results for au:Xu_B in:math

- J. Nitsche proved that an isolated singularity of a conformal hyperbolic metric is either a conical singularity or a cusp one. We prove by developing map that there exists a complex coordinate $z$ centered at the singularity where the metric has the expression of either $\displaystyle{\frac{4\alpha^2\vert z \vert^{2\alpha-2}}{(1-\vert z \vert ^{2\alpha})^2}\vert \mathrm{d} z \vert^2}$ with $\alpha>0$ or $\displaystyle{\vert z \vert ^{-2}\big(\ln|z|\big)^{-2}|dz|^{2}}$.
- Oct 31 2017 math.OC arXiv:1710.10514v1Battery participants in performance-based frequency regulation markets must consider the cost of battery aging in their operating strategies to maximize market profits. In this paper we solve this problem by proposing an optimal control policy and an optimal bidding policy based on realistic market settings and an accurate battery aging model. The proposed control policy has a threshold structure and achieves near-optimal performance with respect to an offline controller that has complete future information. The proposed bidding policy considers the optimal control policy to maximize market profits while satisfying the market performance requirement through a chance-constraint. It factors the value of performance and supports a trade-off between higher profits and a lower risk of violating performance requirements. We demonstrate the optimality of both policies using simulations. A case study based on the PJM regulation market shows that our approach is effective at maximizing operating profits.
- We study the optimal control of battery energy storage under a general "pay-for-performance" setup such as providing frequency regulation and renewable integration. Batteries need to carefully balance the trade-off between following to the instruction signals and their degradation costs in real-time. Existing battery control strategies either do not consider the uncertainty of future signals, or cannot accurately account for battery cycle aging mechanism during operation. In this work, we take a different approach to the optimal battery control problem. Instead of attacking the complexity of battery degradation function or the lack of future information one at a time, we address these two challenges together in a joint fashion. In particular, we present an electrochemically accurate and trackable battery degradation model called the rainflow cycle-based model. We prove the degradation cost is convex. Then we propose an online control policy with a simple threshold structure and show it achieve near-optimal performance with respect to an offline controller that has complete future information. We explicitly characterize the optimality gap and show it is independent to length of the time of operations. Simulation results with both synthetic and real regulation traces are conducted to illustrate the theoretical results.
- We characterize a conformal hyperbolic metric with finitely many singularities on a compact Riemann surface by some bounded projective function on the surface compatible with the singularities.
- Aug 23 2017 math.CV arXiv:1708.06535v2Cone spherical metrics are conformal metrics with constant curvature one and finitely many conical singularities on compact Riemann surfaces. By using Strebel differentials as a bridge, we construct a new class of cone spherical metrics on compact Riemann surfaces by drawing on the surfaces some class of connected metric ribbon graphs.
- Jul 17 2017 math.OC arXiv:1707.04567v2When participating in electricity markets, owners of battery energy storage systems must bid in such a way that their revenues will at least cover their true cost of operation. Since cycle aging of battery cells represents a substantial part of this operating cost, the cost of battery degradation must be factored in these bids. However, existing models of battery degradation either do not fit market clearing software or do not reflect the actual battery aging mechanism. In this paper we model battery cycle aging using a piecewise linear cost function, an approach that provides a close approximation of the cycle aging mechanism of electrochemical batteries and can be incorporated easily into existing market dispatch programs. By defining the marginal aging cost of each battery cycle, we can assess the actual operating profitability of batteries. A case study demonstrates the effectiveness of the proposed model in maximizing the operating profit of a battery energy storage system taking part in the ISO New England energy and reserve markets.
- This paper begins the project of adapting the 1992 book by Adams, Barbasch and Vogan on the Langlands classification of admissible representations of real groups, to $p$-adic groups, continuing in the direction charted by Vogan in his 1993 paper on the Langlands correspondence. This paper presents three theorems in that direction. The first theorem shows how Lusztig's work on perverse sheaves arising from graded Lie algebras may be brought to bear on the problem; the second theorem proves that Arthur parameters determine strongly regular conormal vectors to a parameter space of certain Langlands parameters; the third theorem shows how to replace the microlocalisation functor as it appears in the work of Adams, Barbasch and Vogan with a functor built from Deligne's vanishing cycles functor. The paper concludes with three conjectures, the first of which is the prediction that Arthur packets are Adams-Barbasch-Vogan packets for $p$-adic groups. This paper is the first in a series.
