2013
http://publications.mfo.de/handle/mfo/1345
Fri, 05 Mar 2021 23:10:43 GMT2021-03-05T23:10:43ZRight Simple Singularities in Positive Characteristic
http://publications.mfo.de/handle/mfo/201
Right Simple Singularities in Positive Characteristic
Greuel, Gert-Martin; Nguyen, Hong Duc
We classify isolated singularities $f \in K[[x_1,...,x_n]]$, which are simple, i.e. have no moduli, w.r.t. right equivalence, where $K$ is an algebraically closed field of characteristic $p>0$. For $K=\mathbb{R}$ or $\mathbb{C}$ this classification was initiated by Arnol'd, resulting in the famous ADE-series. The classification w.r.t. contact equivalence for $p>0$ was done by Greuel and Kröning with a result similiar to Arnol'd's. It is surprising that w.r.t. right equivalence and any given $p>0$ we have only finitely many simple singularities, i.e. there are only finitely many $k$ such that $A_k$ and $D_k$ are right simple, all the others have moduli. A major point of this paper is the generalization of the notion of modality to the algebraic setting, its behaviour under morphisms, and its relations to formal deformation theory. As an application we show that the modality is semicontinuous in any characteristic.
OWLF 2013
Tue, 01 Jan 2013 00:00:00 GMThttp://publications.mfo.de/handle/mfo/2012013-01-01T00:00:00ZGreuel, Gert-MartinNguyen, Hong DucWe classify isolated singularities $f \in K[[x_1,...,x_n]]$, which are simple, i.e. have no moduli, w.r.t. right equivalence, where $K$ is an algebraically closed field of characteristic $p>0$. For $K=\mathbb{R}$ or $\mathbb{C}$ this classification was initiated by Arnol'd, resulting in the famous ADE-series. The classification w.r.t. contact equivalence for $p>0$ was done by Greuel and Kröning with a result similiar to Arnol'd's. It is surprising that w.r.t. right equivalence and any given $p>0$ we have only finitely many simple singularities, i.e. there are only finitely many $k$ such that $A_k$ and $D_k$ are right simple, all the others have moduli. A major point of this paper is the generalization of the notion of modality to the algebraic setting, its behaviour under morphisms, and its relations to formal deformation theory. As an application we show that the modality is semicontinuous in any characteristic.On the autonomous metric on groups of Hamiltonian diffeomorphisms of closed hyperbolic surfaces
http://publications.mfo.de/handle/mfo/199
On the autonomous metric on groups of Hamiltonian diffeomorphisms of closed hyperbolic surfaces
Brandenbursky, Michael
Let $\Sigma_g$ be a closed hyperbolic surface of genus $g$ and let Ham($\Sigma_ g$) be the group of Hamiltonian diffeomorphisms of $\Sigma_g$. The most natural word metric on this group is the autonomous metric. It has many interesting properties, most important of which is the bi-invariance of this metric. In this work we show that Ham($\Sigma_g$) is unbounded with respect to this metric.
OWLF 2013
Tue, 23 Jul 2013 00:00:00 GMThttp://publications.mfo.de/handle/mfo/1992013-07-23T00:00:00ZBrandenbursky, MichaelLet $\Sigma_g$ be a closed hyperbolic surface of genus $g$ and let Ham($\Sigma_ g$) be the group of Hamiltonian diffeomorphisms of $\Sigma_g$. The most natural word metric on this group is the autonomous metric. It has many interesting properties, most important of which is the bi-invariance of this metric. In this work we show that Ham($\Sigma_g$) is unbounded with respect to this metric.Supertropical Quadratic Forms I
http://publications.mfo.de/handle/mfo/200
Supertropical Quadratic Forms I
Knebusch, Manfred; Rowen, Louis; Izhakian, Zur
We initiate the theory of a quadratic form q over a semiring $R$. As customary, one can write $q(x+y)=q(x)+q(y)+b(x,y)$, where b is a companion bilinear form. But in contrast to the ring-theoretic case, the companion bilinear form need not be uniquely defined. Nevertheless, q can always be written as a sum of quadratic forms $q=\kappa+\rho$, where $\kappa$ is quasilinear in the sense that $\kappa(x+y)=\kappa(x)+\kappa(y)$, and $\rho$ is rigid in the sense that it has a unique companion. In case that $R$ is a supersemifield (cf. Definition 4.1 below) and $q$ is defined on a free $R$-module, we obtain an explicit classification of these decompositions $q=\kappa+\rho$ and of all companions $b$ of $q$. As an application to tropical geometry, given a quadratic form $q:V \to R$ on a free module $V$ over a commutative ring $R$ and a supervaluation $\rho$: $R \to U$ with values in a supertropical semiring [5], we define - after choosing a base $\mathcal{L}=(v_i|i \in I)$ of $V$- a quadratic form $q^\varphi:U^{(I)} \to U$ on the free module $U^{(I)}$ over the semiring $U$. The analysis of quadratic forms over a supertropical semiring enables one to measure the “position” of $q$ with respect to $\mathcal{L}$ via ${\varphi}$.
