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The number zero holds a special status among numbers, indispensable for developing a comprehensive number theory. 1,2,3,4 Despite its importance in mathematics, the neuronal foundation of zero in the human brain is unknown. We conducted single-neuron recordings in neurosurgical patients 5,6,7 while they made judgments involving nonsymbolic number representations (dot numerosity), including the empty set, and symbolic numbers (Arabic numerals), including numeral zero. Neurons showed responsiveness to either the empty set or numeral zero, but not both. Neuronal activity to zero in both nonsymbolic and symbolic formats exhibited a numerical distance effect, indicating that zero representations are integrated together with countable numerosities and positive integers at the low end of the number line. 8,9 A boundary in neuronal coding existed between the nonsymbolic empty set and small numerosities, correlating with the relative difficulty in discriminating numerosity zero behaviorally. Conversely, no such boundary was found for symbolic zero activity, suggesting that symbolic representations integrate zero with other numerals along the number line, reconciling its outlier role. The status of zero as a special nonsymbolic numerical quantity is reflected in the activity of neurons in the human brain, which seems to serve as a scaffold for more advanced representations of zero as a symbolic number. • Single-neuron recordings in the medial temporal lobe of patients • Neurons signal both nonsymbolic and symbolic zero • Neuronal representations of zero integrated along the number line • Nonsymbolic empty sets distinctly encoded from other small numbers Kutter et al. identify and characterize single-neuron activity in the human brain correlated with the representation of both nonsymbolic (dot numerosities) and symbolic (Arabic numerals) number zero. [ABSTRACT FROM AUTHOR]

In this paper, a novel maximum likelihood sequence estimation (MLSE) optimum receiver design is proposed to reduce the complexity of the conventional optimum receiver. We first propose a multipath channel classification method according to the zero distribution of the channel impulse response; that is, the multipath channel is divided into a minimum phase channel and a non-minimum phase channel. Then, an MLSE optimum receiver design with channel identification based on zero distribution is also proposed. In this design, the distribution of zero can be obtained directly by calculating the Z transform of the channel. Thus, the complexity is reduced. The number of the multiplication reduced is N (2 N + 1) . In addition, 2 N + 1 memory units storing the autocorrelation sequence are reduced. The simulation results indicate that this channel classification method effectively represents the characteristics of the channel. Moreover, the MLSE optimum receiver proposed performs similarly and is less complex than the conventional one. The performance is greatly improved compared with the minimum mean square error (MMSE) equalizer. [ABSTRACT FROM AUTHOR]

We prove that for even integer k, almost all of zeros of the period polynomial associated to a cusp form of weight k on Γ 0 (N) are on the circle | z | = 1 ∕ N under some conditions. [ABSTRACT FROM AUTHOR]

Let ζ = (ζn) be an arbitrary complex sequence such that 0 < |ζ1| ≤ |ζ2| ≤... and ζn → ∞ as n → ∞, let nζ(r) and Nζ(r) be the counting function and the integrated counting function of this sequence, respectively. By Eζ we denote the class of all entire functions whose zeros are precisely the ζn, where a complex number that occurs m times in the sequence ζ corresponds to a zero of multiplicity m. Suppose that Φ is a convex function on R such that Φ(σ)/σ → +∞ as σ → +∞. It is proved that there exists an entire function f ∈ Eζ such that... where Mf (r) denotes the maximum modulus of the function f, and it is shown that the above inequality implies the inequality... The formulated result is a consequence of the following more general statement: if the righthand derivative Φ′+ of the function Φ assumes only integer values and... then there exists an entire function f ∈ Eζ such that lnMf(r) = o(eΦ(ln r)) as r → +∞. [ABSTRACT FROM AUTHOR]

In this paper, we establish inequalities concerning the growth of polynomials involving the polar derivative of the polynomials. [ABSTRACT FROM AUTHOR]

In this article some results on the value distribution theory of analytic functions defined in angles of C , due mainly to B. Ja. Levin and A. Pfluger, will be extended to the more general situation where the functions are defined in angles of C \ { 0 } . More precisely, angles S (θ 1 , θ 2) with vertex at the origin will be considered and where a singularity at zero is allowed. An special class of these functions are those of completely regular growth for which it is proved a basic result which yields an expression of the density of its zeros in terms of the indicator function. [ABSTRACT FROM AUTHOR]

