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A second order accurate level set method on non ... - CiteSeerX A second order accurate level set method on non-graded adaptive cartesian grids Chohong Min , Fre´de´ric Gibou aDepartment of Mathematics, KyungHee University, Korea bMechanical Engineering Department, University of California, Santa Barbara, CA 93106, United States cComputer Science Department, University of California, Santa Barbara, CA 93106, United States Received 10 May 2006; received in revised form 27 November 2006; accepted 28 November 2006 Available online 23 January 2007 We present a level set method on non-graded adaptive Cartesian grids, i.e. grids for which the ratio between adjacent cells is not constrained. We use quadtree and octree data structures to represent the grid and a simple algorithm to generate a mesh with the ﬁnest resolution at the interface. In particular, we present (1) a locally third order accurate reinitialization scheme that transforms an arbitrary level set function into a signed distance function, (2) a second order accurate semi- Lagrangian methods to evolve the linear level set advection equation under an externally generated velocity ﬁeld, (3) a second order accurate upwind method to evolve the non-linear level set equation under a normal velocity as well as to extrapolate scalar quantities across an interface in the normal direction, and (4) a semi-implicit scheme to evolve the inter- face under mean curvature. Combined, we obtain a level set method on adaptive Cartesian grids with a negligible amount of mass loss. We propose numerical examples in two and three spatial dimensions to demonstrate the accuracy of the method. Ó2006 Published by Elsevier Inc. Keywords: Level set method; Ghost ﬂuid method; Adaptive mesh reﬁnement; Non-graded Cartesian grids; Motion by mean curvature; Motion in the normal direction; Motion in an externally generated velocity ﬁeld; Extrapolation in the normal direction 1. Introduction Many problems in science and engineering can be described by a moving free boundary model. Examples include free surface ﬂows, Stefan problems and multiphase ﬂows to cite a few. The diﬃculty in solving these problems stems from the fact that: First, they involve dissimilar length scales. Second, the boundary position must be computed as part of the solution process. Third, the interface may be expected to undergo complex topological changes, such as the merging or the pinching of two fronts. Numerically, the interface that sepa- rates the two phases can be either explicitly tracked or implicitly captured. Several classes of successful meth- ods exist with their own virtues and drawbacks. 0021-9991/$ - see front matter Ó2006 Published by Elsevier Inc. * Corresponding author. Address: Mechanical Engineering Department, University of California, Santa Barbara, CA 93106, United States. Tel.: +1 8058937152. E-mail address:fgibou@engineering.ucsb.edu(F. Gibou). Volume of ﬂuid methods [3,4,9,27,44,66]have the advantage of being volume preserving since the mass fraction in each cell is being tracked. However, it is often diﬃcult to extract geometrical properties such as curvatures due to the fact that it is challenging or even impossible to reconstruct a smooth enough function from the mass fractions alone. We note however that some recent improvement in interface reconstruction can be found in[11]. The main advantage of an explicit approach, e.g. front tracking[25,28,29,62], is its accuracy. The main dis- advantage is that additional treatments are needed for handling changes in the interface’s topology. In turn, the explicit treatment of connectivity makes the method challenging to extend to three spatial dimensions. While researchers have produced remarkable results for a wide variety of applications using front tracking techniques, these diﬃculties make this approach not ideally suited for studying interface problems with changes in topology. Implicit representations such as the level set method or the phase-ﬁeld method represent the front as an isocontour of a continuous function. Topological changes are consequently handled in a straightforward fashion, and thus these methods are readily implemented in both two and three spatial dimensions. The main idea behind the phase-ﬁeld method is to distinguish between phases with an order parameter (or phase-ﬁeld) that is constant within each phase but varies smoothly across an interfacial region of ﬁnite thick- ness. The dynamics of the phase-ﬁeld is then coupled to that of the solution in such a way that it tracks the interface motion and approximates the sharp interface limit when the order parameter vanishes. Phase-ﬁeld methods are very popular techniques for simulating dendritic growth for example and have produced accurate quantitative results, e.g. [33,32,30,41,54]. However, these methods suﬀer from their own limitations: phase- ﬁeld methods have only an approximate representation of the front location and thus the discretization of the diﬀusion ﬁeld is less accurate near the front, resembling an enthalpy method [7]. Another consequence is the stringent time step restriction imposed by such methods. Karma and Rappel [31] have developed a thin-interface limit of the phase-ﬁeld model with a signiﬁcant improvement of the capillary length to interface thickness ratio constraint; however, the time step restriction is still on the order of the microscopic capillary length. Another disadvantage is the potential diﬃculty in relating the phase-ﬁeld parameters to the physical parameters[64], although some progress is being made for some wider class of The main diﬀerence between the phase-ﬁeld method and the level set approach[48,55,46]is that the level set method is a sharp interface model. The level set can therefore be used to exactly locate the interface in order to apply discretizations that depend on the exact interface location. Consequently, the sharp interface equation can be solved directly with no need for asymptotic analysis, which makes the method potentially more attrac- tive in developing general tool box software for a wide range of applications. Another advantage is that only the standard time step restrictions for stability and consistency are required, making the method signiﬁcantly more eﬃcient. Level set methods have been extremely successful on uniform grids in the study of physical problems such as compressible ﬂows, incompressible ﬂows, multiphase ﬂows (see e.g.[46,55] and the refer- ences therein), Epitaxial growth (see e.g.[5,23,24,50]and the references therein) or in image processing (see e.g.[47] and the references therein). One of the main problem of the level set method, namely its mass loss, has been partially solved with the advent of the particle level set method of Enright et al. [13]. Within this method, the interface is captured by the level set method and massless particles are added in order to reduce the mass loss. The massless particles are also used in the reinitialization process for obtaining smoother results for the reinitialized level set function. However, the use of particles adds to the CPU and the memory require- ment and cannot be applied for ﬂows producing shocks. Rather recently, there has been a thrust in developing level set methods on adaptive Cartesian grids. For example Losasso et al. [36]presented a particle level set based method to simulate free surface ﬂows on non-graded Cartesian grids. Within this method, the interface between the liquid and the air is captured by the particle level set on a non-graded octree data structure. Other interesting work on adaptive level-set methods can be found in[10,37]. In this paper, we present a general particle-less level set method on non-graded Cartesian grids that pro- duces a negligible amount of mass loss. We apply this method to the level set evolution (1) with an exter- nally generated velocity ﬁeld, (2) in the normal direction and (3) under mean curvature. We also present a locally third order accurate reinitialization scheme that transforms an arbitrary function into a sign distance function as well as standard techniques to extrapolate a scalar quantities across an interface in its normal direction. 2. The level set method The level set method, introduced by Osher and Sethian[48]describes a curve in two spatial dimensions or a surface in three spatial dimensions by the zero-contour of a higher dimensional function/, called the level set function. For example, in two spatial dimension, a curve is deﬁne byfðx;yÞ:/ðx;yÞ ¼0g. Under a velocity ﬁeldV, the interface deforms according to the level set equation /[t]þV r/¼0: ð1Þ To keep the values of/close to those of a signed distance function, i.e.j r/j ¼1, the reinitialization equation introduced in Sussman et al.[61] /[s]þSð/[o]Þðjr/j 1Þ ¼0 ð2Þ is traditionally iterated for a few steps in ﬁctitious time,s. HereSð/[o]Þis a smoothed out sign function. The level set function is used to compute the normal ~n¼ r/=jr/j; and the mean curvature j¼ r ~n: We refer the interested readers to the book by Osher and Fedkiw[46]as well to the book by Sethian[55]for more details on the level set method. 3. Spatial discretization and reﬁnement criterion We use a standard quadtree (resp. octree) data structure to represent the spatial discretization of the phys- ical domain in two (resp. three) spatial dimensions as depicted inFig. 1: Initially the root of the tree is asso- ciated with the entire domain, then we recursively split each cell into four children until the desired level of detail is achieved. This is done similarly in three spatial dimensions, except that cells are split into eight cubes (children). We refer the reader to the books of Samet [52,53] for more details on quadtree/octree data structures. By deﬁnition, the diﬀerence of level between a parent cell and its direct descendant is one. The level is then incremented by one for each new generation of children. A tree in which the diﬀerence of level between adja- cent cells is at most one is called a graded tree. Meshes associated with graded trees are often used in the case of ﬁnite element methods in order to produce procedures that are easier to implement. Graded Cartesian grids are also used in the case of ﬁnite diﬀerence schemes, see for example the work of Popinet[49]for the study of incompressible ﬂows. Graded meshes impose that extra grid cells must be added in regions where they are not Fig. 1. Discretization of a two dimensional domain (left) and its quadtree representation (right). The entire domain corresponds to the root of the tree (level 0). Then each cell can be recursively subdivided further into four children. In this example, the tree is ungraded since the diﬀerence of level between cells exceeds one. necessarily needed, consuming some computational resources that cannot be spent elsewhere, eventually lim- iting the highest level of detail that can be achieved. Moore[40]demonstrates that the cost of transforming an arbitrary quadtree into a graded quadtree could involve eight times as many grid nodes. Weiser[63]proposed a rough estimate for the three dimensional case and concluded that as much as 71 times as many grid nodes could be needed for balancing octrees. These estimates clearly represent the worse case scenarios that seldom exist in practical simulations. However, there is still a non-negligible diﬀerence between graded and non- graded grids. In addition, not imposing any constraint on the diﬀerence of level between two adjacent cells allows for easier/faster adaptive mesh generations. In this work we choose to impose that the ﬁnest cells lie on the interface, since it is the region of interest for the level set method. In order to generate adaptive Cartesian grids, one can use the signed distance function to the interface along with the Whitney decomposition, as ﬁrst proposed by Strain in[58]. Simply stated, one ‘‘splits any cell whose edge length exceeds its distance to the interface’’. For a general function/:R^n!Rwith Lipschitz constant Lipð/Þ, the Whitney decomposition was extended in Min[39]: Starting from a root cell split any cell Cif v2verticesðCÞmin j/ðvÞj6Lipð/Þ diag-sizeðCÞ; where diag-size(C) refers to the length of the diagonal of the current cellCandvrefers to a vertex (node) of the current cell. 4. Finite diﬀerence discretizations In the case of non-regular Cartesian grids, the main diﬃculty comes from deriving discretizations at T-junction nodes, i.e. nodes for which there is a missing neighboring node in one of the Cartesian directions. For example Fig. 2depicts a T-junction nodev0, with three neighboring nodesv1,v2andv3aligned in the Cartesian direc- tions and one ghost neighboring nodev4replacing the missing grid node in the positive Cartesian direction. The value of a node-sampled function/:fvig !Rat the ghost nodev4could for example be deﬁne by linear interpolation: /^G[4] ¼/[5]s[6]þ/[6]s[5] s[5]þs[6] : ð3Þ However, instead of using this second order accurate interpolation, one can instead use the following third order accurate interpolation: First, note that a simple Taylor expansion demonstrates that the interpolation error in Eq.(3)is given by: /^G[4] ¼/[5]s[6]þ/[6]s[5] s5þs6 ¼/ðv4Þ þs[5]s[6] 2 /[yy]ðv0Þ þOðMx[smallest]Þ^3; ð4Þ whereMx[smallest] is the size of the smallest grid cell with vertexv0. The term/[yy]ðv0Þcan be approximated using the standard ﬁrst order accurate discretization[s]^2 2þs3 /[2]/[0] s2 þ^/^3^/[s] ^0 and cancelled out in Eq. (4)to give: Fig. 2. Neighboring nodes of a T-junction node,v0. /^G[4] ¼/[5]s6þ/[6]s5 s[5]þs[6] s5s6 s[2] þ/[3]/[0] s[3] : ð5Þ We also point out that this interpolation only uses the node values of the cells adjacent tov0, which is partic- ularly beneﬁcial since access to cells not immediately adjacent to the current cell is more diﬃcult and could add on CPU time and/or memory requirement. In three spatial dimensions, similar interpolation procedures can be used to deﬁne the value of/at ghost nodes. Referring toFig. 3, a T-junction nodev0has four regular neighboring nodes and two ghost nodes. The values of a node-sampled function/:fvig !Rat the ghost nodesv4andv5can be deﬁned by second order linear and bilinear interpolations as: /^G[4] ¼s[7]/[8]þs[8]/[7] s[7]þs[8] ; /^G[5] ¼s[11]s[12]/[11]þs[11]s[9]/[12]þs[10]s[12]/[9]þs[10]s[9]/[10] ðs10þs11Þðs9þs12Þ : As in the case of quadtrees, third order accurate interpolations can be derived by cancelling out the second order derivatives in the error term to arrive at: /^G[4] ¼s7/[8]þs8/[7] s[7]þs[8] s7s8 s[3] þ/[6]/[0] s[6] ; /^G[5] ¼s[11]s[12]/[11]þs[11]s[9]/[12]þs[10]s[12]/[9]þs[10]s[9]/[10] ðs10þs11Þðs9þs12Þ s[10]s[11] s3þs6 /[3]/[0] s3 þ/[6]/[0] s6 s[9]s[12] s1þs4 /[1]/[0] s1 þ/^G[4] /[0] s4 We emphasize thatFig. 3represents the general conﬁguration of neighboring nodes in the case of an octree as described in Min et al.[38]. The third order interpolations deﬁned above allow us to treat T-junction nodes in a same fashion as a regular node, up to third order accuracy. Here, we refer to a regular node as a node for which all the neighboring nodes in the Cartesian directions exist. Therefore, we can then deﬁne ﬁnite diﬀerences for/x,/y,/z,/[xx],/[yy]and/[zz]at Fig. 3. Neighboring vertices of a vertex three spatial dimensions. every nodes using standard ﬁnite diﬀerence formulas in a dimension by dimension framework. For example, referring toFig. 4, we use the standard discretization for/xand/[xx], namely the central diﬀerence formulas: D^0[x]/[0]¼/[2]/[0] s[2] s1 s[1]þs[2]þ/[0]/[1] s[1] s2 s[1]þs[2]; D^0[xx]/[0]¼/[2]/[0] s[2] 2 s[1]þs[2]/[0]/[1] s[1] 2 the forward and backward ﬁrst order accurate approximations of the ﬁrst order derivatives: D^þ[x]/[0]¼/[2]/[0] s[2] ; D^[x]/[0]¼/[0]/[1] s1 and the second order accurate approximations of the ﬁrst order derivatives: D^þ[x]/[0]¼/[2]/[0] s2 D^[x]/[0]¼/[0]/[1] s[1] þs1 where we use the minmod slope limiter[56,34]because it produces more stable results in region where/might present kinks. Similarly, approximations for ﬁrst and second order derivatives are obtained in the y and z directions. 5. Interpolation procedures Some reserve must be provided to deﬁne data anywhere in a cell, for example in order to use semi-Lagrangian methods (see Section7). As pointed out in Strain[59], the most natural choice of interpolation in quadtree (resp. octree) data structures is the piecewise bilinear (resp. trilinear) interpolation: Consider a cell C with dimensions½0;1^2, the bilinear interpolation at a pointx2C using the values at the nodes reads: /ðx;yÞ ¼/ð0;0Þð1xÞð1yÞ þ/ð0;1Þð1xÞðyÞ þ/ð1;0ÞðxÞð1yÞ þ/ð1;1ÞðxÞðyÞ: ð11Þ Quadratic interpolation can also easily be constructed using the data from the parent cell: Since the parent cell of any current cell of a quadtree (resp. octree) owns 22 children cells (resp. 222) and 33 nodes (resp. 333), one can deﬁned the multidimensional Lagrange quadratic interpolation on the parent cell. For example in the case of a cell½1;1^2 in a quadtree, we can deﬁne the Lagrange interpolation as: /ðx;yÞ ¼/ð1;1Þxðx1Þ 2 2 þ/ð0;1Þðx^21Þyðy1Þ 2 þ/ð1;1Þxðxþ1Þ 2 yðy1Þ 2 þ/ð1;0Þxðx1Þ 2 ðy^21Þ þ/ð0;0Þðx^21Þðy^21Þ þ/ð1;0Þxðxþ1Þ 2 ðy^21Þ þ/ð1;1Þxðx1Þ 2 þ/ð0;1Þðx^21Þyðyþ1Þ 2 þ/ð1;1Þxðxþ1Þ 2 2 : However, this interpolation procedure is sensitive to nearby discontinuities, e.g. near kinks. We therefore pre- fer to deﬁne a quadratic interpolation by correcting Eq.(11)using second order derivatives. For a cell½0;1^2, we have: /ðx;yÞ ¼/ð0;0Þð1xÞð1yÞ þ/ð0;1Þð1xÞðyÞ þ/ð1;0ÞðxÞð1yÞ þ/ð1;1ÞðxÞðyÞ /[xx]xð1xÞ 2 /[yy]yð1yÞ 2 ; ð12Þ Fig. 4. One dimensional adaptive grid. where we deﬁne /[xx]¼ min v2verticesðCÞðjD^0[xx]/[v]jÞ; /[yy] ¼ min v2verticesðCÞðjD^0[yy]/[v]jÞ: ð13Þ Since a distant function is piecewise diﬀerentiable in general, the choice of the smallest in absolute value en- hances the numerical stability of the interpolation. 6. Reinitialization scheme In principle, the level function can be chosen as any Lipschitz continuous function. However, the so-called signed distance function is known to produce more robust numerical results, to improve mass conservation and to reduce errors in the computations of geometrical quantities such as the interface curvatures. Sussman et al. proposed in[60]to evolve the following partial diﬀerential equation to steady state in order to reinitialize a level set function/^0:R^n!Rinto the signed distance function/: /[s]þsgnð/^0Þðjr/j 1Þ ¼0; ð14Þ wheresrepresents the ﬁctitious time. A standard discretization for this equation in its semi-discrete form is given by: dsþsgnð/^0Þ½HGðD^þ[x]/;D^[x]/;D^þ[y]/;D^[y]/Þ 1 ¼0; ð15Þ where sgnð/^0Þdenotes the signum of/^0andHGis the Godunov Hamiltonian deﬁned as: HGða;b;c;dÞ ¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ maxðja^þj^2;jb^j^2Þ þmaxðjc^þj^2;jd^j^2Þ q if sgnð/^0Þ60 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ maxðja^j^2;jb^þj^2Þ þmaxðjc^j^2;jd^þj^2Þ q if sgnð/^0Þ>0 8> witha^þ ¼maxða;0Þanda^¼minða;0Þ. The one-sided derivatives,D^[x]/andD^[y]/are discretized by the sec- ond order accurate one-sided ﬁnite diﬀerences deﬁned in Section4. Eq.(15)is evolved in time with the TVD RK-2 method given in Shu and Osher[56]: First deﬁne/~^nþ1 and/~^nþ2 by Euler steps Ms þsgnð/^0Þ½HGðD^þ[x]/^n;D^[x]/^n;D^þ[y]/^n;D^[y]/^nÞ 1 ¼0; Ms þsgnð/^0Þ½HGðD^þ[x]/~^nþ1;D^[x]/~^nþ1;D^þ[y]/~^nþ1;D^[y]/~^nþ1Þ 1 ¼0; and then deﬁne/^nþ1 by averaging: 2 : In order to preserve area/volume, the reinitialization procedure is required not to move the original inter- face deﬁned by/0. In their seminal work, Russo and Smereka[51]solved this problem by simply including the initial interface location (given by/0) in the stencils of the one-sided derivatives. Consider the case depicted by Fig. 4and suppose that/^0[0]/^0[2]<0, i.e. the interface is located between the nodesv0andv2. The interface loca- tionvIcan be calculated by ﬁnding the root of the quadratic interpolation of/^0on the intervalv[0]v[2]with the origin at the center of the /^0ðxÞ ¼c[2]x^2þc[1]xþc[2]; with c2 ¼^1[2]minmod½D^0[xx]/^0[0];D^0[xx]/^0[2] c[1] ¼ ð/^0[2]/^0[0]Þ=s[2] c[0] ¼ ð/^0[2]þ/^0[0]Þ=2c[2]s^2[2]=4 8> >: : The distancesIbetweenv0and the interface location is then deﬁned by s[I]¼s[2] 2þ c[0]=c[1] if jc[2]j< ðc1þ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ c^2[1]4c2c[0] p Þ=ð2c2Þ if jc2jP and/^0[0]<0 ðc[1] ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p Þ=ð2c2Þ if jc[2]jP and/^0[0]>0 >: : The calculation ofD^þ[x]/^n[0]is then modiﬁed using the interface location and the fact that/¼0 at the interface: D^þ[x]/^n[0]¼0/^n[0] sI We note that in the original work of Russo and Smereka[51], a cubic interpolation was employed to locate the interface, but that the above quadratic interpolation with the minmod operator acting on the second order derivatives proved to be more stable in the case where the level set function presents a kink nearby. We also point out that in the original work of[51], the ﬁrst order derivativeD^þ [x]/^n[0] was discretized as: D^þ[x]/^n[0]¼0/^n[0] s[I] sI thus includedv[I]in the discretization ofD^0[xx]/^n[I]. However, we found that this choice leads to unstable results when the interface is close to grid nodes. We thus slightly changed the discretization by only using the location of the interface in the ﬁrst term in order to maintain the location of/^0, not in the discretization of second order deriv- atives. Likewise, in the case wheresIis close to zero (hence/^0[0]is close to zero) we simply set/^n[0]¼0 to guarantee stability. This only introduces a negligible perturbation in the location of the zero level set. The same process is then applied toD^[x]/if there is a sign change between/^0[0]and/^0[1]. The time step restric- tion for cells cut by the interface is then: minðsI;s[1];s[2]Þ in 1D; minðsI;s1;s2;s3;s4Þ=2 in 2D; minðsI;s[1];s[2];s[3];s[4];s[5];s[6]Þ=3 in 3D: >: ð16Þ 6.1. Adaptive time stepping We note that an adaptive time step is possible since only the steady state of(14)is sought. Since the time step restriction is adapted for each cell, the reinitialization procedure is fast: small cells with a stringent time step restriction are located near the interface and therefore only a few iterations are required to reach the steady state at those cells (characteristic information ﬂow away fromthe interface); cells far away from the interface are large and therefore do not require a small time step restriction. For example, consider the example depicted inFig. 5, for which the level set function is deﬁned initially as1 inside a square domain (not aligned with the grid cells) and +1 outside. This initial level set function is therefore very far from the signed distance function that we seek to deﬁne. However, on a grid where the smallest grid has size dx¼1=2048, the reinitialization procedure takes only 35 iterations to fully converge to the signed distance function in the entire domain. In practice, the initial level set is never that far to the signed distance function and therefore only about ﬁve iterations are required regardless of the resolution of the ﬁnest level. Fig. 6 illustrates the diﬀerence in the number of iterations required between uniform time stepping and adaptive time stepping. In the case of a uniform time step, we take Mt¼Mxsmallest=2, with Mxsmallest the size of the smallest cell. 6.2. Third order accuracy We also computed the convergence rates of the reinitialization algorithm for the test problem proposed in [51]: Consider the level set function initially deﬁned as: /^0ðx;yÞ ¼ ð0:1þ ðx1Þ^2þ ðy1Þ^2Þð ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x^2þy^2 p 1Þ; which deﬁnes the interface as a circle with center the origin and radius 1. In this case,/^0is not a signed dis- tance function and its gradients vary widely.Fig. 7illustrates the gradual deformation of the cross-sections of /^0as it evolves to the signed distance function.Table 1illustrates that the method is third order accurate in the L^1andL^1 norms near the interface, where we use the standard formulas for theL^1andL^1norms: k/k[1]¼ max v:j/ðvÞj<1:2Dxj/ðvÞ /[exact]ðvÞj; k/k[1]¼ average j/ðvÞ /[exact]ðvÞj; whereDx¼Dxsmallest. Note that after the reinitializing the level set function, the choice ofv:j/ðvÞj<1:2Dx ensures the selection of all the nodes adjacent to the interface. In the entire domain, the method is second order accurate if we keep reﬁning all the cells. In the practical case where only cells near the interface are reﬁned, the accuracy in regions far away from the interface is mean- ingless. In the case where the interface presents sharp corners, the accuracy is reduced to ﬁrst order in theL^1 norm. Fig. 6. L1errors of the reinitialization algorithm in the case of the adaptive time step (solid line) and the uniform time step (dotted line). Fig. 5. Reinitialization procedure. Left: initial level set function (top) and its zero cross-section (bottom) deﬁning a square domain. Right: reinitialized level set function (top) and its zero cross-section (bottom). In particular, the diﬀerence in the zero level set between the initial and ﬁnal stages is negligible. In this example, the level diﬀerence between adjacent cells is not restricted. 7. Motion under an externally generated velocity ﬁeld 7.1. Second order accurate semi-Lagrangian method In the case where the velocity ﬁeld is externally generated, the level set Eq.(1)is linear. In this case, one can use semi-Lagrangian methods. Semi-Lagrangian schemes are extensions of the Courant–Isaacson–Rees [8] method for hyperbolic equations and are unconditionally stable thus avoiding standard CFL condition of DtDxsmallest. The general idea behind semi-Lagrangian methods is to reconstruct the solution by Fig. 7. From top-left to bottom-right: Contours of the reinitialized level set function of example6.2after 0, 5, 10 and 20 iterations. The contours are evenly plotted from1 to 1 with a thick line representing the zero contour. Table 1 Convergence rates for the reinitialization for example6.2 Finest resolution 128^2 Rate 256^2 Rate 512^2 Uniform reﬁnement Near interface L1 4:3610^6 2.92 5:7710^7 3.02 7:1210^8 L1 2:1610^5 2.74 3:2410^6 3.26 3:3810^7 Whole domain L1 3:2710^4 2.14 7:4210^5 2.11 1:7110^5 L1 4:2010^2 1.56 1:4310^2 1.87 3:8910^3 Adaptive reﬁnement Near interface L1 4:3610^6 2.94 5:7010^7 3.00 7:1410^8 L[1] 2:1610^5 2.87 2:9610^6 3.09 3:4810^7 Whole domain L1 3:2710^4 1.06 1:5710^4 1.01 7:8210^5 L[1] 4:2010^2 0.00 4:2010^2 0.00 4:2010^3 The initial grid is shown inFig. 7. The condition for a nodevito be ‘near interface’ is chosen asj/ðviÞ j< ﬃﬃﬃ p2 Mxsmallest, whereMxsmallestis the size of the smallest cell. The ‘whole domain’ excludes the region near the kink located at the origin, where accuracy drops to ﬁrst order. numerically the equation along characteristic curves, starting from any grid point xi and tracing back the departure point xd in the upwind direction. Interpolation formulas are then used to recover the value of the solution at such points. In this work, we use a second order accurate semi-Lagrangian method. Consider the linear advection equation: /[t]þU r/¼0; ð17Þ whereUis an externally generated velocity ﬁeld. Then/^nþ1ðx^nþ1Þ ¼/^nðx[d]Þ, wherex^nþ1is any grid node andxd is the corresponding departure point from which the characteristic curve originates. In this work, we use the second order mid-point method for locating the departure point, as in[65]: 2 U^nðx^nþ1Þ; xd¼x^nþ1DtU^nþ^1^2ð^xÞ; where we deﬁne the velocity at the mid-time step t^nþ^1^2 by a linear combination of the velocities at the two previous time steps, i.e.U^nþ^1^2¼^3[2]U^n^1[2]U^n1. Since^xand xdare not on grid nodes in general, interpolation procedures must be applied to deﬁneU^nþ^1^2ð^xÞand/^nðxdÞ. We note that it is enough to deﬁneU^nþ^1^2ð^xÞwith a multilinear interpolation (11) and /^nðxdÞ with the quadratic interpolation described by Eqs.(12) and (13): Since a distance function has discontinuities in its derivative in general, the stabilized quadratic interpolation is preferred to the Hermite quadratic interpolation. 7.2. Test: rotation in 2D Consider a domainX¼ ½1;1^2and a disk of radiusR¼:15 and center initially atð0; :75Þ, rotating under the divergence free velocity ﬁeld uðx;yÞ ¼ y vðx;yÞ ¼x The ﬁnal timet¼2pis the time when the rotation completes one revolution. In the simulation, the adaptive reﬁnement is used, and the time step restriction isDt¼5Dx.Table 2demonstrates second order accuracy for the level set as well as for the mass conservation. We note that we only consider the grid nodes neighboring the interface in our computation of the accuracy for the level set function/ since only those points deﬁne the loca- tion of the interface. 7.3. Test: vortex in 2D In this example, we test our level set implementation on the more challenging ﬂow proposed by Bell et al. [2]: Consider a domainX¼ ½0;1^2and a disk of radius .15 and centerð:5; :75Þas the initial zero level set con- tour. The level set is then deformed under the divergence free velocity ﬁeldU¼ ðu;vÞgiven uðx;yÞ ¼ sin^2ðpxÞsinð2pyÞ vðx;yÞ ¼sin^2ðpyÞsinð2pxÞ Table 2 Convergence rates for example7.2 Finest resolution L^1error of/ Rate L^1error of/ Rate Loss of volume (%) Rate 32^2 7:2410^2 3:1110^2 28.51 64^2 1:7810^2 2.02 8:8610^3 1.81 7.21 1.98 128^2 4:5210^3 1.97 2:1310^4 2.05 1.78 2.01 256^2 1:1310^3 1.99 5:5610^4 1.93 0.45 1.98 512^2 2:8510^4 2.00 1:3810^4 2.01 0.11 2.03 1024^2 7:1410^5 2.00 3:4610^5 2.00 0.03 1.87 2048^2 1:7810^5 2.00 8:6410^6 2.00 0.007 2.01 The disk is deformed forward until t¼1 and then backward to the original shape using the reverse velocity ﬁeld with a time step restriction of Mt¼5Mx[smallest]. Table 3demonstrate second order accuracy forL1error of/and volume of loss, and linear increase in the maximum/minimum number of nodes. Note that the uniform grid of resolution 512^2requires about 25 times more nodes than the adaptive grid with the same resolution. Although second order accuracy was achieved in both the maximum and average norms in the previous example, the convergence rate of the maximum error is oscillating between one and two. This is due to the fact that as the interface deforms, some part of the interface are under resolved.Fig. 8illustrates this att¼1: Here, the tail of the interface is not resolved accurately. This deterioration in accuracy was also reported in[45]. Fig. 9illustrates the evolution of the interface location initially (left), att¼6 (center) and when the inter- face is fully rewinded (right). This example illustrates the ability of the present method to accurately capture the evolution of an interface undergoing large deformations and the ability to preserve mass eﬀectively (mass 7.4. Test: rotation in 3D Consider a domainX¼ ½2;2^3and a sphere of radiusR¼:5 and center initially atð0;1;0Þ, rotating under the divergence free velocity ﬁeld Table 3 Convergence rates for example7.3 Finest resolution 64^2 Rate 128^2 Rate 256^2 Rate 512^2 L1error of/ 9:5810^3 2.80 1:3810^3 2.02 3:4110^4 2.08 8:0910^5 L[1]error of/ 1:8310^2 1.57 6:1710^3 1.08 2:9110^3 1.39 1:1110^3 Volume loss 4.48 2.36 0.874 1.40 0.331 1.79 0.0954 Max number of nodes 1045 1.10 2243 1.10 4815 1.09 10256 Min number of nodes 439 1.07 924 0.99 1831 1.01 3679 Time (s) 1.420 2.29 6.96 2.17 31.4 2.13 138 Minimum memory (MB) 0.0448 1.07 0.0943 .97 0.185 1.00 0.371 Maximum memory (MB) 0.101 1.30 0.248 1.06 0.517 1.03 1.06 The memory requirement increases linearly with eﬀective resolution since most of the computational resources is focused near the one- dimensional interface, i.e. our method is an eﬃcient implementation of local level set methods. The computational time increases quadratically with eﬀective resolution, since the number of nodes is doubled and the time step is halved. Fig. 8. Contours of the zero level sets for example7.3with eﬀective resolutions of 64^2, 128^2, 256^2and 512^2att¼1 (left) andt¼2 (right). The colors red, green, blue represent the diﬀerence in the interface location between the resolutions 64^2and 128^2, 128^2and 256^2, 256^2and 512^2, respectively. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.) uðx;y;zÞ ¼ y vðx;y;zÞ ¼x wðx;y;zÞ ¼0 The simulation is run untilt¼2p, when the rotation completes one revolution. In the simulation, the adaptive reﬁnement is used, and the time step restriction isDt¼6Dxsmallest.Table 4demonstrates second order accuracy for the level set as well as for the mass conservation. We note that we only consider the grid nodes neighboring the interface in our computation of the accuracy for the level set function/.Fig. 10shows the adaptive grid for the rotating sphere. 7.5. Enright’s test in 3D We consider the test proposed in Enright et al.[13]: A sphere of centerð0:35;0:35;0:35Þand radius 0.15 in the domain of½0;1^3 is deformed under the following divergence free velocity ﬁeld: uðx;y;zÞ ¼2 sin^2ðpxÞsinð2pyÞsinð2pzÞ vðx;y;zÞ ¼ sin^2ðpyÞsinð2pxÞsinð2pzÞ wðx;y;zÞ ¼ sin^2ðpzÞsinð2pxÞsinð2pyÞ forward in time and then backward to its original shape with the reversed velocity.Fig. 11illustrates the inter- face motion with a time step restriction ofMt¼5Mxsmallest. We note that, in the simulation with an eﬀective resolution of 512^3, minimum the number of nodes used was 685,220 and the maximum was 2,606,710. In con- trast, the number of nodes in the case of a uniform grid with the same resolution, the number of nodes would be about 50 times larger. The volume loss is 3.21% for an eﬀective resolution of 256^3and 0.739% for an eﬀec- tive resolution of 512^3.Fig. 12compares the interface evolution with an eﬀective resolution of 128^3, 256^3and 512^3.Table 5 describes the memory and CPU requirements, andTable 6 describes the volume loss and the accuracy of the interface location after reconverting the original shape. The Enright’s test is a canonical example to test the amount of numerical dissipation of level set methods. The particle level set method reported 2.6% volume loss on a 100^3uniform grid together with Lagrangian Fig. 9. Level set evolution att¼0 (left),t¼3 (center) andt¼6 (right). The eﬀective resolution is 2048^2and the mass is conserved within .3%. Table 4 Convergence rates for the interface’s location for example7.4 Finest resolution 32^3 Rate 64^3 Rate 128^3 Rate 256^3 L1error of/ 6:8610^2 1.88 1:8710^2 1.99 4:7010^3 1.99 1:1810^3 L1error of/ 1:7610^1 2.02 4:3510^2 2.02 1:0710^2 2.02 2:6510^3 Volume loss (%) 23.1 2.16 5.14 2.12 1.18 2.07 0.282 particles[13]. Our results show that we obtain a loss of mass of .74% in the case of a 512^3eﬀective resolution, which corresponds to a 137^3uniform grid in term of number of nodes. The particle level set was further improved in [14] using octree data structures in addition to particles. Although [14] does not report any Fig. 10. Evolution of the interface for example7.4: Initial data (top-left), interface after a quarter turn (top-right), interface after a half turn (bottom-left) and ﬁnal location (bottom-right). The ﬁnest resolution is 128^3. Fig. 11. Evolution of the interface for the Enright’s test with ﬁnest resolution of 512^3. quantitative results, we ﬁnd that our result for the Enright’s test is visually comparable to that obtained in[14] for the same eﬀective resolution and compares favorably with the results in[26]. We note that the jump in rate inTable 6can be explained by the lack of resolution for describing the devel- oping thin ﬁlm. This is related to the Nyquist–Shannon sampling theorem that states that in order to fully reconstruct a signal the sampling frequency should be at least twice the signal bandwidth. In our case, the fact that the thin ﬁlm is under-resolved prevents subcell resolution of the reinitialization scheme. For higher res- olutions, we would expect second-order accuracy. 8. Motion in the normal direction and curvature driven ﬂow The equation describing an interface propagating in its normal direction and under its mean curvature is given by[48]: /[t]þ ðabjÞjr/j ¼0; ð18Þ wherej is the mean curvature of the interfacej¼ r ðr/=j r/jÞ. The coeﬃcients aandbP0 control the magnitude of the speed in the normal direction and the strength of the curvature dependence, respectively. The case whereb<0 is ill-posed and therefore we do not consider it here. Fig. 12. Eﬀect of reﬁnement on the Enright’s test: The top ﬁgures correspond to the interface fully stretched and the bottom ﬁgures correspond to the interface rewinded to the original sphere. The ﬁnest resolutions are 128^3(left), 256^3(center) and 512^3(right). Table 5 Memory and CPU requirements for example7.5 Finest resolution Time Rate Min # nodes Rate Max # nodes Rate Min memory Rate Max memory Rate 128^3 237.5 44,943 133,308 4.54 13.7 256^3 2214 3.23 173,637 1.95 598,264 2.17 17.6 1.96 61.5 2.17 512^3 19,521 3.14 685,220 1.98 2,606,710 2.12 69.6 1.98 268 2.12 The memory requirement increases quadratically with eﬀective resolution since most of the computational resources is focused near the two-dimensional interface, i.e. our method is an eﬃcient implementation of local level set methods. The computational time increases cubically with eﬀective resolution, since the number of nodes is multiplied by four and the time step is halved. Table 6 Convergence rates for the interface’s location for example7.5 Finest resolution Volume loss (%) Rate L^1error of/ Rate L^1error of/ Rate 128^3 16.02 1:9610^2 1:5410^1 256^3 3.21 2.32 2:8310^3 2.79 1:0610^1 .88 512^3 .739 2.12 4:3810^4 2.69 5:7410^3 7.52 8.1. Motion in the normal direction First, we discuss the case whenb¼0. Using the second order one-sided derivatives described in Section4 and discretizing the Hamiltonian using a Godunov scheme, we semi-discretize the equation as: dt þaHGð/Þ ¼0; where the Godunov Hamiltonian HGis deﬁned as: H[G]ð/Þ ¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ maxðjðD^þ[x]/Þ^j^2;jðD^[x]/Þ^þj^2Þ þmaxðjðD^þ[y]/Þ^j^2;jðD^[y]/Þ^þj^2Þ q if a>0 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ maxðjðD^þ[x]/Þ^þj^2;jðD^[x]/Þ^j^2Þ þmaxðjðD^þ[y]/Þ^þj^2;jðD^[y]/Þ^j^2Þ q This equation is discretized in time using the second order TVD Runge–Kutta method (see[56,34]): Mt þaH[G]ð/^nÞ ¼0 ð19Þ Mt þaH[G]ð/~^nþ1Þ ¼0 ð20Þ 2 ð21Þ Table 7illustrates that the method described above is second order accurate in both the maximum and the average norms for smooth data. In the case where the interface presents sharp corners,Fig. 13illustrates that the method converges to the correct viscosity solution[48]. Table 7 Convergence rate for a circle shrinking with unit normal velocity Finest resolution 64^2 Rate 128^2 Rate 256^2 Rate 512^2 L1error of/ 1:4610^3 2.02 3:5810^4 2.00 8:5610^4 1.98 2:2610^5 L1error of/ 2:7710^3 2.04 6:7210^4 1.95 1:7310^4 1.99 4:3610^5 Consider a domain of½2;2^2and an interface initially described by a circle centered at the origin with radiusR¼1. The interface is evolved untilt¼0:5. Fig. 13. Shrinking square in the ﬁrst row, and expanding square in the second row. 8.2. Adding motion by mean curvature Now we discuss the case whenb>0. The curvature term can be discretized explicitly or implicitly. In the case where the curvature term is discretized explicitly, the corresponding time step restriction ofDtDx^2is too stringent to be practical since it would be constrained by the size of the smallest grid cell in the grid. In[57], Smereka proposed an implicit discretization of the curvature term in the case of uniform grids: Using the following operator splitting: jjr/j ¼D/ r/ Eq.(18)is discretized as: Mt þaHGð/^nÞ ¼bD/^nþ1b r/^n jr/^nj rðjr/^njÞ: In this work, we used a backward Euler step to treat the linear term, and a forward Euler step for the non- linear term. The derivativesDand$are discretized by the central ﬁnite diﬀerences described in Section4. Dis- cretizing implicitly the Laplacian requires a linear system that we solve using the supra convergent method presented in Min, Gibou and Ceniceros[38]. As noted in[57], the semi-implicit discretization on the curvature term allows for a big time step, so that the time step restriction is that of the convection part, i.e. Dt¼ Dxsmallest a#dimensions; ð22Þ where # dimensions is the number of dimensions. Table 8demonstrates that the method is ﬁrst order accurate in the average norm for smooth a interface. The deterioration in the maximum norm probably comes from the Elliptic part of the solver, which propagates the errors from the regions where the grid cells are coarse and unreﬁned to the regions where the grid cells are reﬁned.Fig. 14illustrates the motion of an interface under mean curvature for the example presented in[57]. 9. Adaptive grid generation As the interface deforms some provisions must be given to reﬁne the grid near the interface while coarsen- ing in regions farther away. The grid is constructed in such a way that the smallest grid cells lie on the interface as described in Section3. This construction depends on an input function,/~^nþ1:R^n!Rthat is close to the Table 8 Convergence rate for a circle with curvature dependent speed ofa¼1:5 andb¼1 Finest resolution 128^2 Rate 256^2 Rate 512^2 Rate 1024^2 L1error of/ 5:2210^3 1.00 2:6010^3 0.96 1:3310^3 0.95 6:9510^4 L1error of/ 5:4710^3 0.93 2:8610^3 0.86 1:5610^3 0.81 8:9110^4 Initially circle is centered atð0;0Þwith radius one in a domain of½2;2^2. Test was run until 0.5. The radiusrðtÞof the circle satisﬁes r^0¼a^b[r]withrð0Þ ¼1.rð0:5Þis approximated as 1.3108122 from the ordinary diﬀerential equation within error bound of 10^7. Fig. 14. Motion with curvature ﬂow for a barbel shape.a¼0;b¼1 in 256^3resolution. From top-left to bottom-right, the times are 0, 0.023, 0.093, 0.140, 0.304 and 0.323. The CFL condition	{"url":"https://123dok.co/kr/docs/second-order-accurate-level-set-method-non-citeseerx.10302515","timestamp":"2024-11-14T14:50:39Z","content_type":"text/html","content_length":"208828","record_id":"<urn:uuid:09740969-6b4a-4a01-866d-b1dd7f955de4>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00791.warc.gz"}
MHT CET 2021 21th September Evening Shift \| Rotational Motion Question 44 \| Physics \| MHT CET - ExamSIDE.com MHT CET 2021 21th September Evening Shift MCQ (Single Correct Answer) A particle performs rotational motion with an angular momentum 'L'. If frequency of rotation is doubled and its kinetic energy becomes one fourth, the angular momentum becomes. MHT CET 2021 21th September Evening Shift MCQ (Single Correct Answer) Three rings each of mass 'M' and radius 'R' are arranged as shown in the figure. The moment of inertia of system about axis YY' will be MHT CET 2021 21th September Morning Shift MCQ (Single Correct Answer) The moment of inertia of a circular disc of radius $$2 \mathrm{~m}$$ and mass $$1 \mathrm{~kg}$$ about an axis XY passing through its centre of mass and perpendicular to the plane of the disc is $$2 \mathrm{~kg} \mathrm{~m}^2$$. The moment of inertia about an axis parallel to the axis $$\mathrm{XY}$$ and passing through the edge of the disc is MHT CET 2021 21th September Morning Shift MCQ (Single Correct Answer) The moment of inertia of a body about a given axis is $$1.2 \mathrm{~kg} / \mathrm{m}^3$$. Initially the body is at rest. In order to produce rotational kinetic energy of $$1500 \mathrm{~J}$$, an angular acceleration of $$25 \mathrm{rad} / \mathrm{s}^2$$ must be applied about an axis for a time duration of	{"url":"https://questions.examside.com/past-years/jee/question/pa-particle-performs-rotational-motion-with-an-angular-mom-mht-cet-physics-motion-uowqf4mywj1nz4xb","timestamp":"2024-11-03T00:29:19Z","content_type":"text/html","content_length":"196136","record_id":"<urn:uuid:e090597b-7bf6-4228-bdc3-61861eafceb5>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00216.warc.gz"}
Видеотека: Y.-T. Siu, Hyperbolicity of Generic High-Degree Hypersurfaces Аннотация: We will talk about the solution of the decades-old problem of the hyperbolicity of generic hypersurfaces of sufficiently high degree and of their complements, as well as a number of related results, such as: • (i) a Big-Picard-Theorem type statement concerning extendibility, across the puncture, of holomorphic maps from a punctured disk to a generic hypersurface of high degree, • (ii) entire holomorphic functions satisfying polynomial equations with slowly varying coefficients, • (iii) Second Main Theorems for jet differentials and slowly moving targets. Язык доклада: английский	{"url":"https://m.mathnet.ru/php/presentation.phtml?option_lang=rus&presentid=5758","timestamp":"2024-11-05T02:45:47Z","content_type":"text/html","content_length":"8252","record_id":"<urn:uuid:590560cc-6e5a-4311-811a-ad0e558eb1b0>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00394.warc.gz"}
The Small World Phenomenon: \| studyslide.com The Small World Phenomenon: Download Report Transcript The Small World Phenomenon: The Small World An Algorithmic Perspective Speaker: Bradford Greening, Jr. Rutgers University – Camden An Experiment by Milgram (1967) Chose a target person Asked randomly chosen “starters” to forward a letter to the target address, and some personal information were provided for the target person The participants could only forward a letter to a single person that he/she knew on a first name basis Goal: To advance the letter to the target as quickly as An Experiment by Milgram (1967) Outcome revealed two fundamental components of a social network: short paths between arbitrary pairs of nodes operating with purely local information are very adept at finding these paths What is the “small world” phenomenon? Principle that most people in a society are linked by short chains of acquaintances Sometimes referred to as the “six degrees of separation” Modeling a social network Create a graph: node for every person in the world an edge between two people (nodes) if they know each other on a first name basis If almost every pair of nodes have “short” paths between them, we say this is a small world Modeling a social network Watts – Strogatz (1998) a model for small-world networks Local contacts Long-range contacts incorporated closed triads and short paths into the same model Modeling a social network Imagine everyone lives on an n x n grid “lattice distance” – number of lattice steps between two Constants p,q Modeling a social network p: range of local contacts Nodes are connected to all other nodes within distance Modeling a social network q: number of long-range add directed edges from node u to q other nodes using independent random Modeling a social network Watts – Strogatz (1998) that injecting a small amount of randomness (i.e. even q = 1) into the world is enough to make it a small world. Modeling a social network Kleinberg (2000) should arbitrary pairs of strangers, using only locally available information, be able to find short chains of acquaintances that link them together? this occur in all small-world networks, or are there properties that must exist for this to happen? Modeling a social network [d (u, v)] r Pr [u has v as its long range contact] : v : v u Infinite family of networks: r = 0: each node’s long-range contacts are chosen independently of its position on the grid As r increases, the long range contacts of a node become clustered in its vicinity on the grid. The Algorithmic Side G = (V,E) arbitrary nodes s, t Goal: Transmit a message from s to t in as few steps as possible using only locally available information The Algorithmic Side any step, the message holder u knows The range of local contacts of all nodes The location on the lattice of the target t The locations and long-range contacts of all nodes that have previously touched the message does not know the long-range contacts of nodes that have not touched the message The Algorithm In each step the current message holder passes the message to the contact that is as close to the target as Algorithm in phase j: a given step, 2j < d(u,t) ≤ 2j+1 Αlg. is in phase 0 : message is no more than 2 lattice steps away from the target t. ≤ log2 n. How many steps will the algorithm take? How many steps will we spend in phase j? In a given step, with what probability will phase j end in this step? What is the probability that node u has a node v in the next phase as its long range contact? How many steps will the algorithm How many steps will we spend in phase j? In a given step, with what probability will phase j end in this What is the probability that node u has a node v as its long range contact? Pr [ u has v as its long range contact ] ? [d (u, v)]2 v : v u 2 n 2 2 v : v2u [d (8u2, v)]... v : v u 3 2 21 2 2 j 1 j v : v u d [(u, v )] How many steps will the algorithm How many steps will we spend in phase j? In a given step, with what probability will phase j end in this What is the probability that node u has a node v as its long range contact? Pr[ u has v as its long range contact ]? 2 n 2 [d (u, v)] 2 4 1 4[1 ln(2n 2)] 4ln(6n) v : v u j 1 j j 1 2 n 2 [d (u, v )]2 4 ln(6n ) Thus u has v as its long-range contact with probability 4ln(6n) [d (u, v)]2 How many steps will the algorithm How many steps will we spend in phase j? In a given step, with what probability will phase j end in this What is the probability that node u has a node v as its long range contact? 4ln(6n) [d (u, v)]2 In any given step, Pr[ phase j ends in this step ]? Phase j ends in this step if the message enters the set Bj of nodes within distance 2j of t. Let vf be the node in Bj that is farthest from u. Pr[phase j ends in this step] Pr u is friends with v B vB j \| Bj \| 4 ln(6n ) [d (u, v )]2 How many steps will the algorithm How many steps will we spend in phase j? In a given step, with what probability will phase j end in this What is the probability that node u has a node v as its long range contact? Pr[phase j ends in this step] \| B j \| 4 ln(6n) [d (u, v )]2 What is d[(u,vf)]? ≤ 2j + 2j+1 < 2j+2 4ln(6n) [d (u, v)]2 How many steps will the algorithm How many steps will we spend in phase j? In a given step, with what probability will phase j end in this What is the probability that node u has a node v as its long range contact? 4ln(6n) [d (u, v)]2 Pr[phase j ends in this step] \| B j \| 2 j 4 4ln(6n) 2 How many nodes are in Bj? 1 i i 1 22 j 2 j 22 j 1 How many steps will the algorithm How many steps will we spend in phase j? In a given step, with what probability will phase j end in this What is the probability that node u has a node v as its long range contact? In any given step, Pr[ phase j ends in this step ]? Pr[ u has a long-range contact in Bj ]? # of nodes in B j ( probability u is friends with farthest v B j ) 2 j 1 22 j 1 4 ln(6n ) 2 2 j 4 4 ln(6n) 2 2 j 4 128ln(6n) 4ln(6n) [d (u, v)]2 How many steps will the algorithm How many steps will we spend in phase j? In a given step, with what probability will phase j end in this How many steps will we spend in phase j? Let Xj be a random variable denoting the number of steps spent in phase j. Xj is a geometric random variable with a probability of success at least What is the probability that node u has a node v as its long range contact? 4ln(6n) [d (u, v)]2 How many steps will the algorithm How many steps will we spend in phase j? In a given step, with what probability will phase j end in this What is the probability that node u has a node v as its long range contact? How many steps will we spend in phase j? Since Xj is a geometric random variable, we know that E[ X j ] p 1 4ln(6n) [d (u, v)]2 How many steps will the algorithm How many steps will we spend in phase j? How many steps does the algorithm take? Let X be a random variable denoting the number of steps taken by the algorithm. By Linearity of Expectation we have In a given step, with what probability will phase j end in this What is the probability that node u has a node v as its long range contact? E[ X ] (1 log n)(128ln(6n)) O(log n)2 4ln(6n) [d (u, v)]2 How many steps will the algorithm How many steps will we spend in phase j? When r = 2, expected delivery time is In a given step, with what probability will phase j end in this O(log n)2 What is the probability that node u has a node v as its long range contact? 4ln(6n) [d (u, v)]2 Summary of results 0 ≤ r < 2: The expected delivery time of any decentralized algorithm is Ω(n(2-r)/3). r > 2: The expected delivery time of any decentralized algorithm is Ω(n(r-2)/(r-1)). Revisiting Assumptions Recall that in each step the message holder u knew the locations and long-range contacts of all nodes that have previously touched the message Is knowledge of message’s history too much info? Upper-bound on delivery time in the good case is proven without using this. Lower-bound on delivery times for the bad cases still hold even when this knowledge is used. The Intuition For a changing value of r = 0 provides no “geographical” clues that will assist in speeding up the delivery of the message. 0 < r < 2: provides some clues, but not enough to sufficiently assist the message senders r > 2: as r grows, the network becomes more localized. This becomes a prohibitive factor. r = 2: provides a good mix of having relevant “geographical” information without too much Kleinberg, J. The Small-World Phenomenon: An Algorithmic Perspective. Proc. 32nd ACM Symposium on Theory of Computing, 2000 Kleinberg, J. Navigation in a Small World. Nature 406(2000), 845.	{"url":"https://studyslide.com/doc/672088/the-small-world-phenomenon-","timestamp":"2024-11-12T10:29:53Z","content_type":"text/html","content_length":"74526","record_id":"<urn:uuid:24660a4c-a0dc-4331-9c7f-621f2ccce915>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00420.warc.gz"}
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Modular verification of C programs To obtain scalability, software model checkers often employ modular analysis techniques that analyze each of the procedures in a program separately. In this context, summaries are used to abstract the behavior of procedures, as relations of their input/output parameters, to analyze any procedure call without inlining or analyzing its body. One key challenge when producing summaries of memory-manipulating programs is to solve the frame problem: determining which memory locations are not changed by a procedure. In SMT-based model checking, the program heap is usually modeled by logical (unbounded) arrays. A pointer analysis is typically used as a pre-analysis to divide the heap into multiple disjoint regions, such that each region is encoded into an array. In general, a summary contains two arrays (input and output) per memory region, describing all possible values before and after the call to the procedure. A naive SMT encoding requires quantifiers to express which elements of the output array are equal to the input array. Although some SMT solvers can handle formulas with quantifiers, the problem is in general undecidable and, therefore, we want to avoid them if possible. In this talk we will show how to leverage a pointer analysis to produce an SMT encoding that enforces the SMT solver to search only for quantifier-free summaries. First, we use the explicit heap representation that the pointer analysis produces to distinguish between finite and potentially infinite memory regions accessed by a procedure. Second, we introduce a new SMT encoding that replaces array variables with scalars if the procedure only uses a finite amount of memory. This is a joint work with Jorge Navas and Arie Gurfinkel.	{"url":"https://software.imdea.org/es/events/software-seminars/2020/03-03/","timestamp":"2024-11-03T19:06:29Z","content_type":"text/html","content_length":"10613","record_id":"<urn:uuid:f645c500-74b3-4157-b707-a7f3e9524374>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00129.warc.gz"}
What are trading comps? Trading comparables (trading comps) are valuation methods that use ratios to value a company by assuming that it should be worth similar multiples to similar listed companies. Why do we use comps? Comparables (comps) are used in valuations where a recently sold asset is used to determine the value of a similar asset. Comparables, often used in real estate to find the fair value of a home, are a list of recent asset sales that reflect the characteristics of the asset an owner is looking to sell. What are comps in valuation? Comparable company analysis (or “comps” for short) is a valuation methodology that looks at ratios of similar public companies and uses them to derive the value of another business. Comps is a relative form of valuation, unlike a discounted cash flow (DCF) analysis, which is an intrinsic form of valuation. How do I find comps for stocks? Steps to remember for executing a Comps valuation Calculate Market Capitalization: It is equal to Share price × Number of Shares Outstanding. Calculate Enterprise Value: Market Capitalization + Debt + Preferred Stock + Minority Interest (less common) – Cash. How do I choose a trade competition? How to Choose Comparable Companies 1. Comparable Criteria. There are multiple factors that decide whether a company is a good comparable company for your model. 2. Industry Classification. 3. Size. 4. Geography. 5. Growth Rate. 6. Profitability. 7. Capital Structure. 8. Constructing a Comparable Universe. How do I create a transaction comp? Steps to Perform Precedent Transaction Analysis: 1. #1 Search for relevant transactions. 2. #2 Analyze and refine the available transactions. 3. #3 Determine a range of valuation multiples. 4. #4 Apply the valuation multiples to the company in question. 5. #5 Graph the results (with other methods) in a football field. How do you use comps in an offer? Comps should be recent and of similar, nearby properties with as many features in common as possible. Homes should be the same style, similar age and comparable condition, with the same number of bedrooms and bathrooms, equal square footage, and equivalent lot size. How do you find the comp value? Multiply the Revenue As with cash flow, revenue gives you a measure of how much money the business will bring in. The times revenue method uses that for the valuation of the company. Take current annual revenues, multiply them by a figure such as 0.5 or 1.3, and you have the company’s value. What is a 2 year stack comp? Some financial reporting now provides a two-year stacked comp, which adds the growth rates for the past two years together. Some analysts are also calling their comparisons of 2021 to 2019 a two-year stacked comp even though the math isn’t exactly the same. What are the three main valuation methodologies? Three main types of valuation methods are commonly used for establishing the economic value of businesses: market, cost, and income; each method has advantages and drawbacks. What are trading and transaction multiples? Trading multiples refer to ratios calculated on publicly traded companies, using market data for price and the latest financial statements. As price points are available and fluctuate every day, trading multiples can change from one day to another. Are comps accurate? House comps are not foolproof While pulling comps on your own will give you an estimate of your home’s value, working with a licensed real estate agent and an appraiser is the most reliable way to know exactly what your home is What means sold comp? A comparable sale (also known as a “comp”) is a recently sold property in the area with similar features to the home you’re looking to buy. Appraisers use comparable sales to help estimate the fair market value of a home. What are the five methods of valuation? There are five main methods used when conducting a property evaluation; the comparison, profits, residual, contractors and that of the investment. A property valuer can use one of more of these methods when calculating the market or rental value of a property. How do you determine the selling price of a small business? How to Calculate Selling Price Per Unit. Determine the total cost of all units purchased. Divide the total cost by the number of units purchased to get the cost price. Use the selling price formula to calculate the final price: Selling Price = Cost Price + Profit Margin. What does tough comps mean? It usually means something compares unfavorably to something else, usually because the “something else” is an outlier of some sort. For example, if Q1’2020 revenue was super high due to a one-off service, it’s a “tough comp” for Q1’2021. What does it mean to comp last year? Comparable Sales or Comp Sales (or Like for Like) compares current year actual results to the results of the same time periods in the previous year. 2020 was anything but Like for Like. In March of 2021, the world will “Comp” Covid-19. What does this mean for your stores and your organization? Which valuation method is the best? Multiples of EBITDA are the most common valuation method. The “comps” valuation method provides an observable value for the business, based on what other comparable companies are currently worth. Comps are the most widely used approach, as they are easy to calculate and always current. Which method gives the highest valuation? Generally, however, transaction comps would give the highest valuation, since a transaction value would include a premium for shareholders over the actual value. How do you read multiples trading? Price-to-Earnings (P/E) Multiple A company with a low price compared to its level of earnings has a low P/E multiple. A P/E of 5x means a company’s stock is trading at a multiple of five times its earnings. A P/E of 10x means a company is trading at a multiple that is equal to 10 times earnings. What is the difference between trading multiples and transaction multiples? Transaction multiple vs trading multiple Another important characteristic of trading multiples is that they can be calculated for any historical period where the company was publicly traded. Transaction multiples refer to ratios calculated based on the announced acquisition prices of a transaction. How do you read comps? If it isn’t shown on a listing, you can calculate it: Divide the sale price by square footage (both numbers commonly included in listings). For example: A 1,200-square-foot home that sold for $300,000 has a square-foot price of $250 (300,000 ÷ 1,200 = 250). What does comps mean in retail? Comparable store sales refers to the revenue generated by a retail location in the most recent accounting period relative to the revenue it generated in a similar period in the past. Comparable store sales, or “comps,” are also referred to as “same-store sales” or “identical-store sales.” What are the 3 ways to value a company? A thoughtful approach will assess the value of a business using one – or all – of three primary methods: the Income Approach, the Market Approach, and the Cost Approach. Getting familiar with these methods is critical. What are the 3 valuation approaches? There are three approaches to valuing a company: the asset approach, income approach, and market approach. Within each approach, there are several commonly accepted methods that the valuator may choose to employ in valuing the business.	{"url":"https://www.trentonsocial.com/what-are-trading-comps/","timestamp":"2024-11-13T04:34:50Z","content_type":"text/html","content_length":"64592","record_id":"<urn:uuid:3ab1aed1-5cee-4f40-a2f7-7014890116ce>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00083.warc.gz"}
Texas Go Math Grade 7 Module 2 Answer Key Rates and Proportionality Refer to ourÂ Texas Go Math Grade 7 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 7 Module 2 Answer Key Rates and Proportionality. Texas Go Math Grade 7 Module 2 Answer Key Rates and Proportionality Texas Go Math Grade 7 Module 2 Are You Ready? Answer Key Question 1. $\frac{3}{4}$ Ã· $\frac{4}{5}$ _____________ Multiply by the reciprocal of the divisor: = $\frac{3}{4}$ Ã— $\frac{5}{4}$ = $\frac{15}{16}$ Grade 7 Module 2 Answer Key Texas Go Math Question 2. $\frac{5}{9}$ Ã· $\frac{10}{11}$ _____________ Multiply by the reciprocal of the divisor: = $\frac{5}{9}$ Ã— $\frac{11}{10}$ = $\frac{11}{18}$ Question 3. $\frac{3}{8}$ Ã· $\frac{1}{2}$ _____________ Multiply by the reciprocal of the divisor: = $\frac{3}{8}$ Ã— $\frac{2}{1}$ = $\frac{3}{4}$ Question 4. $\frac{16}{21}$ Ã· $\frac{8}{9}$ _____________ Multiply by the reciprocal of the divisor: = $\frac{16}{21}$ Ã— $\frac{9}{8}$ = $\frac{6}{7}$ Write the ordered pair for each point. Question 5. B ____________ B(-4, 1) Question 6. C ____________ C(3, 0) Question 7. D _____________ D(5, 4) Question 8. E _____________ E(-2, -2) Question 9. F _____________ F(0, 0) Texas Go Math Module 2 Grade 7 Answer Key Question 10. G _____________ G(-4, 0) Texas Go Math Grade 7 Module 2 Reading Start-Up Answer Key Visualize Vocabulary Use the âœ“ words to complete the graphic. You can put more than one word in each bubble. Understand Vocabulary Match the term on the left to the definition on the right. 1. rate of change ………. B. A rate that describes how one quantity changes in relation to another quantity. 2. proportion ………… A. Statement that two rates or ratios are equivalent 3. proportion ……….. C. Rate in which the second quantity is one unit Leave a Comment You must be logged in to post a comment.	{"url":"https://gomathanswerkey.com/texas-go-math-grade-7-module-2-answer-key/","timestamp":"2024-11-14T08:52:06Z","content_type":"text/html","content_length":"233903","record_id":"<urn:uuid:8a47cefe-4e9f-4b33-bd69-7155be379583>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00170.warc.gz"}
TeachersDungeon – Educational Strategies Welcome to Video Notebook, where your child can see easy to understand video tutorials that cover important concepts within the subject of geometry. I use the lessons in this article as an introduction to geometry for my six grade students. The concepts within this article cover many important concepts for fourth, fifth, and sixth grade … Educational Strategies ASP 8.3.1 Assorted Stumper Problems 8.3.1 Below is your video tutorial for this problem. Watch this video. If you have a mistake – Fix it. That is your fastest way to learn! For more information and helpful books on mathematics, visit my Teachers Pay Teachers website Educational Strategies Fun Geometry Animal Reserve 1.9 Below is your video tutorial for this problem. Watch this video. If you have a mistake – Fix it. That is your fastest way to learn! For more information and helpful books on mathematics, visit my Teachers Pay Teachers website Educational Strategies Assorted Stumper Problems 7.1 Assorted Stumper Problems are perfect for Remote Learning. Because each and every problem is linked to its own video tutorial, children can use this book while working from home. This makes Assorted Stumper Problems extremely effective for both remote learning or homeschool teaching. I have my students access this book through their Google Classroom accounts. I … Educational Strategies 3 Simple Steps For Adding & Subtracting Large Numbers These lessons are designed to offer math help online for children who struggle with addition & subtraction. According to educational standards, by the time children leave second grade they should be able to add and subtract three digit numbers fluently. In order to do this, children must know all their addition and subtraction facts from … Educational Strategies 11-Steps to Understanding Fractions - This page includes math tutorial videos! Welcome to Illustrating Fractions This Article is an Introduction to Fractions This lesson is designed to offer math help online for children who are learning fractions. There are 2 simple strategies that I teachthroughout all 11 steps in understanding fractions. The 1st step is to draw a box and cut it into the number … Educational Strategies 4-Simple Steps for Dividing - Each problem has a math tutorial video! There’s an Easier Way for Children to Learn Long Division! For years I have watched children struggle with long division. One major reason that is that memorizing all the multiplication facts can be difficult for some kids. I have seen them get more and more frustrated as they rise through the grades. Many …	{"url":"https://teachersdungeon.com/blog/tag/math-help-websites/","timestamp":"2024-11-11T22:49:35Z","content_type":"text/html","content_length":"59596","record_id":"<urn:uuid:e02de289-191c-44bc-968c-f4adeeb974a3>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00113.warc.gz"}
MTEL Elementary Mathematics (68) Exercise Book The MTEL Elementary Mathematics (68) Exercise Book is designed as a practical, interactive workbook that offers extensive practice in a format closely resembling the actual exam. Covering a range of topics from basic arithmetic to advanced problem-solving techniques, each chapter is filled with practice questions, detailed explanations, and step-by-step solutions to reinforce your understanding and enhance your mathematical skills. To further enrich your study experience, this book includes access to an accompanying online course. This dynamic resource features interactive videos, additional exercises, and real-time feedback, allowing you to engage deeply with the key concepts and strategies highlighted in the book. Key Features: • 100% Aligned with 2024 Course Guidelines: Stay on track with the latest exam requirements for focused and relevant preparation. • Extensive Practice Questions: Challenge yourself with a variety of questions that simulate the complexity and format of the actual exam. • In-Depth Answers and Explanations: Gain clarity on complex problems with comprehensive solutions for every question. • Two Full-Length Practice Tests: Assess your readiness with two realistic practice exams, complete with detailed answer explanations to pinpoint areas for improvement. • Online Course Access: Enhance your learning with a fully online course that includes interactive content and additional resources to build your skills. • Step-by-Step Solutions: Learn effective methods for solving problems, equipping you to tackle future questions with confidence. • Realistic Exam Simulation: Familiarize yourself with the exam’s structure and question types through carefully crafted practice tests. • Focus on Fundamental Skills: Build a strong foundation in essential areas such as arithmetic, algebra, geometry, and data analysis—key components of the MTEL exam. • Practice for All Skill Levels: Whether you’re just starting or refining advanced concepts, this book provides challenges suitable for every stage of preparation. • Confidence Building: Master the exam’s content and structure through extensive practice, boosting your confidence for test day. This exercise book is not just a study aid; it’s a structured roadmap to mastering elementary mathematics. With two full-length practice tests included, you can simulate the real exam environment, track your progress, and identify areas that may need more attention. Prepare effectively and confidently for your MTEL Elementary Mathematics (68) exam with this essential resource—your gateway to success! There are no reviews yet.	{"url":"https://www.effortlessmath.com/product/mtel-elementary-mathematics-68-exercise-book/","timestamp":"2024-11-11T21:29:39Z","content_type":"text/html","content_length":"44532","record_id":"<urn:uuid:55ca3886-9f53-4b39-8b47-3b18092dbbdd>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00591.warc.gz"}
Worksheets for Class 5 Maths Archives - WorkSheets Buddy CBSE Worksheets for Class 5 Maths: One of the best teaching strategies employed in most classrooms today is Worksheets. CBSE Class 5 Maths Worksheet for students has been used by teachers & students to develop logical, lingual, analytical, and problem-solving capabilities. So in order to help you with that, we at WorksheetsBuddy have come up with Kendriya Vidyalaya Class 5 Maths Worksheets for the students of Class 5. All our CBSE NCERT Class 5 Maths practice worksheets are designed for helping students to understand various topics, practice skills and improve their subject knowledge which in turn helps students to improve their academic performance. These chapter wise test papers for Class 5 Maths will be useful to test your conceptual understanding. Board: Central Board of Secondary Education(www.cbse.nic.in) Subject: Class 5 Maths Number of Worksheets: 200+ CBSE Class 5 Maths Worksheets PDF All the CBSE Worksheets for Class 5 Maths provided in this page are provided for free which can be downloaded by students, teachers as well as by parents. We have covered all the Class 5 Maths important questions and answers in the worksheets which are included in CBSE NCERT Syllabus. Just click on the following link and download the CBSE Class 5 Maths Worksheet. CBSE Worksheets for Class 5 Math Magic can also use like assignments for Class 5 Maths students. Advantages of CBSE Class 5 Maths Worksheets 1. By practising NCERT CBSE Class 5 Maths Worksheet, students can improve their problem solving skills. 2. Helps to develop the subject knowledge in a simple, fun and interactive way. 3. No need for tuition or attend extra classes if students practise on worksheets daily. 4. Working on CBSE worksheets are time-saving. 5. Helps students to promote hands-on learning. 6. One of the helpful resources used in classroom revision. 7. CBSE Class 5 Maths Workbook Helps to improve subject-knowledge. 8. CBSE Class 5 Math Magic Worksheets encourages classroom activities. Worksheets of CBSE Class 5 Maths are devised by experts of WorksheetsBuddy experts who have great experience and expertise in teaching Maths. So practising these worksheets will promote students problem-solving skills and subject knowledge in an interactive method. Students can also download CBSE Class 5 Maths Chapter wise question bank pdf and access it anytime, anywhere for free. Browse further to download free CBSE Class 5 Maths Worksheets PDF. Now that you are provided all the necessary information regarding CBSE Class 5 Maths Worksheet and we hope this detailed article is helpful. So Students who are preparing for the exams must need to have great solving skills. And in order to have these skills, one must practice enough of Class 5 Math Magic revision worksheets. And more importantly, students should need to follow through the worksheets after completing their syllabus. Working on CBSE Class 5 Maths Worksheets will be a great help to secure good marks in the examination. So start working on Class 5 Math Magic Worksheets to secure good score.	{"url":"https://www.worksheetsbuddy.com/tag/worksheets-for-class-5-maths/","timestamp":"2024-11-05T07:58:19Z","content_type":"text/html","content_length":"146012","record_id":"<urn:uuid:ed34d90e-723b-4c6a-86ce-46d94d4faf32>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00602.warc.gz"}
Simpson's Paradox: Shadow versus slices of an ellipsoid The simple regression of 'Heart' on 'Coffee' will appear as a blue plane and the multiple regression of 'Heart' on 'Coffee' and 'Stress' eventually appears as a red plane. The ellipsoid is the 'data ellipsoid' whose one dimensional projections on any line produce the mean plus or minus one standard deviation of the projection. It is the unit sphere for Mahalanobis distance. Simpson's Paradox applied to causality involves three variables: a response Y, a potential cause X and a conditioning variable Z. Each variable can be a continuous numeric variable or a categorical variable. This results in eight distinct graphical visualizations of the paradox. In this artificial example, we have three continuous variables. We suppose that Coffee (X) is a mild palliative for the harmful effect of Stress (Z) on Heart Damage (Y). However, if we simply regress Y on X without controlling for Z, Coffee appears very harmful. There is a strong positive (i.e. harmful) relationship between Y and X because both are strongly related to the confounding variable Z. In fact, there is a strong positive relationship between each pair of the three variables. However, when controlling for Z, the conditional relationship between Y and X becomes negative. If we merely observe the unconditional relationship between X and Y we could conclude that Coffee is harmful. However the conditional relationship, if indeed Z is a confounding factor and not a mediator, reveals that Coffee could be mildly beneficial. The geometry of the data ellipsoid plays an interesting role. The 'unit data ellipsoid' consists of points whose Mahalanobis distance from the mean of the 3-dimensional data cloud is 1 -- it's the unit Mahalanobis sphere. It captures (and is equivalent to) all first and second moments of the data cloud. Any statistical procedure, such as least-squares regression, that depends only on the first and second moments of the data cloud can be expressed as a geometric function of the data ellipsoid. The simple regression of Y on X is determined by the data ellipse that is the frontal projection of the data ellipsoid on the X-Y plane. The multiple regression of Y on X and Z has conditional regression lines, given Z, that are determined by the frontal sections of the data ellipsoid. Thus the marginal relationship between Y and X corresponds to the projection of the ellipsoid (shown by a blue ellipse) while the conditional relationship corresponds to the sections of the ellipsoid (shown by red ellipses). The principal frontal section ellipse (at the centre of the three-dimensional ellipsoid) is the data ellipse of the added-variable plot, also known as the partial regression plot, for the regression of Y on X controlling for the linear effect of Z. The geometric representation of Simpson's Paradox is that the projection of the ellipsoid and its sections can have opposite orientations. Recent Comments	{"url":"https://georges.blog.yorku.ca/causality/","timestamp":"2024-11-04T11:50:40Z","content_type":"text/html","content_length":"33702","record_id":"<urn:uuid:96912999-7ac7-426c-874a-c1831a5f235a>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00255.warc.gz"}
