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Article
Open AccessMonotone Nondecreasing Sequences of the Euler Totient Function
Let M(x) denote the largest cardinality of a subset of \(\{n \in \mathbb {N}: n \le x\}\) ...
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Article
Open AccessPerfectly Packing a Square by Squares of Nearly Harmonic Sidelength
A well-known open problem of Meir and Moser asks if the squares of sidelength 1/n for \(n\ge 2\) ...
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Article
Sums of GUE matrices and concentration of hives from correlation decay of eigengaps
Associated to two given sequences of eigenvalues \(\lambda _1 \ge \cdots \ge \lambda _n\) ...
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Article
Infinite partial sumsets in the primes
We show that there exist infinite sets A = (a1, a2, …} and B = {b1, b2, …} of natural numbers such that ai + bj is prime whenever 1 ≤ i < j.
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Article
Open AccessUndecidable Translational Tilings with Only Two Tiles, or One Nonabelian Tile
We construct an example of a group \(G = \mathbb {Z}^2 \times G_0\) G ...
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Article
Open AccessLarge prime gaps and probabilistic models
We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigor...
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Article
Optimal Sine and Sawtooth Inequalities
We determine the optimal inequality of the form \(\sum _{k=1}^m a_k\sin kx\le 1\) ...
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Chapter
Series
Now that we have developed a reasonable theory of limits of sequences, we will use that theory to develop a theory of infinite series $$\begin{aligned...
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Chapter
Fourier Series
In the previous two chapters, we discussed the issue of how certain functions (for instance, compactly supported continuous functions) could be approximated by polynomials. Later, we showed how a different cla...
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Chapter
Integers and Rationals
In Chapter 2 we built up most of the basic properties of the natural number system, but we have reached the limits of what one can do with just addition and multiplicati...
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Chapter
Continuous Functions on Metric Spaces
In the previous chapter we studied a single metric space (X, d), and the various types of sets one could find in that space. While this is already quite a rich subject, the theory of metric spaces becomes even ri...
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Chapter
Differentiation of Functions
We can now begin the rigorous treatment of calculus in earnest, starting with the notion of a derivative. We can now define derivatives analytically, using limits, in contrast to the geometric definition of de...
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Chapter
Lebesgue Integration
In Chap. 11, we approached the Riemann integral by first integrating a particularly simple class of functions, namely the piecewise constant functions. Among other things, piecewise constant functions only attain...
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Chapter
Introduction
This text is an honors-level undergraduate introduction to real analysis: the analysis of the real numbers, sequences and series of real numbers, and real-valued functions. This is related to, but is distinct fro...
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Chapter
Continuous Functions on \({{\textbf{R}}}\)
In previous chapters we have been focusing primarily on sequences. A sequence \((a_n)_{n=0}^\infty \) can be viewed as a function from ...
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Chapter
Metric Spaces
In Definition 6.1.5 we defined what it meant for a sequence \((x_n)_{n=m}^\infty \) of real numbers to converge to another real number x;...
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Chapter
Uniform Convergence
In the previous two chapters we have seen what it means for a sequence \((x^{(n)})_{n=1}^\infty \) of points in a metric space ...
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Book
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Chapter
Lebesgue Measure
In the previous chapter we discussed differentiation in several variable calculus. It is now only natural to consider the question of integration in several variable calculus. The general question we wish to a...
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Chapter
The Riemann Integral
In the previous chapter we reviewed differentiation—one of the two pillars of single variable calculus. The other pillar is, of course, integration, which is the focus of the current chapter. More precisely, we w...