Some results on the sharpening and generalizations of an inequality due to Rivlin are obtained. Our results give bounds sharper than the bounds given by all the earlier results in this direction, and also improve upon the recent works of Kumar and Milovanović. [ABSTRACT FROM AUTHOR]

Let (ζn) be a sequence of complex numbers such that ζn → ∞ as n → ∞, N(r) be the integrated counting function of this sequence, and let α be a positive continuous and increasing to +∞ function on R for which α(r) = o(log(N(r)/log r)) as r → +∞. It is proved that for any set E ⊂ (1, +∞) satisfying ∫E rα(r)dr = +∞, there exists an entire function f whose zeros are precisely the ζn, with multiplicities taken into account, such that the relation lim inf r∈E, r→+∞ log log M(r) / log r log(N(r)/log r) = 0 holds, where M(r) is the maximum modulus of the function f. It is also shown that this relation is best possible in a certain sense. [ABSTRACT FROM AUTHOR]

The COVID-19 pandemic posed substantial challenges to healthcare systems globally and severely disrupted essential health services, including routine immunization programs. In India, these disruptions were exacerbated due to the sudden emergence of the pandemic and lockdown measures, leading to mass migrations and a shortage of healthcare workers. Caregivers' concerns about routine immunization sessions further compounded the problem, resulting in a sharp increase in zero-dose children. This review paper examines India's strategies for conducting one of the world's largest COVID-19 vaccination programs while effectively restoring and perpetuating its Universal Immunization Program (UIP). The UIP played a pivotal role in sustaining immunization services during the pandemic, ultimately improving immunization coverage compared to pre-pandemic levels. India's accomplishments in this regard are highlighted through key performance indicators, the reach of immunization services, a reduction in zero-dose children, and antigen-wise coverage. The paper also discusses the successful integration of COVID-19 vaccination within the UIP framework, underscoring the significance of existing infrastructure, technology, and capacity building. India's dedication to concurrently managing routine immunization and COVID-19 vaccination showcases the adaptability and resilience of its healthcare system. India's journey serves as a global example of efficient mass immunization during challenging times, emphasizing the importance of political will, healthcare infrastructure investment, skilled healthcare workforces, and comprehensive vaccination programs. In a world grappling with the dual challenge of COVID-19 and routine immunization, India's experience provides a roadmap for strengthening healthcare systems and promoting public health as the critical agenda in challenging times. [ABSTRACT FROM AUTHOR]

The urgent need to reduce greenhouse gas emissions in line with the Paris Agreement is a compelling reason for the entire veterinary profession to act on climate change because of its impact on animal health. The aim of this perspective is to provide a business framework that veterinary practices can use to implement the path to Net Zero carbon emissions. Practice management is identified as a key stakeholder capable of implementing significant change within the sector. Climate related business opportunities and challenges are identified and integrated into a stepwise process for practices to follow. The pathway requires establishing a culture of sustainability within the veterinary team, measuring and reporting emissions, setting targets and systematically prioritizing reductions. Practices can begin this process immediately by reducing emissions under direct control of the business (Scope 1) and emissions from electricity purchases (Scope 2). To complete the pathway, emissions from all other activities (Scope 3) will need to be reduced and offset. Reduction of Scope 3 emissions is more challenging and will require collaboration between all supply chain stakeholders. The progression of climate change is now inevitable and a proactive approach from veterinary leaders, in particular practice management, will provide new opportunities, manage risks and inspire the broader veterinary sector to join their efforts to achieve a better future for animal health.

