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How do you write (9y^6) ^(3/2) as a radical form? \| Socratic How do you write #(9y^6) ^(3/2)# as a radical form? 1 Answer ${\left(9 {y}^{6}\right)}^{\frac{3}{2}}$ can be written as ${\sqrt{\left(9 {y}^{6}\right)}}^{3}$ It simplifies to $27 {y}^{9}$ One of the laws of indices states that ${x}^{\frac{p}{q}} = {\sqrt[q]{x}}^{p}$ (The denominator or the index is the root and the numerator is the power) ${\left(9 {y}^{6}\right)}^{\frac{3}{2}}$ can be written as ${\sqrt{\left(9 {y}^{6}\right)}}^{3}$ This can be simplified: $= {\left(3 {y}^{3}\right)}^{3}$ =$27 {y}^{9}$ Impact of this question 1342 views around the world	{"url":"https://socratic.org/questions/how-do-you-write-9y-6-3-2-as-an-radical-form-1","timestamp":"2024-11-15T00:59:18Z","content_type":"text/html","content_length":"32838","record_id":"<urn:uuid:970a7e96-4860-4ab5-821e-c2508c472839>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00139.warc.gz"}
Burst synchronization Burst synchronization From Scholarpedia Jonathan E. Rubin (2007), Scholarpedia, 2(10):1666. doi:10.4249/scholarpedia.1666 revision #126750 [link to/cite this article] General definition Neurons engage in various forms of activity, characterized by the generation of particular patterns of action potentials. One such activity pattern is a burst, which consists of a group of at least two action potentials that occur relatively close together in time (the active phase ), separated from all other action potentials by sufficiently large time intervals (silent phases ). Various classifications of bursting patterns for individual neurons have been established, depending on the minimal models that produce them (e.g. Rinzel, 1987) or on the bifurcation events leading to the onset and offset of the active phase (Izhikevich, 2000). Some of these, along with a variety of biological examples and further issues, are reviewed elsewhere in this encyclopedia (see Bursting). A variety of different definitions of synchronization appear in the neuroscience literature. In a weak sense, neurons can be considered to exhibit synchrony if there is some consistent temporal relationship between some aspects of their respective activity patterns. Following this notion, synchronization of neurons refers to the establishment of some such relationship between them. Synchronization may, for example, be achieved through direct coupling or through common inputs. Some authors impose stronger definitions of synchrony, requiring the maintenance of a specific phase relationship between events (e.g. neuronal action potentials) or even a precise temporal coincidence of events. Commonly mentioned phase relationships include in-phase, which refers to events that happen together, and anti-phase, which is used to describe events that alternate in time. Based on these ideas, burst synchronization naturally refers to the introduction of a temporal relationship between the bursts produced by two or more neurons. This definition includes some ambiguity, because the active phase of each burst consists of multiple spikes. Hence, the phrase burst synchronization is typically used to refer to a temporal relationship between active phase onset or offset times across neurons, while spike synchronization characterizes a temporal relationship between the spikes fired by different bursting neurons within their respective active phases. For instance, a pair of cells can exhibit in-phase burst synchronization with anti-phase spiking, meaning that they enter and exit the active phase together, yet during their shared active phase, they take turns spiking. This form of synchronization can be observed, for example, when either excitatory synaptic coupling or diffusive coupling is introduced between a pair of model respiratory neurons (de Vries and Sherman, 2005; Butera et al., 2005); see Figure Experimental relevance Experimentally, burst synchronization, in one sense or another, has been considered within cell cultures, where the interaction of spontaneous bursts, stimulation-induced bursts, and propagation of activity can be conveniently studied (e.g. Maeda et al., 1995). Synchronization of bursts may also be particularly relevant in responses to novel stimuli (Sherman, 2001), in thalamocortical interactions in the context of sleep rhythms (Steriade et al., 1997), and in pathological conditions. For example, an increase in burstiness and synchronization has been observed in various areas of the basal ganglia in Parkinson's disease, and it has been hypothesized that these developments contribute to the motor complications, particularly resting tremor, that are characteristic of this condition (Bergman et al., 1998; Bevan et al., 2002). Bursting also has been observed in non-neuronal cell types. In particular, significant theoretical work has focused on the analysis of bursting in insulin-secreting pancreatic $\beta$-cells. Elements contributing to burst synchronization The activity pattern that develops when coupling is introduced between two or more neurons depends on their intrinsic dynamics as well as the nature of the coupling. There are a large variety of possible burst mechanisms that can arise intrinsically in single cell models. These mechanisms can be classified by extracting the fastest evolving equations within a model (the fast subsystem) and considering the bifurcation mechanisms in the dynamics of these equations that yield the onset and offset of the active phase (Rinzel, 1987; Izhikevich, 2000a). For example, the most commonly studied form of bursting is known as square-wave bursting. While the name square-wave bursting refers to the approximately constant spike amplitude seen during the active phase (see Figure 2 below), this solution is also known as fold/homoclinic bursting, since the onset of the active phase occurs at a fold bifurcation for the fast subsystem and the offset of the active phase occurs at a homoclinic bifurcation for the fast subsystem. Coupling between cells, in general terms, can be diffusive, via gap junctions or electrical synapses, or synaptic, through chemical synapses. Synaptic coupling can be excitatory or inhibitory, fast or slow, and depressing, facilitating, or neither. In every one of these cases, parameters can be tuned such that some form of burst synchronization results when coupling is introduced between two identical, intrinsically bursting model cells. Moreover, under some conditions, model cells that do not burst in the absence of coupling can be induced by coupling to burst in a synchronized way, and heterogeneity or noise can promote such emergent bursting. In the next part of this article, several examples are presented to illustrate these theoretical results. Burst synchronization in particular settings A general theory or classification scheme on burst synchronization has not been developed. The examples in the following sections represent particular results that have been observed. It is important to note that most of these results depend on model details or even parameter choices within models, and different outcomes may occur when these specifics are varied. Diffusive coupling A neuron in a diffusively coupled network may, for example, be modeled as \[ \begin{array}{rcl} C\dot{V_i} & = & f(V_i,h_i) + \sum_{j \neq i}^N g_{ji} (V_j-V_i), \\ \dot{h_i} & = & g(V_i,h_i) \end {array} \] where `V[i] denotes voltage, the terms being summed represent the coupling from other cells in the network, indexed by j and scaled with constants g[ji], and h[i] is a vector encompassing a collection of ion channel activation and inactivation variables. In a pair of identical intrinsically square-wave bursting cells, the introduction of such coupling, with g[12]=g[21]=g>0, leads to the co-existence of two solutions exhibiting in-phase burst synchronization, one with in-phase spikes and one with anti-phase spikes. In some cases, for small g, the solution with anti-phase spikes is stable and the other is not. As g is increased, however, stability switches to the in-phase branch (Sherman and Rinzel, 1992). In other cases, the anti-phase spiking solution is never stable (de Vries and Sherman, 1998). It is also possible that diffusive coupling could appear in an equation other than the voltage equation, such as an equation for the intracellular concentration of calcium or some other ion, representing a diffusive exchange of that ion between cells. Weak diffusive coupling of calcium, for example, has been shown to enhance burst synchronization in a pancreatic $\beta$-cell model, but stronger coupling was found to yield a death of oscillations through a pitchfork bifurcation. This mechanism arises in models in which calcium drives a negative feedback, and it carries over even if diffusive ionic coupling and diffusive voltage coupling are both present, as long as the latter is not too strong relative to the former (Tsaneva-Atanasova et al., 2006). Relaxation oscillations Models exhibiting bursting often can be decomposed into subsets of variables that evolve on highly disparate timescales. For example, given \[\tag{1} \begin{array}{rcl} \dot{V} & = & f(V,h), \\ \dot {h} & = & \epsilon g(V,h), \end{array} \] if $ \epsilon>0 $ is sufficiently small, then V is the fast variable and h is the slow variable. The fast subsystem corresponding to system (1) is the equation $ \dot{V} = f(V,h) $ with h constant. The slow subsystem for (1) is $ h'=g(V(h),h)\ ,$ where $ f(V(h),h)=0 $ and differentiation is with respect to rescaled time. To simplify analysis of burst synchronization in such systems, it is sometimes useful to eliminate terms responsible for fast spiking, as long as this can be done in a way that preserves oscillations between silent and active phases at burst frequency ( Figure ) . Oscillatory solutions of the resulting reduced model that make such transitions are called relaxation oscillations. Cells in model neuronal networks consisting of relaxation oscillators can be coupled diffusively or synaptically. If the variable $ V $ in system (1) denotes voltage, then synaptic coupling may be modeled through the addition to the $ V $ equation of either • an explicitly time-dependent term depending on the spike times of presynaptic cells, such as $ \sum_j \alpha(t-t_j) $ for spike times t[j], where $ \alpha(t) $ is some function that is zero for t<0, • a term depending directly on presynaptic voltage, or • a term depending on an auxiliary variable that obeys its own differential equation that is driven by the presynaptic voltage. In each of the last two cases, we can write the synaptic term in (1) as $c(V_{pre},V)\ ,$ since in the third case the auxiliary variable depends implicitly on $V_{pre}\ ;$ however, unlike the second case, the third case introduces an additional timescale, associated with the auxiliary variable, into the system. A key mechanism affecting synchronization when synaptic coupling turns on quickly is fast threshold modulation (FTM) (Somers and Kopell, 1993). In relaxation oscillations, the transition of a cell between its silent and active phases occurs when the cell reaches a saddle-node bifurcation (or knee) in a manifold of equilibrium points of its fast subsystem, under the flow of its slow subsystem. Suppose that when synaptic coupling is introduced, system (1) becomes \[ \begin{array}{rcl} \dot{V} & = & f(V,h)+c(V_{pre},V), \\ \dot{h} & = & \epsilon g(V,h), \end{array} \] where $ c(V_{pre},V) $ represents a coupling function of the second or third type described above. If the value of $ c(V_{pre},V) $ changes abruptly, then the manifold of equilibrium points where $ \dot{V}=0 $ may also jump abruptly in phase space. In particular, the knee that forms the threshold for transitions between phases moves rapidly, hence FTM occurs. In FTM, for example, a cell that is below a knee for $ c(V_{pre_1},V)=c_1 $ may lie above the corresponding knee for $ c(V_{pre_2},V)=c_2, $ such that if the coupling term quickly changes from c[1] to c[2], the cell quickly jumps between phases ( Figure ) . FTM can lead to very rapid synchronization of relaxation oscillators, although it does not guarantee that the synchronization will be in-phase. Indeed, it is interesting to note that in-phase and out-of-phase synchronization of relaxation oscillators can both be induced by both excitatory and inhibitory synaptic coupling, depending on model details, including synaptic timescales (see Rubin and Terman, 2002, for a partial review). Synaptic coupling A classical example of phase-locked bursting, specifically anti-phase burst synchronization, is the half-center oscillation (Brown, 1914). In this rhythm, one burster is in the active phase while the other is in the silent phase, and at some point the bursters switch roles. A half-center oscillation may be achieved by coupling two bursters with inhibitory synapses, which ensures that their active phases do not overlap. Alternatively, a half-center oscillation may emerge from coupling two continuously spiking cells with synaptic inhibition, if either 1) each cell undergoes some form of adaptation while it is spiking, such as spike slowing through a gradually augmenting outward current, that allows the suppressed cell to become active, or 2) each cell includes some feature, such as the hyperpolarization-induced deinactivation of an inward current, that allows it to escape from inhibition and become active after a period of suppression. When a pair of identical square-wave bursters are coupled with fast excitatory synapses, burst synchronization has been found to occur. Analogously to the case of diffusive coupling, the stable solution in this setting may feature anti-phase spikes. The effect of increasing the coupling strength is apparently model-dependent; the anti-phase spiking state may remain stable until a transition to tonic spiking occurs (Best et al., 2005) or the in-phase spiking state may take over (de Vries and Sherman, 2005). In several models featuring bursts composed of small numbers of spikes as well as nonzero synaptic delays, stable in-phase spike synchronization within bursts has also been observed. It was proposed that the key to this result is arrival of an input in between spikes, when the trajectory of the postsynaptic cell is close to its voltage nullcline (or nullsurface) and hence a large phase advance can be induced (Takekawa et al., 2007). The precise role of synaptic delays in this result remains to be established. Applying a sufficiently strong excitatory synaptic input with constant conductance to an uncoupled square-wave burster can switch its activity to tonic spiking. Given this, a seemingly paradoxical result is that after such an input is applied to two identical square-wave bursters and induces tonic spiking, the introduction of excitatory synaptic coupling between the neurons can in fact cause the pair to switch back to bursting, in a synchronized manner (Butera et al., 1999). This observation can be explained in terms of a fast-slow decomposition and bifurcation analysis (Best et al., 2005). Such promotion of bursting by excitatory synaptic coupling may be relevant for respiratory rhythms (Butera et al., 2005). In networks of bursters, existence and stability results for particular synchronized solutions can be obtained using consistency conditions based on phase resetting curves (PRC). A basic PRC is a function $ \Delta(\phi) \ ,$ such that if a perturbation to an oscillator occurs when that oscillator is at phase $ \phi $ of its oscillation, then its phase is shifted by $ \Delta(\phi)\ .$ In a network of coupled bursters, however, an input to one burster may shift the duration of its active phase, and hence the duration of the input to all cells postsynaptic to that one, which complicates the analysis. Nonetheless, this analysis can be carried out in small networks as long as the effects of each input to a cell are confined to the two burst cycles following its occurrence (Canavier, 2005). Larger networks and arbitrary coupling dynamics In larger coupled networks of bursters, clustered solutions may exist, in which cells within each cluster are in-phase synchronized, while different clusters take turns entering the active phase. If the cells within each cluster are in fact synchronized with zero phase differences, then the number of cells in each cluster is relevant for solution existence only inasmuch as this feature affects the strength of coupling between clusters. To analyze the stability of a particular clustered solution, it is necessary to consider the robustness to perturbations of both the synchronization of the cells within each cluster and the phase differences between clusters. Clustered bursting oscillations, with in-phase synchrony within each cluster, have been proposed as a binding mechanism. According to this idea, neurons that encode a particular stimulus feature synchronize in the same cluster. For example, if a red vertical bar were observed, then cells responding to the color red and cells responding to vertical bars would engage in in-phase synchronized bursting oscillations together (Terman and Wang, 1995). Certain results about synchronization can be derived based only on the topology of the coupling architecture in networks composed of identical neurons or of multiple classes of identical neurons (e.g. Belykh et al., 2005; Golubitsky et al., 2005). Typically, these findings concern perfectly in-phase synchronization or clustered oscillations with exact in-phase synchrony within clusters and some form of symmetry of phase relations between clusters. For example, a synchronous solution will exist if a network is balanced, in the sense that in the synchronous state, the total input strength to each cell in the network from other cells in the network is the same. Since these results do not depend on the particular dynamics of the identical cells, as long as they oscillate, they apply to bursting oscillations in particular, but not exclusively. In contrast, one result that is bursting-dependent but is still independent of the details of synaptic coupling has been obtained for a pair of elliptic bursters with sufficiently similar spike frequencies. Analysis in this setting, based on a normal form for a Bautin bifurcation, shows that weak coupling can be sufficient to lead to rapid burst synchronization, regardless of the form of the coupling, through either FTM or elimination of a delayed bifurcation effect. In the normal form analysis, a two-timescale canonical model is derived in which the coupling terms are linear combinations of the fast variables. In this setting, certain additional details, such as the phase relationships between spikes within the synchronized bursts, do depend on the form of coupling, which determines the signs of the coupling coefficients in the canonical model(Izhikevich, 2000b). Alternatively, connection architectures lacking symmetry may produce more complex dynamics featuring some degree of burst synchronization. For example, switching a network of Morris-Lecar cells, driven by a pacemaking core, from a local to a small-world to a random connectivity was observed to induce a switch from propagating waves to somewhat synchronized bursting to synchronized spiking in response to each pacemaker oscillation (Shao et al., 2006). Heterogeneous networks Heterogeneous networks can be formed by coupling together cells with different forms of intrinsic dynamics. With diffusive or synaptic coupling, under the right conditions, a wide variety of heterogeneous networks can each engage in synchronized bursting. Even a network composed of an intrinsically quiescent cell and an intrinsically tonically spiking cell can exhibit synchronized bursting if either diffusive or excitatory synaptic coupling is introduced between the cells, although the dynamic mechanisms differ in the two cases. In the synaptic case, the synchrony may fail to be precise, in the sense that the cells may enter and exit the active phase at different times, but synchrony is achieved in that a consistent temporal relationship between the cells' activity patterns develops. When the introduction of coupling induces bursting in a network of cells that do not burst when uncoupled, this phenomenon is referred to as emergent bursting (de Vries and Sherman, 2005). While the presence of noise was found to enhance emergent bursting in a coupled network (de Vries and Sherman, 2000), the mechanism underlying this finding is effectively that noise introduces another source of heterogeneity into the system (Pedersen, 2005). Burst synchronization can also be considered in networks featuring combinations of excitatory and inhibitory synaptic coupling (E-I networks). For example, suppose a synchronized group E[1] of excitatory bursters becomes active and induces activity in a collection I[1] of inhibitory cells to which they send synaptic inputs. If the pattern of inhibitory connections is off-register with the excitatory coupling architecture, then group I[1] will inhibit a set E[2] of excitatory cells that is disjoint from E[1]. Eventually, E[2], along with an associated group I[2] of inhibitory cells, may replace E[1] and I[1] in the active phase, and a clustered bursting solution results. E-I networks can engage in a variety of different bursting activity patterns, featuring different degrees of synchronization, depending on the relative coupling strengths, coupling architecture, and intrinsic dynamics represented in the network. As noted earlier, changes in these patterns may be relevant to changes in sleep states, in the setting of thalamocortical networks, or to changes associated with parkinsonian dopamine depletion, in the indirect pathway of the basal ganglia. In networks composed of segregated columns of intrinsic bursters together with columns of regular or tonically spiking cells, it has been observed that the introduction of a small set of long range connections enhances burst synchronization across the network (French and Gruenstein, 2006). To a large extent, however, bursting in heterogeneous networks of cells with complex coupling architectures remains to be analyzed. Burst synchronization via common inputs Results have begun to emerge on synchronization of uncoupled neurons through common or correlated inputs. This work has not focused on bursting neurons. In other fields, such as the study of laser dynamics, this topic has received some attention, including experimental studies. For example, a common noise input has been shown to induce some degree of burst synchronization in similar, but non-identical, bursting lasers, and this effect was enhanced by sinusoidal modulation of the input signal (DeShazer et al., 2004). A variety of maps have been used as computationally efficient representations of neuronal dynamics. These maps may be designed, or derived, to allow the possibility of bursting solutions. For example, in square-wave bursting in a differential equation model (e.g. Chay and Rinzel, 1985), a slow variable may increase during the silent phase and then show a net decrease during each oscillation in the active phase. In an analogous burst solution in a map, the slow variable would gradually decrease over a sequence of iterations, each corresponding to one spike in the active phase. When the slow variable became sufficiently small, its value would jump up, providing a condensed representation of reinjection into the active phase after passage through the silent phase (see Figure ; see also Medvedev, 2005) . As with differential equation models, coupling may be introduced between map-based bursters, and burst synchronization, featuring various phase relationships, may occur (de Vries, 2005). It has not yet been established, however, what the precise transformations are from forms of coupling studied in differential equation models to coupling terms appearing in maps. Open Questions The consideration of burst synchronization in networks of more than two cells, particularly featuring connection architectures other than nearest-neighbor or all-to-all, remains as a largely open area of research. The many open challenges in the study of burst synchronization include • the analysis of burst synchronization in networks of synaptically coupled non-square-wave bursters, • the rigorous establishment of conditions for the stability of synchronized bursting solutions with anti-phase or in-phase spikes in networks of two or more cells with diffusive or excitatory synaptic coupling, • the exploration of how coupling delays interact with other network features to affect burst synchronization • the systematic analysis of burst synchronization in heterogeneous networks of more than two cells, including the interaction of intrinsic dynamics and coupling architecture, • the rigorous analysis of how burst characteristics, such as period and duty cycle, depend on coupling strength and other parameters in coupled networks with synchronized bursting solutions, • the analysis of the effects of noise on coupled networks of bursters and on emergent bursting, • the exploration of burst synchronization in uncoupled or weakly coupled cells sharing a common external input or correlated external inputs, and • the interaction of multiple mechanisms, each of which is sufficient to induce synchronized bursting on its own, within a coupled neuronal network. • Belykh I., de Lange E., and Hasler M. (2005) Synchronization of bursting neurons: What matters in the network topology. Phys. Rev. Lett. 94: 188101. • Bergman H., Feingold A., Nini A., Raz A., Slovin H., Abeles M., and Vaadia E. (1998) Physiological aspects of information processing in the basal ganglia of normal and parkinsonian primates. Trends in Neurosci. 21:32-38. • Best J., Borisyuk A., Rubin J., Terman D., and Wechselberger M. (2005) The dynamic range of bursting in a model respiratory pacemaker network. SIAM J. Appl. Dyn. Sys. • Bevan M., Magill P., Terman D., Bolam J., and Wilson C. (2002) Move to the rhythm: oscillations in the subthalamic nucleus-external globus pallidus network. Trends Neurosci. 25:525-31. • Brown T.B. (1914) On the nature of the fundamental activity of the nervous centers; together with an analysis of the conditioning of rhythmic activity in progression, and a theory of the evolution of function in the nervous system. J. of Physiology. 48: 18-46. • Butera R., Rinzel J., and Smith J. (1999) Models of respiratory rhythm generation in the pre-Bötzinger complex. II. Populations of coupled pacemaker neurons. J. Neurophysiol. 81:398-415. • Butera R., Rubin J., Terman D., and Smith J. (2005) Oscillatory bursting mechanisms in respiratory pacemaker neurons and networks. In: S. Coombes and P. Bressloff, eds. Bursting: The genesis of rhythm in the nervous system. World Scientific, Singapore, 303-346. • Canavier C. (2005) Analysis of circuits containing bursting neurons using phase resetting curves. In: S. Coombes and P. Bressloff, eds. Bursting: The genesis of rhythm in the nervous system. World Scientific, Singapore, 175-200. • Chay T. and Rinzel J. (1985) Bursting, beating, and chaos in an excitable membrane model. Biophys. J. 47:357-366. • DeShazer D., Tighe B., Kurths J., and Roy R. (2004) Experimental observation of noise-induced synchronization of bursting dynamical systems. IEEE Jour. Sel. Top. Quant. Elect. 10: 906-910. • de Vries G. (2005) Modulatory effects of coupling on bursting maps. In: S. Coombes and P. Bressloff, eds. Bursting: The genesis of rhythm in the nervous system. World Scientific, Singapore, • de Vries G., Sherman A., and Zhu H.-R. (1998) Diffusively coupled bursters: effects of cell heterogeneity. Bull. Math. Biol. 60: 1167-1200. • de Vries G. and Sherman A. (2005) Beyond synchronization: modulatory and emergent effects of coupling in square-wave bursting. In: S. Coombes and P. Bressloff, eds. Bursting: The genesis of rhythm in the nervous system. World Scientific, Singapore, 243-272. • French D.A. and Gruenstein E.I. (2006) An integrate-and-fire model for synchronized bursting in a network of cultured cortical neurons. J. Comp. Neurosci. 21:227-241. • Golubitsky M., Josic K. and Shiau L. (2005) Bursting in coupled cell systems. In: S. Coombes and P. Bressloff, eds. Bursting: The genesis of rhythm in the nervous system. World Scientific, Singapore, 201-222. • Izhikevich E.M. (2000a) Neural excitability, spiking, and bursting. Int. J. Bif. and Chaos, 10:1171-1266. • Izhikevich E.M. (2000b) Synchronization of elliptic bursters. SIAM Review, 43:315-344. • Maeda E., Robinson H., and Kawana A. (1995) The mechanisms of generation and propagation of synchronized bursting in developing networks of cortical neurons. J. Neurosci. 15:6834-6845. • Medvedev G. (2005) Reduction of a model of an excitable cell to a one-dimensional map. Physica D, 202:37-59. • Pedersen M. (2005) A comment on enhanced bursting in pancreatic beta-cells. J. Theor. Biol. 235: 1-3. • Rinzel J. (1987) A formal classification of bursting mechanisms in excitable systems. In: A. Gleason, ed. Proceedings of the International Congress of Mathematicians, American Mathematical Society, Providence, RI, 1578-1594. • Rubin J. and Terman D. (2002) Geometric singular perturbation analysis of neuronal dynamics. In: B. Fiedler, ed. Handbook of Dynamical Systems, vol. 2: Towards Applications. Elsevier, Amsterdam, • Shao J., Tsao T.-H., and Butera R. (2006) Bursting without slow kinetics: A role for small world? Neural Comput. 18: 2029-2035. • Sherman A. and Rinzel J. (1992) Rhythmogenic effects of weak electrotonic coupling in neuronal models. Proc Natl Acad Sci U S A. 89:2471-4. • Sherman S.M. (2001) Tonic and burst firing: dual modes of thalamocortical relay. Trends in Neurosci., 24:122-126. • Somers D. and Kopell N. (1993). Rapid synchrony through fast threshold modulation. Biol. Cybern. 68: 393-407. • Steriade M., Jones E. and McCormick D. (editors, 1997) Thalamus, Volume II, Elsevier, Amsterdam. • Takekawa T., Aoyagi T., and Fukai T. (2007). Synchronous and asynchronous bursting states: Role of intrinsic neural dynamics. J. Comp. Neurosci. 23:189-200. • Terman D. and Wang D. L. (1995). Global competition and local cooperation in a network of neural oscillators. Physica D 81: 148-176. • Tsaneva-Atanasova K., Zimliki C., Bertram R., and Sherman A. (2006) Diffusion of calcium and metabolites in pancreatic islets: Killing oscillations with a pitchfork. Biophys. J. 90:3434-3446. Internal references See also Bursting, Synchronization, Fast Threshold Modulation, LEGION, Neural Synchrony Measures	{"url":"http://www.scholarpedia.org/article/Burst_synchronization","timestamp":"2024-11-05T04:10:48Z","content_type":"text/html","content_length":"67953","record_id":"<urn:uuid:4ca55586-b7ee-4aa3-9487-513a85113fe8>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00232.warc.gz"}
