A function \( f: (a, b) \to \mathbb{R} \) is said to be logarithmically convex on the interval \( (a, b) \) if for any \( x, y \in (a, b) \) and \( \lambda \in [0, 1] \), the following inequality holds: \[ f(\lambda x + (1 - \lambda) y) \leq (f(x)^{\lambda}

In mathematics, a **homomorphism** is a structure-preserving map between two algebraic structures of the same type. More specifically, it is a function that respects the operation(s) defined on those structures. The concept of homomorphism is widely used in various branches of mathematics, including group theory, ring theory, and linear algebra. ### Types of Homomorphisms 1.

Janusz Grabowski is not a widely recognized public figure or concept as of my last knowledge update in October 2023.

Abraham de Moivre (1667–1754) was a French mathematician known for his work in probability theory and for his contributions to the development of the theory of complex numbers.

A function \( f: \mathbb{R}^n \to \mathbb{R} \) is said to be radially unbounded if it behaves in a way such that, as you move further away from the origin in all directions, the function's output tends to infinity.

Dušan Repovš is a notable figure in the field of mathematics, particularly known for his work in logic, set theory, and foundational mathematics. He has made significant contributions to mathematical education and research.

Gregory Eskin is a notable figure in the field of mathematics and mathematical biology. He has made contributions primarily in areas related to applied mathematics, mathematical modeling, and complex systems. However, it is essential to note that there may be multiple individuals with the name Gregory Eskin, so the specific context in which you are asking about him could influence the answer.

Paolo Marcellini is an Italian mathematician known for his work in functional analysis, calculus of variations, and mathematical analysis. His contributions to the field often involve the study of nonlinear partial differential equations and optimization problems. He has published numerous research papers and is recognized in academic circles for his expertise in these areas.

Shoshana Kamin is a notable Jewish-American artist, primarily recognized for her work as an abstract painter. She is known for her unique approach to form, color, and texture within her artwork, which often reflects her cultural heritage and personal experiences.

Albert-László Barabási is a prominent Hungarian-American physicist known for his work in network science. He is particularly renowned for his research on complex networks, which has applications in various fields including sociology, biology, and computer science. Barabási is best known for the development of the Barabási-Albert model, which describes how networks grow and evolve over time, emphasizing the role of "preferential attachment" where nodes with higher connectivity are more likely to attract new connections.

C. R. Rao, or Calyampudi Radhakrishna Rao, is an Indian-American mathematician and statistician regarded as one of the leading figures in the field of statistics. He was born on September 10, 1920, in Hadagali, Karnataka, India. Rao is best known for his significant contributions to statistical theory, especially in the areas of estimation theory, multivariate analysis, and design of experiments.

Elchanan Mossel is a prominent mathematician and computer scientist known for his work in areas such as probability theory, graphical models, statistical physics, and information theory. He has made significant contributions to the understanding of random processes, combinatorial structures, and machine learning. Mossel is often associated with research on the dynamics of social networks and the analysis of algorithms, particularly in the context of information spread and consensus in networks.

Elliott Waters Montroll (1911–2004) was an American mathematician and physicist known for his contributions to statistical mechanics, mathematical biology, and the field of operations research. He is particularly recognized for his work in the area of random walks and their applications in various scientific fields, including physics and biology. Montroll's research often focused on the mathematical modeling of systems with a strong emphasis on probabilistic methods and stochastic processes.

Eugene Lukacs does not appear to refer to a widely recognized figure or concept as of my last update in October 2023. It's possible that the name could relate to a specific individual who is not widely known or may have emerged in a particular field after that date. If you could provide more context or clarify the specific area you are referring to (such as art, science, literature, etc.

Franz Thomas Bruss does not appear to be a widely recognized figure, concept, or term within the general knowledge up to October 2023. It's possible that he could represent a specific individual with local significance or relevance in a certain field, such as academia, arts, or business, but additional context would be helpful to provide a more accurate response.

Hans Frauenfelder was a physicist known for his contributions to the field of condensed matter physics, specifically in areas such as magnetism and the study of glasses and disordered systems. He worked extensively on the properties of complex materials and made significant contributions to the understanding of molecular dynamics and the behavior of systems at a microscopic level. In addition to his research, Frauenfelder was also recognized for his involvement in scientific education and communication.

Henry McKean is an Irish journalist and broadcaster known for his work in radio and television. He has been involved in various media endeavors, often focusing on investigative journalism and human interest stories. McKean has worked with different broadcasting organizations, including making significant contributions to the Irish news landscape.

Magda Peligrad is a prominent mathematician known for her work in the field of probability theory, particularly in relation to stochastic processes and statistical inference. She has contributed significantly to the understanding of limit theorems, mixing processes, and properties of random walks, among other topics. Her research often involves the interplay of probability theory with other areas of mathematics, such as statistics and ergodic theory.

Nikolai Smirnov (1900–1974) was a prominent Russian mathematician known for his contributions to various areas of mathematics, particularly in statistics and probability theory. He is best known for the Smirnov tests, which are statistical methods used for assessing the goodness of fit of a distribution, specifically the Kolmogorov-Smirnov test that compares a sample distribution to a reference probability distribution or compares two sample distributions.

Ole Barndorff-Nielsen is a Danish statistician known for his contributions to the fields of statistics and mathematical finance. He has made significant advancements in various areas, including statistical theory, distribution theory, and stochastic processes. Barndorff-Nielsen is notably recognized for his work on the Barndorff-Nielsen distribution, which is a family of probability distributions that extends the concept of the exponential family.