scholarly journals A Formula for Popp’s Volume in Sub-Riemannian Geometry

2013 ◽  
Vol 1 ◽  
pp. 42-57 ◽  
Author(s):  
Davide Barilari ◽  
Luca Rizzi
Keyword(s):  
Riemannian Manifold ◽  
Explicit Formula ◽  
General Formula ◽  
Riemannian Geometry ◽  
Isometry Group ◽  
Natural Generalization ◽  
Riemannian Structure ◽  
The One

Abstract For an equiregular sub-Riemannian manifold M, Popp’s volume is a smooth volume which is canonically associated with the sub-Riemannian structure, and it is a natural generalization of the Riemannian one. In this paper we prove a general formula for Popp’s volume, written in terms of a frame adapted to the sub-Riemannian distribution. As a first application of this result, we prove an explicit formula for the canonical sub- Laplacian, namely the one associated with Popp’s volume. Finally, we discuss sub-Riemannian isometries, and we prove that they preserve Popp’s volume. We also show that, under some hypotheses on the action of the isometry group of M, Popp’s volume is essentially the unique volume with such a property.

Quasi-cyclic codes over the ring 𝔽p[u]/〈u2 − u〉

2015 ◽  
Vol 08 (04) ◽  
pp. 1550085
Author(s):  
Sukhamoy Pattanayak ◽  
Abhay Kumar Singh
Keyword(s):  
Structural Properties ◽  
Cyclic Codes ◽  
Natural Generalization ◽  
Spanning Set ◽  
The One ◽  
Quasi Cyclic Codes

Quasi-cyclic (QC) codes are a natural generalization of cyclic codes. In this paper, we study some structural properties of QC codes over [Formula: see text], where [Formula: see text] is a prime and [Formula: see text]. By exploring their structure, we determine the one generator QC codes over [Formula: see text] and the size by giving a minimal spanning set. We discuss some examples of QC codes of various length over [Formula: see text].

Re-measuring Twenty-first Century Population Ageing

2017 ◽  
Author(s):  
Sergei Scherbov ◽  
Warren C. Sanderson
Keyword(s):  
Life Cycle ◽  
Old Age ◽  
Human Life ◽  
Natural Generalization ◽  
Population Ageing ◽  
First Century ◽  
Greek City ◽  
Study Population ◽  
Twenty First Century ◽  
The One

Probably the most famous demographic riddle of all time is the one that the Sphinx was said to have posed to travellers outside the Greek city of Thebes: ‘Which creature walks on four legs in the morning, two at noon, and three in the evening?’ Unfortunate travellers who could not answer the riddle correctly were immediately devoured. Oedipus, fresh from killing his father, was the first to have got the answer right. The correct answer was ‘humans’. People crawl on their hands and knees as infants, walk on two feet in adulthood, and walk with a cane in old age. We easily recognize the three ages of humans. Humans are born dependent on the care of others. As they grow, their capacities and productivities generally increase, but eventually these reach a peak. After a while, capacities and productivities decline and, eventually, if they are lucky enough to survive, people become elderly, often again requiring transfers and care from others. The human life cycle is the basis of all studies of population ageing, and so we cannot begin to study population ageing without first answering the Sphinx’s riddle. However, answering the Sphinx’s riddle is not enough to get us started on a study of population ageing. We must take two more steps before we begin. First, we must recognize that not all people age at the same rate. As seen in Chapter 5, nowadays more educated people tend to have longer life expectancies than less educated people. Second, we must realize that there is no natural generalization of the Sphinx’s riddle to whole populations. Populations cannot be categorized into the stages of infancy, adulthood, and old age. Indeed, if the Sphinx was reborn today, we might find her sitting near another city and posing an equally perplexing riddle, one especially relevant for our times: ‘What can grow younger as it grows older?’ Answering this riddle correctly is the central challenge of this chapter and the key to understanding population ageing in the twenty-first century.

Induced Representations and Invariants

1950 ◽  
Vol 2 ◽  
pp. 334-343 ◽  
Author(s):  
G. DE B. Robinson
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Explicit Formula ◽  
Linear Group ◽  
Schur Functions ◽  
The Other ◽  
Induced Representations ◽  
Direct Sum ◽  
Invariant Matrices ◽  
The One

1. Introduction. The problem of the expression of an invariant matrix of an invariant matrix as a direct sum of invariant matrices is intimately associated with the representation theory of the full linear group on the one hand and with the representation theory of the symmetric group on the other. In a previous paper the author gave an explicit formula for this reduction in terms of characters of the symmetric group. Later J. A. Todd derived the same formula using Schur functions, i.e. characters of representations of the full linear group.

