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c_uwg2wlf36qmm | Thus sexual reproduction, as a form of natural selection, has an effect on evolution. Sexual dimorphism is where the basic phenotypic traits vary between males and females of the same species. Dimorphism is found in both sex organs and in secondary sex characteristics, body size, physical strength and morphology, biolo... | Sexual reproduction |
c_dujfe1mky44o | In order to resolve issues brought by Han unification, a Unicode Technical Standard known as the Unicode Ideographic Variation Database have been created to resolve the problem of specifying specific glyph in plain text environment. By registering glyph collections into the Ideographic Variation Database (IVD), it is p... | Han unification |
c_q03tm6inp2ne | In order to resolve the challenges where it is hard to determine the dual-threshold value empirically, Otsu's method can be used on the non-maximum suppressed gradient magnitude image to generate the high threshold. The low threshold is typically set to 1/2 of the high threshold in this case. Since the gradient magnitu... | Canny edge detection |
c_ql3ijzmdd7bo | In order to restore joint cartilage after injury due to chondrocyte loss, cell therapy and chondrocyte replenishment has been shown to work in certain studies. Lying self-assembled MSCs (mesenchymal stem cells) on top of chondrocyte-laden hydrogel scaffolds has shown cell-mediated regeneration of hyaline-like cartilage... | Artificial cartilage |
c_0rr4kvlf0m5c | In order to resume the profitable coin export on their terms, the Nepalese invaded Tibet in 1788 and again in 1790/91. When the Tibetan Government turned to China for help, an imperial army was sent to Tibet. Together with the Tibetan army they managed to drive out the Nepalese by the autumn of 1792.The Chinese took th... | Historical money of Tibet |
c_wb88hj0o1y8t | In order to solve temporarily the shortage of coins in Tibet when the Chinese army arrived in 1791, the Chinese had allowed the striking of the so-called "Kong-par tangkas" which were produced from alloyed silver and had a design copied from Nepalese prototypes. These tangkas which first were produced in the Kongpo pro... | Historical money of Tibet |
c_71mm1mo187ke | These had both Tibetan and Chinese inscriptions. Meanwhile, the striking of the Kong-par tangkas continued through the year 1792 and in early 1793. Both types of coins were authorised by the Chinese, and struck under joint Chinese and Tibetan supervision, but they were not part of the Chinese currency system, as silver... | Historical money of Tibet |
c_fijdv941nh74 | In 1791 it was originally planned by the Chinese authorities to cast copper cash coins in Tibet. Had this plan been carried out, the Tibetan coinage could have become part of the Chinese currency system. But this plan was abandoned because it was found to be too expensive to transport copper from China to Tibet in orde... | Historical money of Tibet |
c_ml13m2uw1inq | In order to return to the normal arrangement of half the state Senators being contested at each election, following a double dissolution, section 13 of the Australian Constitution requires the senate to divide the state senators into two classes, with three-year and six-year terms. This has traditionally been done by a... | Double dissolution |
c_bzgt8hezd9k2 | In order to reverse the decline of the high brown fritillary, conservation plans focusing on bracken habitat management are the primary course of action. Bracken sites are in the process of being assessed to see which steps could be taken next. There are tentative plans to implement winter bracken litter clearance and ... | High brown fritillary |
c_iuxe32zsxod0 | The Wilder Blean project, headed up by the Wildwood Trust and Kent Wildlife Trust, is introducing European bison to the UK for the first time in 6000 years. Their actions create open, light-flooded patches that encourage the growth of cow wheat which the fritillary depends upon. The herd of 3 females and 1 male will be... | High brown fritillary |
