Visual Proof Research Articles

Tri-Hepta-Penta

Summary We offer a visual proof of a relationship between triangular, pentagonal, and heptagonal numbers.

The Radius of an Internally Tangent Circle

Summary We offer a visual proof for a theorem in Euclidean geometry regarding the radius of the incircle of an arbelos.

A generalisation of b a < a b when e ≤ a < b

In this short note we give a simple proof of a generalization of the inequality obtained in Chakraborty [(2022). A visual proof that International Journal of Mathematical Education in Science and Technology, 55(5), 1291–1292], as well as of similar inequalities.

Single-cell polymer coating improves the desiccation tolerance of Metarhizium brunneum blastospores

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

Sum of Cubes

Summary We give yet another visual proof for the sum of cubes formula using two stacks of the sum of the first n cubes.

Prevalence of nano-sized coal mine dust in North and Central Appalachian coal mines – Insights from SEM-EDS imaging

The increasing prevalence of coal mine dust-related lung diseases in coal miners calls for urgent and meticulous scrutiny of airborne respirable coal mine dust (RCMD), specifically focusing on particles at the nano-level. This necessity is driven by expanding research, including the insights revealed in this paper, that establish the presence and significantly increased toxicity of nano-sized coal dust particles in contrast to their larger counterparts. This study presents an incontrovertible visual proof of these tiny particulates in samples collected from underground mines, utilizing advanced techniques such as scanning electron microscopy (SEM) and energy-dispersive spectroscopy (EDS). The intricate elemental composition of nano-sized coal dust identified through EDS analysis reveals the presence of elements such as silica and iron, which are known to contribute to lung pathologies when inhaled over prolonged periods. The outcomes of the statistical analyses reveal significant relationships between particle size and elemental composition, highlighting that smaller particles tend to have higher carbon content, while larger particles exhibit increased concentrations of elements like silica and aluminum. These analyses underscore the complex interactions within nano-sized coal dust, providing critical insights into their behavior, transport, and health impacts. The nano-sized coal dust could invade the alveoli, carrying these toxic elements from where they are impossible to exhale. The revelation of nano-sized coal dust's existence and the associated health hazards necessitate their incorporation into the regulatory framework governing the coal mining industry. This study lays the groundwork for heightened protective measures for miners, urging the invention of state-of-the-art sampling instruments, comprehensive physicochemical profiling of RCMD nanoparticles, and the pursuit of groundbreaking remedies to neutralize their toxic impact. These findings advocate for a paradigm shift in how the coal mining industry views and handles particulate matter, proposing a re-evaluation of occupational health standards and a call to action for protecting coal miners worldwide.

Proof Without Words: Lagrange’s Cosine Identity

Summary I provide a visual proof of a special case of Lagrange’s Cosine Identity. The idea of the proof is based on a sequence of isosceles triangles.

Development of a Nanomarker for In Vivo Monitoring of Dopamine in Plants.

Dopamine, alongside norepinephrine and epinephrine, belongs to the catecholamine group, widely distributed across both plant and animal kingdoms. In mammals, these compounds serve as neurotransmitters with roles in glycogen mobilization. In plants, their synthesis is modulated in response to stress conditions aiding plant survival by emitting these chemicals, especially dopamine that relieves their resilience against stress caused by both abiotic and biotic factors. In present studies, there is a lack of robust methods to monitor the operations of dopamine under stress conditions or any adverse situations across the plant's developmental stages from cell to cell. In our study, we have introduced a groundbreaking approach to track dopamine generation and activity in various metabolic pathways by using the simple nitrogen and sulfur co-doped carbon quantum dots (N, S-CQDs). These CQDs exhibit dominant biocompatibility, negligible toxicity, and environmentally friendly characteristics using a quenching process for fluorometric dopamine detection. This innovative nanomarker can detect even small amounts of dopamine within plant cells, providing insights into plant responses to strain and anxiety. Confocal microscopy has been used to corroborate this occurrence and to provide visual proof of the process of binding dopamine with these N, S-CQDs inside the cells.

