T-norm

In mathematics, a t-norm (also T-norm or, unabbreviated, triangular norm) is a kind of binary operation used in the framework of probabilistic metric spaces and in multi-valued logic, specifically in fuzzy logic. A t-norm generalizes intersection in a lattice and conjunction in logic. The name triangular norm refers to the fact that in the framework of probabilistic metric spaces t-norms are used to generalize the triangle inequality of ordinary metric spaces. == Definition == A t-norm is a function T: [0, 1] × [0, 1] → [0, 1] that satisfies the following properties: Commutativity: T(a, b) = T(b, a) Monotonicity: T(a, b) ≤ T(c, d) if a ≤ c and b ≤ d Associativity: T(a, T(b, c)) = T(T(a, b), c) The number 1 acts as identity element: T(a, 1) = a Since a t-norm is a binary algebraic operation on the interval [0, 1], infix algebraic notation is also common, with the t-norm usually denoted by ∗ {\displaystyle } . The defining conditions of the t-norm are exactly those of a partially ordered abelian monoid on the real unit interval [0, 1]. (Cf. ordered group.) The monoidal operation of any partially ordered abelian monoid L is therefore by some authors called a triangular norm on L. === Classification of t-norms === A t-norm is called continuous if it is continuous as a function, in the usual interval topology on [0, 1]2. (Similarly for left- and right-continuity.) A t-norm is called strict if it is continuous and strictly monotone. A t-norm is called nilpotent if it is continuous and each x in the open interval (0, 1) is nilpotent, that is, there is a natural number n such that x ∗ {\displaystyle } ... ∗ {\displaystyle } x (n times) equals 0. A t-norm ∗ {\displaystyle } is called Archimedean if it has the Archimedean property, that is, if for each x, y in the open interval (0, 1) there is a natural number n such that x ∗ {\displaystyle } ... ∗ {\displaystyle } x (n times) is less than or equal to y. The usual partial ordering of t-norms is pointwise, that is, T1 ≤ T2 if T1(a, b) ≤ T2(a, b) for all a, b in [0, 1]. As functions, pointwise larger t-norms are sometimes called stronger than those pointwise smaller. In the semantics of t-norm fuzzy logics, however, the larger a t-norm, the weaker (in terms of logical strength) conjunction it represents. == Prominent examples == Minimum t-norm ⊤ m i n ( a , b ) = min { a , b } , {\displaystyle \top _{\mathrm {min} }(a,b)=\min\{a,b\},} also called the Gödel t-norm, as it is the standard semantics for conjunction in Gödel fuzzy logic. Besides that, it occurs in most t-norm based fuzzy logics as the standard semantics for weak conjunction. It is the pointwise largest t-norm (see the properties of t-norms below). Product t-norm ⊤ p r o d ( a , b ) = a ⋅ b {\displaystyle \top _{\mathrm {prod} }(a,b)=a\cdot b} (the ordinary product of real numbers). Besides other uses, the product t-norm is the standard semantics for strong conjunction in product fuzzy logic. It is a strict Archimedean t-norm. Łukasiewicz t-norm ⊤ L u k ( a , b ) = max { 0 , a + b − 1 } . {\displaystyle \top _{\mathrm {Luk} }(a,b)=\max\{0,a+b-1\}.} The name comes from the fact that the t-norm is the standard semantics for strong conjunction in Łukasiewicz fuzzy logic. It is a nilpotent Archimedean t-norm, pointwise smaller than the product t-norm. Drastic t-norm ⊤ D ( a , b ) = { b if a = 1 a if b = 1 0 otherwise. {\displaystyle \top _{\mathrm {D} }(a,b)={\begin{cases}b&{\mbox{if }}a=1\\a&{\mbox{if }}b=1\\0&{\mbox{otherwise.}}\end{cases}}} The name reflects the fact that the drastic t-norm is the pointwise smallest t-norm (see the properties of t-norms below). It is a right-continuous Archimedean t-norm. Nilpotent minimum ⊤ n M ( a , b ) = { min ( a , b ) if a + b > 1 0 otherwise {\displaystyle \top _{\mathrm {nM} }(a,b)={\begin{cases}\min(a,b)&{\mbox{if }}a+b>1\\0&{\mbox{otherwise}}\end{cases}}} is a standard example of a t-norm that is left-continuous, but not continuous. Despite its name, the nilpotent minimum is not a nilpotent t-norm. Hamacher product ⊤ H 0 ( a , b ) = { 0 if a = b = 0 a b a + b − a b otherwise {\displaystyle \top _{\mathrm {H} _{0}}(a,b)={\begin{cases}0&{\mbox{if }}a=b=0\\{\frac {ab}{a+b-ab}}&{\mbox{otherwise}}\end{cases}}} is a strict Archimedean t-norm, and an important representative of the parametric classes of Hamacher