c_fqiakembl46r | In order to rewrite Eq.(1) in more familiar form we use the general formula Here Sm is the molar entropy and Vm the molar volume. With Eq.(2) μ(p,T) can be found by a line integration in the p-T plane. First we integrate from the origin (0,0) to (p,0), so at T =0. Next we integrate from (p,0) to (p,T), so with constant... | Maxon excitation |
c_u4717vj27sao | In the first integral dT=0 and in the second dp=0. With Eq.(2) we obtain We are interested only in cases where p is small so that Vm is practically constant. So where Vm0 is the molar volume of the liquid at T =0 and p =0. | Maxon excitation |
c_v9f3rbdqr0fd | The other term in Eq.(3) is also written as a product of Vm0 and a quantity pf which has the dimension of pressure The pressure pf is called the fountain pressure. It can be calculated from the entropy of 4He which, in turn, can be calculated from the heat capacity. For T =Tλ the fountain pressure is equal to 0.692 bar... | Maxon excitation |
c_0m6zjlxu19zk | With a density of liquid helium of 125 kg/m3 and g = 9.8 m/s2 this corresponds with a liquid-helium column of 56 meter height. So, in many experiments, the fountain pressure has a bigger effect on the motion of the superfluid helium than gravity. With Eqs. | Maxon excitation |
c_a6k8lchcv910 | (4) and (5), Eq.(3) obtains the form Substitution of Eq.(6) in (1) gives with ρ0 = M4/Vm0 the density of liquid 4He at zero pressure and temperature. Eq.(7) shows that the superfluid component is accelerated by gradients in the pressure and in the gravitational field, as usual, but also by a gradient in the fountain pr... | Maxon excitation |
c_tmbqke1qmuwx | Figure 7 shows two vessels both containing He-II. The left vessel is supposed to be at zero kelvins (Tl=0) and zero pressure (pl = 0). The vessels are connected by a so-called superleak. | Maxon excitation |
c_i49xzb9bs29k | This is a tube, filled with a very fine powder, so the flow of the normal component is blocked. However, the superfluid component can flow through this superleak without any problem (below a critical velocity of about 20 cm/s). In the steady state vs=0 so Eq.(7) implies where the index l (r) applies to the left (right)... | Maxon excitation |
c_9xnqy9vkpd9b | In this particular case pl = 0, zl = zr, and pfl = 0 (since Tl = 0). Consequently, 0 = p r − p f r . | Maxon excitation |
c_5e0yylgx7hyr | {\displaystyle 0=p_{r}-p_{fr}.} This means that the pressure in the right vessel is equal to the fountain pressure at Tr. In an experiment, arranged as in figure 8, a fountain can be created. The fountain effect is used to drive the circulation of 3He in dilution refrigerators. | Maxon excitation |
c_ywaxnxsvsods | In order to route calls to a phone, cell towers listen for a signal sent from the phone and negotiate which tower is best able to communicate with the phone. As the phone changes location, the antenna towers monitor the signal, and the phone is "roamed" to an adjacent tower as appropriate. By comparing the relative sig... | Smart phone tracking |
c_8tn00hdzfp27 | Other means make use of the antenna pattern, which supports angular determination and phase discrimination. Newer phones may also allow the tracking of the phone even when turned on but not active in a telephone call. This results from the roaming procedures that perform hand-over of the phone from one base station to ... | Smart phone tracking |
c_2kx3120gtg7p | In order to run safely on multiprocessor machines, access to shared resources (like files, data structures) must be serialized so that threads or processes do not attempt to modify the same resource at the same time. In order to prevent multiple threads from accessing or modifying a shared resource simultaneously, Drag... | Dragonfly BSD |
c_rt03sl8w0yp7 | Much of the system's core, including the LWKT subsystem, the IPI messaging subsystem and the new kernel memory allocator, are lockless, meaning that they work without using mutexes, with each process operating on a single CPU. Critical sections are used to protect against local interrupts, individually for each CPU, gu... | Dragonfly BSD |
c_mae79h90qzal | Among other things, the use of serializing tokens prevents many of the situations that could result in deadlocks and priority inversions when using mutexes, as well as greatly simplifying the design and implementation of a many-step procedure that would require a resource to be shared among multiple threads. The serial... | Dragonfly BSD |
