Order-independent transparency (OIT) is a class of techniques in rasterisational computer graphics for rendering transparency in a 3D scene, which do not require rendering geometry in sorted order for alpha compositing. == Description == Commonly, 3D geometry with transparency is rendered by blending (using alpha compositing) all surfaces into a single buffer (think of this as a canvas). Each surface occludes existing color and adds some of its own color depending on its alpha value, a ratio of light transmittance. The order in which surfaces are blended affects the total occlusion or visibility of each surface. For a correct result, surfaces must be blended from farthest to nearest or nearest to farthest, depending on the alpha compositing operation, over or under. Ordering may be achieved by rendering the geometry in sorted order, for example sorting triangles by depth, but can take a significant amount of time, not always produce a solution (in the case of intersecting or circularly overlapping geometry) and the implementation is complex. Instead, order-independent transparency sorts geometry per-pixel, after rasterisation. For exact results this requires storing all fragments before sorting and compositing. == History == The A-buffer is a computer graphics technique introduced in 1984 which stores per-pixel lists of fragment data (including micro-polygon information) in a software rasteriser, REYES, originally designed for anti-aliasing but also supporting transparency. More recently, depth peeling in 2001 described a hardware accelerated OIT technique. With limitations in graphics hardware the scene's geometry had to be rendered many times. A number of techniques have followed, to improve on the performance of depth peeling, still with the many-pass rendering limitation. For example, Dual Depth Peeling (2008). In 2009, two significant features were introduced in GPU hardware/drivers/Graphics APIs that allowed capturing and storing fragment data in a single rendering pass of the scene, something not previously possible. These are, the ability to write to arbitrary GPU memory from shaders and atomic operations. With these features a new class of OIT techniques became possible that do not require many rendering passes of the scene's geometry. The first was storing the fragment data in a 3D array, where fragments are stored along the z dimension for each pixel x/y. In practice, most of the 3D array is unused or overflows, as a scene's depth complexity is typically uneven. To avoid overflow the 3D array requires large amounts of memory, which in many cases is impractical. Two approaches to reducing this memory overhead exist. Packing the 3D array with a prefix sum scan, or linearizing, removed the unused memory issue but requires an additional depth complexity computation rendering pass of the geometry. The "Sparsity-aware" S-Buffer, Dynamic Fragment Buffer, "deque" D-Buffer, Linearized Layered Fragment Buffer all pack fragment data with a prefix sum scan and are demonstrated with OIT. Storing fragments in per-pixel linked lists provides tight packing of this data and in late 2011, driver improvements reduced the atomic operation contention overhead making the technique very competitive. == Exact OIT == Exact, as opposed to approximate, OIT accurately computes the final color, for which all fragments must be sorted. For high depth complexity scenes, sorting becomes the bottleneck. One issue with the sorting stage is local memory limited occupancy, in this case a SIMT attribute relating to the throughput and operation latency hiding of GPUs. Backwards memory allocation (BMA) groups pixels by their depth complexity and sorts them in batches to improve the occupancy and hence performance of low depth complexity pixels in the context of a potentially high depth complexity scene. Up to a 3× overall OIT performance increase is reported. Sorting is typically performed in a local array, however performance can be improved further by making use of the GPU's memory hierarchy and sorting in registers, similarly to an external merge sort, especially in conjunction with BMA. == Approximate OIT == Approximate OIT techniques relax the constraint of exact rendering to provide faster results. Higher performance can be gained from not having to store all fragments or only partially sorting the geometry. A number of techniques also compress, or reduce, the fragment data. These include: Stochastic Transparency: draw in a higher resolution in full opacity but discard some fragments. Downsampling will then yield transparency. Adaptive Transparency, a two-pass technique where the first constructs a visibility function which compresses on the fly (this compression avoids having to fully sort the fragments) and the second uses this data to composite unordered fragments. Intel's pixel synchronization avoids the need to store all fragments, removing the unbounded memory requirement of many other OIT techniques. Weighted Blended Order-Independent Transparency replaced the over operator with a commutative approximation. Feeding depth information into the weight produces visually-acceptable occlusion. == OIT in Hardware == The Sega Dreamcast games console included hardware support for automatic OIT.