OWLF 2013
Tue, 01 Jan 2013 00:00:00 GMThttp://publications.mfo.de/handle/mfo/2002013-01-01T00:00:00ZKnebusch, ManfredRowen, LouisIzhakian, ZurWe initiate the theory of a quadratic form q over a semiring $R$. As customary, one can write $q(x+y)=q(x)+q(y)+b(x,y)$, where b is a companion bilinear form. But in contrast to the ring-theoretic case, the companion bilinear form need not be uniquely defined. Nevertheless, q can always be written as a sum of quadratic forms $q=\kappa+\rho$, where $\kappa$ is quasilinear in the sense that $\kappa(x+y)=\kappa(x)+\kappa(y)$, and $\rho$ is rigid in the sense that it has a unique companion. In case that $R$ is a supersemifield (cf. Definition 4.1 below) and $q$ is defined on a free $R$-module, we obtain an explicit classification of these decompositions $q=\kappa+\rho$ and of all companions $b$ of $q$. As an application to tropical geometry, given a quadratic form $q:V \to R$ on a free module $V$ over a commutative ring $R$ and a supervaluation $\rho$: $R \to U$ with values in a supertropical semiring [5], we define - after choosing a base $\mathcal{L}=(v_i|i \in I)$ of $V$- a quadratic form $q^\varphi:U^{(I)} \to U$ on the free module $U^{(I)}$ over the semiring $U$. The analysis of quadratic forms over a supertropical semiring enables one to measure the “position” of $q$ with respect to $\mathcal{L}$ via ${\varphi}$.Obtaining Error-Minimizing Estimates and Universal Entry-Wise Error Bounds for Low-Rank Matrix Completion
http://publications.mfo.de/handle/mfo/196
Obtaining Error-Minimizing Estimates and Universal Entry-Wise Error Bounds for Low-Rank Matrix Completion
Király, Franz J.; Theran, Louis
We propose a general framework for reconstructing and denoising single entries of incomplete and noisy entries. We describe: effective algorithms for deciding if and entry can be reconstructed and, if so, for reconstructing and denoising it; and a priori bounds on the error of each entry, individually. In the noiseless case our algorithm is exact. For rank-one matrices, the new algorithm is fast, admits a highly-parallel implementation, and produces an error minimizing estimate that is qualitatively close to our theoretical and the state-of-the-art Nuclear Norm and OptSpace methods.
OWLF 2013
Mon, 10 Jun 2013 00:00:00 GMThttp://publications.mfo.de/handle/mfo/1962013-06-10T00:00:00ZKirály, Franz J.Theran, LouisWe propose a general framework for reconstructing and denoising single entries of incomplete and noisy entries. We describe: effective algorithms for deciding if and entry can be reconstructed and, if so, for reconstructing and denoising it; and a priori bounds on the error of each entry, individually. In the noiseless case our algorithm is exact. For rank-one matrices, the new algorithm is fast, admits a highly-parallel implementation, and produces an error minimizing estimate that is qualitatively close to our theoretical and the state-of-the-art Nuclear Norm and OptSpace methods.On Concentrators and Related Approximation Constants
http://publications.mfo.de/handle/mfo/198
On Concentrators and Related Approximation Constants
Bondarenko, A. V.; Prymak, A.; Radchenko, D.
Pippenger ([Pip77]) showed the existence of (6m, 4m, 3m, 6)-concentrator for each positive integer m using a probabilistic method. We generalize his approach and prove existence of (6m, 4m, 3m, 5.05)-concentrator (which is no longer regular, but has fewer edges). We apply this result to improve the constant of approximation of almost additive set functions by additive set functions from 44.5 (established by Kalton and Roberts in [KR83]) to 39. We show a more direct connection of the latter problem to the Whitney type estimate for approximation of continuous functions on a cube in Rd by linear functions, and improve the estimate of this Whitney constant from 802 (proved by Brudnyi and Kalton in [BK00]) to 73.
