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https://encyclopedia2.thefreedictionary.com/regular+hexahedron	[ "# cube\n\n(redirected from regular hexahedron)\nAlso found in: Dictionary, Thesaurus, Wikipedia.\n\n## cube,\n\nin geometry, regular solid bounded by six equal squares. All adjacent faces of a cube are perpendicular to each other; any one face of a cube may be its base. The dimensions of a cube are the lengths of the three edges which meet at any vertex. The volume of a cube is equal to the product of its dimensions, and since its dimensions are equal, the volume is equal to the third power, or cube, of any one of its dimensions. Hence, in arithmetic and algebra, the cube of a number or letter is that number or letter raised to the third power. For example, the cube of 4 is 43=4×4×4=64. The problem of constructing a cube with a volume equal to twice that of a given cube using only a compass and a straightedge is known as the problem of the duplication of the cube and is one of the famous geometric problems of antiquitygeometric problems of antiquity,\nthree famous problems involving elementary geometric constructions with straight edge and compass, conjectured by the ancient Greeks to be impossible but not proved to be so until modern times.\n. The cube, or hexahedron, is one of only five regular polyhedra (see polyhedronpolyhedron\n, closed solid bounded by plane faces; each face of a polyhedron is a polygon. A cube is a polyhedron bounded by six polygons (in this case squares) meeting at right angles.\n).\n\n## Cube\n\nA solid figure, bounded by six squares, and hence also called a hexahedron.\n\n## Cube\n\n(1) One of five types of regular polyhedrons, having six square faces, 12 edges, and eight vertices; three mutually perpendicular edges meet at each vertex. A cube is sometimes called a hexahedron.\n\n(2) The cube of the number a is the third power of the number, that is, the product a • a • a = a3. It is so named because it expresses the volume of a cube whose edge is equal to a.\n\n## cube\n\n[kyüb]\n(mathematics)\nRegular polyhedron whose faces are all square.\nFor a number a, the new number obtained by taking the threefold product of a with itself: a × a × a.\n\n## cube\n\n1\n1. a solid having six plane square faces in which the angle between two adjacent sides is a right angle\n2. the product of three equal factors: the cube of 2 is 2 × 2 × 2 (usually written 23)\n\n## cube\n\n2\nany of various tropical American plants, esp any of the leguminous genus Lonchocarpus, the roots of which yield rotenone\n\n## Cube\n\n(1)\nThree-dimensional visual language for higher-order logic.\n\n\"The Cube Language\", M. Najork et al, 1991 IEEE Workshop on Visual Langs, Oct 1991, pp.218-224.\n\n## cube\n\n(2)\n[short for \"cubicle\"] A module in the open-plan offices used at many programming shops. \"I've got the manuals in my cube.\"\n\n## cube\n\n(3)\nA NeXT machine (which resembles a matte-black cube).\n\n## cube\n\n(1) See OLAP cube and OLAP.\n\n(2) Apple's earlier Cube computer. See G4.\nSite: Follow: Share:\nOpen / Close" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.91543585,"math_prob":0.99233216,"size":2986,"snap":"2019-43-2019-47","text_gpt3_token_len":744,"char_repetition_ratio":0.1167002,"word_repetition_ratio":0.057361376,"special_character_ratio":0.24313463,"punctuation_ratio":0.13893376,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9975303,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-12T13:17:36Z\",\"WARC-Record-ID\":\"<urn:uuid:886c62eb-8f26-4fd9-a032-89502fda34fd>\",\"Content-Length\":\"45553\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0c15fc4d-ce56-48e1-af50-b23bb0799bdc>\",\"WARC-Concurrent-To\":\"<urn:uuid:66f5fc62-6861-495c-b869-d4bc3992ec33>\",\"WARC-IP-Address\":\"45.34.10.165\",\"WARC-Target-URI\":\"https://encyclopedia2.thefreedictionary.com/regular+hexahedron\",\"WARC-Payload-Digest\":\"sha1:CMQ7YI66SRPMF6CEJUIC7LYA7G4ESD7S\",\"WARC-Block-Digest\":\"sha1:UDAH2VQBRODS3YWAXXHCLDYB5CEN5FT4\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496665573.50_warc_CC-MAIN-20191112124615-20191112152615-00452.warc.gz\"}"}
https://math.answers.com/Q/What_is_5a_-_1_plus_a_equals_11	[ "", null, "", null, "", null, "", null, "0\n\n# What is 5a - 1 plus a equals 11?\n\nUpdated: 12/15/2022", null, "Wiki User\n\n14y ago\n\nA= 2", null, "Wiki User\n\n14y ago", null, "", null, "", null, "Earn +20 pts\nQ: What is 5a - 1 plus a equals 11?\nSubmit\nStill have questions?", null, "", null, "Related questions\n\n### What does a equal in this problem 5a-1 plus a equals 11?\n\nIf: 5a-1+a = 11 Then: a = 2\n\n### What is the solution of 6a plus 5a equals -11?\n\n6a + 5a = -11 11a = -11 a = -11/11 = -1\n\n### How do you solve 6a plus 5a equals -11?\n\n6a + 5a = -11 Combining like terms: 11a = -11 Dividing by 11: a = -1\n\n### What is the answer to 5a plus 1 equals a-3?\n\nAnswer to: 5a + 1 = a - 3a = -1\n\n### Solve the equation 2a plus 13 equals 5a 1?\n\nIf: 2a+13 = 5a+1 Then: a = 4\n\n### 5a equals -5a plus 5?\n\n5a = -5a + 5Add 5a to each side:10a = 5Divide each side by 10:a = 5/10 = 1/2\n\n### Does 1 plus 1 equals 11 I am 6 And i want to know?\n\nNo it equals 2 10 plus 1 equals 11\n\n### How do you solve the equation 5a plus 2 equals 7?\n\n5a + 2 = 7 subtract 2 from both sides: 5a = 5 divide both sides by 5: a = 1\n\n### 2a-4b plus 4b-6a equals a plus b?\n\n2a-6a-a=b+4b-4b -5a=5b-4b -5a=1b a=-5 b=1\n\n### What is the solution to 5a plus 11 plus 3a - 7 equals -4?\n\nSo we have 5a + 11 + 3a - 7 = -4 Start by combining common terms such as the 5a and 3a to give you 8a and 11 and -7 to give you 4. So now your equation is 8a + 4 = -4. Subtract 4 from both sides gives you 8a = -8. Divide -8 by 8 to give you a = -1, your solution.\n\n### 3a plus 5 equals 5a plus 3?\n\nIf 3a+5=5a+3, then: 3a+5 - 3a = 5a+3 - 3a 5=2a+3 5 - 3=2a+3 - 3 2=2a a=1\n\n5a(2b + 1)" ]	[ null, "https://math.answers.com/icons/searchIcon.svg", null, "https://math.answers.com/icons/searchGlassWhiteIcon.svg", null, "https://math.answers.com/icons/notificationBellIcon.svg", null, "https://math.answers.com/icons/coinIcon.svg", null, "https://math.answers.com/images/avatars/default.png", null, "https://math.answers.com/images/avatars/default.png", null, "https://math.answers.com/images/avatars/default.png", null, "https://math.answers.com/icons/sendIcon.svg", null, "https://math.answers.com/icons/coinIcon.svg", null, "https://math.answers.com/icons/searchIcon.svg", null, "https://st.answers.com/html_test_assets/imp_-_pixel.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.89488673,"math_prob":0.9999273,"size":1219,"snap":"2023-40-2023-50","text_gpt3_token_len":609,"char_repetition_ratio":0.1473251,"word_repetition_ratio":0.652459,"special_character_ratio":0.52912223,"punctuation_ratio":0.08125,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99964166,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-10-02T19:20:55Z\",\"WARC-Record-ID\":\"<urn:uuid:c0997793-df08-452b-b529-3d9af991dfb1>\",\"Content-Length\":\"165341\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d51cab67-a817-4f39-988d-c9df6bc0e13e>\",\"WARC-Concurrent-To\":\"<urn:uuid:40586f39-6a39-43bd-bad8-57ddc77137fb>\",\"WARC-IP-Address\":\"146.75.36.203\",\"WARC-Target-URI\":\"https://math.answers.com/Q/What_is_5a_-_1_plus_a_equals_11\",\"WARC-Payload-Digest\":\"sha1:CWA35535LLUGPOOXK426KWNWPNKC56SB\",\"WARC-Block-Digest\":\"sha1:2OZSJUMNM22PWT6ITLRWIFMR2UXITDM5\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233511002.91_warc_CC-MAIN-20231002164819-20231002194819-00687.warc.gz\"}"}
https://byjus.com/questions/liquid-pressure-is-measured-by/	[ "", null, "# How is Liquid Pressure Measured?\n\nTo measure liquid pressure, a manometer is used.\n\nLiquid pressure is defined as the increase in the pressure when there is an increase in the depth of the liquid. Liquid pressure is given by the following equation:\n\n P = ρgh\n\nWhere,\n\n• P is the liquid pressure\n• ρ is the density of the liquid\n• g is the acceleration due to gravity\n• h is the depth\n\nWorking of manometer\nA manometer is used for measuring the liquid pressure with respect to an outside source which is usually considered to be the earth’s atmosphere.\n\nLiquid such as mercury is used for the measurement of the pressure. The other end of the U-tube is filled with the gas for which the pressure needs to be calculated. The end where the gas is filled is sealed while the other end is kept open. Now the atmospheric pressure, as well as the gas pressure, acts on the liquid.\n\n• The air in the tube is said to be equal to the outside air pressure when the liquid is straight level in both the tubes.\n• The air in the tube is said to be lighter than the out air’s pressure when the liquid rises above the straight level.\n• The air in the tube is said to be heavier than the outside air’s pressure when the liquid is below the straight level.", null, "(22)", null, "(6)" ]	[ null, "https://www.facebook.com/tr", null, "https://cdn1.byjus.com/wp-content/uploads/2021/08/upvote-lineart.svg", null, "https://cdn1.byjus.com/wp-content/uploads/2021/08/downvote-lineart.svg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9489305,"math_prob":0.98509365,"size":1197,"snap":"2022-27-2022-33","text_gpt3_token_len":263,"char_repetition_ratio":0.20284995,"word_repetition_ratio":0.097345136,"special_character_ratio":0.21470343,"punctuation_ratio":0.062240664,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99064726,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-07-04T06:30:13Z\",\"WARC-Record-ID\":\"<urn:uuid:fd99d3bf-bd0a-4f5a-9efc-6c8161b09bc5>\",\"Content-Length\":\"187895\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:36237716-a125-4e9a-8c88-8f17ba1f87ff>\",\"WARC-Concurrent-To\":\"<urn:uuid:a8537e09-f5cf-4cde-8b9f-6fa4a7348bf9>\",\"WARC-IP-Address\":\"162.159.130.41\",\"WARC-Target-URI\":\"https://byjus.com/questions/liquid-pressure-is-measured-by/\",\"WARC-Payload-Digest\":\"sha1:VF6XWERUNESBVIJNJTXFX4FJY2UPXY5E\",\"WARC-Block-Digest\":\"sha1:LDGAJHHEZGUEZXSL5AGX4FXVXJTDAMSA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656104354651.73_warc_CC-MAIN-20220704050055-20220704080055-00096.warc.gz\"}"}
http://www.s-pay.com.ua/index.php/kindle/differential-geometry-and-mathematical-physics-part-i-manifolds-lie-groups	[ "# Download e-book for iPad: Differential Geometry and Mathematical Physics: Part I. by Rudolph, G. and Schmidt, M.", null, "By Rudolph, G. and Schmidt, M.\n\nISBN-10: 9400753454\n\nISBN-13: 9789400753457\n\nRanging from undergraduate point, this booklet systematically develops the fundamentals of - research on manifolds, Lie teams and G-manifolds (including equivariant dynamics) - Symplectic algebra and geometry, Hamiltonian structures, symmetries and relief, - Integrable platforms, Hamilton-Jacobi conception (including Morse households, the Maslov category and caustics). the 1st merchandise is suitable for almost all components of mathematical physics, whereas the second one merchandise offers the foundation of Hamiltonian mechanics. The last thing introduces to special unique components. invaluable heritage wisdom on topology is prov\n\nRead or Download Differential Geometry and Mathematical Physics: Part I. Manifolds, Lie Groups and Hamiltonian Systems PDF\n\nSimilar differential geometry books\n\nGet Connections, curvature and cohomology. Vol. III: Cohomology PDF\n\nGreub W. , Halperin S. , James S Van Stone. Connections, Curvature and Cohomology (AP Pr, 1975)(ISBN 0123027039)(O)(617s)\n\nRudolph, G. and Schmidt, M.'s Differential Geometry and Mathematical Physics: Part I. PDF\n\nRanging from undergraduate point, this publication systematically develops the fundamentals of - research on manifolds, Lie teams and G-manifolds (including equivariant dynamics) - Symplectic algebra and geometry, Hamiltonian platforms, symmetries and aid, - Integrable platforms, Hamilton-Jacobi concept (including Morse households, the Maslov type and caustics).\n\nGet A treatise on the geometry of surfaces PDF\n\nThis quantity is made out of electronic photographs from the Cornell collage Library ancient arithmetic Monographs assortment.\n\nMeant for a twelve months direction, this article serves as a unmarried resource, introducing readers to the \\$64000 ideas and theorems, whereas additionally containing sufficient historical past on complex subject matters to attract these scholars wishing to specialise in Riemannian geometry. this can be one of many few Works to mix either the geometric components of Riemannian geometry and the analytic points of the speculation.\n\nAdditional info for Differential Geometry and Mathematical Physics: Part I. Manifolds, Lie Groups and Hamiltonian Systems\n\nSample text\n\n3) may be taken as the extension of the tangent mapping from Tm M to Dm M. 2. 2 we conclude that for local charts (U, κ) on M at m and (V , ρ) on N at Φ(m) one has Φm X m ρ,i φ(m) = ρ ◦ Φ ◦ κ −1 i κ,j κ(m) j Xm . 4) That is, locally the tangent mapping of Φ is given by the derivative (matrix of partial derivatives) of the local representative Φκ,ρ = ρ ◦ Φ ◦ κ −1 at κ(m). 4 Let M and N be open subsets of the finite-dimensional real vector spaces V and W , respectively. Let Φ ∈ C k (M, N ) and v ∈ M.\n\nNote that, here, continuity of f need not be required. Indeed, for every pair of charts (Ui , κi ), (Vj , ρj ) such that Φ(Ui ) ⊂ Vj we have Φ Ui = ρj−1 ◦ Φκi ,ρj ◦ κi , which is continuous as a composition of continuous mappings. 2. 1 iff it is of class C k in the sense of classical calculus. To see this, choose global charts corresponding to two chosen bases. In particular, multilinear mappings between finite-dimensional real vector spaces are smooth. 3. Let (Ui , κi ) and (Vi , ρi ), i = 1, 2, be local charts on M and N , respectively, such that W := U1 ∩ U2 ∩ Φ −1 (V1 ∩ V2 ) = ∅.\n\n6 Submanifolds Let k ≥ 1 and let N be a C k -manifold. 1 (Submanifold) A C k -submanifold of N is a pair (M, ϕ), where M is a C k -manifold and ϕ : M → N is an injective immersion of class C k . Submanifolds (M1 , ϕ1 ) and (M2 , ϕ2 ) are said to be equivalent if there exists a diffeomorphism ψ : M1 → M2 such that ϕ2 ◦ ψ = ϕ1 . 2 1. Let (M, ϕ) be a C k -submanifold of N and let ϕ˜ : M → ϕ(M) denote the induced mapping. Since ϕ˜ is bijective, one can use it to carry over the topological and differentiable structure from M to ϕ(M), thus making ϕ˜ into a diffeomorphism." ]	[ null, "https://images-na.ssl-images-amazon.com/images/I/41WojgGYYiL._SX331_BO1,204,203,200_.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8543602,"math_prob":0.91173357,"size":4125,"snap":"2020-34-2020-40","text_gpt3_token_len":1106,"char_repetition_ratio":0.10264499,"word_repetition_ratio":0.11188811,"special_character_ratio":0.2489697,"punctuation_ratio":0.14713217,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9940865,"pos_list":[0,1,2],"im_url_duplicate_count":[null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-09-30T02:43:38Z\",\"WARC-Record-ID\":\"<urn:uuid:3f4d8fc6-8bdd-4ed2-bd29-70348d96a3ba>\",\"Content-Length\":\"27929\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0e1b3431-e2d7-4a7e-ae25-bf6a9327fde1>\",\"WARC-Concurrent-To\":\"<urn:uuid:00f6d13d-1ceb-4e81-bd1d-157f7d83daf0>\",\"WARC-IP-Address\":\"93.190.41.65\",\"WARC-Target-URI\":\"http://www.s-pay.com.ua/index.php/kindle/differential-geometry-and-mathematical-physics-part-i-manifolds-lie-groups\",\"WARC-Payload-Digest\":\"sha1:OZ2AERR6OECCZHSP53KX6YAJSNFKVGJE\",\"WARC-Block-Digest\":\"sha1:ADLVYAO5XLTDOSKJHZFEPKTVVEL66ATA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600402101163.62_warc_CC-MAIN-20200930013009-20200930043009-00662.warc.gz\"}"}
https://asmedigitalcollection.asme.org/appliedmechanics/article/84/3/031008/422436/Revisiting-the-Instability-and-Bifurcation	[ "Development of soft electromechanical materials is critical for several tantalizing applications such as human-like robots, stretchable electronics, actuators, energy harvesting, among others. Soft dielectrics can be easily deformed by an electric field through the so-called electrostatic Maxwell stress. The highly nonlinear coupling between the mechanical and electrical effects in soft dielectrics gives rise to a rich variety of instability and bifurcation behavior. Depending upon the context, instabilities can either be detrimental, or more intriguingly, exploited for enhanced multifunctional behavior. In this work, we revisit the instability and bifurcation behavior of a finite block made of a soft dielectric material that is simultaneously subjected to both mechanical and electrical stimuli. An excellent literature already exists that has addressed the same topic. However, barring a few exceptions, most works have focused on the consideration of homogeneous deformation and accordingly, relatively fewer insights are at hand regarding the compressive stress state. In our work, we allow for fairly general and inhomogeneous deformation modes and, in the case of a neo-Hookean material, present closed-form solutions to the instability and bifurcation behavior of soft dielectrics. Our results, in the asymptotic limit of large aspect ratio, agree well with Euler's prediction for the buckling of a slender block and, furthermore, in the limit of zero aspect ratio are the same as Biot's critical strain of surface instability of a compressed homogeneous half-space of a neo-Hookean material. A key physical insight that emerges from our analysis is that soft dielectrics can be used as actuators within an expanded range of electric field than hitherto believed.\n\nIntroduction\n\nSoft materials, such as polymers and many soft biological materials, play an important role in our daily life. They can be easily deformed to large strain values due to intrinsically low elastic stiffness. Meanwhile, surface instabilities like wrinkles [1,2] and creases are often observed under mechanical compression or constrained swelling. Soft dielectrics, an important subclass of soft materials, can achieve significantly large deformation when they are subject to electrical stimuli. Soft dielectrics find applications in human-like robots [6,7], stretchable electronics , actuators , energy harvesters , among others. Large deformations of soft dielectrics are often accompanied by electromechanical instabilities including pull-in instability , wrinkling and the creasing , the electro-creasing to cratering instability , electro-cavitation , among others.\n\nHistorically, instabilities are often thought to cause “failure” and usually avoided. The pull-in instability, for example, is suppressed in order to enhance the actuation strain and the electrical energy density of soft dielectrics. More recently, research has increasingly also been directed at how electromechanical instabilities of soft dielectrics can be harnessed for various applications such as giant actuation strain, dynamic surface patterning, and energy harvesting [28,29].\n\nA commonly used actuator, for example, is a film of dielectric elastomer coated with compliant electrodes on its surfaces. Upon application of a voltage difference between the two electrodes, the Maxwell stress from the electric field compresses the film in the thickness direction, causes expansion in the plane, and creates a large actuation strain. The thinning of the film increases the intensity of the electric field in the material. When the film thickness decreases to a certain threshold value, the film is unable to sustain the Maxwell stress and the pull-in instability occurs. Exploitation of soft dielectric films in applications requires a thorough understanding of large deformation mechanics and the electromechanical instabilities induced by voltages and mechanical forces. To this end, numerous theoretical analyses [10,16,22,3033] have been carried on this subject matter.\n\nIn a prior work , Zhao and Suo analyzed the electromechanical stability of a film of dielectric elastomer subject to tensile forces in its plane and a voltage difference across its thickness. From the principle of minimum energy, they studied the stability of the homogeneously2 deformed film by examining the positive definiteness of the Hessian matrix. They showed that prestress can significantly enhance the stability of the homogeneously deformed film and markedly increase the actuation stretch. We remark that Zhao and Suo assumed a homogeneously deformed film throughout their equilibrium state and stability analysis. Subsequently, this assumption of homogeneous deformation has been widely used in other works [10,24,25,30,31,34,35].\n\nThe aforementioned assumption of a homogeneous deformation imposes the restriction that the upper and bottom surfaces of the dielectric thin film remain perfectly plane. Hence, nonhomogeneous deformation and the effects of the geometry of the dielectric film, like the thickness or the aspect ratio, on the electromechanical instability are excluded. In a recent work, Dorfmann and Ogden investigated the instability (buckling) of an infinite plate of electroelastic material by analyzing its incremental elastic deformation. In another work, Dorfmann and Ogden studied the surface instability of a half-space subject to both mechanical compression and an electric field normal to its surface.\n\nIn this work, we present a complete linearized bifurcation analysis for electromechanical instability in a finite block of a soft dielectric material subject to physically reasonable boundary conditions,3 and compare it with the response of a thin film and half-space. An elastic finite block is often used to study the mechanical behavior of elastic materials at finite strain, such as the instability and post-buckling of a mechanical compressed elastic block, and the buckling of a compressible magnetoelastic block . Unlike a half-space [1,36] or an infinite long plate , a finite block has measurable length quantities, such as the aspect ratio, and allows for physically well-defined boundary conditions on all its surfaces. Hence, the effect of the boundary conditions due to finite dimensions on the electromechanical instability can be addressed in the present work by analyzing a finite block. Compared to the in-plane biaxial dead loads on the dielectric film, as used in past works , we employ displacement-controlled boundary conditions on the two sides of the finite block which allows a facile consideration of both tension and compression.4 Based on the implicit function theorem [40,41], we present an analysis of the onset of bifurcation from the trivial solution of a finite block of dielectric elastomer subject to mechanical loads (compression or extension) and a voltage across its thickness. Although our analysis of electromechanical instability is applicable to a general elastic dielectric elastomer, we present closed-form expressions for the special case of ideal neo-Hookean dielectrics.\n\nThe paper is organized as follows. In Sec. 2, we present the general formulation for the electrostatic problem of a finite block of a dielectric elastomer subject to electromechanical loads. The linear bifurcation analysis is presented in Sec. 3, where the incremental boundary-value problem is obtained by linearizing the equations of equilibrium with respect to deformation and the polarization. In Sec. 4, we obtain the solutions of the homogeneous deformation and the incremental boundary-value problem, and discuss the onset of bifurcation from the trivial solution. Finally, in Sec. 5, we compare our analytical results with Euler's predictions for the buckling of both mechanically and electromechanically compressed slender block and discuss the pertinent physical insights.\n\nFormulation\n\nDomain and Boundary Conditions.\n\nConsider a finite block of an elastic dielectric (see Fig. 1). Assuming plane-strain condition in the X3 direction, the dielectric block in the reference configuration can be represented by\n\n$ΩR={X∈ℝ2:0≤X1≤l1,−l22≤X2≤l22}$\n(1)\nwhere X1 and X2 are the Cartesian coordinates, l1 is the length, and l2 is the height of the dielectric block. The boundary $∂ΩR$ of ΩR consists of four parts\n$Sl={X∈ΩR:X1=0}, Sr={X∈ΩR:X1=l1}Su={X∈ΩR:X2=l22}, Sb={X∈ΩR:X2=−l22}$\n(2)\nThe deformation of the block is expressed by a smooth function $x:ΩR→ℝ2$, and the constraint of incompressibility requires a unit Jacobian, such that\n$J=detF=1$\n(3)\n\nwhere $F=∇x=(∂x/∂X1)e1+(∂x/∂X2)e2$ is the deformation gradient in two dimensions, and $ei$, i = 1, 2, are the unit vectors in the Xi directions.\n\nA few comments regarding the boundary conditions are in order. For a dielectric elastomer film, a voltage is usually applied across the top and bottom surfaces and in-plane tensile dead loads are introduced. Such kinds of boundary conditions are used for dielectric elastomers that work in uniaxial actuation mode, such as in spring-roll actuators . In this paper, we apply a potential difference between the upper and lower surfaces, but mimic the physical situation where the dielectric block is controlled by a loading device that stretches or compresses the block in the X1 direction. For example, the loading device can be made of two well-lubricated, rigid plates (see Fig. 1) that are in contact with the side surfaces $Sl∪Sr$ with rollers. During the compression/tension process, there is no rotation of the two plates in order to ensure that the compressive/tensile stresses on the rigid plates are always along the X1 direction. The mechanical and electrostatic boundary conditions on the side surfaces $Sl∪Sr$ are defined by\n$x·e1=λX1, D̃·e1=0 on Sl∪Sr$\n(4)\nwhere $λ>0$ is the prescribed stretch along the X1 direction and $D̃$ is the nominal electric displacement. Meanwhile, the mechanical and electrostatic boundary conditions on the upper and lower surfaces $Su∪Sb$ are\n$t̃e=0, ξ=ξb on Su∪Sb$\n(5)\n\nwhere $t̃e$ is the surface traction and ξ is the voltage. Here, the prescribed voltages are $ξb=V$ on the upper surface $Su$ and $ξb=0$ on the lower surface $Sb$ (see Fig. 1).\n\nEquations of Electrostatics of a Deformable Media.\n\nIn the deformed domain Ω, the true electric field is denoted by e, the true electric displacement by d, and the polarization by p. In the absence of free charges, currents, and magnetic fields, the Maxwell equations reduce to\n$curl e=0, div d=0, d=ϵ0e+p in Ω$\n(6)\n\nwhere ϵ0 is the vacuum permittivity. The curl, the divergence, and the gradient operators in the current configuration are denoted by “curl,” “div,” and “grad,” respectively. In contrast, the corresponding operators in the reference configuration are denoted by “Curl,” “Div,” and “$∇$.” The equality, $curl e=0$ in Eq. (6), indicates that there exists a scalar potential (voltage) ξ such that $e=−grad ξ$.\n\nTo represent e, d, and p in the undeformed domain ΩR, we use the composition maps \n$E=e°x, D=d°x, P=p°x$\n(7)\nIn the undeformed domain ΩR, the nominal electric field is denoted by $Ẽ$, the nominal electric displacement by $D̃$, and the nominal polarization by $P̃$. We define the relations of the true fields in Eq. (7) and the nominal fields in ΩR as [42,43]\n$Ẽ=FTE, D̃=JF−1D, P̃=JP$\n(8)\nwhere $FT$ and $F−1$ are the transpose and the inverse of the deformation gradient F, respectively, and $J=detF$ is the Jacobian. Maxwell's equations (Eq. (6)) in ΩR are\n$Curl Ẽ=0, Div D̃=0, D̃=F−1(ϵ0JF−TẼ+P̃)$\n(9)\n\nwhere $Ẽ=−∇ξ$.\n\nCombining Eq. (8), an alternative form of Eq. (9) in ΩR can be written as\n$FTE=−∇ξ, Div D̃=0, FD̃=ϵ0JE+P̃$\n(10)\n\nFree Energy of the System.\n\nElectromechanics of deformable dielectrics can be formulated in a variety of ways cf. Refs. [36,42,43] for just a few examples. In this paper, we follow the energy formulation of continuum magneto-electro-elasticity as described by Liu . The notion of invoking a minimum energy principle with Maxwell's equations as a constraint has roots in an earlier work on micromagnetism and ferroelectrics .\n\nSubject to both mechanical and electrical loads, the total free energy (see, for example, Refs. [42,46]) of the system in Fig. 1 is given by\n$F[x,P̃]=U[x,P̃]+Eelect[x,P̃]$\n(11)\nHere, $U[x,P̃]$ is the internal energy\n$U[x,P̃]=∫ΩRΨ(F,P̃)$\n(12)\nwhere $F=∇x$ and $Ψ(F,P̃)$ is the internal energy density. The electric energy, $Eelect[x,P̃]$, in Eq. (11) is\n$Eelect[x,P̃]=ϵ02∫ΩRJ\|E\|2+∫Su∪SbξD̃·N$\n(13)\n\nwhere $J=det∇x$ is the Jacobian and N is the unit normal to the surfaces $Su∪Sb$. The relations among E, $D̃$, and $P̃$ in Eq. (13) are given by Eq. (10). Note that the mechanical work done by the loading device on the side surfaces $Sl∪Sr$ is not included into the total free energy due to the nominal displacement-controlled boundary condition (Eq. (4)1).\n\nFirst Variation of the Free Energy.\n\nWhen the aforementioned electromechanical system is in equilibrium at a deformation x and a polarization $P̃$, the first variation of the energy functional $F[x,P̃]$ must vanish (subject to the constraint imposed by Maxwell's equations). Since there exist two functions $x:ΩR→Ω$ and $P̃:ΩR→ℝ2$ in $F[x,P̃]$, the vanishing of the first variation requires that both the first variations with respect to x and $P̃$ must be zero (see Appendix A for details).\n\nVariation of Polarization.\n\nThe first variation of $F[x,P̃]$ in Eq. (11) with respect to the polarization $P̃$ gives\n$∂Ψ∂P̃−E=0 in ΩR$\n(14)\n\nwhere $E=−F−T∇ξ$. The detailed derivation of Eq. (14) is given in Appendix A1.\n\nVariation of Deformation.\n\nVanishing of the first variation of $F[x,P̃]$ in Eq. (11) with respect to the deformation x yields the Euler–Lagrange equation (see Appendix A2 for details)\n$Div(∂Ψ∂F+Σ̃−qF−T)=0 in ΩR$\n(15)\nand the natural boundary conditions\n$(∂Ψ∂F+Σ̃−qF−T)e1=se1 on Sl∪Sr$\n(16)\n\n$(∂Ψ∂F+Σ̃−qF−T)e2=0 on Su∪Sb$\n(17)\nwhere q is the hydrostatic pressure required by the incompressibility constraint (Eq. (3)), s is the normal stress on the side surfaces $Sl∪Sr$, and $Σ̃$ is the so-called Piola–Maxwell stress defined by\n$Σ̃=E⊗D̃−ϵ0J2\|E\|2F−T$\n(18)\nEquations (10), (14)(17), along with the constraint of incompressibility (Eq. (3)) and the boundary conditions (Eqs. (4) and (5)), form a boundary-value problem, whose solution includes all the possible equilibrium solutions for a finite block of soft dielectric subject to mechanical and electrical loads. The aforementioned boundary-value problem can be compactly summarized as:\n$DivT=0, ∂Ψ∂P̃−E=0, detF=1FTE=−∇ξ, DivD̃=0 FD̃=ϵ0JE+P̃} in ΩR$\n(19)\n\n$x1=λX1, D̃·e1=0, Te1=se1 on Sl∪Sr$\n(20)\n\n$ξ=ξb, Te2=0 on Su∪Sb$\n(21)\nwhere the total nominal stress T is\n$T=∂Ψ∂F+Σ̃−qF−T$\n(22)\n\nOnset of Electromechanical Buckling\n\nOf interest here is the condition for the onset of buckling of the dielectric block. Mathematically, buckling is governed by the onset of bifurcation in the trivial solution to the boundary-value problem (Eqs. (19)(21)). Based on the implicit function theorem [40,41], the equilibrium equations have a nontrivial solution bifurcating from its trivial solution only if the linearized equations of equilibrium possess a nonzero solution. It is obvious that the onset of bifurcation depends on the applied mechanical and electrical loads. The linearized equations describe the response of the dielectric block, in a state of equilibrium, to infinitesimal increments of the deformation and the polarization.\n\nLinearization With Respect to Both the Deformation and the Polarization.\n\nLet $x$ and $P̃$ be the infinitesimal increments of the deformation x and the polarization $P̃$, respectively. Other increments depend on $x$ and $P̃$ at $(x,P̃)$. We denote other linearized increments (omitting higher terms) by taking advantage of the superscripts. For example, the total linearized increment of a general field Θ is denoted by $Θ$ (see Appendix B for details). $Θ$ is usually the sum of two increments $Θ†$ and $Θ‡$, which denote, respectively, the increments related to the deformation and the polarization. Since the deformation gradient F and the Jacobian J are independent of the polarization, we have the linearized increments\n$F=∇x, J=JF−T·∇x,(FT)=(∇x)T, (F−T)=−F−T(∇x)TF−T$\n(23)\nIn contrast, the total linearized increments of the electric field E, the nominal electric displacement $D̃$, and the Piola–Maxwell stress $Σ̃$ consist of two parts:\n$E=E†+E‡, D̃=D̃†+D̃‡, Σ̃=Σ̃†+Σ̃‡$\n(24)\nCombining Eq. (24) and $Σ̃$ in Eq. (18), we have the relation between the linearized increments (see Appendix B2)\n$Σ̃=E⊗D̃+E⊗D̃ −ϵ02{2J(E·E)F−T+\|E\|2[J(F−T)+JF−T]}$\n(25)\nSimilarly, the linearization of the boundary-value problem (Eqs. (19)(21)) can be written as\n$DivT=0, ∂2Ψ∂P̃∂F[F]+∂2Ψ∂P̃2P̃−E=0F−T·F=0, (FT)E+FTE=−∇ξDivD̃=0, FD̃+FD̃=ϵ0JE+ϵ0JE+P̃}in ΩR$\n(26)\n\n$x·e1=0, D̃·e1=0, Te1=se1 on Sl∪Sr$\n(27)\n\n$ξ=0, Te2=0 on Su∪Sb$\n(28)\nwhere $T$ is the linearized increment of the total nominal stress T in Eq. (22), given by\n$T=∂2Ψ∂F2[F]+∂2Ψ∂F∂P̃P̃+Σ̃−qF−T+qF−T(FT)F−T$\n(29)\n\nWe remark that the linearized boundary-value problem (Eqs. (26)(28)) considers the total increments including both the incremental deformation and the incremental polarization. The condition of the nonzero solution of $(x,P̃)$ in Eqs. (26)(28) determines the onset of the electrical buckling—and more precisely the onset of bifurcation from the solution $(x,P̃)$ —of a finite block of dielectric elastomer subject to electromechanical loads.