c_epm70wgg38lx | In order to sail upwind, sailing craft must zig-zag across the direction of the oncoming wind, called beating to windward. The higher a vessel that can point into the wind, the shorter its "course made good" to an upwind destination. Beating upwind, a vessel alternates between having the wind come on the port and starb... | Broad reach |
c_7vy52l6ig445 | In order to satisfy needs of a VLDB the database storage needs to have low access latency and contention, high throughput, and high availability. | Very large database |
c_9bk16kufe2uw | In order to save costs, an increasing number of polyester intermediate producers like spinning mills, strapping mills, or cast film mills are working on the direct use of the PET-flakes, from the treatment of used bottles, with a view to manufacturing an increasing number of polyester intermediates. For the adjustment ... | Plastic bottle recycling |
c_9p7cdplgcln9 | When using directly secondary materials such as bottle flakes in spinning processes, there are a few processing principles to obtain.High-speed spinning processes for the manufacture of partially oriented yarn ("POY") normally need an IV of 0.62–0.64. Starting from bottle flakes, the viscosity can be set via the degree... | Plastic bottle recycling |
c_u3dngk2jpztg | In order to protect the spinnerets, an efficient filtration of the melt is, in any case is necessary. For the time-being, the amount of POY made of 100% recycling polyester is rather low because this process requires high purity of spinning melt. Most of the time, a blend of virgin and recycled pellets is used.Staple f... | Plastic bottle recycling |
c_s7e3quiuqkmc | In this case, too, the required viscosity can be adjusted via drying or vacuum adjustment in case of vacuum extrusion. For adjusting the viscosity, however, an addition of chain length modifier like ethylene glycol or diethylene glycol can also be used.Spinning non-woven—in the fine titer field for textile applications... | Plastic bottle recycling |
c_q5uvhan8zhxe | In order to say something about the security properties of the above explained XTR encryption scheme, first it is important to check the security of the XTR group, which means how hard it is to solve the Discrete Logarithm problem there. The next part will then state the equivalency between the Discrete Logarithm probl... | XTR |
c_nb23fq6u4et8 | In order to scale up the perovskite layer while maintaining high efficiency, various techniques have been developed to coat the perovskite film more uniformly. For example, some physical approaches are developed to promote supersaturation through rapid solvent removal, thus getting more nucleations and reducing grain g... | Perovskite solar cell |
c_tjed97rlvuac | And chemical additives, such as chloride additives, Lewis base additives, surfactant additive, and surface modification, can influence the crystal growth to control the film morphology. For example, a recent report of surfactant additive, such as L-α-phosphatidylcholine (LP), demonstrated the suppression of solution fl... | Perovskite solar cell |
c_ll7kt9w8wtax | In order to scroll a character screen, the VIC-II is set to 38-column and/or 24-line mode via the registers at $D011 and $D016. This creates an off-screen buffer where the row of characters to be scrolled is placed. By adjusting the scroll bits in the above-mentioned registers, one row may be moved on-screen after whic... | MOS Technology VIC-II |
c_80wjce8pvpoa | VIC-II scrolling is a relatively complicated, CPU intensive task, although it is not uncommon for C64 game programmers to cheat by designing graphics so that the color RAM can remain static. Another standard trick is to use a section of the screen (perhaps the bottom or top 4 or 5 character rows) as a game status area ... | MOS Technology VIC-II |
c_7ll54iq876rd | Two 1k blocks of screen ram are reserved; one is displayed while another is written to, then during vblank they are quickly swapped through manipulation of the VIC-II registers. Unfortunately this cannot be done with color RAM. Late in the C64's commercial lifespan, an exploit known as VSP (Virtual Screen Positioning) ... | MOS Technology VIC-II |