ElabFTW
eLabFTW is a web application written by Nicolas Carpi in PHP which can be used to create personal and common logbooks. It has been developed at the Curie Institute originally. Besides there, it is used on universities around the world eLabFTW is licensed under the GNU Affero General Public License as free software. It is translated into seven languages. == Description == eLabFTW is a free and open-source lab book. It is written in PHP and uses a MySQL database. Docker containers are also available. Among the various features are Secure. Entries and transmission are encrypted Timestamps. RFC 3161 compliant timestamping of experiments. Inventory management. Apart from experience logs, it also can manage the inventory Import and export. Entries can be imported and exported == Platforms == eLabFTW is a PHP package with Mysql database. Therefore, it can be executed on most servers. Furthermore, the docker containers allow to run it almost everywhere. == Usage == eLabFTW is used by various universities, like University of Alberta, Berkeley University, Hanover Medical School, Cardiff University and UMC Utrecht
Community cloud
A community cloud in computing is a collaborative effort in which infrastructure is shared between several organizations from a specific community with common concerns (security, compliance, jurisdiction, etc.), whether managed internally or by a third party and hosted internally or externally. This is controlled and used by a group of organizations that have shared interests. The costs are spread over fewer users than a public cloud (but more than a private cloud), so only some of the cost savings potential of cloud computing are realized. The community cloud is provisioned for use by a group of consumers from different organizations who share the same concerns (e.g., application, security, policy, and efficiency demands).
Web application
A web application (or web app) is application software that is created with web technologies and runs via a web browser. Web applications emerged during the late 1990s and allowed for the server to dynamically build a response to the request, in contrast to static web pages. Web applications are commonly distributed via a web server. There are several different tier systems that web applications use to communicate between the web browsers, the client interface, and server data. Each system has its own uses as they function in different ways. However, there are many security risks that developers must be aware of during development; proper measures to protect user data are vital. Web applications are often constructed with the use of a web application framework. Single-page applications (SPAs) and progressive web apps (PWAs) are two architectural approaches to creating web applications that provide a user experience similar to native apps, including features such as smooth navigation, offline support, and faster interactions. Web applications are often fully hosted on remote cloud services, can require a constant connection to them, and can replace conventional desktop applications for operating systems such as Microsoft Windows, thus facilitating the operation of software as a service as it grants the developer the power to tightly control billing based on use of the remote services as well as vendor lock-in by hosting data remotely. Modern browsers such as Chrome offer sandboxing for every browser tab which improves security and restricts access to local resources. No software installation is required as the app runs within the browser which reduces the need for managing software installations. With the use of remote cloud services, customers do not need to manage servers as that can be left to the developer and the cloud service and can use the software with a relatively low power, low-resource PC such as a thin client. The source code of the application can stay the same across operating systems and devices of users with the use of responsive web design, since it only needs to be compatible with web browsers which adhere to web standards, making the code highly portable and saving on development time. Numerous JavaScript frameworks and CSS frameworks facilitate development. == History == The concept of a "web application" was first introduced in the Java language in the Servlet Specification version 2.2, which was released in 1999. At that time, both JavaScript and XML had already been developed, but the XMLHttpRequest object had only been recently introduced on Internet Explorer 5 as an ActiveX object. Beginning around the early 2000s, applications such as "Myspace (2003), Gmail (2004), Digg (2004), [and] Google Maps (2005)," started to make their client sides more and more interactive. A web page script is able to contact the server for storing/retrieving data without downloading an entire web page. The practice became known as Ajax in 2005. Eventually this