- The objective of this note is to provide an interpretation of the discrete version of Morse inequalities, following Witten's approach via supersymmetric quantum mechanics, adapted to finite graphs, as a particular instance of Morse-Witten theory for cell complexes. We describe the general framework of graph quantum mechanics and we produce discrete versions of the Hodge theorems and energy cut-offs within this formulation.
- Apr 26 2017 math.AG arXiv:1704.07583v1For $n\ge5$, it is well known that the moduli space $\mathfrak{M_{0,\:n}}$ of unordered $n$ points on the Riemann sphere is a quotient space of the Zariski open set $K_n$ of $\mathbb C^{n-3}$ by an $S_n$ action. The stabilizers of this $S_n$ action at certain points of this Zariski open set $K_n$ correspond to the groups fixing the sets of $n$ points on the Riemann sphere. Let $\alpha$ be a subset of $n$ distinct points on the Riemann sphere. We call the group of all linear fractional transformations leaving $\alpha$ invariant the stabilizer of $\alpha$, which is finite by observation. For each non-trivial finite subgroup $G$ of the group ${\rm PSL}(2,{\Bbb C})$ of linear fractional transformations, we give the necessary and sufficient condition for finite subsets of the Riemann sphere under which the stabilizers of them are conjugate to $G$. We also prove that there does exist some finite subset of the Riemann sphere whose stabilizer coincides with $G$. Next we obtain the irreducible decompositions of the representations of the stabilizers on the tangent spaces at the singularities of $\mathfrak{M_{0,\:n}}$. At last, on $\mathfrak{M_{0,\:5}}$ and $\mathfrak{M_{0,\:6}}$, we work out explicitly the singularities and the representations of their stabilizers on the tangent spaces at them.
- Mar 24 2017 math.OC arXiv:1703.07824v1When providing frequency regulation in a pay-for-performance market, batteries need to carefully balance the trade-off between following regulation signals and their degradation costs in real-time. Existing battery control strategies either do not consider mismatch penalties in pay-for-performance markets, or cannot accurately account for battery cycle aging mechanism during operation. This paper derives an online control policy that minimizes a battery owner's operating cost for providing frequency regulation in a pay-for-performance market. The proposed policy considers an accurate electrochemical battery cycle aging model, and is applicable to most types of battery cells. It has a threshold structure, and achieves near-optimal performance with respect to an offline controller that has complete future information. We explicitly characterize this gap and show it is independent of the duration of operation. Simulation results with both synthetic and real regulation traces are conducted to illustrate the theoretical results.
- A vital aspect in energy storage planning and operation is to accurately model its operational cost, which mainly comes from the battery cell degradation. Battery degradation can be viewed as a complex material fatigue process that based on stress cycles. Rainflow algorithm is a popular way for cycle identification in material fatigue process, and has been extensively used in battery degradation assessment. However, the rainflow algorithm does not have a closed form, which makes the major difficulty to include it in optimization. In this paper, we prove the rainflow cycle-based cost is convex. Convexity enables the proposed degradation model to be incorporated in different battery optimization problems and guarantees the solution quality. We provide a subgradient algorithm to solve the problem. A case study on PJM regulation market demonstrates the effectiveness of the proposed degradation model in maximizing the battery operating profits as well as extending its lifetime.
- We consider using a battery storage system simultaneously for peak shaving and frequency regulation through a joint optimization framework which captures battery degradation, operational constraints and uncertainties in customer load and regulation signals. Under this framework, using real data we show the electricity bill of users can be reduced by up to 15\%. Furthermore, we demonstrate that the saving from joint optimization is often larger than the sum of the optimal savings when the battery is used for the two individual applications. A simple threshold real-time algorithm is proposed and achieves this super-linear gain. Compared to prior works that focused on using battery storage systems for single applications, our results suggest that batteries can achieve much larger economic benefits than previously thought if they jointly provide multiple services.