OWLF 2013
Mon, 10 Jun 2013 00:00:00 GMThttp://publications.mfo.de/handle/mfo/1982013-06-10T00:00:00ZBondarenko, A. V.Prymak, A.Radchenko, D.Pippenger ([Pip77]) showed the existence of (6m, 4m, 3m, 6)-concentrator for each positive integer m using a probabilistic method. We generalize his approach and prove existence of (6m, 4m, 3m, 5.05)-concentrator (which is no longer regular, but has fewer edges). We apply this result to improve the constant of approximation of almost additive set functions by additive set functions from 44.5 (established by Kalton and Roberts in [KR83]) to 39. We show a more direct connection of the latter problem to the Whitney type estimate for approximation of continuous functions on a cube in Rd by linear functions, and improve the estimate of this Whitney constant from 802 (proved by Brudnyi and Kalton in [BK00]) to 73.Mesh Ratios for Best-Packing and Limits of Minimal Energy Configurations
http://publications.mfo.de/handle/mfo/197
Mesh Ratios for Best-Packing and Limits of Minimal Energy Configurations
Bondarenko, A. V.; Hardin, Douglas P.; Saff, Edward B.
For $N$-point best-packing configurations $\omega_N$ on a compact metric
space $(A, \rho)$, we obtain estimates for the mesh-separation ratio $\gamma(\rho_N , A)$, which
is the quotient of the covering radius of $\omega_N$ relative to $A$ and the minimum pairwise
distance between points in $\omega_N$ . For best-packing configurations $\omega_N$ that
arise as limits of minimal Riesz $s$-energy configurations as $s \to \infty$, we prove that
$\gamma(\omega_N , A) ≤ 1$ and this bound can be attained even for the sphere. In the particular
case when $N = 5$ on $S^1$
with $\rho$ the Euclidean metric, we prove our main result that among the infinitely many 5-point best-packing configurations there is a unique
configuration, namely a square-base pyramid $\omega^*_5$, that is the limit (as $s \to \infty$) of
5-point $s$-energy minimizing configurations. Moreover, $\gamma(\omega^*_5, S^2) = 1$.
OWLF 2013
Mon, 10 Jun 2013 00:00:00 GMThttp://publications.mfo.de/handle/mfo/1972013-06-10T00:00:00ZBondarenko, A. V.Hardin, Douglas P.Saff, Edward B.For $N$-point best-packing configurations $\omega_N$ on a compact metric
space $(A, \rho)$, we obtain estimates for the mesh-separation ratio $\gamma(\rho_N , A)$, which
is the quotient of the covering radius of $\omega_N$ relative to $A$ and the minimum pairwise
distance between points in $\omega_N$ . For best-packing configurations $\omega_N$ that
arise as limits of minimal Riesz $s$-energy configurations as $s \to \infty$, we prove that
$\gamma(\omega_N , A) ≤ 1$ and this bound can be attained even for the sphere. In the particular
case when $N = 5$ on $S^1$
with $\rho$ the Euclidean metric, we prove our main result that among the infinitely many 5-point best-packing configurations there is a unique
configuration, namely a square-base pyramid $\omega^*_5$, that is the limit (as $s \to \infty$) of
5-point $s$-energy minimizing configurations. Moreover, $\gamma(\omega^*_5, S^2) = 1$.The algebraic combinatorial approach for low-rank matrix completion
http://publications.mfo.de/handle/mfo/195
The algebraic combinatorial approach for low-rank matrix completion
Király, Franz J.; Theran, Louis; Ryota,Tomioka; Uno, Takeaki
We propose an algebraic combinatorial framework for the problem of completing partially observed low-rank matrices. We show that the intrinsic properties of the problem, including which entries can be reconstructed, and the degrees of freedom in the reconstruction, do not depend on the values of the observed entries, but only on their position. We associate combinatorial and algebraic objects, differentials and matroids, which are descriptors of the particular reconstruction task, to the set of observed entries, and apply them to obtain reconstruction bounds. We show how similar techniques can be used to obtain reconstruction bounds on general compressed sensing problems with algebraic compression constraints. Using the new theory, we develop several algorithms for low-rank matrix completion, which allow to determine which set of entries can be potentially reconstructed and which not, and how, and we present algorithms which apply algebraic combinatorial methods in order to reconstruct the missing entries.
