Higher Dimensional Discrete Cheeger Inequalities

For graphs there exists a strong connection between spectral and combinatorial expansion properties. This is expressed, e.g., by the discrete Cheeger inequality, the lower bound of which states that $\lambda(G) \leq h(G)$, where $\lambda(G)$ is the second smallest eigenvalue of the Laplacian of a graph $G$ and $h(G)$ is the Cheeger constant measuring the edge expansion of $G$. We are interested in generalizations of expansion properties to finite simplicial complexes of higher dimension (or uniform hypergraphs). Whereas higher dimensional Laplacians were introduced already in 1945 by Eckmann, the generalization of edge expansion to simplicial complexes is not straightforward. Recently, a topologically motivated notion analogous to edge expansion that is based on $\mathbb{Z}_2$-cohomology was introduced by Gromov and independently by Linial, Meshulam and Wallach. It is known that for this generalization there is no higher dimensional analogue of the lower bound of the Cheeger inequality. A different, combinatorially motivated generalization of the Cheeger constant, denoted by $h(X)$, was studied by Parzanchevski, Rosenthal and Tessler. They showed that indeed $\lambda(X) \leq h(X)$, where $\lambda(X)$ is the smallest non-trivial eigenvalue of the ($(k-1)$-dimensional upper) Laplacian, for the case of $k$-dimensional simplicial complexes $X$ with complete $(k-1)$-skeleton. Whether this inequality also holds for $k$-dimensional complexes with non-complete $(k-1)$-skeleton has been an open question. We give two proofs of the inequality for arbitrary complexes. The proofs differ strongly in the methods and structures employed, and each allows for a different kind of additional strengthening of the original result.Comment: 14 pages, 2 figure

A Presheaf Interpretation of the Generalized Freyd Conjecture

We give a generalized version of the Freyd conjecture and a way to think about a possible proof. The essential point is to describe an elementary formal reduction of the question that holds in any triangulated category. There are no new results, but at least one known example drops out quite trivially.Comment: 8 pages; formerly titled "Thinking about the Freyd conjecture

Program for girls' junior high school (mission) in Shanghai, China

This item was digitized by the Internet Archive. Thesis (M.A.)--Boston Universityhttps://archive.org/details/programforgirlsj00wes

A master list of reading materials for enriching courses in mathematics.

Thesis (M.A.)--Boston Universit

Assessment in the Early Years: The Perspectives and Practices of Early Childhood Educators

In recent years there has been growing attention on the importance of assessment in early childhood education, especially in relation to supporting children’s learning. The present study aimed to investigate early childhood educators’ perspectives and practices regarding assessment in the early years. In particular, the meanings and values which educators ascribe to assessment were explored. Moreover, the study focused on strategies educators employed, along with the associated support and challenges relating to their assessment practice. Adopting a qualitative design, in-depth interviews were conducted with eight educators from different settings, and thematic analysis was used to identify emergent themes. Subsequently, information from assessment tools that educators used in practice were collected and analysed. Findings show that educators hold diverse views and have varying approaches to assessment, using different tools and methods. Nevertheless, participants agree that assessment is important for supporting children’s learning and development. Data suggests that collaboration plays some role in aiding assessment practice, particularly collaborating with colleagues and parents; however, findings also indicate that children have limited participation in the assessment process. The study also suggests that time, structural factors, qualification and training contribute to the ease in which assessment is carried out. Delving into educators’ perspectives and practices on early years’ assessment can offer insight on what actually happens in settings and the thoughts and attitudes that direct them, while shedding light on different issues they are faced with. The author hopes that the findings of the study can direct future research investigating issues surrounding assessment practice, greater collaboration with families, and children’s agency in assessment

The Effects of Parent Factors on Children\u27s Separation Anxiety

Parental anxiety is a well-known factor that contributes to the development of anxiety in children. However, little is known about how specific parental factors influence the development and trajectory of childhood anxiety. There is also a paucity of research on separation anxiety disorder specifically. Complicating matters, children who suffer from clinically significant separation anxiety tend to be younger and thus it is harder to obtain accurate information from this age group. The purpose of the present study was to determine the mechanism through which parental factors such as somatization, anxiety sensitivity, and separation anxious behaviors might influence the development of anxiety in young children (specifically separation anxiety). As a result, 50 children and their mothers completed measures asking about demographic variables, internalizing problems, anxiety sensitivity, and somatization. The results showed that only parental separation anxious behaviors uniquely influenced separation anxious behaviors in children. Results and implications of these findings may be used to direct further examination of influential parental factors and inform treatment by targeting anxious parental behaviors

Disentangling the effects of phonation and articulation: Hemispheric asymmetries in the auditory N1m response of the human brain

BACKGROUND: The cortical activity underlying the perception of vowel identity has typically been addressed by manipulating the first and second formant frequency (F1 & F2) of the speech stimuli. These two values, originating from articulation, are already sufficient for the phonetic characterization of vowel category. In the present study, we investigated how the spectral cues caused by articulation are reflected in cortical speech processing when combined with phonation, the other major part of speech production manifested as the fundamental frequency (F0) and its harmonic integer multiples. To study the combined effects of articulation and phonation we presented vowels with either high (/a/) or low (/u/) formant frequencies which were driven by three different types of excitation: a natural periodic pulseform reflecting the vibration of the vocal folds, an aperiodic noise excitation, or a tonal waveform. The auditory N1m response was recorded with whole-head magnetoencephalography (MEG) from ten human subjects in order to resolve whether brain events reflecting articulation and phonation are specific to the left or right hemisphere of the human brain. RESULTS: The N1m responses for the six stimulus types displayed a considerable dynamic range of 115–135 ms, and were elicited faster (~10 ms) by the high-formant /a/ than by the low-formant /u/, indicating an effect of articulation. While excitation type had no effect on the latency of the right-hemispheric N1m, the left-hemispheric N1m elicited by the tonally excited /a/ was some 10 ms earlier than that elicited by the periodic and the aperiodic excitation. The amplitude of the N1m in both hemispheres was systematically stronger to stimulation with natural periodic excitation. Also, stimulus type had a marked (up to 7 mm) effect on the source location of the N1m, with periodic excitation resulting in more anterior sources than aperiodic and tonal excitation. CONCLUSION: The auditory brain areas of the two hemispheres exhibit differential tuning to natural speech signals, observable already in the passive recording condition. The variations in the latency and strength of the auditory N1m response can be traced back to the spectral structure of the stimuli. More specifically, the combined effects of the harmonic comb structure originating from the natural voice excitation caused by the fluctuating vocal folds and the location of the formant frequencies originating from the vocal tract leads to asymmetric behaviour of the left and right hemisphere