Classical and Discrete Functional Analysis with Measure Theory

Functional analysis deals with infinite-dimensional spaces. Its results are among the greatest achievements of modern mathematics and it has wide-reaching applications to probability theory, statistics, economics, classical and quantum physics, chemistry, engineering, and pure mathematics. This book deals with measure theory and discrete aspects of functional analysis, including Fourier series, sequence spaces, matrix maps, and summability. Based on the author's extensive teaching experience, the text is accessible to advanced undergraduate and first-year graduate students. It can be used as a basis for a one-term course or for a one-year sequence, and is suitable for self-study for readers with an undergraduate-level understanding of real analysis and linear algebra. More than 750 exercises are included to help the reader test their understanding. Key background material is summarized in the Preliminaries.

On Injectivity of an Integral Operator Connected to Riemann Hypothesis

Using the equivalent formulation of RH given by Beurling ([4],1955), Alcantara-Bode showed ([2], 1993) that Riemann Hypothesisholds if and only if the integral operator on the Hilbert space L2(0; 1)having the kernel defined by fractional part function of the expressionbetween brackets {y/x}, is injective.Since then, the injectivity of the integral operator used in equivalentformulation of RH has not been addressed nor has been dissociatedfrom RH and, a pure mathematics solution for RH is not ready yet.Here is a numerical analysis approach of the injectivity of the linearbounded operators on separable Hilbert spaces addressing the problemslike the one presented in [2]. Apart of proving the injectivity of theBeurling - Alcantara-Bode integral operator, we obtained the followingresult: every linear bounded operator (or its associated Hermitian),strict positive definite on a dense family of including approximationsubspaces in L2(0,1) built on simple functions, is injective if the rateof convergence to zero of its unbounded sequence of inverse conditionnumbers on approximation subspaces is o(n-s) for some s ≥ 0. Whens = 0, the sequence is inferior bounded by a not null constant, that isthe case in the Beurling - Alcantara-Bode integral operator.In the Theorem 4.1 we addressed with numerical analysis toolsthe injectivity of the integral operator in [2] claiming that - even if asolution in pure mathematics is desired, together with the Theorem 1,pg. 153 in [2], the RH holds.

Purifying applied mathematics and applying pure mathematics: how a late Wittgensteinian perspective sheds light onto the dichotomy

In this work we argue that there is no strong demarcation between pure and applied mathematics. We show this first by stressing non-deductive components within pure mathematics, like axiomatization and theory-building in general. We also stress the “purer” components of applied mathematics, like the theory of the models that are concerned with practical purposes. We further show that some mathematical theories can be viewed through either a pure or applied lens. These different lenses are tied to different communities, which endorse different evaluative standards for theories. We evaluate the distinction between pure and applied mathematics from a late Wittgensteinian perspective. We note that the classical exegesis of the later Wittgenstein’s philosophy of mathematics, due to Maddy, leads to a clear-cut but misguided demarcation. We then turn our attention to a more niche interpretation of Wittgenstein by Dawson, which captures aspects of the aforementioned distinction more accurately. Building on this newer, maverick interpretation of the later Wittgenstein’s philosophy of mathematics, and endorsing an extended notion of meaning as use which includes social, mundane uses, we elaborate a fuzzy, but more realistic, demarcation. This demarcation, relying on family resemblance, is based on how direct and intended technical applications are, the kind of evaluative standards featured, and the range of rhetorical purposes at stake.

Proof that Einstein’s field equations are invalid: Exposition of the unimodular defect

Albert Einstein first presented his gravitational field equations in unimodular coordinates. In these coordinates, the field equations can be written explicitly in terms of the Einstein pseudotensor for the energy-momentum of the gravitational field. Since this pseudotensor produces, by contraction, a first-order intrinsic differential invariant, it violates the laws of pure mathematics. This is sufficient to prove that Einstein’s unimodular field equations are invalid. Since the unimodular form must hold in the general theory of relativity, it follows that the latter is also physically and mathematically unsound, lacking a proper mathematical foundation.