c_l7tzlwa07nsk | In order to search a document, Apache Solr performs the following operations in sequence: Indexing: converts the documents into a machine-readable format. Querying: understanding the terms of a query asked by the user. These terms can be images or keywords, for example. Mapping: Solr maps the user query to the document... | Apache Solr |
c_r36kq82l87gs | In order to secretly gain knowledge of the value of the top card, second dealing is always used in conjunction with other cheating methods. One of the easiest, and most common methods of gaining knowledge of the top card is to employ the use of marked cards. When marked cards are not in use the cheat can still gain kno... | Second dealing |
c_po2pv0brcuw1 | While holding the deck in the hand, the dealer will either push the top card to the side - revealing a small portion of the back of the second card (called 'the brief'), and strike the second card instead (this method is known as the strike second deal), or by pushing two top cards as one, in almost perfect alignment, ... | Second dealing |
c_3bu1038xi6vg | In this case the dealer will push the top card upwards and strike the partially exposed second card. Although this method is usually employed with the use of a gaffed dealing shoe, an expert card mechanic can do it with a regular dealing shoe as well. The second deal can be performed using either one or two hands, and ... | Second dealing |
c_vr7tpotortaf | In order to see adenomatoid tumors multiple imaging modalities are used. These tumors appear to be small, solid, and circumscribed with a color that usually ranges from white to tan.Ultrasound, MRI, and CT scans are all used to identify whether a patient has adenomatoid tumors. The shape, location, and activity of the ... | Adenomatoid tumor |
c_7fhahvi4pliv | Dynamic contrast enhanced MRI can differentiate tumors from disorders in the testicular region. Misdiagnosing something can sometimes cause more damage then doing nothing. One weakness of adenomatoid tumor ultrasound imaging is that it is only possible if the tumor is hyperechoic. | Adenomatoid tumor |
c_3vau958xe9b0 | Ultrasonically can help identify if it fits this category or not. This specific characteristic in adenomatoid tumors is variable and can determine whether or not ultrasound is the right technique for the job. Having other options available to identify the tumor helps fill in the gap when ultrasound is not possible. MRI... | Adenomatoid tumor |
c_jbs7oweu7d7g | In order to see how the various typeface aspects work together for typeface selection, look at the Times New Roman typeface. Mackiewicz notes that its letterforms "display complexity and perfection". She also lists features of the Times New Roman typeface that make it professional in personality: "moderate weight, mode... | Form and document creation |
c_u2ylekuy1oqf | Reid Goldsborough, a syndicated newspaper columnist, provides the history of the Times New Roman typeface. Times New Roman "was commissioned by the British newspaper The Times in 1931", and in 2004, the U.S. State Department "mandated that all U.S. | Form and document creation |
c_ys9vb9cuegls | diplomatic documents use Times New Roman instead of the previous Courier New". In Jo Mackiewicz’s study of typefaces, "one participant said that Times New Roman could be used in ‘any lengthy passages that need good readability". Finally, in a study that evaluated Times New Roman against the newer ClearType typefaces, i... | Form and document creation |
c_kvuw5fn0vkmk | In order to see whether p53 re-expression in cancer cells (that have lost p53 expression) has the suppressive effect on the promoter of the survivin gene, a luciferase reporter construct was made. The isolated survivin promoter was placed upstream of the luciferase reporter gene. In a luciferase reporter assay, if the ... | Survivin |
c_bjs4o0khbz6t | High luciferase activity was measured in the cells with mutant p53 and significantly lower luciferase levels were measured for cells with wild-type p53.Transfection of different cell types with wild-type p53 was associated with a strong repression of the survivin promoter. Transfection with mutant p53 was not shown to ... | Survivin |
