Number Of Symbols Research Articles

(Analysis of Children's Cognitive Abilities Through Web-Based Educational Games at TKN Pembina Gondang)

This research is motivated by the use of games which are inseparable from the world of children, it could even be said to be very important for the growth of children's intelligence, the importance of these games is that wherever they are, both at home and in the school environment, various games are now available for children so that the aim of This research is 1) to analyze children's cognitive abilities through web-based educational games and 2) the impact of these games on children's cognitive abilities. The location of this research is TKN Pembina Gondang which is located in the North Lombok area using research methods, namely using qualitative research with a case study approach and the data obtained is analyzed using data analysis, namely data transcription, data reduction, data validation or triangulation, presentation. data and drawing conclusions. So the research results obtained are: 1) The use of web-based educational games is able to influence the cognitive abilities of young children, especially in the aspects of problem solving, logical thinking and symbolic thinking; 2) The use of web-based educational game media, especially Wordwall Net, has a positive impact on children's cognitive abilities, because through this game children behave in a way that shows curiosity, have initiative in choosing games, are able to solve problems creatively, use number symbols to calculate, follow the rules, and be able to use technology correctly.

(Dis)similarities between non-symbolic and symbolic number representations: Insights from vector space models

Empirical evidence in support of a shared system for non-symbolic and symbolic number processing has been inconclusive. The current study aims to address this question in a novel way, specifically by testing whether the efficient coding principle based on co-occurrence of number symbols in natural language holds for both non-symbolic and symbolic number processing. The efficient coding principle postulates that perception is optimized when stimuli frequently co-occur in a natural environment. We hypothesized that both numerical ratios and co-occurrence frequencies of symbolic numbers would significantly influence participants' performance on a non-symbolic and symbolic number comparison task. To test this hypothesis, we employed latent semantic analysis on a TASA corpus to quantify number co-occurrence in natural language and calculate language similarity estimates. We engaged 73 native English speakers (mean age = 19.36, standard deviation = 1.83) with normal or corrected vision and no learning disorders in a number comparison task involving non-symbolic (dot arrays) and symbolic stimuli (Arabic numerals and English number words). Results showed that numerical ratios significantly predicted participants' performances across all number formats (ps < 0.001). Language similarity estimates derived from everyday language also predicted performance on the non-symbolic task and the symbolic task involving number words (ps < 0.007). Our results highlight the complex nature of numerical processing, pointing to the co-occurrence of number symbols in natural language as an auxiliary factor in understanding the shared characteristics between non-symbolic and symbolic number representations. Given that our study focused on a limited number range (5 to 16) and a specific task type, future studies should explore a wider range of tasks and numbers to further test the role of the efficient coding principle in number processing.

Multi-IRSs enhanced UAV secure semantic communication system

This paper delves into a unmanned aerial vehicle (UAV) semantic communication system featuring multiple intelligent reflecting surfaces (IRSs). The UAV transmits semantic symbols to users in the presence of eavesdropping, and thus has inevitable security problems. In order to measure secure communication performance in semantic communication, the security semantic transmission rate (SSR) based on the semantic similarity is employed. An optimization problem is formulated to enhance the average SSR by optimizing UAV trajectory, IRS reflecting coefficients, number of semantic symbols, and transmission power, while satisfying position, speed, anti-collision, and power constraints. Due to the highly dynamic nature of the environment and the difficulties in solving complex modeling problems, a novel approach based on hybrid soft actor–critic (H-SAC) deep reinforcement learning (DRL) is proposed to address this non-convex optimization problem. The algorithm addresses the challenge of dynamic environments in UAV scenarios by seamlessly accommodating both discrete and continuous action spaces such as the trajectory and the IRS reflecting coefficient within the network. This enables effective processing of actions taken by intelligent agents, ensuring adaptability to the diverse requirements of the environment without the need for separate handling mechanisms. Simulation results show that the proposed intelligent algorithm exhibits excellent convergence and achieves a well-balanced pairing of appropriate IRSs and selection of flight paths, significantly improving the average SSR of the system. Specifically, our proposed scheme enhances the secure semantic transfer rate by more than fifty percent compared to when the number of semantic symbols is fixed at two.