was replaced by web APIs using JSON, accessed via JavaScript asynchronously on the client side. In earlier computing models like client-server, the processing load for the application was shared between code on the server and code installed on each client locally. In other words, an application had its own pre-compiled client program which served as its user interface and had to be separately installed on each user's personal computer. An upgrade to the server-side code of the application would typically also require an upgrade to the client-side code installed on each user workstation, adding to the support cost and decreasing productivity. Additionally, both the client and server components of the application were bound tightly to a particular computer architecture and operating system, which made porting them to other systems prohibitively expensive for all but the largest applications. Later, in 1995, Netscape introduced the client-side scripting language called JavaScript, which allowed programmers to add dynamic elements to the user interface that ran on the client side. Essentially, instead of sending data to the server in order to generate an entire web page, the embedded scripts of the downloaded page can perform various tasks such as input validation or showing/hiding parts of the page. "Progressive web apps", the term coined by designer Frances Berriman and Google Chrome engineer Alex Russell in 2015, refers to apps taking advantage of new features supported by modern browsers, which initially run inside a web browser tab but later can run completely offline and can be launched without entering the app URL in the browser. == Structure == Traditional PC applications are typically single-tiered, residing solely on the client machine. In contrast, web applications inherently facilitate a multi-tiered architecture. Though many variations are possible, the most common structure is the three-tiered application. In its most common form, the three tiers are called presentation, application and storage. The first tier, presentation, refers to a web browser itself. The second tier refers to any engine using dynamic web content technology (such as ASP, CGI, ColdFusion, Dart, JSP/Java, Node.js, PHP, Python or Ruby on Rails). The third tier refers to a database that stores data and determines the structure of a user interface. Essentially, when using the three-tiered system, the web browser sends requests to the engine, which then services them by making queries and updates against the database and generates a user interface. The 3-tier solution may fall short when dealing with more complex applications, and may need to be replaced with the n-tiered approach; the greatest benefit of which is how business logic (which resides on the application tier) is broken down into a more fine-grained model. Another benefit would be to add an integration tier, which separates the data tier and provides an easy-to-use interface to access the data. For example, the client data would be accessed by calling a "list_clients()" function instead of making an SQL query directly against the client table on the database. This allows the underlying database to be replaced without making any change to the other tiers. There are some who view a web application as a two-tier architecture. This can be a "smart" client that performs all the work and queries a "dumb" server, or a "dumb" client that relies on a "smart" server. The client would handle the presentation tier, the server would have the database (storage tier), and the business logic (application tier) would be on one of them or on both. While this increases the scalability of the applications and separates the display and the database, it still does not allow for true specialization of layers, so most applications will outgrow this model. == Security == Security breaches on these kinds of applications are a major concern because it can involve both enterprise information and private customer data. Protecting these assets is an important part of any web application, and there are some key operational areas that must be included in the development process. This includes processes for authentication, authorization, asset handling, input, and logging and auditing. Building security into the applications from the beginning is sometimes more effective and less disruptive in the long run. == Development == Writing web applications is simplified with the use of web application frameworks. These frameworks facilitate rapid application development by allowing a development team to focus on the parts of their application which are unique to their goals without having to resolve common development issues such as user management. In addition, there is potential for the development of applications on Internet operating systems, although currently there are not many viable platforms that fit this model.