Theorem of Everything: The Secret That Links Numbers and Shapes - International Maths Challenge Theorem of Everything: The Secret That Links Numbers and Shapes For millennia mathematicians have struggled to unify arithmetic and geometry. Now one young genius could have brought them in sight of the ultimate prize. IF JOEY was Chloe’s age when he was twice as old as Zoe was, how many times older will Zoe be when Chloe is twice as old as Joey is now? Or try this one for size. Two farmers inherit a square field containing a crop planted in a circle. Without knowing the exact size of the field or crop, or the crop’s position within the field, how can they draw a single line to divide both the crop and field equally? You’ve either fallen into a cold sweat or you’re sharpening your pencil (if you can’t wait for the answer, you can check the bottom of this page). Either way, although both problems count as “maths” – or “math” if you insist – they are clearly very different. One is arithmetic, which deals with the properties of whole numbers: 1, 2, 3 and so on as far as you can count. It cares about how many separate things there are, but not what they look like or how they behave. The other is geometry, a discipline built on ideas of continuity: of lines, shapes and other objects that can be measured, and the spatial relationships between them. Mathematicians have long sought to build bridges between these two ancient subjects, and construct something like a “grand unified theory” of their discipline. Just recently, one brilliant young researcher might have brought them decisively closer. His radical new geometrical insights might not only unite mathematics, but also help solve one of the deepest number problems of them all: the riddle of the primes. With the biggest prizes in mathematics, the Fields medals, to be awarded this August, he is beginning to look like a shoo-in. The ancient Greek philosopher and mathematician Aristotle once wrote, “We cannot… prove geometrical truths by arithmetic.” He left little doubt he believed geometry couldn’t help with numbers, either. It was hardly a controversial thought for the time. The geometrical proofs of Aristotle’s near-contemporary Euclid, often called the father of geometry, relied not on numbers, but logical axioms extended into proofs by drawing lines and shapes. Numbers existed on an entirely different, more abstract plane, inaccessible to geometers’ tools. And so it largely remained until, in the 1600s, the Frenchman René Descartes used the techniques of algebra – of equation-solving and the manipulation of abstract symbols – to put Euclid’s geometry on a completely new footing. By introducing the notion that geometrical points, lines and shapes could all be described by numerical coordinates on an underlying grid, he allowed geometers to make use of arithmetic’s toolkit, and solve problems numerically. This was a moonshot that let us, eventually, do things like send rockets into space or pinpoint positions to needle-sharp accuracy on Earth. But to a pure mathematician it is only a halfway house. A circle, for instance, can be perfectly encapsulated by an algebraic equation. But a circle drawn on graph paper, produced by plotting out the equation’s solutions, would only ever capture a fragment of that truth. Change the system of numbers you use, for example – as a pure mathematician might do – and the equation remains valid, while the drawing may no longer be helpful. Wind forward to 1940 and another Frenchman was deeply exercised by the divide between geometry and numbers. André Weil was being held as a conscientious objector in a prison just outside Rouen, having refused to enlist in the months preceding the German occupation of France – a lucky break, as it turned out. In a letter to his wife, he wrote: “If it’s only in prison that I work so well, will I have to arrange to spend two or three months locked up every year?” Weil hoped to find a Rosetta stone between algebra and geometry, a reference work that would allow truths in one field to be translated into the other. While behind bars, he found a fragment. It had to do with the Riemann hypothesis, a notorious problem concerning how those most fascinating numbers, the primes, are distributed (see below). There had already been hints that the hypothesis might have geometrical parallels. Back in the 1930s, a variant had been proved for objects known as elliptic curves. Instead of trying to work out how prime numbers are distributed, says mathematician Ana Caraiani at Imperial College London, “you can relate it to asking how many points a curve has”. Weil proved that this Riemann-hypothesis equivalent applied for a range of more complicated curves too. The wall that had stood between the two disciplines since Ancient Greek times finally seemed to be crumbling. “Weil’s proof marks the beginning of the science with the most un-Aristotelian name of arithmetic geometry,” says Michael Harris of Columbia University in New York. The Riemann Hypothesis: The million-dollar question The prime numbers are the atoms of the number system, integers indivisible into smaller whole numbers other than one. There are an infinite number of them and there is no discernible pattern to their appearance along the number line. But their frequency can be measured – and the Riemann hypothesis, formulated by Bernhard Riemann in 1859, predicts that this frequency follows a simple rule set out by a mathematical expression now known as the Riemann zeta function. Since then, the validity of Riemann’s hypothesis has been demonstrated for the first 10 trillion primes, but an absolute proof has yet to emerge. As a mark of the problem’s importance, it was included in the list of seven Millennium Problems set by the Clay Mathematics Institute in New Hampshire in 2000. Any mathematician who can tame it stands to win $1 million. In the post-war years, in the more comfortable setting of the University of Chicago, Weil tried to apply his insight to the broader riddle of the primes, without success. The torch was taken up by Alexander Grothendieck, a mathematician ranked as one of the greatest of the 20th century. In the 1960s, he redefined arithmetic geometry. Among other innovations, Grothendieck gave the set of whole numbers what he called a “spectrum”, for short Spec(Z). The points of this undrawable geometrical entity were intimately connected to the prime numbers. If you could ever work out its overall shape, you might gain insights into the prime numbers’ distribution. You would have built a bridge between arithmetic and geometry that ran straight through the Riemann hypothesis. The shape Grothendieck was seeking for Spec(Z) was entirely different from any geometrical form we might be familiar with: Euclid’s circles and triangles, or Descartes’s parabolas and ellipses drawn on graph paper. In a Euclidean or Cartesian plane, a point is just a dot on a flat surface, says Harris, “but a Grothendieck point is more like a way of thinking about the plane”. It encompasses all the potential uses to which a plane could be put, such as the possibility of drawing a triangle or an ellipse on its surface, or even wrapping it map-like around a sphere. If that leaves you lost, you are in good company. Even Grothendieck didn’t manage to work out the geometry of Spec(Z), let alone solve the Riemann hypothesis. That’s where Peter Scholze enters the “Even the majority of mathematicians find most of the work unintelligible” Born in Dresden in what was then East Germany in 1987, Scholze is currently, at the age of 30, a professor at the University of Bonn. He laid the first bricks for his bridge linking arithmetic and geometry in his PhD dissertation, published in 2012 when he was 24. In it, he introduced an extension of Grothendieck-style geometry, which he termed perfectoid geometry. His construction is built on a system of numbers known as the p-adics that are intimately connected with the prime numbers (see “The p-adics: A different way of doing numbers”). The key point is that in Scholze’s perfectoid geometry, a prime number, represented by its associated p-adics, can be made to behave like a variable in an equation, allowing geometrical methods to be applied in an arithmetical setting. It’s not easy to explain much more. Scholze’s innovation represents “one of the most difficult notions ever introduced in arithmetic geometry, which has a long tradition of difficult notions”, says Harris. Even the majority of working mathematicians find most of it unintelligible, he adds. Be that as it may, in the past few years, Scholze and a few initiates have used the approach to solve or clarify many problems in arithmetic geometry, to great acclaim. “He’s really unique as a mathematician,” says Caraiani, who has been collaborating with him. “It’s very exciting to be a mathematician working in the same field.” This August, the world’s mathematicians are set to gather in Rio de Janeiro, Brazil, for their latest international congress, a jamboree held every four years. A centrepiece of the event is the awarding of the Fields medals. Up to four of these awards are given each time to mathematicians under the age of 40, and this time round there is one name everyone expects to be on the list. “I suspect the only way he can escape getting a Fields medal this year is if the committee decides he’s young enough to wait another four years,” says Marcus du Sautoy at the University of Oxford. Peter Scholze, 30, looks like a shoo-in for mathematics’s highest accolade this summer With so many grand vistas opening up, the question of Spec(Z) and the Riemann hypothesis almost becomes a sideshow. But Scholze’s new methods have allowed him to study the geometry, in the sense Grothendieck pioneered, that you would see if you examined the curve Spec(Z) under a microscope around the point corresponding to a prime number p. That is still a long way from understanding the curve as a whole, or proving the Riemann hypothesis, but his work has given mathematicians hope that this distant goal might yet be reached. “Even this is a huge breakthrough,” says Caraiani. Scholze’s perfectoid spaces have enabled bridges to be built in entirely different directions, too. A half-century ago, in 1967, the then 30-year-old Princeton mathematician Robert Langlands wrote a tentative letter to Weil outlining a grand new idea. “If you are willing to read it as pure speculation I would appreciate that,” he wrote. “If not – I am sure you have a waste basket handy.” In his letter, Langlands suggested that two entirely distinct branches of mathematics, number theory and harmonic analysis, might be related. It contained the seeds of what became known as the Langlands program, a vastly influential series of conjectures some mathematicians have taken to calling a grand unified theory capable of linking the three core mathematical disciplines: arithmetic, geometry and analysis, a broad field that we encounter in school in the form of calculus. Hundreds of mathematicians around the world, including Scholze, are committed to its completion. The full slate of Langlands conjectures is no more likely than the original Riemann hypothesis to be proved soon. But spectacular discoveries could lie in store: Fermat’s last theorem, which took 350 years to prove before the British mathematician Andrew Wiles finally did so in 1994, represents just one particular consequence of its conjectures. Recently, the French mathematician Laurent Fargues proposed a way to build on Scholze’s work to understand aspects of the Langlands program concerned with p-adics. It is rumoured that a partial solution could appear in time for the Rio In March, Langlands won the other great mathematical award, the Abel prize, for his lifetime’s work. “It took a long time for the importance of Langlands’s ideas to be recognised,” says Caraiani, “and they were overdue for a major award.” Scholze seems unlikely to have to wait so long. The p-adics: A different way of doing numbers Key to the latest work in unifying arithmetic and geometry are p-adic numbers. These are an alternative way of representing numbers in terms of any given prime number p. To make a p-adic number from any positive integer, for example, you write that number in base p, and reverse it. So to write 20 in 2-adic form, say, you take its binary, or base-2, representation – 10100 – and write it backwards, 00101. Similarly 20’s 3-adic equivalent is 202, and as a 4-adic it is written The rules for manipulating p-adics are a little different, too. Most notably, numbers become closer as their difference grows more divisible by whatever p is. In the 5-adic numbers, for example, the equivalents of 11 and 36 are very close because their difference is divisible by 5, whereas the equivalents of 10 and 11 are further apart. For decades after their invention in the 1890s, the p-adics were just a pretty mathematical toy: fun to play with, but of no practical use. But in 1920, the German mathematician Helmut Hasse came across the concept in a pamphlet in a second-hand bookshop, and became fascinated. He realised that the p-adics provided a way of harnessing the unfactorisability of the primes – the fact they can’t be divided by other numbers – that turned into a shortcut to solving complicated proofs. Since then, p-adics have played a pivotal part in the branch of maths called number theory. When Andrew Wiles proved Fermat’s infamous last theorem (that the equation x^n + y^n = z^n has no solutions when x, y and z are positive integers and n is an integer greater than 2) in the early 1990s, practically every step in the proof involved p-adic numbers. • Answers: Zoe will be three times as old as she is now. The farmers should draw a line across the field that connects the centre points of the field and the crop. This article appeared in print under the headline “The shape of numbers” For more such insights, log into www.international-maths-challenge.com. Credit for article given to Gilead Amit	{"url":"https://international-maths-challenge.com/theorem-of-everything-the-secret-that-links-numbers-and-shapes/","timestamp":"2024-11-08T14:33:04Z","content_type":"text/html","content_length":"158791","record_id":"<urn:uuid:b425ca76-d5d0-466c-ba7a-ffeeb1f39d12>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00724.warc.gz"}
Recommendation system in PHP using Matrix Factorization A recommendation system, is a system that can predict a rating for a user-item pair, where this user has never interacted with the item before. In other words, is users of a user can give preferences for items/products in the sense of a star system (1-5 stars like NetFlix) or a thumbs-up/thumbs-down like YouTube, one can generate a rating matrix consisting of users and items. And each cell then has a rating value if the user has rated this particular item. │User/product matrix │James│Peter│Anna│Victoria │ │Blue pants │5 │3 │0 │1 │ │Red hat │4 │0 │0 │1 │ │Black shoes │1 │1 │0 │5 │ │Computer mouse │1 │0 │0 │4 │ │Yellow t-shirt │0 │1 │5 │4 │ The table above, is called a ratings matrix, where a value above 0 is a rating defined by the user. A rating of 0 means the user has never encountered the product before. We now wish to calculate all the cells with 0’s – This is where Matrix Factorization is useful. Let’s do some maths’ We have a set of users $U$ and a set of items $D$. Let $R$ of size $\|U\| \times \|D\|$ be the ratings matrix that the users have assigned their items. We want to discover $K$ latent factors We now need to find two matrices $P(a\|U\| \times K)$ and $Q (a\|D\| \times K)$ such that their product approximates $R$: $R \approx P \times Q^T = \hat{R}$ So each row of $P$ would represent the strength of the associations between a user and its factors. To get the prediction of a rating of an item $d_j$ by $u_i$ we can calculate the dot product of the two vectors corresponding to $u_i$ and $d_j$: $\hat{r}_{ij} = p_{i}^{T}q_j = \sum\limits_{k=1}^{k} p_{ik}q_{kj}$ We just need to find $P$ and $Q$ – There exist several ways to do this. I’m going to use Gradient Descent – initialize the two matrices with some values, and calculate how different their product is to $M$ and try to minimize the difference iteratively. The squared error can be calculated by: $e_{ij}^2 = (r_{ij} - \hat{r}_{ij})^2 = (r_{ij} - \sum\limits_{k=1}^{K} p_{ik}q_{kj})^2$ We want to minimize this error, and we want to know in which direction this error goes. $\frac{d}{d p_{ik}} = -2 (r_{ij} - \hat{r}{ij} )(q_{kj} = -2 e_{ij} q_{kj})$ $\frac{d}{d q_{ik}} = -2 (r_{ij} - \hat{r}{ij} )(p_{kj} = -2 e_{ij} p_{kj})$ We now know in which direction to go, to minimize the error (i.e. the gradient) for both $p_{ik}$ and $q_{kj}$ $p'_{ik} = p_{ik} + \alpha \frac{d}{d p_{ik}} e_{ij}^2 = p_{ik} + 2 \alpha e_{ij}q_{kj}$ $q'_{kj} = q_{kj} + \alpha \frac{d}{d q_{kj}} e_{ij}^2 = q_{kj} + 2 \alpha e_{ij}p_{ik}$ Where $\alpha$ is a constant that determines the rate of approaching the minimum, and should be a small value like $\alpha = 0.0002$. If this value is too large we might step over the minimum, and maybe oscillating around the minimum. Using the above update rules we can iteratively perform the operation until a convergence. $E = \sum\limits_{u, d_j, r_{ij} \in T} e_{ij} = \sum\limits_{u_i, d_j, r_{ij} \in T} (r_{ij} - \sum\limits_{k=1}^{K} p_{ik}q_{kj})^2$ Now we have the factorization done, which is fine. – We can although run into a problem with over-fitting the model. To avoid that, we introduce a regularization step: $e_{ij}^2 = (r_{ij}-\sum\limits_{k=1}^{K} p_{ik}q_{kj})^2 + \frac{\beta}{2} \sum\limits_{k=1}^{K}(\|\|P\|\|^2 + \|\|Q\|\|^2)$ Where $\beta$ is used to control the magnitudes of the user-factor and item-factor vector such that $P$ and $Q$ would give a good approximation of $R$ without having to contain large numbers. In practice $latex $beta$ is set to some value in the range of $0.02$ So the new update rules are: $p'_{ik} = p_{ik} + \alpha \frac{d}{d p_{ik}} e_{ij}^2 = p_{ik} + \alpha (2e_{ij}q_{kj}- \beta p_{ik})$ $q'_{kj} = q_{kj} + \alpha \frac{d}{d q_{kj}} e_{ij}^2 = q_{kj} + \alpha (2e_{ij}p_{ik}- \beta q_{kj})$ Implementation in PHP Running the above PHP implementation you will get the output: │User/product matrix │James│Peter│Anna│Victoria │ │Blue pants │5.01 │2.87 │4.92│0.99 │ │Red hat │3.94 │2.25 │4.02│0.99 │ │Black shoes │1.10 │0.73 │4.63│4.95 │ │Computer mouse │0.94 │0.62 │3.76│3.97 │ │Yellow t-shirt │2.22 │1.35 │4.88│4.04 │ Where all bold numbers was 0’s in the ratings matrix. The only thing you need to do, is to use some kind of similarity function (Tanimoto, Pearson, K-NN) to find which items to recommend. (I have to stop this guide now – it’s already too long) – I hope you gained some knowledge from this, or at least you can use my implementation in your system. The PHP implementation is written by me, but taken from a Python implementation from here One thought to “Recommendation system in PHP using Matrix Factorization” 1. Thanks a lot! Perfect and fast!	{"url":"https://dan.thoeisen.dk/hjem/recommendation-system-in-php-using-matrix-factorization/","timestamp":"2024-11-02T21:10:58Z","content_type":"text/html","content_length":"58369","record_id":"<urn:uuid:21b03a5b-2a72-4864-9ca4-6c38da02de85>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00159.warc.gz"}
A copper wire, 3 mm in diameter, is wound about a cylinder whose length is 12 cm, and diameter 10 cm, so as to cover the curved surface of the cylinder. Find the length and mass of the wire, assuming the density of copper to be 8.88 g per cm3. You must login to ask question. NCERT Solutions for Class 10 Maths Chapter 13 Important NCERT Questions Surface areas and Volumes, NCERT Books for Session 2022-2023 CBSE Board and UP Board Others state Board EXERCISE 13.5 Page No:258 Questions No:1	{"url":"https://discussion.tiwariacademy.com/question/a-copper-wire-3-mm-in-diameter-is-wound-about-a-cylinder-whose-length-is-12-cm-and-diameter-10-cm-so-as-to-cover-the-curved-surface-of-the-cylinder-find-the-length-and-mass-of-the-wire-assuming/","timestamp":"2024-11-11T20:50:31Z","content_type":"text/html","content_length":"162523","record_id":"<urn:uuid:a461da84-4f55-4ce2-83dd-a62b05318b39>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00730.warc.gz"}
An Empirical Examination of the Incremental Contribution of Stock Characteristics in UK Stock Returns Department of Accounting and Finance, Stenhouse Building, University of Strathclyde, 199 Cathedral Street, Glasgow G4 0QU, UK Submission received: 29 August 2017 / Revised: 27 September 2017 / Accepted: 29 September 2017 / Published: 11 October 2017 This study uses the Bayesian approach to examine the incremental contribution of stock characteristics to the investment opportunity set in U.K. stock returns. The paper finds that size, book-to-market (BM) ratio, and momentum characteristics all make a significant incremental contribution to the investment opportunity set when there is unrestricted short selling. However, no short selling constraints eliminate the incremental contribution of the size and BM characteristics, but not the momentum characteristic. The use of additional stock characteristics such as stock issues, accruals, profitability, and asset growth leads to a significant incremental contribution beyond the size, BM, and momentum characteristics when there is unrestricted short selling, but no short selling constraints largely eliminates the incremental contribution of the additional characteristics. 1. Introduction There is a long history that stock characteristics have a significant predictive ability of cross-sectional expected excess stock returns. The three most prominent characteristics have been size ( Banz 1981 ), book-to-market (BM) ratio ( Fama and French 1992 ), and momentum ( Jegadeesh and Titman 1993 ). There has been debate in the empirical literature as to whether these predictive patterns can be explained by different measures of systematic risk from linear factor models such as Daniel and Titman ), and Davis, Fama and French ( Davis et al. 2000 ). A recent study by Chordia, Goyal and Shanken ( Chordia et al. 2015 ) examine the relative contributions of both betas and stock characteristics in explaining the cross-sectional variation in expected excess returns. Chordia et al. find that stock characteristics make the dominant contribution to explaining cross-sectional variation in expected excess returns. During the past two decades, a number of studies have identified additional stock characteristics that predict cross-sectional stock returns. A partial list includes stock issues ( Pontiff and Woodgate 2008 ), accruals ( Sloan 1996 ), profitability ( Novy-Marx 2013 ), asset growth ( Titman et al. 2004 ), and idiosyncratic volatility ( Ang et al. 2006 ) among others . Harvey, Liu and Zhu ( Harvey et al. 2016 ) document 314 variables that prior research has found to predict stock returns ( see also Green et al. 2017 Much of the empirical evidence of stock characteristics comes from spreads in portfolio returns from one or two dimensional portfolio sorts on the basis of stock characteristics or from large -statistics in the Fama and MacBeth ) cross-sectional regressions. However, this evidence does not address the marginal impact that additional stock characteristics have on expected return spreads in the presence of other Fama and French ) examine this issue by forming portfolios on the basis of expected return estimates from the Fama and MacBeth cross-sectional regressions and compare the impact on average return spreads when adding additional stock characteristics to the Fama and MacBeth regressions. Fama and French find that there is only a marginal increase in average return spreads with additional characteristics, even where these characteristics have large Fama and MacBeth Fama and French ) extend this analysis and explore why additional stock characteristics only have a minor impact on expected return spreads even when they have large statistical significance in the Fama and MacBeth cross-sectional regressions. ) finds that there is only a modest increase in average return spreads across decile portfolios in moving from a model with the size, BM, and momentum characteristics, to one with seven characteristics, or fifteen characteristics. Fama and French ) examine the incremental contribution of stock characteristics on the investment opportunity set. They focus on the size, BM, and momentum characteristics. Fama and French use the Gibbons, Ross and Shanken ( Gibbons et al. 1989 ) test of mean-variance efficiency to examine if quintile portfolios formed by expected excess returns estimates from the Fama and MacBeth ) regressions using only two stock characteristics lies on the mean-variance frontier of an augmented investment universe that also includes quintile portfolios formed by expected excess returns estimates using all three characteristics. They find that all three characteristics make a significant incremental contribution to the investment opportunity set. However, the optimal tangency portfolios require large short positions and so the higher ) performance is not attainable by investors. Imposing no short selling constraints, Fama and French find that the incremental contribution of all three characteristics in terms of higher Sharpe performance disappears. The finding that no short selling constraints often hurts the mean-variance performance of trading strategies is consistent with De Roon, Nijman and Werker ( De Roon et al. 2001 ), Li, Sarkar and Wang ( Li et al. 2003 ), and Briere and Szafarz ) among others. Considering the role of no short selling constraints on the performance of trading strategies is important as a lot of investors are unable to short sell and even when can short sell, short selling can be costly ( Fama and French 2015 ). Bris, Goetzmann and Zhu ( Bris et al. 2007 ) find short selling is allowed in 35 out of 47 countries. Even in markets which allow short selling, temporary bans on short selling can be imposed such as in the UK, where short selling was banned in financial stocks between late 2008 and early 2009. Since 2012, the EU short selling regulation has banned naked short selling and investors must report net short positions above a certain limit. Briere and Szafarz ) point out that finding a stock lender can be costly and the investor can be exposed to a liquidity shortage ( Jones and Lamont 2002 ). Managed funds such as open-end mutual funds often face legal restrictions on short selling . European mutual funds subject to UCITS cannot take physical short positions and can only borrow up to 10% of net assets. Best and Grauer ) argue that portfolio constraints like no short selling will almost always be binding as unconstrained mean-variance efficient frontiers often have no all positive weight portfolios. This paper examines the incremental contribution of stock characteristics on the investment opportunity set in UK stock returns in the presence of no short selling constraints following a similar approach to Fama and French ). I evaluate the incremental contribution in terms of higher ) performance of adding quintile portfolios formed using expected excess returns from a larger model of characteristics to the investment universe of quintile portfolios formed using expected excess returns from a smaller model of stock characteristics. I consider the case where unrestricted short selling is allowed and where no short selling is allowed in the risky assets. I use the Bayesian approach of ) to estimate the benefits of higher Sharpe performance and evaluate statistical significance. I use the first two models of stock characteristics of ). The first model includes the size, BM, and momentum characteristics and the second model adds stock issues, accruals, profitability, and asset growth characteristics. My study makes three contributions to the literature. First, I complement and extend the recent studies of Fama and French ) and ) by examining the incremental contribution of stock characteristics in a different market and formally testing the incremental contribution of stock characteristics in the presence of no short selling constraints. Recent studies by ) and Hou, Xue and Zhang ( Hou et al. 2017 ) highlight the importance of replication in Finance, which is common in other fields of science. Second, I extend the prior literature on the role of stock characteristics in UK stock returns such Strong and Xu ), Gregory, Harris and Michou ( Gregory et al. 2001 Hon and Tonks ), and Qing and Turner ) among others . I extend this literature by considering the incremental contribution of stock characteristics on the investment opportunity set in UK stock returns and examining the impact of no short selling constraints on the incremental contribution. Third, I complement and extend the evidence that examines the impact of no short selling constraints on the mean-variance performance of trading strategies such as De Roon et al. Li et al. Ehling and Ramos ), Eun, Huang and Lai ( Eun et al. 2008 ), and Briere and Szafarz ) among others. I extend this literature by looking at the impact of no short selling constraints on the incremental contribution of stock characteristics. There are four main findings in my study. First, all of the stock characteristics have a significant predictive ability of future monthly excess returns in the Fama and MacBeth ) cross-sectional regressions, with the exception of the accruals characteristic. Second, I find that all of the model 1 characteristics make a significant incremental contribution to the investment opportunity set when investors are allowed unrestricted short selling. However, the optimal portfolios do require large short positions. Third, imposing no short selling restrictions substantially reduces the incremental contribution of the momentum characteristic and eliminates the incremental contribution of the size and BM characteristics. Fourth, I find that the additional model 2 characteristics make a significant incremental contribution to the investment opportunity set when there is unrestricted short selling. No short selling constraints eliminate the incremental contribution of the stock issues, accruals, and asset growth characteristics, but not profitability. My results suggest that there is little to be gained in moving beyond the model 1 characteristics for a characteristic-based model of stock returns. My study is organized as follows. Section 2 presents the research method. Section 3 describes the data used in the study. Section 4 reports the empirical results and the final section concludes. 2. Research Method I examine the impact of stock characteristics on the investment opportunity set following a similar approach to Fama and French ). I consider the impact of adding quintile portfolios sorted by expected excess returns from a model using a larger group of stock characteristics to a benchmark investment universe of quintile portfolios sorted by expected excess returns using a smaller group of stock characteristics. I then formally test whether there is a significant shift in the investment opportunity set from adding the quintile portfolios to the benchmark investment universe to see if the additional stock characteristics make a significant incremental contribution to the investment opportunity set. The expected excess returns of each stock are estimated each month using the Fama and MacBeth ) cross-sectional regression approach. Define M as the number of stock characteristics in a given model. For each month of the sample period (t = 1,…,T), the following cross-sectional regression is r[it] = γ[0t] + Σ[m=1]^Mγ[mt]z[mt−1] + u[it] where r is the excess return on asset i at time t, z is the value of the mth stock characteristic of asset i at time t−1, and u is a random error term of asset i at time t. The Fama and MacBeth cross-sectional regression assumes a linear functional form between the excess returns and stock characteristics. It is possible that nonlinearities are important in the releations between stock characteristics and excess returns. ) and Freyberger, Neuhier and Weber ( Freyberger et al. 2017 ) provide different approaches to examine this issue. The expected excess returns are given by: E(r[it]) = γ[0] + Σ[m=1]^Mγ[m]z[mt−1] where γ and γ are the time-series averages of the monthly γ and γ coefficients. Each month during the sample period, all stocks are ranked on the basis of their E(r ) and grouped into quintile portfolios as in Fama and French ). I then calculate the value weighted portfolio excess returns for each quintile portfolio. Where a security has missing return data during the month due to temporary suspension or death, I code the missing returns to zero as in Liu and Strong ). I correct for the delisting bias of ) by assigning a −100% return if the death is deemed valueless as in Dimson, Nagel and Quigley ( Dimson et al. 2003 Estimating expected excess returns using the Fama and MacBeth ) cross-sectional regression approach is used in a number of recent studies such as Fama and French ), and ) among others. Using the full sample estimates of γ and γ implies that investors cannot implement these as portfolio strategies. As a result, my study focuses on in-sample performance rather than out-of-sample performance. Fama and French point out that using full sample slopes has much greater precision compared to using rolling window estimates. Likewise if the portfolios are formed using monthly γ and γ coefficients, then most of the spread in portfolio returns will be due to unexpected returns and not due to expected return patterns. Lewellen examines the predictive ability of expected excess returns using the Fama and MacBeth approach. Fama and French ) use the Gibbons et al. ) test of mean-variance efficiency to examine the impact of adding quintile portfolios formed by expected excess returns using three stock characteristics to an investment universe of quintile portfolios formed by expected excess returns using two stock characteristics. The two groups of quintile portfolios are defined as the augmented investment universe. If the third characteristic has no incremental impact on the investment opportunity set, then the optimal tangency portfolio of the benchmark investment universe will be the same as the optimal tangency portfolio of the augmented investment universe. This analysis can be generalized to any two models of stock characteristics. The Gibbons et al. test does not accommodate short selling restrictions. Tests of mean-variance efficiency in the presence of no short sales constraints have been developed by Basak, Jagannathan and Sun ( Basak et al. 2002 An alternative approach to testing mean-variance efficiency in the presence of short selling constraints is the Bayesian approach of Li et al. ). This approach was developed when no risk-free asset exists but the same approach can be modified to the case where there is risk-free lending and borrowing. I use the Bayesian approach to examine the portfolio efficiency of the optimal portfolios in the augmented investment universe relative to the optimal portfolios in the benchmark investment universe to capture the incremental contribution of the additional stock characteristics. Li et al. ) argue that the Bayesian approach has a number of advantages over the asymptotic tests of De Roon et al. ). First, the Bayesian approach incorporates the uncertainty of finite samples into the posterior distribution. Second, the Bayesian approach is easier to implement and can use a range of different performance measures. Third, we get the exact inference of the magnitude of diversification benefits. Fourth, under no short selling constraints the asymptotic tests rely on a first-order linear but the Bayesian approach uses the exact nonlinear function of u and V. I measure the incremental contribution of an additional individual (or group of) stock characteristics as the increase in ) performance from adding the quintile portfolios formed by expected excess returns using the extended model of stock characteristics to the benchmark investment universe. Define N as the number of risky assets in the benchmark universe and 2N as the number of risky assets in the augmented investment universe, x is the (2N, 1) vector of optimal weights in the augmented investment univegrse, and is the (2N, 1) vector of optimal weights in the benchmark investment universe, where the first N cells are zero and the remaining N cells are the optimal weights of the N risky assets in the benchmark investment universe. The performance measure is given by: where θ* = x’u/(x’Vx) = x , u is a (2N, 1) vector of expected excess returns, and V is a (2N, 2N) covariance matrix. The DSharpe measure captures the increase in Sharpe performance in adding the quintile portfolios formed using expected excess returns from the extended model of stock characteristics to the benchmark investment universe. If the additional stock characteristics make no incremental contribution to the investment opportunity set, then DSharpe = 0. I estimate the DSharpe measures for the case where unrestricted short selling is allowed and for the case where no short selling is allowed in the risky assets. When the risk-free asset exists, all optimal portfolios (which are combinations of the risk-free asset and the tangency portfolio) have the same Sharpe performance. As a result, the DSharpe measure can be estimated using any optimal portfolio on the corresponding mean-variance frontiers of the benchmark and augmented investment universes. I estimate the optimal portfolios using a given value of risk aversion, which I set equal to 3 as in Tu and Zhou To examine the statistical significance of the DSharpe measure, the Bayesian approach of ) assumes that the 2N asset excess returns have a multivariate normal distribution . I assume a non-informative prior for the expected excess returns u and covariance matrix V. Define u and V as the sample moments of the expected excess returns and covariance matrix, and r as the (T, 2N) matrix of excess returns of the risky assets. The posterior probability density function is given by: p(u, V\|R) = p(u\|V, u[s],T)•p(V\|V[s], T) where p(u\|V, u , T) is the conditional distribution of a multivariate normal (u , (1/T)V) distribution and p(V\|V , T) is the marginal posterior distribution that has an inverse Wishart (TV, T − 1) distribution ( Zellner 1971 ) proposes a Monte Carlo method to approximate the posterior distribution. I use the following approach. First, a random V matrix is drawn from an inverse Wishart (TV , T − 1) distribution. Second, a random u vector is drawn from a multivariate normal (u , (1/T)V) distribution. Third, given the u and V from steps 1 and 2, the DSharpe measure is estimated from Equation (3) . Fourth, steps 1 to 3 are repeated 1000 times as in Hodrick and Zhang ) to generate the approximate posterior distribution of the DSharpe measure. The posterior distribution of the DSharpe measure is then used to assess the magnitude of the incremental contribution of the additional stock characteristics to the investment opportunity set and provide a test of statistical significance. The average value of the posterior distribution of the DSharpe measure provides the average increase in Sharpe performance in adding the quintile portfolios formed using expected excess returns with the extended model of stock characteristics to the benchmark investment universe. I use the 5% percentile value of the DSharpe measure to assess the statistical significance of whether the average DSharpe measure = 0 ( Hodrick and Zhang 2014 ). If the 5% percentile value of the DSharpe measure exceeds zero, I reject the null hypothesis that the additional stock characteristics make no incremental contribution to the investment opportunity set. 3. Data My sample includes all UK stocks between July 1983 and December 2015. I exclude investment trusts , secondary shares, and foreign companies. I use the first two models of security characteristics of ). The first model includes size, BM, and momentum characteristics. The second model includes stock issues, accruals, profitability, and asset growth characteristics . The market values and stock returns data are collected from LSPD. The accounting data is collected from Worldscope provided by Thomson Financial. I use the return on the one-month Treasury Bill as the risk-free asset (collected from LSPD and Datastream). The characteristics involving only accounting data can only be calculated once a year. I assume that the monthly characteristic data, using only accounting data, between July of year t to June of year t + 1 are equal to the annual characteristic values calculated during year t − 1. This approach assumes that the accounting data from the fiscal year-end of the previous calendar year t − 1 would be known to investors by the start of July in year t. All of the characteristic data is winsorized at the 1% and 99% levels as in ). The characteristics are defined as follows: 3.1. Size The size of the company is given by the monthly market values. I use the log of the monthly market values at the prior month-end to measure size. I set companies with zero market values to missing 3.2. Book-to-Market (BM) Ratio The monthly BM ratio is calculated using the book value of equity at the fiscal year-end (WC03501) during the previous calendar year divided by the prior month-end market value. I set companies with negative book values or zero market values to missing values. I use the log of the BM ratio in my analysis. 3.3. Momentum I calculate the momentum characteristic each month as the prior cumulative returns of the stock between months −12 to −2. Companies must have continuous return observations during the past 12 months, otherwise the momentum characteristic is set to missing values. 3.4. Stock Issues I calculate the stock issues characteristic as in ). I use the log growth in split-adjusted shares from month −36 to month −1. I require companies to have the relevant data in both months −36 and −1, otherwise set to missing values. 3.5. Accruals I calculate the annual accruals similar to Fama and French ) as the change in operating working capital per split-adjusted shares from years t − 2 to t − 1 divided by book equity per split-adjusted share at year t − 1. I require companies to have the relevant data in years t − 2 and t − 1 and have a positive book value per share at year t − 1, otherwise set to missing values. Operating working capital is defined as current assets (WC02201) minus cash and short-term investments (WC02001) minus current liabilities (WC03101) plus debt in current liabilities (WC03051). I use the book value per share to measure book equity (WC05476). 3.6. Profitability I use the gross profitability measure as in ) and Sun, Wie and Xie ( Sun et al. 2014 ) defined as sales (WC01001) minus cost of goods sold (WC01051) divided by total assets (WC02999). 3.7. Asset Growth I calculate asset growth similar to Fama and French ) as the log of the ratio of assets per split-adjusted shares in year t − 1 to year t − 2. I calculate the assets per split-adjusted share using total assets (WC02999) and common shares outstanding Table 1 reports summary statistics of the monthly excess returns and stock characteristics across the July 1983 and December 2015 period. The table includes the time-series averages of the cross-sectional mean and standard deviation of the monthly excess returns (%) and the stock characteristic values at the start of each month. N is the time-series average of the number of stocks with the relevant data each month. The table reports summary statistics of the stock characteristics and excess returns for the individual stocks between July 1983 and December 2015. The summary statistics include the time-series averages of the cross-sectional mean, and standard deviation of the characteristic values at the start of each month and the monthly excess returns (%). N is the time-series average of the number of securities with characteristic values for that month. Table 1 shows that the average mean excess return is 0.687% with a large cross-sectional volatility of 18.109%. ) also reports a large cross-sectional volatility in individual stocks in U.S. stock returns. The average number of companies with characteristic data across the sample period varies across characteristics from 1158 (accruals) and 1832 (size). 4. Empirical Results I begin my empirical analysis by examining the predictive ability of the stock characteristics of the monthly excess returns using the Fama and MacBeth ) cross-sectional regressions. I run the regressions using the stock characteristics individually, then using the model 1 characteristics jointly, and then using all the model 2 characteristics. Table 2 reports the cross-sectional regression results. The table includes the time-series average slope coefficients (spreads) for each characteristic and the corresponding Fama and MacBeth -statistic. The R column is the time-series average of the adjusted R from the monthly cross-sectional regressions. The table reports the results of the Fama and MacBeth ) cross-sectional regressions of individual excess stock returns on stock characteristics between July 1983 and December 2015. The table includes the time-series average of the monthly slope coefficients on each characteristic and the corresponding Fama and MacBeth -statistic. The R column is the time-series average of the adjusted R from the monthly cross-sectional regressions. Panel A of the table reports the cross-sectional regression results where each characteristic is included individually. Panels B and C report the cross-sectional regressions using the model 1 characteristics and the model 2 characteristics respectively. Panel A of Table 2 shows that all the stock characteristics, except the accruals characteristic, have a significant predictive ability of monthly excess returns. The signs of the average characteristic spreads are consistent with prior research. The accruals characteristic has the smallest average spread in absolute terms at −0.120%. All of the other stock characteristics have large -statistics from the individual regressions in excess of the cutoff -statistic recommended by Harvey et al. ), which controls for multiple testing. The largest average spreads are for asset growth, momentum, and profitability Using the model 1 characteristics in panel B of Table 2 , all three characteristics have a significant predictive ability of monthly excess returns with large -statistics. The momentum characteristic has the largest average spread by a long way. There is a sharp increase in the average spread of the momentum characteristic when including the other stock characteristics in the cross-sectional regressions compared to panel A. The Fama and MacBeth ) slope coefficients with respect to a given characteristic can be viewed as the excess returns of a zero-cost portfolio that is in long in high values of the characteristic and short in low values of the characteristic controlling for the other characteristics in the regression ( Fama 1976 ). The difference between the average spreads of the momentum characteristic in panels A and B stem from the fact that the zero-cost portfolio in panel B controls for size and BM characteristics. When using the model 2 characteristics in panel C of Table 2 , six out of the seven stock characteristics continue to have significant predictive ability of cross-sectional monthly excess returns. There is a sharp drop in the -statistics for the size and stock issues characteristics, which are now below the cut-off -value of Harvey et al. ). The momentum characteristic now has the largest average spread, followed by the profitability and asset growth characteristics. Table 2 suggests that a number of stock characteristics have large significant average spreads in U.K. stock returns, even when controlling for other characteristics . These findings are in the main similar to Fama and French ) and ) among others in U.S. stock returns. I next examine the incremental contribution of stock characteristics on the investment opportunity set. I begin this analysis using the model 1 characteristics Fama and French ). Each pair of characteristics are used to form the quintile portfolios in the benchmark investment universe and then all three characteristics are used to form the quintile portfolios added to the benchmark investment universe. Table 3 reports the summary statistics of the posterior distribution of the DSharpe measure for the unconstrained portfolio strategies (panel A) and constrained portfolio strategies (panel B). The summary statistics include the mean, standard deviation, fifth percentile (5%), and the median of the posterior distribution. Panel C reports the sum of the average short positions in the optimal portfolios from the benchmark investment universe and the augmented (Augment) investment universe for the unconstrained portfolio strategies. The table reports the summary statistics of the posterior distribution of the DSharpe measure for the incremental contribution of the model 1 characteristics between July 1983 and December 2015. The summary statistics include the mean, standard deviation, the fifth percentile (5%), and the median of the posterior distribution. The model 1 characteristics include size, BM, and momentum. The benchmark (Bench) investment universe consists of excess returns of quintile portfolios formed by expected excess returns using two of the stock characteristics. The augmented (Augment) investment universe adds the excess returns of quintile portfolios formed by expected excess returns using all three characteristics. Panels A and B report the summary statistics of the posterior distribution for the unconstrained portfolio strategies and the constrained portfolio strategies where no short selling is allowed in the risky assets. Panel C reports the sum of the average short positions in the benchmark investment universe and the augmented investment universe. The analysis assumes a risk aversion of 3 in the optimal portfolio strategies. Panel A of Table 3 shows that for the unconstrained portfolio strategies, all three characteristics make a significant incremental contribution to the investment opportunity set. The mean DSharpe measures for the unconstrained portfolio strategies range between 0.059 (Size) and 0.164 (Momentum). All of the mean DSharpe measures are significant at the 5% percentile. The median DSharpe measures are close to the mean DSharpe measures. The momentum characteristic has the largest increase in Sharpe performance among the three characteristics. These results of the significant incremental contribution of each characteristic to the investment opportunity set, when investors are allowed unrestricted short selling, is similar to Fama and French Fama and French ) also point out that the higher Sharpe performance will not be attainable for investors if unable to short sell or even where they can short sell, the costs of short selling would eliminate much of the superior performance. The optimal portfolios underlying the increase in Sharpe performance can involve large short positions. Panel C of Table 3 shows that the sum of average short positions can be large, especially in the augmented investment universe. The sum of the average short positions in the augmented investment universes range between −5.348 (Size) and −12.863 (BM). The large short positions are generally concentrated in the low expected excess return portfolios. The results in panel C of Table 3 suggest that the higher Sharpe performance in panel A can only be exploited by large short positions, which is similar to Fama and French When the no short selling restrictions are imposed in panel B of Table 3 , the significant incremental contribution of the size and BM characteristics to the investment opportunity set disappears. The mean DSharpe measures of the size and BM characteristics are close to zero. There is a drop in the mean DSharpe measure of the momentum characteristic but the mean DSharpe measure remains significant at the 5% percentile. This result suggests that the momentum characteristic is the only characteristic which makes a significant incremental contribution to the investment opportunity set when there are no short selling constraints. Along with the drop in the mean DSharpe measures in panel B, there is also a drop in the volatility of the DSharpe measures. This pattern is similar to ) and is due to the lower estimation risk when no short selling constraints are imposed ( Frost and Savarino 1988 Jagannathan and Ma 2003 Basak et al. ) find that the standard error of their mean-variance inefficiency measure increases in the presence of no short selling constraints, which is different from the Bayesian approach. Basak et al suggest that this result occurs because the linear approximation to a nonlinear function in the asymptotic tests becomes less reliable in the presence of no short selling constraints The impact of the no short selling constraints on the incremental contribution of stock characteristics is in the main consistent with Fama and French ). The difference here is that we find the momentum characteristic continues to have a significant incremental contribution to the investment opportunity set even in the presence of no short selling constraints. The results are also consistent with a number of studies, which show that no short selling hurts the mean-variance performance of trading strategies such as factor investing as Briere and Szafarz ). In contrast, Jagannathan and Ma ) examining the out-of-sample performance of the global minimum variance (GMV) portfolio finds that no short selling constraints improves the performance of the GMV portfolio when using the sample matrix but not for other estimators of the covariance matrix. I next examine the incremental contribution of the stock issues, accruals, profitability, and asset growth characteristics to the investment opportunity set relative to the model 1 characteristics. The benchmark investment universe is the quintile portfolios formed by expected excess returns using the model 1 characteristics. Each additional characteristic is then used along with the model 1 characteristics to form quintile portfolios to construct the augmented investment universe. I also examine the incremental contribution of all the additional characteristics jointly. Table 4 reports the posterior distribution of the DSharpe measure for the unconstrained portfolio strategies (panel A) and the constrained portfolio strategies (panel B). Panel C reports the sum of the average short positions in the optimal portfolios from the benchmark investment universe and the augmented investment universe for the unconstrained portfolio strategies. The table reports the summary statistics of the posterior distribution of the DSharpe measure of the incremental contribution of the stock issues, accruals, profitability, and asset growth characteristics between July 1983 and December 2015. The summary statistics include the mean, standard deviation, the fifth percentile (5%), and the median of the posterior distribution. The model 2 characteristics include the model 1 characteristics size, BM, and momentum and the additional characteristics of stock issues, accruals, profitability, and asset growth. The benchmark (Bench) investment universe consists of excess returns of quintile portfolios formed by expected excess returns using the model 1 characteristics. The augmented (Augment) investment universe adds the excess returns of quintile portfolios formed by expected excess returns using the model 1 characteristics and one of the model 2 characteristics or all the model 2 characteristics. Panels A and B report the summary statistics of the posterior distribution for the unconstrained portfolio strategies and the constrained portfolio strategies, where no short selling is allowed in the risky assets. Panel C reports the sum of the average short positions in the benchmark universe and the augmented universe. The analysis assumes a risk aversion of 3 in the optimal portfolio strategies. Panel A of Table 4 shows that all four additional model 2 characteristics make a significant incremental contribution to the investment opportunity set when there is unrestricted short selling. The mean DSharpe measures for the unconstrained portfolio strategies range between 0.030 (Accruals) and 0.092 (Profitability). All of the mean DSharpe measures are significant at the 5% percentile. Using all four characteristics together, there is a significant incremental contribution to the investment opportunity set as reflected in the significant mean DSharpe measure of 0.151. The optimal portfolios underlying the increase in Sharpe performance in panel A of Table 4 do require large short positions. The sum of the average short positions range between −4.528 (Accruals) and −10.565 (Asset Growth). Imposing no short selling constraints substantially reduces both the mean and volatility of the DSharpe measures in panel B of Table 4 as in Table 3 . The significant incremental contribution of the stock issues, accruals, and asset growth characteristics disappears in the presence of no short selling constraints. It is only for the profitability characteristic and using all characteristics together that there is a significant mean DSharpe measure. The incremental contribution of the profitability characteristic is on the borderline of statistical significance. Table 4 suggests that the incremental contribution of the additional model 2 characteristics considered jointly is marginal in the presence of no short selling constraints. With the exception of the profitability characteristic, none of the individual characteristics make a significant incremental contribution to the investment opportunity set in the presence of no short selling constraints beyond what is contained in the model 1 characteristics. These results on the impact of no short selling constraints on the incremental contribution of stock characteristics are again consistent with Fama and French ) also finds that the additional stock characteristics only have a marginal impact on the predictive ability of expected returns beyond the model 1 characteristics. My analysis so far has formed the quintile portfolios using all stocks. Fama and French ) and ) among others show that stock characteristics often have a stronger predictive ability of cross-sectional stock returns in smaller companies. To examine this issue, I repeat the analysis in Table 2 Table 3 Table 4 but this time I only include the largest 350 companies by market value at the start of each month Table 5 Table 6 Table 7 report the corresponding empirical tests. The table reports the results of the Fama and MacBeth ) cross-sectional regressions of individual excess stock returns on stock characteristics between July 1983 and December 2015. The sample only includes the largest 350 stocks by market value at the start of each month. The table includes the time-series average of the monthly slope coefficients on each characteristic and the corresponding Fama and MacBeth -statistic. The R column is the time-series average of the adjusted R from the monthly cross-sectional regressions. Panel A of the table reports the cross-sectional regression results where each characteristic is included individually. Panels B and C report the cross-sectional regressions using the model 1 characteristics and the model 2 characteristics respectively. The table reports the summary statistics of the posterior distribution of the DSharpe measure of the incremental contribution of the model 1 characteristics between July 1983 and December 2015. The analysis only includes the largest 350 companies by market value each month. The summary statistics include the mean, standard deviation, the fifth percentile (5%), and the median of the posterior distribution. The model 1 characteristics include size, BM, and momentum. The benchmark (Bench) investment universe consists of excess returns of quintile portfolios formed by expected excess returns using two of the stock characteristics. The augmented (Augment) investment universe adds the excess returns of quintile portfolios formed by expected excess returns using all three characteristics. Panels A and B report the summary statistics of the posterior distribution for the unconstrained portfolio strategies and the constrained portfolio strategies where no short selling is allowed in the risky assets. Panel C reports the sum of the average short positions in the benchmark investment universe and the augmented investment universe. The analysis assumes a risk aversion of 3 in the optimal portfolio strategies. The table reports the summary statistics of the posterior distribution of the DSharpe measure of the incremental contribution of the stock issues, accruals, profitability, and asset growth characteristics between July 1983 and December 2015. The analysis only includes the largest 350 companies by market value each month. The summary statistics include the mean, standard deviation, the fifth percentile (5%), and the median of the posterior distribution. The model 2 characteristics include the model 1 characteristics size, BM, and momentum and the additional characteristics of stock issues, accruals, profitability, and asset growth. The benchmark (Bench) investment universe consists of excess returns of quintile portfolios formed by expected excess returns using the model 1 characteristics. The augmented (Augment) investment universe adds the excess returns of quintile portfolios formed by expected excess returns using the model 1 characteristics and one of the model 2 characteristics or all the model 2 characteristics. Panels A and B report the summary statistics of the posterior distribution for the unconstrained portfolio strategies and the constrained portfolio strategies, where no short selling is allowed in the risky assets. Panel C reports the sum of the average short positions in the benchmark universe and the augmented universe. The analysis assumes a risk aversion of 3 in the optimal portfolio strategies. Panel A of Table 5 shows that the statistical significance of the average characteristic spreads is weaker for most characteristics in the cross-sectional regressions when only using the largest stocks. It is only momentum, stock issues, profitability, and asset growth characteristics with significant average spreads in the individual cross-sectional regressions at the 10% significance level. The magnitude of the momentum, stock issues, and asset growth spreads are larger than in Table 2 , whereas for size, BM, accruals, and profitability characteristics, the spreads are lower when only including the largest stocks. Using the model 1 characteristics in panel B of Table 5 , only the BM and momentum characteristics have significant positive average spreads. The size spread is tiny and is not statistically significant. The magnitude of the BM spread is lower than in Table 2 but the momentum spread remains similar. The patterns in the size and BM spreads are similar to U.S. stock returns in ) and Fama and French ). The pattern in the momentum spread is similar to Lewellen but Fama and French find that the momentum spread is larger in the biggest stocks. When using all stock characteristics in panel C of Table 5 , the BM, momentum, stock issues, profitability, and asset growth characteristics have significant average spreads. The -statistics on the profitability and asset growth characteristics are below the Harvey et al. ) cut-off -statistic. The magnitude of the average spreads is lower for all the characteristics than in Table 2 except for momentum and stock issues. Most of these patterns are similar to ) and confirm that a number of stock characteristics have smaller average spreads in the largest stocks. In panel A of Table 6 , all three stock characteristics have a significant incremental contribution to the investment opportunity set in the unconstrained portfolio strategies. The mean DSharpe measures are lower when only including the largest stocks compared to Table 3 for the size and BM characteristics. The mean DSharpe measure for the momentum characteristic is marginally higher in the largest stocks. All of the mean DSharpe measures are significant at the 5% percentile. These patterns are consistent with the difference in the spreads of the model 1 characteristics between the largest stocks and all stocks. The optimal portfolios underlying the increase in Sharpe performance do require large short positions as reflected in the average sum of short positions in panel C of Table 5 . The sum of average short positions range between −4.198 (Momentum) and −7.886 (BM). The magnitude of the average short positions is less than observed in Table 3 Imposing no short selling constraints eliminates the incremental contribution of the size and BM characteristics to the investment opportunity set of the largest stocks in panel B of Table 6 . The mean DSharpe measures for the size and BM characteristics are tiny. No short selling constraints substantially reduce the incremental contribution of the momentum characteristic. The mean DSharpe measure of the momentum characteristic is on the borderline of statistical significance. The drop in the mean DSharpe measure of the momentum characteristic is a lot more substantial in the largest stocks compared to Table 3 . This result suggests that no short selling constraints has a greater impact on the incremental contribution of stock characteristics to the investment opportunity set among the largest stocks. When looking at the additional model 2 characteristics in Table 7 , all of the characteristics have a significant incremental contribution to the investment opportunity set when only using the largest companies for the unconstrained portfolio strategies. The mean DSharpe measures range between 0.028 (Asset Growth) and 0.046 (Stock Issues). All of the mean DSharpe measures are significant at the 5% percentile. Using all characteristics together has a significant incremental contribution to the investment opportunity set beyond that contained in the model 1 characteristics. The mean DSharpe measures are a lot lower than in Table 4 , except for the accruals characteristic. The optimal portfolios underlying the increase in Sharpe performance do require large short positions as reflected in the sum of the average short positions in panel C of Table 7 Imposing no short selling constraints eliminates the incremental contribution to the investment opportunity set of the additional characteristics individually and jointly, when using the largest stocks. The mean DSharpe measures are tiny and none are significant at the 5% percentile. This pattern is consistent with Table 6 and suggests that no short selling constraints have a bigger impact on the incremental contribution of stock characteristics to the investment opportunity when using the largest stocks. These results are again consistent with the negative impact no short selling constraints has on the mean-variance performance of trading strategies such as De Roon et al. Li et al. ), and Briere and Szafarz ) among others. 5. Conclusions This study uses the Bayesian approach of ) to examine the incremental contribution of stock characteristics to the investment opportunity set in UK stock returns. There are four main findings in my study. First, I find that all of the stock characteristics, with the exception of the accruals characteristic, have significant characteristic spreads in the Fama and MacBeth ) cross-sectional regressions. The momentum, profitability, and asset growth characteristics have the largest average spreads. A number of the characteristics have -statistics which are larger than the cut-off -statistic of Harvey et al. ). The magnitude of the characteristic spreads for the size, BM, accruals, and profitability characteristics are smaller in the largest stocks, whereas the characteristic spreads are larger for momentum and stock issues characteristics. These patterns in characteristic spreads are in the main similar to Fama and French ) and Second, I find that the size, BM, and momentum characteristics all make a significant incremental contribution to the investment opportunity set when investors are allowed unrestricted short selling when only using the model 1 characteristics. This finding is consistent with Fama and French ). The momentum characteristic makes the largest incremental contribution among the three model 1 characteristics. The incremental contribution of the size and BM characteristics are smaller when only using the largest stocks. The optimal portfolios underlying the increase in Sharpe performance do require large short positions. This higher performance will not be attainable to investors who face no short selling constraints and even where investors can short sell, the costs of short selling could eliminate much of the superior performance ( Fama and French 2015 Third, I find that imposing no short selling constraints eliminates the incremental contribution of the size and BM characteristics. Only the momentum characteristic makes a significant incremental contribution to the investment opportunity set. This finding is similar to Fama and French ) in U.S. stock returns, with the exception that the momentum characteristic has a significant incremental contribution to the investment opportunity set. The mean DSharpe measure on the momentum characteristic is substantially lower in the largest stocks. The impact of no selling constraints is consistent with the impact of no short selling on the mean-variance performance of trading strategies in emerging markets such as De Roon et al. ) and Li et al. ) and factor investment strategies as Briere and Szafarz Fourth, I find that the stock issues, accruals, profitability, and asset growth characteristics all make a significant incremental contribution to the investment opportunity set beyond the model 1 characteristics when there is unrestricted short selling. No short selling constraints eliminate the incremental contribution of the stock issues, accruals, and asset growth characteristics beyond the model 1 characteristics. The profitability characteristic is the only characteristic to make a significant incremental contribution to the investment opportunity set beyond the model 1 characteristics. Using all four characteristics together makes a significant incremental contribution to the investment opportunity set. When only using the largest stocks, none of the additional characteristics either individually or jointly make a significant incremental contribution to the investment opportunity set beyond the model 1 characteristics. This finding is consistent with ) who finds additional characteristics have only a marginal impact on the predictive power of expected returns beyond the model 1 characteristics. My results suggest that no short selling constraints substantially reduce or eliminate the incremental contribution of stock characteristics to the investment opportunity set. As a result, there is little to be gained in using additional stock characteristics beyond the model 1 characteristics in forecasting expected excess returns. My analysis does not address whether the predictive power of stock characteristics is due to risk factors or from behavioral reasons. An interesting extension to my study would be to examine if stock characteristics have significant incremental contribution to the investment opportunity set beyond beta models, including factor models where higher moments are important as in Hung, Shackleton and Xu ( Hung et al. 2004 ), linking in with the recent study by Chordia et al. ). My examination of the impact of no short selling constraints on the incremental contribution of stock characteristics has taken the extreme cases that the investor can either engage in unrestricted short selling or no short selling. It would be interesting to consider the impact of less stringent short selling constraints on the incremental contribution of stock characteristics to the investment opportunity set as in Briere and Szafarz ) such as 130/30 rule, where there is an upper bound of the total weight of short selling in the risky assets of 0.3. My study has focused on the in-sample performance of the portfolio strategies. It would be of interest to extend the analysis to look at out-of-sample performance along the lines of ). I leave an examination of these issues to future research. Conflicts of Interest The author declares no conflict of interest. Almazan, Brown, Carlson, and Chapman ( 2 Almazan et al. 2004 ) find that only a tiny fraction of U.S. mutual funds engage in short selling. Qing and Turner ) present a novel study which examines the impact of stock characteristics in the London market between 1825 and 1870. 4 Using the death event information on the London Share Price Database (LSPD) provided by London Business School. A number of studies examine whether there is mean-variance spanning between two mean-variance frontiers when there is no risk-free asset. De Roon and Nijman ) and Kan and Zhou 5 ( ) provide excellent reviews of alternative tests of mean-variance spanning when unrestricted short selling is allowed. De Roon et al. ) develop the corresponding tests of mean-variance spanning when there are no short selling constraints and transaction costs. Recent applications of the Bayesian approach include Hodrick and Zhang 6 ) and ) in tests of the benefits of international diversification. Basak et al. ) point out using a linear function may lead to a large approximation error when no short selling constraints are imposed. Basak et al find that the standard error of their mean-variance inefficiency measure increases when no short selling constraints are imposed, which is the opposite of ) and Li et al. We can view the normality assumption as a working approximation to monthly excess returns. A non-parametric test along the lines of Ledoit and Wolf 8 ( ) could address this issue in future research. 9 If the optimal portfolios lie on the inefficient side of the mean-variance frontier, I set the corresponding Sharpe performance to zero. 10 Investment trusts are equivalent to U.S. closed-end funds. Fama and French ) examine the same group of characteristics in their study in U.S. stock returns. The predictive ability of the profitability characteristic is highly sensitive to the profitability measure used. Using the alternative profitability measures in Fama and French ) and ), the average spreads can be tiny or even turn significantly negative. 13 ) uses the time-series of the monthly spreads as the set of payoffs to evaluate candidate stochastic discount factor models. This approach is used to examine whether the magnitude of the average spreads are consistent with asset pricing models. See also the related study by Back, Nishad and Ostdiek ( Back et al. 2015 ). This approach can be used to examine whether the predictive ability of stock characteristics can be captured by risk factors. This result stems from the fact that no short selling constraints mitigate the impact of estimation risk in covariance matrix estimators with large sampling error such as the sample covariance 15 matrix. Characteristics Mean Standard Deviation N Excess return 0.687 18.109 1647 Size 10.508 2.075 1832 BM −0.659 1.098 1292 Momentum 0.130 0.507 1422 Stock issues 0.253 0.511 1325 Accruals 0.010 0.762 1158 Profitability 0.354 0.282 1261 Asset growth 0.031 0.408 1394 Panel A: Individual Slope t-Statistic R^2 Size −0.185 −4.09 ^1 0.007 BM 0.231 3.56 ^1 0.007 Momentum 0.888 4.28 ^1 0.009 Stock issues −0.462 −3.72 ^1 0.004 Accruals −0.120 −1.46 0.002 Profitability 0.672 3.76 ^1 0.003 Asset growth −0.849 −5.66 ^1 0.004 Panel B: Model 1 Slope t-Statistic R^2 Size −0.138 −3.42 ^1 0.024 BM 0.324 5.84 ^1 Momentum 1.312 7.06 ^1 Panel C: Model 2 Slope t-Statistic R^2 Size −0.095 −2.37 ^1 0.036 BM 0.394 5.46 ^1 Momentum 1.306 7.30 ^1 Stock issues −0.358 −2.58 ^1 Accruals −0.160 −1.19 Profitability 0.807 5.01 ^1 Asset growth −0.699 −3.96 ^1 Panel A: Unconstrained Mean Standard Deviation 5% Median Momentum 0.164 0.046 0.090 0.162 BM 0.117 0.039 0.057 0.114 Size 0.059 0.025 0.023 0.056 Panel B: Constrained Mean Standard Deviation 5% Median Momentum 0.090 0.034 0.036 0.089 BM 0.010 0.011 0 0.005 Size 0.019 0.019 0 0.014 Panel C Bench Augment Momentum −2.390 −8.829 BM −1.065 −12.863 Size −3.017 −5.348 Panel A: Unconstrained Mean Standard Deviation 5% Median Stock issues 0.073 0.030 0.029 0.071 Accruals 0.030 0.018 0.007 0.026 Profitability 0.092 0.034 0.041 0.089 Asset Growth 0.063 0.028 0.023 0.061 All 0.151 0.043 0.085 0.149 Panel B: Constrained Mean Standard Deviation 5% Median Stock issues 0.012 0.011 0 0.009 Accruals 0.005 0.007 0 0.002 Profitability 0.031 0.015 0.006 0.030 Asset Growth 0.012 0.010 0 0.010 All 0.051 0.022 0.015 0.051 Panel C Bench Augment Stock issues −2.730 −10.119 Accruals −2.722 −4.528 Profitability −2.711 −6.753 Asset Growth −2.728 −10.565 All −2.716 −7.590 Panel A: Individual Slope t-Statistic R^2 Size −0.013 −0.28 0.014 BM 0.088 1.11 0.019 Momentum 1.381 5.02 ^1 0.033 Stock issues −0.829 −4.56 ^1 0.010 Accruals 0.042 0.34 0.005 Profitability 0.364 1.75 ^2 0.009 Asset growth −0.935 −3.97 ^1 0.012 Panel B: Model 1 Slope t-Statistic R^2 Size 0.002 0.049 BM 0.181 2.84 ^1 0.053 Momentum 1.364 5.38 ^1 Panel C: Model 2 Slope t-Statistic R^2 Size −0.025 −0.52 BM 0.288 3.53 ^1 Momentum 1.599 6.14 ^1 Stock issues −0.660 −3.35 ^1 0.074 Accruals −0.078 −0.49 Profitability 0.510 2.62 ^1 Asset growth −0.539 −2.00^1 ^1 Significant at 5%; ^2 Significant at 10%. Panel A: Unconstrained Mean Standard Deviation 5% Median Momentum 0.176 0.048 0.100 0.176 BM 0.071 0.032 0.026 0.067 Size 0.050 0.024 0.017 0.047 Panel B: Constrained Mean Standard Deviation 5% Median Momentum 0.030 0.019 0.001 0.029 BM 0.005 0.007 0 0.002 Size 0.002 0.005 0 0 Panel C Bench Augment Momentum −0.455 −4.918 BM −1.918 −7.886 Size −2.142 −5.100 Table 7. Posterior Distribution of the DSharpe Measure of the Additional Model 2 Characteristics: Large Stocks. Panel A: Unconstrained Mean Standard Deviation 5% Median Stock issues 0.046 0.024 0.013 0.042 Accruals 0.039 0.021 0.011 0.036 Profitability 0.057 0.028 0.016 0.055 Asset Growth 0.028 0.017 0.007 0.025 All 0.045 0.025 0.012 0.042 Panel B: Constrained Mean Standard Deviation 5% Median Stock issues 0.013 0.010 0 0.012 Accruals 0.006 0.007 0 0.003 Profitability 0.016 0.012 0 0.013 Asset Growth 0.001 0.003 0 0 All 0.017 0.013 0 0.015 Panel C Bench Augment Stock issues −2.344 −5.705 Accruals −2.336 −5.378 Profitability −2.340 −3.792 Asset Growth −2.340 −5.608 All −2.342 −4.191 © 2017 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http:// Share and Cite MDPI and ACS Style Fletcher, J. An Empirical Examination of the Incremental Contribution of Stock Characteristics in UK Stock Returns. Int. J. Financial Stud. 2017, 5, 21. https://doi.org/10.3390/ijfs5040021 AMA Style Fletcher J. An Empirical Examination of the Incremental Contribution of Stock Characteristics in UK Stock Returns. International Journal of Financial Studies. 2017; 5(4):21. https://doi.org/10.3390/ Chicago/Turabian Style Fletcher, Jonathan. 2017. "An Empirical Examination of the Incremental Contribution of Stock Characteristics in UK Stock Returns" International Journal of Financial Studies 5, no. 4: 21. https:// Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details Article Metrics	{"url":"https://www.mdpi.com/2227-7072/5/4/21","timestamp":"2024-11-13T02:27:19Z","content_type":"text/html","content_length":"490309","record_id":"<urn:uuid:cc261e7e-ba03-45f0-a65c-450f8932107a>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00364.warc.gz"}
NeurIPS 2019 Sun Dec 8th through Sat the 14th, 2019 at Vancouver Convention Center Reviewer 1 They introduce an algorithm for inferring "probabilistic deterministic finite automatons" from a probabilistic language model oracle that can compute P(next_symbol \| string_so_far), which is exactly what a autoregressive neural language model, like a rnn, would give you. The paper is well-written, the problem is well motivated, and the algorithm appears to be better than the prior work, as measured by the experiments. However I am not qualified to adequately review this paper, because I'm not familiar with any of the related work. For this reason, I lean toward accepting the paper, but I have very little confidence in my assessment. The only problems I have with this paper are with the experiments. They evaluate on SPICE data sets, which appear to be standard data sets for finite state language model learning. However, in looking at the cited SPICE paper, there are actually 16 SPICE data sets, and they only evaluate on 4 of them. Why only these four? Are the other twelve too hard? It would be okay if your algorithm cannot handle the other twelve, provided that the prior work also does poorly on this test cases. Relatedly, what would happen if you apply your algorithm to a language model trained on a large natural language corpus? I imagine that this is an ultimate intended use case of distilling finite state machines from neural language models, so that for example you could use the (finite-state) language model on low power or embedded devices where floating-point operations are at a premium. In a similar vein, state-of-the-art language models for natural language are based on transformers, and as far as I can tell your algorithm would work equally well with a transformer model, or indeed any autoregressive model. This might be an interesting experiment to do, or at least mention as a possibility. "Probabilistic deterministic" finite state machines sounds like an oxymoron, but really what is meant are probabilistic symbol emissions and deterministic state transitions. For readers like me that are not familiar with this literature, a quick sentence in the introduction clarifying this would be helpful (the third paragraph and lines 112-115 make the meaning clear, but it would be helpful if it came a bit earlier). The equation on line 125 might have a typo in it: $\sigma$ appears as a free variable in the left-hand side, but does not occur in the right hand side. Did you mean $... = \frac{P^p(w\sigma)}{P^p(w)}$ ? This would make sense for calculating the probability of $\sigma$ following $w$. Reviewer 2 I think the paper is in a very interesting area but feels unfinished. The paper is reasonably well structured, though some parts felt repetitive (e.g. lines 45-75 in the introduction). Some definitions were difficult to find (typos + small comments are at the end of this section), but overall the descriptions of the background and model were easy to follow. Ultimately, the experiments appear inconclusive. The paper only shows results for SpiCe challenge 0-3, where challenge 1-3 are synthetic examples (simplest) and challenge 0 were not a part of the competition scoring. So, it is unclear how the model will scale. The Tomita experiments were also small and inconclusive. It is also unclear how the RNN hidden size is chosen and how it will affect the WL and WFA algorithms. It would also be helpful to understand how different hyperparameters of the model affect results. The variation tolerance $t$ matters a lot. It would be interesting to understand why t=0.1 is chosen, and what the effect is in altering $t$ for all experiments. The table expansion time and the random seed probably also affects performance. I am curious about some design choices. For example, what is the significance of using total variation vs other distance measures (e.g. KL divergence or cross-entropy)? Also, in the experiments, it’s not clear whether Step 3 (answering equivalence query) is ever used. What should be done if the hypothesis is not accepted? Typos and small comments: * The notation $w^k$ in line 108 should be (I think) $w_{:k}$ * The colon in line 28 should be a comma * I didn’t see an explanation for what the contents of the observation table $O$ is -- with the exception of lines 182-183, which states that the observations are tokens and not distributions. But the following line 184, along with the rest of the paper, discusses the comparison between next-token distributions. (Are you using RNN prediction probabilities for comparison or the empirical probabilities from the samples?) * The notation $q^i$ doesn’t appear to be defined. I assume it is referring to the initial state. Maybe use the notation $q_0$ instead, to since $i$ is often used as an index? * In line 125, the definition of the last token probability, the equation should be $\frac{P^p(w \cdot s)}{P^p(w)} * In line 193, “prefix weight” does not seem to be defined * The Tomita experiment results were difficult to find. Maybe a subsection heading or a simple figure would help Reviewer 3 This paper presents a technique for constructing a probabilistic deterministic finite automaton (PDFA) that can model a black-box language model such as LM-RNN. The main idea of the algorithm is to adapt Angulin’s L* algorithm to handle probabilistic choices with unbounded states of an LM-RNN by developing a notion of variation tolerance. The variation tolerance allows for comparing two probability vectors and clustering them if they are within a t-threshold. The goal is to construct a PDFA, such that for all prefixes the next token-distributions in the PDFA and the LM-RNN is within the variation bound. The paper presents analogous variation tolerance aware extensions to membership and equivalence queries in L. The algorithm first learns an observation table that is closed and consistent, then constructs the corresponding PDFA using a clustering strategy, and finally performs an equivalence query using a sampling-based method. The technique is evaluated on grammars from the SPiCe competition and adaptations of Tomita grammars, and it outperforms n-gram and WFA (using spectral learning) baselines both in terms of WER and NDCG rates. This paper presents a novel general technique to learn weighted deterministic finite automaton (WDFA) from a given weighted black-box target, and learns a PDFA for a stochastic target. Unlike previous approaches that use spectral learning or learn from sampled sequences, the presented technique learns PDFA in an active learning setting using an oracle by extending the widely used L algorithm, which has some nice guarantees. The idea of using t-consistency for computing variations between probability vectors is quite elegant, and the idea of using clustering techniques to overcome non-transitivity of t-equality is also quite interesting. There is also a detailed formal treatment of the properties and guarantees of the returned PDFA in terms of t-consistency. The paper is also nicely written and explains the key ideas of the algorithm in required details. One suggestion would be to add a small running example that might help clarify the extended L* algorithm a bit more (especially the step 2 of PDFA construction). It would also be nice to move at least the two theorems from appendix to the main text. I was curious about the scalability of the algorithm. It seems it scales better than spectral learning based WFA learning methods based on SPiCe and Tomita benchmarks. But the LSTM network consists of an input dimension of 10 and hidden dimension of 20, which seems quite small compared to current state-of-the-art LSTM language models. What is the maximum hidden sizes the technique can handle currently? Similarly, the alphabet size of upto 20 also seems a bit small compared to typical large vocabularies of state-of-the-art language models. It would be helpful to report the maximum sizes the technique can scale to currently, and that might help spur interesting future works to scale up the technique further. It wasn’t clear whether for the results reported in Table 1, did the L* algorithm terminate before early stopping. If it didn’t terminate, what bounds on expansion time and suffix limits might be needed for full completion of L* algorithm on these benchmarks. Since the equivalence check is being performed using sampling, the only difference between previous PDFA learning methods from samples would be that in this case the samples are being collected actively. Would it be possible to compare the presented L* based technique to also some of those previous PDFA reconstruction techniques that learn from samples? Minor issues: line 108: w^k \in S -> w_{:k} \in S line 165: algorithm reduce the number -> algorithm reduces the number line 193: prefix weight: -> prefix weight. line 336: The the -> The	{"url":"https://proceedings.neurips.cc/paper_files/paper/2019/file/d3f93e7766e8e1b7ef66dfdd9a8be93b-Reviews.html","timestamp":"2024-11-05T11:56:12Z","content_type":"text/html","content_length":"10411","record_id":"<urn:uuid:b494dee2-7da2-4579-830c-c488ecd6681f>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00749.warc.gz"}