Pictorial Number Sandpaper Media in Recognizing Number Symbols and Training Children's Fine Motor Skills

Pictorial Number Sandpaper Media in Recognizing Number Symbols and Training Children's Fine Motor Skills

Modular forms and congruences for k-marked odd Durfee symbols

In 2007, Andrews introduced the k-marked odd Durfee symbols and studied their parities. Let Dk0(n) denote the number of k-marked odd Durfee symbols of n. He proved that Dk0(n) is even for k≥1 and n≡k−1(mod2). Inspired by Andrews' work, many authors investigated congruences for Dk0(n) modulo 2 or powers of 2. For example, Wang obtained D20(8n+4)≡D20(8n+6)≡0(mod2) which were first conjectured by Andrews. More recently, Xia proved that D2mk0(2n+1)≡0(mod2m+1) with m,k≥1 and n≥0. In this paper, we first study the modular properties of the generating function of D20(n) which is completed to be a weakly modular form of weight 52. As application, we utilize the Hecke operators to establish families of congruences for D20(n) modulo powers of primes ℓ≥5.

Method for measuring radiation patterns of digital antenna arrays

Formulation of the problem. The Digital Beamforming Antenna Arrays (DBAA) are widely used as the directional antennas in radio systems today. However, such arrays typically do not have a radiofrequency output. Consequently, the standard methods for measuring the Radiation Patterns (RP), based on the transmitting of a monochromatic probing signals, reception of the signals in the far field, as well as measuring the output power of the antenna array, cannot be used to measure the RP of the DBAA. The method considered in this work involves the transmitting a probing signal modulated by a pseudonoise sequence and processing the DBAA output signal at the zero intermediate frequency (base-band) using a matched filter or a correlator. Target. A method for measuring the radiation pattern of the DBAA is considered, based on the correlation processing of its output signal. Results. The paper presents a description of the setup for measuring the DBAA RP, as well as a mathematical explanation of the measurement procedure. It is shown that the accuracy of the RP measurement depends on the Signal-to-Noise Ratio (SNR) at the outputs of the receivers of the DBAA channels, and the number of array channels and the number of symbols of the pseudonoise sequence used for the modulation of the radiated signal. The error in the RP measurement decreases by 3 dB each time as the number of the DBAA channels or the number of symbols of the pseudonoise sequence used in the measurements is doubled. This result is confirmed by simulation the measurement of RP of the eight- and sixteen-channel linear DBAA using pseudo-noise sequences of maximum length (M-sequences) with a number of symbols equal to 1023 and 2047. Practical significance. The proposed method does not require the usage of the up-converter to convert the base-band DBAA output signal from the base-band to a carrier frequency, and it also does not require the usage of a power meter for measurement the analog signal at the output of this converter.

Pengaruh Penggunaan Media Roulette Angka Terhadap Kemampuan Kognitif Anak dalam Mengenal Lambang Bilangan pada Kelompok B Usia 5-6 Tahun TK Pelangi Medan T.A 2023/2024

&lt;em&gt;The results of this study indicated that there was a difference in the level of cognitive ability of children in recognizing number symbols in Group B, aged 5-6 years, at TK Pelangi Medan, where learning activities were conducted using number roulette media. The children were given 2 treatment periods using the number roulette media and 2 treatment periods without using the roulette media. In the first period (X&lt;sub&gt;1&lt;/sub&gt;O&lt;sub&gt;1&lt;/sub&gt;), children who received treatment scored an average of 9.43, which falls into the fairly good category. In the second period (X&lt;sub&gt;0&lt;/sub&gt;O&lt;sub&gt;2&lt;/sub&gt;), without treatment using the roulette media, the children scored an average of 6.12, which falls into the less good category. In the third period (X&lt;sub&gt;1&lt;/sub&gt;O&lt;sub&gt;3&lt;/sub&gt;), children who received treatment scored an average of 17.62, which falls into the very good category, and in the fourth period (X&lt;sub&gt;0&lt;/sub&gt;O&lt;sub&gt;4&lt;/sub&gt;), without treatment using the roulette media, the children scored an average of 14.37, which falls into the good category.The data scores indicate that the use of number roulette media has a significant effect on the cognitive ability of children in Group B, aged 5-6 years, at TK Pelangi in recognizing number symbols. This is evidenced by the SPSS output, which shows that the Asymp. Sig (2-tailed) value is 0.000. Since the value 0.000 is less than 0.005, Ha is accepted, meaning there is a significant difference in the average learning outcomes before and after treatment. Therefore, it can be said that "there is a significant effect through the use of number roulette media on the cognitive ability of children in recognizing number symbols in Group B, aged 5-6 years, at TK Pelangi Medan during the 2023/2024 academic year&lt;/em&gt;