Focus recovery based on the linear canonical transform
For digital image processing, the Focus recovery from a defocused image is an ill-posed problem since it loses the component of high frequency. Most of the methods for focus recovery are based on depth estimation theory. The Linear canonical transform (LCT) gives a scalable kernel to fit many well-known optical effects. Using LCTs to approximate an optical system for imaging and inverting this system, theoretically permits recovery of a defocused image. == Depth of field and perceptual focus == In photography, depth of field (DOF) means an effective focal length. It is usually used for stressing an object and deemphasizing the background (and/or the foreground). The important measure related to DOF is the lens aperture. Decreasing the diameter of aperture increases focus and lowers resolution and vice versa. == The Huygens–Fresnel principle and DOF == The Huygens–Fresnel principle describes diffraction of wave propagation between two fields. It belongs to Fourier optics rather than geometric optics. The disturbance of diffraction depends on two circumstance parameters, the size of aperture and the interfiled distance. Consider a source field and a destination field, field 1 and field 0, respectively. P1(x1,y1) is the position in the source field, P0(x0,y0) is the position in the destination field. The Huygens–Fresnel principle gives the diffraction formula for two fields U(x0,y0), U(x1,y1) as following: U ( x 0 , y 0 ) = 1 j λ ∫ ∫ U ( x 1 , y 1 ) e j k r 01 r 01 cos θ d x 1 d y 1 {\displaystyle \mathbf {U} (x_{0},y_{0})={\frac {1}{j\lambda }}\int \!\int \mathbf {U} (x_{1},y_{1}){\frac {e^{jkr_{01}}}{r_{01}}}\cos \theta dx_{1}dy_{1}} where θ denotes the angle between r 01 {\displaystyle r_{01}} and z {\displaystyle z} . Replace cos θ by r 01 z {\displaystyle {\frac {r_{01}}{z}}} and r 01 {\displaystyle r_{01}} by [ ( x 0 − x 1 ) 2 + ( y 0 − y 1 ) 2 + z 2 ] 1 / 2 {\displaystyle [(x_{0}-x_{1})^{2}+(y_{0}-y_{1})^{2}+z^{2}]^{1/2}} we get U ( x 0 , y 0 ) = 1 j λ z ∫ ∫ U ( x 1 , y 1 ) exp ( j k z [ 1 + ( x 0 − x 1 z ) 2 + ( y 0 − y 1 z ) 2 ] 1 / 2 ) 1 + ( x 0 − x 1 z ) 2 + ( y 0 − y 1 z ) 2 d x 1 d y 1 {\displaystyle \mathbf {U} (x_{0},y_{0})={\frac {1}{j\lambda z}}\int \!\int \mathbf {U} (x_{1},y_{1}){\frac {\exp(jkz[1+({\frac {x_{0}-x_{1}}{z}})^{2}+({\frac {y_{0}-y_{1}}{z}})^{2}]^{1/2})}{1+({\frac {x_{0}-x_{1}}{z}})^{2}+({\frac {y_{0}-y_{1}}{z}})^{2}}}dx_{1}dy_{1}} The further distance z or the smaller aperture (x1,y1) causes a greater diffraction. A larger DOF can lead to a more effective focused wave distribution. This seems to be a conflict. Here are the notations: Diffraction In a real imaging environment, the depths of objects comparing to the aperture are usually not enough to lead to serious diffraction. However, a long enough depth of the object can truly blurs the image. Effective Focus Small aperture, small blurring radius, few wave information. Loses details in comparing to a large aperture. In conclusion, diffraction explains a micro behavior whereas DOF shows a macro behavior. Both of them are related to aperture size. == Linear canonical transform == As the meaning of "canonical", the linear canonical transform (LCT) is a scalable transform that connects to many important kernels such as the Fresnel transform, Fraunhofer transform and the fractional Fourier transform. It can be easily controlled by its four parameters, a, b, c, d (3 degrees of freedom). The definition: L M ( f ( u ) ) = ∫ L M ( u , u ′ ) f ( u ′ ) d u ′ {\displaystyle L_{M}(f(u))=\int L_{M}(u,u')f(u')du'} where L M ( u , u ′ ) = { 1 b e − j π / 4 e [ j π ( d b u 2 ) − 2 1 b u u ′ + a b u ′ 2 ] , if b ≠ 0 d e j 2 c d u 2 δ ( u ′ − d u ) , if b = 0 {\displaystyle L_{M}(u,u')={\begin{cases}{\sqrt {\frac {1}{b}}}e^{-j\pi /4}e^{[j\pi ({\frac {d}{b}}u^{2})-2{\frac {1}{b}}uu'+{\frac {a}{b}}u'^{2}]},&{\mbox{if }}b\neq 0\\{\sqrt {d}}e^{{\frac {j}{2}}cdu^{2}}\delta (u'-du),&{\mbox{if }}b=0\end{cases}}} Consider a general imaging system with object distance z0, focal length of the thin lens f and an imaging distance z1. The effect of the propagation in freespace acts as nearly a chirp convolution, that is, the formula of diffraction. Besides, the effect of the propagation in thin lens acts as a chirp multiplication. The parameters are all simplified as paraxial approximations while meeting the freespace propagation. It does not consider aperture size. From the properties of the LCT, it is possible to obtain those 4 parameters for this optical system as: [ 1 − z 1 f λ z 0 − λ z 0 z 1 f + λ z 1 − 1 λ f 1 − z 0 f ] {\displaystyle {\begin{bmatrix}1-{\frac {z_{1}}{f}}\quad &\lambda z_{0}-{\frac {\lambda z_{0}z_{1}}{f}}+\lambda z_{1}\\-{\frac {1}{\lambda f}}\quad &1-{\frac {z_{0}}{f}}\end{bmatrix}}} Once the values of z1, z0 and f are known, the LCT can simulate any optical system.