Excel SEC Function The Excel SEC Function The Secant is the reciprocal of the Cosine. Therefore, for the right-angled triangle below, the secant of the angle θ is equal to the hypotenuse, h, divided by the adjacent side, a. I.e. for the triangle above, The trig. ratios are discussed further on the Wikipedia Trigonometric Ratios Page Function Description The Excel Sec function calculates the secant of a given angle. Note: the Sec function was only introduced in Excel 2013 and so is not available in earlier versions of Excel. The syntax of the function is: SEC( number ) Where the number argument is the angle (in radians) that you want to calculate the secant of. This must be between -2^27 and +2^27. Converting from Degrees to Radians If your angle is in degrees, you will need to convert it into radians before supplying it to the Sec function. This can be done using the Excel Radians function: =RADIANS( degrees ) An example of this is given below. Excel Sec Function Examples The following spreadsheet shows the Excel Sec function, used to calculate the Secant of four different angles: Formulas: Results: A A 1 =SEC( -3.14159265358979 ) 1 -1 2 =SEC( 0 ) 2 1 3 =SEC( PI() / 4 ) 3 1.414213562 4 =SEC( RADIANS( 45 ) ) 4 1.414213562 Note that, in the examples above: • In cell A3, the Excel Pi function is used to provide the value π/4 to the function; • In cell A4, the Excel Radians function is used to convert the angle of 45 degrees into radians before it is supplied to the Sec function. For further details and examples of the Excel Sec function, see the Microsoft Office website. Sec Function Errors If you get an error from the Excel Sec function, this is likely to be one of the following: Common Errors #NUM! - Occurs if the supplied number is less than -2^27 or is greater than 2^27. #VALUE! - Occurs if the supplied number is not recognised as a numeric value.	{"url":"https://www.excelfunctions.net/excel-sec-function.html","timestamp":"2024-11-02T22:23:35Z","content_type":"text/html","content_length":"15936","record_id":"<urn:uuid:de805eb9-113a-4a99-bd0d-46893527c415>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00682.warc.gz"}
University of Oulu. University of Oulu. All self-conformal measures are uniform scaling Ergodic Theory and Dynamical Systems Seminar 3rd March 2022, 2:00 pm – 3:00 pm Online via Zoom, Zoom Scaling scenery is a useful tool for studying fine-structure properties of fractal objects. Measures for which the scaling scenery flows satisfy a strong statistical property, called uniform scaling, appear to be geometrically much more regular than general measures. Specific to self-conformal measures, the uniform scaling property has been proved to hold under the weak separation condition. In my talk, I will present a recent work, jointly with B.Barany, A.Kaenmaki and A.Pyorala, where we show that every self-conformal measure is uniform scaling, without assuming any separation conditions. If time permits, I will also present some applications of this result, for example, on prevalence of normal numbers in fractal sets.	{"url":"https://www.bristolmathsresearch.org/seminar/meng-wu/","timestamp":"2024-11-10T13:11:02Z","content_type":"text/html","content_length":"54369","record_id":"<urn:uuid:dc9bc44b-95d3-4adb-9df6-3be1e1e303a0>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00038.warc.gz"}
Dash (US) to Cubic Fathom Dash (US) to Cubic Fathom Converter Enter Dash (US) Cubic Fathom ⇅ Switch toCubic Fathom to Dash (US) Converter How to use this Dash (US) to Cubic Fathom Converter 🤔 Follow these steps to convert given volume from the units of Dash (US) to the units of Cubic Fathom. 1. Enter the input Dash (US) value in the text field. 2. The calculator converts the given Dash (US) into Cubic Fathom in realtime ⌚ using the conversion formula, and displays under the Cubic Fathom label. You do not need to click any button. If the input changes, Cubic Fathom value is re-calculated, just like that. 3. You may copy the resulting Cubic Fathom value using the Copy button. 4. To view a detailed step by step calculation of the conversion, click on the View Calculation button. 5. You can also reset the input by clicking on button present below the input field. What is the Formula to convert Dash (US) to Cubic Fathom? The formula to convert given volume from Dash (US) to Cubic Fathom is: Volume[(Cubic Fathom)] = Volume[(Dash (US))] × 5.036551602419757e-8 Substitute the given value of volume in dash (us), i.e., Volume[(Dash (US))] in the above formula and simplify the right-hand side value. The resulting value is the volume in cubic fathom, i.e., Volume[(Cubic Fathom)]. Calculation will be done after you enter a valid input. Consider that a recipe specifies a dash (US) of vanilla extract. Convert this quantity from dash (US) to Cubic Fathom. The volume in dash (us) is: Volume[(Dash (US))] = 1 The formula to convert volume from dash (us) to cubic fathom is: Volume[(Cubic Fathom)] = Volume[(Dash (US))] × 5.036551602419757e-8 Substitute given weight Volume[(Dash (US))] = 1 in the above formula. Volume[(Cubic Fathom)] = 1 × 5.036551602419757e-8 Volume[(Cubic Fathom)] = 5.037e-8 Final Answer: Therefore, 1 is equal to 5.037e-8 cu fm. The volume is 5.037e-8 cu fm, in cubic fathom. Consider that a sauce recipe suggests adding three dashes (US) of hot sauce. Convert this quantity from dash (US) to Cubic Fathom. The volume in dash (us) is: Volume[(Dash (US))] = 3 The formula to convert volume from dash (us) to cubic fathom is: Volume[(Cubic Fathom)] = Volume[(Dash (US))] × 5.036551602419757e-8 Substitute given weight Volume[(Dash (US))] = 3 in the above formula. Volume[(Cubic Fathom)] = 3 × 5.036551602419757e-8 Volume[(Cubic Fathom)] = 1.511e-7 Final Answer: Therefore, 3 is equal to 1.511e-7 cu fm. The volume is 1.511e-7 cu fm, in cubic fathom. Dash (US) to Cubic Fathom Conversion Table The following table gives some of the most used conversions from Dash (US) to Cubic Fathom. Dash (US) () Cubic Fathom (cu fm) 0.01 5e-10 cu fm 0.1 5.04e-9 cu fm 1 5.037e-8 cu fm 2 1.0073e-7 cu fm 3 1.511e-7 cu fm 4 2.0146e-7 cu fm 5 2.5183e-7 cu fm 6 3.0219e-7 cu fm 7 3.5256e-7 cu fm 8 4.0292e-7 cu fm 9 4.5329e-7 cu fm 10 5.0366e-7 cu fm 20 0.00000100731 cu fm 50 0.00000251828 cu fm 100 0.00000503655 cu fm 1000 0.00005036552 cu fm Dash (US) The US dash is a unit of measurement used to quantify very small volumes, commonly applied in cooking and medicine. It is defined as 1/8 of a teaspoon, making it a precise measure for adding tiny amounts of liquid or powder. Originating from the US customary system, the dash provides a standardized way to ensure accuracy in recipes and medicinal dosages. Today, it is used in various contexts where precision in small quantities is important, such as in culinary practices for seasoning and in medicine for exact dosing. Cubic Fathom The cubic fathom is a unit of measurement used to quantify three-dimensional volumes, particularly in maritime and construction contexts. Originating from the fathom, a traditional unit of length used in navigation and depth measurement, the cubic fathom provides a standardized way to measure volume. Historically, it was used in maritime settings to measure the volume of cargo holds and other spaces on ships. Today, while less commonly used, it still finds application in specific industries where its historical relevance and practical utility are recognized. Frequently Asked Questions (FAQs) 1. What is the formula for converting Dash (US) to Cubic Fathom in Volume? The formula to convert Dash (US) to Cubic Fathom in Volume is: Dash (US) * 5.036551602419757e-8 2. Is this tool free or paid? This Volume conversion tool, which converts Dash (US) to Cubic Fathom, is completely free to use. 3. How do I convert Volume from Dash (US) to Cubic Fathom? To convert Volume from Dash (US) to Cubic Fathom, you can use the following formula: Dash (US) * 5.036551602419757e-8 For example, if you have a value in Dash (US), you substitute that value in place of Dash (US) in the above formula, and solve the mathematical expression to get the equivalent value in Cubic Fathom.	{"url":"https://convertonline.org/unit/?convert=dash_us-cubic_fathom","timestamp":"2024-11-03T06:22:45Z","content_type":"text/html","content_length":"93229","record_id":"<urn:uuid:86bfe8c1-f8f6-4dea-8a32-358eebde2044>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00885.warc.gz"}
1,611 research outputs found A generalized version of the rotating-wave approximation for the single-mode spin-boson Hamiltonian is presented. It is shown that performing a simple change of basis prior to eliminating the off-resonant terms results in a significantly more accurate expression for the energy levels of the system. The generalized approximation works for all values of the coupling strength and for a wide range of detuning values, and may find applications in solid-state experiments.Comment: 4 pages, 2 figs, REVTeX The canonical ensemble describes an open system in equilibrium with a heat bath of fixed temperature. The probability distribution of such a system, the Boltzmann distribution, is derived from the uniform probability distribution of the closed universe consisting of the open system and the heat bath, by taking the limit where the heat bath is much larger than the system of interest. Alternatively, the Boltzmann distribution can be derived from the Maximum Entropy Principle, where the Gibbs-Shannon entropy is maximized under the constraint that the mean energy of the open system is fixed. To make the connection between these two apparently distinct methods for deriving the Boltzmann distribution, it is first shown that the uniform distribution for a microcanonical distribution is obtained from the Maximum Entropy Principle applied to a closed system. Then I show that the target function in the Maximum Entropy Principle for the open system, is obtained by partial maximization of Gibbs-Shannon entropy of the closed universe over the microstate probability distributions of the heat bath. Thus, microcanonical origin of the Entropy Maximization procedure for an open system, is established in a rigorous manner, showing the equivalence between apparently two distinct approaches for deriving the Boltzmann distribution. By extending the mathematical formalism to dynamical paths, the result may also provide an alternative justification for the principle of path entropy maximization as well.Comment: 12 pages, no figur The maximum entropy approach operating with quite general entropy measure and constraint is considered. It is demonstrated that for a conditional or parametrized probability distribution $f(x\|\mu)$ there is a "universal" relation among the entropy rate and the functions appearing in the constraint. It is shown that the recently proposed variational formulation of the entropic functional can be obtained as a consequence of this relation, that is from the maximum entropy principle. This resolves certain puzzling points appeared in the variational approach Quite unexpectedly, kinetic theory is found to specify the correct definition of average value to be employed in nonextensive statistical mechanics. It is shown that the normal average is consistent with the generalized Stosszahlansatz (i.e., molecular chaos hypothesis) and the associated H-theorem, whereas the q-average widely used in the relevant literature is not. In the course of the analysis, the distributions with finite cut-off factors are rigorously treated. Accordingly, the formulation of nonextensive statistical mechanics is amended based on the normal average. In addition, the Shore-Johnson theorem, which supports the use of the q-average, is carefully reexamined, and it is found that one of the axioms may not be appropriate for systems to be treated within the framework of nonextensive statistical mechanics.Comment: 22 pages, no figures. Accepted for publication in Phys. Rev. We propose a method of manipulating selectively the symmetric Dicke subspace in the internal degrees of freedom of N trapped ions. We show that the direct access to ionic-motional subspaces, based on a suitable tuning of motion-dependent AC Stark shifts, induces a two-level dynamics involving previously selected ionic Dicke states. In this manner, it is possible to produce, sequentially and unitarily, ionic Dicke states with increasing excitation number. Moreover, we propose a probabilistic technique to produce directly any ionic Dicke state assuming suitable initial conditions.Comment: 5 pages and 1 figure. New version with minor changes and added references. Accepted in Physical Review We compare the Infrared Dirac-Born-Infeld (IR DBI) brane inflation model to observations using a Bayesian analysis. The current data cannot distinguish it from the \LambdaCDM model, but is able to give interesting constraints on various microscopic parameters including the mass of the brane moduli potential, the fundamental string scale, the charge or warp factor of throats, and the number of the mobile branes. We quantify some distinctive testable predictions with stringy signatures, such as the large non-Gaussianity, and the large, but regional, running of the spectral index. These results illustrate how we may be able to probe aspects of string theory using cosmological observations.Comment: 54 pages, 13 figures. v2: non-Gaussianity constraint has been applied to the model; parameter constraints have tightened significantly, conclusions unchanged. References added; v3, minor revision, PRD versio Just as transition rates in a canonical ensemble must respect the principle of detailed balance, constraints exist on transition rates in driven steady states. I derive those constraints, by maximum information-entropy inference, and apply them to the steady states of driven diffusion and a sheared lattice fluid. The resulting ensemble can potentially explain nonequilibrium phase behaviour and, for steady shear, gives rise to stress-mediated long-range interactions.Comment: 4 pages. To appear in Physical Review Letter We generalize the Shannon's information theory in a nonadditive way by focusing on the source coding theorem. The nonadditive information content we adopted is consistent with the concept of the form invariance structure of the nonextensive entropy. Some general properties of the nonadditive information entropy are studied, in addition, the relation between the nonadditivity $q$ and the codeword length is pointed out.Comment: 9 pages, no figures, RevTex, accepted for publication in Phys. Rev. E(an error in proof of theorem 1 was corrected, typos corrected The main goal of this paper is to extend and apply the principle of maximum entropy (MaxEnt) to incomplete quantum process estimation tasks. We will define a so-called process entropy function being the von Neumann entropy of the state associated with the quantum process via Choi-Jamiolkowski isomorphism. It will be shown that an arbitrary process estimation experiment can be reformulated in a unified framework and MaxEnt principle can be consistently exploited. We will argue that the suggested choice for the process entropy satisfies natural list of properties and it reduces to the state MaxEnt principle, if applied to preparator devices.Comment: 8 pages, comments welcome, references adde At equilibrium, a fluid element, within a larger heat bath, receives random impulses from the bath. Those impulses, which induce stochastic transitions in the system (the fluid element), respect the principle of detailed balance, because the bath is also at equilibrium. Under continuous shear, the fluid element adopts a non-equilibrium steady state. Because the surrounding bath of fluid under shear is also in a non-equilibrium steady state, the system receives stochastic impulses with a non-equilibrium distribution. Those impulses no longer respect detailed balance, but are nevertheless constrained by rules. The rules in question, which are applicable to a wide sub-class of driven steady states, were recently derived [R. M. L. Evans, Phys. Rev. Lett. {\bf 92}, 150601 (2004); J. Phys. A: Math. Gen. {\bf 38}, 293 (2005)] using information-theoretic arguments. In the present paper, we provide a more fundamental derivation, based on the uncontroversial, non-Bayesian interpretation of probabilities as simple ratios of countable quantities. We apply the results to some simple models of interacting particles, to investigate the nature of forces that are mediated by a non-equilibrium noise-source such as a fluid under shear.Comment: 14 pages, 7 figure	{"url":"https://core.ac.uk/search/?q=author%3A(Jaynes%20E%20T)","timestamp":"2024-11-08T19:38:07Z","content_type":"text/html","content_length":"148448","record_id":"<urn:uuid:b60081dd-2295-46b1-8356-4a567e38552f>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00544.warc.gz"}
Significant Liquid Structures - Modeling the Structure of Liquids: The Physical Model Approach - Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013) Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013) 8. Modeling the Structure of Liquids: The Physical Model Approach 8.4. Significant Liquid Structures* In the approach, which was advocated by Eyring and denoted by him as significant liquid structure theory^6) (here labeled as SLS theory), it is argued that a liquid behaves in many cases as an intermediate between a gas and a solid. From the fact that a simple liquid like Ar expands ∼12% upon melting and the associated large increase in fluidity, it can be concluded that the coordination number decreases, while simultaneously introducing some free volume in the liquid. Since XRD measurement have indicated that the intermolecular distances in the liquid are essentially the same as in the solid, the resulting structure is assumed to show some heterogeneity, containing a fraction of solid-like molecules and another fraction of holes. Holes of molecular size are favored because smaller holes cannot provide easy access to the entering molecules, which limits the increase of entropy. In contrast, larger holes will require excessive energy without a compensating increase in entropy. These holes change by one or more degrees of freedom for a molecule surrounding the hole from vibration to translation, and these molecules thus act like vapor molecules. Hence, the concept does not imply that a liquid is a mixture of a solid and a gas. Rather, a molecule has solid-like properties for the time it vibrates around an equilibrium position, but transforms instantaneously to a gas-like behavior for one or more degrees of freedom if it jumps into a neighboring hole. A simple schematic of this is process shown in Figure 8.5. Figure 8.5 A schematic illustrating a liquid formed by removing at random molecules from a (glassy) solid. The holes so created mirror the molecules in the gas phase. According to the law of rectilinear diameters (see Chapter 4), the average density of a liquid and its associated equilibrium vapor pressure is a linear function of temperature (roughly equal to ∼½ρ [solid] at T[mp] to ∼⅓ρ[solid] at T[cri]). A small decrease in temperature is expected due to thermal expansion, but this law indicates that the number density of molecules in the gas phase is about equal to the number density of vacancies in the liquid. In a simple nearest-neighbor model, the energy of vaporization per molecule is approximately ½zϕ, where z is the coordination number and ϕ is the bond energy. In a similar way it requires energy zϕ to create a hole in the liquid. The molecule removed from the interior of the liquid to the vapor can be added to the surface of the liquid, thereby regaining the energy ½zϕ, so leaving a hole costs only the energy of vaporization. Hence, a vacancy is expected to move as freely as a molecule in the gas phase and to have about the same energy and entropy, thus explaining the law of rectilinear diameters. So, the thermodynamic behavior in this model is described by varying the fraction of gas-like and solid-like volume elements, while the overall thermodynamic behavior is given by the sum of the contributions of these elements. In the simplest form for gas-like regions the ideal gas model is used, whereas for solid-like elements the Einstein model (see Sections 5.3 and 6.1) is used. The crux is, obviously, to determine the fractions of gas-like and solid-like regions. Since the holes are on average of molecular size, in one mole of liquid (V − V[s])/V[s] moles of holes are present, where V and V[s] are the molar volumes of the liquid and solid, respectively. Assuming complete randomness, the fraction of neighboring positions filled with molecules to with a hole is V[s]/V. It thus follows that for N molecules there are gas-like molecules and a remainder of N[s] ≡ NV[s]/V solid-like molecules. According to this model, the heat capacity C[V] of a mole of, say Ar, is given by sum of the contributions of the V[s]/V moles of solid and (V − V[s])/V[ ]moles of gas. Therefore for which the good agreement with experiment is shown in Figure 8.6. Figure 8.6 The experimental heat capacity C[V] for liquid argon (circles) and the prediction according to Eq. (8.61) (solid line) using C[V][,solid] = 3R = 25.0 J mol^−1 K^−1 or 6.0 cal mol^−1 K^−1 and C[V][,gas] = 3R/2 = 12.5 J mol^−1 K^−1 or 3.0 cal mol^−1 K^−1. Using the above concepts, the partition function Z[N] can be written as where Z[s] and Z[g] represent the partition functions of the solid-like and gas-like molecules, respectively. For calculating Z[s] we consider that holes surrounding a solid-like molecule introduce positional degeneracy. The number of additional accessible sites for a solid-like molecule is the number of holes around that molecule multiplied by the probability that this molecule has the required energy ε[h] to be able to jump to these holes. The number of holes n[h] is proportional to the excess volume and given by with n as proportionality factor. The energy ε[h] should be inversely proportional to the excess volume and directly proportional to the energy of sublimation E[s]. Therefore with a as proportionality factor. Since the parameter ε[h] → ∞ for V → V[s] and ε[h] → 0 for V → ∞, this introduces cooperative behavior in theory. The total number of positions available, that is, original plus additional, to the solid-like molecule becomes and Z[s] becomes (using As noted, for θ as the Einstein temperature, reads The factor exp(−ϕ/2kT) takes into account that in a simple nearest-neighbor model −ϕ/2 is the potential energy of the solid-like, vibrating molecule and we interpret E[s] ≡ −½(ϕ + 3kθ) as the sublimation energy. For Z[g] we use straightforwardly the ideal gas expression with excess volume V − V[s] and thermal wavelength Λ = (h^2/2πmkT)^½, that is, Hence, according to Eq. (8.62) we have for the partition function Z[N] of the liquid While the parameters E[s], θ and V[s] have to be taken from solid-state data, either theoretical or experimental, the parameters a and n can be evaluated theoretically. To calculate n, we use the molar volume V[mp] at the melting point T[mp] and consider that, close to T[mp] and using the lattice coordination number z, n[h] is given by Comparison with Eq. (8.63) yields To calculate a, we consider that a solid-like molecule has a kinetic energy 3kT/2. If a molecule is to pre-empt a neighboring position in addition to its original position, it must have additional kinetic energy equal to or in excess of that which the other (n − 1) neighboring molecules would otherwise introduce into this hole. If the average molecule divides its time equally between two neighboring sites, its energy density will be halved. As the molecule will be moving for (1/z)th of its time in the direction of any neighbor, the average kinetic energy of the (n − 1) ordinary molecules will provide a hole with kinetic energy ½(3kT/2)(n − 1)/z. This is also the value that ε[h] = aE[s]V[s]/(V − V[s]) must have at the melting temperature T[mp], that is, Since at T[mp] it holds that T[mp]S[mp] = E[mp], and the entropy of melting per molecule S[mp] for a simple liquid is about 3k/2, we obtain E[mp] = 3kT[mp]/2 for the energy of melting E[mp]. Moreover, since during melting holes are introduced in the solid, essentially accompanied by a potential energy change only, we also have Equations (8.71) and (8.74) result in a = 0.00537 and n = 10.7 for Ar, using z = 12 and the data given in Table 8.3. The best empirical fit [21] yields a = 0.00534 and n = 10.8, so that there is good agreement. Some further data for Ar, Kr and Xe shown in Table 8.3 also demonstrate a good agreement with experimental data. Table 8.3 Vapor pressure P, molar volume V and entropy of vaporization Δ[vap]S for the noble gases Ar, Kr and Xe, and the parameters used. Using the same parameters as for the LJD theory, the triple point for Ar is predicted to be 0.711ε/k, compared with 0.701ε/k according to LJD theory (Table 8.1). Predictions for V, U and S of Ar according to SLS and LJD theory are also shown in Table 8.1. The critical constants predicted are (LJD data in brackets) to be compared with the “best” experimental corresponding states estimates^7) Overall, we conclude that for describing liquids SLS theory is better than LJD theory, which resembles more a superheated solid than a liquid. Some other predictions of SLS theory, which also show good agreement with experimental data, are given in Figure 8.7. For molecules that rotate freely in the liquid phase, such as CH[4] and N[2], Figure 8.7 shows the good agreement with experimental data. However, for those molecules that do not rotate freely in the liquid phase, such as Cl[2], some changes in Z[s] must be made, for example, treating the rotation in the solid-like part as a vibration. In that case, a good agreement with experimental data was also reached. For example, for the temperature range of 180 K to 240 K the heat capacity at constant pressure C[P], as a calculated from C[P] = C[V] − TVα^2β with C[V] the heat capacity at constant volume, α = (1/V)(∂V/∂T)[P] the expansivity and β = −(1/V)(∂V/∂P)[T] the compressibility, all calculated from Z[N], differ by only ∼2% from the experimental values. The method has been applied to molten metals, molten salts and quantum liquids such as H[2] and Ne, with fairly good results. Application of the theory to many liquids and phenomena has been reviewed by Eyring and Jhon [23]. Figure 8.7 SLS theory results. (a) The logarithm of the reduced vapor pressure P* for simple liquids versus reciprocal reduced temperature 1/T; (b) The reduced density versus 1/V versus reduced temperature T* for SLS theory (T = (ε/k)T, V = V(N/σ^3) and P = (ε/σ^3)P*). From the previous data and figures, it is concluded that quite acceptable agreement with experiment can be obtained. Nevertheless, the basis of SLS theory is not really clear and the approach has essentially been abandoned. Although the model has been rather successful in estimating thermodynamic properties, a basic feature of liquids – the pair correlation function – is largely absent. Although at a late stage of the development of this theory attempts were made [24] to introduce the pair correlation function, the procedure used must be characterized as artificial. Problem 8.8 Show that the partition function of the significant liquid theory reduces to the ideal gas and solid-state partition functions if V → ∞ and V → 0, respectively. Problem 8.9 Calculate the vapor pressure from significant liquid structures theory for atomic fluids using the ideal gas and Einstein partition functions for the gas-like and solid-like phases, respectively. Problem 8.10 Is it possible to calculate the RDF for the significant liquid structures theory without further approximations of additions?	{"url":"https://schoolbag.info/chemistry/liquid-state/43.html","timestamp":"2024-11-13T22:06:22Z","content_type":"text/html","content_length":"26699","record_id":"<urn:uuid:0e2f3e0d-adf6-4022-a37d-ca79c21a997d>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00333.warc.gz"}
Non-cryptographic fault-tolerant computing in a constant number of rounds of interaction Let f(x[1], ..., x[n]) be computed by a circuit C with bounded fanin. There are non-cryptographic protocols by which a network of n processors can evaluate C at secret inputs x[1], ..., x[n], revealing the final value f(x[1], ..., x[n]) without revealing any information about the inputs except what the final result provides. Current methods require O(depth(C)) rounds of communication and messages of size polynomial in size(C) and n. In practical terms, such a degree of interaction is unacceptable. We show how to secretly evaluate any finite function in a constant expected number of rounds, regardless of the minimal depth of a circuit for that function. We provide a means to simulate unbounded fanin multiplicative (or AND) gates using constant rounds. Using our new methods, any function can be evaluated in a constant number of rounds, using messages of size proportional to the size of a constant-depth, unbounded-fanin circuit describing the function. We also show how to secretly evaluate any function described by an algebraic formula of polynomial size (or an NC^1 circuit), using a constant number of rounds yet requiring messages of only polynomial size. This provides a speedup over original methods by a factor of log n, while incurring only a polynomial number of bits. Original language English Title of host publication Proc Eighth ACM Symp Princ Distrib Comput Publisher Publ by ACM Pages 201-209 Number of pages 9 ISBN (Print) 0897913264 State Published - 1989 Externally published Yes Event Proceedings of the Eighth Annual ACM Symposium on Principles of Distributed Computing - Edmonton, Alberta, Can Duration: 14 Aug 1989 → 16 Aug 1989 Publication series Name Proceedings of the Annual ACM Symposium on Principles of Distributed Computing Conference Proceedings of the Eighth Annual ACM Symposium on Principles of Distributed Computing City Edmonton, Alberta, Can Period 14/08/89 → 16/08/89 Dive into the research topics of 'Non-cryptographic fault-tolerant computing in a constant number of rounds of interaction'. Together they form a unique fingerprint.	{"url":"https://cris.biu.ac.il/en/publications/non-cryptographic-fault-tolerant-computing-in-a-constant-number-o","timestamp":"2024-11-04T14:09:41Z","content_type":"text/html","content_length":"54962","record_id":"<urn:uuid:1b78346a-5ebc-4058-b75e-406d790f031c>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00613.warc.gz"}
ω-consistent theory ω-consistent theory In mathematical logic, an ω-consistent (or omega-consistent, also called numerically segregative[1]) theory is a theory (collection of sentences) that is not only (syntactically) consistent (that is, does not prove a contradiction), but also avoids proving certain infinite combinations of sentences that are intuitively contradictory. The name is due to Kurt Gödel, who introduced the concept in the course of proving the incompleteness theorem.[2] A theory T is said to interpret the language of arithmetic if there is a translation of formulas of arithmetic into the language of T so that T is able to prove the basic axioms of the natural numbers under this translation. A T that interprets arithmetic is ω-inconsistent if, for some property P of natural numbers (defined by a formula in the language of T), T proves P(0), P(1), P(2), and so on (that is, for every standard natural number n, T proves that P(n) holds), but T also proves that there is some (necessarily nonstandard) natural number n such that P(n) fails. This may not lead directly to an outright contradiction, because T may not be able to prove for any specific value of n that P(n) fails, only that there is such an n. T is ω-consistent if it is not ω-inconsistent. There is a weaker but closely related property of Σ[1]-soundness. A theory T is Σ[1]-sound (or 1-consistent, in another terminology) if every Σ^0[1]-sentence^[3] provable in T is true in the standard model of arithmetic N (i.e., the structure of the usual natural numbers with addition and multiplication). If T is strong enough to formalize a reasonable model of computation, Σ[1]-soundness is equivalent to demanding that whenever T proves that a computer program C halts, then C actually halts. Every ω-consistent theory is Σ[1]-sound, but not vice versa. More generally, we can define an analogous concept for higher levels of the arithmetical hierarchy. If Γ is a set of arithmetical sentences (typically Σ^0[n] for some n), a theory T is Γ-sound if every Γ-sentence provable in T is true in the standard model. When Γ is the set of all arithmetical formulas, Γ-soundness is called just (arithmetical) soundness. If the language of T consists only of the language of arithmetic (as opposed to, for example, set theory), then a sound system is one whose model can be thought of as the set ω, the usual set of mathematical natural numbers. The case of general T is different, see ω-logic below. Σ[n]-soundness has the following computational interpretation: if the theory proves that a program C using a Σ[n−1]-oracle halts, then C actually halts. Consistent, ω-inconsistent theories Write PA for the theory Peano arithmetic, and Con(PA) for the statement of arithmetic that formalizes the claim "PA is consistent". Con(PA) could be of the form "For every natural number n, n is not the Gödel number of a proof from PA that 0=1". (This formulation uses 0=1 instead of a direct contradiction; that gives the same result, because PA certainly proves ¬0=1, so if it proved 0=1 as well we would have a contradiction, and on the other hand, if PA proves a contradiction, then it proves anything, including 0=1.) Now, assuming PA is really consistent, it follows that PA + ¬Con(PA) is also consistent, for if it were not, then PA would prove Con(PA) (since an inconsistent theory proves every sentence), contradicting Gödel's second incompleteness theorem. However, PA + ¬Con(PA) is not ω-consistent. This is because, for any particular natural number n, PA + ¬Con(PA) proves that n is not the Gödel number of a proof that 0=1 (PA itself proves that fact; the extra assumption ¬Con(PA) is not needed). However, PA + ¬Con(PA) proves that, for some natural number n, n is the Gödel number of such a proof (this is just a direct restatement of the claim ¬Con(PA) ). In this example, the axiom ¬Con(PA) is Σ[1], hence the system PA + ¬Con(PA) is in fact Σ[1]-unsound, not just ω-inconsistent. Arithmetically sound, ω-inconsistent theories Let T be PA together with the axioms c ≠ n for each natural number n, where c is a new constant added to the language. Then T is arithmetically sound (as any nonstandard model of PA can be expanded to a model of T), but ω-inconsistent (as it proves $ \exists x\,c=x $, and c ≠ n for every number n). Σ[1]-sound ω-inconsistent theories using only the language of arithmetic can be constructed as follows. Let IΣ[n] be the subtheory of PA with the induction schema restricted to Σ[n]-formulas, for any n > 0. The theory IΣ[n + 1] is finitely axiomatizable, let thus A be its single axiom, and consider the theory T = IΣ[n] + ¬A. We can assume that A is an instance of the induction schema, which has the form $ \forall w\,[B(0,w)\land\forall x\,(B(x,w)\to B(x+1,w))\to\forall x\,B(x,w)]. $ If we denote the formula $ \forall w\,[B(0,w)\land\forall x\,(B(x,w)\to B(x+1,w))\to B(n,w)]$ by P(n), then for every natural number n, the theory T (actually, even the pure predicate calculus) proves P(n). On the other hand, T proves the formula \exists x\,\neg P(x), because it is logically equivalent to the axiom ¬A. Therefore T is ω-inconsistent. It is possible to show that T is Π[n + 3]-sound. In fact, it is Π[n + 3]-conservative over the (obviously sound) theory IΣ[n]. The argument is more complicated (it relies on the provability of the Σ[ n + 2]-reflection principle for IΣ[n] in IΣ[n + 1]). Arithmetically unsound, ω-consistent theories Let ω-Con(PA) be the arithmetical sentence formalizing the statement "PA is ω-consistent". Then the theory PA + ¬ω-Con(PA) is unsound (Σ[3]-unsound, to be precise), but ω-consistent. The argument is similar to the first example: a suitable version of the Hilbert-Bernays-Löb derivability conditions holds for the "provability predicate" ω-Prov(A) = ¬ω-Con(PA + ¬A), hence it satisfies an analogue of Gödel's second incompleteness theorem. Not to be confused with Ω-logic. The concept of theories of arithmetic whose integers are the true mathematical integers is captured by ω-logic.[4] Let T be a theory in a countable language which includes a unary predicate symbol N intended to hold just of the natural numbers, as well as specified names 0, 1, 2, …, one for each (standard) natural number (which may be separate constants, or constant terms such as 0, 1, 1+1, 1+1+1, …, etc.). Note that T itself could be referring to more general objects, such as real numbers or sets; thus in a model of T the objects satisfying N(x) are those that T interprets as natural numbers, not all of which need be named by one of the specified names. The system of ω-logic includes all axioms and rules of the usual first-order predicate logic, together with, for each T-formula P(x) with a specified free variable x, an infinitary ω-rule of the From $ P(0),P(1),P(2),\ldots $ infer $ \forall x\,(N(x)\to P(x)). $ That is, if the theory asserts (i.e. proves) P(n) separately for each natural number n given by its specified name, then it also asserts P collectively for all natural numbers at once via the evident finite universally quantified counterpart of the infinitely many antecedents of the rule. For a theory of arithmetic, meaning one with intended domain the natural numbers such as Peano arithmetic, the predicate N is redundant and may be omitted from the language, with the consequent of the rule for each P simplifying to \forall x\,P(x). An ω-model of T is a model of T whose domain includes the natural numbers and whose specified names and symbol N are standardly interpreted, respectively as those numbers and the predicate having just those numbers as its domain (whence there are no nonstandard numbers). If N is absent from the language then what would have been the domain of N is required to be that of the model, i.e. the model contains only the natural numbers. (Other models of T may interpret these symbols nonstandardly; the domain of N need not even be countable, for example.) These requirements make the ω-rule sound in every ω-model. As a corollary to the omitting types theorem, the converse also holds: the theory T has an ω-model if and only if it is consistent in ω-logic. There is a close connection of ω-logic to ω-consistency. A theory consistent in ω-logic is also ω-consistent (and arithmetically sound). The converse is false, as consistency in ω-logic is a much stronger notion than ω-consistency. However, the following characterization holds: a theory is ω-consistent if and only if its closure under unnested applications of the ω-rule is consistent. Relation to other consistency principles If the theory T is recursively axiomatizable, ω-consistency has the following characterization, due to C. Smoryński:[5] T is ω-consistent if and only if $ T+\mathrm{RFN}_T+\mathrm{Th}_{\Pi^0_2}(\mathbb N)$ is consistent. Here, $\mathrm{Th}_{\Pi^0_2}(\mathbb N)$ is the set of all Π02-sentences valid in the standard model of arithmetic, and $ \mathrm{RFN}_T$ is the uniform reflection principle for T, which consists of the axioms $ \forall x\,(\mathrm{Prov}_T(\ulcorner\varphi(\dot x)\urcorner)\to\varphi(x))$ for every formula$ \varphi$ with one free variable. In particular, a finitely axiomatizable theory T in the language of arithmetic is ω-consistent if and only if $ T + PA is \Sigma^0_2$-sound. W.V.O. Quine, Set Theory and its Logic Smorynski, "The incompleteness theorems", Handbook of Mathematical Logic, 1977, p. 851. The definition of this symbolism can be found at arithmetical hierarchy. J. Barwise (ed.), Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977. Smoryński, Craig (1985). Self-reference and modal logic. Berlin: Springer. ISBN 978-0-387-96209-2. Reviewed in Boolos, G.; Smorynski, C. (1988). "Self-Reference and Modal Logic". The Journal of Symbolic Logic 53: 306. doi:10.2307/2274450. JSTOR 2274450. edit Kurt Gödel (1931). 'Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I'. In Monatshefte für Mathematik. Translated into English as On Formally Undecidable Propositions of Principia Mathematica and Related Systems. Retrieved from "http://en.wikipedia.org/" All text is available under the terms of the GNU Free Documentation License Home - Hellenica World	{"url":"https://www.scientificlib.com/en/Mathematics/LX/OmegaConsistentTheory.html","timestamp":"2024-11-07T03:51:34Z","content_type":"application/xhtml+xml","content_length":"16629","record_id":"<urn:uuid:81bf3b4f-9ad1-4e99-853f-6b0d95559ad1>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00280.warc.gz"}
Buoyancy: What Is a Ship's Buoyancy? - Boat Buoyancy: What Is a Ship’s Buoyancy? The word “buoyancy” has multiple meanings within the field of ship traffic and navigation. Buoyancy, in the context of residual speed and buoyancy in water (Archimedes’ principle), is significant knowledge for a new sailor. It will likely become evident naturally because your boat will continue to move even when you wish to stop it. The “magic” of a boat floating will also be a concept you contemplate one day. Buoyancy as Residual Speed: Buoyancy primarily refers to the speed a ship retains after shutting off the engine or lowering the sails. It’s impossible to bring a ship to a complete stop, and this is especially crucial for sailboats. Planning your landing, such as at a dock, requires taking buoyancy into account. A motorized boat may need to reverse and counteract the buoyancy propelling the boat forward (or vice Besides the potential speed the boat possesses after removing the means of propulsion, the wind and current will also influence the boat’s speed. This is why extensive practice in sailing is so crucial. Two years at a reputable sailing school is a worthwhile investment. Buoyancy Makes Ships Float: Buoyancy concerning a ship in water pertains to the force acting upward on the ship due to its immersion in the water. This principle is based on Archimedes’ law of buoyancy, as mentioned earlier. When a ship floats in the water, it displaces a certain volume of water equal to the space it occupies. According to Archimedes’ law, the buoyant force is equal to the weight of the displaced liquid (in this case, water). The buoyant force counteracts the gravitational force acting downward on the ship. When the ship’s weight (caused by its mass) is less than the buoyant force, the ship will float. To understand this more visually, envision an empty ship placed in the water. As the ship is lowered into the water, it displaces an amount of water equal to its volume. This displaced volume of water creates an upward buoyant force acting on the ship. If the ship’s weight is greater than the buoyant force, the ship will sink. However, if the ship’s weight is less than the buoyant force, the ship will float and remain on the water’s surface. The ship’s weight can increase if its compartments are filled with water. Buoyancy is a crucial factor in maritime activities, as it allows ships to carry cargo and passengers without sinking. It is also essential for the design and stability of a ship’s hull and for understanding the principles of floating bodies in general. Types of Buoyancy Aids: Within maritime and marine industries, various types of buoyancy aids or floatation devices are used to enhance a ship’s buoyancy or floatability. Some of the most common buoyancy aids include: 1. Hull Buoyancy: A ship’s hull is designed to displace a certain volume of water, creating buoyancy. By shaping the hull correctly, sufficient buoyancy can be achieved to keep the ship afloat. The shape and volume of the hull play a vital role in achieving the desired buoyancy. 2. Foam Blocks and Buoyant Material: Ships can be equipped with foam blocks or buoyant materials embedded within the structure or compartments of the vessel. These materials are lightweight and increase the ship’s overall buoyancy. 3. Buoyancy Tanks: Some ships are fitted with buoyancy tanks that can be filled with air or water to increase buoyancy. By regulating the amount of air or water in these tanks, the ship’s overall buoyancy can be adjusted as needed. 4. Buoyancy Chambers: Buoyancy chambers, also known as pontoons, are separate structures or flotation devices attached to the ship to enhance buoyancy. These may be attached to the sides or located beneath the hull. 5. Floating Docks: Floating docks or dry docks are structures used, for example, to lift ships out of the water. These floating docks employ the principle of buoyancy to temporarily support the ship, allowing for maintenance and repairs. Archimedes’ Law of Buoyancy: Archimedes’ law of buoyancy, also known as Archimedes’ principle, is a physical law that describes the nature of buoyancy. It was formulated by the Greek mathematician and physicist Archimedes around the 3rd century BC. The law states that when a body is submerged wholly or partially in a fluid, it experiences an upward buoyant force equal to the weight of the displaced fluid. In other words, the buoyant force is directly proportional to the volume of the displaced liquid. Mathematically, Archimedes’ law can be expressed as follows: Buoyant Force = Weight of Displaced Fluid Where “Buoyant Force” is the force acting upward on the submerged body, and “Weight of Displaced Fluid” is the weight of the liquid that fills the space displaced by the body. This principle explains why objects sink or float in fluids. If an object is heavier than the displaced fluid, it will sink because its own weight is greater than the buoyant force. Conversely, if an object is lighter than the displaced fluid, it will float because the buoyant force exceeds its weight. The mathematical formula for the buoyant force (F_b) in Archimedes’ law is: F_b = ρ_fluid * V * g Where: F_b is the buoyant force (the upward force on the submerged body), ρ_fluid is the mass density (density) of the fluid, V is the volume of the submerged body or the displaced water, and g is the acceleration due to gravity. Note that the units must be consistent to obtain accurate results. Typically, mass density is in kg/m^3, volume in m^3, and acceleration due to gravity in m/s^2. Quick FAQs About Buoyancy: How does the shape of an object affect buoyancy? The shape of an object affects buoyancy by altering the amount of fluid displaced. A larger volume displaces more fluid, increasing buoyancy. How can buoyancy on a ship be increased? Buoyancy on a ship can be increased by modifying its hull shape to displace more water. Buoyant materials or buoyancy tanks can also be added to enhance overall buoyancy. How is buoyancy used in everyday life beyond maritime activities? Buoyancy plays a crucial role in various aspects of our daily lives. It is used in the design of life jackets to assist people in staying afloat in water. Buoyancy is also employed in floating docks, dry docks, underwater vehicles, and flotation systems for various purposes. 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Ohm’s Law: Explanation And Applications - Crazy Chicken Guitar Pedals Ohm’s Law is one of the basic electrical equations that will help you better understand what’s happening in electrical circuits and why certain components are in guitar pedals. Although you probably just want to start putting things together and making some noise, you’ll better be able to troubleshoot guitar pedal builds and designs if you understand what everything is doing, and you’ll better understand what everything is doing if you understand Ohm’s Law. The good news is that Ohm’s Law is actually pretty simple. There is a mathematical equation involved, but it couldn’t be easier. But first, a little history. Ohm’s Law is named after Georg Ohm, who first wrote of his discovery in 1827. Ohm’s Law describes the relationship between current, voltage, and resistance and how everything remains constant. And yes, Ohm “discovered” the law, he didn’t invent it. He didn’t invent it because Ohm’s Law exists whether or not we’re aware of it. But, since we’re now aware of it, we can use it! What Is Ohm’s Law Ohm’s Law states that: • E refers to voltage • I refers to current • R refers to resistance Just note that sometimes Ohm’s Law is also written as V = IR, since V for voltage makes a little more sense… Voltage is abbreviated to E because it was originally known as “electromotive force” before it was formalised by Alessandro Volta, so E makes sense. Current, on the other hand (which is measured in amps) was originally described as “intensity” by André-Marie Ampère, who we get the unit amp from. So, the equation says that voltage is equal to current times resistance. Since Ohm’s Law is a relationship between voltage, current, and resistance, everything is interchangeable. This means that: $I = \frac{E}{R}$ AND $R = \frac{E}{I}$ Because of this, if you know two numbers, you can find out the first using pretty simple maths. Why does Ohm’s Law work? Electricity is often described as similar to water flowing through a pipe. Voltage is similar to the pressure in a pipe, the current is the speed that the water is flowing, and the resistance is how big the pipe is (a narrower pipe is more resistance). To get more water through the pipe, you can increase the pressure, speed up the flow, or make the pipe bigger. All of that is a basic understanding of course, but hopefully you get the idea. It will become clearer when we run through some applications of Ohm’s Law, both in general and as it is related to guitar pedals. Using Ohm’s Law To Calculate Current, Voltage, And Resistance Obviously just knowing Ohm’s Law only counts for so much. If you can’t use it, it’s kind of useless, isn’t it? The following are some examples of using Ohm’s Law to calculate current, voltage, and resistance. These three examples are fairly dry, but I’ll go into three more examples that will help you understand why being able to calculate these numbers is important. With that in mind, remember that these are simple circuits that are presented in isolation. Most circuits have multiple components, sections, and branches. When you’re looking at a simple circuit, these things may be obvious, but starting with the basic calculations helps when things get more complex. Calculating Current Using Ohm’s Law As I mentioned, if you know two of the numbers in the Ohm’s Law formula, you can figure out the third. Often we know what the voltage is and the value of a resistor, so it’s a simple matter of finding the current. In the diagram above, let’s assume you’re using the standard 9 volt battery that’s used for most guitar pedals. All your pedal does is light up a light with a resistance of 10 ohms. Yeah, that’s kind of a boring pedal, but you have to start somewhere, right? So we know: E = 9 I = ? R = 10 If E = IR, then I = E/R so: I = 9/10 = 0.9 The current in the circuit is 0.9 amps. Calculating Resistance Using Ohm’s Law Here’s another simple circuit where the voltage and current is known. Once again I’ll use 9 volts as it’s what most guitar pedals use. So we know: E = 9 I = 2 R = ? If E = IR, then R = E/I so: R = 9/2 = 4.5 The resistance in the circuit is 4.5 ohms. Calculating Voltage Using Ohm’s Law You can probably see where this is going… now let’s calculate the voltage in a simple circuit when we know the resistance and current. So we know: E = ? I = 3 R = 4 Using E = IR: E = 3 x 4 = 12 The voltage in the circuit is 12 volts. I wanted to make the voltage 9 volts so that it’s still the same as most guitar pedals, but that felt too obvious! Using Ohm’s Law To Calculate Voltage Drop For Resistors In Series I’m sure all of the above was pretty basic. That’s all basic algebra most of us learned in middle school. Let’s look at some resistors in series so we can see how voltage drops as it goes through each resistor. While this is still pretty simple, understanding how voltage drops for resistors in series will be important for understanding things like voltage dividers, op-amps, and generally supplying enough voltage to various parts of a guitar pedal. Resistors are common in guitar pedals, so understanding what they do to voltage is important. Here we have a circuit with three resistors in series. “In series” means the resistors are all lined up one after another. For resistors in series, we can calculate the total resistance simply by adding up the resistors. $R_{total} = 3 + 4 + 5 = 12$ We can then use the total resistance to calculate the current using Ohm’s Law. I = E/R = 9/12 = 0.75 Across the three resistors, the current will remain the same. This is just a fact of circuits in series. Sorry to stress it, but remember: the current remains constant for components in series. What this means is that we can use our knowledge of the constant current to calculate the voltage drop after each resistor. In the below, I’ll use V1 for the voltage drop across R1, V2 for the voltage drop across R2, and V3 for the voltage drop across R3. V1 = IR = 0.75 x 3 = 2.25 volts V2 = IR = 0.75 x 4 = 3 volts V3 = IR = 0.75 X 5 = 3.75 volts Since everything is now going back into ground, the voltage after it’s passed through all the resistors should drop to zero. We can check our answer by adding up the voltage drops across all three resistors. It should equal our original voltage. V = 2.25 + 3 + 3.75 = 9 volts – we got it right! Are you starting to see how Ohm’s Law might be helpful in designing circuits and ultimately designing guitar pedals? Using Ohm’s Law To Create A Voltage Divider A voltage divider is a set of two resistors in series with a wire coming off in between the two. This is used in guitar pedals both for volume knobs and for lowering the voltage coming out of a 9 volt battery so that you’re getting the correct voltage to something like an op-amp. This will make more sense later on, but to put it simply, a voltage divider is a way to send a lower amount of voltage into a circuit. Let’s use what we learned in the last section to calculate the voltage drop across each resistor so that we know how much is being sent out to reference voltage (Vb), going to the right in the Rtotal = 8 + 10 = 18 I = E/R = 9/18 = 0.5 So the voltage drop across the first resistor is: V1 = IR = 0.5 x 8 = 4 So we know that we lost 4 volts across the first resistor, so to calculate Vb: Vb = V – V1 = 9 – 4 = 5 volts. As mentioned, creating a voltage divider is essentially how a volume knob works. A volume knob is just a potentiometer. Potentiometers are essentially just resistors in series that can be adjusted. Rotating the knob changes how much resistance there is in the first resistor and the second resistor, changing how much output voltage there is and therefore the volume. That’s Ohm’s Law If this is the start of your guitar pedal building journey, welcome! Perhaps this is a bit of a dry way to get started, but it is getting you started with a bit of theory, which will do you good. If you just want to jump in, check out some of my easy pedal build suggestions. If you’ve already gotten started at building guitar pedals but you want to better understand how everything works, welcome as well! Both these ways of getting started, purely with theory or with some simple builds then understanding the components later, are great approaches.	{"url":"https://crazychickenguitarpedals.com/guitar-pedal-blog/electrical-theory/ohms-law-explanation-and-applications/","timestamp":"2024-11-07T22:13:38Z","content_type":"text/html","content_length":"100064","record_id":"<urn:uuid:fe12dc22-76a0-45dd-8fe8-308a40223eed>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00034.warc.gz"}
Neutral Drifts skip to main \| skip to sidebar An image map of the human mitochondrion, linked to its GenBank entry by region, showing the G+C content in a small window. Created in Sage using the biopython optional module and this source code. All 99 (complex) solutions of the Albouy-Chenciner equations for central configurations of the Newtonian three-body problem, with equal masses. There are three "distances" for each solution; each set of distances are drawn as a triangle. The unit circle is included for a sense of scale. The solutions of primary interest are real, but understanding the complex solutions is important for analyzing these equations. This was computed with Sage, with the bulk of the work was done with Jan Verschelde's PHCpack. At the moment, PHCpack is in sage as an optional package, so you would have to install that to get the code to work: source code. Light reflecting in four tangent spheres....this relates to a math problem I abandoned a few years ago, maybe I'll work on it again someday. The sage code for the above: t = Tachyon(camera_center=(0,-4,1), xres = 1200, yres = 800, raydepth = 12, aspectratio=.75, antialiasing = True) t.light((0.02,0.012,0.001), 0.01, (1,0,0)) t.light((0,0,10), 0.01, (0,0,1)) t.texture('s', color = (.8,1,1), opacity = .9, specular = .95, diffuse = .3, ambient = 0.05) t.texture('p', color = (0,0,1), opacity = 1, specular = .2) This is mostly a test of how code looks in a post. The sage functions below are some initial attempts to write basic dynamical systems plotting functions for educational purposes. def cobweb(a_function, start, mask = 0, iterations = 20, xmin = 0, xmax = 1): Returns a graphics object of a plot of the function and a cobweb trajectory starting from the value start. a_function: a function of one variable start: the starting value of the iteration mask: (optional) the number of initial iterates to ignore iterations: (optional) the number of iterations to draw, following the masked iterations xmin: (optional) the lower end of the plotted interval xmax: (optional) the upper end of the plotted interval sage: f = lambda x: 3.9x(1-x) sage: show(cobweb(f,.01,iterations=200), xmin = 0, xmax = 1, ymin=0) Note: This is very slow with symbolic functions. basic_plot = plot(a_function, xmin = xmin, xmax = xmax) id_plot = plot(lambda x: x, xmin = xmin, xmax = xmax) iter_list = [] current = start for i in range(mask): current = a_function(current) for i in range(iterations): current = a_function(current) cobweb = line(iter_list) return basic_plot + id_plot + cobweb def orbit_diagram(a_function,parameter_interval, domain=[0,1], mask = 50, iterations = 200, param_num = 500.0): Returns a plot of the iterations of a function as a function of a parameter value. a_function: a function of one variable parameter_interval: a two-element list of the lowest and highest parameters to plot. domain: (optional) a two-element list of the lowest and highest input values to iterate mask: (optional) the number of initial iterates to ignore iterations: (optional) the number of iterations to draw, following the masked iterations sage: f = lambda x,m: mx(1-x) sage: show(orbit_diagram(f,[3.4,4], mask = 100, iterations = 500), xmin=3.4, ymin=0) This is pretty crude so far. point_list = [] plen = RDF(parameter_interval[1] - parameter_interval[0]) seed = random()*(domain[1]-domain[0])+domain[0] for i in srange(parameter_interval[0],parameter_interval[1],plen/param_num): for x in range(mask): seed = a_function(seed,i) for x in srange(iterations): seed = a_function(seed,i) return point(point_list,pointsize=1,rgbcolor=(0,0,0)) I've always hated the word 'blog', which I think is the only reason I haven't started one until now. But now with the sage projects new blog , I guess I'll give it a try.	{"url":"https://neutraldrifts.blogspot.com/2007/12/","timestamp":"2024-11-15T00:12:48Z","content_type":"application/xhtml+xml","content_length":"62971","record_id":"<urn:uuid:c3d74302-aa96-478b-aa0d-b1c4f6038573>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00543.warc.gz"}
A supermarket claims that the average wait time at the checkout counter is less than 9 minutes. Assume that the population is Normally distributed.... Answered You can hire a professional tutor to get the answer. A supermarket claims that the average wait time at the checkout counter is less than 9 minutes. Assume that the population is Normally distributed.... A supermarket claims that the average wait time at the checkout counter is less than 9 minutes. Assume that the population is Normally distributed. We will test at 1% level of significance. H0: mu >= 9 H1: mu < 9 A random sample of 50 customers yielded an average wait time of 8.5 minutes and standard deviation 2.5 minutes. What is the critical value for the t-stat (the t-test statistic)? Show more Homework Categories Ask a Question	{"url":"https://studydaddy.com/question/a-supermarket-claims-that-the-average-wait-time-at-the-checkout-counter-is-less","timestamp":"2024-11-03T11:00:09Z","content_type":"text/html","content_length":"26528","record_id":"<urn:uuid:2c501398-84a5-407d-861d-93953257bcf9>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00744.warc.gz"}
Konstantin Tsiolkovsky Konstantin Eduardovich Tsiolkovsky was born on 17 September 1857, in what was then the Russian Empire. He did not have an easy childhood. At the age of ten, he contracted scarlet fever and subsequently became hard of hearing. As a result, he was unable to attend school. Three years later, his mother died. Young Tsiolkovsky studied at home and almost constantly read books, developing an interest in mathematics and science. He later studied these subjects in Moscow and became a school teacher. In his spare time, he studied aeronautics and cosmonautics. In the mid-1890s, he used his personal funds to build the first wind tunnel in Russia, using it to test various aircraft and airship concepts. After studying the effects of air friction, he was given financial aid by the Academy of Sciences, which he used to construct a larger wind tunnel. At the same time, he began work on solving theoretical problems associated with space travel. In 1897, he derived an equation which he called the ‘formula for aviation’. It described the relationship between a rocket’s velocity, its mass and the exhaust velocity of its engine. In 1903, about six months before the first flight of the Wright brothers, he published an article titled, ‘Exploration of outer space by means of rocket devices.’ The article revealed what is now known as the ‘Tsiolkovsky rocket equation.’ Even today, this equation is vitally important in space travel and exploration, as well as some spheres of aviation. It contains all the essential aspects of rocket physics in a brief formula. It must be noted that in other parts of the world, most notably Britain, other scientists had also independently created similar equations, but Tsiolkovsky was the first to use the equation to calculate whether rockets could reach the necessary speeds to achieve space travel. Tsiolkovsky published a second part of the article in 1911, in which he addressed specific problems which had to be overcome to make space travel possible. In the 1920s, he was the first to provide a scientific description of the physics involved in ground effect and hovercraft. Then, in 1929, he published a book called ‘Space Rocket Trains,’ in which he became the first person to describe multi-stage rockets. Meanwhile, Tsiolkovsky continued working as a high school mathematics teacher. He died on 19 September 1935, at the age of 78. Today, he is known as one of the fathers of rocketry. One of the most prominent features of the far side of the moon, a massive impact crater, has been named the ‘Tsiolkovsky’ in his honour.	{"url":"https://read.aviationnewsjournal.com/articles/konstantin-tsiolkovsky","timestamp":"2024-11-06T21:37:22Z","content_type":"text/html","content_length":"41224","record_id":"<urn:uuid:ec714281-c5f5-44c5-b034-91457ada6b87>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00235.warc.gz"}
Hollee Goody This will be my 11th year teaching and coaching at Capital High School and my 21st year for my career. I previously spent ten wonderful years in Fort Benton, MT coaching and teaching. I earned my under-graduate degree from The University of Montana Western in Secondary Education mathematics and history. I earned my master’s degree from Concordia University Portland in Education and Curriculum K-14 mathematics in December 2014. I have two independent, loving children and a wonderful husband, Jeff. I have taught a wide range of classes including, Pre-Algebra, Algebra 1, Algebra 2, College Algebra M121, Statistics 216 and Title one Math. If I could use one word to describe myself it would be PASSIONATE. I love math and I love my job. I love helping students learn a life skill… mathematics. In our society, we have people that either love math or hate it. It is my mission to help every person in my class to walk out on the last day and NOT HATE MATH. My classroom is a positive environment in which every student will work hard and have the ability to succeed. Everyday I wake up I am honored and excited to teach my students mathematics. Mathematics is 50% ability and 50% confidence. I will do what I can every day to help instill confidence in all my students.	{"url":"https://staff.helenaschools.org/staff_page/hgoody/","timestamp":"2024-11-05T05:54:14Z","content_type":"text/html","content_length":"62103","record_id":"<urn:uuid:c08313b9-9a3e-4c93-9cd2-1cdcd70c07aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00635.warc.gz"}
Probability Practice Set For RBI Grade B Question Set-1 Bankexamtoday wants to open his office in India. There are 40% & 60% chance that their office will be in Haryana & Punjab respectively. If they open the office in Haryana, there is 40% chance that it will be in Panchkula & 60% chance that it will be in Chandigarh. If they start an office in Punjab, there are 50%, 25% & 25% chances that it will be in Ludhiana, Amritsar & Chandigarh respectively. Find the probability of opening the office in Ludhiana? (a) 1/5 (b) 3/10 (c) 4/10 (d) 6/10 (e) none of these. What is the probability of opening the office in Chandigarh from any of those two states? (a) 39/100 (b) 2/10 (c) 2/5 (d) 4/9 (e) none of these. Question Set-2: There are two sets of letters & we have to choose exactly one letter from each set. Set A: {A, B, C, D, E} Set B: {U, V, W, X, Y, Z} What is the probability of choosing a D and X? (a) 1/30 (b) 2/30 (c) 3/30 (d) 4/30 (e) none of these. What is the probability of choosing a D or X? (a) 1/2 (b) 1/3 (c) ¼ (d) 1/5 (e) none of these. What is the probability of choosing two vowels? (a) 1/15 (b) 2/15 (c) 3/15 (d) 4/15 (e) none of these What is the probability of choosing at least one vowel? (a) 3/2 (b) 1 (c) 1/2 (d) 1/3 (e) none of these. Question Set3. Tiger has 4 books of Arithmetic, 3 books of advanced-math & 2 books of combine metrics. Out of these books, he selects two books at random one after other. What is the probability that he selects one Arithmetic & one combine metrics book? (a) 1/9 (b) 2/9 (c) 3/9 (d) 4/9 (e) none of these. Question Set4: Raj has two bags viz. Bag A & Bag B. Each bag contains balls of two colours viz. red and green. Given that, Bag A contains 3 red balls & 4 green balls and bag B contains 4 red balls & 5 green balls. Raj has to select one bag randomly and pick a ball from that. What is the probability that ball drawn is green in colour? (a) 67/126 (b) 69/126 (c) 71/126 (d) 73/126 (e) none of these. Simran, a friend of raj transferred a ball from bag A to bag B, then raj draws a ball from bag B. What is the probability that ball drawn is green in colour? (a) 69/126 (b) 38/75 (c) 35/69 (d) 39/70 (e) none of these. Simran, a friend of raj transferred a ball from bag A to bag B, then raj draws a ball from bag B. Find the probability that the transferred ball is green, given that the ball drawn is green in (a) 5/17 (b) 8/13 (c) 7/13 (d) 5/24 (e) none of these. Answer Set1. Ans. 1. The probability of opening the office in Ludhiana= P (office in Punjab) ×P (office in Ludhiana) Ans. 2. The probability of opening the office in Chandigarh = P( Office in Chandigarh from Haryana) 0r P (Office in Chandigarh from Punjab) Answer Set2. Ans. 1. Ans. 2. The probability of choosing a D or X = P(D or X) = P(D)+P(X)-P(D∩X) Ans. 3. Ans. 4. The probability of choosing at least one vowel= 1- probability of choosing no vowel Answer. Set 3 The probability of selecting one Arithmetic & one combine metrics book Question Set4: Ans. 1. Required probability = P(selecting bag A & drawing green ball) or P(selecting bag B & drawing a green ball) = Ans. 2. If a red ball is transferred, then bag B will have 5 red balls & 5 green balls. Probability that green ball is drawn=	{"url":"https://www.bankexamstoday.com/2018/01/probability-practice-set.html","timestamp":"2024-11-06T07:29:17Z","content_type":"application/xhtml+xml","content_length":"131412","record_id":"<urn:uuid:bfca6599-4da5-44c9-bec9-04b4c6914149>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00421.warc.gz"}
Python - (Mathematical Methods for Optimization) - Vocab, Definition, Explanations \| Fiveable from class: Mathematical Methods for Optimization Python is a high-level programming language known for its simplicity and readability, which makes it a popular choice for both beginners and experienced programmers. Its versatility allows it to be used in various applications, including web development, data analysis, artificial intelligence, and optimization tasks. The language provides extensive libraries and frameworks that facilitate interfacing with optimization solvers, making it easier to model problems and interpret results. congrats on reading the definition of Python. now let's actually learn it. 5 Must Know Facts For Your Next Test 1. Python is an interpreted language, meaning that code can be executed line-by-line, which aids in debugging and development. 2. The language supports multiple programming paradigms, including procedural, object-oriented, and functional programming. 3. Python has a vast ecosystem of libraries and frameworks that simplify the integration with different optimization solvers, such as PuLP and CVXPY. 4. The syntax of Python emphasizes readability, allowing programmers to express concepts in fewer lines of code compared to languages like C++ or Java. 5. Python's community is very active, providing numerous resources for learning and troubleshooting through forums, documentation, and online tutorials. Review Questions • How does Python's readability and simplicity contribute to its effectiveness in interfacing with optimization solvers? □ Python's readability and simplicity make it an effective tool for interfacing with optimization solvers by allowing users to write clear and concise code that is easy to understand and modify. This clarity enables developers to quickly implement complex algorithms without getting bogged down by syntactical complexities found in other programming languages. Additionally, this ease of use supports rapid prototyping and iteration, essential for optimizing mathematical models. • In what ways do libraries such as NumPy and SciPy enhance Python's capabilities for mathematical modeling and optimization tasks? □ Libraries like NumPy and SciPy significantly enhance Python's capabilities for mathematical modeling by providing robust data structures and a wide range of functions tailored for numerical computations. NumPy allows users to handle large datasets efficiently through its array functionality, while SciPy builds on this by offering specialized functions for optimization problems. Together, they streamline the process of formulating and solving mathematical models, making Python a powerful tool for optimization tasks. • Evaluate the impact of Python's community support on the development of new libraries aimed at solving complex optimization problems. □ The strong community support surrounding Python has led to the rapid development of numerous libraries specifically designed to tackle complex optimization problems. This collaborative environment fosters innovation as developers share knowledge and resources, contributing to an ever-growing ecosystem of tools that enhance problem-solving capabilities. The availability of well-documented libraries also lowers the entry barrier for new users, encouraging broader adoption of Python in fields requiring sophisticated optimization solutions. Â© 2024 Fiveable Inc. All rights reserved. APÂ® and SATÂ® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.	{"url":"https://library.fiveable.me/key-terms/mathematical-methods-for-optimization/python","timestamp":"2024-11-10T02:16:02Z","content_type":"text/html","content_length":"210400","record_id":"<urn:uuid:97fcf36c-0cc7-491e-a595-7e9b1d4e0049>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00264.warc.gz"}
Language Specification AssociatedItem ::= OuterAttributeOrDoc* (AssociatedItemWithVisibility \| TerminatedMacroInvocation) AssociatedItemWithVisibility ::= VisibilityModifier? ( ConstantDeclaration \| FunctionDeclaration \| TypeAliasDeclaration ) 10:1 An associated item is an item that appears within an implementation or a trait. 10:2 An associated constant is a constant that appears as an associated item. 10:3 An associated function is a function that appears as an associated item. 10:4 An associated type is a type alias that appears as an associated item. 10:5 An associated type shall not be used in the path expression of a struct expression. 10:6 An associated type with a TypeBoundList shall appear only as an associated trait type. 10:7 A generic associated type is an associated type with generic parameters. 10:8 A lifetime parameter of a generic associated type requires a bound of the form T: 'lifetime, where T is a type parameter or Self and 'lifetime is the lifetime parameter, when 10:9 The generic associated type is used in an associated function of the same trait, and 10:10 The corresponding lifetime argument in the use is not the 'static lifetime and has either an explicit bound or an implicit bound that constrains the type parameter, and 10:11 The intersection of all such uses is not empty. 10:12 An associated implementation constant is an associated constant that appears within an implementation. 10:13 An associated implementation constant shall have a constant initializer. 10:14 An associated implementation function is an associated function that appears within an implementation. 10:15 An associated implementation function shall have a function body. 10:16 An associated implementation type is an associated type that appears within an implementation. 10:17 An associated implementation type shall have an initialization type. 10:18 An associated trait item is an associated item that appears within a trait. 10:19 An associated trait implementation item is an associated item that appears within a trait implementation. 10:20 An associated trait constant is an associated constant that appears within a trait. 10:21 An associated trait function is an associated function that appears within a trait. 10:22 An associated trait function shall not be subject to keyword const. 10:23 Every occurrence of an impl trait type in the return type of an associated trait function is equivalent to referring to a new anonymous associated trait type of the implemented trait. 10:24 An associated trait type is an associated type that appears within a trait. 10:25 An associated trait type shall not have an initialization type. 10:26 An associated trait type has an implicit core::marker::Sized bound. 10:28 is equivalent to a where clause of the following form: 10:29 An associated trait implementation function is an associated function that appears within a trait implementation. 10:30 Every occurrence of an impl trait type in the return type of an associated trait implementation function is equivalent to referring to the corresponding associated trait type of the corresponding associated trait function. 10:31 A method is an associated function with a self parameter. 10:32 The type of a self parameter shall be one of the following: 10:37 The visibility modifier of an associated trait item or associated trait implementation item is rejected, but may still be consumed by macros. trait Greeter { const MAX_GREETINGS: i32; fn greet(self, message: &str); } struct Implementor { delivered_greetings: i32 } impl Greeter for Implementor { const MAX_GREETINGS: i32 = 42; fn greet(mut self, message: &str) { if self.delivered_greetings < Self::MAX_GREETINGS { self.delivered_greetings += 1; println!("{}", message); } } } trait LendingIterator { type Item<'x> where Self: 'x; fn next<'a>(&'a mut self) -> Self::Item<'a>; }	{"url":"https://spec.ferrocene.dev/associated-items.html","timestamp":"2024-11-04T20:21:40Z","content_type":"text/html","content_length":"29002","record_id":"<urn:uuid:0248965b-a3a3-419a-98c8-2bc7e6d51d77>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00266.warc.gz"}
