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Integration by parts is a fundamental technique in calculus for evaluating integrals involving the product of two functions. It is particularly useful when direct integration is not feasible. The method is based on the product rule for differentiation, which states that the derivative of a product equals the derivative of the first function times the second, plus the first function times the derivative of the second. By integrating this identity and rearranging terms, the integration by parts...
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Integration by Parts: Definite Integrals01:23

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Definite integrals involving the product of two functions over a fixed interval can be evaluated using integration by parts. This method rewrites the integral as the difference of a product evaluated at the endpoints and a remaining definite integral that is often simpler to compute.A representative example is the definite integral of the inverse tangent function. Since there is no direct integration formula for arctan ⁡x, the integrand is rewritten as a product of arctan⁡ x and the...
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Depth perception is the ability to perceive objects three-dimensionally. It relies on two types of cues: binocular and monocular. Binocular cues depend on the combination of images from both eyes and how the eyes work together. Since the eyes are in slightly different positions, each eye captures a slightly different image. This disparity between images, known as binocular disparity, helps the brain interpret depth. When the brain compares these images, it determines the distance to an object.
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Consider a real-valued function defined on a closed interval. One of the fundamental objectives in calculus is to determine the area under the graph of such a function. When an exact computation is not readily available, this area can be estimated by dividing the interval into a finite number of equal subintervals. Each subinterval corresponds to a rectangle whose width is the length of the subinterval and whose height is determined by the value of the function at a selected point within that...
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SPAC: una plataforma empresarial escalable e integrada para el análisis espacial de células únicas

Fang Liu1, Rui He2, Thomas Sheeley3

  • 1Biomedical and Computational Science Directorate, Frederick National Laboratory for Cancer Research, Rockville, MD, USA.

BMC bioinformatics
|January 29, 2026
PubMed
Resumen

SPAC es una nueva plataforma basada en web que hace que el análisis espacial de células únicas sea más fácil y rápido para todos los investigadores. Acelera significativamente el procesamiento de datos y permite la exploración detallada de los microambientes tisulares.

Palabras clave:
computación de alto rendimientovisualización interactivaimagen multiplexadaanálisis escalableanálisis de células únicasómicas espacialesproteómica espacialmicroambiente tumoral

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Área de la Ciencia:

  • Biología espacial de células únicas
  • Biología computacional
  • Bioinformática

Sus antecedentes:

  • Las tecnologías de células únicas resueltas espacialmente ofrecen información transformadora sobre la arquitectura tisular y la enfermedad.
  • El análisis de conjuntos de datos espaciales de alta dimensionalidad es computacionalmente intensivo y carece de herramientas fáciles de usar para los investigadores.

Objetivo del estudio:

  • Desarrollar una plataforma escalable basada en web para el análisis de datos espaciales de células únicas eficiente y reproducible.
  • Cerrar la brecha entre los investigadores experimentales y computacionales proporcionando una herramienta accesible.

Principales métodos:

  • SPAC utiliza una arquitectura de cuatro niveles: motor de análisis de Python, integración HPC/GPU, interfaz de navegador sin código y Shiny para visualización de Python.
  • La plataforma admite roles de usuario distintos, lo que permite la personalización para científicos de datos y la facilidad de uso para científicos de laboratorio.
  • Las características incluyen reproducibilidad incorporada y soporte de flujo de trabajo colaborativo.

Principales resultados:

  • SPAC logró una aceleración de más de 20 veces en la agrupación no supervisada en un conjunto de datos grande (2,6 millones de células), reduciendo el tiempo de ~3 horas a <10 minutos con aceleración de GPU.
  • Permitió el perfilado espacial de grano fino de los compartimentos del microambiente tumoral.
  • Demostró la escalabilidad y el rendimiento de la plataforma en datos biológicos complejos.

Conclusiones:

  • SPAC supera los desafíos clave en el análisis espacial de células únicas al combinar una interfaz intuitiva con computación de alto rendimiento.
  • La plataforma agiliza análisis complejos, fomenta la colaboración y acelera la traducción de datos espaciales en información biológica.