ashvin rajan - Academia.edu (original) (raw)
Papers by ashvin rajan
RePEc: Research Papers in Economics, Aug 1, 1996
We adapt an elegant piece of reasoning by Balasko (1979) to the implete markets modelled by Duffi... more We adapt an elegant piece of reasoning by Balasko (1979) to the implete markets modelled by Duffie and Shafer (1985) and prove that on compact sets of such markets the Lebesgue measure of economies with m equilibria is 0 (1/m)
Journal of Mathematical Economics, Mar 1, 1999
... Department of Mathematics, The Johns Hopkins University, Baltimore, MD 21218, USA. Received 2... more ... Department of Mathematics, The Johns Hopkins University, Baltimore, MD 21218, USA. Received 26 August 1996; accepted 2 May 1997. Available online 21 January 1999. Abstract. We adapt an elegant piece of reasoning by Y. Balasko [Balasko, Y., 1979. ...
RePEc: Research Papers in Economics, Mar 1, 1996
We present a unified mathematical framework within which pure exchange economies with a finite se... more We present a unified mathematical framework within which pure exchange economies with a finite set of agents as well as those with a continuum of traders as mathematically modelled by Aumann (1964) can be analyzed simultaneously We prove that the number of equilibrium price vectors of our economies are generically finite Hence for markets with a continuum of traders the equilibrium allocations (which by the celebrated theorem of Aumann (1964) coincide with the core allocations) are finite for an open dense set of such markets This presents a limiting case result that complements similar asymptotic theorems about cores of large economies that have been proved by Bewley (1973) and Dierker (1975) If we assume that the measure on the space of agents is one with a finite number of atoms of equal weight our reasoning recovers the classical theorems proved about equilibria by Debreu (1970) for economies with a finite number of agents
1. Introduction. A rational right triangle is a right triangle whose sides are all positive ratio... more 1. Introduction. A rational right triangle is a right triangle whose sides are all positive rational numbers. Such a triangle is denoted {a, b, c} where a and b are the legs, and c is the hypotenuse. Throughout this paper, a squarefree integer is understood to be a positive integer which is not divisible by the square of an integer greater than 1. A congruent number is a square-free integer which is the area of a rational right triangle. A square-free integer N is a congruent number if and only if the elliptic curve N y 2 = (x 2 − 1)x has positive rank. For details, see Koblitz [7]. In the spirit of Euclid's proof of the infinitude of prime numbers, one can also show that there are infinitely many congruent numbers as follows: If there were only finitely many of them, say N 1 ,. .. , N r , all greater than 1, then consider N = N 1 • • • N r. Elementary number theory shows that sqf(N 3 − N), the square-free part of N 3 − N, cannot be 1. Moreover, it is a congruent number which cannot be any of the N i 's. Indeed, if it were N 1 , say, let M = N 2 • • • N r and d = gcd(N 1 , M). Writing N 1 = dn and M = dm with gcd(m, n) = 1, one sees that sqf(m) sqf(N 2 1 M 2 − 1) = d. This last equality implies that sqf(N 2 1 M 2 − 1) divides d, and hence M, but at the same time, since it divides N 2 1 M 2 − 1, it must be 1, and this is impossible. Chahal [2] established that the residue classes of 1, 2, 3, 5, 6, 7 modulo 8 contain infinitely many congruent numbers. Bennett [1] extended Chahal's result by showing that if a and m are positive integers such that gcd(a, m) is square-free, then the residue class of a modulo m contains infinitely many congruent numbers.
Journal of Mathematical Economics, 1997
We present a unified mathematical framework within which pure exchange economies with a finite se... more We present a unified mathematical framework within which pure exchange economies with a finite set of agents as well as those with a continuum of traders as mathematically modelled by Aumann (1964) can be analyzed simultaneously We prove that the number of equilibrium price vectors of our economies are generically finite Hence for markets with a continuum of traders the equilibrium allocations (which by the celebrated theorem of Aumann (1964) coincide with the core allocations) are finite for an open dense set of such markets This presents a limiting case result that complements similar asymptotic theorems about cores of large economies that have been proved by Bewley (1973) and Dierker (1975) If we assume that the measure on the space of agents is one with a finite number of atoms of equal weight our reasoning recovers the classical theorems proved about equilibria by Debreu (1970) for economies with a finite number of agents
Bulletin of the Australian Mathematical Society, 2022
We show that there exists a continuous function from the unit Lebesgue interval to itself such th... more We show that there exists a continuous function from the unit Lebesgue interval to itself such that for any epsilongeq0\epsilon \geq 0epsilongeq0 and any natural number k, any point in its domain has an epsilon\epsilon epsilon -neighbourhood which, when feasible, contains k mutually disjoint extremally scrambled sets of identical Lebesgue measure, homeomorphic to each other. This result enables a satisfying generalisation of Li–Yorke (topological) chaos and suggests an open (difficult) problem as to whether the result is valid for piecewise linear functions.
