Using Continued Fractions to Solve Diophantine Equation

Diophantine Equation

Given a diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers [Hilbert, 1902].

From: Handbook of the History of Logic , 2009

Introduction

T.K. Puttaswamy , in Mathematical Achievements of Pre-Modern Indian Mathematicians, 2012

One of the most significant contributions of ancient Indians to mathematics is in providing a systematic method of solution in integers of the indeterminate equation of the first degree, "Kuttaka" (Pulveriser) by−ax=c , presently known as "Diophantine equation," after the great Greek mathematician, Diophantus, who lived probably in the latter half of the third century ad, as well as of the indeterminate equation of the second degree, "Vargaprakriti" ("Varga" meaning square and "Prakrti" referring to the coefficient N) Nx ²+k=y ². The former has been treated for the first time in the history of mathematics by Aryabhata I, while the latter is a phenomenal contribution of Brahmagupta. Brahmagupta solved completely to find the rational solutions of the equation Nx ²+k=y ². He also exhibited how to find the integral solutions of the equation Nx ²+1=y ², provided an integral solution (x y)=(α β) can be found for the equation Nx ²+k=y ², when k=−1, ±2, ±4. The whole modern world would have to admit that Brahmagupta must have been a genius to accomplish this marvelous feat in ad 628, without using continued fractions, which is the method now employed to solve "Vargaprakriti." The equation Nx ²+1=y ² at present bears the name "Pell's equation" after the English mathematician John Pell (ad 1610–1689), although his association with it consists of merely the publication of solutions of it in his edition of Brounker's translation of Rhonius Algebra in ad 1668. It was an accident that the celebrated Swiss mathematician Leonhard Euler (ad 1707–1783) referred to it as "Pell's equation." But this has no historical justification whatsoever. Pell has made no contribution to this topic.

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Stream Ciphers and Number Theory

In North-Holland Mathematical Library, 2004

4.3.7 The Case d = 10

The cyclotomic numbers of order ten were attacked by Dickson [106], Bruck [43] and Whiteman [455] with different approaches. The complete tables of the cyclotomic constants of order ten have been given by Whiteman [455].

Dickson showed that if p is a prime of the form 5k + 1, then there are exactly four integral simultaneous solutions of the pair of diophantine equations

(4.3) $16 p = x^{2} + 50 u^{2} + 50 v^{2} + 125 w^{2}, x w = v^{2} - 4 u v - u^{2},$

with x uniquely determined by the condition x ≡ 1 (mod 5). The four solutions are given by (x, u, v, w), (x, v, −u, −w), (x, −u, −v, w), (x, −v, u, −w).

The 100 constants (h, k) have at most 22 different values for a given p, which are expressible in terms of p, x, u, v. Tables 4.4 and 4.5 summarize the relations of the constants in two cases, where (ij) denotes (i, j). There are ten sets of formulas depending on the parity of f and the quintic residue character of 2 modulo p. The 22 essentially different formulas of each set are given in the accompanying four tables, i.e., Tables B.5–B.8.

Table 4.4. The relations of the cyclotomic numbers of order 10 for even f.

	0	1	2	3	4	5	6	7	8	9
0	$(00)$	$(01)$	$(02)$	$(03)$	$(04)$	$(05)$	$(06)$	$(07)$	$(08)$	$(09)$
1	$(01)$	$(09)$	$(12)$	$(13)$	$(14)$	$(15)$	$(16)$	$(17)$	$(18)$	$(12)$
2	$(02)$	$(12)$	$(08)$	$(18)$	$(24)$	$(25)$	$(26)$	$(27)$	$(24)$	$(13)$
3	$(03)$	$(13)$	$(18)$	$(07)$	$(17)$	$(27)$	$(36)$	$(36)$	$(25)$	$(14)$
4	$(04)$	$(14)$	$(24)$	$(17)$	$(06)$	$(16)$	$(26)$	$(36)$	$(26)$	$(15)$
5	$(05)$	$(15)$	$(25)$	$(27)$	$(16)$	$(05)$	$(15)$	$(25)$	$(27)$	$(16)$
6	$(06)$	$(16)$	$(26)$	$(36)$	$(26)$	$(15)$	$(04)$	$(14)$	$(24)$	$(17)$
7	$(07)$	$(17)$	$(27)$	$(36)$	$(36)$	$(25)$	$(14)$	$(03)$	$(13)$	$(18)$
8	$(08)$	$(18)$	$(24)$	$(25)$	$(26)$	$(27)$	$(24)$	$(13)$	$(02)$	$(01)$
9	$(09)$	$(12)$	$(13)$	$(14)$	$(15)$	$(16)$	$(17)$	$(18)$	$(12)$	$(01)$

Table 4.5. The relations of the cyclotomic numbers of order 10 for odd f.