Uniform convergence in probability
Uniform convergence in probability is a form of convergence in probability in statistical asymptotic theory and probability theory. It means that, under certain conditions, the empirical frequencies of all events in a certain event-family uniformly converge to their theoretical probabilities. Uniform convergence in probability has applications to statistics as well as machine learning as part of statistical learning theory. Specifically, the Glivenko-Cantelli theorem and the homonymous classes of functions are fundamentally related to uniform convergence. The law of large numbers says that, for each single event A {\displaystyle A} , its empirical frequency in a sequence of independent trials converges (with high probability) to its theoretical probability. In many application however, the need arises to judge simultaneously the probabilities of events of an entire class S {\displaystyle S} from one and the same sample. Moreover, it, is required that the relative frequency of the events converge to the probability uniformly over the entire class of events S {\displaystyle S} . The Uniform Convergence Theorem gives a sufficient condition for this convergence to hold. Roughly, if the event-family is sufficiently simple (its VC dimension is sufficiently small) then uniform convergence holds. == Definitions == For a class of predicates H {\displaystyle H} defined on a set X {\displaystyle X} and a set of samples x = ( x 1 , x 2 , … , x m ) {\displaystyle x=(x_{1},x_{2},\dots ,x_{m})} , where x i ∈ X {\displaystyle x_{i}\in X} , the empirical frequency of h ∈ H {\displaystyle h\in H} on x {\displaystyle x} is Q ^ x ( h ) = 1 m | { i : 1 ≤ i ≤ m , h ( x i ) = 1 } | . {\displaystyle {\widehat {Q}}_{x}(h)={\frac {1}{m}}|\{i:1\leq i\leq m,h(x_{i})=1\}|.} The theoretical probability of h ∈ H {\displaystyle h\in H} is defined as Q P ( h ) = P { y ∈ X : h ( y ) = 1 } . {\displaystyle Q_{P}(h)=P\{y\in X:h(y)=1\}.} The Uniform Convergence Theorem states, roughly, that if H {\displaystyle H} is "simple" and we draw samples independently (with replacement) from X {\displaystyle X} according to any distribution P {\displaystyle P} , then with high probability, the empirical frequency will be close to its expected value, which is the theoretical probability. Here "simple" means that the Vapnik–Chervonenkis dimension of the class H {\displaystyle H} is small relative to the size of the sample. In other words, a sufficiently simple collection of functions behaves roughly the same on a small random sample as it does on the distribution as a whole. The Uniform Convergence Theorem was first proved by Vapnik and Chervonenkis using the concept of growth function. == Uniform Convergence Theorem == The statement of the Uniform Convergence Theorem is as follows: If H {\displaystyle H} is a set of { 0 , 1 } {\displaystyle \{0,1\}} -valued functions defined on a set X {\displaystyle X} and P {\displaystyle P} is a probability distribution on X {\displaystyle X} then for ε > 0 {\displaystyle \varepsilon >0} and m {\displaystyle m} a positive integer, we have: P m { | Q P ( h ) − Q x ^ ( h ) | ≥ ε for some h ∈ H } ≤ 4 Π H ( 2 m ) e − ε 2 m / 8 . {\displaystyle P^{m}\{|Q_{P}(h)-{\widehat {Q_{x}}}(h)|\geq \varepsilon {\text{ for some }}h\in H\}\leq 4\Pi _{H}(2m)e^{-\varepsilon ^{2}m/8}.