c_sfyj0m3hn84f | At one point, there was deletion that caused the survivin levels to be indifferent to the presence of the p53 over-expression plasmid, indicating that there is a specific region proximal to the transcription start site that is needed for p53 suppression of survivin. Although it has been found that two p53 binding sites... | Survivin |
c_ocxgkeibgh73 | In order to see while hunting for food at night, the swallow-tailed gull's eyes are larger in size and volume than those of any other gull. They also possess a tapetum lucidum in the back of the eye that reflects light back through the retina, increasing the amount of light available to the photoreceptor cells. | Swallow-tailed gull |
c_ogga50cq9p6j | In order to select suitable equipment and media, it should be known also which amount and which throughput has to be handled. In larger factories, little amounts are virtually ever cleaned economically. Additionally, the pricing method needs to be determined. Sensitive parts sometimes need to be fixed in boxes. When de... | Parts cleaning |
c_2cismw6tcjyj | In order to select the best quality males and maximize their offspring survival rate, females will pool their eggs with other females into a single male's territory and create a larger brood. Studies have shown that pooling to create larger broods serves to not only decrease predatory effects, but also increases patern... | Scissortail sergeant |
c_epmpxww5s13w | It has also been hypothesized that egg pooling is a simple imitation tactic by which females confirm the quality of the male. By copying the spawning behavior of other females, females can minimize the search costs of finding a good mate. In order to increase the propagation of their offspring, female scissortail serge... | Scissortail sergeant |
c_fywz43mm7d7b | Female fish utilize multiple strategies in order to select for male parental quality: important factors include mate size, courtship rates, and male cannibalism. Some females will monitor a male's parental quality by laying small clutches of “test eggs” in a male's territory. This unusual tactic is a direct way for fem... | Scissortail sergeant |
c_6m0sqytwmpof | Females only lay one test clutch and return shortly after laying it. Capable males prove their parental quality by defending the brood and not cannibalizing the eggs. Test eggs are energetically expensive to create, so this strategy is typically only used by large females at the beginning of the mating phase. | Scissortail sergeant |
c_8enevzsatjxl | In order to send a packet that is larger than the path MTU, the sending node splits the packet into fragments. The Fragment extension header carries the information necessary to reassemble the original (unfragmented) packet. Next header (8 bits) Identifies the type of the next header. Reserved (8 bits) Initialized to a... | IPv6 packet |
c_s55qiu9p9tcq | Fragment offset (13 bits) Offset, in 8-octet units, relative to the start of the fragmentable part of the original packet. Res (2 bits) Reserved; initialized to zeroes. | IPv6 packet |
c_2hrge063q7eq | M Flag (1 bit) 1 means more fragments follow; 0 means last fragment. Identification (32 bits) Packet identification value, generated by the source node. Needed for reassembly of the original packet. | IPv6 packet |
c_iyzdhs5vt5nd | In order to send data, a modulation of light is required. A modulation is the form in which the light signal varies in order to represent different symbols. In order for the data to be decoded. | Visible light communications |
c_dpgf4iala4ey | Unlike radio transmission, a VLC modulation requires the light signal to be modulated around a positive dc value, responsible for the lighting aspect of the lamp. The modulation will thus be an alternating signal around the positive dc level, with a high-enough frequency to be imperceptible to the human eye.Due to this... | Visible light communications |
c_gq5buab1aao1 | In order to send spam, spammers need to obtain the email addresses of the intended recipients. To this end, both spammers themselves and list merchants gather huge lists of potential email addresses. Since spam is, by definition, unsolicited, this address harvesting is done without the consent (and sometimes against th... | Spam emails |
