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Hadjicostas's formula is a mathematical formula used in the field of number theory, specifically in relation to the sum of binomial coefficients. It provides a method for calculating the sum of the squares of binomial coefficients.

The Hasse–Weil zeta function is a mathematical tool used in number theory and algebraic geometry, particularly in the study of algebraic varieties over finite fields and their properties. It generalizes the classical Riemann zeta function and serves as an important object in understanding the distribution of points on algebraic varieties defined over finite fields.

The Hilbert–Pólya conjecture is an unproven hypothesis in mathematics that suggests a connection between the zeros of the Riemann zeta function and the eigenvalues of certain self-adjoint operators.

The Matsumoto zeta function is a mathematical function that arises in the study of certain types of number-theoretic problems, particularly those related to generalizations of classical zeta functions. It is typically associated with an extension of the classical Riemann zeta function and can be defined for various types of number systems.

The Ramanujan–Petersson conjecture is a significant result in number theory, specifically in the theory of modular forms and automorphic forms. It was formulated by mathematicians Srinivasa Ramanujan and Hans Petersson and deals with the growth rates of the coefficients of certain types of modular forms.

Crystallography awards are recognitions given to individuals or groups who have made significant contributions to the field of crystallography, which is the study of crystal structures and their properties. These awards can be presented by various organizations, societies, or institutions involved in scientific research and education. The awards often aim to highlight breakthroughs in crystallographic techniques, methods, and applications, as well as promote the importance of crystallography in understanding material science, biology, chemistry, and physics.

The Nobel Prize in Physics is one of the six Nobel Prizes established by the will of Alfred Nobel, the inventor of dynamite, in 1895. It is awarded annually by the Royal Swedish Academy of Sciences to individuals or groups who have made significant contributions to the field of physics. The prize recognizes outstanding achievements, including groundbreaking discoveries, innovations, or theoretical advancements.

Georges Friedel (1898–1977) was a French crystallographer known for his work in the field of crystallography and material science. He is particularly recognized for contributions related to the understanding of crystal structures and the development of techniques for the study of solid materials. Friedel is also known for his work on the Friedel law in crystallography, which describes certain symmetry in the diffraction patterns of crystals.

The Langlands–Deligne local constant is a fundamental concept in the theory of automorphic forms and number theory, particularly in the context of the Langlands program. It arises in the study of the local Langlands correspondence, which connects representations of p-adic groups to Galois representations.

Li's criterion is a mathematical result that gives conditions for the non-existence of solutions to certain types of differential equations, particularly for higher-order linear differential equations. It is named after the mathematician Li, Chen, and Zhang, who contributed to the understanding of oscillation theory in the context of differential equations. Specifically, in the context of second-order linear differential equations, Li's criterion can relate to the oscillatory behavior of solutions.

The list of zeta functions typically refers to various mathematical functions that generalize the classical Riemann zeta function. These functions have applications in number theory, mathematical physics, and other areas of mathematics.

The multiple zeta function is a generalization of the classical Riemann zeta function, which plays a significant role in number theory and mathematical analysis. The classical Riemann zeta function is defined for complex numbers \( s \) with real part greater than 1 as: \[ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}. \] The multiple zeta function extends this idea to multiple variables.

The prime zeta function is a mathematical function related to prime numbers and is defined as the infinite series: \[ P(s) = \sum_{p \text{ prime}} \frac{1}{p^s} \] where \( p \) runs over all prime numbers and \( s \) is a real number greater than 1.

A tensor is a mathematical object that generalizes scalars, vectors, and matrices to higher dimensions. Tensors are used in various fields such as physics, engineering, and machine learning to represent data and relationships in a structured manner. ### Basic Definitions: 1. **Scalar**: A tensor of rank 0, which is a single number (e.g., temperature, mass).

The virial theorem is a powerful result in classical mechanics and astrophysics, particularly useful for systems of particles bound by forces, such as stars in a galaxy or gas molecules in a container. It relates the average total kinetic energy of a system to its average total potential energy.

Computational mathematics is a branch of applied mathematics that focuses on numerical methods and algorithms for solving mathematical problems. It involves the development, analysis, and implementation of algorithms that solve mathematical problems on computers. This field combines mathematics, computer science, and engineering to address various problems in science, engineering, finance, and other areas.

General topology, also known as point-set topology, is a branch of topology dealing with the basic set-theoretic definitions and constructions used in topology. Here’s a list of key topics typically covered in a general topology course: 1. **Topological Spaces** - Definition of topological spaces - Basis for a topology - Subspace topology - Product topology - Quotient topology 2.

Super Bloch oscillations refer to a phenomenon observed in quantum mechanics, particularly in the context of ultracold atoms and optical lattices. This effect is an extension of the more basic concept of Bloch oscillations, which occur when charged particles, such as electrons, are subjected to an oscillating electric field while in a lattice potential.

Applied mathematics is a branch of mathematics that deals with mathematical methods and techniques that are used in practical applications across various fields such as science, engineering, business, and industry. Unlike pure mathematics, which is focused on abstract concepts and theoretical constructs, applied mathematics emphasizes the development and application of mathematical models and tools to solve real-world problems.