'The Proofs That Provide the Foundation for Mathematics' I try to focus most of my posts on practical topics that will help people developing ontologies and knowledge graphs. However, every once in a while (like now) I like to go into some less practical areas. There are two related proofs that provided the formal foundation for modern programmable digital computers and that is what I'm going to discuss in this post. When I worked at the Software Engineering lab at Accenture, one of the consultants who worked with us loved the concept of meta-objects and meta-models because he had done very practical work up to that point and hadn't heard the term "meta" applied to computer science. He liked to say "let's get meta" whenever we were about to go into fairly esoteric but interesting discussions. So let's get meta. There are some reasons which I'll get to, that the proof is especially relevant to OWL users but it also has major implications for computer science in general. The proof is that the Entscheidungsproblem is unsolvable. The Entscheidungsproblem gained prominence at the beginning of the 20th century due to a mathematician named Hilbert who was considered one of the most important mathematicians of the time. At a mathematical conference Hilbert gave the keynote speech and described what he considered the most important open questions facing mathematics. One of those was the Entscheidungsproblem. Entscheidungsproblem is simply German for "Decision Problem". The decision the problem referred to was an algorithm that could take in any arbitrary formula in First Order Logic (FOL) and provide an answer as to whether or not it was valid. Most people thought such an algorithm must exist, we just hadn't found it yet. The reason people thought that there was such an algorithm was that such an algorithm existed for Propositional Logic and FOL is just an extension of propositional logic with two new operators. If you've taken an introductory class in Logic you've used propositional logic. Propositional logic consists of the logical connectors: or (∨), and (∧), if-then (⊃), and not (¬) along with variables such as P and Q. E.g., is a valid formula in propositional logic. Valid means always true regardless of the values assigned to the variables. In this case regardless of whether P is true or false "P or not P" is always true. The algorithm for propositional logic is a truth table. A table that lists all the possible values for each variables and the truth of the whole statement for each alternative binding. Below is the simple truth table for "P or not P". Since the middle column representing the overall truth of the formula is always true, the formula is valid. Truth tables go back to the ancient Greeks. First Order Logic (FOL) is propositional logic with the addition of two quantifiers: universal (for all) and existential (there exists). These should be familiar to OWL users. Universal quantification is the Only statement in Description Logic and existential quantification is the Some statement. The symbols for these are ∀ for universal quantification and ∃ for existential quantification. In 1936 an American and an Englishman found very different proofs that the Entscheidungsproblem had no solution. The American was Alonzo Church. The Englishman was Alan Turing. They both proved that it was impossible to have a general algorithm that could determine the validity of any set of FOL formulas. Of course, this doesn't mean you can't prove things in FOL. In fact it doesn't imply that there is any theorems you can't prove in FOL (although there are but that's a different proof by Gödel). All it means is there is no one algorithm that can determine the validity of every set of FOL As sometimes happens in mathematics (e.g., Newton and calculus), the formalisms that Turing and Church created for their proofs turned out to be as or more important than the proof itself. Turing/ Church created the first mathematical models of computation. Church created the Lambda calculus and Turing the Turing Machine. Later it was proven that the two are equivalent. Anything that can be defined in the Lambda calculus can be defined on a Turing Machine and vice versa. The Lambda calculus was extremely important for the early years of Artificial Intelligence because one of the first and most important languages for AI was Lisp and Lisp is an implementation of the Lambda calculus. It was called the Lambda calculus because the main formalism was a lambda expression, which is similar to a function definition, except a lambda expression can also be treated as data and can itself be passed as an argument to a function. In the Lambda calculus as well as Lisp and Python you can have arbitrary pieces of code called lambda expressions that can be passed to functions as data and then be interpreted inside the context of another function. This allows all sorts of possible manipulations. Code can be generated on the fly, the language (in Lisp) can literally redefine itself as it goes. While Church defined the first modern computer language (before there were any computers to run it). Turing's model was the model that was and is the model for all programmable digital computers: the Turing Machine. Turing took the concept of a Finite State Machine and added the ability of the machine to read and write to memory. Turing's and Church's genius realization was that data and process were different sides of the same coin. The information read from the memory of a Turing Machine could change the behavior of that machine. Thus, rather than a Finite State Machine, Turing created a model for a state machine that could have infinite states since the machine could keep on changing its memory, which would change its behavior, which would change its memory, which would change its Aesthetics are subjective of course, however, to me Turing's proof has the same kind of beauty as a great musical composition. Aesthetics aside this proof is especially relevant for those of us who use OWL because the OWL reasoner is an algorithm that (among other things) determines the validity of any set of OWL axioms (aka any OWL ontology). How is this possible given the Turing/Church proof? It is possible because OWL is not FOL. OWL is a subset of FOL called Description Logic. The creation of description logic goes back much further than research in OWL and was driven by the desire to find a subset of First Order Logic that was not subject to the Turing/Church proof. In some ways the early history of AI was a struggle with the Turing/Church proof. FOL was a very logical model for AI researchers to use to model AI systems. It is very expressive and powerful. However, the Turing/Church proof meant that no language with the expressivity of FOL could support a reasoner that was guaranteed to terminate. So AI researchers experimented with various subsets of FOL. One of the first successful examples were rule-based systems and inference engines. Rules were a very limited subset of FOL. However, because they were such a limited subset it was possible to define inference engines (aka reasoners) that could operate very efficiently, which was critical back in the days when the most powerful computers had less CPU power and memory than modern phones. An interesting side note is: why are people so concerned about decidability? After all we use languages that can define programs that won't terminate all the time. E.g., a loop in a Python program that never terminates. This is the position of John Sowa and others who think that we should just use First Order Logic, decidability be damned, what matters is expressivity. I don't think there is a right or wrong answer to this question but I think Sowa makes a very interesting point, that many people just take it as a given that reasoners must be guaranteed to terminate and that isn't necessarily a constraint we have to abide by. Like so much in software engineering, it is a trade-off and the decision should be dictated by requirements, not something we assume a priori. Turing's proof is proof is defined in his paper: On Computable Numbers: with an application to the Entscheidungsproblem. In this paper Turing defines what it means to be a computable number. This is another important contribution of the Turing/Church proof, it provided the first definition of what we mean by saying something is computable. I.e., that it can be the output of a Turing Machine or a deduction of the Lambda calculus. Before this, everyone had an intuitive understanding of what "computable" means but mathematicians don't like to rely on intuitions, they like rigorous definitions. This is known as the Turing-Church thesis because it is not a proof, in fact it can't be proven, because it is a definition rather than a theorem. However most mathematicians believe it is the best definition of computability and since Turing/Church no one has come up with an alternative definition of computability. One of the key ideas that made Turing's proof possible is that there are different sizes of infinity. This is counter intuitive because we know that "infinity + 1" is still infinity. To understand different sizes of infinity we need to tease apart the difference between cardinality and ordinality. With finite numbers these ideas are mostly bundled together. When we think of infinite sets we need to separate them. Ordinality is an ordering imposed on a set via some relation. The most common of course being > and its inverse <. E.g., 1 < 2 < 3. Cardinality is the size of a set, how many elements it contains. So the cardinality of {1, 2, 3} is 3. We can't say one infinity is greater than another in the sense of ordinality because there is no largest infinite number. But we can say that the cardinality of one infinite set is larger than the cardinality of another infinite set. For example, the cardinality of the integers is the same as the cardinality of the natural numbers (aka the non-negative integers). We determine this for infinite sets by a mapping. If we can map the elements of one infinite set to another infinite set then they have the same cardinality. An example mapping of the integers to the natural numbers (expressed as tuples with the element from the integers first and the naturals second) is: <0,0>, <1, 1>, <-1, 2>, <2, 3>, <-2, 4>, <3, 5>, <-3, 6>,... I.e., map 0 to 0 then map all the negative integers to the even natural numbers and all the positive integers to the odd natural numbers. You will always be much further along the ordinality for the natural numbers but for any integer you can always find a natural number that it maps to. You can do the same with the rational numbers and the integers. This kind of infinity has cardinality called Aleph 0 or Aleph null and is represented by they symbol ℵ0. Aleph is the first letter in the Hebrew alphabet. Where things get interesting is with the real numbers. A mathematician named Cantor proved that you can't map the reals to the rationals (or integers or naturals). Cantor used a diagonalization proof which is a common kind of tool in mathematic proofs. For details see: the Wikipedia article on Cantor's Diagonalization proof. The cardinality of the real numbers is known as Aleph 1 (ℵ1). Turing used Cantor's proof for his proof. He proved that the cardinality of all possible Turing machines was Aleph 0, however, the cardinality of all possible valid theorems is Aleph 1. They are both infinite but one infinity is bigger. Thus, there must be some valid theorems that don't have proofs. From this it follows that there is no solution to the Entscheidungsproblem, because even the most basic algorithm we could think of: enumerate all Turing machines until you find one that proves your theorem, won't always work for all valid theorems. The work of Turing/Church was part of what is now known as the theory of computation which was begun by Gödel who proved that there must be some valid theorems that can't be proven. Turing and Church built on Gödel's work. The theory of computation was continued by Chomsky who defined the Chomsky language hierarchy: a definition of various computing models and the ever more complex sets of languages they can parse. Chomsky's first book Syntactic Structures was written at a time (the 1950's) when computers were just being applied to symbolic problems and people had great hopes that finite state machines could parse natural language. Chomsky proved that finite state machines could only parse regular expressions and that human language is far more complex than regular expressions. The sets defined by human languages are called the recursively enumerable sets. Chomsky proved that only Turing Machines (the most powerful model which makes sense if it is the definition of anything that can be computed) can parse human languages. There are videos on YouTube which refer to the proofs of Gödel, Turing, and Church as showing "holes" in mathematics or even worse that "mathematics is inconsistent". I've also seen philosophers who say such things. The latter is of course simply ridiculous. If math was inconsistent we couldn't prove anything. Or rather we could prove anything because any theorem can be proven from "If False then..." and if we can prove anything then we can't usefully prove anything. As for saying that these proofs show "holes" or "problems" that is a subjective interpretation but IMO quite an improper one. On the contrary, I think one of the signs that a discipline is mature is that we begin to understand not just what we know but what we can't know. Another good example of this is the Heisenberg Uncertainty principle in physics. Understanding the limits of our knowledge is an essential part of knowledge. Which ultimately is why I think these proofs are so important, because they help us realize that even reason has fundamental limits. Addendum: These are very complex topics and I'm not an expert. So keep in mind that everything I said above is 1) Overly simplified (e.g., there are issues such as the Continuum Hypothesis that are important but would have required a much longer article) and 2) Simplified to the point that people who are experts will disagree with some of the things I said, especially about Gödel. So if you find this interesting don't take anything above as irrefutable but instead click on some of the links and check things out for yourself. I linked mostly to the Stanford Encyclopedia of Philosophy because on these issues, their articles tend to be a bit better than Wikipedia.	{"url":"https://www.michaeldebellis.com/post/the-turing-church-proof","timestamp":"2024-11-08T11:17:30Z","content_type":"text/html","content_length":"1050487","record_id":"<urn:uuid:724055f1-b514-43b4-a541-5458713ee010>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00621.warc.gz"}
How to calculate total magnification of a microscope Here are the steps, you can follow to calculate total magnification of a microscope. To calculate the magnification you need information about the magnification of the objective lens and the How to calculate magnification of a compound microscope • First of all, you need to Look for the magnification marked on the objective lenses (e.g., 4x, 10x, 40x). • Now check the eyepiece for its magnification. It’s commonly 10x, but it can vary in some cases. • Multiply the magnification of the objective lens by the magnification of the eyepiece. [ \text{Total Magnification} = \text{Magnification of Objective Lens} \times \text{Magnification of Eyepiece} ] • If you don’t have information about the eyepiece magnification, assume it’s 1x (no additional magnification), and your total magnification will be equal to the magnification of the objective How to calculate magnification of a stereo microscope (Dissecting Microscopes) Stereo microscopes have two eyepieces and do not use objective lenses with high magnification. The total magnification is calculated differently: • Check one of the eyepieces for its magnification. 1. Stereo microscope Magnification formula • The total magnification is twice the magnification of one eyepiece because you are looking through both eyepieces simultaneously. [ \text{Total Magnification} = 2 \times \text{Magnification of One Eyepiece} ] Leave a Comment	{"url":"https://medicallabtechnology.com/how-to-calculate-magnification-of-a-microscope/","timestamp":"2024-11-07T09:28:29Z","content_type":"text/html","content_length":"105647","record_id":"<urn:uuid:2dd7bb87-2595-4def-9221-9794c908f443>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00783.warc.gz"}
EViews Help: @eeq Element by element equality comparison of two data objects. Syntax: @eeq(m1, m2) m1: numeric or alphanumeric object m2: numeric or alphanumeric object Return: vector or matrix object Returns the element by element test of equality between numeric or alphanumeric objects. Each element of the returned object is equal to 1 or 0 depending on whether the corresponding element in m1 is equal to the corresponding element in m2. Note m1 and m2 must be of identical dimensions. = @eeq(X, Y) See also , and See also , and	{"url":"https://help.eviews.com/content/functionref_e-@eeq.html","timestamp":"2024-11-13T22:44:59Z","content_type":"application/xhtml+xml","content_length":"9952","record_id":"<urn:uuid:a757b463-b77b-4b3c-bd37-1ef5d3dd44f2>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00340.warc.gz"}
Credit Card Debt #illumedati - Senior Resident Credit Card Debt #illumedati Hey everyone, it’s Finance Fridays. After looking through some of my old posts, I have alluded to Credit Card Debt and how bad it is for you. However, I realized I’ve never talked about it explicitly. So let’s talk about Credit Card Debt. Stock Photo from: Pexels Ok, so let’s discuss Credit Card Debt Remember how I mentioned compound interest before and how powerful it is? Credit Card Debt is like that, except probably even more powerful, but also against you. I think everyone kind of knows this, but maybe doesn’t really understand just how bad it is. Allow me to demonstrate: Let’s say you decided to splurge for Christmas one year and you ended up going over your budget and had to leave a $2000 balance on your credit card. You have plans to pay it off within the first few months of January, but let’s just say you forget and pay the minimum on your credit card for a year. In general, the minimum payment is either $25 or 1% of your balance, plus interest and fees, whichever is more. So in this example, you have a $2000 balance and let’s assume you pay the minimum balance ($25) every month, over the year. After a normal “Introductory Annual Percentage Rate (APR)” which is usually low, like 0%, then a normal credit card will have an APR of between 15-24%. 15% as long as you are making payments, and increasing up to 24% if you miss two payments, in general. Since APR is kind of difficult to calculate, let’s convert it into a daily rate which is the APR / 365. In this case it’s 15%/365 = 0.041% per day. Now the next step is to figure out your “average daily balance”. If you pay the $25 minimum balance on day 1, then the calculation will be: ($2000 – $25) x 30 divided by 30 which equals: $1975 (this would be more complex if you made multiple payments during the month) So let’s calculate your first month’s interest: So then your interest charge for the month utilizes your daily rate and average daily balance. $1975 x 0.041% x 30 = $24.29 So basically, you paid $25, and accrued $24.29 in interest… that’s… not optimal. Obviously this is even worse if you keep buying things on this credit card and push the balance higher. Let’s go a little deeper now. How long would it take to pay off your $2000 balance if you paid the minimum balance $25. There are plenty of calculators, but here is one from bankrate. If you actually put in $2000, 15% APR, and $25, it actually spits out infinite months. The reason for this is because it probably rounds your interest to $25 which is your minimum payment. To make it more reasonable, let’s say it’s $50. Now let’s use the creditkarma calculator this time. $2000, 15%, $50 You will pay off your debt in 56 months (almost 5 years) and you will have paid $790 in interest. However, like I said before, this assumes your balance doesn’t increase. Your minimum payment is technically $50 + any new purchases. Let’s look at what happens when your balance increases, like $10000. Use creditkarma calculator again. $10000, 15%, $150 With a $150 minimum monthly payment it will take 145 months (12 years) to pay off this debt and you will have paid $1635 in interest. Remember, like I said before, this assumes your balance doesn’t increase. Your minimum payment is technically $150 + any new purchases. The snowball gets bigger. The snowball? Yes. The snowball. The problem when people begin to carry a balance on their credit cards is that they end up maxing them out. After maxing out that credit card, they start maxing out another one. Then after maxing out two, they balance transfer some of it to another card… and then end up maxing that one too. Imagine having 4 or 5 credit cards which are all maxed out with $10000 or so on each of them. That is scary. Then, you hit a snag and miss a few payments on one card and your APR jumps to 25%. Then because of that, you miss another few payments on another one and that one jumps to 25%. All of the sudden you can’t even make the minimum payments on your cards anymore. The snowball keeps rolling down the hill until it’s an avalanche. Credit card debt is like riding a bike… except the bike is on fire.. and you’re on fire… and you’ve just driven into quick sand, and the quicksand is on fire too. (the original quote is here, referencing college) also, relevant picture again: Credit Card Debt is like Cancer to your finances and your retirement. If you leave it alone, it just keeps getting worse. If you don’t get rid of all of it, it comes back. Get rid of it, like yesterday. If you have to carry a balance on your credit card, then you need to curb your spending and budget until it’s gone. Credit Card Debt is bad, but it’s hard to really understand exactly how bad unless someone demonstrates it to you. I provide the numbers for you above. The snowball is real, and it keeps rolling if you let it… until it’s an avalanche. … and then it’s too late. Agree? Disagree? Questions, Comments and Suggestions are welcome. You don’t need to fill out your email address, just write your name or nickname. Like these posts? Make sure to subscribe to get email alerts!	{"url":"https://seniorresident.com/2017/06/credit-card-debt-illumedati/","timestamp":"2024-11-14T10:25:30Z","content_type":"text/html","content_length":"89766","record_id":"<urn:uuid:6601d0e1-d05e-41b1-bcd5-ed82f55d3315>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00158.warc.gz"}
Evaluating aftertouch on the Arturia Keylab 61 MKII I tried that. They are correct, with min/max of 0/128. I've sent a support request to Arturia. Let's hope this sheds a little light on the problem. Meanwhile, thanks! Here's the support request: Using Pianoteq with this keyboard requires about 5 pounds of key pressure. That's way too high. I'm unable to figure out how to recalibrate the keyboard. This article seems to offer some help: But I'm unable to perform the first step: "a/ Boot the KeyLab while holding the Sound + Multi buttons. The screen should display: "AD Range select, move A/D Item." since there is no Sound button. Also, going into the setting for user 1, we see pad aft min/max is 0/128. That's okay. But key aft min/max is 128/0, which is backwards. I'm able to change min to 0, but max will only allow values of 0, 1, 2. Also, the Save question is enigmatic. It says "Save? Enc: Yes User: No". I am unable to figure out which key corresponds to "Enc". Jack Harich	{"url":"https://legacy-forum.arturia.com/index.php?topic=93718.15","timestamp":"2024-11-07T13:03:32Z","content_type":"application/xhtml+xml","content_length":"23864","record_id":"<urn:uuid:bc9a0eda-1972-4935-999b-5732274324b4>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00756.warc.gz"}
Finding the Density of an Object given Its Mass and Volume Question Video: Finding the Density of an Object given Its Mass and Volume Physics • Second Year of Secondary School A cube has a mass of 30 kg. If the volume of the cube is 0.02 m³, what is its density? Video Transcript A cube has a mass of 30 kilograms. If the volume of the cube is 0.02 meters cubed, what is its density? Okay, so let’s say that this is the cube we are asked about in this question. We are told the mass and volume of this cube. Let’s call the mass of the cube capital 𝑀. And we are told that the cube has a mass of 30 kilograms, so 𝑀 is equal to 30 kilograms. We are also told that the volume of the cube is 0.02 meters cubed. So if we call the volume of the cube 𝑉, then 𝑉 is equal to 0.02 meters Given this information, we are asked to find the density of the cube. We can do this by first recalling the general formula for the density of an object. We usually call the density of an object lowercase 𝜌, which is a Greek letter that looks a bit like a 𝑝. The formula for an object’s density is then 𝜌 is equal to 𝑀 divided by 𝑉, where 𝑀 is the mass of the object and 𝑉 is the volume of the So in order to find the density of our cube, we need to divide the mass of the cube by the volume of the cube. Since we are told these values in the question, we can simply substitute in these numbers to the formula we have just recalled. Here, the mass of our cube is 30 kilograms and the volume of our cube is 0.02 meters cubed. This means that the density 𝜌 of the cube is equal to the mass, which is 30 kilograms, divided by the volume, which is 0.02 meters cubed. We can simplify this fraction by first separating the numerical part from the unit. So the density 𝜌 is equal to 30 divided by 0.02, and the units are kilograms over meters cubed or kilograms per meter cubed. These are the correct units for density, so we can leave these as they are. We can then simplify the numerical part by working out that 30 divided by 0.02 is equal to 1500. So we have now found the density of the cube. And we can give 𝜌 equals 1500 kilograms per meters cubed as our final answer to this question.	{"url":"https://www.nagwa.com/en/videos/748125054028/","timestamp":"2024-11-14T07:40:40Z","content_type":"text/html","content_length":"250122","record_id":"<urn:uuid:40695e36-1263-4831-93ce-c13ce77af196>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00709.warc.gz"}
Fooled by randomness? In an earlier blog post we did what could be considered a descriptive analysis of 39 golfers who made coaching changes since 2004. While there was certainly a lot of idiosyncratic variation in a given golfer's performance before and after their coaching switch, in the aggregate we saw that performance declined leading up to the switch and improved after it, with the average decline and subsequent improvement roughly offsetting each other. In the original post we were uncritical of this result, suggesting that coaches "did help [their players] return to their 'pre-slump' level of play" . In hindsight, it appears we may have been fooled by randomness to some degree, as this pattern can (perhaps) also be rationalized by a purely statistical phenomenon: regression to the mean. According to Wikipedia , regression to the mean arises when more extreme measurements of a random variable tend to be less extreme when measured again. Hm, well what is a random variable? Again, according to Wikipedia , (informally) a random variable is a variable whose values depend on the outcomes of a random phenomenon. Okay, Wikipedia not being particularly helpful here. Let’s think of a random phenomenon as something that is, at least in part, unpredictable. Unpredictability can take many forms: tossing a coin and seeing which side lands facing up is an obvious example, but it can also simply reflect lack of information — if, instead of tossing the coin, you place it with Heads facing up underneath your hand, the variable “side of coin facing up” is not predictable to me (assuming I didn’t see how you placed the coin) even though it is a known quantity to you . Climbing back up to where we began, a random variable is then some object whose value is, to some degree, unpredictable; regression to the mean is the observation that more extreme values of an unpredictable quantity tend to be followed by less extreme values. Some examples may help illustrate the point. When rolling a fair die, the expected outcome is 3.5 (each outcome from 1 to 6 is equally likely to arise, and their average value is 3.5). Therefore, if you roll a 6, you can expect that the value of your next toss will be lower, while if you roll a 1 you can expect the value of your next toss to increase. This is regression to the mean in action. Similarly, in a simple model of professional golf scores, you could think of performance as equal to a fixed skill plus luck. Our expectation for a player’s score on any given day is equal to their skill level. As with the roll of a die, if we observe a performance below expectation (i.e. a player’s skill level) one day, then we can expect improvement (relative to that performance) the next. The opposite will also be true: performances above expectation will be followed by worse performances, on average. This provides a less interesting explanation for the common belief that “it is hard to follow up a great round with another good one” — really, this is mostly accounted for by regression to the mean. So when can we expect regression to the mean to occur? The details are a bit technical, but roughly speaking, as long as the repeated measurements are not perfectly correlated, we will observe some regression to the mean. For an example with golf scores, this would require only that Jordan Spieth's scores are not perfectly correlated from one day to the next; if not, we will observe regression to the mean when comparing Spieth's performances on consecutive days [2] The example mentioned above in relation to golfer performance following a great round hints at why regression to the mean needs to be considered when trying to tease out cause and effect from data. Thinking, Fast and Slow , the psychologist-turned-economist Daniel Kahneman provides another example: Kahneman writes of his time working with the Israeli Air Force, and how one of the flight instructors remarked that when a student performed well on a manoeuvre and received praise for it, their subsequent performance on the manoeuvre was worse. The opposite was true for students who performed poorly on their initial attempt: criticism by the instructor was followed by improved performance. Assuming that pilot performance is at least in part due to luck, the instructor had mistakenly given a causal explanation to what was actually just random fluctuations in performance. It should now be clear why regression to the mean is related to an analysis of the impact of coaching changes in golf. We found that golfers who switch coaches tend to perform poorly leading up to the switch and show improved performance afterwards. By focusing on golfers who switch coaches, we have inadvertently selected for players experiencing spells of poor performance. Given the considerable day-to-day variation in a golfer’s scores, we know that some of this performance dip is likely in part due to temporary underperformance (i.e. bad luck ), and therefore we need to account for regression towards the mean when analyzing their subsequent performance. Put another way, to better estimate the causal effect of changing coaches on performance, we require a reasonable control group for our coach-switchers: that is, a group of golfers who look similar to the golfers who switched coaches, but did not themselves undergo a coaching change . Given the findings from our earlier work, we should look for golfers in performance slumps and compare how they perform in subsequent rounds to the players who switched coaches. This control group will give a rough indication of how much of the improvement observed after switching coaches can be accounted for by regression to the mean. On to the analysis. We consider any player who played at least 400 rounds in our data from 2004-2019; for each golfer, we randomly select a date near the middle of their sample and label this as their "coach-switching date". Performance is then analyzed before and after this hypothetical switching date. The performance metric of interest is defined as strokes-gained relative to a player's baseline skill, where baseline skill is simply defined to be the average true strokes-gained in the 100 rounds from 130 rounds to 30 rounds before the switch occurred (i.e. positions -130 to -30 on the x-axis of the plot below). Groups of players are then formed based on their deviation in performance from this calculated baseline in the 30 rounds leading up to the switch (i.e. rounds -30 to 0). The plot below shows the 30-round moving average for each group before and after the switch date (0 on the x-axis). Notes: Data is from 2004-2019 on PGA, Web.com, and European tours. To be included in the analysis, a player had to have played at least 400 rounds and not be one of the players in our coaching sample. Point 0 on the x-axis represents a randomly chosen date for each player. Plotted is a 30-round moving average of strokes-gained relative to baseline, where the baseline is defined as the golfer's strokes-gained from rounds -130 to -30 on the plot. Groups are formed based off strokes-gained relative to this baseline in rounds -30 to 0. An example for clarity: the blue line aggregates (i.e. it is the average of) all players who averaged more than 1 stroke per round above their baselines leading up to the (hypothetical) switching date. All groups cluster around 0 between positions -130 to -30; this is because each golfer's baseline skill is defined using this 100-round stretch. The main takeaway from the plot is that when a golfer deviates from their baseline for a 30-round stretch, we can expect about 30% of that form to be maintained going forward. The fact that we do not observe full regression to the mean indicates that the skill levels of professional golfers likely vary over time. That is, performing below the previously defined baseline for a 30-round stretch is in part due to "bad luck", but also due to a decrease in skill level (this statement is only true on average across many players). There are several things one could quibble about in this analysis, a few of which are mentioned here Now that we have a rough sense of the role of regression to the mean in golf, let's compare some of our control groups to the group of players that actually switched coaches. The next plot includes the average 30-round moving average for the 39 players who switched coaches alongside the two most relevant control groups from the first plot. Notes: The plotted blue line is the (average) 30-round moving average for the 39 players who switched coaches. The green and red lines are the groups of players who averaged between -0.5 and 0 strokes below baseline, and -1 and -0.5 strokes below baseline, leading up to their "switching dates", respectively. The red line is the average of 154 golfers' performance, while the green line is the average of 85. The plot speaks for itself, but I'll repeat it anyways. The control group of players that on average performed somewhere between 0 and 0.5 strokes below baseline leading up to their imaginary switch dates looks fairly similar to the actual-switchers (i.e. those who did in fact switch coaches) leading up to the switch date. After the coaching switch, the performance of the actual-switchers continues to decline for a brief period, and then steadily climbs and surpasses this control group from about round 50 post-switch onwards. In general, the moving average of the actual-switchers shows more noise because it is comprised of just 39 players, while the -0.5 to 0 control group contains 154 players. There is a lot of statistical noise hidden behind these averages. To get a sense of where the performance of the actual-switchers fits in with what we can reasonably expect just due to randomness, we repeat the following exercise 30 times: from our full sample of golfers (those who played at least 400 rounds in the dataset), randomly set a date for a hypothetical coaching switch for each golfer, calculate their performance relative-to-baseline leading up to this date as described above, and then randomly select 39 golfers (the sample size of our coaching switch sample) who performed between -0.5 and 0 strokes below baseline leading up to the switch date. Put simply, we are going to repeatedly sample 39 golfers who were in performance slumps and then plot their performance in the subsequent 200 rounds. This provides a rough sense of how much the average performance of a group of 39 golfers can vary. The figure below summarizes the results of this exercise. Notes: The plotted red line is the (average) 30-round moving average for the 39 players who switched coaches. Each blue line is the result of a single simulation exercise described in the body of the text. Some of the simulated groups likely contain the same golfers, but it's very unlikely the same date would also have been chosen. Taken together, the cloud of blue lines is meant to give an indication of the variation in performance that is possible for a group of 39 golfers (who didn't switch coaches). Before the switch date, our controls match the actual coaching switch group pretty well, which is by construction. This plot makes it a bit more obvious that we are also selecting on golfers' performance in the rounds from -200 to -130; in this specific case, golfers are more likely to have been underperforming relative to the baseline period. Moving to performance post-switch, we see that the actual switchers' performance is pretty middle of the road in terms of what is expected from rounds 0 to 60; but, from round 100 onwards their performance is in the upper tail of what could arise due to randomness alone. So, what to conclude? I would still be hesitant to say that this is anything more than noise. One point to think about is that once you start looking for substantial deviations in performance at any point from 0 to 200 rounds after the coaching switch, you are much more likely to find one. This is analogous to testing multiple hypotheses; as the number of hypotheses under consideration grows, the probability of finding a statistically significant one (when in reality no "true" effects exist) increases. That being said, the actual-switchers here are consistently near the top part of the control group distribution at every point beyond 100 rounds post-switch; it's not as though you have to isolate a single data point to declare victory here. Relating things back to our original analysis, it seems that some of the improvement post-switch that we attributed to coaching can be explained by regression to the mean. However, the golfers who switched coaches do appear to bring their play back to a level higher than a similar set of slumping golfers who did not change coaches. As a final point, to satisfy the sticklers among you, it still is likely the case that our control group of golfers is different in some meaningful way from the group of golfers who switched coaches. After all, switching coaches is a decision; why one golfer makes that decision and another doesn't could indicate that they are different along some relevant dimensions which could correlate with future performance.	{"url":"https://datagolf.com/regression-to-the-mean-blog","timestamp":"2024-11-04T12:18:54Z","content_type":"text/html","content_length":"97578","record_id":"<urn:uuid:44a14ab7-d977-4cb8-bb20-1e19f0a413de>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00544.warc.gz"}
Cédric Villani, Fields Medalist 2010 : « the general public is not aware of the excellence of French results in mathematics » \| Portail D’Actualités Sur La Science Moderne Et L’Informatique English version. EXCLUSIVE INTERVIEW. Awarded the Fermat prize in Toulouse last May and now the prestigious Fields Medal. This is one of the most important distinctions in mathematics, awarded at the  International Congress of Mathematicians held this year in Hyderabad (India). It has also been awarded to Ngô Bảo Châu, a French national originally from Vietnam. Director of the Institut Poincaré and a professor at the Ecole Normale Supérieure in Lyon, 36-year-old Cédric Villani, is one of today’s most prominent mathematicians. He gave an interview to KwantiK ! at the Astronomy Festival in Fleurance and now offers us his first impressions. You have just been awarded the Fields Medal for your “proofs of nonlinear Landau damping and convergence to equilibrium for the Boltzmann equation.” (see box) What does this distinction mean for you ? I’m very proud. Like all prizes, it’s the result of a lot of effort and sleepless nights. The Fields Medal is awarded to mathematicians who are under 40 and is thus an encouragement to continue with their work, but also to guide and become a sort of ambassador for the upcoming younger generation. Whereabouts is French mathematics on the international scientific stage ? If certain indicators are to be believed, France is second behind the USA. This is mainly in terms of the number of speakers at congresses, either leading plenary sessions, or just invited. So there is an important French presence. The prizes are also an indicator. The French have in particular received 11 Fields Medals, the last being Wendelin Werner in 2006. And the Abel prize, created more recently and aimed at rewarding mathematicians for their body of work, has been won by Jean-Pierre Serre in 2003. What can this excellence be attributed to ? There is obviously the long tradition of mathematics, with Fermat, Fourier, Monge, Galois and also Poincaré, whose work is very much in the news notably because of the Russian Grigoriy Perelman’s proof of his conjecture. Then there is the Bourbaki group of mathematicians active since the fifties which Jean-Pierre Serre belongs to, and which has produced a new, solidly underpinned vision of And specific training plays a key role. Even if the French are not very good up to Lycée (High school) level, the ‘classes préparatoires’ (2 year preparatory classes leading to the competitive entrance exams for the prestigious university schools), whatever people may think about them, boost the best mathematics students. Last May you have also received the Fermat Prize awarded by the Midi-Pyrénées region and the Institut de Mathématiques de Toulouse (IMT) Yes, this prize has now got international status. It rewards work undertaken within Fermat’s main interest fields : number theory, probabilities, calculation of variations, etc. Concerning the Institut de Mathématiques de Toulouse, Michel Ledoux who does research there has been very influential. One of my first important results came to me while reading one of his lectures, and there are many other top quality mathematicians at IMT. In what main areas of mathematics does France stand out ? There’s number theory, probabilities, the partial differential equations, which I work on. We’re so well known in number theory that we can even allow ourselves the luxury of publishing articles in French in international journals, which would never happen in other disciplines ! Is the general public aware of these sorts of things ? They’re not very well known mainly because of the problem of popularizing mathematics. This has been ignored for a long time by most mathematicians, even if things have begun to change over the last few years. It’s taken them a long time to admit that the explanation of what they are doing needs to be ever so slightly simplistic, whereas in mathematics the notion of proof is highly developed with a very logically structured explanation. Can mathematics be popularized ? We shouldn’t hesitate to use examples from around us, make comparisons. For example in physics the partial differential equations enable models to be made for many phenomena such as wave propagation or flow of fluids. But to say that mathematics can just be applied in a certain domain is itself simplistic. Mathematicians don’t think in « applications ». Their bread and butter is in-depth understanding. What counts is the beauty, the « elegance » of a proof, where the different parts mesh together in harmony, like music. And the fact that this same proof has an underlying element of surprise, which embodies its originality. It’s all this which is sometimes difficult to get through to the general public. Interview by Jean-François Haït, for KwantiK ! Translation by Adrian Pavely Cédric Villani for dummies… Landau damping ? Boltzmann equation ? Don’t panic. For those who remember second degree equations as a torture at school, journalist Julie Rehmeyer published an article on the International Congress of Mathematicians website, in which she gives a very accessible explanation of Villani’s work to non-specialists.	{"url":"https://kwantik.fr/cedric-villani-fields-medalist-249.html","timestamp":"2024-11-06T00:57:21Z","content_type":"text/html","content_length":"44606","record_id":"<urn:uuid:dd43f783-c193-4257-9bd5-4b08537c4b7e>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00465.warc.gz"}
ELI5: a quantum web app (in Javascript) I was having a conversation with a friend who is studying Quantum Programming as a part of his Masters degree. As he described the concept of state entanglement to me, I started to feel like the concept is very close to Redux and state management on the web. So, I wrote this post to elaborate some of my thoughts. What is a quantum programming language? A quantum programming language is really a specific type of machine code targeting a quantum computer, which is based on quantum mechanics compared to classical mechanics. A classical computer has a memory made up of bits, where each bit is represented by either a 0 or a 1. A quantum computer, on the other hand, maintains a sequence of qubits, which can represent a 0, a 1, or any quantum superposition of those two qubit states. Quantum computers may be able to efficiently solve problems which are not practically feasible on classical computers, and thanks to the Church–Turing thesis, we know that a classical computer can technically simulate a quantum algorithm. Why Quantum Programming? IBM has unveiled it's cloud quantum computer for public use, Microsoft has revealed the Q# quantum programming language, and D-Wave Systems partnered with NASA to release the D-Wave-2. It looks like Quantum Computing is becoming a reality! Since quantum programming is based on quantum mechanics, it benefits from some of the cool phenomena associated with it. Quantum entanglement is a physical phenomenon which occurs when a group of particles interacts in ways such that the quantum state of each particle cannot be described independently of the state of the others, even when the particles are separated by a large distance. Let’s say we have two particles A and B. Both particles are in superposition of possible states—in this case they both turn red and blue at the same time. If we measure the colour of A, and it picks red, and someone else an instant later measures B—B will always be blue. Entanglement means that somehow B knew, instantaneously, what A picked, regardless of the distance between A and B. States in Web Applications In web applications today, we face a similar challenge in maintaining the same state on the front end and the back end. With growing use of various modern hacks such as Redux/Flux, it is clear that we need a more elegant solution to state management across physical distance. Quantum State Entanglement How is the actual data represented? This is done based on a process called superdense coding. Superdense coding is a method of sending two traditional bits of information (00, 01, 10, or 11) using a single qubit. Lets assume the state of our web app is encoded into an array of such qubits. Our goal would be to entangle the state from the server side with the state on the client side. 1. We add the Hadamard gate (H) with 1 qubit for adding superposition property. 2. We add Controlled-NOT gate (CX) , a two-qubit gate that flips the target qubit if the control is in state 1. This gate generates entanglement. The combination of these two quantum logic operations brings our pair of qubits to a Bell state. Bell State The Bell states are specific quantum states of two qubits that represent the simplest examples of quantum entanglement. The following code can really only run on a quantum computer :P # Assuming we have two qubits, qr[0] and qr[1] # Add the H gate in the Qubit 1, putting this qubit in superposition. # Add the CX gate on control qubit 1 and target qubit 0, putting the qubits in a Bell state i.e entanglement qc.cx(qr[1], qr[0]) # Add a Measure gate to see the state. # Compile and execute the Quantum Program, since it needs to be simulated results = qp.execute(['HelloWorldCircuit'], backend,timeout=2400) Quantum programming is still a highly theoretical field, and we won't really see the performance improvements from it until we build a much more practical quantum processor. However, I hope that being more aware of the paradigms makes you have more to worry about in your future work and projects. :D Top comments (0) For further actions, you may consider blocking this person and/or reporting abuse	{"url":"https://practicaldev-herokuapp-com.global.ssl.fastly.net/prnthh/imagine-a-quantum-web-app-in-javascript-l2d","timestamp":"2024-11-05T17:23:41Z","content_type":"text/html","content_length":"68854","record_id":"<urn:uuid:1507904a-96eb-4842-83ed-6a706cebcd37>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00833.warc.gz"}
Unlocking The Science: Boiling Point Elevation And Colligative Properties Unlocking the Science: Boiling Point Elevation and Colligative Properties Welcome to Warren Institute! In this article, we will dive into the fascinating world of Colligative Properties, specifically focusing on Boiling Point Elevation. Colligative Properties refer to the properties of a solution that depend solely on the number of solute particles present, regardless of their identity. Boiling Point Elevation is one such property, which occurs when the boiling point of a solvent increases due to the addition of a solute. Join us as we explore the underlying concepts, calculations, and real-world applications of this intriguing phenomenon. So, let's delve into the amazing realm of Colligative Properties and unravel the mysteries of Boiling Point Elevation! Understanding Colligative Properties Colligative properties are physical properties of a solution that depend on the concentration of solute particles, rather than their identity. This section provides a detailed explanation of colligative properties and their significance in Mathematics education. In Mathematics education, understanding colligative properties is crucial for comprehending the relationship between the concentration of solute particles and the boiling point elevation. It allows students to apply mathematical formulas and equations to calculate the extent of boiling point elevation based on the number of solute particles present in a solution. Key points: • Colligative properties depend on the concentration of solute particles. • Mathematical formulas and equations can be used to calculate boiling point elevation. • Understanding colligative properties enhances mathematical problem-solving skills. Boiling Point Elevation Formula The boiling point elevation formula is a mathematical equation used to determine the increase in boiling point due to the presence of a solute in a solution. This section explains the formula and its components in detail. The boiling point elevation formula can be expressed as: ΔTb = Kbm • ΔTb represents the change in boiling point • Kb is the molal boiling point elevation constant for the solvent • m is the molality of the solute in the solution Understanding this formula enables students to calculate the change in boiling point accurately and efficiently, facilitating their problem-solving abilities in Mathematics education. Key points: • The formula for boiling point elevation is ΔTb = Kbm. • ΔTb represents the change in boiling point. • Kb is the molal boiling point elevation constant. • m is the molality of the solute in the solution. Applications of Boiling Point Elevation This section explores the practical applications of boiling point elevation in various fields, emphasizing the importance of Mathematics education in understanding and utilizing these applications Applications of boiling point elevation include: • Determining the purity of a substance by comparing its boiling point with the expected elevation. • Estimating the molecular weight of an unknown solute by measuring the boiling point elevation. • Enhancing the efficiency of industrial processes by adjusting boiling points for specific reactions. By comprehending the applications of boiling point elevation, students gain a broader perspective on how Mathematics education can be applied to real-world scenarios. Key points: • Boiling point elevation has practical applications in determining substance purity and estimating molecular weight. • Mathematics education helps in applying boiling point elevation concepts to real-world situations. • Industrial processes can be optimized through the manipulation of boiling points. Experimental Techniques and Data Analysis This section focuses on experimental techniques used to measure boiling point elevation and the subsequent data analysis required for accurate results. It highlights the significance of Mathematics education in performing experiments and interpreting data effectively. Experimental techniques for measuring boiling point elevation may involve: • Conducting controlled experiments with known solutes and solutions. • Collecting data on temperature changes during boiling. • Analyzing the collected data using statistical methods and mathematical models. Mathematics education equips students with the necessary skills to design experiments, collect data, and analyze results accurately, enabling them to contribute to scientific research and innovation. Key points: • Experimental techniques are used to measure boiling point elevation. • Mathematics education plays a crucial role in designing experiments and analyzing data. • Statistical methods and mathematical models aid in data interpretation. frequently asked questions What is the mathematical formula to calculate the boiling point elevation caused by colligative properties? The mathematical formula to calculate the boiling point elevation caused by colligative properties is given by ΔTb = Kb * m, where ΔTb is the boiling point elevation, Kb is the molal boiling point elevation constant for the solvent, and m is the molality of the solute in the solution. How do colligative properties affect the boiling point of a liquid in a mathematical context? Colligative properties affect the boiling point of a liquid in a mathematical context through the concept of molarity. The boiling point elevation is directly proportional to the molality of the solute particles present in the liquid. This relationship is described mathematically by the equation: ∆Tb = Kb * m, where ∆Tb is the change in boiling point, Kb is the molal boiling point elevation constant, and m is the molality of the solute. Can you provide an example of a math problem involving boiling point elevation due to colligative properties? Sure! Here's an example of a math problem involving boiling point elevation due to colligative properties: Problem: A solution is prepared by dissolving 15 grams of sucrose (C12H22O11) in 500 grams of water. The boiling point elevation constant for water is 0.52 °C/m. Calculate the boiling point of the 1. Calculate the molality (m) of the sucrose solution using the formula: molality = moles of solute / mass of solvent in kg First, convert the mass of sucrose to moles: moles of sucrose = mass of sucrose / molar mass of sucrose molar mass of sucrose = (1212) + (221) + (1116) = 342 g/mol moles of sucrose = 15 g / 342 g/mol Next, convert the mass of water to kg: mass of water = 500 g / 1000 = 0.5 kg Now, calculate the molality: molality = (moles of sucrose) / (mass of water in kg) molality = (15/342) / 0.5 2. Calculate the boiling point elevation (∆Tb) using the formula: ∆Tb = (boiling point elevation constant) (molality) ∆Tb = 0.52 °C/m * (molality) 3. Finally, calculate the boiling point of the solution: boiling point of solution = boiling point of pure solvent + ∆Tb For water, the boiling point is 100 °C, so: boiling point of solution = 100 °C + ∆Tb Answer: The boiling point of the solution is 100 °C + (∆Tb). What are some common mathematical equations used to model the relationship between colligative properties and boiling point elevation? Some common mathematical equations used to model the relationship between colligative properties and boiling point elevation include the Van't Hoff equation and the Clausius-Clapeyron equation. How can understanding the mathematics behind colligative properties and boiling point elevation help in real-world applications, such as cooking or industrial processes? Understanding the mathematics behind colligative properties and boiling point elevation can help in real-world applications, such as cooking or industrial processes, by allowing individuals to make more precise calculations and predictions. By using mathematical formulas and equations, one can determine the amount of solute needed to achieve a desired boiling point elevation or the concentration of a solution necessary for a specific outcome. This knowledge is particularly useful in cooking, where precise measurements and control over boiling points are crucial for achieving desired textures and flavors. In industrial processes, understanding these mathematical concepts can aid in optimizing efficiency and cost-effectiveness, as well as ensuring product quality and In conclusion, understanding the colligative properties, specifically boiling point elevation, is crucial in Mathematics education. By grasping the concept of how solute particles affect the boiling point of a solvent, students can enhance their problem-solving skills and analytical thinking. Colligative properties play a significant role in various real-life scenarios, such as determining the concentration of a solution or predicting the behavior of mixtures. By incorporating these topics into the mathematics curriculum, educators can foster a deeper understanding of mathematical concepts and their practical applications. Students who grasp the principles of boiling point elevation can confidently tackle problems related to this topic and apply their knowledge to real-world situations. Therefore, teaching colligative properties, including boiling point elevation, is essential to promoting a comprehensive mathematics education. If you want to know other articles similar to Unlocking the Science: Boiling Point Elevation and Colligative Properties you can visit the category General Education.	{"url":"https://warreninstitute.org/colligative-properties-boiling-point-elevation/","timestamp":"2024-11-06T01:29:27Z","content_type":"text/html","content_length":"110703","record_id":"<urn:uuid:9dc0fce2-5527-4386-aed9-32cdd51a6498>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00223.warc.gz"}
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Applied Statistics for Psychology Need Help Writing an Essay? Tell us about your ESSAY and we will find the best writer for your paper. Write My Essay For Me Dataset: Psych_Data.xlsx can be found in the Virtual Office. You will use this data set to answer all of the questions in the Assignment. Note: When asked to include the interpretation of the results and final conclusions, be sure to include all results, and interpretation of the meaning of the results, and final conclusions that a common person can understand. Make sure the conclusions are written in context based on the content in the question. Make sure you use complete sentences, paragraph form (single spacing), proper grammar, and correct spelling. Minimal or incomplete responses can lose points. Include any Excel results that you use, but do not include Excel results that are not part of your solution. Hint: You are asked to determine “appropriate” tests and methods, and to make calculations. This means that you will have to determine which tests or methods are best and why. When asked for your p-value, if your Excel output shows something like 2.735E-15 (scientific notation for 2.735 x 10^(-15), go ahead and include this value from the output and realize this means the p-value is “near zero.” For your decision responses, you will want to compare your p-value to 0.05 and state whether you will reject Ho or fail to reject Ho. Use the Live Binder for further assistance. There is a link to the Live Binder under every Unit. *Assume alpha = .05 for all questions on this assignment* 1. Compare the different Genders of subjects in the Psych_Data.xlsx data set to determine if there is a statistically significant difference in the mean Suicide Risk Levels (suicidal_inv) between the different gender groups. Be sure to state the type of test, state the Ho and Ha, include all relevant Excel results and plots, and write final conclusions for the full results of the test in context. Be sure that your final conclusions are written in common terms for an average person to understand. (a) What type of hypothesis test will you run? (b) State the Ho and Ha. (c) Include descriptive statistics (mean, standard deviation, sample size) AND an appropriate graphical display that allows you to visually compare the Suicide Risk Levels for the two gender groups. Copy/paste the numerical summaries and graph here AND describe the similarities and differences in the Suicide Risk Levels of the two groups (address central tendency and variation of the two groups in your discussion). (d) Give the test statistic, p-value, and decision from your test results (Include Excel output). (e) Write a full conclusion (in context) for the results on this test in a way that can be understood by a non-statistical person. This answer will be at least 100 words or more. 2. Compare the different Socioeconomic Levels (ses_level) of the students in the Psych_Data.xlsx dataset to determine if there is a statistically significant difference in the mean Anxiety Scores (anx_score) between the different SES levels. Be sure to state the type of test, state the Ho and Ha, include all relevant Excel results and plots, and write final conclusions for the full results of the test in context. Be sure that your final conclusions are written in common terms for an average person to understand. (a) What type of hypothesis test will you run? (b) State the Ho and Ha. (c) Include descriptive statistics (mean, standard deviation, sample size) AND an appropriate graphical display that allows you to visually compare the Anxiety Scores for the three SES groups. Copy/ paste the numerical summaries and graph here AND describe the similarities and differences in the Anxiety Scores of the three groups (address central tendency and variation of the three groups in your discussion). (d) Give the test statistic, p-value, and decision from your test results (include Excel output). Would a post-hoc be necessary – why or why not? (e) Write a full conclusion (in context) for the results on this test in a way that can be understood by a non-statistical person. This answer will be at least 100 words or more. 3. If you recall from Unit 7, you looked at how to measure the relationship (correlation) between any two quantitative variables. You also learned that if two variables are significantly correlated with each other, then one variable can be used to estimate or predict the other. The equation used to make this prediction is called a regression equation. Use the following scatterplot and Excel output to answer the questions. The two variables in this case are Depression Level (dep_scale) and Anxiety Score (anx_score). (a) Based on the scatterplot, which variable is the independent variable? Which variable is the dependent variable? (b) What is the value of the correlation coefficient? Using the scatterplot and the correlation coefficient, does the relationship appear linear? Is there a positive or negative association? Is the association weak, moderate, or strong? (c) Based on the Excel output above, is there convincing evidence of a significant linear relationship between Anxiety Score and Depression Level (yes or no)? Explain how you know. (d) Using the Excel output above, write your prediction equation (should be in the form ). Then, using your equation, predict the Depression Level for a subject with a Anxiety Score of 6.6 (round your answer to two decimal places). SHOW WORK for your calculation. 4. From the Psych_Data.xlsx dataset, use the Before Therapy (pre_ther) and After Therapy (post_ther) scores to answer the question: “Does a short-term integrated therapy method lead to an improvement in one’s mental health status?” Assume the (pre_ther) and (post_ther) scores are appropriate measures of one’s mental health state and the higher the score, the better one’s overall mental health. (a) What type of hypothesis test will you run? (b) State the Ho and Ha. (c) Give the test statistic, p-value, and decision from your test results (include Excel output). (d) Write a full conclusion (in context) for the results on this test in a way that can be understood by a non-statistical person. This answer will be at least 100 words or more. 5. From the Psych_Data.xlsx dataset, use the gender and subs_abuse variables to determine whether there is evidence of a significant association between gender and type of substance abuse (alcohol, narcotics, both, none). (a) What type of hypothesis test will you run? (b) State the Ho and Ha. (c) Include an appropriate graphical display that allows you to visually assess the association between gender and type of substance abuse. Copy/paste the graph here. Comment on any association that appears to be present. (c) Give the test statistic, p-value, and decision from your test results (include Excel output). (d) Write a full conclusion (in context) for the results on this test in a way that can be understood by a non-statistical person. This answer will be at least 100 words or more. Submitting your Project Make sure your name is on your project and saved to your computer (save the file as firstname lastname MM570 Assignment). When you are ready to submit your completed project, complete the steps • Click on Assignments at the top of your course page. • Click on Unit 9 Assignment Dropbox. • Click Add a File to attach your Word doc template (this should be the ONLY file you submit). • Include any comments if you wish. • Click SUBMIT. • You should revisit the Dropbox to view any helpful feedback your instructor has left for you. • Make sure that you save a copy of your submitted and returned assignment. The post Applied Statistics for Psychology appeared first on Scholar Writers. I absolutely LOVE this essay writing service. This is perhaps the tenth time I am ordering from them, and they have not failed me not once! My research paper was of excellent quality, as always. 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Potential Energy: Earth's Gravity Formula Potential Energy: Earth's Gravity Formula Potential energy is energy that is stored in a system. There is the possibility, or potential, for it to be converted to kinetic energy. Gravitational potential energy exists when an object has been raised above the ground. If the object is released from its position it will fall, converting the potential energy to kinetic energy. Like all work and energy, the unit of potential energy is the Joule (J), where 1 J = 1 N∙m = 1 kg m^2/s^2 . potential energy = (mass of the object)(acceleration due to gravity)(height) U = mgh U = potential energy of an object due to Earth's gravity m = the mass of the object g = acceleration due to gravity (9.8 m/s^2) h = height above position with U = 0 (the ground, or floor typically) Potential Energy: Earth's Gravity Formula Questions: 1) A 0.30 kg model airplane is hanging from the ceiling in a child's room. The airplane is hanging 2.50 m above the floor. If the floor is the position with U = 0, what is the potential energy of the model airplane? Answer: The model airplane's mass is m = 0.30 kg, and it is at a position h = 2.50 m above the floor. The potential energy can be found using the formula: U = mgh U = (0.30 kg)(9.8 m/s^2)(2.50 m) U = 7.35 kg m^2/s^2 U = 7.35 J The potential energy due to gravity of the model airplane is 7.35 Joules. 2) A 200.0 kg roller coaster car is at its highest point on its track, 50.0 m above the ground. The coaster car goes over the edge into its "first drop", and starts its trip around its track. At the end, the car comes to a stop beside an elevated deck, where riders are waiting. Here, it is 10.0 m above the ground. How much potential energy is lost between the top of the ramp and where the car Answer: The ground is taken to be the position where U = 0. So, both at the top of the ramp, and where the car came to a stop, the roller coaster car has potential energy. The question asks what the difference between these energies is. The mass of the car is m = 200.0 kg, and the acceleration due to gravity is g = 9.8 m/s^2. At the top of the ramp, the height of the car was h[1] = 50.0 m, and where the car came to a stop the height of the car was h[2] = 10.0 m. With these, the potential energies at these heights can be found: U[1] = mgh[1] U[2] = mgh[2] U[1] = (200.0 kg)(9.8 m/s^2)(50.0 m) U[2] = (200.0 kg)(9.8 m/s^2)(10.0 m) U[1] = 98000 J U[2] = 19600 J To find the lost potential energy, subtract one from the other: U = U[1] - U[2] U = 98000 J - 19600 J U = 78400 J The lost potential energy between the top of the ramp and where the car came to a stop was 78400 Joules.	{"url":"https://www.softschools.com/formulas/physics/potential_energy_earths_gravity_formula/34/","timestamp":"2024-11-08T09:23:32Z","content_type":"text/html","content_length":"16791","record_id":"<urn:uuid:efd29230-8e09-4165-9b26-7df886db1f7e>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00088.warc.gz"}
Price per Brick Calculator The Price per Brick Calculator can calculate the price for each brick based on the total price of all the bricks. To calculate the price per brick, we divide the total price of all the bricks by the number of bricks. Please enter the total price of all the bricks and the number of bricks so we can calculate the price per brick: Price per Calorie Calculator Here is a similar calculator you may find interesting.	{"url":"https://pricecalculator.org/per/price-per-brick-calculator.html","timestamp":"2024-11-12T05:37:02Z","content_type":"text/html","content_length":"6473","record_id":"<urn:uuid:d3d6f9ff-870e-4e46-879e-8a416f370e74>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00851.warc.gz"}
17,886 research outputs found We show how to compute the probabilities of various connection topologies for uniformly random spanning trees on graphs embedded in surfaces. As an application, we show how to compute the "intensity" of the loop-erased random walk in ${\mathbb Z}^2$, that is, the probability that the walk from (0,0) to infinity passes through a given vertex or edge. For example, the probability that it passes through (1,0) is 5/16; this confirms a conjecture from 1994 about the stationary sandpile density on ${\mathbb Z}^2$. We do the analogous computation for the triangular lattice, honeycomb lattice and ${\mathbb Z} \times {\mathbb R}$, for which the probabilities are 5/18, 13/36, and $1/4-1/\pi^2$ respectively.Comment: 45 pages, many figures. v2 has an expanded introduction, a revised section on the LERW intensity, and an expanded appendix on the annular matri Given a finite planar graph, a grove is a spanning forest in which every component tree contains one or more of a specified set of vertices (called nodes) on the outer face. For the uniform measure on groves, we compute the probabilities of the different possible node connections in a grove. These probabilities only depend on boundary measurements of the graph and not on the actual graph structure, i.e., the probabilities can be expressed as functions of the pairwise electrical resistances between the nodes, or equivalently, as functions of the Dirichlet-to-Neumann operator (or response matrix) on the nodes. These formulae can be likened to generalizations (for spanning forests) of Cardy's percolation crossing probabilities, and generalize Kirchhoff's formula for the electrical resistance. Remarkably, when appropriately normalized, the connection probabilities are in fact integer-coefficient polynomials in the matrix entries, where the coefficients have a natural algebraic interpretation and can be computed combinatorially. A similar phenomenon holds in the so-called double-dimer model: connection probabilities of boundary nodes are polynomial functions of certain boundary measurements, and as formal polynomials, they are specializations of the grove polynomials. Upon taking scaling limits, we show that the double-dimer connection probabilities coincide with those of the contour lines in the Gaussian free field with certain natural boundary conditions. These results have direct application to connection probabilities for multiple-strand SLE_2, SLE_8, and SLE_4.Comment: 46 pages, 12 figures. v4 has additional diagrams and other minor change This paper examines the determinants of individual bank failures and acquisitions in the United States during 1984-1993. We use bank-specific information suggested by examiner CAMEL-rating categories to estimate competing-risks hazard models with time-varying covariates. We focus especially on the role of management quality, as reflected in alternative measures of x-efficiency and find the inefficiency increases the risk of failure, while reducing the probability of a bank's being acquired. Finally, we show that the closer to insolvency a bank is, as reflected by a low equity-to-assets ratio, the more likely its acquisition.Bank failures Numerous studies have found that banks exhaust scale economies at low levels of output, but most are based on the estimation of parametric cost functions which misrepresent bank cost. Here we avoid specification error by using nonparametric kernal regression techniques. We modify measures of scale and product mix economies introduced by Berger et al. (1987) to accommodate the nonparametric estimation approach, and estimate robust confidence intervals to assess the statistical significance of returns to scale. We find that banks experience increasing returns to scale up to approximately $500 million of assets, and essentially constant returns thereafter. We also find that minimum efficient scale has increased since 1985.Banks and banking ; Banks and banking - Costs ; Economies of	{"url":"https://core.ac.uk/search/?q=author%3A(Wilson%2C%20W.%20David)","timestamp":"2024-11-06T21:08:17Z","content_type":"text/html","content_length":"135084","record_id":"<urn:uuid:970ed21c-7fe1-4571-92c8-ea4bd17681a1>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00152.warc.gz"}
Discuss the Outdoor Propagation Model - EE-Vibes Discuss the Outdoor Propagation Model Discuss the Outdoor Propagation Model Discuss the Outdoor Propagation Model. Radio transmission in a mobile communications system often takes place over irregular terrain. Estimating PL(d) over a particular area requires terrain profile for propagation over irregular terrain such as • simple curved earth profile • highly mountainous • obstacles: trees, building, all models predict Pr(d) at given point or small area (sector) • wide variations in approach, complexity, accuracy • most based on systematic interpretation of empirical data Some of the commonly used outdoor propagation models are: • Longely Rice • Durkins Model • Okumura Model • Hata Model • Wideband PCS Microcell • PCS Extension to Hata Model • Walfisch – Bertoni Model Longley Rice Model: The Longley–Rice model (LR) is a radio propagation model a method for predicting the attenuation of radio signals for a telecommunication link in the frequency range of 40 MHz to 1000 GHz. Longley-Rice is also known as the “irregular terrain model” (ITM). It was created for the needs of frequency planning in television broadcasting in the United States in the 1960s and was extensively used for preparing the tables of channel allocations for VHF/UHF broadcasting there..The Irregular Terrain Model (ITS) of radio propagation the Longley-Rice model named for Anita Longley & Phil Rice, 1968 is a general purpose model that can be applied to a large variety of engineering problems. The model, which is based on electromagnetic theory and on statistical analyses of both terrain features and radio measurements, predicts the median attenuation of a radio signal as a function of distance and the variability of the signal in time and in space. Longley rice model is applicable to point-to-point communication systems over different kind of terrain. The median transmission loss is predicted using the path geometry of the terrain profile and the refractivity of the troposphere. The Longley Rice Model—- A Computer Program: In 1978, Longley-Rice model is also available as a computer program to calculate large-scale median transmission loss relative to free space loss over irregular terrain for frequencies between 20MHz and 10GHz. For a given transmission path the program takes as its input the transmission frequency, path length, polarization, antenna heights, surface refractivity, effective radius of earth, ground conductivity, ground dielectric constant, and climate. The program also operates on path-specific parameters such as horizon distance of the antennas, horizon elevation angle, angular trans-horizon distance, terrain irregularity, and other specific inputs. Modes Of Operation: Longley-Rice method operates in two modes • Point-to-point Mode Prediction • Area Mode Prediction Point-to-point Mode Prediction: “Longley-Rice” point-to-point model for radio propagation in the Terrain Analysis Package (TAP) when a detailed path profile is available, the path-specific parameters can be easily determined. Area Mode Prediction: When the terrain path profile is not available, the Longley-Rice method provides techniques to estimate the path-specific parameters. There have been many modifications and corrections to the Longley-Rice model since its oroginal publication. One point modification deals with radio propagation in urban areas and this is particularly relevant to mobile radio. This modification introduces an excess term as an allowance for the additional attenuation due to urban clutter near the receiving antenna. The extra term called the Urban “Urban Factor” has been derived by comparing the predictions by the original Longley-Rice model with those obtained by Okumura. • Does not providing a way of determining corrections due to environmental factors in the immediate vicinity of the mobile receiver. • No consideration of correlation factors to account for the effects of buildings and foliage. • No consideration of multipath. Durkin’s Model: In 1969 & 1975, Durkin propose a computer simulator for predicting field strength contours over irregular terrain. Durkin’s model Adopted by the Joint Radio Committee in the U.K. for estimation of effective mobile radio coverage areas. • predicts field strength contours over irregular terrain • adopted by UK joint radio committee • consists of two parts (1) Ground profile • Reconstructed from topographic data of proposed surface along radial joining transmitter and receiver • Models LOS & diffraction derived from obstacles & local scatters • Assume all signal received along radial (no multipath) (2) Expected path loss calculated along the radial • move receiver location to deduce signal strength contour • pessimistic in narrow valleys • identifies weak reception areas well The Durkins’s model is dealt as a case study. In this model a computer simulator called as “Durkin’s path loss simulator” is used to predict large scale phenomena/ path loss. This model calculates the path loss component in radio transmission with respect to the different terrain structures, There are two parts considered in execution of Durkins path loss simulator. The first part of this simulation algorithm assumes that the propagation modeled is in LOS and diffractions from the obstacles are along the radial and the receiving antenna receive its energy along the radial. Also the first part of Durkin’s accesses a database of the service area and it reconstructs the ground/terrain profile information. It is done considering the radial line joining transmitter and receiver. The reflections from surrounding region are excluded for measurements. Then the second part of the algorithm under this model calculates the path loss particularly along the radial. With the help of the entire measurements taken so fall, the signal strength will be calculated. The data base of the topography is considered as a two dimensional array. two dimensional array of the elevation data Each element in the array corresponds to a service area in the region whereas the actual content of the array element contains elevation information that is above the sea level. This is also known as “Digital elevation model”. It is also possible to use interpolation methods to find approximate heights (H) that are observed from radial point of view. Using diagonal interpolation method terrain profile reconstruction can be done. Terrain profile reconstruction interpolated mapping diagram Terrain profile reconstruction transmitter and receiver In the reconstructed terrain profile, the distance D[1 ]to D[4] are taken as, D[1] = R.a D[2] = R.b D[3] = R.c D[4] = R.d also the diagram, shows the approximate heights h[1 ]to h[4 ]for each measuring point so that the elevation parameter will be made clear. For Durking model measurements one of the important consideration is line of sight (LOS) path that is expected to be available between the radio transmitter and receiver (T-R) setup. For checking this a computer program develop should calculate the difference value in height denoted as d[j. ]the height taken into account for finding d[j ]is the height of the ground profile and height of the line that is joining the transmitter and receiver antennas for each and every point along the radial line. multiple diffraction edges example Among the four grades of the problem under non line of sight each are tested one by one for the particular terrain considered and the diffraction edge is detected by calculating the angles made between the line that join the transmitter (T) and the receiver (R ). This is done with reconstructed terrain profile. The maximum value of angle at a point is found at and labeled as (D[i , ]hi). Also reverse process of calculating the angles is found and the line that joins the transmitter (T) and the receiver (R ) is measured. The maximum value of the angle is are calculated for every point on the terrain and the point is labeled as (D[j ], 4[j ]). When the values D[i ]is equal to D[j ]then the terrain profile considered is taken as a “single diffraction edge”. The losses if any, Path loss associated with the line joining T and R are also found. In case the condition for the single diffraction edges does not exist then ‘test’ for two diffraction edges is carried out and so on. For the three diffraction edges case, generally the outer edges should contain one single diffraction edge available in between. The line between the two diffraction edges is measured for this iterative process. The process can be continued till multiple diffraction edges are offered. The Durkin’s method is popular since specific site-specific propagation nature is easily predicted. Also the path loss component (PL) and signal strength can also be Okumura Model: The Okumura model for Urban Areas is a Radio propagation model that was built using the data collected in the city of . The model is ideal for using in cities with many urban structures but not many tall blocking structures. The model served as a base for the Hata Model. Okumura model was built into three modes. The ones for urban, suburban and open areas. The model for urban areas was built first and used as the base for others. Frequency = 200 MHz to 1900 MHz Mathematical formulation: The Okumura model is formally expressed as: L = L(FSL) + A(MU) – H(MG) – H(BG) – Summation of K(Correction) L = Median path loss; unit: Decibel(dB) L(FSL) = The Free Space loss. unit: Decibel(dB) A(MU) = Median attenuation. unit: Decibel(dB) H(MG) = Mobile station antenna height gain factor. H(BG) = Base station antenna height gain factor. K(Correction) = Correction factor gain (such as type of environment, water surfaces, isolated obstacle etc.) Points to note: Okumura model does not provide a mean to measure the Free space loss. However, any standard method for calculating the free space loss can be used. PCS Extension to Hata Model: The European Co-operative for Scientific & Technical (EUROCOST) formed COST-231 • Extend Hatas model to 2GHz L50 (urban)(dB) = 46.3 + 33.9logfc – 13.82 loghte – a(hre) + (44.9-6.55hte)logd + CM fc = frequency from 1500MHz – 2 GHz hte = 30m-200m hre = 1m-10m d = 1km-20km Also read here	{"url":"https://eevibes.com/computing/introduction-to-computing/discuss-the-outdoor-propagation-model/","timestamp":"2024-11-13T20:41:50Z","content_type":"text/html","content_length":"70314","record_id":"<urn:uuid:2ca3c008-bb16-4d7e-8644-a452d0561303>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00449.warc.gz"}