t-norms and Schweizer–Sklar t-norms. == Properties of t-norms == The drastic t-norm is the pointwise smallest t-norm and the minimum is the pointwise largest t-norm: ⊤ D ( a , b ) ≤ ⊤ ( a , b ) ≤ ⊤ m i n ( a , b ) , {\displaystyle \top _{\mathrm {D} }(a,b)\leq \top (a,b)\leq \mathrm {\top _{min}} (a,b),} for any t-norm ⊤ {\displaystyle \top } and all a, b in [0, 1]. In particular, we have that: ⊤ D ( a , b ) ≤ ⊤ L u k ( a , b ) ≤ ⊤ p r o d ( a , b ) ≤ ⊤ m i n ( a , b ) , {\displaystyle \top _{\mathrm {D} }(a,b)\leq \top _{\mathrm {Luk} }(a,b)\leq \top _{\mathrm {prod} }(a,b)\leq \mathrm {\top _{min}} (a,b),} for all a, b in [0, 1]. For every t-norm T, the number 0 acts as null element: T(a, 0) = 0 for all a in [0, 1]. A t-norm T has zero divisors if and only if it has nilpotent elements; each nilpotent element of T is also a zero divisor of T. The set of all nilpotent elements is an interval [0, a] or [0, a), for some a in [0, 1]. === Properties of continuous t-norms === Although real functions of two variables can be continuous in each variable without being continuous on [0, 1]2, this is not the case with t-norms: a t-norm T is continuous if and only if it is continuous in one variable, i.e., if and only if the functions fy(x) = T(x, y) are continuous for each y in [0, 1]. Analogous theorems hold for left- and right-continuity of a t-norm. A continuous t-norm is Archimedean if and only if 0 and 1 are its only idempotents. A continuous Archimedean t-norm is strict if 0 is its only nilpotent element; otherwise it is nilpotent. By definition, moreover, a continuous Archimedean t-norm T is nilpotent if and only if each x < 1 is a nilpotent element of T. Thus with a continuous Archimedean t-norm T, either all or none of the elements of (0, 1) are nilpotent. If it is the case that all elements in (0, 1) are nilpotent, then the t-norm is isomorphic to the Łukasiewicz t-norm; i.e., there is a strictly increasing function f such that ⊤ ( x , y ) = f − 1 ( ⊤ L u k ( f ( x ) , f ( y ) ) ) . {\displaystyle \top (x,y)=f^{-1}(\top _{\mathrm {Luk} }(f(x),f(y))).} If on the other hand it is the case that there are no nilpotent elements of T, the t-norm is isomorphic to the product t-norm. In other words, all nilpotent t-norms are isomorphic, the Łukasiewicz t-norm being their prototypical representative; and all strict t-norms are isomorphic, with the product t-norm as their prototypical example. The Łukasiewicz t-norm is itself isomorphic to the product t-norm undercut at 0.25, i.e., to the function p(x, y) = max(0.25, x ⋅ y) on [0.25, 1]2. For each continuous t-norm, the set of its idempotents is a closed subset of [0, 1]. Its complement—the set of all elements that are not idempotent—is therefore a union of countably many non-overlapping open intervals. The restriction of the t-norm to any of these intervals (including its endpoints) is Archimedean, and thus isomorphic either to the Łukasiewicz t-norm or the product t-norm. For such x, y that do not fall into the same open interval of non-idempotents, the t-norm evaluates to the minimum of x and y. These conditions actually give a characterization of continuous t-norms, called the Mostert–Shields theorem, since every continuous t-norm can in this way be decomposed, and the described construction always yields a continuous t-norm. The theorem can also be formulated as follows: A t-norm is continuous if and only if it is isomorphic to an ordinal sum of the minimum, Łukasiewicz, and product t-norm. A similar characterization theorem for non-continuous t-norms is not known (not even for left-continuous ones), only some non-exhaustive methods for the construction of t-norms have been found. == Residuum == For any left-continuous t-norm ⊤ {\displaystyle \top } , there is a unique binary operation ⇒ {\displaystyle \Rightarrow } on [0, 1] such that ⊤ ( z , x ) ≤ y {\displaystyle \top (z,x)\leq y} if and only if z ≤ ( x ⇒ y ) {\displaystyle z\leq (x\Rightarrow y)} for all x, y, z in [0, 1]. This operation is called the residuum of the t-norm. In prefix notation, the residuum of a t-norm ⊤ {\displaystyle \top } is often denoted by ⊤ → {\displaystyle {\vec {\top }}} or by the letter R. The interval [0, 1] equipped with a t-norm and its residuum forms a residuated lattice. The relation between a t-norm T and its residuum R is an instance of adjunction (specifically, a Galois connection): the residuum forms a right adjoint R(x, –) to the functor T(–, x) for each x in the lattice [0, 1] taken as a poset category. In the standard semantics of t-norm based fuzzy logics, where conjunction is interpreted by a t-norm, the residuum plays the role of implication (often