MORC2 phosphorylation fine tunes its DNA compaction activity.

Variants in the poorly characterised oncoprotein, MORC2, a chromatin remodelling ATPase, lead to defects in epigenetic regulation and DNA damage response. The C-terminal domain (CTD) of MORC2, frequently phosphorylated in DNA damage, promotes cancer progression, but its role in chromatin remodelling remains unclear. Here, we report a molecular characterisation of full-length, phosphorylated MORC2, demonstrating its preference for binding open chromatin and functioning as a DNA sliding clamp. We identified a phosphate interacting motif within the CTD that dictates ATP hydrolysis rate and cooperative DNA binding. The DNA binding impacts several structural domains within the ATPase region. We provide the first visual proof that MORC2 induces chromatin remodelling through ATP hydrolysis-dependent DNA compaction, regulated by its phosphorylation state. These findings highlight phosphorylation of MORC2 CTD as a key modulator of chromatin remodelling, presenting it as a potential therapeutic target.

Bridged Pt-OH-Mn Mediator in N-coordinated Mn Single Atoms and Pt Nanoparticles for Electrochemical Biomolecule Oxidation and Discrimination.

The rational design of efficient catalysts for uric acid (UA) electrooxidation, as well as the establishment of structure-activity relationships, remains a critical bottleneck in the field of electrochemical sensing. To address these challenges, herein, a hybrid catalyst that integrates carbon-supported Pt nanoparticles and nitrogen-coordinated Mn single atoms (PtNPs/MnNC) is developed. The metal-metal interaction during annealing affords the construction of metallic-bonded Pt-Mn pairs between PtNPs and Mn single atoms, facilitating the electron transfer from PtNPs to the support and thereby optimizing the electronic structure of catalysts. More importantly, experiments and theoretical calculations provide visual proof for the 'incipient hydrous oxide adatom mediator' mechanism for UA oxidation. The Pt-Mn pairs first adsorb OH* to construct the bridged Pt-OH-Mn mediators to serve as a highly active intermediate for N-H bond dissociation and proton transfer. Benefiting from the unique electronic and geometric structure of the catalytic center and reactive intermediates, PtNPs/MnNC exhibits superior electrooxidation performance. The electrochemical sensor based on PtNPs/MnNC enables sensitive detection and discrimination of UA and dopamine in serum samples. This work offers new insights into the construction of novel electrocatalysts for sensitive sensing platforms.

Bridged Pt−OH−Mn Mediator in N‐coordinated Mn Single Atoms and Pt Nanoparticles for Electrochemical Biomolecule Oxidation and Discrimination

AbstractThe rational design of efficient catalysts for uric acid (UA) electrooxidation, as well as the establishment of structure‐activity relationships, remains a critical bottleneck in the field of electrochemical sensing. To address these challenges, herein, a hybrid catalyst that integrates carbon‐supported Pt nanoparticles and nitrogen‐coordinated Mn single atoms (PtNPs/MnNC) is developed. The metal‐metal interaction during annealing affords the construction of metallic‐bonded Pt−Mn pairs between PtNPs and Mn single atoms, facilitating the electron transfer from PtNPs to the support and thereby optimizing the electronic structure of catalysts. More importantly, experiments and theoretical calculations provide visual proof for the ‘incipient hydrous oxide adatom mediator’ mechanism for UA oxidation. The Pt−Mn pairs first adsorb OH* to construct the bridged Pt−OH−Mn mediators to serve as a highly active intermediate for N−H bond dissociation and proton transfer. Benefiting from the unique electronic and geometric structure of the catalytic center and reactive intermediates, PtNPs/MnNC exhibits superior electrooxidation performance. The electrochemical sensor based on PtNPs/MnNC enables sensitive detection and discrimination of UA and dopamine in serum samples. This work offers new insights into the construction of novel electrocatalysts for sensitive sensing platforms.

Proof Without Words: Candido’s Identity

Summary We provide a visual proof to Candido’s identity.