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Wilson on the Evolution of Social Behaviors Jim Gunn on Building Astronomical Instruments Ecologist Jennifer Dunne on Humans’ Place in Food Webs CRISPR Pioneer Jennifer Doudna on Its Research Promise Priyamvada Natarajan: How Black Holes Shape Galaxies Carolina Araujo on Supporting Women in Mathematics Been Kim: A New Approach to Understanding How Machines Think Meenakshi Wadhwa on Meteorites and the Solar System Martin Rees on the Future of Science and Humanity Why Different Parts of a Coffee Mug Produce Different Pitches Valeria Pettorino on Learning About Dark Energy With the Euclid Satellite Mario Jurić on How Astronomy Is Changing Renee Reijo Pera on the Importance of Timing in Embryo Development Tomas Bohr on Performing the Double-Slit Experiment with Bouncing Droplets Rosaly Lopes on Volcanoes Throughout the Solar System Cohl Furey on the Octonions and Particle Physics Jessica Whited on Limb Regeneration and the Axolotl Genome Carina Curto on How Physicists Can Think About Neuroscience Lisa Manning on the Dynamics of Glasses and Embryos Michela Massimi: Defending the Philosophy of Science Donald Richards: A Revealer of Secrets in the Data of Life and the Universe Günter Ziegler Seeks God’s Perfect Math Proofs Barbara Engelhardt on How to Improve Statistical Analyses of Genomes Daniel Goldman and His Smart Robots Gil Kalai: Why Quantum Computers Won’t Work Erich Jarvis on Theories About the Origin of Vocal Learning Ed Boyden on the Promise of Expansion Microscopy Richard Schwartz: In Praise of Simple Problems Corina Tarnita: First Understand Nature’s Rules Minhyong Kim: Connecting Number Theory to Physics Federico Ardila: A Mathematician Who Dances to the Joys and Sorrows of Discovery Michael Assis: The Atomic Theory of Origami Rebecca Goldin: Why Math Is the Best Way to Make Sense of the World Nigel Goldenfeld: Seeing Emergent Physics Behind Evolution Neil Johnson: A Physicist Who Models ISIS and the Alt-Right Svitlana Mayboroda: Taming Rogue Waves Jay Pasachoff: Eclipse Hunter Reveals the Science That Can Only Be Done in the Dark Andrea Ghez: Black-Hole Hunter Takes Aim at Einstein Jessica Flack: How Nature Solves Problems Through Computation Purvesh Khatri: More Data — the Dirtier the Better Tim Maudlin: A Defense of the Reality of Time John Novembre: A Map of Human History, Hidden in DNA Sharon Glotzer: ‘Digital Alchemist’ Seeks Rules of Emergence Sylvia Serfaty: In Mathematics, ‘You Cannot Be Lied To’ Francis Su: Math and the Good Life Francis Su: Math Is for Everybody Marcus Feldman: In Search of Actions That Alter Evolution Elena Aprile: In the Deep, a Drive to Find Dark Matter Janet Conrad: On a Hunt for a Ghost of a Particle Erik Verlinde: The Case Against Dark Matter Cynthia Dwork: How to Force Our Machines to Play Fair Richard Lenski: A Conductor of Evolution’s Subtle Symphony Michael Costanzo: Giant Genetic Map Reveals Life’s Hidden Links Peter and Rosemary Grant: Watching Evolution Happen in Two Lifetimes Tracy Slatyer: A Seeker of Dark Matter’s Hidden Light Miranda Cheng: A Moonshine Master Toys With String Theory Suchitra Sebastian: An Explorer of Quantum Borderlands Ken Ono: A Life Inspired by an Unexpected Genius Janna Levin on Science and Culture Tiny Tests Seek the Universe’s Big Mysteries David Moore: Tabletop Physics The Evolutionary Argument Against Reality David Deamer: How We’re Studying the Origins of Life Michael Atiyah’s Imaginative State of Mind Leslie Valiant: Searching for the Algorithms Underlying Life Richard Dawid: Why Trust a Theory? Christoph Adami: The Information Theory of Life Joan Strassmann: The Woman Who Stared at Wasps Gabriela González: Searching the Sky for the Wobbles of Gravity Nima Arkani-Hamed’s Visions of Future Physics Nancy Moran: An Explorer of Life’s Deepest Partnerships James Bullock: The Case for Complex Dark Matter Hiranya Peiris: How to Test If We Live in a Multiverse A ‘Rebel’ Without a Ph.D.	{"url":"https://www.quantamagazine.org/videos/alan-guth-how-many-two-headed-cows-in-a-multiverse/","timestamp":"2024-11-04T17:24:50Z","content_type":"text/html","content_length":"670354","record_id":"<urn:uuid:2fd13dd1-bbef-4083-9a64-9a77e9f8c1e0>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00807.warc.gz"}
Q@TN-workshop details Schedule on Monday 13 Hours Speaker Talks 9:30-10:30 De Bernardis Tutorial 1 10:30-11:00 – Break 11:00-11:30 Mazzola Talk 1 11:30-12:00 Giacomelli Talk 2 12:00-12:30 Trillo Talk 3 12:30-14:00 – Lunch 14:00-14:30 Balducci Talk 4 14:30-15:00 De Lazzari Talk 5 15:00-15:30 – Break 15:30-16:00 Ludescher Talk 6 16:00-16:30 Aw & Prodhon (40mins) Talk 7 Schedule on Thursday 14 Hours Speaker Talks 9:30-10:30 Scarani Tutorial 2 10:30-11:00 – Break 11:00-11:30 Foletto Talk 8 11:30-12:00 Sighinolfi Talk 9 12:00-12:30 Zaw Talk 10 12:30-14:00 – Lunch 14:00-14:30 Tarabunga Talk 11 14:30-15:00 Roccuzzo Talk 12 15:00-15:30 – Break 15:30-16:00 Sidajaya Talk 13 16:00-16:30 Leone Talk 14 1: Giulia Mazzola, ETH Zurich Title: On the Black Hole War: A quantum information view to reconcile Hawking’s prediction of information loss with unitarity of quantum theory Abstract: A major discovery by Hawking was that the interplay between general relativity and quantum theory leads to the prediction that black holes must radiate. In Hawking’s original calculation however, the black hole radiation was found to be of thermal character thus leaving behind a mixed state describing the radiation as soon as the black hole has fully evaporated. This conclusion stands in apparent contradiction to the reversibility of time evolution in quantum theory which predicts a pure final state of the radiation, thereby giving rise to the famous black hole information puzzle. This discrepancy can be more concretely illustrated in terms of (von Neumann) entropy: Hawking’s prediction leads to a continuously increasing entropy during the black hole evaporation while quantum theory instead dictates that the entropy should decrease in the final stages of the evaporation. Recent calculations based on random unitary models or using the gravitational path-integral integral formalism have shed new light on this puzzle, supporting the latter behavior of the radiation entropy in favor of the unitary evolution picture. Hence, was Hawking’s conclusion wrong? And if so, what was wrong in his calculation? In analyzing this question, it turns out that information-theoretic tools such as the Quantum de Finetti theorem allow us to interpret the different behaviors of the black hole radiation entropy during evaporation and therefore might help in understanding the relations between recent results and Hawking’s conclusions. In this talk, I will mainly focus on the random unitary model for black hole evaporation and, as an outlook, present a recently suggested framework in which Hawking’s original predictions and the unitary picture of quantum theory may be reconciled. Main references: [1] P. Hayden, J. Preskill, Black holes as mirrors: quantum information in random subsystems (2007), arXiv:0708.4025 [2] R. Renner, J. Wang, The black hole information puzzle and the quantum de Finetti theorem (2021), arXiv:2110.14653 2: Luca Giacomelli, INO-CNR BEC Trento Title: Confirmed coming, talk soon 3: David Trillo, IQOQI University of Vienna Title: Time translations on uncontrolled quantum systems Abstract: A time translation is an operation which takes an unperturbed quantum system from an initial state to some state lying on its evolution curve. We show that it is possible to effect many kinds of time translations on a system without the need for any control or information of any kind other than its dimension. In particular, up to some constraints which depend on dimension, one can probabilistically reset, fast-forward or rewind the natural evolution of a system with a heralded protocol, which works universally. For qubits, the probability of success of some of these protocols can be made as close to 1 as wanted. 4: Federico Balducci, SISSA Trieste Title: Signatures of many-body localization in the dynamics of two-level systems in glasses Abstract: In this talk, I will discuss some consequences of the interplay of disorder and quantum effects in structural glasses at low temperatures. To this end, I will briefly introduce the two-level systems (TLS) model of impurities coupled to phonons. By integrating out the phonons within the the framework of the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation, interactions among TLS are generated, as well as dissipation terms. I will show how the unitary dynamics of the TLS bears clear signatures of many-body localization physics, and that also in presence of dissipation there is a long, experimentally accessible, localized transient. This in turn entails that assuming ergodicity when discussing TLS physics might not be justified for all kinds of experiments on low-temperature glasses. I substantiate the analytical picture with numerical results on the time behavior of the concurrence, which measures pairwise entanglement also in nonisolated systems. For a couple of TLS, the concurrence presents a power-law decay both in the absence and in the presence of dissipation, which is again a signature of localization. 5: Claudia De Lazzari, Q@TN, University of Trento Title: Dimension of Tensor Network Varieties Abstract: A Tensor Network variety is an algebraic variety of tensors associated to a graph and two sets of positive integer weights on its edges and on its vertices respectively, called bond and physical dimensions. In quantum many-body physics they are used as a variational ansatz class to describe strongly correlated quantum systems whose entanglement structure is encoded by the underlying graph. In the talk I will present an upper bound on the dimension of the Tensor Network variety. I will discuss a refined upper bound in cases relevant for applications such as Matrix Product States, and highlight some examples where the bound is not sharp. This is based on a joint work with Alessandra Bernardi and Fulvio Gesmundo. 6: Stefan Ludescher, IQOQI, University of Vienna Title: Entanglement/Asymmetry correspondence for internal quantum reference frames Abstract: In recent years, the notion of quantum reference frames was vividly discussed throughout different communities. I will give a brief introduction to the quantum foundational and especially, to the perspective neutral approach to quantum reference frames. I will introduce the notions of perfect and imperfect reference frames. For imperfect quantum reference frames, the question “what is a good quantum reference frame?”, arises. I will propose a solution to the problem, if the underlying symmetry group is a compact Lie group. 7: AW Cenxin Clive & Julien PRODHON, CQT, National University of Singapore Title: Remembering to take a Bath: Retrodiction, Dilation & Memory Abstract: It was recently noticed that classical and quantum fluctuation relations can naturally be derived from Bayesian retrodiction [Buscemi and Scarani, PRE 103, 052111 (2011)], and by the same token expanded beyond their usual domain of application in thermal processes. This approach is a new bridge connecting the foundations of information theory and thermodynamics, both classical and quantum. In this talk we (1) introduce this approach and (2) extend it to scenarios with encoded memory. This uncovers in-roads to speaking of non-Markovian retrodiction with arbitrary priors and illuminates the logical foundation behind the thermodynamic 2nd Law and its generalizations. 8: Giulio Foletto, University of Padova Title: Weak measurements in quantum information protocols Abstract: The quantum weak measurements, first introduced in 1988, have proven useful in a broad range of applications, from the amplification of feeble signals to the study of paradoxes, through quantum state reconstruction. Although they can only extract partial information from a system, they perturb it less than standard projections, sometimes allowing sequences of meaningful measurements on a single state. In turn, this has enabled protocols that extract a valuable quantum resource repeatedly. In this talk, some of these protocols will be discussed, also showing proof-of-concept experimental implementations with single photons. We will see how weak measurements enable sequential quantum random access codes, the sequential certification of entanglement and nonlocality, and the sequential generation of device-independent random numbers. In all these cases, weak measurements enrich or boost the performance of well-known quantum information protocols. 9: Matteo Sighinolfi, Q@TN, University of Trento Title: Stochastic dynamics of impurities in a Fermi bath Abstract: Dynamics of impurities embedded in an ultra-cold Fermi gas is investigated by using a Generalized Langevin equation. The latter — derived by means of influence functional theory — describes the stochastic classical dynamics of the impurities and the quantum nature of the fermionic bath manifests in the emergent interaction between the impurities and in the viscosity tensor. By focusing on the two-impurity case, existence of bound states, in different conditions of coupling and temperature, is predicted and their life-time is analytically estimated. 10: Zaw Lin Htoo, CQT, National University of Singapore Title: Generating entangled states with identical bosons Abstract: A symmetrised state is required to describe a system containing many identical bosons, which results in a state that appears to be highly entangled. However, as these particles cannot be distinguished from one another, this entanglement cannot be accessed directly. In this talk, the idea of extracting entanglement from identical bosons is explored. This idea was first introduced with ideal mode splitting, which maps the entanglement structure of the symmetrised state onto distinguishable modes, converting it into useful entanglement. As ideal mode splitting requires non-destructive particle-number measurements, the more practical scenario of destructive measurement is considered in the context of boson subtraction. Finally, an experimental proposal to generate entangled states with a trapped-ion phonon setup is presented to demonstrate how a boson subtraction scheme might be carried out in the lab. 11: Poetri Tarabunga, SISSA Trieste Title: Quantum Correlations at Finite-Temperature Critical Points Abstract: It is well-known that long-range correlations play an important role in phase transitions. It is then of vital interest to study the nonlocal quantum correlations involved in phase transitions of quantum systems. Entanglement has been the most prominent measure of quantum correlations, but even separable states can also exhibit nonclassical behavior. A more basic form of quantum correlation, the quantum discord, captures more basic aspects of a quantum system, namely that measuring a quantum system necessarily disturbs it. In recent decades, quantum discord has been shown to be able to identify and characterize quantum critical points. However, the fate of quantum correlations at finite-temperature critical points is considerably less understood. Finite-temperature phase transitions are typically governed by classical field theories, and are driven by thermal fluctuations instead of quantum fluctuations. Therefore, it is unclear whether quantum correlations play any role in such transitions. In our recent work, we show that the two-body quantum discord can display genuine signatures of critical behavior at thermal critical points, in contrast to entanglement which does not display any long-range critical behavior. 12: Santo Maria Roccuzzo, Q@TN, University of Trento Title: Supersolidity in a dipolar Bose gas Abstract: Supersolids are exotic materials combining the frictionless flow of a superfluid with the crystal-like periodic density modulation of a solid. The supersolid phase of matter was predicted 50 years ago for solid helium, for which, despite decades of investigation, has not yet been demonstrated. Quantum gases with spin–orbit coupling, cavity-mediated interactions and long-range dipolar interactions are emerging as interesting alternatives. In this talk, I will present recent results on the supersolid properties exhibited by dipolar quantum gases, focusing in particular on the appearance of additional mechanisms for sound propagation, the emergence of non-classical rotational inertia, and the occurrence of quantized vortices. 13: Peter Sidajaya, CQT, National University of Singapore Title: Simulation of non-maximally entangled states using one bit of communication Abstract: From Bell’s theorem we know that local hidden variables could not simulate the behaviour of an entangled state measured using projective measurements. However, it is known that by adding one bit of communication between the parties, it is possible to simulate the behaviour of a maximally entangled state (singlet). We examine the case for a non-maximally entangled state. We use an Artificial Neural Network (ANN) constrained by locality and supplemented by one bit of communication to generate a protocol mimicking the behaviour of a non-maximally entangled state as closely as possible. Our results suggest that it might be possible to simulate the behaviour using only one bit. 14: Nicolo Leone, Q@TN, University of Trento Title: Certifying quantum randomness using Single-Photon Entanglement Abstract: Single-Photon Entanglement(SPE) is a particular type of entanglement in which two degrees of freedom of the same photon are correlated. This type of entanglement does not require non-linear optics to be generated: it is possible to obtain entangled single photon states of momentum and polarization just employing an attenuated light source like a laser or an LED, and linear optical components as beam splitters, mirrors and half-wave plates. Here we report on a new semi-device-independent quantum random number generator (QRNG) based on SPE. Entanglement is indeed an important resource for random number generation since it can be used to guarantee the randomness. Our certification scheme is based on Bell inequality in the CHSH form: every time a violation of the latter is observed for a generated sequence of random numbers, it is possible to lower bound the min-entropy of the produced bits. The min-entropy is an important parameter for RNG, since it quantifies the effectiveness of the best strategy that an adversary can apply to guess the generated sequence. It is important to point out that, since SPE is a local phenomenon, no coincidence measurements are directly necessary to test Bell inequality as only single detection events are required. As a consequence of that, commercial single photon avalanche diodes are used to detect single photon events. The proposed protocol is semi-device independent since it requires a characterization of some non-idealities of the experimental setup, like the memory effects introduced by detectors (dead time, dark counts and afterpulsing) and the polarization dependence of the optical components (i.e. beam splitters and mirrors in the Rotation stage). Moreover two additional hypothesis has to be introduced: firstly, the stationarity of the input state and of the measurement conditions is assumed and, secondly, the provider of the devices is considered trusted. The aim of the proposed protocol is indeed not to fight against a malicious eavesdropper, but essentially to provide robustness against unwanted and undetectable flaws of the system. Under these hypothesis we were able to obtain a kHz-rate certified semi-device independent QRNG, based on commercial linear optical components. Joint work with Stefano Azzini, Sonia Mazzucchi, Valter Moretti, Lorenzo Pavesi. Other details can be found here on these posters	{"url":"https://quantumtrento.eu/workshop-details/","timestamp":"2024-11-05T19:53:50Z","content_type":"text/html","content_length":"95890","record_id":"<urn:uuid:fd26acc6-5516-4592-a74e-d445601a3948>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00221.warc.gz"}
Linear Combination Calculator - calculator Home Calculator Linear Combination Calculator Linear Combination Calculator Linear Combination Calculator The Linear Combination Calculator helps solve systems of linear equations. By using a combination of coefficients from two equations, it allows users to find values for variables such as x and y. This method is useful in various fields, including engineering and economics, where multiple equations must be solved simultaneously. How to Use the Linear Combination Calculator To use the calculator, input values for a₁, b₁, c₁, a₂, b₂, and c₂ from your equations. Click 'Calculate' to find the values of x and y. The calculator will display the results and provide a step-by-step solution using the linear combination method. Linear Combination Calculator Frequently Asked Questions What is a linear combination? A linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results. In the context of equations, it allows us to manipulate and solve systems efficiently. When do we use linear combinations? Linear combinations are used in solving systems of linear equations, vector spaces, and in various applications across physics, computer science, and economics where relationships between variables need to be explored. What are the benefits of using a calculator for linear combinations? Using a calculator simplifies the solving process, reduces manual errors, and allows for quick computation of results. This is particularly useful for complex systems or when dealing with larger Can this calculator handle more than two equations? This specific calculator is designed for two-variable systems. For more equations, consider using methods like matrix algebra or dedicated software tools that can handle higher dimensions. Is this method always applicable? The linear combination method is applicable when you have a consistent system of equations. If the equations are inconsistent or dependent, other methods may be required. What is the least common multiple (LCM)? The least common multiple of two integers is the smallest number that is a multiple of both. It's crucial in the linear combination method to eliminate variables effectively. What if I input invalid numbers? The calculator should handle invalid inputs gracefully, providing error messages or defaulting to zero. Always check your inputs before calculation. How accurate are the results? The results are accurate as long as the input values are correct. Ensure to use precise numbers to avoid rounding errors in your calculations. Can I use negative numbers? Yes, negative numbers are valid inputs in the calculator. They can represent various scenarios, including debt or decrease in quantity, which are common in real-world problems. Where can I find more resources on linear combinations? Many educational websites, textbooks, and online courses cover linear combinations. Look for resources focusing on linear algebra or system solving techniques for deeper insights.	{"url":"https://calculatordna.com/linear-combination-calculator/","timestamp":"2024-11-10T06:27:18Z","content_type":"text/html","content_length":"84694","record_id":"<urn:uuid:151ab73f-c50a-47c6-9850-1dc83a1ba8be>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00858.warc.gz"}
Favorite Restaurants? I'm always up on finding good food joints, so post your favorite restaurant. :dribble: Mine's called Bova's, best Italian place I've ever found. Great sauce, giant (and I DO meant GIANT) calzones, and homemade spaghetti are not to be missed in particular. Found HERE mine is also an italian, called al dente... i have to admit, the food is not the BEST i've had from an italian, but ive eaten at loads of italien's in central london which have amazing food, but this one is on the borders of greater london so obviously not as good. but the waiters there make up for the food quality, and obviously the price is better. italian restaurants are always my favourite. or steak houses. yum. website and location i like italian and japan/china kitchen. My favorite place is IL Patio. Because that place have really big portions and normal worth of it I'm a freak for Chinese food, and there's a place about 40 minutes from where I live called Sampan. I go there when I can, and leave full every time. They gots feckin' awesome tofu and sesame Olive Garden FUCK YEA. i like chirese food like sweet n sour chikn. chens wok or lucky cafe is wer id usually eat that. but my fav restruant is Zaxby's. My favorite restaurant is called Miyabi's. It's a Japanese hibachi joint and I simply love it for two reasons: the show and the taste. Those chefs are like magicians, but the magic happens on a stove instead of a stage. They also have a very fine hostess who happens to be half Asian (Japanese) and half Caucasian (American) btw and her name is Christen. That is all. Mine's called Bravo, it's an Italian place with the most amazing raviolis ever. Mines a sub shop just outside of Boston, MA. Best cheesesteaks i've ever had.. in all my life. In matter of fact.. all Mom and Pop shops make the best food. Atleast back east. Tucson sucks when it comes to food. As much as I'm a freak for chinese food, and I am, my favorite restaurant has to be a tie between Red Robin and Tully's. I'm pretty sure Tully's is local, but it's a sports bar basically. Famous Tully's Tenders. They soooooo good. Red Robin is more like a sports bar best burgers around joint. They soooooo gooooood. They also have AWESOME steak fries with their special seasoning :dribble: best stake ive had in aus. Mine's called Bravo, it's an Italian place with the most amazing raviolis ever. I often eat at reptards house, when he's not home. Just me and his mother. i got 99 bitches and a problem aint one • 2 weeks later... I like Seafood from this place called Red Lobster, Some ppl over seas may of never heard of it but its great food. But sadly to say once I took my wife there and she puked lobster all over the new car on the way home one day. :sweat: I like Seafood from this place called Red Lobster, Some ppl over seas may of never heard of it but its great food. But sadly to say once I took my wife there and she puked lobster all over the new car on the way home one day. :sweat: Let's not get into all the gory details!!! Besides it was the bad peanut butter! I sacrificed myself and gave the family the good stuff and gave myself the stuff that wasn't supposed to be put on the shelves. So nice of me! Oh you Voltrons -- so silleh. burger king. wat? its a restaurant burger king. wat? its a restaurant -8 Reputation points for Hardy My favorite place is called (hang on..): Mario's Pizza Best deep-pans in Denmark... Else im going to a place called: 7-Møllen They haz nice burgers. -8 Reputation points for Hardy +9001 rep points for Hardy This topic is now archived and is closed to further replies.	{"url":"https://jediphoenix.ipbhost.com/topic/560-favorite-restaurants/","timestamp":"2024-11-04T05:45:54Z","content_type":"text/html","content_length":"253461","record_id":"<urn:uuid:2c348e27-8aa8-442a-a474-a55dc48f127c>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00316.warc.gz"}
