Coordination number refers to the number of ligand atoms or ions that are directly bonded to a central atom or ion in a coordination complex. It is an important concept in coordination chemistry and helps in understanding the structure and stability of coordination compounds. For example, in a metal complex such as [Co(NH₃)₆]³⁺, the cobalt ion (Co³⁺) is surrounded by six ammonia (NH₃) ligands.
VSEPR theory, or Valence Shell Electron Pair Repulsion theory, is a model used in chemistry to predict the three-dimensional shapes of molecules based on the repulsion between electron pairs in the valence shell of atoms. The fundamental concept behind VSEPR theory is that electron pairs, whether they are bonding pairs (shared between atoms) or lone pairs (non-bonding electrons that belong to a single atom), repel each other due to their negative charge.
A molecular sensor is a type of sensor that detects and measures specific molecular substances, such as gases, ions, or biological molecules, based on their unique properties. These sensors typically employ various biochemical, optical, or electronic techniques to identify the presence, concentration, or changes in the targeted molecules.
The Kenmotsu manifold is a specific type of Riemannian manifold known in the context of differential geometry. It is characterized by having certain curvature properties and is considered in the study of submanifolds and their embeddings. To be more precise, a Kenmotsu manifold is a type of 3-dimensional (or higher-dimensional) contact metric manifold that satisfies certain conditions relating to its contact structure and the metric.
Applied and Computational Harmonic Analysis (ACHA) is an interdisciplinary field that focuses on the analysis and application of harmonic analysis techniques using computational methods. The field blends concepts from harmonic analysis, applied mathematics, and numerical analysis to solve problems in various scientific and engineering domains. ### Key Components: 1. **Harmonic Analysis**: This is the study of functions and their representations as superpositions of basic waves, often using tools like Fourier analysis.
Edward S. Reed could refer to multiple individuals, but one notable person by that name is Edward S. Reed (b. 1934), who is an American scientist and educator known for his contributions in the field of music education and school administration. However, it is essential to clarify the context in which you are asking about Edward S. Reed, as there may be other individuals with similar names in various fields.
The Kulkarni–Nomizu product is a mathematical operation used in the context of differential geometry, particularly for constructing new geometric structures on manifolds. Specifically, it is a way to combine two Riemannian manifolds using their cotangent bundles to create a new manifold, often involving the introduction of a new metric.
The Lanczos tensor, often referred to in the context of numerical linear algebra and more specifically in the Lanczos algorithm, is associated with the process of reducing large symmetric matrices to tridiagonal form. The Lanczos algorithm is used to find the eigenvalues and eigenvectors of large, sparse symmetric matrices, which often arise in various fields like quantum mechanics, structural engineering, and machine learning.
The last geometric statement of Jacobi, often referred to as Jacobi's last theorem, pertains to the geometry of curves and is essentially connected to elliptic functions and their relation to algebraic curves. In its simplest form, Jacobi's last theorem asserts that if a non-singular algebraic curve can be parameterized by elliptic functions, then the degree of the curve must be 3 (a cubic curve).
The Lebrun manifold, also known as the Lebrun-Simpson manifold, is an important example in the study of Riemannian geometry and in the context of \(4\)-manifolds. It is a complex manifold that can be described as a Kähler surface. Specifically, it is notable for being a non-Kähler symplectic manifold, and it can be constructed as a particular type of complex algebraic surface.
The term "HABU" can refer to different things in different contexts, but it is most commonly associated with a type of venomous snake found in Southeast Asia, particularly the "Habus" of Japan, such as the Okinawa habu (Protobothrops flavoviridis).
Bent's rule is a principle in chemistry that pertains to the hybridization of atomic orbitals in heteroatomic molecules, particularly those containing a central atom bonded to different substituents. Formulated by Linus Pauling and named after the chemist Robert S. Bent, the rule states that: "In a molecule, the more electronegative atoms will tend to occupy positions that allow for greater p-character in the hybrid orbitals formed by the central atom.
Bond softening refers to a phenomenon observed in the context of materials science and solid-state physics, particularly in the study of mechanical properties of materials. It denotes a reduction in the strength of atomic or molecular bonds in a material, which can lead to a decrease in its overall mechanical strength and stiffness.
A Gaussian orbital is a type of mathematical function used to represent atomic orbitals in quantum chemistry and computational chemistry.
The center of mass (COM) is a point in a system of particles or a continuous mass distribution where the total mass of the system can be considered to be concentrated for the purpose of analyzing motion. It is the balance point of the mass distribution, meaning that if a system were to be suspended at this point, it would remain in equilibrium.
Empedocles was a pre-Socratic Greek philosopher who lived around 495-435 BCE. He is best known for his work in philosophy and natural science, particularly for introducing the idea that all matter is composed of four fundamental elements: earth, water, air, and fire. He proposed that these elements are in constant motion and interact through two opposing forces he called Love (philia), which brings things together, and Strife (neikos), which separates them.
Clairaut's relation, also known as Clairaut's theorem, is a fundamental result in differential geometry that relates the curvature of a surface to the derivatives of the surface's height function. Specifically, it applies to surfaces of revolution, which are surfaces generated by rotating a curve about an axis.
A **differentiable stack** is a concept arising from the fields of differential geometry, algebraic topology, and category theory, particularly in the context of homotopy theory and advanced mathematical frameworks like derived algebraic geometry. In general, a **stack** is a categorical structure that allows for the systematic handling of "parametrized" objects, facilitating the study of moduli problems in algebraic geometry and related fields.