- Feb 17 2017 math.OC arXiv:1702.04816v1Reduced installation and operating costs give energy storage systems an opportunity to participate actively and profitably in electricity markets. In addition to providing ancillary services, energy storage systems can also arbitrage temporal price differences. Congestion in the transmission network often accentuates these price differences and will under certain circumstances enhance the profitability of arbitrage. On the other hand, congestion may also limit the ability of a given storage device to take advantage of arbitrage opportunities. This paper analyzes how transmission congestion affects the profitability of arbitrage by storage devices in markets with perfect and imperfect competition. Imperfect competition is modeled using a bilevel optimization where the offers and bids submitted by the storage devices can alter the market outcome. Price-taker and price-maker assumptions are also investigated through market price duration curves. This analysis is based on simulating an entire year of market operation on the IEEE Reliability Test system.
- Dynamic line rating (DLR) models the transmission capacity of overhead lines as a function of ambient conditions. It takes advantage of the physical thermal property of overhead line conductors, thus making DLR less conservative compared to the traditional worst-case oriented nominal line rating (NLR). Employing DLR brings potential benefits for grid integration of variable Renewable Energy Sources (RES), such as wind and solar energy. In this paper, we reproduce weather conditions from renewable feed-ins and local temperature records, and calculate DLR in accordance with the RES feed-in and load demand data step. Simulations with high time resolution, using a predictive dispatch optimization and the Power Node modeling framework, of a six-node benchmark power system loosely based on the German power system are performed for the current situation, using actual wind and PV feed-in data. The integration capability of DLR under high RES production shares is inspected through simulations with scaled-up RES profiles and reduced dispatchable generation capacity. The simulation result demonstrates a comparison between DLR and NLR in terms of reductions in RES generation curtailments and load shedding, while discussions on the practicality of adopting DLR in the current power system is given in the end.
- Nov 01 2016 math.OC arXiv:1610.09413v2Energy storage can facilitate the integration of renewable energy resources by providing arbitrage and ancillary services. Jointly optimizing energy and ancillary services in a centralized electricity market reduces the system's operating cost and enhances the profitability of energy storage systems. However, achieving these objectives requires that storage be located and sized properly. We use a bi-level formulation to optimize the location and size of energy storage systems which perform energy arbitrage and provide regulation services. Our model also ensures the profitability of investments in energy storage by enforcing a rate of return constraint. Computational tractability is achieved through the implementation of a primal decomposition and a subgradient-based cutting-plane method. We test the proposed approach on a 240-bus model of the Western Electricity Coordinating Council (WECC) system and analyze the effects of different storage technologies, rate of return requirements, and regulation market policies on ES participation on the optimal storage investment decisions. We also demonstrate that the proposed approach outperforms exact methods in terms of solution quality and computational performance.
- We investigate genuine multipartite nonlocality of pure permutationally invariant multimode Gaussian states of continuous variable systems, as detected by the violation of Svetlichny inequality. We identify the phase space settings leading to the largest violation of the inequality when using displaced parity measurements, distinguishing our results between the cases of even and odd total number of modes. We further consider pseudospin measurements and show that, for three-mode states with asymptotically large squeezing degree, particular settings of these measurements allow one to approach the maximum violation of Svetlichny inequality allowed by quantum mechanics. This indicates that the strongest manifestation of genuine multipartite quantum nonlocality is in principle verifiable on Gaussian states.
- Aug 31 2016 math.RA arXiv:1608.08463v2We classify the RBA-bases of $6$-dimensional noncommutative semisimple algebras for which the algebra has a positive degree map. We show that these RBAs are parametrized by seven real numbers, the first four of which are positive and the remaining three arbitrary. Our classification gives formulas for their standard bases and structure constants. Using these we give a list of all noncommutative integral table algebras of rank 6 with order up to 150. Four in the list are primitive, but we show these cannot be realized as adjacency algebras of association schemes. In the last section of the paper we apply our methods to give a precise description of the noncommutative integral table algebras of rank 6 for which the multiplicity of both linear characters is 1.