scholarly journals On Geodesics of a Modified Riemannian Manifold

1960 ◽  
Vol 3 (3) ◽  
pp. 255-261 ◽  
Author(s):  
D. K. Sen
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Geometry ◽  
Sufficient Condition

In Riemannian geometry the autoparallels associated with the affine connexion coincide with the geodesies which arise from the metric. This is not the case in a modification of Riemannian geometry suggested by Lyra. A sufficient condition that the two classes of curves coincide is obtained.

scholarly journals The super-Sasaki metric on the antitangent bundle

2020 ◽  
Vol 17 (08) ◽  
pp. 2050122
Author(s):  
Andrew James Bruce
Keyword(s):  
Riemannian Manifold ◽  
Tangent Bundle ◽  
Symplectic Form ◽  
Riemannian Metric ◽  
Riemannian Structure ◽  
Sasaki Metric

We show how to lift a Riemannian metric and almost symplectic form on a manifold to a Riemannian structure on a canonically associated supermanifold known as the antitangent or shifted tangent bundle. We view this construction as a generalization of Sasaki’s construction of a Riemannian metric on the tangent bundle of a Riemannian manifold.

scholarly journals A cut-off tubular geometry of loop space

2017 ◽  
Vol 29 (03) ◽  
pp. 1750006
Author(s):  
Partha Mukhopadhyay
Keyword(s):  
Quantum Mechanics ◽  
Riemannian Manifold ◽  
Loop Space ◽  
Indirect Method ◽  
Infinite Sequence ◽  
Cartesian Product ◽  
Riemannian Metrics ◽  
Expansion Theorem ◽  
Killing Equation ◽  
The One

Motivated by the computation of loop space quantum mechanics as indicated in [14], here we seek a better understanding of the tubular geometry of loop space [Formula: see text] corresponding to a Riemannian manifold [Formula: see text] around the submanifold of vanishing loops. Our approach is to first compute the tubular metric of [Formula: see text] around the diagonal submanifold, where [Formula: see text] is the Cartesian product of [Formula: see text] copies of [Formula: see text] with a cyclic ordering. This gives an infinite sequence of tubular metrics such that the one relevant to [Formula: see text] can be obtained by taking the limit [Formula: see text]. Such metrics are computed by adopting an indirect method where the general tubular expansion theorem of [21] is crucially used. We discuss how the complete reparametrization isometry of loop space arises in the large-[Formula: see text] limit and verify that the corresponding Killing equation is satisfied to all orders in tubular expansion. These tubular metrics can alternatively be interpreted as some natural Riemannian metrics on certain bundles of tangent spaces of [Formula: see text] which, for [Formula: see text], is the tangent bundle [Formula: see text].

scholarly journals PARTIAL ISOMETRIES OF A SUB-RIEMANNIAN MANIFOLD

2012 ◽  
Vol 23 (02) ◽  
pp. 1250043
Author(s):  
MAHUYA DATTA
Keyword(s):  
Riemannian Manifold ◽  
Tangent Bundle ◽  
Smooth Manifold ◽  
Partial Isometry ◽  
Riemannian Metric ◽  
Riemannian Structure ◽  
Partial Isometries ◽  
Smooth Map

In this article, we obtain the following generalization of isometric C1-immersion theorem of Nash and Kuiper. Let M be a smooth manifold of dimension m and H a rank k subbundle of the tangent bundle TM with a Riemannian metric gH. Then the pair (H, gH) defines a sub-Riemannian structure on M. We call a C1-map f : (M, H, gH) → (N, h) into a Riemannian manifold (N, h) a partial isometry if the derivative map df restricted to H is isometric, that is if f*h|H = gH. We prove that if f0 : M → N is a smooth map such that df0|H is a bundle monomorphism and [Formula: see text], then f0 can be homotoped to a C1-map f : M → N which is a partial isometry, provided dim N > k. As a consequence of this result, we obtain that every sub-Riemannian manifold (M, H, gH) admits a partial isometry in ℝn, provided n ≥ m + k.

scholarly journals Execute Elementary Row and Column Operations on the Partitioned Matrix to Compute M-P InverseA†

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Xingping Sheng
Keyword(s):  
Explicit Formula ◽  
New Method ◽  
New Approach ◽  
Elimination Process ◽  
Rectangular Matrix ◽  
The One

We first study the complexity of the algorithm presented in Guo and Huang (2010). After that, a new explicit formula for computational of the Moore-Penrose inverseA†of a singular or rectangular matrixA. This new approach is based on a modified Gauss-Jordan elimination process. The complexity of the new method is analyzed and presented and is found to be less computationally demanding than the one presented in Guo and Huang (2010). In the end, an illustrative example is demonstrated to explain the corresponding improvements of the algorithm.

