Browsing by Author "Kang, Ming-Hsuan"

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Artin L-Functions on Finite Quotients of PGL(3)
(2015-01-01)
Kang, Ming-Hsuan
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Li, Wen-Ching Winnie
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應用數學系
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Department of Applied Mathematics
Let XG be a finite 2D complex arising from PGL3. In this paper, we study Artin Lfunctions attached to finite- dimensional representations. of the fundamental group of XG. Some key properties, including rationality, functional equation, and invariance under induction, of these functions are proved. Moreover, using a cohomological argument, we establish a connection between the Artin L- functions and the L- function of., generalizing the identity on zeta functions of XG obtained in [ 9, 10].
Building lattices and zeta functions
(2020-04-01)
Deitmar, Anton
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Kang, Ming-Hsuan
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McCallum, Rupert
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應用數學系
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Department of Applied Mathematics
We give a Lefschetz formula for tree lattices and apply it to geometric zeta functions. We further generalize Bass's approach to Ihara zeta functions to the higher-dimensional case of a building.
ON SPHERICAL MONTE CARLO SIMULATIONS FOR MULTIVARIATE NORMAL PROBABILITIES
(2015-09-01)
Teng, Huei-Wen
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Kang, Ming-Hsuan
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Fuh, Cheng-Der
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交大名義發表
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National Chiao Tung University
The calculation of multivariate normal probabilities is of great importance in many statistical and economic applications. In this paper we propose a spherical Monte Carlo method with both theoretical analysis and numerical simulation. We start by writing the multivariate normal probability via an inner radial integral and an outer spherical integral using the spherical transformation. For the outer spherical integral, we apply an integration rule by randomly rotating a predetermined set of well-located points. To find the desired set, we derive an upper bound for the variance of the Monte Carlo estimator and propose a set which is related to the kissing number problem in sphere packings. For the inner radial integral, we employ the idea of antithetic variates and identify certain conditions so that variance reduction is guaranteed. Extensive Monte Carlo simulations on some probabilities confirm these claims.
Riemann Hypothesis and strongly Ramanujan complexes from GL(n)
(2016-04-01)
Kang, Ming-Hsuan
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應用數學系
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Department of Applied Mathematics
We investigate the Riemann Hypothesis on combinatorial zeta functions associated to finite quotients of the affine building of GL(n). We prove that if the quotient complex is strongly Ramanujan then these zeta functions satisfy the Riemann Hypothesis. On the other hand, we show that the converse statement is also true provided the extra generic condition. In the end, we give an example to show that this generic condition is indeed necessary. (C) 2015 Elsevier Inc. All rights reserved.
Toroidal fullerenes with the Cayley graph structures
(2011-11-06)
Kang, Ming-Hsuan
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應用數學系
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Department of Applied Mathematics
We classify all possible structures of fullerene Cayley graphs. We give each one a geometric model and compute the spectra of its finite quotients. Moreover, we give a quick and simple estimation for a given toroidal fullerene. Finally, we provide a realization of those families in three-dimensional space. (C) 2011 Elsevier B.V. All rights reserved.
Tree-Lattice Zeta Functions and Class Numbers
(2018-01-01)
Deitmar, Anton
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Kang, Ming-Hsuan
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應用數學系
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Department of Applied Mathematics
We extend the theory of Ihara zeta functions to noncompact arithmetic quotients of Bruhat-Tits trees. This new zeta function turns out to be a rational function despite the infinite-dimensional setting. In general, it has zeros and poles in contrast to the compact case. The determinant formulas of Bass and Ihara hold if we define the determinant as the limit of all finite principal minors. From this analysis we derive a prime geodesic theorem, which, applied to special arithmetic groups, yields new asymptotic assertions on class numbers of orders in global fields.
Twisted Poincare series and zeta functions on finite quotients of buildings
(2019-05-01)
Kang, Ming-Hsuan
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McCallum, Rupert
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應用數學系
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Department of Applied Mathematics
In the case where G = SL2(F) for a non-archimedean local field F and Gamma is a discrete torsion-free cocompact subgroup of G, there is a known relationship between the Ihara zeta function for the quotient of the Bruhat-Tits tree of G by the action of Gamma, and an alternating product of determinants of twisted Poincare series for parabolic subgroups of the affine Weyl group of G. We show how this can be generalized to other split simple algebraic groups of rank two over F and formulate a conjecture about how this might be generalized to groups of higher rank.
Zeta and L-functions of finite quotients of apartments and buildings
(2018-10-01)
Kang, Ming-Hsuan
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Li, Wen-Ching Winnie
;
Wang, Chian-Jen
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應用數學系
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Department of Applied Mathematics
In this paper, we study relations between Langlands L-functions and zeta functions of geodesic walks and galleries for finite quotients of the apartments of G =PGL(3) and PGSp(4) over a nonarchimedean local field with q elements in its residue field. They give rise to an identity (Theorem 5.3) which can be regarded as a generalization of Ihara's theorem for finite quotients of the Bruhat-Tits trees. This identity is shown to agree with the q = 1 version of the analogous identities for finite quotients of the building of G established in [KL14, KLW10, FLW13], verifying the philosophy of the field with one element by Tits. A new identity for finite quotients of the building of PGSp(4) involving the standard L-function (Theorem 6.3), complementing the one in [FLW13] which involves the spin L-function, is also obtained.
Zeta functions of complexes arising from PGL(3)
(2014-05-01)
Kang, Ming-Hsuan
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Li, Wen-Ching Winnie
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應用數學系
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Department of Applied Mathematics
In this paper we obtain a closed form expression of the zeta function Z(Xr,u) of a finite quotient Xr of the Bruhat-Tits building of PGL(3) over a nonarchimedean local field F by a discrete cocompact torsion-free subgroup Gamma of PGL3. Analogous to a graph zeta function, Z(Xr, u) is a rational function with two different expressions and it satisfies the Riemann hypothesis if and only if Xr is a Ramanujan complex. (C) 2014 Elsevier Inc. All rights reserved.
Zeta functions of F-1-buildings
(2016-04)
Deitmar, Anton
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Kang, Ming-Hsuan
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應用數學系
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Department of Applied Mathematics
The analogue of the Bruhat-Tits building of a p-adic group in F1-geometry is a single apartment. In this setting, the trace formula gives rise to a several variable zeta function analogously to the p-adic case. The analogy carries on to the fact that the restriction to certain lines yield zeta functions which are defined in geometrical terms. Also, the classical formula of Ihara has an analogue in this setting.