Drawing an analogy among the splits in US Constitutional democracy, in Communist totalitarianism, in our field, and in many of our patients' psychic foundations, the essay advances the idea that the disruption of fundamental splits—often fastened together by keystone selfobjects—is vital for an emancipatory psychoanalysis. My argument is that to heal our democracy, and to help many of our patients heal, our praxis must cultivate a generative rupturing of the psyche/social status quo that goes beyond the incremental work that often characterizes what we do, and builds upon the promise of Winnicott's "breakdown experience" and Fanon's decolonial praxis. The author further considers resonances across her conception of "feminine law" and the vaginal signifier as the zero of the signifying chain, Koichi Togashi's "psychoanalytic zero", and Hortense Spillers' reading of the black feminine as the zero of the social order. This zero, she argues, acts as a site of rupture and radical transformation, enabling alterity, contradiction, and the operations of a disordered apres coup temporality. Clinically, this manifests as desperation and desire coalescing as a liberatory force, infused with ethical charge, with the potential to dismantle archaic oppressive structures and alliances. The essay concludes that that we are at the brink of a collective, paradoxical experiment in constituting ourselves as fully human in which we reclaim an alienated (material) dimension of human subjectivity, surrendering to the other-than-human, for the sake of joining liberty and love, present and ancestral truths, in a relationship that doesn't require splitting. [ABSTRACT FROM AUTHOR]

Existing databases on conflict-related sexual violence (CRSV) feature a disproportionate number of "zero" observations, which conflate a true absence of CRSV with an unknown presence of CRSV. Empirical studies model such zeros as solely indicating a lack of CRSV, thereby obscuring what needs to be known about its determinants and patterns. In this article, I present a comprehensive meta-reanalysis of quantitative CRSV studies that have disregarded the two-fold nature of excessive zeros in their data. To redress this neglect, I differentiate the two types of zeros probabilistically and then estimate the prevalence of CRSV conditional on the statistical partitioning of its two zero types. My meta-reanalysis refines previous findings, restoring confidence in intuitive theoretical expectations and resolving inconclusive and unexpected results in the field. It also dispels potential misconceptions by providing alternative evidence and explanations. Overall, distinguishing the two classes of zeros in estimation yields more informed results, making substantial contributions to scholarly understanding of CRSV while avoiding spurious confirmations and disconfirmations. [ABSTRACT FROM AUTHOR]

In this chapter, the question is addressed what the world would be like if the number zero had never been invented. This anti-historical question is dodged by asking a slightly less difficult question, namely: what would the world have been like if the so-called complex numbers (ubiquitous in many fields of science and engineering) had never been invented? [ABSTRACT FROM AUTHOR]

Zero is a very versatile entity, making its appearance in a variety of environments. It can appear as a representation of the absence of items in arithmetic to the absence of sound in the merging of words, to the zero as a number in arithmetical operations. All these aspects of zero are fascinating in their own right. However, while the idea of arithmetical zero came to the human mind very early in its intellectual journey, the human response to derivative ideas has been driven by cultures. While, on the one hand, European cultures came to the conclusion that the universe exists and therefore, by definition, a non-existent entity cannot represent reality; on the other hand, the Indian philosophers, used to the ideas of non-existent entities in other contexts, especially in the grammar of conjunction of words and sounds, were fascinated by the null and assumed that it was the null that held the key. They therefore traced the idea of the origin of the universe to this null entity, which by definition is not knowable since it is formless. This divergence meant that intellectual perspectives and studies were driven by this differing approach to zero. It allowed Indian mathematicians to explore zero much earlier than their European counterparts. Modern science, having embraced the zero in all its glory, then went on to twist its existence in different ways. In the present article we discuss some of the aspects of this journey of zero. [ABSTRACT FROM AUTHOR]

Not dividing by zero is one of the strongest prohibitions that we learn as children. Yet, as we grow in age and in competence, we realize that many mathematical constructions would be much simpler if we could divide by zero. But this implies enlarging the family of numbers and making space for an infinite number, ∞, as in the Riemann sphere, or even many infinite numbers, as the founders of calculus envisioned. [ABSTRACT FROM AUTHOR]

We investigate if philosophical ideas of nothing could have given rise to the mathematical zero, which is distinct from nothing and is an independent number with a set of basic operations defined. This investigation should be complementary to other studies, e.g., those based on epigraphic and paleographic evidence in order to probe the origin of zero. First, we argue that a practical need did not give rise to the mathematical zero. Moreover, neither a place-value number system nor a placeholder in such a system was required to conceptualize zero. Therefore, we insist that zero and its basic operations were intellectually premised, which likely required a philosophical base of nonexistence, and not that of emptiness or void as commonly assumed. The prevalence of a philosophy of nothing in ancient India and the appearance of the first clear prescription of basic operations of zero in an Indian book of the first millennium CE point to a development of the concept of zero in India. We back this up with 'living evidence' of word-numerals developed in India, and propose that the comments of the Syrian Bishop Severus Sebokht, which are often used to argue against the Indian origin of zero, do not provide evidence against such an origin. [ABSTRACT FROM AUTHOR]

