scholarly journals Can We Make a Finsler Metric Complete by a Trivial Projective Change?

2012 ◽  
pp. 231-242 ◽  
Author(s):  
Vladimir S. Matveev
Keyword(s):  
Finsler Metric ◽  
Projective Change

scholarly journals β-Change of Riemannian Space and Special Cases of one-form β

2018 ◽  
Vol 1 (1) ◽  
pp. 30-37
Author(s):  
Khageshwar Mandal
Keyword(s):  
Riemannian Space ◽  
Homogeneous Function ◽  
Finsler Metric ◽  
Special Cases ◽  
Projective Change

In this paper, we consider β-change of Finsler metric L given by L = f(L; β), where f is any positively homogeneous function of degree one in L and β. The objectives of the paper are to obtain the β-change of a Riemannian space, projective change of Finsler metric and to discuss the special cases of one-form β.

A geometric approach to quantum circuit lower bounds

2006 ◽  
Vol 6 (3) ◽  
pp. 213-262 ◽  
Author(s):  
M.A. Nielsen
Keyword(s):  
Lower Bounds ◽  
Geometric Approach ◽  
Quantum Circuit ◽  
Geodesic Equation ◽  
Initial Position ◽  
Minimal Length ◽  
Finsler Metric ◽  
Finsler Metrics ◽  
Minimal Size ◽  
Circuit Lower Bounds

What is the minimal size quantum circuit required to exactly implement a specified n-qubit unitary operation, U, without the use of ancilla qubits? We show that a lower bound on the minimal size is provided by the length of the minimal geodesic between U and the identity, I, where length is defined by a suitable Finsler metric on the manifold SU(2^n). The geodesic curves on these manifolds have the striking property that once an initial position and velocity are set, the remainder of the geodesic is completely determined by a second order differential equation known as the geodesic equation. This is in contrast with the usual case in circuit design, either classical or quantum, where being given part of an optimal circuit does not obviously assist in the design of the rest of the circuit. Geodesic analysis thus offers a potentially powerful approach to the problem of proving quantum circuit lower bounds. In this paper we construct several Finsler metrics whose minimal length geodesics provide lower bounds on quantum circuit size. For each Finsler metric we give a procedure to compute the corresponding geodesic equation. We also construct a large class of solutions to the geodesic equation, which we call \emph{Pauli geodesics}, since they arise from isometries generated by the Pauli group. For any unitary U diagonal in the computational basis, we show that: (a) provided the minimal length geodesic is unique, it must be a Pauli geodesic; (b) finding the length of the minimal Pauli geodesic passing from I to U is equivalent to solving an exponential size instance of the closest vector in a lattice problem (CVP); and (c) all but a doubly exponentially small fraction of such unitaries have minimal Pauli geodesics of exponential length.

On Randers change of a Finsler space with mth-root metric

2014 ◽  
Vol 11 (10) ◽  
pp. 1450087 ◽  
Author(s):  
Bankteshwar Tiwari ◽  
Manoj Kumar
Keyword(s):  
Finsler Space ◽  
Riemannian Metrics ◽  
Finsler Metric

In this paper, we find a condition under which a Finsler space with Randers change of mth-root metric is projectively related to a mth-root metric and also we find a condition under which this Randers transformed mth-root Finsler metric is locally dually flat. Moreover, if transformed Finsler metric is conformal to the mth-root Finsler metric, then we prove that both of them reduce to Riemannian metrics.

LAPLACIAN COMPARISON ON COMPLEX FINSLER MANIFOLDS AND ITS APPLICATIONS

2013 ◽  
Vol 24 (05) ◽  
pp. 1350034
Author(s):  
JINXIU XIAO ◽  
CHUNHUI QIU ◽  
QUN HE ◽  
ZHIHUA CHEN
Keyword(s):  
Maximum Principle ◽  
Distance Function ◽  
Finsler Metric ◽  
Finsler Manifolds ◽  
Complex Finsler Metric ◽  
Bonnet Theorem

By defining the Rund Laplacian, we obtain the first and the second holomorphic variation formulas for the strongly pseudoconvex complex Finsler metric. Using the holomorphic variation formulas, we get an estimate for the Levi forms of distance function on complex Finsler manifolds. Further, an estimate for the Rund Laplacians of distance function on strongly pseudoconvex complex Finsler manifolds is obtained. As its applications, we get the Bonnet theorem and maximum principle on complex Finsler manifolds.