	0	1	2	3	4	5	6	7	8	9
0	$(00)$	$(01)$	$(02)$	$(03)$	$(04)$	$(05)$	$(06)$	$(07)$	$(08)$	$(09)$
1	$(10)$	$(11)$	$(12)$	$(13)$	$(14)$	$(06)$	$(04)$	$(14)$	$(18)$	$(19)$
2	$(20)$	$(21)$	$(22)$	$(23)$	$(18)$	$(07)$	$(14)$	$(03)$	$(13)$	$(23)$
3	$(22)$	$(31)$	$(31)$	$(20)$	$(19)$	$(08)$	$(18)$	$(13)$	$(02)$	$(12)$
4	$(11)$	$(21)$	$(31)$	$(21)$	$(10)$	$(09)$	$(19)$	$(23)$	$(12)$	$(01)$
5	$(00)$	$(10)$	$(20)$	$(22)$	$(11)$	$(00)$	$(10)$	$(20)$	$(22)$	$(11)$
6	$(10)$	$(09)$	$(19)$	$(23)$	$(12)$	$(01)$	$(11)$	$(21)$	$(31)$	$(21)$
7	$(20)$	$(19)$	$(08)$	$(18)$	$(13)$	$(02)$	$(12)$	$(22)$	$(31)$	$(31)$
8	$(22)$	$(23)$	$(18)$	$(07)$	$(14)$	$(03)$	$(13)$	$(23)$	$(20)$	$(21)$
9	$(11)$	$(12)$	$(13)$	$(14)$	$(06)$	$(04)$	$(14)$	$(18)$	$(19)$	$(10)$

It has been proven that the set of tenth power residues or the set of tenth power residues together with zero modulo a prime p = 10f + 1 cannot form a difference set [455]. However, (4.3) and the constants in Tables B.5–B.8 show that the cyclotomic numbers of order 10 are roughly flat in each of the cases. This means that the corresponding F(x) defined in Section 4.3 in this case is cryptographically attractive from the difference property and nonlinearity viewpoints. The actual a(x) can be easily calculated by hand.

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String Theory and the Real World: From Particle Physics to Astrophysics

Frederik Denef , in Les Houches, 2008

7.2 The elliptic fibration over $M_{n}$

We consider now the CY elliptic fibration with as base manifold B our example (5.4), which we denoted by $M_{n}$ , the $ℂ ℙ^{1}$ bundle over $ℂ ℙ^{2}$ with twist n ≥ 0. ³³ This was defined by five fields, which we will now call u_i , and U(1) × U(1) gauge group, with charges

and positive FI parameters (ξ ¹ ξ ²). Recall from (5.33) that the volume of $M_{n}$ , n > 0 is indeed of Swiss cheese type.

We consider again a CY elliptic fibration of the form

(7.3) $Z : y^{2} = x^{3} + f (\vec{u}) x z^{4} + g (\vec{u}) z^{6} = 0$

over $M_{n}$ , and the Calabi–Yau condition Σ_i D_i = [Z] fixes the charges of the fields and polynomials to be

u₁	u₂	u₃	u₄	u₅	x	y	z	f	g
1	1	1	—n	0	0	0	n — 3	4(3 — n)	6(3 — n)
0	0	0	1	1	0	0	—2	8	12
0	0	0	0	0	2	3	1	0	0

The corresponding D-term constraints are, explicitly:

(7.4) ${| u_{1} |}^{2} + {| u_{2} |}^{2} + {| u_{3} |}^{2} - n {| u_{4} |}^{2} + (n - 3) {| z |}^{2} = ξ^{1},$

(7.5) ${| u_{4} |}^{2} + {| u_{5} |}^{2} - 2 {| z |}^{2} = ξ^{2},$

(7.6) $2 {| x |}^{2} + 3 {| y |}^{2} + {| z |}^{2} = ξ^{3} \cdot$

In accord with the F-theory limit of vanishing elliptic fiber, we take the third FI parameter ξ ³ much smaller than ξ ¹ ξ ².

