Number Theory

Precompact abelian groups and topological annihilators

by Gabor Lukacs

Published in Journal of Pure and Applied Algebra 208 (2007), no. 3, 1159-1168.

For a compact Hausdorff abelian group K and its subgroup H, one defines the g-closure g(H) of H in K as the subgroup... more For a compact Hausdorff abelian group K and its subgroup H, one defines the g-closure g(H) of H in K as the subgroup consisting of \chi \in K such that \chi(a_n) --> 0 in T=R/Z for every sequence {a_n} in \hat K (the Pontryagin dual of K) that converges to 0 in the topology that H induces on \hat K. We prove that every countable subgroup of a compact Hausdorff group is g-closed, and thus give a positive answer to two problems of Dikranjan, Milan and Tonolo. We also show that every g-closed subgroup of a compact Hausdorff group is realcompact. The techniques developed in the paper are used to construct a close relative of the closure operator g that coincides with the G_\delta-closure on compact Hausdorff abelian groups, and thus captures realcompactness and pseudocompactness of subgroups.

A note on odd perfect numbers

by Arnie Dris

To appear in Matimyas Matematika (official publication of the Mathematical Society of the Philippines)

In this note, we show that if $N$ is an odd perfect number and $q^{\alpha}$ is some prime power exactly dividing it,... more In this note, we show that if $N$ is an odd perfect number and $q^{\alpha}$ is some prime power exactly dividing it, then $\sigma(N/q^{\alpha})/q^{\alpha}>5$. In general, we also show that if $\sigma(N/q^{\alpha})/q^{\alpha}<K$, where $K$ is any constant, then $N$ is bounded by some function depending on $K$.

The Abundancy Index of Divisors of Odd Perfect Numbers

by Arnie Dris

Journal of Integer Sequences, Vol. 15 (2012), Article 12.4.4

http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Dris/dris6.html

If $N = {q^k}{n^2}$ is an odd perfect number, where $q$ is the Euler prime, then we show that $n < q$ is sufficient... more If $N = {q^k}{n^2}$ is an odd perfect number, where $q$ is the Euler prime, then we show that $n < q$ is sufficient for Sorli's conjecture that $k = \nu_{q}(N) = 1$ to hold. We also prove that $q^k < \frac{2}{3}{n^2}$, and that $I(q^k) < I(n)$, where $I(x)$ is the abundancy index of $x$.

Solving the Odd Perfect Number Problem: Some Old and New Approaches

by Arnie Dris

Outstanding Graduate Thesis (2008), Master of Science in Mathematics, De La Salle University, Manila, Philippines

A perfect number is a positive integer $N$ such that the sum of all the positive divisors of $N$ equals 2N, denoted by... more A perfect number is a positive integer $N$ such that the sum of all the positive divisors of $N$ equals 2N, denoted by $\sigma(N) = 2N$. The question of the existence of odd perfect numbers (OPNs) is one of the longest unsolved problems of number theory. This thesis presents some of the old as well as new approaches to solving the OPN Problem. In particular, a conjecture predicting an injective and surjective mapping $X = \frac{\sigma(p^k)}{p^k}, Y = \frac{\sigma(m^2)}{m^2}$ between OPNs $N = {p^k}{m^2}$ (with Euler factor $p^k$) and rational points on the hyperbolic arc $XY = 2$ with $1 < X < 1.25 < 1.6 < Y < 2$ and $2.85 < X + Y < 3$, is disproved. Various results on the abundancy index and solitary numbers are used in the disproof. Numerical evidence against the said conjecture will likewise be discussed. We will show that if an OPN $N$ has the form above, then $p^k < {2/3}{m^2}$ follows from \cite{D10}. We will also attempt to prove a conjectured improvement of this last result to $p^k < m$ by observing that ${\frac{\sigma(p^k)}{m}} \neq 1$ and ${\frac{\sigma(p^k)}{m}} \neq {\frac{\sigma(m)}{p^k}}$ in all cases. Lastly, we also prove the following generalization: If $N = \displaystyle\prod_{i=1}^r {{p_i}^{{\alpha}_i}}$ is the canonical factorization of an OPN $N$, then $\sigma({p_i}^{{\alpha}_i}) \leq {2/3}{\frac{N}{{p_i}^{{\alpha}_i}}}$ for all $i$. This gives rise to the inequality $N^{2 - r} \leq (1/3)(2/3)^{r - 1}$, which is true for all $r$, where $r = \omega(N)$ is the number of distinct prime factors of $N$.