This chapter is an overview of the themes regarding the concept of śūnya ('void') in Indian thought, mainly mādhyamika, with reference to apoha semantic theory, and in relation with another couple of problematic relationships, the metaphoric use of words (diffused in the milieu of poetics) and the paradox of ineffability (current in Vedantic Brahmanical circles). Did the mathematical, the grammatical, or the Buddhist philosophical meaning of śūnya come first? The chapter tries to illustrate the great deal of debate current on this question, but does not offer a unique answer. [ABSTRACT FROM AUTHOR]

This article argues that the search for a philosophy that gave rise to the number zero is misguided. No philosophy is required to invent this number. The article further shows that there are good reasons to accept that Buddhism did not play a role in this invention. It further points out that the notion of number as developed in Indian philosophy had no place for zero. [ABSTRACT FROM AUTHOR]

The world of numbers keeps us intrigued for one reason or the other. From negative numbers to imaginary numbers and from various constants to peculiar ratios, they never cease to surprise us. And our curiosity and imagination keeps on inspiring us to look for these patterns in nature. Consequently leading the way for discoveries that are important to meeting not only our material goals but also spiritual ones at the same time. However, due to an extraordinary reliance of modern applied science on numbers, we have generated more scholarly interest, especially in the last few centuries, around the application of numbers to the material realm as compared to the spiritual. Amidst our attention to the peculiar or special numbers and ratios, there are some numbers -- or rather concepts -- that still look in our face, inviting us to explore a whole new world of possibilities all over again. Though these concepts laid the foundation of modern day mathematics as we use it, yet at times we take them for granted and are not usually interested in revisiting our understanding about their nature, their past and future possibilities. In this chapter, we revisit three of these foundational concepts -- Zero, One and Infinity (ZOI) from an integrative spiritual perspective. [ABSTRACT FROM AUTHOR]

Notions such as śūnyata, catuskoti, and Indra's net, which figure prominently in Buddhist philosophy, are difficult to readily accommodate within our ordinary thinking about everyday objects. Famous Buddhist scholar Nāgārjuna considered two levels of reality: one called conventional reality, and the other ultimate reality. Within this framework, śūnyata refers to the claim that at the ultimate level objects are devoid of essence or 'intrinsic properties', but are interdependent by virtue of their relations to other objects. Catuskoti refers to the claim that four truth values, along with contradiction, are admissible in reasoning. Indra's net refers to the claim that every part of a whole is reflective of the whole. Here we present category theoretic constructions that are reminiscent of these Buddhist concepts. The universal mapping property definition of mathematical objects, wherein objects of a universe of discourse are defined not in terms of their content, but in terms of their relations to all objects of the universe is reminiscent of śūnyata. The objective logic of perception, with perception modeled as [a category of] two sequential processes (sensation followed by interpretation), and with its truth value object of four truth values, is reminiscent of the Buddhist logic of catuskoti. The category of categories, wherein every category has a subcategory of sets with zero structure within which every category can be modeled, is reminiscent of Indra's net. Our thorough elaboration of the parallels between Buddhist philosophy and category theory can facilitate better understanding of Buddhist philosophy, and bring out the broader philosophical import of category theory beyond mathematics. [ABSTRACT FROM AUTHOR]

Although the presence of zero has been traced to many ancient civilizations including India, the question of its origin is still a moot point among historians and mathematicians. As such, in the present chapter a humble attempt will be made to gather data from the Vedic and post-Vedic Sanskrit literature to understand the origin of zero; the philosophy behind its origin; various denominations of zero used in various phases of Indian civilization; and the philosophy and epistemology behind the symbolic representation of zero (0) in India. [ABSTRACT FROM AUTHOR]