It may seem like we have constructed an infinite number of Calabi–Yau four-folds, labeled by n. This is not true. We should keep in mind that we have made the implicit assumption (by using the formula c ₁ = Σ_i D_i — [Z]) that Z is smooth. If this is not the case, we should in principle first resolve the singularities before applying this formula, or use a modification of the formula appropriate for singular spaces. Now, from the U(1)³ charges of the polynomials f and g given above, we see that if n > 3, f and g become negatively charged under the first U(1) and so must necessarily contain an overall factor equal to a power of u ₄. More precisely $f (u) = u_{4}^{k} \tilde{f} (u), g (u) = u_{4}^{l} \tilde{g} (u)$ where k is the smallest integer $\geq 4 (1 - \frac{3}{n})$ and l the smallest integer $\geq 6 (1 - \frac{3}{n})$ . So in this case f, g and the discriminant Δ = 27 g ² + 4 f ³ vanish as some power of u ₁ on the divisor D ₄: u ₁ = 0, and hence the fourfold is singular along the locus Δ = 0. For n not too large, the singularities are harmless in the sense that they can be resolved while preserving the c ₁ = 0 condition, and moreover they have a clean physical interpretation as loci of enhanced gauge symmetry, as mentioned in Section 3.10. For example for n = 4, we generically have f ˜ u ₄, $g \sim u_{4}^{2}$ and $Δ \sim u_{4}^{3}$ , so from the table in Section 3.10 we read off that we get an SU(2) gauge group enhancement. For n = 18, we have $f \sim u_{4}^{4}, g \sum u_{4}^{5}, Δ \sim u_{4}^{10}$ and we get an E ₈ gauge group enhancement. For n > 18, we fall off the table; at this point the singularity becomes so bad that it cannot be resolved preserving the CY condition. This puts a cutoff on n.

At any rate, we will focus on the cases without gauge symmetry enhancement, i.e. n ≤ 3, for which the analysis is most straightforward.

From the charge assignments above, we read off the following relations between the divisors:

(7.7) $D_{1} = D_{2} = D_{3}, D_{5} - D_{4} = n D_{1},$

(7.8) $[Z] = 3 D_{x} = 2 D_{y} = 6 D_{z} + (3 + n) D_{1} + 2 D_{4},$

where in the last line [Z] is the homology class of our Calabi–Yau Z.

An independent set of divisors is given for instance by D ₄, D ₅, D_z . Their pullbacks to Z are denoted by ${\tilde{D}}_{4}, {\tilde{D}}_{5}, {\tilde{D}}_{z}$ . The first two are divisors wrapped on the elliptic fiber and a divisor in the base. The third one is a section of the elliptic fibration, i.e. the base itself. Using the techniques of Section 5, we find the following nonzero intersection numbers between these divisors:

(7.9) $\begin{matrix} {\tilde{D}}_{4}^{3} {\tilde{D}}_{z} = n^{2}, & {\tilde{D}}_{4}^{2} {\tilde{D}}_{z}^{2} = (3 - n) n, & {\tilde{D}}_{4} {\tilde{D}}_{z}^{3} = {(3 - n)}^{2}, \end{matrix}$

(7.10) $\begin{matrix} {\tilde{D}}_{5}^{3} {\tilde{D}}_{z} = n^{2}, & {\tilde{D}}_{5}^{2} {\tilde{D}}_{z}^{2} = - (3 + n) n, & {\tilde{D}}_{5} {\tilde{D}}_{z}^{3} = {(3 + n)}^{2}, \end{matrix}$

(7.11) ${\tilde{D}}_{z}^{4} = - 2 (n^{2} + 24) .$

This data allows us to compute volumes, characteristic classes, indices and so on. (A basis for the Kähler cone is given by ${\tilde{K}}_{1} = {\tilde{D}}_{1}, {\tilde{K}}_{2} = {\tilde{D}}_{5}, {\tilde{K}}_{3} = {[Z]}_{Z}$ , but we will continue to work in the above divisor basis in what follows.)

Again we need some nonperturbative contributions to W, associated to holomorphic M5 instantons wrapping the elliptic fiber and a divisor in $M_{n}$ . The most general such divisor D is given by some polynomial equation $P (\vec{u}) = 0$ , so

(7.12) $D = a D_{4} + b D_{5} = (a + b) D_{4} + b n D_{1},$

where $a + b \in ℤ^{+}, b n \in ℤ^{+}$ .