c_uxmm9d2i8unx | In order to separate CMB from the galactic emission, the measurements will cover 40 GHz to 400 GHz during a 3-year full sky survey using two telescopes on LiteBIRD. The Low Frequency Telescope (LFT) covers 40 GHz to 235 GHz, and the High Frequency Telescope (HFT) covers 280 GHz to 400 GHz. LFT has a 400 mm aperture Cro... | LiteBIRD |
c_vxklfxxswojw | In order to set and stow the square sails, the crew must climb aloft and spread out along the yards. To do this, they stand in footropes suspended beneath the yard and balance themselves between that and the yard itself. The person working on the end of the yardarm has a separate footrope known as the flemish horse. Ja... | Yard (sailing) |
c_qoyn8xuvn5vk | These are usually steel rods, but stiff cordage stretched between the yardarms was used in the past. Almost all ships used in modern times are fitted with steel safety wires (sometimes erroneously called jackstays) along the yards to which sailors attach themselves using a harness. This is a relatively recent innovatio... | Yard (sailing) |
c_fih2pxw8ryl6 | In order to show how time may not be the only schema, Professor Walsh suggested that there is "… the possibility of making sense of the categories in organic as opposed to mechanical terms." He hypothesized that "Elements in an organic complex would here take the place of elements in a temporal situation. Substance mig... | Kantian architectonics |
c_g4foufb4hrsm | In order to show some particularly interesting results about characters, it is rewarding to consider a more general type of functions on groups: Definition (Class functions). A function φ: G → C {\displaystyle \varphi :G\to \mathbb {C} } is called a class function if it is constant on conjugacy classes of G {\displayst... | Complex representations of finite groups |
c_g6zod0jgioxi | The set of all class functions is a C {\displaystyle \mathbb {C} } –algebra and is denoted by C class ( G ) {\displaystyle \mathbb {C} _{\text{class}}(G)} . Its dimension is equal to the number of conjugacy classes of G . {\displaystyle G.} | Complex representations of finite groups |
c_p6bus5f10zeq | Proofs of the following results of this chapter may be found in , and . An inner product can be defined on the set of all class functions on a finite group: ( f | h ) G = 1 | G | ∑ t ∈ G f ( t ) h ( t ) ¯ {\displaystyle (f|h)_{G}={\frac {1}{|G|}}\sum _{t\in G}f(t){\overline {h(t)}}} Orthonormal property. If χ 1 , … , ... | Complex representations of finite groups |
c_o9pa0gb9sesc | {\displaystyle (\chi _{i}|\chi _{j})={\begin{cases}1{\text{ if }}i=j\\0{\text{ otherwise }}\end{cases}}.} Every class function f {\displaystyle f} may be expressed as a unique linear combination of the irreducible characters χ 1 , … , χ k {\displaystyle \chi _{1},\ldots ,\chi _{k}} .One might verify that the irreducibl... | Complex representations of finite groups |
c_zvtuzs34knzo | {\displaystyle \rho _{f}=\sum _{g}f(g)\rho (g).} Then for ρ {\displaystyle \rho } irreducible, we have ρ f = | G | n ⟨ f , χ V ∗ ⟩ ∈ E n d ( V ) {\displaystyle \rho _{f}={\frac {|G|}{n}}\langle f,\chi _{V}^{*}\rangle \in End(V)} from Schur's lemma. Suppose f {\displaystyle f} is a class function which is orthogonal to ... | Complex representations of finite groups |
c_nluayrk02oe5 | Then by the above we have ρ f = 0 {\displaystyle \rho _{f}=0} whenever ρ {\displaystyle \rho } is irreducible. But then it follows that ρ f = 0 {\displaystyle \rho _{f}=0} for all ρ {\displaystyle \rho } , by decomposability. Take ρ {\displaystyle \rho } to be the regular representation. | Complex representations of finite groups |
c_rur49j9ejava | Applying ρ f {\displaystyle \rho _{f}} to some particular basis element g {\displaystyle g} , we get f ( g ) = 0 {\displaystyle f(g)=0} . Since this is true for all g {\displaystyle g} , we have f = 0. {\displaystyle f=0.} | Complex representations of finite groups |