This is an exploration of the mathematics, history, and philosophy of zero. It looks at zero and its development from three perspectives: purely as a sign (syntactic); its meaning (semantics); and its social contexts, roles, and uses (pragmatic). Adopting the meaning as use theory, it argues that the meaning of zero was not fully established until the time of Brahmagupta, at around 628 CE, for Brahmagupta established the role of zero within the network of number relations in the integers. Most notably, zero is the additive identity. This synthesis is analogous for number theory with Euclid's geometry. Thus, it is posited that there are stages in the development of the concept of zero. First, its predecessor serves as a placeholder in an empty column. Second, zero is understood as a sign for nothing (nought), resting on the concept of the empty void. Third, its role as additive identity is understood and accepted. The conjecture that the idea of void is a prerequisite for the development of the number zero is confirmed positively in Indian and Mayan cultures. But the case of Ancient Egypt, which has the concept of void but not zero shows that having the concept of void appears, not of itself, sufficient. [ABSTRACT FROM AUTHOR]

The aim of this chapter is to highlight some conceptual connections between the concept of zero and the philosophical concept of śūnyatā. Both ideas first arose in the Indian subcontinent, and I will try to suggest that this geographical congruence is more than a coincidence. As such, this will likely be a more speculative contribution to this volume than others, but I will attempt to substantiate my proposal by stressing the conceptual advantages deriving from such a syncretic enterprise. I will begin with an overview of the historical development of the number and concept of zero, followed by some considerations of its treatment in contemporary mathematics, as well as an examination of its less-evident semiotic properties. I will then proceed to re-examine the concept of śūnyatā, identifying its central meontological significance in the philosophy of Nāgārjuna. In this context, I will examine the idea of the 'trace' in the thought of Jacques Derrida, viewing it as another actor in the assemblage of concepts that I attempt in this chapter. Finally, I undertake a synthetic reading of these three 'voids' -- of zero, of śūnyatā, and of the trace -- aimed at a speculative rearrangement of these heterogeneous concepts, in a philosophical exercise that I see as both a possible and a necessary evolution of 'comparative philosophy'. [ABSTRACT FROM AUTHOR]

Since recent datings of the Bakhshali manuscript confirm our understanding that the zero is indeed an Indian creation, what is it in Indian culture that could have influenced an invention of this kind? From data relating to the Bakhshali manuscript, it is possible to recognize the influence of Buddhism and, from this, to try to see how this movement could have played a role in the theory leading to the inception of the placeholder zero. Moreover, with Brahmagupta this first form and use of zero was developed to the point of becoming a number, and thus adopting an operative role. While the writer of the Brāhmasphuṭasiddhānta was not a Buddhist, it is still possible to show in that document how Buddhist thinking, particularly its speculative developments in the meaning of śūnyatā as described by the great writers of Mahāyāna Buddhism, may provide a favorable setting for theories about the numerical concept of zero, even to the point where it is perceived as a condition of its 'thinkability' or conceptualization. [ABSTRACT FROM AUTHOR]

Who invented zero? Interestingly, there is no one-line answer. And the acceptability of any attempted answer generally depends on the perception of the seeker, as the connotation of the word 'zero' can be perceived from several different but interrelated perspectives. Many ancient civilizations, including India, had their own version of zero or a zero-like concept or symbol as a representative of 'nothingness'; some as a philosophical conundrum, elsewhere some others even in a practical sense, such as a filler or a gap on their counting board; but except in India, none of these early and somewhat hesitant initial concepts did ultimately mature to its true mathematical potential. Today it is generally accepted worldwide that this peerless concept of a decimal place value system of enumeration in tandem with the true zero of our present-day mathematics, evolved in ancient India. Initially philosophically nurtured and analyzed during the early periods of oral tradition in India by several different schools of thought, eventually at some point of time the concept reflected in the Sanskrit word śūnya was transformed into a numeral for mathematical expression of 'nothing'. Perhaps at a later period, it bloomed into its full potential as a number, kha, an integer on which mathematical operations can be performed. Indeed, this unique feature makes kha, the Indian zero, the true progenitor of our modern mathematical zero. However, the exact time frame of this gradual evolution is still hotly debated, a recent controversy in this direction being the outcome of attempted radiocarbon dating of the famous Bakhshali manuscript by the Bodleian Library of Oxford. Going in the other direction, some scholars suggest that a trace of this concept, if not in total operational perspective, might have had a Greek origin that traveled to India during the Greek invasion of the northern part of the country in the pre-Mauryan period. A relatively recent third view professes the Chinese origin of the concept of zero as a placeholder, which might have traveled with the traders from China to the far eastern parts of Asia, to places like Cambodia, then under Buddhist influence that spread from mainland India, where it got the shape of a 'bold dot', the earliest known written form of zero. Scholars belonging to this school of thought want to credit mainland India only for 'garlanding' this concept of zero toward its modern shape, sometime around the eighth or ninth century CE. Embracing all these paradigms of claims and counterclaims about the origin of zero made till date, this article is a nutshell version of an odyssey, an unparalleled journey from a concept to a number in its own right, perhaps the most significant creation of the human mind ever. [ABSTRACT FROM AUTHOR]