As in the previous example, we can compute the holomorphic Euler characteristic χ ₀ and find (assisted by Mathematica to do the series expansions of characteristic classes and to substitute the intersection numbers):

(7.13) $χ_{0} (D) = - \frac{1}{2} n ((n - 3) a^{2} + (n + 3) b^{2}) .$

When n = 0 or n ≥ 3, this is nonpositive, and therefore even the weak necessary condition χ ₀ ≥ 1 is not satisfied. ³⁴ On the other hand the Diophantine equation χ ₀(D) = 1 has infinitely many solutions for n = 1, 2. For definiteness let us specialize to

(7.14) $n \equiv 1$

from now on. Then to find divisors of arithmetic genus one, we have to solve a ² — 2b ² = 1 for a + b, b nonnegative integers. This is explicitly solved as

(7.15) $\begin{matrix} a = \frac{{(3 + 2 \sqrt{2})}^{k} + {(3 - 2 \sqrt{2})}^{k}}{2}, & b = \frac{{(3 + 2 \sqrt{2})}^{k} - {(3 - 2 \sqrt{2})}^{k}}{2 \sqrt{2}}, \end{matrix}$

k ≥ 0. The first few solutions are (a, b) = {(1, 0), (3, 2), (17, 12), (99, 70), · · ·}.

In particular for (a, b) = (1, 0), i.e. D = D ₄: u ₄ = 0, the instanton is completely rigid and has exactly two zeromodes, i.e. h ^1,0 = h ^2,0 = h ^3,0 = 0. This can be seen as follows. First, it is clear from the charge assignments of the fields that u ₄ = 0 is the unique holomorphic representative in its homology class—there are no other polynomials with the same charges as u ₁. From the one to one correspondence between holomorphic deformations and elements of H ^3,0 (D), this implies h ^3,0 = 0. Furthermore, from the Lefschetz hyperplane theorem, it follows that b ₁(D) = b ₁(Z) = 0, and therefore h ^1,0 = 0. Hence 1 = χ ₀ = 1 + h ^2,0, and therefore also h ^2,0 = 0.

Thus, flux or no flux, D will always contribute to the superpotential, and given (5.33) this is moreover exactly the kind of contribution we need for the large volume scenario to work! Note also that unlike in the KKLT scenario, we only need one instanton correction to stabilize all Kähler moduli. ³⁵

For completeness we give some further topological data for this model, obtained in a way similar to what we did for the example of the elliptic fibration over $ℂ ℙ^{3}$ . The Hodge data of Z|_n=1 is

(7.16) $h^{1, 1} = 3, h^{2, 1} = 0, h^{3, 1} = 3397, h^{2, 2} = 13644 \cdot$

This implies in particular b ₄ = 20440 and a curvature induced D3 tadpole

(7.17) $Q_{c} = \frac{χ (Z)}{24} = 852 \cdot$

According to the estimate (6.51), this yields a discretuum of about 10³⁰⁰⁰ flux vacua.

Following Section 3.8, we find that the IIB weak coupling limit is a v → –v orientifold of the Calabi–Yau hypersurface:

(7.18) $X : v^{2} = h (\vec{u}),$

in a toric variety with fields (u ₁, …, u ₅,v) and the following charge assignments:

u₁	u₂	u₃	u₄	u₅	v	h
1	1	1	—1	0	2	4
0	0	0	1	1	2	4

Computing the third Chern class in the usual way, we find χ(X) = —260, so (using h ^1,1 = 2) h ^2,1 = 132. This also determines the number ξ defined in (4.93): ξ ≈ 0.315.

Thus we conclude that in this model, the large volume scenario can indeed be realized.

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Handbook of Computability Theory

Richard A. Shore , in Studies in Logic and the Foundations of Mathematics, 1999

1 Introduction

Decision problems were the motivating force in the search for a formal definition of algorithm that constituted the beginnings of recursion (computability) theory. In the abstract, given a set A the decision problem for A consists of finding an algorithm which, given input n, decides whether or not n is in A. The classic decision problem for logic is whether a particular sentence is a theorem of a given theory T. Other examples arise in almost all branches of mathematics. In most settings one is almost immediately confronted by the notion of a recursively (or computably) enumerable (r.e.) set (the sets which can be listed (i.e. enumerated) by a computable (i.e. recursive) function): the theorems of a axiomatized theory, the solvable Diophantine equations, the true equations between words in a finitely presented group, etc. Typically, such decision problems amount to deciding if a particular r.e. set is computable (recursive). Indeed, the first examples of unsolvable decision problems provided examples of nonrecursive r.e. sets: the theorems of predicate logic, the word problem for groups, the halting problem. (For technical convenience, we code all expressions in formal languages, groups, etc. as natural numbers and so restrict our attention to sets of natural numbers.)