\n\nLinearization With Respect to Only the Deformation.\n\nA further simplification of Eqs. (26)(28) may be made by considering the linearization with respect to only the deformation. That is, we introduce an infinitesimal increment $x$ of the deformation but a zero increment of the polarization when we linearize the boundary-value problem (Eqs. (19)(21)) at $(x,P̃)$. This simplification, of course, will yield a narrower solution space that is a subspace of the solution space considering both the incremental deformation and polarization; however, it significantly simplifies the analysis and also provides important results for electromechanical buckling. Therefore, by letting $P̃=0$ the total increments (with superscript “”) in Eqs. (24) and (25) reduce to the increments (with superscript “”) with respect to the deformation, the linearized boundary-value problem (Eqs. (26)(28)) can be reduced to\n$DivT†=0, ∂2Ψ∂P̃∂F[F]−E†=0F−T·F=0, (FT)E+FTE†=−∇ξ†DivD̃†=0, FD̃†+FD̃=ϵ0JE†+ϵ0JE} in ΩR$\n(30)\n\n$x·e1=0, D̃†·e1=0, T†e1=s†e1 on Sl∪Sr$\n(31)\n\n$ξ†\|SbSu=0, T†e2=0 on Su∪Sb$\n(32)\nwhere\n$T†=∂2Ψ∂F2[F]+Σ̃†−qF−T+qF−T(FT)F−T$\n(33)\nand\n$Σ̃†=E†⊗D̃+E⊗D̃†−ϵ02{2J(E·E†)F−T+\|E\|2[J(F−T)+JF−T]}$\n(34)\n\nWe remark here that the total incremental boundary-value problem (Eqs. (26)(28)) and the reduced incremental boundary-value problem (Eqs. (30)(32)) are valid for all incompressible elastic soft dielectrics. In the following, we will adopt the neo-Hookean constitutive law to generate specific results.\n\nNeo-Hookean Dielectrics\n\nIn the following, we consider incompressible neo-Hookean dielectrics, whose strain energy function [42,46] under the plane strain assumption is given by\n$Ψ(F,P̃)=μ2(\|F\|2−2)+\|P̃\|22J(ϵ−ϵ0)$\n(35)\n\nwhere μ is the shear modulus, and ϵ and ϵ0 are, respectively, the permittivities of the dielectric elastomer and the vacuum. Note that the second term on the right-hand side of Eq. (35) reflects the usual linear dielectric behavior, that is, the permittivity ϵ of the dielectric elastomer is independent of the deformation.\n\nThe derivatives of the strain energy function (Eq. (35)) are given by\n$∂Ψ∂F=μF−\|P̃\|22J(ϵ−ϵ0)F−T, ∂Ψ∂P̃=P̃J(ϵ−ϵ0)∂2Ψ∂F2=μI4+\|P̃\|22J(ϵ−ϵ0)(F−T⊗F−T−∂F−T∂F)∂2Ψ∂P̃2=I2J(ϵ−ϵ0),∂2Ψ∂F∂P̃=−F−T⊗P̃J(ϵ−ϵ0), ∂2Ψ∂P̃∂F=−P̃⊗F−TJ(ϵ−ϵ0)}$\n(36)\n\nwhere $I4$ and $I2$ are, respectively, the fourth- and second-order identity tensors in two dimensions.\n\nHomogeneous Deformation.\n\nSubstituting Eq. (36) into the boundary-value problem (Eqs. (19)(21)), a trivial solution that corresponds to homogeneous deformation is given by\n$x0(X)=λX1e1+λ−1X2e2, P̃0(X)=−(ϵ−ϵ0)λẼ0e2$\n(37)\nwhere $Ẽ0=V/l2$ and other corresponding quantities are\n$F0(X)=∇x0(X)=λe1⊗e1+λ−1e2⊗e2E0(X)=−λẼ0e2, D̃0(X)=−ϵλ2Ẽ0e2}$\n(38)\nThe Piola–Maxwell stress tensor (Eq. (18)) is\n$Σ̃0:=[−ϵ02λẼ0200(ϵ−ϵ02)λ3Ẽ02]$\n(39)\nthe hydrostatic pressure (i.e., the Lagrange multiplier) is\n$q0=μλ−2+ϵ2λ2Ẽ02$\n(40)\nthe total stress tensor (Eq. (22)) is\n$T0:=[μ(λ−λ−3)−ϵλẼ02000]$\n(41)\nand the compressive/tensile stress on $Sl∪Sr$ is\n$s0=μ(λ−λ−3)−ϵλẼ02$\n(42)\n\nNote that a negative/positive s0 in Eq. (42) represents the nominal compressive/tensile stress on the side surfaces $Sl∪Sr$ in the reference configuration. In contrast, the true compressive/tensile stress in the current configuration is given by $λs0$.\n\nFigure 2 shows how the electric field affects the mechanical behavior of the homogeneous deformation of a finite block. The dimensionless compressive/tensive stress $s0/μ$ and electric field $Ẽ0ϵ/μ$ are used. In the absence of the electric field (i.e., $Ẽ0ϵ/μ=0$) in Fig. 2(a) (or in Eq. (42)), for example, a prescribed stretch $λ>1$ corresponds to a tensile stress $s0>0$, while a stretch $λ<1$ corresponds to a compressive stress $s0<0$ on the side surfaces $Sl∪Sr$ of the block.\n\nThe electric field in Eq. (42) will decrease the nominal stress s0 on the side surfaces. This is because the Maxwell stress in Eq. (39) will make the block decrease its height l2 (due to a positive component of the Maxwell stress in the X2 direction) and increase its length l1 (due to a negative component of the Maxwell stress in the X1 direction). However, the two lubricated rigid plates exert an additional compressive stress on the side surfaces to hinder the extension of the block. Therefore, at a prescribed stretch λ in Fig. 2(a), the electric field $Ẽ0ϵ/μ$ can decrease the nominal stress vector $s0/μ$ (or the true stress vector $λs0/μ$). For example, at a prescribed stretch $λ>1$, the increase of the electric field $Ẽ0ϵ/μ$ can cause the nominal stress $s0/μ$ in Fig. 2(a) (or the true stress $λs0/μ$ in Fig. 2(b)) decrease from positive (tensile stress) to negative (compressive stress). If we were to ignore electrical breakdown, the continually increasing compressive stress will eventually force the block to buckle.\n\nTo further illustrate the effects of the electric field on deformation, we consider two special cases: a prescribed stretch λ = 1 and a zero nominal stress $s0=0$.\n\nIn the first case, the block is undeformed prior to electromechanical buckling. This is because of the constraint of incompressibility and the plane strain assumption in our model, leading to the stretch ratios $λ1=λ=1, λ2=λ−1=1$, and $λ3=1$. Although the block is undeformed under the electric field, it is no longer a stress-free state. The nominal compressive stress s0 in Eq. (42) is $s0=−ϵẼ02$, which is a quadratic function of the nominal electric field (see Fig. 3(a)). Under zero electric field, the block is stress-free, corresponding to the origin $(Ẽ0ϵ/μ,s0/μ)=(0,0)$. It is clear from Fig. 3(a) that the parabola opens downward and the axis of symmetry is $Ẽ0ϵ/μ=0$. In this case, the electric field always induces a compressive state in the block. With a continuously increasing electric field, the block eventually will buckle. Note that only electromechanical buckling is considered in this paper for the constrained deformation—other instabilities including the electrical breakdown, the electro-creasing to cratering instabilities, and the electrocavitation instability [14,1820] are beyond the scope of this paper.\n\nIn the second case, the homogeneously deformed block is stress-free. With $s0=0$ in Eq. (42), the relation between the stretch and the electric field becomes $λ=(1−ϵẼ02/μ)−1/4$ (see Fig. 3(b)). Without considering the electrical breakdown and the pull-in instability , the stretch λ, mathematically, can increase from one to infinity as the dimensionless electric field $Ẽ0ϵ/μ$ increases from zero to one.\n\nIn the following, we will analyze electromechanical buckling by studying the solution of the incremental boundary-value problem (Eqs. (30)(32)) at the homogeneous solution (Eqs. (37)(42)).\n\nIncremental Boundary-Value Problem.\n\nLet us first address Eq. (30) in ΩR. The increments of the Jacobian $J$ and the electric field $E†$ in Eq. (30) must vanish due to the constraint of incompressibility, $F−T·∇x=0$, namely\n$J=JF−T·∇x=0$\n(43)\n\n$E†=−P̃⊗F−TJ(ϵ−ϵ0)[∇x]=−(F−T·∇x)P̃J(ϵ−ϵ0)=0$\n(44)\nand the two relations, $(FT)E+FTE†=−∇ξ†$ and $FD̃†+FD̃=ϵ0JE†+ϵ0JE$ in Eq. (30), reduce to\n$∇ξ†=−(∇x)TE, D̃†=−F−1FD̃$\n(45)\nThen, the incremental Piola-Maxwell stress (Eq. (34)) reduces to\n$Σ̃†=E⊗D̃†−ϵ0J\|E\|22(F−T)$\n(46)\nNote that Eqs. (43)(46) hold for all kinematically admissible deformations of incompressible neo-Hookean dielectrics, including the homogeneous deformation. Moreover, the constraint makes the divergence of $D̃†$ in Eq. (45) vanish automatically in the case of homogeneous deformation, such that\n$DivD̃†=−Div(F0−1FD̃0)=−D̃0·∇(F0−T·F)=0$\n(47)\nBased on Eqs. (43)(47) and the homogeneous solution (Eqs. (37)(42)), the linearized boundary-value problem (Eqs. (30)(32)) reduces to\n$DivT†=0, F0−T·∇x=0(∇x)TE0=−∇ξ†, D̃†=−F0−1(∇x)D̃0} in ΩR$\n(48)\n\n$x·e1=0, D̃†·e1=0, T†e1=s†e1 on Sl∪Sr$\n(49)\n\n$ξ†\|SbSu=0, T†e2=0 on Su∪Sb$\n(50)\nwhere the incremental voltage $ξ†$ and the incremental nominal electric displacement $D̃†$ are\n$ξ†=λẼ0x2+ξ0, D̃†=ϵλẼ0(x1,2e1+λ2x2,2e2)$\n(51)\nand the incremental nominal stress $T†$ is\n$T†:=μ[T11†T12†T21†T22†]$\n(52)\nwith\n$T11†=[1+λ−4+(Ẽ0ϵ/μ)2]x1,1−λ−1μ−1qT12†=x1,2+[λ−2+λ2(Ẽ0ϵ/μ)2]x2,1T21†=x2,1+λ−2x1,2, T22†=2x2,2−λμ−1q$\n\nGoverning Equation.\n\nFrom the constraint of incompressibility, $F0−T·∇x=0$, in Eq. (48), we introduce a stream function $ϕ(X1,X2)$\n$x1=λϕ,2(X1,X2), x2=−λ−1ϕ,1(X1,X2)$\n(53)\nwhere the subscript denotes the partial derivative. With Eqs. (52) and (53), $DivT†=0$ in Eq. (48) gives\n$λ2(ϕ,112+ϕ,222)−μ−1q,1=0$\n(54a)\n\n$λ−2(ϕ,111+ϕ,122)+μ−1q,2=0$\n(54b)\nEliminating $q$ in Eq. (54) yields\n$ϕ,1111+(λ4+1)ϕ,1122+λ4ϕ,2222=0$\n(55)\n\nBoundary Conditions on $Sl∪Sr$ and on $Su∪Sb$⁠.\n\nSubstituting Eq. (53) into the boundary conditions on $Sl∪Sr$ in Eq. (49), we have\n$ϕ,2=0, ϕ,22=0, ϕ,22−ϕ,11=0 on Sl∪Sr$\n(56)\n\nwhere the first and the third equations come from the mechanical boundary conditions (i.e., the controlled-nominal displacement $x·e1=0$ and the free shear stress $T†e1=s†e1$), while the second equation corresponds to the electrostatic boundary condition (i.e., $D̃†·e1=0$ in Eq. (49)).\n\nSimilarly, substituting Eq. (53) into the electrostatic boundary condition on $Su∪Sb$ in Eq. (50), we have\n$ϕ,1(X1,l22)=ϕ,1(X1,−l22)$\n(57)\nMoreover, substituting Eq. (53) into the mechanical boundary conditions on $Su∪Sb$ in Eq. (50) yields\n$[λ−4+(Ẽ0ϵ/μ)2]ϕ,11−ϕ,22=0 2μϕ,12+λ2q=0} on Su∪Sb$\n(58)\n\nSolution of the Incremental Boundary-Value Problem.\n\nDue to the boundary conditions on $Sl∪Sr$ in Eq. (56), the solution of $ϕ(X1,X2)$ in Eq. (55) admits the series form\n$ϕ(X1,X2)=∑m=1∞Ym(X2)sin(kmX1)+BX1+C,$\n(59)\nwhere $km=(mπ/l1), m=1,2,3,…$, B and C are constants. C is physically irrelevant and may be chosen to be zero. B is related to rigid body motion of the deformed dielectric block. Substituting Eq. (59) into Eq. (53), we have\n$x1=λ∑m=1∞Ym′(X2)sin(kmX1)$\n(60a)\n\n$x2=−λ−1{∑m=1∞kmYm(X2)cos(kmX1)+B}$\n(60b)\nwhere the prime denotes the derivative with respect to X2. Substituting Eq. (59) into Eq. (54) and performing the integration with respect to X1, we obtain\n$q=−μλ2∑m=1∞km−1[Ym‴(X2)−km2Ym′(X2)]cos(kmX1)$\n(61)\nThen the increment $s†$ of the nominal stress in Eq. (49) is\n$s†=μλ{∑m=1∞[km−2Ym‴(X2) +(λ−4+(Ẽ0ϵ/μ)2)Ym′(X2)]km cos(kmX1)}$\n(62)\nSubstituting Eq. (59) into the electrostatic boundary conditions on $Su∪Sb$ in Eq. (57) and using the orthogonality relation of Fourier series yields\n$Ym(l22)=Ym(−l22)$\n(63)\nMoreover, substituting Eqs. (53), (59), and (61) into the mechanical boundary conditions on $Su∪Sb$ in Eq. (58) and using again the orthogonality relation of Fourier series, we obtain\n$Ym″(X2)+[λ−4+(Ẽ0ϵ/μ)2]km2Ym(X2)=0Ym‴(X2)−(1+2λ−4)km2Ym′(X2)=0} on Su∪Sb$\n(64)\nFinally, substituting Eq. (59) into the governing equation (55), we find that $Ym(X2)$ yields the following fourth-order ordinary differential equation:\n$Ym(4)(X2)−km2(1+λ−4)Ym″(X2)+km4λ−4Ym(X2)=0$\n(65)\nThe general solution of $Ym(X2)$ in Eq. (65) is\n$Ym(X2)={C1mcosh(kmλ−2X2)+C2msinh(kmλ−2X2) +C3mcosh(kmX2)+C4msinh(kmX2) for λ≠1(C¯1m+C¯2mX2)cosh(kmX2) +(C¯3m+C¯4mX2)sinh(kmX2) for λ=1$\n(66)\n\nwhere $Cim$ and $C¯im, i=1,2,3,4$, are constant coefficients.\n\nBifurcation at Varying Stretch $λ≠1$⁠.\n\nSubstituting the general solution to $Ym(X2)$ in Eq. (66)$1$ for $λ≠1$ into Eq. (64), we obtain a system of four linear equations in four unknowns $Cim, i=1,2,3,4$. The system of four equations can be rewritten in a matrix form of $MCm=0$, where M is the 4 × 4 coefficient matrix and $Cm=(C1m,C2m,C3m,C4m)T$. The nonzero solution of $Cm$ requires a zero determinant of M, namely\n$detM=\|M110M1300M220M240M320M34M410M430\|=0$\n(67)\nwhich can be decomposed into a product of two 2 × 2 determinants, such that\n$\|M11M13M41M43\|•\|M22M24M32M34\|=0$\n(68)\nwhere\n$M11=[2λ−4+(Ẽ0ϵ/μ)2]coshmπl22λ2l1M13=[1+λ−4+(Ẽ0ϵ/μ)2]coshmπl22l1M22=(1+λ−4)coshmπl22λ2l1, M24=2λ−2coshmπl22l1M32=[2λ−4+(Ẽ0ϵ/μ)2]sinhmπl22λ2l1M34=[1+λ−4+(Ẽ0ϵ/μ)2]sinhmπl22l1M41=(1+λ−4)sinhmπl22λ2l1, M43=2λ−2sinhmπl22l1$\n\nEquation (68) holds if either of the 2 × 2 determinant vanishes, indicating the possibility of two types of buckling.\n\nFor the first type, the vanishing of the left 2 × 2 determinant in Eq. (68) admits nonzero $C1m$ and $C3m$ but zero $C2m$ and $C4m$, leaving two hyperbolic cosine functions of $X2∈[−l2/2,l2/2]$ in $Ym(X2)$ in Eq. (66)1 and making it become an even function of X2. This type of electrical buckling, of course, satisfies the electrostatic boundary conditions, Eq. (63), since $Ym(X2)$ is an even function. Moreover, the even function, $Ym(X2)$ in Eq. (66)1, makes the perturbed displacement $x1(X1,X2)$ in Eq. (60) become an odd function of X2 and $x2(X1,X2)$ become an even function of X2, such as $x1(X1,X2)=−x1(X1,−X2)$ and $x2(X1,X2)=x2(X1,−X2)$. It is assumed that the constant B in Eq. (60) for the coordinates is appropriately chosen to make $x2(0,0)=0$. Then, the buckling modes of the first type are antisymmetric with respect to the X1 axis. This type of buckling is called an antisymmetric buckling about the X1 axis. For instance, Figs. 4(a) and 5(a) are antisymmetric bifurcation modes with m = 1 and m = 2, respectively.\n\nFor the second type, the right 2 × 2 determinant in Eq. (68) vanishes and then $Ym(X2)$ in Eq. (66)1 has nonzero $C2m$ and $C4m$ but zero $C1m$ and $C3m$. Thus, $Ym(X2)$ only contains two hyperbolic sine functions of X2 (i.e., $Ym(X2)$ becomes an odd function of X2). The perturbed displacement $x1(X1,X2)$ in Eq. (60) is an even function of X2, such that $x1(X1,X2)=x1(X1,−X2)$, while the perturbed displacement $x2(X1,X2)$ in Eq. (60) has the property $x2(X1,X2)+x2(X1,−X2)=−2λ−1B$. If the constant B in Eq. (60) is chosen to be zero for an appropriate fixity condition of coordinates, the perturbed displacement $x2(X1,X2)$ in Eq. (60) becomes an odd function of X2, such as $x2(X1,X2)=−x2(X1,−X2)$. This type of electrical buckling satisfies the mechanical boundary conditions (Eq. (64)), however, it does not satisfy the electrostatic boundary conditions (Eq. (63)) since $Ym(X2)$ is an odd function now. For the mechanical compression, the buckling modes of the second type are symmetric with respect to the X1 axis. This type of buckling is called a symmetric buckling about the X1 axis. The symmetric bifurcation modes with m = 1 and m = 2 are shown, respectively, in Figs. 4(b) and 5(b).\n\nThe critical conditions for the two types of buckling can be explicitly written as\n\nType (i): Antisymmetric\n$(1+λ−4)[1+λ−4+(Ẽ0ϵ/μ)2]tanhmπl22λ2l1 −2λ−2[2λ−4+(Ẽ0ϵ/μ)2]tanhmπl22l1=0$\n(69)\nType (ii): Symmetric\n$(1+λ−4)[1+λ−4+(Ẽ0ϵ/μ)2]tanhmπl22l1 −2λ−2[2λ−4+(Ẽ0ϵ/μ)2]tanhmπl22λ2l1=0$\n(70)\n\nBifurcation at Fixed Stretch λ = 1.\n\nSimilarly, substituting the general solution to $Ym(X2)$ in Eq. (66)2 for λ = 1 into Eq. (64), we obtain a system of four linear equations in four unknowns $C¯im, i=1,2,3,4$. The system of four equations can be written in a matrix form as $M¯C¯m=0$, where $M¯$ is the 4 × 4 coefficient matrix and $C¯m=(C¯1m,C¯2m,C¯3m,C¯4m)T$. A nonzero solution of $C¯m$ requires $detM¯=0$, which, similar to Eq. (68), can be reduced to a product of two 2 × 2 determinants, such that\n$\|M¯11M¯14M¯41M¯44\|•\|M¯22M¯23M¯32M¯33\|=0$\n(71)\nwhere\n$M¯11=[2+(Ẽ0ϵ/μ)2]coshmπl22l1M¯14=2l1mπcoshmπl22l1+l22[2+(Ẽ0ϵ/μ)2]sinhmπl22l1M¯22=l22sinhmπl22l1, M¯23=coshmπl22l1M¯32=2l1mπsinhmπl22l1+l22[2+(Ẽ0ϵ/μ)2]coshmπl22l1M¯33=[2+(Ẽ0ϵ/μ)2]sinhmπl22l1M¯41=sinhmπl22l1, M¯44=l22coshmπl22l1$\nSimilar to the analysis of Eq. (68), the left 2 × 2 determinant in Eq. (71) might correspond to the type of an antisymmetric buckling, while the right one might corresponds to the type of a symmetric buckling. However, the right 2 × 2 determinant in Eq. (71) is always nonzero, indicating the nonexistence of the type of symmetric buckling at λ = 1 regardless of how large the electric field is. While both symmetric and antisymmetric buckling satisfy all the mechanical boundary conditions for purely mechanical compression, only antisymmetric buckling satisfies both the mechanical and electrostatic boundary conditions for the combined electromechanical loading. The critical condition for the antisymmetric buckling of a dielectric block under electric field at λ = 1 in plane strain comes from the vanishing of the left 2 × 2 determinant in Eq. (71), and yields\n$[1+12(Ẽ0ϵ/μ)2]mπl2l1−sinhmπl2l1=0$\n(72)\n\nDiscussion and Conclusions\n\nComparison With Euler's Prediction for the Mechanical Compression.\n\nThe buckling of Euler's column studied by Leonhard Euler in 1757 is one of the classical problems in engineering. The formula derived by Euler gives the critical load at which a long, slender, ideal column is in a state of unstable equilibrium (i.e., even an infinitesimal lateral force will make the column buckle).\n\nConsidering the conditions of end support of the column, Euler's formula can be expressed as\n$F=π2EeIe(Kl1)2$\n(73)\n\nwhere F is the critical force, Ee is the plane strain elastic Young's modulus, Ie is the area moment of inertia of the cross section, l1 is the length of the column, and K is column effective length factor that depends on the conditions of end support. For example, the factor K is 0.5 for both fixed ends while it is 1 for both pinned ends.\n\nThe effective Young's modulus under plane strain is $Ee=4μ$ in Eq. (73) for incompressible neo-Hookean materials with shear modulus μ, and the area moment of inertia is $Ie=l23/12$ for a block with height l2 and unit width. Thus, the critical nominal stress sc from Eq. (73) is\n$sc=Fl2×1=μπ23(Kl1/l2)2$\n(74)\n\nIn contrast to Euler's formula for slender structures, our buckling analysis is valid for a finite compressed elastic block with any aspect ratio $l1/l2$. In the absence of an electric field, Eq. (69) gives the critical stretch λc for the buckling of an incompressible neo-Hookean block subject to a purely mechanical compression. With the relation between the stretch and the nominal stress in Eq. (42), we can obtain the critical nominal stress that corresponds to the critical stretch λc. We remark here that Eq. (69) determines the critical stretch of antisymmetric buckling and Eq. (42) is used to transform the critical stretch into the critical nominal stress for the purpose of a direct comparison with Euler's prediction (Eq. (74)). Note that only the results of the antisymmetric buckling are used to compare with Euler's prediction (Eq. (74)) since the antisymmetric buckling always occurs prior to symmetric buckling. Moreover, only antisymmetric buckling satisfies all the mechanical and electrostatic boundary conditions, while the symmetric buckling satisfies only the mechanical boundary conditions. The detailed discussion of the difference between antisymmetric and symmetric buckling is given in Secs. 5.3 and 5.4.\n\nFigure 6(b) shows the critical nominal stress for the antisymmetric buckling mode m = 2 in Eq. (69) of a finite block, whose buckling pattern is shown schematically in Fig. 5(a). The buckling pattern and the boundary condition on the left and the right surfaces in Fig. 5(a) are very similar to that of the buckling of Euler's column with fixed-fixed ends. Compared with Euler's prediction, the two predicted critical loads of buckling agree well with each other only at a sufficiently large aspect ratio (i.e., $l1/l2>5$). The obvious discrepancy at small aspect ratios is because Euler's analysis is only valid for a slender column.\n\nComparison of Euler's Prediction for Electroelastic Buckling at a Fixed Stretch λ = 1.\n\nCompared with the mechanical compressive stress, the electrostatic Maxwell stress can also make the dielectric block buckle in our model. The special case of a prescribed stretch λ = 1 under an electric field in our model corresponds to zero strain but nonzero stress in the homogeneous solution. The magnitude of the compressive stress in Eq. (42) at λ = 1 is\n$\|s0\|=ϵẼ02$\n(75)\nFrom Euler's prediction (Eq. (74)), when $\|s0\|$ in Eq. (75) increases to the critical value sc in Eq. (74), the dielectric block begins to buckle and the critical nominal electric field for the electromechanical buckling is given by\n$Ẽc=π(Kl1/l2)μ3ϵ$\n(76)\n\nwhere the factor K = 0.5 is for both fixed ends, while K = 1 is for both pinned ends of the Euler column.\n\nIn contrast to the approximation (Eq. (76)) from Euler's formula, our analytical prediction of the critical nominal electric field from Eq. (72) is obtained as\n$Ẽc=(l1mπl2sinhmπl2l1−1)2μϵ$\n(77)\n\nwhere the modes m = 1 and m = 2 are related to two different boundary conditions corresponding K = 1 and K = 0.5 in Eq. (76).\n\nIn Fig. 7, we plot the variation of the dimensionless critical electric field $Ẽcϵ/μ$ with respect to the aspect ratio $l1/l2$ from both Euler's approximation (Eq. (76)) and our analytical prediction (Eq. (77)). The critical electric field decreases monotonously with the increase of the aspect ratio $l1/l2$. This trend agrees with intuition that a more slender block (i.e., a larger aspect ratio $l1/l2$) is more likely to become unstable under external stimuli such as an electric field. In the limiting case $l1/l2→∞$, the critical electric field approaches zero and an exceedingly small electric field can make the block buckle.\n\nBuckling of a Mechanically Compressed Block.\n\nIn contrast to the critical force for Euler's column, the critical stretch (or strain) is often used to define the critical conditions for the buckling of finite blocks or surface instability of soft materials [1,33,3638,]. In Biot's half-space problem , the critical stretch for surface instability of a homogeneous neo-Hookean half-space under plane strain compression is 0.544 at which all the wavelengths become unstable. Later, Levinson studied the stability of a compressed block in the current configuration. Recently, Dorfmann and Ogden studied the surface instability of the homogeneous deformation of a half-space subject to both mechanical and electrical loads by solving the incremental boundary-value problem.\n\nIn our work, the neo-Hookean block is compressed under plane strain by changing the stretch λ. The critical condition of the buckling is determined by either Eq. (69) for antisymmetric buckling or Eq. (70) for symmetric buckling in the absence of electric fields. The critical stretch λc for the mechanical buckling of the compressed block with different aspect ratios $l1/l2$ is plotted in Fig. 8. The critical stretches for antisymmetric/symmetric buckling with different modes $m=1,2,3,5$ are plotted in solid/dashed lines. In particular, the critical stretches for all modes approach 0.544 when the aspect ratio $l1/l2$ decreases to zero (i.e., $l2/l1$ increases to infinity). The critical stretch $λc=0.544$ of this limiting case ($l2/l1→∞$) coincides with Biot's prediction since the limiting case ($l2/l1→∞$) of block is that of a half-space.\n\nFigure 8 also shows that the critical stretch for antisymmetric buckling is always larger than that of symmetric buckling. This means that the antisymmetric buckling in a compressed block occurs prior to symmetric buckling. Indeed, symmetric buckling cannot occur unless the passive constraints are considered to be acting until $λ<0.544$. Therefore, only the antisymmetric buckling is compared with Euler's prediction in the preceding discussion.\n\nBuckling of an Electromechanically Compressed Block.\n\nIn Secs. 5.1, 5.2, and 5.3, we have shown that either the mechanical compression in Figs. 6 and 8 or the electric field in Fig. 7 can make the dielectric block buckle. An obvious extension to these notions is that the combined electromechanical loading ought to make buckling of the block yet easier. For purely mechanical loading, the block in our problem can only buckle under compression ($λ<1$) rather than extension ($λ>1$). Since the electric field can make the block buckle at λ = 1 in Fig. 7, the block may also become buckle in extension (i.e., $λ>1$) under an electric field.\n\nUsing Eqs. (69) and (70), we plot Fig. 9 the critical stretch λc as a function of the aspect ratio $l1/l2$ for the buckling mode m = 2 under different electric fields $Ẽ0ϵ/μ$. The solid lines denote antisymmetric buckling, while the dashed lines represent symmetric buckling. Note that antisymmetric buckling satisfies all the boundary conditions, while the symmetric buckling satisfies all the boundary conditions other than the electrostatic boundary conditions of the perturbed voltage on the upper and lower surfaces. Furthermore, the critical stretches for the antisymmetric buckling (solid lines) rather than the symmetric buckling (dashed lines) in Fig. 9 are very sensitive to the electric fields. When the aspect ratio $l1/l2$ is larger than five, for example, the differences of the critical stretches for the symmetric buckling between the mechanical compression and the electromechanical loading are negligible. On the other hand, since the occurrence of the symmetric buckling is always later than the onset of antisymmetric buckling, in practice only the effects of the electric fields on antisymmetric buckling are of interest.\n\nCompared with the critical stretch for buckling of a mechanically compressed block, the critical stretch that accounts for the electric field is shifted upward for a small aspect ratio $l1/l2$. For example, the critical stretch for the buckling of a mechanically compressed block in the limiting case $l1/l2→0$ is 0.544 while it increases to 0.628 at an electric field $Ẽ0ϵ/μ=2$ in Fig. 9.\n\nIt is clear from Fig. 9 that the electric field can cause the block to buckle more easily in a compressed state ($λ<1$). Moreover, the electric field can make the block buckle even if the block is in extension ($λ>1$).\n\nWe know that both mechanical compression and the electric field can make the block buckle. For antisymmetric buckling with mode m = 2, the variation of the critical stretch λc with respect to the critical electric field $Ẽcϵ/μ$ is plotted in Fig. 10. It is obvious that a slender block (i.e., with high aspect ratio $l1/l2$) is more likely to buckle when it is subject to a combined loading. For example, at a zero electric field (i.e., $Ẽ0ϵ/μ=0$), the critical stretch is slightly less than one in the case of a large aspect ratio $l1/l2=10$, while it approaches 0.544 at an aspect ratio $l1/l2=1$. For each aspect ratio $l1/l2$ in Fig. 10, the critical stretch λc increases monotonically with the increase of $Ẽcϵ/μ$. It clearly shows how the electric field makes the block buckle in an extended state (i.e., $λc>1$). We finally remark that for actual applications, electric breakdown should also be considered and the comparison of the critical electric fields between the electric breakdown and the electrical buckling is needed for a safe design of electrical devices.\n\nIn summary, for a mechanical compression without electric field, the block mathematically exhibits two types of buckling modes, i.e., antisymmetric and the symmetric buckling, however, the antisymmetric buckling will always precede the other. Our results, in the asymptotic limit of large aspect ratio, agree well with Euler's prediction for the buckling of a slender block and, furthermore, at a zero aspect ratio are the same as Biot's critical strain of surface instability of a compressed homogeneous half-space of a neo-Hookean material. For the case where electric fields are included, aside from similar interesting asymptotic connection to Euler's formula, we find that the electric field can cause the block to buckle more easily in a compressed state, and the electric field can even cause the block to buckle in a state of tension.\n\nAcknowledgment\n\nFinancial support from the M.D. Anderson Professorship, NSF CMMI Grant No. 1463339 and NPRP award [NPRP 6-282-2-119] from the Qatar National Research Fund (a member of The Qatar Foundation).\n\nAppendix A: First Variation of the Energy Functional\n\nThe infinitesimal variations of the deformation $x=x(X)$ and the polarization $P̃=P̃(X)$ are denoted, respectively, by $δx$ and $δP̃$, such that\n$δx=η1xd, δP̃=η2P̃p$\n(A1)\n\nwhere $η1,η2∈ℝ$ and $max{\|η1\|,\|η2\|}≪1$, and $xd$ and $P̃p$ are two smooth variations.\n\nThe deformation $x=x(X)$ and the polarization $P̃=P̃(X)$ may not appear directly in the energy functional including the deformation gradient $F=∇x$, the Jacobian $J=det∇x$, the voltage ξ, and the nominal electric displacement $D̃$, among others. Thus, we perform their first variations implicitly and only keep the first-order terms of η1 and η2, such that\n$F→F+η1Fd, J→J+η1Jd, ξ→ξ+η1ξd+η2ξp,D̃→D̃+η1D̃d+η2D̃p, E→E+η1Ed+η2Ep$\n(A2)\nwhere the subscripts “d” and “p” denote, respectively, the variations related to the deformation and the polarization. For example, $Fd$ and Jd in Eq. (A2) are\n$Fd=ddη1∇(x+η1xd)\|η1=0=∇xd,Jd=ddη1det(F+η1Fd)\|η1=0=JF−T·∇xd$\n(A3)\nSubstituting Eq. (A2) into the Maxwell equation (Eq. (10)) and taking partial derivatives with respect to η1 and η2 at $η1=η2=0$, respectively, we have\n$DivD̃d=DivD̃p=0 in ΩR$\n(A4)\nand the relations\n$FTEd+(FT)dE=−∇ξd, FTEp=−∇ξp,FD̃d+FdD̃=ϵ0JEd+ϵ0JdE, FD̃p=ϵ0JEp+P̃p$\n(A5)\nWith Eqs. (A1) and (A2), the variations of Eqs. (4) and (5) are\n$xd·e1=0, D̃d·e1=D̃p·e1=0 on Sl∪Sr$\n(A6)\n\n$ξd=ξp=0 on Su∪Sb$\n(A7)\n\nFirst Variation With Respect to the Polarization\n\nThe first variation of the energy functional (Eq. (11)) with respect to the polarization $P̃$ is\n$ddη2F[x,P̃+η2P̃p]\|η2=0=ddη2U[x,P̃+η2P̃p]\|η2=0+ddη2Eelect[x,P̃+η2P̃p]\|η2=0=∫ΩR∂Ψ∂P̃·P̃p+ϵ0∫ΩRJE·Ep +∫Su∪Sb(ξpD̃·N+ξD̃p·N)$\n(A8)\nWith Eqs. (A4)(A7) and the divergence theorem, Eq. (A8) becomes\n$ddη2F[x,P̃+η2P̃p]\|η2=0=∫ΩR(∂Ψ∂P̃·P̃p+ϵ0JE·Ep)+∫∂ΩRξD̃p·N=∫ΩR(∂Ψ∂P̃·P̃p+ϵ0JE·Ep)+∫ΩR(ξDivD̃p+D̃p·∇ξ)=∫ΩR(∂Ψ∂P̃·P̃p+ϵ0JE·Ep)−∫ΩRE·FD̃p=∫ΩR(∂Ψ∂P̃·P̃p+E·(ϵ0JEp−FD̃p))=∫ΩR(∂Ψ∂P̃−E)·P̃p$\n(A9)\n\nBased on the basic lemma of calculus of variations, vanishing of Eq. (A9) gives Eq. (14).\n\nFirst Variation With Respect to the Deformation\n\nWe introduce a Lagrange multiplier function $q:ΩR→ℝ2$ to address the variation of a constrained problem such that the deformation x is subject to the constraint of incompressibility $J=det∇x=1$. The modified energy functional of Eq. (11), including the Lagrangian multiplier q to enforce incompressibility, is\n$F̂[x,P̃]=∫ΩR(Ψ(∇x,P̃)+ϵ02J\|E\|2−q(J−1))+∫Su∪SbξD̃·N$\n(A10)\nThe first variation of Eq. (A10) with respect to the deformation x is\n$ddη1F̂[x+η1xd,P̃]\|η1=0=∫ΩR(∂Ψ∂F·∇xd+ϵ02Jd\|E\|2+ϵ0JE·Ed−qJd) +∫Su∪Sb(ξdD̃·N+ξD̃d·N)$\n(A11)\nWith Eqs. (A3)(A7) and the divergence theorem, Eq. (A11) becomes\n$ddη1F̂[x+η1xd,P̃]\|η1=0=∫ΩR(∂Ψ∂F·∇xd+ϵ02Jd\|E\|2+ϵ0JE·Ed−qJd) +∫∂ΩRξD̃d·N=∫ΩR(∂Ψ∂F·∇xd+ϵ02Jd\|E\|2+ϵ0JE·Ed−qJd) +∫ΩR(ξDivD̃d+D̃d·∇ξ)=∫ΩR(∂Ψ∂F·∇xd+ϵ02Jd\|E\|2−qJd+E·(ϵ0JEd−FD̃d))=∫ΩR(∂Ψ∂F·∇xd+ϵ02Jd\|E\|2−qJd+E·(FdD̃−ϵ0JdE))=∫ΩR(∂Ψ∂F+E⊗D̃−ϵ0J2\|E\|2F−T−qJF−T)·∇xd=∫ΩR(∂Ψ∂F+Σ̃−qJF−T)·∇xd=∫∂ΩRxd·(∂Ψ∂F+Σ̃−qJF−T)N −∫ΩRxd·Div(∂Ψ∂F+Σ̃−qJF−T)$\n(A12)\n\nwhere $Σ̃$ is the Piola–Maxwell stress defined by Eq. (18). Similar derivations of the Piola–Maxwell stress can also be found in the work [42,46] and many other references. With the boundary condition of $xd$ in Eq. (A6), the vanishing of Eq. (A12) gives Eqs. (15)(17).\n\nAppendix B: Linearized Analysis\n\nSuppose that a deformation x and a polarization $P̃$ have infinitesimal increments $x$ and $P̃, \|\|x\|\|,\|\|P̃\|\|≪1$. For a general field $Θ(x,P̃)$ that is (Fréchet-) differentiable at $(x,P̃)$, we have the expansion in the neighborhood of $(x,P̃)$, such that\n$Θ(x+x,P̃+P̃)=Θ(x,P̃)+∂Θ∂x·x+∂Θ∂P̃·P̃+o(\|\|x\|\|,\|\|P̃\|\|)$\n(B1)\nWe define\n$Θ=Θ†+Θ‡$\n(B2)\nwhere\n$Θ†=∂Θ∂x·x, Θ‡=∂Θ∂P̃·P̃$\n(B3)\n\nHere, $Θ$ denotes the total linearized increment, and $Θ†$ and $Θ‡$ denote the linearized increments with respect to the deformation and the polarization.\n\nWith Eqs. (B1)(B3) and the chain-rule, the linearized increments of the deformation gradient $F=∇x$ and the Jacobian $J=detF=det∇x$ are\n$F=∇x, (F−T)=−F−T(∇x)TF−T,J=∂J∂F·F=JF−T·∇x$\n(B4)\nSimilarly, the linearized increments of other fields at $(x,P̃)$ can be written implicitly as\n$(qsξ⋮ED̃Σ̃)Linearization→(q=q†s=s†+s‡ξ=ξ†+ξ‡⋮E=E†+E‡D̃=D̃†+D̃‡Σ̃=Σ̃†+Σ̃‡)$\n(B5)\n\nWe remark that the linearized increments of the deformation gradient F, the Jacobian J and the Lagrange multiplier q only depend on the increment $x$ at $(x,P̃)$.