- We prove that the pressure metric on the Teichmüller space of a bordered surface is incomplete and its partial completion can be given by the moduli space of metric graphs for a fat graph associated to the same bordered surface equipped with pressure metric. As a corollary, we show that the pressure metric is not a constant multiple of the Weil-Petersson metric which is different from the closed surface case.
- A celebrated and deep theorem in the theory of Riemann surfaces states the existence and uniqueness of the Jenkins-Strebel differentials on a Riemann surface under some conditions, but the proof is non-constructive and examples are difficult to find. This paper deals with an example of a simple case, namely Jenkins-Strebel differentials on the Riemann sphere with four fixed simple poles. We will give explicit expressions of these Jenkins-Strebel differentials by means of the Weierstrass $\wp$ function and expose a simple algorithm determining the correspondence between these differentials and some classes of simple closed curves on the Riemann sphere with four points removed.
- Energy storage in data centers has mainly been used as devices to backup generators during power outages. Recently, there has been a growing interest in using energy storage devices to actively shape power consumption in data centers to reduce their skyrocketing electricity bills. In this paper, we consider using energy storage in data centers for two applications in a joint fashion: reducing peak demand charges and enabling data centers to participate in regulation markets. We develop an optimization framework that captures the cost of electricity, degradation of energy storage devices, as well as the benefit from regulation markets. Under this frame- work, using real data Microsoft data center traces and PJM regulation signals, we show the electricity bill of a data center can be reduced by up to 20%. Furthermore, we demonstrate that the saving from joint optimization can be even larger than the sum of individually optimizing each component. We quantify the particular aspects of data center load profiles that lead to this superlinear gain. Compared to prior works that consider using energy storage devices for each single application alone, our results suggest that energy storage in data centers can have much larger impacts than previously thought possible.
- We develop a general procedure to study the combinatorial structure of Arthur packets for $p$-adic quasisplit $Sp(N)$ and $O(N)$ following the works of M\oeglin. This allows us to answer many delicate questions concerning the Arthur packets of these groups, for example the size of the packets.
- Mar 24 2016 math.AP arXiv:1603.07175v1We consider the problem $$ \epsilon^2 ∆u-V(y)u+u^p\,=\u20090,~~u>0~~\quad\mboxin\quad\Omega,~~\quad\frac ∂u∂\nu\,=\u20090\quad\mboxon~~~∂\Omega, $$ where $\Omega$ is a bounded domain in $\mathbb R^2$ with smooth boundary, the exponent $p>1$, $\epsilon>0$ is a small parameter, $V$ is a uniformly positive, smooth potential on $\bar{\Omega}$, and $\nu$ denotes the outward normal of $\partial \Omega$. Let $\Gamma$ be a curve intersecting orthogonally with $\partial \Omega$ at exactly two points and dividing $\Omega$ into two parts. Moreover, $\Gamma$ satisfies stationary and non-degeneracy conditions with respect to the functional $\int_{\Gamma}V^{\sigma}$, where $\sigma=\frac {p+1}{p-1}-\frac 12$. We prove the existence of a solution $u_\epsilon$ concentrating along the whole of $\Gamma$, exponentially small in $\epsilon$ at any positive distance from it, provided that $\epsilon$ is small and away from certain critical numbers. In particular, this establishes the validity of the two dimensional case of a conjecture by A. Ambrosetti, A. Malchiodi and W.-M. Ni(p.327, [4]).
- Let $G$, $H$ be finite groups and let $X$ be a finite $G$-set. $G$-perfect nonlinear functions from $X$ to $H$ have been studied in several papers. They have more interesting properties than perfect nonlinear functions from $G$ itself to $H$. By introducing the concept of a $(G, H)$-related difference family of $X$, we obtain a characterization of $G$-perfect nonlinear functions on $X$. When $G$ is abelian, we characterize a $G$-difference set of $X$ by the Fourier transform on a normalized $G$-dual set $\widehat X$. We will also investigate the existence and constructions of $G$-perfect nonlinear functions and $G$-bent functions. Several known results in [2,6,10,17] are direct consequences of our results.