One can say that all these sets are simply noncomputable. Another view sees them as more complicated or harder to compute than the recursive sets. This is the view that leads to the notion of relative computability (reducibility) introduced by Turing [1936, 1939] and Post [1936, 1944]. The equivalence classes under this notion of relative computability were first called the degrees of recursive unsolvability. As Church's Thesis became widely accepted the word "recursive" was dropped and they became simply the degrees of unsolvability. As Turing's model of computation became the standard one, they became the Turing degrees. In view of the centrality of Turing's notion as the basic general definition of computability, the unqualified notion of degree eventually became that of Turing degrees. Other notions of relative computability whether stronger or weaker, from one-one to truth-table to arithmetic to constructibility, are refereed to by specifying the reducibility.

The starting point for the investigation of this fundamental notion of relative computability was the r.e. degrees (those equivalence classes containing r.e. sets). The classic results of logic (such as Göudel's incompleteness theorem, Church's proof of the undecidability of predicate logic and Turing's unsolvability of the Halting problem) each proved that there was a nonrecursive r.e. degree. All the natural examples, however, of nonrecursive r.e. sets supplied by standard theories which could be proven undecidable (e.g., Peano arithmetic) or from other natural definitions of noncomputable r.e. sets, turned out to have the same complexity. They were all complete, i.e. of the same degree as the halting problem K = {e | ϕ _e (e) ↓}. The obvious question, first proposed by Post [1944], was whether there are any other classes of decision problems under the equivalence of relative computability, i.e. are there any r.e. degrees other than those of the recursive sets, 0, and 0′, the degree of K?

Post attacked this problem by trying to define set theoretic properties of r.e. sets such as simplicity or hypersimplicity which would guarantee incompleteness as well as nonrecursiveness. Post concentrated on thinness properties of the complement of the r.e. set. This particular approach was doomed to failure (Yates [1965]), but Post's work initiated the study of the structure of the r.e. sets under set inclusion and the connections between their set-theoretic structure and computational complexity (see Soare [1999]). The solution to Post's problem, however, came from another approach.

Friedberg [1957] and Muchnik [1956] independently solved the problem by constructing intermediate r.e. degrees. The construction technique they introduced is called the priority method. It has been extensively studied, expanded and developed over the years. The priority method has proven useful in many areas of recursion theory and in applications to other areas of logic as well. Indeed, this technique has been called the hallmark of recursion theory. The principle arena of both its development and application has been in the study of r.e. sets and in particular of the r.e. degrees. The great strides that have been made in the past forty years in the understanding of the structure of the r.e. degrees have gone hand in hand with the development of new types of priority arguments. In this chapter we try to present the important results contributing to our overall picture of the structure R of the r.e. degrees with just a word or two about the associated proof techniques followed by appropriate references.

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Statistical inference and a mixture model

N. Balakrishnan , ... Fotios S. Milienos , in Reliability Analysis and Plans for Successive Testing, 2021

6.3.2 Multistate case

A general MLE procedure or an EM algorithm, in the case of i.i.d. multistate trials and when the available data contain only the number of trials, may also be adopted here along the lines of Section 5.2.2. Specifically, let us assume that $Z_{i} \in Γ = {γ_{0}, γ_{1}, \dots, γ_{r}}$ with $P (Z_{i} = γ_{j}) = θ_{j}$ , for $j = 0, 1, \dots, r, and i = 1, \dots, n$ . Let us also denote by $k_{j}$ the number of trials resulting in an outcome of Type j in n trials, for $j = 0, 1, \dots, r$ ; this means that the vector $k = (k_{0}, k_{1}, \dots, k_{r})$ is a solution of the Diophantine equation

(6.3.6) $k_{0} + k_{1} + \dots + k_{r} = n, with k_{j} \in {0, 1, \dots, n} .$

Note that there are $(\begin{matrix} n + r \\ r \end{matrix})$ different vectors k satisfying (6.3.6). Denoting by $V_{i}$ the set of vectors of the space $Γ^{n}$ with $k_{i} = (k_{0 i}, k_{1 i}, \dots, k_{r i})$ determining the frequencies of each outcome, for $i = 1, \dots, (\begin{matrix} n + r \\ r \end{matrix})$ , we have

$P (X > n; θ) = \sum_{i = 1}^{(\begin{matrix} n + r \\ r \end{matrix})} c_{n i} θ_{0}^{k_{0 i}} θ_{1}^{k_{1 i}} \dots θ_{r}^{k_{r i}}, P (X = n; θ) = \sum_{i = 1}^{(\begin{matrix} n + r \\ r \end{matrix})} v_{n i} θ_{0}^{k_{0 i}} θ_{1}^{k_{1 i}} \dots θ_{r}^{k_{r i}},$

with $θ = (θ_{0}, θ_{1}, \dots, θ_{r - 1})$ , $θ_{r} = 1 - \sum_{j = 0}^{r - 1} θ_{j}$ and $k_{r i} = n - \sum_{j = 0}^{r - 1} k_{j i}$ . Moreover, the conditional probabilities $P (S_{n}^{j} = s | X = n; θ)$ and $P (S_{n}^{j} = s | X > n; θ)$ , where $S_{n}^{j}$ denotes the number of outcomes equaling j among the n trials, are given by

