain spaces of weight 0 weakly holomorphic modular forms and consider the integrality of Fourier coefficients of the modular forms. We use the results to construct automorphic correction of the rank 2 hyperbolic Kac-Moody algebras H(a), a=4,5,6, through Hilbert modular forms explicitly given by Borcherds lifts of the weakly holomorphic modular forms. We also compute asymptotic of the Fourier coefficients as they are related to root multiplicities of the rank 2 hyperbolic Kac-Moody algebras. This work is a continuation of the previous paper, where automorphic correction was constructed for H(a), a=3, 11, 66.

math.NT↗
arXiv2013

Weyl Group Multiple Dirichlet Series for Symmetrizable Kac-Moody Root Systems

Weyl group multiple Dirichlet series, introduced by Brubaker, Bump, Chinta, Friedberg and Hoffstein, are expected to be Whittaker coefficients of Eisenstein series on metaplectic groups. Chinta and Gunnells constructed these multiple Dirichlet series for all the finite root systems using the method of averaging a Weyl group action on the field of rational functions. In this paper, we generalize Chinta and Gunnells' work and construct Weyl group multiple Dirichlet series for the root systems associated with symmetrizable Kac-Moody algebras, and establish their functional equations and meromorphic continuation.

math.NT↗
arXiv2013

Rumor Evolution in Social Networks

Social network is a main tunnel of rumor spreading. Previous studies are concentrated on a static rumor spreading. The content of the rumor is invariable during the whole spreading process. Indeed, the rumor evolves constantly in its spreading process, which grows shorter, more concise, more easily grasped and told. In an early psychological experiment, researchers found about 70% of details in a rumor were lost in the first 6 mouth-to-mouth transmissions \cite{TPR}. Based on the facts, we investigate rumor spreading on social networks, where the content of the rumor is modified by the individuals with a certain probability. In the scenario, they have two choices, to forward or to modify. As a forwarder, an individual disseminates the rumor directly to its neighbors. As a modifier, conversely, an individual revises the rumor before spreading it out. When the rumor spreads on the social networks, for instance, scale-free networks and small-world networks, the majority of individuals actually are infected by the multi-revised version of the rumor, if the modifiers dominate the networks. Our observation indicates that the original rumor may lose its influence in the spreading process. Similarly, a true information may turn to be a rumor as well. Our result suggests the rumor evolution should not be a negligible question, which may provide a better understanding of the generation and destruction of a rumor.

physics.soc-ph↗
arXiv2011

Traffic Fluctuations on Weighted Networks

Traffic fluctuation has so far been studied on unweighted networks. However many real traffic systems are better represented as weighted networks, where nodes and links are assigned a weight value representing their physical properties such as capacity and delay. Here we introduce a general random diffusion (GRD) model to investigate the traffic fluctuation in weighted networks, where a random walk's choice of route is affected not only by the number of links a node has, but also by the weight of individual links. We obtain analytical solutions that characterise the relation between the average traffic and the fluctuation through nodes and links. Our analysis is supported by the results of numerical simulations. We observe that the value ranges of the average traffic and the fluctuation, through nodes or links, increase dramatically with the level of heterogeneity in link weight. This highlights the key role that link weight plays in traffic fluctuation and the necessity to study traffic fluctuation on weighted networks.

cond-mat.stat-mech↗
arXiv2011

Diffusion-annihilation proecesses in weighted scale-free networks with identical degree sequence

The studies based on $A+A \rightarrow \emptyset$ and $A+B\rightarrow \emptyset$ diffusion-annihilation processes have so far been studied on weighted uncorrelated scale-free networks and fractal scale-free networks. In the previous reports, it is widely accepted that the segregation of particles in the processes is introduced by the fractal structure. In this paper, we study these processes on a family of weighted scale-free networks with identical degree sequence. We find that the depletion zone and segregation are essentially caused by the disassortative mixing, namely, high-degree nodes tend to connect with low-degree nodes. Their influence on the processes is governed by the correlation between the weight and degree. Our finding suggests both the weight and degree distribution don't suffice to characterize the diffusion-annihilation processes on weighted scale-free networks.

cond-mat.dis-nn↗
arXiv2010

Deterministic weighted scale-free small-world networks

We propose a deterministic weighted scale-free small-world model for considering pseudofractal web with the coevolution of topology and weight. In the model, we have the degree distribution exponent $γ$ restricted to a range between 2 and 3, simultaneously tunable with two parameters. At the same time, we provide a relatively complete view of topological structure and weight dynamics characteristics of the networks: weight and strength distribution; degree correlations; average clustering coefficient and degree-cluster correlations; as well as the diameter. We show that our model is particularly effective at mimicing weighted scale-free small-world networks with a high and relatively stable clustering coefficient, which rapidly decline with the network size in most previous models.

cond-mat.stat-mech↗
arXiv2009

Analytic Solution to Clustering Coefficients on Weighted Networks

Clustering coefficient is an important topological feature of complex networks. It is, however, an open question to give out its analytic expression on weighted networks yet. Here we applied an extended mean-field approach to investigate clustering coefficients in the typical weighted networks proposed by Barrat, Barthélemy and Vespignani (BBV networks). We provide analytical solutions of this model and find that the local clustering in BBV networks depends on the node degree and strength. Our analysis is well in agreement with results of numerical simulations.

cond-mat.dis-nn↗
arXiv2009

The rigorous solution for the average distance of a Sierpinski network

The closed-form solution for the average distance of a deterministic network--Sierpinski network--is found. This important quantity is calculated exactly with the help of recursion relations, which are based on the self-similar network structure and enable one to derive the precise formula analytically. The obtained rigorous solution confirms our previous numerical result, which shows that the average distance grows logarithmically with the number of network nodes. The result is at variance with that derived from random networks.

cond-mat.stat-mech↗
arXiv2009

Influences of degree inhomogeneity on average path length and random walks in disassortative scale-free networks

Various real-life networks exhibit degree correlations and heterogeneous structure, with the latter being characterized by power-law degree distribution $P(k)\sim k^{-γ}$, where the degree exponent $γ$ describes the extent of heterogeneity. In this paper, we study analytically the average path length (APL) of and random walks (RWs) on a family of deterministic networks, recursive scale-free trees (RSFTs), with negative degree correlations and various $γ\in (2,1+\frac{\ln 3}{\ln 2}]$, with an aim to explore the impacts of structure heterogeneity on APL and RWs. We show that the degree exponent $γ$ has no effect on APL $d$ of RSFTs: In the full range of $γ$, $d$ behaves as a logarithmic scaling with the number of network nodes $N$ (i.e. $d \sim \ln N$), which is in sharp contrast to the well-known double logarithmic scaling ($d \sim \ln \ln N$) previously obtained for uncorrelated scale-free networks with $2 \leq γ<3$. In addition, we present that some scaling efficiency exponents of random walks are reliant on degree exponent $γ$.

physics.soc-ph↗