How Many Ounces Of Water Are In One Bottle? A Guide To Understanding The Measurement - JNV Exam Test Answers Hub How Many Ounces Of Water Are In One Bottle? A Guide To Understanding The Measurement Being mindful of the amount of water we consume is important for staying healthy, but figuring out exactly how many ounces of water are in a bottle can be challenging. We’re here to help you understand the basic measurements and how to calculate it so you can stay hydrated all day long. What Is An Ounce? An ounce is a unit of measurement for both liquid and dry substances. It is equal to about 28 grams and is used in the US and UK to measure weight and volume. A fluid ounce is the unit of measurement for liquids, and it is equal to about 29.6 milliliters. How Many Ounces Are In A Bottle? The amount of water in a bottle can vary, depending on the size of the container. In general, most bottles of water contain 16 fluid ounces of water, which is equal to 473 milliliters. That means that one bottle of water contains approximately half a liter of water. Calculating Ounces In A Bottle To calculate the amount of ounces in a bottle, you will need to know the size of the container. For example, if you have a bottle the size of a standard water bottle (24 ounces), then you will have 24 fluid ounces of water in the bottle. To calculate the amount of ounces in liters, you will need to use a conversion equation. The Conversion Equation The conversion equation for determining the amount of ounces in a bottle is: • 1 liter = 33.814 ounces • 1 ounce = 0.0296 liters To calculate, simply divide the amount of liters by 33.814, or multiply the amount of ounces by 0.0296. Knowing how many ounces of water are in a bottle is important for hydration, and now you know how many ounces are in a standard bottle and how to calculate the amount of ounces in different containers. We hope this guide was helpful in understanding this essential measurement. Leave a Comment	{"url":"https://jnvetah.org/how-many-ounces-of-water-are-in-one-bottle-a-guide-to-understanding-the-measurement/","timestamp":"2024-11-05T09:33:54Z","content_type":"text/html","content_length":"54837","record_id":"<urn:uuid:770070b1-d0c8-4014-a9ad-eaa682fd9a99>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00748.warc.gz"}
Scale invariance in particle physics and cosmology Models with classical scale invariance (CSI) provides us with a dynamical origin for all masses (via dimensional transmutation) and can account for all evidence of beyond the standard model physics. Furthermore, a general theory with CSI is renormalizable (even in the gravity sector) and can solve the hierarchy problem. The price to pay is a classical ghost. The theory, however, admits quantizations that preserve unitarity and a Hamiltonian bounded from below. The solution of the hierarchy problem implies that the theory can be tested through inflationary data (indeed it predicts a (gravitational) isocurvature mode that could be observed in the next future). I will give an overview of CSI and introduce the subsequent talks on this subject.	{"url":"https://indico.cern.ch/event/740038/contributions/3283558/","timestamp":"2024-11-15T00:27:28Z","content_type":"text/html","content_length":"111401","record_id":"<urn:uuid:d706fd0d-666f-49c1-98ac-a3f1e2b553ce>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00527.warc.gz"}
Mathematics and Computing: Exploring the Intersection of Numbers and Technology The synergy between mathematics and computing is unlocking unprecedented possibilities in the ever-evolving landscape of technology and science. By harnessing mathematical theories with computational power, they advance fields such as Artificial Intelligence, Quantum Computing, and Big Data Analytics to be used by varied industries for good. This article explains the dynamic interplay of these two faculties, which drives innovation and enables to tackle complex challenges and optimize systems across various domains. The Foundation of Mathematics Let’s explore the foundational elements of mathematics which is crucial for advancing in various fields. Numbers and Operations This is a fundamental area to understand different types of numbers like integers, variational numbers and mastering options like addition, subtraction, multiplication and division. These basics are essential for high level math and everyday calculations. Algebra and Equations Algebra involves solving equations and understanding functions, which includes manipulating variables and constants to find unknown values. This forms the basis for advanced studies in Mathematic Science and Engineering. Geometry and Shapes Geometry focuses on properties in relationships of shapes, sizes and relative positions of figures in space. It includes studying points, surfaces, lines and solids, which are crucial for fields like architecture, engineering and computer graphics. The Evolution of Computing Let’s explore the journey of computing which has been transforming evolving from simple tools sophisticated machines. Early Computational Devices Abacus and slide rule are the early tools in performing basic arithmetic operations. It’s only after 17th century mechanical calculators such as Blaise Pascal’s Pascaline marked a significance in advancing the automated calculations. Digital Revolution The 20th century saw the birth of electronic computers. It started with ENIAC and leading to development of transistors and degraded circuits. Personal computers, internet, smartphones and others have revolutionized the communication and information processing at the dawn of the digital revolution. Today, the new frontier is quantum computing, promising exponential increases in computational power for solving complex problems. Convergence of Mathematics and Computing The following are the distinct fields where the convergence of mathematics and computing is evident. 1. Science and Engineering Use mathematical models and algorithms to simulate physical systems. Ex: Weather forecasting fluid dynamic simulations, and structural analysis Application of numerical techniques to solve complex equations. Ex: Computational fluid dynamics finite element analysis 2. Finance and Economics Development of Mathematical Models for Pricing Derivatives and Managing Risks Ex: Option Pricing Models, Value at Risk Models Usage of Algorithms to Execute Trades at Optimal Prices. Integration of Machine Learning for Predictive Analysis and Market Forecasting 3. Biology and Medicine Mathematical algorithms for analyzing biological data such as DNA sequencing. Ex: Sequence alignment algorithms. Protein structure prediction. Use of mathematical algorithms in image processing for diagnostics. Ex: CT scan image and MRI techniques. 4. Computer Science and Artificial Intelligence Development of algorithms inspired by mathematical theories. Ex: Vector machines, clustering algorithms, and neural networks. • Data analysis and big data Use of computational and statistical techniques to analyze large data sets. Ex: Data mining, pattern recognition, and predictive modeling. 5. Cryptography and Security Development of secure communication algorithms based on number theory and algebra. Ex: RSA, ECC, Use of algorithms and magical proofs to ensure data security integrity. Computational Mathematics: Bridging Theory and Practice Computational Mathematics is a field that combines mathematical theory, computational techniques, and numerical analysis to solve complex problems. This concept encompasses a wide range of applications, from engineering to scientific research, finance, and data science. The following are the key components of Computational Mathematics. 1. Mathematical Theory – This theory involves developing mathematical models and theorems in fundamental areas such as Linear Algebra, Differential Equations, Calculus, and Probability Theory. 2. Numerical Analysis – This is a study of algorithms for numerical approximation, ensuring the accuracy, efficiency, and stability of computational methods. 3. Computational Techniques – These techniques are used for algorithms and numerical methods to approximate solutions, which include iterative methods, Monte Carlo simulations, and finite element Bridging the Gap Following are the ways how comparison mathematics bridges theory and practice: Modeling real-world problems Translating physical, biological, and financial problems into mathematical models. Ex: Modeling heat transfer using partial data equations predicting the fall and rise of the trades in the form of market analysis models. Algorithm development Use of sufficient algorithms to solve mathematical models. Ex: Partial Products Algorithm. Simulation and experimentation Use of computational tools to simulate scenarios and validate models. Ex: Simulating climatic models to predict weather patterns. Challenges and Solutions The following are a few of the challenges with innovative solutions and how combination of mathematics and computing can more effectively harness to solve real-world problems and drive technological Challenges Solutions 1. Complexity and computationally intensive Use optimization techniques and methods to simplify models without losing accuracy. Employ approximation methods and techniques like perturbation theory mathematical models and asymptotic analysis. Leverage high-performing community to handle complex computations efficiently. 2. Scalability Issues in handling large Design algorithms efficiently to lower computational complexity, enhancing scalability. Utilize big data technologies and frameworks like Hadoop and Spark datasets and complex simulations for handling large-scale data processing. Explore cloud computing to scale computations dynamically. 3. Accuracy and Precision in computational Conduct thorough error analysis to minimize computational errors. Develop adaptive algorithms to adjust the parameters based on error estimates. Regularly results validate and verify models against analytical solutions and empirical data. 4. Data Quality and Availability Develop robust data cleaning techniques to preprocess and enhance data quality. Create synthetic data when real data is insufficient or unavailable. Promote collaborative data-sharing techniques across organizations to improve data availability. 5. Real-time processing for applications Utilize frameworks like Apache Kafka and Apache Flink for real-time data processing. Develop efficient algorithms to optimize real-time performance. like finance and healthcare Deploy edge computing closer to the source and reduce latency in processing data. 6. Interpreting Complex Results for Use data visualization tools to present results intuitively. Develop explainable models to provide insights for the decision-making process. Integrate decision-making user training to interpret and understand complex computational results. Educational Perspectives Following are the educational perspectives where students can learn and leverage both mathematical and computational skills, fostering innovation and addressing complex problems. Interdisciplinary Curriculum • Enroll in courses that cover both mathematical concepts and computational techniques. Example computational mathematics or applied mathematical computing. Integrate Core Disciplines • Integrate courses on Discrete Mathematics, Linear Algebra, and Calculus Theory for Computer Science Applications and courses on Algorithm Design, Numerical Methods, and Software Tools for Hands-On Learning • Enroll in a lab work, where all students can use computational tools to solve mathematical problems. Get hands-on experience with software like MATLAB, Mathematica, Python libraries, and R. Collaborative Learning • Encourage teamwork and form teams comprising students from different disciplines to tackle complex problems. Join workshops and seminars to foster collaboration and exchange of ideas. Professional Development • Collaborate with industries for internships and real-world problem-solving and get additional training and certifications in relevant software and technologies to equip with skills for both academic and industrial. Research & Innovation • Participation in research projects involving mathematical modeling, algorithm development, and data analysis can bridge the gap between mathematics and computing. Continuous Improvement • Join professional development programs to stay updated with advancements and access online courses focusing on emerging topics and technologies in both fields. Future Trends and Technologies 1. Quantum Computing Uses advanced mathematical concepts such as linear algebra, quantum mechanics, and complex power order theory. Promises cryptography optimization and simulations of complex systems. 2. Advanced Data Analysis and Big Data Statistical methods, optimization techniques, and machine learning algorithms for data analysis. Improves decision-making in industries like healthcare, finance, and marketing through predictive analytics, real-time insights, and pattern recognition. 3. AI and ML Involves linear algebra for neural network computations, probability theory for making predictions, and calculus for optimization. Enhances capabilities in natural language processing, autonomous systems, computer vision, and personalized recommendations. 4. Smart Cities and IoT Mathematical modeling, optimization, algorithms, and statistical analysis to manage resources, infrastructure, and traffic. Enhances urban planning, transportation systems, and energy management to more efficient and sustainable cities. 5. Bioinformatics and Computational Biology Statistical models, graph theory, and algorithms for sequence, alignment, structure prediction, and systems biology. Advances personalized in medicine, drug discovery, and genomics research by providing insights into genetic variations in the biological process. The synergy between mathematics and computing is fueling rapid advancements in technology and science. Integrating mathematical theories with computational power drives innovation in fields like Artificial Intelligence, Quantum Computing, and Big Data Analytics. This collaboration enhances problem-solving and optimizes systems, addressing challenges such as complexity and data quality. As we progress from early computational tools to cutting-edge technologies, this dynamic interplay continues to transform industries and shape the future of technology. FAQs (Frequently Asked Questions) What are the practical applications of mathematics in computing? Mathematics is used for algorithmic design, data analysis, machine learning, computer graphics, and cryptography. How does computational mathematics differ from theoretical mathematics? Computational mathematics focuses on problem-solving with algorithms, while theoretical mathematics emphasizes abstract concepts and proofs. What role does mathematics play in artificial intelligence and machine learning? Mathematics underpins algorithms for optimization, model training, neural network architecture, and data analysis in artificial intelligence. How can studying mathematics benefit a career in computer science? Problem-solving skills, algorithmic design, understanding of complex computational systems and computer science, and data analysis are the benefits of studying mathematics for computer science. What are some emerging technologies that combine mathematics and computing? Quantum computing, big data analytics, bioinformatics, and the Internet of Things integrated with advanced mathematics and computational methods in artificial intelligence are emerging technologies combining both mathematics and computing.	{"url":"https://mvjce.edu.in/blog/unlocking-the-future-the-power-of-mathematics-and-computing/","timestamp":"2024-11-04T10:33:05Z","content_type":"text/html","content_length":"286393","record_id":"<urn:uuid:5ef2e9b3-b9d4-4b2d-85e7-5e5c9686f105>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00637.warc.gz"}
Frequency dependent characterization of electrical components LCR Meter – SinePhase Impedance Analyzer with SMD Adapter holding capacitor In an electrical circuit, a potential difference between two points induces the flow of current. Impedance is defined as the sum of resistance and, in the case of alternating current supplies, reactance; the effect is to reduce current flow. If the circuit comprises only resistors, then the impedance is constant at all frequencies and has what is known as ohms law, defined by: V = IR (where V = voltage, I = current and R = resistance). Therefore, R = V / I. In the case of a DC (direct current) circuit that contains a capacitor, current will flow through the capacitor until it becomes fully charged. This rate of charge reduces exponentially as the capacitor charge nears the target voltage. Conversely, if the circuit is powered by AC (alternating current) such as a mains supply, the current response will oscillate in phase with the voltage and can be seen as a sine wave on an oscilloscope. This current response is at its highest rate early in the capacitor charging cycle, but the constant reversal of current flow that is characteristic of AC supplies actually causes the circuit to behave almost as if the capacitor were not present. This is especially so at high frequencies. Therefore, the extent and effect of capacitive impedance depend on the AC supply frequency measured in Hertz (cycles per second), the capacitance (measured in micro or millifarads) and any other components connected either in series or in parallel. The type of impedance produced is said to be reactive (as opposed to purely resistive) and is a linear function of the frequency, both for pure capacitance and inductance. Notably, in a purely capacitive circuit, the current response is ninety degrees ahead of the AC voltage. However, in practice, some resistance is always present; capacitors are often also combined with resistors such as in purpose-designed signal noise filters. In addition, inductors are components that have a reactive response and cause impedance. In a sense, they are the opposite of capacitors in that the current flow through an inductive load lags the voltage supply, with a phasing difference of ninety degrees. If the supply frequency is changed, the current response will also increase or decrease. If we extend the example capacitive circuit further with the addition of an inductor, the different components will all cause phase differences between the voltage applied and the resultant current flow. The net result will be either a capacitive or an inductive circuit load or dynamic impedance. Additionally, if the frequency of the supply is varied, the two opposing effects of leading and lagging responses combine to cancel each other out at a certain point in the frequency range, due to the rising and falling response of the different components at varying frequencies. This distinct point is known as the resonant frequency and is where the maximum response is obtained. In these circuits, the terms R, L and C refer resistance, inductance and capacitance (respectively). At maximum dynamic impedance, current flow is minimum and the maximum voltage is produced across the load. This type of electronic engineering can be employed to design and configure high pass and low pass Impedance analysers measure precisely these frequency-dependent impedance characteristics, between two test points in an electrical system. They measure the phase angle, i.e. how much the current lags or leads the voltage curve. Using trigonometry, an inverse tangent function is employed in algorithms (in a similar manner to vector mathematics calculations) to determine the dynamic impedance, referred to in electrical engineering by the symbol X. Whether the electrical impedance in the circuit or medium being tested is resistive, capacitive or inductive, its measurement and the recording of any changes in response over time have a number of applications. Impedance measurement is used in geology and soil science for electrical resistivity tomography (ERT) or imaging (ERI), as well as construction engineering and spectroscopy in laboratories. Similarly, in the world of medicine, electrical impedance myography (EIM) is a non-invasive procedure used to assess muscular health, atrophy or hypertrophy along with the diagnosis of neuromuscular disease. Here, the test analyses impedance to low currents flowing through cells and fluids, which have a reactive (capacitive) response. The results across a spectrum of frequencies give a good indication of muscle integrity. Other medical uses include cardiography and phlebography (vein analysis and the diagnosis of thrombosis). Modern impedance analysers can also be used for the rapid, efficient testing of materials such as polymers, plastics, ceramics and glass in addition to semiconductors and electrolytes. On production lines, pass or fail test parameters can be set, including for components and circuits containing piezoelectric devices which operate at a range of frequencies. A specially designed impedance meter may be required for certain test applications. The speed and accuracy of the response will depend on the type of meter and its cost. Typically, handheld impedance analysers might measure frequencies ranging from 100 Hertz up to 100 Kilohertz, whereas bench top devices are able to sweep through larger frequency curves at various voltage levels with ease. They can also measure the Q (quality) factor, which is the ratio of resistance to reactivity. These more advanced meters also give greater accuracy when testing DC circuits. In summary, modern impedance meters are capable of testing multiple parameters and offer the advantage of replacing several separate instruments that were previously necessary to provide a similar range of test functionality. In this section you find following videos (just scroll down) regarding RLC. 1. Impedance, AC Circuites and Phasors 2. Intro to Frequency-Dependent Impedance \| Capacitors in Alternating Currents 3. Physics – RCL Circuits With Reactance and Impedance (1 of 2) 4. Physics – RCL Circuits With Reactance and Impedance (2 of 2) Resonance Frequency 1. Impedance, AC Circuites and Phasors Introduction to Phasors, Impedance and AC Circuits. Explain the relationship between frequency and the overall impedance. Videl length: 3 minutes 53 seconds \| Resolution: 720p 2. Intro to Frequency-Dependent Impedance \| Capacitors in Alternating Currents Capacitors, which are cool in their own special way, become even more interesting if you shake them around electrically. Never shake a baby, though. More phasors in here. Video length: 9 minutes 43 seconds \| Resolution: 720p (HD) 3. Physics - RCL Circuits With Reactance and Impedance (1 of 2) Problem: Find the a) Inductor reach; b) Capacitor reach; c) Total reactance; d) Impedance; e) Phase angle; f) Current; g) Power consumed by the resistor; and others of the RCL circuit with reactance and Impedance. Video length: 14 minutes 10 seconds \| Resolution: 1080p (HD) 4. Physics - RCL Circuits With Reactance and Impedance (2 of 2) Resonance Frequency Problem: a) Find the resonance frequency (fo); b) What is the impedance @ fo? c) What is the phase angle @ fo? d) Sketch Z vs f; of the RCL circuit with reactance and Impedance. Video length: 9 minutes 25 seconds \| Resolution: 1080p (HD) Do you have any questions regarding our Impedance Analyzers \| LCR Meters? Please feel free to contact us. You can call us (telephone number on top of website) or just use this simple contact form below. SinePhase Impedance Analyzer and LCR Meter are designed for measurements up to 16777 kHz. • Model 16777k for measurements between 1 kHz and 16777 kHz. • Model 2097k for measurements between 1 kHz and 2097 kHz. • Model 262k for measurements between 1 kHz and 262 kHz. In addition, Impedance Analyzer Models 2097k and 262k are offered in two versions covering individual impedance precision ranges. They can be operated directly from any standard Laptop or PC without the need for additional battery power pack or main supply. USB Power & Control The USB power and control concept does not only turn the instrument into the smallest and most mobile of its type, but also features fully integrated PC control and data acquisition as the standard mode of operation. Impedance Analyzer \| LCR Meter For measurements between 1 kHz and 16777 kHz. Easy-to-use self-explanatory All instruments are supplied with easy-to-use self-explanatory measurement software. In addition to the wide range of functions provided with the standard measurement software, SinePhase also offers optional tools like the Fitting Tool or the Correlation Tool. Based on feedback by our customers, SinePhase is commited to continuously extend its library of readily available optional tools. Impedance Analyzer \| LCR Meter For measurements between 1 kHz and 2097 kHz. Individual Analyzer Systems SinePhase impedance analysis technology is not limited to our standard product family. Our experienced specialists will work with you to specify, design and manufacture customized hardware solutions that are based on our basic analyzer hardware platform. Combined with customized software there are no limits to realize individual analyzer systems that fit your particular application environment. Impedance Analyzer \| LCR Meter For measurements between 1 kHz and 262 kHz. OEM & Embedded Solutions For industrial customers SinePhase impedance analyzer technology is also available on an OEM basis, by embedding SinePhase core technology into your product design. Please contact us for further information.	{"url":"https://sinephase.com/tag/reactive/","timestamp":"2024-11-13T23:59:24Z","content_type":"text/html","content_length":"102164","record_id":"<urn:uuid:509ff593-fde3-4296-b06e-0947576daae7>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00795.warc.gz"}
Unlike a permutation, in a combination the order of the objects selected does not matter. For example, if you were choosing a team of three students from a class of ten, the order you said "Joe, Sally, John" would not matter to who is in the group- saying "Sally, Joe, John" instead doesn't change anything. How to solve a combination The number of ways you can make a combination of r objects out of a set of n objects is made from the formula below: An Explanation You may be asking (like I did when I learned this material), "What? That formula doesn't make sense!".- However, it is based on the formula for a If you think about it, the n! makes perfect sense- thats how many options you'd have if you picked in order from all of them. Removing the (n-r)! gets it down to just the numbers you want (n-r equals the last number you arn't picking. If you pick 3 from 5, it'll be 5!/(5-3)!, so 54321/21, which, after cancelling it out to 543, accomplishes the "pick x of 5" part.) The extra r! removes all the redundant options. For example, say you have 6 different balls, labeled A through F. If you pick three, you can pick ABC, ABD, ABE, ABF, etc. You'll end up with six of the same- ABC, ACB, BAC, BCA, CAB, CBA- all of which only count as one in a combination. As you'll notice, since we picked 3, 3!=32*1=6- the same number we have to divide out! An Example of a Combination Lets take the example above. Out of a class of ten students, how many ways could you make a team of three students? We know n is ten because the set (the students) has ten objects in it. We know r is 3 because the team will have three students on it. To solve, first we set up the equation: Then we fill in the values: By taking apart the factorials, we can simplify the 10! and 7!. Alternativley, just start at 10 and multiply down, stopping right before 7. C= 120 There you have it, there are 120 ways to pick a team of 3 people out of ten.	{"url":"https://www.abcofmath.com/2013/10/combinations.html","timestamp":"2024-11-03T16:04:46Z","content_type":"application/xhtml+xml","content_length":"86528","record_id":"<urn:uuid:d3d8f881-35c0-4479-960d-68b4315024d7>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00014.warc.gz"}