Advancing mathematics by guiding human intuition with AI

The practice of mathematics involves discovering patterns and using these to formulate and prove conjectures, resulting in theorems. Since the 1960s, mathematicians have used computers to assist in the discovery of patterns and formulation of conjectures1, most famously in the Birch and Swinnerton-Dyer conjecture2, a Millennium Prize Problem3. Here we provide examples of new fundamental results in pure mathematics that have been discovered with the assistance of machine learning—demonstrating a method by which machine learning can aid mathematicians in discovering new conjectures and theorems. We propose a process of using machine learning to discover potential patterns and relations between mathematical objects, understanding them with attribution techniques and using these observations to guide intuition and propose conjectures. We outline this machine-learning-guided framework and demonstrate its successful application to current research questions in distinct areas of pure mathematics, in each case showing how it led to meaningful mathematical contributions on important open problems: a new connection between the algebraic and geometric structure of knots, and a candidate algorithm predicted by the combinatorial invariance conjecture for symmetric groups4. Our work may serve as a model for collaboration between the fields of mathematics and artificial intelligence (AI) that can achieve surprising results by leveraging the respective strengths of mathematicians and machine learning.

The Riemann Hypothesis is most likely true

The Riemann hypothesis has been considered the most important unsolved problem in pure mathematics. The David Hilbert's list of 23 unsolved problems contains the Riemann hypothesis. Besides, it is one of the Clay Mathematics Institute's Millennium Prize Problems. The Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)< e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n> 5040$, where $\sigma(x)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1} > e^{\gamma} \times \log\theta(q_{n})$ is satisfied for all primes $q_{n}> 2$, where $\theta(x)$ is the Chebyshev function. Using both inequalities, we show that the Riemann hypothesis is most likely true.

An introduction to the superunified theory of quantum fields & fundamental interactions (Discoveries in pure mathematics)

This is intended to describe the physical Universe as self-excited and self-organized mathematical continuum. There does exist the universal pure (not applied) mathematical machine perceived by the intelligent observers in a capacity of certain material world. In this short article we are able to indicate only some key points of the theory which suggests practically infinite amount of combinatorics.

The Riemann Hypothesis is most likely true

The Riemann hypothesis has been considered the most important unsolved problem in pure mathematics. The David Hilbert's list of 23 unsolved problems contains the Riemann hypothesis. Besides, it is one of the Clay Mathematics Institute's Millennium Prize Problems. The Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)< e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n> 5040$, where $\sigma(x)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1} > e^{\gamma} \times \log\theta(q_{n})$ is satisfied for all primes $q_{n}> 2$, where $\theta(x)$ is the Chebyshev function. Using both inequalities, we show that the Riemann hypothesis is most likely true.

The Riemann Hypothesis is True

The Riemann hypothesis has been considered the most important unsolved problem in pure mathematics. The David Hilbert's list of 23 unsolved problems contains the Riemann hypothesis. Besides, it is one of the Clay Mathematics Institute's Millennium Prize Problems. The Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)< e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n> 5040$, where $\sigma(x)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1} > e^{\gamma} \times \log\theta(q_{n})$ is satisfied for all primes $q_{n}> 2$, where $\theta(x)$ is the Chebyshev function. Using both inequalities, we show that the Riemann hypothesis is true.

A commentary on the Special Issue “Innovations in measuring and fostering mathematical modelling competencies”

This is a commentary on the ESM 2021 Special Issue on Innovations in Measuring and Fostering Mathematical Modelling Competencies. We have grouped the ten studies into three themes: competencies, fostering, and measuring. The first theme and the papers therein provide a platform to discuss the cognitivist backgrounds to the different conceptualizations of mathematical modelling competencies, based on the modelling cycle. We suggest theoretical widening through a competence continuum and enriching of the modelling cycle with overarching, analytic dimensions for creativity, tool use, metacognition, and so forth. The second theme and the papers therein showcase innovative ideas on fostering and on the definition and analysis thereof. These reveal the need for a social turn in modelling research in order to capture aspects of student collaboration and agency, as well as tensions in fostering when tasks are derived from real-world scenarios, but socio-mathematical norms come from the (pure) mathematics classroom. The third theme, measuring, and the papers therein offer insights into the challenges of positivist research that aims to develop innovative measurement instruments that are both reliable and valid, particularly in light of student group work, cultural background, and other socio-cultural aspects. Drawing on the three discussions, we go on to make recommendations for further research.