Generating functions

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Discrete Time ◽  
Generating Functions ◽  
Final Section ◽  
Lagrangian Submanifolds ◽  
Variational Problems ◽  
Action Functional ◽  
Arnold Conjecture ◽  
Cotangent Bundles ◽  
Symplectic Action

This chapter discusses generating functions in more detail. It shows how generating functions give rise to discrete-time analogues of the symplectic action functional and hence lead to discrete variational problems. The results of this chapter form the basis for the proofs in Chapter 11 of the Arnold conjecture for the torus and in Chapter 12 of the existence of the Hofer–Zehnder capacity. The final section examines generating functions for exact Lagrangian submanifolds of cotangent bundles.

The arnold conjecture

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Fixed Points ◽  
Generating Functions ◽  
Floer Homology ◽  
Lagrangian Submanifolds ◽  
Index Theory ◽  
Intersection Problem ◽  
Arnold Conjecture ◽  
Conley Index Theory ◽  
Cotangent Bundles ◽  
Symplectic Action

This chapter contains a proof of the Arnold conjecture for the standard torus, which is based on the discrete symplectic action. The symplectic part of this proof is very easy. However, for completeness of the exposition, one section is devoted to a fairly detailed discussion of the relevant Conley index theory and of Ljusternik–Schnirelmann theory. Closely related to the problem of finding symplectic fixed points is the Lagrangian intersection problem. The chapter outlines a proof of Arnold’s conjecture for cotangent bundles that again uses the discrete symplectic action, this time to construct generating functions for Lagrangian submanifolds. The chapter ends with a brief outline of the construction and applications of Floer homology.

scholarly journals Asymptotic results for the first and second moments and numerical computations in discrete-time bulk-renewal process

2019 ◽  
Vol 29 (1) ◽  
pp. 135-144
Author(s):  
James Kim ◽  
Mohan Chaudhry ◽  
Abdalla Mansur
Keyword(s):  
Discrete Time ◽  
Renewal Process ◽  
Generating Functions ◽  
Continuous Time ◽  
Constant Term ◽  
Numerical Results ◽  
Asymptotic Results ◽  
Numerical Computations ◽  
Second Moments ◽  
Simplified Solution

This paper introduces a simplified solution to determine the asymptotic results for the renewal density. It also offers the asymptotic results for the first and second moments of the number of renewals for the discrete-time bulk-renewal process. The methodology adopted makes this study distinguishable compared to those previously published where the constant term in the second moment is generated. In similar studies published in the literature, the constant term is either missing or not clear how it was obtained. The problem was partially solved in the study by Chaudhry and Fisher where they provided a asymptotic results for the non-bulk renewal density and for both the first and second moments using the generating functions. The objective of this work is to extend their results to the bulk-renewal process in discrete-time, including some numerical results, give an elegant derivation of the asymptotic results and derive continuous-time results as a limit of the discrete-time results.

scholarly journals Two-Sex Branching Processes with Offspring and Mating in a Random Environment

2009 ◽  
Vol 46 (04) ◽  
pp. 993-1004
Author(s):  
S. Ma ◽  
M. Molina
Keyword(s):  
Mathematical Model ◽  
Probability Distribution ◽  
Discrete Time ◽  
Random Environment ◽  
Generating Functions ◽  
Branching Processes ◽  
Random Vectors ◽  
Environmental Process ◽  
Mating Function ◽  
The Mathematical Model

We introduce a class of discrete-time two-sex branching processes where the offspring probability distribution and the mating function are governed by an environmental process. It is assumed that the environmental process is formed by independent but not necessarily identically distributed random vectors. For such a class, we determine some relationships among the probability generating functions involved in the mathematical model and derive expressions for the main moments. Also, by considering different probabilistic approaches we establish several results concerning the extinction probability. A simulated example is presented as an illustration.

scholarly journals Exact Lagrangian submanifolds in simply-connected cotangent bundles

2007 ◽  
Vol 172 (1) ◽  
pp. 1-27 ◽  
Author(s):  
Kenji Fukaya ◽  
Paul Seidel ◽  
Ivan Smith
Keyword(s):  
Lagrangian Submanifolds ◽  
Cotangent Bundles ◽  
Simply Connected

scholarly journals Leakage of rank-dependent functionally generated trading strategies

2020 ◽  
Vol 16 (4) ◽  
pp. 573-591
Author(s):  
Kangjianan Xie
Keyword(s):  
Discrete Time ◽  
Generating Functions ◽  
Trading Strategy ◽  
Trading Strategies ◽  
Finite Variation ◽  
Leakage Effect

AbstractThis paper investigates the so-called leakage effect of trading strategies generated functionally from rank-dependent portfolio generating functions. This effect measures the loss in wealth of trading strategies due to renewing the portfolio constituent stocks. Theoretically, the leakage effect of a trading strategy is expressed explicitly by a finite-variation term. The computation of the leakage is different from what previous research has suggested. The method to estimate leakage in discrete time is then introduced with some practical considerations. An empirical example illustrates the leakage of the corresponding trading strategies under different constituent list sizes.

Chern–Simons gauge theory and symplectic quantum mechanics

2015 ◽  
Vol 93 (9) ◽  
pp. 971-973
Author(s):  
Lisa Jeffrey
Keyword(s):  
Quantum Mechanics ◽  
Gauge Theory ◽  
Partition Function ◽  
Partition Functions ◽  
Action Functional ◽  
Symplectic Action ◽  
Chern Simons

We describe the relation between the Chern–Simons gauge theory partition function and the partition function defined using the symplectic action functional as the Lagrangian. We show that the partition functions obtained using these two Lagrangians agree, and we identify the semiclassical formula for the partition function defined using the symplectic action functional. We also compute the semiclassical formulas for the partition functions obtained using the two different Lagrangians: the Chern–Simons functional and the symplectic action functional.

scholarly journals Discrete-time fractional variational problems

2011 ◽  
Vol 91 (3) ◽  
pp. 513-524 ◽  
Author(s):  
Nuno R.O. Bastos ◽  
Rui A.C. Ferreira ◽  
Delfim F.M. Torres
Keyword(s):  
Discrete Time ◽  
Variational Problems ◽  
Fractional Variational Problems

ON THE DISCRETE-TIME G/GI/∞ QUEUE

2008 ◽  
Vol 22 (4) ◽  
pp. 557-585 ◽  
Author(s):  
Iddo Eliazar
Keyword(s):  
Fixed Points ◽  
Discrete Time ◽  
Generating Functions ◽  
Random Variables ◽  
Output Process ◽  
Input Output ◽  
Input Process ◽  
Probability Generating Functions ◽  
Queue Model ◽  
Multidimensional Probability

The discrete-time G/GI/∞ queue model is explored. Jobs arrive to an infinite-server queuing system following an arbitrary input process X; job sizes are general independent and identically distributed random variables. The system's output process Y (of job departures) and queue process N (tracking the number of jobs present in the system) are analyzed. Various statistics of the stochastic maps X↦ Y and X↦ N are explicitly obtained, including means, variances, autocovariances, cross-covariances, and multidimensional probability generating functions. In the case of stationary inputs, we further compute the spectral densities of the stochastic maps, characterize the fixed points (in the L2 sense) of the input–output map, precisely determine when the output and queue processes display either short-ranged or long-ranged temporal dependencies, and prove a decomposition result regarding the intrinsic L2 structure of general stationary G/GI/∞ systems.

Sign in / Sign up

Export Citation Format

Share Document