Adjective
/lɒɡ ˈkɒnvɛks/
The term "log-convex" refers to a property of a function or a sequence, where the logarithm of the function is convex. More formally, a function ( f(x) ) is log-convex if for any two points ( x_1 ) and ( x_2 ) and for any ( \lambda \in [0, 1] ):
[ f(\lambda x_1 + (1 - \lambda)x_2) \leq f(x_1)^\lambda \cdot f(x_2)^{1 - \lambda} ]
In a broader sense, log-convexity implies that the multiplicative growth of a function is slower than its additive growth, making it a significant concept in various mathematical fields such as convex analysis, optimization, and probability theory.
Frequency of Use: "Log-convex" is primarily used in mathematical and statistical contexts, often appearing in academic papers, textbooks, and specialized discussions. It is less common in everyday conversation.
Context of Use: The term is predominantly found in written texts within mathematics and statistics.
The sequence of the coefficients of the power series is log-convex, indicating a certain stability in its growth.
El conjunto de los coeficientes de la serie de potencias es log-convexo, lo que indica una cierta estabilidad en su crecimiento.
For the analysis of random variables, we need to check if the probability distribution function is log-convex.
Para el análisis de variables aleatorias, necesitamos verificar si la función de distribución de probabilidad es log-convexa.
In convex analysis, many important functions exhibit log-convexity, leading to useful mathematical properties.
En el análisis convexo, muchas funciones importantes exhiben log-convexidad, lo que conduce a propiedades matemáticas útiles.
The term "log-convex" does not have widespread idiomatic expressions associated with it due to its specialized mathematical context. However, one could find phrases in the realm of mathematics that reference its properties. Here are some relevant usages:
When discussing optimization problems, we often rely on the log-convex property to ensure convergence of solutions.
Al discutir problemas de optimización, a menudo dependemos de la propiedad log-convexa para garantizar la convergencia de soluciones.
In statistical mechanics, the log-convexity of the partition function guarantees the existence of a thermodynamic limit.
En la mecánica estadística, la log-convexidad de la función de partición garantiza la existencia de un límite termodinámico.
The log-convex behavior of the likelihood function in estimations implies a consistent estimator.
El comportamiento log-convexo de la función de verosimilitud en estimaciones implica un estimador consistente.
The term "log-convex" is derived from two parts: - "log" refers to the logarithm function, which is a mathematical function with widespread applications in various fields. - "convex" comes from the Latin word "convexus," meaning "arched" or "vaulted," which relates to functions that curve upwards.
Synonyms: - Logarithmically convex (in some contexts) - Multiplicatively convex
Antonyms: - Log-concave (the opposite property, where the logarithm of the function is concave)
This information provides a comprehensive overview of the term "log-convex," detailing its mathematical context, example usages, and related linguistic features.