Economic Theory, 1997
We present a unified mathematical framework within which among others pure exchange economies wit... more We present a unified mathematical framework within which among others pure exchange economies with a finite set of agents as well as those with a continuum of traders may be studied simultaneously We prove that the reasoning presented by Balasko (1975) on the equilibrium set for finite economies generalizes very naturally to our setting His results may therefore be recovered as a special case of those presented in this note
International Journal of Bifurcation and Chaos
For any fixed positive real number [Formula: see text], we generalize a construction devised by K... more For any fixed positive real number [Formula: see text], we generalize a construction devised by Kan [1984] to construct continuous chaotic functions which possess scrambled sets whose Lebesgue measure is exactly equal to [Formula: see text].
Bulletin of the Australian Mathematical Society
We present a family of continuous piecewise linear maps of the unit interval into itself that are... more We present a family of continuous piecewise linear maps of the unit interval into itself that are all chaotic in the sense of Li and Yorke [‘Period three implies chaos’, Amer. Math. Monthly 82 (1975), 985–992] and for which almost every point (in the sense of Lebesgue) in the unit interval is an eventually periodic point of period p,pgeq3p,p\geq 3p,pgeq3 , for a member of the family.
Economics Working Paper Archive, Aug 1, 1996
We adapt an elegant piece of reasoning by Balasko (1979) to the implete markets modelled by Duffi... more We adapt an elegant piece of reasoning by Balasko (1979) to the implete markets modelled by Duffie and Shafer (1985) and prove that on compact sets of such markets the Lebesgue measure of economies with m equilibria is 0 (1/m)
Using relations between Lambert series, Eisenstein series, and the Weierstrass ℘- function, we ev... more Using relations between Lambert series, Eisenstein series, and the Weierstrass ℘- function, we evaluate the sums of certain infinite series in terms of Bernoulli numbers, Hurwitz numbers, and the arc length of the lemniscate.
Economic Theory, 1996
We present a unified mathematical framework within which, among others, pure exchange economies ... more We present a unified mathematical framework within which, among others, pure exchange economies with a finite set of agents, as well as those with a continuum of traders may be studied simultaneously. We prove that the reasoning presented by Balasko (1975) on the equilibrium set for finite economies generalizes very naturally to our setting. His results may therefore be recovered as a special case of those presented in this note.
Journal of Mathematical Economics, 1996
We adapt an elegant piece of reasoning by Y. Balasko [Balasko, Y., 1979. Economies with a finite ... more We adapt an elegant piece of reasoning by Y. Balasko [Balasko, Y., 1979. Economies with a finite but large number of equilibria. J. Math. Econ. 6, 145–147.] to the incomplete markets modelled by D. Duffie and W. Shafer [Duffie, D., Shafer, W., 1985. Equilibrium in incomplete markets: I. A basic model of generic existence. J. Math. Econ. 14, 285–300.], and prove that on compact sets of such markets, the Lebesgue measure of economies with m equilibria is O().
Using relations between Lambert series, Eisenstein series, and the Weierstrass ℘- function, we ev... more Using relations between Lambert series, Eisenstein series, and the Weierstrass ℘- function, we evaluate the sums of certain infinite series in terms of Bernoulli numbers, Hurwitz numbers, and the arc length of the lemniscate.
Rocky Mountain Journal of Mathematics, 2009
... TERUTAKE ABE, ASHVIN RAJAN AND FRANC OIS RAMAROSON Dedicated to Professor Takashi Ono ... is ... more ... TERUTAKE ABE, ASHVIN RAJAN AND FRANC OIS RAMAROSON Dedicated to Professor Takashi Ono ... is a triangle whose three sides all have rational lengths.) We call ( 2)-congruent numbers "congruent numbers." Our inspiration was the following pretty density argument ...
Journal of Mathematical Economics, 1996
We show that for an open dense set of markets with a continuum of traders the number of equilibri... more We show that for an open dense set of markets with a continuum of traders the number of equilibrium allocations [which by the celebrated theorem of Aumann (Econometrica. 1964, 32. 39–50) coincide with the core allocations for such markets are finite. This presents a limiting case result that complements similar asymptotic theorems for cores of large economies proved by Bewley (Econometrica 1973, 41, 425–454), and Dierker (Journal of Mathematical Economics 1975, 2. 155–169). If we require that the measure on the space of agents be one with a finite number of atoms of equal weight, our reasoning recovers the classical results obtained by Debreu (Econometrica. 1970. 38, 387–392) for economies with a finite number of agents.