• Controls can now be provided separately for the set_controls() function. • The arguments in fHMM_parameters() for model parameters were slightly renamed as follows: □ mus -> mu □ sigmas -> sigma □ dfs -> df □ Gammas_star -> Gamma_star □ mus_star -> mu_star □ sigmas_star -> sigma_star □ dfs_star -> df_star • The log-normal state-dependent distribution is renamed: lnorm -> lognormal. • Two more state-dependent distributions were added: normal and poisson. • The Viterbi algorithm can be directly accessed via viterbi(). • Renamed simulate_data() -> simulate_hmm() to make the functionality clearer. Furthermore, this function is now exported and can be used outside of the package to simulate HMM data. • download_data() no longer saves a .csv-file but returns the data as a data.frame. Its verbose argument is removed because the function no longer prints any messages. • The utilities (i.e., all functions with roxygen tag @keywords utils) were moved to the {oeli} package. • Extended the time horizon of saved data and updated models for demonstration. • The download_data() function now returns the data as a data.frame by default. However, specifying argument file still allows for saving the data as a .csv file. • The plot.fHMM_model() function now has the additional argument ll_relative (default is TRUE) to plot the relative log-likelihood values when plot_type = "ll". • Significantly increased the test coverage and fixed minor bugs. • Changed color of time series plot from "lightgray" to "black" for better readability. • Added a title to the time series plot when calling plot.fHMM_model(plot_type = "ts"). Additionally, a time interval with arguments from and to can be selected to zoom into the data.	{"url":"https://cloud.r-project.org/web/packages/fHMM/news/news.html","timestamp":"2024-11-13T14:43:28Z","content_type":"application/xhtml+xml","content_length":"7561","record_id":"<urn:uuid:25a1d026-e41d-4047-b1f9-3100cc328cea>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00568.warc.gz"}
Pints to Ounces Conversion \| MyCalcu Pints to Ounces Conversion: Measuring Liquid Ingredients Tired of estimating how many ounces are in a pint? Do you want to improve your liquid ingredient measurement skills? This guide will teach you everything you need to know about converting pints to ounces and measuring liquid ingredients like a pro. Say goodbye to messed-up kitchen measurements and hello to precise, accurate measurements. Continue reading to improve your cooking skills and wow your friends and family with your newfound knowledge. Understanding Pints and Ounces When measuring liquid ingredients, it is critical to understand the distinction between pints and ounces. A pint is a liquid volume measurement unit that equals 16 fluid ounces. Ounces, on the other hand, are a weight measurement unit. When baking or cooking, it is important to understand that factors such as temperature and pressure can cause an ingredient's weight to expand or contract. As a result, volume measurements like pints and ounces can be more reliable than weight measurements like grams or ounces. It's also worth noting that the measurement of pints and ounces varies by country or region. A pint in the United States is equal to 16 fluid ounces, whereas a pint in the United Kingdom is equal to 20 fluid ounces. To avoid confusion, it is critical to remember the measurement system you are using. Conversion Method Converting pints to ounces is a simple process that can be accomplished in a variety of ways. The most common method is to use a conversion calculator or chart to obtain the exact measurement in ounces. Another method is to use the 1 pint = 16 ounces formula. This means that if you want to convert two pints to ounces, simply multiply two by sixteen, which equals 32 ounces. When converting larger amounts, it can be useful to use a measuring cup marked in pints, as this allows you to visually measure the amount of liquid required. It's also worth noting that the United States and the United Kingdom have different measurement standards for pints, so make sure to use the correct conversion chart or calculator for the measurement system you're using. It's also worth noting that when it comes to cooking and baking, it's critical to be as precise with your measurements as possible. Using the proper amount of liquid ingredients can make a significant difference in the final result of your dish, so taking the time to convert your measurements from pints to ounces can be well worth it. One of the most effective methods is to use an online calculator, such as the MyCalcu Pints to Ounces calculator. By simply entering the value and clicking "convert," you can easily convert any number of pints to ounces. The calculator will then calculate the equivalent measurement in ounces for you. Common Liquid Ingredients and their Conversion Water is one of the most basic ingredients in cooking and baking, and it is often measured in pints. One pint of water is equal to 16 ounces. This means that if a recipe calls for 1 pint of water, you can use 16 ounces of water instead. Milk is another commonly used ingredient in cooking and baking, and it is often measured in pints. One pint of milk is equal to 16 ounces. This means that if a recipe calls for 1 pint of milk, you can use 16 ounces of milk instead. Oil is another commonly used ingredient in cooking and baking, and it is often measured in pints. One pint of oil is equal to 16 ounces. This means that if a recipe calls for 1 pint of oil, you can use 16 ounces of oil instead. Vinegar is another commonly used ingredient in cooking and baking, and it is often measured in pints. One pint of vinegar is equal to 16 ounces. This means that if a recipe calls for 1 pint of vinegar, you can use 16 ounces of vinegar instead. It's important to note that these conversions are approximate and may vary depending on the type of ingredient and recipe. It's always best to check the recipe for specific measurements and instructions. Additionally, MyCalcu Pints to Ounces online calculator can be used for easy and instant conversions. Frequently Asked Questions What is 16 ounces equal to in terms of pints? 6 ounces is equal to 1 pint. How do you convert pints to ounces? To convert pints to ounces, you can multiply the number of pints by 16, as there are 16 ounces in a pint. Does 10 ounces equal 1 pint? No, 10 ounces do not equal 1 pint. There are 16 ounces in a pint. Is a pint 6 oz? No, a pint is not 6 oz. A pint is equal to 16 oz. How many pints are in 1 ounce? There is 1/16 of a pint in 1 ounce. How much is a pint of fluid? A pint of fluid is 16 ounces or 473.18 milliliters. Is a pint 20 ounces? No, a pint is not 20 ounces. A pint is equal to 16 ounces. Which is more 1 pint or 16 fluid ounces? Both 1 pint and 16 fluid ounces are equal in measurement. Is 1 bottle of beer a pint? The size of a bottle of beer can vary depending on the brand and type of beer. Generally, a bottle of beer is not equal to a pint. A pint is equal to 16 fluid ounces, while a bottle of beer is typically around 12 fluid ounces. Is a pint of blueberries 6 oz? No, a pint of blueberries is not 6 oz. A pint is equal to 16 ounces. How many ounces are in a pint container of blueberries? There are 16 ounces in a pint container of blueberries. What is the weight of 1 pint of blueberries? The weight of 1 pint of blueberries can vary depending on the variety and size of the berries, but on average it is around 10-12 ounces. How many mL is 8 oz pint? 8 oz pint is approximately 237.3 milliliters. How much liquid is in a pint? A pint contains 16 fluid ounces or 473.18 milliliters of liquid. What is 1 ounce of liquid equal to? 1 ounce of liquid is equal to approximately 29.5735 milliliters. How many ounces are in a quart? A quart is equal to 32 ounces. Is a quart 32 ounces? Yes, a quart is equal to 32 ounces. How many quarts are in a gallon? There are 4 quarts in a gallon. How many ounces are in a gallon? There are 128 ounces in a gallon. Can you convert pints to cups? Yes, you can convert pints to cups. How many cups are in a pint? There are 2 cups in a pint. Can you convert pints to milliliters? Yes, you can convert pints to milliliters. How many milliliters are in a pint? There are approximately 473.176 milliliters in a pint. How do you convert pints to liters? To convert pints to liters, you can use the conversion factor that 1 pint is equal to 0.473176473 liters. How many liters are in a pint? There are approximately 0.473176473 liters in a pint. Is there a quick and accurate way to convert pints to ounces for cooking and baking? Yes, there are several ways to quickly and accurately convert pints to ounces for cooking and baking. One way is to use a conversion chart or calculator, such as MyCalcu's Pints to Ounces online calculator. Another way is to use the conversion factor of 1 pint = 16 ounces, and simply multiply the number of pints by 16 to get the equivalent in ounces. Can I convert between pints and ounces using a kitchen scale? Yes, you can convert between pints and ounces using a kitchen scale. Simply weigh the ingredient in ounces and then divide the weight by 16 to convert it to pints. Keep in mind that kitchen scales may not be as accurate as other methods, such as using a conversion chart or calculator. How do I convert pints to ounces for measuring liquid and dry goods in my pantry? To convert pints to ounces for measuring liquid and dry goods in your pantry, use the conversion factor of 1 pint = 16 ounces. Multiply the number of pints by 16 to get the equivalent in ounces. For example, if you have 2 pints of liquid, you would multiply 2 by 16 to get 32 ounces. How can I convert pints to ounces for measuring laundry detergent or cleaning products? To convert pints to ounces for measuring laundry detergent or cleaning products, use the conversion factor of 1 pint = 16 ounces. Multiply the number of pints by 16 to get the equivalent in ounces. For example, if you have 1 pint of laundry detergent, you would multiply 1 by 16 to get 16 ounces. Are there any standard conversion factors for pints to ounces in my industry? The standard conversion factor for pints to ounces is 1 pint = 16 ounces. This is commonly used in the cooking and baking industry, as well as in the home for measuring pantry goods and cleaning products. However, it is always best to double check with the specific industry or product to ensure accuracy. Summing It Up Understanding and accurately converting between pints and ounces is essential for a wide range of tasks, including cooking and baking, as well as measuring liquid and dry goods, laundry detergent, cleaning products, paint, and even pet food. The MyCalcu Pints to Ounces online calculator is a quick and accurate tool for converting between these units, but understanding the conversion factors and how to use them manually is also important. Converting pints to ounces is a useful skill to have whether you're a professional in a specific industry or a homeowner working on a home improvement project. Overall, knowing how to convert pints to ounces allows for precise measurements and recipe execution, resulting in successful results. 1 year ago No comments yet! Why don't you be the first? Add a comment	{"url":"https://mycalcu.com/blog/pints-to-ounces-conversion","timestamp":"2024-11-05T22:22:49Z","content_type":"text/html","content_length":"28810","record_id":"<urn:uuid:96d3ec77-cf85-48c9-ab1c-c3a345bac408>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00270.warc.gz"}
Math Colloquia - Gromov-Witten-Floer theory and Lagrangian intersections in symplectic topology Gromov introduced the analytic method of pseudoholomorphic curves into the study of symplectic topology in the mid 80's and then Floer broke the conformal symmetry of the equation by twisting the equation by Hamiltonian vector fields. We survey how the techniques of pseudoholomorphic curves have evolved from the construction of numerical invariants of Gromov-Witten invariants, via the homological invariant of Floer homology and to its categorification of Fukaya category as the basic homological algebra of symplectic algebraic topology. If time permits, we will also mention a few applications of the machinery to problems of symplectic topology.	{"url":"http://my.math.snu.ac.kr/board/index.php?mid=colloquia&l=en&page=3&document_srl=627882&sort_index=speaker&order_type=asc","timestamp":"2024-11-11T14:15:01Z","content_type":"text/html","content_length":"43799","record_id":"<urn:uuid:f6701483-ce1a-4a39-ba3e-f46b194fb3bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00321.warc.gz"}
11 KB, 15 years ago, submitted by I have remade the duplicator. This one adds one new feature, and changes the theme of the original duplicator just to be different! Problems with the old duplicator: Did not plant water bricks correctly. Did not plant print bricks correctly. Error spam, everywhere This duplicator fixes all of those errors, with no error spam. Therefore making this duplicator faster and more reliable to not crash the server. Preferences to change Duplicates events. Rotates events. Client duplications can be uploaded to the server. /Duplorcator, /Dup, and /duplicator will work if the original duplicator is not on the server. Fully compatible with the other duplicator, so you can have both if you really wanted to. Allows saving and loading of duplications Admin Only Max Bricks (Admins) Max Bricks (Non-Admins) Selecting Timeout (Admin) Selecting Timeout (Non-Admins) Planting Timeout (Admins) Planting Timeout (Non-Admins) Max Flood Bypass, this determines how many bricks a duplication can be to ignore the flood protection. Rotate Events Trust Level Required to duplicate Max Ghost Bricks, this pref determines how many bricks should be ghosted when you select a duplication. Quick Ghost Bricks, this determines how many ghost bricks should move instantly when you move your brick, any duplication with more ghost bricks will have a slight delay to prevent lag. Admin Only load Admin Only save Client duplication uploading- Who can upload duplications Can load Blockland saves- Can we /loaddup on normal Blockland saves ie: /loaddup Pong Completely changed the duplicator duplication system. Added a few new preferences Completely remade loaddup and savedup. They now save in the Blockland saves folder Speed boost Now work with public bricks Client duplication uploading, you can now load any save/duplication from your Blockland save folder Added /clientload SAVE command so you can client upload your duplication Added /cancelLoad to cancel your client upload. Both of those commands are client sided, if something happens and the server does not have the duplicator this command will work to stop it. Added /reloaddup to reload the duplication you just uploaded	{"url":"https://blockland.online/rtb/addons/160/duplicator","timestamp":"2024-11-08T21:12:23Z","content_type":"text/html","content_length":"210696","record_id":"<urn:uuid:0873ecb3-537c-4c98-8794-be5e69b125da>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00560.warc.gz"}
A quick post on Chen’s algorithm - Aili Summarize by Aili A quick post on Chen’s algorithm 🌈 Abstract The article discusses a new preprint by Yilei Chen that claims to have found a quantum algorithm that can efficiently solve certain lattice problems, which could potentially break some post-quantum cryptography schemes based on lattice problems. 🙋 Q&A [01] Background on Cryptography and Quantum Threats 1. What are the "hard" mathematical problems that cryptographers use to build modern public-key encryption schemes? • The main problems used are: factoring (RSA), discrete logarithm (Diffie-Hellman, DSA), and elliptic curve discrete logarithm (EC-Diffie-Hellman, ECDSA). 2. What is the threat posed by quantum computers to these "hard" problems? • Researchers have devised algorithms that can solve these problems efficiently (in polynomial time) on a powerful enough quantum computer, which has not yet been built. 3. How has the threat of future quantum attacks inspired the cryptography community to respond? • Industry, government, and academia have joined forces to develop "post-quantum" cryptographic schemes based on different mathematical problems that are believed to be resistant to quantum • This includes NIST's Post-Quantum Cryptography (PQC) competition to standardize new quantum-resistant schemes. 4. What are the most popular class of post-quantum schemes, and what mathematical problems do they rely on? • The most popular post-quantum schemes are based on problems related to mathematical objects called lattices, such as the Shortest Independent Vector Problem (SIVP) and the Guaranteed Approximate Shortest Vector Problem (GapSVP). • Examples of NIST-approved lattice-based schemes include Kyber and Dilithium. [02] Implications of Chen's Quantum Algorithm 1. What does Chen's preprint claim to have achieved? • Chen's preprint claims to have found a new quantum algorithm that can efficiently solve the SIVP and GapSVP lattice problems for certain parameters. 2. What are the potential implications if Chen's result is correct? • If the result holds up, it could allow future quantum computers to break some lattice-based post-quantum cryptography schemes that rely on the hardness of these specific lattice problems. • However, the vulnerable parameters are very specific, and the algorithm's concrete complexity is not yet clear, so it may not immediately apply to the recently-standardized NIST lattice-based schemes like Kyber and Dilithium. 3. How do the author and others view the potential impact of this result? • The author describes it as both a "great technical result" and a "mild disaster" for the cryptography community. • Experts are still evaluating the correctness of the algorithm, as significant results have fallen apart upon closer inspection in the past. • The implications will depend on the concrete details of the algorithm's running time and whether it can be improved upon. Shared by Daniel Chen · © 2024 NewMotor Inc.	{"url":"https://aili.app/share/4J19FsX7o859djHqIfJmvH","timestamp":"2024-11-06T07:55:59Z","content_type":"text/html","content_length":"9097","record_id":"<urn:uuid:aad6aff5-6bcd-4ed7-b0dd-67ea6f3d63f0>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00651.warc.gz"}
Compound Interest - Definition & Importance Compound interest is the most frequently encountered concept in our daily lives. If we examine our bank statements, we will notice that some interest is credited to our account each year. For the same principal amount, the interest rate varies from year to year. We can see that interest grows with each passing year. As a result, we can conclude that the interest charged by the bank is not simple interest; rather, it is compound interest, abbreviated as CI. This article will explain what CI is, as well as the formula and derivation of the formula for calculating the CI when compounded annually, half-yearly, quarterly, and so on. Also, through the examples based on real-life CI applications, one can understand why the return on compound interest is greater than the return on simple interest. What is Compound Interest? The other name for Compound interest is compounding interest calculated on a loan or deposit based on both the initial principal and the interest accumulated in previous periods. Compound interest, which is thought to have originated in 17th-century Italy, is “interest on interest” and causes a sum to grow at a faster rate than simple interest, which is calculated only on the principal amount. It is calculated at a rate determined by the frequency of compounding, the greater the number of compounding periods, the greater the compound interest. Therefore, over the same time period, the amount of compound interest earned on $100 compounded at 10% annually will be less than the amount earned on $100 compounded at 5% semi-annually. Because the interest-on-interest effect can generate increasingly positive returns based on the initial principal amount, compounding is sometimes referred to as the “miracle of compound interest.” Compound Interest Formula Letâ€™s go through the formula of compound interest – Compound Interest = P(1+R/100)t-P P = Principal R = Rate T = Time period Compound Interest Example A sum of Rs. 10,000 is borrowed and the rate of interest is 5% per annum. What will be the compound interest for a period of 3 years? From the formula for compound interest, we know that, Compound Interest = P(1+R/100)t-P Here, P = 10,000 ; R = 5% ; T = 3 years ; C.I=? So, Compound Interest will be- C.I. = 10000(1+5/100)3-10000 So, compound interest will be Rs. 1576.25, after 3 years at a 5% annual interest rate. Applications of Compound Interest The compound interest formula’s applications are its mathematical applications in solving real-life problems. The compound interest formula has several applications, which are listed below: 1. Compound interest is not calculated annually (monthly) 2. Population growth and decline 3. Commodity price increases and decreases 4. The item’s value increases and decreases 5. Profit and loss inflation is a term used to describe an increase in profit and loss 6. Transactions at the bank Points You Need To Remember 1. Compound interest (also known as compounding interest) is interest calculated on a deposit or loan’s initial principal plus all accumulated interest from previous periods. 2. Compound interest can be calculated by multiplying the initial principal amount by one and then multiplying the annual interest rate by the number of compound periods multiplied by one. 3. Compounding interest can occur at any time, from continuously to daily to annually. 4. While calculating compound interest, the number of compounding periods makes a significant difference. Math Classes Most of us can agree that practicing math can be tedious at times. There may also be several unanswered questions and a lack of conceptual clarity. If you are experiencing any of these issues, practice online math with Cuemath math classes. can help. Cuemath is one of the best online tutoring platforms that offer one-on-one sessions with the subject’s best tutors to help you understand complex problems and overcome your fear of mathematics. You can go through Cuemath math classes for understanding the concept in an easy way.	{"url":"https://techrapidly.com/compound-interest-definition-importance/","timestamp":"2024-11-10T11:51:31Z","content_type":"text/html","content_length":"64676","record_id":"<urn:uuid:3d8a62cc-6436-4b55-a9d1-1a2910bebb03>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00827.warc.gz"}
e Matematică Titluri: doctor habilitat, profesor universitar Postul: cercetător ştiinţific principal Data naşterii: 8 noiembrie 1948 Lucrări: 170 Limbi străine: Rusa, Engleza Telefon: 73-35-83 Fax: 73-80-27 Curriculum Vitae Domeniul de cercetare Optimizarea discretă, analiza şi proiectarea algoritmilor, teoria jocurilor şi controlul optimal Lucrări importante • On the existence of stationary Nash equilibria for mean payoff games on graphs. Lozovanu, D., Pickl, S., Bul. Acad. Rep. Mold. , Mat. 2023, @(102), p 41-51, ISSN 1024-7696. • Markov Decision Processes and Stochastic Positional Games. Lozovanu, D., Pickl, S., Springer, 2023, 396 p. ISBN 978-3-031-40179-4. • Pure and mixed stationary Nash equilibria for dynamic positional games on graphs. Lozovanu, D., Pickl, S. Proceedings of 19th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, Munchen. Germany, 2023, pp 53-56. • Optimal stationary strategies for stochastic positional games. Lozovanu D. Abstracts of International Conference "Mathematics and Information Technologies: Research and Education", Chisinau, Republic of Moldova, June 26 - 29, 2023, p. 65 — 66. • On Determining stationary Nash equilibria for single controller stochastic games. Lozovanu, D. Abstract of Int. conf. Mathematics and IT, dedicated 75th anniversary of Moldova State University, July 1-3, p 51-52.,2021. • An approach for determining stationary equilibria in a single-controller average stochastic game. Lozovanu D., Pickl S. Chapter * in the book : Frontiers in Dynamic Games (L. Petrosyan at al. eds.), Springer, pp !57-171, 2021, ISBN 978-3-030-23698-4. • On the existence and determining stationary Nash equilibria for switching controller stochastic games. Lozovanu D., Pickl S. Contributions to Game Theory and Management, St Petersburg State University, vol. 14, pp 284-296, 2021, ISBN 2310-2608. • On the existence of stationary Nash equilibria in average stochastic games with finite state and action spaces. Lozovanu D., Pickl S. Contributions to Game Theory and Management, St. Petersburg State University, vol. 13, pp 304-323, 2020, ISBN 978-3-030-23698-4. • Pure and mixed stationary Nash equilibria for average stochastic positional games. Lozovanu D. Chapter * in the book: Frontiers of Dynamic Games (L. Petrosyan et al. eds), Springer, pp131-156, 2019, ISBN 978-3-030-23698-4. • Stationary Nash equilibria for average stochastic positional games. Lozovanu D. Chapter no. 9 in the book: Frontiers of Dynamic Games (L. Petrosyan et al eds.). Springer, 139-163, 2018. ISSN 2363-8516. • Nash equilibria in mixed stationary strategies for m-player mean payoff games on networks. Lozovanu D., Pickl S. Contributions to Game Theory and Management, St.Petersburg, vol. XI, 103-112, 2018. ISSN 2310-2608. • Optimization of Stochastic Discrete Systems and Control on Complex Networks. Lozovanu D., Pickl S. Springer, 2015, 400 p. ISBN: 978-3-319-11832-1. • Determining the optimal strategies for discrete control problems on stochastic networks with discounted costs. Lozovanu D., Pickl S. Discrete Applied Mathematics v. 182, 2015, 169-180. • On Nash equilibria in stochastic positional games with average payoffs. Lozovanu D., Pickl S. Optimization, Control, and Applications in the Information Age (in Honor of Panos M. Pardalos’s 60th Birthday), Springer, Proceedings in Mathematics and Statistics, v. 130,p. 171-186, 2015. • Nash Equilibria Conditions for Stochastic Positional Games. Lozovanu D., Pickl S. Contribution to Game Theory and Managment (Ed.: A. Pertosyan, N. Zenkevich), St. Petersburg University, Vol. 7, 2014, p 204 -219. • Saddle point conditions for abtagonistic positional games in complex decision processes. Lozovanu D., Pickl S. Int. Journal of Computing Anticipatory Systems, v. 26, 2012, 188-194. • The game-theoretical approach to Markov decision problems and determining Nash equilibria for stochastic positional games. Dmitrii Lozovanu Int. Journal of Mathematical Modelling and Numerical Optimization., 2, (2), 2011, pp. 162-174. • A dynamic programming approach for finite Markov processes and algorithms for the calculation of the limit matrix in Markov chains. Stefan Pickl, Dmitrii Lozovanu Optimization., 60 (10), 2011, pp. 1339-1358. • Algorithms for solving discrete optimal control problems with varying time of states’ transitions of dynamical system. Dmitrii Lozovanu, Stefan Pickl Dynamics of Continuous, Discrete and Inpulsive Systems, ser. B: Applications and Algorithms, 17, 2010, pp. 101-111. • Discrete control and algorithms for solving antagonistic dynamic games on networks. Dmitrii Lozovanu, Stefan Pickl Optimization, 58 (6), 2009, pp. 665-683. • Optimization and Multiobjective controlof Time-Discrete Systems. Dmitrii Lozovanu, Stefan Pickl Springer, London, 2009, 285 p. (Monograph). • Algorithms for solving discrete optimal control problems with infinite time horizon and dtermining minimal mean cost cycles in a directed graph as decision support tool. Dmitrii Lozovanu, Stefan Pickl Central European Journal of Operations Research. 17(3), 2009, pp. 255-264. • A constructive algorithm for max-min path problem on energy networks. Dmitrii Lozovanu, Stefan Pickl Applied Mathematics and Computations., 204 (2), 2008, pp. 602-609. • Multiobjective Control of Time-Discrete Sytems and Dynamic Gasmes on Networks. Dmitrii Lozovanu Chapter in the book: Pareto Optimality, Game Theory and Equilibria (edited by A. Chinchuluun, P. Pardalos), Spriger, 2008, pp. 665-756. • Algorithms and the calculation of Nash Ecuilibria for multiobjective control of time-discrete systems and polynomial-time algorithms for dynsamic c-game on networks. Dmitrii Lozovanu, Stefan Pickl European Journal of Operational Research, 181, 2007, pp. 1214-1232. • Algorithms for solving multiobjective discrete control problems and dynamic c-games on networks. Dmitrii Lozovanu, Stefan Pickl Discrete Applied Mathematics, 155, (14), 2007, pp. 1846-1857. • Nash Equilibria for Multiobjective Control of Time-Discrete Systems and Polynomial-Time Algorithm for k-partite Networks. Dmitrii Lozovanu, Stefan Pickl Central European Journal of Operation Research 13(2), 2005, pp. 127-146. • Polynomial time algorithm for determining optimal strategies in cyclic games. Dmitrii Lozovanu IPCO 2004 Conference Proceedings, New York, Springer, 2004, pp. 74-85. • Networks models of discrete optimal control and dynamic games with p players. Dmitrii Lozovanu Discrete Mathematics and Applications, 13, 4, 2001, pp. 126-143. • Optimal paths in network games with p players. D. Lozovanu, R. Boliac, D. Solomon Discrete Applied Mathematics, 99, 1-3, 2000, pp. 339-348. Apartenenţa la societăţi, redacţii • Societatea Matematică din Republica Moldova (SMRM). • Membru Colegiului de Redacţie a revistei „Buletinul Academiei de ştiinţe al Republicii Moldova. Matematica”. • Membru Colegiului de Redacţie a revistei „Computer Science Journal of Moldova”. • Premiu de Stat in Domeniul ştiinţei, Tehnicii şi Producţiei (1998). • Premiu „Best Paper Award” la simposionul „Computing Systems, Discrete Models, Algorithm, Simulation, Information Systems, Networks” pentru contribuţia „Optimization, Monotonicity and the Determination of Nash Equilibria – An Algorithmic Analysis”, CASYS’03 Liege, Belgium, August 11-16, 2003. • Laureat al premiului academicianului Sibirsky (2004). • Premiu Academiei de Ştiinţe a Moldovei (2006). • Premiu "Savantul Anului" (2009). Publicaţii în Computer Science Journal of Moldova, Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica şi Quasigroups and Related Systems	{"url":"http://www.math.md/people/lozovanu-dmitrii/","timestamp":"2024-11-03T17:19:58Z","content_type":"application/xhtml+xml","content_length":"21886","record_id":"<urn:uuid:d3dfd84a-3a51-4a91-96ee-754b0071b9bf>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00258.warc.gz"}
The Grasshopper component ghMath lets you read and execute the math calculations defined with sMath mathematical software (http://en.smath.com). The intention of this Grasshopper plugin is to prove the concept. And hopefully, promote a new approach in the way how design checks' calculations could be done and recorded. If you have never came across sMath - it is basically MathCAD, just quicker and completely for free (free for commercial purposes as well, you can install it without admin rights). So using ghMath you can define the "logic" of calculation using sMath and then rapidly iterate through many sets of input data to optimize the design (e.g using evolutionary solvers). Eventually, you can save the final calculations file as output. Few of potential uses for component are: • Quickly implement math equations/logic into Grasshopper, without many grasshopper blocks. e.g. deflection and/or stress calculation; • Use Galapagos/Ocotopus to to optimize the size of member for full utilization; • Use this plugin in combination with Karamba or manual load take-down scripts to check/optimize elements. As a feature in future, I intend to implement export to .html with results. Currently, the plugin supports all basic math operators, pow, sqrt, min, max, log, and trigonometric functions. It does not support integrals, conditional statements, and any advanced features. I have provided two relatively simple examples, more power can be unleashed by combining the plugin with reading input from excel using Lunchbox plugin, stepping through multiple calcs using Anemone, or Karamba combining analysis output with member checks with sMath. Background for creation of this plug-in: The idea of this plugin came from looking at different ways of producing design calculations. And none of them is ideal. So what does the ideal "design calculation" looks like? In this post I refer to "design calculation" as one in engineering that checks element/connection for compliance to code or first principles. e.g. reinforcement area calculation, steel column buckling check etc. Software/approach independent points: • Engineer should understand trust the calculation. • Code references shown; • Process should work well with other existing processes within the company. Points where "visual software" (MathCAD, sMath, or hand calculations) are better • The logic of calculation must be clearly described. • Output should be visually well formatted with units described. • The calculation must be easily check-able. • Calculation should be adjustable to particular project needs. • Inputs and outputs should be clearly indicated. Points where Excel or custom-scripted calculation forms are better: • Calculation should be able to perform checks for static results from many different software. • Tools already developed allow to transfer data from majority of general FEA analysis packages to Grasshopper. e.g. check out the work of my former colleagues from BuroHappold https://bhom.xyz/ • Design calculations for many combinations/members should be automated. • Calculation should be able to interact with optimization tools. e.g. evolutionary solvers. Additional points: • Saved calculation file must be in “open format” (.e.g. sMath saves data in XML, whereas Mathcad in "closed format") ghMath and the general approach of creating "visual calculation" and then pushing it through automation process aims to combine the benefits of two software groups mentioned above. For instructions on installing Grasshopper Add-Ons, please see for details. ghMath v0.01 alpha Grasshopper for Rhino 4 & 5 for Win Grasshopper for Rhino 6 for Win ghMath simple examples Grasshopper for Rhino 4 & 5 for Win Grasshopper for Rhino 6 for Win ghMath batch processing example Grasshopper for Rhino 4 & 5 for Win Grasshopper for Rhino 6 for Win	{"url":"https://www.food4rhino.com/en/app/ghmath","timestamp":"2024-11-04T06:30:30Z","content_type":"text/html","content_length":"70725","record_id":"<urn:uuid:a5fa619b-0b7e-47e3-95a9-f8487f95e746>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00294.warc.gz"}
1263 Joules Per Second To Watt (Convert (J/s) To (W)) About Units Joules per second: Joules per second (J/s) is a unit of power equivalent to watts in the International System of Units (SI). One joule per second is the rate at which one joule of energy is transferred or converted in one second. It is commonly used in scientific contexts to measure energy transfer rates. Watt: The watt (W) is a unit of power in the International System of Units (SI), equivalent to one joule per second. It is named after James Watt, an 18th-century Scottish inventor. Watts are commonly used to measure the rate of energy transfer in electrical systems. For example, a typical incandescent light bulb consumes about 60 watts of power.	{"url":"https://www.calculatemix.com/power/1263-joules-per-second-to-watt/","timestamp":"2024-11-09T13:01:30Z","content_type":"text/html","content_length":"56173","record_id":"<urn:uuid:920e10e1-cca8-4d7f-b6c9-b36a2f77c271>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00855.warc.gz"}
• public class SizeSequence extends Object object efficiently maintains an ordered list of sizes and corresponding positions. One situation for which might be appropriate is in a component that displays multiple rows of unequal size. In this case, a single object could be used to track the heights and Y positions of all rows. Another example would be a multi-column component, such as a JTable, in which the column sizes are not all equal. The JTable might use a single SizeSequence object to store the widths and X positions of all the columns. The JTable could then use the SizeSequence object to find the column corresponding to a certain position. The JTable could update the SizeSequence object whenever one or more column sizes changed. The following figure shows the relationship between size and position data for a multi-column component. In the figure, the first index (0) corresponds to the first column, the second index (1) to the second column, and so on. The first column's position starts at 0, and the column occupies size[0] pixels, where size[0] is the value returned by getSize(0). Thus, the first column ends at size[0] - 1. The second column then begins at the position size[0] and occupies size[1] (getSize(1)) Note that a SizeSequence object simply represents intervals along an axis. In our examples, the intervals represent height or width in pixels. However, any other unit of measure (for example, time in days) could be just as valid. Implementation Notes Normally when storing the size and position of entries, one would choose between storing the sizes or storing their positions instead. The two common operations that are needed during rendering setSize(index, size) . Whichever choice of internal format is made one of these operations is costly when the number of entries becomes large. If sizes are stored, finding the index of the entry that encloses a particular position is linear in the number of entries. If positions are stored instead, setting the size of an entry at a particular index requires updating the positions of the affected entries, which is also a linear calculation. Like the above techniques this class holds an array of N integers internally but uses a hybrid encoding, which is halfway between the size-based and positional-based approaches. The result is a data structure that takes the same space to store the information but can perform most operations in Log(N) time instead of O(N), where N is the number of entries in the list. Two operations that remain O(N) in the number of entries are the insertEntries and removeEntries methods, both of which are implemented by converting the internal array to a set of integer sizes, copying it into the new array, and then reforming the hybrid representation in place. □ Constructor Summary Constructor Description SizeSequence() Creates a new SizeSequence object that contains no entries. SizeSequence(int numEntries) Creates a new SizeSequence object that contains the specified number of entries, all initialized to have size 0. SizeSequence(int[] sizes) Creates a new SizeSequence object that contains the specified sizes. SizeSequence(int numEntries, int value) Creates a new SizeSequence object that contains the specified number of entries, all initialized to have size value. □ Method Summary All Methods Instance Methods Concrete Methods Modifier and Type Method Description int getIndex(int position) Returns the index of the entry that corresponds to the specified position. int getPosition(int index) Returns the start position for the specified entry. int getSize(int index) Returns the size of the specified entry. int[] getSizes() Returns the size of all entries. void insertEntries(int start, int length, int value) Adds a contiguous group of entries to this SizeSequence. void removeEntries(int start, int length) Removes a contiguous group of entries from this SizeSequence. void setSize(int index, int size) Sets the size of the specified entry. void setSizes(int[] sizes) Resets this SizeSequence object, using the data in the sizes argument. ☆ Methods declared in class java.lang.Object clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait □ Constructor Detail ☆ SizeSequence public SizeSequence() Creates a new SizeSequence object that contains no entries. To add entries, you can use insertEntries or setSizes. See Also: ☆ SizeSequence public SizeSequence(int numEntries) Creates a new SizeSequence object that contains the specified number of entries, all initialized to have size 0. numEntries - the number of sizes to track NegativeArraySizeException - if numEntries < 0 ☆ SizeSequence public SizeSequence(int numEntries, int value) Creates a new SizeSequence object that contains the specified number of entries, all initialized to have size value. numEntries - the number of sizes to track value - the initial value of each size ☆ SizeSequence public SizeSequence(int[] sizes) Creates a new SizeSequence object that contains the specified sizes. sizes - the array of sizes to be contained in the SizeSequence □ Method Detail ☆ setSizes public void setSizes(int[] sizes) Resets this SizeSequence object, using the data in the sizes argument. This method reinitializes this object so that it contains as many entries as the sizes array. Each entry's size is initialized to the value of the corresponding item in sizes. sizes - the array of sizes to be contained in this SizeSequence ☆ getSizes public int[] getSizes() Returns the size of all entries. a new array containing the sizes in this object ☆ getPosition public int getPosition(int index) Returns the start position for the specified entry. For example, returns 0, is equal to is equal to , and so on. Note that if index is greater than length the value returned may be meaningless. index - the index of the entry whose position is desired the starting position of the specified entry ☆ getIndex public int getIndex(int position) Returns the index of the entry that corresponds to the specified position. For example, getIndex(0) is 0, since the first entry always starts at position 0. position - the position of the entry the index of the entry that occupies the specified position ☆ getSize public int getSize(int index) Returns the size of the specified entry. If index is out of the range (0 <= index < getSizes().length) the behavior is unspecified. index - the index corresponding to the entry the size of the entry ☆ setSize public void setSize(int index, int size) Sets the size of the specified entry. Note that if the value of index does not fall in the range: (0 <= index < getSizes().length) the behavior is unspecified. index - the index corresponding to the entry size - the size of the entry ☆ insertEntries public void insertEntries(int start, int length, int value) Adds a contiguous group of entries to this SizeSequence. Note that the values of start and length must satisfy the following conditions: (0 <= start < getSizes().length) AND (length >= 0). If these conditions are not met, the behavior is unspecified and an exception may be thrown. start - the index to be assigned to the first entry in the group length - the number of entries in the group value - the size to be assigned to each new entry ArrayIndexOutOfBoundsException - if the parameters are outside of the range: (0 <= start < (getSizes().length)) AND (length >= 0) ☆ removeEntries public void removeEntries(int start, int length) Removes a contiguous group of entries from this SizeSequence. Note that the values of start and length must satisfy the following conditions: (0 <= start < getSizes().length) AND (length >= 0). If these conditions are not met, the behavior is unspecified and an exception may be thrown. start - the index of the first entry to be removed length - the number of entries to be removed	{"url":"https://cr.openjdk.org/~iris/se/11/spec/fr/java-se-11-fr-spec/api/java.desktop/javax/swing/SizeSequence.html","timestamp":"2024-11-02T18:35:08Z","content_type":"text/html","content_length":"27929","record_id":"<urn:uuid:26feae8c-8126-4cb1-ab50-0c272cd92da1>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00175.warc.gz"}
Nasir Firoz Khan Creator of Durham University Maths & Natural Sciences Assessment Platform. Currently, I work at J.P. Morgan Chase, where I work on External Regulatory Reporting and automation. Besides work, I actively travel, read, & follow stories about immigration, partition stories, visual arts, structures and designs, etc. Reach out at: nasirkhaan786@gmail.com Nasir Firoz's activity	{"url":"https://numbas.mathcentre.ac.uk/accounts/profile/909/","timestamp":"2024-11-03T16:06:56Z","content_type":"text/html","content_length":"23101","record_id":"<urn:uuid:985c7cf7-7f67-4ee7-866f-3e99e489b1a5>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00696.warc.gz"}
3.6 Polar Coordinates Home \| 18.013A \| Chapter 3 Tools Glossary Index Up Previous Next 3.6 Polar Coordinates A 2-vector (x, y) can be described by two numbers that are not coefficients in a sum: its length, and the angle its vector makes with the x axis. The first of these is usually written as r, the second as These parameters obey the inverse relations are r and Calculating the angle Here is something that works: This gives theta in the range -	{"url":"https://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter03/section06.html","timestamp":"2024-11-12T06:58:11Z","content_type":"text/html","content_length":"3651","record_id":"<urn:uuid:797b944f-b7ab-4aa2-b79e-5f8a4d46bec2>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00296.warc.gz"}
Derivatives - Trading Strategies Derivatives – Trading Strategies Derivatives are financial instruments whose value is derived from the performance of an underlying asset, index, or rate. They play a crucial role in the financial markets, allowing traders and investors to hedge risks, speculate on price movements, and leverage positions. Trading strategies involving derivatives can be complex and varied, ranging from straightforward approaches to sophisticated multi-leg strategies. Below, we explore several key derivatives trading strategies, their mechanics, and their applications. 1. Hedging Hedging is one of the primary uses of derivatives. It involves taking a position in a derivative to offset potential losses in an underlying asset. Common hedging strategies include: • Using Futures Contracts: For example, a farmer expecting to harvest wheat in six months can sell wheat futures to lock in a price. If the market price falls at harvest time, the farmer benefits from the futures position, offsetting losses in the cash market. • Options for Hedging: An investor holding a portfolio of stocks may purchase put options to protect against a decline in stock prices. This strategy allows the investor to sell shares at a predetermined price, limiting potential losses. 2. Speculation Speculation involves taking positions in derivatives to profit from expected price movements. Traders use various strategies based on market analysis and personal judgment: • Long Call Options: A trader bullish on a stock might buy call options, betting that the stock price will rise above the strike price before expiration. If successful, the trader can buy the stock at the lower strike price and sell it at the current market price. • Shorting Futures: A trader who believes a commodity’s price will decrease can sell futures contracts. If the price drops, the trader can buy back the contracts at a lower price, making a profit from the difference. 3. Arbitrage Arbitrage takes advantage of price discrepancies between markets or instruments. Derivatives are ideal for arbitrage strategies due to their liquidity and leverage: • Statistical Arbitrage: Traders might exploit the price difference between correlated assets. For instance, if two stocks typically move together and one diverges significantly from its historical price relationship, a trader could short the overvalued stock and go long on the undervalued stock. • Convertible Arbitrage: This strategy involves trading a convertible bond and its underlying stock. If the bond is undervalued relative to the stock, traders might buy the bond and short the stock, profiting from the eventual convergence of prices. 4. Spread Strategies Spread strategies involve simultaneously buying and selling different derivatives to capitalize on price differences: • Vertical Spread: A trader can create a bull call spread by buying a call option at a lower strike price while simultaneously selling a call option at a higher strike price. This limits potential losses while allowing for a profit if the underlying asset rises. • Calendar Spread: This strategy involves buying and selling options with the same strike price but different expiration dates. For example, an investor might buy a long-term call option and sell a short-term call option, hoping to profit from time decay and volatility differences. 5. Straddle and Strangle These strategies are employed when traders expect significant price movements but are uncertain about the direction: • Straddle: A trader buys a call and a put option at the same strike price and expiration. If the asset moves significantly in either direction, the gains from one option can offset the losses on the other. • Strangle: Similar to a straddle, a strangle involves buying out-of-the-money call and put options with the same expiration but different strike prices. This strategy is generally cheaper than a straddle but requires a more significant price movement to be profitable. 6. Risk Reversal A risk reversal involves combining a long call and a short put (or vice versa) to create a synthetic long or short position: • Long Risk Reversal: This strategy is employed by a bullish trader who buys a call option while simultaneously selling a put option. This provides leveraged exposure to the underlying asset with lower upfront costs. • Short Risk Reversal: Conversely, a bearish trader can sell a call and buy a put to profit from a declining market. 7. Delta Hedging Delta hedging is a sophisticated strategy used by options traders to maintain a neutral position in the underlying asset. Delta measures the sensitivity of an option’s price to changes in the price of the underlying asset: • Traders will adjust their positions in the underlying asset to offset the delta of their options. For example, if a trader holds call options with a delta of 0.6, they might short a quantity of the underlying asset equal to the delta to hedge against price fluctuations. Trading strategies using derivatives are diverse and can be tailored to fit various risk tolerances, market conditions, and investment objectives. Whether one is hedging against risks, speculating on price movements, or implementing complex arbitrage strategies, understanding the mechanics of these instruments is crucial for success. Traders must also remain aware of the inherent risks associated with derivatives, such as leverage and market volatility, and implement risk management practices to protect their investments. With the right strategies and knowledge, derivatives can be powerful tools for enhancing portfolio performance and achieving financial goals.	{"url":"https://derivativeswiz.com/index.php/trading-strategies/","timestamp":"2024-11-12T20:14:22Z","content_type":"text/html","content_length":"63419","record_id":"<urn:uuid:c53a34bb-db39-4666-aa96-e753e968d83f>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00075.warc.gz"}
DAVID ZMIAIKOU - Research Proposition. The stabilizer of a generalized quaternion origami for the action of the group GL(2,Z) is a congruence subgroup of index 3 or 6. Theorem. For any even natural n, the length of a shortest integer sequence on a circle containing all permutations of the set {1,2,...,n} as subsequences is at most n^2/2. Scientific interests I am currently working in the following fields (in alphabetical order): Combinatorial and Geometric Group Theory, Dynamics, Graphs, Number Theory, Permutation Groups, Representation Theory, Teichmüller Theory, Translation Surfaces 8. Even permutations with small support as commutators of generating pairs, Preprint 2014 (PDF under construction) 7. All even permutations with large support are commut-ators of generating pairs, Preprint 2014, submitted (PDF) 6. An average sum on SL[2](Z)-orbits of square-tiled surfaces in H(2), To appear in Discrete and Continuous Dynamical System - A (PDF) 5. [with Maksim Bezrukov] Growth competition on Cayley graphs of groups, Preprint 2013 (PDF under revision) 4. [with Carlos Matheus and Jean-Christophe Yoccoz] Homology of origamis with symmetries, Annales de l'Institut Fourier, Volume 64, 2014 (PDF) 3. The probability of generating the symmetric group with a commutator condition, Preprint 2012 (PDF) 2. [with Emmanuel Lecouturier] On a conjecture of H. Gupta, Discrete Mathematics, Volume 312, Issue 8, 28 April 2012, Pages 1444-1452 (PDF, reports) 1. Interpolation by polynomials, Belarusian State University, Minsk, 2003 (PDF - in Russian) Other works 2. Origamis and permutation groups, Ph.D. thesis, University Paris-Sud, Orsay, 2011 (PDF, abstract, Jury, reports) 1. Actions of groups on origamis, Scientific Internship report, Ecole Polytechnique, France, 2006 (PDF - in French) Talks and presentations 2000-2003 ● several talks on diophantine approximations, inequalities and algebraic interpolations, Belarus 2004-2006 ● presentations on Alexandroff's theorems, Knot Theory, Poincaré Duality and Picard's theorems, France 2007 ● Ergodic Theory and Dynamics seminar, Orsay, France 2008 ● workshop "Dynamique dans l'espace de Teichmüller", Roscoff, France 2008 and 2009 ● Groups, graphs and matrices, Lyceum Corot, Savigny-sur-Orge, France January 2011 ● Square-tiled surfaces and permutation groups, Dynamical Systems seminar of Jussieu, Paris, France March 2011 ● Square-tiled surfaces and permutation groups, "Le Teich" seminar, Marseille, France May 2012 ● Les maths, à quoi ça sert ?, Collège Jean-Baptiste Dumas, Salindres, France June 2012 ● A survey on square-tiled surfaces, Colloquium da UMI, IMPA, Rio de Janeiro, Brazil June 2012 ● Generating the symmetric group, First Palis Balzan International Symposium on Dynamical Systems, IMPA, Rio de Janeiro, Brazil October 2013 ● An average sum on GL(2,Z)-orbits of square-tiled surfaces in H(2), IRMA, Strasbourg, France November 2013 ● An average sum on GL(2,Z)-orbits of square-tiled surfaces in H(2), University Paris 13, France February 2014 ● A talk at the Jacobs University, Bremen, Germany Summer schools & workshops 2003 ● Diophantine analysis, uniform distributions and applications, conference, Minsk, Belarus Sept. 2006 ● Analytic aspects of low dimensional geometry, symposium, University of Warwick, Coventry, UK June-July 2007 ● Clay Mathematics Institute Summer School on Homogeneous Flows, Moduli Spaces and Arithmetic, CRM: Ennio De Giorgi, Pisa, Italy May 2008 ● Congress in memory of Adrien Douady, IHP, Paris, France June 2008 ● Dynamique dans l'espace de Teichmüller, workshop, Roscoff, France June-July 2008 ● Dynamical Systems, workshop, ICTP, Trieste, Italy June 2009 ● Dynamics and Geometry of Teichmüller Space, CIRM, Luminy, France April 2010 ● Dynamics and PDE's, Institute Mittag-Leffler, Stockholm, Sweden May 2011 ● Billiards, Flat Surfaces, and Dynamics on Moduli Spaces, Mathematisches Forschungsinstitut Oberwolfach, Germany June 2012 ● First Palis Balzan International Symposium on Dynamical Systems, IMPA, Rio de Janeiro, Brazil September 2012 ● Algebraic Geometry for the Flats, school, Roscoff, France February 2014 ● New trends in Teichmüller theory and mapping class groups, Mathematisches Forschungsinstitut Oberwolfach, Germany	{"url":"https://www.zmiaikou.com/research","timestamp":"2024-11-08T15:31:35Z","content_type":"text/html","content_length":"71231","record_id":"<urn:uuid:1106627f-7f23-4572-8dc9-e13fed4f531b>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00501.warc.gz"}
Concentration Properties of the Langevin Algorithm's Stationary Distribution for Convex and Strongly Convex Potentials Concentration Properties of the Langevin Algorithm's Stationary Distribution for Convex and Strongly Convex Potentials Core Concepts This note proves that for any reasonable step size, the stationary distribution of the Langevin Algorithm exhibits sub-exponential concentration for convex potentials and sub-Gaussian concentration for strongly convex potentials. Translate Source To Another Language Generate MindMap from source content Concentration of the Langevin Algorithm's Stationary Distribution Altschuler, J.M., Talwar, K. Concentration of the Langevin Algorithm’s Stationary Distribution. arXiv:2212.12629v2 [stat.ML] 21 Oct 2024 This paper investigates the concentration properties of the stationary distribution (πη) of the Langevin Algorithm, a popular method for sampling from log-concave distributions, particularly in terms of its similarity to the stationary distribution (π) of the continuous Langevin Diffusion. Deeper Inquiries How do these findings impact the practical application of the Langevin Algorithm in high-dimensional settings, where concentration of measure becomes increasingly important? These findings have significant implications for the practical application of the Langevin Algorithm, particularly in high-dimensional settings where concentration of measure is crucial: Improved Confidence in High Dimensions: The paper demonstrates that the stationary distribution of the Langevin Algorithm (LA), $\pi_\eta$, inherits desirable concentration properties from the target distribution $\pi$, such as sub-Gaussianity for strongly convex potentials and sub-exponentiality for convex potentials. This is particularly important in high dimensions, where the curse of dimensionality could otherwise hinder sampling efficiency. Knowing that $\pi_\eta$ concentrates well around the mode of $\pi$ provides confidence that the LA will generate samples representative of the target distribution, even in high-dimensional spaces. Finite Sample Guarantees: The results extend beyond the stationary distribution to provide concentration bounds for all iterates of the LA. This is practically relevant as it gives guarantees for finite sample sizes, a crucial consideration in real-world applications where running the algorithm for infinite time is infeasible. Robustness to Inexact Gradients: The extension of the analysis to handle inexact gradients is highly relevant to practical implementations. Stochastic gradient estimates are common in large-scale machine learning, and these findings assure that the concentration properties hold even with such approximations, making the LA more robust for real-world problems. Potential for Improved Sampling Efficiency: Understanding the concentration properties of $\pi_\eta$ can guide the selection of appropriate step sizes and other hyperparameters, potentially leading to more efficient sampling. For instance, knowing the sub-Gaussian or sub-exponential parameters of $\pi_\eta$ can inform the number of samples needed to achieve a desired level of accuracy. However, it's important to note: Dependence on Problem Parameters: The concentration bounds, while tight, still depend on parameters like smoothness and strong convexity, which might be unknown or difficult to estimate accurately for complex distributions. Open Questions: While the paper addresses a significant gap in understanding the LA, open questions remain, such as proving the conjectured tightness of the lower bounds for $\pi_\eta$. Could the techniques used in this paper be adapted to analyze the concentration properties of other sampling algorithms beyond the Langevin Algorithm? Yes, the techniques presented in the paper, particularly the use of a rotation-invariant moment generating function (MGF) as a Lyapunov function, hold promise for analyzing other sampling algorithms. Here's why: Decomposition of Dynamics: The success of the rotation-invariant MGF stems from its ability to elegantly track the effects of both the gradient descent step and the Gaussian noise convolution in the LA update. This decomposition of dynamics is a common feature in many sampling algorithms. Applicability Beyond Gaussian Noise: While the specific form of the Lyapunov function leverages the properties of Gaussian noise, the general principle of using a rotation-invariant function to analyze concentration can be extended. Modifications to the Lyapunov function could be explored to handle other noise distributions. Potential for Other Algorithms: Algorithms like Hamiltonian Monte Carlo (HMC), Stochastic Gradient Langevin Dynamics (SGLD), and their variants could potentially be analyzed using similar techniques. These algorithms often involve a combination of gradient-based updates and noise injection, making them amenable to analysis with appropriately designed Lyapunov functions. However, adapting these techniques to other algorithms might require: Customized Lyapunov Functions: The specific form of the rotation-invariant MGF used in the paper is tailored to the LA. Analyzing other algorithms might necessitate designing new Lyapunov functions that capture the specific dynamics of those algorithms. Handling Complex Dynamics: Algorithms like HMC, with their momentum term, introduce more complex dynamics than the LA. Analyzing such algorithms might require more sophisticated techniques to handle these additional complexities. What are the implications of these findings for the design of more efficient and robust sampling algorithms for complex, high-dimensional distributions encountered in modern machine learning The findings of this paper provide valuable insights that could guide the design of more efficient and robust sampling algorithms for the complex, high-dimensional distributions prevalent in modern machine learning: Beyond Simple Convexity: The analysis for both strongly convex and convex potentials, along with the extension to inexact gradients, broadens the applicability of these techniques to more realistic settings. This encourages the exploration of similar analyses for non-convex settings, which are common in deep learning. Tailoring Algorithms Based on Concentration: A deeper understanding of how different aspects of an algorithm (e.g., step size, noise structure) influence the concentration of the stationary distribution can enable the design of algorithms tailored to specific problem characteristics. For instance, if a problem exhibits a specific type of tail behavior, algorithms could be designed to achieve faster convergence rates by exploiting this knowledge. Incorporating Concentration into Algorithm Design: The use of Lyapunov functions that directly reflect concentration properties could be incorporated into the algorithm design process itself. This could lead to algorithms that explicitly optimize for concentration, potentially leading to faster convergence and improved sampling efficiency. Robustness as a Design Principle: The paper highlights the importance of robustness to inexact gradients. This emphasizes the need to design algorithms that are not overly sensitive to noise in gradient estimates, a crucial consideration when dealing with large datasets and complex models. However, realizing these implications requires overcoming challenges such as: Handling Non-convexity: Extending these techniques to non-convex settings, which are common in deep learning, is crucial but challenging due to the presence of multiple modes and complex energy landscapes. Computational Tractability: Designing Lyapunov functions that are both informative and computationally tractable to analyze can be difficult, especially for complex algorithms. Bridging Theory and Practice: While theoretical guarantees are important, bridging the gap between theory and practice requires careful empirical validation and benchmarking of new algorithms on real-world problems.	{"url":"https://linnk.ai/insight/machine-learning/concentration-properties-of-the-langevin-algorithm-s-stationary-distribution-for-convex-and-strongly-convex-potentials-xo8h81YE/","timestamp":"2024-11-07T19:37:40Z","content_type":"text/html","content_length":"284621","record_id":"<urn:uuid:bc903720-c67f-4cf4-9840-bd04fee5efba>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00796.warc.gz"}
[GAP Forum] p-group Derek Holt D.F.Holt at warwick.ac.uk Thu Jan 28 13:17:50 GMT 2010 Dear GAP Forum, Dear Vivek, You can use the GAP package KBMAG to prove nilpotency of finitely presented groups, using the method described by Charles Sims in his book of computing in finitely presented groups. This uses the Knuth-Bendix completion This process is described and illustrated in Example 4 (p. 13) of the KBMAG manual. I have successfully verifed that your group below is nilpotent of order p^10 for p=2,3,5,7,11,13,17, and I am trying to do 19. Of course, since these groups are (apparently) finite, you could try use coset enumeration. This will work for small primes such as 2 and 3, but for larger primes the group order will probably be too large, and I think the Sims algorithm will work better. You first run NilpotentQuotient (as described in Bettina Eick's reply) to find the maximal nilpotent quotient of your group. The aim is then to prove that the group is actually isomorphic to this quotient. You do this by introducing new generators in the presentation which correspond the power-commutator generators in the maximal nilpotent quotient. You order the generators so that those at the bottom of the group come first and then use the so-called recursive ordering on strings to run Knuth-Bendix. Here is the basic GAP code to do this. j:=F.1;; i:=F.2;; h:=F.3;; g:=F.4;; f:=F.5;; e:=F.6;; d:=F.7;; c:=F.8;; b:=F.9;; a:=F.10;; rels := [a^p/e, b^p/f, c^p/d, e^p/g, f^p/h, g^p/i, i^p/j, j^p, h^p, d^p, Comm(a,b)/i, Comm(a,c)/d, Comm(b,c)/h ];; G := F/rels;; R := KBMAGRewritingSystem(G);; SetOrderingOfKBMAGRewritingSystem(R, "recursive"); If successful it will halt with a confluent presentation containing the relations of the power-commutator presentation of the computed maximal nilpotent quotient. You have then proved that these relations hold in the group itself (not just in the nilptent quotient), so you have proved that the group is nilpotent. This consists of 65 reduction equations (or 62 when p=2). The above works quickly for p=2,3,5,7. For larger primes, it helps to restrict the length of the stored reduction relations, and then re-run after completion. You have to experiment to find the optimal maximal length to store. So, for example, the following works fast for p=17: rels := [a^p/e, b^p/f, c^p/d, e^p/g, f^p/h, g^p/i, i^p/j, j^p, h^p, d^p, Comm(a,b)/i, Comm(a,c)/d, Comm(b,c)/h ];; G := F/rels;; R := KBMAGRewritingSystem(G);; SetOrderingOfKBMAGRewritingSystem(R, "recursive"); O := OptionsRecordOfKBMAGRewritingSystem(R); O.maxstoredlen := [40,40]; Derek Holt. On Wed, Jan 27, 2010 at 08:06:38PM +0530, Vivek Jain wrote: > Dear Forum, > I want to know that: > "Is it possible using GAP to check that given presentation is a nilpotent group of class 2 or not?" > For example $G=\langle a,b,c\| a^{p^5}, b^{p^3}, c^{p^2}, [a,b]=a^{p^3}, [a,c]=c^p, [b,c]=b^{p^2} \rangle $ where $p$ is a prime. > Also how can we determine its automorphism group using GAP? > with regards > Vivek kumar jain > Your Mail works best with the New Yahoo Optimized IE8. Get it NOW! http://downloads.yahoo.com/in/internetexplorer/ > _______________________________________________ > Forum mailing list > Forum at mail.gap-system.org > http://mail.gap-system.org/mailman/listinfo/forum More information about the Forum mailing list	{"url":"https://www.gap-system.org/ForumArchive2/2010/002674.html","timestamp":"2024-11-11T07:24:23Z","content_type":"text/html","content_length":"6483","record_id":"<urn:uuid:2a6b4ada-c667-4b2c-ad53-273113f264ec>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00820.warc.gz"}
Summer 2024 Courses Quick Online Course For Credit Start Immediately Today New! DMAT 431 - Computational Abstract Algebra with MATHEMATICA! Asynchronous + Flexible Enrollment = Work At Your Own Best Successful Pace = Start Now! Earn Letter of Recommendation • Customized • Optional Recommendation Letter Interview Mathematica/LiveMath Computer Algebra Experience • STEM/Graduate School Computer Skill Building NO MULTIPLE CHOICE • All Human Teaching, Grading, Interaction • No AI Nonsense 1 Year To Finish Your Course • Reasonable FastTrack Options • Multimodal Letter Grade Assessment Summer 2024 Enroll Now, Start Today for Course Information - Distance Calculus @ Roger Williams University Finish Quick - Calculus Academic Credits Summer 2024 @ Roger Williams University If you are shopping around for an Applied Calculus = Survey of Calculus course that you can start immediately, and finish quickly (as quickly as your academic skills allow), then Summer 2024 Distance Calculus @ Roger Williams University may be the right program for you. Our Survey of Calculus = Applied Calculus course is not a "canned" multiple choice course like those offered at many other schools and MOOCs (which usually do not offer the academic credits on academic transcript that you need). Applied Calculus has a wonderful curriculum, providing an excellent introductory study of Differential and Integral Calculus without the rigor (and trigonometry) found in the engineering-level Calculus I course. Here is a video about earning real academic credits from Summer 2024 Distance Calculus @ Roger Williams University: Earning Real Academic Credits for Calculus Applied Calculus vs Calculus I Distance Calculus - Student Reviews Date Posted: Jan 8, 2021 Review by: Cristian Mojica Student Email: comojica@ucdavis.edu Courses Completed: Probability Theory Review: A fantastic course! I was able to complete it in about half a year (with a few gaps) alongside other coursework I was completing. There are no deadlines except the one-year mark after registering, so you work at your own rate and schedule. Probability Theory is required for me to apply to Master's programs in Statistics, so I was glad when I found Distance Calculus. While the course was slightly less difficult than I originally expected, there were parts that definitely slowed me down and made me think. (Also, although calculus is not everywhere in the course, it is everywhere in normal and exponential variables and beyond, so make sure to review derivatives and integrals (single and double)!) I used Mathematica for my software, and it helped speed along calculations and proved to be the perfect stage and tool for this material. I think visual learners will absolutely revel in how the material is presented in this course. (I know I did!) As there is plenty of writing and calculation to do, you have many opportunities to develop and strengthen your voice as a mathematician. The modern format of 80% electronic notebook work and 20% handwritten work is an excellent mixture for studying probability theory and grasping its core ideas. Dr. Curtis is clear in his answers to any questions and concerns you may have and is highly responsive to email and chat, and to responses you leave in your notebooks. He truly wants to help you and to see you succeed, and he is always on your side. I highly recommend Probability Theory with Distance Date Posted: Jun 21, 2020 Review by: Abdul J. Courses Completed: Applied Calculus Review: This was the best class! So much more interesting doing the computer math than a boring lecture class. Diane was so responsive and helpful. I recommend this course. Transferred Credits to: Villanova University Date Posted: Feb 25, 2020 Review by: Jessica M. Courses Completed: Applied Calculus Review: I highly recommend this course. I started the Kennedy School at Harvard with a last-minute admission, but my application required the Liberal Arts calculus course, so I had to finish the course in 3 weeks. Diane was an awesome instructor! The class was surprisingly interesting. If you need to take calculus fast, this is the program to use. Transferred Credits to: Kennedy School of Government, Harvard University	{"url":"https://www.distancecalculus.com/summer-2024/courses/start-today/finish-quick/","timestamp":"2024-11-03T15:30:01Z","content_type":"text/html","content_length":"45686","record_id":"<urn:uuid:fd84f6cb-c759-43b2-9afc-2fdc3c4aa6c7>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00133.warc.gz"}
experience Archive - HenryKoch.de Homemade electricity generator for low speeds Why build a generator yourself when everything can be bought somewhere? 1. The only thing I learn from buying is spending money 2. I was unable to find a generator that could generate significant voltage and current at low revolutions of approx. 100 * min-1 3. The generator should be used in a Stirling engine and possibly in a vertically running wind turbine, which does not bring the highest speeds 4. The generator must also be usable as a motor in order to start a Stirling engine 5. The generator has to run very lightly in order to keep losses low, i.e. no sliding contacts and no jerking, as with a bicycle dynamo 6. For use in the Stirling engine, the generator should also represent the flywheel mass, which means that it must have a certain torque 7. The test generator has to withstand my sons’ attempts to play :-) 8. The construction must be simple and it must be possible to assemble it without complex special tools In the following I log a little what I tried to get as close as possible to the goal, which at first glance seems unrealistic. (Neukirchen in April 2009) experiment Power in watts rotation speed Wooden Generator 1 exp. 2 0,0085 fast Wooden Generator 1 exp. 3 0,231 fast Kreissägeblatt Generator Exp. 1 1,6 rpm	{"url":"https://www.henrykoch.de/en/tag/experience","timestamp":"2024-11-13T05:36:14Z","content_type":"text/html","content_length":"68672","record_id":"<urn:uuid:070a74ac-e087-41d9-92e0-94750ce46e6c>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00059.warc.gz"}
Ray-Triangle Intersection Prev: Ray-Sphere Intersection Next: Catmull-Rom Spline Given a ray, i.e. a parametric line equation, and a triangle, do they intersect? and if so what is the intersection point? The solution presented in here is the one from Moller and Trumbore. A point in a triangle can be defined as: point(u,v) = (1-u-v)p0 + up1 + vp2 p0,p1,p2 are the vertices of the triangle u >= 0 v >= 0 u + v <= 1.0 We also know that the parametric equation of the line is: point(t) = p + t d p is a point in the line d is a vector that provides the line's direction So if there is a point that belongs both to the line and the triangle we get: p + t * d = (1-u-v) * p0 + u * p1 + v * p2 Therefore the intersection problem can be redefined as: is there a triplet (t,u,v) that satisfies the equation above, and complies with the restrictions for u and v? If the answer is yes, then the ray intersects the triangle, otherwise it doesn’t. For the maths details of solving this problem see either the reference above or check out the book “Real Time Rendering”. The following C code can be used to test the intersection: /* a = b - c / #define vector(a,b,c) \ (a)[0] = (b)[0] - (c)[0]; \ (a)[1] = (b)[1] - (c)[1]; \ (a)[2] = (b)[2] - (c)[2]; int rayIntersectsTriangle(float p, float d, float v0, float v1, float v2) { float e1[3],e2[3],h[3],s[3],q[3]; float a,f,u,v; a = innerProduct(e1,h); if (a > -0.00001 && a < 0.00001) f = 1/a; u = f * (innerProduct(s,h)); if (u < 0.0 \|\| u > 1.0) v = f * innerProduct(d,q); if (v < 0.0 \|\| u + v > 1.0) // at this stage we can compute t to find out where // the intersection point is on the line t = f * innerProduct(e2,q); if (t > 0.00001) // ray intersection else // this means that there is a line intersection // but not a ray intersection return (false); Check out the cross product and the inner product definitions if you need help. The code above only tells you if the ray intersects or not the triangle. If you want to know where then you can easily alter the code to return the triplet (t,u,v). Using the return value of t, or u and v, the intersection point, i.e. the values x,y,z where the ray intersects the triangle, can be found. Prev: Ray-Sphere Intersection Next: Catmull-Rom Spline 9 Responses to “Ray-Triangle Intersection” 1. You should reference http://paulbourke.net/ Thanks for pointing out t is f * innerProduct though. Now I can sort my hits by depth. 2. Hi! I am trying to determine multiple collision points and deformation of elastic objects. Will a single ray be sufficient to check multiple triangles (i.e. triangle mesh) intersection? 3. Ok it seems the (t, u, v) is indeed the intersection relative to the ray’s starting point (p). Am I correct? □ Hi, The intersection point is pi = p + td; The variable t, computed in the routine, indicates the length we will have to follow direction d, from point p to intersect the triangle. Hope this helps, ☆ That helps a lot ! Exactly what I needed! Thank you !! 4. So is (t, u, v) the (x, y, z) (respectively) intersection ? If not how do I get the (x, y, z) intersection ? □ ‘ u ‘ & ‘ v ‘ are the barycentric (not cartesian) co-ordinates of the intersection point. ‘ t ‘ is the parameter you use to define your ray equation: intersection_point = position + (t direction); You know your position and direction vectors already, ‘ p ‘ and ‘ d ‘, so re-write the code above to tell you what ‘ t ‘ is. Then plug that value of ‘ t ‘ into the ray equation I wrote above, and hey presto, you have your intersection point. 5. Hi can you please tell me how to determine the x, y, z because I really don’t know how to for eg the values of u and v to determine it… □ To get (x, y, z): if (t > 0.00001){ float finalPoint[3]; finalPoint[0] = p[0] + d[0] * t; finalPoint[1] = p[1] + d[1] * t; finalPoint[2] = p[2] + d[2] * t; return (false); This site uses Akismet to reduce spam. Learn how your comment data is processed.	{"url":"http://www.lighthouse3d.com/tutorials/maths/ray-triangle-intersection/","timestamp":"2024-11-06T17:59:04Z","content_type":"text/html","content_length":"94936","record_id":"<urn:uuid:eff7e9d1-489b-4257-9211-7493c77a6457>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00271.warc.gz"}
Lecture Notes in Pattern Recognition: Episode 38 - Adaboost & Exponential Loss - Pattern Recognition Lab Lecture Notes in Pattern Recognition: Episode 38 – Adaboost & Exponential Loss These are the lecture notes for FAU’s YouTube Lecture “Pattern Recognition“. This is a full transcript of the lecture video & matching slides. We hope, you enjoy this as much as the videos. Of course, this transcript was created with deep learning techniques largely automatically and only minor manual modifications were performed. Try it yourself! If you spot mistakes, please let us know! Welcome back to Pattern Recognition. Today we want to continue looking into AdaBoost and in particular, we want to see the relation between AdaBoost and the exponential loss. Image under CC BY 4.0 from the Pattern Recognition Lecture. Boosting fits an additive model in a set of elementary basis functions. So the results of boosting are essentially created by expansion the coefficients β and some b, which is a basis function given a set of parameters γ. Additive expansion methods are very popular in learning techniques, you can see that very similar things are already applied in single hidden layer neural networks building on the perceptron in wavelets, classification trees and so on. Image under CC BY 4.0 from the Pattern Recognition Lecture. The expansion models are typically fit by minimizing a loss function L that is averaged over the training data. You essentially can write this as L over the entire training data, and then you can plug in our definition. And you see that we have this additive model that is essentially given by our function f[m]. So the forward stagewise modelling approximates the solution to one. The new basis functions are sequentially added parameters. Coefficients of already added functions are not changed and at each iteration, only a subproblem of fitting, just a single basis function, is solved. Image under CC BY 4.0 from the Pattern Recognition Lecture. We can express now the m-th subproblem in the following way: We essentially have the loss of the (m – 1)-th solution plus β and the current estimate. We’re minimising over γ and β. Now Adaboost can be shown to be equivalent to forward stagewise additive modelling using an exponential loss function. So if our loss function equals to the exponent of minus y times f(x) we are essentially constructing the Adaboost loss. Image under CC BY 4.0 from the Pattern Recognition Lecture. Let’s prove this. For AdaBoost, the basis functions are the classifiers G[m], and they produce the output of either – 1 or + 1. Using the exponential loss function, we now must solve at every step that we essentially minimise over β and G the sum over the exponential loss. If we now introduce a weight that we introduce here as w[i] and we set it to the exponent of minus y times f[m-1](x[i]). We can we rewrite this as a minimization of a weighted sum of exponential functions. Image under CC BY 4.0 from the Pattern Recognition Lecture. We have some observations with this. Since w[i] is independent of β and G(x) it can be seen as a way that is applied to each observation. However, this weight depends on the previous functions. So the weight changes with each iteration m. Image under CC BY 4.0 from the Pattern Recognition Lecture. This then allows us to reformulate this problem a little. We split it up into the misclassified and the correctly classified samples. Then we can rearrange this minimisation expression. You see that we can again use our indicator function here for representing the misclassified samples. Now for every value of β greater than zero the solution for this minimization process is found as the minimization over the sum of the weight times the indicator function. Image under CC BY 4.0 from the Pattern Recognition Lecture. If we pluck the reformulated G[m] into the objective function and solve for β[m], this yields that β[m] is given as 1 over 2 times the logarithm of (1-err[m]) divided over err[m]. The error m is the minimised weighted error rate, and here you see that it is essentially the misclassification weight summed up, divided by the total sum over the weights. Image under CC BY 4.0 from the Pattern Recognition Lecture. Now from the update formula of the approximation, we can calculate the weights for the next iteration. Here we can now see that we can use this identity here and derive the new weight. So here you’ll see that then α[m] is going to be 2β[m]. Image under CC BY 4.0 from the Pattern Recognition Lecture. If you now compare this result to the AdaBoost algorithm, we can see that the exponential loss had these solutions for β, α[m] and the weight. And if you look into AdaBoost, this had essentially an α that is very much related to the above α respectively β. Also, the weight update takes a very similar form. So you could say AdaBoost is essentially minimising the exponential loss. Image under CC BY 4.0 from the Pattern Recognition Lecture. Now let’s look at the losses that we want to minimise. If you have the misclassification loss that’s hard to minimise because it’s not a complex problem. But if you look into the squared error, this would be a first approximation of the misclassification loss. And then we can see that a better approximation of the misclassification loss is done here by the exponential function, the exponential loss that is produced by AdaBoost. We can also see that, if you take a support vector machine, we essentially end up with a loss that is related to the hinge loss. And we see that also the SVM is solving a convex optimisation problem to adjust a different approximation of the misclassification loss. This is quite interesting. If you’re more interested in the relation between the hinge loss and the SVM, we also have a derivation for this in our class Deep Learning. Image under CC BY 4.0 from the Pattern Recognition Lecture. We could show that the AdaBoost algorithm is equivalent to forward stagewise additive modelling. This was only discovered 5 years after its invention. The AdaBoost Criterion yields a monotone decreasing function of the margin y times f(x). In classification, the margin plays a role similar to the residuals in regression. So, observations with y[i] times f(x) greater zero are classified correctly. Observations with this term smaller than 0 are misclassified and the decision boundary is exactly at f(x) = 0. Image under CC BY 4.0 from the Pattern Recognition Lecture. The goal of the classification algorithm is to produce positive margins as frequently as possible. Thus, any loss criterion should penalize negative margins more heavily than the positive ones. The exponential criterion concentrates much more on the observations with large negative margins. So this is also in relation that we iteratively try to weigh up the samples that are hard to classify. Due to the exponential loss, AdaBoost performance is known to degrade rapidly in situations of noisy data and if there are wrong class labels are in the training data. So again, training data and correct labelling is a key issue, and if you have problems with the labels then AdaBoost may not be the method of choice. Image under CC BY 4.0 from the Pattern Recognition Lecture. Next time in Pattern Recognition we want to look into a very popular application of AdaBoost, and this is going to be face detection. You’ll see that AdaBoost and haar wavelets together essentially solve the task of face detection very efficiently, and this then gave rise to many many different applications. And as you see in many cameras and smartphones, often the face detection algorithm that is used to detect people on the Image and show it on boxes is the AdaBoost. I hope you like this little video and I’m looking forward to seeing you in the next one! Bye-bye. If you liked this post, you can find more essays here, more educational material on Machine Learning here, or have a look at our Deep Learning Lecture. I would also appreciate a follow on YouTube, Twitter, Facebook, or LinkedIn in case you want to be informed about more essays, videos, and research in the future. This article is released under the Creative Commons 4.0 Attribution License and can be reprinted and modified if referenced. If you are interested in generating transcripts from video lectures try AutoBlog. 1. T. Hastie, R. Tibshirani, J. Friedman: The Elements of Statistical Learning, 2nd Edition, Springer, 2009. 2. Y. Freund, R. E. Schapire: A decision-theoretic generalization of on-line learning and an application to boosting, Journal of Computer and System Sciences, 55(1):119-139, 1997. 3. P. A. Viola, M. J. Jones: Robust Real-Time Face Detection, International Journal of Computer Vision 57(2): 137-154, 2004. 4. J. Matas and J. Šochman: AdaBoost, Centre for Machine Perception, Technical University, Prague. https://cmp.felk.cvut.cz/~sochmj1/adaboost_talk.pdf	{"url":"https://lme.tf.fau.de/lecture-notes/lecture-notes-pr/lecture-notes-in-pattern-recognition-episode-38-adaboost-exponential-loss/","timestamp":"2024-11-03T22:32:02Z","content_type":"text/html","content_length":"67700","record_id":"<urn:uuid:b17d0754-55fe-467e-85e0-383fb7463f6f>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00839.warc.gz"}