$P (S_{n}^{j} = s | X = n; θ) = \frac{\sum_{i} v_{n i} θ_{0}^{k_{0 i}} θ_{1}^{k_{1 i}} \dots θ_{r}^{k_{r i}}}{P (X = n)}, P (S_{n}^{j} = s | X > n; θ) = \frac{\sum_{i} c_{n i} θ_{0}^{k_{0 i}} θ_{1}^{k_{1 i}} \dots θ_{r}^{k_{r i}}}{P (X > n)},$

where the sums in the numerators are over the integer solutions of (6.3.6) of the form $(k_{0 i}, \dots, k_{j - 1, i}, s, k_{j + 1, i}, \dots, k_{r i})$ . In an analogous way, we can also express, for example, the conditional probabilities of the form $P (S_{n}^{j} = s_{j}, S_{n}^{i} = s_{i} | X = n; θ)$ , for $i \neq j$ .

In this case, the complete likelihood is given by

$L (θ) \propto \prod_{i = 1}^{m} θ_{0}^{k_{0 i}} θ_{1}^{k_{1 i}} \dots θ_{r}^{k_{r i}},$

and so, if we wish to adopt the EM algorithm, the $(k + 1)$ -th estimate of $θ_{j}$ can be determined by the recursive scheme

$θ_{j}^{(k + 1)} = \frac{\sum_{i = 1}^{m} w_{j i} {(θ^{(k)})}^{δ_{i}} w_{j i}^{⁎} {(θ^{(k)})}^{1 - δ_{i}}}{\sum_{i = 1}^{m} n_{i}}, j = 0, \dots, r - 1$

(recall that $δ_{i} = 1$ if $X = n_{i}$ and $δ_{i} = 0$ if $X > n_{i}$ ), where

$w_{j i} (θ^{(k)}) = E (S_{n_{i}}^{j} | X = n_{i}; θ^{(k)}), w_{j i}^{⁎} (θ^{(k)}) = E (S_{n_{i}}^{j} | X > n_{i}; θ^{(k)}) .$

Denoting by $B (θ)$ the negative of the matrix of second derivatives of $\ln L (θ)$ and by $S (θ)$ the gradient vector of $\ln L (θ)$ , the observed Fisher information matrix will be given by $I (\hat{θ})$ , where $\hat{θ}$ is the EM estimate of θ and

(6.3.7) $I (θ) = E (B (θ)) - E (S {(θ)}^{'} S (θ)) + E {(S (θ))}^{'} E (S (θ)) .$

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Logic from Russell to Church

Jens Erik. Fenstad , Hao. Wang , in Handbook of the History of Logic, 2009

4.6 Foundational Matters

The words "Begrundung", "Grundlagenfragen", and "foundational research" occur frequently in the titles of Skolem's papers. But to which extent did Skolem have a complete and coherent view of the foundation of mathematics?

The young Skolem was a mathematician in the tradition of Kronecker; see e.g. his discussion of Kronecker in Skolem [1955c]. The active professors of mathematics in Oslo in Skolem's student days were A. Thue and C. Størmer, both experts in number theory, in particular, in solving diophantine equations. Skolem continued in this tradition, in fact, it is possible, as argued in Jervell [1996], to see this theme, solving equations, as a common concern in much of Skolem's work, including logic. His fellow student was Viggo Brun, who later became famous for his work in combinatorics (on the Goldbach conjecture). As later colleagues, Skolem and Brun edited a new edition of the standard text on combinatorics, Netto's Lehrbuch der Combinatorik.

The young Skolem was as mathematician interested in structures and algorithms, working on the algebra of logic and on combinatorics. He knew, of course, the new and exiting work of Zermelo on the axiom of choice and the foundation of set theory. But he was skeptical as can be seen from his critical attitude to the Zermelo axiomatization of set theory expressed during his Göttingen winter of 1915–16.