- Feb 16 2016 math.OC arXiv:1602.04420v2Because energy storage systems have better ramping characteristics than traditional generators, their participation in frequency regulation should facilitate the balancing of load and generation. However, they cannot sustain their output indefinitely. System operators have therefore implemented new frequency regulation policies to take advantage of the fast ramps that energy storage systems can deliver while alleviating the problems associated with their limited energy capacity. This paper contrasts several U.S. policies that directly affect the participation of energy storage systems in frequency regulation and compares the revenues that the owners of such systems might achieve under each policy.
- Dec 17 2015 math.CO arXiv:1512.04995v1We consider the thickness $\theta (G))$ and outerthickness $\theta _o(G)$ of a graph G in terms of its orientable and nonorientable genus. Dean and Hutchinson provided upper bounds for thickness of graphs in terms of their orientable genus. More recently, Concalves proved that the outerthickness of any planar graph is at most 2. In this paper, we apply the method of deleting spanning disks of embeddings to approximate the thickness and outerthickness of graphs. We first obtain better upper bounds for thickness. We then use a similar approach to provide upper bounds for outerthickness of graphs in terms of their orientable and nonorientable genera. Finally we show that the outerthickness of the torus (the maximum outerthickness of all toroidal graphs) is 3. We also show that all graphs embeddable in the double torus have thickness at most 3 and outerthickness at most 5.
- Let $\Lambda$ be a collection of partitions of a positive integer $d$ of the form $$(a_1,⋯, a_p),\,(b_1,⋯, b_q),\,(m_1+1,1,⋯,1),⋯, (m_l+1,1,⋯,1),$$ where $(m_1,\cdots, m_l)$ is a partition of $p+q-2>0$. We prove that there exists a rational function on the Riemann sphere $\overline{\mathbb{C}}$ with branch data $\Lambda$ if and only if $$\max\bigl(m_1,⋯,m_l\bigr) < \fracd\rm GCD(a_1,⋯, a_p,b_1,⋯, b_q).$$ As an application, we give a new class of branch data which can be realized by Belyi functions on the Riemann sphere.
- Oct 16 2015 math.AP arXiv:1510.04355v1We give negative answers to Lin-Ni's conjecture for any four and six dimensional domains. No condition on the symmetry, geometry nor topology of the domain is needed.
- Jul 30 2015 math.RT arXiv:1507.08024v3We give a survey on M\oeglin's construction of representations in the Arthur packets for $p$-adic quasisplit symplectic and orthogonal groups. The emphasis is on comparing M\oeglin's parametrization of elements in the Arthur packets with that of Arthur.
- Jun 19 2015 math.RA arXiv:1506.05476v2Given a finite-dimensional noncommutative semisimple algebra $A$ with involution, we show that $A$ always has an RBA-basis. We look for an RBA-basis that has integral or rational structure constants, and ask if the RBA admits a positive degree map. For RBAs that have a positive degree map, we try to find an RBA-basis with nonnegative structure constants to determine if there is a generalized table algebra structure. We settle these questions for the algebras $\mathbb{C} \oplus M_n(\mathbb{C})$, $n \ge 2$.
- May 01 2015 math.RT arXiv:1504.08364v3Zelevinsky's classification theory of discrete series of $p$-adic general linear groups has been well known. M\oeglin and Tadic gave the same kind of theory for $p$-adic classical groups, which is more complicated due to the occurrence of nontrivial structure of L-packets. Nonetheless, their work is independent of the endoscopic classification theory of Arthur (also Mok in the unitary case), which concerns the structure of L-packets in these cases. So our goal in this paper is to make more explicit the connection between these two very different types of theories. To do so, we reprove the results of M\oeglin and Tadic in the case of quasisplit symplectic groups and orthogonal groups by using Arthur's theory.
- Mar 18 2015 math.RT arXiv:1503.04897v4In his monograph (2013) Arthur characterizes the L-packets of quasisplit symplectic groups and orthogonal groups. By extending his work, we characterize the L-packets for the corresponding similitude groups with desired properties. In particular, we show these packets satisfy the conjectural endoscopic character identities.
