Sum of Cubes

Proof is one of the main components of both mathematics and mathematics education. However, students and teachers at every level have difficulties in proof and proof teaching. Presenting alternative proof methods to students will not only provide a rich learning environment but also allow the students to have different perspectives by providing alternative ways they can apply for constructing a proof. The visual proofs are one of the alternative proof methods. The visual proofs are seen as valuable tools for mathematics education; it is planned to investigate the views of high school students and their teacher about visual proof. Case study method is used in the study. Participants of the study consisted of four high school students who were nineth-graders and their teacher. The teacher said that after the visual proof activities, the students started to wonder where the formulas came from, in other words, they started questioning, their interest levels increased, and the formulas were kept in their minds more. The high school students stated that they found the visual proof activities enjoyable, and they were happy when they understood the visual proofs and unhappy when they could not understand them.

Proposals for large-scale technical interventions into the Earth system to mitigate global warming – or climate engineering – have sparked considerable debate about their potential implications for international security and global governance. The article furthers this debate by bringing it into dialogue with the literature on visual global politics to develop a more ‘imagistic’ concept of climate engineering imaginaries. Based on a novel visual dataset, three major visual clusters in the public discourse on climate engineering are identified: images of the human–nature relationship, of climate engineering as tangible infrastructure, and of the actors involved in climate engineering projects. The analysis shows how images and other visuals do not only shape the dominant understanding of climate engineering but also competing imaginaries of future political orders in which such approaches might be deployed. Three main results of this analysis stand out. First, dominant ways of seeing climate engineering can further reinforce already dominant discursive frames by adding ‘visual proof’ to their underlying claims. Second, climate engineering visuality can also enable the politicisation of climate engineering by rendering concrete projects visible and hence contestable. Third, climate engineering images can paradoxically limit the scope of imagination as they often revolve around powerful visual icons and symbols of the past and present.

Seeing Women Photographed in Revolutionary Mexico Horacio Legrás (bio) Pure beholding, even if it were to penetrate to the innermost core of the Being of something present-at-hand, could never discover anything that is threatening. —Martin Heidegger, Being and Time In the matter of the visible, everything is a trap. —Jacques Lacan, Seminar XI: The Four Fundamental Concepts of Psychoanalysis The Mexican Revolution is associated with the names of Francisco “Pancho” Villa and Emiliano Zapata, Álvaro Obregón and Francisco Madero, and Diego Rivera and to a lesser extent Frida Kahlo. This overt domination of male figures is perhaps predictable given the historical context of the revolt and the overwhelmingly patriarchal nature of Mexican society at the time. And yet, some of the most recognizable and reprinted photos of the Mexican Revolution are photos of women. The image of two apprehensive waitresses serving breakfast to Zapatista revolutionary troops at Sanborns in 1914 (Figure 1) is as emblematic of the occupation of the capital by the peasant armies as the iconic photograph Villa and Zapata [End Page 3] at the Presidential Chair. Hugo Brehme’s Soldaderas on Top of a Train (Figure 2; Brehme’s original German title reads simply Zapatista Camp on Top of a Train) and the candid figure of an Adelita hanging from a train-coach rival the iconic value of Villa’s gallop in front of the camera or Zapata’s grave gesture while posing with a presidential band crossing his chest, a rifle on his right arm and his left hand resting on the saber’s guard. In Photography and Memory in Mexico, Andrea Noble rejects the vindication of women suggested by the popularity of these images. Visual proof that women were also engaged in the revolution and they too were historical participants is instead evidence of a subtle subalternizing logic made manifest in the words “also” and “too.” Besides, Noble argues, even when women are visible, visibility is often constructed in terms of invisibility.1 Confronted with this disagreement—to use Jacques Rancière’s apposite word—Noble urges being on guard against the temptation of a merely indexical reading that would preempt and disavow the whole problem of conceiving of women as historical actors in their own right. Perhaps the lack of representation of women as historical actors is due to women (and of course there are already too many inside this signifier) having been kept in a subaltern, silenced position in an affair that was, after all, marked by the macho bravado of the refrain “If they are going to kill me tomorrow, better that they kill me right away.” Such an assumption cannot withstand even the most modest historical scrutiny. The mobilization of women is perhaps the most salient trait of the transformations brought about by the Mexican Revolution in the sphere of the social.2 The array of revolutionary destruction that took place primarily in the countryside uprooted provincial life, sending hundreds of thousands of women from their homes into different urban centers. In these centers they encountered a modernization that was changing the landscape of the feminine in its own way, populating the city with the unheard of figures of the chica moderna, the flapper, and the griseta.3 The photographic record testifies to this movement, and yet the force of the testimony remains suspect. Against John Mraz, for whom photographs need to be contextualized but not interpreted, Noble calls attention to the fact that the indexical evidence already poses unique “interpretive challenges.”4 How far should we go in our vigilant approach to the phenomenality of the image? For Noble, the criticism of evidence and indexicality remains to a large extent inside the sphere of validity of these notions. In this essay, I argue that no criticism of evidence (or derivatively truth) can be obtained without tackling in a radical [End Page 4] Click for larger view View full resolution Figure 1. Zapatistas at Sanborns restaurant, December 1914. Photo by Agustín Casasola. Casasola Archive, Mexico. Click for larger view View full resolution Figure 2. Soldaderas on top of a train. Photo by Hugo Brehme. [End Page 5] way and in the same movement the spheres of...

