Automorphism Group of the Rank-decreasing Graph Over the Semigroup of Upper Triangular Matrices

Let F_q be a finite field with q elements, n ≥ 2 a positive integer, and T(n, q) the semigroup of all n × n upper triangular matrices over F_q. The rank-decreasing graph 𝕋 of T(n, q) is a directed graph which has T(n, q) as vertex set, and there is a directed edge from A ∈ T(n, q) to B ∈ T(n, q) if and only if r(AB) < r(B). The zero-divisor graph 𝒯 of T(n, q), with vertex set of all nonzero zero-divisors of T(n, q) and there is a directed edge from a vertex A to a vertex B if and only if AB = 0, can be viewed as a subgraph of 𝕋. In [16 Wang, L. (2015). A note on automorphisms of the zero-divisor graph of upper triangular matrices. Lin. Alg. Appl. 465:214–220.[Crossref], [Web of Science ®] , [Google Scholar]], L. Wang has determined the automorphisms of the zero-divisor graph 𝒯 of T(n, q). In this article, by applying the main result of [17 Wong, D., Ma, X., Zhou, J. (2014). The group of automorphisms of a zero-divisor graph based on rank one upper triangular matrices. Lin. Alg. Appl. 460:242–258.[Crossref], [Web of Science ®] , [Google Scholar]] we determine the automorphisms of the rank-decreasing graph 𝕋 of T(n, q).     

Local and Global Existence for an Aggregation Equation

The purpose of this work is to develop a satisfactory existence theory for a general class of aggregation equations. An aggregation equation is a non-linear, non-local partial differential equation that is a regularization of a backward diffusion process. The non-locality arises via convolution with a potential. Depending on how regular the potential is, we prove either local or global existence for the solutions. Aggregation equations have been used recently to model the dynamics of populations in which the individuals attract each other (Bodnar and Velazquez, 2005 Bodnar , M. , Velazquez , J. J. L. ( 2005 ). Derivation of macroscopic equations for individual cell-based models: a formal approach . Math. Methods Appl. Sci. 28 ( 15 ): 1757 – 1779 .[Crossref], [Web of Science ®] , [Google Scholar]; Holm and Putkaradze, 2005 Holm , D. D. , Putkaradze , V. ( 2005 ). Aggregation of finite size particles with variable mobility . Phys. Rev. Lett. 95 : 226106 . [Google Scholar]; Mogilner and Edelstein-Keshet, 1999 Mogilner , A. , Edelstein-Keshet , L. ( 1999 ). A non-local model for a swarm . J. Math. Biol. 38 ( 6 ): 534 – 570 .[Crossref], [Web of Science ®] , [Google Scholar]; Morale et al., 2005 Morale , D. , Capasso , V. , Oelschläger , K. ( 2005 ). An interacting particle system modelling aggregation behavior: from individuals to populations . J. Math. Biol. 50 ( 1 ): 49 – 66 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]; Topaz and Bertozzi, 2004 Topaz , C. M. , Bertozzi , A. L. ( 2004 ). Swarming patterns in a two-dimensional kinematic model for biological groups . SIAM J. Appl. Math. 65 ( 1 ): 152 – 174 (electronic) .[Crossref], [Web of Science ®] , [Google Scholar]; Topaz et al., 2006 Topaz , C. M. , Bertozzi , A. L. , Lewis , M. A. ( 2006 ). A nonlocal continuum model for biological aggregation . Bull. Math. Biol. 68 ( 7 ): 1601 – 1623 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]).     

Best Proximity Points and Fixed Point Results for Certain Maps in Banach Spaces

In this article, we establish some new existence theorems for best proximity point and fixed point problems for certain mappings in Banach spaces. The main results of this article improve and extend the results presented by Wong [25 C. Wong ( 1974 ). Fixed points and characterizations of certain maps . Pacific J. Math. 54 : 305 – 312 .[Crossref], [Web of Science ®] , [Google Scholar]]. Examples are given to support the usability of our main conclusions.     

