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The '''[[Dirac delta function]]''' is a function introduced in 1930 by Paul Adrien Maurice Dirac in his seminal book on quantum mechanics. A physical model that visualizes a delta function is a mass distribution of finite total mass ''M''—the integral over the mass distribution.  When the distribution becomes smaller and smaller,  while ''M'' is constant, the mass distribution shrinks to a ''point mass'', which by definition has zero extent and yet has a finite-valued integral equal to total mass ''M''. In the limit of a point mass the distribution becomes a Dirac delta function.
{{:{{FeaturedArticleTitle}}}}
 
<small>
Heuristically, the Dirac delta function can be seen as an extension of the Kronecker delta from integral indices (elements of <font style="vertical-align: 13%"> <math>\mathbb{Z}</math></font>) to real indices (elements of <font style="vertical-align: 13%"><math>\mathbb{R}</math></font>). Note that the Kronecker delta acts as a "filter" in a summation:
==Footnotes==
:<math>
{{reflist|2}}
\sum_{i=m}^n \; f_i\; \delta_{ia} =
</small>
\begin{cases}
f_a & \quad\hbox{if}\quad  a\in[m,n] \sub\mathbb{Z}  \\
0  & \quad \hbox{if}\quad a \notin [m,n].
\end{cases}
</math>
 
In analogy, the Dirac delta function &delta;(''x''&minus;''a'')  is defined by (replace ''i'' by ''x'' and the summation over ''i'' by an integration over ''x''),
:<math>
\int_{a_0}^{a_1} f(x)  \delta(x-a) \mathrm{d}x =
\begin{cases}
f(a) & \quad\hbox{if}\quad  a\in[a_0,a_1] \sub\mathbb{R},  \\
0  & \quad \hbox{if}\quad a \notin [a_0,a_1].
\end{cases}
</math>
 
The Dirac delta function is ''not'' an ordinary well-behaved map  <font style="vertical-align: 12%"><math>\mathbb{R} \rightarrow \mathbb{R}</math></font>, but a distribution, also known as an ''improper'' or ''generalized function''. Physicists express its special character by stating that the Dirac delta function makes only sense as a factor in an integrand ("under the integral"). Mathematicians say that the delta function is a linear functional on a space of test functions.
 
==Properties==
Most commonly one takes the lower and the upper bound in the definition of the delta function equal to <math>-\infty</math> and <math> \infty</math>, respectively. From here on this will be done.
:<math>
\begin{align}
\int_{-\infty}^{\infty} \delta(x)\mathrm{d}x &= 1, \\
\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{ikx} \mathrm{d}k &= \delta(x) \\
\delta(x-a) &= \delta(a-x), \\
(x-a)\delta(x-a) &= 0, \\
\delta(ax) &= |a|^{-1} \delta(x) \quad (a \ne 0), \\
f(x) \delta(x-a) &= f(a) \delta(x-a), \\
\int_{-\infty}^{\infty} \delta(x-y)\delta(y-a)\mathrm{d}y &= \delta(x-a)
\end{align}
</math>
The physicist's proof of these properties proceeds by making proper substitutions into the integral and using the ordinary rules of integral calculus. The delta function as a Fourier transform of the unit function ''f''(''x'') = 1 (the second property) will be proved below.
The last property is the analogy of the multiplication of two identity matrices,
:<math>
\sum_{j=1}^n \;\delta_{ij}\;\delta_{jk} = \delta_{ik}, \quad i,k=1,\ldots, n.
</math>
''[[Dirac delta function|.... (read more)]]''

Latest revision as of 10:19, 11 September 2020

Categories of smart home devices shown on Amazon's website in April 2023.

The phrase smart home refers to home automation devices that have internet access. Home automation, a broader category, includes any device that can be monitored or controlled via wireless radio signals, not just those having internet access. Whether the device is powered by the electrical grid or by battery, if it uses the home Wi-Fi network and if an internet logon needs to be created to use it, then it is smart home technology.

Collectively, all the smart home devices on every home's Wi-Fi network helps to make up what is called the Internet of Things (IoT), a huge sea of sensors and control devices across the world that are capable of being accessed from afar via the internet. One of the key reasons such devices need internet access is so that the manufacturer can periodically download updated firmware to the device to keep it up-to-date. However, being available via the internet also means that such devices are, potentially, available for spying or hacking. Today, homes may contain dozens or even hundreds of such devices, and consumers may enjoy their benefits while knowing little about how they work, or even realizing that they are present.

Not all home automation is "smart"

Many remotely controllable devices do not require internet access. They may instead have physical control devices that use either RF (“Radio Frequency”) or IR (“Infrared”) beams, two different kinds of energy used in remote controls to communicate commands. Non-"smart" home automation may also present security risks, because the control signals can be hijacked by bad actors with the right signaling equipment. Garage door openers are of particular note in this regard. Modern automobiles, in fact, are full of automation similar to home automation, and cars are hackable by bad actors in a number of ways. See Wikipedia's Automotive hacking article for more information.

Incompatibility hassles

At present, consumers must make sure that the smart device they wish to use is specified to be compatible whichever phone/tablet operating system they use (Apple vs. Android). Since smart home products emerged in the absence of any standard, a morass of competing methods for networking, control and monitoring now exist. For some products, consumers may need to buy an expensive hub, or bridge, a device that is specific to one vendor. Products made by different manufacturers but performing the same function are typically not interoperable. Consumers often need to open a different app on their smartphone or tablet in order to control devices by each manufacturer. This may make it too expensive and awkward to try out competing devices, leaving consumers stuck with the product they bought originally or else having to add yet more apps to their phones.

Security concerns

Security for smart home products has been uneven and sometimes seriously inadequate. Smart thermostats which can monitor whether a home's occupants are present or not, entry-way locks, robotic vacuums that work with a map of the house, and other smart home devices can present very real dangers if hackers can access their data.

Matter, an emerging standard

Matter is emerging standard in 2023 intended to increase security, reliability and inter-operability of smart-home devices. About ten years ago, industry consortiums formed to work on standards for smart home device communications, and their underlying wireless communications, which would make it possible for products from all vendors to work together seamlessly and provide fast performance, privacy, and security and would work even if there is not connection to the outside internet (i.e., no connection to "the cloud" or to servers).

Footnotes

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