The capture time of grids

We consider the game of Cops and Robber played on the Cartesian product of two trees. Assuming the players play perfectly, it is shown that if there are two cops in the game, then the length of the game (known as the 2-capture time of the graph) is equal to half the diameter of the graph. In particular, the 2-capture time of the m x n grid is proved to be floor ((m+n-2)/2).Comment: 7 page

Mitigating the Problem of Strong Priors in LMs with Context Extrapolation

Language models (LMs) have become important tools in a variety of applications, from data processing to the creation of instruction-following assistants. But despite their advantages, LMs have certain idiosyncratic limitations such as the problem of `strong priors', where a model learns to output typical continuations in response to certain, usually local, portions of the input regardless of any earlier instructions. For example, prompt injection attacks can induce models to ignore explicit directives. In some cases, larger models have been shown to be more susceptible to these problems than similar smaller models, an example of the phenomenon of `inverse scaling'. We develop a new technique for mitigating the problem of strong priors: we take the original set of instructions, produce a weakened version of the original prompt that is even more susceptible to the strong priors problem, and then extrapolate the continuation away from the weakened prompt. This lets us infer how the model would continue a hypothetical strengthened set of instructions. Our technique conceptualises LMs as mixture models which combine a family of data generation processes, reinforcing the desired elements of the mixture. Our approach works at inference time, removing any need for retraining. We apply it to eleven models including GPT-2, GPT-3, Llama 2, and Mistral on four tasks, and find improvements in 41/44. Across all 44 combinations the median increase in proportion of tasks completed is 40%.Comment: 12 pages, 4 figure

Sorting by Swaps with Noisy Comparisons

We study sorting of permutations by random swaps if each comparison gives the wrong result with some fixed probability $p<1/2$. We use this process as prototype for the behaviour of randomized, comparison-based optimization heuristics in the presence of noisy comparisons. As quality measure, we compute the expected fitness of the stationary distribution. To measure the runtime, we compute the minimal number of steps after which the average fitness approximates the expected fitness of the stationary distribution. We study the process where in each round a random pair of elements at distance at most $r$ are compared. We give theoretical results for the extreme cases $r=1$ and $r=n$, and experimental results for the intermediate cases. We find a trade-off between faster convergence (for large $r$) and better quality of the solution after convergence (for small $r$).Comment: An extended abstract of this paper has been presented at Genetic and Evolutionary Computation Conference (GECCO 2017

Floor Heating of the Family House with the Condensing Technics

Import 23/07/2015Gavenčiak Jan, Podlahové vytápění rodinného domu s kondenzační technologií. Bakalářská práce na fakultě stavební VŠB-TU Ostrava na katedře prostředí staveb a TZB, 2015, 125 stran. Vedoucí diplomové práce Ing. Zdeněk Galda, Ph.D. Bakalářská práce se zabývá návrhem podlahového vytápění rodinného domu s kondenzační technologií, návrhem a posouzením solárních kolektorů pro ohřev teplé vody, tepelně technickým posouzením vybraných detailů zejména z hlediska teplotního faktoru vnitřního povrchu a jeho bezpečnosti. Bakalářská práce zahrnuje i energetickou bilanci potřeby tepla vč. energetického štítku obálky budovy. Zpracována byla projektová dokumentace v rozsahu pro provádění stavby.Gavenčiak Jan, Floor Heating of the Family House with the Condensing Technics, 125 pages. Bachelor thesis at the Faculty of Civil Engineering VŠB-TU Ostrava at Department of Indoor Environmental Engineering and construction HVAC. Thesis supervisor Ing. Zdeněk Galda, Ph.D. Bachelor thesis deals with floor heating of the Family house with the condensing technology, design and assessment of solar collectors for hot water, heat and technical assessment of the selected detail especially in terms of the temperature factor of the inner surface and its safety. Bachelor thesis includes the energy balance of the heat requirement incl. label of the building envelope. Project documentation was prepared within the scope of the execution of the project.229 - Katedra prostředí staveb a TZBvýborn