Hasil untuk "Mathematics"

W. Fleming, H. Mete Soner, H. M. Soner et al.

Xin Ma, N. Kishor

T. Nunes, A. Schliemann, David W. Carraher

T. P. Carpenter

J. Abedi, Carol Lord

E. Gutstein

Shama Sadiq, Nadeem Azhar, N. Myrzakulov et al.

This research explores the rapid expansion period of the early universe by applying the Chaplygin gas model, an alternative cosmological framework, to analyze the dynamics of inflationary processes. This study assesses the compatibility of three widely studied scalar field potentials with the latest observational constraints derived from the Planck datasets. Our analysis includes inflationary parameters such as slow-roll parameters, scalar power spectrum PR, scalar spectral index ns, dissipative ratio R, tensor-to-scalar ratio r and running of the scalar spectral index dnsdln⁡k, within the theoretical frameworks of canonical scalar field dynamics and the Chaplygin gas cosmological model. These parameters help to paint a comprehensive picture of the inflationary epoch and its impact on the observable Universe. We also address the generalized ratio of the swampland de-Sitter conjecture through the expression of T′VV′T for three different potentials. We analyze inflation driven by a scalar field ϕ with decay rate Γ(ϕ,T)=CϕTaϕa−1, where Cϕ is a dimensionless coupling and a controls dissipation strength. Working in the strong dissipative regime (R≫1), we systematically investigate the background evolution and perturbation spectrum, deriving inflationary observables.

Rieke Löper, Lennart Abels, Daniel Otero Baguer et al.

Lentigo maligna (LM) is a melanoma in situ with high cumulative sun damage. Histological evaluation of resection margins is difficult and time-consuming. Melanocyte density (MD) is a suitable, quantifiable, and reproducible diagnostic criterion. In this retrospective single-centre study, we investigated whether an artificial intelligence (AI) tool can support the assessment of LM. Training and evaluation were based on MD in Sox-10-stained digitalised slides. In total, 86 whole slide images (WSIs) from LM patients were annotated and used as a training set. The test set consisted of 177 slides. The tool was trained to detect the epidermis, measure its length, and determine the MD. A cut-off of ≥30 melanocytes per 0.5 mm of epidermis length was defined as positive. Our AI model automatically recognises the epidermis and measures the MD. The model was trained on nuclear immunohistochemical signals and can also be applied to other nuclear stains, such as PRAME or MITF. The WSI is automatically visualised by a three-colour heat map with a subdivision into low, borderline, and high melanocyte density. The cut-offs can be adjusted individually. Compared to manually counted ground truth MD, the AI model achieved high sensitivity (87.84%), specificity (72.82%), and accuracy (79.10%), and an area under the curve (AUC) of 0.818 in the test set. This automated tool can assist (dermato) pathologists by providing a quick overview of the WSI at first glance and making the time-consuming assessment of resection margins more efficient and more reproducible. The AI model can provide significant benefits in the daily routine workflow.

Lisa Carbone

The main drawback of using generative AI models for advanced mathematics is that these models are not primarily logical reasoning engines. However, Large Language Models, and their refinements, can pick up on patterns in higher mathematics that are difficult for humans to see. By putting the design of generative AI models to their advantage, mathematicians may use them as powerful interactive assistants that can carry out laborious tasks, generate and debug code, check examples, formulate conjectures and more. We discuss how generative AI models can be used to advance mathematics research. We also discuss their integration with neuro-symbolic solvers, Computer Algebra Systems and formal proof assistants such as Lean.