\n\nLinearized Relation\n\nConsider the linearization of the Maxwell equation (Eq. (10)1) as an example. Substituting the sum of the fields and their linearized increments defined in Eqs. (B4) and (B5) into Eq. (10)1, and ignoring the higher order terms, we obtain\n$FTE→[FT+(FT)](E+E)→FTE+FTE+(FT)E$\n(B6a)\n\n$−∇ξ→−∇(ξ+ξ)=−∇ξ−∇ξ$\n(B6b)\nthen we have\n$FTE+(FT)E=−∇ξ$\n(B7)\n\nOther linearized relations can also be obtained in a similar manner.\n\nLinearized Piola–Maxwell Stress\n\nSubstituting the sum of the fields and their linearized increments defined in Eqs. (B4) and (B5) into the Piola–Maxwell stress (Eq. (18)), and ignoring higher order terms, such that\n$→(E+E)⊗(D̃+D̃) −ϵ0(J+J)2\|E+E\|2[F−T+(F−T)]→E⊗D̃−ϵ0J2\|E\|2F−T+(E⊗D̃+E⊗D̃) −ϵ02{2J(E·E)F−T+\|E\|2[J(F−T)+JF−T]}=Σ̃+Σ̃$\n(B8)\n\nthen we have the linearized increment of the Piola–Maxwell stress $Σ̃*$ in Eq. (25).\n\n2\n\nThe assumption of homogeneous deformation restricts their analysis essentially to tensile loading to avoid buckling instability.\n\n3\n\nBy “physically reasonable,” we imply conditions that are easily realizable in an experimental setup.\n\n4\n\nWe explicitly allow for inhomogeneous deformation modes to study buckling under compression.\n\nReferences\n\nReferences\n1.\nBiot\n,\nM.\n,\n1963\n, “\nSurface Instability of Rubber in Compression\n,”\nAppl. Sci. Res.\n,\n12\n(\n2\n), pp.\n168\n182\n.\n2.\nYang\n,\nS.\n,\nKhare\n,\nK.\n, and\nLin\n,\nP.-C.\n,\n2010\n, “\nHarnessing Surface Wrinkle Patterns in Soft Matter\n,”\n,\n20\n(\n16\n), pp.\n2550\n2564\n.\n3.\nGent\n,\nA.\n, and\nCho\n,\nI.\n,\n1999\n, “\nSurface Instabilities in Compressed or Bent Rubber Blocks\n,”\nRubber Chem. Technol.\n,\n72\n(\n2\n), pp.\n253\n262\n.\n4.\nHong\n,\nW.\n,\nZhao\n,\nX.\n, and\nSuo\n,\nZ.\n,\n2009\n, “\nFormation of Creases on the Surfaces of Elastomers and Gels\n,”\nAppl. Phys. 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J.\n,\n2007\n, “\nDielectric Elastomers as Next-Generation Polymeric Actuators\n,”\nSoft Matter\n,\n3\n(\n9\n), pp.\n1116\n1129\n.\n10.\nMoscardo\n,\nM.\n,\nZhao\n,\nX.\n,\nSuo\n,\nZ.\n, and\nLapusta\n,\nY.\n,\n2008\n, “\nOn Designing Dielectric Elastomer Actuators\n,”\nJ. Appl. Phys.\n,\n104\n(\n9\n), p.\n093503\n.\n11.\nKeplinger\n,\nC.\n,\nKaltenbrunner\n,\nM.\n,\nArnold\n,\nN.\n, and\nBauer\n,\nS.\n,\n2010\n, “\nRöntgen's Electrode-Free Elastomer Actuators Without Electromechanical Pull-in Instability\n,”\nProc. Natl. Acad. Sci.\n,\n107\n(\n10\n), pp.\n4505\n4510\n.\n12.\nKoh\n,\nS. J. A.\n,\nZhao\n,\nX.\n, and\nSuo\n,\nZ.\n,\n2009\n, “\nMaximal Energy That Can Be Converted by a Dielectric Elastomer Generator\n,”\nAppl. Phys. 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Lett.\n,\n95\n(\n3\n), p.\n031904\n.\n18.\nWang\n,\nQ.\n, and\nZhao\n,\nX.\n,\n2013\n, “\nCreasing-Wrinkling Transition in Elastomer Films Under Electric Fields\n,”\nPhys. Rev. E\n,\n88\n(\n4\n), p.\n042403\n.\n19.\nWang\n,\nQ.\n,\nZhang\n,\nL.\n, and\nZhao\n,\nX.\n,\n2011\n, “\nCreasing to Cratering Instability in Polymers Under Ultrahigh Electric Fields\n,”\nPhys. Rev. Lett.\n,\n106\n(\n11\n), p.\n118301\n.\n20.\nWang\n,\nQ.\n,\nSuo\n,\nZ.\n, and\nZhao\n,\nX.\n,\n2012\n, “\nBursting Drops in Solid Dielectrics Caused by High Voltages\n,”\nNat. Commun.\n,\n3\n, p.\n1157\n.\n21.\nHa\n,\nS. M.\n,\nYuan\n,\nW.\n,\nPei\n,\nQ.\n,\nPelrine\n,\nR.\n, and\nStanford\n,\nS.\n,\n2006\n, “\nInterpenetrating Polymer Networks for High-Performance Electroelastomer Artificial Muscles\n,”\n,\n18\n(\n7\n), pp.\n887\n891\n.\n22.\nZhao\n,\nX.\n, and\nSuo\n,\nZ.\n,\n2007\n, “\nMethod to Analyze Electromechanical Stability of Dielectric Elastomers\n,”\nAppl. Phys. Lett.\n,\n91\n(\n6\n), p.\n061921\n.\n23.\nKofod\n,\nG.\n,\n2008\n, “\nThe Static Actuation of Dielectric Elastomer Actuators: How Does Pre-Stretch Improve Actuation?\n,”\nJ. Phys. D: Appl. Phys.\n,\n41\n(\n21\n), p.\n215405\n.\n24.\nLi\n,\nB.\n,\nZhou\n,\nJ.\n, and\nChen\n,\nH.\n,\n2011\n, “\nElectromechanical Stability in Charge-Controlled Dielectric Elastomer Actuation\n,”\nAppl. Phys. Lett.\n,\n99\n(\n24\n), p.\n244101\n.\n25.\nAkbari\n,\nS.\n,\nRosset\n,\nS.\n, and\nShea\n,\nH. R.\n,\n2013\n, “\nImproved Electromechanical Behavior in Castable Dielectric Elastomer Actuators\n,”\nAppl. Phys. Lett.\n,\n102\n(\n7\n), p.\n071906\n.\n26.\nNiu\n,\nX.\n,\nStoyanov\n,\nH.\n,\nHu\n,\nW.\n,\nLeo\n,\nR.\n,\nBrochu\n,\nP.\n, and\nPei\n,\nQ.\n,\n2013\n, “\nSynthesizing a New Dielectric Elastomer Exhibiting Large Actuation Strain and Suppressed Electromechanical Instability Without Prestretching\n,”\nJ. Polym. Sci. Part B: Polym. Phys.\n,\n51\n(\n3\n), pp.\n197\n206\n.\n27.\nJiang\n,\nL.\n,\nBetts\n,\nA.\n,\nKennedy\n,\nD.\n, and\nJerrams\n,\nS.\n,\n2016\n, “\nEliminating Electromechanical Instability in Dielectric Elastomers by Employing Pre-Stretch\n,”\nJ. Phys. D: Appl. Phys.\n,\n49\n(\n26\n), p.\n265401\n.\n28.\nKeplinger\n,\nC.\n,\nLi\n,\nT.\n,\nBaumgartner\n,\nR.\n,\nSuo\n,\nZ.\n, and\nBauer\n,\nS.\n,\n2012\n, “\nHarnessing Snap-Through Instability in Soft Dielectrics to Achieve Giant Voltage-Triggered Deformation\n,”\nSoft Matter\n,\n8\n(\n2\n), pp.\n285\n288\n.\n29.\nShivapooja\n,\nP.\n,\nWang\n,\nQ.\n,\nOrihuela\n,\nB.\n,\nRittschof\n,\nD.\n,\nLópez\n,\nG. P.\n, and\nZhao\n,\nX.\n,\n2013\n, “\nBioinspired Surfaces With Dynamic Topography for Active Control of Biofouling\n,”\n,\n25\n(\n10\n), pp.\n1430\n1434\n.\n30.\nDíaz-Calleja\n,\nR.\n,\nRiande\n,\nE.\n, and\nSanchis\n,\nM.\n,\n2008\n, “\nOn Electromechanical Stability of Dielectric Elastomers\n,”\nAppl. Phys. Lett.\n,\n93\n(\n10\n), p.\n101902\n.\n31.\nXu\n,\nB.-X.\n,\nMueller\n,\nR.\n,\nKlassen\n,\nM.\n, and\nGross\n,\nD.\n,\n2010\n, “\nOn Electromechanical Stability Analysis of Dielectric Elastomer Actuators\n,”\nAppl. Phys. Lett.\n,\n97\n(\n16\n), p.\n162908\n.\n32.\nBertoldi\n,\nK.\n, and\nGei\n,\nM.\n,\n2011\n, “\nInstabilities in Multilayered Soft Dielectrics\n,”\nJ. Mech. Phys. Solids\n,\n59\n(\n1\n), pp.\n18\n42\n.\n33.\nDorfmann\n,\nL.\n, and\nOgden\n,\nR. W.\n,\n2014\n, “\nInstabilities of an Electroelastic Plate\n,”\nInt. J. Eng. Sci.\n,\n77\n, pp.\n79\n101\n.\n34.\nLeng\n,\nJ.\n,\nLiu\n,\nL.\n,\nLiu\n,\nY.\n,\nYu\n,\nK.\n, and\nSun\n,\nS.\n,\n2009\n, “\nElectromechanical Stability of Dielectric Elastomer\n,”\nAppl. Phys. Lett.\n,\n94\n(\n21\n), p.\n211901\n.\n35.\nSuo\n,\nZ.\n,\n2010\n, “\nTheory of Dielectric Elastomers\n,”\nActa Mech. Solida Sin.\n,\n23\n(\n6\n), pp.\n549\n578\n.\n36.\nDorfmann\n,\nA.\n, and\nOgden\n,\nR.\n,\n2010\n, “\nNonlinear Electroelastostatics: Incremental Equations and Stability\n,”\nInt. J. Eng. Sci.\n,\n48\n(\n1\n), pp.\n1\n14\n.\n37.\nLevinson\n,\nM.\n,\n1968\n, “\nStability of a Compressed Neo-Hookean Rectangular Parallelepiped\n,”\nJ. Mech. Phys. Solids\n,\n16\n(\n6\n), pp.\n403\n408\n.\n38.\nTriantafyllidis\n,\nN.\n,\nScherzinger\n,\nW.\n, and\nHuang\n,\nH.-J.\n,\n2007\n, “\nPost-Bifurcation Equilibria in the Plane-Strain Test of a Hyperelastic Rectangular Block\n,”\nInt. J. Solids Struct.\n,\n44\n(\n11\n), pp.\n3700\n3719\n.\n39.\nKankanala\n,\nS.\n, and\nTriantafyllidis\n,\nN.\n,\n2008\n, “\nMagnetoelastic Buckling of a Rectangular Block in Plane Strain\n,”\nJ. Mech. Phys. Solids\n,\n56\n(\n4\n), pp.\n1147\n1169\n.\n40.\nGolubitsky\n,\nM.\n, and\nSchaeffer\n,\nD. G.\n,\n1985\n,\nSingularities and Groups in Bifurcation Theory\n, Vol. 1,\nSpringer\n,\nBerlin\n.\n41.\nChen\n,\nY.-C.\n,\n2001\n, “\nSingularity Theory and Nonlinear Bifurcation Analysis\n,”\nNonlinear Elasticity: Theory and Applications\n, Y. B. Fu and R. W. Ogden, eds.,\nCambridge University Press\n,\nCambridge. UK\n.\n42.\nLiu\n,\nL.\n,\n2014\n, “\nAn Energy Formulation of Continuum Magneto-Electro-Elasticity With Applications\n,”\nJ. Mech. Phys. Solids\n,\n63\n, pp.\n451\n480\n.\n43.\nSuo\n,\nZ.\n,\nZhao\n,\nX.\n, and\nGreene\n,\nW. H.\n,\n2008\n, “\nA Nonlinear Field Theory of Deformable Dielectrics\n,”\nJ. Mech. Phys. Solids\n,\n56\n(\n2\n), pp.\n467\n486\n.\n44.\nJames\n,\nR. D.\n, and\nKinderlehrer\n,\nD.\n,\n1990\n, “\nFrustration in Ferromagnetic Materials\n,”\nContinuum Mech. Thermodyn.\n,\n2\n(\n3\n), pp.\n215\n239\n.\n45.\nShu\n,\nY. C.\n, and\nBhattacharya\n,\nK.\n,\n2001\n, “\nDomain Patterns and Macroscopic Behaviour of Ferroelectric Materials\n,”\nPhilos. Mag. Part B\n,\n81\n(\n12\n), pp.\n2021\n2054\n.\n46.\nDeng\n,\nQ.\n,\nLiu\n,\nL.\n, and\nSharma\n,\nP.\n,\n2014\n, “\nFlexoelectricity in Soft Materials and Biological Membranes\n,”\nJ. Mech. Phys. Solids\n,\n62\n, pp.\n209\n227\n." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9032073,"math_prob":0.9934964,"size":29047,"snap":"2019-43-2019-47","text_gpt3_token_len":6104,"char_repetition_ratio":0.18376201,"word_repetition_ratio":0.068538584,"special_character_ratio":0.20394532,"punctuation_ratio":0.11231185,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99809444,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-14T22:54:42Z\",\"WARC-Record-ID\":\"<urn:uuid:8a25616f-7244-4cae-9471-66d2f3133ab6>\",\"Content-Length\":\"642569\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e6368f51-628a-427b-bcc2-808185a5bbe6>\",\"WARC-Concurrent-To\":\"<urn:uuid:fcfe6bfe-e5bd-4611-8745-8cfb4d8d8bec>\",\"WARC-IP-Address\":\"173.254.190.160\",\"WARC-Target-URI\":\"https://asmedigitalcollection.asme.org/appliedmechanics/article/84/3/031008/422436/Revisiting-the-Instability-and-Bifurcation\",\"WARC-Payload-Digest\":\"sha1:43OFE3CELTDNAZMDZCYYDSVCTBUR2SWA\",\"WARC-Block-Digest\":\"sha1:VCHB2AWAM6VLGDLTVQI2Z2CN7AHC6LSX\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570986655554.2_warc_CC-MAIN-20191014223147-20191015010647-00205.warc.gz\"}"}
http://neurochannels.blogspot.com/2008/02/	[ "## Monday, February 25, 2008\n\n### Large-scale thamamocortical model", null, "While the Blue Brain folk want to construct an incredibly detailed model of a single cortical column, a recent paper by Izhikevich and Edelman (Large-scale model of mammalian thalamocortical systems) reports on a less detailed model of the entire human thalamocortical system.\n\nSome of the details of their model (roughly from large-scale to lower scale) include:\n1. The cortical sheet's geometry was constructed from human MRI data.\n2. Projections among cortical regions were modeled using data from diffusion tensor MRI of the human brain (above image is Figure 1 of the paper showing a subset of such connections).\n3. Synaptic connectivity patterns among neurons within and between cortical layers are based on detailed studies of cat visual cortex (and iterated to all of cortex).\n4. Individual neurons are not modelled using the relatively computationally intensive Hodgkin-Huxely models, but a species of integrate-and-fire neuron that included a variable threshold, short-term synaptic plasticity, and long term spike-timing dependent plasticity.\n5. The only subcortical structure included in the model is the thalamus, but the model does include simple simulated neuromodulatory influences (dopamine, acetylcholine).\n\nTheir model exhibited some very interesting behavior. First, larger-scale oscillatory activity that we see in real brains emerged in the model (e.g., as you would observe via EEG). Also like real brains, the model exhibited ongoing spontaneous activity in the absence of inputs (note this only occurred after an initial 'setup' period in which they simulated random synaptic release events: the learning rule seemed to take care of the rest and push the brain into a regime in which it would exhibit spontaneous activity). Quite surprisingly, they also found that when a single spike was removed from a single neuron, the state of the entire brain would diverge compared to when that spike was kept in the model. There is a lot more, so if this sounds interesting check out the paper. They also mention in the paper that they are currently examining how things change when they add sensory inputs to the model.\n\nOf course, a great deal of work is yet to be done, a great deal of thinking through the implications (and biological relevance) of some of the model's behavior (especially its global sensitivity to single spikes, which to me sounds biologically dubious). However, I find it quite amazing that by simply stamping the basic cortical template onto a model of the entire cortical sheet, and adding the rough inter-area connections, they observed many of the qualitative features of actual cortical activity. We tend to focus so much on local synaptic connections in our models of cortex, it is easy to miss the fact that the long-range projections could have similarly drastic influences on the global behavior of the system.\n\nThis paper is just fun. First, it is a great example of how to write a modeling paper for nonmathematicians. It had enough detail to give the modeler a sense for what they did, but not so much detail that your average systems neuroscientist would instinctively throw it in the trash (as is the case with too many modelling papers). Second, it provides a beautiful example of how people interested in systems-level phenomena can build biology into their model without making the model so computationally expensive that it would take fifty years to simulate ten milliseconds of cortical activity. It will be very interesting in the future as the hyper-realist Blue Brain style models make contact with these middle-level theories. I don't see conflict, but a future of productive theory co-evolution.\n\n## Monday, February 18, 2008\n\n### Visualizing the SVD\n\nWarning: this post isn't directly about neuroscience, but a mathematical tool that is used quite a bit by researchers.\n\nOne of the most important operations in linear algebra is the singular value decomposition (SVD) of a matrix. Gilbert Strang calls the SVD the climax of his linear algebra course, while Andy Long says, \"If you understand SVD, you understand linear algebra!\" Indeed, it ties about a dozen central concepts from linear algebra into one elegant theorem.\n\nThe SVD has many applications, but the point of this message is to examine the SVD itself, to massage intuitions about what is going on mathematically. To help me build intuitions, I wrote a Matlab function to visualize what is happening in each step of the decomposition (svd_visualize.m, which you can click to download). I have found it quite helpful to play around with the function. It takes in two arguments: a 3x3 matrix (A) and a 3xN 'data' matrix in which each of the N columns is a 'data' point in 3-D space. The function returns the three matrices in the SVD of A, but more importantly it generates four plots to visualize what each factor in the SVD is doing.\n\nTo refresh your memory, the SVD of an mxn matrix A is a factorization of A into three matrices, U, S, and V' such that:\nA=USV'\nwhere V' means the transpose of V. Generally, A is an mxn matrix, U is mxm, S is mxn, and V' is nxn.\n\nOne cool thing about the SVD is that it breaks up the multiplication of a matrix A and a vector x into three simpler matrix transformations which can be easily visualized. To help with this visualization, the function svd_visualize generates four graphs: the first figure plots the original data and the next three plots show how those data are transformed via sequential multiplication by each of the matrices in the SVD.\n\nIn what follows, I explore the four plots using a simple example. The matrix A is a 3x3 rank 2 matrix, and the data is a 'cylinder' of points (a small stack of unit circles each in the X-Y plane at different heights). The first plot of svd_visualize simply shows this data in three-space:", null, "In the above figure, the black lines are the standard basis vectors in which the cylinder is initially represented. The green and red lines are the columns of V, which form an orthogonal basis for the same space (more about this anon).\n\nWhen the first matrix in the SVD (V') is applied to the data, this serves to rotate the data in three-space so that the data is represented relative to the V-basis. Spelling this out a bit, the columns of V form an orthogonal basis for three-space. Multiplying a vector by V' changes the coordinate system in which that vector is represented. The original data is represented in the standard basis, multiplication by V' produces that same vector represented in the V-basis. For example, if we multiply V1 (the first column of V) by V, this rotates V1 so that V1 is represented as the point [1 0 0]' relative to the V-basis. Application of this rotation matrix V' to the above data cylinder yields the following:", null, "As promised, the data is the exact same as in Figure 1, but our cylinder has been rotated in three-space so that the V-basis vectors lie along the main axes of the plot. The two green vectors are the first two columns of V, which now lie along the two horizontal axes in the figure (for aficianados, they span the row space of A, or the set of all linear combinations of the rows of A). The red vertical line is the third column of V (its span is the null space of A, where the null space of A is the set of all vectors x such that Ax=0). So we see that the V' matrix rotates the data into a coordinate system in which the null space and row space of A can be more readily visualized.\n\nThe second step in the SVD is to multiply our rotated data by the 'singular matrix' S, which is mxn (in this case 3x3). S is a \"diagonal\" matrix that contains nonnegative 'singular values' of A sorted in descending order (technically, the singular values are the square roots of the eigenvalues of A'A that correspond to its eigenvectors, which are the columns of V). In this case, the singular values are 3 and 1, while the third diagonal elment in S is zero.\n\nWhat does this mean? Generally, multiplying a vector x=(x1,....xn)' by a diagonal matrix with r nonzero elements on the diagonal s1,....sr simply yields b=(s1x1, s2x2, .... srxr, 0 .. 0). That is, it stretches or contracts the components of x by the magnitude of the the singular values and zeroes out those elements of x that correspond to the zeros on the diagonal. Note that SV1 (where V1 is the first column of V) would yield b=(s1, 0 0 0 0 0). That is, it yields a vector whose first entry is s1 and the rest zero. Recall this is because S acts on vectors represented in the V-basis, and in the V-basis, V1 is simply (1,0, ..., 0).\n\nApplication of our singular matrix to the above data yields the following:", null, "This 3-D space represents the outputs space (range) of the A transformation. In this case, the range happens to be three-space, but if A had been Tx3, the input data in three-space would be sent to a point in a T-dimensional space. The figure shows the columns of the matrix U (in green and red) are aligned with the main axes: so the transform S returns values that are in the range of A, but represented in the orthogonal basis set in U. The green basis vectors are the first two columns of U (and they span the column space of A), while the red vector is the third column of U (which spans the null space of A').\n\nSince the column space of A (for this example) is two dimensional, any point in 3-D space in the input space (the original data) is constrained to be projected onto a plane in the output space.\n\nNotice that the individual circles that made up the cylinder have all turned into ellipses in the column space of A. This is due to the disproportionate stretching action of the singular values: the stretching is maximum for the vectors in the direction of V1. Also note that in the U-basis, SV1 lies on the same axis as U1 (U1, in the U-basis, is of course (1, 0, 0)), but s1 units along that axis for reasons discussed in the text after Figure 2.\n\nOne way to look at S is that it implements the same linear transformation as the matrix A, but with the inputs and outputs represented in different basis sets. The inputs to S are the data represented in the V-basis, while the outputs from S are the data represented in the U-basis. That makes it clear why we first multiply the data by V': this changes the basis of the input space to that which is appropriate for S. As you might guess, the final matrix, U, simply transforms the output of the S transform from a representation in the U-basis back into the standard basis.\n\nHence, we shouldn't be surprised that the final step in the SVD is to apply the mxm (in this case, 3x3) matrix U to the transformed data represented in the U-basis. Just like V', U is a rotation matrix: it transforms the data from the U-basis (above picture) back to the standard basis. The standard basis vectors are in black in the above picture, and we can see that the U transformation brings them back into alignment with the main axes of the plot:", null, "Pretty cool. The SVD lets you see, fairly transparently, the underlying transformations implicitly lurking in any matrix. Unlike many other decompositions (such as a diagonalization), it doesn't require A to have any special properties (e.g., A doesn't have to be square, symmetric, have linearly independent columns, etc). Any matrix can be decomposed into a change of basis (rotation by V'), a simple scaling (and \"flattening\") operation (by the singular matrix S), and a final change of basis (rotation by U).\n\nPostscript: I realize this article uses a bit of technical jargon, so I will post a linear algebra primer someday that explains the terminology. For the aficianados, I have left out some details that would have complicated things and made this post too long. In particular, I focused on how the SVD factors act on \"data\" vectors, but little on the properties of the SVD itself (e.g., how to compute U, S, and V; comparison to orthogonal diagonalization, and tons of other things).\n\nIf you have suggestions for improving svd_visualize, please let me know in the comments or email me (thomson ~at~ neuro [dot] duke -dot- edu)." ]	[ null, "http://1.bp.blogspot.com/_IFzDPHUxHI0/R8MvAMnbGxI/AAAAAAAAACM/enthyviVBtk/s200/edelman_model.gif", null, "http://4.bp.blogspot.com/_IFzDPHUxHI0/R7seN8nbGqI/AAAAAAAAABU/YDsldRtDY3k/s400/Initial_data.jpg", null, "http://1.bp.blogspot.com/_IFzDPHUxHI0/R7sigMnbGsI/AAAAAAAAABk/5cpKuXgOxqU/s400/data_in_V.gif", null, "http://1.bp.blogspot.com/_IFzDPHUxHI0/R7s0GMnbGtI/AAAAAAAAABs/CnQoqwnF66Q/s400/transformed_data_in_U.gif", null, "http://3.bp.blogspot.com/_IFzDPHUxHI0/R7s0LsnbGuI/AAAAAAAAAB0/Zz6R5WfNpik/s400/transformed_data_in_E.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9251621,"math_prob":0.94424266,"size":20165,"snap":"2020-34-2020-40","text_gpt3_token_len":4584,"char_repetition_ratio":0.14964536,"word_repetition_ratio":0.81680673,"special_character_ratio":0.2211753,"punctuation_ratio":0.10252018,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9908373,"pos_list":[0,1,2,3,4,5,6,7,8,9,10],"im_url_duplicate_count":[null,2,null,5,null,5,null,5,null,5,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-09-23T15:45:43Z\",\"WARC-Record-ID\":\"<urn:uuid:0ce52270-e3c3-4ddc-a13d-cb052e11490e>\",\"Content-Length\":\"79682\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f2e4d878-13be-49f9-8db3-261a69573bbf>\",\"WARC-Concurrent-To\":\"<urn:uuid:6757d5a6-625a-425a-bc72-ed2c9efbc959>\",\"WARC-IP-Address\":\"172.217.164.161\",\"WARC-Target-URI\":\"http://neurochannels.blogspot.com/2008/02/\",\"WARC-Payload-Digest\":\"sha1:JYJO4Y3DNXFJKKDXBJC5H7AKQQFP3TKY\",\"WARC-Block-Digest\":\"sha1:TQB4ZIGHR6VFS5MR2SLIK77Z2JIFHO6C\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600400211096.40_warc_CC-MAIN-20200923144247-20200923174247-00328.warc.gz\"}"}
https://answers.everydaycalculation.com/add-fractions/4-50-plus-3-96	[ "Solutions by everydaycalculation.com\n\n4/50 + 3/96 is 89/800.\n\n1. Find the least common denominator or LCM of the two denominators:\nLCM of 50 and 96 is 2400\n2. For the 1st fraction, since 50 × 48 = 2400,\n4/50 = 4 × 48/50 × 48 = 192/2400\n3. Likewise, for the 2nd fraction, since 96 × 25 = 2400,\n3/96 = 3 × 25/96 × 25 = 75/2400\n192/2400 + 75/2400 = 192 + 75/2400 = 267/2400\n5. 267/2400 simplified gives 89/800\n6. So, 4/50 + 3/96 = 89/800\n\nMathStep (Works offline)", null, "Download our mobile app and learn to work with fractions in your own time:" ]	[ null, "https://answers.everydaycalculation.com/mathstep-app-icon.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7044635,"math_prob":0.9983994,"size":301,"snap":"2020-45-2020-50","text_gpt3_token_len":116,"char_repetition_ratio":0.16835018,"word_repetition_ratio":0.0,"special_character_ratio":0.4551495,"punctuation_ratio":0.072463766,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99842036,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-21T10:15:51Z\",\"WARC-Record-ID\":\"<urn:uuid:31ff1c63-7258-42e2-a336-1ba792a01193>\",\"Content-Length\":\"7666\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f79b57a8-f3cd-4ad3-ab6b-a0e71fa0f15d>\",\"WARC-Concurrent-To\":\"<urn:uuid:94e56aa9-5e6e-48bb-9840-dadccaa0dfaa>\",\"WARC-IP-Address\":\"96.126.107.130\",\"WARC-Target-URI\":\"https://answers.everydaycalculation.com/add-fractions/4-50-plus-3-96\",\"WARC-Payload-Digest\":\"sha1:7XN4MUDMQBLS3SXONFL2OJTK5NDJCC4M\",\"WARC-Block-Digest\":\"sha1:3NH44JYH2ZFKKHCCWI6323XHJ47JPMFQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107876307.21_warc_CC-MAIN-20201021093214-20201021123214-00440.warc.gz\"}"}
https://piping-designer.com/index.php/disciplines/systems/project-management/3291-point-of-total-assumption	[ "# Point of Total Assumption\n\non . Posted in Project Management Engineering\n\nPoint of total assumption, abbreviatef as PTA, is the difference between the ceiling and target prices, divided by the buyer’s portion of the share ratio for that price range, plus the target cost.\n\n## Point of Total Assumption Formula\n\n$$\\large{ PTA = \\frac{ CP \\;-\\; TP }{ BSR} + TC }$$\nSymbol\n$$\\large{ PTA }$$ = point of total assumption\n$$\\large{ BSR }$$ = buyer's share ratio\n$$\\large{ CP }$$ = ceiling price\n$$\\large{ TC }$$ = target cost\n$$\\large{ TP }$$ = target price", null, "" ]	[ null, "https://piping-designer.com/images/Piping%20Designer%20Gallery/Piping-Designer_Logo_1.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.87816507,"math_prob":0.9997009,"size":507,"snap":"2023-14-2023-23","text_gpt3_token_len":144,"char_repetition_ratio":0.18290259,"word_repetition_ratio":0.0,"special_character_ratio":0.3195266,"punctuation_ratio":0.086419754,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.999984,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-05-29T12:21:57Z\",\"WARC-Record-ID\":\"<urn:uuid:ed6ca4a2-9dc0-433c-894c-a6e8295995b0>\",\"Content-Length\":\"26437\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:328acd15-e53a-4700-85cd-2ee33a6735fe>\",\"WARC-Concurrent-To\":\"<urn:uuid:9fe0ad93-e5ae-4b74-a11a-d59db543e015>\",\"WARC-IP-Address\":\"200.225.40.42\",\"WARC-Target-URI\":\"https://piping-designer.com/index.php/disciplines/systems/project-management/3291-point-of-total-assumption\",\"WARC-Payload-Digest\":\"sha1:XMDJ2EDTC6TM7JNTNZTL5HBSGKQF4T6L\",\"WARC-Block-Digest\":\"sha1:ZAQI6IAWYJUHGBOSP76EHTOFJTYDU47O\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-23/CC-MAIN-2023-23_segments_1685224644855.6_warc_CC-MAIN-20230529105815-20230529135815-00358.warc.gz\"}"}
https://www.tensorflow.org/tfx/transform/api_docs/python/tft/deduplicate_tensor_per_row	[ "# tft.deduplicate_tensor_per_row\n\nDeduplicates each row (0-th dimension) of the provided tensor.\n\n`input_tensor` A two-dimensional `Tensor` or `SparseTensor`. The first dimension is assumed to be the batch or \"row\" dimension, and deduplication is done on the 2nd dimension. If the Tensor is 1D it is returned as the equivalent `SparseTensor` since the \"row\" is a scalar can't be further deduplicated.\n`name` Optional name for the operation.\n\nA `SparseTensor` containing the unique set of values from each row of the input. Note: the original order of the input may not be preserved." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7324404,"math_prob":0.7898276,"size":616,"snap":"2020-45-2020-50","text_gpt3_token_len":159,"char_repetition_ratio":0.16503268,"word_repetition_ratio":0.0,"special_character_ratio":0.21915585,"punctuation_ratio":0.10185185,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9607357,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-28T03:26:55Z\",\"WARC-Record-ID\":\"<urn:uuid:66c763c4-d2e3-44fb-8b95-e2ea607d058b>\",\"Content-Length\":\"358691\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:eebadfac-736e-43b6-9289-db29d7a6052a>\",\"WARC-Concurrent-To\":\"<urn:uuid:95d1b32a-7643-46f3-a55d-2069e3a600dc>\",\"WARC-IP-Address\":\"172.217.164.174\",\"WARC-Target-URI\":\"https://www.tensorflow.org/tfx/transform/api_docs/python/tft/deduplicate_tensor_per_row\",\"WARC-Payload-Digest\":\"sha1:NO5G47GT46PNLFZBNGFRDVBUIJ2HW32H\",\"WARC-Block-Digest\":\"sha1:NHZQXIRXBZMFEWWNXKLSZ6G45JMDBA5S\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107896048.53_warc_CC-MAIN-20201028014458-20201028044458-00559.warc.gz\"}"}
https://se.mathworks.com/matlabcentral/profile/authors/6657407?detail=all	[ "Community Profile", null, "# Mausmi Verma\n\nLast seen: 4 månader ago Active since 2021\n\n#### Statistics\n\n•", null, "#### Content Feed\n\nView by\n\nQuestion\n\nHow to store double value in a matrix\nI have a variable as A <1x3 double> = 1 9 3 I want to store it in a predefined matrix B and want answer to look like B=4...\n\n### 1\n\nQuestion\n\nnested cell array into single cell array\nA={<1x1 cell> <1x3 cell> 4 <1x4 cell> } <1x1>= A{1, 1}{1, 1}{1, 1} <1x3 double> =[2 3 4] <1x3>= A{1, 2}{1, 1} <0x0 doub...\n\n### 1\n\nQuestion\n\nhow to convert array of cells within array of cells in single cell array\nA= {<1x1 cell> <1x3 cell> [4,0] <1x4 cell>} where <1x1>= [2,3,4] <1x3>= [ ] [3,4] [3,8,13] <1x4>= [9,4] [9,8,13] [ ]...\n\n### 1\n\nQuestion\n\nConvert nested cell array into single cell array\nI am getting variable value in workspace in the form of nested cell array A={{ } { }} but i want answer in...\n\n### 1\n\nQuestion\n\nmatching the element of matrix with an array\nSuppose i have an array as, A={ [2,3;2,7] [3,2;3,4;3,8] [4,3;4,5] [5,4;5,10] [7,2;7,8;7,12] } and a matrix as B=[17,8,14,4...\n\n### 1\n\nQuestion\n\nhow to combine two array of cells into one cellwise\nI have two cell arrays with cells of different dimension and i want to combine them into one as, A={ [2,3 ; 2,7] [3,2 ;3,4 ;3,...\n\n### 1\n\nQuestion\n\nRemoving specific value from cell array\nLets suppose i have a cell array as: A={[1 2 4 6 7]; [1 2 5 7 9 8]; [3 4 6 8]; [1 2 3 4 5 6]] now i want to remove the element...\n\n### 1\n\nQuestion\n\nI want to generate a matrix of random numbers between 0 and 2.5 with length=8 and the sum of generated numbers be 20\nI want to generate a matrix of random numbers between 0 and 2.5 The length of generated matrix should be 8 and the total sum of..." ]	[ null, "https://se.mathworks.com/responsive_image/150/150/0/0/0/cache/matlabcentral/profiles/6657407_1571713676492_DEF.jpg", null, "https://se.mathworks.com/matlabcentral/profile/badges/Thankful_5.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6901101,"math_prob":0.94512254,"size":2550,"snap":"2022-05-2022-21","text_gpt3_token_len":937,"char_repetition_ratio":0.15710919,"word_repetition_ratio":0.16666667,"special_character_ratio":0.36941177,"punctuation_ratio":0.14026402,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9733905,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,1,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-28T14:46:59Z\",\"WARC-Record-ID\":\"<urn:uuid:15acbbd0-d5b8-45e5-9853-ecabbc99945c>\",\"Content-Length\":\"85728\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:18971c08-f8ce-4c6c-8a46-877faf401298>\",\"WARC-Concurrent-To\":\"<urn:uuid:2e269fce-cef3-4654-8fbc-5ddb27232cc6>\",\"WARC-IP-Address\":\"104.69.217.80\",\"WARC-Target-URI\":\"https://se.mathworks.com/matlabcentral/profile/authors/6657407?detail=all\",\"WARC-Payload-Digest\":\"sha1:KTTDRAEPRWIKWDJVO56QC4XSY4O6CJVO\",\"WARC-Block-Digest\":\"sha1:R2I6RV7YCE3FPU2PLMYWQMHLDVZYUKDW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652663016853.88_warc_CC-MAIN-20220528123744-20220528153744-00413.warc.gz\"}"}