We describe a picture proof of the Alternating Series Test which uses simple comparisons of areas of rectangles to establish convergence. This article describes a short and compelling visual proof of the Alternating Series Test that uses comparisons of areas of rectangles to visualise and prove convergence and some related estimates. It can be quickly sketched on the board or given to a class as a handout. The Alternating Series Test states that an alternating series a1 � a2 þ a3 � a4 þ a5 � a6 þ��� of real numbers converges provided that a15a25 ��� 50 and limn!1 an ¼ 0. The standard calculus text proof is this: even partial sums s2k ¼ a1 � a2 þ a3� a4 þ���� a2k form an increasing sequence bounded above; odd partial sums s2kþ1 ¼ a1 � a2 þ a3 � a4 þ���� a2k þ a2kþ1 form a decreasing sequence bounded below. Therefore, both partial sum sequences converge. Since s2k þ 1 ¼ s2k þ a2k þ 1 and a2k þ 1 ! 0. It follows that

In a recent Mathematical Intelligencer note, B. Chakraborty [A visual proof that π e < e π . Mathematical Intelligencer, 41(1), 56] provided a visual proof of the inequality π e < e π . The aim of this classroom capsule is to visualize the more general inequality b a < a b , where e ≤ a < b in an amazing way.

In traditional scholarship, portraiture is generally understood as symptomatic of fundamentally modern interests in individuality and selfhood. Scholars have described the practice as having emerged over the course of the fourteenth century and have pointed to portraits as visual proof that a shift from medieval to modern mentalities occurred at that time. This article seeks to recast our account of the "earliest portraits" by analyzing such images in accordance with the artistic, social, and theoretical preoccupations that prevailed during the era. It concentrates on a specific setting often cited as a point of origin for the modern practice of portraiture: the French royal court in the late fourteenth and early fifteenth century. To recover that period's understanding of the concept of "likeness," it explores a variety of late medieval written sources, ranging from works of "natural philosophy" to literary texts. It also considers images themselves, discussing works in media privileged by modern scholarship—paintings and sculpture—but also addressing lost images, accessible only through their description in archival sources, as well as works in less exalted material, such as wax votive figures. It demonstrates that late medieval artists and audiences understood likeness as a highly complex quality, and that they therefore saw images as important mnemonic tools. This created a situation in which artists and patrons alike deployed a variety of forms of likeness in images, enabling them to play a major role in courtly displays of devotion and loyalty.

This research seeks to examine how teamwork, reflective methods, and digital records in field-based learning enhance academic literacy in postgraduate students. The program was carried out via a field study in Semarang, engaging key partner organizations such as Universitas PGRI Semarang (UPGRIS), Semesta School, the Kota Tua heritage area, and the Grand Mosque of Central Java (MAJT). Using a descriptive qualitative method, data were collected via participatory observation, interviews, digital records, and reflective group discussions. The results indicate that collaboration among institutions offers valuable empirical environments that allow students to link theoretical ideas with practical applications, especially in fields like educational governance, digital innovation, cultural preservation, and spiritual leadership. Reflective practices act as vital tools that enable students to interpret field experiences, participate in analytical processing, and convert observations into a more profound academic comprehension. Simultaneously, digital records improve observational precision, deepen reflective involvement, and bolster academic claims with visual and multimodal proof. The research finds that combining collaboration, reflection, and digital documentation creates a comprehensive learning structure that effectively enhances students’ academic writing abilities, critical thinking, and overall academic literacy in the context of Education 5.0. The research suggests the necessity for more organized, technology-enhanced field learning frameworks to maximize academic and professional learning results for graduate students.