NumPy

NumPy (pronounced NUM-py) is a library for the Python programming language, adding support for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays. The predecessor of NumPy, Numeric, was originally created by Jim Hugunin with contributions from several other developers. In 2005, Travis Oliphant created NumPy by incorporating features of the competing Numarray into Numeric, with extensive modifications. NumPy is open-source software and has many contributors. NumPy is fiscally sponsored by NumFOCUS. == History == === matrix-sig === The Python programming language was not originally designed for numerical computing, but attracted the attention of the scientific and engineering community early on. In 1995 the special interest group (SIG) matrix-sig was founded with the aim of defining an array computing package; among its members was Python designer and maintainer Guido van Rossum, who extended Python's syntax (in particular the indexing syntax) to make array computing easier. === Numeric === An implementation of a matrix package was completed by Jim Fulton, then expanded to support multi-dimensional arrays by Jim Hugunin and called Numeric (also variously known as the "Numerical Python extensions" or "NumPy"), with influences from the APL family of languages, Basis, MATLAB, FORTRAN, S and S+, and others. Hugunin, a graduate student at the Massachusetts Institute of Technology (MIT), joined the Corporation for National Research Initiatives (CNRI) in 1997 to work on JPython, leaving Paul Dubois of Lawrence Livermore National Laboratory (LLNL) to take over as maintainer. Other early contributors include David Ascher, Konrad Hinsen and Travis Oliphant. === Numarray === A new package called Numarray was written as a more flexible replacement for Numeric. Like Numeric, it too is now deprecated. Numarray had faster operations for large arrays, but was slower than Numeric on small ones, so for a time both packages were used in parallel for different use cases. The last version of Numeric (v24.2) was released on 11 November 2005, while the last version of numarray (v1.5.2) was released on 24 August 2006. There was a desire to get Numeric into the Python standard library, but Guido van Rossum decided that the code was not maintainable in its state then. === NumPy === In early 2005, NumPy developer Travis Oliphant wanted to unify the community around a single array package and ported Numarray's features to Numeric, releasing the result as NumPy 1.0 in 2006. This new project was part of SciPy. To avoid installing the large SciPy package just to get an array object, this new package was separated and called NumPy. Support for Python 3 was added in 2011 with NumPy version 1.5.0. In 2011, PyPy started development on an implementation of the NumPy API for PyPy. As of 2023, it is not yet fully compatible with NumPy. == Features == NumPy targets the CPython reference implementation of Python, which is a non-optimizing bytecode interpreter. Mathematical algorithms written for this version of Python often run much slower than compiled equivalents due to the absence of compiler optimization. NumPy addresses the slowness problem partly by providing multidimensional arrays and functions and operators that operate efficiently on arrays; using these requires rewriting some code, mostly inner loops, using NumPy. Using NumPy in Python gives functionality comparable to MATLAB since they are both interpreted, and they both allow the user to write fast programs as long as most operations work on arrays or matrices instead of scalars. In comparison, MATLAB boasts a large number of additional toolboxes, notably Simulink, whereas NumPy is intrinsically integrated with Python, a more modern and complete programming language. Moreover, complementary Python packages are available; SciPy is a library that adds more MATLAB-like functionality and Matplotlib is a plotting package that provides MATLAB-like plotting functionality. Although MATLAB can perform