Statistical Applets and Calculators Statistical Applets for Visualization of Statistical Concepts List of simulations/demonstrations in Java Some Java applets written by Jeffrey Rosenthal Java applet in Galton-Watson branching processes by Jonathan Jordan A site with many Java applets Java applet in M/M/1 queues by Jonathan Jordan A list of sites with statistical applets A normal approximation to the binomial distribution Java applet Linear regression Java applet Confidence intervals Java applet The let's make a deal Java Applet Histogram applet Central limit theorem Java applet Interactive Demonstration of the Power of a Hypothesis Test Java applet Simulation of Simple Symmetric Random Walk in Excel by Michael Zazanis (in Greek) Simple Interactive Statistical Analysis This calculator will find critical values or areas for the standard normal, t, chi-square, and F distributions Hypothesis Testing Assesment by James Jones Simulation by James Jones Mann-Whitney test calculator On-line notes on Statistics and calculators Stattucino online spreadsheet for Statistical Calculations Statistical Quality Control Online Statistical Analyses Graphs, Charts and Plots Distribution Calculators	{"url":"https://statlink.tripod.com/id17.html","timestamp":"2024-11-08T08:38:26Z","content_type":"text/html","content_length":"50198","record_id":"<urn:uuid:c330c7a5-12d7-432d-9066-047c8bd183b5>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00134.warc.gz"}
How to avoid for loops when generating index arrays? I often find myself coding nested for loops to generate vectors of integer indices. For example: n = 4; i = 1; for L = 0:n for M = -L:L l(i) = L; m(i) = M; i = i+1; All I need are the vectors "l" and "m". I can preallocate to save some speed, but my real problem is having to use the for loops as sometimes the index vectors I need to create have many more nested for loops whose (note that the inner loop index depends on the outer loop index). Is there a simple way to avoid using loops to generate index vectors like these? Accepted Answer Here's another method, less memory consuming than NDGRID idx=nonzeros(bsxfun(@times,map,1:length(ll) )); idx=nonzeros(bsxfun(@times,map,(1:length(mm)).')) ; More Answers (5) For your particular problem you can do this: M = (0:n(n+2))'; L = floor(sqrt(M)); M = M-L.(L+1); (I've used uppercase letters, 'L' and 'M', in place of your lowercase 'l' and 'm'.) As with Matt Kindig, I am not sure this will be any faster than your for-loops. Time it with a large value for n and see. Here's a way to do it using NDGRID. It's not apriori obvious whether for loops would or would not be faster. It depends what you plan to reuse. 6 Comments I'm starting to think Sean's advice about sticking with for-loops is the best one. There can definitely be ways to cut down on the loop nesting (see my newest Answer based on cell arrays), but the required form would depend on the body of the original set of for-loops. doc meshgrid doc ndgrid %? And of course, depending on your application, two nested for-loops or bsxfun() might be better. 2 Comments Just use the for-loops, they'll be the fastest by far. If you want to disguise it, write a function that takes L and M and returns l and m. cellfun and arrayfun are slow and converting between cells and numeric types is slow. The above with preallocation will be pretty quick. It's kind of hack-y, but it gives the same output as your original posting: l = cell2mat(arrayfun(@(x) xones(1,2x+1), 0:n, 'uni', false)); m= cell2mat( arrayfun(@(x) (-x:1:x), 0:n, 'uni', false)); Keep in mind that this may very well be slower than for-loops--I haven't done any timing comparisons. Here's a way to eliminate one nested loop for L=0:n 2 Comments I'd be surprised if this is faster due to the cell array conversions. I guess one of us will have to run a timing test. For n=1000 I get this, Original Approach: Elapsed time is 0.093130 seconds. Cell-Based Approach Elapsed time is 0.027393 seconds. I think the vectorization inherent in trumps the overhead from the cell conversion. See Also Community Treasure Hunt Find the treasures in MATLAB Central and discover how the community can help you! Start Hunting!	{"url":"https://www.mathworks.com/matlabcentral/answers/74137-how-to-avoid-for-loops-when-generating-index-arrays?s_tid=prof_contriblnk","timestamp":"2024-11-06T20:16:26Z","content_type":"text/html","content_length":"237553","record_id":"<urn:uuid:8e6c5183-b9b6-4bf9-b6bf-dd61910936a8>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00786.warc.gz"}
Symmetric Key algorithms At the base of the symmetric key algorithm figures, of those that use for encryption a simple, secret key, are the elementary figures the transposition and the substitution.[4] The transposition figures realize a permutation of the characters of the cleartext. The encryption key is the pair K = (d, f), where d represents the length of the successive blocks of characters that will be encrypted according to the permutation f. The decryption is obtained by performing the inverse permutation. Substitution cipher schemes replace each character in the message alphabet A with a character in the C cryptogram alphabet. Traditional coding techniques are based on the sender's and recipient's knowledge of the encryption key. The sender encodes the message with a particular encoding system using the secret encryption key, and the recipient decodes that information using the same secret key. No other user needs to know the coding/decoding key. There are two types of symmetrical encoding: stream-level encryption and block-level encryption. Bit-level encryption consists of encoding each bit of information, while at the block level a certain number of message bits are encoded simultaneously (for example 64 bits), called a block. Symmetrical encoding is faster than asymmetric encoding. A number of symmetric algorithms can be implemented in hardware. In this way, an algorithm becomes faster in operation. There are two types of symmetric encryption algorithms: 1. Block algorithms. Set lengths of bits are encrypted in blocks of electronic data with the use of a specific secret key. As the data is being encrypted, the system holds the data in its memory as it waits for complete blocks. 2. Stream algorithms. Data is encrypted as it streams instead of being retained in the system’s memory. Some examples of symmetric encryption algorithms include: · AES (Advanced Encryption Standard) · DES (Data Encryption Standard) · IDEA (International Data Encryption Algorithm) · Blowfish (Drop-in replacement for DES or IDEA) · RC4 (Rivest Cipher 4) · RC5 (Rivest Cipher 5) · RC6 (Rivest Cipher 6) AES, DES, IDEA, Blowfish, RC5, and RC6 are block ciphers. RC4 is stream cipher. The most commonly used symmetric algorithm is the Advanced Encryption Standard (AES), which was originally known as Rijndael. This is the standard set by the U.S. National Institute of Standards and Technology in 2001 for the encryption of electronic data announced in U.S. FIPS PUB 197[1]. This standard supersedes DES, which had been in use since 1977. Under NIST, the AES cipher has a block size of 128 bits, but can have three different key lengths as shown with AES-128, AES-192 and AES-256. Symmetrical cryptography also has some disadvantages, such as: - Does not ensure the authentication of the sender. This security gap does not allow the electronic verification of certain transactions; - The transmission of the secret key between correspondents must be carried out on very secure channels. - When used between network users, a large number of secret keys are required to communicate between two users. One of the more popular and widely adopted symmetric encryption algorithms likely to be encountered nowadays is the Advanced Encryption Standard (AES). It is found at least six times faster than triple DES. A replacement for DES was needed as its key size was too small. With increasing computing power, it was considered vulnerable against exhaustive key search attacks. Triple DES was designed to overcome this drawback but it was found slow. The features of AES are: · Symmetric key symmetric block cipher · 128-bit data, 128/192/256-bit keys · Stronger and faster than Triple-DES · Provide full specification and design details · Software implementable in C and Java AES is an iterative scheme. It is based on ‘substitution–permutation network’. It comprises a series of linked operations, some of which involve replacing inputs with specific outputs (substitutions) and others involve shuffling bits around (permutations). AES performs all its computations on bytes rather than bits. Hence, AES treats the 128 bits of a plaintext block as 16 bytes. These 16 bytes are arranged in four columns and four rows for processing as a matrix − The number of rounds in AES are variable and depend on the length of the key. AES uses 10 rounds for 128-bit keys, 12 rounds for 192-bit keys, and 14 rounds for 256-bit keys. Each of these rounds uses a different 128-bit round key, which is calculated from the original AES key. The schematic of AES structure is illustrated in figure 2: Figure 2. AES scheme Encryption Process Here, we restrict to description of a typical round of AES encryption. Each round comprise four sub-processes, as follows: AddRoundKey, SubBytes, ShiftRows, and MixColumns[1]. Pseudo Code for the AES cipher is: Cipher(byte in[4Nb], byte out[4Nb], word w[Nb(Nr+1)]) byte state[4,Nb] state = in AddRoundKey(state, w[0, Nb-1]) // See Sec. 5.1.4 for round = 1 step 1 to Nr–1 AddRoundKey(state, w[roundNb, (round+1)Nb-1]) end for AddRoundKey(state, w[NrNb, (Nr+1)Nb-1]) out = state Byte Substitution (SubBytes) The 16 input bytes are substituted by looking up a fixed table (S-box) given in design. The result is in a matrix of four rows and four columns. Each of the four rows of the matrix is shifted to the left. Any entries that ‘fall off’ are re-inserted on the right side of the row. The shift is carried out as follows: □ The first row is not shifted. □ Second row is shifted one (byte) position to the left. □ The third row is shifted two positions to the left. □ The fourth row is shifted three positions to the left. □ The result is a new matrix consisting of the same 16 bytes but shifted with respect to each other. Each column of four bytes is now transformed using a special mathematical function. This function takes as input the four bytes of one column and outputs four completely new bytes, which replace the original column. The result is another new matrix consisting of 16 new bytes. It should be noted that this step is not performed in the last round. The 16 bytes of the matrix are now considered as 128 bits and are XORed to the 128 bits of the round key. If this is the last round then the output is the ciphertext. Otherwise, the resulting 128 bits are interpreted as 16 bytes and we begin another similar round. Decryption Process The process of the decryption of an AES ciphertext is similar to the encryption process in the reverse order. Each round consists of the four processes conducted in the reverse order : Since sub-processes in each round are in a reverse manner, the encryption and decryption algorithms need to be separately implemented, although they are very closely related, as follow: InvCipher(byte in[4Nb], byte out[4Nb], word w[Nb(Nr+1)]) byte state[4,Nb] state = in AddRoundKey(state, w[NrNb, (Nr+1)Nb-1]) for round = Nr-1 step -1 downto 1 AddRoundKey(state, w[roundNb, (round+1)Nb-1]) end for AddRoundKey(state, w[0, Nb-1]) out = state In present-day cryptography, AES is widely adopted and supported in both hardware and software. Additionally, AES has built-in flexibility of key length, which allows a degree of ‘future-proofing against progress in the ability to perform exhaustive key searches. However, just as for DES, the AES security is assured only if it is correctly implemented and good key management is employed. AES has three different key lengths. The main difference is the number of rounds that the data goes through in the encryption process, 10, 12, and 14 respectively. In essence, 192-bit and 256-bit provide a greater security margin than 128-bit. In the current technological landscape, 128-bit AES is enough for most practical purposes. Highly sensitive data handled by those with an extreme threat level should probably be processed with either 192 or 256-bit AES. [1] https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.197.pdf	{"url":"https://platform.blocks.ase.ro/mod/page/view.php?id=86","timestamp":"2024-11-03T15:45:24Z","content_type":"text/html","content_length":"79015","record_id":"<urn:uuid:46aae037-9a5f-4105-b6ac-c8febafc6abd>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00825.warc.gz"}
Dec 13, 2010 03:55 PM Dec 10, 2010 07:49 AM Dec 07, 2010 02:58 PM Nov 24, 2010 04:48 AM I believe you can find an example in the Help files, but here you go: Create a Curve, By Equation, select a Coordinate System with Z on the axis of the helix, and choose the Cylindrical option. Now enter the following equations: These are examples of true parametric equations. In this case t is the governing parameter, and by definition it is the independent variable which goes from 0 to 1 to generate the dependent values. Thus, the 8 gives you 8 full turns, and the 5 gives you a z-height of 5, radius of 2. This can also be done with the Cartesian option using sin and cos, but Cylindrical is the easiest.	{"url":"https://community.ptc.com/t5/Creo-Modeling-Questions/helix/td-p/289381","timestamp":"2024-11-07T03:22:22Z","content_type":"text/html","content_length":"229532","record_id":"<urn:uuid:a264c161-c589-43c1-abe5-a7c5103c5bf7>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00250.warc.gz"}
Vector-valued holomorphic functions in several variables Functiones et Approximatio Commentarii Mathematici 63 (2): 247-275 (2020) Adam Mickiewicz University, Faculty of Mathematics and Computer Science In the present paper we give some explicit proofs for folklore theorems on holomorphic functions in several variables with values in a locally complete locally convex Hausdorff space E over C. Most of the literature on vector-valued holomorphic functions is either devoted to the case of one variable or to infinitely many variables whereas the case of (finitely many) several variables is only touched or is subject to stronger restrictions on the completeness of E like sequential completeness. The main tool we use is Cauchy's integral formula for derivatives for an E-valued holomorphic function in several variables which we derive via Pettis-integration. This allows us to generalise the known integral formula, where usually a Riemann-integral is used, from sequentially complete E to locally complete E. Among the classical theorems for holomorphic functions in several variables with values in a locally complete space E we prove are the identity theorem, Liouville's theorem, Riemann's removable singularities theorem and the density of the polynomials in the E-valued polydisc algebra. weakly holomorphic several variables locally complete	{"url":"https://tore.tuhh.de/entities/publication/d5f43966-886c-478d-b8d1-8d1d0a4b34c0","timestamp":"2024-11-03T23:30:32Z","content_type":"text/html","content_length":"890507","record_id":"<urn:uuid:9f8fc101-ef7a-4d91-b1e6-4bfafd27e357>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00488.warc.gz"}
Day 101: Calculating the inverse of a matrix Day 101: Calculating the inverse of a matrix In the Day 101 Casey introduced a lot of math and tried to explain everything in details. He talked about calculating the inverse of a matrix by solving the linear system but that would involve a lot of math and it would be easy to make mistakes. Then he introduced Gaussian elimination and explained the idea behind that. But the thing he tried to explain is "why if you apply the elimination steps on an identity matrix you will get the inverse?". At 55:50: He talked about the idea. At 1:03:25 and 1:17:30 He tried to explain that but without success. At 1:23:30 He explained why it make sense. Actually Casey did very good job explaining and it would take me days if I want to search for this information on my own, Math is hard to understand and it is even harder if you tried to explain it. But the explanation at the end was not satisfying even to him and there got to be a proof why that is true. So while I was studying Linear Algebra, I found the explanation to that. I did some search on the forums to see if someone did explain that, but I didn't find anything and I thought it would be nice to share that Sorry but I don't know other way to insert equations. Wow, that's a pretty good runthrough! I think I was moderating at the time but I remember being kinda confused by Casey's explanation, and this made sense. I don't think this forum software supports LaTeX notation (there are some that do! crazy!!) so embedded image was probably the most sane route.	{"url":"https://hero.handmade.network/forums/code-discussion/t/682-day_101__calculating_the_inverse_of_a_matrix#3893","timestamp":"2024-11-13T16:05:03Z","content_type":"text/html","content_length":"29157","record_id":"<urn:uuid:687feee5-8817-4799-b367-b4b598524971>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00250.warc.gz"}
Generate and return bin, the index of the element of the input ascending-ordered array, such that array(bin) <= value < array(bin+1) holds for the input value. The following conditions hold. • If value < array(1), then bin = 0_IK. • If array(size(array)) <= value, then bin = size(array, kind = IK). • The less-than < comparison operator can be also customized by a user-defined comparison supplied as the external function isLess input argument for arrays that are not ascending-ordered. In such a case, the output bin satisfies both .not.(value < array(bin)) and value < array(bin+1) where the < binary operator is defined by the isLess() user-supplied function. : The input contiguous array of shape (:) of either [in] array • a scalar assumed-length character of kind any supported by the processor (e.g., SK, SKA, SKD , or SKU), whose elements will have to be searched for the largest value smaller than the input value. The input array must be sorted in ascending-order unless an appropriate isLess external comparison function is supplied according to which the array effectively behaves as if it is If the array is of type complex, then only its real component will be compared unless the comparison is defined by the external user-specified input comparison function isLess. [in] value : The input scalar of the same type and kind as the input array whose value is to be searched in array. : The external user-specified function that takes two input scalar arguments of the same type and kind as the input array. It returns a scalar logical of default kind LK that is .true. if the two input arguments meet the user-defined comparison criterion. Otherwise, it is .false.. If array is a scalar character, the two input arguments to isLess() will be scalar character values of the same length as the input value (which could be larger than 1). The following illustrates the generic interface of isLess(), function isLess(value, segment) result(isLess) use pm_kind TYPE(KIND) , intent(in) :: value, segment logical(LK) :: isLess end function isLess pm_kind This module defines the relevant Fortran kind type-parameters frequently used in the ParaMonte librar... integer, parameter LK The default logical kind in the ParaMonte library: kind(.true.) in Fortran, kind(.... where TYPE(KIND) represents the type and kind of the input argument array, which can be one of the following, integer(IK) , intent(in) :: value, segment complex(CK) , intent(in) :: value, segment real(RK) , intent(in) :: value, segment character(len(value),SK), intent(in) :: value, segment This external function is extremely useful where a user-defined comparison check other than < is desired, for example, when the array segments should match the input value only within a given threshold or, when the case-sensitivity in character comparisons do not matter, or when the input array can be considered as an ascending-ordered sequence under certain conditions, for example, when only the magnitudes of the array elements are considered or when the array elements are multiplied by -1 or inverted (that is, when the array is in descending-order). See below for example use cases. (optional, the default comparison operator is <.) bin : The output scalar integer of default kind IK representing the index of the element of array for which array(bin) <= value < array(bin+1) holds or if isLess() is specified, the following conditions hold, isLess(value, array(bin+1)) .and. .not. isLess(value, array(bin)) ! evaluates to .true._LK Possible calling interfaces ⛓ = getBin (array, value, isLess) Generate and return bin, the index of the element of the input ascending-ordered array,... This module contains procedures and generic interfaces for finding the specific array index whose ele... The procedures under this generic interface are impure when the user-specified external procedure isLess is specified as input argument. Note that in Fortran, trailing blanks are ignored in character comparison, that is, "Fortran" == "Fortran " yields .true.. The input array must be a non-empty sequence of values that is sorted according to the criterion specified by the external function isLess() or if it is missing, then the values are sorted in ascending order. These conditions are verified only if the library is built with the preprocessor macro CHECK_ENABLED=1. Be mindful of scenario where there are duplicate values in the input array. In such cases, the returned bin is not necessarily the index of the first or the last occurrence of such value. See below for illustrative examples. The pure procedure(s) documented herein become impure when the ParaMonte library is compiled with preprocessor macro CHECK_ENABLED=1. By default, these procedures are pure in release build and impure in debug and testing builds. See also Example usage ⛓ Example Unix compile command via Intel ifort compiler ⛓ ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe Example Windows Batch compile command via Intel ifort compiler ⛓ ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte.lib /exe:main.exe Example Unix / MinGW compile command via GNU gfortran compiler ⛓ gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte -o main.exe Example output ⛓ 4! Find the index of the largest element in the ascending-ordered `array` that is smaller than or equal to `value` via Binary Search. = getBin (string_SK, strval_SK) ! index of the largest value in `array` that is smaller than or equal to the input `value`. = getBin (string_SK, strval_SK) ! index of the largest value in `array` that is smaller than or equal to the input `value`. = getBin (string_SK, strval_SK) ! index of the largest value in `array` that is smaller than or equal to the input `value`. = getBin (Array_SK, value_SK) ! index of the largest value in `array` that is smaller than or equal to the input `value`. = getBin (Array_IK, value_IK) ! index of the largest value in `array` that is smaller than or equal to the input `value`. = getBin (Array_RK, value_RK) ! index of the largest value in `array` that is smaller than or equal to the input `value`. = getBin (Array_IK, value_IK) ! index of the largest value in `array` that is smaller than or equal to the input `value`. = getBin (Array_IK, value_IK) ! index of the largest value in `array` that is smaller than or equal to the input `value`. = getBin (Array_IK, value_IK, isLess_IK) ! index of the smallest value in `array` that is larger than or equal to the input `value`. = getBin (Array_RK, value_RK, isLess_RK) ! index of the largest absolute value in `array` that is smaller than or equal to the input `value`. = getBin (Array_IK, value_IK) ! index of the largest value in `array` that is smaller than or equal to the input `value`. = getBin (Array_IK, value_IK) ! index of the largest value in `array` that is smaller than or equal to the input `value`. Final Remarks ⛓ If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub. For details on the naming abbreviations, see this page. For details on the naming conventions, see this page. This software is distributed under the MIT license with additional terms outlined below. 1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library. 2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library. This software is available to the public under a highly permissive license. Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it. Amir Shahmoradi, September 1, 2017, 12:00 AM, Institute for Computational Engineering and Sciences (ICES), The University of Texas Austin Definition at line 174 of file pm_arraySearch.F90.	{"url":"https://www.cdslab.org/paramonte/fortran/latest/interfacepm__arraySearch_1_1getBin.html","timestamp":"2024-11-11T01:45:32Z","content_type":"application/xhtml+xml","content_length":"162293","record_id":"<urn:uuid:6ffee8fa-207f-47f0-97cc-b8c61a685b40>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00821.warc.gz"}