Homogenized Down-Up Algebras

Abstract

This paper studies two homogenizations of the down-up algebras introduced in Benkart and Roby (Benkart, G., Roby, T. (1998 Benkart, G. and Roby, T. 1998. Down-up algebras. J. Algebra, 209: 305–344. [Crossref], [Web of Science ®] , [Google Scholar]). Down-up Algebras. J. Algebra 209:305–344). We show that in all cases the homogenizing variable is not a zero-divisor, and that when the parameter β is non-zero, the homogenized down-up algebra is a Noetherian domain and a maximal order, and also Artin-Schelter regular, Auslander regular, and Cohen-Macaulay. We show that all homogenized down-up algebras have global dimension 4 and Gelfand-Kirillov dimension 4, and with one exception all homogenized down-up algebras are prime rings. We also exhibit a basis for homogenized down-up algebras and provide a necessary condition for a Noetherian homogenized down-up algebra to be a Hopf algebra.     

Asymptotic Solutions of Viscous Hamilton–Jacobi Equations with Ornstein–Uhlenbeck Operator

We study the long time behavior of solutions of the Cauchy problem for semilinear parabolic equations with the Ornstein–Uhlenbeck operator in ? ^N . The long time behavior in the main results is stated with help of the corresponding to ergodic problem, which complements, in the case of unbounded domains, the recent developments on long time behaviors of solutions of (viscous) Hamilton–Jacobi equations due to Namah (1996 Namah , G. ( 1996 ). Asymptotic solution of a Hamilton–Jacobi equation . Asymptotic Anal. 12 ( 4 ): 355 – 370 . [CSA] [Web of Science ®] , [Google Scholar]), Namah and Roquejoffre (1999 Namah , G. , Roquejoffre , J.-M . ( 1999 ). Remarks on the long-time behavior of the solutions of Hamilton–Jacobi equations . Comm. PDE 24 ( 5–6 ): 883 – 893 . [CSA] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Roquejoffre (1998 Roquejoffre , J.-M . ( 1998 ). Comportement asymptotique des solutions d’équations de Hamilton–Jacobi monodimensionnelles . C. R. Acad. Sci. Paris Sér. I Math. 326 ( 2 ): 185 – 189 . [CSA] [Crossref] , [Google Scholar]), Fathi (1998 Fathi , A. ( 1998 ). Sur la convergence du semi-groupe de Lax–Oleinik semigroup . C. R. Acad. Sci. Paris Sér. I Math. 327 ( 3 ): 267 – 270 . [CSA] [Crossref] , [Google Scholar]), Barles and Souganidis (2000 Barles , G. , Souganidis , P. E. ( 2000 ). On the large time behavior of solutions of Hamilton–Jacobi equaitons . SIAM J. Math. Anal. 31 ( 4 ): 925 – 939 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar] 2001 Barles , G. , Souganidis , P. E. ( 2001 ). Space-time periodic solutions and long-time behavior of solutions to quasi-periodic parabolic equations . SIAM J. Math. Anal. 32 ( 6 ): 1311 – 1323 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]). We also establish existence and uniqueness results for solutions of the Cauchy problem and ergodic problem for semilinear parabolic equations with the Ornstein–Uhlenbeck operator.     

Eigenfunctions for Partially Rectangular Billiards

In this note, we further develop the methods of Burq and Zworski (2005 Burq , N. , Zworski , M. ( 2005 ). Bouncing ball modes and quantum chaos . SIAM Review 47 ( 5 ): 43 – 49 [CROSSREF] [CSA] [Crossref] , [Google Scholar]) to study eigenfunctions for billiards which have rectangular components: these include the Bunimovich billiard, the Sinai billiard, and the recently popular pseudointegrable billiards (Bogomolny et al., 1999 Bogomolny , E. , Gerland , U. , Schmit , C. ( 1999 ). Models of intermediate spectral statistics . Phys. Rev. E 59 : 1315 – 1318 [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]). The results are an application of a “black-box” point of view as presented in Burq and Zworski (2004 Burq , N. , Zworski , M. ( 2004 ). Geometric control in the presence of a black box . JAMS 17 : 443 – 471 [CROSSREF] [CSA] [Web of Science ®] , [Google Scholar]).     