c_tczv50o8cjvk | It follows from the orthonormal property that the number of non-isomorphic irreducible representations of a group G {\displaystyle G} is equal to the number of conjugacy classes of G . {\displaystyle G.} Furthermore, a class function on G {\displaystyle G} is a character of G {\displaystyle G} if and only if it can be ... | Complex representations of finite groups |
c_6i1qnkjsg256 | {\displaystyle \chi _{j}.} Conversely, it is always possible to write any character as a sum of irreducible characters. The inner product defined above can be extended on the set of all C {\displaystyle \mathbb {C} } -valued functions L 1 ( G ) {\displaystyle L^{1}(G)} on a finite group: ( f | h ) G = 1 | G | ∑ t ∈ G f... | Complex representations of finite groups |
c_dc3b1a1jnf9a | If there is no danger of confusion the index of both forms ( ⋅ | ⋅ ) G {\displaystyle (\cdot |\cdot )_{G}} and ⟨ ⋅ | ⋅ ⟩ G {\displaystyle \langle \cdot |\cdot \rangle _{G}} will be omitted. Let V 1 , V 2 {\displaystyle V_{1},V_{2}} be two C {\displaystyle \mathbb {C} } –modules. Note that C {\displaystyle \mathbb {C}... | Complex representations of finite groups |
c_esnkbhaaik8i | Since the orthonormal property yields the number of irreducible representations of G {\displaystyle G} is exactly the number of its conjugacy classes, then there are exactly as many simple C {\displaystyle \mathbb {C} } –modules (up to isomorphism) as there are conjugacy classes of G . {\displaystyle G.} We define ⟨ V... | Complex representations of finite groups |
c_8i3zq2p0iqci | This form is bilinear with respect to the direct sum. In the following, these bilinear forms will allow us to obtain some important results with respect to the decomposition and irreducibility of representations. For instance, let χ 1 {\displaystyle \chi _{1}} and χ 2 {\displaystyle \chi _{2}} be the characters of V 1 ... | Complex representations of finite groups |
c_rpr6a2nw2c8v | Then ⟨ χ 1 , χ 2 ⟩ G = ( χ 1 | χ 2 ) G = ⟨ V 1 , V 2 ⟩ G . {\displaystyle \langle \chi _{1},\chi _{2}\rangle _{G}=(\chi _{1}|\chi _{2})_{G}=\langle V_{1},V_{2}\rangle _{G}.} | Complex representations of finite groups |
c_ftggh0uzi9fz | It is possible to derive the following theorem from the results above, along with Schur's lemma and the complete reducibility of representations. Theorem. | Complex representations of finite groups |
c_uui1n0gakv4g | Let V {\displaystyle V} be a linear representation of G {\displaystyle G} with character ξ . {\displaystyle \xi .} Let V = W 1 ⊕ ⋯ ⊕ W k , {\displaystyle V=W_{1}\oplus \cdots \oplus W_{k},} where W j {\displaystyle W_{j}} are irreducible. | Complex representations of finite groups |
c_pl46x28m5b3g | Let ( τ , W ) {\displaystyle (\tau ,W)} be an irreducible representation of G {\displaystyle G} with character χ . {\displaystyle \chi .} Then the number of subrepresentations W j {\displaystyle W_{j}} which are isomorphic to W {\displaystyle W} is independent of the given decomposition and is equal to the inner produc... | Complex representations of finite groups |
c_na0hpklqrgep | We also get: ( ξ | χ ) = dim ( V ( τ ) ) dim ( τ ) = ⟨ V , W ⟩ {\displaystyle (\xi |\chi )={\frac {\dim(V(\tau ))}{\dim(\tau )}}=\langle V,W\rangle } and thus dim ( V ( τ ) ) = dim ( τ ) ( ξ | χ ) . {\displaystyle \dim(V(\tau ))=\dim(\tau )(\xi |\chi ).} Corollary. | Complex representations of finite groups |
c_epltrhm0nxfe | Two representations with the same character are isomorphic. This means that every representation is determined by its character.With this we obtain a very useful result to analyse representations: Irreducibility criterion. Let χ {\displaystyle \chi } be the character of the representation V , {\displaystyle V,} then we... | Complex representations of finite groups |
c_fcga3kd2xpsx | {\displaystyle (\chi |\chi )\in \mathbb {N} _{0}.} The case ( χ | χ ) = 1 {\displaystyle (\chi |\chi )=1} holds if and only if V {\displaystyle V} is irreducible. Therefore, using the first theorem, the characters of irreducible representations of G {\displaystyle G} form an orthonormal set on C class ( G ) {\displayst... | Complex representations of finite groups |
c_yj6lq5i9zblf | Corollary. Let V {\displaystyle V} be a vector space with dim ( V ) = n . {\displaystyle \dim(V)=n.} | Complex representations of finite groups |