Acta Arithmetica, 2007
ABSTRACT
RePEc: Research Papers in Economics, Aug 1, 1996
We adapt an elegant piece of reasoning by Balasko (1979) to the implete markets modelled by Duffi... more We adapt an elegant piece of reasoning by Balasko (1979) to the implete markets modelled by Duffie and Shafer (1985) and prove that on compact sets of such markets the Lebesgue measure of economies with m equilibria is 0 (1/m)
Journal of Mathematical Economics, Mar 1, 1999
... Department of Mathematics, The Johns Hopkins University, Baltimore, MD 21218, USA. Received 2... more ... Department of Mathematics, The Johns Hopkins University, Baltimore, MD 21218, USA. Received 26 August 1996; accepted 2 May 1997. Available online 21 January 1999. Abstract. We adapt an elegant piece of reasoning by Y. Balasko [Balasko, Y., 1979. ...
RePEc: Research Papers in Economics, Mar 1, 1996
We present a unified mathematical framework within which pure exchange economies with a finite se... more We present a unified mathematical framework within which pure exchange economies with a finite set of agents as well as those with a continuum of traders as mathematically modelled by Aumann (1964) can be analyzed simultaneously We prove that the number of equilibrium price vectors of our economies are generically finite Hence for markets with a continuum of traders the equilibrium allocations (which by the celebrated theorem of Aumann (1964) coincide with the core allocations) are finite for an open dense set of such markets This presents a limiting case result that complements similar asymptotic theorems about cores of large economies that have been proved by Bewley (1973) and Dierker (1975) If we assume that the measure on the space of agents is one with a finite number of atoms of equal weight our reasoning recovers the classical theorems proved about equilibria by Debreu (1970) for economies with a finite number of agents
1. Introduction. A rational right triangle is a right triangle whose sides are all positive ratio... more 1. Introduction. A rational right triangle is a right triangle whose sides are all positive rational numbers. Such a triangle is denoted {a, b, c} where a and b are the legs, and c is the hypotenuse. Throughout this paper, a squarefree integer is understood to be a positive integer which is not divisible by the square of an integer greater than 1. A congruent number is a square-free integer which is the area of a rational right triangle. A square-free integer N is a congruent number if and only if the elliptic curve N y 2 = (x 2 − 1)x has positive rank. For details, see Koblitz [7]. In the spirit of Euclid's proof of the infinitude of prime numbers, one can also show that there are infinitely many congruent numbers as follows: If there were only finitely many of them, say N 1 ,. .. , N r , all greater than 1, then consider N = N 1 • • • N r. Elementary number theory shows that sqf(N 3 − N), the square-free part of N 3 − N, cannot be 1. Moreover, it is a congruent number which cannot be any of the N i 's. Indeed, if it were N 1 , say, let M = N 2 • • • N r and d = gcd(N 1 , M). Writing N 1 = dn and M = dm with gcd(m, n) = 1, one sees that sqf(m) sqf(N 2 1 M 2 − 1) = d. This last equality implies that sqf(N 2 1 M 2 − 1) divides d, and hence M, but at the same time, since it divides N 2 1 M 2 − 1, it must be 1, and this is impossible. Chahal [2] established that the residue classes of 1, 2, 3, 5, 6, 7 modulo 8 contain infinitely many congruent numbers. Bennett [1] extended Chahal's result by showing that if a and m are positive integers such that gcd(a, m) is square-free, then the residue class of a modulo m contains infinitely many congruent numbers.
Journal of Mathematical Economics, 1997
We present a unified mathematical framework within which pure exchange economies with a finite se... more We present a unified mathematical framework within which pure exchange economies with a finite set of agents as well as those with a continuum of traders as mathematically modelled by Aumann (1964) can be analyzed simultaneously We prove that the number of equilibrium price vectors of our economies are generically finite Hence for markets with a continuum of traders the equilibrium allocations (which by the celebrated theorem of Aumann (1964) coincide with the core allocations) are finite for an open dense set of such markets This presents a limiting case result that complements similar asymptotic theorems about cores of large economies that have been proved by Bewley (1973) and Dierker (1975) If we assume that the measure on the space of agents is one with a finite number of atoms of equal weight our reasoning recovers the classical theorems proved about equilibria by Debreu (1970) for economies with a finite number of agents
Bulletin of the Australian Mathematical Society, 2022
We show that there exists a continuous function from the unit Lebesgue interval to itself such th... more We show that there exists a continuous function from the unit Lebesgue interval to itself such that for any epsilongeq0\epsilon \geq 0epsilongeq0 and any natural number k, any point in its domain has an epsilon\epsilon epsilon -neighbourhood which, when feasible, contains k mutually disjoint extremally scrambled sets of identical Lebesgue measure, homeomorphic to each other. This result enables a satisfying generalisation of Li–Yorke (topological) chaos and suggests an open (difficult) problem as to whether the result is valid for piecewise linear functions.
