Conditions on the existence of unambiguous morphisms

A morphism α is (strongly) unambiguous with respect to a word α if there is no other morphism τ that maps α to the same image as σ. Moreover, α is said to be weakly unambiguous with respect to a word α if σ is the only nonerasing morphism.....

Unambiguous 1-uniform morphisms

A morphism σ is unambiguous with respect to a word α if there is no other morphism τ that maps α to the same image as σ. In the present paper we study the question of whether, for any given word, there exists an unambiguous 1-uniform morphism, i.e., a morphism that maps every letter in the word to an image of length 1

Morphic primitivity and alphabet reductions

An alphabet reduction is a 1-uniform morphism that maps a word to an image that contains a smaller number of dfferent letters. In the present paper we investigate the effect of alphabet reductions on morphically primitive words, i. e., words that are not a fixed point of a nontrivial morphism. Our first main result answers a question on the existence of unambiguous alphabet reductions for such words, and our second main result establishes whether alphabet reductions can be given that preserve morphic primitivity. In addition to this, we study Billaud's Conjecture - which features a dfferent type of alphabet reduction, but is otherwise closely related to the main subject of our paper - and prove its correctness for a special case

Weakly Unambiguous Morphisms

A nonerasing morphism sigma is said to be weakly unambiguous with respect to a word w if sigma is the only nonerasing morphism that can map w to sigma(w), i.e., there does not exist any other nonerasing morphism tau satisfying tau(w) = sigma(w). In the present paper, we wish to characterise those words with respect to which there exists such a morphism. This question is nontrivial if we consider so-called length-increasing morphisms, which map a word to an image that is strictly longer than the word. Our main result is a compact characterisation that holds for all morphisms with ternary or larger target alphabets. We also comprehensively describe those words that have a weakly unambiguous length-increasing morphism with a unary target alphabet, but we have to leave the problem open for binary alphabets, where we can merely give some non-characteristic conditions

Weakly unambiguous morphisms

A nonerasing morphism σ is said to be weakly unambiguous with respect to a word s if σ is the only nonerasing morphism that can map s to σ(s), i. e., there does not exist any other nonerasing morphism τ satisfying τ(s) = σ(s). In the present paper, we wish to characterise those words with respect to which there exists such a morphism. This question is nontrivial if we consider so-called length-increasing morphisms, which map a word to an image that is strictly longer than the word. Our main result is a compact characterisation that holds for all morphisms with ternary or larger target alphabets. We also comprehensively describe those words that have a weakly unambiguous length-increasing morphism with a unary target alphabet, but we have to leave the problem open for binary alphabets, where we can merely give some non-characteristic conditions

Weakly unambiguous morphisms

Weakly unambiguous morphism

How new readers perform as compared to more experienced readers on the PERFORMS scheme

How new readers perform as compared to more experienced readers on the PERFORMS schem

How quickly do breast screeners learn their skills?

The UK’s Breast Screening Programme is 27 years old and many experienced breast radiologists are now retiring, coupled with an influx of new screening personnel. It is important to the ongoing Programme that new mammography readers are quickly up to the skill level of experienced readers. This raises the question of how quickly the necessary cancer detection skills are learnt. All breast screening radiologists in the UK read educational training sets of challenging FFDM images (the PERFORMS® scheme) yearly to maintain and improve their performance in real life screening. Data were examined from the PERFORMS® annual scheme for 54 new screeners, 55 screeners who have been screening for one year and also for more experienced screeners (597 screeners). Not surprisingly, significant differences in cancer detection rate were found between new readers and both of the other groups. Additionally, the performance of 48 new readers who have now been screening for about a year and have taken part twice in the PERFORMS® scheme were further examined where again a significant difference in cancer detection was found. These data imply that cancer detection skills are learnt quickly in the first year of screening. Information was also examined concerning the volume of cases participants read and other factors

The international use of PERFORMS mammographic test sets

To examine the utility of employing breast screening test sets internationally the data of 1,009 radiologists from the USA, UK and other European countries were examined as they inspected 20 carefully selected difficult recent screening cases. Some 720 UK radiologists, 247 American and 42 European radiologists took part. Whilst similar sensitivity scores between the three groups were found, the main difference was the lower specificity of the American radiologists reflecting their different recall clinical practice. It is argued that using test sets internationally provides participants with useful comparative performance information whilst also providing data on how the same cases are interpreted by radiologists from different countries

A perceptual aid to delineating the extent of potential mammographic abnormalities

Being able to accurately determine the extent of a possible malignancy on a mammogram is an important task as this can affect the potential follow up surgical treatment that a woman receives after breast screening. It is known that this can be a difficult task, particularly where the lesion has diffuse abnormalities. A potential computer-aided approach is to employ Hierarchical Clustering-based Segmentation (HCS) and this interactive educational exhibit dynamically demonstrates this technique. HCS is an unsupervised segmentation process that when applied to an image yields a hierarchy of segmentations based on image pixel dissimilarities and so can be used to highlight areas in the mammographic image to aid interpretation