Items where Subject is "55 Algebraic topology" Number of items at this level: 48. Al-Zamil, Qusay (2010) $X_M$-Harmonic Cohomology and Equivariant Cohomology on Riemannian Manifolds With Boundary. [MIMS Preprint] Al-Zamil, Qusay and Montaldi, James (2012) Generalized Dirichlet to Neumann operator on invariant differential forms and equivariant cohomology. Topology and Applications, 159. pp. 823-832. ISSN Al-Zamil, Qusay and Montaldi, James (2010) Generalized Dirichlet to Neumann operator on invariant differential forms and equivariant cohomology. [MIMS Preprint] Al-Zamil, Qusay and Montaldi, James (2010) Witten-Hodge theory for manifolds with boundary. [MIMS Preprint] Al-Zamil, Qusay and Montaldi, James (2012) Witten-Hodge theory for manifolds with boundary and equivariant cohomology. Differential Geometry and its Applications, 30 (2). pp. 179-194. ISSN 0926-2245 Al-Zamil, Qusay Soad Abdul-Aziz (2011) Algebraic Topology of PDEs. Doctoral thesis, Manchester Institute for Mathematical Sciences, The University of Manchester. Bahri, Anthony and Franz, Matthias and Notbohm, Dietrich and Ray, Nigel (2011) Classifying weighted projective spaces. [MIMS Preprint] Bahri, Anthony and Franz, Matthias and Ray, Nigel (2011) Weighted projective spaces and iterated Thom spaces. [MIMS Preprint] Bahri, Anthony and Franz, Matthias and Ray, Nigel (2007) The equivariant cohomology of weighted projective spaces. arXiv:0708.1581. (Unpublished) Boote, Yumi (2016) On the symmetric square of quaternionic projective space. Doctoral thesis, Manchester Institute for Mathematical Sciences, The University of Manchester. Buchstaber, V. and Panov, T. and Ray, N. (2007) Spaces of polytopes and cobordism of quasitoric manifolds. Moscow Mathematical Journal, 7 (2). pp. 219-242. ISSN 1609-4514 Buchstaber, V. M and Ray, N (2007) The universal equivariant genus and Krichever's formula. Russian Mathematical Surveys, 62 (1). pp. 178-180. ISSN 1468-4829 Buchstaber, VIctor (2010) Ring of polytopes and the Rota-Hopf algebra. In: Topology Seminar, University of Manchester, 8 Jan 2010, Manchester. (Unpublished) Buchstaber, Victor (2007) Toric Topology of Stasheff Polytopes. In: Topology Seminar, Manchester, 12 Nov 2007, Manchester, UK. (Unpublished) Buchstaber, Victor and Panov, Taras (2002) Torus Actions and their Applications in Topology and Combinatorics. University Lecture Series number 24; OUP . American Mathematical Society. ISBN 10: Buchstaber, Victor and Shorina, S. Yu. (2004) w-function of the KDV hierarchy. Geometry, Topology and Mathematical Physics, 212. pp. 41-46. ISSN 0065-9290 Buchstaber, Victor M (2005) Circle actions on toric manifolds and their applications. In: Pure Mathematics Colloquium, University of Leicester, 17 November 2005, Leicester, England. (Unpublished) Buchstaber, Victor M and Ray, Nigel (2008) An Invitation to Toric Topology: Vertex Four of a Remarkable Tetrahedron. Proceedings of the International Conference in Toric Topology: Osaka 2006. (In Buchstaber, Victor M. (2008) Combinatorics of simple polytopes and differential equations. [MIMS Preprint] Buchstaber, Victor M. and Ray, Nigel (2001) Tangential structures on toric manifolds, and connected sums of polytopes. International Mathematics Research Notices, 2001. pp. 193-219. ISSN 1073-7928 Civan, Yusuf and Ray, Nigel (2004) Homotopy Decompositions and K-theory of Bott Towers. K-theory, 34 (1). pp. 1-33. ISSN 0920-3036 Eccles, Peter J and Grant, Mark (2006) Bordism classes represented by multiple point manifolds of immersed manifolds. Proceedings of the Steklov Instutute of Mathematics, 252. pp. 47-52. ISSN Eccles, Peter J and Grant, Mark (2006) Bordism groups of immersions and classes represented by self-intersections. [MIMS Preprint] Eccles, Peter J. and Grant, Mark (2012) SELF-INTERSECTIONS OF IMMERSIONS AND STEENROD OPERATIONS. Acta Mathematica Hungarica. pp. 1-10. ISSN 1588-2632 Eccles, Peter J. and Zare, Hadi (2011) The Hurewicz image of the $\eta_i$ family. [MIMS Preprint] Estrada, Sergio and Guil Asensio, Pedro A. and Prest, Mike and Trlifaj, Jan (2009) Model category structures arising from Drinfeld vector bundles. [MIMS Preprint] Garkusha, Grigory and Prest, Mike (2006) Classifying Serre subcategories of finitely presented modules. [MIMS Preprint] Garkusha, Grigory and Prest, Mike (2006) Classifying thick subcategories of perfect complexes. [MIMS Preprint] Grbić, Jelena (2006) Universal homotopy associative, homotopy commutative $H$-spaces and the EHP spectral sequence. Mathematical Proceedings of the Cambridge Philosophical Society, 140 (3). pp. 377-400. ISSN 0305-0041 Grbić, Jelena (2006) Universal homotopy associative, homotopy commutative H-spaces and the EHP spectral sequences. Mathematical Proceedings of the Cambridge Philosophical Society, 140 (3). pp. 377-400. ISSN 0305-0041 Grbić, Jelena (2006) The cohomology of certain 2-local finite groups. Manuscripta Mathematica, 120 (3). pp. 307-318. ISSN 0025-2611 Grbić, Jelena and Theriault, Stephen (2007) The homotopy type of the complement coordinate subspace arrangement. Topology, 46 (4). pp. 377-400. ISSN 0040-9383 Grbić, Jelena and Wu, Jie (2006) Natural transformations of tensor algebras and representations of combinatorial groups. Algebraic and Geometric Topology, 6. pp. 2189-2228. ISSN 1472-2747 Harada, Megumi and Holm, Tara and Ray, Nigel and Williams, Gareth (2013) The equivariant K-theory and cobordism rings of divisive weighted projective spaces. Tohoku Mathematical Journal. (In Press) Kuber, Amit (2013) Grothendieck Rings of Theories of Modules. [MIMS Preprint] Notbohm, Dietrich and Ray, Nigel (2008) On Davis-Januszkiewicz Homotopy Types II; completion and globalisation. Algebraic and Geometric Topology, To app. ISSN 1472-2747 (In Press) Notbohm, Dietrich and Ray, Nigel (2005) On Davis-Januszkiewicz homotopy types I; formality and rationalisation. Algebraic and Geometric Topology, 5. pp. 31-51. ISSN 1472-2747 Panov, Taras and Ray, Nigel and Vogt, Rainer (2004) Colimits, Stanley-Reisner algebras, and loop spaces. In: Categorical Decomposition Techniques in Algebraic Topology. Progress in Mathematics, 215 . Birkhauser Verlag, Basel, pp. 261-291. ISBN 3-7643-0400-6 Robinson, Daniel Mark (2012) The Homotopy Exponent Problem For Certain Classes Of Polyhedral Products. Doctoral thesis, Manchester Institute for Mathematical Sciences, The University of Manchester. Sandling, Robert (2011) Centralisers in the unit group of the Steenrod algebra. [MIMS Preprint] Sandling, Robert (2011) Endomorphisms of the Steenrod algebra and of its odd subalgebra. [MIMS Preprint] Sandling, Robert (2011) The lattice of column 2-regular partitions in the Steenrod algebra. [MIMS Preprint] Sandling, Robert (2011) A presentation of the Steenrod algebra using Kristensen's operator. [MIMS Preprint] Steckles, Katrina (2011) Loop Spaces and Choreographies in Dynamical Systems. Doctoral thesis, Manchester Institute for Mathematical Sciences, The University of Manchester. Symonds, Peter (2005) The bredon cohomology of subgroup complexes. Journal of Pure and Applied Algebra, 199 (1-3). pp. 261-298. ISSN 0022-4049 Zare, Hadi (2009) On Spherical Classes in HQSn. Doctoral thesis, Manchester Institute for Mathematical Sciences, The University of Manchester. Zare, Hadi (2011) On spherical classes in $H_QX$. [MIMS Preprint] Zare, Hadi (2011) On the Bott periodicity, J-homomorphisms, and $H_*Q_0S^{-k}$. [MIMS Preprint]	{"url":"https://eprints.maths.manchester.ac.uk/view/subjects/MSC=5F55.html","timestamp":"2024-11-05T21:56:03Z","content_type":"application/xhtml+xml","content_length":"25248","record_id":"<urn:uuid:371b3ef2-b7f8-4b14-bf99-e6dc2c79f9ed>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00417.warc.gz"}
Symmetries of a class of nonlinear third-order partial differential equations Clarkson, Peter, Mansfield, Elizabeth L., Priestley, T.J. (1997) Symmetries of a class of nonlinear third-order partial differential equations. Mathematical and Computer Modelling, 25 (8-9). pp. 195-212. ISSN 0895-7177. (doi:10.1016/s0895-7177(97)00069-1) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:18356) The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided. Official URL: In this paper, we study symmetry reductions of a class of nonlinear third-order partial differential equations (1) U-t - epsilon u(xxt) + 2 kappa u(x) = uu(xxx) + alpha uu(x) + beta u(x)u(xx), where epsilon, kappa, alpha, and beta are arbitrary constants. Three special cases of equation (1) have appeared in the literature, up to some rescalings. In each case, the equation has admitted unusual travelling wave solutions: the Fornberg-Whitham equation, for the parameters epsilon = 1, alpha = -1, beta = 3, and kappa = 1/2, admits a wave of greatest height, as a peaked limiting form of the travelling wave solution; the Rosenau-Hyman equation, for the parameters epsilon = 0, alpha = 1, beta = 3, and kappa = 0, admits a ''compacton'' solitary wave solution; and the Fuchssteiner-Fokas-Camassa-Holm equation,for the parameters epsilon = 1, alpha = -3, and beta = 2, has a ''peakon'' solitary wave solution. A catalogue of symmetry reductions for equation (1) is obtained using the classical Lie method and the nonclassical method due to Bluman and Cole. • Depositors only (login required):	{"url":"https://kar.kent.ac.uk/18356/","timestamp":"2024-11-03T05:47:40Z","content_type":"application/xhtml+xml","content_length":"35055","record_id":"<urn:uuid:124fb0b8-1963-4efa-8f17-f9f4abe62678>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00799.warc.gz"}
Volume of a Prism - Formula, Derivation, Definition, Examples - Grade Potential Canton, OH Volume of a Prism - Formula, Derivation, Definition, Examples A prism is a vital figure in geometry. The shape’s name is derived from the fact that it is made by taking a polygonal base and extending its sides till it creates an equilibrium with the opposing This article post will discuss what a prism is, its definition, different types, and the formulas for surface areas and volumes. We will also offer examples of how to use the data given. What Is a Prism? A prism is a 3D geometric figure with two congruent and parallel faces, called bases, which take the form of a plane figure. The other faces are rectangles, and their number rests on how many sides the similar base has. For example, if the bases are triangular, the prism would have three sides. If the bases are pentagons, there would be five sides. The characteristics of a prism are astonishing. The base and top each have an edge in parallel with the other two sides, creating them congruent to each other as well! This means that every three dimensions - length and width in front and depth to the back - can be broken down into these four entities: 1. A lateral face (meaning both height AND depth) 2. Two parallel planes which constitute of each base 3. An illusory line standing upright through any provided point on either side of this shape's core/midline—also known collectively as an axis of symmetry 4. Two vertices (the plural of vertex) where any three planes join Kinds of Prisms There are three primary types of prisms: • Rectangular prism • Triangular prism • Pentagonal prism The rectangular prism is a regular type of prism. It has six faces that are all rectangles. It resembles a box. The triangular prism has two triangular bases and three rectangular sides. The pentagonal prism comprises of two pentagonal bases and five rectangular faces. It appears a lot like a triangular prism, but the pentagonal shape of the base makes it apart. The Formula for the Volume of a Prism Volume is a calculation of the total amount of area that an item occupies. As an essential shape in geometry, the volume of a prism is very important for your learning. The formula for the volume of a rectangular prism is V=Bh, where, V = Volume B = Base area h= Height Finally, given that bases can have all types of shapes, you have to learn few formulas to determine the surface area of the base. However, we will go through that later. The Derivation of the Formula To obtain the formula for the volume of a rectangular prism, we need to observe a cube. A cube is a three-dimensional item with six sides that are all squares. The formula for the volume of a cube is V=s^3, assuming, V = Volume s = Side length Right away, we will take a slice out of our cube that is h units thick. This slice will create a rectangular prism. The volume of this rectangular prism is Bh. The B in the formula implies the base area of the rectangle. The h in the formula implies the height, that is how thick our slice was. Now that we have a formula for the volume of a rectangular prism, we can use it on any kind of prism. Examples of How to Use the Formula Considering we know the formulas for the volume of a pentagonal prism, triangular prism, and rectangular prism, now let’s use them. First, let’s work on the volume of a rectangular prism with a base area of 36 square inches and a height of 12 inches. V=432 square inches Now, consider another question, let’s work on the volume of a triangular prism with a base area of 30 square inches and a height of 15 inches. V=450 cubic inches Provided that you have the surface area and height, you will calculate the volume without any issue. The Surface Area of a Prism Now, let’s talk regarding the surface area. The surface area of an object is the measurement of the total area that the object’s surface consist of. It is an crucial part of the formula; thus, we must understand how to find it. There are a several varied ways to figure out the surface area of a prism. To figure out the surface area of a rectangular prism, you can use this: A=2(lb + bh + lh), assuming, l = Length of the rectangular prism b = Breadth of the rectangular prism h = Height of the rectangular prism To compute the surface area of a triangular prism, we will utilize this formula: b = The bottom edge of the base triangle, h = height of said triangle, l = length of the prism S1, S2, and S3 = The three sides of the base triangle bh = the total area of the two triangles, or [2 × (1/2 × bh)] = bh We can also utilize SA = (Perimeter of the base × Length of the prism) + (2 × Base area) Example for Calculating the Surface Area of a Rectangular Prism First, we will determine the total surface area of a rectangular prism with the following information. l=8 in b=5 in h=7 in To figure out this, we will plug these values into the respective formula as follows: SA = 2(lb + bh + lh) SA = 2(85 + 57 + 87) SA = 2(40 + 35 + 56) SA = 2 × 131 SA = 262 square inches Example for Computing the Surface Area of a Triangular Prism To calculate the surface area of a triangular prism, we will find the total surface area by ensuing similar steps as earlier. This prism will have a base area of 60 square inches, a base perimeter of 40 inches, and a length of 7 inches. Hence, SA=(Perimeter of the base × Length of the prism) + (2 × Base Area) SA = (407) + (2*60) SA = 400 square inches With this data, you will be able to work out any prism’s volume and surface area. Test it out for yourself and observe how easy it is! Use Grade Potential to Enhance Your Mathematical Abilities Today If you're having difficulty understanding prisms (or any other math subject, consider signing up for a tutoring session with Grade Potential. One of our professional instructors can help you study the [[materialtopic]187] so you can nail your next exam.	{"url":"https://www.cantoninhometutors.com/blog/volume-of-a-prism-formula-derivation-definition-examples","timestamp":"2024-11-05T19:12:51Z","content_type":"text/html","content_length":"78383","record_id":"<urn:uuid:2be31a1d-2bb5-4b6b-90a8-6db40f4e2c71>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00243.warc.gz"}
EViews Help: @vec Vectorize (stack columns of) matrix. Syntax: @vec(m) m: matrix, sym Return: vector Creates a vector from the columns of the given matrix stacked one on top of each other. The vector will have the same number of elements as the source matrix. matrix m1 = @mrnd(10, 3) vector v1 = @vec(m1) See also , and	{"url":"https://help.eviews.com/content/functionref_v-@vec.html","timestamp":"2024-11-12T10:20:03Z","content_type":"application/xhtml+xml","content_length":"8332","record_id":"<urn:uuid:e7097c27-5985-4a4f-a177-69a3ee774bd5>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00218.warc.gz"}
\| What is Piping What Are The Pump Affinity Laws? Explanation with a Sample Calculation What are Pump Affinity Laws? Pump affinity laws or Pump Laws are used during hydraulic design to calculate the volume, capacity, head, or power consumption of the centrifugal pumps with changing speeds or wheel diameters. These laws are very important in pump design as they express the relationship between the variables that decide the pump performance. The Pump affinity laws are basically a set of formulas that predict the impact of Change in Pump impeller Diameter and its Rotational Speed on the Pump Head (h), Pump Flow (Q), and Power Demand of the Pump (P). The affinity laws are applied both to centrifugal and axial flows. These laws can also be applied to fans and hydraulic turbines. Uses of Pump Affinity Laws Pump Affinity laws are widely used to predict the pump performance at a different speed or with different impeller diameters provided the pump performance curve at a certain speed or impeller diameter is known. Using the series of ratios that pump affinity laws establish, a pump engineer can predict the performance of the pump subject to changing pump conditions. So, Pump affinity laws are a great tool to help pump engineers. You can refer to our video article on the same subject by clicking here. To get an update regarding more similar videos please click here and subscribe to our channel. Types of pump affinity law? As mentioned above these laws study the impact of impeller diameter & rotational speed of the pump. Hence the affinity laws are framed • Once keeping the diameter constant (D=Constant) and • Once keeping the rotational speed constant (N=Constant). So based on these conditions below mentioned are the cases and relations on flow, head & power consumption of the pump. Pump Affinity law keeping the impeller diameter Constant As per the first set of pump affinity laws for a given pump with a fixed impeller diameter: • The flow of the pump is directly proportional to the change in the speed of the pump. This means there will be the same amount of change in the flow of the pump as the rotational speed of the pump changes. • The Head of the pump is proportional to the square of speed. This means a change in pump rotational speed will lead to a change in the square root of the pump head. • Power is proportional to the cube of the speed. This means that when the rotational speed of the pump is increased the power consumption of the pump will be increased by 8 times. The above points can be presented mathematically as given below Fig. 1: Affinity Laws with constant impeller diameter Fig. 2: Graphical representation of Pump Head vs Flow Rate Normally, within the range of normal pump operational speeds, there is no appreciable change in efficiency. Because of this, the first Set of Pump Affinity Laws is reasonably accurate and reliable. Pump Affinity laws keeping the pump speed constant As per the second set of pump affinity laws for a given pump with a constant speed: • The flow of the pump (Pump Capacity) is directly proportional to the change in the impeller diameter of the pump. This means there will be the same amount of change in the flow of the pump as the impeller diameter of the pump is changed. • The Head of the pump is proportional to the square of the impeller diameter. This means a change in the pump impeller diameter will lead to a change in the square root of the pump head • Power is proportional to the cube of the impeller diameter. So, when the diameter of the pump is increased the power consumption of the pump will be increased by 8 times. Mathematically the aforementioned points can be represented as given in Fig. 3. Fig. 3: Affinity Laws with constant pump speed Fig. 4: A typical pump performance curve Note that, inaccuracies can be introduced into the predictions of the second set of pump affinity laws as efficiency changes with changes in impeller diameter. Combined Pump Affinity Law The flow, head, and power consumption of a pump change can be combined as shown in Fig. 5 when both rotational speed & impeller diameter change. Fig. 5: Combined Pump Affinity Law Example Calculation Explaining Pump Affinity Law Consider a pump of flow 1000 m^3/hr is designed for 1500 rpm. The same pump is required to be operated at 3000 rpm. Find the changed flow of the pump. Assume the diameter of the pump is kept As per the pump affinity law when the impeller diameter is kept constant: Q∝N or Q1/Q2=N1/N2; So, Q1 = 1000m^3/hr, N1 = 1500rpm, Q2 =? , N2 = 3000 rpm. Hence, 1000/Q2=1500/3000; Q2=2000 m^3/hr Hence for the same pumps if the RPM of the pump is doubled flow will also get doubled. In the same example let’s take Power consumption = 100Kw, so find P2 at the changed RPM. P∝N^3 or P1/P2=(N1^3)/(N2^3 ) So, 100/P2=(1500^3)/(2000^3 ) Or P2 = 800 kW. Hence a change in rpm will increase the power consumption of the pump 8 times. Want to learn more. Here are a few Handpicked articles to add value to your learning process. NPSH for Pumps: Explanation and Effect Cause and Effect of Pump Cavitation Pump Suction Intake Design with Sample Calculation Types of Pumps used in Process Plants Major Factors Affecting the Pump Performance Other pump-related articles Compressor related articles Other Mechanical Design related articles One thought on “What Are The Pump Affinity Laws? Explanation with a Sample Calculation” 1. I didn’t think the Flow rate for axial flow pumps is proportional to the diameter (as above) given Q=VA and A is proportional to Diameter squared Wikipedia gives Q is proportional to Diameter	{"url":"https://whatispiping.com/pump-affinity-laws/","timestamp":"2024-11-04T04:49:23Z","content_type":"text/html","content_length":"86422","record_id":"<urn:uuid:1b05a565-f519-43f3-92b7-d8fa079a7773>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00403.warc.gz"}
dlagtm: performs a matrix-vector product of the form B := alpha * A * X + beta * B where A is a tridiagonal matrix of order N, B and X are N by NRHS matrices, and alpha and beta are real scalars, each of which may be 0., 1., or -1 - Linux Manuals (l) dlagtm (l) - Linux Manuals dlagtm: performs a matrix-vector product of the form B := alpha * A * X + beta * B where A is a tridiagonal matrix of order N, B and X are N by NRHS matrices, and alpha and beta are real scalars, each of which may be 0., 1., or -1 DLAGTM - performs a matrix-vector product of the form B := alpha * A * X + beta * B where A is a tridiagonal matrix of order N, B and X are N by NRHS matrices, and alpha and beta are real scalars, each of which may be 0., 1., or -1 TRANS, N, NRHS, ALPHA, DL, D, DU, X, LDX, BETA, B, LDB ) CHARACTER TRANS INTEGER LDB, LDX, N, NRHS DOUBLE PRECISION ALPHA, BETA DOUBLE PRECISION B( LDB, * ), D( * ), DL( * ), DU( * ), X( LDX, * ) DLAGTM performs a matrix-vector product of the form TRANS (input) CHARACTER1 Specifies the operation applied to A. = aqNaq: No transpose, B := alpha A * X + beta * B = aqTaq: Transpose, B := alpha * Aaq* X + beta * B = aqCaq: Conjugate transpose = Transpose N (input) INTEGER The order of the matrix A. N >= 0. NRHS (input) INTEGER The number of right hand sides, i.e., the number of columns of the matrices X and B. ALPHA (input) DOUBLE PRECISION The scalar alpha. ALPHA must be 0., 1., or -1.; otherwise, it is assumed to be 0. DL (input) DOUBLE PRECISION array, dimension (N-1) The (n-1) sub-diagonal elements of T. D (input) DOUBLE PRECISION array, dimension (N) The diagonal elements of T. DU (input) DOUBLE PRECISION array, dimension (N-1) The (n-1) super-diagonal elements of T. X (input) DOUBLE PRECISION array, dimension (LDX,NRHS) The N by NRHS matrix X. LDX (input) INTEGER The leading dimension of the array X. LDX >= max(N,1). BETA (input) DOUBLE PRECISION The scalar beta. BETA must be 0., 1., or -1.; otherwise, it is assumed to be 1. B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) On entry, the N by NRHS matrix B. On exit, B is overwritten by the matrix expression B := alpha * A * X + beta * B. LDB (input) INTEGER The leading dimension of the array B. LDB >= max(N,1).	{"url":"https://www.systutorials.com/docs/linux/man/l-dlagtm/","timestamp":"2024-11-12T23:41:10Z","content_type":"text/html","content_length":"10342","record_id":"<urn:uuid:f05d771e-b1ac-40e7-8572-a4969ab8c79a>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00680.warc.gz"}
Pls Help Me Rn Bec Sue Tmr Pls Pls Number of 1-cm cube 24. Volume of solid 24 cm^2 Step-by-step explanation: Steps to multiply using Long Multiplication Multiplying 2-Digit by 2-Digit Numbers Let us multiply 47 by 63 using the long multiplication method. 1. Write the two numbers one below the other as per the places of their digits. Write the bigger number on top and a multiplication sign on the left. Draw a line below the numbers. 2. Multiply ones digit of the top number by the ones digit of the bottom number. Write the product as shown. 3. Multiply the tens digit of the top number by the ones digit of the bottom number. This is our first partial product which we got on multiplying the top number by the ones digit of the bottom number. 4. Write a 0 below the ones digit as shown. This is because we will now be multiplying the digits of the top number by the tens digit of the bottom number. Hence, we write a 0 in the ones place. 5. Multiply the ones digit of the top number by the tens digit of the bottom number. 6. Multiply the tens digit of the top number by the tens digit of the bottom number. This is the second partial product obtained on multiplying the top number by the tens digit of the bottom number. 7. Add the two partial products. In long multiplication method, the number on the top is called the multiplicand. The number by which it is multiplied, that is, the bottom number is called the multiplier. So, a long division problem will have: We follow the same method for multiplying numbers greater than 2-Digits. The figure below shows the long division method to multiply 357 by 23	{"url":"https://smart.gov.qa/question-answers/pls-help-me-rn-bec-sue-tmr-pls-pls-q4ov","timestamp":"2024-11-09T17:16:51Z","content_type":"text/html","content_length":"67933","record_id":"<urn:uuid:97da88ab-730c-401e-92e1-307abf2a3a64>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00740.warc.gz"}
How many seconds in 152 days How many seconds in 152 days How many seconds in 152 days? There are 13132800 seconds in 152 days. You may also interested in: How many weeks in a year? You can use our calculators and converters to convert days to seconds and other time units such as minutes, seconds, millisecond and more. Seconds in Day table More references for	{"url":"https://www.rocknets.com/calculator/time-and-date/how-many-seconds-in-152-days","timestamp":"2024-11-07T13:19:01Z","content_type":"text/html","content_length":"66437","record_id":"<urn:uuid:2b1c739e-59da-42f0-b9e6-c8e7735a5869>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00713.warc.gz"}
The 5 Commandments of Calculus Karen Sleno Karen Sleno is an expert teacher at Flushing High School where she has taught everything from Algebra 1 through AP Calculus and serves as the department chair. Additionally, she is an adjunct instructor at Mott Community College and at the Center for Talented Youth at Johns Hopkins University. In 2022, she won the Michigan Department of Education Regional Teacher of the Year award in Region 5 for her teaching expertise gained over 30 years in the profession. She is a College Board consultant for AP Calculus and has held various roles (including question/exam leader) at the annual AP reading. Her efforts in education and in the AP program earned her recognition as Educator of the Year for her district in 2015. It’s not surprising that the thought of grading AP Calculus exams for 8 hours a day for 7 consecutive days may sound like the worst punishment one could suffer. I might have thought that myself once. However, once I experienced the AP reading, that opinion took a sharp 180. Not only is my time there rewarding due to the people that surround me, but I also learn so much about how the current AP exam is scored that the week really doesn’t feel like work at all. In fact, those 7 days have become the highlight of my year! But this isn’t a post about the AP reading; it’s about something that happened near its conclusion one year. So often, student responses fail to earn points not due to a lack of calculus knowledge, but rather a lack of communication skills. Over and over, we see unequal expressions that are set equal or a missing parenthesis that changes the meaning of an expression completely. Those types of errors are painful to see because they could have been so easily avoided. One year, I said to a colleague, “I wish I could share these ‘fixes’ with every math teacher out there!” And now, here is my opportunity to do just that! May I present…my 5 Commandments of Calculus (or any math, really)! I. THOU SHALT NOT USE VAGUE PHRASES SUCH AS “THE GRAPH”, “THE SLOPE”, OR “IT”. We’ve all seen it. “Decreasing because it’s negative”. What exactly is negative? “The parabola opens up because it’s positive.” What is? Allowing such explanations in our classroom means that students will use those explanations on the AP exam, too, and if there is any uncertainty about what “it” is, full points will likely not be earned. Instead, insist that explanations use names of functions consistently and correctly, even if it sounds silly to student ears. “The function f is decreasing because f’(x) is negative”...yes! II. THOU SHALT NOT CLAIM THAT TWO THINGS ARE EQUAL WHEN THEY ARE NOT. “Stream of consciousness math” is what I’m referring to here…the tendency of students to use the equal sign as an indicator of their next step. The problem with this is that if the two expressions they are linking are not truly equal, a penalty will occur. For example, suppose a student is finding the average value of a function f(x). Here is what we might see: The result is correct mathematically but not in presentation. The student realized that average value requires division by the length of the interval, but in this case, it came as an afterthought. This is not just a calculus difficulty, however; any process (quadratic formula, area/volume to name two) opens the door for the equal sign issue to rear its ugly head. III. THOU SHALT NOT SAY TOO MUCH. This may be the saddest of them all. The student writes a beautiful response that would earn full points, and then says something incorrect because the response didn’t end when it should have! How to remedy this? Remind students that short, concise answers that meet the requirements are all that are necessary to earn full points. Save the essay writing for AP English! IV. THOU SHALT ALWAYS LABEL THY ANSWER. Forgetting to use correct units is one of the most universal errors in any math class; even students in my consumer math class sometimes forget to include the $ with their results. Fortunately, it’s also one of the easiest to remedy with consistent reinforcement. While the College Board is often kind enough to remind students with the words “using correct units” in the question prompt, we should expect this as an automatic signature to any contextual problem, whether in finance, algebra, geometry, or yes, calculus. What? Grammar in math? Oh yes, indeed! Consider this treatment of a logarithm: What is the argument of this logarithm? It should be x based on what was written but sometimes students use this notation to mean ln (x+2). Here is another great example: See it? That first quantity in the numerator should have parentheses around it, but they are missing, so what we read is only 2(3), not (4x + 2)(3). Expecting good grammar/punctuation in class means that students will “perform” what they “practice”. Someday, math classes everywhere will enforce the 5 Commandments every day with every student, but until then, I will preach the word to all teachers who will listen. And maybe, just maybe, in a not so distant future, those AP Calculus students whose work we read will remember what they “shalt” or “shalt not”.	{"url":"https://blog.mathmedic.com/post/the-5-commandments-of-calculus","timestamp":"2024-11-08T04:45:03Z","content_type":"text/html","content_length":"880715","record_id":"<urn:uuid:54969123-4011-468b-9a37-b18291222b2c>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00674.warc.gz"}
