Johann Carl Friedrich Gauss (;, ) ( 30 April 177723 February 1855 ) was a German mathematician and physical scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.

The system analysis is carried out in time domain using differential equations, in complex-s domain with Laplace transform or in frequency domain by transforming from the complex-s domain.

In contrast to the frequency domain analysis of the classical control theory, modern control theory utilizes the time-domain state space representation, a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations.

In mathematics, the Cauchy – Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which must be satisfied if we know that a complex function is complex differentiable.

The equations are one way of looking at the condition on a function to be differentiable in the sense of complex analysis: in other words they encapsulate the notion of function of a complex variable by means of conventional differential calculus.

Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory, and more.

Around 1909, Hilbert dedicated himself to the study of differential and integral equations ; his work had direct consequences for important parts of modern functional analysis.

Many concepts and techniques of analysis and differential equations have been generalized to the setting of Riemannian manifolds.

The analysis of linear systems is possible because they satisfy a superposition principle: if u ( t ) and w ( t ) satisfy the differential equation for the vector field ( but not necessarily the initial condition ), then so will u ( t ) + w ( t ).

Consequently, it is important to counteract side channel attacks ( e. g., timing or simple / differential power analysis attacks ) using, for example, fixed pattern window ( aka.

The 17-year-old Enrico Fermi chose to derive and solve the partial differential equation for a vibrating rod, applying Fourier analysis.

Fourier analysis has many scientific applications – in physics, partial differential equations, number theory, combinatorics, signal processing, imaging, probability theory, statistics, option pricing, cryptography, numerical analysis, acoustics, oceanography, sonar, optics, diffraction, geometry, protein structure analysis and other areas.

He had made advances in the areas of analysis, foundations and logic, made many contributions to the teaching of calculus and also contributed to the fields of differential equations and vector analysis.

In mathematical analysis, the Laplace transform is an isomorphism mapping hard differential equations into easier algebraic equations.

Prior to the emergence of quantum mechanics as a separate theory, the mathematics used in physics consisted mainly of formal mathematical analysis, beginning with calculus ; and, increasing in complexity up to differential geometry and partial differential equations.

Ordinary differential equations appear in the movement of heavenly bodies ( planets, stars and galaxies ); optimization occurs in portfolio management ; numerical linear algebra is important for data analysis ; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.

From the point of view of functional analysis, calculus is the study of two linear operators: the differential operator, and the indefinite integral operator.

Oliver Heaviside FRS ( ( 18 May 1850 – 3 February 1925 ) was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques to the solution of differential equations ( later found to be equivalent to Laplace transforms ), reformulated Maxwell's field equations in terms of electric and magnetic forces and energy flux, and independently co-formulated vector analysis.

In 1884 he recast Maxwell's mathematical analysis from its original cumbersome form ( they had already been recast as quaternions ) to its modern vector terminology, thereby reducing twelve of the original twenty equations in twenty unknowns down to the four differential equations in two unknowns we now know as Maxwell's equations.

If the density of the chain is variable then the analysis above can be adapted to produce equations for the curve given the density, or given the curve to find the density.

These equations can be used as the starting point in the analysis of a flexible chain acting under any external force.

The reason why Euler and some other authors relate the Cauchy – Riemann equations with analyticity is that a major theorem in complex analysis says that holomorphic functions are analytic and viceversa.

Dimensional analysis is routinely used to check the plausibility of derived equations and computations.

Today Diophantine analysis is the area of study where integer ( whole number ) solutions are sought for equations, and Diophantine equations are polynomial equations with integer coefficients to which only integer solutions are sought.

A quantitative analysis of these effects is based on the Fresnel equations, but with additional calculations to account for interference.

By the 1920s Lewis Fry Richardson's interest in weather prediction led him to propose human computers and numerical analysis to model the weather ; to this day, the most powerful computers on Earth are needed to adequately model its weather using the Navier – Stokes equations.

I knew that a conversation with the author would not settle such questions, because a man is not the same as his writing: in the last analysis, the questions had to be settled by the work itself.

The grammatical description of each occurrence in the text must be retrieved from the dictionary to permit such an analysis.

Still, even in such languages tone analysis has not been as simple as one might expect.

Moreover, whereas in Interstate Commerce Commission parlance `` variable cost '' means a cost deemed to vary in direct proportion to changes in rate of output, in the type of analysis now under review `` variable cost '' has been used more broadly, so as to cover costs which, while a function of some one variable ( such as output of energy, or number of customers ), are not necessarily a linear function.

These axioms are sufficient for many proofs in elementary mathematical analysis, and are consistent with some principles, such as the Lebesgue measurability of all sets of reals, that are disprovable from the full axiom of choice.

Basic theories, such as arithmetic, real analysis and complex analysis are often introduced non-axiomatically, but implicitly or explicitly there is generally an assumption that the axioms being used are the axioms of Zermelo – Fraenkel set theory with choice, abbreviated ZFC, or some very similar system of axiomatic set theory like Von Neumann – Bernays – Gödel set theory, a conservative extension of ZFC.

* Requirements analysis – encompasses those tasks that go into determining the needs or conditions to meet for a new or altered product, taking account of the possibly conflicting requirements of the various stakeholders, such as beneficiaries or users.

* Multivariate analysis – analysis of data involving several variables, such as by factor analysis, regression analysis, or principal component analysis

It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory.

In his analysis of Aalto, Giedion gave primacy to qualities that depart from direct functionality, such as mood, atmosphere, intensity of life and even national characteristics, declaring that " Finland is with Aalto wherever he goes ".

Mathematical considerations, such as symmetry and complexity, are used for analysis in theoretical aesthetics.

In computer science, the analysis of algorithms is the determination of the number of resources ( such as time and storage ) necessary to execute them.

This analysis neglects other potential bottlenecks such as memory bandwidth and I / O bandwidth, if they do not scale with the number of processors ; however, taking into account such bottlenecks would tend to further demonstrate the diminishing returns of only adding processors.

Classical methods ( also known as wet chemistry methods ) use separations such as precipitation, extraction, and distillation and qualitative analysis by color, odor, or melting point.

Although there are few examples of such systems competitive with traditional analysis techniques, potential advantages include size / portability, speed, and cost.

This form of data manipulation allows for rapid computer visualisation and analysis, with data presented as point cloud data with additional information, such as each ion's mass to charge ( as computed from the velocity equation above ), voltage or other auxiliary measured quantity or computation therefrom.

Business statistics is the science of good decision making in the face of uncertainty and is used in many disciplines such as financial analysis, econometrics, auditing, production and operations including services improvement, and marketing research.

These techniques allowed for the discovery and detailed analysis of many molecules and metabolic pathways of the cell, such as glycolysis and the Krebs cycle ( citric acid cycle ).

Significant progress in Big Bang cosmology have been made since the late 1990s as a result of advances in telescope technology as well as the analysis of data from satellites such as COBE, the Hubble Space Telescope and WMAP.