https://www.whorld.org/Help/Parameters/Even_Shear.htm	[ "### Even Shear\n\nEven Shear makes the curvature at even vertices asymmetrical. Curve asymmetry at odd vertices is controlled separately, via the Odd Shear parameter. The two curve control points at each vertex are normally contrained to be equidistant from the vertex. Removing this constraint yields two distinct distances, one between the counterclockwise point (A) and the vertex, and the other between the clockwise point (B) and the vertex. Shear changes the ratio of these two distances. Shear is normalized so that at zero, the control points are equidistant as usual, while at −1, the counterclockwise point coincides with the vertex, eliminating curvature on that side. Note that for Even Shear to have any effect, Even Curve must have a nonzero value.", null, "" ]	[ null, "https://www.whorld.org/Help/images/even-shear.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9219409,"math_prob":0.9904472,"size":744,"snap":"2023-14-2023-23","text_gpt3_token_len":158,"char_repetition_ratio":0.1418919,"word_repetition_ratio":0.0,"special_character_ratio":0.18682796,"punctuation_ratio":0.1119403,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9814487,"pos_list":[0,1,2],"im_url_duplicate_count":[null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-05-31T00:29:03Z\",\"WARC-Record-ID\":\"<urn:uuid:54a4f183-f3ad-4baa-8b8b-ccdf4cca9385>\",\"Content-Length\":\"1624\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3a80ee91-203f-41bf-a099-2ede15ecc6f9>\",\"WARC-Concurrent-To\":\"<urn:uuid:20f4f0a1-564a-4351-a664-b171ac7c648e>\",\"WARC-IP-Address\":\"107.180.40.20\",\"WARC-Target-URI\":\"https://www.whorld.org/Help/Parameters/Even_Shear.htm\",\"WARC-Payload-Digest\":\"sha1:NGI64JQOQX6JYJBF26DNKNL3XI67PLLX\",\"WARC-Block-Digest\":\"sha1:DP2QYN2NDNIGILHD2HQRKF3BW57IKCCN\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-23/CC-MAIN-2023-23_segments_1685224646181.29_warc_CC-MAIN-20230530230622-20230531020622-00172.warc.gz\"}"}
https://www.bristolmathsresearch.org/seminar/min-xu/	[ "# Min Xu\n\nRutgers University\n\n### High-dimensional nonparametric density estimation via symmetry and shape constraints\n\nStatistics Seminar\n12th March 2021, 4:00 pm – 5:00 pm\n,\n\nAbstract: We tackle the problem of high-dimensional nonparametric density estimation by taking the class of log-concave densities on R^p and incorporating within it symmetry assumptions, which facilitate scalable estimation algorithms and can mitigate the curse of dimensionality. Our main symmetry assumption is that the super-level sets of the density are K-homothetic (i.e. scalar multiples of a convex body K ⊆ R^p). When K is known, we prove that the K-homothetic log-concave maximum likelihood estimator based on n independent observations from such a density achieves the minimax optimal rate of convergence with respect to, e.g., squared Hellinger loss, of order n^(- 4/5), independent of p. Moreover, we show that the estimator is adaptive in the sense that if the data generating density admits a special form, then a nearly parametric rate may be attained. We also provide worst-case and adaptive risk bounds in cases where K is only known up to a positive definite transformation, and where it is completely unknown and must be estimated nonparametrically. Our estimation algorithms are fast even when n and p are on the order of hundreds of thousands, and we illustrate the strong finite-sample performance of our methods on simulated data. Joint work with Richard Samworth (Cambridge).\n\nOrganiser: Henry Reeve", null, "" ]	[ null, "https://www.bristolmathsresearch.org/wp-content/plugins/cookies-for-comments/css.php", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8993299,"math_prob":0.9776748,"size":1461,"snap":"2021-43-2021-49","text_gpt3_token_len":304,"char_repetition_ratio":0.111873716,"word_repetition_ratio":0.0,"special_character_ratio":0.19575633,"punctuation_ratio":0.101886794,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.978688,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-06T04:54:03Z\",\"WARC-Record-ID\":\"<urn:uuid:4b4508b2-ab20-4642-82be-64b7316f205c>\",\"Content-Length\":\"45778\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:74d302d7-622e-4a5e-9063-35545d44d555>\",\"WARC-Concurrent-To\":\"<urn:uuid:ab0c0df3-9a2d-4ede-820e-77d650ac1e23>\",\"WARC-IP-Address\":\"89.145.92.29\",\"WARC-Target-URI\":\"https://www.bristolmathsresearch.org/seminar/min-xu/\",\"WARC-Payload-Digest\":\"sha1:ZDFJ74OUJLUWTSGRD4B3KY5KR2O3H2EM\",\"WARC-Block-Digest\":\"sha1:VG7O2MCROFL6QLGBTRQYYV5TNNSXY254\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964363290.39_warc_CC-MAIN-20211206042636-20211206072636-00054.warc.gz\"}"}
https://math.stackexchange.com/questions/3257128/am-gm-inequality-involving-squares-and-proof	[ "# AM-GM Inequality Involving Squares and Proof\n\nProve:\n\n$$(a^2 + b^2 + c^2)/3 \\geq ((a + b + c)/3)^2$$ OR $$(a^2 + b^2 + c^2)/3 \\leq ((a + b + c)/3)^2$$\n\nfor all $$a, b, c \\geq 0.$$ The problem wants me to find which inequality is correct and then provide a proof for it.\n\nI'm pretty sure the first inequality is correct (I assumed this after substituting random values as $$a, b,$$ and $$c$$). I've tried factoring the inequalities and this is what I ended up with:\n\n$$9(a^2 + b^2 + c^2) \\geq 3(a^2 + b^2 + c^2 - 2ab + 2ac + 2bc)$$\n\nHowever, I don't know where to go from here. I'm also meant to be applying the AM-GM theorem to solve this problem but I'm unsure where and how to apply it in this situation. Any help would be extremely appreciated :)\n\nPedestrian:\n\n$$a,b,c \\ge 0.$$\n\nThe first inequality is equivalent to (Dr. Graubner):\n\n$$a^2+b^2+c^2 \\ge ab+bc +ac$$.\n\nAM-GM:\n\n$$a^2+b^2 \\ge 2ab$$; $$a^2+c^2 \\ge 2ac$$; $$b^2+c^2\\ge 2bc$$;\n\nAdding LHS and RHS of these inequalities:\n\n$$2(a^2+b^2+c^2) \\ge 2(ab+ac+bc),$$\n\nand we are done.\n\n• Can you please explain how you got to the equivalent first inequality? – Alexander B Jun 10 at 9:04\n• Alexander: $3(a^2+b^2+c^2) \\ge (a^2+b^2+c^2+2ab+2ac+2bc)$. Subtract the RHS square terms : $2a^+2b^2+2c^2 \\ge 2(ab+ac+bc)$OK? – Peter Szilas Jun 10 at 9:59\n\nYour first inequality is equivalent to $$a^2+b^2+c^2\\geq ab+bc+ca$$ and this is $$(a-b)^2+(b-c)^2+(c-a)^2\\geq 0$$ which is true.\n\nYes, we can use AM-GM here: $$\\frac{a^2+b^2+c^2}{3}-\\left(\\frac{a+b+c}{3}\\right)^2=\\frac{1}{9}\\sum_{cyc}(2a^2-2ab)=$$ $$=\\frac{1}{9}\\sum_{cyc}(a^2+b^2-2ab)\\geq\\frac{1}{9}\\sum_{cyc}(2\\sqrt{a^2b^2}-2ab)=\\frac{2}{9}\\sum_{cyc}(\|ab\|-ab)\\geq0.$$ Also, we can use AM-GM for three variables.\n\nIndeed, $$a^3+b^3+c^3-3abc=(a+b+c)\\sum_{cyc}(a^2-ab)=$$ $$=\\frac{9}{2}(a+b+c)\\left(\\frac{a^2+b^2+c^2}{3}-\\left(\\frac{a+b+c}{3}\\right)^2\\right)$$ and since by AM-GM $$a^3+b^3+c^3-3abc\\geq0$$ and $$a+b+c\\geq0,$$ we are done!\n\n• Is there a simpler way to prove the inequality but still use AM-GM? – Alexander B Jun 10 at 8:12\n• @Alexander B I posted another way, but I think the first was much more easier. – Michael Rozenberg Jun 10 at 8:16" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9376174,"math_prob":1.0000019,"size":692,"snap":"2019-26-2019-30","text_gpt3_token_len":234,"char_repetition_ratio":0.12645349,"word_repetition_ratio":0.02962963,"special_character_ratio":0.3684971,"punctuation_ratio":0.08280255,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000061,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-07-17T04:17:24Z\",\"WARC-Record-ID\":\"<urn:uuid:f6ae2955-301c-497a-bf9a-f830266302a1>\",\"Content-Length\":\"155037\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:94cddb31-50b0-4d68-b4c6-8ddb90e4b891>\",\"WARC-Concurrent-To\":\"<urn:uuid:a4cdb0ef-5046-4fc2-8aee-845b2bf76fa0>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/3257128/am-gm-inequality-involving-squares-and-proof\",\"WARC-Payload-Digest\":\"sha1:LZ5W46K24KIDQCC4ATP6EUTXKPYVWX6K\",\"WARC-Block-Digest\":\"sha1:DY7KZWGRP6PGT2SOQP5OBQL4FXIIWRHR\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-30/CC-MAIN-2019-30_segments_1563195525046.5_warc_CC-MAIN-20190717041500-20190717063500-00558.warc.gz\"}"}
https://www.colorhexa.com/798880	[ "# #798880 Color Information\n\nIn a RGB color space, hex #798880 is composed of 47.5% red, 53.3% green and 50.2% blue. Whereas in a CMYK color space, it is composed of 11% cyan, 0% magenta, 5.9% yellow and 46.7% black. It has a hue angle of 148 degrees, a saturation of 5.9% and a lightness of 50.4%. #798880 color hex could be obtained by blending #f2ffff with #001101. Closest websafe color is: #669999.\n\n• R 47\n• G 53\n• B 50\nRGB color chart\n• C 11\n• M 0\n• Y 6\n• K 47\nCMYK color chart\n\n#798880 color description : Dark grayish cyan - lime green.\n\n# #798880 Color Conversion\n\nThe hexadecimal color #798880 has RGB values of R:121, G:136, B:128 and CMYK values of C:0.11, M:0, Y:0.06, K:0.47. Its decimal value is 7964800.\n\nHex triplet RGB Decimal 798880 `#798880` 121, 136, 128 `rgb(121,136,128)` 47.5, 53.3, 50.2 `rgb(47.5%,53.3%,50.2%)` 11, 0, 6, 47 148°, 5.9, 50.4 `hsl(148,5.9%,50.4%)` 148°, 11, 53.3 669999 `#669999`\nCIE-LAB 55.31, -7.105, 2.437 20.585, 23.232, 23.821 0.304, 0.343, 23.232 55.31, 7.511, 161.067 55.31, -7.856, 4.525 48.199, -8.115, 4.438 01111001, 10001000, 10000000\n\n# Color Schemes with #798880\n\n• #798880\n``#798880` `rgb(121,136,128)``\n• #887981\n``#887981` `rgb(136,121,129)``\nComplementary Color\n• #7a8879\n``#7a8879` `rgb(122,136,121)``\n• #798880\n``#798880` `rgb(121,136,128)``\n• #798888\n``#798888` `rgb(121,136,136)``\nAnalogous Color\n• #88797a\n``#88797a` `rgb(136,121,122)``\n• #798880\n``#798880` `rgb(121,136,128)``\n• #887988\n``#887988` `rgb(136,121,136)``\nSplit Complementary Color\n• #888079\n``#888079` `rgb(136,128,121)``\n• #798880\n``#798880` `rgb(121,136,128)``\n• #807988\n``#807988` `rgb(128,121,136)``\n• #818879\n``#818879` `rgb(129,136,121)``\n• #798880\n``#798880` `rgb(121,136,128)``\n• #807988\n``#807988` `rgb(128,121,136)``\n• #887981\n``#887981` `rgb(136,121,129)``\n• #55605a\n``#55605a` `rgb(85,96,90)``\n• #616d67\n``#616d67` `rgb(97,109,103)``\n• #6d7b73\n``#6d7b73` `rgb(109,123,115)``\n• #798880\n``#798880` `rgb(121,136,128)``\n• #87948d\n``#87948d` `rgb(135,148,141)``\n• #94a09a\n``#94a09a` `rgb(148,160,154)``\n• #a2aca6\n``#a2aca6` `rgb(162,172,166)``\nMonochromatic Color\n\n# Alternatives to #798880\n\nBelow, you can see some colors close to #798880. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #79887c\n``#79887c` `rgb(121,136,124)``\n• #79887e\n``#79887e` `rgb(121,136,126)``\n• #79887f\n``#79887f` `rgb(121,136,127)``\n• #798880\n``#798880` `rgb(121,136,128)``\n• #798881\n``#798881` `rgb(121,136,129)``\n• #798883\n``#798883` `rgb(121,136,131)``\n• #798884\n``#798884` `rgb(121,136,132)``\nSimilar Colors\n\n# #798880 Preview\n\nThis text has a font color of #798880.\n\n``<span style=\"color:#798880;\">Text here</span>``\n#798880 background color\n\nThis paragraph has a background color of #798880.\n\n``<p style=\"background-color:#798880;\">Content here</p>``\n#798880 border color\n\nThis element has a border color of #798880.\n\n``<div style=\"border:1px solid #798880;\">Content here</div>``\nCSS codes\n``.text {color:#798880;}``\n``.background {background-color:#798880;}``\n``.border {border:1px solid #798880;}``\n\n# Shades and Tints of #798880\n\nA shade is achieved by adding black to any pure hue, while a tint is created by mixing white to any pure color. In this example, #010101 is the darkest color, while #f6f7f6 is the lightest one.\n\n• #010101\n``#010101` `rgb(1,1,1)``\n• #0a0b0b\n``#0a0b0b` `rgb(10,11,11)``\n• #131615\n``#131615` `rgb(19,22,21)``\n• #1d201e\n``#1d201e` `rgb(29,32,30)``\n• #262b28\n``#262b28` `rgb(38,43,40)``\n• #2f3532\n``#2f3532` `rgb(47,53,50)``\n• #383f3c\n``#383f3c` `rgb(56,63,60)``\n• #424a45\n``#424a45` `rgb(66,74,69)``\n• #4b544f\n``#4b544f` `rgb(75,84,79)``\n• #545f59\n``#545f59` `rgb(84,95,89)``\n• #5d6963\n``#5d6963` `rgb(93,105,99)``\n• #66736c\n``#66736c` `rgb(102,115,108)``\n• #707e76\n``#707e76` `rgb(112,126,118)``\n• #798880\n``#798880` `rgb(121,136,128)``\n• #83918a\n``#83918a` `rgb(131,145,138)``\n• #8e9a94\n``#8e9a94` `rgb(142,154,148)``\n• #98a49e\n``#98a49e` `rgb(152,164,158)``\n``#a3ada7` `rgb(163,173,167)``\n``#adb6b1` `rgb(173,182,177)``\n• #b7bfbb\n``#b7bfbb` `rgb(183,191,187)``\n• #c2c9c5\n``#c2c9c5` `rgb(194,201,197)``\n• #ccd2cf\n``#ccd2cf` `rgb(204,210,207)``\n• #d7dbd9\n``#d7dbd9` `rgb(215,219,217)``\n• #e1e4e2\n``#e1e4e2` `rgb(225,228,226)``\n• #ebedec\n``#ebedec` `rgb(235,237,236)``\n• #f6f7f6\n``#f6f7f6` `rgb(246,247,246)``\nTint Color Variation\n\n# Tones of #798880\n\nA tone is produced by adding gray to any pure hue. In this case, #798880 is the less saturated color, while #04fd78 is the most saturated one.\n\n• #798880\n``#798880` `rgb(121,136,128)``\n• #6f927f\n``#6f927f` `rgb(111,146,127)``\n• #669b7f\n``#669b7f` `rgb(102,155,127)``\n• #5ca57e\n``#5ca57e` `rgb(92,165,126)``\n• #52af7d\n``#52af7d` `rgb(82,175,125)``\n• #48b97d\n``#48b97d` `rgb(72,185,125)``\n• #3fc27c\n``#3fc27c` `rgb(63,194,124)``\n• #35cc7b\n``#35cc7b` `rgb(53,204,123)``\n• #2bd67b\n``#2bd67b` `rgb(43,214,123)``\n• #21e07a\n``#21e07a` `rgb(33,224,122)``\n• #18e97a\n``#18e97a` `rgb(24,233,122)``\n• #0ef379\n``#0ef379` `rgb(14,243,121)``\n• #04fd78\n``#04fd78` `rgb(4,253,120)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #798880 is perceived by people affected by a color vision deficiency. This can be useful if you need to ensure your color combinations are accessible to color-blind users.\n\nMonochromacy\n• Achromatopsia 0.005% of the population\n• Atypical Achromatopsia 0.001% of the population\nDichromacy\n• Protanopia 1% of men\n• Deuteranopia 1% of men\n• Tritanopia 0.001% of the population\nTrichromacy\n• Protanomaly 1% of men, 0.01% of women\n• Deuteranomaly 6% of men, 0.4% of women\n• Tritanomaly 0.01% of the population" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.50974303,"math_prob":0.59309727,"size":3710,"snap":"2020-34-2020-40","text_gpt3_token_len":1620,"char_repetition_ratio":0.12250405,"word_repetition_ratio":0.007352941,"special_character_ratio":0.5692722,"punctuation_ratio":0.23378076,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99290437,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-08-03T12:32:21Z\",\"WARC-Record-ID\":\"<urn:uuid:3fed9222-336e-49a7-ba47-6c7d4fb46d7d>\",\"Content-Length\":\"36340\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e5b93df2-bfd5-47ad-9588-826d547a83f7>\",\"WARC-Concurrent-To\":\"<urn:uuid:12a19d83-5a6a-458c-8c75-5e27eada7ea1>\",\"WARC-IP-Address\":\"178.32.117.56\",\"WARC-Target-URI\":\"https://www.colorhexa.com/798880\",\"WARC-Payload-Digest\":\"sha1:CV7NBEO3TP2MTIZ5I37BOIGCQ6PH3MYV\",\"WARC-Block-Digest\":\"sha1:TWAGD6O5YFK77UMTXSAV2SK6HTFS4OBI\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-34/CC-MAIN-2020-34_segments_1596439735810.18_warc_CC-MAIN-20200803111838-20200803141838-00532.warc.gz\"}"}
http://ristoranteclass.it/11-23/8961.html	[ "# first 20 digits of pi\n\n### Here are the first 10,000 digits of Pi in honor of Pi Day ...\n\nOne way to remember the first few digits of pi is to count the letters in the words of this phrase: “How I need a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.”\n\n### Digits Of Pi\n\nToday is Pi Day — the day each year, March 14, that follows the first three digits of pi (3.14). And this year’s Pi Day is a special one: Since — in the U.S. — the date is represented as 3 ...\n\n### One billion digits of pi - Massachusetts Institute of ...\n\n· Pi is the ratio of the circumference to the diameter (the diameter is 2 times the radius) of a circle. Computing pi is a common way to judge the computing power of supercomputers, and mathematicians now know approximately 10 trillion digits of pi.\n\n### What are the first twenty digets of pie? \| Yahoo Answers\n\nFirst 200 Million Digits of Pi Activity: Pi goes on forever in random numbers never repeating itself. It is both an irrational and transcendental number. Pi has been computed to over one trillion digits …\n\n### The First Thousand Digits of Pi - Fact Monster\n\n1 Million Digits of Pi The first 10 digits of pi (π) are 3.1415926535. The first million digits of pi (π) are below, got a good memory? Then recite as many digits as you can in our quiz!! Why not calculate the circumference of a circle using pi here. Or simply learn about pi here.Maximize the fun you can have this Pi Day by checking out our Pi Day Stuff, Pi Day Deals and Pi Day Celebrations!\n\n### Pi to 20 decimal places! - Eve Astrid Andersson\n\nMore digits: Scroll down to see the first 10,000 digits of Pi at the bottom of this page, or grab even more using the links below. Files containing digits: 10 50 100 1000 10000 100000; 1 million digits of Pi (Might take a while to download) The Pi searcher can show digits of Pi anywhere in the first 200 million digits, using the second line in ...\n\n### Analyzing the first 10 million digits of pi: Randomness ...\n\nFor pi and e, there are no “half to even” cases, since their binary expansions are infinite. This makes the rounding rule simple: if the rounding bit is 0, round down; if the rounding bit is 1, round up. I will show the correctly rounded approximations of pi and e in these five formats. Pi (π) Here are the first 50 decimal digits of pi:\n\n### One Million Digits of Pi On One Page!- [Plus Guides And ...\n\nHowever, Pi starts with 3 which is also a digit. Thus, if you start at 3, then the twenty-fifth digit of Pi is 3. First 25 digits of Pi: The Pi number above gives you \"3.\" followed by 25 digits after the decimal point. If you want to just memorize, learn or see the first 25 digits of Pi …\n\n### First 200 Million Digits of Pi - Pi Across America\n\nOne of the challenges on w3resources is to print pi to 'n' decimal places. Here is my code: from math import pi fraser = str(pi) length_of_pi = [] number_of_places = raw_input(\"Enter the number...\n\n### The First 25 Million Digits of Pi Book - Optional ...\n\nSome spent their lives calculating the digits of Pi, but until computers, less than 1,000 digits had been calculated. In 1949, a computer calculated 2,000 digits and the race was on. Millions of digits have been calculated, with the record held (as of September 1999) by a supercomputer at the University of Tokyo that calculated 206,158,430,000 digits.\n\n### How Many Digits Of Pi You Have To Have Memorized To …\n\nWhat is the first 20 digits of Pi :) - 3329671 metric mean of 5 and 125. A. 65 B. 25 C. 2√30 D. √130 4. Suppose the altitude to the hypotenuse of a right triangle bisects the hypotenuse.\n\n### Pi to 20 decimal places\n\n3. 1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679 ...\n\n### Digits of Pi - Up to 1 Million Digits\n\nThe digits to the right of its decimal point can keep going forever, and there is absolutely no pattern to these digits. A team of researchers at Tokyo University in Japan calculated the digits of pi to 1.24 trillion places. Chances are, you'll never need to know even the first ten digits, but just for fun, here are the first thousand: π = 3.\n\n### First 20 Digits of e - Miniwebtool\n\n20 Digits of Pi. STUDY. Flashcards. Learn. Write. Spell. Test. PLAY. Match. Gravity. Created by. Person4525. Created for Geometry. Terms in this set (18) What are the first 3 digits of pi? 3.14. What are the first 4 digits of pi? 3.141. What are the first 5 digits of pi? 3.1415. What are the first 6 digits of pi? 3.14159. What are the first 7 ...\n\n### Print pi to a number of decimal places - Stack Overflow\n\nPi to 20 decimal places! collected by Eve Andersson : Home: Pi: Digits: 20 Decimal Places 3. 14159265358979323846: Great Pi Day Gift! Los Boludos Made with original vintage vacuum tubes! ...\n\n### The Square Root of Two to 1 Million Digits\n\n· The Square Root of Two to 1 Million Digits What follows are the first 1 million digits of the square root of 2. Actually there are slightly more than 1M digits here. These digits were computed by Robert Nemiroff (George Mason University and NASA Goddard Space Flight Center) and checked by Jerry Bonnell (University Space Research Association and ...\n\n### Pi Day tip on how to remember mathematical constant …\n\nAs we will show, reaching 10 trillion digits of Pi is much more difficult than 5 trillion digits using our current methods and 2010 computer hardware. Hardware: Shigeru Kondo's Desktop The machine we used is mostly the same as the previous computation.\n\n### 2000 places of Pi - MacTutor History of Mathematics\n\nThe First 25 Million Digits of Pi Book makes a coffee table book worthy of conversation, or treasure it on your book shelf along with other math or science classics. Whether you are an expert or just a Pi fan, this is a book you will enjoy having.\n\n### pi to 20 digits - Wolfram\|Alpha\n\n10000 digits of e 2.718281828459045235360287471352662497757247093699959574966967627724076630353 ...\n\n### Pi - Wikipedia\n\n· 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510\n\n### 10000 digits of e - MacTutor History of Mathematics\n\nHowever, Pi starts with 3 which is also a digit. Thus, if you start at 3, then the twentieth digit of Pi is 4. First 20 digits of Pi: The Pi number above gives you \"3.\" followed by 20 digits after the decimal point. If you want to just memorize, learn or see the first 20 digits of Pi including the 3, then you can omit the last digit. Pi Decimal ...\n\n### What are the first 20 digits of pi - Answers\n\nSee the digits of pi, top 10, 20, 30, 40, 50, 60, 100, 200, 300, 400, 1000, 10000, 100000 digits of pi\n\n### First 500 Digits of Pi - Mrs. Jackson's Math Class\n\nOne billion digits of π. One billion (10^9) digits of pi (actually 1,000,000,001 digits if you count the initial \"3\") are in the file pi-billion.txt. The MD5 checksum is in pi-billion.md5. JA0HXV has calculated 100 billion digits of pi and posted them at the website: ...\n\n### First 20 Digits of Pi - Miniwebtool\n\nThe first 20 digits in hexadecimal (base 16) ... Pi Day in 2015 was particularly significant because the date and time 3/14/15 9:26:53 reflected many more digits of pi. In parts of the world where dates are commonly noted in day/month/year format, ...\n\n### Pi and e In Binary - Exploring Binary\n\nWelcome to Mrs. Jackson's Math Class. Learn the digits of Pi for the Pi contest. Here are 500 digits of Pi\n\n### Top 10000 Digits Of Pi\n\n· Here are the first 10,000 digits of Pi in honor of Pi Day. Share this article share tweet text email link Luke Kerr-Dineen. like March 14, 2017 7:32 am. It’s Pi Day ...\n\n### Pi - 10 Trillion Digits - numberworld.org\n\n· The first 51 digits of pi The important ones. For all practical purposes no more than 10 digits of pi are required for mathematic computation, however even with astronomically precise calculations no more than fifty are really necessary.\n\n### what is the first 25 digits to pi? \| Yahoo Answers\n\nCompute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history ...\n\nsitemap" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.86752075,"math_prob":0.8814128,"size":6629,"snap":"2020-24-2020-29","text_gpt3_token_len":1712,"char_repetition_ratio":0.16890566,"word_repetition_ratio":0.0677392,"special_character_ratio":0.30743703,"punctuation_ratio":0.15997182,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9938169,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-06T20:18:41Z\",\"WARC-Record-ID\":\"<urn:uuid:89fd0459-7590-4e43-baa4-07f108732220>\",\"Content-Length\":\"18134\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e74bbf5c-ba52-470a-bcf7-8888315e5860>\",\"WARC-Concurrent-To\":\"<urn:uuid:0c11197d-5002-4447-874d-6156b5553dd3>\",\"WARC-IP-Address\":\"104.27.177.86\",\"WARC-Target-URI\":\"http://ristoranteclass.it/11-23/8961.html\",\"WARC-Payload-Digest\":\"sha1:YPFLCJ4RQOVC5OVXWNRQJDLM2ONCHDY6\",\"WARC-Block-Digest\":\"sha1:6YVUP2X6IULUWY6TBDTRCZ2W64RCEB7I\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593655890181.37_warc_CC-MAIN-20200706191400-20200706221400-00400.warc.gz\"}"}
https://scholarsarchive.byu.edu/etd/1332/	[ "## Theses and Dissertations\n\n### How Eighth-Grade Students Estimate with Fractions\n\n#### Abstract\n\nThis study looked at what components are in student solutions to computational estimation problems involving fractions. Past computational estimation research has focused on strategies used for estimating with whole numbers and decimals while neglecting those used for fractions. An extensive literature review revealed one study specifically directed toward estimating with fractions (Hanson & Hogan, 2000) that researched adult estimation strategies and not children's strategies. Given the lack of research on estimation strategies that children use to estimate with fractions, this study used qualitative research methods to find which estimation components were in 10 eighth-grade students' solutions to estimation problems involving fractions. Analysis of this data differs from previous estimation studies in that it considers actions as the unit of analysis, providing a smaller grain size that reveals the components used in each estimation solution. The analysis revealed new estimation components as well as a new structure for categorizing the components. The new categories are whole number and decimal estimation components, fraction estimation components, and components used with either fractions or whole numbers and decimals. The results from this study contribute to the field of mathematics education by identifying new components to consider when conducting future studies in computational estimation. The findings also suggest that future research on estimation should use a smaller unit of analysis than a solution response to a task, the typical unit of analysis in previous research. Additionally, these results contribute to mathematics teaching by suggesting that all components of an estimation solution be considered when teaching computational estimation, not just the overarching strategy.\n\nMA\n\n#### College and Department\n\nPhysical and Mathematical Sciences; Mathematics Education\n\n2008-03-13\n\nThesis\n\n#### Handle\n\nhttp://hdl.lib.byu.edu/1877/etd2294\n\n#### Keywords\n\nmathematics education, computational estimation, estimation\n\nEnglish\n\nCOinS" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.86570215,"math_prob":0.67123663,"size":2324,"snap":"2023-40-2023-50","text_gpt3_token_len":418,"char_repetition_ratio":0.16896552,"word_repetition_ratio":0.0,"special_character_ratio":0.17211704,"punctuation_ratio":0.1,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9580133,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-11-28T19:59:35Z\",\"WARC-Record-ID\":\"<urn:uuid:6649f3a2-4a17-45ea-81b3-de3c6b165146>\",\"Content-Length\":\"39279\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:be72bb4d-e045-46b5-98cf-4fdffd7d5b3b>\",\"WARC-Concurrent-To\":\"<urn:uuid:aad6d704-b344-4a74-995c-75c0c65e0aac>\",\"WARC-IP-Address\":\"13.57.92.51\",\"WARC-Target-URI\":\"https://scholarsarchive.byu.edu/etd/1332/\",\"WARC-Payload-Digest\":\"sha1:TWURGHBHNGLSKSIZDSCD7JDG7ACGFJSY\",\"WARC-Block-Digest\":\"sha1:6U5MXDCLDTB6BKD46BC2AZCBU5QAV5WV\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679099942.90_warc_CC-MAIN-20231128183116-20231128213116-00206.warc.gz\"}"}
https://cstheory.stackexchange.com/questions/31007/information-complexity-of-query-algorithms	[ "# Information complexity of query algorithms?\n\nInformation complexity has been a very useful tool in communication complexity, mainly used to lower bound the communication complexity of distributed problems.\n\nIs there an analogue of information complexity for query complexity? There are many parallels between query complexity and communication complexity; oftentimes (but not always!) a lower bound in one model gets translated to a lower bound in the other model. Sometimes this translation is quite nontrivial.\n\nIs there a notion of information complexity that is useful for lower bounding the query complexity of problems?\n\nA first pass seems to indicate that information complexity is not very useful; for example, the query complexity of computing the OR of $N$ bits is $\\Omega(N)$ for randomized algorithms and $\\Omega(\\sqrt{N})$ for quantum algorithms, whereas the most straightforward adaption of the notion of information complexity indicates that the information learned by any query algorithm is at most $O(\\log N)$ (because the algorithm stops when it sees the first $1$ in the input)." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9084732,"math_prob":0.9558377,"size":1049,"snap":"2019-35-2019-39","text_gpt3_token_len":195,"char_repetition_ratio":0.2076555,"word_repetition_ratio":0.0,"special_character_ratio":0.1877979,"punctuation_ratio":0.068571426,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98669636,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-09-20T09:32:41Z\",\"WARC-Record-ID\":\"<urn:uuid:14ebd543-2d52-4959-b98c-bd9dd94c95c7>\",\"Content-Length\":\"136948\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:01101515-e2d4-4f56-99cb-7bd1743cad26>\",\"WARC-Concurrent-To\":\"<urn:uuid:721cdce1-ff2b-4351-a2c4-e1780f21ee92>\",\"WARC-IP-Address\":\"151.101.129.69\",\"WARC-Target-URI\":\"https://cstheory.stackexchange.com/questions/31007/information-complexity-of-query-algorithms\",\"WARC-Payload-Digest\":\"sha1:VUF2W6RG7RL3OY5YGJNCSE6MALQLCWA6\",\"WARC-Block-Digest\":\"sha1:NMXH3OJYRI5IAAAWBDQDGAN25IHQUUZU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-39/CC-MAIN-2019-39_segments_1568514573988.33_warc_CC-MAIN-20190920092800-20190920114800-00515.warc.gz\"}"}
https://gitlab.xiph.org/xiph/aom-rav1e/-/commit/75e513f126adb60061e02856bef8ae3078769d26?view=inline	[ "### Set spatial neighbor search resolution 16x16 for block size 64x64\n\n```When the block has width/height above or equal to 64, use 16x16\nblock search step for reference motion vector search in the non-\nimmediate rows and columns.\n\nChange-Id: If11ce97a9328b879f30ef87115086aa0cd985a2f```\nparent 883c63ca\n ... ... @@ -161,22 +161,26 @@ static uint8_t scan_row_mbmi(const AV1_COMMON cm, const MACROBLOCKD xd, for (i = 0; i < xd->n8_w && refmv_count < MAX_REF_MV_STACK_SIZE;) { POSITION mi_pos; const int use_step_16 = (xd->n8_w >= 8); mi_pos.row = row_offset; mi_pos.col = i; if (is_inside(tile, mi_col, mi_row, &mi_pos)) { const MODE_INFO const candidate_mi = xd->mi[mi_pos.row * xd->mi_stride + mi_pos.col]; const MB_MODE_INFO const candidate = &candidate_mi->mbmi; const int len = int len = AOMMIN(xd->n8_w, num_8x8_blocks_wide_lookup[candidate->sb_type]); if (use_step_16) len = AOMMAX(2, len); newmv_count += add_ref_mv_candidate( candidate_mi, candidate, rf, refmv_count, ref_mv_stack, cm->allow_high_precision_mv, len, block, mi_pos.col); i += len; } else { ++i; if (use_step_16) i += 2; else ++i; } } ... ... @@ -193,22 +197,26 @@ static uint8_t scan_col_mbmi(const AV1_COMMON cm, const MACROBLOCKD xd, for (i = 0; i < xd->n8_h && refmv_count < MAX_REF_MV_STACK_SIZE;) { POSITION mi_pos; const int use_step_16 = (xd->n8_h >= 8); mi_pos.row = i; mi_pos.col = col_offset; if (is_inside(tile, mi_col, mi_row, &mi_pos)) { const MODE_INFO const candidate_mi = xd->mi[mi_pos.row xd->mi_stride + mi_pos.col]; const MB_MODE_INFO *const candidate = &candidate_mi->mbmi; const int len = int len = AOMMIN(xd->n8_h, num_8x8_blocks_high_lookup[candidate->sb_type]); if (use_step_16) len = AOMMAX(2, len); newmv_count += add_ref_mv_candidate( candidate_mi, candidate, rf, refmv_count, ref_mv_stack, cm->allow_high_precision_mv, len, block, mi_pos.col); i += len; } else { ++i; if (use_step_16) i += 2; else ++i; } } ... ...\nMarkdown is supported\n0% or .\nYou are about to add 0 people to the discussion. Proceed with caution.\nFinish editing this message first!" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6635705,"math_prob":0.96737075,"size":350,"snap":"2021-31-2021-39","text_gpt3_token_len":99,"char_repetition_ratio":0.11849711,"word_repetition_ratio":0.0,"special_character_ratio":0.28857142,"punctuation_ratio":0.05357143,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.973744,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-07-23T20:14:29Z\",\"WARC-Record-ID\":\"<urn:uuid:66b35515-5b5b-42a7-a3aa-60f039b85ee7>\",\"Content-Length\":\"187458\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:76be79cb-3ec4-49c4-8e45-a2e4122d1199>\",\"WARC-Concurrent-To\":\"<urn:uuid:68ad6ed7-a063-449d-a005-3097966d64e9>\",\"WARC-IP-Address\":\"140.211.166.4\",\"WARC-Target-URI\":\"https://gitlab.xiph.org/xiph/aom-rav1e/-/commit/75e513f126adb60061e02856bef8ae3078769d26?view=inline\",\"WARC-Payload-Digest\":\"sha1:SEMKRCWYHI6I6U7LGHKD2LSOTUSUDJGU\",\"WARC-Block-Digest\":\"sha1:MDZMZSOAVEAPIO7CER75PIM75FQNOORQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046150000.59_warc_CC-MAIN-20210723175111-20210723205111-00325.warc.gz\"}"}