Octopus – Visions in Math I’ve recently been going back to the 3D printed models my students designed in my Math 341 Introduction to Topology class in Fall 2014. I know so much more now than I did then when I first started on this journey. So, I decided to go back to the designs, check over them, edit them (as necessary), then reprint them and publish them on Thingiverse. The first two models I looked at where Candace Bethea’s (’15) $6_2$ knot and Hayley Archer-McClelland’s (’15) three interlocking trefoils. In a previous class using 3D printing, Professor Aaron Abrams contacted the developer of the program Seifert View, and arranged for the computed curves and surfaces to be able to be exported as a file. (This free software normally doesn’t allow you to do this.) We have since found this to be an invaluable tool for 3D printing knots and knots with Seifert surfaces. The $6_2$ knot was first adjusted in, then exported from SeifertView, and then opened in Cinema4D. The surface of the knot needed fixing due to overlapping polygons. This was fixed by deleting overlapping polygons, then filling in the gaps using the Stitch and Sew tool in Edge mode. The knot was originally printed in orange on the Projet-260 3D Systems printer. Later, I printed it on the FormLabs 1+ printer in clear resin. You can find the model here on Thingiverse. The three interlocking trefoils were designed entirely in Cinema4D. The parametric equations of a trefoil knot are $x(t)= (2+\cos(3t))\cos(2t), y=(2+\cos(3t))\sin(2t), z=-\sin(3t)$ for $t\in[-\pi, \pi]$. We first made the curve with the Formula tool for $ t\in[-3.14, 3.14]$. We added a SweepNurb (with no caps) consisting of a circle with radius 0.2 cm around the curve. The choice of $t$ meant there was a small gap between the ends of the tube. We again sealed this gap using the Stitch and Sew tool in Edge mode. The knots were originally printed in pink, pale green and blue on the Projet-260 3D Systems printer. Later, I printed it on the FormLabs 1+ printer in clear resin. You can find the model here on Thingiverse. Mithra Muthukrishnan (’16) used Seifert View to design a figure-8 knot and its Seifert surface. After tweaking in Seifert View, she exported the surface to Cinema4D. It ended up being a complex design process, since we wanted to color the knot and two sides of the surface with different colors. There, the surface was extruded in both directions creating the two sides of the Seifert surface. Finally, three copies of the surface were made. In one, all the surfaces were deleted leaving the knot. In another the knot and one side of the extrusion was deleted. In the final copy, the knot and the other side of the extrusion was deleted. This left three pieces which could each be given their own color before printing. The knot was colored red/pink, the different sides of the Seifert surface were colored white and yellow. The model was originally printed in color on the Projet-260 3D Systems printer. Later, I printed it on the MakerBot 2x printer in bright white. You can find the model here on Thingiverse. The octopus model was designed by DanJoesph Quijada (’15) entirely in Cinema4D. The legs were made using parametric equations like $x(t) = a\sqrt{2}t, y(t)=0, z(t) = b^2 cos^2(ct)e^{-dt^2}$ for various constants $a, b, c,$ and $d$, and bounded time $t$. These curves were then thickened using a SweepNurbs. To make the octopus head, we first made a box, then extruded parts of the sides to alter the shape, then applied the Subdivision Surface tool to it. Finally we made minor adjustments, such as adding the eyes and changing the dimensions of the octopus to make it look more realistic. The model was originally printed in pink on the Projet-260 3D Systems printer. Later, I printed it on the MakerBot 2x printer in bright white, though I had some trouble with the legs. You can find the model here on Thingiverse.	{"url":"https://mathvis.academic.wlu.edu/tag/octopus/","timestamp":"2024-11-05T10:22:31Z","content_type":"text/html","content_length":"43680","record_id":"<urn:uuid:bd5f81a7-1b5f-4d2d-88b1-7d0c769c2394>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00645.warc.gz"}
PROGRAMMING ASSIGNMENT 1: Slide or Jump? solved As you wait impatiently for the next Algorithms lecture, a kid approaches you and challenges you to play a game called Slide or Jump? The game is as follows: you stand at one end of a board consisting of several cells, with each cell containing a non-negative integer that represents the cost of visiting that cell (as in the example below): From any cell, you have one of two possible moves: either slide to the adjacent cell or jump over the adjacent cell. The objective of the game is to reach the last cell while minimizing the sum of the costs of the visited cells. For the sample board above, the cheapest strategy is to slide, jump, and jump, with overall cost of 23 (5 + 7 + 11). Despite your best efforts, it seems as if your opponent is consistently beating you. So you decide to take matters into your own hands by writing a program that determines the optimal sequence of moves. After some thought, you realize that the overall cost reaching the goal can be defined recursively. First, consider the base cases: • If you are in the last cell, the overall cost is equal the cost of that cell (already at the end!!) • If you are in the cell adjacent to the last cell, the overall cost is the sum of the costs of the last two cells (since your only possible move is to slide) • If you are in the third cell from the end, the overall cost is the sum of the cost of that cell and the cost of the last cell (since jumping allows you to reach the end with a lower overall cost) In the general case, let totalCost(n) be the minimum cost of reaching the goal from cell n, then: totalCost(n) = cost of cell n + either the cost of sliding or the cost of jumping (whichever is smaller), or in other words totalCost(n) = cost of cell n + min(cost(n + 1), cost(n + 2)) After coding the recursive routine, you realize that it takes too long for long boards, mainly because of the many overlapping recursive calls required. Therefore, you decide to try a dynamic programming approach. When using this approach, your solution should store partial solutions in an n-element array, so that the array element at location k contains the value of totalCost(k). The algorithm can then compute and store the values of totalCost(k) without the need to recalculate partial results (starting at the end of the board and working towards the first cell). Your assignment is to write a Java class named SlideOrJump which provides public methods with the following signatures and functionality: SlideOrJump(int[] board) // board will have at least two elements, the first one always being zero long recSolution() // computes overall cost recursively (required for A-B-C credit) long dpSolution() // computes overall cost using dynamic programming (required for A-B credit) String getMoves() // returns sequence of moves required (required for A credit, see format below) You should initially test your methods using the SlideJumpTest class provided. Here is a sample sequence of method calls and return values: SlideOrJump game = new SlideOrJump(array); // array = {0, 5, 75, 7, 43, 11} game.recSolution(); // returns 23 game.dpSolution(); // returns 23 game.getMoves(); // returns “SJJ” representing slide, jump, jump For full credit, include code to count and display the number of operations performed by each of the solution methods. In the recursive solution, count method calls. In the dynamic programming solution, count loop iterations. Collect enough data to produce a report that includes a summary of the data presented in both tabular and graphical formats. Analyze the data and form a conclusion regarding the relative efficiency of the methods. Your report should be submitted via Moodle in a file named sj.pdf. The source code for SlideOrJump.java should be submitted to Mimir.	{"url":"https://codeshive.com/questions-and-answers/programming-assignment-1-slide-or-jump-solved/","timestamp":"2024-11-03T03:21:16Z","content_type":"text/html","content_length":"102994","record_id":"<urn:uuid:72e42336-ac8e-42d1-bd17-78fa8e932926>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00679.warc.gz"}
Understanding Mathematical Functions: How To Prove Two Functions Are E Proving two functions are equal is a critical aspect of mathematical analysis as it allows us to understand and compare the behavior of different functions. In this blog post, we will delve into the importance of proving equality between functions and provide a brief overview of mathematical functions. Understanding the process of proving equality between functions is vital for anyone studying mathematics or working with functions in real-world applications. Key Takeaways • Proving equality between functions is essential for comparing their behavior and understanding mathematical analysis. • Mathematical functions play a crucial role in various real-world applications, and understanding their equality is vital for professionals in fields such as engineering, physics, and economics. • Methods for proving equality between functions include direct substitution, algebraic manipulation, and graphical analysis. • Understanding key properties of functions, such as symmetry, periodicity, and asymptotes, is important in proving their equality. • Applying theorems and properties, such as function composition and inverse function properties, is integral in function equality proofs. Understanding Mathematical Functions Mathematical functions are a fundamental concept in the field of mathematics, playing a crucial role in various mathematical theories and applications. In this blog post, we will delve into the definition of a mathematical function, the concept of equal functions, and the different types of mathematical functions. Definition of a Mathematical Function A mathematical function is a relation between a set of inputs (the domain) and a set of outputs (the range), such that each input is related to exactly one output. In other words, for every input, there is a unique corresponding output. This relationship is often represented using function notation, such as f(x), where 'f' is the name of the function and 'x' is the input value. Explanation of the Concept of Equal Functions Two functions are considered equal if they produce the same output for every input in their respective domains. In other words, if the outputs of two functions are identical for all possible input values, then the functions are deemed equal. This concept of equality is crucial in various mathematical analyses and proofs. Types of Mathematical Functions Mathematical functions can be classified into different types based on their properties and characteristics. Some common types of mathematical functions include: • Linear functions: Functions that produce a straight line when graphed, and can be represented in the form f(x) = mx + b, where 'm' is the slope and 'b' is the y-intercept. • Quadratic functions: Functions that produce a parabola when graphed, and can be represented in the form f(x) = ax^2 + bx + c, where 'a', 'b', and 'c' are constants. • Exponential functions: Functions that have a constant base raised to the power of the input value, and can be represented in the form f(x) = a^x, where 'a' is the base. • Trigonometric functions: Functions that are based on the trigonometric ratios of angles in right-angled triangles, such as sine, cosine, and tangent functions. Methods for proving two functions are equal When it comes to understanding mathematical functions, it is crucial to be able to prove the equality of two functions. There are several methods to do so, each with its own advantages and applications. In this post, we will explore three common methods for proving the equality of two functions: direct substitution, algebraic manipulation, and graphical analysis. A. Direct substitution method The direct substitution method involves evaluating both functions at the same point or set of points to demonstrate that they produce the same output. This method is straightforward and can be applied to any type of function, making it a versatile tool for proving equality. Steps for using the direct substitution method: • Evaluate both functions at the same point or set of points • Compare the results to show that they are equal B. Algebraic manipulation method The algebraic manipulation method involves manipulating one or both of the functions through algebraic operations to show that they are equivalent. This method is particularly useful for functions with complex expressions or multiple terms. Steps for using the algebraic manipulation method: • Perform algebraic operations on one or both functions to simplify their expressions • Show that the simplified expressions are equal C. Graphical method The graphical method involves plotting the graphs of both functions on the same set of axes and examining their behavior to confirm their equality. This method provides a visual representation of the functions and can be particularly useful for functions with complex or non-standard forms. Steps for using the graphical method: • Plot the graphs of both functions on the same set of axes • Examine the graphs to show that they coincide, indicating equality By employing these methods, mathematicians and scientists can confidently prove the equality of two functions, furthering their understanding of mathematical relationships and paving the way for new discoveries and applications. Identifying key properties of functions When trying to prove two functions are equal, it is important to identify key properties that can help establish their equivalence. Three important properties to consider are symmetry, periodicity, and asymptotes. A. Symmetry Symmetry is a critical property to consider when comparing two functions. A function is said to be symmetric if its graph remains unchanged after a certain transformation. There are three main types of symmetry to consider: • Even symmetry: A function f(x) is even if f(x) = f(-x) for all x in the domain. This means the graph of the function is symmetric with respect to the y-axis. • Odd symmetry: A function f(x) is odd if f(x) = -f(-x) for all x in the domain. This means the graph of the function is symmetric with respect to the origin. • Periodicity Periodicity is another important property to consider when comparing functions. A function is periodic if it exhibits repetitive behavior at regular intervals. This can be expressed mathematically as f(x + T) = f(x), where T is the period of the function. When comparing two functions, it is important to determine if they share the same period or if one function is a multiple of the other. C. Asymptotes Asymptotes are imaginary lines that a graph approaches but never touches. When comparing functions, it is important to consider their asymptotic behavior. Two common types of asymptotes to consider are: □ Vertical asymptotes: A vertical line x = a is a vertical asymptote of the graph of a function f if the graph approaches the line as the value of x gets close to a from either side, but does not cross it. □ Horizontal asymptotes: A horizontal line y = b is a horizontal asymptote of the graph of a function f if the values of f(x) get close to b as x approaches positive or negative infinity. Applying Theorems and Properties in Function Equality Proofs When proving that two functions are equal, it is important to apply theorems and properties that are related to function composition, properties of inverse functions, and limit properties. These tools can help simplify the proof process and provide a solid foundation for demonstrating the equality of functions. Theorems Related to Function Composition □ Composition of Functions Theorem: This theorem states that if two functions f and g are defined such that the range of g is contained in the domain of f, then the composition of f and g, denoted as f(g(x)), is also a function. □ Associative Property of Function Composition: This property states that the composition of functions is associative, meaning that the order in which functions are composed does not matter. In mathematical terms, (f ∘ g) ∘ h = f ∘ (g ∘ h). Properties of Inverse Functions □ Definition of Inverse Functions: Two functions, f and g, are inverses of each other if and only if the composition of f and g yields the identity function, and vice versa. Symbolically, if f (g(x)) = x and g(f(x)) = x, then f and g are inverses. □ Properties of Inverse Functions: Inverse functions have the property that (f ∘ g)(x) = x and (g ∘ f)(x) = x, which is essential in proving the equality of functions. Utilizing Limit Properties in Function Equality Proofs □ Limit Laws: The properties of limits, such as the sum, difference, product, and quotient laws, can be used to simplify expressions involving functions and their limits. These laws can help establish equality between functions by manipulating their limits. □ Limit Properties of Composite Functions: Understanding how limits behave with composite functions is crucial in proving function equality. Utilizing properties such as the limit of a composite function being the composite of the limits can aid in the proof process. Real-world applications of function equality proofs Mathematical function equality proofs have wide-ranging applications in various real-world fields. Some of the key areas where these proofs are essential include: □ Designing and analyzing systems: Function equality proofs are crucial in engineering for designing and analyzing systems. Engineers often use mathematical models to describe the behavior of systems, and proving that two functions are equal helps ensure the accuracy and reliability of these models. □ Control systems: In areas such as electrical engineering, function equality proofs play a vital role in the design and analysis of control systems. These proofs help engineers verify the equivalence of different control algorithms or system behaviors. □ Quantum mechanics: In the field of physics, function equality proofs are used to establish the equivalence of different mathematical formulations in quantum mechanics. This is critical for ensuring the consistency and validity of theoretical predictions. □ Fluid dynamics: Function equality proofs are applied in fluid dynamics to demonstrate the equivalence of different mathematical models used to describe the behavior of fluids. This is essential for accurately predicting fluid flow and behavior in various practical scenarios. □ Financial modeling: In economics and finance, function equality proofs are used to validate different mathematical models and financial equations. This is crucial for ensuring the accuracy of financial predictions and investment strategies. □ Market analysis: Function equality proofs are also employed in the field of economics to establish the equivalence of different market analysis models and forecasting techniques. Proving the equality of functions helps economists make more reliable predictions and decisions. Proving that two functions are equal is crucial in mathematical analysis and problem-solving. It allows us to verify the accuracy of mathematical models and make confident deductions based on their equality. Understanding mathematical functions and their equality is essential for anyone working in fields like engineering, physics, economics, and more. It provides a solid foundation for reasoning and decision-making in various real-world situations. Final thoughts In conclusion, grasping the concepts of mathematical functions and equality not only enhances our problem-solving abilities but also equips us with a valuable skill set applicable in a wide range of professions. As we delve deeper into the world of mathematics, the significance of understanding and proving function equality becomes increasingly apparent, shaping our understanding of the world around us. ONLY $99 Immediate Download MAC & PC Compatible Free Email Support	{"url":"https://dashboardsexcel.com/blogs/blog/understanding-mathematical-functions-prove-two-functions-equal","timestamp":"2024-11-14T17:49:54Z","content_type":"text/html","content_length":"217515","record_id":"<urn:uuid:2b3554a5-a01c-48f2-9e25-c06dfb594a48>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00841.warc.gz"}
seminars - Approximation Methods of Multivariate Functions for Homomorphic Data Ordering Homomorphic Encryption (HE) is a cryptographic primitive that enables computations between encrypted data without decryption. HE allows operations that use sensitive data to be delegated to others who service outsourced computations while data information is not exposed. With these characteristics, HE is considered one of the important technologies for privacy preservation. Most HE schemes, however, support few operations only, mainly multiplication and addition. Thus, non-polynomial operations between word-wisely encrypted data require much more computational cost than between plain data. Although many approximation algorithms have been suggested to solve this problem, these works mainly focus on the one-variable functions and cannot be directly generalized to the multivariable functions because of the algorithmic limit or the growth of computational cost. In this thesis, we propose new approximation methods of three fundamental multivariate functions: sorting, max index, and softmax. First, We propose an efficient sorting method for encrypted data that works with approximate comparison. Using our method as a building block, we exploit k-way sorting network algorithm to show the implementation result that sorting 5^6=15625 data using 5-way sorting network which is about 23.3% faster than sorting 2^14=16384 data using the general 2-way method. Second, we propose a polynomial approximation method of the multivariate max function that inherits the method of Cheon emph{et al.} (ASIACRYPT 2020). Our algorithm is the generalization of the previous two-variable approach of approximating sign function, analyzing that our algorithm requires 30% less depths to find the largest element than using a state-of-the-art two-variable comparison. Lastly, we suggest the approximation method for softmax activation for a neural network model. By exploiting the algorithm, we develop a secure multi-label tumor classification method using the CKKS scheme, the approximate homomorphic encryption scheme.	{"url":"https://www.math.snu.ac.kr/board/index.php?mid=seminars&page=65&sort_index=Time&order_type=desc&document_srl=828279","timestamp":"2024-11-13T00:56:28Z","content_type":"text/html","content_length":"48388","record_id":"<urn:uuid:2957992b-c1cc-40d2-81ca-1a63bb709dc6>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00029.warc.gz"}
C++ cmath log() - Calculate Natural Logarithm \| Vultr Docs The log() function in the C++ cmath library computes the natural logarithm (base e) of a given input. Natural logarithm calculations are critical in various scientific, engineering, and financial computations where exponential growth or decay is involved. This function is essential for analyzing trends and changes that follow a logarithmic scale. In this article, you will learn how to use the log() function effectively in C++. You will explore how to handle different types of inputs, deal with potential errors, and see practical examples that help illustrate its use in real-world applications. Applying the log() Function Basic Usage of log() 1. Include the <cmath> header in your C++ program to access the log() function. 2. Call the log() function with a numeric argument. #include <iostream> #include <cmath> int main() { double value = 2.718; double result = log(value); std::cout << "Natural logarithm of " << value << " is " << result; return 0; This code computes the natural logarithm of approximately e (2.718), which should ideally return a value close to 1. The log() function is part of the cmath library and calculates the natural Working with Special Values 1. Understand how log() behaves with different types of special input values, such as negatives and zero. 2. Implement checks before using log() to avoid domain errors. #include <iostream> #include <cmath> #include <limits> int main() { double values[] = {0.0, -1.0, 1.0}; for(double value : values) { if(value <= 0) { std::cout << "Log not defined for " << value << std::endl; } else { double result = log(value); std::cout << "Natural logarithm of " << value << " is " << result << std::endl; return 0; Here, the code iterates through an array containing 0.0, -1.0, and 1.0. The log() function is not defined for non-positive values, and using log() with such values without checks could lead to undefined behavior or domain errors. Usage in Complex Calculations 1. Incorporate the log() function into more complex mathematical formulas where the natural logarithm is a part. 2. Use log() to solve real-world problems involving exponential growth or decay, such as calculating compound interest or half-life periods. #include <iostream> #include <cmath> int main() { double rate = 0.05; // 5% growth per year double time = 20; // 20 years double initialAmount = 1000; // initial amount of $1000 // Calculate final amount after 20 years of continuous growth double finalAmount = initialAmount * exp(rate * time); std::cout << "Final amount after 20 years: $" << finalAmount << std::endl; // Calculate time required to double the initial amount double doubleTime = log(2) / rate; std::cout << "Time required to double the initial amount: " << doubleTime << " years" << std::endl; return 0; In this example, exp() is used together with log() to model continuous growth and calculate the time required to double an investment at a constant annual growth rate. The log() function helps convert the goal of doubling the investment into the time needed at a given rate. The log() function from the C++ cmath library is indispensable for computations involving natural logarithms. Mastering its usage, along with understanding the considerations of different input values, enriches the capability to tackle various computational problems in science, finance, and engineering. By integrating the knowledge gained here, boost your efficiency in solving complex calculations that involve logarithmic functions.	{"url":"https://docs.vultr.com/cpp/standard-library/cmath/log","timestamp":"2024-11-10T22:40:31Z","content_type":"text/html","content_length":"419184","record_id":"<urn:uuid:d59a310a-9ca0-4e29-96fe-d15a2347a75d>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00772.warc.gz"}