Snakes and Ladders Game: Developing the Early Childhoods’ Cognitive Ability at Cendrawasih II Kindergarten

A game is a tool functions to develop various aspects of early childhoods’ development. One of games is the snakes and ladders which can develop the early childhoods’ cognitive ability. This study aims to determine the development on the early childhoods’ aged 5-6 years cognitive ability through the use of the snakes and ladders game. This study employs qualitative research method under descriptive approach. The findings of this study show that the snakes and ladders game is beneficial to develop the early childhoods’ cognitive ability. The evidence is proved by the early childhoods’ response that through playing the snakes and ladders game they are able to recognize the symbols of numbers, know and memorize numbers from 1 to 56 that is written in the board of snakes and ladders game, recognize various types of colors, shapes, and understand the characteristics of the animals contained in the snakes and ladders game. In conclusion, the snakes and ladders game can be an effective game to develop the early childhoods’ cognitive ability, besides that the game makes them feel happy to play.

Optimized DM-RS Configuration for Improved 5G New Radio Network Capacity and Performance

Network load in mobile networks is continuously growing, putting pressure on mobile operators to deliver target network performance and user experience. This paper focuses on network capacity improvement through increased spectral efficiency, which can be achieved with overhead reduction by optimizing the demodulation reference signal (DM-RS) configuration, which is one of the channels/reference signals with the largest contribution to overhead. In a high-speed scenario, the Doppler effect noticeably influences the temporal nature of the channel such that a channel interpolation process is required within a slot. However, increasing the number of symbols used for DM-RS has a negative impact on capacity. The Doppler effect was analyzed for various 5G NR configurations of operating frequency and subcarrier spacing (SCS), and various use cases were considered using user equipment (UE) speed as the main parameter. For suitable use cases, the DM-RS configuration was optimized in networks with live traffic. The impact of the configuration change on 5G/NR spectral efficiency, user experience and link adaptation performance was assessed through a deep-dive analysis of active measurements, available performance Management (PM) counters and key performance indicators. An optimized DM-RS configuration is proposed, and it is demonstrated to achieve gains of 5–15%, depending on the metric used, use case analyzed, network load, traffic mix and other relevant network characteristics such as topology and clutter type.

Mapping skills between symbols and quantities in preschoolers: The role of finger patterns.

Mapping skills between different codes to represent numerical information, such as number symbols (i.e., verbal number words and written digits) and non-symbolic quantities, are important in the development of the concept of number. The aim of the current study is to investigate children's mapping skills by incorporating another numerical code that emerges at early stages in development, finger patterns. Specifically, the study investigates (i) the order in which mapping skills develop and the association with young children's understanding of cardinality; and (ii) whether finger patterns are processed similarly to symbolic codes or rather as non-symbolic quantities. Preschool children (3-year-olds, N=113, Mage=40.8months, SDage=3.6months; 4-year-olds, N=103, Mage=52.9months, SDage=3.4months) both cardinality knowers and subset-knowers, were presented with twelve tasks that assessed the mappings between number words, Arabic digits, finger patterns, and quantities. The results showed that children's ability to map symbolic numbers precedes the understanding that such symbols reflect quantities, and that children recognize finger patterns above their cardinality knowledge, suggesting that finger patterns are symbolic in essence. RESEARCH HIGHLIGHTS: Children are more accurate in mapping between finger patterns and symbols (number words and Arabic digits) than in mapping finger patterns and quantities, indicating that fingers are processed holistically as symbolic codes. Children can map finger patterns to symbols above their corresponding cardinality level even in subset-knowers. Finger patterns may play a role in the process by which children learn to map symbols to quantities. Fingers patterns' use in the classroom context may be an adequate instructional and diagnostic tool.

Short-term number sense training recapitulates long-term neurodevelopmental changes from childhood to adolescence.