Economic Theory, 1997
We present a unified mathematical framework within which among others pure exchange economies wit... more We present a unified mathematical framework within which among others pure exchange economies with a finite set of agents as well as those with a continuum of traders may be studied simultaneously We prove that the reasoning presented by Balasko (1975) on the equilibrium set for finite economies generalizes very naturally to our setting His results may therefore be recovered as a special case of those presented in this note
International Journal of Bifurcation and Chaos
For any fixed positive real number [Formula: see text], we generalize a construction devised by K... more For any fixed positive real number [Formula: see text], we generalize a construction devised by Kan [1984] to construct continuous chaotic functions which possess scrambled sets whose Lebesgue measure is exactly equal to [Formula: see text].
Bulletin of the Australian Mathematical Society
We present a family of continuous piecewise linear maps of the unit interval into itself that are... more We present a family of continuous piecewise linear maps of the unit interval into itself that are all chaotic in the sense of Li and Yorke [‘Period three implies chaos’, Amer. Math. Monthly 82 (1975), 985–992] and for which almost every point (in the sense of Lebesgue) in the unit interval is an eventually periodic point of period p,pgeq3p,p\geq 3p,pgeq3 , for a member of the family.
Economics Working Paper Archive, Aug 1, 1996
We adapt an elegant piece of reasoning by Balasko (1979) to the implete markets modelled by Duffi... more We adapt an elegant piece of reasoning by Balasko (1979) to the implete markets modelled by Duffie and Shafer (1985) and prove that on compact sets of such markets the Lebesgue measure of economies with m equilibria is 0 (1/m)
Using relations between Lambert series, Eisenstein series, and the Weierstrass ℘- function, we ev... more Using relations between Lambert series, Eisenstein series, and the Weierstrass ℘- function, we evaluate the sums of certain infinite series in terms of Bernoulli numbers, Hurwitz numbers, and the arc length of the lemniscate.
Economic Theory, 1996
We present a unified mathematical framework within which, among others, pure exchange economies ... more We present a unified mathematical framework within which, among others, pure exchange economies with a finite set of agents, as well as those with a continuum of traders may be studied simultaneously. We prove that the reasoning presented by Balasko (1975) on the equilibrium set for finite economies generalizes very naturally to our setting. His results may therefore be recovered as a special case of those presented in this note.
Journal of Mathematical Economics, 1996
We adapt an elegant piece of reasoning by Y. Balasko [Balasko, Y., 1979. Economies with a finite ... more We adapt an elegant piece of reasoning by Y. Balasko [Balasko, Y., 1979. Economies with a finite but large number of equilibria. J. Math. Econ. 6, 145–147.] to the incomplete markets modelled by D. Duffie and W. Shafer [Duffie, D., Shafer, W., 1985. Equilibrium in incomplete markets: I. A basic model of generic existence. J. Math. Econ. 14, 285–300.], and prove that on compact sets of such markets, the Lebesgue measure of economies with m equilibria is O().
Using relations between Lambert series, Eisenstein series, and the Weierstrass ℘- function, we ev... more Using relations between Lambert series, Eisenstein series, and the Weierstrass ℘- function, we evaluate the sums of certain infinite series in terms of Bernoulli numbers, Hurwitz numbers, and the arc length of the lemniscate.
Rocky Mountain Journal of Mathematics, 2009
... TERUTAKE ABE, ASHVIN RAJAN AND FRANC OIS RAMAROSON Dedicated to Professor Takashi Ono ... is ... more ... TERUTAKE ABE, ASHVIN RAJAN AND FRANC OIS RAMAROSON Dedicated to Professor Takashi Ono ... is a triangle whose three sides all have rational lengths.) We call ( 2)-congruent numbers "congruent numbers." Our inspiration was the following pretty density argument ...
Journal of Mathematical Economics, 1996
We show that for an open dense set of markets with a continuum of traders the number of equilibri... more We show that for an open dense set of markets with a continuum of traders the number of equilibrium allocations [which by the celebrated theorem of Aumann (Econometrica. 1964, 32. 39–50) coincide with the core allocations for such markets are finite. This presents a limiting case result that complements similar asymptotic theorems for cores of large economies proved by Bewley (Econometrica 1973, 41, 425–454), and Dierker (Journal of Mathematical Economics 1975, 2. 155–169). If we require that the measure on the space of agents be one with a finite number of atoms of equal weight, our reasoning recovers the classical results obtained by Debreu (Econometrica. 1970. 38, 387–392) for economies with a finite number of agents.
Acta Arithmetica, 2007
ABSTRACT