c_b7ese99v6hze | A given irreducible representation V {\displaystyle V} of G {\displaystyle G} is contained n {\displaystyle n} –times in the regular representation. In other words, if R {\displaystyle R} denotes the regular representation of G {\displaystyle G} then we have: R ≅ ⊕ ( W j ) ⊕ dim ( W j ) , {\displaystyle R\cong \oplus... | Complex representations of finite groups |
c_i1quy068036f | Recall that e {\displaystyle e} denotes the neutral element of the group. This formula is a "necessary and sufficient" condition for the problem of classifying the irreducible representations of a group up to isomorphism. It provides us with the means to check whether we found all the isomorphism classes of irreducible... | Complex representations of finite groups |
c_sgmlujsfy2tj | Similarly, by using the character of the regular representation evaluated at s ≠ e , {\displaystyle s\neq e,} we get the equation: 0 = χ R ( s ) = ∑ j dim ( W j ) ⋅ χ W j ( s ) . {\displaystyle 0=\chi _{R}(s)=\sum _{j}\dim(W_{j})\cdot \chi _{W_{j}}(s).} | Complex representations of finite groups |
c_rbbhdeeow8ln | Using the description of representations via the convolution algebra we achieve an equivalent formulation of these equations: The Fourier inversion formula: f ( s ) = 1 | G | ∑ ρ irr. rep. of G dim ( V ρ ) ⋅ Tr ( ρ ( s − 1 ) ⋅ f ^ ( ρ ) ) . | Complex representations of finite groups |
c_vy6kyc380lmr | {\displaystyle f(s)={\frac {1}{|G|}}\sum _{\rho {\text{ irr. rep. of }}G}\dim(V_{\rho })\cdot {\text{Tr}}(\rho (s^{-1})\cdot {\hat {f}}(\rho )).} | Complex representations of finite groups |
c_legbm085vjt5 | In addition, the Plancherel formula holds: ∑ s ∈ G f ( s − 1 ) h ( s ) = 1 | G | ∑ ρ irred. rep. of G dim ( V ρ ) ⋅ Tr ( f ^ ( ρ ) h ^ ( ρ ) ) . | Complex representations of finite groups |
c_nj2u96qjdvlz | {\displaystyle \sum _{s\in G}f(s^{-1})h(s)={\frac {1}{|G|}}\sum _{\rho \,\,{\text{ irred. }}{\text{ rep. }}{\text{ of }}G}\dim(V_{\rho })\cdot {\text{Tr}}({\hat {f}}(\rho ){\hat {h}}(\rho )).} | Complex representations of finite groups |
c_800ropa663ct | In both formulas ( ρ , V ρ ) {\displaystyle (\rho ,V_{\rho })} is a linear representation of a group G , s ∈ G {\displaystyle G,s\in G} and f , h ∈ L 1 ( G ) . {\displaystyle f,h\in L^{1}(G).} The corollary above has an additional consequence: Lemma. | Complex representations of finite groups |
c_lkcy0rs0tz5k | Let G {\displaystyle G} be a group. Then the following is equivalent: G {\displaystyle G} is abelian. Every function on G {\displaystyle G} is a class function. All irreducible representations of G {\displaystyle G} have degree 1. {\displaystyle 1.} | Complex representations of finite groups |
c_loquc3mlsayd | In order to show that PSPACE is a subset of IP, we need to choose a PSPACE-complete problem and show that it is in IP. Once we show this, then it clear that PSPACE ⊆ IP. The proof technique demonstrated here is credited to Adi Shamir. | IP (complexity) |
c_oyo8dzk9jokv | We know that TQBF is in PSPACE-Complete. So let ψ be a quantified boolean expression: ψ = Q 1 x 1 … Q m x m {\displaystyle \psi ={\mathsf {Q}}_{1}x_{1}\dots {\mathsf {Q}}_{m}x_{m}} where φ is a CNF formula. | IP (complexity) |
c_fa5izetzhcy6 | Then Qi is a quantifier, either ∃ or ∀. Now fi is the same as in the previous proof, but now it also includes quantifiers. f i ( a 1 , … , a i ) = { f i ( a 1 , … , a m ) = 1 Q i + 1 x i + 1 … Q m x m is true 0 otherwise {\displaystyle f_{i}(a_{1},\dots ,a_{i})={\begin{cases}f_{i}(a_{1},\dots ,a_{m})=1&{\mathsf {Q}}_{... | IP (complexity) |
c_p07q7km6phud | Thus f0 is the truth value of ψ. In order to arithmetize ψ we must use the following rules: f i ( a 1 , … , a i ) = { f i + 1 ( a 1 , … , a i , 0 ) ⋅ f i + 1 ( a 1 , … , a i , 1 ) Q i + 1 = ∀ f i + 1 ( a 1 , … , a i , 0 ) ∗ f i + 1 ( a 1 , … , a i , 1 ) Q i + 1 = ∃ {\displaystyle f_{i}(a_{1},\dots ,a_{i})={\begin{cases... | IP (complexity) |
c_p7qpyhgml97b | So f0 is still the truth value of ψ, but the Rx value produces a result that is linear in x. Also after any Q i x i {\displaystyle {\mathsf {Q}}_{i}x_{i}} we add R x 1 … R x i {\displaystyle \mathrm {R} _{x_{1}}\dots \mathrm {R} _{x_{i}}} in ψ′ in order to reduce the degree down to 1 after arithmetizing Q i {\displayst... | IP (complexity) |
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