https://pythonexamples.org/pandas-dataframe-pop/	[ "# How to Delete Column from Pandas DataFrame?\n\n## Pandas DataFrame.pop() – Delete Column\n\nPandas DataFrame.pop() function is used to delete a column from the DataFrame.\n\nIn this tutorial, we shall go through examples to learn how to use pop() to delete a column from Pandas DataFrame.\n\n### Example 1: Delete a column using pandas pop() function\n\nIn this example, we deleted a specific column, using column name, from the DataFrame with pop(). pandas pop() function updates the original dataframe. The data in the deleted column is lost.\n\nPython Program\n\n``````import pandas as pd\n\nmydictionary = {'names': ['Somu', 'Kiku', 'Amol', 'Lini'],\n'physics': [68, 74, 77, 78],\n'chemistry': [84, 56, 73, 69],\n'algebra': [78, 88, 82, 87]}\n\n#create dataframe\ndf_marks = pd.DataFrame(mydictionary)\nprint('Original DataFrame\\n--------------')\nprint(df_marks)\n\n#delete column\ndf_marks.pop('algebra')\nprint('\\n\\nDataFrame after deleting column\\n--------------')\nprint(df_marks)``````\nRun\n\nOutput\n\n### Example 2: Delete a non-existing column using pandas pop() function\n\nIn this example, we will try deleting a column that is not present in the DataFrame.\n\nWhen you try to delete a non-existing column of DataFrame using pop(), the function pop() throws KeyError.\n\nPython Program\n\n``````import pandas as pd\n\nmydictionary = {'names': ['Somu', 'Kiku', 'Amol', 'Lini'],\n'physics': [68, 74, 77, 78],\n'chemistry': [84, 56, 73, 69],\n'algebra': [78, 88, 82, 87]}\n\n#create dataframe\ndf_marks = pd.DataFrame(mydictionary)\nprint('Original DataFrame\\n--------------')\nprint(df_marks)\n\n#delete column that is not present\ndf_marks.pop('geometry')\nprint('\\n\\nDataFrame after deleting column\\n--------------')\nprint(df_marks)``````\nRun\n\nOutput" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6750509,"math_prob":0.8714238,"size":1775,"snap":"2022-40-2023-06","text_gpt3_token_len":444,"char_repetition_ratio":0.18068887,"word_repetition_ratio":0.37083334,"special_character_ratio":0.30704224,"punctuation_ratio":0.18373494,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99455875,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-10-04T01:15:00Z\",\"WARC-Record-ID\":\"<urn:uuid:37c081d1-a7dc-4b4d-aa62-8203b18924a3>\",\"Content-Length\":\"31393\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:135504d9-36d0-450d-8be6-22813b56f1c6>\",\"WARC-Concurrent-To\":\"<urn:uuid:daeb5fe0-aec5-4822-8011-526ab3244d27>\",\"WARC-IP-Address\":\"99.84.208.77\",\"WARC-Target-URI\":\"https://pythonexamples.org/pandas-dataframe-pop/\",\"WARC-Payload-Digest\":\"sha1:A5RWUA6YGJABK3TFOHTZM5Z5H76SATJ4\",\"WARC-Block-Digest\":\"sha1:OTDG5WAKGINFYJWSYUYM54A6ZZ6MU7ZV\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030337446.8_warc_CC-MAIN-20221003231906-20221004021906-00312.warc.gz\"}"}
https://wikidev.in/wiki/matlab/image_processing/imwrite	[ "You are here : matlabImage Processingimwrite\n\nimwrite() - Image Processing\n\n`imwrite(A,filename) writes image data A to the file specified by filename, inferring the file format from the extension. imwrite creates the new file in your current folder. The bit depth of the output image depends on the data type of A and the file format. For most formats:If A is of data type uint8, then imwrite outputs 8-bit values.If A is of data type uint16 and the output file format supports 16-bit data (JPEG, PNG, and TIFF), then imwrite outputs 16-bit values. If the output file format does not support 16-bit data, then imwrite returns an error.If A is a grayscale or RGB color image of data type double or single, then imwrite assumes that the dynamic range is [0,1] and automatically scales the data by 255 before writing it to the file as 8-bit values. If the data in A is single, convert A to double before writing to a GIF or TIFF file.If A is of data type logical, then imwrite assumes that the data is a binary image and writes it to the file with a bit depth of 1, if the format allows it. BMP, PNG, or TIFF formats accept binary images as input arrays.If A contains indexed image data, you should additionally specify the map input argument.`\n\nSyntax\n\n```imwrite(A,filename)\nimwrite(A,map,filename)\nimwrite(___,fmt)\nimwrite(___,Name,Value)```\n\nExample\n\n```%Write a 50-by-50 array of grayscale values to a PNG file in the current folder.%\nA = rand(50);\nimwrite(A,'myGray.png')```" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7873224,"math_prob":0.70853835,"size":1427,"snap":"2019-26-2019-30","text_gpt3_token_len":345,"char_repetition_ratio":0.15741391,"word_repetition_ratio":0.033755273,"special_character_ratio":0.24246672,"punctuation_ratio":0.13099042,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9709759,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-06-25T09:28:51Z\",\"WARC-Record-ID\":\"<urn:uuid:f589d112-9d03-4276-b6b1-98d259f33ec0>\",\"Content-Length\":\"8448\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:64237e59-739c-48bd-9060-859571d826b1>\",\"WARC-Concurrent-To\":\"<urn:uuid:a0c06976-14a4-4724-9d26-d26309a0631a>\",\"WARC-IP-Address\":\"104.27.133.94\",\"WARC-Target-URI\":\"https://wikidev.in/wiki/matlab/image_processing/imwrite\",\"WARC-Payload-Digest\":\"sha1:Z3WRSNIWHYIT7OVWTAT4J3ZRJZK4ZTVX\",\"WARC-Block-Digest\":\"sha1:EG2ERDX6BO74UMEB4YSPEPAGGFBZBBBK\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-26/CC-MAIN-2019-26_segments_1560627999817.30_warc_CC-MAIN-20190625092324-20190625114324-00087.warc.gz\"}"}
https://homework.cpm.org/category/CON_FOUND/textbook/ac/chapter/13/lesson/13.OF1-S/problem/7-134	[ "", null, "", null, "### Home > AC > Chapter 13 > Lesson 13.OF1-S > Problem7-134\n\n7-134.\n\nWrite the equation of each circle graphed below.\n\nRecall the general equation of a circle: $\\left(x − h\\right)² + \\left(y − k\\right)² = r²$where $\\left(h, k\\right)$ is the center of the circle and $r$ is the radius.\n\n1.", null, "Write in graphing form:\n($(x − 1)^{2} + y^{2} = 9$\n\n1.", null, "" ]	[ null, "https://homework.cpm.org/dist/7d633b3a30200de4995665c02bdda1b8.png", null, 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https://arxiv.org/abs/1608.01512	[ "math.LO\n\n# Title:Strong failures of higher analogs of Hindman's theorem\n\nAbstract: We show that various analogs of Hindman's Theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets:\nTheorem 1: There exists a colouring $c:\\mathbb R\\rightarrow\\mathbb Q$, such that for every $X\\subseteq\\mathbb R$ with $\|X\|=\|\\mathbb R\|$, and every colour $\\gamma\\in\\mathbb Q$, there are two distinct elements $x_0,x_1$ of $X$ for which $c(x_0+x_1)=\\gamma$. This forms a simultaneous generalization of a theorem of Hindman, Leader and Strauss and a theorem of Galvin and Shelah.\nTheorem 2: For every Abelian group $G$, there exists a colouring $c:G\\rightarrow\\mathbb Q$ such that for every uncountable $X\\subseteq G$, and every colour $\\gamma$, for some large enough integer $n$, there are pairwise distinct elements $x_0,\\ldots,x_n$ of $X$ such that $c(x_0+\\cdots+x_n)=\\gamma$. In addition, it is consistent that the preceding statement remains valid even after enlarging the set of colours from $\\mathbb Q$ to $\\mathbb R$.\nTheorem 3: Let $\\circledast_\\kappa$ assert that for every Abelian group $G$ of cardinality $\\kappa$, there exists a colouring $c:G\\rightarrow G$ such that for every positive integer $n$, every $X_0,\\ldots,X_n \\in[G]^\\kappa$, and every $\\gamma\\in G$, there are $x_0\\in X_0,\\ldots, x_n\\in X_n$ such that $c(x_0+\\cdots+x_n)=\\gamma$. Then $\\circledast_\\kappa$ holds for unboundedly many uncountable cardinals $\\kappa$, and it is consistent that $\\circledast_\\kappa$ holds for all regular uncountable cardinals $\\kappa$.\n Comments: Final accepted version. For several of the earlier results that were stated only for regular cardinals, there is now a treatment of the singular cardinal case. Also, a new partition relation for the Real Line was obtained, see Theorem C3 Subjects: Logic (math.LO); Combinatorics (math.CO) MSC classes: 03E02 (Primary), 03E75, 03E35, 05D10, 05A17, 11P99, 20M14 (Secondary) Journal reference: Transactions of the American Mathematical Society 369 no. 12 (2017), 8939-8966 DOI: 10.1090/tran/7131 Cite as: arXiv:1608.01512 [math.LO] (or arXiv:1608.01512v4 [math.LO] for this version)\n\n## Submission history\n\nFrom: David Fernández Bretón [view email]\n[v1] Thu, 4 Aug 2016 12:35:07 UTC (9 KB)\n[v2] Sun, 28 Aug 2016 17:33:20 UTC (20 KB)\n[v3] Sat, 24 Sep 2016 11:55:16 UTC (24 KB)\n[v4] Tue, 22 Nov 2016 14:59:44 UTC (28 KB)" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.78147185,"math_prob":0.9910084,"size":2005,"snap":"2019-13-2019-22","text_gpt3_token_len":623,"char_repetition_ratio":0.11794103,"word_repetition_ratio":0.0,"special_character_ratio":0.30174562,"punctuation_ratio":0.14285715,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9991167,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-19T15:01:34Z\",\"WARC-Record-ID\":\"<urn:uuid:c4cbc8d9-5aee-4c80-b6c2-e78b545d29e3>\",\"Content-Length\":\"19579\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b782ca6b-f4ea-4a50-b4f4-84803acfc5c3>\",\"WARC-Concurrent-To\":\"<urn:uuid:d9871170-8337-48a1-a3e0-400aca9b983e>\",\"WARC-IP-Address\":\"128.84.21.199\",\"WARC-Target-URI\":\"https://arxiv.org/abs/1608.01512\",\"WARC-Payload-Digest\":\"sha1:HQ7PNZXB44UDQCIR6ACW7FL6PI2U7XAH\",\"WARC-Block-Digest\":\"sha1:WCO57N6BSHG34NSYUXU7D2MKJUNWCZ7D\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912201996.61_warc_CC-MAIN-20190319143502-20190319165502-00262.warc.gz\"}"}
https://www.geeksforgeeks.org/tag/two-pointer-algorithm/page/7/	[ "Skip to content\n\n# Tag Archives: two-pointer-algorithm\n\nGiven an array of distinct integers and a sum value. Print all triplets with sum smaller than given sum value. Expected Time Complexity is O(n2).… Read More\nGiven an array of N numbers where a subarray is sorted in descending order and rest of the numbers in the array are in ascending… Read More\nGiven two sorted arrays of distinct elements, we need to print those elements from both arrays that are not common. The output should be printed… Read More\nGiven an array of integers, you have to find three numbers such that the sum of two elements equals the third element.Examples: Input : {5,… Read More\nWrite a program to reverse the given string while preserving the position of spaces. Examples: Input : \"abc de\" Output : edc ba Input :… Read More\nGiven an array of distinct elements. The task is to find triplets in the array whose sum is zero. Examples : Input : arr[] =… Read More\nGiven two sorted arrays and a number x, find the pair whose sum is closest to x and the pair has an element from each… Read More\nGiven a sorted array and a number x, find a pair in array whose sum is closest to x.Examples: Input: arr[] = {10, 22, 28,… Read More\nGiven an array of integers, find all combination of four elements in the array whose sum is equal to a given value X. For example, if… Read More\nGiven an array and a value, find if there is a triplet in array whose sum is equal to the given value. If there is… Read More" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.77639365,"math_prob":0.98701304,"size":1315,"snap":"2021-21-2021-25","text_gpt3_token_len":308,"char_repetition_ratio":0.17009915,"word_repetition_ratio":0.10288066,"special_character_ratio":0.23574145,"punctuation_ratio":0.11913358,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99444216,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-24T23:01:29Z\",\"WARC-Record-ID\":\"<urn:uuid:83bcb759-0365-40b7-9742-4203bd024a29>\",\"Content-Length\":\"93332\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6435a63e-f5a5-4d97-abc2-d7c1027acbb1>\",\"WARC-Concurrent-To\":\"<urn:uuid:55baa73a-d8ac-454d-91fb-79e5e6afe0ea>\",\"WARC-IP-Address\":\"23.12.145.61\",\"WARC-Target-URI\":\"https://www.geeksforgeeks.org/tag/two-pointer-algorithm/page/7/\",\"WARC-Payload-Digest\":\"sha1:OCAEZIHFRA57Y3M7PU74GYBJK5PLOHS5\",\"WARC-Block-Digest\":\"sha1:P3RO4H7UGHMZYOTER5QFCE7TNLPFN4V5\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623488559139.95_warc_CC-MAIN-20210624202437-20210624232437-00423.warc.gz\"}"}
https://www.meritnation.com/ask-answer/question/how-to-calculate-the-number-of-binary-operations-on-any-set/relations-and-functions/2466697	[ "how to calculate the number of binary operations on any set A , say of 4 elements?\n\nLet S be a finite set having n elements.\n\nThen S × S has n2 elements.\n\nSince a binary operation on S is a function from S × S to S.\n\n∴ Total number of binary operations on S is equal to the number of functions from S × S to S.\n\nAlso, cardinality of domain (i.e. S × S) is n2 and cardinality of co-domain (i.e. S) is n.\n\nSo, total number of functions from S × S to S are", null, "∴ Total number of binary operations on set S having n elements is", null, ".\n\nNow, you are saying that on any set A (i.e. S is given to be A here) having 4 elements (i.e. value of n is 4)\n\nUsing the formula derived above, The number of binary operations on\n\nSet A =", null, "=", null, "• 41\nWhat are you looking for?" ]	[ null, "https://s3mn.mnimgs.com/img/shared/discuss_editlive/3021597/2012_06_13_15_47_51/mathmlequation8402978672626898654.png", null, "https://s3mn.mnimgs.com/img/shared/discuss_editlive/3021597/2012_06_13_15_47_51/mathmlequation8402978672626898654.png", null, "https://s3mn.mnimgs.com/img/shared/discuss_editlive/3021597/2012_06_13_15_47_51/mathmlequation2349482958611467009.png", null, "https://s3mn.mnimgs.com/img/shared/discuss_editlive/3021597/2012_06_13_15_47_51/mathmlequation3249821803862957466.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9355226,"math_prob":0.99809337,"size":703,"snap":"2019-43-2019-47","text_gpt3_token_len":192,"char_repetition_ratio":0.15593705,"word_repetition_ratio":0.12666667,"special_character_ratio":0.27596018,"punctuation_ratio":0.11904762,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9994592,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,null,null,null,null,6,null,6,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-19T13:51:22Z\",\"WARC-Record-ID\":\"<urn:uuid:1b1ed2ee-0419-4ed9-be54-c08d4f23b8e9>\",\"Content-Length\":\"29457\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6ed3c024-7e3f-4f53-92e2-7331a591b9ce>\",\"WARC-Concurrent-To\":\"<urn:uuid:f6ad7d69-686c-481b-aefe-1240d3b3b775>\",\"WARC-IP-Address\":\"165.254.45.201\",\"WARC-Target-URI\":\"https://www.meritnation.com/ask-answer/question/how-to-calculate-the-number-of-binary-operations-on-any-set/relations-and-functions/2466697\",\"WARC-Payload-Digest\":\"sha1:QVCRRIJCFP2PLH5ONNUTGFK5ZLEZH63D\",\"WARC-Block-Digest\":\"sha1:GIQRKZEYKEPGNBBLWW3S2BKHI4R7YJI2\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570986693979.65_warc_CC-MAIN-20191019114429-20191019141929-00002.warc.gz\"}"}
https://mathoverflow.net/questions/198039/is-there-a-formula-for-the-total-chern-class-of-the-tangent-space-of-a-projectiv	[ "# Is there a formula for the total Chern Class of the tangent space of a projectivized vector bundle?\n\nLet $V\\rightarrow M$ be a complex vector bundle (of rank $k$) over a complex manifold $M$ (you can assume $M$ is compact if that helps, but it may not be relevant to my question). Let $\\pi:\\mathbb{P}V \\rightarrow M$ be the projectivization of $V$.\n\n$\\textbf{Question}:$ Is there a formula for $c(T\\mathbb{P}V)$, the total Chern class of the Tangent space of $\\mathbb{P}V$?\n\nMy naive guess would be that it should be $\\pi^(c(TM))(1+c_1(\\gamma^))^{k+1}$, where $\\gamma \\rightarrow \\mathbb{P}V$ is the tautological line bundle over $\\mathbb{P}V$. I think my guess is correct if $M$ was just a point, or more generally if $V$ was a trivial bundle. But I do not know if this is correct in general.\n\nThe specific case for which I need an answer is when $M:= \\mathbb{P}^1 \\times \\mathbb{P}^1$ and $V:= \\mathcal{O}(d_1) \\oplus \\mathcal{O}(d_2)$.\n\n$\\textbf{Added Later}:$ It has been pointed out my guess is wrong in general. The correct answer is $$\\pi^(c(TM))c(\\pi^V \\otimes \\gamma^).$$\n\nNo, your formula is not correct. You have to take into account the Chern classes of $V$. The relative tangent bundle $T_{\\mathbb{P}V/M}$ is given by the so-called Euler exact sequence $$0\\rightarrow \\mathscr{O}_{\\mathbb{P}V}\\rightarrow \\pi ^V\\otimes \\gamma^* \\rightarrow T_{\\mathbb{P}V/M}\\rightarrow 0\\ ,$$ while $$0\\rightarrow T_{\\mathbb{P}V/M}\\rightarrow T_{\\mathbb{P}V}\\rightarrow \\pi ^T_M\\rightarrow 0\\ .$$Putting things together we find $c(T_{\\mathbb{P}V})=c(\\pi ^V\\otimes \\gamma^* )\\,\\pi ^c(T_M)$.\n\nThen use the standard formula for $c(\\pi ^V\\otimes \\gamma^* )$.\n\nFor any smooth fiber bundle\n\n$$F\\hookrightarrow P \\stackrel{\\pi}{\\to} M$$\n\nwe have a short exact sequence of vector bundles over $P$\n\n$$0\\to VTP\\to TP \\to \\pi^* TM\\to 0,$$\n\nwhere $VTP$ denotes the vertical tangent bundle defined as the kernel of the differential of $\\pi$. If the bundle is holomorphic then the above is a short exact sequence of complex vector bundles and we deduce\n\n$$c(TP)= c(VTP)\\cdot \\pi^* c(TM).$$\n\nThe classical Euler exact sequence argument shows that when $P=\\mathbb{P}(V)$ that $\\newcommand{\\bC}{\\mathbb{C}}$\n\n$$\\gamma^\\otimes \\pi^V \\cong \\underline{\\bC}\\oplus VTP,$$\n\nwhere $\\underline{\\bC}$ denotes the trivial line bundle. Hence\n\n$$c(TP)= c(\\gamma^\\otimes \\pi^V)\\cdot \\pi^* c(TP).$$\n\nIn Section I.3 of Fulton-Lang Riemann-Roch algebra you can find an explicit formula for $c_k(L\\otimes E)$, $L$ line bundle and $E$ vector bundle of rank $m$. More precisely\n\n$$c_k(L\\otimes E)=\\sum_{j=1}^k \\binom{m-j}{k-j} c_j(E)c_1(L)^{k-j}.$$ Note. The original answer had an error that I have now corrected. (Hat tip to abx)." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.82665956,"math_prob":0.9998031,"size":981,"snap":"2020-24-2020-29","text_gpt3_token_len":322,"char_repetition_ratio":0.114636645,"word_repetition_ratio":0.0,"special_character_ratio":0.31396535,"punctuation_ratio":0.08695652,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000083,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-02T13:24:10Z\",\"WARC-Record-ID\":\"<urn:uuid:7bdae083-d717-4393-b674-4ffc98cfbdfa>\",\"Content-Length\":\"131670\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:68410815-6946-4e55-b648-881f9f6b9abe>\",\"WARC-Concurrent-To\":\"<urn:uuid:046967c8-9303-45ef-bd2a-e0c52f7fe777>\",\"WARC-IP-Address\":\"151.101.193.69\",\"WARC-Target-URI\":\"https://mathoverflow.net/questions/198039/is-there-a-formula-for-the-total-chern-class-of-the-tangent-space-of-a-projectiv\",\"WARC-Payload-Digest\":\"sha1:YKLV3XZNRNWQPZQJG53PB6AD6GSGIGHU\",\"WARC-Block-Digest\":\"sha1:GAZEMCNH4KVGSCX3VQIRJ343OPPVDQ24\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593655878753.12_warc_CC-MAIN-20200702111512-20200702141512-00464.warc.gz\"}"}
https://www.vedantu.com/maths/tan-theta-formula	[ "# Tan Theta Formula\n\n## Tan Theta\n\nThe tangent is defined in right triangle trigonometry as the ratio of the opposite side to the adjacent side (It is applicable for acute angles only because it's only defined this way for right triangles). To find values of the tangent function at different angles when evaluating the tangent function, we first define the reference angle created by the terminal side and the x-axis. Then we calculate the tangent of this reference angle and determine whether it is positive or negative based on which quadrant the terminal side is in. In the first and third quadrants, the tangent is positive. In the second and fourth quadrants, the tangent is negative. The slope of the terminal side is also equal to the tangent.\n\nLet us discuss an introduction to Trigonometry in detail before looking at the formula. Trigonometry is a branch of mathematics concerned with the application of specific functions of angles to calculations. In trigonometry, there are six functions of an angle that are widely used. Sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc) are their names and abbreviations. In relation to a right triangle, these six trigonometric functions. The sine of A, or sin A, is defined as the ratio of the side opposite to A and the side opposite to the right angle (the hypotenuse) in a triangle. The other trigonometric functions are defined similarly. These functions are properties of the angle that are independent of the triangle's size, and measured values for several angles were tabulated before computers made trigonometry tables outdated. In geometric figures, trigonometric functions are used to calculate unknown angles and distances from known or measured angles. Trigonometry has a wide range of applications, from specific fields such as oceanography, where it is used to measure the height of tides in oceans, to the backyard of our home, where it can be used to roof a building, make the roof inclined in the case of single independent bungalows, and calculate the height of the roof etc. Here, we will discuss the tan theta formula in detail.\n\n### How to Find the Tangent?\n\nYou must first locate the hypotenuse to find the tangent. The hypotenuse is typically the right triangle's longest side. The next task is to decide the angle. There are only two angles to choose from. You cannot choose the right angle. After you've chosen an angle, you will mark the sides. The side opposite to this angle will be the opposite side and the side next to the angle is the adjacent side. After labelling the sides, you can take the required ratio. Let’s discuss ratios, what is tan theta and it’s practical applications?\n\n### What is Tan Theta?\n\nThe length of the opposite side to the length of the adjacent side of a right-angled triangle is known as the tangent function or tangent ratio of the angle between the hypotenuse and the base.\n\nAs discussed, the tangent function is one of the three most common trigonometric functions, along with sine and cosine. The tangent of an angle in a right triangle is equal to the length of the opposite side (O) divided by the length of the adjacent side (A). It is written simply as 'tan' in a formula.\n\n⇒ tan x = O/A\n\ntan(x) is the symbol for the tangent function which is also called the tan x formula. It is one of the six trigonometric functions that are commonly used. Sine and cosine are most often associated with the tangent. In trigonometry, the tangent function is a periodic function that is very useful.\n\nThe tan formula is as follows:\n\nWhat is tan theta in terms of sine and cos?\n\n⇒ tan x = sin x/cos x\n\nor, tan theta = sin theta/cos theta (here, theta is an angle)\n\nThe sine of an angle is equal to the length of the opposite side divided by the length of the hypotenuse side, while the cosine of an angle is the ratio of the adjacent side to the hypotenuse side.\n\nHence, sin x = Opposite Side/Hypotenuse Side\n\ncos x = Adjacent Side/Hypotenuse Side\n\nTherefore, (tan formula) tan x = Opposite Side/Adjacent Side\n\n### Finding the Tangent of the Triangle\n\nAngles A and B are the two angles we will deal with in this triangle. To find our tangent, we must first find out hypotenuse. Our right angle is clear to me. Can you see it? That means our hypotenuse is direct across it, and the side that measures 5 is the hypotenuse. Can you see it? Okay, as we have our hypotenuse, let's choose an angle to work with. We'll choose angle B. As B is our angle, our opposite side is the side that measures 3. Our adjacent side is the one that measures 4 because it is the only side next to angle B which is not the hypotenuse.\n\nThis means that our tangent of angle B will be the ratio of the opposite side over the adjacent side or we can write it as 3/4 which will be equal to 0.75. Similarly, If we chose angle A, our sides will change and the tangent will be 4/3 which will be equal to 1.33.\n\n### Tangent to Find the Missing Side\n\nIn a few problems, we will have to find the missing side of the triangle. This particularly has real-life applications like when construction companies are building on hills. In these kinds of problems, an angle is given to us. To solve these problems, we first have to locate the side that is missing. In the triangle given above, the side missed is the adjacent side to the angle given. Right? What we have to do now is just to write the equation from the definition of the tangent. Once we have the equation, we can move ahead, use algebra to solve for the variable which will be missing.\n\nBy multiplying both sides by x and then dividing both sides by tan 66, we've identified our variable. And by using the calculator to measure tan 66, we get the answer as 2.22.\n\n### Arctan - Inverse Tangent Function\n\nThere is an inverse function for any trigonometry function, for example, tan has arctan, which works in a reverse manner. These inverse functions have the same name as the originals, but with the word 'arc' added to the start. So, arctan is the opposite of tan. If we know the tangent of an angle and want to know the actual angle, we use the inverse function.\n\n### Large & Negative Angles\n\nThe two variable angles in a right triangle are always less than 90 degrees. However, we can find the tangent of any angle, irrespective of its height, as well as the tangent of negative angles. We can also graph the tangent function.\n\n### Tangents Will Be Used to Calculate the Height of a Building or a Mountain:\n\nYou can easily find the height of a building if you know the distance from where you observe it and also the angle of elevation. Similarly, you can find another side of the triangle if you know the value of one side and the angle of depression from the top of the house. All you need to know is a side and an angle of the triangle.\n\n### Conclusion\n\nIn fields like astronomy, mapmaking, surveying, and artillery range finding, trigonometry emerged from the need to compute angles and distances. Plane trigonometry deals with issues involving angles and distances in a single plane. Spherical trigonometry considers applications of related problems in more than one plane of three-dimensional space." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.92787665,"math_prob":0.9794985,"size":8762,"snap":"2021-43-2021-49","text_gpt3_token_len":1943,"char_repetition_ratio":0.17743777,"word_repetition_ratio":0.04737184,"special_character_ratio":0.20851403,"punctuation_ratio":0.10438293,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99912256,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-18T22:56:10Z\",\"WARC-Record-ID\":\"<urn:uuid:4fa50ff2-cbb9-441c-b7c1-36f865034d47>\",\"Content-Length\":\"90869\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ab1d3d8d-bc40-48ee-b33e-02814786ddc7>\",\"WARC-Concurrent-To\":\"<urn:uuid:58b54166-d5b2-4be1-85a2-38b445274037>\",\"WARC-IP-Address\":\"13.32.208.96\",\"WARC-Target-URI\":\"https://www.vedantu.com/maths/tan-theta-formula\",\"WARC-Payload-Digest\":\"sha1:Y44DAWEJMSIPQVODFO4QEAMOUKGILPON\",\"WARC-Block-Digest\":\"sha1:S2IOM7QP5MF3JAR4YWELISJHXUAWJ23H\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585215.14_warc_CC-MAIN-20211018221501-20211019011501-00087.warc.gz\"}"}
https://code.orgmode.org/bzg/org-mode/commit/8ecc966292f322ec6d0d0fb29e1087a55d22975f	[ "### Implement \"delay\" cookies for scheduled items.\n\n```* org-agenda.el (org-agenda-skip-scheduled-delay-if-deadline):\nNew option. The structure of the possible values is copied\n(org-agenda-get-scheduled): Honor the two new option,\n`org-scheduled-delay-days' and\nscheduled entry has a delay cookie like \"-2d\" (similar to the\nneeded.\n\n* org.el (org-deadline-warning-days): Small docstring fix.\n(org-scheduled-delay-days): New option (see\n(org-get-wdays): Use the new option.\n\nThanks to Andrew M. Nuxoll and Michael Brand for this idea.\n\nYou can now use a \"delay cookie\" in scheduled items. For example,\n\n* TODO Sleep\nSCHEDULED: <2013-02-06 mer. -3d>\n\nwill not be shown on 06/02 but on 09/02, three days later.\n\nThe value of the cookie overrides any value of `org-scheduled-delay-days',\nunless `org-scheduled-delay-days' is negative (same logic than for\n\nAlso check org-agenda-skip-scheduled-delay-if-deadline, which does for", null, "Bastien Guerry 6 years ago\nparent\ncommit\n8ecc966292\n2 changed files with 89 additions and 24 deletions\n1. 51 8\nlisp/org-agenda.el\n2. 38 16\nlisp/org.el\n\n#### + 51 - 8 lisp/org-agenda.el View File\n\n ``@@ -843,6 +843,21 @@ because you will take care of it on the day when scheduled.\"`` `` (const :tag \"Remove prewarning if entry is scheduled\" t)`` `` (integer :tag \"Restart prewarning N days before deadline\")))`` `` `` ``+(defcustom org-agenda-skip-scheduled-delay-if-deadline nil`` ``+ \"Non-nil means skip scheduled delay when entry also has a deadline.`` ``+This variable may be set to nil, t, the symbol `post-deadline',`` ``+or a number which will then give the number of days after the actual`` ``+scheduled date when the delay should expire. The symbol `post-deadline'`` ``+eliminates the schedule delay when the date is posterior to the deadline.\"`` ``+ :group 'org-agenda-skip`` ``+ :group 'org-agenda-daily/weekly`` ``+ :version \"24.3\"`` ``+ :type '(choice`` ``+ (const :tag \"Always honor delay\" nil)`` ``+ (const :tag \"Ignore delay if posterior to the deadline\" post-deadline)`` ``+ (const :tag \"Ignore delay if entry has a deadline\" t)`` ``+ (integer :tag \"Honor delay up until N days after the scheduled date\")))`` ``+`` `` (defcustom org-agenda-skip-additional-timestamps-same-entry nil`` `` \"When nil, multiple same-day timestamps in entry make multiple agenda lines.`` `` When non-nil, after the search for timestamps has matched once in an`` ``@@ -5331,7 +5346,13 @@ the documentation of `org-diary'.\"`` `` (setq results (append results rtn))))))))`` `` results))))`` `` `` ``+(defsubst org-em (x y list)`` ``+ \"Is X or Y a member of LIST?\"`` ``+ (or (memq x list) (memq y list)))`` ``+`` `` (defvar org-heading-keyword-regexp-format) ; defined in org.el`` ``+(defvar org-agenda-sorting-strategy-selected nil)`` ``+`` `` (defun org-agenda-get-todos ()`` `` \"Return the TODO information for agenda display.\"`` `` (let* ((props (list 'face nil`` ``@@ -6143,7 +6164,8 @@ FRACTION is what fraction of the head-warning time has passed.\"`` `` deadline-results))`` `` d2 diff pos pos1 category category-pos level tags donep`` `` ee txt head pastschedp todo-state face timestr s habitp show-all`` ``- did-habit-check-p warntime inherited-tags ts-date)`` ``+ did-habit-check-p warntime inherited-tags ts-date suppress-delay`` ``+ ddays)`` `` (goto-char (point-min))`` `` (while (re-search-forward regexp nil t)`` `` (catch :skip`` ``@@ -6162,12 +6184,38 @@ FRACTION is what fraction of the head-warning time has passed.\"`` `` warntime (get-text-property (point) 'org-appt-warntime))`` `` (setq pastschedp (and todayp (< diff 0)))`` `` (setq did-habit-check-p nil)`` ``+ (setq suppress-delay`` ``+ (let ((ds (and org-agenda-skip-scheduled-delay-if-deadline`` ``+ (let ((item (buffer-substring (point-at-bol) (point-at-eol))))`` ``+ (save-match-data`` ``+ (and (string-match`` ``+ org-deadline-time-regexp item)`` ``+ (match-string 1 item)))))))`` ``+ (cond`` ``+ ((not ds) nil)`` ``+ ;; The current item has a deadline date (in ds), so`` ``+ ;; evaluate its delay time.`` ``+ ((integerp org-agenda-skip-scheduled-delay-if-deadline)`` ``+ ;; Use global delay time.`` ``+ (- org-agenda-skip-scheduled-delay-if-deadline))`` ``+ ((eq org-agenda-skip-scheduled-delay-if-deadline`` ``+ 'post-deadline)`` ``+ ;; Set delay to no later than deadline.`` ``+ (min (- d2 (org-time-string-to-absolute`` ``+ ds d1 'past show-all (current-buffer) pos))`` ``+ org-scheduled-delay-days))`` ``+ (t 0))))`` ``+ (setq ddays (if suppress-delay`` ``+ (let ((org-scheduled-delay-days suppress-delay))`` ``+ (org-get-wdays s t t))`` ``+ (org-get-wdays s t)))`` `` ;; When to show a scheduled item in the calendar:`` `` ;; If it is on or past the date.`` ``- (when (or (and (< diff 0)`` ``+ (when (or (and (> ddays 0) (= diff (- ddays)))`` ``+ (and (zerop ddays) (= diff 0))`` ``+ (and (< diff 0)`` `` (< (abs diff) org-scheduled-past-days)`` `` (and todayp (not org-agenda-only-exact-dates)))`` ``- (= diff 0)`` `` ;; org-is-habit-p uses org-entry-get, which is expansive`` `` ;; so we go extra mile to only call it once`` `` (and todayp`` ``@@ -6578,7 +6626,6 @@ The modified list may contain inherited tags, and tags matched by`` `` s))`` `` `` `` (defvar org-agenda-sorting-strategy) ;; because the def is in a let form`` ``-(defvar org-agenda-sorting-strategy-selected nil)`` `` `` `` (defun org-agenda-add-time-grid-maybe (list ndays todayp)`` `` \"Add a time-grid for agenda items which need it.`` ``@@ -6893,10 +6940,6 @@ without respect of their type.\"`` `` (cond ((and ha (not hb)) -1)`` `` ((and (not ha) hb) +1))))`` `` `` ``-(defsubst org-em (x y list)`` ``- \"Is X or Y a member of LIST?\"`` ``- (or (memq x list) (memq y list)))`` ``-`` `` (defun org-entries-lessp (a b)`` `` \"Predicate for sorting agenda entries.\"`` `` ;; The following variables will be used when the form is evaluated.``\n\n#### + 38 - 16 lisp/org.el View File\n\n ``@@ -2864,7 +2864,7 @@ is used.\"`` `` (string :tag \"Format string\")))))`` `` `` `` (defcustom org-deadline-warning-days 14`` ``- \"No. of days before expiration during which a deadline becomes active.`` ``+ \"Number of days before expiration during which a deadline becomes active.`` `` This variable governs the display in sparse trees and in the agenda.`` `` When 0 or negative, it means use this number (the absolute value of it)`` `` even if a deadline has a different individual lead time specified.`` ``@@ -2874,6 +2874,20 @@ Custom commands can set this variable in the options section.\"`` `` :group 'org-agenda-daily/weekly`` `` :type 'integer)`` `` `` ``+(defcustom org-scheduled-delay-days 0`` ``+ \"Number of days before a scheduled item becomes active.`` ``+This variable governs the display in sparse trees and in the agenda.`` ``+The default value (i.e. 0) means: don't delay scheduled item.`` ``+When negative, it means use this number (the absolute value of it)`` ``+even if a scheduled item has a different individual delay time`` ``+specified.`` ``+`` ``+Custom commands can set this variable in the options section.\"`` ``+ :group 'org-time`` ``+ :group 'org-agenda-daily/weekly`` ``+ :version \"24.3\"`` ``+ :type 'integer)`` ``+`` `` (defcustom org-read-date-prefer-future t`` `` \"Non-nil means assume future for incomplete date input from user.`` `` This affects the following situations:`` ``@@ -16216,21 +16230,29 @@ If SECONDS is non-nil, return the difference in seconds.\"`` `` (and (< (org-time-stamp-to-now timestamp-string) ndays)`` `` (not (org-entry-is-done-p))))`` `` `` ``-(defun org-get-wdays (ts)`` ``- \"Get the deadline lead time appropriate for timestring TS.\"`` ``- (cond`` ``- ((<= org-deadline-warning-days 0)`` ``- ;; 0 or negative, enforce this value no matter what`` ``- (- org-deadline-warning-days))`` ``- ((string-match \"-\\\\([0-9]+\\\\)\\\\([hdwmy]\\\\)\\\\(\\\\'\\\\\|>\\\\\| \\\\)\" ts)`` ``- ;; lead time is specified.`` ``- (floor (* (string-to-number (match-string 1 ts))`` ``- (cdr (assoc (match-string 2 ts)`` ``- '((\"d\" . 1) (\"w\" . 7)`` ``- (\"m\" . 30.4) (\"y\" . 365.25)`` ``- (\"h\" . 0.041667)))))))`` ``- ;; go for the default.`` ``- (t org-deadline-warning-days)))`` ``+(defun org-get-wdays (ts &optional delay zero-delay)`` ``+ \"Get the deadline lead time appropriate for timestring TS.`` ``+When DELAY is non-nil, get the delay time for scheduled items`` ``+instead of the deadline lead time. When ZERO-DELAY is non-nil`` ``+and `org-scheduled-delay-days' is 0, enforce 0 as the delay,`` ``+don't try to find the delay cookie in the scheduled timestamp.\"`` ``+ (let ((tv (if delay org-scheduled-delay-days`` ``+ org-deadline-warning-days)))`` ``+ (cond`` ``+ ((or (and delay (< tv 0))`` ``+ (and delay zero-delay (<= tv 0))`` ``+ (and (not delay) (<= tv 0)))`` ``+ ;; Enforce this value no matter what`` ``+ (- tv))`` ``+ ((string-match \"-\\\\([0-9]+\\\\)\\\\([hdwmy]\\\\)\\\\(\\\\'\\\\\|>\\\\\| \\\\)\" ts)`` ``+ ;; lead time is specified.`` ``+ (floor (* (string-to-number (match-string 1 ts))`` ``+ (cdr (assoc (match-string 2 ts)`` ``+ '((\"d\" . 1) (\"w\" . 7)`` ``+ (\"m\" . 30.4) (\"y\" . 365.25)`` ``+ (\"h\" . 0.041667)))))))`` ``+ ;; go for the default.`` ``+ (t tv))))`` `` `` `` (defun org-calendar-select-mouse (ev)`` `` \"Return to `org-read-date' with the date currently selected.``" ]	[ null, "https://secure.gravatar.com/avatar/beca86d6416e66d0b9d5b25699995b98", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6265436,"math_prob":0.81846863,"size":4838,"snap":"2019-51-2020-05","text_gpt3_token_len":1510,"char_repetition_ratio":0.21617708,"word_repetition_ratio":0.13440861,"special_character_ratio":0.3978917,"punctuation_ratio":0.095607236,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9781571,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-07T10:23:56Z\",\"WARC-Record-ID\":\"<urn:uuid:d92356d2-8c10-4584-9470-d4de06a88232>\",\"Content-Length\":\"83575\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:28fe6f0d-555b-410b-a084-aa15002456a6>\",\"WARC-Concurrent-To\":\"<urn:uuid:d1f904c9-2e1d-4fc4-8ca7-e2da37815dc0>\",\"WARC-IP-Address\":\"45.77.206.30\",\"WARC-Target-URI\":\"https://code.orgmode.org/bzg/org-mode/commit/8ecc966292f322ec6d0d0fb29e1087a55d22975f\",\"WARC-Payload-Digest\":\"sha1:YFJNIEO6TALC37FYAMS7BAE6HOVZGPFF\",\"WARC-Block-Digest\":\"sha1:Q5DBPXVUH7OIFW4STREX2KU2UHLQ6WGQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575540497022.38_warc_CC-MAIN-20191207082632-20191207110632-00326.warc.gz\"}"}