- The existence of the well-known Jacquet-Langlands correspondence was established by Jacquet and Langlands via the trace formula method in 1970. An explicit construction of such a correspondence was obtained by Shimizu via theta series in 1972. In this paper, we extend the automorphic descent method of Ginzburg-Rallis-Soudry to a new setting. As a consequence, we recover the classical Jacquet-Langlands correspondence for PGL(2) via a new explicit construction.
- Jan 06 2015 math.RT arXiv:1501.00763v6Let $G \subseteq \tilde{G}$ be two quasisplit connected reductive groups over a local field of characteristic zero and $G_{der} = \tilde{G}_{der}$. Although the existence of L-packets is still conjectural in general, it is believed that the L-packets of $G$ should be the restriction of that of $\tilde{G}$. Motivated by this, we hope to construct the L-packets of $\tilde{G}$ from that of $G$. The primary example in our mind is when $G = Sp(2n)$, whose L-packets have been determined by Arthur (2013), and $\tilde{G} = GSp(2n)$. As a first step, we need to consider some well-known conjectural properties of L-packets. In this paper, we show how they can be deduced from the conjectural endoscopy theory. As an application, we obtain some structural information about L-packets of $\tilde{G}$ from that of $G$.
- The central extension of mapping class groups of punctured surfaces of finite type that arises in Chekhov-Fock quantization is 12 times of the Meyer class plus the Euler classes of the punctures, which agree with the one arising in the Kashaev quantization.
- Let $G$ be a finite abelian group acting faithfully on a finite set $X$. As a natural generalization of the perfect nonlinearity of Boolean functions, the $G$-bentness and $G$-perfect nonlinearity of functions on $X$ are studied by Poinsot et al. [6,7] via Fourier transforms of functions on $G$. In this paper we introduce the so-called $G$-dual set $\widehat X$ of $X$, which plays the role similar to the dual group $\widehat G$ of $G$, and the Fourier transforms of functions on $X$, a generalization of the Fourier transforms of functions on finite abelian groups. Then we characterize the bent functions on $X$ in terms of their own Fourier transforms on $\widehat X$. Bent (perfect nonlinear) functions on finite abelian groups and $G$-bent ($G$-perfect nonlinear) functions on $X$ are treated in a uniform way in this paper, and many known results in [4,2,6,7] are obtained as direct consequences. Furthermore, we will prove that the bentness of a function on $X$ can be determined by its distance from the set of $G$-linear functions. In order to explain the main results clearly, examples are also presented.
- Let $e_\l(x)$ be a Neumann eigenfunction with respect to the positive Laplacian $\Delta$ on a compact Riemannian manifold $M$ with boundary such that $\Delta\, e_\l=\l^2 e_\l$ in the interior of $M$ and the normal derivative of $e_\l$ vanishes on the boundary of $M$. Let $\chi_\lambda$ be the unit band spectral projection operator associated with the Neumann Laplacian and $f$ a square integrable function on $M$. We show the following gradient estimate for $\chi_\lambda\,f$ as $\lambda\geq 1$: $\|\nabla\ \chi_\l\ f\|_\infty\leq C\l \|\chi_\l\f\|_\infty+\l^{-1}\|\Delta\ \chi_\l\ f\|_\infty$, where $C$ is a positive constant depending only on $M$. As a corollary, we obtain the gradient estimate of $e_\l$: for every $\l\geq 1$, there holds $\|\nabla e_\l\|_\infty\leq C\,\l\, \|e_\l\|_\infty$.