UNDERSTANDING DEEPFAKE TECHNOLOGY: IMPLICATIONS, CHALLENGES AND CASE STUDIES

The rise in deepfake technology has sparked serious worries about how it may distort the facts, sway public opinion, and jeopardise personal privacy. This study explores the complex world of deepfakes, looking at its technological underpinnings, social effects, and moral ramifications. This study investigates the diverse uses of deepfake technology, spanning from political misinformation operations to entertainment, through an analysis of academic literature and case studies. Moreover, it explores the difficulties that deepfakes provide for media legitimacy, dependability, and the accuracy of visual proof. This research delves into the legal and regulatory frameworks pertaining to deepfakes and suggests some approaches to alleviate their adverse effects. Finally, by fostering a thorough knowledge of deepfakes, our research hopes to enable people, decision-makers, and technology developers to properly navigate this quickly changing digital terrain.

From Cell Populations to Molecular Complexes: Multiplexed Multimodal Microscopy to Explore p53-53BP1 Molecular Interaction.

Surpassing the diffraction barrier revolutionized modern fluorescence microscopy. However, intrinsic limitations in statistical sampling, the number of simultaneously analyzable channels, hardware requirements, and sample preparation procedures still represent an obstacle to its widespread diffusion in applicative biomedical research. Here, we present a novel pipeline based on automated multimodal microscopy and super-resolution techniques employing easily available materials and instruments and completed with open-source image-analysis software developed in our laboratory. The results show the potential impact of single-molecule localization microscopy (SMLM) on the study of biomolecules' interactions and the localization of macromolecular complexes. As a demonstrative application, we explored the basis of p53-53BP1 interactions, showing the formation of a putative macromolecular complex between the two proteins and the basal transcription machinery in situ, thus providing visual proof of the direct role of 53BP1 in sustaining p53 transactivation function. Moreover, high-content SMLM provided evidence of the presence of a 53BP1 complex on the cell cytoskeleton and in the mitochondrial space, thus suggesting the existence of novel alternative 53BP1 functions to support p53 activity.

An Interesting Geometric Series

Summary We present a visual proof that ∑ n = 1 ∞ ( 1 2 ) n = 1 + 2 .

Political public relations and crisis communication of the government: The case study of the earthquake of 6 February 2023 in Türkiye

On February 6, 2023, a massive 7.8 earthquake struck southeast Türkiye and caused killing thousands of people and leaving devastation across a wide swath of the country. In this context, this paper examines the government communications and crisis communication in the context of the disaster responses of the emergency management agencies during the earthquake of February 2023 in Türkiye in terms of image repair strategy and aims to reveal how the government responds to the opposing political circles' claims. It was concluded that the ruling party used communication channels intensively to refute the opposition's arguments and presented its own efforts and actions through visual and numerical proof and the main character of the government's crisis communication was based on highlighting what they made well.

Novel Cs-Co3O4@g-C3N4 nanocomposite constructed for malachite green dye adsorption

The present study focuses on synthesizing a novel nanosorbent, Cs-Co3O4@g-C3N4, via the sol–gel method assisted by ultrasonic power. The synthesized ternary composite was found to include the Cs2O, Co3O4 and g-C3N4 phases, Cs, Co, O, N and C elements, 28.68 m2/g surface area as provided by the XRD, EDX –XPS and nitrogen adsorption techniques. In addition, the images of metal oxide nanoparticles dispersed within the g-C3N4 sheets, as evidenced in the TEM results provided visual proof for the composite formation. The effectiveness of this nanosorbent in adsorbing a commonly used dye, Malachite green (MG) was thoroughly studied. The MG dye adsorption performance was investigated using equilibrium and kinetic investigations. The results indicated that the adsorption process followed the Langmuir monolayer isotherm and exhibited second-order kinetics. The adsorption capacity of MG dye was found to be 1472 mg/g, with the primary interaction being attributed to electrostatic attraction. This interaction is enhanced by the smaller size and increased surface area of the Cs-Co3O4@g-C3N4 nanosorbent, leading to superior adsorption. The findings of this study advocate that the newly developed Cs-Co3O4@g-C3N4 nanosorbent has the ability to effectively remove MG dye and other dyes, making it highly relevant for applications in industrial wastewater treatment.