scholarly journals Julia sets of complex Hénon maps

2018 ◽  
Vol 29 (07) ◽  
pp. 1850047
Author(s):  
Lorenzo Guerini ◽  
Han Peters
Keyword(s):  
Julia Set ◽  
A Priori ◽  
Natural Generalization ◽  
Dominated Splitting ◽  
Henon Maps ◽  
Additional Hypothesis ◽  
Hénon Maps ◽  
Open Questions ◽  
The One ◽  
Weaker Conditions

There are two natural definitions of the Julia set for complex Hénon maps: the sets [Formula: see text] and [Formula: see text]. Whether these two sets are always equal is one of the main open questions in the field. We prove equality when the map acts hyperbolically on the a priori smaller set [Formula: see text], under the additional hypothesis of substantial dissipativity. This result was claimed, without using the additional assumption, in [J. E. Fornæss, The julia set of hénon maps, Math. Ann. 334(2) (2006) 457–464], but the proof is incomplete. Our proof closely follows ideas from [J. E. Fornæss, The julia set of hénon maps, Math. Ann. 334(2) (2006) 457–464], deviating at two points, where substantial dissipativity is used. We show that [Formula: see text] also holds when hyperbolicity is replaced by one of the two weaker conditions. The first is quasi-hyperbolicity, introduced in [E. Bedford and J. Smillie, Polynomial diffeomorphisms of [Formula: see text]. VIII. Quasi-expansion. Amer. J. Math. 124(2) (2002) 221–271], a natural generalization of the one-dimensional notion of semi-hyperbolicity. The second is the existence of a dominated splitting on [Formula: see text]. Substantially dissipative, Hénon maps admitting a dominated splitting on the possibly larger set [Formula: see text] were recently studied in [M. Lyubich and H. Peters, Structure of partially hyperbolic hénon maps, ArXiv e-prints (2017)].

scholarly journals An explicit formula for ndinv, a new statistic for two-shuffle parking functions

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Angela Hicks ◽  
Yeonkyung Kim
Keyword(s):  
Explicit Formula ◽  
Parking Functions ◽  
Recursive Definition ◽  
Classical Definition ◽  
Inversion Number ◽  
International Audience ◽  
New Interpretation ◽  
The One ◽  
Definition Of ◽  
New Statistics

International audience In a recent paper, Duane, Garsia, and Zabrocki introduced a new statistic, "ndinv'', on a family of parking functions. The definition was guided by a recursion satisfied by the polynomial $\langle\Delta_{h_m}C_p1C_p2...C_{pk}1,e_n\rangle$, for $\Delta_{h_m}$ a Macdonald eigenoperator, $C_{p_i}$ a modified Hall-Littlewood operator and $(p_1,p_2,\dots ,p_k)$ a composition of n. Using their new statistics, they are able to give a new interpretation for the polynomial $\langle\nabla e_n, h_j h_n-j\rangle$ as a q,t numerator of parking functions by area and ndinv. We recall that in the shuffle conjecture, parking functions are q,t enumerated by area and diagonal inversion number (dinv). Since their definition is recursive, they pose the problem of obtaining a non recursive definition. We solved this problem by giving an explicit formula for ndinv similar to the classical definition of dinv. In this paper, we describe the work we did to construct this formula and to prove that the resulting ndinv is the same as the one recursively defined by Duane, Garsia, and Zabrocki. Dans un travail récent Duane, Garsia et Zabrocki ont introduit une nouvelle statistique, "ndinv'' pour une famille de Fonctions Parking. Ce "ndinv" découle d'une récurrence satisfaite par le polynôme $\langle\Delta_{h_m}C_p1C_p2...C_{pk}1,e_n\rangle$, oú $\Delta_{h_m}$ est un opérateur linéaire avec fonctions propres les polynômes de Macdonald, les $C_{p_i}$ sont des opérateurs de Hall-Littlewood modifiés et $(p_1,p_2,\dots ,p_n)$ est un vecteur à composantes entières positives. Par moyen de cette statistique, ils ont réussi à donner une nouvelle interprétation combinatoire au polynôme $\langle\nabla e_n, h_j h_n-j\rangle$ on remplaçant "dinv'" par "ndinv". Rappelons nous que la conjecture "Shuffle"' exprime ce même polynôme comme somme pondérée de Fonctions Parking avec poids t à la "aire'" est q au "dinv". Puisque il donnent une définition récursive du "ndinv" il posent le problème de l'obtenir d'une façon directe. On rèsout se problème en donnant une formule explicite qui permet de calculer directement le "ndinv" à la manière de la formule classique du "dinv". Dans cet article on décrit le travail qu'on a fait pour construire cette formule et on démontre que nôtre formule donne le même "ndinv" récursivement construit par Duane, Garsia et Zabrocki.

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