scholarly journals Jacobi-Maupertuis Randers-Finsler metric for curved spaces and the gravitational magnetoelectric effect

2019 ◽  
Vol 60 (12) ◽  
pp. 122501 ◽  
Author(s):  
Sumanto Chanda ◽  
G. W. Gibbons ◽  
Partha Guha ◽  
Paolo Maraner ◽  
Marcus C. Werner
Keyword(s):  
Magnetoelectric Effect ◽  
Finsler Metric ◽  
Curved Spaces

scholarly journals The Β-Change by Finsler Metric of C-Reducible Finsler Spaces in Finsler Geometry

2019 ◽  
Vol 33 (1) ◽  
pp. 1-10
Author(s):  
Khageswar Mandal
Keyword(s):  
Finsler Space ◽  
Homogeneous Function ◽  
Finsler Geometry ◽  
Finsler Metric ◽  
Finsler Spaces

 This paper considered about the β-Change of Finsler metric L given by L*= f(L, β), where f is any positively homogeneous function of degree one in L and β and obtained the β-Change by Finsler metric of C-reducible Finsler spaces. Also further obtained the condition that a C-reducible Finsler space is transformed to a C-reducible Finsler space by a β-change of Finsler metric.

GEODESICS AND OPERATOR MEANS IN THE SPACE OF POSITIVE OPERATORS

1993 ◽  
Vol 04 (02) ◽  
pp. 193-202 ◽  
Author(s):  
GUSTAVO CORACH ◽  
HORACIO PORTA ◽  
LÁZARO RECHT
Keyword(s):  
Geometric Mean ◽  
Geodesic Segment ◽  
Finsler Metric ◽  
Natural Structure ◽  
Operator Inequalities ◽  
C Algebra ◽  
Operator Means ◽  
Natural Notion ◽  
Invertible Elements ◽  
Classical Convexity

The set A+ of positive invertible elements of a C*-algebra has a natural structure of reductive homogeneous manifold with a Finsler metric. Because pairs of points can be joined by uniquely determined geodesics and geodesics are "short" curves, there is a natural notion of convexity: C ⊂ A+ is convex if the geodesic segment joining a, b ∈ C is contained in C. We show that this notion is related to the classical convexity of real and operator valued functions. Several results about convexity are proved in this paper. The expressions of these results are closely related to the operator means of Kubo and Ando, in particular to the geometric mean of Pusz and Woronowicz, and they produce several norm estimations and operator inequalities.

scholarly journals METRIC COMPATIBLE OR NON-COMPATIBLE FINSLER–RICCI FLOWS

2012 ◽  
Vol 09 (05) ◽  
pp. 1250041 ◽  
Author(s):  
SERGIU I. VACARU
Keyword(s):  
Ricci Flow ◽  
Evolution Equations ◽  
Finsler Geometry ◽  
Flow Theory ◽  
Finsler Metric ◽  
Geometric Evolution ◽  
Ricci Flows ◽  
Canonical Metric ◽  
Self Consistent ◽  
Math Phys

There were elaborated different models of Finsler geometry using the Cartan (metric compatible), or Berwald and Chern (metric non-compatible) connections, the Ricci flag curvature, etc. In a series of works, we studied (non)-commutative metric compatible Finsler and non-holonomic generalizations of the Ricci flow theory [see S. Vacaru, J. Math. Phys. 49 (2008) 043504; 50 (2009) 073503 and references therein]. The aim of this work is to prove that there are some models of Finsler gravity and geometric evolution theories with generalized Perelman's functionals, and correspondingly derived non-holonomic Hamilton evolution equations, when metric non-compatible Finsler connections are involved. Following such an approach, we have to consider distortion tensors, uniquely defined by the Finsler metric, from the Cartan and/or the canonical metric compatible connections. We conclude that, in general, it is not possible to elaborate self-consistent models of geometric evolution with arbitrary Finsler metric non-compatible connections.

ON KROPINA-RANDERS CHANGE OF mth-ROOT FINSLER METRIC

2017 ◽  
Vol 20 (2) ◽  
pp. 115-128
Author(s):  
Gauree Shanker ◽  
Sruthy Asha Baby
Keyword(s):  
Finsler Metric
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