https://physics.stackexchange.com/questions/78121/textbook-problem-free-energy-expression	[ "Textbook Problem: Free energy expression\n\nIn Klotz, Introduction to Chemical Thermodynamics, Ex. 8.2 requires me to derive $$dG = V \\left( \\frac{\\partial p}{\\partial V} \\right)_V dV + \\left[ V \\left( \\frac{\\partial p}{\\partial T} \\right)_V - S \\right]dT,$$ where $G$ is the Gibbs free energy, $S$ is the entropy and all the other variables have their ordinary common-sense meaning.\n\nI'd appreciate feedback on my solution, which I find just very simple, maybe too simple to be right.\n\nFirst of all, when recognizing that $G$ is a function of $V$ and $T$, we can write the total differential as\n\n$$dG = \\left( \\frac{\\partial G}{\\partial V} \\right)_T dV + \\left( \\frac{\\partial G}{\\partial T} \\right)_V dT$$\n\nIn Eq. 8.19, we learn that\n\n$$dG = V dp - S dT$$\n\nand so, when multiplying both sides by $1/\\partial V$ at constant temperature, we find\n\n$$\\left( \\frac{\\partial G}{\\partial V} \\right)_T = V \\left( \\frac{\\partial p}{\\partial V} \\right)_T - S \\left( \\frac{\\partial T}{\\partial V} \\right)_T$$\n\nwhere $(\\partial T /\\partial V)_T$ is zero, because we're at constant $T$. A similar procedure yields the bracket expression, this time instead we just multiply both sides with $1/\\partial T$ and using constant $V$, we arrive at\n\n$$\\left( \\frac{\\partial G}{\\partial T} \\right)_V = V \\left( \\frac{\\partial p}{\\partial T} \\right)_V - S \\left( \\frac{\\partial T}{\\partial T} \\right)_V = V \\left( \\frac{\\partial p}{\\partial T} \\right)_V - S$$\n\nand from comparison with the total differential expression, one immediately recognizes the above very first expression.\n\nIs this a valid solution/Ansatz?\n\n• Comment 1: Thanks for the edit. Comment 2: Can we add equation numbers? Sep 22 '13 at 18:24\n\nSeems OK to me. Although I would just write $dp$ in $dG=Vdp−SdT$ in terms of $dV$ and $dT$.\nEDIT (09/22/2013): In more detail: substitute $dp=(\\frac{\\partial p}{\\partial V})_T dV+(\\frac{\\partial p}{\\partial T})_V dT$ into $dG=Vdp-SdT$.\n• @TMOTTM: We start with what you have to proof an rewrite it once, just by setting the brackets differently: $dG = V \\left( \\frac{\\partial p}{\\partial V} \\right)_V dV + \\left[ V \\left( \\frac{\\partial p}{\\partial T} \\right)_V - S \\right]dT = V\\ \\left[ \\left( \\frac{\\partial p}{\\partial V} \\right)_V dV + \\left( \\frac{\\partial p}{\\partial T} \\right)_V \\ dT \\right] - S \\ dT$. Username akhmeteli points out that the expression in the bracket is $dp$, and then you're done. Oct 22 '13 at 20:48" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7782423,"math_prob":0.9999417,"size":1548,"snap":"2022-05-2022-21","text_gpt3_token_len":464,"char_repetition_ratio":0.2623057,"word_repetition_ratio":0.114285715,"special_character_ratio":0.3113695,"punctuation_ratio":0.082474224,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000058,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-01-27T09:16:34Z\",\"WARC-Record-ID\":\"<urn:uuid:01047681-60cd-4013-a642-b318bbb2a6f9>\",\"Content-Length\":\"137139\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7b944760-f07f-482d-a0f7-2f35a551e532>\",\"WARC-Concurrent-To\":\"<urn:uuid:1a2d0bdb-c5d2-4abc-a4dd-177336402d1e>\",\"WARC-IP-Address\":\"151.101.129.69\",\"WARC-Target-URI\":\"https://physics.stackexchange.com/questions/78121/textbook-problem-free-energy-expression\",\"WARC-Payload-Digest\":\"sha1:P3UY6N5AXS6H5Y5NPBDVOTNRQWOINRZB\",\"WARC-Block-Digest\":\"sha1:3TGLVDKJ4UV2PEKURIVQF7ZKXBHDVYRI\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-05/CC-MAIN-2022-05_segments_1642320305242.48_warc_CC-MAIN-20220127072916-20220127102916-00487.warc.gz\"}"}
https://www.hpmuseum.org/forum/thread-4503-post-40483.html	[ "HP-41 Programs\n08-09-2015, 07:16 PM\nPost: #1\n Tim Walker", null, "Junior Member Posts: 2 Joined: Aug 2015\nHP-41 Programs\nHi: I' am looking for a program to calculate the volume of laminar flow of water in a pipe using Manning's formula. Inputs would be: Slope of the pipe, diameter of the pipe, Manning's roughness factor, and measured depth of flow. Barcode of the program would be great, or mag cards (I could provide the cards if I can find some), or even the program listing if that is all you have. Email me direct at [email protected] Thanks, Tim\n08-09-2015, 07:44 PM\nPost: #2\n Thomas Klemm", null, "Senior Member Posts: 1,804 Joined: Dec 2013\nRE: HP-41 Programs\nBut as far as I remember the slope of the pipe wasn't involved. Maybe it's still useful for you.\nSome of the attachments in my posts contain the barcode of the programs. Thus it should be easy for you to load them into your calculator.\n\nKind regards\nThomas\n08-11-2015, 07:32 PM\nPost: #3", null, "SlideRule", null, "Senior Member Posts: 1,325 Joined: Dec 2013\nRE: HP-41 Programs\nCircular X-Section, Partially Full, Gravity Flow, SOLVER approach\n\nProblem\nA 4-foot-diameter finished concrete pipe is laid at a slope of 0.2%. The water depth is 3 feet. Calculate the flow rate and flow velocity using Manning's equation.\n\nSolution\n\nStep 1. Calculate the flow rate (ft3/sec).\n- Q for Partially Full Circular X-Section\nQ=C÷N((0.5×SIN(ACOS((H−D÷2)÷(D÷2))×2)+π(360−ACOS((H−D÷2)÷(D÷2))×2)÷360)\n×(D÷2)^2)^(5÷3)÷(π×D(360−ACOS((H−D÷2)÷(D÷2))×2)÷360)^(2÷3)×S^0.5\n\nWhere:\nQ = flow rate (ft3/sec, m3/s)\nC = 1.49 for English units, 1 for Metric units\nN = Manning roughness coefficient\nH = depth of water (ft, m)\nD = diameter of pipe (ft, m)\nS = slope of energy line (decimal)\n\nStep 2. Compute the velocity of flow.\n- V for Partially Full Circular X-Section\nV=C÷N((0.5×SIN(ACOS((H−D÷2)÷(D÷2))×2)+π(360−ACOS((H−D÷2)÷(D÷2))\n×2)÷360)×(D÷2)^2)÷(π×D(360−ACOS((H−D÷2)÷(D÷2))×2)÷360)^(2÷3)×S^0.5\n\nWhere:\nV = velocity of flow (ft/sec, m/s)\nC = 1.49 for English units, 1 for Metric units\nN = Manning roughness coefficient\nH = depth of water (ft, m)\nD = diameter of pipe (ft, m)\nS = slope of energy line (decimal)\n\nsee attached PDF for Equations & Illustrations.\n[attachment=2421]\nBEST!\nSlideRule\n08-18-2015, 03:55 PM\nPost: #4\n Tim Walker", null, "Junior Member Posts: 2 Joined: Aug 2015\nRE: HP-41 Programs\nTo Sliderule:\n\nThanks again!\n\nTim Walker\n « Next Oldest \| Next Newest »\n\nUser(s) browsing this thread: 1 Guest(s)" ]	[ null, "https://www.hpmuseum.org/forum/images/buddy_offline.gif", null, "https://www.hpmuseum.org/forum/images/buddy_offline.gif", null, "https://www.hpmuseum.org/forum/uploads/avatars/avatar_187.gif", null, "https://www.hpmuseum.org/forum/images/buddy_offline.gif", null, "https://www.hpmuseum.org/forum/images/buddy_offline.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.81168735,"math_prob":0.95277196,"size":2308,"snap":"2022-40-2023-06","text_gpt3_token_len":831,"char_repetition_ratio":0.106770836,"word_repetition_ratio":0.18598382,"special_character_ratio":0.3544194,"punctuation_ratio":0.122137405,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9890049,"pos_list":[0,1,2,3,4,5,6,7,8,9,10],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-01-27T01:31:29Z\",\"WARC-Record-ID\":\"<urn:uuid:61f1f9d3-799e-4eb3-a4d9-f88b2b07294a>\",\"Content-Length\":\"25000\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:111584b2-d11f-441c-8776-366456345bad>\",\"WARC-Concurrent-To\":\"<urn:uuid:797d12d3-31a1-48a6-979f-7eb5bebf7f88>\",\"WARC-IP-Address\":\"209.197.117.170\",\"WARC-Target-URI\":\"https://www.hpmuseum.org/forum/thread-4503-post-40483.html\",\"WARC-Payload-Digest\":\"sha1:K2QQT653FGQF5FLXNXMYB6GYEV445DQ3\",\"WARC-Block-Digest\":\"sha1:E6D6GMJGKELHSWIY6SH6UPDM6ZMGKHPH\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764494852.95_warc_CC-MAIN-20230127001911-20230127031911-00023.warc.gz\"}"}
https://vcs.vera-visions.com/eukara/hl-pak0-gen/src/branch/master/ccase.sh	[ "You can not select more than 25 topics Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.\n\n#### 39 lines 486 B Raw Permalink Blame History\n\n ```#!/bin/sh ``` ``` ``` ```usage () ``` ```{ ``` ``` echo Usage: `basename \\$0` [-r ] file... >&2 ``` ``` exit 2 ``` ```} ``` ``` ``` ```if [ \\$# -lt 1 ] ``` ```then ``` ``` usage ``` ```fi ``` ``` ``` ```if [ \"\\$1\" = \"-r\" ] ``` ```then ``` ``` recursive=1 ``` ``` shift ``` ``` if [ \\$# -lt 1 ] ``` ``` then ``` ``` usage ``` ``` fi ``` ```else ``` ``` recursive=0 ``` ```fi ``` ``` ``` ```for i in \"\\$@\" ``` ```do ``` ``` new=`echo \\$i \| tr \"[:upper:]\" \"[:lower:]\"` ``` ``` if [ \"\\$new\" != \"\\$i\" ] ``` ``` then ``` ``` echo \\$i \"->\" \\$new >&2 ``` ``` mv \"\\$i\" \"\\$new\" ``` ``` fi ``` ``` if [ \\$recursive = 1 -a -d \"\\$new\" ] ``` ``` then ``` ``` \\$0 -r \"\\$new\"/* ``` ``` fi ``` ```done ```" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.59507895,"math_prob":0.56583846,"size":635,"snap":"2023-40-2023-50","text_gpt3_token_len":249,"char_repetition_ratio":0.15689382,"word_repetition_ratio":0.12738854,"special_character_ratio":0.52283466,"punctuation_ratio":0.10169491,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98873794,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-03T21:18:57Z\",\"WARC-Record-ID\":\"<urn:uuid:64815090-4e2b-4a98-abc4-500c94aaae6d>\",\"Content-Length\":\"39035\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4ceabea8-5487-4630-bba6-2bcaa10adb4d>\",\"WARC-Concurrent-To\":\"<urn:uuid:f5c37fe8-532b-44a6-ade1-3e90f5c719ef>\",\"WARC-IP-Address\":\"97.90.117.47\",\"WARC-Target-URI\":\"https://vcs.vera-visions.com/eukara/hl-pak0-gen/src/branch/master/ccase.sh\",\"WARC-Payload-Digest\":\"sha1:RW55WKNFHWSN577EAZELZ2DCZOPLQU4I\",\"WARC-Block-Digest\":\"sha1:XIHRLYHKMDB7OHSGYHOX66IIBW3I4PB3\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100508.53_warc_CC-MAIN-20231203193127-20231203223127-00722.warc.gz\"}"}
http://zdyboke.com/qspevdutipx-34-760-1.iunm	[ "• 产品中心\n\n• 舒比拓\n• 欧志姆\n• 纳弗拉\n• 皓蓝\n• 芜琼花\n• 哈皮卡\n• 皓齿清川西\n• 纳齿健\n• 优益洁\n• 乐益齿\n• 惠百施\n• 玖耀(中石油专属)\n\n• 猫小菲\n• 恩芝\n• 阿莎娜\n• 樱恋\n• 尼奈\n• 珂尼娜\n• 优香水滴\n\n• 清妍萃\n• 蝶印\n• 莱清菲\n• 卡蓓诺\n• 伊植贝\n• 迈森\n• 零可琳\n• 斐绿蔓\n• 津纯美\n• 百沐乐\n• 艾丝塔\n• 乐诗黎\n• 乐途\n• 赫丝町\n\n• 虹丝克润\n• 克林汉\n• 虹克林畔\n• 臻图\n• 悠蓓静\n• 葆色\n• 雅力新\n\n• 普玛兰得\n• 五色墨\n• 裴丽\n• 萩原\n• 艾兰\n• 棠印\n\n• 芜琼花\n• 希官羽\n• 魅儿\n• 葆色\n• 芳幸\n• 虹克林畔\n• 克林汉\n• 卫丫\n• 臻图\n• 诺因(中石油专属)\n\n• 翡皙\n• 千佰果\n• 芜琼花\n• 品颂\n• 露善\n• 卜赛维\n• 麦虎\n• 茉斐琳(中石油专属)\n• 安沐舒\n• 安沐舒\n\n• 迈森\n• 瑷微丹\n• 露善\n• 蝶印\n• 赫丝町\n• 艾佩丽可\n• 藤美姬\n• 植密社\n\n• 怡馥利\n• 纽碧缇\n• 桃沢子\n• 芒乐\n\n• 葆帝薇\n• 德露宝\n• 多顺\n• ### 家居日杂\n\n• 诺因(中石油专属)\n• 简沐\n• 新闻中心\n• 集团新闻\n• 产品新闻\n• 媒体报道\n•", null, "• 企业文化\n• 集团文化\n• 集团活动\n•", null, "• 和麦秀\n• 线上销售\n• 加入我们\n• 校园招聘\n• 社会招聘\n• 薪酬福利\n•", null, "• 联系我们\n• 联系我们\n•", null, "• ### 纳齿健牌钛合托玛琳去垢牙刷\n\n•", null, "• #### 纳齿健牌钛合托玛琳去垢牙刷\n\n纳齿健钛合托玛琳去垢牙刷\n\n采用双层细软纤丝细毛，可以更好的清洁牙间隙和牙垢。添加电气石和二氧化钛成分，增加纤细弹性。有效析出牙齿沉淀色素，长期使用，使牙齿沉淀牙垢，逐步脱落。并且在刷头达到防止细菌滋生的作用，保证牙刷的洁净卫生，达到牙齿洁净美白的最佳状态。刷柄如宝石般通透晶莹，让你在刷牙时有高品质的生活体验。" ]	[ null, "http://www.hemaiheda.com/uploads/image/20151209/1449637815.jpg", null, "http://www.hemaiheda.com/uploads/image/20151209/1449642402.jpg", null, "http://www.hemaiheda.com/uploads/image/20170410/1491808356.jpg", null, "http://www.hemaiheda.com/uploads/image/20151209/1449639744.jpg", null, "http://www.hemaiheda.com/uploads/image/20151228/1451292049.png", null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.93665886,"math_prob":0.41606408,"size":358,"snap":"2020-34-2020-40","text_gpt3_token_len":548,"char_repetition_ratio":0.14124294,"word_repetition_ratio":0.25,"special_character_ratio":0.10335196,"punctuation_ratio":0.0,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9734184,"pos_list":[0,1,2,3,4,5,6,7,8,9,10],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-08-15T21:15:53Z\",\"WARC-Record-ID\":\"<urn:uuid:1533af0a-434b-4bbc-878e-be50fbc98548>\",\"Content-Length\":\"39735\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:09cb0105-2635-46b8-a102-d004134c89f6>\",\"WARC-Concurrent-To\":\"<urn:uuid:7a218e7e-0a4e-48a6-b9f0-19a077e6373e>\",\"WARC-IP-Address\":\"175.29.246.119\",\"WARC-Target-URI\":\"http://zdyboke.com/qspevdutipx-34-760-1.iunm\",\"WARC-Payload-Digest\":\"sha1:WJS3IFBNP6YCJHAEDFHCPDUVXTLNE5YK\",\"WARC-Block-Digest\":\"sha1:A7O6EZXU23X3GHKHHXBMTU5A4PN6HFJ2\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-34/CC-MAIN-2020-34_segments_1596439741154.98_warc_CC-MAIN-20200815184756-20200815214756-00233.warc.gz\"}"}
https://fawalltweakunes.gq/avo-course-notes-part-4-three-term-avo-inversion.php	[ "### AVO Course Notes, Part 4. Three-term AVO Inversion\n\nShuey R. Williams M. Etris, Nick J. Crabtree and Jan Dewar Errors and Omissions A large volume of data is being converted to make this online archive.\n\n• PEH:Reservoir Geophysics!\n• AVO Modeling in Seismic Processing and Interpretation Part 1. Fundamentals?\n• Vita: Life in a Zone of Social Abandonment.\n• Spin 2004: 16th International Spin Physics Symposium; Workshop On Polerized Electron Sources and Polarimeters.\n• Zoeppritz-based AVO inversion using an improved Markov chain Monte Carlo method \| SpringerLink.\n• Machine Learning Methodology;\n• Technical article: Gardner's relations and AVO inversion.\n\nIf you notice any problems with an article examples: incorrect or missing figures, issue with rendering of formulas etc. The CSEG does not endorse or warrant the information printed. Article References Print. Examples of Azimuthal AVO The literature abounds with numerous excellent examples and observations of AVAZ some of which are shown here in order to better understand the expectation and potential in pursuing this type of analysis Lynn et al.\n\nFigure 1. Figure 2. AVAZ variation: parallel is strong flat gradient and perpendicular is weak -ve gradient. Figure 3. Figure 4. Figure 5. Figure 6. Figure 7a. Figure 7b. Note that unlike figure 6, the relative stack response from these azimuthally anisotropic gradients would be stronger than the isotropic equivalent. Figure 8a. Figure 8b. Table 1. Figure 9a. Figure Figure 11a. Figure 11b. Figure 11c. Zoom of figure 11a for 3 term vs. Figure 11d. Figure 12a. Figure 12b. Figure 13a.\n\n## AVO Modeling in Seismic Processing and Interpretation Part 1. Fundamentals \| CSEG RECORDER\n\nHTI vertical fracture model. Figure 13b. VTI horizontal layering model. Pre-drill MuRho inversion from 3D at proposed Well A location showing highest rigidity Colorado B zone hot colours that is most likely to support natural or induced fractures. In our first example,the methodis tested on a simple synthetic. This example was usedinitially becauseit truly represents a \"blocky\" impedance and therefor. In this casewe haveuseda smoothedversion of the sonic velocities to provide the constraint.\n\nA visual comparisonwoulU indicate that the extracteU velocity profile corresponds very well to the input. A moredetailed comparisonof the two figures showsthat the original and extracted logs do not matchperfectly. It is doubtful that a perfect match could ever be obtai neU. At the' top of the figure we see a sonic log with 'its reflectivity sequencebelow.\n\nIn this example,we have assumedthat the density is constant, but this is not a necessary restriction. The reflectivity wascbnvolvedwith a zero-phasewavelet,bandlimitedfrom10 to 60 Hz, andthe final syntheticis shownat the bottomof the figure. The results of the maximum-likelihood inversion method are sbown in Figure 6. In this calculation, the waveletwasassumed known. Notethe blocky nature of the estimatedvelocity profile compared with the actual sonic log profile. Again, the input and output logs do not matchperfectly. The fact that the two do not perfectly matchis due to slight errors in the reflectivity sizes whichare amplified by the integration process,and is partially the effect of the constaintused.\n\nTheconstraintshownin Figure 6. In practice, this information could be derived from stacking velocities or from nearby well control. This blocky impedance canbe contrastedwith the more traditional narrow-band. Finally, Figure 6. In summary, maximum-likelihoodinversion is a procedurewhich extracts a broad-band estimate of the seismic reflectivity and, by the introduction of 1inear constraints, al lows us to invert to an acoustic impedancesection which retains the major geological features of boreholelog data.\n\nAnother method of- recursive, single trace inversion which uses a \"sparse-spike\" assumption is the L1 normmethod, developed primarilyby Dr. DougOldenburgof UBC. This method is also often referred to as the linear programming method,and this can lead to confusion.\n\nActually, the two namesrefer to separateaspectsof the method. Themathematical modelusedin the construction of the algorithm is the minimizationof the L1 norm. However,the methodusedto solve the problem is linear programming. The basic theory of this methodis found in a paper by Oldenburg, et el The authors point out that if a high-resolution aleconvolution is performedon the seismictrace, the resulting estimateof the reflectivity can be thought of as an averagedversion of the original reflectivity, as shownat the top of Figure6.\n\nNow, the layered earth model equates to a \"blocky\" impedancefunction, which in turn equates to a \"sparse-spiKe\" reflectivity function. The above constraint will thus restrict our inverted result to a \"sparse\" structure so that extremely fine structure, such as very small reflection coefficients, will not be fully inverted. The other key difference in the linear programmingmethod is that the L1 norm is minimized rather than the L2 norm. The L1 norm is defined as the sum of the absolute values of the seismic trace.\n\nThe two norms are shownbelow, applied to the trace x: x1 : xi and x2: xi i i:1 The fact that the L1 norm favours a \"sparse\" structure is shown in the following simple example. Taken from the notes to Dr.\n\nOldenburg's CSEG convention course' \"Inverse theory with application to aleconvolution and seismograminversion\". Hence, minimizing the L1 norm would reveal that g is a \"preferred\" seismic trace based on it's sparseness. Oldenburget al. That is, the reliable frequencyband is honored whileat the same timea sparsereflectivity is created.\n\n• A Day and a Night and a Day: A Novel.\n• Practical applications of P-wave AVO for unconventional gas Resource Plays!\n• Babys First Book of Seriously Fucked-up Shit.\n\nThe results of their. The data consist of 49 traces with a sample rate of 4 msecand a Hz bandwidth. The figure showsthe linear programming reflectivity and impedanceestimates below the input seismic section. It should be pointed out that a three trace spatial smootherhas been applied to the final results in both cases. Finally, let us consider a dataset fromAlberta which has been processeU through the LP inversion method. The input seismic is shownin Figure 6. The constraints useU here were from well log data.\n\nIn the final inversion notice that the impedance has been superimposed on the final reflectivity estimate using a grey level scale. The sequence! BaseUon the these data handouts, do the following interpretation exerc i se: [ Tie the synthetic to the seismicline at SP Hint- use reverse polari ty syntheti c. Use a blocked off version of the sonic log. As the time separation between reflection coefficients becomessmaller, the interference between overlapping wavelets becomesmore severe.\n\nIndeed, in Figure 6. In fact, the effect is more of a differentiation of the wavelet, which alters the amplitude spectrumas wel1 as the phase spectrum. In this section we will look closer at the effect of wavelets on thin beds and how. The first comprehensivel'ook at thin bed effects was done by Widess In this paper he used a model which has becomethe standard for discussing thin beds, the wedgemodel. That is, consider a high velocity laye6 encasedin a low velocity layer or vice versa and allow the thickness of the layer to pinch out to zero.", null, "Next create the reflectivity responsefrom the impedance,and convolvewith a wavelet. The thickness of the layer is given in terms of two-waytime through the layer and is then related to the dominantperiod of the wavelet. The usual wavelet used is a Ricker becauseof the simpl i city of its shape. Figure 7.\n\n### Lithology and fluid prediction from amplitude versus offset (AVO) seismic data\n\nNote that what is refertea to as a wavelengthin his plot i s actually twice the dominantperiod. A few important points can be noted from Figure 7. First, the wavelets start interfering with eackotherat a thicknessjust below two dominant periods,but remain Clistinguishable down to about one period." ]	[ null, "https://www.researchgate.net/profile/Paul_Veeken/publication/269490374/figure/fig1/AS:668993115402248@1536511796666/a-The-AVO-response-on-a-refl-ection-in-a-CDP-gather-The-amplitude-changes-clearly-with.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8556537,"math_prob":0.8420299,"size":8828,"snap":"2020-24-2020-29","text_gpt3_token_len":1988,"char_repetition_ratio":0.118880324,"word_repetition_ratio":0.026105873,"special_character_ratio":0.19324875,"punctuation_ratio":0.117981076,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9624548,"pos_list":[0,1,2],"im_url_duplicate_count":[null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-12T15:53:05Z\",\"WARC-Record-ID\":\"<urn:uuid:74a1a736-7c9c-4ae2-ac58-12699e245e5d>\",\"Content-Length\":\"21153\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7cc62a09-0b6b-4258-bc35-e72027240cfe>\",\"WARC-Concurrent-To\":\"<urn:uuid:74d8c2b4-1c0f-4746-aff5-e12b5e344566>\",\"WARC-IP-Address\":\"104.27.168.38\",\"WARC-Target-URI\":\"https://fawalltweakunes.gq/avo-course-notes-part-4-three-term-avo-inversion.php\",\"WARC-Payload-Digest\":\"sha1:JY6G7NJOWRVEQDN752QGYXNJ2U26N6CJ\",\"WARC-Block-Digest\":\"sha1:UEAIKU62X2NIRGZYY44SY5Z7T2MK2RVF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593657138752.92_warc_CC-MAIN-20200712144738-20200712174738-00197.warc.gz\"}"}
https://www.colorhexa.com/cf0053	[ "# #cf0053 Color Information\n\nIn a RGB color space, hex #cf0053 is composed of 81.2% red, 0% green and 32.5% blue. Whereas in a CMYK color space, it is composed of 0% cyan, 100% magenta, 59.9% yellow and 18.8% black. It has a hue angle of 335.9 degrees, a saturation of 100% and a lightness of 40.6%. #cf0053 color hex could be obtained by blending #ff00a6 with #9f0000. Closest websafe color is: #cc0066.\n\n• R 81\n• G 0\n• B 33\nRGB color chart\n• C 0\n• M 100\n• Y 60\n• K 19\nCMYK color chart\n\n#cf0053 color description : Strong pink.\n\n# #cf0053 Color Conversion\n\nThe hexadecimal color #cf0053 has RGB values of R:207, G:0, B:83 and CMYK values of C:0, M:1, Y:0.6, K:0.19. Its decimal value is 13566035.\n\nHex triplet RGB Decimal cf0053 `#cf0053` 207, 0, 83 `rgb(207,0,83)` 81.2, 0, 32.5 `rgb(81.2%,0%,32.5%)` 0, 100, 60, 19 335.9°, 100, 40.6 `hsl(335.9,100%,40.6%)` 335.9°, 100, 81.2 cc0066 `#cc0066`\nCIE-LAB 44.08, 70.912, 15.106 27.295, 13.893, 9.428 0.539, 0.274, 13.893 44.08, 72.503, 12.025 44.08, 123.633, 3.06 37.274, 65.482, 11.095 11001111, 00000000, 01010011\n\n# Color Schemes with #cf0053\n\n• #cf0053\n``#cf0053` `rgb(207,0,83)``\n• #00cf7c\n``#00cf7c` `rgb(0,207,124)``\nComplementary Color\n• #cf00bb\n``#cf00bb` `rgb(207,0,187)``\n• #cf0053\n``#cf0053` `rgb(207,0,83)``\n• #cf1500\n``#cf1500` `rgb(207,21,0)``\nAnalogous Color\n• #00bbcf\n``#00bbcf` `rgb(0,187,207)``\n• #cf0053\n``#cf0053` `rgb(207,0,83)``\n• #00cf15\n``#00cf15` `rgb(0,207,21)``\nSplit Complementary Color\n• #0053cf\n``#0053cf` `rgb(0,83,207)``\n• #cf0053\n``#cf0053` `rgb(207,0,83)``\n• #53cf00\n``#53cf00` `rgb(83,207,0)``\n• #7c00cf\n``#7c00cf` `rgb(124,0,207)``\n• #cf0053\n``#cf0053` `rgb(207,0,83)``\n• #53cf00\n``#53cf00` `rgb(83,207,0)``\n• #00cf7c\n``#00cf7c` `rgb(0,207,124)``\n• #830034\n``#830034` `rgb(131,0,52)``\n• #9c003f\n``#9c003f` `rgb(156,0,63)``\n• #b60049\n``#b60049` `rgb(182,0,73)``\n• #cf0053\n``#cf0053` `rgb(207,0,83)``\n• #e9005d\n``#e9005d` `rgb(233,0,93)``\n• #ff0368\n``#ff0368` `rgb(255,3,104)``\n• #ff1d77\n``#ff1d77` `rgb(255,29,119)``\nMonochromatic Color\n\n# Alternatives to #cf0053\n\nBelow, you can see some colors close to #cf0053. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #cf0087\n``#cf0087` `rgb(207,0,135)``\n• #cf0076\n``#cf0076` `rgb(207,0,118)``\n• #cf0064\n``#cf0064` `rgb(207,0,100)``\n• #cf0053\n``#cf0053` `rgb(207,0,83)``\n• #cf0042\n``#cf0042` `rgb(207,0,66)``\n• #cf0030\n``#cf0030` `rgb(207,0,48)``\n• #cf001f\n``#cf001f` `rgb(207,0,31)``\nSimilar Colors\n\n# #cf0053 Preview\n\nThis text has a font color of #cf0053.\n\n``<span style=\"color:#cf0053;\">Text here</span>``\n#cf0053 background color\n\nThis paragraph has a background color of #cf0053.\n\n``<p style=\"background-color:#cf0053;\">Content here</p>``\n#cf0053 border color\n\nThis element has a border color of #cf0053.\n\n``<div style=\"border:1px solid #cf0053;\">Content here</div>``\nCSS codes\n``.text {color:#cf0053;}``\n``.background {background-color:#cf0053;}``\n``.border {border:1px solid #cf0053;}``\n\n# Shades and Tints of #cf0053\n\nA shade is achieved by adding black to any pure hue, while a tint is created by mixing white to any pure color. In this example, #0b0004 is the darkest color, while #fff6fa is the lightest one.\n\n• #0b0004\n``#0b0004` `rgb(11,0,4)``\n• #1e000c\n``#1e000c` `rgb(30,0,12)``\n• #320014\n``#320014` `rgb(50,0,20)``\n• #46001c\n``#46001c` `rgb(70,0,28)``\n• #590024\n``#590024` `rgb(89,0,36)``\n• #6d002c\n``#6d002c` `rgb(109,0,44)``\n• #810034\n``#810034` `rgb(129,0,52)``\n• #94003b\n``#94003b` `rgb(148,0,59)``\n• #a80043\n``#a80043` `rgb(168,0,67)``\n• #bb004b\n``#bb004b` `rgb(187,0,75)``\n• #cf0053\n``#cf0053` `rgb(207,0,83)``\n• #e3005b\n``#e3005b` `rgb(227,0,91)``\n• #f60063\n``#f60063` `rgb(246,0,99)``\n• #ff0b6d\n``#ff0b6d` `rgb(255,11,109)``\n• #ff1e78\n``#ff1e78` `rgb(255,30,120)``\n• #ff3284\n``#ff3284` `rgb(255,50,132)``\n• #ff4690\n``#ff4690` `rgb(255,70,144)``\n• #ff599c\n``#ff599c` `rgb(255,89,156)``\n• #ff6da7\n``#ff6da7` `rgb(255,109,167)``\n• #ff81b3\n``#ff81b3` `rgb(255,129,179)``\n• #ff94bf\n``#ff94bf` `rgb(255,148,191)``\n• #ffa8cb\n``#ffa8cb` `rgb(255,168,203)``\n• #ffbbd6\n``#ffbbd6` `rgb(255,187,214)``\n• #ffcfe2\n``#ffcfe2` `rgb(255,207,226)``\n• #ffe3ee\n``#ffe3ee` `rgb(255,227,238)``\n• #fff6fa\n``#fff6fa` `rgb(255,246,250)``\nTint Color Variation\n\n# Tones of #cf0053\n\nA tone is produced by adding gray to any pure hue. In this case, #6f6066 is the less saturated color, while #cf0053 is the most saturated one.\n\n• #6f6066\n``#6f6066` `rgb(111,96,102)``\n• #775864\n``#775864` `rgb(119,88,100)``\n• #7f5063\n``#7f5063` `rgb(127,80,99)``\n• #874861\n``#874861` `rgb(135,72,97)``\n• #8f4060\n``#8f4060` `rgb(143,64,96)``\n• #97385e\n``#97385e` `rgb(151,56,94)``\n• #9f305c\n``#9f305c` `rgb(159,48,92)``\n• #a7285b\n``#a7285b` `rgb(167,40,91)``\n• #af2059\n``#af2059` `rgb(175,32,89)``\n• #b71858\n``#b71858` `rgb(183,24,88)``\n• #bf1056\n``#bf1056` `rgb(191,16,86)``\n• #c70855\n``#c70855` `rgb(199,8,85)``\n• #cf0053\n``#cf0053` `rgb(207,0,83)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #cf0053 is perceived by people affected by a color vision deficiency. This can be useful if you need to ensure your color combinations are accessible to color-blind users.\n\nMonochromacy\n• Achromatopsia 0.005% of the population\n• Atypical Achromatopsia 0.001% of the population\nDichromacy\n• Protanopia 1% of men\n• Deuteranopia 1% of men\n• Tritanopia 0.001% of the population\nTrichromacy\n• Protanomaly 1% of men, 0.01% of women\n• Deuteranomaly 6% of men, 0.4% of women\n• Tritanomaly 0.01% of the population" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.53172874,"math_prob":0.8585422,"size":3655,"snap":"2020-45-2020-50","text_gpt3_token_len":1594,"char_repetition_ratio":0.14023556,"word_repetition_ratio":0.011111111,"special_character_ratio":0.55376196,"punctuation_ratio":0.23276836,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98400325,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-28T08:55:42Z\",\"WARC-Record-ID\":\"<urn:uuid:1c0f60a9-c6af-483f-b7d4-b680b97eed60>\",\"Content-Length\":\"36214\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e9513d38-a645-466c-9399-7077f8652e60>\",\"WARC-Concurrent-To\":\"<urn:uuid:3dbdd26c-e65f-44ab-b4bd-368bdc52d370>\",\"WARC-IP-Address\":\"178.32.117.56\",\"WARC-Target-URI\":\"https://www.colorhexa.com/cf0053\",\"WARC-Payload-Digest\":\"sha1:F7GVJWLY6CNPQLIIP3F7OD5R6ASHLDR4\",\"WARC-Block-Digest\":\"sha1:RI3UMZMA2TC5TZHTJI4SPL6FHGPOIJAT\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107897022.61_warc_CC-MAIN-20201028073614-20201028103614-00155.warc.gz\"}"}