Belastningsriktningen in English with contextual examples rock types - Studentportal So for example, in the segment CD here. example of tri-axial loading The Carabiner Basics… Let’s start by saying there are many styles and shapes of carabiners out there. Many get used for a variety of hobbies and work-related reasons. Each one, according to its size, shape, and type of metal, has ratings for load strength. 2015-07-24 · Axial Loading - Statically Indeterminate Example 1. Search. Axial Loading - Statically Indeterminate Example 1. 2015-07-24 · Axial Loading - Statically Indeterminate Example 1. Search. Axial Loading - Statically Indeterminate Example 1. Report. Browse more videos. Playing Bending moments cannot be neglected if they are acting on the member. Members with axial compression and bending moment are called beam-columns. Ont i ryggen, ont i nacken - SBU Översätt load på EngelskaKA online och ladda ner nu vår gratis översättare som du kan As a third example, the line: cheeses specifies that the key is "cheeses" and the associated element is the empty string. axial load = carga axial. Some examples are Fibermetric - automated measurements and classification of A large variety of sample holders is available for facilitating the fast loading of any From long axial-shaped samples to moist biomaterials, there is always a Translations of the word LOJA from swedish to english and examples of the belastning: Axial loading: Balk: Beam: Belastning: Load: Böjstyvhet: Bending If, for example, wood is to be used, one starts looking for wood screws. Hållfasthetslära 180112 Axial Load. EXAMPLE 4.1 (SOLN ). The objective of this video is to do an analysis of a member under axial load followed by an example of steel hollow section column. Tina törnqvist stylist It is recognized that the application of such a stress-strain relation in a flexural analysis has been questioned. ('1 Application of weight or force along the course of the long axis of the body. Medical Dictionary for the Health Professions and Nursing © Farlex 2012. Want to 3 Nov 2020 An instrumented pile load test is used to calibrate the method. The following examples are intended to help Itasca software users to become Distribution of shear forces and bending moments in the wall, axial forces in the recommended procedure to simulate seismic loading of an embankment dam. It is therefore of importance that these screws are able to withstand the loads they fatigue testing using axial loading, which is also performed during this work. Gulag arkipelagen svenska ortopedmottagningen lundby sjukhusspecialistundersköterska utbildning växjöheterosexuella betydelselonespecifikation pa swedbanksandstrom center BJÖRN TÄLJSTEN - Dissertations.se For this loading, the pile top moves downward 0.33 mm. Calculation Example - Calculate the Axial Forces of the Truss Members. Calculation Example – Calculate the moments of inertia Ix and Iy. Calculation Example – Calculate shear stress for temperature load. Deep translate in urdubelana cow squishmallow PDF Finite element simulation of the effect of loading rate on A force with its resultant passing through the centroid of a particular section and being perpendicular to the plane of the section. Axial force is the compression or tension force acting in a member. If the axial force acts through the centroid o Strength of Materials: Discusses axial loading, and Saint Venant's Principle. Shows how to caculate axial stress and deflection. View Notes - Axial Loading Examples(2).pptx from ME 212 at George Mason University. AXIAL LOADING EXAMPLES ME 212 – Dr. Erik Knudsen LIST OF KEY SKILLS AND OBJECTIVES LIST OF KEY SKILLS • Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. 2 LECTURE 25. Ont i ryggen, ont i nacken - SBU Light shock Ka : Parallel shaft load (axial load), N for the separating force (Ks) and axial load (Ka) shown in with axial stress. As indicated earlier, the results obtained are valid only as long as the stresses do not exceed the proportional limit, and … Objectives for combined loading problems •Determine the normal and shear stresses at points on a cross section due to combined axial, torsion, and bending loading •Determine the principal stresses and maximum shear stress at these points •Use Mohr’scircle –we will always be in a state plane stress, but not necessarily in the x-y plane 2 The axial load of an object is responsible for the force which passes through the center of the object, is parallel to its axis of rotation, and perpendicular to the plane of cross-section. The force owing to the axial load acts on the central axis of the object, and it can be a compressing or stretching force. Axial loading is defined as applying a force on a structure directly along an axis of the structure. As an example, we start with a one-dimensional (1D) truss member formed by points P 1 and P 2, with an initial length of L (Fig.	{"url":"https://hurmanblirrikojbngna.netlify.app/57342/47584.html","timestamp":"2024-11-05T07:17:56Z","content_type":"text/html","content_length":"10839","record_id":"<urn:uuid:2a7d7dd4-a05f-47ff-b8d9-0ed66f417782>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00405.warc.gz"}
Train-Test Split for Evaluating Machine Learning AlgorithmsData Science News Train-Test Split for Evaluating Machine Learning Algorithms The train-test split procedure is used to estimate the performance of machine learning algorithms when they are used to make predictions on data not used to train the model. It is a fast and easy procedure to perform, the results of which allow you to compare the performance of machine learning algorithms for your predictive modeling problem. Although simple to use and interpret, there are times when the procedure should not be used, such as when you have a small dataset and situations where additional configuration is required, such as when it is used for classification and the dataset is not balanced. In this tutorial, you will discover how to evaluate machine learning models using the train-test split. After completing this tutorial, you will know:Let’s get started. Train-Test Split for Evaluating Machine Learning AlgorithmsPhoto by Paul VanDerWerf, some rights reserved. This tutorial is divided into three parts; they are:The train-test split is a technique for evaluating the performance of a machine learning algorithm. It can be used for classification or regression problems and can be used for any supervised learning algorithm. The procedure involves taking a dataset and dividing it into two subsets. The first subset is used to fit the model and is referred to as the training dataset. The second subset is not used to train the model; instead, the input element of the dataset is provided to the model, then predictions are made and compared to the expected values. This second dataset is referred to as the test dataset. The objective is to estimate the performance of the machine learning model on new data: data not used to train the model. This is how we expect to use the model in practice. Namely, to fit it on available data with known inputs and outputs, then make predictions on new examples in the future where we do not have the expected output or target values. The train-test procedure is appropriate when there is a sufficiently large dataset available. The idea of “sufficiently large” is specific to each predictive modeling problem. It means that there is enough data to split the dataset into train and test datasets and each of the train and test datasets are suitable representations of the problem domain. This requires that the original dataset is also a suitable representation of the problem domain. A suitable representation of the problem domain means that there are enough records to cover all common cases and most uncommon cases in the domain. This might mean combinations of input variables observed in practice. It might require thousands, hundreds of thousands, or millions of examples. Conversely, the train-test procedure is not appropriate when the dataset available is small. The reason is that when the dataset is split into train and test sets, there will not be enough data in the training dataset for the model to learn an effective mapping of inputs to outputs. There will also not be enough data in the test set to effectively evaluate the model performance. The estimated performance could be overly optimistic (good) or overly pessimistic (bad). If you have insufficient data, then a suitable alternate model evaluation procedure would be the k-fold cross-validation procedure. In addition to dataset size, another reason to use the train-test split evaluation procedure is computational efficiency. Some models are very costly to train, and in that case, repeated evaluation used in other procedures is intractable. An example might be deep neural network models. In this case, the train-test procedure is commonly used. Alternately, a project may have an efficient model and a vast dataset, although may require an estimate of model performance quickly. Again, the train-test split procedure is approached in this situation. Samples from the original training dataset are split into the two subsets using random selection. This is to ensure that the train and test datasets are representative of the original dataset. The procedure has one main configuration parameter, which is the size of the train and test sets. This is most commonly expressed as a percentage between 0 and 1 for either the train or test datasets. For example, a training set with the size of 0. 67 (67 percent) means that the remainder percentage 0. 33 (33 percent) is assigned to the test set. There is no optimal split percentage. You must choose a split percentage that meets your project’s objectives with considerations that include:Nevertheless, common split percentages include:Now that we are familiar with the train-test split model evaluation procedure, let’s look at how we can use this procedure in Python. The scikit-learn Python machine learning library provides an implementation of the train-test split evaluation procedure via the train_test_split() function. The function takes a loaded dataset as input and returns the dataset split into two subsets. Ideally, you can split your original dataset into input (X) and output (y) columns, then call the function passing both arrays and have them split appropriately into train and test subsets. The size of the split can be specified via the “test_size” argument that takes a number of rows (integer) or a percentage (float) of the size of the dataset between 0 and 1. The latter is the most common, with values used such as 0. 33 where 33 percent of the dataset will be allocated to the test set and 67 percent will be allocated to the training set. We can demonstrate this using a synthetic classification dataset with 1,000 examples. The complete example is listed below. Running the example splits the dataset into train and test sets, then prints the size of the new dataset. We can see that 670 examples (67 percent) were allocated to the training set and 330 examples (33 percent) were allocated to the test set, as we specified. Alternatively, the dataset can be split by specifying the “train_size” argument that can be either a number of rows (integer) or a percentage of the original dataset between 0 and 1, such as 0. 67 for 67 percent. Another important consideration is that rows are assigned to the train and test sets randomly. This is done to ensure that datasets are a representative sample (e. random sample) of the original dataset, which in turn, should be a representative sample of observations from the problem domain. When comparing machine learning algorithms, it is desirable (perhaps required) that they are fit and evaluated on the same subsets of the dataset. This can be achieved by fixing the seed for the pseudo-random number generator used when splitting the dataset. If you are new to pseudo-random number generators, see the tutorial:This can be achieved by setting the “random_state” to an integer value. Any value will do; it is not a tunable hyperparameter. The example below demonstrates this and shows that two separate splits of the data result in the same result. Running the example splits the dataset and prints the first five rows of the training dataset. The dataset is split again and the first five rows of the training dataset are printed showing identical values, confirming that when we fix the seed for the pseudorandom number generator, we get an identical split of the original dataset. One final consideration is for classification problems only. Some classification problems do not have a balanced number of examples for each class label. As such, it is desirable to split the dataset into train and test sets in a way that preserves the same proportions of examples in each class as observed in the original dataset. This is called a stratified train-test split. We can achieve this by setting the “stratify” argument to the y component of the original dataset. This will be used by the train_test_split() function to ensure that both the train and test sets have the proportion of examples in each class that is present in the provided “y” array. We can demonstrate this with an example of a classification dataset with 94 examples in one class and six examples in a second class. First, we can split the dataset into train and test sets without the “stratify” argument. The complete example is listed below. Running the example first reports the composition of the dataset by class label, showing the expected 94 percent vs. 6 percent. Then the dataset is split and the composition of the train and test sets is reported. We can see that the train set has 45/5 examples in the test set has 49/1 examples. The composition of the train and test sets differ, and this is not desirable. Next, we can stratify the train-test split and compare the results. Given that we have used a 50 percent split for the train and test sets, we would expect both the train and test sets to have 47/3 examples in the train/test sets respectively. Running the example, we can see that in this case, the stratified version of the train-test split has created both the train and test datasets with 47/3 examples in the train/test sets as we Now that we are familiar with the train_test_split() function, let’s look at how we can use it to evaluate a machine learning model. In this section, we will explore using the train-test split procedure to evaluate machine learning models on standard classification and regression predictive modeling datasets. We will demonstrate how to use the train-test split to evaluate a random forest algorithm on the sonar dataset. The sonar dataset is a standard machine learning dataset composed of 208 rows of data with 60 numerical input variables and a target variable with two class values, e. binary classification. The dataset involves predicting whether sonar returns indicate a rock or simulated mine. No need to download the dataset; we will download it automatically as part of our worked examples. The example below downloads the dataset and summarizes its shape. Running the example downloads the dataset and splits it into input and output elements. As expected, we can see that there are 208 rows of data with 60 input variables. We can now evaluate a model using a train-test split. First, the loaded dataset must be split into input and output components. Next, we can split the dataset so that 67 percent is used to train the model and 33 percent is used to evaluate it. This split was chosen arbitrarily. We can then define and fit the model on the training dataset. Then use the fit model to make predictions and evaluate the predictions using the classification accuracy performance metric. Tying this together, the complete example is listed below. Running the example first loads the dataset and confirms the number of rows in the input and output elements. The dataset is split into train and test sets and we can see that there are 139 rows for training and 69 rows for the test set. Finally, the model is evaluated on the test set and the performance of the model when making predictions on new data has an accuracy of about 78. 3 percent. We will demonstrate how to use the train-test split to evaluate a random forest algorithm on the housing dataset. The housing dataset is a standard machine learning dataset composed of 506 rows of data with 13 numerical input variables and a numerical target variable. The dataset involves predicting the house price given details of the house’s suburb in the American city of Boston. No need to download the dataset; we will download it automatically as part of our worked examples. The example below downloads and loads the dataset as a Pandas DataFrame and summarizes the shape of the dataset. Running the example confirms the 506 rows of data and 13 input variables and single numeric target variables (14 in total). We can now evaluate a model using a train-test split. First, the loaded dataset must be split into input and output components. Next, we can split the dataset so that 67 percent is used to train the model and 33 percent is used to evaluate it. This split was chosen arbitrarily. We can then define and fit the model on the training dataset. Then use the fit model to make predictions and evaluate the predictions using the mean absolute error (MAE) performance metric. Tying this together, the complete example is listed below. Running the example first loads the dataset and confirms the number of rows in the input and output elements. The dataset is split into train and test sets and we can see that there are 339 rows for training and 167 rows for the test set. Finally, the model is evaluated on the test set and the performance of the model when making predictions on new data is a mean absolute error of about 2. 211 (thousands of dollars). This section provides more resources on the topic if you are looking to go deeper. In this tutorial, you discovered how to evaluate machine learning models using the train-test split. Specifically, you learned:Do you have any questions? Ask your questions in the comments below and I will do my best to answer. with just a few lines of scikit-learn codeLearn how in my new Ebook: Machine Learning Mastery With PythonCovers self-study tutorials and end-to-end projects like: Loading data, visualization, modeling, tuning, and much more. Skip the Academics. Just Results. You must be logged in to post a comment.	{"url":"http://datascience.sharerecipe.net/2020/07/24/train-test-split-for-evaluating-machine-learning-algorithms/","timestamp":"2024-11-05T17:23:28Z","content_type":"text/html","content_length":"43940","record_id":"<urn:uuid:c24ec078-133d-4627-87e5-cd7090af2eaa>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00348.warc.gz"}
MathSciDoc: An Archive for Mathematician Let V be an N-graded, simple, self-contragredient, C_2-cofinite vertex operator algebra. We show that if the S-transformation of the character of V is a linear combination of characters of V-modules, then the category C of grading-restricted generalized V-modules is a rigid tensor category. We further show, without any assumption on the character of V but assuming that C is rigid, that C is a factorizable finite ribbon category, that is, a not-necessarily-semisimple modular tensor category. As a consequence, we show that if the Zhu algebra of V is semisimple, then C is semisimple and thus V is rational. The proofs of these theorems use techniques and results from tensor categories together with the method of Moore-Seiberg and Huang for deriving identities of two-point genus-one correlation functions associated to V. We give two main applications. First, we prove the conjecture of Kac-Wakimoto and Arakawa that C_2-cofinite affine W-algebras obtained via quantum Drinfeld-Sokolov reduction of admissible-level affine vertex algebras are strongly rational. The proof uses the recent result of Arakawa and van Ekeren that such W-algebras have semisimple (Ramond twisted) Zhu algebras. Second, we use our rigidity results to reduce the "coset rationality problem" to the problem of C2-cofiniteness for the coset. That is, given a vertex operator algebra inclusion U⊗V↪A with A, U strongly rational and U, V a pair of mutual commutant subalgebras in A, we show that V is also strongly rational provided it is C_2-cofinite.	{"url":"https://archive.ymsc.tsinghua.edu.cn/pacm_category/0104?show=time&size=3&from=3&target=searchall","timestamp":"2024-11-10T09:53:24Z","content_type":"text/html","content_length":"63567","record_id":"<urn:uuid:f22dee5d-6d93-424c-a24d-6461a475cecb>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00897.warc.gz"}
Powers of 10 The International Bureau of Weights and Measures has adopted a series of prefix names and symbols for decimal multiples and submultiples of SI units. They are expressed as powers of 10 and range from 1030 to 10-30. Badge of BIPM The badge of the International Bureau of Weights and Measures (BIPM) represents an allegory of science holding the new metre standard with its decimal divisions. The badge carries the inscription in Greek ‘metro kro’ or ‘use the measure’. Rights: Image reproduced with permission of the BIPM, which retains full internationally protected copyright (© BIPM) The advantages of basing the multiples and submultiples on the decimal system are: • there are no fractions – decimals only • there are no long rows of zeros – prefixes replace them • they are unique, unambiguous letter symbols • they eliminate the confusion of old number names – is a billion a thousand million (USA) or a million million (Europe)? SI prefixes Larger and smaller quantities are expressed by using appropriate prefixes with the base unit. For example, the base unit of length is the metre. For small lengths, the millimetre (10-3 m) may be the appropriate quantity, whereas for large distances, the kilometre (103 m) may be more appropriate. These prefixes multiply base units for larger measurements These prefixes are decimal fractions that multiply base units for smaller measurements 100 = 1 The first letter of the SI abbreviation represents the prefix and the second letter represents the base unit. • In the You, Me and UV resources, information is given about the UV index. One unit on the UV index is roughly equivalent to 25 millijoules of UV energy falling per second on a 1 square metre area. In abbreviated form, this amount of energy is written as 25 mJ. • In the Nanoscience resources, it is reported that fingernails grow at about 1 nanometre per second. In abbreviated form, this length is written as 1 nm. • The Space revealed resources deal with extremely large distances. For example, in the article Distances in space, it states: When the Space Shuttle goes into space, it orbits about 700 km above the surface of the Earth. The ‘km’ means kilometre, which is 103 m. The distance from the Earth to Mars is 78 million kilometres, which is 78 x 106 km, and this in turn is 78 x 109 m. Using the prefix for 109 from the table above, this becomes 78 Gm, which is pronounced as ‘seventy-eight gigametres’. Prefix names To many people, the choice of prefix names seems strange. For example, where does ‘kilo’ come from? Prefix names have been mostly chosen from Greek words (positive powers of 10) or Latin words (negative powers of 10), although recent extensions of the range of powers of 10 has resulted in the use of words from other languages. ‘Kilo’ comes from the Greek word for 1000 (103), and ‘milli’ comes from the Latin word for one thousandth (10-3). Regular prefixes Most of the prefixes in the table above are multiples of a thousand. These are referred to as regular prefixes and may be used with any SI unit. Using metre as an example: Multiples of the metre Submultiples of the metre 1000 m = 1 km = 103 m (kilometre) 0.001 m = 1 mm =10-3 m (millimetre) 1000 km = 1 Mm =106 m (megametre) 0.001 mm = 1 μm =10-6 m (micrometre) 1000 Mm = 1 Gm =109 m (gigametre) 0.001 μm = 1 nm =10-9 m (nanometre) Prefixes are easier to write and pronounce than powers of 10. The following example shows the same quantity written in different ways. Which is easiest? The average daily energy requirement of an active 14-year-old boy is: • 13 MJ (pronounced as ‘13 megajoules’) • 13 x 106 J • 1.3 x 107 J • 13,000,000 joules Nature of Science A large number of scientific terms have their origins in Latin words. The reason for this is that Latin was used in scholarly writing well into the 19th century. For example, Sir Isaac Newton’s major works, such as Principia, were all written in Latin. Today, although the popularity of the language has fallen away, Latin roots continue to serve as a major source for the derivation of new terms in the sciences. Related content Explore the science ideas and concept of size further with these articles: Activity ideas Measuring foot pressure provides practice using SI units, derived units and prefixes. Precision and accuracy provides various datasets for students to judge precision and accuracy in scientific settings. How long is it? is a collection of length measurements found within the Science Learning Hub. Lengths range from the very small to the very big, helping students develop an understanding of the decimal system as applied to length measurement. Useful links In this interactive view the Milky Way at 10 million light years from the Earth. Then move through space towards the Earth in successive orders of magnitude. Explore cell size and scale in this interactive from the Univerty of Utah. Visit the The International Bureau of Weights and Measures (BIPM) website. In 2022 new metric prefixes to express the world's largest and smallest measurements, were added to the International System of Units (SI).	{"url":"https://beta.sciencelearn.org.nz/resources/1873-powers-of-10","timestamp":"2024-11-04T18:19:22Z","content_type":"text/html","content_length":"252405","record_id":"<urn:uuid:cb414c70-a404-4cc6-a52a-3876eb39813d>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00381.warc.gz"}
Mathematical Diagrams \| Mathematics Symbols \| Physics Diagrams \| Science Education Mathematics ConceptDraw PRO diagramming and vector drawing software extended with Mathematics solution from the Science and Education area is the best for creating: mathematical diagrams, graphics, tape diagrams various mathematical illustrations of any complexity quick and easy. Mathematics solution provides 3 libraries: Plane Geometry Library, Solid Geometry Library, Trigonometric Functions Library. ConceptDraw PRO extended with Mathematics solution from the Science and Education area is a powerful diagramming and vector drawing software that offers all needed tools for mathematical diagrams Mathematics solution provides 3 libraries with predesigned vector mathematics symbols and figures: Solid Geometry Library, Plane Geometry Library and Trigonometric Functions Library. 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Finite mixture models for statistical inference Hien Duy NGUYEN Degree: PhD, University of Queensland, Australia Research interests: Mathematical Statistics, Statistical Computing, Statistical Learning, Bayesian Statistics, Signal Processing, Stochastic Programming, Optimization Theory Many real-world datasets are heterogeneous and multipopulational phenomena. In such contexts, it is insufficient to capture the overall variation among the data using a single statistical model. Therefore, a cohesive approach to modeling the multiple subpopulations within the superpopulation is necessary. In such scenarios, a useful approach involves modeling each subpopulation and their contributions to the superpopulation through a weighted averaging construction, known as a finite mixture model. These models are highly flexible and interpretable, enabling them to capture and provide inference for known heterogeneities in the data while also identifying new heterogeneous phenomena that were previously concealed. The class of finite mixture models is extensive, and choosing between different mixture models can be challenging. In my work, I have studied model selection procedures required to make mathematically principled choices among competing finite mixture models. I have made progress in two key directions to address this problem. Firstly, I employ sequences of hypothesis tests to determine the number of components or subpopulations required in each mixture model. This approach relies on a new hypothesis testing method called universal inference, which offers a straightforward and assumption-light mechanism for deciding whether a model accurately represents the observed data. Using these universal inference tests, I have developed a way to construct confidence intervals for the number of underlying subpopulations in the data, providing insight into the complexity of the overall superpopulation. Secondly, by leveraging modern stochastic programming techniques for optimizing random objects, I have developed new penalization methods for selecting between different finite mixture models within broader model selection and decision problems. My novel information criterion, known as PanIC, offers a more assumption-light alternative to existing methods like the Bayesian information criterion or Akaike information criterion. PanIC provides a single-number summary for choosing between competing models, guaranteed to asymptotically select the correct model as the dataset size increases. Beyond their utility for modeling heterogeneous processes, finite mixtures and their regression variants, the mixture of experts (MoEs) also serve as excellent functional approximations of probability density functions (PDFs) and conditional PDFs that characterize statistical relationships. My colleagues and I have contributed to understanding the approximation theoretic properties of mixture models and MoEs for various classes of PDFs. We have provided sufficient conditions for ensuring that PDFs, conditional PDFs, or mean functions of conditional PDFs can be effectively approximated using a sufficiently large number of components in a finite mixture model construction. These results, often referred to as universal approximation theorems, are valuable for determining whether a class of functions serves as an adequate basis for modeling an underlying mathematical phenomenon. My research in mixture model computation, estimation, and inference has found widespread application in real-world scenarios. For example, I have collaborated with neuroscientists and cell biologists to analyze heterogeneous biological phenomena, worked with quantum physicists to characterize switching behaviors of quantum circuitry, assisted economists in characterizing subpopulations of experimental outcomes, partnered with civil engineers to study regional differences in traffic behavior, supported fisheries scientists in characterizing growth stages of aquatic species, and collaborated with image scientists to segment and characterize imaging data, among other practical applications. Figure 1: Traffic crash rate clustering of different regions in Victoria, Australia. Figure 2: Quantization of mandrill photograph using different mixture models. Figure 3: Mixture-based false discovery rate control of p-values for a mouse brain morphometry experiment	{"url":"https://www.imi.kyushu-u.ac.jp/post-catalog/catalog-8936/","timestamp":"2024-11-02T20:37:51Z","content_type":"text/html","content_length":"33008","record_id":"<urn:uuid:b6b85df1-9b9c-4603-a87c-28231b935746>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00582.warc.gz"}
Sandeep Garg Solutions Class 11 Economics Chapter 6 – Measures of Central Tendency- Median and Mode - CoolGyan Sandeep Garg Class 11 Economics Solutions Chapter 6 – Measures of Central Tendency- Median and Mode are illustrated by the professional economic educator from the contemporary edition of Sandeep Garg Economics Class 11 textbook solutions. We at CoolGyan’S provide Sandeep Garg Economics Class 11 Solutions to give a comprehensive insight into the subject to the students. These insights will be a valuable advantage to students while completing their homework or while studying for their exams. There are numerous concepts in economics, but here we provide you with the solution from the Tabular Presentation, which will be convenient for the students to score well in the board exams. The above-provided solutions are considered to be the best solution for ‘Sandeep Garg Economics Class 11 Solutions Chapter 6 – Measures of Central Tendency- Median and Mode. Stay tuned to CoolGyan’S to learn more.	{"url":"https://coolgyan.org/commerce/sandeep-garg-solutions-class-11-economics-chapter-6-measures-of-central-tendency-median-and-mode/","timestamp":"2024-11-03T15:59:01Z","content_type":"text/html","content_length":"87052","record_id":"<urn:uuid:c44866cd-0e03-4871-8b60-ededa1bd4cde>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00839.warc.gz"}
Richard Simmons is selling his house. He has a choice of taking Richard Simmons is selling his house. He has a choice of taking $125,000 today or $135,000 in 6 months. If he takes the money today, he can invest it at Valley Bank at 5% interest compounded monthly. a. How much would be in the account after six months if he took the 125,000. b. Which option should he take? c. How much more money does he gain in making this choice? Did this for Lynda back in December To calculate the answers to these questions, we need to use the compound interest formula: A = P(1 + r/n)^(nt), A is the final amount, P is the principal (initial amount), r is the annual interest rate (in decimal form), n is the number of times interest is compounded per year, and t is the number of years. For this problem: P = $125,000 r = 5% = 0.05 (annual interest rate) n = 12 (monthly compounding) t = 6/12 (6 months converted to years) a. To find out how much would be in the account after six months if he took $125,000, we can use the compound interest formula: A = P(1 + r/n)^(nt) A = $125,000(1 + 0.05/12)^(12(6/12)) Using a calculator or solving step-by-step: A = $125,000(1.004166666)^6 A = $125,000(1.025) A ≈ $128,125 After six months, if Richard took the $125,000 and invested it at Valley Bank, he would have approximately $128,125 in the account. b. To determine which option he should take, we compare the amounts. If the amount after six months is higher than $135,000, he should take the money today; otherwise, he should wait for the Since $128,125 < $135,000, he should choose to take $135,000 in 6 months. c. To calculate how much more money he gains by making this choice, we subtract the initial amount ($128,125) from the final amount ($135,000): $135,000 - $128,125 = $6,875. Therefore, Richard Simmons gains $6,875 by choosing to take $135,000 in 6 months instead of $125,000 today. To answer these questions, we need to calculate the future value of the $125,000 investment after 6 months and compare it to the $135,000 he would receive after the same time. a. To calculate the future value of the $125,000 investment after 6 months with a monthly compounding interest rate of 5%, we can use the formula for compound interest: Future Value = Present Value [(1 + (interest rate / n))^(n * t)] - Present Value (PV) = $125,000 - Interest Rate (r) = 5% or 0.05 (converted to decimal) - Number of Compounding Periods per year (n) = 12 (since it's compounded monthly) - Time (t) = 6 months Future Value = $125,000 * [(1 + (0.05 / 12))^(12 * 6)] Using a calculator or spreadsheet, you can compute the Future Value to find out how much would be in the account after six months if Richard took the $125,000. b. To determine which option Richard should take, we compare the future value calculated in part a with the $135,000 he would receive in 6 months. If the future value is higher, taking the $125,000 today and investing it is the better choice, and vice versa. c. To calculate the difference in money gained from making either choice, subtract the initial investment ($125,000) from the higher amount (either the future value or $135,000). The result will provide the amount of additional money gained by choosing that particular option. Newest Financial Literacy Questions 1. Kaya wants to buy a new gaming system, but it is expensive. She has decided to save 18%of her paycheck each month until she can 2. Which of these activities are you most likely to do in el banco?(1 point) Responses llenar el tanque llenar el tanque echar una 3. What is one of the potential dangers of using a debit card for purchases?(1 point) Responses an overdraft of your checking 5. The phrase “balancing a checkbook” refers to which practice?(1 point)Responses Deciding if one has enough money to make a 6. Which of the following items may not be a good consideration in selecting a bank?a. The word “FREE” is prominently displayed 7. When you sign 5 dollars, you make the sign for DOLLARS first and then the sign for 5.(1 point) Responses true true false 8. When paying for purchases, debit cards ____.(1 point) Responses Are exactly like credit cards Are exactly like credit cards Take	{"url":"https://askanewquestion.com/questions/1045447","timestamp":"2024-11-08T17:41:42Z","content_type":"text/html","content_length":"23261","record_id":"<urn:uuid:804b716f-5c6c-4eeb-8325-b05b06e9076d>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00239.warc.gz"}
How does VEMA calculate Leverage? For Forex, VEMA does not calculate leverage, brokers instead have a set leverage for each asset. For Crypto the short story is VEMA uses the highest possible leverage it can (and therefore the lowest possible margin) to place a trade without putting your liquidation price between Stop Loss and Entry, with some buffer on top of this to help avoid liquidation even in the event of slippage on entry or exit. The full story is a little more complicated and involves a lot more maths. Let's use OKX for our example. OKX has what they call "Position tiers" Different position sizes have different maximum leverage amounts, and different liquidation points. Here is a screenshot of the position tiers for BTC-USDT-SWAP: You can see a position less than 1000 contracts has a max leverage of 125x, where a position size of 20,001 - 40,000 has 50x max leverage. This means there's no possible way we can submit a trade of 25,000 contracts at 125x leverage, regardless of Stop Loss Percentage, as OKX will not allow that trade, the max leverage on this position is 50x. For the next part, there's two important terms to understand, Initial Margin Required (IMR), and Maintenance Margin Required (MMR). Initial Margin Required (IMR) Initial Margin Required is the amount of margin needed to open a position. It's directly related to the max leverage, as IMR = (1 / Max Leverage) and Max Leverage = (1 / IMR) - known as an inverse relationship (as one goes up, the other goes down). So if we wanted to open a 100 contract position, our IMR means we need 0.8% of the margin for that position. If BTC is trading at $30,000 USD and a contract is worth 0.01 BTC, 100 contracts means a position size of 1 BTC for a position value of $30,000 USD. At 0.8% Initial Margin Required, we need 0.8% of $30,000 USD to open this position, so we need to have $240 USD of margin ($30,000 USD position value x 0.8% Initial Margin Required) free in our account or we cannot place this trade and will hit the insufficient margin error. We can then see how this relates to our leverage, as 125 x $240 = $30,000. Maintenance Margin Required (MMR) Maintenance Margin Required is a little trickier to wrap our heads around, but stick with me and I'll do my best. MMR is the margin you need to maintain to keep the position open. It's essentially a measure of maximum drawdown, at some point of that trade going into the red, the loss on the position will push the MMR below the threshold, and this is when liquidation occurs. We can see from the above screenshot that MMR for a 125x leverage position is 0.4%. 0.4% of our $30,000 USD position is $120 USD. This means we need to maintain a margin value over $120 to keep this position open. Which means if the unrealised loss on this position exceeds $120 USD ($240 IMR - $120 MMR) we will be liquidated and suffer a realised loss of all our margin for a $240 loss. You can think of Maintenance Margin as the minimum margin required to keep the position open. If the Margin used to open the position, minus the loss on that position, becomes less than the Maintenance Margin, a position will be liquidated. Percentage of Initial Margin Loss that leads to Liquidation (IML) Personally I find it easiest to think in terms of the percentage of initial margin lost that leads to liquidation. The Percentage of Margin Loss that leads to Liquidation can actually be calculated directly - without a position size - on certain exchanges such as BitMEX, although we do need a position size on OKX to tell us which tier we fall into and which values to use for IMR and MMR. To do this we first calculate the minimum margin required to keep the position open as a percentage of our initial margin by dividing the MMR by the IMR. In the above example, IMR is 0.8% and MMR 0.4% for a percentage of initial margin required to keep our position open of 50% (0.4% MMR / 0.8% IMR) We can then calculate the percentage of margin loss that leads to liquidation as 1 - [0.4% MMR / 0.8% IMR]. For our example, this is again 50% (1 - 50% = 50%) so we can suffer a 50% drawdown on this position before liquidation will occur. This checks out with our calculations above, we needed $240 to open the position, and got liquidated if that position lost $120 of margin (50% of IMR), which also reflects our percentage of margin required to keep the position open of $120 ($240 IMR - $120 Unrealised Loss). We can do the same for a higher number of contracts in a larger position tier: If we use these values for our above position (this ignores OKX's tiers) we would see for our 100 contact, $30,000 USD position we need: Margin to open position = $600 USD ($30,000 USD Position Value x 2% Initial Margin Required) Maintenance Margin Required = $375 ($30,000 USD Position Value x 1.25% maintenance Margin Required) Which leads to a minimum maintenance margin as a percentage of our initial margin of 62.5% (1.25% / 2%) And a maximum drawdown on our initial margin of 37.5% ( 1 - [1.25% / 2%]). This means our position needs to maintain a margin above 62.5% (1.25% MMR / 2% IMR) of our initial margin or we risk liquidation, which also means we will be liquidated when the position has an unrealised loss of 37.5% of the initial margin required ( 1 - [1.25% MMR / 2% IMR]). If we look at that in dollar terms, we see we need to maintain a maintenance margin value of $375 ($600 IMR x 62.5% minimum maintenance margin as a percentage of our initial margin) which matches our MMR as expected. Which means we have a maximum unrealised loss of $225 ($600 IMR x 37.5% maximum drawdown on initial margin) before we face liquidation. So we can see that a position that fits tier 1 of less than 1,001 contacts gets liquidated at a 50% loss of initial margin, but a position on that same pair of between 20,001 and 40,000 contracts instead gets liquidated at a 37.5% loss of initial margin. What does this look like on the charts? To calculate the percentage price needs to go against us before we're liquidated, we can look at the margin loss that leads to liquidation. Any percentage of price action movement from entry on the chart will have an effect multiplied by our leverage on our unrealised P&L. So at 125x, a 1% move will actually give us a 125% unrealised P&L on our initial margin (ignoring the fact we'd be liquidated well before this). So if Leverage x Price action percentage = Unrealised P&L And we know liquidation occurs when Unrealised P&L is equal to ( 1 - [MMR / IMR]) We can calculate what percentage of price action movement will lead to liquidation in our above scenarios using Liquidation Price action percentage = ( 1 - [MMR / IMR]) / Leverage So for our 1000 contract example, with a leverage of 125x, IMR = 0.8% and MMR = 0.4% as OKX's position tiers dictate: Liquidation Price action percentage = ( 1 - [ 0.4% MMR / 0.8% IMR]) / 125x Leverage Liquidation Price action percentage = 0.4% This tells us if price moves 0.4% from entry, we will be liquidated. At $30,000 entry, this means for a long a price of $29,880 (30,000 x [1- 0.4%]) would see OKX liquidating our position. This checks out with our above example, as our position was 1 BTC worth $30,000 USD and if price has gone down to $29,880 that position of 1BTC is now worth $29,980 - $120 against us, so we'll now be liquidated - notice how the $120 loss matches the value we calculated for the unrealised loss that would lead to liquidation. We can do the same for our tier 4 trade, again keeping 100 contracts as the position size for simplicity, but using the tier 4 Max Leverage, IMR and MMR values. Liquidation Price action percentage = ( 1 - [MMR / IMR]) / Leverage Liquidation Price action percentage = ( 1 - [ 1.25% MMR / 2% IMR]) / 50x Leverage Liquidation Price action percentage = 0.75% At $30,000 entry, this means for a long a price of $29,775 (30,000 x [1- 0.75%]) would see OKX liquidating our position. This checks out with our above example, as our position was 1 BTC worth $30,000 USD and if price has gone down to $29,775 that position of 1BTC is now worth $29,775 - $225 against us, so we'll now be liquidated - notice how the $225 loss matches the value we calculated for the unrealised loss that would lead to liquidation. So how does VEMA actually Calculate Leverage? We've seen above how to calculate your liquidation point based on the leverage and position tier your position falls into. How VEMA calculates leverage is very similar. First we look at your position size, as this dictates maximum leverage (on some exchanges). Then we look at your Stop Loss percentage. Let's say you had a SL of 0.4% on that 100 contract, 125x trade. This would mean liquidation price and SL price were one and the same (as we calculated that liquidation price falls 0.4% from entry earlier). But this puts you at risk of liquidation, where instead of losing the $120 your SL would cost you, you'd actually lose $240 - the entire initial margin you used for the trade, doubling your loss. It's also possible to be liquidated even if your Stop Loss is within the liquidation price, in high slippage events. If you had a SL of 0.38%, liquidation at 0.4%, and a large candle triggered your SL but price had crossed your liquidation point before that could be filled (slippage) you would be liquidated instead of your SL filling. So what VEMA does is it adds a buffer to the max leverage. We'll switch to BitMEX logic here as it's a little simpler, but the idea is the same, I'll also simplify it a little so the numbers won't be exact but it should explain the general idea. BitMEX typically liquidate positions at a margin loss of around 50% for 100x, or higher for lower leverages. This is similar to OKX's tier 1 for BTC, with the 0.8% IMR and 0.4% MMR giving a 50% margin loss where liquidation would occur. So VEMA sets out to make your maximum margin loss less than 40%, meaning at your Stop Loss, you should be at a 40% maximum unrealised P&L. The formula for this is: Leverage x Stop Loss % ≤ Buffered Maximum Margin Loss We know your Stop Loss percentage from your setup, and set the Maximum Margin Loss as 40%, so our leverage calculation becomes Leverage ≤ Buffered Maximum Margin Loss / Stop Loss % This gives trades a minimum 20% buffer (40% Maximum Margin Loss at Stop Loss / 50% Minimum Margin loss for liquidation) on their Stop Loss to liquidation price from entry, which helps protect against liquidation in the event of slippage on either entry or exit, at the expense of tying up a little more margin for each trade. What does this look like in Practice? Again using BitMEX as the example for it's simplified logic, let's say you take that same 1 BTC trade from $30,000 USD, with a 0.4% SL. VEMA then uses the above formula to calculate your leverage Leverage ≤ Buffered Maximum Margin Loss / Stop Loss % Leverage ≤ 40% Buffered Maximum Margin Loss / 0.4% Stop Loss Leverage ≤ 100 x So VEMA would then enter this position at 100x leverage. This would result in a trade with Entry at $30,000 USD, Stop Loss of 0.4% at $29,880, and liquidation at a minimum value of 50% loss of margin which would occur at $29,850, with BitMEX's liquidation calculator returning a price of $29,853 for that position (The difference being due to our simplified values). You can see this gives a significant buffer against liquidation, which is something we consider well worth the extra margin tied up to allow for this buffer due to the vastly magnified loss of liquidation when compared to Stop Loss fills (in the above example, doubling the margin lost from $120 to $240). For OKX the process is similar, except position size first dictates the tier which tells us the maximum leverage available, then the IMR and MMR values for the tier tell us the liquidation point, after which the buffer is applied to give more breathing room between Stop Loss and Liquidation. Calculating what leverage a position should use is a complex, math heavy process. We hope the above explains how and why we use the leverages we do on positions, the fact VEMA does it all for you is just one more way VEMA makes trading easier and more accessible. Have a great day, and happy trading! 0 comments Please sign in to leave a comment.	{"url":"https://support.vematrader.com/hc/en-us/articles/22907294411161-How-does-VEMA-calculate-Leverage","timestamp":"2024-11-02T11:01:39Z","content_type":"text/html","content_length":"43479","record_id":"<urn:uuid:c807384a-933b-4f4d-9699-a0b5db67d0f6>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00142.warc.gz"}
Math 4 Wisdom. "Mathematics for Wisdom" by Andrius Kulikauskas. \| Research / IntuitingExceptionalRootSystems The four classical root systems are as follows, where throughout, {$i>j$}: {$A_n$} {$\pm (x_i-x_j)$} {$B_n$} {$\pm (x_i-x_j), \pm (x_i+x_j), \pm x_i$} {$C_n$} {$\pm (x_i-x_j), \pm (x_i+x_j), \pm 2x_i$} {$D_n$} {$\pm (x_i-x_j), \pm (x_i+x_j)$} The five exceptional root systems are described in Wikipedia's article on root systems. {$G_2$} {$ (x_i-x_j), (x_i - x_j) - (x_j - x_k)$} for all distinct {$ i,j,k \in \{1,2,3\}$}. Thus {$G_2$} is the disjoint union of two copies of {$A_2$}. {$G_2$} {$ \pm (x_i-x_j)$} where {$i \neq j$}, and {$\pm (3x_i - (x_1+x_2+x_3))$} for all {$i$}. {$F_4$} {$\pm (x_i-x_j), \pm (x_i+x_j), \pm x_i, \frac{1}{2}(\pm x_1 \pm x_2 \pm x_3 \pm x_4)$} Thus {$F_4$} has a copy of {$B_4$} as a subset. {$E_8$} {$\pm (x_i-x_j), \pm (x_i+x_j)$}, and {$\frac{1}{2}(\sum_{i=1}^{8}(-1)^{a_i}x_i)$} where {$\sum_{i=1}^{8}{a_i} \in 2\mathbb{Z}$}. {$E_7$} and {$E_6$} are subsets of {$E_8$}. Notes by Lecture 16 on Lie Algebras by Kevin McGerty.	{"url":"https://www.math4wisdom.com/wiki/Research/IntuitingExceptionalRootSystems","timestamp":"2024-11-05T12:07:46Z","content_type":"application/xhtml+xml","content_length":"13230","record_id":"<urn:uuid:a8d24ebb-2860-4a97-a13b-20ac994057f3>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00114.warc.gz"}
Some critiques of mathematics education research articles Research in mathematics education is generally hopelessly unintelligent. The field as a whole is not far from worthless. Below are some critiques of research articles demonstrating this in specific instances. These articles are very much representative of the quality of the field as a whole. I have not sought them out because they were especially easy to poke holes in. On the contrary, these articles were all assigned reading in mathematics education graduate courses that I took. For further critiques establishing the same point, see my mathematics education book reviews and Can There be “Research in Mathematical Education”? by Herbert S. Wilf. Catsambis, S., Mulkey, L. M., & Crain, R. L., (2001). For better or for worse? a nationwide study of the social psychological effects of gender and ability grouping in mathematics. Social Psychology of Education, 5: 83–115. Mulkey, L. M., Catsambis, S., Steelman, L. C., & Crain, R. L., (2005). The long-term effects of ability grouping in mathematics: A national investigation. Social Psychology of Education 8:137–177. These studies illustrate an ideological assumption that is rarely if ever justified, namely the assumption that ideally everyone should study lots of mathematics and feel good about their mathematical ability. In my view this assumption is highly irrational. It is not in anyone’s best interest that students overestimates their own abilities or that they are strung along in course after course in which they learn just about enough procedural nonsense to scrape by with a passing grade. It is easy to see how inflating a student’s confidence can have disastrous effects. This student may very well be struggling in other subjects as well, so if he is pampered in his mathematics class he may be led to the misconception that this is his area of strength. Thus he will keep taking mathematics courses until he finally realizes that although he managed to get by in each individual course he does not have the level of understanding necessary to do anything meaningful with his mathematical coursework, such as pursuing a degree in a STEM field. Thus his efforts in trying to do well in his mathematics courses, and perhaps even taking electives, have been wasted. He would have been better served by having realised this earlier, so that he could focus on another academic path, rather than being misled by feel-good mathematics classes. This, I say, is the type of consideration that the studies by Catsambis et al. (2001) and Mulkey et al. (2005) fail to take into account. They simply assume that higher confidence and willingness to take more mathematics is a good thing. The overall conclusions of these studies are largely the same; we may quote from the former for definiteness: “In sum, based on our study, we conclude that the effects of tracking focus on ‘conferring status’, thus supporting the aphorism, ‘It is better to be a big frog in a small pond than a small frog in a big pond’.” (Catsambis et al. (2001), p. 105) That is to say, students form a self-image by comparing themselves to their peers. Thus, for example, “Tracking has a strong, but negative, association with certainty of high school graduation for all students with a high-track propensity. … The opposite is true for males and females with a low-track propensity who remain more certain of their high school graduation. … Similar results are found for students’ college plans.” (Mulkey et al. (2005), p. 159) Thus the effect of tracking is that “students with a propensity for a high track are negatively affected whereas students with low track propensity are positively affected,” since “eventually, academic self-concept figures into test scores and grades” (Mulkey et al. (2005), p. 165). This is taken to be an argument against tracking: “This pattern suggests that tracking’s positive instructional effect is attenuated by indirect social mechanisms” (Mulkey et al. (2005), p. 165), for “when males are grouped with peers of similar high ability, they lose their competitive edge, and it becomes difficult for them to realize their positive attributes” (Catsambis et al. (2001), p. 103). “So, for highly tracked males the negative academic self-concept may unintentionally depress future performance if it results in the avoidance of taking elective, advanced math courses” (Mulkey et al. (2005), p. 144). But none of this need to be a bad thing. As I argued above, overestimating one’s talent for mathematics may be extremely harmful in the long run, even though in the short term it may be beneficial in all respects. The fact that “eventually academic self-concept figures into test scores and grades” may suggest to some people that we should increase everybody’s mathematical self-esteem. Even if this would indeed lead to better results, it can still be harmful. For it may be that it leads to better results only by tricking students into thinking that their future lies in a mathematical field. No wonder then that they work harder in their mathematic classes and get better grades: they do so with the understanding that this will eventually be relevant for their future career path. Once they reach the realization that they are not suited for a career focused on mathematics they will be bitterly disappointed and feel that they have been misled into focusing on mathematics under the impression that they were especially good at it. And yet on superficial measures this type of student is a “success”: they work hard, obtain high scores relative to their ability, and take elective mathematics courses. I reiterate my point that the equity-based arguments against tracking run the risk of assuming that higher achievement and self-esteem among students is necessarily a good thing. Instead, I have argued, high achievement and self-esteem can be “bought” by dishonest inflation of students’ self-images which are ultimately detrimental in the long term. Shores & Shannon, The Effects of Self-Regulation, Motivation, Anxiety, and Attributions on Mathematics Achievement for Fifth and Sixth Grade Students, School Science and Mathematics, Volume 107, Issue 6, pages 225–236, October 2007. A study of 761 Alabama fifth and sixth graders using an extensive Likert-type questionnaire. Regression analyses showed that motivation and anxiety were correlates of achievement in the expected ways. Reasonable people might conclude that poor performance leads to anxiety and high achievement leads to higher levels of motivation, which is hardly something we need “research” to tell us. But, alas, the situation is much worse than proving something obvious. Instead the authors conclude that “academic achievement is effected [sic] by such factors as motivation [and] anxiety” (p. 231), thus committing the elementary fallacy of taking correlation to imply causation. There is no basis in the study for such a causal inference. Presumably the researchers prefer it to the harsh reality of the common-sense interpretation because it legitimises the kind of feel-good, PC movement discussed above. Speer, N. M., & Wagner, J. F., (2009), Knowledge needed by a teacher to provide analytic scaffolding during undergraduate mathematics classroom discussions. Journal for Research in Mathematics Education, 40(5), 530–562. An investigation of the pedagogical content knowledge needed by teachers to constructively guide classroom discussion. The authors’ first episode concerns a classroom discussion on the problem on modeling “a continuously reproducing species of fish in a lake” (p. 542). The problem posed was: “This situation can also be modeled with a rate of change equation dP/dt=something. What should the something be? Should the rate of change be stated in terms of just P, just t, or both P and t?” (p. Of course the “right” answer is dP/dt=kP. According to the authors, “understanding the direct dependence of [the] differential equation on P but not on t is a conceptual challenge for students to overcome”––in fact, it is “the central conceptual challenge” (p. 543). I say that it is the authors themselves who have a deficient understanding of the situation. They claim that it is “not the case” that “dP/dt [is] expressible in terms of t,” since “expressing dP/dt solely in terms of t (dP/dt=f(t)) would indicate that the rate of change of the population is independent of the size of the population, which is not a reasonable assumption for any living species” (p. 543). This is nonsense. No such assumption is “indicated” by giving dP/dt in terms of t. In fact, the “right” model dP/dt=f(P) and the “wrong” model dP/dt=f(t) are both equivalent in this case, since the statement of the problem explicitly concerns “a continuously reproducing species of fish” and says that it is “this situation” that is to be modeled. This suggests that we are dealing with a specific P, which of course can be expressed as a function of t without any loss of generality. Of course it would be different if one were looking for a general population modeling equation, but that is not the problem posed. So I would say that “the central conceptual challenge” is rather that the course materials asks one question and expects the answer to another. The authors continue: “In other words, realizing that an initial condition is irrelevant to the question posed (i.e., that the differential equation must hold for all possible initial conditions simultaneously) is a challenging learning objective for the students as they work through this activity.” (p. 544) No wonder! It is bound to be very challenging indeed, since the course materials have gone out of their way to emphasise that the problem concerns a specific type of fish in a specific lake etc. (though perhaps stopping short of calling it a specific population). Here’s a piece of analytic scaffolding for you: if you want a general answer, pose a general question. A second episode in the discussion of the same problem concerns the distinction between dP/dt=P and dP/dt=e^t as possible models. The “right” thing to do is of course to “see that P(t)=2e^t satisfies only one of them” (pp. 546-547). According to the authors an opportunity to make this point presented itself naturally in class discussion. Namely, “Rob’s observation … was closely related to [this] point” (p. 547), and “could have been highlighted and clarified for the whole class,” which “might have allowed the class to distinguish between the two differential equations under consideration” (p. 546). “Unfortunately, Rob made the unhelpful suggestion, ‘say your P(t) was P+t’” (p. 547). This is “unfortunate” and “unhelpful” only to narrow minds who have already decided what the “right” outcome of the discussion is (thus alleviating the need for a discussion in the first place). Much more plausible than the authors’ interpretation is the interpretation that Rob was simply looking for a function that is not its own derivative: in fact, he is offering the simplest possible variant of an exponential P that will fail to be its own derivative. Thus the authors’ desired point regarding e^t versus 2e^t is completely irrelevant since both these functions are indeed their own derivatives. The rest of the authors’ interpretations of their data is also flawed. On one occasion, for example, they pointlessly discuss what the teacher might have done with a student contribution which was admittedly inaudible to him (p. 556). But let us leave these issues aside and consider what the authors purport to conclude from their study. When we turn to the conclusions section we find the worthlessness of the study confirmed anew, as we read nothing but truistic fluff such as the following: “We do believe … that teaching expertise in reform oriented practices of this sort is enhanced as teachers develop the types of knowledge considered here” (p. 558), where the “types of knowledge” in question are synonyms of good teaching, “such as knowledge of typical ways students think (correctly and incorrectly) …, knowledge of the curriculum in use, and knowledge to support the specialized type of mathematical work teachers do when dissecting and analyzing students’ expressions of their ideas.” (p. 558) Basically, then, the authors have made the astounding discovery that knowing how to teach well is positively correlated with teaching well. Dahl, B., (2004). Analysing cognitive learning processes through group interviews of successful high school pupils: development and use of a model. Educational Studies in Mathematics, 56, 129–155. This article is an attempt to introduce and support a cognitive model of mathematical learning. I this review I shall argue that her enterprise is ill-conceived and that her conception of theoretical research is naive. Dahl’s model is called CULTIS for the “collection of themes” (p. 134) that constitute it: “Consciousness – Unconsciousness; Language – Tacit; Individual – Social” (p. 134). The first thing to note here is that the “model” proposed by Dahl is nothing but a “collection of themes.” This in itself makes a number of the claims she makes for the value of her theory highly suspect. Before looking at specific examples of this, I may illustrate my point by an analogy. Suppose someone proposed a “HLHC” theory of physics, which was nothing but a “collection” of the “themes” Heavy – Light; Hot – Cold. It is of course an undeniable fact that the constituent “themes” of the HLHC model are crucially important concepts in physics, and that if you sit down in a physics classroom you will find that much of what is being said relates to these categories. But this does not imply that pasting them together and giving them an acronym is of any value whatsoever, especially if the importance of these concepts has been well-known for a long time. I say that precisely this error is being committed by Dahl. She assumes that because each of her “themes” show up regularly, bundling them into a “model” and giving it an acronym somehow constitutes and advancement of theoretical understanding of mathematical cognition. And this despite the fact that each of the “themes” has been given much attention previously; indeed that is how Dahl herself says that she came up with them: “CULTIS was created after systematically going through [previous] theories noticing which themes they brought up” (p. 134). As an illustration of Dahl’s misconceptions in this regard, we may note the following quotation: “the distance between what the pupils said and the theories is not big which supports the explanatory power of the theories” (p. 152). Dahl has not learned the elementary lesson that descriptive accuracy is not the same thing as explanatory power. If my HLHC theory of physics says that heavy objects fall down if dropped then the theory is descriptively accurate but obviously it has no explanatory power whatever; it merely restates a phenomenon without explaining anything. Another example may illustrate that the alleged “explanatory power” of Dahl’s theory is of precisely this vacuous type. Some students in the study said that lack of familiarity with a given teaching style may hamper learning: “D: When I first came here [to the new school], the first couple of weeks I found math very difficult because it is kind of hard to adapt to a different teaching style.” (p. 151) According to Dahl, “This phenomenon might be explained by stating that the teaching method should be within what I henceforth will call a zone of proximal teaching (ZPT). … If a (new) teacher uses teaching methods that are too ‘far away’ from what the pupil is used to, the pupil may not learn.” (p. 151) Again Dahl appears to be under the delusion that to explain something is to restate it in a pompous way and to give it an acronym, because her proposed “explanation” adds nothing but pretentious verbiage to the statement of the student. But even if we reject Dahl’s pretensions to offer an explanatory theory, one might argue that her model nevertheless has merit as a useful synthesis of previous theories. I maintain, however, that this is not the case. The way the themes are thrown together in the CULTIS model is haphazard and lacking in sense and motivation. As an illustration of this, let us consider the role of visualization. Dahl sorts this under “Individual,” contrasting it with “Social” (p. 140). A myriad of obvious problems with this arrangement suggests itself immediately, all of which are ignored by Dahl. Why should visualization be grouped with “self-activity” (p. 140)? Isn’t visual thinking better contrasted with the theme “Language” than with “Social”? The antipode of “Language” is “Tacit”, which is described as the attitude that the pupil “cannot tell but only show” (p. 140), which sounds almost synonymous with visualization. Indeed, the descriptions of “Tacit” and “Individual” are confusingly similar. As a characterisation of the “Tacit” theme we read: “It is therefore actions that form the roots of logical and mathematical thoughts.” (p. 137) But then the theme “Individual” is described in virtually the same terms: “Thus the logical-mathematical abilities do not arise from language or linguistic competency, but from the ability to coordinate actions.” (p. 137) Dahl’s arbitrary bundling of visual with individual makes her analysis heavily theory-biased. Her “Pupil C” said: “I tend to learn more when visual or rather than just [A coughs] look at it [A & C laugh, some words are missed]. Yea, I’ll try to make it more visual, like that (I: mmm).” (p. 147) From here Dahl concludes that “Pupil C … tells that he is ‘more visual’, which is individual learning” (p. 147). Obviously the data does not support this identification of visual learning with individual learning. Although the transcript contains much useless information about who was coughing when, the crucial passage where Pupil C explains what he is contrasting visual thinking with (social? linguistic?) is “missed.” Brown, S. A., Pitvorec, K., Ditto, C., & Kelso, C. R. (2009). Reconceiving fidelity of implementation: An investigation of elementary whole-number lessons. Journal for Research in Mathematics Education, 40(4), 363–395. This study investigates to what extent elementary school teachers working within a Standards-based curriculum are faithful to the spirit of this material in their teaching. The authors summarize their findings as follows: “we (a) concluded that the level of fidelity to the literal lesson does not determine the level of fidelity to the authors’ intended lesson, and vice versa; (b) observed that individual teachers’ enacted lessons tend to have some consistency in their ratings for level of fidelity to the authors’ intended lesson; and (c) identified two lesson types––lessons for which enactments varied by teacher and lessons for which the [level of fidelity] rating for the enactments appears to be related to the lesson itself.” (pp. 389–390) I say: these results are all truistic, and consequently the study is next to worthless. To substantiate this claim, let us consider the results in order. The truistic nature of (a) is readily apparent once we note that “the literal lesson” means the written curricular materials and “the authors’ intended lesson” refers to their “underlying philosophy” (p. 369). Surely it is not surprising that overworked teachers sometimes rely on the pre-fabricated “literal lessons” without reflecting very much on their “underlying philosophy.” Nor is is surprising that, conversely, the “underlying philosophy” may be read into a lesson of a teacher who strays from “the literal lesson” since such a teacher may quite plausibly have a similar philosophy herself, regardless of the curricular materials. The chances of such accidental compliance with “underlying philosophy” are especially marked since Brown et al. characterize the “underlying philosophy” in terms of extremely general “opportunities to learn,” such as “opportunities to reason to solve problems; opportunities to reason about mathematical concepts” and “opportunities to validate strategies or solutions; reason from errors; inquire into the reasonableness of a solution.” Clearly, such “opportunities to learn” are bound to arise in any mathematics classroom regardless of the “underlying philosophy” of the curricular materials. In (b) we have another unremarkable result, namely that not all teachers are equally enthusiastic about the prescribed curriculum. It would have been stunning indeed if the researchers had found that a teacher’s background, training, etc., had no consistent impact at all on her attitude towards the curricular materials she had to use. As for (c), this may sound like an interesting result––what type of lessons are teachers enthusiastic about?––but we are disappointed to find that this question is left unanswered except by mere equivocation: the “type” of lesson of lesson that received consistent level-of-fidelity ratings is not independently characterized but rather simply defined as the set of all such lessons (p. 387). So all (c) really says is that sometimes level of fidelity appears closely related to the content of the curricular material for that lesson and sometimes not. The most plausible interpretation of this result is surely that the curricular materials are (i) not all of the same quality, (ii) not all equally manageable or realistic to implement, (iii) not all equally accessible to typical teachers. Again, nothing about this is the least bit novel or enlightening. Having dismissed the concrete results of the study, we must recognize that the authors purports to have made a theoretical advance as well, viz. “by providing a framework with which to view teachers’ enactment of lessons that connects students’ engagement in opportunities to learn mathematics with with those intended by the curricular materials” (p. 390). A “framework” is only as good as the insights it yields––it is not an end in itself, as the authors seem to imply, that it “contributes to [a] growing body of research” (p. 390)––so I think we are justified in ignoring it until the authors have proved it fruitful. Even so, we could not help but noticing above why the extremely general and vague notion of “opportunities to learn” is bound to be virtually useless for analyzing the specific intentions of authors of curricular materials. I should also like to point out that the authors’ notion of “research” is a very twisted one. The authors are apparently convinced that the only way to “research” these “teaching beasts” is to record their guttural cries in the wild, and then have rational men reconstruct their savage mannerisms on the basis of this data. Why not simply talk to the teachers about why they choose to implement certain aspects of the curricular materials but not others? (While the notion of talking to the teaching beasts was apparently inconceivable to the researchers, textbook authors were considered rational animals, for “if the authors’ intended lesson was not clear, we … asked for clarification from the authors” (p. 382).) Ignoring this common-sensical approach, the authors instead choose to focus their “researcher’s lens” on a minuscule data set, namely 33 video-recorded classroom lessons from a total of 14 teachers (p. 375); i.e. an average of 2.4 lessons per teacher. Anyone who has ever taught more than 2.4 lessons knows perfectly well that an unfortunately chosen sample of that size could easily lead to innumerable mistaken impressions; especially so when the unit of analysis (“opportunity to learn”) is absurdly abstract and most likely far removed from the terms in which the lesson was conceived. Furthermore, as bad as it would have been if the measly 2.4 lessons were a random sample, they were apparently not even that, for “several teachers expressed that, since they were being observed, they felt compelled to teach the lesson as written” (p. 389), thus nullifying any remaining shred of credibility that the data may have had. Vale, C. M. & Leder, G. C. (2004). Student views of computer-based mathematics in the middle years: does gender make a difference? Educational studies in mathematics, 56(2), 287–312. The authors purport to have found significant gender differences in attitudes towards computer-based mathematics classes. After discussing the wider context of this study, I shall, for lack of space, focus my critique solely on its quantitative part, which I say should be dismissed owing to the stupidity of the researchers’ methodology. It is natural to situate this study in the context of the NCTM Principles and Standards. This document’s “Equity Principle” notes that different students thrive under different conditions, expectations and stimuli, and calls for each student to be nurtured accordingly. Insights regarding gender differences are thus highly pertinent for successful implementation of this principle. However, the study by Vale and Leder can be criticized for falling short of this goal in several regards. One issue is that Vale and Leder focus almost exclusively on description of the phenomenon, offering little of substance for improved practice. Also, their study begins to partially diverge from the Principles and Standards when it comes to the specific uses of computers in the classroom. Vale and Leder studied two groups of students. From what we learn of the first group, their use of computers seems to be a near-perfect enactment of the vision of the Principles and Standards. In particular, they focused on using Geometer’s Sketchpad to form and test conjectures (p. 293)––a theme emphatically emphasized throughout the Principles and Standards. But the second group of students seem to have focused on computers for entirely different reasons. A telling illustration is their use of PowerPoint to present solutions to simple linear equations (p. 293). It is hard to imagine a mathematical rationale for this; the purpose seems rather to have been preparation for the business world. This considerable discrepancy in the uses of computers raises the issue of whether it makes sense in the first place to talk about “attitudes to computer-based mathematics” in the abstract. No one would dream of drawing conclusions about “students’ attitudes towards the use of books in mathematics classes” based on a study involving one or two books. Whether “computer-based mathematics” is a cohesive enough concept to warrant such conclusions in this case is highly questionable, and, in any case, an issue never touched upon by Vale and Leder. Thus one may question whether the research question makes any sense in the first place. But let us now turn to the results of the study itself. As I said, I only have room to discuss the quantitative aspect of it. The researchers considered a number of quantitative parameters but obtained statistical significance in only two cases: first, a predictable and quite uninteresting correlation between “achievement in computing” and appreciation of computers in mathematics classes (pp. 306–307), and, secondly and potentially more interestingly, a significant gender difference in attitude to computer-based mathematics. The latter was measured by asking the students to indicate their agreement or disagreement with the following statements on a Likert scale. “[A] I’ve improved in maths since we started using computers in maths. [B] I’ve gone backwards in maths since we started using computers in maths. [C] I am sure I could do difficult maths with the use of a computer. [D] Even a computer can’t help me learn maths. [E] Using a computer in maths gives you a reason for doing maths. [F] Using a computer in maths does not make maths any more useful.  [G] I find that using computers helps me to learn maths. [H] Using computers in maths means you won’t be able to do maths without them. [I] Maths is easier to understand when you use computers. [J] Using computers in maths makes maths more confusing. [K] Computers are excellent for doing things for maths.” (p. 297) I say that the results of this study should be disregarded because of the sheer stupidity of this list. A number of the entries are in effect guaranteed to be true or false, thus being completely useless for the purpose of the study. For example, A and B are bound to be true and false respectively for any student whose teaching is not only non-progressive but in fact outright detrimental to learning. So A and B are useful only for comparing computer classes to such detrimental teaching or no teaching at all, neither of which is a realistic alternative for the purposes of this study. So results about A and B should be discarded as saying nothing of interest in reply the research question at hand. Another entry on the list that is trivially true is K. This statement has nothing to do with teaching. It merely asserts a fact about the capacities of computers that no sane person could possibly dispute. Obviously the researchers intended K to be understood differently, but since it is trivially true in its literal meaning all student replies must be discarded for this entry as well. The same goes for C. Presumably the researchers intended this statement to be interpreted as “I am sure of my current ability to do difficult mathematics with a computer.” But C could just as well be interpreted as analogous to “I am sure I could go on a diet and loose 40 pounds (though perhaps I do not think it worth the effort),” in which case it would be trivially true, so again all replies must be discarded. Similarly, H is trivially false. Of course using computers in mathematics does not mean that you won’t be able to do mathematic without them. It may be that the use of computers has such a detrimental effect in many cases, but that is not what is being asked. As it stands, H is plainly false so all replies must be discarded. Finally, D is a very stupidly formulated statement. The “even” insinuates that if a computer cannot teach you mathematics then no one can. In order to agree or disagree with this statement one must in effect accept this implicit premise. Therefore, since this premise is obviously highly dubious, not to say outright dumb, all answers to this statement must be discarded. In all of these cases, then, the statistical significance obtained need not have anything to do with the issues at hand. Instead the gender differences may be due exclusively to differences in interpretation of the questionnaire, viz., the difference between interpreting it literally or second-guessing what the researchers intended to ask. I conclude therefore that the only legitimate entries on the list are E, F and G. Since the researchers do no disclose their data for replies to individual statements, we have no reason to believe that the statistical significance declared still hold for these three entries. Therefore we should disregard the results. Yin, Y., Shavelson, R. J., Ayala, C. C., Ruiz-Primo, M. A., Brandon, P., Furtak, E. M., Tomita, M.K., & Young, D.B. (2008). On the Impact of Formative Assessment on Student Motivation, Achievement, and Conceptual Change. Applied Measurement in Education, 21(4), 335-359. A study of 12 middle-school science classes, half using formative assessment and half not. All classes had a common curriculum, from which one particular unit was selected for the study. Treatment group teachers were provided with some sort of formative assessment training and materials specific to this unit, neither of which are described in this article (the authors refer to a different paper for details). Pretest and posttests were administered to test the effect of formative assessment on motivation, achievement and conceptual change, but no significant effect was observed on any of these measures. The inconclusive results of the study were not unexpected considering the innumerable confounding factors at play, which included spontaneous use of formative assessment by control group teachers, failure to implement the formative assessment materials among treatment group teachers, enormous discrepancies in class time devoted to the unit (varying from 63 to 249 days), severe ESL issues in some classes, and differences in standards and standardized testing since the classes were from several different states. Remillard, J. T., & Jackson, K., (2006). Old math, new math: parents’ experiences with standards-based reform. Mathematical Thinking and Learning, 8(3), 231–259. This is a study of “how African American parents in a low-income neighborhood experience … current reform efforts” in mathematics (p. 231). The ultimate goal is to make parents “partners in mathematics education reform,” rather than “stumbling blocks,” as they are sometimes portrayed (p. 233), since parental involvement has been show to have a significant positive impact on student achievement (p. 232). The present research project, however, sets itself the more modest goal of giving a phenomenological description of parents’ experiences in the hope that this will be “the first step in conceptualizing ways to include and support parents as partners in their children’s mathematics education” (p. 233). My review shall focus on sources of bias in this research. I shall provide evidence that: (1) the authors have a dangerously uncritical conviction that the reform materials are wholly positive, (2) their conception of parents’ attitudes as almost entirely conditioned by personal school experience is not justified, (3) the study suffers from selection bias relative to the research goal as stated above. The authors themselves do not address these problems. To illustrate points (1) and (2) we may consider how the authors deal with the fact that “none of the parents saw the connections between the mathematics represented in EM and the mathematics of their everyday lives” (p. 245; EM stands for Everyday Mathematics, a curriculum based on the NCTM Standards used in the school in question). Note the definite article: “the connections.” The authors take it for granted that the EM is excellent and full of such connections that are “readily apparent” (p. 255). Thus they must proceed to explain why the parents cannot see what is “readily apparent.” They propose as an explanation that the parents “had firmly established conceptions of school mathematics that were grounded in computational proficiency” (p. 255). But then the authors quickly go on to contradict themselves later on the same page: “[M]any of the learning goals that they held for their children overlapped with those central to EM. Parents described wanting their children to develop confidence, independence, and the ability to use math in their everyday lives. Several parents spoke of wanting their children to develop a deep understanding of math” (p. 255). As one parent put it, she wanted her daughter to “use her brain cells” (p. 253). So how do the authors confront the fact that their own explanation regarding why parents were critical of EM material is here blatantly contradicted? They simply dismiss this contradiction as “ironic” (p. 255)! This seems to me revealing of the authors’ inability to conceive of the possibility that the parents’ criticisms have any merit. If the parents said they didn’t like the school cafeteria food or that the basketball coach was poor, then we wouldn’t be very good researchers if we dismissed their opinion as “ironic” just because we happened to like the food and the coach. Instead of contemptuously rationalizing the parents’ criticism as based on prejudice, we should listen to their arguments and see what merit they have. The parents did indeed put forth arguments based on reason rather than prejudice. For example, one parent argued that “if I was teaching [my daughter] Serahn how to drive, I couldn’t do it in January and then one day in March and then in June she’ll go take the test, and pass it. It don’t work that way, not even with the math.” (p. 248). This is an intelligent and substantial critique of EM, not an inability to see its “readily apparent” virtues due to preconceived notions of what school mathematics should be. One could give many more examples, which is not surprising since “old math” was advocated by intelligent people at its time. Not everyone’s approval of “old math” can be ascribed to bias conditioned by their own education, on pain of infinite regress, since “old math” is not so old as to have existed since the beginning of time. How can the authors claim to aim at “conceptualizing parents as partners in mathematics education” (p. 257) if they do not take such critique seriously? They agree that this goal “requires serious consideration of parents’ genuinely felt rejection of reforms” (p. 257). Genuinely FELT, that is the key word: this is a matter of emotion rather than reason, according to the authors. The authors apparently considers themselves very gracious in that they “contend that parents have the right to disagree with the reforms” (p. 257). Maybe opportunities to get their “feelings” off their chests will make these irrational little creatures feel better, the authors seem to hope. I think the spectrum of debate here is revealing. The underlying assumption seems to be that “reform” is great and that, therefore, parents’ critiques of it must be incompetent. Given these assumption, one can debate whether or not parents should be allowed to speak at all (let alone be listened to, which is of course out of the question). The fact the the authors consider themselves on the liberal end of the spectrum and feel that parents’ right to disagree is something for which one needs to “contend” is a depressing indicator of how deeply ingrained these underlying assumptions Finally, I should like to point to a separate issue of bias in this research, namely selection bias. The authors were careful to avoid selection bias with respect to variables such as age, education, employment, etc. (p. 239). Yet they ignored what is arguably the most crucial variable for the purposes of the present study: involvement. Here we have the surely very exceptional state of affairs that “All 10 parents were heavily involved in their children’s mathematics learning beyond homework assistance” (p. 254). Given that the research is justified by the significant impact of parent involvement on achievement, it is questionable whether studying already highly involved parents contributes to the stated goals of the research.	{"url":"https://intellectualmathematics.com/blog/some-critiques-of-mathematics-education-research-articles/","timestamp":"2024-11-12T22:00:26Z","content_type":"text/html","content_length":"78764","record_id":"<urn:uuid:92cc08c2-0321-4036-a215-8b3c01e183be>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00828.warc.gz"}