The minimal non-Koszul A(Γ)

The algebras A(Γ), where Γ is a directed layered graph, were first constructed by Gelfand et al. [5 Gelfand, I., Serconek, S., Retakh, V., Wilson, R. L. (2005). On a class of algebras associated to directed graphs. Selecta Math. (N.S.) 11(2):281–295.[Crossref], [Web of Science ®] , [Google Scholar]]. These algebras are generalizations of the algebras Q_n, which are related to factorizations of non-commutative polynomials. It was originally conjectured that these algebras were Koszul. In 2008, Cassidy and Shelton found a counterexample to this claim, a non-Koszul A(Γ) corresponding to a graph Γ with 18 edges and 11 vertices. We produce an example of a directed layered graph Γ with 13 edges and 9 vertices, which produces a non-Koszul A(Γ). We also show this is the minimal example with this property.     

Propagation of Singularities for Schrödinger Equations on the Euclidean Space with a Scattering Metric

Given a scattering metric on the Euclidean space. We consider the Schrödinger equation corresponding to the metric, and study the propagation of singularities for the solution in terms of the “homogeneous wavefront set”. We also prove that the notion of the homogeneous wavefront set is essentially equivalent to that of the quadratic scattering wavefront set introduced by Wunsch (1999 Wunsch , J. ( 1999 ). Propagation of singularities and growth for Schrödinger operators . Duke Math. J. 98 : 137 – 186 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]). One of the main results in Wunsch (1999 Wunsch , J. ( 1999 ). Propagation of singularities and growth for Schrödinger operators . Duke Math. J. 98 : 137 – 186 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) follows on the Euclidean space with a weaker, almost optimal condition on the potential.     

Almost Exponential Decay Near Maxwellian

ABSTRACT

By direct interpolation of a family of smooth energy estimates for solutions near Maxwellian equilibrium and in a periodic box to several Boltzmann type equations in Guo (2002 Guo , Y. ( 2002 ). The Landau equation in a periodic box . Comm. Math. Phys. 231 ( 3 ): 391 – 434 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar] 2003a Guo , Y. ( 2003a ). Classical solutions to the Boltzmann equation for molecules with an angular cutoff . Arch. Ration. Mech. Anal. 169 ( 4 ): 305 – 353 . [CSA] [CROSSREF]  [Google Scholar] b Guo , Y. ( 2003b ). The Vlasov-Maxwell-Boltzmann system near maxwellians . Invent. Math. 153 ( 3 ): 593 – 630 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) and Strain and Guo (2004 Strain , R. M. , Guo , Y. ( 2004 ). Stability of the relativistic Maxwellian in a collisional plasma . Comm. Math. Phys. 251 ( 2 ): 263 – 320 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]), we show convergence to Maxwellian with any polynomial rate in time. Our results not only resolve the important open problem for both the Vlasov-Maxwell-Boltzmann system and the relativistic Landau-Maxwell system for charged particles, but also lead to a simpler alternative proof of recent decay results (Desvillettes and Villani, 2005 Desvillettes , L. , Villani , C. ( 2005 ). On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation . Invent. Math. 159 ( 2 ): 245 – 316 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) for soft potentials as well as the Coulombic interaction, with precise decay rate depending on the initial conditions.     

ℤ3-Twisted Representations of Lattice Vertex Operator Algebras

Abstract

Let L be a positive definite even lattice and let g ∈ Aut L be a fixed point free automorphism of order 3. We determine the twisted Zhu's algebra A _? (V _L ) for the lattice vertex operator algebra V _L , where ? is an automorphism of V _L induced from g. As a result, we show that the set of all irreducible ?-twisted modules for V _L (up to isomorphism) are exactly those constructed by Dong and Lepowsky (1996 Dong, C. and Lepowsky, J. 1996. The algebraic structure of relative twisted vertex operators. J. Pure and Applied Algebra, 110: 259–295. [Crossref], [Web of Science ®] , [Google Scholar]) and Lepowsky (1985 Lepowsky, J. 1985. Calculus of twisted vertex operators. Proc. Natl. Acad. Sci. USA, 82: 8295–8299. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]).     

