Materials with memory, often referred to as "shape memory materials," are a class of advanced materials that can undergo significant changes in shape or properties in response to external stimuli, such as temperature, stress, or electric/magnetic fields. The most well-known examples of shape memory materials include shape memory alloys (SMAs) and shape memory polymers (SMPs).

Band bending is a phenomenon that occurs in semiconductor physics and materials science, particularly at the interface between two different materials, such as a semiconductor and a metal or between two different semiconductors. It describes the change in energy band structure, specifically the bending of the energy bands (valence band and conduction band) in response to an electric field, charge distribution, or the presence of interfaces.

In mathematics, a locus (plural: loci) is a set of points that satisfy a particular condition or a set of conditions. It can be thought of as a geometric shape or figure that represents all possible locations in a given space that meet specified criteria. For example: 1. **Circle**: The locus of all points that are a fixed distance (radius) from a given point (the center) defines a circle.

Metal-induced gap states (MIGS) are electronic states that can form in the band gap of a semiconductor when a metal is in contact with it. These states emerge due to the interaction between the metal and the semiconductor's surface, which can modify the electronic structure. When a metal is deposited on a semiconductor, the Fermi level of the metal aligns with the energy levels in the semiconductor, creating an interface.

An algebraic fraction is a fraction in which the numerator and/or the denominator are algebraic expressions. An algebraic expression is an expression that can include numbers, variables (like \(x\) or \(y\)), and algebraic operations such as addition, subtraction, multiplication, and division. For example, the following are algebraic fractions: 1. \(\frac{x^2 + 2x + 1}{x - 1}\) 2.

Algebraic operations refer to mathematical processes that manipulate algebraic expressions and equations using rules of algebra. The primary algebraic operations include: 1. **Addition**: Combining two or more algebraic expressions. For example, \(a + b\) or \(2x + 3x = 5x\). 2. **Subtraction**: Removing one algebraic expression from another.

"Cancelling out," in a general context, refers to the process of nullifying or counteracting something so that it no longer has an effect or significance. This term can be applied in various fields, including mathematics, science, and everyday situations. Here are a few examples: 1. **Mathematics**: In algebra, cancelling out often refers to the process of simplifying fractions or equations.

In mathematics, the term "conjugate" can refer to different concepts depending on the context, particularly in complex numbers and algebraic expressions.

Equating coefficients is a mathematical technique often used to solve polynomial equations or to find relationships between different algebraic expressions. This method is particularly useful in situations where you have two polynomials that are set equal to each other, and you want to find values for their coefficients or variables. Here's how it generally works: 1. **Setup Equations**: Start with two polynomials that are equal to each other.

A worm's-eye view is a perspective used in photography, art, and visual storytelling that depicts a scene from a low angle, as if the viewer were at the level of a worm looking up. This perspective can emphasize the height of objects, such as buildings or trees, creating a sense of grandeur or immensity. It often conveys feelings of vulnerability or insignificance, as the viewer sees the world from a position that is usually not encountered in everyday life.

Factorization is the process of breaking down an expression, number, or polynomial into a product of its factors. Factors are numbers or expressions that can be multiplied together to obtain the original number or expression. Factorization is a fundamental concept in mathematics, used in various areas such as arithmetic, algebra, and number theory.

The term "formula" can have different meanings depending on the context in which it is used: 1. **Mathematics and Science**: In mathematics and science, a formula is a concise way of expressing information symbolically. It consists of mathematical symbols and numbers that represent a relationship or rule.

A linear equation is a mathematical equation that represents a straight line when graphed on a coordinate plane. It typically takes the form: \[ ax + by + c = 0 \] or in slope-intercept form: \[ y = mx + b \] where: - \( x \) and \( y \) are the variables. - \( a \), \( b \), and \( c \) are constants (with \( a \) and \( b \) not both zero).

Solving quadratic equations using continued fractions is a method linked to the approximation of the solutions of these equations through the use of continued fractions. Quadratic equations typically take the form: \[ ax^2 + bx + c = 0 \] where \(a\), \(b\), and \(c\) are coefficients, and \(x\) is the variable we want to solve for.

A relation \( R \) on a set is called a transitive relation if, for all elements \( a, b, c \) in that set, whenever \( a \) is related to \( b \) (denoted \( aRb \)) and \( b \) is related to \( c \) (denoted \( bRc \)), then \( a \) must also be related to \( c \) (denoted \( aRc \)).

Two-element Boolean algebra, also known as Boolean algebra of two values, is a mathematical structure that deals with binary variables that can take on one of two values: typically represented as 0 and 1. This framework is foundational to digital logic and computer science.

In mathematics, the term "sign" refers to the indication of whether a number is positive, negative, or zero. It is typically represented using the following symbols: - Positive numbers: Represented by a plus sign (+) or no sign at all (e.g., +5 or 5). - Negative numbers: Represented by a minus sign (−) (e.g., −3). - Zero: The number 0 is neutral and does not carry a sign.

In mathematics, equality is a fundamental relationship that asserts that two expressions represent the same value or entity. It is typically denoted by the equality symbol "=". When we say that two things are equal, we mean that they have the same mathematical value or that they are identical in a specific context.

Multiplication is one of the four fundamental arithmetic operations in mathematics, alongside addition, subtraction, and division. It involves combining equal groups of items to find the total number of items. In simpler terms, multiplication can be thought of as repeated addition.

Euclidean plane geometry is a branch of mathematics that studies the properties and relationships of points, lines, angles, surfaces, and shapes in a two-dimensional plane. It is named after the ancient Greek mathematician Euclid, who is often referred to as the "father of geometry" due to his influential work, "Elements," which systematically presented the principles and proofs of geometry.