Choriocarcinoma presented as a vaginal tumor

SummaryA visual proof that 1 − (1/2) + (1/4) − (1/8) + ⋯ converges to 2/3.

The motivation of this note is to explore a generalization of a problem posed in [1], and our invitation to a visual proof (in Exercise below) comes after reading [2].Theorem. If and , then Proof. ...

Equipping cells with artificial shells or coats has been explored throughout the last decade, with goals such as immunomasking, in vivo tracing, and imparting tolerance to various biotic and abiotic stressors. One stressor, however, drying, has curiously been overlooked. In an industrial setting, the drying of cells is relevant when a satisfactory product shelf life must be achieved at a low cost. The drying of entomopathogenic organisms for biocontrol is a prime example of this. Here, the thin-walled blastospores of the entomopathogenic fungus Metarhizium brunneum are a great model organism for testing whether thin-cell surface polyelectrolyte layers may increase desiccation tolerance. In this study, we coated single M. brunneum blastospores with alternating layers of chitosan and alginate and assessed their effects on blastospore viability after drying. The desiccation tolerance improved with increasing layer numbers from 6.9% to a maximum of 27.5%. In addition, as the polymer chain length decreased, the desiccation tolerance further increased to 33.1%. Furthermore, we provided visual proof of the coating surrounding the blastospores via the use of fluorescent polymers and scanning electron microscopy. Finally, an investigation of differences in water absorption into coated and uncoated cells revealed that water absorbed faster into coated cells when alginate was on the surface of the structure but slower when the outermost layer was composed of chitosan. We conclude that, via polyelectrolyte multilayering on thin-walled blastospores, desiccation tolerance can be significantly increased, but a deeper understanding is necessary to extract the full potential from this technique.Graphical

An Xcalibur X-ray diffractometer with a Sapphire 3 CCD detector (Rigaku Oxford Diffraction) has been calibrated to remove damage effects. Visual proof of the validity of the detector calibration map is obtained for the first time. The calibration has significantly improved the reliability of the structural data on Ba3TaFe3Si2O14 single crystal.

Sistem keamanan akses yang andal tidak hanya membedakan antara pengguna sah dan tidak sah, tetapi juga mencatat setiap percobaan akses sebagai bukti digital. Penelitian ini mengembangkan sistem kunci pintu pintar berbasis multi-sensor yang mengintegrasikan RFID, sensor sidik jari, dan sensor getaran, serta dilengkapi dokumentasi visual otomatis menggunakan ESP32-CAM. Arduino Mega digunakan sebagai pengendali utama yang berkomunikasi secara serial dengan dua modul ESP32 untuk akuisisi data dan pengiriman gambar ke server Laravel. Sistem memberikan respons real-time terhadap autentikasi yang berhasil, percobaan akses yang gagal, dan deteksi gangguan fisik. Ketika kondisi tertentu terpenuhi, sistem secara otomatis mengambil gambar, mencatat waktu, dan mengirim data ke dashboard web. Hasil pengujian menunjukkan respons rata-rata 4–5 detik dari pemicu hingga unggahan gambar. Sistem ini menawarkan kombinasi keamanan fungsional dan visual, menjadikannya solusi praktis dan adaptif untuk aplikasi keamanan berbasis IoT.

Microplastics are persistent anthropogenic pollutants which have become a global concern owing to their widespread existence and unfamiliar threats to the environment and living organisms. This study demonstrates the degradation of fragmented microplastics particularly low-density polyethylene (LDPE) film in water, through visible light-induced plasmonic photocatalysts comprising of platinum nanoparticles deposited on zinc oxide (ZnO) nanorods (ZnO-Pt). The ZnO-Pt nanocomposite photocatalysts were observed to have better degradation kinetics for a model organic dye (methylene blue) compared to bare ZnO nanorods, attributed to the plasmonic effects leading to better interfacial exciton separation and improved hydroxyl radical activity along with a 78% increase in visible light absorption. These demonstrations of the plasmonically enhanced photocatalyst enabled it to effectively degrade microplastic fragments as confirmed following the changes in carbonyl and vinyl indices in infrared absorption. In addition, visual proof of physical surface damage of the LDPE film establishes the efficacy of using plasmonically enhanced nanocomposite photocatalytic materials to tackle the microplastic menace using just sunlight for a clean and green approach towards mitigation of microplastics in the ecosystem.

Probabilistic graphical model identifies clusters of EEG patterns in recordings from neonates