Number sense is fundamental to the development of numerical problem-solving skills. In early childhood, children establish associations between non-symbolic (e.g., a set of dots) and symbolic (e.g., Arabic numerals) representations of quantity. The developmental estrangement theory proposes that the relationship between non-symbolic and symbolic representations of quantity evolves with age, with increased dissociation across development. Consistent with this theory, recent research suggests that cross-format neural representational similarity (NRS) between non-symbolic and symbolic quantities is correlated with arithmetic fluency in children but not in adolescents. However, it is not known if short-term training (STT) can induce similar changes as long-term development. In this study, children aged 7-10 years underwent a theoretically motivated 4-week number sense training. Using multivariate neural pattern analysis, we investigated whether short-term learning could modify the relation between cross-format NRS and arithmetic skills. Our results revealed a significant correlation between cross-format NRS and arithmetic fluency in distributed brain regions, including the parietal and prefrontal cortices, prior to training. However, this association was no longer observed after training, and multivariate predictive models confirmed these findings. Our findings provide evidence that intensive STT during early childhood can promote behavioral improvements and neural plasticity that resemble and recapitulate long-term neurodevelopmental changes that occur from childhood to adolescence. More generally, our study contributes to our understanding of the malleability of number sense and highlights the potential for targeted interventions to shape neurodevelopmental trajectories in early childhood. RESEARCH HIGHLIGHTS: We tested the hypothesis that short-term number sense training induces the dissociation of symbolic numbers from non-symbolic representations of quantity in children. We leveraged a theoretically motivated intervention and multivariate pattern analysis to determine training-induced neurocognitive changes in the relation between number sense and arithmetic problem-solving skills. Neural representational similarity between non-symbolic and symbolic quantity representations was correlated with arithmetic skills before training but not after training. Short-term training recapitulates long-term neurodevelopmental changes associated with numerical problem-solving from childhood to adolescence.

Complexity-calibrated benchmarks for machine learning reveal when prediction algorithms succeed and mislead

Recurrent neural networks are used to forecast time series in finance, climate, language, and from many other domains. Reservoir computers are a particularly easily trainable form of recurrent neural network. Recently, a “next-generation” reservoir computer was introduced in which the memory trace involves only a finite number of previous symbols. We explore the inherent limitations of finite-past memory traces in this intriguing proposal. A lower bound from Fano’s inequality shows that, on highly non-Markovian processes generated by large probabilistic state machines, next-generation reservoir computers with reasonably long memory traces have an error probability that is at least ∼60%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sim 60\\%$$\\end{document} higher than the minimal attainable error probability in predicting the next observation. More generally, it appears that popular recurrent neural networks fall far short of optimally predicting such complex processes. These results highlight the need for a new generation of optimized recurrent neural network architectures. Alongside this finding, we present concentration-of-measure results for randomly-generated but complex processes. One conclusion is that large probabilistic state machines—specifically, large ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon$$\\end{document}-machines—are key to generating challenging and structurally-unbiased stimuli for ground-truthing recurrent neural network architectures.

Social Determinants of Health, Risk and Resilience against Alzheimer's Disease and Related Dementias: The Healthy Brain Initiative.

Few studies have investigated associations between perceived social determinants of health (SDOH) and Alzheimer's disease and related dementia (ADRD) biomarkers or between SDOH and resilience against ADRD. To examine associations between perceived and objective SDOH and ADRD-related outcomes. We used cross-sectional data on≥50-year-olds without dementia in the Healthy Brain Initiative (n = 162). Questionnaires captured trust in neighbors and indices of perceived neighborhood greenspace access, time spent in neighborhood greenspaces, and interpersonal discrimination. Residential addresses were linked to 2021 Area Deprivation Index scores. The Vulnerability Index (VI) is based on 12 dementia risk factors (e.g., age, race/ethnicity, diabetes) and Resilience Index (RI) is based on 6 protective factors (e.g., diet, mindfulness, physical activity). Cognitive measured included number symbol coding task and Montreal Cognitive Assessment. Biomarkers included Aβ42/40 and pTau-217/npTau-217, hippocampal and white matter hyperintensity volume, lipoprotein A, and high-sensitivity c-reactive protein. Perceived greater access to greenspaces (estimate = 2.83, 95% CI = 1.40-4.26) and greater time in neighborhood greenspaces were associated with greater RI scores (estimate = 2.30, 95% CI = 1.24-3.35). Reporting greater discrimination (estimate = 0.10, 95% CI = 0.04-0.16) and living in higher deprivation neighborhoods were associated with greater VI scores (estimate = 0.017, 95% CI = 0.003-0.032). Greater discrimination was associated with greater white matter hyperintensity volume (estimate = 0.27, 95% CI = 0.04-0.51). Perceived greenspace access and time spent in greenspaces were associated with resilience against ADRD, and interpersonal discrimination was associated with vulnerability to ADRD. Future work needs to validate perceived SDOH measures, examine associations in racially/ethnic diverse populations, and investigate longitudinal associations between SDOH and ADRD-related biomarkers.

A Modified Hybrid Brain-Computer Interface Speller Based on Steady-State Visual Evoked Potentials and Electromyogram.