DECOMPOSITION OF A HIGHER DEGREE FORM

ABSTRACT

We give another proof of Harrison's decomposition result,[2] Harrison, D.K. 1975. A Grothendieck Ring of Higher Degree Forms. Journal of Algebra, 35: 123–138. [Crossref], [Web of Science ®] , [Google Scholar] Prop. 2.3 for higher degree forms over a noetherian ring, exploiting an earlier introduction of the centre. We generalise to higher degree forms over a noetherian scheme: we extend the notion of centre; we prove a decomposition result; we extend Harrison's result,[2] Harrison, D.K. 1975. A Grothendieck Ring of Higher Degree Forms. Journal of Algebra, 35: 123–138. [Crossref], [Web of Science ®] , [Google Scholar] Prop. 4.3 on the behaviour of the centre under a flat base extension; and we improve his result,[2] Harrison, D.K. 1975. A Grothendieck Ring of Higher Degree Forms. Journal of Algebra, 35: 123–138. [Crossref], [Web of Science ®] , [Google Scholar] Prop. 4.2, giving conditions on the base scheme under which the centre of the tensor product of two higher degree forms is isomorphic to the tensor product of their centres.     

Global Stability and Bounds for Coarsening Rates Within the Mean-Field Theory for Domain Coarsening

We investigate the long-time behavior of solutions to the classical mean-field model for coarsening by Lifshitz–Slyozov and Wagner (LSW). In the original work (Lifshitz and Slyozov, 1961 Lifshitz , I. M. , Slyozov , V. V. ( 1961 ). The kinetics of precipitation from supersaturated solid solutions . J. Phys. Chem. Solids 19 : 35 – 50 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]; Wagner 1961 Wagner , C. ( 1961 ). Theorie der Alterung von Niederschlägen durch Umlösen . Z. Elektrochemie 65 : 581 – 594 . [CSA]  [Google Scholar]) convergence of solutions to a uniquely determined self-similar solution was predicted. However, it is by now well known (Giron et al., 1998 Giron , B. , Meerson , B. , Sasorov , V. P. ( 1998 ). Weak selection and stability of localized distributions in Ostwald ripening . Phys. Rev. E 58 : 4213 – 4216 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]; Niethammer and Pego 1999 Niethammer , B. , Pego , R. L. ( 1999 ). Non-self-similar behavior in the LSW theory of Ostwald ripening . J. Stat. Phys. 95 ( 5/6 ): 867 – 902 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar] 2001 Niethammer , B. , Pego , R. L. ( 2001 ). The LSW model for domain coarsening: Asymptotic behavior for total conserved mass . J. Stat. Phys. 104 ( 5/6 ): 1113 – 1144 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) that the long-time behavior of solutions depends sensitively on the initial data. In Niethammer and Pego (1999 Niethammer , B. , Pego , R. L. ( 1999 ). Non-self-similar behavior in the LSW theory of Ostwald ripening . J. Stat. Phys. 95 ( 5/6 ): 867 – 902 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar] 2001 Niethammer , B. , Pego , R. L. ( 2001 ). The LSW model for domain coarsening: Asymptotic behavior for total conserved mass . J. Stat. Phys. 104 ( 5/6 ): 1113 – 1144 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) a necessary criterion for convergence to any self-similar solution which behaves like a finite power at the end of its (compact) support is given. It says that the data have to be regularly varying at the end of their support with the same power. This criterion is also shown to be sufficient if the power is sufficiently small and for data which are close to self-similar.

In this article we extend the local stability result to the whole range of self-similar solutions with compact support. Our first main result establishes global stability of self-similar solutions with not too large power. The proof relies on a global contraction argument for the spreading of characteristics. In addition, we also establish upper and lower bounds for the coarsening rates of the system for a suitable class of initial data whose variation is bounded at the end of the support but not necessarily regular.     

