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Daniel Katz

Publications and source records attributed to Daniel Katz.

10 recordsLinked to original sources
arXiv2025

Joint reductions and mixed Buchsbaum-Rim multiplicities of modules and a joint-reduction-number-zero theorem

We offer new definitions of joint reductions and mixed Buchsbaum-Rim multiplicity for certain collections of modules over a Noetherian local ring and illustrate their application to give two different proofs of a joint-reduction-number-zero theorem for integrally closed modules over two-dimensional regular local rings. We also relate the mixed Buchsbaum-Rim multiplicity of modules to the Euler-Poincar\'{e} characteristic of a natural Koszul complex and relate it to the mixed Buchsbaum-Rim multiplicity of ideals by generalising a lemma from intersection theory.

math.AC↗
arXiv2024

On Abelian extensions in mixed characteristic and ramification in codimension one

A theorem of Paul Roberts states that the integral closure of a regular local ring in a generically abelian extension is Cohen-Macaulay, provided the characteristic of the residue field does not divide the order of the Galois group. An example of Koh shows the conclusion is false in the modular case. After a modification to the statement concerning ramification over $p$ in codimension one, we give an extension of Roberts's theorem to the modular case for unramified regular local rings in mixed characteristic when the $p$-torsion of the Galois group is annihilated by $p$.

math.AC↗
arXiv2018

Big Jets Model with CPT Invariance and Dynamics of Expansion with Quantum Yang-Mills Gravity

Based on particle physics, the fundamental CPT invariance suggests a Big Jets model for the beginning of the universe, in which two oppositely directed jets evolved into a gigantic "matter half-universe" and a gigantic "antimatter half-universe" after annihilation and decay processes. In the geometric-optics limit, quantum Yang-Mills gravity with $T_4$ translational gauge symmetry in flat spacetime leads to an effective metric tensor in the Hamilton-Jacobi equation for macroscopic objects. This effective metric tensor does not exist in the wave equations of quantum particles. For cosmological expansion, we assume that an "effective metric tensor" for spacetime geometry based on Yang-Mills gravity corresponds to the usual FLRW form. Dynamical equations of expansion for the matter half-universe are obtained and solved. The time-dependent scale factors and the estimated age of the universes, $t^{YM}_o \approx 15.3 \times 10^9 yr$, based on Yang-Mills gravity are consistent with experiments. CPT invariance implies that the same evolution process and dynamics of cosmic expansion also hold for the distant `antimatter half-universe.'

physics.gen-ph↗
arXiv2018

A New Quantum Operator for Distance

We introduce a new semi-relativistic quantum operator for the length of the worldline a particle traces out as it moves. In this article the operator is constructed in a heuristic way and some of its elementary properties are explored. The operator ends up depending in a very complicated way on the potential of the system it is to act on so as a proof of concept we use it to analyze the expected distance traveled by a free Gaussian wavepacket with some initial momentum. It is shown in this case that the distance such a particle travels becomes light-like as its mass vanishes and agrees with the classical result for macroscopic masses. This preliminary result has minor implications for the Weak Equivalence Principle (WEP) in quantum mechanics. In particular it shows that the logical relationship between two formulations of the WEP in classical mechanics extends to quantum mechanics. That our result is qualitatively consistent with the work of others emboldens us to start the task of evaluating the new operator in non-zero potentials. However, we readily acknowledge that the looseness in the definition of our operator means that all of our so-called results are highly speculative. Plans for future work with the new operator are discussed in the last section.

quant-ph↗
arXiv2015

Recombination of H and He in Yang-Mills Gravity

We investigate some aspects of the thermal history of the early universe according to Yang-Mills Gravity (YMG); a gauge theory of gravity set in flat spacetime. Specifically, equations for the ionization fractions of hydrogen and singly ionized helium during the recombination epoch are deduced analytically and then solved numerically. By considering several approximations we find that the presence of primordial helium and its interaction with Lyman series photons has a much stronger effect on the overall free electron density in YMG than it does in the standard, General Relativity (GR) based, model. Compared to the standard model recombination happens over a much larger range of temperatures, although there is still a very sharp temperature of last scattering around 2000 K. Since the ionization history of the universe is not directly observable we discuss how one may use it to predict the CMB power spectrum and thus test YMG. This topic will be explored in detail in an upcoming paper.

gr-qc↗
arXiv2014

A New Analytic Galactic Luminosity Profile Function

In 2010 Spergel introduced an alternative to the traditional Sersic form for galactic luminosity profiles based on modified Bessel functions of the second kind. His motivation was the desire for an accurate one-parameter profile form with a simple Fourier transform (in contrast to the Fourier transform of the Sersic profile which can't be written in closed form), but we have found that the Spergel profile almost universally makes integrals easier when it replaces the Sersic profile in the integrand. In the original paper on the subject Spergel noted that his profile seems to fit galaxies on average just as well as Sersic's. Here we make this observation quantitative by comparing the residuals from fitting Sersic and Spergel forms to data. We find that the Spergel profile actually fits better than the Sersic for a random sample of 16 galaxies.

astro-ph.GA↗
arXiv2013

Quasi-finite modules and asymptotic prime divisors

Let $A$ be a Noetherian ring, $J\subseteq A$ an ideal and $C$ a finitely generated $A$-module. In this note we would like to prove the following statement. Let $\{I_n\}_{n\geq 0}$ be a collection of ideals satisfying : (i) $I_n\supseteq J^n$, for all $n$, (ii) $J^s\cdot I_s \subseteq I_{r+s}$, for all $r,s\geq 0$ and (iii) $I_n\subseteq I_m$, whenever $m\leq n$. Then $\Ass_A(I_nC/J^nC)$ is independent of $n$, for $n$ sufficiently large. Note that the set of prime ideals $\cup_{n\geq 1} \Ass_A(I_nC/J^nC)$ is finite, so the issue at hand is the realization that the primes in $\Ass_A(I_nC/J^nC)$ \textit{do not} behave periodically, as one might have expected, say if $\bigoplus _{n\geq 0}I_n$ were a Noetherian $A$-algebra generated in degrees greater than one. We also give a multigraded version of our results.

math.AC↗
arXiv2008

On sumfree subsets of hypercubes

We consider the possible sizes of large sumfree sets contained in the discrete hypercube $\{1,...,n\}^k$, and we determine upper and lower bounds for the maximal size as $n$ becomes large. We also discuss a continuous analogue in which our lower bound remains valid and our upper bound can be strengthened, and we consider the generalization of both problems to $l$-fold-sumfree sets.

math.NT↗
arXiv2004

Integral closure of ideals and annihilators of homology

This article outgrew from an effort to understand our basic question: Are the annihilators of the non-zero Koszul homology modules $H_i$ of an unmixed ideal $I$ contained in the integral closure $\bar{I}$ of $I$? We also obtain some variations on a result of Burch, which continue the theme of the paper in that they deal with annihilators of homology and integrally closed ideals.

math.AC↗