} In the above, for any x ∈ X m , {\displaystyle x\in X^{m},} Q P ( h ) = P { ( y ∈ X : h ( y ) = 1 } , {\displaystyle Q_{P}(h)=P\{(y\in X:h(y)=1\},} Q ^ x ( h ) = 1 m | { i : 1 ≤ i ≤ m , h ( x i ) = 1 } | {\displaystyle {\widehat {Q}}_{x}(h)={\frac {1}{m}}|\{i:1\leq i\leq m,h(x_{i})=1\}|} and | x | = m . {\displaystyle |x|=m.} P m {\displaystyle P^{m}} indicates that the probability is taken over x {\displaystyle x} consisting of m {\displaystyle m} i.i.d. draws from the distribution P . {\displaystyle P.} Finally, the growth function Π H {\displaystyle \Pi _{H}} is defined in the following way, for any { 0 , 1 } {\displaystyle \{0,1\}} -valued functions H {\displaystyle H} over X {\displaystyle X} and for any natural number m {\displaystyle m} : Π H ( m ) = max | { h ∩ D : D ⊆ X , | D | = m , h ∈ H } | . {\displaystyle \Pi _{H}(m)=\max |\{h\cap D:D\subseteq X,|D|=m,h\in H\}|.} From the point of view of Learning Theory one can consider H {\displaystyle H} to be the Concept/Hypothesis class defined over the instance set X {\displaystyle X} . Crucially, the Sauer–Shelah lemma implies that Π H ( m ) ≤ m d {\displaystyle \Pi _{H}(m)\leq m^{d}} , where d {\displaystyle d} is the VC dimension of H {\displaystyle H} . == Proof of the Uniform Convergence Theorem == and are the sources of the proof below. Before we get into the details of the proof of the Uniform Convergence Theorem we will present a high level overview of the proof. Symmetrization: We transform the problem of analyzing | Q P ( h ) − Q ^ x ( h ) | ≥ ε {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{x}(h)|\geq \varepsilon } into the problem of analyzing | Q ^ r ( h ) − Q ^ s ( h ) | ≥ ε / 2 {\displaystyle |{\widehat {Q}}_{r}(h)-{\widehat {Q}}_{s}(h)|\geq \varepsilon /2} , where r {\displaystyle r} and s {\displaystyle s} are i.i.d samples of size m {\displaystyle m} drawn according to the distribution P {\displaystyle P} . One can view r {\displaystyle r} as the original randomly drawn sample of length m {\displaystyle m} , while s {\displaystyle s} may be thought as the testing sample which is used to estimate Q P ( h ) {\displaystyle Q_{P}(h)} . Permutation: Since r {\displaystyle r} and s {\displaystyle s} are picked identically and independently, so swapping elements between them will not change the probability distribution on r {\displaystyle r} and s {\displaystyle s} . So, we will try to bound the probability of | Q ^ r ( h ) − Q ^ s ( h ) | ≥ ε / 2 {\displaystyle |{\widehat {Q}}_{r}(h)-{\widehat {Q}}_{s}(h)|\geq \varepsilon /2} for some h ∈ H {\displaystyle h\in H} by considering the effect of a specific collection of permutations of the joint sample x = r | | s {\displaystyle x=r||s} . Specifically, we consider permutations σ ( x ) {\displaystyle \sigma (x)} which swap x i {\displaystyle x_{i}} and x m + i {\displaystyle x_{m+i}} in some subset of 1 , 2 , . . . , m {\displaystyle {1,2,...,m}} . The symbol r | | s {\displaystyle r||s} means the concatenation of r {\displaystyle r} and s {\displaystyle s} . Reduction to a finite class: We can now restrict the function class H {\displaystyle H} to a fixed joint sample and hence, if