Sharp estimates for solutions of mean field equations with collapsing singularity

The pioneering work of Brezis-Merle [7 Brezis, H., Merle, F. (1991). Uniform estimates and blow-up behavior for solutions of ?Δu = V(x)e^u in two dimensions. Commun. Partial Differential Equation 16:1223–1254.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]], Li-Shafrir [27 Li, Y.Y., Shafrir, I. (1994). Blow-up analysis for solutions of ?Δu = V(x)e^u in dimension two. Indiana Univ. Math. J. 43:1255–1270.[Crossref], [Web of Science ®] , [Google Scholar]], Li [26 Li, Y.Y. (1999). Harnack inequality: the method of moving planes. Commun. Math. Phys. 200:421–444.[Crossref], [Web of Science ®] , [Google Scholar]], and Bartolucci-Tarantello [3 Bartolucci, D., Tarantello, G. (2002). Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory. Commun. Math. Phys. 229:3–47.[Crossref], [Web of Science ®] , [Google Scholar]] showed that any sequence of blow-up solutions for (singular) mean field equations of Liouville type must exhibit a “mass concentration” property. A typical situation of blowup occurs when we let the singular (vortex) points involved in the equation (see (1.1) below) collapse together. However in this case, Lin-Tarantello in [30 Lin, C.S., Tarantello, G. (2016). When “blow-up” does not imply “concentration”: A detour from Brezis-Merle’s result. C. R. Math. Acad. Sci. Paris 354:493–498.[Crossref], [Web of Science ®] , [Google Scholar]] pointed out that the phenomenon: “bubbling implies mass concentration” might not occur and new scenarios open for investigation. In this paper, we present two explicit examples which illustrate (with mathematical rigor) how a “nonconcentration” situation does happen and its new features. Among other facts, we show that in certain situations, the collapsing rate of the singularities can be used as blow-up parameter to describe the bubbling properties of the solution-sequence. In this way, we are able to establish accurate estimates around the blow-up points which we hope to use toward a degree counting formula for the shadow system (1.34) below.     

A ring with FT-rank two

We construct a ring which admits a 2-generated, faithful torsion module but lacks a cyclic faithful torsion module. This answers a question by Oman and Schwiebert [1 Oman, G., Schwiebert, R. (2012). Rings which admit faithful torsion modules. Commun. Algebra 40(6):2184–2198.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar], 2 Oman, G., Schwiebert, R. (2012). Rings which admit faithful torsion modules II. J. Algebra Appl. 11(3):1250054 (12 p.).[Crossref], [Web of Science ®] , [Google Scholar]].     

Conformal Compactification of Asymptotically Locally Hyperbolic Metrics II: Weakly ALH Metrics

In this paper we pursue the work initiated in [6 Bahuaud , E. ( 2009 ). Intrinsic characterization for Lipschitz asymptotically hyperbolic metrics . Pacific J. Math. 239 : 231 – 249 .[Crossref], [Web of Science ®] , [Google Scholar], 7 Bahuaud , E. , Gicquaud , R. ( 2011 ). Conformal compactification of asymptotically locally hyperbolic metrics . J. Geom. Anal. 21 : 1085 – 1118 .[Crossref], [Web of Science ®] , [Google Scholar]]: study the extent to which conformally compact asymptotically hyperbolic metrics can be characterized intrinsically. We show how the decay rate of the sectional curvature to ?1 controls the Hölder regularity of the compactified metric. To this end, we construct harmonic coordinates that satisfy some Neumann-type condition at infinity. Combined with a new integration argument, this permits us to recover to a large extent our previous result without any decay assumption on the covariant derivatives of the Riemann tensor.     

2 × 2 Permanental Ideals of Hypermatrices

We analyze the structure of ideals generated by some classes of 2 × 2 permanents of hypermatrices, generalizing [9 Laubenbacher , R. C. , Swanson , I. ( 2000 ). Permanental ideals . J. Symbolic Comput. 30 : 195 – 205 .[Crossref], [Web of Science ®] , [Google Scholar]] on 2 × 2 permanental ideals of generic matrices. We compare the obtained structure to that of the corresponding determinantal ideals in [11 Swanson , I. , Taylor , A. ( 2013 ). Minimal primes of ideals arising from conditional independence statements . J. Algebra 392 : 299 – 314 .[Crossref], [Web of Science ®] , [Google Scholar]]: as expected, the permanental ideals have many more (minimal) components. In the last two sections, we examine a few related classes of permanental ideals.     