sparse matrix operations, NumPy alone cannot perform such operations and requires the use of the scipy.sparse library. Internally, both MATLAB and NumPy rely on BLAS and LAPACK for efficient linear algebra computations. Python bindings of the widely used computer vision library OpenCV utilize NumPy arrays to store and operate on data. Since images with multiple channels are simply represented as three-dimensional arrays, indexing, slicing or masking with other arrays are very efficient ways to access specific pixels of an image. The NumPy array as universal data structure in OpenCV for images, extracted feature points, filter kernels and many more vastly simplifies the programming workflow and debugging. Importantly, many NumPy operations release the global interpreter lock, which allows for multithreaded processing. NumPy also provides a C API, which allows Python code to interoperate with external libraries written in low-level languages. === The ndarray data structure === The core functionality of NumPy is its "ndarray", for n-dimensional array, data structure. These arrays are strided views on memory. In contrast to Python's built-in list data structure, these arrays are homogeneously typed: all elements of a single array must be of the same type. Such arrays can also be views into memory buffers allocated by C/C++, Python, and Fortran extensions to the CPython interpreter without the need to copy data around, giving a degree of compatibility with existing numerical libraries. This functionality is exploited by the SciPy package, which wraps a number of such libraries (notably BLAS and LAPACK). NumPy has built-in support for memory-mapped ndarrays. === Limitations === Inserting or appending entries to an array is not as trivially possible as it is with Python's lists. The np.pad(...) routine to extend arrays actually creates new arrays of the desired shape and padding values, copies the given array into the new one and returns it. NumPy's np.concatenate([a1,a2]) operation does not actually link the two arrays but returns a new one, filled with the entries from both given arrays in sequence. Reshaping the dimensionality of an array with np.reshape(...) is only possible as long as the number of elements in the array does not change. These circumstances originate from the fact that NumPy's arrays must be views on contiguous memory buffers. Algorithms that are not expressible as a vectorized operation will typically run slowly because they must be implemented in "pure Python", while vectorization may increase memory complexity of some operations from constant to linear, because temporary arrays must be created that are as large as the inputs. Runtime compilation of numerical code has been implemented by several groups to avoid these problems; open source solutions that interoperate with NumPy include numexpr and Numba. Cython and Pythran are static-compiling alternatives to these. Many modern large-scale scientific computing applications have requirements that exceed the capabilities of the NumPy arrays. For example, NumPy arrays are usually loaded into a computer's memory, which might have insufficient capacity for the analysis of large datasets. Further, NumPy operations are executed on a single CPU. However, many linear algebra operations can be accelerated by executing them on clusters of CPUs or of specialized hardware, such as GPUs and TPUs, which many deep learning applications rely on. As a result, several alternative array implementations have arisen in the scientific python ecosystem over the recent years, such as Dask for distributed arrays and TensorFlow or JAX for computations on GPUs. Because of its popularity, these often implement a subset of NumPy's API or mimic it, so that users can change their array implementation with minimal changes to their code required. A library named CuPy, accelerated by Nvidia's CUDA framework, has also shown potential for faster computing, being a 'drop-in replacement' of NumPy. == Examples == NumPy is conventionally imported as np. === Basic operations === === Universal functions === === Linear algebra === === Multidimensional arrays === === Incorporation with OpenCV === === Nearest-neighbor search === Functional Python and vectorized NumPy version. === F2PY === Quickly wrap native code for faster scripts.