- A conformal metric $g$ with constant curvature one and finite conical singularities on a compact Riemann surface $\Sigma$ can be thought of as the pullback of the standard metric on the 2-sphere by a multi-valued locally univalent meromorphic function $f$ on $\Sigma\backslash \{{\rm singularities}\}$, called the \it developing map of the metric $g$. When the developing map $f$ of such a metric $g$ on the compact Riemann surface $\Sigma$ has reducible monodromy, we show that, up to some Möbius transformation on $f$, the logarithmic differential $d\,(\log\, f)$ of $f$ turns out to be an abelian differential of 3rd kind on $\Sigma$, which satisfies some properties and is called a \it character 1-form of $g$. Conversely, given such an abelian differential $\omega$ of 3rd kind satisfying the above properties, we prove that there exists a unique conformal metric $g$ on $\Sigma$ with constant curvature one and conical singularities such that one of its character 1-forms coincides with $\omega$. This provides new examples of conformal metrics on compact Riemann surfaces of constant curvature one and with singularities. Moreover, we prove that the developing map is a rational function for a conformal metric $g$ with constant curvature one and finite conical singularities with angles in $2\pi\,{\Bbb Z}_{>1}$ on the two-sphere.
- In a laboratory experiment, round by round, individual interactions should lead to the social evolutionary rotation in population strategy state space. Successive switching the incentive parameter should lead to successive change of the rotation ---- both of its direction and its strength. In data from a switching payoff matrix experiment of extended 2x2 games (Binmore, Swierzbinski and Proulx, 2001 [1]), we find the changing of the social evolutionary rotation can be distinguished quantitatively. The evolutionary rotation can be captured by evolutionary dynamics. With eigenvalue from the Jacobian of a constrained replicator dynamics model, an interpretation for observed rotation strength is given. In addition, equality-of-populations rank test shows that relative response coefficient of a group could persist cross the switching parameter games. The data has successively been used to support Von Neumann's minimax theory. Using the old data, with observed evolutionary rotation, this report provides a new insight into evolutionary game theory and experimental social dynamics.
- Aug 14 2010 math.CO arXiv:1008.2228v1We give a full description of the algebraic structures of the Bose-Mesner algebra and Terwilliger algebra of the wreath product of one-class association schemes.
- Let $e_\l(x)$ be an eigenfunction with respect to the Dirichlet Laplacian $\Delta_N$ on a compact Riemannian manifold $N$ with boundary: $\Delta_N e_\l=\l^2 e_\l$ in the interior of $N$ and $e_\l=0$ on the boundary of $N$. We show the following gradient estimate of $e_\l$: for every $\l\geq 1$, there holds $\l\|e_\l\|_\infty/C\leq \|\nabla e_\l\|_\infty\leq C{\l}\|e_\l\|_\infty$, where $C$ is a positive constant depending only on $N$. In the proof, we use a basic geometrical property of nodal sets of eigenfunctions and elliptic apriori estimates.
- Let $e_\l(x)$ be an eigenfunction with respect to the Laplace-Beltrami operator $\Delta_M$ on a compact Riemannian manifold $M$ without boundary: $\Delta_M e_\l=\l^2 e_\l$. We show the following gradient estimate of $e_\l$: for every $\l\geq 1$, there holds $\l\|e_\l\|_\infty/C\leq \|\nabla e_\l\|_\infty\leq C{\l}\|e_\l\|_\infty$, where $C$ is a positive constant depending only on $M$.
- Oct 09 2008 math.CO arXiv:0810.1437v1Borodin et al figured out a gap of the paper published at J. Combinatorial Theory Ser. B (Vol.96 (2006) 958--963), and gave a new proof with the similar technique. The purpose of this note is to fix the gap by slightly revising the definition of special faces, and adding a few lines of explanation in the proofs (new added text are all in black font).
- Jan 08 2008 math.DG arXiv:0801.0948v3Sixty years ago, S. B. Myers and N. E. Steenrod (\it Ann. of Math. \bf 40 (1939), 400-416) showed that the isometry group of a Riemannian manifold without boundary has a structure of Lie group. Recently A. V. Bagaev and N. I. Zhukova (\it Siberian Math. J. \bf 48 (2007), 579-592) proved the same result for a Riemannian orbifold. In this paper, we firstly show that the isometry group of a Riemannian manifold $M$ with boundary has dimension at most ${1/2} \dim M (\dim M-1)$. Then we completely classify such Riemannian manifolds with boundary that their isometry groups attain the preceding maximal dimension.