https://electronics.stackexchange.com/questions/479479/how-do-i-find-specific-transfer-functions-from-a-plant-of-a-mimo-system	[ "# How do I find specific transfer functions from a plant of a MIMO system?\n\nSuppose I have a physical system, for example a mass-spring-damper system, which has been written in state space. Now, suppose that the matrices describing the system are $$\\A,B,C,D\\$$.\n\nMoreover, let' s say it is a MIMO system, and I have $$\\6\\$$ inputs, which are $$\\u_{1},u_{2},d_{1},d_{2},n_{1},n_{2}\\$$, where the first two are the outputs of the controller, the second pair are the disturbances and the last ones are the noises.\n\nAns there are $$\\4\\$$ outputs, which are the displacement of a mass, defined as $$\\z_{1},z_{2}\\$$ each of which has two components.\n\nSo, I define the plant as:\n\nG = ss(A,B,C,D)\n\n\nnow, I have my plant defined. Suppose now I want to find here the transfer functions of the systems, such as the sensitivity function, the complementary sensitivity function and the control sensitivity function, how could I do?\n\nI know that I will have smething like:\n\n$$\\\\begin{bmatrix} z_{11}\\\\ z_{12}\\\\ z_{21}\\\\ z_{2,2} \\end{bmatrix} = \\begin{bmatrix} & & & \\\\ & & & \\\\ & & & \\\\ & & & \\\\ & & & \\\\ & & & \\end{bmatrix} \\cdot \\begin{bmatrix} d_1\\\\ d_2\\\\ u_1\\\\ u_2\\\\ n_1\\\\ n_2 \\end{bmatrix}\\$$\n\nand to know the transfer function I need to look at the appropriate entry in the transfer matrix, but how do I know that the order of the inputs and of the outputs is this?\n\nSo, what I mean is that I could also have\n\n$$\\\\begin{bmatrix} z_{21}\\\\ z_{22}\\\\ z_{11}\\\\ z_{12} \\end{bmatrix} = \\begin{bmatrix} & & & \\\\ & & & \\\\ & & & \\\\ & & & \\\\ & & & \\\\ & & & \\end{bmatrix} \\cdot \\begin{bmatrix} u_1\\\\ u_2\\\\ d_1\\\\ d_2\\\\ n_1\\\\ n_2 \\end{bmatrix}\\$$\n\nor other combinations, so if I do $$\\G(1,1)\\$$, in the first case I obtain a transfer function, and in the second case I obtain a different transfer function.\n\nSo, how can I do?\n\nFor any linear time-invariant system the transfer function will be\n\n$$G(s) = YU^{-1} = C (sI-A)^{-1}B + D,$$\n\nand the blank matrix in you question will be $$\\G(s) \\in \\mathcal{R}^{4 \\times 6} \\$$, with the top element of the output matrix $$\\ z\\$$\n\n$$z_{1} = ([C_{1i}] (sI-A)^{-1}B + D)U,$$\n\nwhere $$\\ [C_{1i}] \\$$ is the row vector with the elements in the first row of $$\\C\\$$. Do notice that $$\\z_1(u_1,u_2,\\dots,n_2) \\$$.\n\nIf you permute the inputs in $$\\U\\$$ it is the same as applying an invertible matrix $$\\ P\\$$ to have a $$\\ \\hat{U} = PU\\$$. So, as long as $$\\ P\\$$ is just a permutation (a single 1 per line and no columns has more than a single 1), if $$\\G(1,1)\\$$ gives you a TF you will be able to find that same TF somewhere in $$\\ \\hat{G} \\$$.\n\nMoreover, if you have\n\n$$\\begin{bmatrix} d_1\\\\ d_2\\\\ u_1\\\\ u_2\\\\ n_1\\\\ n_2 \\end{bmatrix} \\rightarrow G,$$\n\n$$\\begin{bmatrix} u_1\\\\ u_2\\\\ d_1\\\\ d_2\\\\ n_1\\\\ n_2 \\end{bmatrix} \\rightarrow \\hat{G},$$\n\nThe permutation is\n\n$$P = \\begin{bmatrix} 0 & 0 & 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 & 0 & 0 \\\\ 1 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 \\end{bmatrix}$$\n\n$$G(1,1) = \\hat{G}(1,3).$$\n• Thanks for answering, so if I have understood, if I select $G(1,1)$, doesn' t matter if I do a permutation of the inputs or of the outputs, I will alway get the same result, so always the same transfer function for $G(1,1)$. But how do I know which transfer function is in position $G(1,1)$? For example I know that the transfer function from the disturbance to the output is the sensitivity function, how do I understand if it is in $G(1,1)$? Thank you again.\n• \"if $G(1,1)$ gives you a TF you will be able to find that same TF somewhere in $\\hat{G}$.\" but in many situations $G(1,1) \\neq \\hat{ G}(1,1)$.\n• Not sure what you mean, but if you want to find how $u_4 \\rightarrow y_1$ look at $z_{1,u_4} = z_{y_1,u_4} = ([C_{1,i}] (sI-A)^{-1}[B_{j,4}] + [D_{j,4}])u_4,$" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.860732,"math_prob":0.99992335,"size":1698,"snap":"2022-40-2023-06","text_gpt3_token_len":524,"char_repetition_ratio":0.17591499,"word_repetition_ratio":0.16722408,"special_character_ratio":0.36277974,"punctuation_ratio":0.13294798,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9998764,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-02-09T02:38:12Z\",\"WARC-Record-ID\":\"<urn:uuid:839b3e74-08e9-47c6-96a0-27934e8153af>\",\"Content-Length\":\"152677\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:30d4e80d-d834-44bb-870e-dba6c621b467>\",\"WARC-Concurrent-To\":\"<urn:uuid:0e104eee-32a8-4b5c-a916-4185fe32a7c9>\",\"WARC-IP-Address\":\"151.101.193.69\",\"WARC-Target-URI\":\"https://electronics.stackexchange.com/questions/479479/how-do-i-find-specific-transfer-functions-from-a-plant-of-a-mimo-system\",\"WARC-Payload-Digest\":\"sha1:YYB3MLXSBN6EHAVBNO26YVJUYLUU25RW\",\"WARC-Block-Digest\":\"sha1:PKJ3OLEMZPHCSQXAWLSHMLPNGJHQ45HF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764501066.53_warc_CC-MAIN-20230209014102-20230209044102-00788.warc.gz\"}"}
https://physics.stackexchange.com/questions/350196/show-that-there-are-compressive-forces-in-a-static-fluid-not-by-arguing-that-the	[ "# Show that there are compressive forces in a static fluid not by arguing that there is no shear force\n\nAll textbooks I have read say that there is no shear force in a static fluid because a fluid will flow continuously under the influence of a force parallel to its surface. I understand well this part. However, no textbook really shows why there can be compressive forces in a fluid. The fact that there is no shear force in a static fluid does not support the argument that there are compressive forces. I could as well say there is no compressive force in a static fluid (of course I know this is not the case).\n\nIn fact, I did try to find an example to support the argument that there are compressive forces in a static fluid but I encountered this repeating issue: a compressive force on one plane can be a shear force on another plane if that plane is oriented at right angle to the original plane.\n\nFurthermore, I couldn't find a virtual experiment in my head in which I could exert a compressive force in a fluid. If you think about it, we cannot really \"push\" a liquid.\n\nMy questions are:\n\n1. How do we prove/show that there are compressive forces in a static fluid.\n\n2. How do we prove/show that a fluid element does not move continuously under the influence of compressive forces.\n\n• What is the question?\n– JMac\nAug 4, 2017 at 1:30\n\nIn a static fluid compressive force per unit area is called pressure. First off let me clear a misunderstanding that you have viz. \"...a compressive force on one plane can be a shear force on another plane if that plane is oriented at right angle to the original plane\". This is simply incorrect. Stress is not a vector but a tensor, and pressure is simply the isotropic part of that tensor. Now you must be aware that a tensor is a linear map from vectors to vectors, which is to say that a tensor takes vector as an input and outputs another vector. Now stress tensor $\\tau$, by definition, takes area vector as its input, and outputs force vector (per unit area) acting on that area. Therefore to find force on any oriented plane located at $\\mathbf{x}$ and whose unit normal is $\\mathbf{n}$, you must take the stress tensor at that point $\\tau_\\mathbf{x}$ and feed $\\mathbf{n}$ into it to get the force-per-unit-area vector $\\mathbf{f}=\\tau_\\mathbf{x}(\\mathbf{n})$. The force vector on a plane with a different orientation, say $\\mathbf{n}'$, located at the same point, is $\\mathbf{f}'=\\tau_\\mathbf{x}(\\mathbf{n}')$, and in general $\\mathbf{f}'\\neq \\mathbf{f}$. In static fluid, the force vector (per unit area) is always normal to any plane, and therefore if you take two orthogonal planes the corresponding force vectors will also be orthogonal. That is why compressive force on one plane does not become shear force on another plane; the forces on the two planes are to be computed separately and there is no obvious relation between them.\n\nNow coming to your primary question: We can show that compressive stresses must exist in a static fluid by descending to the molecular level and using the definition of stress as momentum flux per unit area across an oriented plane due to molecules which are perpetually in motion. But I think you are seeking an explanation at the continuum level.\n\nIn a restricted sense, you \"... could as well say there is no compressive force in a static fluid...\". The reason is that usually (though not always) pressure differences are all that matter, and therefore you may consider any value of pressure to be your zero reference (just like in the case of potential energy). Pressure measured with respect to some such (arbitrary) zero reference is called \"gauge pressure\". But I think what you have in mind is \"absolute pressure\", in which the vaccuum is taken to be your reference for measurement of pressure in any other system.\n\nWhenever external forces act on a body it will cause stress inside the body (there could be mean motion/rotation as well, but that does not concern us). In an open container of fluid, even neglecting contribution due to gravity, there is the atmosphere pushing down on the fluid and container walls that are pushing from the sides. What if we go into deep space where there is no gravity and just consider a free blob of liquid not inside any container? There is still surface tension force to reckon. If the fluid blob becomes too big (e.g. stars) there are forces due to self gravitation.\n\nSo to summarize:\n\n1. How do we prove/show that there are compressive forces in a static fluid?\n\nWhen a body is acted upon by external forces it sets up stresses inside the body. If those stresses cannot be shear (because by definition, shear stresses induce motion in a fluid), then they must be compressive stresses.\n\n1. How do we prove/show that a fluid element does not move continuously under the influence of compressive forces?\n\nBecause presence of compressive stress does not imply presence of shear stress.\n\n• I am still not completely satisfied with the answers though. Could you provide a direct observation leading to the conclusion that a fluid element will not move continuously under compressive forces? Aug 4, 2017 at 18:23\n• @Geophysics You may un-accept the answer if you are not satisfied with it. May be you will get other answers. I am not sure what you mean by direct observation. How would you observe such a thing? We define fluids to be those that will deform continuously under shear stress. Can you think of a situation where compressive forces alone can set a fluid into continuous motion?\n– Deep\nAug 5, 2017 at 3:33\n• I tried but could not think of an example and I think that it is my problem: an indirect answer, like yours, though makes complete sense, gives me a hard time accepting. The same thing goes for textbooks. All of them that I have read only provide an example in which a shear stress is applied and the fluid moves and conclude that a fluid does not exert shear forces. However, none of them provide an example for normal stresses. Their conclusion about normal stresses is indirect just like yours. I just find it strange that nobody could provide a direct observation for that property of a fluid. Aug 5, 2017 at 5:40" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.932636,"math_prob":0.9193176,"size":3568,"snap":"2023-40-2023-50","text_gpt3_token_len":765,"char_repetition_ratio":0.1386083,"word_repetition_ratio":0.013355592,"special_character_ratio":0.21580717,"punctuation_ratio":0.08802309,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.989294,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-10T23:50:23Z\",\"WARC-Record-ID\":\"<urn:uuid:b3414169-50a4-4e34-af3b-859391572e4e>\",\"Content-Length\":\"169134\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c16c2bec-3ed7-4e5b-848d-b3e76aee6d06>\",\"WARC-Concurrent-To\":\"<urn:uuid:c7329cf2-8751-4500-b840-0bffae7c3ef1>\",\"WARC-IP-Address\":\"172.64.144.30\",\"WARC-Target-URI\":\"https://physics.stackexchange.com/questions/350196/show-that-there-are-compressive-forces-in-a-static-fluid-not-by-arguing-that-the\",\"WARC-Payload-Digest\":\"sha1:LJLUH2F5GK4MXNXQAWLGE3DVSGC7ZM5H\",\"WARC-Block-Digest\":\"sha1:O4SM4U42W3FU7AU4KR74K5B6UTWVVDGP\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679102697.89_warc_CC-MAIN-20231210221943-20231211011943-00878.warc.gz\"}"}
http://mathonline.wikidot.com/the-group-action-of-conjugation-of-a-subgroup-on-a-group	[ "The Group Action of Conjugation of a Subgroup on a Group\n\n# The Group Action of Conjugation of a Subgroup on a Group\n\nLet $G$ be a group and let $H$ be a subgroup of $G$. We can define a group action of the subgroup $H$ on the set $G$ as follows. For each $h \\in H$ and for each $g \\in G$ let $(h, g) \\to hgh^{-1} \\in G$. We will now check that this is indeed a group action of the group $H$ on the set $G$.\n\nFor all $h_1, h_2 \\in H$ and for all $g \\in G$ we have that:\n\n(1)\n\\begin{align} \\quad h_1(h_2g) = h_1(h_2gh_2^{-1}) = h_1h_2gh_2^{-1}h_1^{-1} = (h_1h_2)g \\end{align}\n\nAnd for all $g \\in G$ we have that (where $e \\in H \\subseteq G$ is the identity):\n\n(2)\n\\begin{align} \\quad ge = geg^{-1} = gg^{-1} = e \\end{align}\n\nThus $(h, g) \\to hgh^{-1}$ is indeed a group action of the group $H$ on the set $G$." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.715195,"math_prob":0.9999883,"size":817,"snap":"2020-45-2020-50","text_gpt3_token_len":308,"char_repetition_ratio":0.1500615,"word_repetition_ratio":0.24528302,"special_character_ratio":0.38678092,"punctuation_ratio":0.05464481,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999937,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-25T02:26:38Z\",\"WARC-Record-ID\":\"<urn:uuid:cbde2902-8fe7-4dfd-a535-4531d9bbb394>\",\"Content-Length\":\"15058\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:498a222d-e073-452f-8e22-383c9c2aed4d>\",\"WARC-Concurrent-To\":\"<urn:uuid:57fe53c7-2d92-42cd-b5b3-2da7eeeaf727>\",\"WARC-IP-Address\":\"107.20.139.176\",\"WARC-Target-URI\":\"http://mathonline.wikidot.com/the-group-action-of-conjugation-of-a-subgroup-on-a-group\",\"WARC-Payload-Digest\":\"sha1:SNB7AMSEE5VQEHW2ANTBAWW3ZAVOTSYG\",\"WARC-Block-Digest\":\"sha1:RGP2A3NDTCOBPYRS7JM24PYM6OEYEWWW\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107885126.36_warc_CC-MAIN-20201025012538-20201025042538-00401.warc.gz\"}"}
http://basesloadedumpiring.com/nfvk5yb/237019-properties-of-real-numbers-worksheet-pdf	[ "Worksheet On Real Numbers For Class 10 Pdf. And c are real numbers. Neucha Love Ya Like A Sister Take any two whole numbers and add them. Basic Number Properties The ideas behind the basic properties of real numbers are rather simple. merely reinforcement of previously learned properties of sets and real numbers. Close. 2. Calculations Using Significant Figures Worksheet A... Fha Streamline Worksheet Without Appraisal 2019, Abundance Of Isotopes Chem Worksheet 4 3 Answer Key, Chemical Bonding Worksheet Fill In The Blanks, Production Possibilities Frontier Worksheet Answers. The numbers increase from left to right and the point labeled 0 is the the point on a number line that corresponds to a real number is the of the number. Algebra Properties Let a, b, and c be real numbers, variables, or algebraic expressions. ... Find the Missing Numbers. Real numbers are simply the combination of rational and irrational numbers, in the number system. Al 2 o 3 22. Naming Ionic And Covalent Com... Polyatomic ions are periodic trends table worksheet periodic trends table worksheet by gemma warner savio staff 10 months ago 6 minutes 18 ... 21 posts related to writing formulas for ionic compounds worksheet 2. It cannot be both. Identify and apply the properties of real numbers closure commutative associative distributive identity inverse 1 which property is illustrated by the equation ax ay a x y. Addition. Holt Algebra 2 1-2 Properties of Real Numbers For all real numbers a and b, WORDS Distributive Property When you multiply a sum by a number, the result is the same whether you add and then multiply or whether you multiply each term by the number and add the products. Order Of Operations With Real Numbers Worksheet. Distributive property of multiplication worksheet - I. Distributive property of multiplication worksheet - II. example: 9 + 6 + 1 = 16 1 + 6 + 9 = 16 Associative Property of Addition You can group addends different ways, and the sum will not change. . Quadratic equations word problems worksheet. The Real Number System - Displaying top 8 worksheets found for this concept.. We define the real number system to be a set R together with an ordered pair of functions from R X R into R that satisfy the seven properties listed in this and the succeeding two sections of this chapter. px, Please allow access to the microphone This is a set of 4 properties of real numbers worksheets.Worksheet 1: Commutative, Associative and Distributive only. This lesson on Properties of Real Numbers is one that gets covered at the beginning of every Algebra course. True means that the statement is true for all real numbers. Commutative, Associatice and Distributive only.Worksheet 3: Matching. A.N.1: Identifying Properties: Identify and apply the properties of real numbers (closure, commutative, associative, distributive, identity, inverse) 1 Which property is illustrated by the equation ax+ay =a(x+y)? Commutative property of addition. U.S. National Standards. Which sentence is an example of the distributive property? Solving Systems Of Equations Algebraically Workshe... Converse Of The Pythagorean Theorem Worksheet. . Question 5 Which property of addition does the following expresion illustrate? Basic Number Properties Commutative Property a. On the other hand, imaginary numbers are the un-real numbers and cannot be represented on the number line. 4 + 5 = 9 (whole number) 8 + 4 = 12 (whole number) 90 + 0 = 90 (whole number) It is clear from the above examples that sum of any two whole numbers results in whole number. With Identity ©W P2p0 s1S2 g 5Keu6t 2aG ESBoPfltew VaermeP uL TL vCC. Special Elite Gurmukhi Just Me Again Down Here c. Some irrational numbers are integers. Focus on understanding solving equations as a process of reasoning and explaining the reasoning. Zero is the additive inverse of itself. Gloria Hallelujah Numbers: Free worksheets, handouts, esl printable exercises pdf and resources. ZIP (176.79 KB) This r squared creation is the complementary worksheet for our 1.6 PowerPoint, which students will complete as homework. Crafty Girls t Worksheet by Kuta Software LLC Kuta Software - Infinite Algebra 1 Name_____ Sets of Real Numbers … TRUE means that the statement is true for all real numbers. Main content: Properties of Real Numbers Other contents: Commutative, Associative Add to my workbooks (1) Download file pdf Embed in my website or blog Add to Google Classroom Add to Microsoft Teams Share through Whatsapp Hello Math Teachers! The printable properties worksheets for 3rd grade and 4th grade kids include commutative and associative properties of addition and multiplication. Observe the sum carefully. VT323 Black Ops One Integers and absolute value worksheets. ID: 1132291 Language: English School subject: Math Grade/level: 7/8 Age: 10-14 Main content: Properties Other contents: Add to my workbooks (4) Download file pdf Embed in … Commutative property of multiplication. When two numbers are multiplied together, the product is the same regardless of the order in which the numbers are multiplied. Sets of numbers in the real number system reals a real number is either a rational number or an irrational number. Addition Properties Commutative Property of Addition You can add numbers in any order. Unkempt Russo One 60 Definite And Indefinite Articles Spanish Worksheet... Qualitative Vs Quantitative Worksheet Answers. Worksheet 1 properties of real numbers instructions. Creepster Yanone Kaffeesatz 2000+ Worksheets available here and free to be downloaded! In general, all the arithmetic operations can be performed on these numbers and they can be represented in the number line, also. Hello Math Teachers! Holt Algebra 1.6 Properties of Real Numbers Worksheet (DOC & PDF) by . Baloo Paaji Addition. Look at the top of your web browser. Properties of Real Numbers When analyzing data or solving problems with real numbers, it can be helpful to understand the properties of real numbers. is a real number. •Ex: x = 5, then y = x + 6 is the same as y = 5 + 6. 2 4 7 0 11 3 rationals a rational number is any number that can be put in the form p q where p and q are integers and 0q. Real Numbers Worksheets and Quizzes Real Numbers Properties of Real Numbers What are Real Numbers? 13 1-4 Online Activities - Properties of Real Numbers. Properties of real numbers worksheet pdf. . Chewy Decimal place value worksheets. Jolly Lodger Fill in the missing numbers and find what property is used. Properties Of Atoms And The Periodic Table Workshe... Naming Other Organic Compounds Worksheet Answers, Fragments And Run On Sentences Worksheet Pdf, Qualitative Vs Quantitative Observations Worksheet, The Fall Of The House Of Usher Worksheet Answers, Genetics Practice Problems Worksheet Answers Pdf, Printable Three Branches Of Government Worksheet Pdf, Newton S Laws Of Motion Review Worksheet Answers, 7 2 Cell Structure Worksheet Answers Biology, Chemistry Temperature Conversion Worksheet. The sets of rational and irrational numbers together make up the set of real numbers.As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Check my answers In the number system, real numbers are the combination of irrational and rational numbers. If a number is rational, then it must be a whole number. 1) associative 2) additive identity Keystone Review { Properties of Real Numbers Name: Date: 1. This quiz and worksheet will gauge your understanding of the properties of real numbers. Properties of Real Numbers Worksheets. 12 5 1 8 3 4 62713 irrationals an irrational number is a nonrepeating nonterminating decimal. Skills Worksheet Active Reading Section 1 Scientif... Six Types Of Chemical Reactions Worksheet, Simple And Compound Interest Worksheet Answers. Name_____ _____ Date_____ Properties of Real Numbers – Practice A Match each expression with one of the properties shown. Covers the following skills: Compare real numbers; locate real numbers on a number line. 16 11. Zero Property C. Commutative property D. Identity property Question 6 Which property of addition is used in the following? Numbers. If you ever need some worksheets to improve your children\\'s skills, download them from here. These imaginary numbers are typically used to describe complex numbers. 11 Fredericka the Great View Homework Help - Algebra 1 Honors Worksheet _1_Properties of Real Numbers _ Order of Operations_.pdf from ENGINEERIN 101 at Young Men's Preparatory Academy. ID: 1142029 Language: English School subject: Math Grade/level: Algebra1 Age: 12+ Main content: Properties of real numbers Other contents: Add to my workbooks (3) Download file pdf Embed in my website or blog Add to Google Classroom Cut and paste or write equations in their correct position in the table.Worksheet 2: Matching. FALSE means that there is at least one set of real numbers that makes it false. Shadows Into Light Two 70 Dancing Script Freckle Face What is the difference between commutative and associative property? The product of 1 a, a6=0, and its reciprocal is A. Additive Identity The sum of any number and is equal to the number. 12 Identify and apply the properties of real numbers closure commutative associative distributive identity inverse 1 which property is illustrated by the equation ax ay a x y. Satisfy Bangers Properties Real Numbers Addition and Multiplication . Real Numbers Worksheets: Operations with Real Numbers Worksheets Quizzes: Integers and Real Numbers Quiz Classifying Numbers Real Numbers and Integers Quiz Identifying Real and Imaginary Numbers Quiz Math Quizzes Integes. For any number , the sum of and is . 24 a. 1-4 Exit Quiz - Properties of Real Numbers. If you see a message asking for permission to access the microphone, please allow. Coming Soon There are specific instructions for each partner to not just \"grade\" each other's work and give the actual What do you want to do? 28 You can easily perform all arithmetic operations on these numbers and can represent them on the number line. (You add the part in parenthesis first.) Escolar Addends are grouped with parenthesis. Integers worksheet – Printable PDFs. Hello Math Teachers! Here is your free content for this lesson! a + 0 = a 6 + 0 = 6. a × 1 = a 6 × 1 = 6 Email my answers to my teacher, Font: Henny Penny This is an activity where two students work independently to add and subtract real numbers, then collaborate to check their answers. $1.95. Annie Use Your Telescope Focus on understanding solving equations as a process of reasoning and explaining the reasoning. d. b. Homework. Addition properties worksheets include special focus in each property of addition. If false, explain why. Kranky 12. THE REAL NUMBER SYSTEM 5 1.THE FIELD PROPERTIES. 1 C. 1 a2 D. 0 13. When two numbers are multiplied together the product is the same regardless of the order in which the numbers are multiplied. Includes key in PDF format.I use this works 1-4 Guide Notes SE - Properties of Real Numbers. Ubuntu We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, … Orbitron 9 Hello Math Teachers! Focus on understanding solving equations as a process of reasoning and explaining the reasoning. A. ab = ba B. a(bc) = (ab)c C. a(b+c) = ab+ac D. a1 = a 2. In Algebra 2 these are of the up-most importance because these properties are not only essential pieces to knowing what to do IN a problem, but they are also a lot of times listed in the Directions of the problem. Real numbers are simply the combination of rational and irrational numbers, in the number system. What is the multiplicative inverse of 1 5? Given any number n, we know that n is either rational or irrational. Classifying Real Numbers Worksheet. 8th Grade Math Worksheets and Answer key, Study Guides. 35 problems, 2-sided worksheet, on identifying Properties of Real Numbers using a fun puzzle and stating reasons for an Algebraic Proof. Terminating And Repeating Decimals Worksheet 8th G... Dna Mutations Practice Worksheet Answer Key Pdf, Secret Of Photo 51 Video Worksheet Answer Key, Thomas Paine Common Sense Worksheet Answers, 4th Grade Photosynthesis Diagram Worksheet. PDF printable integers math worksheets for children in: Pre-K, Kindergarten, 1 st grade, 2 nd grade, 3 rd grade, 4 th grade, 5 th grade, 6 th grade and 7 th grade. Lobster Every year a few more properties are added to the list to master. False means that there is at least one set of real numbers that makes it false. Ribeye Marrow Explore some of them for free! Real Numbers are closed (the result is also a real number) under addition and multiplication: Closure example. A sodium chlorine. Bubblegum Sans 18 1-4 Assignment - Properties of Real Numbers. Algebra 1 Honors Practice Identifying Properties of Real Numbers Identify the property shown. 3 5 8 or 5 3 8 b. Math Worksheet Properties Of Real Numbers. When one group believes it is competent check their work using random questions. 40 + = B. Multiplicative Identity The product of any number and is equal to the number… The LibreTexts libraries are Powered by MindTouch ® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Kindergarten shapes worksheets on identifying simple shapes keywords. Architects Daughter Comic Neue Properties of real numbers worksheet pdf. When two numbers are added, the sum is the same regardless of the order in which the numbers are added. 22 Fredoka One English vocabulary resources online The properties of real numbers are. IDENTITY PROPERTIES A. Decide whether each statement is TRUE or FALSE. Classifying Numbers In The Real Number System Graphic Organizer And Activity Real Number System Real Numbers Math School, Properties Of Real Numbers Handy Reference Poster Gra Real Numbers Solving Equations Algebra, Properties Worksheets Properties Of Mathematics Worksheets Mathematics Worksheets Math Addition Worksheets Math Worksheets, Real Numbers Math Number Sense Real Numbers Number Worksheets, Properties Of Real Numbers Sort Task Cards Real Numbers Task Cards Evaluating Expressions, Properties Of Integers With Worked Solutions Examples Videos Math Properties Associative Property Commutative Property, Properties Worksheets Free Math Worksheets Teaching Math Math Properties, Real Number Properties Real Number System Real Numbers Number System Math, Properties Of Real Numbers Foldable For Interactive Notebooks Math Properties Real Numbers Algebra Interactive Notebooks, Identifying Number Sets Worksheets Scientific Notation Word Problems Number Worksheets Algebra, Properties Worksheets Finding Identity Property Of Multiplication Worksheet Properties Of Multiplication Distributive Property Multiplication Worksheets, Properties Of Real Numbers Algebraic Expressions Distance Learning Algebraic Expressions Real Numbers Real Number System, Following Directions Real Numbers Math Vocabulary Real Numbers Activity Real Numbers Real Number System, Properties Worksheets Properties Of Mathematics Worksheets Mathematics Worksheets Math Properties Properties Of Multiplication, Classifying Real Numbers Coloring Activity Real Numbers Color Activities Common Core Math Middle School, Properties Of Real Numbers Matching Activity Real Numbers Real Numbers Activity Creative Math, Determining Distributive Property Worksheet Distributive Property Worksheets Number Worksheets. Associative Property Of Addition Math For First Graders Math Properties Associative Property. To link to this page, copy the following code to your site: More … Properties of Real Numbers - Word Docs & PowerPoints 1-1 Assignment - Properties of Real Numbers 1-1 Bellwork - Properties of real numbers 1-1 Exit Quiz - Properties of Real Numbers 1-1 Guided Notes SE - Properties of Real Numbers 1-1 Guided Notes TE - Properties of Real […] Covered By Your Grace Properties of Real Numbers identity property of addition_Adding 0 to a number leaves it unchanged identity property of multiplication_Multiplying a number by 1 leaves it unchanged multiplication property of 0_Multiplying a number by 0 gives 0 additive Inverse & definition of opposites_Adding a number to its opposite gives 0 o Every number has an opposite Closure property. Closure property. Includes key in PDF format.I use this works To make the task of counting easier, addition came about. Properties of the Real Numbers The following are the properties of addition and multiplication if x, y, and z are real numbers: Addition Multiplication Commutative x y=y x x⋅y=y⋅x Associative x y z=x y z x⋅y ⋅z=x⋅ y⋅z Identity x 0=x x⋅1=x Inverse There is a unique number −x such that x −x =0 If x≠0, there is a unique number … The pdf exercises best suit students of grade 1 through grade 7. The Real Number System - Displaying top 8 worksheets found for this concept.. They come in many forms, most commonly associated with children's school work assignments, tax forms, and accounting or other business environments. Decide whether each statement is true or false. Which property of real numbers is illustrated by the equation p 3+ p 3 = 0? We have thousands of printable worksheets such as Properties Of Real Numbers Worksheet With Answers Pdf/page/2 that … 20 Pacifico TRUE means that the statement is true for all real numbers. . 1-4 Guide Notes SE - Properties of Real Numbers. Ordering Real Numbers Worksheet 8th Grade Pdf. Atomic Structure Worksheet Teaching Chemistry Chemistry Worksheets Chemistry ... Answer key for chemistry matter 1. 4 + 5 = 9 (whole number) 8 + 4 = 12 (whole number) 90 + 0 = 90 (whole number) It is clear from the above examples that sum of any two whole numbers results in whole number. Additive inverse and identity worksheets included. 1-4 Exit Quiz - Properties of Real Numbers. Aldrich Grand Hotel Find the multiplicative inverse of each number. FALSE means that there is at least one set of real numbers that makes it false. 1-4 Bell Work - Properties of Real Numbers. 