Many cultures contributed to the invention of zero, but how to apportion credit? This article explores a new way to do so: imagine royalties were due for every usage of zero, and then figure out how these should be divided among present-day countries. We report on what such a divided allocation might look like, and give reactions, both positive and negative, to it. We also explore how such ideas might be relevant in terms of potential exploitation of the Indian cyber market by foreign corporations. [ABSTRACT FROM AUTHOR]

Authors of medieval Arabic books on arithmetic often solved problems with 'nothing', and in calculation with Indian (i.e., 'Arabic') numerals, they frequently operated on zero while at the same time claiming that the zero is a sign indicating the absence of a number. We investigate how these authors conceived of nothing and zero from the many remarks they made in their books, and in the context of their understanding of number as an amount or measure of some material or noetic unit. [ABSTRACT FROM AUTHOR]

If mathematics is the language the universe was written in and mathematics is discovered rather than invented, then Brahmagupta's 628 CE Sanskrit text first defines a zero as old as the universe. Brahmagupta's definition of zero as a sum of equal and opposite negative and positive quantities is the first scientific definition of zero found to be consistent with laws of motion and particle physics. However, having originated in the East, the full power of India's symmetric and scientific zero failed to migrate West via the medieval Arabic world to Renaissance Europe. Thus, only a trivial mathematical concept of zero emerged, representing either nothing on its own or an arithmetical placeholder used alongside other numbers. Zero as an arbitrary midpoint for measurement purposes within a single quantity is similarly trivial, whether it be centuries BCE or CE, or a measure above or below a surveyor's foundation line as used for construction in ancient Egypt. Thus, the oldest original extant definition of zero compatible with laws of physics describing our universe originated with Brahmagupta in India 628 CE. Yet long lost in transmission and translation, the sad aftermath in classrooms today is a zero in which negatives to its left are arbitrarily less than positives to its right. [ABSTRACT FROM AUTHOR]

As various articles in this volume demonstrate, zero has many forms and finds expression in a whole host of contexts, each as mystifying as the other. Each culture seems to have a very specific reaction to the number, and the reaction varies significantly from culture to culture. From the great hatred in Europe to the near obsession of their cousins in the Indian subcontinent, the entire spectrum of reactions can be found. Egyptians, Sumerians, and the Chinese managed to do some interesting algebra while circumventing zero completely. In this chapter we will investigate why cultures even with deeply common roots reacted so differently to zero. We take the example of the Indo-European cultures and their diametrically opposite reactions to zero. We suggest that this may be related to their experiences of nature post-separation, which were further amplified by their way of life. We then study the wider context of zero in its different manifestations and explore the relation between its different forms. We show that music seems to make a connecting link between these different aspects. [ABSTRACT FROM AUTHOR]

By utilizing Nevanlinna theory of meromorphic functions, we characterize meromorphic solutions of the following nonlinear differential equation of the form f n f ′ + P (z , f , f ′ , ... , f (t)) = P 1 e α 1 z + P 2 e α 2 z + ⋯ + P m e α m z , where n ≥ 3, t ≥ 0 and m ≥ 1 are integers, n ≥ m, P(z, f, f′, ..., f(t)) is a differential polynomial in f (z) of degree d ≤ n with small functions of f (z) as its coefficients, and αj, Pj (j = 1, 2, ..., m) are nonzero constants such that ∣α1∣ > ∣α2∣ > ... > ∣αm∣. Also we provide the concrete forms of the solutions of the equation above, and present some examples illustrating the sharpness of our results. [ABSTRACT FROM AUTHOR]