On Sorli's Conjecture Regarding Odd Perfect Numbers

by Arnie Dris

draft only

Let $N = {q^k}{n^2}$ be an odd perfect number with Euler prime $q$. Since $\gcd(q,n) = 1$, we know that $q \neq n$. In... more Let $N = {q^k}{n^2}$ be an odd perfect number with Euler prime $q$. Since $\gcd(q,n) = 1$, we know that $q \neq n$. In his M.~Sc. thesis completed in 2008 (available online via \url{http://www.scribd.com/doc/16144034/OPNThesis1}), the author proved that $q^k < n^2$. This implies that, if $n < q$, then Sorli's conjecture that $k = \nu_{q}(N) = 1$ would follow. In this paper, the author proposes a hypothesis that could disprove $n < q$.

On Sorli's Conjecture Regarding Odd Perfect Numbers

by Jose Arnaldo Dris

draft only

Let $N = {q^k}{n^2}$ be an odd perfect number with Euler prime $q$. Since $\gcd(q,n) = 1$, we know that $q \neq n$. In... more Let $N = {q^k}{n^2}$ be an odd perfect number with Euler prime $q$. Since $\gcd(q,n) = 1$, we know that $q \neq n$. In his M.~Sc. thesis completed in 2008 (available online via \url{http://www.scribd.com/doc/16144034/OPNThesis1}), the author proved that $q^k < n^2$. This implies that, if $n < q$, then Sorli's conjecture that $k = \nu_{q}(N) = 1$ would follow. In this paper, the author proposes a hypothesis that could disprove $n < q$.

The Abundancy Index of Divisors of Odd Perfect Numbers

by Jose Arnaldo Dris

Journal of Integer Sequences, Vol. 15 (2012), Article 12.4.4

http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Dris/dris6.html

If $N = {q^k}{n^2}$ is an odd perfect number, where $q$ is the Euler prime, then we show that $n < q$ is sufficient... more If $N = {q^k}{n^2}$ is an odd perfect number, where $q$ is the Euler prime, then we show that $n < q$ is sufficient for Sorli's conjecture that $k = \nu_{q}(N) = 1$ to hold. We also prove that $q^k < \frac{2}{3}{n^2}$, and that $I(q^k) < I(n)$, where $I(x)$ is the abundancy index of $x$.

A Note on Odd Perfect Numbers

by Jose Arnaldo Dris

Elementary number theory, Perfect number, Abundancy index, OPN Conjecture

In this note, we show that if $N$ is an odd perfect number and $q^{\alpha}$ is some prime power exactly dividing it,... more In this note, we show that if $N$ is an odd perfect number and $q^{\alpha}$ is some prime power exactly dividing it, then $\sigma(N/q^{\alpha})/q^{\alpha}>5$. In general, we also show that if $\sigma(N/q^{\alpha})/q^{\alpha}<K$, where $K$ is any constant, then $N$ is bounded by some function depending on $K$.