Skolem was himself an expert in using the axiomatic method in the study of algebraic structures, see his early work on lattice theory, Section 4.2. But he did not believe in the possibility of an axiomatic characterization of basic concepts such as numbers and sets. His skepticism was based on his work on the so-called "set theoretical relativism" and on his construction of nonstandard models for arithmetic. This was his firm conviction, and it is not likely that the current discussion of the Skolem Paradox would have changed his attitude towards foundational matters.

His first publication on recursive mathematics was the 1923 paper on the Begründung der elementaren Arithmetik durch die rekurrierende Denkweise. This was the starting step — and it remained so. Skolem never worked out a complete account similar to what we find in the work of R. Goldstein [1957] and [1961] on recursive arithmetic.

Skolem did, however, return to his "program" on many later occasions, some important references are Skolem [1950a] and [1955c], but then mostly in form of examples and suggestions. He wanted step-wise to build up a "sound" foundation. This is clear from the pioneering Skolem [1923a], and is explicit in his later lecture on The logical background of arithmetic, Skolem [1953a]. His style bears some resemblance to current work in computer science, and some have seen him as an early pioneer of this field, see Jervell [1996].

"Basic objects" were almost always numbers. Skolem did not make any systematic attempt to build up a "constructive" analysis. He made over the years a number of references to Brouwer and intuitionism, but it is unclear how well he was acquainted with Brouwer's actual writings. In his lectures on set theory at Notre Dame, Skolem [1962c], he includes several short sections on alternative approaches such as intuitionism and his own work on ramified type theory. It is further of interest to note that he has a substantial section on the "operative" approach of P. Lorenzen. This was an approach that evidently appealed to Skolem.

Skolem was acquainted with the work of H. Weyl [1918] on predicative analysis. But he never seemed to have made a thorough study of Weyl's work, and there are only some very superficial references to it in his papers. This is, perhaps, due to a difference in style. Both Skolem and Weyl were concerned with the problem of impredicativity and the unrestricted use of quantifiers in the tradition of Cantor and Frege. Skolem wanted to build bottom-up. Weyl, both in his [1910] paper and his [1918] book, continued to work in the axiomatic tradition, but with proper restrictions on the set existence axioms. In particular, he restricted the axiom of comprehension to arithmetical properties, i.e. to properties expressible using only first-order quantifiers. A comprehensive study of Weyl's contributions can be found in Feferman [1998], see, in particular, the chapter Weyl vindicated: Das Kontinuum seventy years later.

Skolem was, as we have argued, not interested in the meta-mathematics of the constructive approach. It is interesting to note a similar attitude on Brouwer's part. He did not e.g. regard the axiomatization of intuitionistic logic to be particularly important, what mattered was the actual working out of the mathematics guided by the correct insights.

Today we have a rather complete view of the meta-mathematics of the many constructive approaches to mathematics, an excellent reference is Feferman [1998]. How would Skolem fit into the picture drawn by Feferman? Skolem worked with primitive recursive arithmetic in his [1923a] paper, he used some form of the Konig lemma in his construction of non-standard models of arithmetic, he studied various extended schemes of recursion. But as he did not develop his ideas into a full account, we cannot know what principles he ultimately would have accepted. But he would have been pleased to see the extent of "real" mathematics that can be developed on a recursive or predicative basis.

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2D Materials

N. Leconte , ... J. Jung , in Semiconductors and Semimetals, 2016

4.2 Graphene on hBN

4.2.1 Hofstadter and Wannier Diagrams

We remind that the Hamiltonian described in Section 2.1 contains three terms, namely the periodic moiré modulation H ₀, the periodic mass term H _Z, and the virtual strain term H _AB. In Figs. 9 and 10, we map out the DOS vs energy and magnetic field to obtain the Hofstadter butterfly. The different panels of Fig. 7 are based on the subsets of these three terms, while Fig. 10 includes all three of them. To observe the experimental electron–hole asymmetry and the gap opening at negative energy, a mixing of terms with the virtual strain term is required. If one were to observe a gap using the mass-term contribution only, the value of C _Z would have to be at least twice as large (not shown here).

The full Hamiltonian maps in Fig. 10 capture well most of the single-particle features in experiment. The lower panel is obtained from the upper panel by integration of the DOS, to replace the energy dependency to a charge carrier density. The complex diamond-like fractal structure at negative energies depicts an intricate structure of secondary features. For visual reference, we draw a selection of lines in the Wannier representation of the data (lower panel). Such lines can be represented by a Diophantine equation, for which the slope t is directly linked to the transverse conductivity quantization by $σ_{x y} = t \frac{e^{2}}{h}$ . Our detailed Wannier maps are expected to provide sound reference for experimentalists performing high magnetic field dissipationless conductivity measurements.