In this paper, a new integral‐type transformation is being introduced, which is the generalization of Laplace and Fourier transform. This integral transform is named as "Fareeha transform." It is successfully applied on ordinary differential equation to show its efficacy and simplicity. Also, the possible applications of Fareeha transform in control theory, electric circuits, and data compression have been discussed in detail. [ABSTRACT FROM AUTHOR]

In this paper we discuss findings of a study that investigated the resources and language that teachers in Malawi use to teach the concept of zero. In Malawi primary schools, textual resources available to teachers are mainly the curriculum materials in the form of syllabus, teacher guides and learner textbooks. The syllabus and teacher guides are in English while the learner textbooks are in Chichewa as teaching is in Chichewa or other local language in the first 4 years of primary school. We used the Mediating Primary Mathematics framework (Venkat and Askew in Educ Stud Math 97:71–92, 2018) and a qualitative case study of two teachers to explore the resources used, how the teachers interacted with the resources and how they moved between the two languages. Our findings include that the language and resources that the teachers used provided affordances as well as constraints for learning the concept of zero. We identified two types of challenges for the teachers; that of naming and that of representing the concept of zero. We discuss what the Malawi context illuminates about teaching zero in post-colonial multilingual settings. [ABSTRACT FROM AUTHOR]

We extend a result proved by Dirr and Wimmer [IEEE Trans. Automat. Control 52(2007)] for polynomials to the matrix valued analytic functions and thereby obtain generalizations of some well-known results concerning the zero free regions of a class of analytic functions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new approach to estimating differentiated product demand systems that allows for products with zero sales in the data. Zeroes in demand are a common problem in differentiated product markets, but fall outside the scope of existing demand estimation techniques. We show that with a lower bound imposed on the expected sales quantities, we can construct upper and lower bounds for the conditional expectation of the inverse demand. These bounds can be translated into moment inequalities that are shown to yield consistent and asymptotically normal point estimators for demand parameters under natural conditions. In Monte Carlo simulations, we demonstrate that the new approach works well even when the fraction of zeroes is as high as 95%. We apply our estimator to supermarket scanner data and find that correcting the bias caused by zeroes has important empirical implications, for example, price elasticities become twice as large when zeroes are properly controlled. [ABSTRACT FROM AUTHOR]

Yupana Inca Tawa Pukllay (YITP) is a ludic didactic resource based on semiotic alternation that, using the reading of numbers in the Inca numeral system, improves its equivalent Indo-Arabic reading. Twelve children from first to fourth grade of a bilingual (Spanish-Quechua), multi-grade elementary school in a small rural Peruvian community were assigned an electronic tablet with YITP and learned autonomously, without teachers during the COVID-19 pandemic. The results obtained show that: (a) they learned in a very short period of time (14 min - 05h 41 min) (b) they improved digit reading accuracy on the first attempt (c) they improved digit reading speed d) they achieved a high percentage of correct reading of numbers containing at least one zero digit. The results suggest the potential of YITP as an educational tool in the teaching-learning process of arithmetic.

Scientific knowledge is acquired according to some paradigm. Galileo wrote that the “book of nature” was written in mathematical language and could not be understood unless one first understood the language and recognized the characters with which it was written. It is argued that Turing planted the seeds of a new paradigm. According to the Turing Paradigm, the “book of nature” is written in algorithmic language, and science aims to learn how the algorithms change the physical, social, and human universe. Some sources of the Turing Paradigm are pointed out, and a few examples of the application of the Turing Paradigm are discussed.

This paper contains the finding of some upper bound estimates for the maximal modulus of a lacunary polynomial of degree n on a circle of radius 0 < r ≤ R ≤ k under the assumption that the polynomial has no zero in a disk of radius k; k > 0. Our result extends some known inequalities concerning derivative of a polynomial into integral analogues and it further generalizes as well as sharpens some other results in this direction. [ABSTRACT FROM AUTHOR]