Wrike

Wrike, Inc. is an American project management application service provider based in San Jose, California. Wrike also has offices in India, Dallas, Tallinn, Nicosia, Dublin, Tokyo, Melbourne, and Prague. == History == Wrike was founded in 2006 by Andrew Filev. Currently CEO at Wrike is Thomas Scott. Filev initially self-funded the company before later obtaining investor funding. Wrike released the beta version of its software (also called Wrike) in December 2006. The company then launched a new "Enterprise" platform in December 2013. In June 2015, Wrike announced the opening of an office in Dublin, Ireland and in 2016, Wrike launched a datacenter there to host data in compliance with local privacy regulations. In July 2016, Wrike announced the launch of Wrike for Marketers. That same year, Wrike's headquarters moved from Mountain View to San Jose, California. In January 2021, Citrix Systems announced its intention to acquire Wrike for $2.25 billion. The acquisition closed in March 2021. On January 31, 2022, it was announced that Citrix had been acquired in a $16.5 billion deal by affiliates of Vista Equity Partners and Evergreen Coast Capital. Citrix would merge with TIBCO Software, a Vista portfolio company to form Cloud Software Group (CSG). In September 2022, Wrike separated from Citrix Systems. In July 2023, Vista transferred ownership to Symphony Technology Group. == Investments == Wrike received $1 million in Angel funding in 2012 from TMT Investments. In October, 2013, Wrike secured $10 million in investment funding from Bain Capital. In May 2015, the company secured $15 million in a new round of funding. Investors included Scale Venture Partners, DCM Ventures, and Bain Capital. At that time, Wrike had 8,000 customers, 200 employees, and 30,000 new users each month. On November 29, 2018, Wrike signed a definitive agreement to receive a majority investment by Vista Equity Partners (“Vista”), a firm focused on software, data and technology-enabled businesses. == Software == The Wrike project management software is a Software-as-a-Service (SaaS) product with tools for managing projects, deadlines, schedules, and workflow processes. It includes collaboration features. The application is available in English, French, Spanish, German, Portuguese, Italian, Japanese and Russian. Wrike has triggers for task automation in workflow management. === Features === Wrike features a multi-pane UI and consists of features in two categories: project management, and team collaboration. According to Wrike, project management features are designed to help teams track dates and dependencies associated with projects, manage assignments and resources, and track time. These include an interactive Gantt chart, a workload view, and a sortable table that can be customized to store project data. The software includes a co-editing tool, discussion threads on tasks, and tools for attaching documents, editing them, and tracking their changes. Wrike uses an "inbox" feature and browser notifications to alert users of updates from their colleagues and dashboards for quick overviews of pending tasks. These updates are also available in Wrike's mobile apps on iOS and Android. Wrike has an optional feature set called "Wrike for Marketers" which has several tools for managing marketing workflows. In May 2012, Wrike announced the launch of a freemium version of its software for teams of up to 5 users. That year also saw the integration of a live text coeditor into its workspace to unify collaboration and task management. In late 2013 Wrike released a new feature set called Wrike Enterprise which included advanced analytics and other tools targeted at large business customers. Since then it has released several major updates to Wrike Enterprise, including a customizable spreadsheet called "Dynamic Platform" in late 2014 and custom workflows for teams in 2015. In July 2016, Wrike was updated with a set of add-on features under the name "Wrike for Marketers," which includes integrations with Adobe Photoshop, a tool for submitting requests, and proofing and approval tools for creative assets like videos and images. Wrike is available as native Android and iOS apps. Mobile apps include an interactive Gantt chart that syncs across devices. The apps are available offline, and sync when connection is restored. === Criticism === Critics said new users may have a learning curve with complex features. Wrike has 2,710 customers for an estimated 0.04% market share. Competitors include Google Workspace, Slack (software), and Quip (software).