An optical lattice of flux A suitable optical lattice for cold atoms could produce a large effective magnetic field in which the atoms would realize analogs to quantum Hall states. Figure 1: The accumulation of phase as a spin-$1/2$ particle moves through an inhomogeneous (quadrupole) magnetic field. The black vectors indicate the component of the normalized effective Zeeman field in the ${e}_{x}-{e}_{y}$ plane and the background color represents the ${e}_{z}$ component. The Bloch spheres show the enclosed solid angle, which sets the particle’s accumulated phase as it moves in a loop about the origin. Ultracold neutral atoms are among the most simple and flexible of quantum many-body systems. As such, they offer the capability to realize strongly interacting systems in their most fundamental form, absent the unwanted complexities that complicate (and sometimes enrich) their solid-state brethren. The question then arises: What classes of systems can be implemented with cold atoms, and of these, which can offer insight beyond that afforded in more conventional systems? In a paper published in Physical Review Letters, Nigel Cooper at the University of Cambridge, UK, proposes an elegant technique to take ultracold atoms to the extreme, where atoms moving about in a lattice potential experience an effective magnetic field with of order a unit flux quanta per lattice site [1]. This is a realm that is inaccessible in conventional materials and promises new types of quantized Hall effects where Landau-level quantization and band-structure effects are intertwined. Magnetic fields are enigmatic. In our studies we learn the Lorentz force law: in a uniform magnetic field, the force on a moving, charged object is perpendicular to both the magnetic field $B$ and the object’s velocity $v$. This doesn’t fit in with our usual intuitive picture of forces derived from gradients of potentials, but instead requires a new type of potential: the electromagnetic vector potential $A$. Like the usual scalar potential $ϕ$, the vector potential is related to the seemingly more physical fields via derivatives: the magnetic field $B=∇×A$. These potentials are not just mathematical sleights-of-hand; in quantum systems, they take center stage. In Schrödinger’s wave mechanics, the evolution of a particle’s wave function can be partially understood in terms of its quantum mechanical phase. Usually this phase can be divided into two parts: the dynamic phase acquired in proportion to the particle’s kinetic energy, and the phase from scalar potentials. Each of these accumulates at a rate proportional to the associated energy. If we consider a particle moving in a closed loop, with zero scalar potential, then the dynamic phase acquired upon traversing the loop will tend to zero as the velocity drops to zero. The phase acquired in a magnetic field is different: it depends on the geometry of the particle’s path. If our loop now encloses a magnetic flux $Φ=∫B⋅da$, then the particle will acquire an additional phase proportional to $Φ$. This interpretation is uncomfortable: somehow the particle has a nonlocal knowledge of the magnetic field everywhere inside the loop. It is more natural to think of the vector potential, in which case the acquired phase is no more than the line-integral $∫A⋅dl$ of the vector potential around the loop. This leads to the celebrated Aharonov-Bohm effect [2,3] where a charged particle acquires a geometric phase as it moves in the completely field-free region outside an infinite solenoid. Other physical situations produce geometric phases in which neutral particles can behave as if magnetic fields were present. This concept was introduced to quantum mechanics as Berry’s phase [4] for particles with internal structure (like spin states) for which the energy is dependent on parameters in the Hamiltonian, such as position or momentum. If a particle starts in an eigenstate, it can acquire a geometric phase upon traversing a closed loop in parameter space, provided the “motion” is sufficiently slow that it adiabatically remains in the same eigenstate. The simplest example of a Berry’s phase is shown in Fig. 1 where a neutral spin- $1/2$ particle is moving in an inhomogenous magnetic field, giving rise to a position-dependent Zeeman shift—the difference in energy between the spin being oriented along or away from the magnetic field. The figure depicts the particle’s trajectory, along with the orientation of its ground state on the Bloch sphere. (This sphere defines the allowed states of a spin- $1/2$ particle.) The particle accumulates a geometric phase of $Ω/2$, one-half the solid area traced out on the Bloch sphere as the particle moves in space. Such a phase can be interpreted as arising from a geometric gauge field, in analogy with the electromagnetic vector potential. Credit: (Bottom) from [ Figure 2: Configurations that give rise to Berry’s phases for cold atoms in an optical lattice that generates an artificial gauge field. (Top) Existing techniques can produce an infinitely precessing Zeeman field along ${e}_{x}$, but not along ${e}_{y}$. (Bottom) Cooper’s proposal remedies this problem by allowing a net positive Berry’s phase in the lattice’s unit cell. Geometric phases are real. A system particularly well suited for observing and studying their effects is a trap of ultracold atoms. In these systems researchers can first construct suitable geometries and then unambiguously measure the result. Geometric phases in neutral atoms were vividly demonstrated at MIT in 2002 when topological phases were directly imprinted into the wave function of a Bose-Einstein condensate (BEC) of sodium atoms in a magnetic field by inverting the orientation of the field. This produced $2π$ or $4π$ phase windings leading to vortices in the final wave function [5]. Going beyond this elegant demonstration, the next step is more powerful: constructing configurations where the atomic system obeys a new Hamiltonian containing steady-state gauge fields [6]. Such a system mimics that of a particle in an effective magnetic field [7]. Instead of using a magnetic field to generate a Berry’s phase, these ideas require a laser to couple different internal (spin) states of atoms. Such coupling is formally equivalent to a Zeeman magnetic field, but spatially structured on the scale of the optical wavelength. In our group at NIST, we followed these theory proposals with a series of experiments demonstrating the effective mapping between the Berry’s phase and the electromagnetic vector potential, leading to artificial magnetic and electric fields (see Refs. [8,9] and references therein). The top of Fig. 2 depicts how these ideas lead to large Berry’s phases, and also highlights a key limitation. The illustrated Bloch vector can wind an unlimited number of times around its equator as an atom moves along $ex$, but it can tip to the $±z$ axes of the Bloch sphere when the atom moves along $ey$. This implies that we can create large “artificial magnetic fields,” but the maximum magnetic flux passing through the system scales as the length of the system, not its area, making it difficult to scale the artificial field to larger systems. Cooper proposes a technique to overcome this limitation [1] by creating a specific effective Zeeman field using standing waves of light—a type of lattice. The essence of his proposal is depicted in the bottom of Fig. 2: atoms moving in the optical unit cell experience a Berry’s phase, both as a function of $x$ and $y$ (see also Ref. [10]). Usually such ideas lead to staggered magnetic fields with equal and opposite sign in neighboring lattice sites, with zero average. The current work overcomes this limitation, allowing for large effective magnetic fields with flux scaling as the system’s area, not its linear extent. How does this work? The effective electromagnetic vector potential created by this technique has a gauge-dependent singularity called a Dirac string that effectively concentrates magnetic field of one sign to isolated points (with no physical effect). As a result, the effective magnetic field is still staggered, but acquires a nonzero average. To more intuitively understand how this gives rise to an effective field of the same sign, consider an atom moving along each of the two closed loops depicted in Fig. 2 (bottom). For the left loop, the Zeeman field traces out a circle at the top of the Bloch sphere with a counterclockwise direction, while the right loop traces a circle at the bottom of the Bloch sphere with a clockwise direction. If the top loop acquired a geometric phase $ϕ$, then the bottom loop acquired a phase of $-(2π-ϕ)$ from the entire top portion of the Bloch sphere, but with the opposite sign. Since the acquired phase is defined only modulo $2π$, these two curves enclose the same effective field. This seemingly simple observation provides a straightforward path to realize larger effective magnetic fields than have hitherto been possible, yet with a comparatively simple set of lasers. (The pioneering proposals for large in-lattice gauge fields require more numerous lasers [11], and lack the simplicity and elegance of this approach.) To complete the story, Cooper studied the properties of the lowest Bloch bands in this optical flux lattice and computed the Chern number. (Here, the Chern number enumerates the number of times the wave function’s phase winds by $2π$ on a path running from one side of the Block band to the other: a loop on the torus). In some cases, the Chern number is $±1$, showing that these bands are topologically equivalent to the lowest Landau level (also with Chern number $±1$). Thus fermions completely filling the lowest band will be an integer quantum Hall state with a quantized Hall resistance $RH=h/e2$. The atoms used in cold-atom experiments are typically bosons and do not have a Fermi energy, but at fillings of around one per-lattice site—about one atom per magnetic flux quanta—they are expected to display interaction-driven bosonic fractional quantum Hall states. I would like to thank V. Gurarie and A. Lamacraft for discussions that prepared me to appreciate the current work. I also acknowledge the financial support of the NSF through the PFC at JQI, and the ARO with funds from both the Atomtronics MURI and the DARPA OLE Program. Correction (29 April 2011): References [8] and [9] were corrected.	{"url":"https://physics.aps.org/articles/v4/35","timestamp":"2024-11-05T16:43:33Z","content_type":"text/html","content_length":"42246","record_id":"<urn:uuid:ae17761f-9110-4825-a208-4f762a241cef>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00529.warc.gz"}
Squeezed Phonons via Second Order Raman Scattering A stimulated second-order Raman scattering process. The subscript k1 (k2) represents the higher- (lower-) energy incident coherent photons. The arrows show the directions of the corresponding photon and phonon momenta. The incoming coherent photon beams are about parallel to each other. The actual physical process is as follows: an incident photon in mode k1 interacts with a solid, producing two acoustic phonons, and leaves in mode k2. For appropriate choice of initial phtoon and phonon states, the acoustic phonons can be in a two-mode squeezed state. Notice that there are also additional incoming photons in mode k2, therefore this mode becomes stronger after the interaction, at the expense of mode k1. Also notice that the vector sum of the two phonon momenta is the difference of the input and output photon momentum vectors. Since photon wave vectors are generally much smaller than the phonon ones \|ki\| << \|qs\|, the two acoustic phonon modes have nearly opposite wave vectors ±qs. This figure is not to scale. The appropriate scale in the figure would make all four lines of photons and phonons vertical and almost parallel with each other. For the sake of clarity, we have increased the angle between them. Image Source: X. Hu, Quantum Fluctuations In Condensed Matter Systems, UM Ph.D. Thesis, 1997, Page 59. X. Hu and F. Nori, UM Preprint, 1995; Physical Review Letters 76, 2294 (1996); Physical Review Letters 79, 4605 (1997).	{"url":"https://public.websites.umich.edu/~nori/squeezed/figure5.htm","timestamp":"2024-11-12T16:50:05Z","content_type":"text/html","content_length":"2631","record_id":"<urn:uuid:27327d96-dfdf-4415-ab7c-5160d3f37c56>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00686.warc.gz"}
Chicken Joggers Problem C Chicken Joggers In the woods of Lill-Jansskogen, there is a network of trails that are often used by joggers. The trails have been much appreciated, and have been specially selected by the professors of the Royal Institute of Technology, enabling university students to take a short break from their studies and refresh their smart minds. Strangely enough, the network of trails actually form a tree. When the trails were selected, the professors of the university took the set of trails that they found in Lill-Jansskogen and created a minimum spanning tree, in order to “encourage and inspire computer science students to participate in physical activities by applying graph theory in the beautiful surroundings of the Royal Institute of Technology”. Unfortunately, the computer science students are not that brave. Winter is approaching, and it is getting darker and darker in the city of Stockholm. Recently, a bunch of programmers from CSC (Community of Scared Cowards) have been complaining that it is too dark in parts of the trail network at night. Some of the trails are lit up by lamps, but sometimes that is not enough for the CSC. They would like to see that all the trails that they might use are lit up properly! You have been tasked with satisfying the cowards by placing lamps at intersections. For economic reasons, it might not be possible to place lights at all intersections, so it will suffice to make sure that there is a lamp in at least one of the two intersections adjacent to a trail that could possibly be used by the joggers. Some intersections already have lamps, and of course, you can keep using those lamps. You don’t know exactly what trails the joggers are using, but you do know that the joggers will always start and finish at the university campus. You also know that joggers are training for an upcoming marathon, so they always run exactly $S$ meters in total. A jogger might turn around at any point in time, even in the middle of a trail and she can do so as many times as she wants, in order to fulfill the requirement of running exactly $S$ meters. You will be given a map of the woods and the jogging trails included in the minimum spanning tree created by the professors. It is guaranteed that there is exactly one route between each pair of intersections, where a route is a set of adjacent trails. Your task is to find the minimum number of additional lamps you needed in order to satisfy the frightened runners, no matter which trails they use (subject to the restrictions above) Input starts with two integers $N$ ($2 \leq N \leq 50\, 000$), and $S$ ($1 \leq S \leq 10^4$), the number of intersections and the total distance in meters that a jogger wants to run, respectively. Then follow $N-1$ lines with three integers $a$ ($1 \leq a \leq N$), $b$ ($1 \leq b \leq N$), $d$ ($1 \leq d \leq 100$), meaning that there is a bidirectional trail between intersection $a$ and $b$ with length $d$ meters. Then follows a line with a single integer $L$ ($0 \leq L \leq N$), the number of lamps that have already been placed. Then follow $L$ distinct integers $\ell _1, \dots , \ell _ L$ on one line, meaning there is already a lamp placed at intersections $\ell _1, \dots , \ell _ L$. The university campus is at intersection number 1. Output contains a single integer, the minimum number of additional lamps you need to place in order to satisfy the joggers’ requirements. Sample Input 1 Sample Output 1 Sample Input 2 Sample Output 2 Sample Input 3 Sample Output 3	{"url":"https://liu.kattis.com/courses/AAPS/AAPS24/assignments/gooe7g/problems/joggers","timestamp":"2024-11-08T12:06:28Z","content_type":"text/html","content_length":"29964","record_id":"<urn:uuid:7c601697-6f74-49f4-a0de-c107c3a021dd>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00659.warc.gz"}
Talk 61 - Horizons are Watching You Talk 61 - Horizons are Watching You (2023). Talk 61 - Horizons are Watching You. Perimeter Institute for Theoretical Physics. https://pirsa.org/23070007 Talk 61 - Horizons are Watching You. Perimeter Institute for Theoretical Physics, Jul. 31, 2023, https://pirsa.org/23070007 @misc{ scivideos_PIRSA:23070007, doi = {10.48660/23070007}, url = {https://pirsa.org/23070007}, author = {}, keywords = {Quantum Fields and Strings, Quantum Foundations, Quantum Information}, language = {en}, title = {Talk 61 - Horizons are Watching You}, publisher = {Perimeter Institute for Theoretical Physics}, year = {2023}, month = {jul}, note = {PIRSA:23070007 see, \url{https://scivideos.org/pirsa/23070007}} Gautam Satishchandran Talk numberPIRSA:23070007 We show that if a massive (or charged) body is put in a quantum superposition of spatially separated states in the vicinity of any (Killing) horizon, the mere presence of the horizon will eventually destroy the coherence of the superposition in a finite time. This occurs because, in effect, the long-range fields sourced by the superposition register on the black hole horizon which forces the emission of entangling “soft gravitons/photons” through the horizon. This enables the horizon to harvest “which path” information about the superposition. We provide estimates of the decoherence time for such quantum superpositions in the presence of a black hole and cosmological horizon. Finally, we further sharpen and generalize this mechanism by recasting the gedankenexperiment in the language of (approximate) quantum error correction. This yields a complementary picture where the decoherence is due to an “eavesdropper” (Eve) in the black hole attempting to obtain "which path" information by measuring the long-range fields of the superposed body. We explicitly compute the quantum fidelity to determine the amount of information such an Eve can obtain and show, by the information-disturbance tradeoff, a direct relationship between the information gained by Eve and the decoherence of the superposition in the exterior. In particular, we show that the decoherence of the superposition corresponds to the "optimal" measurement made by Eve in the black hole interior.	{"url":"https://scivideos.org/pirsa/23070007","timestamp":"2024-11-10T22:05:26Z","content_type":"text/html","content_length":"51661","record_id":"<urn:uuid:15cecfa2-7497-4164-b39c-10afdfff213f>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00323.warc.gz"}
Numbers and notes February 4th, 2012 \| Author: Gina Collecchia Comments Off on Gibbs phenomenon is boring, so this is going to be short. Basically, Gibbs phenomenon was discovered by a guy with the last name “Gibbs” when he saw that the Fourier series of nondifferentiable waves is least-good where the waves are not differentiable. Yeah…okay. So, when a wave has sharp edges/corners like a square wave or the absolute value function, the Fourier series representation will be infinite and have these little tail thingies sticking out at these corners and edges. Honestly, it’s not that surprising. These thingies are called “Gibbs horns” or “Gibbs’ horns” or if the writer just hates checking Wikipedia, “Gibb’s horns,” and the points at which they exist are usually called “jump discontinuities,” i.e., sharp edge corner things.	{"url":"http://numbersandnotes.com/tag/fourier-series/","timestamp":"2024-11-02T12:40:09Z","content_type":"application/xhtml+xml","content_length":"29036","record_id":"<urn:uuid:8d90fe15-35d5-4de3-ba23-33b87e336208>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00709.warc.gz"}
Ask the Teacher: Is 11 a Teen Number? – KindergartenWorks Ask the Teacher: Is 11 a Teen Number? Here’s a great kindergarten math question that I posed to the awesome kindergarten teachers on facebook. “Is 11 a teen number?” Here is your answer straight from the source – what real kindergarten teachers have to say about 11 being a teen number. Plus, I’ll weigh in on the subject too with a pictured explanation. Teachers were polled: Is 11 a teen number? When asked, the overwhelming majority of kindergarten teachers said yes. And yes – as the people who teach these things to five and six-year-olds – we are in agreement with our kinders that these irregularly named numbers indeed should have been named oneteen, twoteen and threeteen… {wink} What the teachers said Here are examples of what other kindergarten teachers had to say: Yes, teen family is numbers 10-19. – Diana B. In the Kindergarten world, the answer is yes. It makes sense to go with teens starting with 10-19 on the hundreds chart when you work with numbers to 100. You just go with a few having oddball names 11 and 12. In the growing up teenager world…no, it’s technically 13-19. – Jeanne S. Bahaha, I just had this conversation with my Kinders yesterday . I told them that even though you don’t say eleven-teen or twelve-teen they are still teen numbers because (like the song says) they start with a 1. Thank you Harry Kindergarten, lol! – Maureen D. Yes, 1 group tens and some ones. Both #’s require you bundle and unbundle to regroup when adding and subtracting. If not a teen then what? – Karri C. Yes, however I tell my students people aren’t teenagers until they are 13 – Jennifer L. A side note: I agree with my peers that mathematically speaking 11 is a teen number – but if you’re categorizing the ages of children – then 11 is a pre-teen number. There is a big difference in the maturity and development that happens in those pre-teen years. Now, here’s a little bit more about why 11 is indeed a teen number when we’re talking all things math. Why 11 is a Teen Number It’s all based on our numbering system (which is called base ten). Our numbers become grouped based on sets/groups of ten. That means every time you hit a new grouping of ten objects, the number range changes in name. If a number has one group of ten and additional ones (single objects), then I believe it qualifies as a teen number. Let’s look at how this works by starting out with the number 9. Nine is nine single objects. It’s not a teen number since it lacks a group of ten. When you add one more, you make a complete group of 10. 10 is one group and zero single objects. The 1 in 10 represents the group and the 0 represents the zero objects. Ten is not a teen number even though it has one group of ten. It lacks the additional ones. When you add one more, you have 11. Let me model how to use the ten frame and a crayon to show how to decompose the numbers into tens and ones. 11 is one group of ten and one single object. The first 1 in 11 represents the group of ten and the second 1 represents the 1 single object. It is the first teen number since it has just one group of ten and additional ones. To hit this point home, lets take a number that says “teen” in the name. Let’s decompose 17 into tens and ones. 17 is one group of ten and seven single objects. The 1 in 17 represents the group of ten and the 7 represents the 7 single objects. It plainly says “teen” but qualifies since it has one group of ten and additional ones. Here is a famous video (in the world of kindergarten anyway) that teaches how the teens all start with a 1. Catchy, isn’t it? I literally didn’t get this at all growing up. I was taught using tally marks and bundles of straws during calendar time. Yeah, that didn’t work. Only once I started teaching the standard of composing and decomposing numbers 11-19 to my own classes of kindergartners using the ten frame did I understand the huge importance this concept plays in the world of understanding numbers. I wish I had been taught this using a tool like the ten frame growing up. It’s so powerful that I made my own ten frame manipulatives to use with my class and we used them all the time! So, yes – 11 is a teen number. So are 12, 13, 14, 15, 16, 17, 18 and 19. Here’s a lesson example to teach how to compose and decompose teen numbers. It can even be used with accelerated kindergartners or even first graders with numbers to 99. If you like what I do here on KindergartenWorks, then be sure to subscribe today. I look forward to sharing ideas with you weekly. Do you work on the teen numbers in your class? Here are great printable tools to teach numbers 11-19:	{"url":"https://www.kindergartenworks.com/guided-math/is-11-a-teen-number/","timestamp":"2024-11-07T23:02:40Z","content_type":"text/html","content_length":"176542","record_id":"<urn:uuid:62122ebc-b14a-4b81-9750-9b82ad43c337>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00118.warc.gz"}
How many Minutes in a Year: Total Minutes in Regular & Leap Year How many Minutes in a Year: Number of Minutes in Regular and Leap Years Wondering there are how many minutes in a year? There are 525,600 minutes in an ordinary year, and 527,040 minutes in a leap year. The extra 1,440 minutes in a leap year are from the extra day that is February 29 (24 hours * 60 minutes = 1440). To calculate the number of minutes in a year, we first obtain the total days in the year. We then multiply the total days by 24 in order to get the total hours. These total hours are then multiplied by 60 to get the total minutes in a year. How many Minutes in a Year: Number of Minutes in Regular and Leap Years How many minutes in a year? The total minutes in an ordinary year is 525,600 minutes. However, there are 366 days in a leap year. As a result, the total minutes in a leap year like 2024 is 527,040 minutes. How many minutes in 2024? There are 527,040 minutes in 2024. Total minutes for each month in 2024 The following are the total minutes for each month in 2024, and the total minutes for the year 2024. Total minutes in each month of 2024 Month Number of Days Total minutes January 31 44,640 minutes February 29 41,760 minutes March 31 44,640 minutes April 30 43,200 minutes May 31 44,640 minutes June 30 43,200 minutes July 31 44,640 minutes August 31 44,640 minutes September 30 43,200 minutes October 31 44,640 minutes November 30 43,200 minutes December 31 44,640 minutes Total minutes in 2024: 527,040 minutes Total weeks, days, hours, minutes and seconds in 2024 The total weeks, days, hours, minutes and seconds in 2024 are: Share On Your Favorite Social Media! Use the following links to spread the word...	{"url":"https://www.mpesacharges.com/how-many-minutes-in-a-year/","timestamp":"2024-11-08T03:07:22Z","content_type":"text/html","content_length":"50059","record_id":"<urn:uuid:a53007d2-9b5e-43a9-aafe-e761abaf8fa4>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00387.warc.gz"}
Calculation of the static forces acting on ACV bag-finger skirts A mathematical model of the geometry formation of an ACV bag-finger skirt is developed to determine skirt shape and its deflection for varying cushion pressures, and calculations for the reactions of supports of the rigid structure on the skirt at the inner and outer attachment points are obtained. The model assumption that the finger triangle of the two-dimensional bag-finger skirt turns around the inner attachment point with changing ratio of cushion pressure to bag pressure is confirmed by experiments using skirt rigs. Good agreement is found between theoretical and experimental results, and it is shown that when the cushion pressure is changing, the pressure ratio is the essential dimensionless parameter for the skirt geometry formation and its deflection, and for the forces acting on it. 1985 Joint International Conference on Air Cushion Technology Pub Date: □ Ground Effect Machines; □ Skirts; □ Aerodynamic Forces; □ Frequency Response; □ Mathematical Models; □ Engineering (General)	{"url":"https://ui.adsabs.harvard.edu/abs/1985act..conf...16X/abstract","timestamp":"2024-11-07T22:57:14Z","content_type":"text/html","content_length":"34826","record_id":"<urn:uuid:b2438edb-4a88-4bec-a12d-304fb6d96fdd>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00427.warc.gz"}