H {\displaystyle H} has finite VC Dimension, it reduces to the problem to one involving a finite function class. We present the technical details of the proof. It should be stressed that this proof glosses over details like the measurability of the events V {\displaystyle V} and R {\displaystyle R} ; measurability is granted in the case of H {\displaystyle H} being finite or countable, but this is not normally the case in standard applications of the theorem (e.g. for statistical learning theory or to prove the Glivenko-Cantelli theorem). To get measurability, one needs to use a notion of separability of the underlying space, possibly related to H {\displaystyle H} . === Symmetrization === Lemma: Let V = { x ∈ X m : | Q P ( h ) − Q ^ x ( h ) | ≥ ε for some h ∈ H } {\displaystyle V=\{x\in X^{m}:|Q_{P}(h)-{\widehat {Q}}_{x}(h)|\geq \varepsilon {\text{ for some }}h\in H\}} and R = { ( r , s ) ∈ X m × X m : | Q r ^ ( h ) − Q ^ s ( h ) | ≥ ε / 2 for some h ∈ H } . {\displaystyle R=\{(r,s)\in X^{m}\times X^{m}:|{\widehat {Q_{r}}}(h)-{\widehat {Q}}_{s}(h)|\geq \varepsilon /2{\text{ for some }}h\in H\}.} Then for m ≥ 2 ε 2 {\displaystyle m\geq {\frac {2}{\varepsilon ^{2}}}} , P m ( V ) ≤ 2 P 2 m ( R ) {\displaystyle P^{m}(V)\leq 2P^{2m}(R)} . Proof: By the triangle inequality, if | Q P ( h ) − Q ^ r ( h ) | ≥ ε {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{r}(h)|\geq \varepsilon } and | Q P ( h ) − Q ^ s ( h ) | ≤ ε / 2 {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{s}(h)|\leq \varepsilon /2} then | Q ^ r ( h ) − Q ^ s ( h ) | ≥ ε / 2 {\displaystyle |{\widehat {Q}}_{r}(h)-{\widehat {Q}}_{s}(h)|\geq \varepsilon /2} . Therefore, P 2 m ( R ) ≥ P 2 m { ∃ h ∈ H , | Q P ( h ) − Q ^ r ( h ) | ≥ ε and | Q P ( h ) − Q ^ s ( h ) | ≤ ε / 2 } = ∫ V P m { s : ∃ h ∈ H , | Q P ( h ) − Q ^ r ( h ) | ≥ ε and | Q P ( h ) − Q ^ s ( h ) | ≤ ε / 2 } d P m ( r ) = A {\displaystyle {\begin{aligned}&P^{2m}(R)\\[5pt]\geq {}&P^{2m}\{\exists h\in H,|Q_{P}(h)-{\widehat {Q}}_{r}(h)|\geq \varepsilon {\text{ and }}|Q_{P}(h)-{\widehat {Q}}_{s}(h)|\leq \varepsilon /2\}\\[5pt]={}&\int _{V}P^{m}\{s:\exists h\in H,|Q_{P}(h)-{\widehat {Q}}_{r}(h)|\geq \varepsilon {\text{ and }}|Q_{P}(h)-{\widehat {Q}}_{s}(h)|\leq \varepsilon /2\}\,dP^{m}(r)\\[5pt]={}&A\end{aligned}}} since r {\displaystyle r} and s {\displaystyle s} are independent. Now for r ∈ V {\displaystyle r\in V} fix an h ∈ H {\displaystyle h\in H} such that | Q P ( h ) − Q ^ r ( h ) | ≥ ε {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{r}(h)|\geq \varepsilon } . For this h {\displaystyle h} , we shall
KeyBase
KeyBase is a database and web application for managing and deploying interactive taxonomic keys for plants and animals developed by the Royal Botanic Gardens Victoria. KeyBase provides a medium where pathway keys which were traditionally developed for print and other classical types of media, can be used more effectively in the internet environment. The platform uses a concept called "keys" which can be easily linked together, joined with other keys, or merged into larger other seamless keys groups, with each still available to be browsed independently. Keys in the KeyBase database can be filtered and displayed in a variety of ways, filters, and formats.