8 Real numbers. Writing electron configuration worksheet answer key. Some of the worksheets for this concept are Sets of numbers in the real number system, Components of the real number system, 6th number grade system, Sets of real numbers date period, Real numbers precalculus, Real numbers, Real numbers and number operations, Introduction to 1 real numbers and algebraic expressions. Holt Algebra 2 1-2 Properties of Real Numbers For all real numbers a and b, WORDS Associative Property The sum or product of three or more real numbers is the same regardless of the way the numbers are grouped. Determine which properties of real numbers that is applied in each statement in exercise 13 30. a×b is real 6 × 2 = 12 is real . Rancho Reenie Beanie Take any two whole numbers and add them. Pinyon Script Identifying equivalent algebraic expressions: Worksheet 8.1 Name ……………………………… Date ……………………………… Score Some of the worksheets displayed are practice 8 1 give the iupac name of each of the following bcpldtpbc note 201306 acc j naming organic c... Label them with their charge. Identify the square root of a perfect square to 400 or, if it is not a perfect square root, locate it as an irrational number between two consecutive positive integers. Schoolbell Amatic SC Remember that the real numbers are made up of all the rational and irrational numbers. . Closure Property of Multiplication The product of two real numbers is a real number. 35 problems, 2-sided worksheet, on identifying Properties of Real Numbers using a fun puzzle and stating reasons for an Algebraic Proof. A worksheet, in the word's original meaning, is a sheet of paper on which one performs work. Free Preschool Worksheets Color By Number Numbers 1 10 Pre Writing Worksheets Line T In 2020 Free Preschool Printables Shapes Worksheets Tracing Worksheets Preschool Shape is not fixed in space and recognise shapes in different orientations e understand the difference … t Worksheet by Kuta Software LLC Kuta Software - Infinite Algebra 1 Name_____ Sets of Real Numbers … •If a = b, then b may be substituted for a in any expression containing a. 10 ©W P2p0 s1S2 g 5Keu6t 2aG ESBoPfltew VaermeP uL TL vCC. 80 1 B. For example, the reciprocal of 5 is$\\frac{1}{5}\\$, and the oppostie number of 5 is -5. Which equation illustrates the multiplicative inverse property? Adding zero leaves the real number unchanged, likewise for multiplying by 1: Identity example. At the same time, the imaginary numbers are the un-real numbers, which cannot be expressed in the number line and is commonly used to represent a complex number. Focus on understanding solving equations as a process of reasoning and explaining the reasoning. This worksheet is practice for the students on how to use the Commutative, Associative, and Distributive Property. e 3 KAUl MlN erJi Hg 0hPt5sc Gr ae 2s Deirfv NeEd z.7 w qMua5d2e w Jw ViGtqhO qI3nvf ti hnziYt3eh FA 2l ug BeTb Wr0ag F1I. Fontdiner Swanky You may even think of it as “common sense” math because no complex analysis is really required. ID: 1273662 Language: English School subject: Math Grade/level: 7-9 Age: 11-15 Main content: Properties of Real Numbers Other contents: Properties of Real Numbers Add to my workbooks (1) Download file pdf Embed in my website or blog Add to Google Classroom 35 problems, 2-sided worksheet, on identifying Properties of Real Numbers using a fun puzzle and stating reasons for an Algebraic Proof. Real Numbers. Cherry Cream Soda 35 problems, 2-sided worksheet, on identifying Properties of Real Numbers using a fun puzzle and stating reasons for an Algebraic Proof. PROPERTIES OF REAL NUMBERS Let , , and be any real numbers 1. Mountains of Christmas 14 Addition. . Luckiest Guy Lobster Two a. Atomic structure with answer. 1 associative 2 commutative 3 distributive 4 identity 2 the statement 2 0 2 is an example of the use of which property of real. Real numbers can be pictured as points on a line called areal number line. The set of counting numbers was formed. Oswald Explore some of them for free! (3 + 9) + 8 = 3 + (9 + 8) b.14 • 1 = 14 SOLUTION a.Associative property of addition b.Identity property of multiplication. ID: 1066822 Language: English School subject: Math Grade/level: 7-12 Age: 12-18 Main content: Real Numbers Other contents: Add to my workbooks (2) Download file pdf Embed in my website or blog Add to Google Classroom You could orally count along with your child or perhaps your students, you could possibly introduce numbers and … All integers are rational. ...accomplish this. 16 Lesson Proper: Recall how the set of real numbers was formed and how the operations are performed. . Ionic Bonds Worksheets Answer Key Chemistry Worksheets... C nh 2n 1. Other Properties. What Are the Properties of Real Numbers? Open Sans SWBAT: identify and apply the commutative, associative, and distributive properties to simplify expressions 4 Algebra Regents Questions 1) The statement is an example of the use of which property of real numbers? Find the multiplicative inverse of each number. In general, all the arithmetic operations can be performed on these numbers and they can be represented in the number line, also. Decide whether each statement is TRUE or FALSE. Here we have discussed the critical properties of real numbers which help solve algebraic problems. Substitution Property of Equality •If numbers are equal, then substituting one in for the another does not change the equality of the equation. Section P.2 Properties of Real Numbers 17 Properties of Real Numbers Let a, b, and c represent real numbers. Kalam Arial a+b is real 2 + 3 = 5 is real. Estimating percent worksheets. Includes key in PDF format.I use this works The or additive inverse, of any number a is ºa. Exo 2 . The quiz will also assess your comprehension of concepts like classification and complex equations. Addition Properties Worksheets . Writing and evaluating expressions worksheet Rock Salt When two numbers are added the sum is the same regardless of the order in which the numbers are added. Shape tracing worksheet trace the shapes. Size: Basic number properties commutative property a. Real numbers 2) Put a check mark for each set that the number is a part of: Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers -7 ¾ 2 5 0.398 3) True or false? NUMBERS Worksheet 1 Properties of Real Numbers Instructions: Assume that a;b; and c are real numbers. 3 + 5 = 8 or 5 + 3 = 8 b. Multiplication. Indie Flower Limiting Reactant Worksheet Stoichiometry 6 Answer... Scientific Method Review Worksheet Fill In The Blank. Thus, is called the additive identity. to any real number, the sum is the number itself. r squared creation . For a real number, it reverses its sign: the opposite to a positive number is negative, and the opposite to a negative number is positive. Partner A has Partner B's worksheet answers, and vice versa. What is the general formula for a noncyclic alkane c h. Organic Chemistry Nomenclature Worksheet Week News Softwares Includes ... properties of real numbers worksheet with answers pdf, Classification Of Matter Worksheet Chemistry Answer Key, Writing Electron Configuration Worksheet Answer Key, Elements Compounds And Mixtures Worksheet Grade 8 Answer Key, Ionic Compound Formula Writing Worksheet Answers, Formulas And Nomenclature Binary Ionic Compounds Worksheet Answers." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.83425283,"math_prob":0.94044787,"size":25688,"snap":"2023-40-2023-50","text_gpt3_token_len":5591,"char_repetition_ratio":0.23294657,"word_repetition_ratio":0.16390291,"special_character_ratio":0.2120056,"punctuation_ratio":0.11531878,"nsfw_num_words":1,"has_unicode_error":false,"math_prob_llama3":0.98634404,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-10-04T12:53:18Z\",\"WARC-Record-ID\":\"<urn:uuid:7ca9d19f-2d11-4f8b-aebd-3460a151c08f>\",\"Content-Length\":\"58662\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:57651c78-deb1-45b0-9d4e-182b6fac1d61>\",\"WARC-Concurrent-To\":\"<urn:uuid:5cdafcd6-ee03-4e91-8591-c4e7e4f9943c>\",\"WARC-IP-Address\":\"160.153.0.24\",\"WARC-Target-URI\":\"http://basesloadedumpiring.com/nfvk5yb/237019-properties-of-real-numbers-worksheet-pdf\",\"WARC-Payload-Digest\":\"sha1:SWZTZW7BRQ2MGMXY5XH3HSMROY7PUWZX\",\"WARC-Block-Digest\":\"sha1:O47NB32DMIIVAGDYHI2K6K23M65DPFTB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233511369.62_warc_CC-MAIN-20231004120203-20231004150203-00293.warc.gz\"}"}
https://www.lation.org/revelation/archives/date/2008/05	[ "May 27, 2008\nHave you ever been so been so?\n\nSir Stephen Treadwell once said “it’s easy to come up with a good idea, but it’s harder not to.”\n\nMay 25, 2008\n\n1+1+1+2+2+2+2+2+1+1+1\n\n1+1+1+2+2+2+2+2+1+1+1\n\n1+1+1+2+2+2+2+2+1+1+1\n\n1+1+1+2+2+2+2+2+1+1+1\n\n1+1+1+2+2+2+2+2+1+1+1\n\n1+1+1+2+2+2+2+2+1+1+1\n\n1+1+1+2+2+2+2+2+1+1+1\n\n1+1+1+2+2+2+2+2+1+1+1\n\n1+1+1+2+2+2+2+2+1+1+1\n\n1+1+1+2+2+2+2+2+1+1+1\n\n1+1+1+2+2+2+2+2+1+1+1\n\n1+1+1+2+2+2+2+2+1+1+1\n\n1+1+1+2+2+2+2+2+1+1+1\n\n1+1+1+2+2+2+2+2+1+1+1\n\n1+1+1+2+2+2+2+2+1+1+1\n\n1+1+1+2+2+2+2+2+1+1+1\n\nMay 23, 2008\n\n### Everybody needs a teacher. People like to talk about it, happens all the time. Just stop and think there’s a reason for it all. Something out of nothing, the only time is now. Real is now, the rest is later. Everyday there’s a way to make things better. Now not never we must work together. All we need is you and we. Caring is the key. Open the door and step into reality. Make consciousness a state of bliss and happiness our greatest gift. Make reason a season that begins without ends. Our time is money and survival our profit. Live the life of our dreams with all of our means.", null, "", null, "", null, "", null, "", null, "", null, "May 22, 2008", null, "May 16, 2008", null, "Vicks Plant, the plant of a raver.", null, "", null, "", null, "May 13, 2008\nbackwards can be used to obvious effect.", null, "May 11, 2008\n(nooone)\n\nI heard that The Big Bads inspired Eminem to do many record office skits. Detroit. Unh." ]	[ null, "http://www.lation.org/revelation/wp-content/uploads/2008/05/thedeadhippies.jpg", null, "http://www.lation.org/revelation/wp-content/uploads/2008/05/mel005.jpg", null, "http://www.lation.org/revelation/wp-content/uploads/2008/05/mel006.jpg", null, "http://www.lation.org/revelation/wp-content/uploads/2008/05/mel007.jpg", null, "http://www.lation.org/revelation/wp-content/uploads/2008/05/mel008.jpg", null, "http://www.lation.org/revelation/wp-content/uploads/2008/05/mel009.jpg", null, "http://www.lation.org/revelation/wp-content/uploads/2008/05/03.jpg", null, "http://www.lation.org/revelation/wp-content/uploads/2008/05/vicks.jpg", null, "http://www.lation.org/revelation/wp-content/uploads/2008/05/friends.jpg", null, "http://www.lation.org/revelation/wp-content/uploads/2008/05/living.jpg", null, "http://www.lation.org/revelation/wp-content/uploads/2008/05/future.jpg", null, "http://www.lation.org/revelation/wp-content/uploads/2008/05/sticky.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7382099,"math_prob":0.9984078,"size":1208,"snap":"2021-31-2021-39","text_gpt3_token_len":563,"char_repetition_ratio":0.32475084,"word_repetition_ratio":0.06818182,"special_character_ratio":0.45943707,"punctuation_ratio":0.07027027,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99674284,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],"im_url_duplicate_count":[null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-08-06T02:09:28Z\",\"WARC-Record-ID\":\"<urn:uuid:a54efe75-3a67-4c32-bd5e-368db0ff2576>\",\"Content-Length\":\"31387\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:699f1f18-6d3b-43b4-863c-a612ff5f545a>\",\"WARC-Concurrent-To\":\"<urn:uuid:a5380552-c365-4027-b593-c2deb8203d09>\",\"WARC-IP-Address\":\"64.90.51.189\",\"WARC-Target-URI\":\"https://www.lation.org/revelation/archives/date/2008/05\",\"WARC-Payload-Digest\":\"sha1:QHW3G6RMEKOOSNZCLMMP4OOIVZMGHIHT\",\"WARC-Block-Digest\":\"sha1:5B2LXCFH7M2EQTRUGLNA2XBV5O6BFU3N\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046152112.54_warc_CC-MAIN-20210806020121-20210806050121-00048.warc.gz\"}"}
https://help.semmle.com/qldoc/javascript/semmle/javascript/Expr.qll/module.Expr.html	[ "# Module Expr\n\nProvides classes for working with expressions.\n\n## Import path\n\n`import semmle.javascript.Expr`\n\n## Imports\n\n javascript Provides classes for working with JavaScript programs, as well as JSON, YAML and HTML.\n\n## Classes\n\n AddExpr An addition or string-concatenation expression. ArithmeticExpr A binary arithmetic expression using `+`, `-`, `/`, `%` or `**`. ArrayComprehensionExpr An array comprehension expression. ArrayExpr An array literal. ArrowFunctionExpr An arrow function expression. AssignAddExpr A compound add-assign expression. AssignAndExpr A compound bitwise-‘and’-assign expression. AssignDivExpr A compound divide-assign expression. AssignExpExpr A compound exponentiate-assign expression. AssignExpr A simple assignment expression. AssignLShiftExpr A compound left-shift-assign expression. AssignModExpr A compound modulo-assign expression. AssignMulExpr A compound multiply-assign expression. AssignOrExpr A compound bitwise-‘or’-assign expression. AssignRShiftExpr A compound right-shift-assign expression. AssignSubExpr A compound subtract-assign expression. AssignURShiftExpr A compound unsigned-right-shift-assign expression. AssignXOrExpr A compound exclusive-‘or’-assign expression. Assignment An assignment expression, either compound or simple. AwaitExpr An `await` expression. BigIntLiteral A BigInt literal. BinaryExpr An expression with a binary operator. BitAndExpr A bitwise ‘and’ expression. BitNotExpr A bitwise negation expression. BitOrExpr A bitwise ‘or’ expression. BitwiseBinaryExpr A bitwise binary expression, that is, either a bitwise ‘and’, a bitwise ‘or’, or an exclusive ‘or’ expression. BitwiseExpr A bitwise expression using `&`, `\|`, `^`, `~`, `<<`, `>>`, or `>>>`. BooleanLiteral A Boolean literal, that is, either `true` or `false`. CallExpr A function call expression. Comparison A comparison expression, that is, either an equality test (`==`, `!=`, `===`, `!==`) or a relational expression (`<`, `<=`, `>=`, `>`). CompoundAssignExpr A compound assign expression. ComprehensionBlock A comprehension block in a comprehension expression. ComprehensionExpr A comprehension expression, that is, either an array comprehension expression or a generator expression. ConditionalExpr A conditional expression. DecExpr A (pre or post) decrement expression. Decoratable A program element to which decorators can be applied, that is, a class, a property or a member definition. Decorator A decorator applied to a class, property or member definition. DeleteExpr A `delete` expression. DivExpr A division expression. DotExpr A dot expression. DynamicImportExpr A dynamic import expression. EqExpr An equality test using `==`. EqualityTest An equality test using `==`, `!=`, `===` or `!==`. ExpExpr An exponentiation expression. Expr An expression. ExprOrType A program element that is either an expression or a type annotation. ForInComprehensionBlock A `for`-`in` comprehension block in a comprehension expression. ForOfComprehensionBlock A `for`-`of` comprehension block in a comprehension expression. FunctionBindExpr A function-bind expression. FunctionExpr A (non-arrow) function expression. FunctionSentExpr A `function.sent` expression. GEExpr A greater-than-or-equal expression. GTExpr A greater-than expression. GeneratorExpr A generator expression. Identifier An identifier. ImmediatelyInvokedFunctionExpr An immediately invoked function expression (IIFE). ImportMetaExpr An `import.meta` expression. InExpr An `in` expression. IncExpr A (pre or post) increment expression. IndexExpr An index expression (also known as computed property access). InstanceofExpr An `instanceof` expression. InvokeExpr An invocation expression, that is, either a function call or a `new` expression. LEExpr A less-than-or-equal expression. LShiftExpr A left-shift expression using `<<`. LTExpr A less-than expression. Label A statement or property label, that is, an identifier that does not refer to a variable. LegacyLetExpr An old-style `let` expression of the form `let(vardecls) expr`. Literal A literal. LogAndExpr A logical ‘and’ expression. LogNotExpr A logical negation expression. LogOrExpr A logical ‘or’ expression. LogicalBinaryExpr A short-circuiting logical binary expression, that is, a logical ‘or’ expression, a logical ‘and’ expression, or a nullish-coalescing expression. LogicalExpr A logical expression using `&&`, `\|\|`, or `!`. MethodCallExpr A method call expression. ModExpr A modulo expression. MulExpr A multiplication expression. NEqExpr An inequality test using `!=`. NegExpr An arithmetic negation expression (also known as unary minus). NewExpr A `new` expression. NonStrictEqualityTest A non-strict equality test using `!=` or `==`. NullLiteral A `null` literal. NullishCoalescingExpr A nullish coalescing ‘??’ expression. NumberLiteral A numeric literal. ObjectExpr An object literal, containing zero or more property definitions. OptionalChainRoot INTERNAL: This class should not be used by queries. OptionalUse A call or member access that evaluates to `undefined` if its base operand evaluates to `undefined` or `null`. ParExpr A parenthesized expression. PlusExpr A unary plus expression. PostDecExpr A postfix decrement expression. PostIncExpr A postfix increment expression. PreDecExpr A prefix decrement expression. PreIncExpr A prefix increment expression. PropAccess A property access, that is, either a dot expression of the form `e.f` or an index expression of the form `e[p]`. Property A property definition in an object literal, which may be either a value property, a property getter, or a property setter. PropertyAccessor A property getter or setter in an object literal. PropertyGetter A property getter in an object literal. PropertySetter A property setter in an object literal. RShiftExpr A right-shift expression using `>>`. RegExpLiteral A regular expression literal. RelationalComparison A relational comparison using `<`, `<=`, `>=`, or `>`. SeqExpr A sequence expression (also known as comma expression). ShiftExpr A shift expression. SpreadElement A spread element. SpreadProperty A spread property in an object literal. StrictEqExpr A strict equality test using `===`. StrictEqualityTest A strict equality test using `!==` or `===`. StrictNEqExpr A strict inequality test using `!==`. StringLiteral A string literal, either single-quoted or double-quoted. SubExpr A subtraction expression. ThisExpr A `this` expression. TypeofExpr A `typeof` expression. URShiftExpr An unsigned right-shift expression using `>>>`. UnaryExpr An expression with a unary operator. UpdateExpr An update expression, that is, an increment or decrement expression. ValueProperty A value property definition in an object literal. VoidExpr A `void` expression. XOrExpr An exclusive ‘or’ expression. YieldExpr A `yield` expression." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.51563156,"math_prob":0.9476068,"size":6032,"snap":"2020-24-2020-29","text_gpt3_token_len":1332,"char_repetition_ratio":0.27422032,"word_repetition_ratio":0.016107382,"special_character_ratio":0.1780504,"punctuation_ratio":0.13691129,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9582884,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-15T01:40:37Z\",\"WARC-Record-ID\":\"<urn:uuid:7bca1c0c-bedb-43f6-b08c-d94497f7033f>\",\"Content-Length\":\"30651\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4dcdacfd-890c-4b7a-948d-a3b91a1668f8>\",\"WARC-Concurrent-To\":\"<urn:uuid:48fd19d3-8d94-4449-9554-da97becf7f5e>\",\"WARC-IP-Address\":\"104.26.8.225\",\"WARC-Target-URI\":\"https://help.semmle.com/qldoc/javascript/semmle/javascript/Expr.qll/module.Expr.html\",\"WARC-Payload-Digest\":\"sha1:F2IVYNGGAGJGNQ73TSGRPQNZFN5D7CC2\",\"WARC-Block-Digest\":\"sha1:6X5CTIKDMZSHUDQSVXS5SNIITQIVW4XF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593657154789.95_warc_CC-MAIN-20200715003838-20200715033838-00487.warc.gz\"}"}
https://www.worldofitech.com/java-programming-concurrenthashmap/	[ "# Java ConcurrentHashMap", null, "Contents\n\n## Java ConcurrentHashMap\n\nIn this tutorial, we will learn about the Java ConcurrentHashMap class and its tasks with the help of examples.\n\nThe ConcurrentHashMap class of the Java collections framework gives a thread-safe map. That is, various strings can get to the map immediately without influencing the consistency of sections on a map.\n\nIt implements the ConcurrentMap interface.\n\n## Make a ConcurrentHashMap\n\nTo make a simultaneous hashmap, we should import the java.util.concurrent.ConcurrentHashMap package first. When we import the package, here is the way we can make simultaneous hashmaps in Java.\n\n``````// ConcurrentHashMap with capacity 8 and load factor 0.6\nConcurrentHashMap<Key, Value> numbers = new ConcurrentHashMap<>(8, 0.6f);``````\n\nIn the above code, we have created a concurrent hashmap named numbers.\n\nHere,\n\n• Key – a unique identifier used to associate each element (value) in a map\n• Value – elements associated by keys in a map\n\nNotice the part new ConcurrentHashMap<>(8, 0.6). Here, the first parameter is capacity and the second parameter is loadFactor\n\n• capacity – The capacity of this map is 8. Meaning, it can store 8 entries.\n• loadFactor – The load factor of this map is 0.6. This means, whenever our hash table is filled by 60%, the entries are moved to a new hash table of double the size of the original hash table.\n\nIt’s possible to create a concurrent hashmap without defining its capacity and load factor. For example,\n\n``````// ConcurrentHashMap with default capacity and load factor\nConcurrentHashMap<Key, Value> numbers1 = new ConcurrentHashMap<>();``````\n\nBy default,\n\n• the capacity of the map will be 16\n• the load factor will be 0.75\n\n## Creating ConcurrentHashMap from Other Maps\n\nHere is how we can create a concurrent hashmap containing all the elements of other maps.\n\n``````import java.util.concurrent.ConcurrentHashMap;\nimport java.util.HashMap;\n\nclass Main {\npublic static void main(String[] args) {\n\n// Creating a hashmap of even numbers\nHashMap<String, Integer> evenNumbers = new HashMap<>();\nevenNumbers.put(\"Two\", 2);\nevenNumbers.put(\"Four\", 4);\nSystem.out.println(\"HashMap: \" + evenNumbers);\n\n// Creating a concurrent hashmap from other map\nConcurrentHashMap<String, Integer> numbers = new ConcurrentHashMap<>(evenNumbers);\nnumbers.put(\"Three\", 3);\nSystem.out.println(\"ConcurrentHashMap: \" + numbers);\n}\n}``````\n\nOutput\n\n``````HashMap: {Four=4, Two=2}\nConcurrentHashMap: {Four=4, Two=2, Three=3}``````\n\n## Methods of ConcurrentHashMap\n\nThe ConcurrentHashMap class provides methods that allow us to perform various operations on the map.\n\n## Insert Elements to ConcurrentHashMap\n\n• put() – inserts the specified key/value mapping to the map\n• putAll() – inserts all the entries from specified map to this map\n• putIfAbsent() – inserts the specified key/value mapping to the map if the specified key is not present in the map]\n\nFor example,\n\n``````import java.util.concurrent.ConcurrentHashMap;\n\nclass Main {\npublic static void main(String[] args) {\n// Creating ConcurrentHashMap of even numbers\nConcurrentHashMap<String, Integer> evenNumbers = new ConcurrentHashMap<>();\n\n// Using put()\nevenNumbers.put(\"Two\", 2);\nevenNumbers.put(\"Four\", 4);\n\n// Using putIfAbsent()\nevenNumbers.putIfAbsent(\"Six\", 6);\nSystem.out.println(\"ConcurrentHashMap of even numbers: \" + evenNumbers);\n\n//Creating ConcurrentHashMap of numbers\nConcurrentHashMap<String, Integer> numbers = new ConcurrentHashMap<>();\nnumbers.put(\"One\", 1);\n\n// Using putAll()\nnumbers.putAll(evenNumbers);\nSystem.out.println(\"ConcurrentHashMap of numbers: \" + numbers);\n}\n}``````\n\nOutput\n\n``````ConcurrentHashMap of even numbers: {Six=6, Four=4, Two=2}\nConcurrentHashMap of numbers: {Six=6, One=1, Four=-4, Two=2}\n``````\n\n## Access ConcurrentHashMap Elements\n\n### 1. Using entrySet(), keySet() and values()\n\n• entrySet() – returns a set of all the key/value mapping of the map\n• keySet() – returns a set of all the keys of the map\n• values() – returns a set of all the values of the map\n\nFor example,\n\n``````import java.util.concurrent.ConcurrentHashMap;\n\nclass Main {\npublic static void main(String[] args) {\nConcurrentHashMap<String, Integer> numbers = new ConcurrentHashMap<>();\n\nnumbers.put(\"One\", 1);\nnumbers.put(\"Two\", 2);\nnumbers.put(\"Three\", 3);\nSystem.out.println(\"ConcurrentHashMap: \" + numbers);\n\n// Using entrySet()\nSystem.out.println(\"Key/Value mappings: \" + numbers.entrySet());\n\n// Using keySet()\nSystem.out.println(\"Keys: \" + numbers.keySet());\n\n// Using values()\nSystem.out.println(\"Values: \" + numbers.values());\n}\n}``````\n\nOutput\n\n``````ConcurrentHashMap: {One=1, Two=2, Three=3}\nKey/Value mappings: [One=1, Two=2, Three=3]\nKeys: [One, Two, Three]\nValues: [1, 2, 3]``````\n\n#### 2. Using get() and getOrDefault()\n\n• get() – Returns the worth related to the predetermined key. Returns null if the key isn’t found.\n• getOrDefault() – Returns the worth related to the predetermined key. Returns the predefined default value if the key isn’t found.\n\nFor instance,\n\n``````import java.util.concurrent.ConcurrentHashMap;\n\nclass Main {\npublic static void main(String[] args) {\n\nConcurrentHashMap<String, Integer> numbers = new ConcurrentHashMap<>();\nnumbers.put(\"One\", 1);\nnumbers.put(\"Two\", 2);\nnumbers.put(\"Three\", 3);\nSystem.out.println(\"ConcurrentHashMap: \" + numbers);\n\n// Using get()\nint value1 = numbers.get(\"Three\");\nSystem.out.println(\"Using get(): \" + value1);\n\n// Using getOrDefault()\nint value2 = numbers.getOrDefault(\"Five\", 5);\nSystem.out.println(\"Using getOrDefault(): \" + value2);\n}\n}``````\n\nOutput\n\n``````ConcurrentHashMap: {One=1, Two=2, Three=3}\nUsing get(): 3\nUsing getOrDefault(): 5``````\n\n## Eliminate ConcurrentHashMap Elements\n\n• remove(key) – returns and eliminates the passage related to the predetermined key from the map\n• remove(key, value) – eliminates the passage from the map just if the predefined key planned to the predetermined worth and return a boolean value\n\nFor instance,\n\n``````import java.util.concurrent.ConcurrentHashMap;\n\nclass Main {\npublic static void main(String[] args) {\n\nConcurrentHashMap<String, Integer> numbers = new ConcurrentHashMap<>();\nnumbers.put(\"One\", 1);\nnumbers.put(\"Two\", 2);\nnumbers.put(\"Three\", 3);\nSystem.out.println(\"ConcurrentHashMap: \" + numbers);\n\n// remove method with single parameter\nint value = numbers.remove(\"Two\");\nSystem.out.println(\"Removed value: \" + value);\n\n// remove method with two parameters\nboolean result = numbers.remove(\"Three\", 3);\nSystem.out.println(\"Is the entry {Three=3} removed? \" + result);\n\nSystem.out.println(\"Updated ConcurrentHashMap: \" + numbers);\n}\n}\n``````\n\nOutput\n\n``````ConcurrentHashMap: {One=1, Two=2, Three=3}\nRemoved value: 2\nIs the entry {Three=3} removed? True\nUpdated ConcurrentHashMap: {One=1}``````\n\n## Bulk ConcurrentHashMap Operations\n\nThe ConcurrentHashMap class provides different bulk operations that can be applied safely to concurrent maps.\n\n### 1. forEach() Method\n\nThe forEach() method iterates over our entries and executes the specified function.\n\nIt includes two parameters.\n\n• parallelismThreshold – It specifies that after how many elements operations in a map are executed in parallel.\n• transformer – This will transform the data before the data is passed to the specified function.\n\nFor example,\n\n``````import java.util.concurrent.ConcurrentHashMap;\n\nclass Main {\npublic static void main(String[] args) {\n\nConcurrentHashMap<String, Integer> numbers = new ConcurrentHashMap<>();\nnumbers.put(\"One\", 1);\nnumbers.put(\"Two\", 2);\nnumbers.put(\"Three\", 3);\nSystem.out.println(\"ConcurrentHashMap: \" + numbers);\n\n// forEach() without transformer function\nnumbers.forEach(4, (k, v) -> System.out.println(\"key: \" + k + \" value: \" + v));\n\n// forEach() with transformer function\nSystem.out.print(\"Values are \");\nnumbers.forEach(4, (k, v) -> v, (v) -> System.out.print(v + \", \"));\n}\n}``````\n\nOutput\n\n``````ConcurrentHashMap: {One = 1, Two = 2, Three = 3}\nkey: One value: 1\nkey: Two value: 2\nkey: Three value: 3\nValues are 1, 2, 3,``````\n\nIn the above program, we have used equal limit 4. This implies if the map contains 4 sections, the activity will be executed in parallel.\n\n#### Variation of forEach() Method\n\n• forEachEntry() – executes the specified function for each entry\n• forEachKey() – executes the specified function for each key\n• forEachValue() – executes the specified function for each value\n\n## 2. search() Method\n\nThe search() technique look through the map dependent on the predefined capacity and returns the coordinated passage.\n\nHere, the predetermined capacity figures out what section is to be looked.\n\nIt likewise incorporates a discretionary boundary parallelThreshold. The equal limit determines that after the number of components in the guide the activity is executed in equal.\n\nFor instance,\n\n``````import java.util.concurrent.ConcurrentHashMap;\n\nclass Main {\npublic static void main(String[] args) {\n\nConcurrentHashMap<String, Integer> numbers = new ConcurrentHashMap<>();\nnumbers.put(\"One\", 1);\nnumbers.put(\"Two\", 2);\nnumbers.put(\"Three\", 3);\nSystem.out.println(\"ConcurrentHashMap: \" + numbers);\n\n// Using search()\nString key = numbers.search(4, (k, v) -> {return v == 3 ? k: null;});\nSystem.out.println(\"Searched value: \" + key);\n\n}\n}``````\n\nOutput\n\n``````ConcurrentHashMap: {One=1, Two=2, Three=3}\nSearched value: Three``````\n##### Variants of search() Method\n• searchEntries() – search function is applied to key/value mappings\n• searchKeys() – search function is only applied to the keys\n• searchValues() – search function is only applied to the values\n\n## 3. Reduce() Method\n\nThe reduce() technique aggregates (gather together) every section in a guide. This can be utilized when we need all the sections to play out a typical assignment, such as adding all the values of a map.\n\nIt incorporates two parameters.\n\n• parallelismThreshold – It indicates that after the number of components, activities in a guide are executed in parallel.\n• transformer – This will change the data before the data is passed to the predetermined capacity.\n\nFor instance,\n\n``````import java.util.concurrent.ConcurrentHashMap;\n\nclass Main {\npublic static void main(String[] args) {\n\nConcurrentHashMap<String, Integer> numbers = new ConcurrentHashMap<>();\nnumbers.put(\"One\", 1);\nnumbers.put(\"Two\", 2);\nnumbers.put(\"Three\", 3);\nSystem.out.println(\"ConcurrentHashMap: \" + numbers);\n\n// Using search()\nint sum = numbers.reduce(4, (k, v) -> v, (v1, v2) -> v1 + v2);\nSystem.out.println(\"Sum of all values: \" + sum);\n\n}\n}``````\n\nOutput\n\n``````ConcurrentHashMap: {One=1, Two=2, Three=3}\nSum of all values: 6``````\n\nIn the above program, notice the statement\n\n``numbers.reduce(4, (k, v) -> v, (v1, v2) -> v1+v2);``\n\nHere,\n\n• 4 is a parallel threshold\n• (k, v) -> v is a transformer function. It transfers the key/value mappings into values only.\n• (v1, v2) -> v1+v2 is a reducer function. It gathers together all the values and adds all values.\n\nVariants of reduce() Method\n\n• reduceEntries() – returns the result of gathering all the entries using the specified reducer function\n• reduceKeys() – returns the result of gathering all the keys using the specified reducer function\n• reduceValues() – returns the result of gathering all the values using the specified reducer function\n\n## ConcurrentHashMap vs HashMap\n\nHere are some of the differences between ConcurrentHashMap and HashMap,\n\n• ConcurrentHashMap is a thread-safe collection. That is, multiple threads can access and modify it at the same time.\n• ConcurrentHashMap provides methods for bulk operations like forEach(), search() and reduce().\n\n## Why ConcurrentHashMap?\n\n• The ConcurrentHashMap class allows numerous strings to get to its entrances simultaneously.\n• By default, the simultaneous hashmap is separated into 16 segments. This is the motivation behind why 16 strings are permitted to simultaneously change the guide simultaneously. Nonetheless, quite a few strings can get to the guide at a time.\n• The putIfAbsent() strategy won’t abrogate the passage in the guide if the predefined key as of now exists.\n• It gives its own synchronization.\n\nThanks for reading! We hope you found this tutorial helpful and we would love to hear your feedback in the Comments section below. 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