- Sep 28 2006 math.CO arXiv:math/0609757v1An $(L,d)^*$-coloring is a mapping $\phi$ that assigns a color $\phi(v)\in L(v)$ to each vertex $v\in V(G)$ such that at most $d$ neighbors of $v$ receive colore $\phi(v)$. A graph is called $(m,d)^*$-choosable, if $G$ admits an $(L,d)^*$-coloring for every list assignment $L$ with $|L(v)|\geq m$ for all $v\in V(G)$. In this note, it is proved that every toroidal graph, which contains no adjacent triangles and contains no 6-cycles and $l$-cycles for some $l \in \{5,7\}$, is $(3,1)^*$-choosable.
- Let $e(x,y,\l)$ be the spectral function and ${\chi}_\l$ the unit band spectral projection operator, with respect to the Laplace-Beltrami operator $\D_M$ on a closed Riemannian manifold $M$. We firstly review the one-term asymptotic formula of $e(x,x,\l)$ as $\l\to\infty$ by Hörmander (1968) and the one of $\p^\al_x\p^\bt_y e(x,y,\l)|_{x=y}$ as $\l\to\infty$ in a geodesic normal coordinate chart by the author (2004) and the sharp asymptotic estimates from above of the mapping norm $\|\chi_\l\|_{L_2\to L_p}$ ($2\leq p\leq\infty$) by Sogge (1988 $&$ 1989) and of the mapping norm $\|\chi_\l\|_{L_2\to {\rm Sobolev} L_p}$ by the author (2004). In the paper we show the one term asymptotic formula for $e(x,y,\l)$ as $\l\to\infty$, provided that the Riemannian distance between $x$ and $y$ is ${\rm O}(1/\l)$. As a consequence, we obtain the sharp estimate of the mapping norm $\|\chi_\l\|_{L_2\to C^\d}$ ($0<\d<1$), where $C^\d(M)$ is the space of Hölder continuous functions with exponent $\d$ on $M$. Moreover, we show a geometric property of the eigenfunction $e_\l$: $\D_M e_\l+\l^2 e_\l=0$, which says that $1/\l$ is comparable to the distance between the nodal set of $e_\l$ (where $e_\l$ vanishes) and the concentrating set of $e_\l$ (where $e_\l$ attains its maximum or minimum) as $\l\to\infty$.
- In this paper, we give the sharp estimates for the degree of symmetry and the semi-simple degree of symmetry of certain four dimensional fiber bundles by virtue of the rigidity theorem of harmonic maps due to Schoen and Yau. As a corollary of this estimate, we compute the degree of symmetry and the semi-simple degree of symmetry of ${\Bbb C}P^2\times V$, where $V$ is closed smooth manifold admitting a real analytic Riemannian metric of non-positive curvature. In addition, by the Albanese map, we obtain the sharp estimate of the degree of symmetry of a compact smooth manifold with some restrictions on its one dimensional cohomology.
- A. Borel proved that, if a finite group $F$ acts effectively and continuously on a closed aspherical manifold $M$ with centerless fundamental group $\pi_1(M)$, then a natural homomorphism $\psi$ from $F$ to the outer automorphism group ${\rm Out} \pi_1(M)$ of $\pi_1(M)$, called the associated abstract kernel, is a monomorphism. In this paper, we investigate to what extent Borel's theorem holds for a compact Lie group $G$ acting effectively and smoothly on a particular orientable aspherical manifold $N$ admitting a Riemannian metric $g_0$ of non-positive curvature in case that $\pi_1(N)$ has a non-trivial center. It turns out that if $G$ attains the maximal dimension equal to the rank of Center $\pi_1(N)$ and the metric $g_0$ is real analytic, then any element of $G$ defining a diffemorphism homotopic to the identity of $N$ must be contained in the identity component $G^0$ of $G$. Moreover, if the inner automorphism group of $\pi_1(N)$ is torsion free, then the associated abstract kernel $\psi: G/G^0\to {\rm Out} \pi_1(N)$ is a monomorphism. The same result holds for the non-orientable $N$'s under certain techical assumptions. Our result is an application of a theorem by Schoen-Yau (Topology, \bf 18 (1979), 361-380) on harmonic mappings.