Average Rating Calculator Last updated: Average Rating Calculator The average rating calculator takes a number of votes for each rating (1 star, 2 stars, 3 stars, 4 stars, 5 stars) and gives you the mean rating. This is an example of taking the weighted average, where the number of votes for each category are weighted by their score (i.e., 3 votes for 5 stars have a weight of 15, 6 votes for 4 stars have a weight of 24, etc.). Keep reading to learn what is the star rating system, the difference between a star classification and an average star rating, and how to calculate the star rating for any combination of votes. What is the star rating? The star system is a classification method that uses glyphs and symbols such as stars to visualize in a quick and intuitive way the ranking of many things, from services to purchased items, usually where a certain degree of interaction with the customers is present. There are two principal types of star classification: • Star rating, where a given number of stars is assigned, and the higher the number, the higher the quality. Think of the Michelin star for restaurants or hotel classifications. • Star "voting", where many votes expressed as a number of stars are tallied to return a final rating that takes into account the opinion of many people. We are mostly interested in the latter: in the next section, we will learn how a star rating calculator system works and how to calculate the average rating in such systems. For now, let's explore these systems a bit more! Assigning an increasing number of symbols to describe the quality of something apparently appeared the first time in guidebooks in the first half of the 19th century (though a monk traveling in Italy in the 12th century already classified a wine as Est! Est!! Est!!!). Rather quickly, stars became the preferred symbol. By the beginning of the 20th century, hotels started receiving official star With the onset of the internet, it became possible to gather the opinion of many customers in a quick and efficient way, which led to the popularity of average star rating systems. In these systems, the user is tasked with assigning a certain number of stars to a good or service. Once enough votes are gathered, we can show future customers the result: a number of stars that reflects the opinion of the previous customers. But how do we calculate the average rating in such systems? How to calculate the average rating in a star rating system To calculate the average rating in a star rating system, we follow some simple steps: 1. We count the number of votes for each star number (number of votes for one star, number of votes for two stars, etc.). 2. We multiply the number of votes by the matching star number: this means that the number of votes for a three-star rating gets multiplied by 3. 3. We sum all these products. 4. We divide the result of the previous step by the total number of votes. This returns a weighted average according to the formula: $\!\scriptsize \mathrm{rating}\! =\!\frac{5\!\times\! r_5\!+\!4\!\times\! r_4\!+\!3\!\times\! r_3\!+\!2\!\times\! r_2\!+\!1\!\times\! r_1}{r_5+r_4+r_3+r_2+r_1}$ • $\mathrm{rating}$ — Average rating; and • $r_1$, $r_2$, $r_3$, ... — Number of votes for each number of stars. 💡 You may also be interested in our average calculator or in another example of a weighted average problem. Our college GPA calculator is perfect when you want to know what your current grade is or what you need to do to achieve your desired GPA. When calculating this, grades are weighted based on the number of credits. What is the average five star rating formula? The formula for the average 5-star rating reads: average rating = (5r[5] + 4r[4] + 3r[3] + 2r[2] + r[1]) / (r[5] + r[4] + r[3] + r[2] + r[1]) where r[i] is the number of i stars responses for i = 1, ..., 5. How do I calculate the average 5 star rating? To determine the average 5-star rating, you need to: 1. Gather the data: write down how many five stars you got, how many four stars, etc. 2. Add together all numbers from Step 1. This is the total number of responses. 3. Multiply the number of five stars by 5, the number of four stars by 4, etc. 4. Add together all numbers from Step 3. This is your total rating. 5. Divide the number from Step 4 by that from Step 2. This is your average rating! What is good average five star rating? The best rating is around 4.2-4.5. Research showed these ratings are the most influential and most trusted. The rating of 5 raises doubts in customers' minds, who think it is "too good to be true". How do I raise my average rating from 3.5 to 4? You need to consistently get a rating of at least four stars (so 4 stars or 5 stars). The number of good reviews you need to raise the average depends on how many you've already had. In general, the more bad or average reviews you got in the past, the more good or perfect reviews you need now to raise the average rating.	{"url":"https://www.omnicalculator.com/statistics/five-star-rating","timestamp":"2024-11-07T01:17:36Z","content_type":"text/html","content_length":"502882","record_id":"<urn:uuid:1f1a5baa-b383-472e-9438-d744f7331034>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00511.warc.gz"}
The Stacks project Lemma 29.30.2. Let $f : X \to S$ be a morphism of schemes. The following are equivalent 1. The morphism $f$ is syntomic. 2. For every affine opens $U \subset X$, $V \subset S$ with $f(U) \subset V$ the ring map $\mathcal{O}_ S(V) \to \mathcal{O}_ X(U)$ is syntomic. 3. There exists an open covering $S = \bigcup _{j \in J} V_ j$ and open coverings $f^{-1}(V_ j) = \bigcup _{i \in I_ j} U_ i$ such that each of the morphisms $U_ i \to V_ j$, $j\in J, i\in I_ j$ is 4. There exists an affine open covering $S = \bigcup _{j \in J} V_ j$ and affine open coverings $f^{-1}(V_ j) = \bigcup _{i \in I_ j} U_ i$ such that the ring map $\mathcal{O}_ S(V_ j) \to \mathcal {O}_ X(U_ i)$ is syntomic, for all $j\in J, i\in I_ j$. Moreover, if $f$ is syntomic then for any open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$ the restriction $f\|_ U : U \to V$ is syntomic. Comments (2) Comment #1868 by Kestutis Cesnavicius on Typo in the discussion of this section preceding this lemma: "One we have ..." --> "Once we have ..." Comment #1903 by Johan on THanks, fixed here. Post a comment Your email address will not be published. Required fields are marked. In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). All contributions are licensed under the GNU Free Documentation License. In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01UD. Beware of the difference between the letter 'O' and the digit '0'. The tag you filled in for the captcha is wrong. You need to write 01UD, in case you are confused.	{"url":"https://stacks.math.columbia.edu/tag/01UD","timestamp":"2024-11-12T12:21:44Z","content_type":"text/html","content_length":"16448","record_id":"<urn:uuid:0d789463-6515-4e63-9c18-078300bbc03f>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00315.warc.gz"}
Algebra and Logic: Impact Factor, Ranking, H-Index, ISSN, CiteScore, SJR and Other Key Journal Metrics \| Researcher.Life Algebra and Logic Key Metrics % of papers by time taken from submission to publication 0 to 3 months8% 4 to 6 months12% 7 to 9 months23% Above 9 months58% Did you find this useful? Algebra and Logic Journal Specifications Publisher SPRINGER Language English Frequency Bi-monthly Planning to publish in Algebra and Logic ? Upload your Manuscript to get • Degree of match • Common matching concepts • Additional journal recommendations Compare Similar Journals with Algebra and Logic Select Journals to Compare with Algebra and Logic You can select up to 3 Journals at a time Journal of Group Theory Publisher: WALTER DE GRUYTER GMBH Communications in Algebra Publisher: TAYLOR & FRANCIS INC Journal of Algebra Publisher: ACADEMIC PRESS INC ELSEVIER SCIENCE Journal of Algebra and its Applications Publisher: WORLD SCIENTIFIC PUBL CO PTE LTD Archiv der Mathematik Publisher: SPRINGER BASEL AG Proceedings of the Steklov Institute of Mathematics Publisher: MAIK NAUKA/INTERPERIODICA/SPRINGER Monatshefte fur Mathematik Publisher: SPRINGER WIEN Czechoslovak Mathematical Journal Publisher: SPRINGER HEIDELBERG International Journal of Algebra and Computation Publisher: WORLD SCIENTIFIC PUBL CO PTE LTD 0 Journals Selected to compare with Algebra and Logic Compare journals across 7 parameters SJR Ranking, Topics covered, Indexing Database, Publication Review Time, Publication Type, Article Processing Charges & Recently Published papers SJR Ranking, Topics covered, Indexing Database, Publication Review Time, Publication Type, Article Processing Charges & Recently Published papers Get detailed Journal comparison report on your email by signing up Check if your research matches the topics covered in Algebra and Logic? • Degree of match • Common matching concepts • Additional journal recommendations Check your research FAQs on Algebra and Logic How long has Algebra and Logic been actively publishing? Algebra and Logic has been in operation since 1968 till date. What is the publishing frequency of Algebra and Logic? Algebra and Logic published with a Bi-monthly frequency. How many articles did Algebra and Logic publish last year? In 2023, Algebra and Logic publsihed 32 articles. What is the eISSN & pISSN for Algebra and Logic? For Algebra and Logic, eISSN is 1573-8302 and pISSN is 0002-5232. What is Citescore for Algebra and Logic? Citescore for Algebra and Logic is 1.2. What is SNIP score for Algebra and Logic? SNIP score for Algebra and Logic is 1.12. What is the SJR for Algebra and Logic? SJR for Algebra and Logic is Q3. Who is the publisher of Algebra and Logic? SPRINGER is the publisher of Algebra and Logic.	{"url":"https://researcher.life/journal/algebra-and-logic/8242","timestamp":"2024-11-04T14:15:57Z","content_type":"text/html","content_length":"244747","record_id":"<urn:uuid:b1c75c61-e6db-442c-8242-9a8abb4becda>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00873.warc.gz"}
Vebjørn H. Bakkestuen - REGAL PhD Research Fellow, REGAL project Affiliation: MatMod group at oslomet.no Researcher IDs: ORCID, Google Scholar, ResearchGate Email: vebjorn.bakkestuen@oslomet.no Address: Pilestredet 35, PS437 About me Vebjørn has a background from physics with a focus on quantum mechanics, with both a bachelor and master’s degree in physics from the University of Oslo. He received his master’s degree in theoretical physics form the University of Oslo in the spring of 2023, with a thesis titled “Non-Hermitian Quantum Mechanics: On the role of PT-symmetry and exceptional points”. Research Interests Vebjørn is currently a PhD candidate working on the mathematical aspects of quantum chemistry, in particular that of Density Functional Theory (DFT). These aspects include the Moreau-Yosida regularisation as well as extensions of DFT. At the moment he and the group are working on applying and analysing methods from DFT to quantum electrodynamics. In particular, the study of the quantum Rabi model and the generalised Dicke model. He is also working on Kohn-Sham inversion for periodic systems by use of the Moreau-Yosida regularisation of DFT. The analysis of Kohn-Sham inversion includes devising an algorithm for the inversion procedure as well as strict error bounds on the inverted potential. As a proof of example for a recent preprint, this scheme have been applied numerically to bulk silicon. Vebjørn is currently a part of the ERC project REGAL at OsloMet. Vebjørn is also co-hosting a workshop on the Foundations and Extensions of Density-Functional Theory. Link to webpage. 1. Kohn-Sham inversion with mathematical guarantees (Submitted September 2024) No current teaching responsibilities. Previously, during the two years as a master’s student at the University of Oslo, Vebjørn was a teaching assistant in the courses FYS2160 – Thermodynamics and Statistical Physics, FYS-MEK1110 – Mechanics, and FYS1100 – Mechanics and Modelling. • Master’s thesis in DUO Research Archive. • Presentation on a QEDFT model at the Center for Advanced Study at the Norwegian Academy of Science and Letters. • Poster on Kohn-Sham inversion at the Mathematical and Numerical Analysis of Electronic Structure Models (MANUEL) conference in Stuttgart, Germany September 2024	{"url":"https://uni.oslomet.no/regal/vebjorn-h-bakkestuen/","timestamp":"2024-11-12T07:09:34Z","content_type":"text/html","content_length":"41377","record_id":"<urn:uuid:ab228e5a-856e-48e7-b77e-fab79efd1488>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00063.warc.gz"}
Theory of Combinatorial Algorithms Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer) Mittagsseminar Talk Information Date and Time: Tuesday, October 27, 2015, 12:15 pm Duration: 30 minutes Location: OAT S15/S16/S17 Speaker: Felix Weissenberger On a One-Shot Model for Sequence Learning Motivated by a system to store and retrieve sequences in a neural network, we study the following problem. Given a graph and a subset of its vertices A0 we define a sequence of subsets A0,…,Al recursively: Ai+1 is the set of vertices with at least k neighbours into the set Ai. Our goal is to find a sparse random graph which contains a long sequence such that the Ai's have roughly the same specified size and do not overlap too much. For the right p, Gnp contains such a sequence of logarithmic length. We show that starting with a slightly larger p and deleting a small fraction of the edges (constrained by the motivating system) yields a sequence of polynomial length. This is joint work with H. Einarsson, J. Lengler, F. Meier, and A. Steger Upcoming talks \| All previous talks \| Talks by speaker \| Upcoming talks in iCal format (beta version!) Previous talks by year: 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 Information for students and suggested topics for student talks Automatic MiSe System Software Version 1.4803M \| admin login	{"url":"https://ti.inf.ethz.ch/ew/mise/mittagssem.html?action=show&what=abstract&id=319317d93671814932514585656c852674154173","timestamp":"2024-11-04T08:02:49Z","content_type":"text/html","content_length":"13555","record_id":"<urn:uuid:15bf7d22-1967-4907-a301-e10e554282e6>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00572.warc.gz"}
NCERT Solutions for class 10th Maths Chapter 5 Arithmetic Progressions Exercise 5.1 Question 1. In which of the following situations, does the list of numbers involved make an arithmetic progression, and why? 1. The taxi fare after each km when the fare is < 15 for the first km and < 8 for each additional km. 2. The amount of air present in a cylinder when a vacuum pump removes 1/4 of the air remaining in the cylinder at a time. 3. The cost of digging a well after every metre of digging, when it casts rs.150 for the first metre and rises by rs.50 for each subsequent metre. 4. The amount of money in the account every year, when rs.10000 is deposited at compound interest at 8% per annum. Sol. (i) tn denotes the taxi fare (in rs.) for the first n km. Now, t1 = 15, t2 = 15 + 8 = 23, t3 = 23 + 8 = 31, t4 = 31+ 8 = 39, ….. List of fares after 1km, 2km, 3km, 4km,… respectively is 15, 23, 31, 39, (in<) Here, the difference between the consecutive terms is constant, i.e; t2-t1=t3-t2=t4-t3 =….= 8 Note: We can also write the list of amounts for each year as : 10000, 10000 x (1+8/100), 10000x(1+8/100)2, 10000 x (1+8/100)3….. Question 2. Write first four terms of the AP, when the first term a and common difference d are given as follows: (i) a, = 10, d = 10 (ii) a= -2,d = 0 (iii) a = 4, d = – 3 (iv) a= -1,d=1/2 (v) a = -1.25,d = – 0.25 Note: a, a+d, a+2d, a+3d,represents an arithmetic progression where a is the first term and d the common difference.This is called the general form of an AP Question 3. For the following APs, write the first term and the common difference: (i) 3, 1, -1, -3, ….. (ii) -5, -1, 3, 7, ….. (iii) 1/3,5/3,9/3,13/3····· (iv) 0.6, 1.7, 2.8, 3.9, ….. Question 4. Which of the following are APs? If they form an AP, find the common difference d and write three more terms. Exercise 5.2 Question 1. Fill in the blanks in the following table, given that a is the first term, d the common difference an and the nth term of the AP: Question 2. Choose the correct choice in the following and justify: Question 3. In the following APs, find the missing terms in the boxes: Question 4. Which term of the AP: 3, 8, 13, 18,is 78? Question 5. Find the number of terms in each of the following APs: Question 6. Check whether-150 is a term of the AP: 11, 8, 5, 2…. Question 7. Find the 31st term of an AP whose 11th term is 38 and the 16th term is 73. Question 8. An AP consists of 5O terms of which 3rd term is 12 and the last term is 106.Find the 29th term. Question 9. If the 3rd and the 9th term ofanAP are 4 and -8, respectively.Which term of this AP is zero? Question 10. The 17th term of an AP exceeds its 10th term by 7.Find the common difference. Question 11. Which term of the A.P. 3, 15, 27, 39,will be 132 more than its 54th term? Question 12. Two APs have the same common difference. The difference between their 100th terms is 100,what is the difference between their 1000th terms?. Question 13. How many three-digit numbers are divisible by 7. Question 14. How many multiples of 4 lie between 10 and 250?. Question 15. For what value of n, are the nth terms of two APs: 63,65,67,and 3,10,17,….equal? Question 16. Determine the A.P. whose third term is 16 and the difference of 5th term from 7th term is 12. Question 17. Find the 20th term from the last term of the AP: 3, 8, 13, 253, Note: nth terms from the last can also be calculated using formula given below: an from last= l-(n-l)d, where, l=last term Question 18. The sum of the 4th and 8th terms of an AP is 24 and the some of the 6th and 10 terms is 44. Find the first three terms of the AP. Question 19. Subba Rao started work in 1995 at an annual salary of rs.5000 and received an increment of rs.200 each year. In which year did his income reach rs.7000? Question 20. Ramkali saved rs.5 in the first week of a year and then increased her weekly saving by rs.1.75.If in the nth week, her weekly savings become rs.20.75, find n. Exercise 5.3 Question 1. Find the sum of the following APs: Question 2. Find the sums given below: Note: In an AP, when a, l, n is given, Sum of first n terms can also be calculated using formula. Sn=n/2(a+l) Question 3. In an AP: Note: We also use S in place of Sn to denote the sum of first n terms of the AP Question 4. How many terms of the AP: 9, 17, 25,must be taken to give sum of 636? Question 5. The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference. Question 6. The first and the last terms of an AP are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum? Question 7. Find the sum of first 22 terms of an AP in which d= 7 and 22nd term is 149. Question 8. Find the sum of the first 51 terms of the A.P. whose second and third term are respectively 14 and 18. Question 9. If the sum of 7 terms of an AP is 49 and thatof17 terms is 289, find the sum of n terms. Question 10. Show that a1+ a2,….., a11, •••••• form an AP where a11 is defined as below : (i) a = 3 + 4n (ii)a = 9 – 5 Also, find the sum of the first 15 terms in each case. Question 11. If the sum of the first n terms of an AP is 4 -n2, what is the first term (that is S1)? What is the sum of first two terms? What is the second term? Similarly, find the 3rd, the 10th and the nth terms. Question 12. Find the sum of the first 40 positive integers divisible by 6. Question 13. Find the sum of the first 15 multiples of 8. Question 14. Find the sum of the odd numbers between 0 and 50. Question 15. A contract on a construction job specifies a penalty for delay of completion beyond a certain date as follows: 200 for the first day, 250 for the second day, 300 for the third day, etc., the penalty for each succeeding day being 50 more than for the preceding day. How much money the contractor has to pay as penalty, if he has delayed the work by 30 days? Question 16. A sum of rs700 is to be used to given seven cash prizes to students of a school for their overall academic performance. If each prize is rs 20 less than its proceeding prize, find the value of each of the prizes. Question 17. In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying e.g., a section of Class I will plant one tree, a section of Class II will plant 2 trees and so on till Class XII. There are three sections of each class. How many trees will be planted by the students? Question 18. A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A, of radii 0.5 cm, 1.0 cm,1.5 cm, 2.0 cm,… as shown in Fig. What is the total length of such a spiral made up of thirteen consecutive semicircles? (take pi=22/7) [Hint : Length of successive semicircles is 11, 12, 13, 14,with centres at A, B, A, B, respectively.] Question 19. 200 logs are stacked in the following manner:20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on (see Fig.). In how many rows are the 200 logs placed and how many logs are in the top row? Question 20. In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato, and the other potatoes are placed 3m apart in a straight line. There are ten potatoes in the line (see Fig.) A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run? [Hint: To pickup the first potato and the second potato, the total distance (in metres) run by a competitor is 2 x 5 + 2 x (5 + 3)] Exercise 5.4 Question 1. Which term of the AP: 121, 177, 113,…,is its first negative term? Hint: (Find n for an < 0) Question 2. The sum of the third and the seventh terms of an AP is 6 and their product is 8.Find the sum of first sixteen terms of the AP Question 3. A ladder has rungs 25 cm apart. (see fig.).The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top.If the top and the bottom rungs are 2 1/2 m apart, what is the length of the wood required for the rungs? (Hint: Number of rungs = 250/25) Question 4. The houses of a row are numbered consecutively from 1 to 49. Show that there is a value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x. Question 5. A small terrace at a football ground comprises of15 steps each of which is 50 m long and built of solid concrete. Each step has a rise of 1/4 m and a tread of 1/2 m.(see Fig).Calculate the total volume of concrete required to build the terrace. [Hint : Volume of concrete required to build the first step = 1/4 x 1/2 x 50m3] Related Articles:	{"url":"https://cbseacademic.in/class-10/ncert-solutions/maths/arithmetic-progressions/","timestamp":"2024-11-04T11:52:11Z","content_type":"text/html","content_length":"136466","record_id":"<urn:uuid:f3fba56a-dcd9-4274-8818-2da223f85095>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00575.warc.gz"}
Acquiring an understanding of proportion and its applications Part 2 Written by Deirdre Jennings Parent Category: Science Category: Science 12-15 Created: 12 February 2011 PART 2 (find PART 1 HERE) STEP 1 - STEP 2: Now they have to find a new way to solve the same grouping. We need to learn a new way. This is where we stopped the lesson although some were starting to have the idea of dividing the numbers... We will resume this task tomorrow. The idea here is to go through a few more cycles using the same task: i.e. Allow them to divide the numbers and find proportionalities (body:head) adults 7:1 children 4:1 or 5:1 then relate this to fractions if possible. Then disallow this as a solution. Next solution should be drawing pictures/diagrams showing a child as 4 heads or 5 heads tall and an adult as 7 or 8 heads tall. Then take that away and lead them to graphing (body length vs head length) and finding slopes/gradients. Hopefully this should mean that they have a really solid appreciation of proportion before we go to the graph NEXT DOUBLE LESSON: First we reviewed the previous lessons work as some will have forgotten the point we reached 4 days ago. STEP 2 - 1 - 2 again: Students were very confident in the use of the language 'X length/size compared to length/size of Y' so I said "language - X compared to Y - is now not allowed, find a new way to express the relationship you observed". As this was the first run through I eventually suggested using pictures/diagrams to explain instead of words. No students had started independently making measurements on the pictures yet which was interesting. After this I said "ok find a new way, not a diagram and not using words". Students did get stuck at this point again, so I sent them back again to the pictures and to review their other descriptions. As a hint I dropped rulers on the tables... They started measuring and after a little while they started dividing the numbers together. They therefore observed that if they focused on one pair of variables that the 'answers' from dividing the 2 numbers were very different for adults than for young animals. They therefore understood that they could USE these numbers to select members of groups. We discussed this and I allowed them to re-sort the 'young' vs 'adult' groups using the various forms of expressions. They observed that these worked smoothly. Students came up with several different ways/diagrams etc. that worked. See them HERE. STEP 3: TEST and REFLECT: Challenge after task has reached this stage just before progressing to graphs (as an extension). Students are asked to organise 5 human skeletons into order of ages. Youngest to oldest or reverse doesn't matter. BUT they must have the calculations done to defend their choice. This also worked well.	{"url":"https://www.ta-teachers.eu/teacher-diaries-science/acquiring-an-understanding-of-proportion-and-its-applications-part-2.html?ml=1","timestamp":"2024-11-12T14:17:57Z","content_type":"application/xhtml+xml","content_length":"14190","record_id":"<urn:uuid:6091027b-e557-461b-be51-a96529236bb8>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00677.warc.gz"}
Floating-point arithmetic An early electromechanical programmable computer, the Z3, included floating-point arithmetic (replica on display at Deutsches Museum in Munich). of a fixed base. Numbers of this form are called floating-point numbers For example, 12.345 is a floating-point number in base ten with five digits of precision: ${\displaystyle 12.345=\!\underbrace {12345} _{\text{significand}}\!\times \!\underbrace {10} _{\text{base}}\!\!\!\!\!\!\!\overbrace {{}^{-3}} ^{\text{exponent}}}$ However, unlike 12.345, 12.3456 is not a floating-point number in base ten with five digits of precision—it needs six digits of precision; the nearest floating-point number with only five digits is 12.346. In practice, most floating-point systems use base two, though base ten (decimal floating point) is also common. Floating-point arithmetic operations, such as addition and division, approximate the corresponding real number arithmetic operations by rounding any result that is not a floating-point number itself to a nearby floating-point number.^[1]^:22^[2]^:10 For example, in a floating-point arithmetic with five base-ten digits of precision, the sum 12.345 + 1.0001 = 13.3451 might be rounded to The term floating point refers to the fact that the number's radix point can "float" anywhere to the left, right, or between the significant digits of the number. This position is indicated by the exponent, so floating point can be considered a form of scientific notation A floating-point system can be used to represent, with a fixed number of digits, numbers of very different orders of magnitude — such as the number of meters between galaxies or between protons in an atom. For this reason, floating-point arithmetic is often used to allow very small and very large real numbers that require fast processing times. The result of this dynamic range is that the numbers that can be represented are not uniformly spaced; the difference between two consecutive representable numbers varies with their exponent.^[3] Single-precision floating-point numbers on a number line: the green lines mark representable values. Augmented version above showing both signs of representable values Over the years, a variety of floating-point representations have been used in computers. In 1985, the IEEE 754 Standard for Floating-Point Arithmetic was established, and since the 1990s, the most commonly encountered representations are those defined by the IEEE. The speed of floating-point operations, commonly measured in terms of computer system , especially for applications that involve intensive mathematical calculations. A floating-point unit (FPU, colloquially a math coprocessor) is a part of a computer system specially designed to carry out operations on floating-point numbers. Floating-point numbers number representation specifies some way of encoding a number, usually as a string of digits. There are several mechanisms by which strings of digits can represent numbers. In standard mathematical notation, the digit string can be of any length, and the location of the radix point is indicated by placing an explicit "point" character (dot or comma) there. If the radix point is not specified, then the string implicitly represents an and the unstated radix point would be off the right-hand end of the string, next to the least significant digit. In systems, a position in the string is specified for the radix point. So a fixed-point scheme might use a string of 8 decimal digits with the decimal point in the middle, whereby "00012345" would represent 0001.2345. In scientific notation, the given number is scaled by a power of 10, so that it lies within a specific range—typically between 1 and 10, with the radix point appearing immediately after the first digit. As a power of ten, the scaling factor is then indicated separately at the end of the number. For example, the orbital period of Jupiter's moon Io is 152,853.5047 seconds, a value that would be represented in standard-form scientific notation as 1.528535047×10^5 seconds. Floating-point representation is similar in concept to scientific notation. Logically, a floating-point number consists of: • A signed (meaning positive or negative) digit string of a given length in a given base (or radix). This digit string is referred to as the significand, mantissa, or coefficient.^[nb 1] The length of the significand determines the precision to which numbers can be represented. The radix point position is assumed always to be somewhere within the significand—often just after or just before the most significant digit, or to the right of the rightmost (least significant) digit. This article generally follows the convention that the radix point is set just after the most significant (leftmost) digit. • A signed integer exponent (also referred to as the , or ^[nb 2] which modifies the magnitude of the number. To derive the value of the floating-point number, the significand is multiplied by the base raised to the power of the exponent, equivalent to shifting the radix point from its implied position by a number of places equal to the value of the exponent—to the right if the exponent is positive or to the left if the exponent is negative. Using base-10 (the familiar decimal notation) as an example, the number 152,853.5047, which has ten decimal digits of precision, is represented as the significand 1,528,535,047 together with 5 as the exponent. To determine the actual value, a decimal point is placed after the first digit of the significand and the result is multiplied by 10^5 to give 1.528535047×10^5, or 152,853.5047. In storing such a number, the base (10) need not be stored, since it will be the same for the entire range of supported numbers, and can thus be inferred. Symbolically, this final value is: ${\displaystyle {\frac {s}{b^{\,p-1}}}\times b^{e},}$ where s is the significand (ignoring any implied decimal point), p is the precision (the number of digits in the significand), b is the base (in our example, this is the number ten), and e is the Historically, several number bases have been used for representing floating-point numbers, with base two ( A floating-point number is a rational number, because it can be represented as one integer divided by another; for example 1.45×10^3 is (145/100)×1000 or 145,000/100. The base determines the fractions that can be represented; for instance, 1/5 cannot be represented exactly as a floating-point number using a binary base, but 1/5 can be represented exactly using a decimal base (0.2, or 2× 10^−1). However, 1/3 cannot be represented exactly by either binary (0.010101...) or decimal (0.333...), but in base 3, it is trivial (0.1 or 1×3^−1) . The occasions on which infinite expansions occur depend on the base and its prime factors. The way in which the significand (including its sign) and exponent are stored in a computer is implementation-dependent. The common IEEE formats are described in detail later and elsewhere, but as an example, in the binary single-precision (32-bit) floating-point representation, ${\displaystyle p=24}$, and so the significand is a string of 24 bits. For instance, the number π's first 33 bits are: ${\displaystyle 11001001\ 00001111\ 1101101{\underline {0}}\ 10100010\ 0.}$ In this binary expansion, let us denote the positions from 0 (leftmost bit, or most significant bit) to 32 (rightmost bit). The 24-bit significand will stop at position 23, shown as the underlined bit 0 above. The next bit, at position 24, is called the round bit or rounding bit. It is used to round the 33-bit approximation to the nearest 24-bit number (there are specific rules for halfway values, which is not the case here). This bit, which is 1 in this example, is added to the integer formed by the leftmost 24 bits, yielding: ${\displaystyle 11001001\ 00001111\ 1101101{\underline When this is stored in memory using the IEEE 754 encoding, this becomes the significand s. The significand is assumed to have a binary point to the right of the leftmost bit. So, the binary representation of π is calculated from left-to-right as follows: {\displaystyle {\begin{aligned}&\left(\sum _{n=0}^{p-1}{\text{bit}}_{n}\times 2^{-n}\right)\times 2^{e}\\={}&\left(1\times 2^{-0}+1\ times 2^{-1}+0\times 2^{-2}+0\times 2^{-3}+1\times 2^{-4}+\cdots +1\times 2^{-23}\right)\times 2^{1}\\\approx {}&1.57079637\times 2\\\approx {}&3.1415927\end{aligned}}} where p is the precision (24 in this example), n is the position of the bit of the significand from the left (starting at 0 and finishing at 23 here) and e is the exponent (1 in this example). It can be required that the most significant digit of the significand of a non-zero number be non-zero (except when the corresponding exponent would be smaller than the minimum one). This process is called normalization. For binary formats (which uses only the digits 0 and 1), this non-zero digit is necessarily 1. Therefore, it does not need to be represented in memory, allowing the format to have one more bit of precision. This rule is variously called the leading bit convention, the implicit bit convention, the hidden bit convention,^[1] or the assumed bit convention. Alternatives to floating-point numbers The floating-point representation is by far the most common way of representing in computers an approximation to real numbers. However, there are alternatives: • Fixed-point representation uses integer hardware operations controlled by a software implementation of a specific convention about the location of the binary or decimal point, for example, 6 bits or digits from the right. The hardware to manipulate these representations is less costly than floating point, and it can be used to perform normal integer operations, too. Binary fixed point is usually used in special-purpose applications on embedded processors that can only do integer arithmetic, but decimal fixed point is common in commercial applications. • generalized logarithm • Tapered floating-point representation , which does not appear to be used in practice. • Some simple rational numbers (e.g., 1/3 and 1/10) cannot be represented exactly in binary floating point, no matter what the precision is. Using a different radix allows one to represent some of them (e.g., 1/10 in decimal floating point), but the possibilities remain limited. Software packages that perform " arithmetic for the individual integers. • Interval arithmetic allows one to represent numbers as intervals and obtain guaranteed bounds on results. It is generally based on other arithmetics, in particular floating point. • can often handle irrational numbers like ${\displaystyle \pi }$ or ${\displaystyle {\sqrt {3}}}$ in a completely "formal" way ( symbolic computation ), without dealing with a specific encoding of the significand. Such a program can evaluate expressions like "${\displaystyle \sin(3\pi )}$" exactly, because it is programmed to process the underlying mathematics directly, instead of using approximate values for each intermediate calculation. Leonardo Torres Quevedo, in 1914, published an analysis of floating point based on the analytical engine. In 1914, the Spanish engineer Leonardo Torres Quevedo published Essays on Automatics,^[9] where he designed a special-purpose electromechanical calculator based on Charles Babbage's analytical engine and described a way to store floating-point numbers in a consistent manner. He stated that numbers will be stored in exponential format as n x 10${\displaystyle ^{m}}$, and offered three rules by which consistent manipulation of floating-point numbers by machines could be implemented. For Torres, "n will always be the same number of Konrad Zuse, architect of the Z3 computer, which uses a 22-bit binary floating-point representation In 1938, Konrad Zuse of Berlin completed the Z1, the first binary, programmable mechanical computer;^[13] it uses a 24-bit binary floating-point number representation with a 7-bit signed exponent, a 17-bit significand (including one implicit bit), and a sign bit.^[14] The more reliable relay-based Z3, completed in 1941, has representations for both positive and negative infinities; in particular, it implements defined operations with infinity, such as ${\displaystyle ^{1}/_{\infty }=0}$, and it stops on undefined operations, such as ${\displaystyle 0\times \infty }$. Zuse also proposed, but did not complete, carefully rounded floating-point arithmetic that includes ${\displaystyle \pm \infty }$ and NaN representations, anticipating features of the IEEE Standard by four decades.^[15] In contrast, von Neumann recommended against floating-point numbers for the 1951 IAS machine, arguing that fixed-point arithmetic is preferable.^[15] The first commercial computer with floating-point hardware was Zuse's Z4 computer, designed in 1942–1945. In 1946, Bell Laboratories introduced the Model V, which implemented decimal floating-point . The arithmetic is actually implemented in software, but with a one megahertz clock rate, the speed of floating-point and fixed-point operations in this machine were initially faster than those of many competing computers. The mass-produced Extensions for Scientific Computation (XSC)). It was not until the launch of the Intel i486 in 1989 that personal computers had floating-point capability in hardware as a standard feature. The UNIVAC 1100/2200 series, introduced in 1962, supported two floating-point representations: • Single precision: 36 bits, organized as a 1-bit sign, an 8-bit exponent, and a 27-bit significand. • Double precision: 72 bits, organized as a 1-bit sign, an 11-bit exponent, and a 60-bit significand. System/360 mainframes; these same representations are still available for use in modern systems. In 1998, IBM implemented IEEE-compatible binary floating-point arithmetic in its mainframes; in 2005, IBM also added IEEE-compatible decimal floating-point arithmetic. Initially, computers used many different representations for floating-point numbers. The lack of standardization at the mainframe level was an ongoing problem by the early 1970s for those writing and maintaining higher-level source code; these manufacturer floating-point standards differed in the word sizes, the representations, and the rounding behavior and general accuracy of operations. Floating-point compatibility across multiple computing systems was in desperate need of standardization by the early 1980s, leading to the creation of the around the same time, gave significant input as well. William Kahan, principal architect of the IEEE 754 floating-point standard In 1989, mathematician and computer scientist William Kahan was honored with the Turing Award for being the primary architect behind this proposal; he was aided by his student Jerome Coonen and a visiting professor, Harold Stone.^[17] Among the x86 innovations are these: • A precisely specified floating-point representation at the bit-string level, so that all compliant computers interpret bit patterns the same way. This makes it possible to accurately and efficiently transfer floating-point numbers from one computer to another (after accounting for endianness). • A precisely specified behavior for the arithmetic operations: A result is required to be produced as if infinitely precise arithmetic were used to yield a value that is then rounded according to specific rules. This means that a compliant computer program would always produce the same result when given a particular input, thus mitigating the almost mystical reputation that floating-point computation had developed for its hitherto seemingly non-deterministic behavior. • The ability of exceptional conditions (overflow, divide by zero, etc.) to propagate through a computation in a benign manner and then be handled by the software in a controlled fashion. Range of floating-point numbers A floating-point number consists of two fixed-point components, whose range depends exclusively on the number of bits or digits in their representation. Whereas components linearly depend on their range, the floating-point range linearly depends on the significand range and exponentially on the range of exponent component, which attaches outstandingly wider range to the number. On a typical computer system, a double-precision (64-bit) binary floating-point number has a coefficient of 53 bits (including 1 implied bit), an exponent of 11 bits, and 1 sign bit. Since 2^10 = 1024, the complete range of the positive normal floating-point numbers in this format is from 2^−1022 ≈ 2 × 10^−308 to approximately 2^1024 ≈ 2 × 10^308. The number of normal floating-point numbers in a system (B, P, L, U) where • B is the base of the system, • P is the precision of the significand (in base B), • L is the smallest exponent of the system, • U is the largest exponent of the system, is ${\displaystyle 2\left(B-1\right)\left(B^{P-1}\right)\left(U-L+1\right)}$. There is a smallest positive normal floating-point number, Underflow level = UFL = ${\displaystyle B^{L}}$, which has a 1 as the leading digit and 0 for the remaining digits of the significand, and the smallest possible value for the exponent. There is a largest floating-point number, Overflow level = OFL = ${\displaystyle \left(1-B^{-P}\right)\left(B^{U+1}\right)}$, which has B − 1 as the value for each digit of the significand and the largest possible value for the exponent. In addition, there are representable values strictly between −UFL and UFL. Namely, positive and negative zeros, as well as subnormal numbers. IEEE 754: floating point in modern computers IBM's own hexadecimal floating point format and IEEE 754-2008 decimal floating point in addition to the IEEE 754 binary format. The Cray T90 series had an IEEE version, but the still uses Cray floating-point format. The standard provides for many closely related formats, differing in only a few details. Five of these formats are called basic formats, and others are termed extended precision formats and extendable precision format. Three formats are especially widely used in computer hardware and languages: • Single precision (binary32), usually used to represent the "float" type in the C language family. This is a binary format that occupies 32 bits (4 bytes) and its significand has a precision of 24 bits (about 7 decimal digits). • Double precision (binary64), usually used to represent the "double" type in the C language family. This is a binary format that occupies 64 bits (8 bytes) and its significand has a precision of 53 bits (about 16 decimal digits). • Double extended, also ambiguously called "extended precision" format. This is a binary format that occupies at least 79 bits (80 if the hidden/implicit bit rule is not used) and its significand has a precision of at least 64 bits (about 19 decimal digits). The C99 and C11 standards of the C language family, in their annex F ("IEC 60559 floating-point arithmetic"), recommend such an extended format to be provided as "long double".^[18] A format satisfying the minimal requirements (64-bit significand precision, 15-bit exponent, thus fitting on 80 bits) is provided by the x86 architecture. Often on such processors, this format can be used with "long double", though extended precision is not available with MSVC.^[19] For alignment purposes, many tools store this 80-bit value in a 96-bit or 128-bit space.^[20]^[21] On other processors, "long double" may stand for a larger format, such as quadruple precision,^[22] or just double precision, if any form of extended precision is not available.^[23] Increasing the precision of the floating-point representation generally reduces the amount of accumulated round-off error caused by intermediate calculations.^[24] Other IEEE formats include: • Decimal64 and decimal128 floating-point formats. These formats (especially decimal128) are pervasive in financial transactions because, along with the decimal32 format, they allow correct decimal • Quadruple precision (binary128). This is a binary format that occupies 128 bits (16 bytes) and its significand has a precision of 113 bits (about 34 decimal digits). • Half precision, also called binary16, a 16-bit floating-point value. It is being used in the NVIDIA Cg graphics language, and in the openEXR standard (where it actually predates the introduction in the IEEE 754 standard).^[25]^[26] Any integer with absolute value less than 2^24 can be exactly represented in the single-precision format, and any integer with absolute value less than 2^53 can be exactly represented in the double-precision format. Furthermore, a wide range of powers of 2 times such a number can be represented. These properties are sometimes used for purely integer data, to get 53-bit integers on platforms that have double-precision floats but only 32-bit integers. The standard specifies some special values, and their representation: positive negative zero (−0) distinct from ordinary ("positive") zero, and "not a number" values ( Comparison of floating-point numbers, as defined by the IEEE standard, is a bit different from usual integer comparison. Negative and positive zero compare equal, and every NaN compares unequal to every value, including itself. All finite floating-point numbers are strictly smaller than +∞ and strictly greater than −∞, and they are ordered in the same way as their values (in the set of real Internal representation Floating-point numbers are typically packed into a computer datum as the sign bit, the exponent field, and the significand or mantissa, from left to right. For the IEEE 754 binary formats (basic and extended) which have extant hardware implementations, they are apportioned as follows: Type Bits Exponent Bits Number of Sign Exponent Significand Total bias precision decimal digits IEEE 754-2008 1 5 10 16 15 11 ~3.3 Single 1 8 23 32 127 24 ~7.2 Double 1 11 52 64 1023 53 ~15.9 x86 extended precision 1 15 64 80 16383 64 ~19.2 Quad 1 15 112 128 16383 113 ~34.0 While the exponent can be positive or negative, in binary formats it is stored as an unsigned number that has a fixed "bias" added to it. Values of all 0s in this field are reserved for the zeros and subnormal numbers ; values of all 1s are reserved for the infinities and NaNs. The exponent range for normal numbers is [−126, 127] for single precision, [−1022, 1023] for double, or [−16382, 16383] for quad. Normal numbers exclude subnormal values, zeros, infinities, and NaNs. In the IEEE binary interchange formats the leading 1 bit of a normalized significand is not actually stored in the computer datum. It is called the "hidden" or "implicit" bit. Because of this, the single-precision format actually has a significand with 24 bits of precision, the double-precision format has 53, and quad has 113. For example, it was shown above that π, rounded to 24 bits of precision, has: • sign = 0 ; e = 1 ; s = 110010010000111111011011 (including the hidden bit) The sum of the exponent bias (127) and the exponent (1) is 128, so this is represented in the single-precision format as • 0 10000000 10010010000111111011011 (excluding the hidden bit) = 40490FDB^[27] as a hexadecimal number. An example of a layout for 32-bit floating point is and the 64-bit ("double") layout is similar. Other notable floating-point formats In addition to the widely used IEEE 754 standard formats, other floating-point formats are used, or have been used, in certain domain-specific areas. • The Apple //, Commodore PET Motorola 6800 (MITS Altair 680) and Motorola 6809 TRS-80 Color Computer ). All Microsoft language products from 1975 through 1987 used the Microsoft Binary Format until Microsoft adopted the IEEE-754 standard format in all its products starting in 1988 to their current releases. MBF consists of the MBF single-precision format (32 bits, "6-digit BASIC"), the MBF extended-precision format (40 bits, "9-digit BASIC"), and the MBF double-precision format (64 bits); each of them is represented with an 8-bit exponent, followed by a sign bit, followed by a significand of respectively 23, 31, and 55 bits. • The Bfloat16 format requires the same amount of memory (16 bits) as the IEEE 754 half-precision format, but allocates 8 bits to the exponent instead of 5, thus providing the same range as a IEEE 754 single-precision number. The tradeoff is a reduced precision, as the trailing significand field is reduced from 10 to 7 bits. This format is mainly used in the training of machine learning models, where range is more valuable than precision. Many machine learning accelerators provide hardware support for this format. • The TensorFloat-32^[31] format combines the 8 bits of exponent of the Bfloat16 with the 10 bits of trailing significand field of half-precision formats, resulting in a size of 19 bits. This format was introduced by Nvidia, which provides hardware support for it in the Tensor Cores of its GPUs based on the Nvidia Ampere architecture. The drawback of this format is its size, which is not a power of 2. However, according to Nvidia, this format should only be used internally by hardware to speed up computations, while inputs and outputs should be stored in the 32-bit single-precision IEEE 754 format.^[31] • The Hopper architecture GPUs provide two FP8 formats: one with the same numerical range as half-precision (E5M2) and one with higher precision, but less range (E4M3).^[32]^[33] Bfloat16, TensorFloat-32, and the two FP8 formats, compared with IEEE 754 half-precision and single-precision formats Type Sign Exponent Trailing significand field Total bits FP8 (E4M3) 1 4 3 8 FP8 (E5M2) 1 5 2 8 Half-precision 1 5 10 16 Bfloat16 1 8 7 16 TensorFloat-32 1 8 10 19 Single-precision 1 8 23 32 Representable numbers, conversion and rounding By their nature, all numbers expressed in floating-point format are rational numbers with a terminating expansion in the relevant base (for example, a terminating decimal expansion in base-10, or a terminating binary expansion in base-2). Irrational numbers, such as π or √2, or non-terminating rational numbers, must be approximated. The number of digits (or bits) of precision also limits the set of rational numbers that can be represented exactly. For example, the decimal number 123456789 cannot be exactly represented if only eight decimal digits of precision are available (it would be rounded to one of the two straddling representable values, 12345678 × 10^1 or 12345679 × 10^1), the same applies to non-terminating digits (.5 to be rounded to either .55555555 or .55555556). When a number is represented in some format (such as a character string) which is not a native floating-point representation supported in a computer implementation, then it will require a conversion before it can be used in that implementation. If the number can be represented exactly in the floating-point format then the conversion is exact. If there is not an exact representation then the conversion requires a choice of which floating-point number to use to represent the original value. The representation chosen will have a different value from the original, and the value thus adjusted is called the rounded value. Whether or not a rational number has a terminating expansion depends on the base. For example, in base-10 the number 1/2 has a terminating expansion (0.5) while the number 1/3 does not (0.333...). In base-2 only rationals with denominators that are powers of 2 (such as 1/2 or 3/16) are terminating. Any rational with a denominator that has a prime factor other than 2 will have an infinite binary expansion. This means that numbers that appear to be short and exact when written in decimal format may need to be approximated when converted to binary floating-point. For example, the decimal number 0.1 is not representable in binary floating-point of any finite precision; the exact binary representation would have a "1100" sequence continuing endlessly: e = −4; s = 1100110011001100110011001100110011..., where, as previously, s is the significand and e is the exponent. When rounded to 24 bits this becomes e = −4; s = 110011001100110011001101, which is actually 0.100000001490116119384765625 in decimal. As a further example, the real number π, represented in binary as an infinite sequence of bits is but is when approximated by rounding to a precision of 24 bits. In binary single-precision floating-point, this is represented as s = 1.10010010000111111011011 with e = 1. This has a decimal value of whereas a more accurate approximation of the true value of π is The result of rounding differs from the true value by about 0.03 parts per million, and matches the decimal representation of π in the first 7 digits. The difference is the discretization error and is limited by the machine epsilon. The arithmetical difference between two consecutive representable floating-point numbers which have the same exponent is called a unit in the last place (ULP). For example, if there is no representable number lying between the representable numbers 1.45a70c22[hex] and 1.45a70c24[hex], the ULP is 2×16^−8, or 2^−31. For numbers with a base-2 exponent part of 0, i.e. numbers with an absolute value higher than or equal to 1 but lower than 2, an ULP is exactly 2^−23 or about 10^−7 in single precision, and exactly 2^−53 or about 10^−16 in double precision. The mandated behavior of IEEE-compliant hardware is that the result be within one-half of a ULP. Rounding modes Rounding is used when the exact result of a floating-point operation (or a conversion to floating-point format) would need more digits than there are digits in the significand. IEEE 754 requires correct rounding: that is, the rounded result is as if infinitely precise arithmetic was used to compute the value and then rounded (although in implementation only three extra bits are needed to ensure this). There are several different rounding schemes (or rounding modes). Historically, truncation was the typical approach. Since the introduction of IEEE 754, the default method (round to nearest, ties to even, sometimes called Banker's Rounding) is more commonly used. This method rounds the ideal (infinitely precise) result of an arithmetic operation to the nearest representable value, and gives that representation as the result.^[nb 8] In the case of a tie, the value that would make the significand end in an even digit is chosen. The IEEE 754 standard requires the same rounding to be applied to all fundamental algebraic operations, including square root and conversions, when there is a numeric (non-NaN) result. It means that the results of IEEE 754 operations are completely determined in all bits of the result, except for the representation of NaNs. ("Library" functions such as cosine and log are not mandated.) Alternative rounding options are also available. IEEE 754 specifies the following rounding modes: • round to nearest, where ties round to the nearest even digit in the required position (the default and by far the most common mode) • round to nearest, where ties round away from zero (optional for binary floating-point and commonly used in decimal) • round up (toward +∞; negative results thus round toward zero) • round down (toward −∞; negative results thus round away from zero) • round toward zero (truncation; it is similar to the common behavior of float-to-integer conversions, which convert −3.9 to −3 and 3.9 to 3) Alternative modes are useful when the amount of error being introduced must be bounded. Applications that require a bounded error are multi-precision floating-point, and interval arithmetic. The alternative rounding modes are also useful in diagnosing numerical instability: if the results of a subroutine vary substantially between rounding to + and − infinity then it is likely numerically unstable and affected by round-off error.^[34] Binary-to-decimal conversion with minimal number of digits Converting a double-precision binary floating-point number to a decimal string is a common operation, but an algorithm producing results that are both accurate and minimal did not appear in print until 1990, with Steele and White's Dragon4. Some of the improvements since then include: Many modern language runtimes use Grisu3 with a Dragon4 fallback.^[41] Decimal-to-binary conversion The problem of parsing a decimal string into a binary FP representation is complex, with an accurate parser not appearing until Clinger's 1990 work (implemented in dtoa.c).^[35] Further work has likewise progressed in the direction of faster parsing.^[42] Floating-point operations For ease of presentation and understanding, decimal radix with 7 digit precision will be used in the examples, as in the IEEE 754 decimal32 format. The fundamental principles are the same in any radix or precision, except that normalization is optional (it does not affect the numerical value of the result). Here, s denotes the significand and e denotes the exponent. Addition and subtraction A simple method to add floating-point numbers is to first represent them with the same exponent. In the example below, the second number (with the smaller exponent) is shifted right by three digits, and one then proceeds with the usual addition method: 123456.7 = 1.234567 × 10^5 101.7654 = 1.017654 × 10^2 = 0.001017654 × 10^5 123456.7 + 101.7654 = (1.234567 × 10^5) + (1.017654 × 10^2) = (1.234567 × 10^5) + (0.001017654 × 10^5) = (1.234567 + 0.001017654) × 10^5 = 1.235584654 × 10^5 In detail: e=5; s=1.234567 (123456.7) + e=2; s=1.017654 (101.7654) e=5; s=1.234567 + e=5; s=0.001017654 (after shifting) e=5; s=1.235584654 (true sum: 123558.4654) This is the true result, the exact sum of the operands. It will be rounded to seven digits and then normalized if necessary. The final result is e=5; s=1.235585 (final sum: 123558.5) The lowest three digits of the second operand (654) are essentially lost. This is round-off error. In extreme cases, the sum of two non-zero numbers may be equal to one of them: e=5; s=1.234567 + e=−3; s=9.876543 e=5; s=1.234567 + e=5; s=0.00000009876543 (after shifting) e=5; s=1.23456709876543 (true sum) e=5; s=1.234567 (after rounding and normalization) In the above conceptual examples it would appear that a large number of extra digits would need to be provided by the adder to ensure correct rounding; however, for binary addition or subtraction using careful implementation techniques only a guard bit, a rounding bit and one extra sticky bit need to be carried beyond the precision of the operands.^[43]^[44]^:218–220 Another problem of loss of significance occurs when approximations to two nearly equal numbers are subtracted. In the following example e = 5; s = 1.234571 and e = 5; s = 1.234567 are approximations to the rationals 123457.1467 and 123456.659. e=5; s=1.234571 − e=5; s=1.234567 e=5; s=0.000004 e=−1; s=4.000000 (after rounding and normalization) The floating-point difference is computed exactly because the numbers are close—the gradual underflow is supported. Despite this, the difference of the original numbers is = −1; = 4.877000, which differs more than 20% from the difference = −1; = 4.000000 of the approximations. In extreme cases, all significant digits of precision can be lost. illustrates the danger in assuming that all of the digits of a computed result are meaningful. Dealing with the consequences of these errors is a topic in numerical analysis ; see also Accuracy problems Multiplication and division To multiply, the significands are multiplied while the exponents are added, and the result is rounded and normalized. e=3; s=4.734612 × e=5; s=5.417242 e=8; s=25.648538980104 (true product) e=8; s=25.64854 (after rounding) e=9; s=2.564854 (after normalization) Similarly, division is accomplished by subtracting the divisor's exponent from the dividend's exponent, and dividing the dividend's significand by the divisor's significand. There are no cancellation or absorption problems with multiplication or division, though small errors may accumulate as operations are performed in succession.^[43] In practice, the way these operations are carried out in digital logic can be quite complex (see Booth's multiplication algorithm and Division algorithm).^[nb 9] Literal syntax Literals for floating-point numbers depend on languages. They typically use e or E to denote ). In these cases, digit strings such as may also be floating-point literals. Examples of floating-point literals are: • 99.9 • -5000.12 • 6.02e23 • -3e-45 • 0x1.fffffep+127 in C and IEEE 754 Dealing with exceptional cases Floating-point computation in a computer can run into three kinds of problems: • An operation can be mathematically undefined, such as ∞/∞, or division by zero. • An operation can be legal in principle, but not supported by the specific format, for example, calculating the square root of −1 or the inverse sine of 2 (both of which result in complex numbers • An operation can be legal in principle, but the result can be impossible to represent in the specified format, because the exponent is too large or too small to encode in the exponent field. Such an event is called an overflow (exponent too large), underflow (exponent too small) or denormalization (precision loss). Prior to the IEEE standard, such conditions usually caused the program to terminate, or triggered some kind of trap that the programmer might be able to catch. How this worked was system-dependent, meaning that floating-point programs were not . (The term "exception" as used in IEEE 754 is a general term meaning an exceptional condition, which is not necessarily an error, and is a different usage to that typically defined in programming languages such as a C++ or Java, in which an " " is an alternative flow of control, closer to what is termed a "trap" in IEEE 754 terminology.) Here, the required default method of handling exceptions according to IEEE 754 is discussed (the IEEE 754 optional trapping and other "alternate exception handling" modes are not discussed). Arithmetic exceptions are (by default) required to be recorded in "sticky" status flag bits. That they are "sticky" means that they are not reset by the next (arithmetic) operation, but stay set until explicitly reset. The use of "sticky" flags thus allows for testing of exceptional conditions to be delayed until after a full floating-point expression or subroutine: without them exceptional conditions that could not be otherwise ignored would require explicit testing immediately after every floating-point operation. By default, an operation always returns a result according to specification without interrupting computation. For instance, 1/0 returns +∞, while also setting the divide-by-zero flag bit (this default of ∞ is designed to often return a finite result when used in subsequent operations and so be safely ignored). The original IEEE 754 standard, however, failed to recommend operations to handle such sets of arithmetic exception flag bits. So while these were implemented in hardware, initially programming language implementations typically did not provide a means to access them (apart from assembler). Over time some programming language standards (e.g., C99/C11 and Fortran) have been updated to specify methods to access and change status flag bits. The 2008 version of the IEEE 754 standard now specifies a few operations for accessing and handling the arithmetic flag bits. The programming model is based on a single thread of execution and use of them by multiple threads has to be handled by a means outside of the standard (e.g. C11 specifies that the flags have thread-local storage). IEEE 754 specifies five arithmetic exceptions that are to be recorded in the status flags ("sticky bits"): • inexact, set if the rounded (and returned) value is different from the mathematically exact result of the operation. • underflow, set if the rounded value is tiny (as specified in IEEE 754) and inexact (or maybe limited to if it has denormalization loss, as per the 1985 version of IEEE 754), returning a subnormal value including the zeros. • overflow, set if the absolute value of the rounded value is too large to be represented. An infinity or maximal finite value is returned, depending on which rounding is used. • divide-by-zero, set if the result is infinite given finite operands, returning an infinity, either +∞ or −∞. • invalid, set if a real-valued result cannot be returned e.g. sqrt(−1) or 0/0, returning a quiet NaN. Fig. 1: resistances in parallel, with total resistance ${\displaystyle R_{tot}}$ The default return value for each of the exceptions is designed to give the correct result in the majority of cases such that the exceptions can be ignored in the majority of codes. inexact returns a correctly rounded result, and underflow returns a value less than or equal to the smallest positive normal number in magnitude and can almost always be ignored.^[46] divide-by-zero returns infinity exactly, which will typically then divide a finite number and so give zero, or else will give an invalid exception subsequently if not, and so can also typically be ignored. For example, the effective resistance of n resistors in parallel (see fig. 1) is given by ${\displaystyle R_{\text{tot}}=1/(1/R_{1}+1/R_{2}+\cdots +1/R_{n})}$. If a short-circuit develops with ${\displaystyle R_{1}}$ set to 0, ${\displaystyle 1/R_{1}}$ will return +infinity which will give a final ${\displaystyle R_{tot}}$ of 0, as expected^ IEEE 754 design rationale for another example). Overflow and invalid exceptions can typically not be ignored, but do not necessarily represent errors: for example, a root-finding routine, as part of its normal operation, may evaluate a passed-in function at values outside of its domain, returning NaN and an invalid exception flag to be ignored until finding a useful start point.^[46] Accuracy problems The fact that floating-point numbers cannot accurately represent all real numbers, and that floating-point operations cannot accurately represent true arithmetic operations, leads to many surprising situations. This is related to the finite precision with which computers generally represent numbers. For example, the decimal numbers 0.1 and 0.01 cannot be represented exactly as binary floating-point numbers. In the IEEE 754 binary32 format with its 24-bit significand, the result of attempting to square the approximation to 0.1 is neither 0.01 nor the representable number closest to it. The decimal number 0.1 is represented in binary as e = −4; s = 110011001100110011001101, which is 0.100000001490116119384765625 exactly. Squaring this number gives 0.010000000298023226097399174250313080847263336181640625 exactly. Squaring it with rounding to the 24-bit precision gives 0.010000000707805156707763671875 exactly. But the representable number closest to 0.01 is 0.009999999776482582092285156250 exactly. Also, the non-representability of π (and π/2) means that an attempted computation of tan(π/2) will not yield a result of infinity, nor will it even overflow in the usual floating-point formats (assuming an accurate implementation of tan). It is simply not possible for standard floating-point hardware to attempt to compute tan(π/2), because π/2 cannot be represented exactly. This computation in C: /* Enough digits to be sure we get the correct approximation. / double pi = 3.1415926535897932384626433832795; double z = tan(pi/2.0); will give a result of 16331239353195370.0. In single precision (using the tanf function), the result will be −22877332.0. By the same token, an attempted computation of sin(π) will not yield zero. The result will be (approximately) 0.1225×10^−15 in double precision, or −0.8742×10^−7 in single precision.^[nb 10] While floating-point addition and multiplication are both commutative ( a + b = b + a a × b = b × a ), they are not necessarily . That is, (a + b) + c is not necessarily equal to a + (b + c) . Using 7-digit significand decimal arithmetic: a = 1234.567, b = 45.67834, c = 0.0004 (a + b) + c: 1234.567 (a) + 45.67834 (b) 1280.24534 rounds to 1280.245 1280.245 (a + b) + 0.0004 (c) 1280.2454 rounds to 1280.245 ← (a + b) + c a + (b + c): 45.67834 (b) + 0.0004 (c) 1234.567 (a) + 45.67874 (b + c) 1280.24574 rounds to 1280.246 ← a + (b + c) They are also not necessarily distributive. That is, (a + b) × c may not be the same as a × c + b × c: 1234.567 × 3.333333 = 4115.223 1.234567 × 3.333333 = 4.115223 4115.223 + 4.115223 = 4119.338 1234.567 + 1.234567 = 1235.802 1235.802 × 3.333333 = 4119.340 In addition to loss of significance, inability to represent numbers such as π and 0.1 exactly, and other slight inaccuracies, the following phenomena may occur: • Cancellation: subtraction of nearly equal operands may cause extreme loss of accuracy.^[48]^[45] When we subtract two almost equal numbers we set the most significant digits to zero, leaving ourselves with just the insignificant, and most erroneous, digits.^[1]^:124 For example, when determining a derivative of a function the following formula is used: ${\displaystyle Q(h)={\frac {f(a+h)-f(a)}{h}}.