Solving the Odd Perfect Number Problem: Some New Approaches

by Jose Arnaldo Dris

http://www.dlsu.edu.ph/conferences/snt_congress/2009/_pdf/A1905.pdf

A conjecture predicting an injective and surjective mapping X = Sigma(p^k)/p^k , Y = Sigma(m^2)/m^2 between OPNs N =... more A conjecture predicting an injective and surjective mapping X = Sigma(p^k)/p^k , Y = Sigma(m^2)/m^2 between OPNs N = (p^k)(m^2) (with Euler factor p^k) and rational points on the hyperbolic arc XY = 2 with 1 < X < 1.25 < 1.6 < Y < 2 and 2.85 < X + Y < 3, is disproved. We will show that if an OPN N has the form above, then p^k < (2/3)(m^2). We then give a somewhat weaker corollary to this last result (m^2 − p^k ≥ 8) and give possible improvements along these lines. We will also attempt to prove a conjectured improvement to p^k < m by observing that Sigma(p^k)/m is not equal to Sigma(m)/p^k and Sigma(p^k)/m is not equal to 1 in all cases. Lastly, we also prove the following generalization: If N = PROD_{i = 1}^{omega(N)} { {p_i}^{\alpha_i} } is the canonical factorization of an OPN N, then { {p_i}^{\alpha_i} } ≤ (2/3)(N/{p_i}^{\alpha_i}) for all i. This gives rise to the inequality N^{2 − r} ≤ (1/3)(2/3)^{r - 1}, which is true for all r, where r = omega(N) is the number of distinct prime factors of N.

Prime Factorisation A New Approach

by Rahulkrishnan Chandrasekharan

In number theory, integer factorization or prime factorization is the decomposition of a composite number into smaller... more In number theory, integer factorization or prime factorization is the decomposition of a composite number into smaller non-trivial divisors, which when multiplied together equals the original integer. A prime factor can be visualized by understanding Euclid's geometric position. He saw a whole number as a line segment, which has a smallest line segment greater than 1 that can divide equally into it. By the fundamental theorem of arithmetic, every positive integer has a unique prime factorization. However, the fundamental theorem of arithmetic gives no insight into how to obtain an integer's prime factorization; it only guarantees its existence.

Science and Math Defeated Glossary of Technical Terminology

by Nikolay Sokolov

Also published online: http://sciencedefeated.wordpress.com/glossary-of-terms/

Several readers, particularly Todd Trimble, repeatedly challenge me to “systematize” my findings, and develop them... more Several readers, particularly Todd Trimble, repeatedly challenge me to “systematize” my findings, and develop them into an actual language. I think this is quite wise, if not impossible. Sometimes I use terms that are idiosyncratic and/or have been explained in previous posts. However, the document will also constitute its own special resource. Where helpful, I have added example sentences that in some way illustrate the meaning of the word. There is a dash of humor throughout, in order to keep the technical stuff more lively.

Agents Network on a Grid: An Approach with the Set of Circulant Operators

by Babiga Birregah

Volumetric diffusers: pseudorandom cylinder arrays on a periodic lattice

by Richard Hughes

R. J. Hughes, J. A. S. Angus, T. J. Cox, O. Umnova, G. A. Gehring, M. A. Pogson, D. M. Whittaker, Journal of the Acoustical Society of America, Vol. 128(5), pp. 2847-2856 (2010)
Download: http://usir.salford.ac.uk/14622/
Permalink: http://dx.doi.org/10.1121/1.3493455

Most conventional diffusers take the form of a surface based treatment, and as a result can only operate in... more Most conventional diffusers take the form of a surface based treatment, and as a result can only operate in hemispherical space. Placing a diffuser in the volume of a room might provide greater efficiency by allowing scattering into the whole space. A periodic cylinder array (or sonic crystal) produces periodicity lobes and uneven scattering. Introducing defects into an array, by removing or varying the size of some of the cylinders, can enhance their diffusing abilities. This paper applies number theoretic concepts to create cylinder arrays that have more even scattering. Predictions using a boundary element method are compared to measurements to verify the model, and suitable metrics are adopted to evaluate performance. Arrangements with good aperiodic autocorrelation properties tend to produce the best results. At low frequency power is controlled by object size and at high frequency diffusion is dominated by lattice spacing and structural similarity. Consequently the operational bandwidth is rather small. By using sparse arrays and varying cylinder sizes, a wider bandwidth can be achieved.

Volume diffusers for architectural acoustics

by Richard Hughes

Ph.D. Thesis (2011)
Download: http://usir.salford.ac.uk/17672/

Most conventional diffusers are used on room surfaces, and consequently can only operate on a hemispherical area.... more Most conventional diffusers are used on room surfaces, and consequently can only operate on a hemispherical area. Placing a diffuser in the volume of a room may provide greater efficiency by allowing scattering into the whole space. There are very few examples of volume diffusers and they tend to be limited in design; subsequently a suitable method for their development is lacking.