OWLF 2012; OWLF 2013
Thu, 14 Mar 2013 00:00:00 GMThttp://publications.mfo.de/handle/mfo/1952013-03-14T00:00:00ZKirály, Franz J.Theran, LouisRyota,TomiokaUno, TakeakiWe propose an algebraic combinatorial framework for the problem of completing partially observed low-rank matrices. We show that the intrinsic properties of the problem, including which entries can be reconstructed, and the degrees of freedom in the reconstruction, do not depend on the values of the observed entries, but only on their position. We associate combinatorial and algebraic objects, differentials and matroids, which are descriptors of the particular reconstruction task, to the set of observed entries, and apply them to obtain reconstruction bounds. We show how similar techniques can be used to obtain reconstruction bounds on general compressed sensing problems with algebraic compression constraints. Using the new theory, we develop several algorithms for low-rank matrix completion, which allow to determine which set of entries can be potentially reconstructed and which not, and how, and we present algorithms which apply algebraic combinatorial methods in order to reconstruct the missing entries.Even-homogeneous supermanifolds on the complex projective line
http://publications.mfo.de/handle/mfo/186
Even-homogeneous supermanifolds on the complex projective line
Vishnyakova, E. G.
The classification of even-homogeneous complex supermanifolds of dimension $1|m$, $m<=3$, on $\mathbb{CP}^1$ up to isomorphism is given. An explicit description of such supermanifolds in terms of local charts and coordinates is obtained.
OWLF 2012
Thu, 14 Mar 2013 00:00:00 GMThttp://publications.mfo.de/handle/mfo/1862013-03-14T00:00:00ZVishnyakova, E. G.The classification of even-homogeneous complex supermanifolds of dimension $1|m$, $m<=3$, on $\mathbb{CP}^1$ up to isomorphism is given. An explicit description of such supermanifolds in terms of local charts and coordinates is obtained.Enhanced Spatial Skin-Effect for Free Vibrations of a Thick Cascade Junction with "Super Heavy" Concentrated Masses
http://publications.mfo.de/handle/mfo/1073
Enhanced Spatial Skin-Effect for Free Vibrations of a Thick Cascade Junction with "Super Heavy" Concentrated Masses
Čečkin, Grigorij A.; Mel'nyk, Taras A.
The asymptotic behavior (as $\varepsilon \to 0$) of eigenvalues and eigenfunctions of a boundary-value problem for the Laplace operator in a thick cascade junction with concentrated masses is studied. This cascade junction consists of the junction’s body and a great number $5N= \mathcal{O}(\varepsilon^{-1})$ of $\varepsilon$-alternating thin rods belonging to two classes. One class consists of rods of finite length and the second one consists of rods of small length of order $\mathcal{O}(\varepsilon)$. The mass density is of order $\mathcal{O}(\varepsilon^{-\alpha})$ on the rods from the second class and $\mathcal{O}(1)$ outside of them. There exist five qualitatively different cases in the asymptotic behavior of eigen-magnitudes as $\varepsilon \to 0$, namely the case of “light” concentrated masses $(a \in (0,1))$, “intermediate” concentrated masses $(\alpha=1)$ and “heavy” concentrated masses ($\alpha \in (1, +\infty")$) that we divide into “slightly heavy” concentrated masses ($\alpha \in (1,2)$), “moderate heavy” concentrated masses ($\alpha=2$), and “super heavy” concentrated masses ($alpha>2$). In the paper we study the influence of the concentrated masses on the asymptotic behavior of the eigen-magnitudes in the cases $\alpha=2$ and $\alpha>2$. The leading terms of asymptotic expansions both for the eigenvalues and eigenfunctions are constructed and the corresponding asymptotic estimates are proved. In addition, a new kind of high-frequency vibrations is found.