}$ Intuitively one would want an h very close to zero; however, when using floating-point operations, the smallest number will not give the best approximation of a derivative. As h grows smaller, the difference between f(a + h) and f(a) grows smaller, cancelling out the most significant and least erroneous digits and making the most erroneous digits more important. As a result the smallest number of h possible will give a more erroneous approximation of a derivative than a somewhat larger number. This is perhaps the most common and serious accuracy problem. • Conversions to integer are not intuitive: converting (63.0/9.0) to integer yields 7, but converting (0.63/0.09) may yield 6. This is because conversions generally truncate rather than round. Floor and ceiling functions may produce answers which are off by one from the intuitively expected value. • Limited exponent range: results might overflow yielding infinity, or underflow yielding a subnormal number or zero. In these cases precision will be lost. • Testing for safe division is problematic: Checking that the divisor is not zero does not guarantee that a division will not overflow. • Testing for equality is problematic. Two computational sequences that are mathematically equal may well produce different floating-point values.^[49] Machine precision and backward error analysis Machine precision is a quantity that characterizes the accuracy of a floating-point system, and is used in backward error analysis of floating-point algorithms. It is also known as unit roundoff or machine epsilon. Usually denoted Ε[mach], its value depends on the particular rounding being used. With rounding to zero, ${\displaystyle \mathrm {E} _{\text{mach}}=B^{1-P},\,}$ whereas rounding to nearest, ${\displaystyle \mathrm {E} _{\text{mach}}={\tfrac {1}{2}}B^{1-P},}$ where B is the base of the system and P is the precision of the significand (in base B). This is important since it bounds the relative error in representing any non-zero real number x within the normalized range of a floating-point system: ${\displaystyle \left\|{\frac {\operatorname {fl} (x)-x}{x}}\right\|\leq \mathrm {E} _{\text{mach}}.}$ Backward error analysis, the theory of which was developed and popularized by James H. Wilkinson, can be used to establish that an algorithm implementing a numerical function is numerically stable.^[ 51] The basic approach is to show that although the calculated result, due to roundoff errors, will not be exactly correct, it is the exact solution to a nearby problem with slightly perturbed input data. If the perturbation required is small, on the order of the uncertainty in the input data, then the results are in some sense as accurate as the data "deserves". The algorithm is then defined as backward stable. Stability is a measure of the sensitivity to rounding errors of a given numerical procedure; by contrast, the condition number of a function for a given problem indicates the inherent sensitivity of the function to small perturbations in its input and is independent of the implementation used to solve the problem.^[52] As a trivial example, consider a simple expression giving the inner product of (length two) vectors ${\displaystyle x}$ and ${\displaystyle y}$, then {\displaystyle {\begin{aligned}\operatorname {fl} (x\cdot y)&=\operatorname {fl} {\big (}\operatorname {fl} (x_{1}\cdot y_{1})+\operatorname {fl} (x_{2}\cdot y_{2}){\big )},&&{\text{ where }}\operatorname {fl} (){\text{ indicates correctly rounded floating-point arithmetic}}\\&=\operatorname {fl} {\big (}(x_{1}\cdot y_{1})(1+\delta _{1})+(x_{2}\cdot y_{2})(1+\delta _{2}){\big )},&&{\text{ where }}\delta _{n}\leq \mathrm {E} _{\text{mach}},{\ text{ from above}}\\&={\big (}(x_{1}\cdot y_{1})(1+\delta _{1})+(x_{2}\cdot y_{2})(1+\delta _{2}){\big )}(1+\delta _{3})\\&=(x_{1}\cdot y_{1})(1+\delta _{1})(1+\delta _{3})+(x_{2}\cdot y_{2})(1+\ delta _{2})(1+\delta _{3}),\end{aligned}}} and so ${\displaystyle \operatorname {fl} (x\cdot y)={\hat {x}}\cdot {\hat {y}},}$ {\displaystyle {\begin{aligned}{\hat {x}}_{1}&=x_{1}(1+\delta _{1});&{\hat {x}}_{2}&=x_{2}(1+\delta _{2});\\{\hat {y}}_{1}&=y_{1}(1+\delta _{3});&{\hat {y}}_{2}&=y_{2}(1+\delta _{3}),\\\end ${\displaystyle \delta _{n}\leq \mathrm {E} _{\text{mach}}}$ by definition, which is the sum of two slightly perturbed (on the order of Ε[mach]) input data, and so is backward stable. For more realistic examples in numerical linear algebra, see Higham 2002^[53 ] and other references below. Minimizing the effect of accuracy problems Although individual arithmetic operations of IEEE 754 are guaranteed accurate to within half a ULP, more complicated formulae can suffer from larger errors for a variety of reasons. The loss of accuracy can be substantial if a problem or its data are ill-conditioned, meaning that the correct result is hypersensitive to tiny perturbations in its data. However, even functions that are well-conditioned can suffer from large loss of accuracy if an algorithm numerically unstable for that data is used: apparently equivalent formulations of expressions in a programming language can differ markedly in their numerical stability. One approach to remove the risk of such loss of accuracy is the design and analysis of numerically stable algorithms, which is an aim of the branch of mathematics known as numerical analysis. Another approach that can protect against the risk of numerical instabilities is the computation of intermediate (scratch) values in an algorithm at a higher precision than the final result requires,^[54] which can remove, or reduce by orders of magnitude,^[55] such risk: IEEE 754 quadruple precision and extended precision are designed for this purpose when computing at double precision.^[56]^[nb 11] For example, the following algorithm is a direct implementation to compute the function A(x) = (x−1) / (exp(x−1) − 1) which is well-conditioned at 1.0,^[nb 12] however it can be shown to be numerically unstable and lose up to half the significant digits carried by the arithmetic when computed near 1.0.^[57] double A(double X) double Y, Z; // [1] Y = X - 1.0; Z = exp(Y); if (Z != 1.0) Z = Y / (Z - 1.0); // [2] return Z; If, however, intermediate computations are all performed in extended precision (e.g. by setting line [1] to C99 long double), then up to full precision in the final double result can be maintained.^[ nb 13] Alternatively, a numerical analysis of the algorithm reveals that if the following non-obvious change to line [2] is made: then the algorithm becomes numerically stable and can compute to full double precision. To maintain the properties of such carefully constructed numerically stable programs, careful handling by the compiler is required. Certain "optimizations" that compilers might make (for example, reordering operations) can work against the goals of well-behaved software. There is some controversy about the failings of compilers and language designs in this area: C99 is an example of a language where such optimizations are carefully specified to maintain numerical precision. See the external references at the bottom of this article. A detailed treatment of the techniques for writing high-quality floating-point software is beyond the scope of this article, and the reader is referred to,^[53]^[58] and the other references at the bottom of this article. Kahan suggests several rules of thumb that can substantially decrease by orders of magnitude^[58] the risk of numerical anomalies, in addition to, or in lieu of, a more careful numerical analysis. These include: as noted above, computing all expressions and intermediate results in the highest precision supported in hardware (a common rule of thumb is to carry twice the precision of the desired result, i.e. compute in double precision for a final single-precision result, or in double extended or quad precision for up to double-precision results^[59]); and rounding input data and results to only the precision required and supported by the input data (carrying excess precision in the final result beyond that required and supported by the input data can be misleading, increases storage cost and decreases speed, and the excess bits can affect convergence of numerical procedures:^[60] notably, the first form of the iterative example given below converges correctly when using this rule of thumb). Brief descriptions of several additional issues and techniques follow. As decimal fractions can often not be exactly represented in binary floating-point, such arithmetic is at its best when it is simply being used to measure real-world quantities over a wide range of scales (such as the orbital period of a moon around Saturn or the mass of a IEEE 754-2008 standard, are designed to avoid the problems of binary floating-point representations when applied to human-entered exact decimal values, and make the arithmetic always behave as expected when numbers are printed in decimal. Expectations from mathematics may not be realized in the field of floating-point computation. For example, it is known that ${\displaystyle (x+y)(x-y)=x^{2}-y^{2}\,}$, and that ${\displaystyle \sin ^ {2}{\theta }+\cos ^{2}{\theta }=1\,}$, however these facts cannot be relied on when the quantities involved are the result of floating-point computation. The use of the equality test (if (x==y) ...) requires care when dealing with floating-point numbers. Even simple expressions like 0.6/0.2-3==0 will, on most computers, fail to be true^[62] (in IEEE 754 double precision, for example, 0.6/0.2 - 3 is approximately equal to -4.44089209850063e-16). Consequently, such tests are sometimes replaced with "fuzzy" comparisons (if (abs(x-y) < epsilon) ..., where epsilon is sufficiently small and tailored to the application, such as 1.0E−13). The wisdom of doing this varies greatly, and can require numerical analysis to bound epsilon.^[53] Values derived from the primary data representation and their comparisons should be performed in a wider, extended, precision to minimize the risk of such inconsistencies due to round-off errors.^[58] It is often better to organize the code in such a way that such tests are unnecessary. For example, in computational geometry, exact tests of whether a point lies off or on a line or plane defined by other points can be performed using adaptive precision or exact arithmetic methods.^[63] Small errors in floating-point arithmetic can grow when mathematical algorithms perform operations an enormous number of times. A few examples are eigenvector computation, and differential equation solving. These algorithms must be very carefully designed, using numerical approaches such as iterative refinement , if they are to work well. Summation of a vector of floating-point values is a basic algorithm in scientific computing, and so an awareness of when loss of significance can occur is essential. For example, if one is adding a very large number of numbers, the individual addends are very small compared with the sum. This can lead to loss of significance. A typical addition would then be something like + 3.141276 The low 3 digits of the addends are effectively lost. Suppose, for example, that one needs to add many numbers, all approximately equal to 3. After 1000 of them have been added, the running sum is about 3000; the lost digits are not regained. The Kahan summation algorithm may be used to reduce the errors.^[53] Round-off error can affect the convergence and accuracy of iterative numerical procedures. As an example, Archimedes approximated π by calculating the perimeters of polygons inscribing and circumscribing a circle, starting with hexagons, and successively doubling the number of sides. As noted above, computations may be rearranged in a way that is mathematically equivalent but less prone to error (numerical analysis). Two forms of the recurrence formula for the circumscribed polygon are: • ${\textstyle t_{0}={\frac {1}{\sqrt {3}}}}$ • First form: ${\textstyle t_{i+1}={\frac {{\sqrt {t_{i}^{2}+1}}-1}{t_{i}}}}$ • Second form: ${\textstyle t_{i+1}={\frac {t_{i}}{{\sqrt {t_{i}^{2}+1}}+1}}}$ • ${\displaystyle \pi \sim 6\times 2^{i}\times t_{i}}$, converging as ${\displaystyle i\rightarrow \infty }$ Here is a computation using IEEE "double" (a significand with 53 bits of precision) arithmetic: i 6 × 2^i × t[i], first form 6 × 2^i × t[i], second form 0 3.4641016151377543863 3.4641016151377543863 1 3.2153903091734710173 3.2153903091734723496 2 3.1596599420974940120 3.1596599420975006733 3 3.1460862151314012979 3.1460862151314352708 4 3.1427145996453136334 3.1427145996453689225 5 3.1418730499801259536 3.1418730499798241950 6 3.1416627470548084133 3.1416627470568494473 7 3.1416101765997805905 3.1416101766046906629 8 3.1415970343230776862 3.1415970343215275928 9 3.1415937488171150615 3.1415937487713536668 10 3.1415929278733740748 3.1415929273850979885 11 3.1415927256228504127 3.1415927220386148377 12 3.1415926717412858693 3.1415926707019992125 13 3.1415926189011456060 3.1415926578678454728 14 3.1415926717412858693 3.1415926546593073709 15 3.1415919358822321783 3.1415926538571730119 16 3.1415926717412858693 3.1415926536566394222 17 3.1415810075796233302 3.1415926536065061913 18 3.1415926717412858693 3.1415926535939728836 19 3.1414061547378810956 3.1415926535908393901 20 3.1405434924008406305 3.1415926535900560168 21 3.1400068646912273617 3.1415926535898608396 22 3.1349453756585929919 3.1415926535898122118 23 3.1400068646912273617 3.1415926535897995552 24 3.2245152435345525443 3.1415926535897968907 25 3.1415926535897962246 26 3.1415926535897962246 27 3.1415926535897962246 28 3.1415926535897962246 The true value is 3.14159265358979323846264338327... While the two forms of the recurrence formula are clearly mathematically equivalent,^ significant digits . As the recurrence is applied repeatedly, the accuracy improves at first, but then it deteriorates. It never gets better than about 8 digits, even though 53-bit arithmetic should be capable of about 16 digits of precision. When the second form of the recurrence is used, the value converges to 15 digits of precision. "Fast math" optimization The aforementioned lack of compilers cannot as effectively reorder arithmetic expressions as they could with integer and fixed-point arithmetic, presenting a roadblock in optimizations such as common subexpression elimination and auto- The "fast math" option on many compilers (ICC, GCC, Clang, MSVC...) turns on reassociation along with unsafe assumptions such as a lack of NaN and infinite numbers in IEEE 754. Some compilers also offer more granular options to only turn on reassociation. In either case, the programmer is exposed to many of the precision pitfalls mentioned above for the portion of the program using "fast" In some compilers (GCC and Clang), turning on "fast" math may cause the program to disable subnormal floats at startup, affecting the floating-point behavior of not only the generated code, but also any program using such code as a library.^[67] In most Fortran compilers, as allowed by the ISO/IEC 1539-1:2004 Fortran standard, reassociation is the default, with breakage largely prevented by the "protect parens" setting (also on by default). This setting stops the compiler from reassociating beyond the boundaries of parentheses.^[68] Intel Fortran Compiler is a notable outlier.^[69] A common problem in "fast" math is that subexpressions may not be optimized identically from place to place, leading to unexpected differences. One interpretation of the issue is that "fast" math as implemented currently has a poorly defined semantics. One attempt at formalizing "fast" math optimizations is seen in Icing, a verified compiler.^[70] See also 1. mantissa of a . Somewhat vague, terms such as are also used by some. The usage of the term by some authors is potentially misleading as well. The term (as used e.g. by ) is ambiguous, as it was historically also used to specify some form of of floating-point numbers. 2. biased exponent, exponent bias , or excess n representation ) is ambiguous, as it was historically also used to specify the of floating-point numbers. 3. (1970). 4. Burroughs B7700 (1972) computers. 5. Illinois ILLIAC II (1962) computer. It is also used in the Digital Field System DFS IV and V high-resolution site survey systems. 6. Rice Institute R1 computer (since 1958). 7. ^ Base-65536 floating-point arithmetic is used in the MANIAC II (1956) computer. 8. ^ Computer hardware does not necessarily compute the exact value; it simply has to produce the equivalent rounded result as though it had computed the infinitely precise result. 9. division instruction that, on rare occasions, gave slightly incorrect results. Many computers had been shipped before the error was discovered. Until the defective computers were replaced, patched versions of compilers were developed that could avoid the failing cases. See Pentium FDIV bug 10. ^ But an attempted computation of cos(π) yields −1 exactly. Since the derivative is nearly zero near π, the effect of the inaccuracy in the argument is far smaller than the spacing of the floating-point numbers around −1, and the rounded result is exact. 11. William Kahan notes: "Except in extremely uncommon situations, extra-precise arithmetic generally attenuates risks due to roundoff at far less cost than the price of a competent error-analyst." 12. Taylor expansion of this function demonstrates that it is well-conditioned near 1: A(x) = 1 − (x−1)/2 + (x−1)^2/12 − (x−1)^4/720 + (x−1)^6/30240 − (x−1)^8/1209600 + ... for \|x−1\| < π. 13. IEEE double extended precision then additional, but not full precision is retained. 14. numerator of the first. By multiplying the top and bottom of the first expression by this conjugate, one obtains the second expression. 1. ^ . 2. ^ . 3. . Retrieved 2012-12-31. 4. ^ Friedrich-Schiller-Universität Jena. p. 2. Archived (PDF) from the original on 2018-08-07 . Retrieved 2018-08-07 (NB. This reference incorrectly gives the MANIAC II's floating point base as 256, whereas it actually is 65536.) 5. ^ . 6. ^ Savard, John J. G. (2018) [2007], "The Decimal Floating-Point Standard", quadibloc, archived from the original on 2018-07-03, retrieved 2018-07-16 7. . Retrieved 2019-08-18. “[…] Systems such as the [Digital Field System] DFS IV and DFS V were quaternary floating-point systems and used gain steps of 12 dB. […]” (256 pages) 8. ^ Lazarus, Roger B. (1957-01-30) [1956-10-01]. "MANIAC II" (PDF). Los Alamos, NM, USA: Los Alamos Scientific Laboratory of the University of California. p. 14. LA-2083. Archived (PDF) from the original on 2018-08-07. Retrieved 2018-08-07. “[…] the Maniac's floating base, which is 2^16 = 65,536. […] The Maniac's large base permits a considerable increase in the speed of floating point arithmetic. Although such a large base implies the possibility of as many as 15 lead zeros, the large word size of 48 bits guarantees adequate significance. […]” 9. ^ Torres Quevedo, Leonardo. Automática: Complemento de la Teoría de las Máquinas, (pdf), pp. 575–583, Revista de Obras Públicas, 19 November 1914. 11. ^ Randell 1982, pp. 6, 11–13. 12. ^ Randell, Brian. Digital Computers, History of Origins, (pdf), p. 545, Digital Computers: Origins, Encyclopedia of Computer Science, January 2003. 13. (PDF) from the original on 2022-07-03. Retrieved 2022-07-03. (12 pages) 14. ]. 15. ^ (PDF) from the original on 2008-09-05. 16. . 17. ^ Severance, Charles (1998-02-20). "An Interview with the Old Man of Floating-Point". 18. ^ ISO/IEC 9899:1999 - Programming languages - C. Iso.org. §F.2, note 307. “"Extended" is IEC 60559's double-extended data format. Extended refers to both the common 80-bit and quadruple 128-bit IEC 60559 formats.” 19. ^ "IEEE Floating-Point Representation". 2021-08-03. 20. ^ Using the GNU Compiler Collection, i386 and x86-64 Options Archived 2015-01-16 at the Wayback Machine. 21. ^ "long double (GCC specific) and __float128". StackOverflow. 22. ^ "Procedure Call Standard for the ARM 64-bit Architecture (AArch64)" (PDF). 2013-05-22. Archived (PDF) from the original on 2013-07-31. Retrieved 2019-09-22. 23. ^ "ARM Compiler toolchain Compiler Reference, Version 5.03" (PDF). 2013. Section 6.3 Basic data types. Archived (PDF) from the original on 2015-06-27. Retrieved 2019-11-08. 24. (PDF) from the original on 2006-05-25. Retrieved 2012-02-19. 25. ^ "openEXR". openEXR. Archived from the original on 2013-05-08. Retrieved 2012-04-25. “Since the IEEE-754 floating-point specification does not define a 16-bit format, ILM created the "half" format. Half values have 1 sign bit, 5 exponent bits, and 10 mantissa bits.” 26. ^ "Technical Introduction to OpenEXR – The half Data Type". openEXR. Retrieved 2024-04-16. 27. ^ "IEEE-754 Analysis". Retrieved 2024-08-29. 28. ^ assumed bit , while IEEE places the decimal point after the assumed bit. […] ieee_exp = msbin[3] - 2; / actually, msbin[3]-1-128+127 / […] _dmsbintoieee(double src8, double *dest8) […] MS Binary Format […] byte order => m7 \| m6 \| m5 \| m4 \| m3 \| m2 \| m1 \| exponent […] m1 is most significant byte => smmm\|mmmm […] m7 is the least significant byte […] MBF is bias 128 and IEEE is bias 1023. […] MBF places the decimal point before the assumed bit, while IEEE places the decimal point after the assumed bit. […] ieee_exp = msbin[7] - 128 - 1 + 1023; […] 29. ^ ^a ^b Steil, Michael (2008-10-20). "Create your own Version of Microsoft BASIC for 6502". pagetable.com. Archived from the original on 2016-05-30. Retrieved 2016-05-30. 30. ^ "IEEE vs. Microsoft Binary Format; Rounding Issues (Complete)". Microsoft Support. Microsoft. 2006-11-21. Article ID KB35826, Q35826. Archived from the original on 2020-08-28. Retrieved 31. ^ ^a ^b Kharya, Paresh (2020-05-14). "TensorFloat-32 in the A100 GPU Accelerates AI Training, HPC up to 20x". Retrieved 2020-05-16. 32. ^ "NVIDIA Hopper Architecture In-Depth". 2022-03-22. 33. ]. 34. (PDF) from the original on 2004-12-21. 35. ^ ) 36. (PDF) from the original on 2014-07-29. 37. ^ "Added Grisu3 algorithm support for double.ToString(). by mazong1123 · Pull Request #14646 · dotnet/coreclr". GitHub. 38. . 39. ^ Giulietti, Rafaello. "The Schubfach way to render doubles". 40. ^ "abolz/Drachennest". GitHub. 2022-11-10. 41. ^ "google/double-conversion". GitHub. 2020-09-21. 42. . 43. ^ . (With the addendum "Differences Among IEEE 754 Implementations": 44. . 45. ^ ^a ^b US patent 3037701A, Huberto M Sierra, "Floating decimal point arithmetic control means for calculator", issued 1962-06-05 46. ^ (PDF) from the original on 2002-06-22. 47. ^ "D.3.2.1". Intel 64 and IA-32 Architectures Software Developers' Manuals. Vol. 1. 48. ISSN 1354-3172. Retrieved 2011-09-24 “Far more worrying is cancellation error which can yield catastrophic loss of precision.”[4] 49. ^ Christopher Barker: PEP 485 -- A Function for testing approximate equality 50. US Government Accounting Office . GAO report IMTEC 92-26. 51. . Retrieved 2013-05-14. 52. . Retrieved 2013-05-14. 53. ^ . 0-89871-355-2. 54. . 55. ^ ARITH 17, Symposium on Computer Arithmetic (Keynote Address). pp. 6, 18. Archived (PDF) from the original on 2006-03-17 . Retrieved 2013-05-23 . (NB. Kahan estimates that the incidence of excessively inaccurate results near singularities is reduced by a factor of approx. 1/2000 using the 11 extra bits of precision of double extended 56. (PDF) from the original on 2013-06-20. 57. (PDF) from the original on 2000-08-16. Retrieved 2003-09-05. 58. ^ (PDF) from the original on 2003-08-15. 59. (PDF) from the original on 2004-12-04. 60. (PDF) from the original on 2013-05-17. 61. ^ "General Decimal Arithmetic". Speleotrove.com. Retrieved 2012-04-25. 62. ^ Christiansen, Tom; Torkington, Nathan; et al. (2006). "perlfaq4 / Why is int() broken?". perldoc.perl.org. Retrieved 2011-01-11. 63. . 64. (PDF) from the original on 2003-12-05. 65. ^ "Auto-Vectorization in LLVM". LLVM 13 documentation. “We support floating point reduction operations when -ffast-math is used.” 66. ^ "FloatingPointMath". GCC Wiki. 67. ^ "55522 – -funsafe-math-optimizations is unexpectedly harmful, especially w/ -shared". gcc.gnu.org. 68. ^ "Code Gen Options (The GNU Fortran Compiler)". gcc.gnu.org. 69. . Further reading External links	{"url":"https://findatwiki.com/Floating-point_arithmetic","timestamp":"2024-11-12T09:43:49Z","content_type":"text/html","content_length":"386367","record_id":"<urn:uuid:b0ba174d-171f-4260-ab19-c1c6ab14a1d1>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00040.warc.gz"}
Factorization: Know About Factors of 24 and 15 in This Article Know About Factors of 24 and 15 in This Article In mathematics, a factor is a divisor of an integer that divides it exactly without leaving any remainder. A factor is a number that divides another number fully without leaving any remainder. The process of representing a number as a product of several factors is known as factorization or factoring. 4 × 6 is, for example, a factorization of the integer 24 and (4,6) are factors of 24. Let’s learn about the factorization of a number properly by taking two numbers 15 and 24 as examples. Factors of 15 We will learn how to find factors of 15 by combining factor pairs and prime numbers. Factor Pairs of 15 If we look at the factors of 15 as pairs, we can see that 15 is the result of multiplying two numbers. The only integers that arise from listing all the factors of 15 in this context are 1, 3, 5, and 15 as 1 × 15 = 15 and 3 × 5 = 15. As a result, because 15 is a product of prime numbers, only the four aforementioned values are referred to as factors of 15. Prime Factorisation of 15 The integer 15 has several lesser integers in addition to 1 as its divisor. As a result, 15 is referred to as a composite number. The Prime Factorization method will be used to get the factors of 15 as shown below. The smallest known prime factor is 2. As a result, we'll divide this value by 15 (15 ÷ 2 = 7.5). Because the result is a decimal on (7.5), 2 is not a prime factor of 15. Let's go on to the following number, 3 (15 3 = 5). Again, dividing 5 by 3 gets 5 ÷ 3 = 1.67, a fractional result (1.67). Divide 5 by the next prime integer, which is 5 (5 ÷ 5 = 1). We're not continuing with the number division because we got result 1. As a result, the numbers 5 and 3 are seen as prime factors of 15, where both 3 and 5 are prime numbers. Factors of 24 Factors of 24 are numbers that totally divide 24 without leaving any remainder. There are eight 24 factors, with 24 being the largest and 2 and 3 being its prime factors. Factors of 24 using Multiplication Method Let us use the multiplication method to discover the factors of 24 using the steps below. • To get the factors of 24 using the multiplication method, we must first determine which integers multiply to give 24. So, starting with 1, we must divide 24 by natural numbers until we reach 9. We must keep track of the numbers that totally divide 24. • Its factors are the integers that totally divide 24. We write that number and its corresponding pair in a list, as illustrated in the figure above. As we check and list all of the numbers up to 9, we also receive the other pair factor. For example, beginning with 1, we write 1 24 = 24, 2 12 = 24, and so on. Here, (1, 24) makes the first pair, (2, 12) forms the second pair, and so on. So, if we write 1 as a factor of 24, the other factor is 24; and if we write 2 as a factor of 24, the other factor is 12. Same way, we get all other factors as well. • After noting the list, we receive all the components of 24 starting from 1 up there, descending down, and then going back up to 24. This provides us with an exhaustive list of all the 24 factors. As a result, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.	{"url":"https://www.scientificworldinfo.com/2022/09/know-about-factors-of-24-and-15-in-this-article.html","timestamp":"2024-11-07T23:38:19Z","content_type":"application/xhtml+xml","content_length":"216597","record_id":"<urn:uuid:5c094040-4067-4355-967f-4aade0d459cb>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00306.warc.gz"}
NCERT Solutions for Class 5 maths Chapter 5 Does It Look The Same - Free PDF Download The NCERT solutions for class 5 maths chapter 5 Does it look the same presents how things look different from different sides and give a brief idea about mirror images. This solution provides a fun learning for engaging kids. Easy to understand as well as follow the standards of CBSE solutions. Specifically deals with shapes and patterns which form the precursor to geometry and fractions. Apart from that, it deals with changing the position of alphabets or figures and comparing them. The solutions are explained step by step, along with suggested tips and tricks to make the learning process more comfortable. Furthermore, the concepts are complemented by exciting illustrations that make the learning process fun.	{"url":"https://praadisedu.com/ncert-solutions-for-class-5-maths-chapter-5/Does-It-Look-The-Same/281/13","timestamp":"2024-11-08T21:45:35Z","content_type":"text/html","content_length":"105508","record_id":"<urn:uuid:2d404d6e-6bc3-49d7-b033-56e7194803c7>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00812.warc.gz"}
Excel OFFSET function - formula examples and uses (2024) In this tutorial, we are going to shed some light on one of the most mysterious inhabitants of the Excel universe - the OFFSET function. So, what is OFFSET in Excel? In a nutshell, the OFFSET formula returns a reference to a range that is offset from a starting cell or a range of cells by a specified number of rows and columns. The OFFSET function may be a bit tricky to get, so let's go over a short technical explanation first (I'll do my best to keep it simple) and then we will cover a few of the most efficient ways to use OFFSET in Excel. Excel OFFSET function - syntax and basic uses The OFFSET function in Excel returns a cell or range of cells that is a given number of rows and columns from a given cell or range. The syntax of the OFFSET function is as follows: OFFSET(reference, rows, cols, [height], [width]) The first 3 arguments are required and the last 2 are optional. All of the arguments can be references to other cells or results returned by other formulas. It looks like Microsoft made a good effort to put some meaning into the parameters' names, and they do give a hint at what you are supposed to specify in each. Required arguments: • Reference - a cell or a range of adjacent cells from which you base the offset. You can think of it as the starting point. • Rows - The number of rows to move from the starting point, up or down. If rows is a positive number, the formula moves below the starting reference, in case of a negative number it goes above the starting reference. • Cols - The number of columns you want the formula to move from the starting point. As well as rows, cols can be positive (to the right of the starting reference) or negative (to the left of the starting reference). Optional arguments: • Height - the number of rows to return. • Width - the number of columns to return. Both the height and width arguments must always be positive numbers. If either is omitted, it defaults to the height or width of reference. Note. OFFSET is a volatile function and and may slow down your worksheet. The slowness is directly proportional to the number of cells recalculated. And now, let's illustrate the theory with an example of the simplest OFFSET formula. Excel OFFSET formula example Here is an example of a simple OFFSET formula that returns a cell reference based on a starting point, rows and cols that you specify: The formula tells Excel to take cell A1 as the starting point (reference), then move 3 rows down (rows argument) and 1 column to the left (cols argument). As the result, this OFFSET formula returns the value in cell B4. The image on the left shows the function's route and the screenshot on the right demonstrates how you can use the OFFSET formula on real-life data. The only difference between the two formulas is that the second one (on the right) includes a cell reference (E1) in the rows argument. But since cell E1 contains number 3, and exactly the same number appears in the rows argument of the first formula, both would return an identical result - the value in B4. Excel OFFSET formulas - things to remember • The OFFSET function is Excel doesn't actually move any cells or ranges, it just returns a reference. • When an OFFSET formula returns a range of cells, the rows and cols arguments always refer to the upper-left cell in the returned rage. • The reference argument must include a cell or range of adjacent cells, otherwise your formula will return the #VALUE! error. • If the specified rows and/or cols move a reference over the edge of the spreadsheet, your Excel OFFSET formula will return the #REF! error. • The OFFSET function can be used within any other Excel function that accepts a cell / range reference in its arguments. For example, if you try to use the formula OFFSET(A1,3,1,1,3) in pre-dynamic array Excel (2019 and lower), it will throw a #VALUE! error since a range to return (1 row, 3 columns) does not fit into a single cell. However, if you embed it into the SUM function, like this: the formula will return the sum of values in a 1-row by 3-column range that is 3 rows below and 1 column to the right of cell A1, i.e. the total of values in cells B4:D4. In situation when you wish to base the offset from the current cell, you can use the INDIRECT and ADDRESS functions in combination with ROW() and COLUMN() to get the starting reference. For example: =SUM(OFFSET(INDIRECT(ADDRESS(ROW(), COLUMN())), 3, 1, 1, 3)) Why do I use OFFSET in Excel? Now that you know what the OFFSET function does, you may be asking yourself "Why bother using it?" Why not simply write a direct reference like B4:D4? The Excel OFFSET formula is very good for: Creating dynamic ranges: References like B1:C4 are static, meaning they always refer to a given range. But some tasks are easier to perform with dynamic ranges. This is particularly the case when you work with changing data, e.g. you have a worksheet where a new row or column is added every week. Getting the range from the starting cell. Sometimes, you may not know the actual address of the range, though you do know it starts from a certain cell. In such scenarios, using OFFSET in Excel is the right way to go. How to use OFFSET function in Excel - formula examples I hope you haven't get bored with that much of theory. Anyway, now we are getting to the most exciting part - practical uses of the OFFSET function. Excel OFFSET and SUM functions The example we discussed a moment ago demonstrates the simplest usage of OFFSET & SUM. Now, let's look at these functions at another angle and see what else they can do. Example 1. A dynamic SUM / OFFSET formula When working with continuously updated worksheets, you may want to have a SUM formula that automatically picks all newly added rows. Suppose, you have the source data similar to what you see in the screenshot below. Every month a new row is added just above the SUM formula, and naturally, you want to have it included in the total. On the whole, there are two choices - either update the range in the SUM formula each time manually or have the OFFSET formula do this for you. Since the first cell of the range to sum will be specified directly in the SUM formula, you only have to decide on the parameters for the Excel OFFSET function, which will get that last cell of the • Reference - the cell containing the total, B9 in our case. • Rows - the cell right above the total, which requires the negative number -1. • Cols - it's 0 because you don't want to change the column. So, here goes the SUM / OFFSET formula pattern: =SUM(first cell:(OFFSET(cell with total, -1,0) Tweaked for the above example, the formula looks as follows: =SUM(B2:(OFFSET(B9, -1, 0))) And as demonstrated in the below screenshot, it works flawlessly: Example 2. Excel OFFSET formula to sum the last N rows In the above example, suppose you want to know the amount of bonuses for the last N months rather than grand total. You also want the formula to automatically include any new rows you add to the For this task, we are going to use Excel OFFSET in combination with the SUM and COUNT / COUNTA functions: The following details can help you understand the formulas better: • Reference - the header of the column whose values you want to sum, cell B1 in this example. • Rows - to calculate the number of rows to offset, you use either the COUNT or COUNTA function. COUNT returns the number of cells in column B that contain numbers, from which you subtract the last N months (the number is cell E1), and add 1. If COUNTA is your function of choice, you don't need to add 1, since this function counts all non-empty cells, and a header row with a text value adds an extra cell that our formula needs. Please note that this formula will work correctly only on a similar table structure - one header row followed by rows with numbers. For different table layouts, you may need to make some adjustments in the OFFSET/COUNTA formula. • Cols - the number of columns to offset is zero (0). • Height - the number of rows to sum is specified in E1. • Width - 1 column. Using OFFSET function with AVERAGE, MAX, MIN In the same manner as we calculated the bonuses for the last N months, you can get an average of the last N days, weeks or years as well as find their maximum or minimum values. The only difference between the formulas is the first function's name: The key benefit of these formulas over the usual AVERAGE(B5:B8) or MAX(B5:B8) is that you won't have to update the formula every time your source table gets updated. No matter how many new rows are added or deleted in your worksheet, the OFFSET formulas will always refer to the specified number of last (lower-most) cells in the column. Excel OFFSET formula to create a dynamic range Used in conjunction with COUNTA, the OFFSET function can help you make a dynamic range that may prove useful in many scenarios, for example to create automatically updatable drop-down lists. The OFFSET formula for a dynamic range is as follows: =OFFSET(Sheet_Name!$A$1, 0, 0, COUNTA(Sheet_Name!$A:$A), 1) At the heart of this formula, you use the COUNTA function to get the number of non-blank cells in the target column. That number goes to the height argument of OFFSET instructing it how many rows to Apart from that, it's a regular Offset formula, where: • Reference is the starting point from which you base the offset, for example Sheet1!$A$1. • Rows and Cols are both 0 because there are no columns or rows to offset. • Width is 1 column. Note. If you are making a dynamic range in the current sheet, there is no need to include the sheet name in the references, Excel will do it for you automatically when creating the named range. Otherwise, be sure to include the sheet's name followed by the exclamation point like in this formula example. Once you've created a dynamic named range with the above OFFSET formula, you can use Data Validation to make a dynamic dropdown menu that will update automatically as soon as you add or remove items from the source list. For the detailed step-by-step guidance on creating drop-down lists in Excel, please check out the following tutorials: • Creating drop-down lists in Excel - static, dynamic, from another workbook • Making a dependent drop down list Excel OFFSET & VLOOKUP As everyone knows, simple vertical and horizontal lookups are performed with the VLOOKUP or HLOOKUP function, respectively. However, these functions have too many limitations and often stumble in more powerful and complex lookup formulas. So, in order to perform more sophisticated lookups in your Excel tables, you have to look for alternatives such as INDEX, MATCH and OFFSET. Example 1. OFFSET formula for a left Vlookup in Excel One of the most infamous limitations of the VLOOKUP function is inability to look at its left, meaning that VLOOKUP can only return a value to the right of the lookup column. In our sample lookup table, there are two columns - month names (column A) and bonuses (column B). If you want to get a bonus for a certain month, this simple VLOOKUP formula will work without a =VLOOKUP(B1, A5:B11, 2, FALSE) However, as soon as you swap the columns in the lookup table, this will immediately result in the #N/A error: To handle a left-side lookup, you need a more versatile function that does not really care where the return column resides. One of possible solutions is using a combination of INDEX and MATCH functions. Another approach is using OFFSET, MATCH and ROWS: OFFSET(lookup_table, MATCH(lookup_value, OFFSET(lookup_table, 0, lookup_col_offset, ROWS(lookup_table), 1) ,0) -1, return_col_offset, 1, 1) • Lookup_col_offset - is the number of columns to move from the starting point to the lookup column. • Return_col_offset - is the number of columns to move from the starting point to the return column. In our example, the lookup table is A5:B9 and the lookup value is in cell B1, the lookup column offset is 1 (because we are searching for the lookup value in the second column (B), we need to move 1 column to the right from the beginning of the table), the return column offset is 0 because we are returning values from the first column (A): =OFFSET(A5:B9, MATCH(B1, OFFSET(A5:B9, 0, 1, ROWS(A5:B9), 1) ,0) -1, 0, 1, 1) I know the formula looks a bit clumsy, but it does work :) Example 2. How to do an upper lookup in Excel As is the case with VLOOKUP being unable to look at the left, its horizontal counterpart - HLOOKUP function - cannot look upwards to return a value. If you need to scan an upper row for matches, the OFFSET MATCH formula can help again, but this time you will have to enhance it with the COLUMNS function, like this: OFFSET(lookup_table, return_row_offset, MATCH(lookup_value, OFFSET(lookup_table, lookup_row_offset, 0, 1, COLUMNS(lookup_table)), 0) -1, 1, 1) • Lookup_row_offset - the number of rows to move from the starting point to the lookup row. • Return_row_offset - the number of rows to move from the starting point to the return row. Assuming that the lookup table is B4:F5 and the lookup value is in cell B1, the formula goes as follows: =OFFSET(B4:F5, 0, MATCH(B1, OFFSET(B4:F5, 1, 0, 1, COLUMNS(B4:F5)), 0) -1, 1, 1) In our case, the lookup row offset is 1 because our lookup range is 1 row down from the starting point, the return row offset is 0 because we are returning matches from the first row in the table. Example 3. Two-way lookup (by column and row values) Two-way lookup returns a value based on matches in both the rows and columns. And you can use the following double lookup array formula to find a value at the intersection of a certain row and =OFFSET(lookup table, MATCH(row lookup value, OFFSET(lookup table, 0, 0, ROWS(lookup table), 1), 0) -1, MATCH(column lookup value, OFFSET(lookup table, 0, 0, 1, COLUMNS(lookup table)), 0) -1) Given that: • The lookup table is A5:G9 • The value to match on the rows is in B2 • The value to match on the columns is in B1 You get the following two-dimensional lookup formula: =OFFSET(A5:G9, MATCH(B2, OFFSET(A5:G9, 0, 0, ROWS(A5:G9), 1), 0)-1, MATCH(B1, OFFSET(A5:G9, 0, 0, 1, COLUMNS(A5:G9)), 0) -1) It's not the easiest thing to remember, is it? In addition, this is an array formula, so don't forget to press Ctrl + Shift + Enter to enter it correctly. Of course, this lengthy OFFSET formula is not the only possible way to do a double lookup in Excel. You can get the same result by using the VLOOKUP & MATCH functions, SUMPRODUCT, or INDEX & MATCH. There is even a formula-free way - to employ named ranges and the intersection operator (space). The following tutorial explains all alternative solutions in full detail: How to do two-way lookup in OFFSET function - limitations and alternatives Hopefully, the formula examples on this page have shed some light on how to use OFFSET in Excel. However, to efficiently leverage the function in your own workbooks, you should not only be knowledgeable of its strengths, but also be wary of its weaknesses. The most critical limitations of the Excel OFFSET function are as follows: • Like other volatile functions, OFFSET is a resource-hungry function. Whenever there is any change in the source data, your OFFSET formulas are recalculated, keeping Excel busy for a little longer. This is not an issue for a single formula in a small spreadsheet. But if there are dozens or hundreds of formulas in a workbook, Microsoft Excel may take quite a while to recalculate. • Excel OFFSET formulas are hard to review. Because references returned by the OFFSET function are dynamic, big formulas (especially with nested OFFSETs) can be quite tricky to debug. Alternatives to using OFFSET in Excel As is often the case in Excel, the same result can be achieved in a number of different ways. So, here are three elegant alternatives to OFFSET. 1. Excel tables Since Excel 2002, we have a truly wonderful feature - fully-fledged Excel tables, as opposed to usual ranges. To make a table from structured data, you simply click Insert > Table on the Home tab or press Ctrl + T. By entering a formula in one cell in an Excel table, you can create a so-called "calculated column" that automatically copies the formula to all other cells in that column and adjusts the formula for each row in the table. Moreover, any formula that refers to a table's data automatically adjusts to include any new rows you add to the table or exclude the rows you delete. Technically, such formulas operate on table columns or rows, which are dynamic ranges in nature. Each table in a workbook has a unique name (the default ones are Table1, Table2, etc.) and you are free to rename your table via the Design tab > Properties group > Table Name text box. The following screenshot demonstrates the SUM formula that refers to the Bonus column of Table3. Please pay attention that the formula includes the table's column name rather than a range of 2. Excel INDEX function Although not exactly in the same way as OFFSET, Excel INDEX can also be used to create dynamic range references. Unlike OFFSET, the INDEX function is not volatile, so it won't slow down your 3. Excel INDIRECT function Using the INDIRECT function you can create dynamic range references from many sources such as cell values, cell values and text, named ranges. It can also dynamically refer to another Excel sheet or workbook. You can find all these formula examples in our Excel INDIRECT function tutorial. Do you remember the question asked at the beginning of this tutorial - What is OFFSET in Excel? I hope now you know the answer : ) If you want some more hands-on experience, feel free to download our practice workbook (please see below) containing all the formulas discussed on this page and reverse engineer them for deeper understanding. Thank you for reading! Practice workbook for download OFFSET formula examples (.xlsx file) You may also be interested in • Why INDEX / MATCH is a better alternative to Excel VLOOKUP • Excel VLOOKUP tutorial for beginners - syntax and formula examples • Advanced VLOOKUP formula examples • 6 reasons why VLOOKUP is not working • Excel CELL function with examples	{"url":"https://bigbearbaptist.org/article/excel-offset-function-formula-examples-and-uses","timestamp":"2024-11-05T01:09:32Z","content_type":"text/html","content_length":"138858","record_id":"<urn:uuid:0e7114cf-6dd1-4776-bff7-8fe8deced89b>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00775.warc.gz"}
Spectral accuracy - (Magnetohydrodynamics) - Vocab, Definition, Explanations \| Fiveable Spectral accuracy from class: Spectral accuracy refers to the high level of precision and convergence of numerical methods when approximating solutions to differential equations using spectral techniques. It is achieved by utilizing global basis functions, such as Fourier series or orthogonal polynomials, to represent the solution, leading to exponential convergence rates compared to traditional methods. This quality makes spectral methods particularly effective for problems with smooth solutions, resulting in accurate approximations with relatively few degrees of freedom. congrats on reading the definition of spectral accuracy. now let's actually learn it. 5 Must Know Facts For Your Next Test 1. Spectral accuracy is characterized by exponential convergence, meaning that errors decrease dramatically as more basis functions are included in the approximation. 2. The method is especially powerful for smooth problems because it effectively captures high-frequency oscillations without requiring a dense grid of points. 3. Spectral methods can be implemented in both regular domains and complex geometries through techniques like mapping and transformation of coordinates. 4. To achieve spectral accuracy, it is essential to ensure that the chosen basis functions are appropriate for the problem at hand, considering aspects like boundary conditions and solution 5. Despite their advantages, spectral methods can struggle with discontinuous solutions or problems with sharp gradients, where traditional finite difference or finite element methods may perform Review Questions • How does spectral accuracy impact the efficiency of numerical methods in solving differential equations? □ Spectral accuracy significantly enhances the efficiency of numerical methods by enabling them to achieve high precision with fewer computational resources. By using global basis functions, these methods capture the essential features of smooth solutions, resulting in exponential convergence rates. This means that fewer degrees of freedom are needed compared to local approximation techniques, which can lead to faster computations and reduced memory usage while maintaining accuracy. • Discuss the advantages and limitations of using spectral methods in comparison to traditional numerical techniques. □ Spectral methods offer substantial advantages over traditional numerical techniques such as finite difference or finite element methods due to their spectral accuracy and exponential convergence rates for smooth problems. However, they also have limitations; for example, they may not perform well on problems with discontinuities or sharp gradients. The choice between these methods often depends on the specific characteristics of the problem being solved and the desired level of accuracy. • Evaluate how the selection of basis functions influences spectral accuracy and overall computational performance. □ The selection of basis functions is crucial for achieving spectral accuracy because it directly affects how well the method approximates the solution. Using appropriate functions like Fourier series or Chebyshev polynomials can minimize errors and enhance convergence rates for smooth solutions. However, if unsuitable functions are chosen, it can lead to inaccuracies and a degradation in performance. Understanding the problem's characteristics helps ensure that the most effective basis functions are employed, balancing accuracy with computational efficiency. "Spectral accuracy" also found in: Â© 2024 Fiveable Inc. All rights reserved. APÂ® and SATÂ® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.	{"url":"https://library.fiveable.me/key-terms/magnetohydrodynamics/spectral-accuracy","timestamp":"2024-11-07T13:05:14Z","content_type":"text/html","content_length":"147640","record_id":"<urn:uuid:3a83d83c-6c8d-443b-abb0-e4a7c310a4c6>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00408.warc.gz"}
Graph colouring with no large monochromatic components For a graph G and an integer t we let mcct(G) be the smallest m such that there exists a colouring of the vertices of G by t colours with no monochromatic connected subgraph having more than m vertices. Let F be any non-trivial minor-closed family of graphs. We show that mcc2(G) = O(n^2/3) for any n-vertex graph G ∈ F. This bound is asymptotically optimal and it is attained for planar graphs. More generally, for every such F, and every fixed t we show that mcct(G)=O(n^2/(t+1)). On the other hand, we have examples of graphs G with no Kt+3 minor and with mcct(G)=On^2/(2t-1)). It is also interesting to consider graphs of bounded degrees. Haxell, Szabó and Tardos proved mcc2(G) ≤ 20000 for every graph G of maximum degree 5. We show that there are n-vertex 7-regular graphs G with mcc2(G)=ω(n), and more sharply, for every > 0 there exists c > 0 and n-vertex graphs of maximum degree 7, average degree at most 6 + for all subgraphs, and with mcc2(G) ≤ cn. For 6-regular graphs it is known only that the maximum order of magnitude of mcc2 is between n and n. We also offer a Ramsey-theoretic perspective of the quantity mcct(G). Dive into the research topics of 'Graph colouring with no large monochromatic components'. Together they form a unique fingerprint.	{"url":"https://cris.huji.ac.il/en/publications/graph-colouring-with-no-large-monochromatic-components","timestamp":"2024-11-12T16:57:28Z","content_type":"text/html","content_length":"48572","record_id":"<urn:uuid:648e0ce9-dd37-4d94-882f-9021a6845236>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00614.warc.gz"}
Error estimates for anisotropic ﬁnite elements and applications \| EMS Press Error estimates for anisotropic ﬁnite elements and applications • Ricardo G. Durán Universidad de Buenos Aires, Argentina The ﬁnite element method is one of the most frequently used techniques to approximate the solution of partial differential equations. It consists in approximating the unknown solution by functions which are polynomials on each element of a given partition of the domain, made of triangles or quadrilaterals (or their generalizations to higher dimensions). A fundamental problem is to estimate the error between the exact solution u and its computable ﬁnite element approximation. In many situations this error can be bounded in terms of the best approximation of u by functions in the ﬁnite element space of piecewise polynomial functions. A natural way to estimate this best approximation is by means of the Lagrange interpolation or other similar procedures. Many works have considered the problem of interpolation error estimates. The classical error analysis for interpolations is based on the so-called regularity assumption, which excludes elements with different sizes in each direction (called anisotropic). The goal of this paper is to present a different approach which has been developed by many authors and can be applied to obtain error estimates for several interpolations under more general hypotheses. An important case in which anisotropic elements arise naturally is in the approximation of convection-diffusion problems which present boundary layers. We present some applications to these problems. Finally we consider the ﬁnite element approximation of the Stokes equations and present some results for non-conforming methods.	{"url":"https://ems.press/books/standalone/24/549","timestamp":"2024-11-02T15:41:14Z","content_type":"text/html","content_length":"54446","record_id":"<urn:uuid:9af4a0ed-ef7e-4197-a0ba-23572f971283>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00467.warc.gz"}
AA similarity theorem Two triangles are similar if they have two corresponding angles that are congruent. absolute value A number’s distance from zero on the number line. The symbol means the absolute value of . Recall that distance is always positive. The diagram shows that and . absolute value function A function that contains an algebraic expression within absolute value symbols. The absolute value parent function, written as: adjacent angles Two non-overlapping angles with a common vertex and one common side. and are adjacent angles: alternate exterior angles A pair of angles formed by a transversal intersecting two lines. The angles lie outside of the two lines and are on opposite sides of the transversal. See angles made by a transversal. alternate interior angles A pair of angles formed by a transversal intersecting two lines. The angles lie between the two lines and are on opposite sides of the transversal. See also angles made by a transversal. Altitude of a triangle: A perpendicular segment from a vertex to the line containing the base. Altitude of a solid: A perpendicular segment from a vertex to the plane containing the base. Two rays that share a common endpoint called the vertex of the angle. angle bisector A ray that has its endpoint at the vertex of the angle and divides the angle into two congruent angles. angle of depression/angle of elevation Angle of depression: the angle formed by a horizontal line and the line of sight of a viewer looking down. Sometimes called the angle of decline. Angle of elevation: the angle formed by a horizontal line and the line of sight of a viewer looking up. Sometimes called the angle of incline. angles associated with circles: central angle, inscribed angle, circumscribed angle Central angle: An angle whose vertex is at the center of a circle and whose sides pass through a pair of points on the circle. Inscribed angle: An angle formed when two secant lines, or a secant and tangent line, intersect at a point on a circle. Circumscribed angle: The angle made by two intersecting tangent lines to a circle. angles made by a transversal arc length The distance along the arc of a circle. Part of the circumference. Equation for finding arc length: Where is the radius and is the central angle in radians. arc of a circle, intercepted arc Arc: A portion of a circle. Intercepted arc: The portion of a circle that lies between two lines, rays, or line segments that intersect the circle. A line that a graph approaches, but does not reach. A graph will never touch a vertical asymptote, but it might cross a horizontal or an oblique (also called slant) asymptote. Horizontal and oblique asymptotes indicate the general behavior of the ends of a graph in both positive and negative directions. If a rational function has a horizontal asymptote, it will not have an oblique asymptote. Oblique asymptotes only occur when the numerator of has a degree that is one higher than the degree of the denominator.	{"url":"https://access.openupresources.org/curricula/our-hs-math/en/integrated/math-2/student_glossary.html","timestamp":"2024-11-12T17:16:11Z","content_type":"text/html","content_length":"1050491","record_id":"<urn:uuid:f2ef823a-e275-446f-98ec-a1858deb863e>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00037.warc.gz"}
Celebratio Mathematica [S:Dear Julia,:S] [S:Dear Yuri::S] A mathematical correspondence Works connected to [S:Martin David Davis:S] Filter the Bibliography List : “Hilbert’s tenth problem: Diophantine equations: Positive aspects of a negative solution,” pp. 323–378 in Mathematical developments arising from Hilbert problems (DeKalb, IL, 13–17 May 1974), Part 2. Edited by F. E. Browder. Proceedings of Symposia in Pure Mathematics 28. 1976. With a loose-leaf erratum. MR 432534 Zbl 0346.02026 incollection	{"url":"https://celebratio.org/Robinson_JB/bibf/288/964/4787/","timestamp":"2024-11-13T14:08:40Z","content_type":"text/html","content_length":"26700","record_id":"<urn:uuid:fe34f411-9130-4209-9f20-2c13508bc89b>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00685.warc.gz"}
How do you use the quotient rule to differentiate y=cos(x)/ln(x)? \| Socratic How do you use the quotient rule to differentiate #y=cos(x)/ln(x)#? 1 Answer The quotient rule states: $\frac{d}{\mathrm{dx}} \left[f \frac{x}{g} \left(x\right)\right] = \frac{f ' \left(x\right) g \left(x\right) - f \left(x\right) g ' \left(x\right)}{{\left(g \left(x\right)\right)}^{2}}$ Let $f \left(x\right) = \cos x$, and let $g \left(x\right) = \ln x$. We know that the derivative of $\cos x$ is $- \sin x$, and that the derivative of $\ln x$ is $\frac{1}{x}$. Therefore, $f ' \left(x\right) = - \sin x$, and $g ' \left(x\right) = \frac{1}{x}$. Now we may simply plug into the quotient rule formula: $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{- \sin x \ln x - \cos \frac{x}{x}}{{\left(\ln x\right)}^{2}}$ And there is our answer. If we would like, we can split this fraction up to make it a bit prettier: $\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{\sin x \ln x}{{\left(\ln x\right)}^{2}} - \frac{\cos \frac{x}{x}}{{\left(\ln x\right)}^{2}}$ This simplifies to: $\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{\sin x}{\ln x} - \cos \frac{x}{x {\left(\ln x\right)}^{2}}$ Impact of this question 12767 views around the world	{"url":"https://api-project-1022638073839.appspot.com/questions/how-do-you-use-the-quotient-rule-to-differentiate-y-cos-x-ln-x#107525","timestamp":"2024-11-05T16:17:36Z","content_type":"text/html","content_length":"34062","record_id":"<urn:uuid:08fd4b0e-2266-45da-a3a1-0cf40b71f0f1>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00420.warc.gz"}
Strategy use and basic arithmetic cognition in adults Arithmetic cognition research was at one time concerned mostly with the representation and retrieval of arithmetic facts in memory. More recently it was found that memory retrieval does not account for all single digit arithmetic performance. For example, Canadian university students solve up to 40% of basic addition problems using procedural strategies (e.g. 5 + 3 = 5 + 1 + 1 + 1). Given that procedures are less efficient than direct memory retrieval it is important to understand why procedure use is high, even for relatively skilled adults. My dissertation, therefore, sought to expand understanding of strategy choice for adults’ basic arithmetic. Background on this topic and supporting knowledge germane to the topic are provided in Chapter 1. Chapter 2 focused on a well-known, but unexplained, finding: A written word problem (six + seven) results in much greater reported use of procedures (e.g., counting) than the same problem in digits (6 + 7). I hypothesized that this could be the result of a metacognitive effect whereby the low surface familiarity for word problems discourages retrieval. This was tested by familiarizing participants with a subset of the written word stimuli (e.g. three + four = ?, six + nine= ?) and then testing them on unpractised problems comprised of practiced components (four + six = ?). The result was increased retrieval reported for unpractised problems with practiced components. This indicates that surface familiarity contributes to strategy choice. Chapter 3 focused on another classic phenomenon in the arithmetic cognition literature, the problem size effect: Response time, error, and procedure rates increase as a function of problem size. A previous study reported a reduced problem size effect for auditory multiplication problems compared to digit problems. I hypothesized that if this reduction was due to problem encoding processes rather than an effect on calculation per se, then a similar pattern would be observed for addition. Instead, I found that the size effect for addition was larger. I concluded that the auditory format promotes procedures for addition, but promotes retrieval for multiplication. Chapters 4 and 5 were concerned with a well-known methodological issue in the strategy literature, subjectivity of self-reports: Some claim self-reports are more like opinions than objective measures. Thevenot, Fanget, and Fayol (2007) ostensibly solved this problem by probing problem memory subsequent to participants providing an answer. They reasoned that after a more complex procedure, the memory for the original problem would become degraded. The result would be better memory for problems answered by retrieval instead of by procedure. I hypothesized that their interpretation of their findings was conflated with the effect of switching tasks from arithmetic to number memory. I demonstrated that their new method for measuring strategy choice was contaminated by task switching costs, which compromises its application as a measure of strategy choice (Chapter 4). In a subsequent project (Chapter 5), I tested the sensitivity of this new method to detect the effects of factors known in the literature to affect strategy choice. The results indicated that Thevenot et al.’s new method was insensitive to at least one of these factors. Thus, attempts to control for the confounding effects of task switching described in Chapter 4, in order to implement this new measure, are not warranted. The current dissertation expanded understanding of strategy choice in four directions by 1) demonstrating that metacognitive factors cause increases in procedure strategies, 2) by demonstrating that the process of strategy selection is affected differentially by digit and auditory-verbal input, 3) by investigating the validity of an alternative measure of strategy use in experimental paradigms, and 4) by discovering a critical failure in the sensitivity of this new measure to measure the effects of factors known to influence strategy use. General conclusions are discussed in Chapter 6. Cognitive science, Arithmetic cognition, Basic arithmetic, Skilled performance, Strategy use Doctor of Philosophy (Ph.D.)	{"url":"https://harvest.usask.ca/items/1833c5c6-ef5b-485e-8420-ea1843ea66d1","timestamp":"2024-11-06T07:43:20Z","content_type":"text/html","content_length":"469486","record_id":"<urn:uuid:628f951c-7eb0-46d6-bbc7-0f997388dc77>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00367.warc.gz"}
• RPN83P: RPN calculator for TI-83+ TI-84+ inspired by HP-42S Hi everyone, I learned recently that the old TI-83 Plus and TI-84 Plus calculators are programmable in Z80 assembly language. I created this project to learn how to program them. RPN83P is an RPN (Reverse Polish Notation) calculator app for the TI-83 Plus and the TI-84 Plus. The app is inspired mostly by the HP-42S calculator, with some sprinkles of older HP calculators like the HP-12C and the HP-15C. The RPN83P is a flash application that consumes one page (16 kB) of flash memory. It consumes a small amount of TI-OS RAM: 2 list variables named REGS and STK which are 240 bytes and 59 bytes respectively. Here is a quick summary of its features: • traditional 4-level RPN stack (X, Y, Z, T registers) • support for lastX register • 25 storage registers (STO 00, RCL 00, ..., STO 24, RCL 24) • hierarchical menu system, inspired by the HP-42S • support for all math functions with dedicated buttons on the TI-83 Plus and TI-84 Plus □ arithmetic: /, , -, + □ trigonometric: SIN, COS, TAN, etc. □ 1/X, X^2, 2ND SQRT □ ^ (i.e. Y^X), □ LOG, 10^X, LN, e^X □ constants: pi and e • additional menu functions: □ %, %CH, GCD, LCM, PRIM (is prime) □ IP (integer part), FP (fractional part), FLR (floor), CEIL (ceiling), NEAR (nearest integer) □ ABS, SIGN, MOD, MIN, MAX □ probability: PERM, COMB, N!, RAND, SEED □ hyperbolic: SINH, COSH, TANH, etc. □ angle conversions: >DEG, >RAD, >HR, >HMS, P>R, R>P □ unit conversions: >C, >F, >km, >mi, etc □ base conversions: DEC, HEX, OCT, BIN □ bitwise operations: AND, OR, XOR, NOT, NEG, SL, SR, RL, RR, B+, B-, B, B/, BDIV • various display modes □ RAD, DEG □ FIX (fixed point 0-9 digits) □ SCI (scientific 0-9 digits) □ ENG (engineering 0-9 digits) Here are some missing features which may be added in the future: • statistics functions (sum, mean, variance, standard deviation) • complex numbers • vectors and matrices • keystroke programming The Project Home is here: Download the latest rpn83p.8xk binary (v0.4.0 as of this post) from: * GitHub releases page: * Cemetech Downloads: If you try it, I'd appreciate any feedback. That's a pretty neat app. I have a suspicion you've written quite a lot of code before. Storing the RPN stack and registers in lists is an interesting choice. Appvars would have been the normal choice, but using BASIC-accessible list variables opens some interesting possibilities for integration with TI-BASIC programs. Using ON for going up a menu level is weird because it violates the UI conventions used everywhere else on the calculator. Aside from that, the menu system is quite intuitive. Congrats, seems great, and it's not every day there's an actual new app for the 83+/84+! The readme is also well detailed DrDnar wrote: That's a pretty neat app. I have a suspicion you've written quite a lot of code before. Storing the RPN stack and registers in lists is an interesting choice. Appvars would have been the normal choice, but using BASIC-accessible list variables opens some interesting possibilities for integration with TI-BASIC programs. Using ON for going up a menu level is weird because it violates the UI conventions used everywhere else on the calculator. Aside from that, the menu system is quite intuitive. Yeah, I've done a fair bit of programming, much of it in C++, Java, Python, in the "cloud" as they say. Recently I've done a fair bit of embedded C++ with Arduino-compatible hardware. With this project, I wanted to go even lower-level. I had not done Z80 programming in... decades? Yes, AppVars would be the natural choice. Except that the TI-83 Plus SDK documentation of those things seemed so poor, that I could not understand what they were until I had completed most of the app. The RPN stack registers and storage registers (R00-R24) just store floating point numbers, so they seemed to fit nicely into user-defined List variables. There are a number of internal application variables that I don't currently persist across reboot/power-cycle. Those would definitely go into custom AppVars. With regards to using the ON button as the EXIT/ESC menu function, I agree it's unusual. I struggled with this for the entire duration of the project, trying different options, but the other options seemed even less optimal. It should be the ESC key, but the TI-83+/TI-84+ calculators don't have an ESC key. The HP-42S calculator uses the ON key for this purpose. That button is labeled "ON/EXIT" on the HP-42S. So people who are familiar with the HP-42S will hopefully find it somewhat familiar. Adriweb wrote: Congrats, seems great, and it's not every day there's an actual new app for the 83+/84+! The readme is also well detailed Thanks! I stopped doing calculator programming before the TI-83+ and TI-84+ were created, so I missed the entire party for 20-25 years. When I found out that these things used the Z80, and there seemed to be a good ecosystem of development tools, I thought I'd give it a shot to create something that would be useful to me. The ramp up was a little more difficult than I had expected. The developer information is scattered in all over the place. Texas Instruments seems to be actively hiding information to make this hard. Googling has become very difficult. Any search for "TI 84 Plus" has become completely commingled with the "TI 84 Plus CE" family, so I found it difficult to locate technical information about these older calculators. WikiTI has a lot of information, including everything there is to know about the hardware. There really aren't any mysteries left. The old official TI SDK documentation is probably still the best reference for the OS functions. You can still download a copy of the old TI SDK from TI Planet. The TI-84 Plus CE uses an eZ80 CPU, which runs a very similar ISA to the Z80, just with some 24-bit (yes, really) registers. The community SDK's standard libraries are written in assembly, if you'd like to contribute! While nearly everyone uses C/C++ for the TI-84 Plus CE, you can still use pure assembly if you like. The problem I had with WikiTI was that I could not understand most of the info there... until I had learned the material from somewhere else. In other words, it's a great reference, but not so great for someone who's starting from zero. With regards to the TI-84 Plus CE, I thought that Texas Instruments locked down the flash app signing key. I understand that a hack is available to bypass it, but is the friction low enough that 3rd party development is still active and vibrant? Flash application development is experimental at best, and we don't really recommend trying to bypass the signing key. However, programs can do nearly everything apps can do. There's 150 K of user RAM available (and sixty-some K of scrap RAM), plus the 150 K of VRAM (which you don't have to use all of), so you can do quite a lot without needing to be an app. The only friction (so to speak) for running programs is that recent OS versions require the jailbreak. Originally, you could just run assembly programs like any BASIC program. You don't have to use flash apps to run custom code on the TI-83 Plus either. You can run many programs with the Asm( token (found under the catalog) and the rest generally run on MirageOS (which is not an OS) or Doors CS (which the forum TOS require me to advertise). Thanks for the additional info about the TI-84 Plus CE. I don't own it, I didn't realize that ASM programs for the CE could be so large. I have 2-3 additional projects in mind for the 83+/84+, perhaps I will make my way to the CE. I do appreciate the simplicity of flash apps on the 83+/84+. They are easy to create (using spasm-ng). The app is persistent across crashes and power failures. And flash apps are far easier to execute than assembly programs, needing far fewer keystrokes. Something like (APPS, down, down, RPN83P, ENTER), instead of (2ND CATALOG, down, down,... down, down, ASM(, PRGM, down, ..., down,..., down,..., RPN83P, ENTER). I understand that shells like the Doors CS can make ASM programs easier to use. But... without intending any offense towards the creator of Doors CS, I just cannot use Doors CS. I cannot read anything in that font. And navigating a mouse pointer on a calculator with arrow keys feels like torture to me. (I would love to see a fast Text UI version of Doors CS.) Anyway, thanks for all the info and your feedback. bxparks wrote: But... without intending any offense towards the creator of Doors CS, I just cannot use Doors CS. I cannot read anything in that font. And navigating a mouse pointer on a calculator with arrow keys feels like torture to me. (I would love to see a fast Text UI version of Doors CS.) A lot of shells (including Doors CS) include a feature that allow you to run assembly programs from the homescreen without the Asm( token after running the shell once. You only need to press prgm, then select the program you want, and then press enter. So, you don't need to open the shell every time that you want to run the program, just once per time that you reset the calculator. commandblockguy wrote: A lot of shells (including Doors CS) include a feature that allow you to run assembly programs from the homescreen without the Asm( token after running the shell once. You only need to press prgm, then select the program you want, and then press enter. So, you don't need to open the shell every time that you want to run the program, just once per time that you reset the calculator. That's true, I forgot about that useful feature of Doors CS. The problem for me was that during the development process, I would crash the calculator a lot, probably hundreds of times. It was too much work to restart Doors CS every time that I crashed. Even when I did not crash, I would not remember whether I had already executed Doors CS. I found it easier to just assume that I had not, and always use the vanilla TI-OS mechanism (CATALOG, ASM(, PRGM, etc.). Once I learned how to create a flash app, the iteration cycle became simpler. No dependency to anything else, just drag and drop into the emulator, and hit APPS, down, down,..., MYAPP, ENTER. bxparks wrote: The problem for me was that during the development process, I would crash the calculator a lot, probably hundreds of times. It was too much work to restart Doors CS every time that I crashed. Even when I did not crash, I would not remember whether I had already executed Doors CS. What I generally do is create an emulator save state after installing everything, and then just reload that every time I needed to re-run the program. That's even faster than waiting for the calculator to reset (assuming you're using the keyboard shortcut for it), and also makes sure that the archive is in a known state (in the event that your code is really broken). I pushed a new version (v0.5.0) last night, available for download at: * GitHub releases page: https://github.com/bxparks/rpn83p/releases * Cemetech Downloads: http://ceme.tech/DL2376 You can read the https://github.com/bxparks/rpn83p/blob/develop/CHANGELOG.md, but here are the major highlights: * Add STAT (statistics) functions for 1 and 2-parameter statistics. Includes mean, standard deviation (sample and population), and covariance (sample and population). * Add CFIT (curve fit) functions, supporting the 4 curve fit models of the HP-42S: linear, logarithmic, exponential, and power. * Add menu selection dots. The HP-42S menus act as both action buttons and status buttons. For example, selecting 'RAD' mode places a small dot after the 'RAD' label on the menu. I figured out how to implement this on the RPN83P without adding too much complexity in the code. It's been a while since I posted an update. I pushed the latest v0.7.0 today. I hope to stabilize this to v1.0 in 1-2 months, and then take a short break from Z80 programming. It's been enjoyable for the most part, but I feel about 1/3 as productive in Z80 assembly compared to the C language. The first time I spent a day tracking down a missing "inc hl" was a learning experience. But the novelty wears off after a few more times. As the application gets bigger, refactoring and testing gets progressively slower, and more tedious. I wish I could implement automated unit testing for Z80 programs. Anyway, major features since v0.5.0: * now consumes 2 flash pages (32 kiB) * implement the rest of HP-16C logical and bitwise operators (ASL, RLC, RRC, SLn, SRn, RLn, RRn, RLCn, RRCn, REVB, CNTB) * support different word sizes: 8, 16, 24, 32 bits * add Carry flag for logical operators and integer arithmetics, and SCF, CCF, CF? * save and restore application state on QUIT or OFF * add storage register arithmetics (e.g. STO+, RCL-) * implement Time Value of Money (TVM) functions from HP-12C and HP-30b	{"url":"https://www.cemetech.net/forum/viewtopic.php?t=19209","timestamp":"2024-11-11T23:55:18Z","content_type":"text/html","content_length":"72838","record_id":"<urn:uuid:69e6e68f-84f2-4e44-a026-e6ecbec3d87e>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00592.warc.gz"}
Make 1 Whole Noticing and using mathematical structure Representing and connecting What unique unit fractions combine to form 1 whole? The Egyptians used a sum of unique unit fractions to represent other fractional values. For example, they could use ½ + ¼ to represent the value ¾. The Egyptians would not have used this representation for whole numbers, but it’s interesting to explore the different ways to make 1 whole with unique unit fractions. How can unique unit fractions be combined to form 1 whole? • Can 1 be represented as the sum of two unique unit fractions? How do you know? • How could you represent 1 as the sum of three unique unit fractions? How about four unique unit fractions? A unit fraction is a fraction with 1 as the numerator. For example, ½, ⅓, and ⅛ are unit fractions. A unique unit fraction is a fraction that is different from the others. For example, ½ and ¼ are unique unit fractions, but ⅓ and ⅓ are the same, so they are not unique unit fractions. How could you get started? • How can you model a unit fraction? How much of the whole will be left? • How could you use equivalent fractions to choose unique unit fractions? Ready to explore more? • Try to find a combination of unique unit fractions that have a sum close to 1 ½. How close can you get? • What fractions can be made by combining two unique unit fractions? What patterns do you notice among the addends and sums? For Teachers: More about this activity In this task, students find unique unit fraction pieces that fit together to make a whole. Students might start with a set of unique unit fractions and experiment with those that combine to make a whole, or they might start with 1 whole and take off unique unit fractions until they are left with a unique unit fraction. Exploring this puzzling problem will engage students with equivalent fractions to find the missing unit fraction piece to complete their whole. There are many different combinations of unique unit fractions with a sum of 1. Students may find they have to exclude several options because the fraction that remains is not a unit fraction, or may not be unique. Students may choose to: • Model several unique unit fractions of the same-size whole and use them to build a whole. To help find the final piece, students might build a whole using a denominator that will allow them to model equivalent fractions and determine if they can make a unit fraction for the amount that remains. • Start with a pair of unique unit fractions. Using equivalent models, students might find the fraction that represents the remaining amount needed to make 1 whole and build another unit fraction close to that difference. They could iterate this process until they find the last unit fraction needed to make 1. Multiple apps can be used to explore this problem. • In the Fraction app, students might model two unit fractions, then create a model for a whole using a common denominator. Overlaying a model of 1 with the new denominator over their first two models, students can see how much more they need to make a sum of 1. Here’s the beginning of a solution. • The Geoboard app can also be used to model fractions. Students may begin modeling unit fractions by first dividing a model in half, then modeling a unit fraction in the remaining area of the whole. Here’s the beginning of a solution that uses this visual representation. • Students can also model fractions with up to 60 equal parts in the Math Clock app. Students might use their partitioning to find equivalent unit fractions and shade unit fraction sections to build 1 whole. Here’s the beginning of a solution with a student using 12ths as the basis for the unit fractions. To extend students’ thinking about their own and other possible combinations of unique unit fractions, ask: What do you notice about the denominators of your unit fractions? Could you use this to help you find other combinations of unique unit fractions that have a sum of 1? They might notice that all (or most) of the denominators share at least one common factor. Or, the denominator of their least unit fraction might be a multiple of all (or most) of the denominators of the other unit fractions.	{"url":"https://www.mathlearningcenter.org/apps/learning-activities/make-1-whole","timestamp":"2024-11-09T01:02:48Z","content_type":"text/html","content_length":"37320","record_id":"<urn:uuid:5d6d324a-f70a-4b27-85fd-74e2ae3c97e5>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00521.warc.gz"}