James Maclaurin - Academia.edu (original) (raw)

Uploads

Papers by James Maclaurin

arXiv: Probability, 2020

This paper concerns the large deviations of a system of interacting particles on a random graph. ... more This paper concerns the large deviations of a system of interacting particles on a random graph. There is no stochasticity, and the only sources of disorder are the random graph connections, and the initial condition. The average number of afferent edges on any particular vertex must diverge to infinity as NtoinftyN\to \inftyNtoinfty, but can do so at an arbitrarily slow rate. These results are thus accurate for both sparse and dense random graphs. A particular application to sparse Erdos-Renyi graphs is provided. The theorem is proved by pushing forward a Large Deviation Principle for a `nested empirical measure' generated by the initial conditions to the dynamics. The nested empirical measure can be thought of as the density of the density of edge connections: the associated weak topology is more coarse than the topology generated by the graph cut norm, and thus there is a broader range of application.

In this chapter we review classical algebraic geometry. 1.1 Algebraic Sets Algebraic Geometry is ... more In this chapter we review classical algebraic geometry. 1.1 Algebraic Sets Algebraic Geometry is the study of the zeros of polynomials. We will study polynomials in C[x 1 , x ,. .. x n ], and their corresponding zero-sets within C n. It is somewhat idiosyncratic to confine ourselves to polynomials over C. Many take a more general perspective and look at polynomials over a closed field k. In any case, the proof of the general case of most of these theorems is straightforward. We denote the algebra of polynomials in n dimensions by C[x 1 , x 2 ,. .. , x n ] Note that C[x 1 ,. .. , x n ] is also a ring. We denote the ideal generated by the polynomials {p 1 ,. .. , p k } as (p 1 ,. .. , p k). Definition 1.1.1. For any S ⊂ C[x 1 ,. .. x n ], the vanishing set of S is V (S) = {(ξ 1 ,. .. ξ n) ∈ C n : ∀f ∈ S f (ξ 1 ,. .. ξ n) = 0} Vanishing sets are geometric objects which may be specified algebraically (as the common zeros of a set of polynomials). For this reason they are also known as algebraic sets. The vanishing set of a single polynomial is known as a hypersurface. Definition 1.1.2. Conversely, for any X ⊂ C n we may define the ideal of all polynomials which are zero on X: I(X) = {f ∈ C[x 1 ,. .. x n : ∀α ∈ X f (α) = 0} Lemma 1.1.3. i. If F ⊂ F ⊂ C[x 1 ,. .. x n ] then V (F) ⊂ V (F). ii. If X ⊆ X ⊂ C n then I(X) ⊂ I(X). Proof. i. Let ξ ∈ V (F). Then ∀f ∈ F , f (ξ) = 0. But since F ⊂ F , it immediately follows that ∀f ∈ F , f (ξ) = 0 and hence ξ ∈ V (F). Thus V (F) ⊂ V (F). ii. If f ∈ I(X) then as X ⊂ X , f (x) = 0 for all x in X. So f ∈ I(X). 2 Corollary 1.1.4. If X ⊂ C m and S ⊂ C[x 1. .. x n ] then i. V (I(X)) ⊇ X ii. I(V (S)) ⊇ S iii. V (I(V (S))) = V (S) iv. I(V (I(X))) = I(X) Proof. (i) follows from applying Lemma 1.1.3 to X twice.

arXiv: Probability, 2020

This paper concerns the large deviations of a system of interacting particles on a random graph. ... more This paper concerns the large deviations of a system of interacting particles on a random graph. There is no stochasticity, and the only sources of disorder are the random graph connections, and the initial condition. The average number of afferent edges on any particular vertex must diverge to infinity as NtoinftyN\to \inftyNtoinfty, but can do so at an arbitrarily slow rate. These results are thus accurate for both sparse and dense random graphs. A particular application to sparse Erdos-Renyi graphs is provided. The theorem is proved by pushing forward a Large Deviation Principle for a `nested empirical measure' generated by the initial conditions to the dynamics. The nested empirical measure can be thought of as the density of the density of edge connections: the associated weak topology is more coarse than the topology generated by the graph cut norm, and thus there is a broader range of application.

In this chapter we review classical algebraic geometry. 1.1 Algebraic Sets Algebraic Geometry is ... more In this chapter we review classical algebraic geometry. 1.1 Algebraic Sets Algebraic Geometry is the study of the zeros of polynomials. We will study polynomials in C[x 1 , x ,. .. x n ], and their corresponding zero-sets within C n. It is somewhat idiosyncratic to confine ourselves to polynomials over C. Many take a more general perspective and look at polynomials over a closed field k. In any case, the proof of the general case of most of these theorems is straightforward. We denote the algebra of polynomials in n dimensions by C[x 1 , x 2 ,. .. , x n ] Note that C[x 1 ,. .. , x n ] is also a ring. We denote the ideal generated by the polynomials {p 1 ,. .. , p k } as (p 1 ,. .. , p k). Definition 1.1.1. For any S ⊂ C[x 1 ,. .. x n ], the vanishing set of S is V (S) = {(ξ 1 ,. .. ξ n) ∈ C n : ∀f ∈ S f (ξ 1 ,. .. ξ n) = 0} Vanishing sets are geometric objects which may be specified algebraically (as the common zeros of a set of polynomials). For this reason they are also known as algebraic sets. The vanishing set of a single polynomial is known as a hypersurface. Definition 1.1.2. Conversely, for any X ⊂ C n we may define the ideal of all polynomials which are zero on X: I(X) = {f ∈ C[x 1 ,. .. x n : ∀α ∈ X f (α) = 0} Lemma 1.1.3. i. If F ⊂ F ⊂ C[x 1 ,. .. x n ] then V (F) ⊂ V (F). ii. If X ⊆ X ⊂ C n then I(X) ⊂ I(X). Proof. i. Let ξ ∈ V (F). Then ∀f ∈ F , f (ξ) = 0. But since F ⊂ F , it immediately follows that ∀f ∈ F , f (ξ) = 0 and hence ξ ∈ V (F). Thus V (F) ⊂ V (F). ii. If f ∈ I(X) then as X ⊂ X , f (x) = 0 for all x in X. So f ∈ I(X). 2 Corollary 1.1.4. If X ⊂ C m and S ⊂ C[x 1. .. x n ] then i. V (I(X)) ⊇ X ii. I(V (S)) ⊇ S iii. V (I(V (S))) = V (S) iv. I(V (I(X))) = I(X) Proof. (i) follows from applying Lemma 1.1.3 to X twice.