A Decagonal Number Theorem

Summary We offer a visual proof of a relationship between decagonal numbers and triangular numbers.

Differential CD38 Expression on Daratumumab-Naïve Vs. Resistant Multiple Myeloma - a Direct Stochastic Optical Reconstruction Microscopy ( dSTORM) Analysis in 53 Patients

Background: The anti-CD38 antibody Daratumumab (Dara) has become the new standard of care for the treatment of newly-diagnosed and relapsed/ refractory multiple myeloma (MM). At the same time, Dara resistance remains poorly understood. Potential resistance mechanisms have been proposed and range from intrinsic downregulation in CD38 dim MM cells to genetic alterations and splicing variants which may affect CD38 epitope expression (Adamia, ASH 2022). Here, we used super-resolution microscopy for high-precision quantification of CD38 in MM and piloted this approach in a large cohort of 53 MM patients with vs. without Dara resistance. Methods: Direct stochastic optical reconstruction microscopy ( dSTORM) is a novel imaging modality based on photoswitching of certain fluorophores coupled to antibodies for molecular representation of surface molecules at unprecedented levels of resolution (&amp;lt;20 nM). To correlate CD38 expression data with clinical outcome, we divided our cohort based on treatment status into two groups: Dara-naïve (n = 31) vs. Dara-resistant (n = 22) patients. Imaging data was complemented by phenotypic assessment using a well-established multi-epitope anti-CD38 antibody (Cytognos), as per standard-of-care. Results: Mean CD38 receptor density as measured with Dara via dSTORM was significantly higher in Dara-naïve (of 48.81 ± 29.65 receptors/µm²) vs. Dara-resistant patients (15.52 ± 3.9 receptors/µm², p = 0.016). This finding was corroborated by multi-epitope phenotypic analysis which revealed mean CD38 receptor expression levels of 32.09 ± 13.14 receptors/µm² in Dara-naïve vs. 21.65 ± 17.73 receptors/µm² in Dara-resistant patients ( p = 0.05). Lower CD38 expression via dSTORM vs. multi-epitope anti-CD38 staining was consistently observed in our cohort and may reflect ongoing blockage of the receptor binding site by Dara in cases with shorter time to last Dara application (median 53 days, range 1-189 days). As there was no correlation with the interval between Dara application and sampling however, splice-site mutations affecting CD38 receptor accessibility as proposed by Adamia et al. may depict a more relevant mechanism of resistance in our Dara-resistant cohort. In a next step, we therefore investigated alternative epitope binding in Dara-resistant patients. To this end, we investigated antibody binding by dSTORM imaging for isatuximab (Isa), an alternative CD38-directed monoclonal antibody that binds to a location in the C-terminal portion of the CD38 extracellular domain which is entirely different from the Dara epitope. Interestingly, there was a trend for increased detection rate with Isa for mean CD38 receptor expression in Dara-resistant patients (22.2 ± 4.6 receptors/µm²) vs. Dara (15.5 ± 3.9 receptors/µm², p = 0.1) and on par with CD38 receptor expression. No data was available to assess Dara binding in Isa pre-exposed patients. Conclusion: Innovative microscopy using dSTORM allows for high-resolution epitope quantification at unprecedented granularity. While this finding enables visual proof of differential epitope binding of Dara vs. Isa on CD38 in a clinically feasible setting, correlation studies to determine the clinical relevance of this observation remain ongoing.

A Visual Proof of Lehmer’s Arctangent Sum

Summary We provide a visual proof to Lehmer’s infinite sum of the arctangents of the inverse of the odd-indexed Fibonacci numbers. A few corollaries follow from the diagram, including Euler’s Machin-like formula and Strassnitzky’s formula.