Thus, it was not until two centuries had passed that in 1872 Karl Weierstrass presented the first definition of a function with a graph that would today be considered fractal, having the non-intuitive property of being everywhere continuous but nowhere differentiable.

In 1975 Mandelbrot solidified hundreds of years of thought and mathematical development in coining the word " fractal " and illustrated his mathematical definition with striking computer-constructed visualizations.

Authorities disagree on the exact definition of fractal, but most usually elaborate on the basic ideas of self-similarity and an unusual relationship with the space a fractal is embedded in.

In other words, the box definition is " external " — one needs to assume the fractal is contained in a Euclidean space, and define boxes according to the external structure " imposed " by the containing space.

One can imagine the fractal disconnected from its environment, define balls using the distance between points on the fractal and calculate the dimension ( to be more precise, the N < sub > covering </ sub > definition is also external, but the other two are internal ).

As with many other fractal s, the stages are obtained via a recursive definition.

Heschel then goes on to explore the problems of doubts and faith ; what Judaism means by teaching that God is one ; the essence of humanity and the problem of human needs ; the definition of religion in general and of Judaism in particular ; and human yearning for spirituality.

He goes on to define error as self-contradiction of definition (" an absurdity, or senselesse Speech ") or conclusions that do not follow the definitions on which they are supposed to be based.

Arguing " creation " goes against the accepted definition of time as a statistic.

It goes beyond architecture ’ s definition of ‘ making buildings ’ and reaches out for global views on architecture and design, broader attitudes to social structures, and creating environments to live in.

By incorporating index of refraction in its definition, NA has the property that it is constant for a beam as it goes from one material to another provided there is no optical power at the interface.

This definition proved adequate for subsequent versions including the ` extended ', ` revised extended ', and ` Government-Binding ' ( GB ) versions of generative grammar, but may no longer be sufficient for the current minimalist grammar in that merge may require a formal definition that goes beyond the tree manipulation characteristic of Move α.

By convention, we assume all possible states and transitions have been included in the definition of the processes, so there is always a next state and the process goes on forever.

Kant referred to these objects as " The Thing in Itself " and goes on to argue that their status as objects beyond all possible experience by definition means we cannot know them.

But by whatever method one goes about it, all consistency proofs would seem to necessitate the primitive notion of contradiction ; moreover, it seems as if this notion would simultaneously have to be " outside " the formal system in the definition of tautology.

Canada has adopted an expansive definition of its maritime law, which goes beyond traditional admiralty law.

Later in the same work, Mill goes on to write that he is proposing “ an arbitrary definition of man, as a being who inevitably does that by which he may obtain the greatest amount of necessaries, conveniences, and luxuries, with the smallest quantity of labour and physical self-denial with which they can be obtained .”

* Modern Orthodoxy is, almost by definition, inhibited from becoming a strong movement, because this would entail organization and authority to a degree " which goes against the very grain of modernity ".

In short, the speed of government goes beyond sentient control ( in this fictional universe, many alien species co-exist, with a common definition of sentience marking their status as equals ).

Majdalany goes on to state that the British simply used the name as a label for the Kikuyu ethnic community without assigning any specific definition.

Starting with the definition of beauty as " the idea in the form of limited appearance ," he goes on to develop the various elements of art ( the beautiful, sublime and comic ), and the various forms of art ( plastic art, music and poetry ) by means of the Hegelian antitheses -- form and content, objective and subjective, inner conflict and reconciliation.

More specifically, he goes on to offer his definition of “ ideograph ”: “ an ideograph is an ordinary-language term found in political discourse.

A more general definition of window functions does not require them to be identically zero outside an interval, as long as the product of the window multiplied by its argument is square integrable, that is, that the function goes sufficiently rapidly toward zero.

The current definition of topos goes back to William Lawvere and Myles Tierney.

A recursive definition of a hereditarily finite set goes as follows:

An example of a quasistatic process that is not reversible is a compression against a system with a piston subject to friction — Although the system is always in thermal equilibrium, the friction ensures the generation of dissipative entropy, which directly goes against the definition of reversible.

A dynamically typed language such as Smalltalk can be seen as a strongly typed language with a very permissive type system where any syntactically correct program is well-typed ; as long as its dynamic semantics ensures that no such program ever " goes wrong " in an appropriate sense, it satisfies the definition above and can be called type-safe.

The Wi-Fi Alliance definition of interoperability goes well beyond the ability to work in a Wi-Fi network.

The User-User specification goes beyond the definition of specific server object classes to define classes local to the client, as well as some of the interaction with other parts of the system.

Carl Jung's definition of abstraction broadened its scope beyond the thinking process to include exactly four mutually exclusive, different complementary psychological functions: sensation, intuition, feeling, and thinking.

For example, one definition of bandwidth could be the range of frequencies beyond which the frequency function is zero.

Alfred Marshall provides a still widely-cited definition in his textbook Principles of Economics ( 1890 ) that extends analysis beyond wealth and from the societal to the microeconomic level:

In a narrower sense, the notion of " francophone " reaches beyond the dictionary definition of " French language speaker ".

In the essay What is it like to be a bat ?, Thomas Nagel famously argued that explaining subjective experience — the " what it is like " to be something — is currently beyond the reach of scientific inquiry, because scientific understanding by definition requires an objective perspective, which, according to Nagel, is diametrically opposed to the subjective first-person point of view.

According to Margaret Bent ( 1997 ), " a piece of music in several parts with words " is as precise a definition of the motet as will serve from the 13th to the late 16th century and beyond.

Research in NLP evaluation has received considerable attention, because the definition of proper evaluation criteria is one way to specify precisely an NLP problem, going thus beyond the vagueness of tasks defined only as language understanding or language generation.

A popular alternative definition of chronic pain, involving no arbitrarily fixed durations is " pain that extends beyond the expected period of healing.

The United Nations Global Compact Cities Programme has defined sustainable political development is a way that broadens the usual definition beyond states and governance.

That particular way of definition of entropy is largely beyond the scope of the present article, but here it may be said that it is entirely derived from the concepts of classical thermodynamics ; in particular, neither flow rates nor changes over time are admitted into the definition of the entropy of the small local region.

For generalized or extended thermodynamics, the definition of the quantity known as the entropy of a small local region is in terms beyond those of classical thermodynamics ; in particular, flow rates are admitted into the definition of the entropy of a small local region.

State law regulates most aspects of electoral law, including primaries, the eligibility of voters ( beyond the basic constitutional definition ), the running of each state's electoral college, and the running of state and local elections.

The 1980s saw the rise of more general-purpose wearable computers that fit the modern definition of " computer " by going beyond task-specific hardware to more general-purpose ( e. g. reprogrammable by the user ) devices.

They assert that DID cannot be accurately diagnosed because of vague and unclear diagnostic criteria in the DSM and undefined concepts such as " personality state " and " identities ", and question the evidence for childhood abuse beyond self-reports, the lack of definition of what would indicate a threshold of abuse sufficient to induce DID and the extremely small number of cases of children diagnosed with DID despite an average age of appearance of the first alter of three years.

Biosafety in agriculture, chemistry, medicine, exobiology and beyond will likely require application of the precautionary principle, and a new definition focused on the biological nature of the threatened organism rather than the nature of the threat.

It has been used by different groups and thinkers for different purposes, with different and sometimes opposing definitions, and the precise definition of this principle, beyond having something to do with normative rules of some sort or another, is controversial.

The most commonly used definition of outer space is everything beyond the Kármán line, which is above the Earth's surface.