To enhance the information transfer rate (ITR) of a steady-state visual evoked potential (SSVEP)-based speller, more characters with flickering symbols should be used. Increasing the number of symbols might reduce the classification accuracy. A hybrid brain-computer interface (BCI) improves the overall performance of a BCI system by taking advantage of two or more control signals. In a simultaneous hybrid BCI, various modalities work with each other simultaneously, which enhances the ITR. In our proposed speller, simultaneous combination of electromyogram (EMG) and SSVEP was applied to increase the ITR. To achieve 36 characters, only nine stimulus symbols were used. Each symbol allowed the selection of four characters based on four states of muscle activity. The SSVEP detected which symbol the subject was focusing on and the EMG determined the target character out of the four characters dedicated to that symbol. The frequency rate for character encoding was applied in the EMG modality and latency was considered in the SSVEP modality. Online experiments were carried out on 10 healthy subjects. The average ITR of this hybrid system was 96.1 bit/min with an accuracy of 91.2%. The speller speed was 20.9 char/min. Different subjects had various latency values. We used an average latency of 0.2 s across all subjects. Evaluation of each modality showed that the SSVEP classification accuracy varied for different subjects, ranging from 80% to 100%, while the EMG classification accuracy was approximately 100% for all subjects. Our proposed hybrid BCI speller showed improved system speed compared with state-of-the-art systems based on SSVEP or SSVEP-EMG, and can provide a user-friendly, practical system for speller applications.

Sensorimotor mechanisms selective to numerosity derived from individual differences.

We have previously shown that after few seconds of adaptation by finger-tapping, the perceived numerosity of spatial arrays and temporal sequences of visual objects displayed near the tapping region is increased or decreased, implying the existence of a sensorimotor numerosity system (Anobile et al., 2016). To date, this mechanism has been evidenced only by adaptation. Here, we extend our finding by leveraging on a well-established covariance technique, used to unveil and characterize 'channels' for basic visual features such as colour, motion, contrast, and spatial frequency. Participants were required to press rapidly a key a specific number of times, without counting. We then correlated the precision of reproduction for various target number presses between participants. The results showed high positive correlations for nearby target numbers, scaling down with numerical distance, implying tuning selectivity. Factor analysis identified two factors, one for low and the other for higher numbers. Principal component analysis revealed two bell-shaped covariance channels, peaking at different numerical values. Two control experiments ruled out the role of non-numerical strategies based on tapping frequency and response duration. These results reinforce our previous reports based on adaptation, and further suggest the existence of at least two sensorimotor number channels responsible for translating symbolic numbers into action sequences.

Sensorimotor mechanisms selective to numerosity derived from individual differences

We have previously shown that after few seconds of adaptation by finger-tapping, the perceived numerosity of spatial arrays and temporal sequences of visual objects displayed near the tapping region is increased or decreased, implying the existence of a sensorimotor numerosity system (Anobile et al., 2016). To date, this mechanism has been evidenced only by adaptation. Here, we extend our finding by leveraging on a well-established covariance technique, used to unveil and characterize ‘channels’ for basic visual features such as colour, motion, contrast, and spatial frequency. Participants were required to press rapidly a key a specific number of times, without counting. We then correlated the precision of reproduction for various target number presses between participants. The results showed high positive correlations for nearby target numbers, scaling down with numerical distance, implying tuning selectivity. Factor analysis identified two factors, one for low and the other for higher numbers. Principal component analysis revealed two bell-shaped covariance channels, peaking at different numerical values. Two control experiments ruled out the role of non-numerical strategies based on tapping frequency and response duration. These results reinforce our previous reports based on adaptation, and further suggest the existence of at least two sensorimotor number channels responsible for translating symbolic numbers into action sequences.

Европейские источники о символах алан

In the 16th century in the written sources of Western Europe, unexpectedly appears information about the visual symbols of the Alans, who have long since disappeared from the historical arena of the region. This information has practically remained outside the scientific attention of both historians and specialists in the field of heraldry and vexillology. Accordingly, the symbols themselves did not receive consideration as historical sources. The current situation determines the scientific relevance of the study. The purpose of the study is to preliminary determine the general range of sources and verify the data presented in them for their further inclusion in the study within such scientific disciplines as heraldry and vexillology. The study is based on the identified original data from written sources about the visual symbols of the Alans, which were practically not involved in the analysis by specialists. The study used methods of textual and logical analysis of sources and based on the principles of historicism and objectivity, systematic presentation. Among the presented zoomorphic symbols are identified images of a cat, marten, dog, lion, winged lioness, dragon and deer. The historical reliability of most of them at the present stage raises objective doubts. The number of such symbols undoubtedly includes those that are clearly the author’s interpretations. The reasons for them in some cases can be determined. At the same time, other symbols that are promising for analysis are outlined as historical sources. The preliminary analysis confirms the stated need to introduce the identified materials into study within such scientific fields as heraldry and vexillology.