Averaging of Backward Stochastic Differential Equations and Homogenization of Partial Differential Equations with Periodic Coefficients

Abstract

We study the limit of the solutions of systems of semi-linear partial differential equations (PDEs) of second order of parabolic type, with rapidly oscillating periodic coefficients, a singular drift, and singular coefficients of the zero and second order terms. Our basic tool is the approach given by Pardoux [14 Pardoux , E. 1999 . Homogenization of linear and semilinear second order parabolic PDEs with periodic coefficients: a probabilistic approach . J. Funct. Anal. 167 : 498 – 520 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]]. In particular, we use the weak convergence of an associated backward stochastic differential equation (BSDE).     

DIRECT SUM DECOMPOSITION OF THE PRODUCT OF PREINJECTIVE MODULES OVER RIGHT PURE SEMISIMPLE HEREDITARY RINGS

ABSTRACT

In his recent work, [1] Simson, D. 2000. An Artin Problem for Division Ring Extensions and the Pure Semisimplicity Conjecture, II. J. Algebra, 227: 670–705. [Crossref], [Web of Science ®] , [Google Scholar] and [2] Simson, D. 2001. On Small Right Pure Semisimple Rings and the Structure of their Auslander-Reiten Quiver. Communic. in Algebra, 29 in press[Web of Science ®] , [Google Scholar], on the pure semisimplicity conjecture Simson raised two problems about the structure of the direct sum decomposition of the direct product modulo the direct sum of indecomposable preinjective modules over right pure semisimple hereditary rings. The main goal of this paper is the proof of a theorem that resolves one of these problems and provides a partial answer to the other.     

Discrete Morse theory and a reformulation of the K(π,1)-conjecture

A recent theorem of Dobrinskaya [20 Dobrinskaya, N.È. (2006). Configuration spaces of labeled particles and finite Eilenberg-MacLane complexes. Proc. Steklov Inst. Math. 252(1):30–46.[Crossref] , [Google Scholar]] states that the K(π,1)-conjecture holds for an Artin group G if and only if the canonical map BM→BG is a homotopy equivalence, where M denotes the Artin monoid associated to G. The aim of this paper is to give an alternative proof by means of discrete Morse theory and abstract homotopy theory. Moreover, we exhibit a new model for the classifying space of an Artin monoid, in the spirit of [13 Charney, R., Meier, J., Whittlesey, K. (2004). Bestvina’s normal form complex and the homology of Garside groups. Geom. Dedicata 105:171–188.[Crossref], [Web of Science ®] , [Google Scholar]], and a small chain complex for computing its monoid homology, similar to the one of [44 Squier, C. C. (1994). The homological algebra of Artin groups. Math. Scand. 75(1):5–43.[Crossref], [Web of Science ®] , [Google Scholar]].     

Self-Dual Hopf Quivers

ABSTRACT

We study self-dual coradically graded pointed Hopf algebras with a help of the dual Gabriel theorem for pointed Hopf algebras (van Oystaeyen and Zhang, 2004 van Oystaeyen , F. , Zhang , P. ( 2004 ). Quiver Hopf algebras . J. Algebra 280 ( 2 ): 577 – 589 . [CSA] [CROSSREF]  [Google Scholar]). The co-Gabriel Quivers of such Hopf algebras are said to be self-dual. An explicit classification of self-dual Hopf quivers is obtained. We also prove that finite dimensional pointed Hopf algebras with self-dual graded versions are generated by group-like and skew-primitive elements as associative algebras. This partially justifies a conjecture of Andruskiewitsch and Schneider (2000 Andruskiewitsch , N. , Schneider , H.-J . ( 2000 ). Finite quantum groups and Cartan matrices . Adv. Math. 154 : 1 – 45 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) and may help to classify finite dimensional self-dual coradically graded pointed Hopf algebras.