Test data

Test data are sets of inputs or information used to verify the correctness, performance, and reliability of software systems. Test data encompass various types, such as positive and negative scenarios, edge cases, and realistic user scenarios, and aims to exercise different aspects of the software to uncover bugs and validate its behavior. Test data is also used in regression testing to verify that new code changes or enhancements do not introduce unintended side effects or break existing functionalities. == Background == Test data may be used to verify that a given set of inputs to a function produces an expected result. Alternatively, data can be used to challenge the program's ability to handle unusual, extreme, exceptional, or unexpected inputs. Test data can be produced in a focused or systematic manner, as is typically the case in domain testing, or through less focused approaches, such as high-volume randomized automated tests. Test data can be generated by the tester or by a program or function that assists the tester. It can be recorded for reuse or used only once. Test data may be created manually, using data generation tools (often based on randomness), or retrieved from an existing production environment. The data set may consist of synthetic (fake) data, but ideally, it should include representative (real) data. == Limitations == Due to privacy regulations such as GDPR, PCI, and the HIPAA, the use of privacy-sensitive personal data for testing is restricted. However, anonymized (and preferably subsetted) production data may be used as representative data for testing and development. Programmers may also choose to generate synthetic data as an alternative to using real or anonymized data. While synthetic data can offer significant advantages, such as enhanced privacy and flexibility, it also comes with limitations. For instance, generating synthetic data that accurately reflects real-world complexity can be challenging. There is also a risk of synthetic data not fully capturing the nuances of real data, potentially leading to gaps in test coverage. == Domain testing == Domain testing is a set of techniques focusing on test data. This includes identifying critical inputs, values at the boundaries between equivalence classes, and combinations of inputs that drive the system toward specific outputs. Domain testing helps ensure that various scenarios are effectively tested, including edge cases and unusual conditions.

Curve (tonality)

In image editing, a curve is a remapping of image tonality, specified as a function from input level to output level, used as a way to emphasize colours or other elements in a picture. Curves can usually be applied to all channels together in an image, or to each channel individually. Applying a curve to all channels typically changes the brightness in part of the spectrum. Light parts of a picture can be easily made lighter and dark parts darker to increase contrast. Applying a curve to individual channels can be used to stress a colour. This is particularly efficient in the Lab colour space due to the separation of luminance and chromaticity, but it can also be used in RGB, CMYK or whatever other colour models the software supports.

