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Aleksandra Kowalska

Publications and source records attributed to Aleksandra Kowalska.

4 recordsLinked to original sources

Counting solutions to quadratic forms in eight prime variables of off-diagonal rank $3$

In this note we count the number of solutions to a non-degenerate quadratic form in eight prime variables of off-diagonal rank $3$. It is a continuation of work by L. Zhao, who counted solutions to forms in at least nine variables, by B. Green, who counted solutions to 'generic' (which implies off-diagonal rank $4$) forms in eight variables, and by J. Dobrowolski, who counted solutions to forms in eight variables of off-diagonal rank at most $2$.

math.NT

Sárközy's theorem in $\mathbf{F}_2[x]$

Green showed that, conditional on GRH, a subset $A \subseteq [N]$ with $\mid A \mid \gg_ε N^{\frac{11}{12}+ε}$ must contain two elements whose difference is $p-1$ for $p$ a prime. We prove an analogous unconditional result for $\mathbf{F}_2[x]$, improving the exponent to $\frac{7}{8}+ε$.

math.NT

A Generalisation of Sperner's Theorem Using Weighted Chains

We find the (unique) largest subset of $\{0, 1, 2\}^n$ such that it contains no two elements, one of which is coordinatewise greater than the other, but strictly greater on at most $k$ coordinates. To do so, we decompose the cube into weighted chains. In Appendix B we present a new proof of Sperner's theorem we found while working on this problem.

math.CO

Is the Symbiotic Recurrent Nova T CrB Late? Recent Photometric Evolution and Comparison with Past Pre-Outburst Behaviour

T CrB is a symbiotic recurrent nova that last erupted in 1946. Given its recurrence timescale of approximately 80 years, the next outburst is eagerly anticipated by the astronomical community. In this work, we analyse the optical light curves of T CrB, comparing recent photometric evolution with historical data to evaluate potential predictive indicators of nova eruptions. Although the "super-active" phases preceding both the 1946 and anticipated eruptions are strikingly similar, the subsequent photometric behaviour differs. We find that the decline in brightness observed in 2023, interpreted by some as a "pre-eruption dip", deviates from the deep minimum recorded prior to the 1946 event and does not reliably predict the eruption timing. Recent photometric and spectroscopic observations indicate that the system is returning to a high-accretion state. Given this, an eruption may be imminent, even without distinct precursors. While the next eruption of T CrB will be a major scientific event, its expected peak brightness of $V \sim 2$ mag highlights the importance of setting realistic public expectations for what will be a visually modest, yet astrophysically very significant, celestial event.

astro-ph.SR