Research in Pairs 2013
Fri, 13 Dec 2013 00:00:00 GMThttp://publications.mfo.de/handle/mfo/10732013-12-13T00:00:00ZČečkin, Grigorij A.Mel'nyk, Taras A.The asymptotic behavior (as $\varepsilon \to 0$) of eigenvalues and eigenfunctions of a boundary-value problem for the Laplace operator in a thick cascade junction with concentrated masses is studied. This cascade junction consists of the junction’s body and a great number $5N= \mathcal{O}(\varepsilon^{-1})$ of $\varepsilon$-alternating thin rods belonging to two classes. One class consists of rods of finite length and the second one consists of rods of small length of order $\mathcal{O}(\varepsilon)$. The mass density is of order $\mathcal{O}(\varepsilon^{-\alpha})$ on the rods from the second class and $\mathcal{O}(1)$ outside of them. There exist five qualitatively different cases in the asymptotic behavior of eigen-magnitudes as $\varepsilon \to 0$, namely the case of “light” concentrated masses $(a \in (0,1))$, “intermediate” concentrated masses $(\alpha=1)$ and “heavy” concentrated masses ($\alpha \in (1, +\infty")$) that we divide into “slightly heavy” concentrated masses ($\alpha \in (1,2)$), “moderate heavy” concentrated masses ($\alpha=2$), and “super heavy” concentrated masses ($alpha>2$). In the paper we study the influence of the concentrated masses on the asymptotic behavior of the eigen-magnitudes in the cases $\alpha=2$ and $\alpha>2$. The leading terms of asymptotic expansions both for the eigenvalues and eigenfunctions are constructed and the corresponding asymptotic estimates are proved. In addition, a new kind of high-frequency vibrations is found.A Relation Between N-Qubit and 2N-1-Qubit Pauli Groups via Binary LGr(N,2N)
http://publications.mfo.de/handle/mfo/1072
A Relation Between N-Qubit and 2N-1-Qubit Pauli Groups via Binary LGr(N,2N)
Holweck, F. G.; Saniga, Metod; Lévay, Péter
Employing the fact that the geometry of the $N$-qubit ($N\geq 2$) Pauli group is embodied in the structure of the symplectic polar space $\mathcal{W}(2N-1, 2)$ and using properties of the Lagrangian Grassmannian $LGr(N, 2N)$ defined over the smallest Galois field, it is demonstrated that there exists a bijection between the set of maximum sets of mutually commuting elements of the $N$-qubit Pauli group and a certain subset of elements of the $2^{N-1}$-qubit Pauli group. In order to reveal finer traits of this correspondence, the cases $N=3$ (also addressed recently by Lévay, Planat and Saniga (JHEP 09 (2013) 037)) and $N=4$ are discussed in detail. As an apt application of our findings, we use the stratification of the ambient projective space $PG(2^N-1, 2)$ of the $2^{N-1}$-qubit Pauli group in terms of $G$-orbits, where $G\equiv SL(2,2) \times SL(2,2) \times \cdot \cdot \cdot \times SL(2,2) \rtimes S_N$, to decompose $\underline{\pi}(LGr(N,2N))$ into non-equivalent orbits. This leads to a partition of $LGr(N, 2N)$ into distinguished classes that can be labeled by elements of the above-mentioned Pauli groups.
Research in Pairs 2013
Mon, 09 Dec 2013 00:00:00 GMThttp://publications.mfo.de/handle/mfo/10722013-12-09T00:00:00ZHolweck, F. G.Saniga, MetodLévay, PéterEmploying the fact that the geometry of the $N$-qubit ($N\geq 2$) Pauli group is embodied in the structure of the symplectic polar space $\mathcal{W}(2N-1, 2)$ and using properties of the Lagrangian Grassmannian $LGr(N, 2N)$ defined over the smallest Galois field, it is demonstrated that there exists a bijection between the set of maximum sets of mutually commuting elements of the $N$-qubit Pauli group and a certain subset of elements of the $2^{N-1}$-qubit Pauli group. In order to reveal finer traits of this correspondence, the cases $N=3$ (also addressed recently by Lévay, Planat and Saniga (JHEP 09 (2013) 037)) and $N=4$ are discussed in detail. As an apt application of our findings, we use the stratification of the ambient projective space $PG(2^N-1, 2)$ of the $2^{N-1}$-qubit Pauli group in terms of $G$-orbits, where $G\equiv SL(2,2) \times SL(2,2) \times \cdot \cdot \cdot \times SL(2,2) \rtimes S_N$, to decompose $\underline{\pi}(LGr(N,2N))$ into non-equivalent orbits. This leads to a partition of $LGr(N, 2N)$ into distinguished classes that can be labeled by elements of the above-mentioned Pauli groups.