2D volumetric diffusers are investigated, considering a number of design concepts; namely arrays of slats, percolation structures and cylinder arrays. An experimental technique is adapted for their measurement, and the results are used to verify prediction models for each type. Diffusive efficacy is assessed through a new metric based on an existing surface diffuser coefficient and a measure of scattered power requiring half of the energy to be back-scattered.

Single layer slat arrays are formed from optimal aperiodic sequences, though due to the directional scattering from individual slats at higher frequencies, performance is heavily dependent on line-of-sight through the array. This limits the operational bandwidth to approximately 1.5 octaves. Multi-layer structures offer improvements by allowing cancellation of the back-scattered lobe, though at high frequency the specular reflection from an individual slat still dominates. Percolation fractals use slats orientated in multiple directions and by scattering laterally can channel sound and diffuse at lower frequencies. Low frequency diffusion however is limited and the best structures are those which provide a broad range of geometric reflection paths.

Through application of number theoretic concepts, arrangements of cylinders are shown to offer more enhanced diffusing abilities than slat and percolation structures. At low frequency scattered power is controlled by cylinder size and at high frequency diffusion is dominated by their spacing. By minimising structural similarity and including cylinders with circumference comparable to wavelength, significant diffusion is achieved over an approximate 5 octave bandwidth.

Algebraic Number Theory, Polygons and Quadratic Reciprocity

by Daniel Fretwell

This was a summer project I undertook after my 3rd undergraduate year, under the supervision of Dr. Neil Dummigan. This was a summer project I undertook after my 3rd undergraduate year, under the supervision of Dr. Neil Dummigan.

Class Field Theory (part 1)

by Daniel Fretwell

The first part of my masters dissertation, completed under the supervision of Dr. Neil Dummigan.

This is... more The first part of my masters dissertation, completed under the supervision of Dr. Neil Dummigan.

This is a quite informal view of global class field theory, viewed from the platform of ideals.

See the second part, "Class Field Theory: Proofs and Applications", for a more detailed view along with proofs, including the introduction of ideles, a bit of cohomology and applications of class field theory to the representation of primes by the quadratic form x^2 + ny^2.

Class Field Theory (part 2): Proofs and Applications

by Daniel Fretwell

The second part of my masters dissertation, done under the supervision of Dr. Neil Dummigan.

This... more The second part of my masters dissertation, done under the supervision of Dr. Neil Dummigan.

This installment proves everything done informally in the first part. This is quite a difficult and lengthy task and many new devices need to be invented, such as the ideles and the Herbrand quotient.

Finally, we apply the theory to the representation of primes by the quadratic form x^2 + ny^2, giving some examples.

The repulsion motif in Diophantine equations

by Thomas Ward

Co-authored with Graham Everest, published in American Mathematical Monthly August-September 2011; see http://www.maa.org/pubs/monthly_augsep11_toc.html

Problems related to the existence of integral and rational points on cubic curves date back at least to Diophantus. A... more Problems related to the existence of integral and rational points on cubic curves date back at least to Diophantus. A significant step in the modern theory of these equations was made by Siegel, who proved that a nonsingular plane cubic equation has only finitely many integral solutions. Examples show that simple equations can have inordinately large integral solutions in comparison to the size of their coefficients. Nonetheless, a conjecture of Hall suggests a bound on the size of integral solutions in terms of the coefficients of the defining equation. It turns out that a similar phenomenon seems, conjecturally, to be at work for solutions which are close to being integral in another sense. We describe this conjecture as an illustration of an underlying motif—repulsion—in the theory of Diophantine equations.

Mahler measure and entropy for commuting automorphisms of compact groups

by Thomas Ward

Co-authored with Douglas Lind and Klaus Schmidt

Mixing automorphisms of compact groups and a theorem of Schlickewei

by Thomas Ward

Co-authored with Klaus Schmidt

We prove that every mixing Z^d-action by automorphisms of a compact, connected, abelian group is mixing of all orders. We prove that every mixing Z^d-action by automorphisms of a compact, connected, abelian group is mixing of all orders.