G. Lakoff, R. Núñez

A. Frenzel, T. Goetz, R. Pekrun et al.

Mary J. Schleppegrell

W. Blum, P. Galbraith, Hans-Wolfgang Henn et al.

Faruk Özger, Merve Temizer Ersoy, Zeynep Ödemiş Özger

Integral equations, which are defined as “the equation containing an unknown function under the integral sign”, have many applications of real-world problems. The second type of Fredholm integral equations is generally used in radiation transfer theory, kinetic theory of gases, and neutron transfer theory. A special case of these equations, known as the quadratic Chandrasekhar integral equation, given by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn><mo>+</mo><mi>λ</mi><mi>x</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><msubsup><mo>∫</mo><mrow><mn>0</mn></mrow><mn>1</mn></msubsup><mfrac><mi>s</mi><mrow><mi>t</mi><mo>+</mo><mi>s</mi></mrow></mfrac><mi>x</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mi>d</mi><mi>t</mi><mo>,</mo></mrow></semantics></math></inline-formula> can be very often encountered in many applications, where <i>x</i> is the function to be determined, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> is a parameter, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>,</mo><mi>s</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>. In this paper, using a fixed-point theorem, the existence conditions for the solution of Fredholm integral equations of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>χ</mi><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow><mo>=</mo><mi>ϱ</mi><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow><mo>+</mo><mi>χ</mi><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow><msubsup><mo>∫</mo><mrow><mi>p</mi></mrow><mi>q</mi></msubsup><mi>k</mi><mrow><mo>(</mo><mi>l</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>V</mi><mi>χ</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mi>d</mi><mi>z</mi></mrow></semantics></math></inline-formula> are investigated in the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>ω</mi></msub><mfenced separators="" open="[" close="]"><mi>p</mi><mo>,</mo><mi>q</mi></mfenced><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>χ</mi></semantics></math></inline-formula> is the unknown function to be determined, <i>V</i> is a given operator, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϱ</mi><mo>,</mo><mspace width="3.33333pt"></mspace><mspace width="3.33333pt"></mspace><mi>k</mi></mrow></semantics></math></inline-formula> are two given functions. Moreover, certain important applications demonstrating the applicability of the existence theorem presented in this paper are provided.

Nancy C. Jordan, J. Glutting, Chaitanya Ramineni

F. Cucker, S. Smale

T. Rowland, P. Huckstep, A. Thwaites

Guojin He, Shengyun Ji, Rongjun Wu et al.

In the field of electronic communication warfare, accurately predicting the range and intensity of shortwave interference signals presents a significant challenge due to the complex interplay between the ionospheric parameters and the electromagnetic environment. To address this challenge, we designed a novel tool to assess the interference impact area of shortwave interference signals in a dynamically changing ionospheric environment. Considering sophisticated ionospheric radio wave propagation models and innovative spatial grid methods, this tool finishes the comprehensive spatial distribution of the interference impact area and delivers grid-based insights into the interference intensity. Furthermore, the test verification of the tool demonstrated a mean error of 8.42 dB between the measured and simulated results, underscoring the efficacy and reliability of this tool. This pioneering work is poised to make substantial contributions to the field of communication electronic warfare and holds significant promise for guiding the development of interference countermeasures.

C.W. Chukwu, E. Bonyah, M.L. Juga et al.

Tuberculosis remains one of the most dangerous diseases globally and has affected many people in Sub-Saharan Africa. In this paper, a fractional stochastic model of tuberculosis disease was formulated and analyzed. The existence and uniqueness of solutions are presented in the new approach far from the deterministic fractional operators. We carry out numerical simulations of the model using three fractional operators. Our numerical results suggest that the Caputo–Fabrizio operator has a more random effect among the three different operators than the Atangana–Baleanu and the Caputo operators. Further, it is envisaged that the fractional-order derivatives significantly impact the dynamics of the disease.

Mathijs Wintraecken

Translation of the paper ``Simplizialzerlegungen von Beschrankter Flachheit'' by Hans Freudenthal (https://doi.org/10.2307/1968813), in which Freudenthal answers ``a question by Brouwer about the construction of an infinite series of subdivisions of a polytope, such that the next element in the sequence is a subdivision of the previous one and such that the subsimplices that arise do not become arbitrarily flat.''