24SevenOffice

24SevenOffice is a Norwegian software company headquartered in Oslo, Norway, with offices in Stockholm, Sweden and London, United Kingdom. Founded in 1997, the company specializes in web-based (SaaS) ERP and CRM systems. == Company history == 24SevenOffice was founded in 1997 in Porsgrunn, Norway, as IKT Interactive AS and marketed as kontorplassen.no. The name "24SevenOffice" was introduced for the company's London branch when the company entered the British market in 2003. The company changed its name to 24SevenOffice in February 2005. Originally based in Skien, the company later moved to Oslo Innovation Center, then to Tjuvholmen in the waterfront Fjord City of Oslo, and now the headquarters are located in Inkognitogaten 33, Solli plass, Oslo. The idea for the company's product was developed in 1996, and 24SevenOffice was an early innovator in the Scandinavian market in web-based enterprise resource planning solutions (ERP). A British office was established at Surrey Business Park in May 2003, with the company launching its web-based (SaaS) utility computing system to the UK SME market in 2004. An office in Chennai, India, was established in 2005, and 24SevenOffice entered the Swedish market when they acquired the leading competitor and ERP-provider Start & Run in a cash deal. In August 2005, the company had an initial public offering that raised NOK 15 million, and the company entered The Norwegian Over the Counter Market list as of 5 October 2005 (the ticker was 24SO), reaching a market value of NOK 175 million, with 5000 customers in Norway. In 2006, the company signed a deal to sponsor rally driver Petter Solberg, at the time the largest private sponsorship in Norwegian sport. Instead of receiving NOK 5 million in cash, Solberg received a 2.9 per cent ownership in the company. The company entered the German-speaking market in April 2006 when an office in Frankfurt am Main was opened. In late August/early September, they established an office with ten sales agents plus a general manager in Stockholm for the Swedish market. 24SevenOffice initiated strategic cooperation with Active 24 in early 2006 to develop a common platform. During the summer, Active 24 was bought by 24SevenOffice's ERP/CRM competitor Mamut (company), and 24SevenOffice terminated the contract with Active 24 in October demanding NOK 200 million in compensation for lost revenue. After a breakdown of settlement negotiations in the Forliksråd in January 2007, 24SevenOffice filed a case against Active 24 for breach of agreement in the Oslo District Court in March. 24SevenOffice lost on all counts in the District Court in December 2007. In January 2008, 24SevenOffice appealed the case to the Borgarting Court of Appeal, reducing the cause of action from NOK 250 to 30 million. 24SevenOffice lost on all counts in the Court of Appeal in December 2008, and was ordered to cover the costs incurred by Active 24 in connection with the dispute totaling NOK 6.91 million. 24SevenOffice appealed the case to the Supreme Court of Norway, but the Supreme Court Appeals Committee in March 2008 unanimously rejected the appeal from 24SevenOffice over the Borgarting Appeal Court's unanimous judgment of December 2008. On a counterclaim from Active 24 and Mamut against 24SevenOffice, the Oslo District Court in May 2010 found, that 24SevenOffice should pay Active 24 NOK 12 million in compensation for wrongfully having terminated the agreement, and a further NOK 360.000 of the opponent's legal costs. 24SevenOffice disagreed with the court ruling, and appealed once again. The Borgarting Court of Appeal in November 2011, ruled to reduce the amount of damages to NOK 4.4 million plus NOK 900.000 in penal interest. With several scrip issues, 24SevenOffice raised 25 million NOK (about $4 million at the time) between October 2005 and July 2006. They entered into a strategic partnership with Bluegarden, who for 30 years had delivered digital services for payroll, human resource planning, recruitment and training, in March 2006, and they made a large-scale agreement in April 2006, with US telecommunications software company Webex, a competitor to Norwegian Tandberg videoconferencing equipment manufacturer. In September 2006, 24SevenOffice signed an agreement with Fokus Bank to provide their customers with extended functionality in Internet banking. 24SevenOffice had by 2007 reportedly 9000 customers, joined the OpenAjax Alliance, and entered into a strategic partnership with Dun & Bradstreet in May 2007, but despite getting listed on Oslo Axess on 22 June (ticker: TFSO), reaching a market capitalization of NOK 120 million, the company was still losing money. The company ended 2007 with a revenue of NOK 21.7 million. In 2008, 24SevenOffice bought 50% of the stocks in telecommunication company Oyatel, partnered with Nets Group to facilitate invoicing for businesses, and telecommunications company Telipol chose 24SevenOffice's second-generation Internet platform for its 8,000 users. They announced an increase in revenues in Q2 to 11.1 million, up from 4.7 million in the same period the year before. 24SevenOffice had a turnover of NOK 37 million in the first half of 2009, a doubling compared to the same period the previous year, and presented its first positive EBITDA in Q2. The Norwegian Association of Auditors signed an agreement with 24SevenOffice in 2011, whereby they only recommend 24SevenOffice as a system for their members to use. On 27 June 2013, the shareholders of 24SevenOffice took off from the stock exchange and privatized the company. In recent years, the company has invested heavily in finance and accounting – and got leading auditing companies such as PwC and KPMG on the customer list. == Products == 24SevenOffice is a web-based (SaaS) ERP system. It includes modules for CRM, accounting, invoicing, e-mail, file/document management and project management. == Awards == 24SevenOffice won the Seal of Excellence in Multimedia Award at the 2004 CeBIT, became Norwegian Gazelle Company of the year 2004, chosen by Dagens Næringsliv and Dun & Bradstreet, won Product of the Year in the Norwegian finance magazine Kapital, and the IKT Grenland Innovation Award in 2008.