AN ANALYSIS OF THE MENTALITY FOR MONGOLIANS’ SYMBOLISM BASED ON THE MYTH OF “KHUKHU NAMJIL”

Analyzing the symbolic mindset from an overview of the content of any myth is an expression of essence from phenomena, and it will be an important tool for determining the characteristics of the nation's mentality. Therefore, many legends of the origin of the Morin huur (Horsehead fiddle) were collected and compiled in the course of my research; then 15 selected myths which are called similar versions of Khuhu Namjil tied to the main motif with the main idea of "A young man who lost his magical flying horse to a villain was depressed and made fiddles for him” were confirmed that it’s the same version of the same myth in terms of their main content, narrative, and characters. Accordingly, this research mainly used historical material, genre division, text analysis, motif, and comparative study methods. In order to analyze the symbolic mentality of the Mongolians from the myths by using some theories of ritual, cultural studies, and aesthetics, it was analyzed and determined that "the characters of the horse with the wings of desire and the mother “Etugen” were created in their world of intellection". These characters are represented by the legendary “Jonon” black horse, flying winged horse, mermaid, and “Altai”s maiden and it was proved from the content that the abstract concept of spirit is believed to be as a winged horse, and the talisman of the owner of mountains and springs has been worshiped as a mermaid and a fairy for ages. In addition, it has been proved that the symbols of colors (such as green color representing “Lus” or the owner of Springs, black color as civilian, bravery, and splendor, and red color is as warmth and joy) and numbers (one, two, three) were created from Mongolian people's devotion and belief are reflected in the legend by using the events and the narratives in it. Basically, by deepening and expanding the scope of the study of the legend of the fiddle, which previous researchers studied, and by clarifying the meaning of the event and the ideology or mindset of the characters, it is not only innovative but also brings one step closer to the further research of the legend related to the origin of the fiddle thus, the results of this research are significant and a pioneer in the study of fiddle mythology.

How Toddlers Use Core and Fringe Vocabulary: What's in an Utterance?

Selecting vocabulary for preliterate individuals who use augmentative and alternative communication presents multiple challenges, as the number of symbols provided must be balanced with cognitive, motoric, and other needs. Prioritizing certain types of vocabulary thus becomes a necessity. For example, prioritizing core vocabulary-that is, words that are commonly used across a group of people and contexts-is a common practice that attempts to address some of these issues. However, most core vocabulary research to date has narrowly focused on individual word counts, ignoring other critical aspects of language development such as how vocabulary aligns with typical development and how children use core and fringe vocabulary within their utterances. Descriptive and inferential statistics were used to analyze 112 transcripts to describe how typically developing toddlers (aged 2.5 years) use core and fringe vocabulary within their utterances, in reference to a range of commonly used core vocabulary lists. Results indicated that the proportion of the toddlers' utterances that consisted of only core, only fringe, or core + fringe vocabulary varied dramatically depending on the size of the core vocabulary list used, with smaller core lists yielding few "core-only" utterances. Furthermore, utterances containing both core and fringe vocabulary were both grammatically and semantically superior to utterances containing only core or only fringe vocabulary, as evidenced by measures such as mean length of utterance and total number of words. Thus, relying on word frequency counts is an insufficient basis for selecting vocabulary for aided preliterate communicators.

Rigidity of pressures of Hölder potentials and the fitting of analytic functions through them

Abstract The first part of this work is devoted to the study of higher derivatives of pressure functions of Hölder potentials on shift spaces with finitely many symbols. By describing the derivatives of pressure functions via the central limit theorem for the associated random processes, we discover some rigid relationships between derivatives of various orders. The rigidity imposes obstructions on fitting candidate convex analytic functions by pressure functions of Hölder potentials globally, which answers a question of Kucherenko and Quas. In the second part of the work, we consider fitting candidate analytic germs by pressure functions of locally constant potentials. We prove that all 1-level candidate germs can be realised by pressures of some locally constant potentials, as long as the number of symbols in the symbolic set is large enough. There are also some results on fitting 2-level germs by pressures of locally constant potentials obtained in the work.