The opening of the experimental gap at n/n ₀ = − 4 at zero field, and subsequent closing at around six roots in the inequivalent K and K′ contributions (Moon and Koshino, 2014). The transport behavior of the charge carriers at the secondary level is predicted to follow the Schrödinger equation, rather than complying to the Dirac equation, unlike the primary charge carriers at zero energy (Chizhova et al., 2014). The linear energy-dependent behavior in the upper panel seems to confirm this, even if slight distortions are apparent. Additional features develop at filling factors below $n / n_{0} = - 4$ , which, by comparison with Fig. 7, are induced by the virtual strain term (most prominent when mixed with the mass term).

A fundamental limitation of periodic supercell simulations restricts the accessible twist angle values to well-defined commensurate numbers. Formally, this is no exception for the present method. However, because our actual supercells contain a repetition of 60 moiré unit cells in both the x and y-directions, the periodic mismatch arising at the boundaries becomes increasingly negligible in our bulk picture with increasing system size. Convergence studies (not shown here) confirm this. We illustrate this capability by giving some zero field and finite magnetic field DOS in Figs. 9 and 10, using the small-angle approximation. Large angles are also accessible, but require the complete expression given in Jung et al. (2014). We will illustrate such large angles in Section 4.5.

4.2.2 Transport Properties

The origin of electron scattering in graphene transport has been attributed mainly to the effects of Coulomb impurity scattering (Adam et al., 2007; Ando, 2006; Nomura and MacDonald, 2006; Sarma et al., 2011) although random strain fields have also been suggested as sources of scattering (Couto et al., 2014). In ultraclean devices such as suspended graphene or ultra clean graphene on G/BN several scattering mechanisms could sensitively impact the mobility. One scenario that we wish to explore through our numerical simulation is to examine the role of the charge scatterers in G/BN that are related to ULR disorder with very large correlation lengths on the order of 20 nm observed in recent measurements (Wong et al., 2015). In order to verify the feasibility of such scenario, we compare in Fig. 11E–G the impact of our disorder models on the dynamic observables of the system. The random strain field behavior is also provided for reference.

Our calculation on ULR correlated onsite disorder (light blue (light gray in the print version) line) for $λ = 20$ and $W = 0.05$ follows almost exactly the curves of RS disorder. Because of this, microscopic characterization of the sample might be required to differentiate between these two possible scenarios in experiment.

Finally, the general shape for the low disorder case agrees with DaSilva et al. (2015), where a fixed mobility of 5000 cm²/V s is used. The relatively higher conductivity for low energies in our curves is rationalized by the increasing mobility toward the Dirac point. Our results again suggest that many-body effects are not necessarily required to explain the hole side minimum of conductivity, as opposed to the claim in the claims in Slotman et al. (2015), where a simplified model is used.

4.2.3 Sublattice Asymmetric Disorder

To further illustrate the freedom that this kind of computer experiments provide us with, we calculate the Hofstadter butterfly (see Fig. 12) for the sublattice-selective functionalization disorder introduced in Section 2.2. What started as a toy in the theorist's playground (Leconte et al., 2011; Lherbier et al., 2013) has since been observed in experiments of sublattice-selective nitrogen substitution (Lv et al., 2012; Telychko et al., 2014; Wang et al., 2012b; Zabet-Khosousi et al., 2014; Zhao et al., 2011), molybdenum substitution (Wan et al., 2013) and hydrogen functionalization (Lin et al., 2015). Here, we combine our realistic model for the G/BN system with the dilute sublattice-specific modification of onsite energies, as introduced in the model and methodology section. The parameters of choice for the latter are $A = 0.05$ for a concentration of 0.01 % of shifted onsite energies. In the lower panel of Fig. 12, we allow for variations in the value of A.

As expected from the zero-field literature (Lherbier et al., 2013), a gap opens at the Dirac point, which scales with concentration and onsite energy shift. The consequence of this gap opening in the finite magnetic field regime is a splitting of the zero LL, as observed in the upper panel. The dashed lines in the lower panel indicate the band gap edges and correspond to the position of two LL0-type features in the upper panel. The reason for the LL0 splitting roots in degeneracy lifting of the valley degree of freedom, which is directly related to the global sublattice symmetry breaking. Additionally, our simulations suggest that controlled sublattice chemical modification of this moiré system at the atomic level, by introducing onsite potential breaking disorders such as hydrogen or vacancies, can lead to a downward shift in the energy position of the superlattice Dirac features.

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Using Continued Fractions to Solve Diophantine Equation

Diophantine Equation

Introduction