Cloud load balancing

Cloud load balancing is a type of load balancing that is performed in cloud computing. Cloud load balancing is the process of distributing workloads across multiple computing resources. Cloud load balancing reduces costs associated with document management systems and maximizes availability of resources. It is a type of load balancing and not to be confused with Domain Name System (DNS) load balancing. While DNS load balancing uses software or hardware to perform the function, cloud load balancing uses services offered by various computer network companies. == Comparison With DNS load balancing == Cloud load balancing has an advantage over DNS load balancing as it can transfer loads to servers globally as opposed to distributing it across local servers. In the event of a local server outage, cloud load balancing delivers users to the closest regional server without interruption for the user. Cloud load balancing addresses issues relating to TTL reliance present during DNS load balancing. DNS directives can only be enforced once in every TTL cycle and can take several hours if switching between servers during a lag or server failure. Incoming server traffic will continue to route to the original server until the TTL expires and can create an uneven performance as different internet service providers may reach the new server before other internet service providers. Another advantage is that cloud load balancing improves response time by routing remote sessions to the best performing data centers. == Importance of Load Balancing == Cloud computing brings advantages in "cost, flexibility and availability of service users." Those advantages drive the demand for Cloud services. The demand raises technical issues in Service Oriented Architectures and Internet of Services (IoS)-style applications, such as high availability and scalability. As a major concern in these issues, load balancing allows cloud computing to "scale up to increasing demands" by efficiently allocating dynamic local workload evenly across all nodes. == Load Balancing Techniques == === Scheduling Algorithms === Opportunistic Load Balancing (OLB) is the algorithm that assigns workloads to nodes in free order. It is simple but does not consider the expected execution time of each node. Load balance Min-Min (LBMM) assigns sub-tasks to the node which requires minimum execution time. === Load Balancing Policies === Workload and Client Aware Policy (WCAP) specifies the unique and special property (USP) of requests and computing nodes. With the information of USP, the schedule can decide the most suitable node to complete a request. WCAP makes the most of computing nodes by reducing their idle time. Also, it reduces performance time through searches based on content information. === A Comparative Study of Algorithms === Biased Random Sampling bases its job allocation on the network represented by a directed graph. For each execution node in this graph, in-degree means available resources and out-degree means allocated jobs. In-degree will decrease during job execution while out-degree will increase after job allocation. Active Clustering is a self-aggregation algorithm to rewire the network. The experiment result is that"Active Clustering and Random Sampling Walk predictably perform better as the number of processing nodes is increased" while the Honeyhive algorithm does not show the increasing pattern. == Client-side Load Balancer Using Cloud Computing == Load balancer forwards packets to web servers according to different workloads on servers. However, it is hard to implement a scalable load balancer because of both the "cloud's commodity business model and the limited infrastructure control allowed by cloud providers." Client-side Load Balancer (CLB) solve this problem by using a scalable cloud storage service. CLB allows clients to choose back-end web servers for dynamic content although it delivers static content.