Movable short-circuit technique to extract the relative permittivity of materials from a coaxial cell In recent years, industrial applications have been based on the use of intrinsic material properties that improve designs, processes, qualities and product controls. To get to those intrinsic parameters, various appropriate techniques are required. In this paper, a new technique has been developed and presented. It essentially puts the emphasis on the dielectric relative permittivity extraction from the principle of a movable short-circuit through the coaxial transmission-line cell. This technique is aimed at drastically reducing the discontinuity impacts at the interface feed line (connector) and ideal line, solving the phase constant frequency limit, stopping the constraints bound to the higher mode propagations and improving the accuracy level when the frequency range has increased. The technique is based on the use of the sum of two different lengths of the cell by removing the first value of the phase constant in the frequency range of interest when it is negative. This new technique can be easily implemented; its focus is not on iterative principles, but on the use of the constant propagation of a Quasi-TEM mode of the transmission-line. The bio-food industry (semolina), environmental field (palm tree) and building trade (aquarium sand) were used to test the validity of the technique in 2-20 GHz. 1. Introduction The material characterization has a wide variety of techniques [1-3], depending on the application domain [4]. These domains help to reveal why different materials show different properties and behaviors [5-7]. A large panel of material characterization techniques is described in the literature. Most of accurate techniques are developed in relation to the fixture’s geometry, the precise measurement apparatus and the kind and shape of the material to be tested [8]. The transmission-line technique is the well-known and popular in the literature. That technique can be used in transmission [7-9] and/or in reflection with open-end configuration [10, 11] or short-circuit [12]. All structures terminated by an open circuit and/or short-circuit use the iterative procedure [13]. Probe technique which is used for liquid through Cole-Cole theory [14], but also through combination of cavity mixed with probe [1] and [6] has a good accuracy according to the material thickness. For thick material, the free-space technique is quite often used [15, 16]. But the resonator technique, through the use of cavity is the best and is well-developed in [17]. The sample to be characterized defines the technique through its state and sharp [18]. All techniques have advantages and inconveniences. One of them is discontinuity that stops the scanned frequency range, promotes the spread of high-order modes [19] and increases measurement errors through the intrinsic material complex parameters [20]. In order to contribute to the techniques improvement, we have developed a new technique called Movable Short-Circuited Line (MSCL). This technique is broadband and belongs to the reflection transmission-line technique [12]. The use of two transmission-lines is a good way to reduce discontinuities instead of choosing an equivalent electric circuit as their representation [21]. Those discontinuities exist at the junction of connector and the ideal test cell interface. Discontinuity represents imperfections of the entire test cell. From the two transmission-line technique using the wave cascade matrix principal [7, 22], we have developed the mathematic formulation for the MSCL. It is based upon the sum of measurements when using two different lengths of the test cell, but in inserting the first value of the studied band. That has allowed sorting out the discontinuity impacts on the extraction of the relative permittivity. Both circular coaxial line fixtures are differentiated only by their lengths while diameters and conductors are identical. The material under test is trapped inside the fixture, and s-parameter measurements are done in two configurations: in presence and absence of the sample under test (SUT). The use of the propagation constant parameter through measurement of the reflection coefficient is the foundation of the technique. This is a quick, simple and highly reliable technique when the fixture is completely full of the material under test (MUT). The technique is suitable for destructive (wafer, etc.) as well as non-destructive (liquid, powder, etc.) samples. 2. Theory and mathematic model 2.1. Propagation constant determination The propagation constant $\gamma$ is the unique parameter we can extract, and it must be applied in this technique. The short-circuit reflection coefficient ${S}_{11}^{sc}$ is given in Eq. (1) as: ${S}_{11}^{sc}=\left\|{S}_{11}^{sc}\right\|{e}^{-2\gamma l}.$ Fig. 1 describes the test fixture, where $Z$ and $L$ are respectively the cell and the connector lengths. Fig. 1A coaxial test fixture design The propagation constant is determined from the reflection coefficient through the following expression: $\gamma l=-\frac{1}{2}\mathrm{l}\mathrm{n}\left(\frac{{S}_{11}^{sc}}{\left\|{S}_{11}^{sc}\right\|}\right).$ The propagation constant is a complex parameter as defined below. $\gamma l=\alpha l+j\beta l,$ where $\theta =\beta l$ is the electric length and $\alpha l$ the attenuation constant. We assume that only a Quasi-TEM mode propagates in and the vacuum configuration is linked to the filled structure by: ${\beta }_{d}={\beta }_{vac}\sqrt{{\mu }_{eff}{\epsilon }_{eff}},$ where ${\beta }_{d}$ and ${\beta }_{vac}$ are respectively the phase constants of the sample under test (SUT) and vacuum filled coaxial. In that case, the material relative parameters (permittivity and/or permeability) are obtained from equation (4) as: ${\mu }_{eff}{\epsilon }_{eff}={\left(\frac{{\beta }_{d}}{{\beta }_{vac}}\right)}^{2}.$ We have easily filled up the circular coaxial test fixture with the non-destructive material, the one we used to validate the technique implementation. 2.2. De-embedding technique The work requires the good knowledge of the connector propagation constant ${\gamma }_{0}$ as it is shown in the Fig. 2. and ${l}_{0}$ the connector length. Fig. 2A simplified fixture roadmap The discontinuity impedance ${Z}_{d}$ appears at the connector-fixture interface. From ${Z}_{in}^{sc}$ and ${Z}_{in}^{oc}$, respectively, the input impedance of the connector when the connector is in short-circuit and in open-circuit configurations. The connector propagation constant is as follows: ${\gamma }_{0}{l}_{0}=\mathrm{a}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left[{\left(\frac{{Z}_{in}^{sc}}{{Z}_{in}^{oc}}\right)}^{\frac{1}{2}}\right].$ If we call by ${S}_{11}^{g}$ the entire reflection coefficient and $\Gamma$ the input cell reflection coefficient, both are linked by Eq. (7) as: $\Gamma ={S}_{11}^{g}{e}^{2{\gamma }_{0}{l}_{0}}.$ Because the main and only goal of this technique is to determine the relative permittivity ${\epsilon }_{r}$ of the material as mentioned in introduction, the focus of our work is to get that parameters by using the phase constant of the structure filled of vacuum and/or material under test (MUT).Whereas it is necessary to neglect the material losses, we have come with a mathematic equation to solve the negative number of the phase constant and discontinuities as in imperfection between the contact interface and ideal connector line as follows: ${\left(\beta l\right)}_{id}={\left(\beta l\right)}_{mes}-{\left(\beta l\right)}_{mes}^{iv},$ where ${\left(\beta l\right)}_{mes}$ is the phase constant of the test cell, measured in any aforementioned configurations, and ${\left(\beta l\right)}_{mes}^{iv}$ its initial value once measured in the work frequency range. ${\left(\beta l\right)}_{id}$ is the ideal phase constant as $\beta l>0$. 3. Technique description Due to the high order mode propagations in the test cell (at interface of connector – ideal cell that leads imperfections) and the phase constant behavior which is the high order mode propagation consequence, the frequency range of study is limited [7-9]. On the other hand, the use of the short-circuit configuration involves the use of iterative method [12] and [21]; we have put forward this new technique. There are two methodologies we have developed for the relative permittivity ${\epsilon }_{r}$ or permeability ${\mu }_{r}$ determination. Both of them are concerned with a model based on the transmission-line propagation constant. The determination can be based on a variety of criteria. 3.1. Use of one transmission-line technique Let us consider a transmission-line cell, terminated by a short circuit in the end of the line. From the input impedance as illustrated below: Fig. 3Ideal transmission-line ended by short-circuit Eq. (9) is provided to calculate the input impedance once the line propagation constant is known. ${Z}_{in}={Z}_{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\gamma l\right).$ Taking into account the reflection coefficient after de-embedding the entire structure, the input impedance is given by: ${Z}_{in}={Z}_{0}\frac{1+\Gamma }{1-\Gamma }.$ If we suppose that the transmission line is lossless, in that case, $\gamma l\approx j\theta$, Eq. (9) will be rewritten. At the same time, we respectively named by ${\left({Z}_{in}^{sc}\right)}_{v}$ and ${\left({Z}_{in}^{sc}\right)}_{MUT}$ the input impedances in absence and presence of the MUT, which are linked to Eq. (9) by: $\frac{{\left({Z}_{in}^{sc}\right)}_{v}}{{\left({Z}_{in}^{sc}\right)}_{MUT}}=\sqrt{\frac{{\epsilon }_{eff}}{{\mu }_{eff}}}\frac{\mathrm{t}\mathrm{a}\mathrm{n}\left({\theta }_{v}\right)}{\mathrm{t}\ mathrm{a}\mathrm{n}\left({\theta }_{v}\sqrt{{\mu }_{eff}{\epsilon }_{eff}}\right)}.$ In this context, the iterative technique may be used. This solution can be watered down by its complexity through time taken to get to the aim. That’s why we have started by using a technique based on the average of two different extracted relative permittivity from two different lengths of a transmission line. If the test cell is homogenous like the one we manufactured and use to validate the technique process, then Eq. (11) becomes ${\mu }_{{r}_{1}}{\epsilon }_{{r}_{1}}={\left[\frac{{\left({\beta }_{1}{l}_{1}\right)}_{mes}^{MUT}-{\left({\left({\beta }_{1}{l}_{1}\right)}_{mes}^{MUT}\right)}_{iv}}{{\left({\beta }_{1}{l}_{1}\ right)}_{mes}^{v}-{\left({\left({\beta }_{1}{l}_{1}\right)}_{mes}^{v}\right)}_{iv}}\right]}^{2},$ and for the second transmission-line, we used the following equation in Eq. (12b), ${\mu }_{{r}_{2}}{\epsilon }_{{r}_{2}}={\left[\frac{{\left({\beta }_{2}{l}_{2}\right)}_{mes}^{MUT}-{\left({\left({\beta }_{2}{l}_{2}\right)}_{mes}^{MUT}\right)}_{iv}}{{\left({\beta }_{2}{l}_{2}\ right)}_{mes}^{v}-{\left({\left({\beta }_{2}{l}_{2}\right)}_{mes}^{v}\right)}_{iv}}\right]}^{2}.$ We have noticed that the real relative permittivity or relative permeability is obtained once the average operation is done, as follows Eq. (13a): ${\mu }_{r}{\epsilon }_{r}=\frac{{\mu }_{{r}_{1}}{\epsilon }_{{r}_{1}}+{\mu }_{{r}_{2}}{\epsilon }_{{r}_{2}}}{2}$ We have named ${\left({\beta }_{k}{l}_{k}\right)}_{mes}^{x}$ and ${\left({\left({\beta }_{k}{l}_{k}\right)}_{mes}^{x}\right)}_{iv}$ the phase constants when the test cell is filled up of MUT or not and measured ($x$ is MUT or vacuum and, $k=1$ or $2$, the number of the cell that we used). If the sample to characterize is not magnetic, the final Eq. (13a) will approximately be reduced to: ${\epsilon }_{r}\cong \frac{{\epsilon }_{{r}_{1}}+{\epsilon }_{{r}_{2}}}{2}$ The combination of both transmission lines can be done in another way. This is the technique we have named “Movable Short-Circuit Line” (MSCL). 3.2. Movable short-circuit transmission-line technique In this section, we combined two transmission lines, having the same characteristic impedance, geometry and dimensions (inner and outer diameters), but different lengths. The short-circuit position is considered movable because of the fixture’s length as shown in Fig. 4. Fig. 4Two transmission-lines with short-circuit at the bottom In this situation, discontinuities are the same and, we assume that higher propagation modes cannot spread or propagate in. The ideal line, which is the length’s difference, $∆l={l}_{2}-{l}_{1}$, has a propagation constant given in Eq. (14). ${\beta }_{r}∆l={\beta }_{2}{l}_{2}+{\beta }_{1}{l}_{1}-2{\beta }_{1}{l}_{1}.$ From Eq. (8) and supposing that ${\left({\beta }_{1}{l}_{1}\right)}_{mes}^{iv}\cong {\left({\beta }_{2}{l}_{2}\right)}_{mes}^{iv}$, which are the initial values of the phase constant when using the test cell corresponding to length ${l}_{1}$ and ${l}_{2}$ in each configuration (with or without MUT), we can write: ${\beta }_{r}∆l={\left({\beta }_{2}{l}_{2}\right)}_{mes}+{\left({\beta }_{1}{l}_{1}\right)}_{mes}-2{\left({\beta }_{1}{l}_{1}\right)}_{mes}^{iv}.$ Associating Eqs. (12) and (15), the relative permittivity or permeability can be estimated through Eq. (16), given below as: ${\mu }_{r}{\epsilon }_{r}={\left[\frac{{{\left({\beta }_{2}{l}_{2}\right)}_{mes}^{MUT}+\left({\beta }_{1}{l}_{1}\right)}_{mes}^{MUT}-{2\left({\left({\beta }_{1}{l}_{1}\right)}_{mes}^{MUT}\right)}_ {iv}}{{\left({\beta }_{2}{l}_{2}\right)}_{mes}^{v}+{\left({\beta }_{1}{l}_{1}\right)}_{mes}^{v}-{2\left({\left({\beta }_{1}{l}_{1}\right)}_{mes}^{v}\right)}_{iv}}\right]}^{2}.$ 4. Technique improvement This technique presents the peculiarity to behave according to the resonances of the waveguide. Electromagnetic simulations allowed us during the de-embedding steps, to redefine the reflection coefficient of Eq. (7) by adding “$j$” as given in the following equation: ${\Gamma }_{j}={S}_{11}^{g}{e}^{2j{\gamma }_{0}{l}_{0}}.$ So, all equations that we got previously are combined to the new formulation of the reflection coefficient. In that case, Eq. (5) will be written as: ${\mu }_{eff}{\epsilon }_{eff}=\mathcal{g}{\left(\frac{{\left({\beta }_{MUT}\right)}_{j}}{{\left({\beta }_{v}\right)}_{j}}\right)}^{2},$ where “$\mathcal{g}$“ is a constant to be determined. We note by $∆{T}_{v}$ and $∆{T}_{m}$ the linearised phase constants when the test cell is in filled up with vacuum and MUT before correction, respectively, and by $∆{T}_{v}^{j}$ and $∆{T}_{m}^{j}$ once having corrected. By the way, we defined two elements $B$ and $D$ such as $D=\frac{∆{T}_{m}^{j}}{∆{T}_{m}}$ and $B=\frac{∆{T}_{v}^{j}}{∆{T} _{v}}$. In that case, Eq. (18) is written as: ${\mu }_{eff}{\epsilon }_{eff}={\left(\frac{\sum B}{\sum D}\right)}^{2}{\left\{\frac{∆{T}_{m}^{j}}{∆{T}_{v}^{j}}\right\}}^{2}.$ Eq. (19) says that $\mathcal{g}={\left(\frac{\sum B}{\sum D}\right)}^{2}$ changes according to the material under test and the study frequency range. In addition, the fixture dimensions limit the study’s bandwidth, especially in frequencies lower than $2$ GHz. In other words, as long as the gap between the inner diameter “$b$” of the outer conductor and the outer diameter “$a$” of the inner conductor is small, low frequencies than 2 GHz are reachable. That is the main reason in the choice of the test cell in terms of shape and dimensions. In point of fact, we validated the technique implementation with aquarium sand, semolina and palm trees with two species: vinifera and laurentiis. Further reasons to be sure that the results are right, we compared them with those obtained by using the two transmission-line technique (2TL) when the test cells are exactly the same. We have named by ${\phi }_{1}={\beta }_{1}{l}_{1}$ and ${\phi }_{2}={\beta }_{2}{l}_{2}$ the linearised phase constant when the technique is used without improvement and by ${\phi }_{1}^{j}={\beta }_{1}^{j}{l}_{1}$ and ${\phi }_{2}^{j}={\beta }_{2}^{j}{l}_{2}$ once improved. In that case: $∆{T}_{v}={\left({\left({\phi }_{2}\right)}_{v}\right)}_{mes}+{\left({\left({\phi }_{1}\right)}_{v}\right)}_{mes}-2{\left({\left({\phi }_{1}\right)}_{v}\right)}_{mes}^{iv},$ $∆{T}_{m}={\left({\left({\phi }_{2}\right)}_{MUT}\right)}_{mes}+{\left({\left({\phi }_{1}\right)}_{MUT}\right)}_{mes}-2{\left({\left({\phi }_{1}\right)}_{MUT}\right)}_{mes}^{iv},$ $∆{T}_{v}^{j}={\left({\left({\phi }_{2}\right)}_{v}^{j}\right)}_{mes}+{\left({\left({\phi }_{1}\right)}_{v}^{j}\right)}_{mes}-2{\left({\left({\phi }_{1}\right)}_{v}^{j}\right)}_{mes}^{iv},$ $∆{T}_{m}^{j}={\left({\left({\phi }_{2}\right)}_{MUT}^{j}\right)}_{mes}+{\left({\left({\phi }_{1}\right)}_{MUT}^{j}\right)}_{mes}-2{\left({\left({\phi }_{1}\right)}_{MUT}^{j}\right)}_{mes}^{iv},$ 5. Technique validation Measurements have been done from 20 MHz to 20 GHz with a particular attention on the frequency range 2-20 GHz. We have validated the technique with experiment on non-magnetic materials such as: palm tree raffia, semolina and aquarium sand. A vector network analyzer (VNA), connected to two circular coaxial test cells, which are manufactured in brass, where $b$ = 14.38 mm and $a$ = 4 mm as it is shown in Fig. 4, with 80 mm and 100 mm of lengths, is the RF measurement equipment we used. Fig. 5The test cell design 5.1. Electromagnetic simulations We have evaluated the technique relative error from the phase constant when the structure is filled up with vacuum. Using the circular coaxial dimensions that we have designed and manufactured, we compared both in good situation (absence of connector) and the real situation (there is a connector placed at a tip of the test cell). Fig. 6Phase constants of ideal line after extraction from electromagnetic simulations: use of Eq. (20a) We have extracted the phase constants in both cases and can notice that both sketch curves are nearly the same and negatives from 20 MHz up to 1 GHz. In that case, we have applied Eq. (8) to solve Fig. 7Relative error’s curve from ideal and de-embedded phase constants Once the incertitude of the error was evaluated, we observed, as the following figure shows that the relative error decreases when frequency increases. Our main goal is to have a technique with an error of less than 10 % in the frequency range 2-20 GHz. The limits generated by the propagation modes now are stopped up to 20 GHz in our case when the MUT is vacuum. 5.2. Experimental validation To validate the movable short-circuit line technique implementation, we applied the extraction procedure to the biotechnology food such as semolina in frequency up to 5 GHz as it shown below. Fig. 8Relative permittivity result comparisons using two different equations The use of Eq. (19) gives good results from 1.7 GHz while both techniques (basic and improvement ones) start to approach each another from 3 GHz. We have compared the improvement technique results to those obtained with the use of two transmission-lines in reflection/transmission [9]. Fig. 9Relative permittivity result comparisons using two different techniques These results are closes to each another and prove that the movable short-circuit line technique, in its improvement version, is suitable for material relative permittivity extraction. The appropriateness or suitability of the low frequency where the technique can be acceptable is around 1.6 GHz. Because of that, we have extracted other material relative permittivities such as aquarium sand and palm raffia in order to certify that assertion. Fig. 10Relative permittivity results using the MSCL technique from Eq. (19) As we focused on frequency range 2-20 GHz, it is observed that the movable short-circuit technique has allowed covering frequencies upper than 10.4 GHz. Theoretically, when only vacuum is inside of this fixture, high order modes are generated from 10.4 GHz. Eq. (20), given in [21] is satisfied. ${f}_{ma{x}_{\left(GHz\right)}}\approx \frac{191}{\left(b+a\right)\sqrt{{\epsilon }_{r}}},$ where “$b$” and “$a$”, inner diameter of the outer conductor and outer diameter of inner conductor, respectively, are in mm. The next sketched curves are used to compare results between two different techniques: the two transmission-lines (2TL) and MSCL. Fig. 11Comparison of relative permittivity of a) aquarium sand from MSCL and 2TL techniques and b) Laurentiis from MSCL and 2TL techniques The two transmission-line results are obtained with 4-6 % of relative error in the entire frequency range. The results are in conformity with those from electromagnetic simulation ones. The correction on the return loss coefficient allowed improving the extraction process and the technique implementation to cover frequencies 1.7-3.5 GHz. 6. Conclusions In spite of the multitude of techniques which suffer from frequency limits because of the high order mode propagations that are generated by the test cell discontinuities, we have presented in this paper how to overcome that difficulty through the movable broadband short-circuit technique. This new way to extract the relative permittivity and/or permeability parameter is well adapted for industrial applications. We avoided using iterative solutions and suggested how to calculate automatically the correction factor according to the MUT and the scanned frequency range. The electromagnetic simulations have proved that the error rate is inversely proportional to frequency. It means that the more the covered frequencies increase, the better is the accuracy, which covers 7% to 3% when using homogenous configuration. We validated the technique implementation with some biotechnology materials: two palm tree species (vinifera and laurentiis), aquarium sand and semolina in the frequency range 2-18 GHz with a good accuracy. We have proved that the technique result can be repeated as long as the calibration of the connector is well made, especially its short circuit in microwave domains. Nevertheless, despite the progress we have made, the technique is not suitable for frequencies lower than 1.7 GHz, due to the test cell (dimensions or shape) and/or the mathematical formulation. This means that developing a new fixture or a new technique in order to reach low frequencies will be suitable. • Zarral L., Djahli F., Ndagijimana F. Technique of coaxial frame in reflection for the characterization of single and multilayer materials with correction of air gap. International Journal of Antennas and Propagation, Vol. 15, Issue 5, 2014, p. 1-9. • Peyman A. Holden S. J., Watts S., Perrott R., Gabriel C. Dielectric properties of porcine cerebrospinal tissues at microwave frequencies: in vivo, in vitro and systematic variation with age. Physics in Medicine and Biology, Vol. 52, Issue 8, 2007, p. 2229-2245. • Hand J. W. Modelling the interaction of electromagnetic fields (10 MHz – 10 GHz) with the human body. Physics in Medicine and Biology, Vol. 53, Issue 16, 2008, p. 243-286. • Cabanes-Sempere M., et al. Characterization method of dielectric properties of free falling drops in a microwave processing cavity and its application in microwave internal gelation. Measurement Science and Technology, Vol. 24, Issue 9, 2013. • La Gioia A., O’Halloran M., Porter E. Modelling the sensing radius of a coaxial probe for dielectric characterization of biological tissues. IEEE Access, Vol. 6, 2018, p. 46516-46526. • Moukanda M., Ndagijimana F., Chilo J., Saguet P. Coaxial probe fixture used for complex permittivity measurement of thin layers. African Physical Review, Vol. 2, Issue 31, 2008, p. 65-67. • Reynoso-Hernandez J. A. Unified method for determining the complex propagation constant of reflecting and non-reflecting transmission lines. IEEE Microwave and Wireless Components Letters, Vol. 13, Issue 8, 2003, p. 351-353. • Takach A. A., Moukanda M. F., Ndagijimana F., Al-Husseini M., Jomaa J. Two-line technique for dielectric material characterization with application in 3D-printing filament electrical parameters extraction. Progress in Electromagnetic Research M., Vol. 85, 2019, p. 195-207. • Moukanda Mbango F., Al Takach A., Ndagijimana F. Complex relative permittivity extraction technique of biotechnology materials in microwave domains. International Journal of Electronics, Communication and Instrumentation Engineering Research and Development, Vol. 9, Issue 1, 2019, p. 33-42. • Mosig J. R., Besson J. C. E., Gex-Farby M., Gardiol F. E. Reflection of an open-ended coaxial line and application to non-destructive measurement of materials. IEEE Transaction on Instrumentation and Measurement, Vol. 30, 1981, p. 46-51. • Seaman R., Burdete E., Dehaan R. Open-ended coaxial exposure device for applying RF/microwave fields to very small biological preparations. IEEE Transaction on Microwave Theory and Techniques, Vol. 37, 1989, p. 102-111. • James Baker-Jarvis Transmission/reflection and short-circuit line permittivity measurements. National Institute of Standards and Technology, 1990, p. 68-76. • Wolfson B. J., Wentworth S. M. Complex permittivity and permeability measurement using a rectangular waveguide. Microwave and Optical Technology Letters, Vol. 27, Issue 3, 2000, p. 180-182. • Marsland T., Evans E. Dielectric measurements with an open-ended coaxial probe. IEEE Proceeding, Vol. 134, 1987, p. 341-349. • Ghodgaonkar D. K., Varadan V. V., Varadan V. K. Free-space measurement of complex permittivity and complex permeability of magnetic materials at microwave frequencies. IEEE Transaction on Instrumentation and Measurement, Vol. 39, Issue 2, 1990, p. 387-394. • Sagnard F., Bentabet F., Vignat C. In situ measurements of the complex permittivity of materials using reflection ellipsometry in the microwave band: Experiments (Part II). IEEE Transaction on Instrumentation and Measurement, Vol. 54, Issue 3, 2005, p. 1274-1282. • Salahum E., Queffelec P., Le Floch M., Gelin Ph. A Broadband Permeameter for in Situ Measurements of Rectangular Samples. IEEE Transactions on Magnetics. Vol. 37, Issue 4, 2001, p. 2743-2745. • Pozar D. M. Microwave Engineering, 3rd Edition. John Wiley and Sons, Inc., Hoboken, 2005. • Aleksandrov N. L., et al. Selective excitation of high order modes in circular waveguides. International Journal of Infrared and Millimeter Waves, Vol. 13, Issue 9, 1992, p. 1369-1385. • Moukanda Mbango F, Ndagijimana F. Electric parameter extractions using a broadband technique from coaxial line discontinuities. International Journal of Scientific Research and Management, Vol. 7, Issue 5, 2019, p. 248-253. • Moukanda Mbango F. Contribution à la Caractérisation Electrique de Matériaux Utilisés en Microélectronique Radiofréquence. Thesis of Université Joseph Fourier, Grenoble, 2008. • Janezic M. D., Jargon J. A. Complex permittivity determination from propagation constant measurements. IEEE Microwave and Guided Wave Letters. Vol. 9, Issue 2, 1999, p. 76-78. About this article Discontinuity impacts movable short-circuit propagation constant relative permittivity Authors would like to thank in a particular way F. Kuhlman for helpful suggestions and given samples to validate the new technique, also the anonymous reviewers for their attentive revision and thorough assessment. Copyright © 2019 M. G. Lountala, et al. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.	{"url":"https://www.extrica.com/article/20925","timestamp":"2024-11-03T17:14:24Z","content_type":"text/html","content_length":"174970","record_id":"<urn:uuid:4873e380-abb5-4e7e-871b-596783897c74>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00748.warc.gz"}
Voltage-Controlled Current Sources Next: Voltage-Controlled Voltage Sources Up: Dependent Sources Previous: Dependent Sources Contents Index Voltage-Controlled Current Sources This is a special case of the general source specification included for backward compatibility. General Form: gname n+ n- nc+ nc- [expr] srcargs gname n+ n- function \| cur [=] expr srcargs gname n+ n- poly poly_spec srcargs where srcargs = [ac table(name)] g1 2 0 5 0 0.1mmho g2 2 0 5 0 log10(x) g3 2 0 function log10(v(5)) The n+ and n- are the positive and negative nodes, respectively. Current flow is from the positive node, through the source, to the negative node. The parameters nc+ and nc- are the positive and negative controlling nodes, respectively. In the first form, if the expr is a constant, it represents the transconductance in siemens. If no expression is given, a unit constant value is assumed. Otherwise, the expr computes the source current, where the variable ``x'' if used in the expr is taken to be the controlling voltage (v(nc+,nc-)). In this case only, the pwl construct if used in the expr takes as its input variable the value of ``x'' rather than time, thus a piecewise linear transfer function can be implemented using a pwl statement. The second form is similar, but ``x'' is not defined. The keywords ``function'' and ``cur'' are equivalent. The third form allows use of the SPICE2 poly construct. More information on the function specification can be found in 2.15, and the poly specification is described in 2.15.2. If the ac parameter is given and the table keyword follows, then the named table is taken to contain complex transfer coefficient data, which will be used in ac analysis (and possibly elsewhere, see below). For each frequency, the source output will be the interpolated transfer coefficient from the table multiplied by the input. The table must be specified with a .table line, and must have the ac keyword given. If an ac table is specified, and no dc/transient transfer function or coefficient is given, then in transient analysis, the source transfer will be obtained through Fourier analysis of the table data. This is somewhat experimental, and may be prone to numerical errors. In ac analysis, the transfer coefficient can be real or complex. If complex, the imaginary value follows the real value. Only constants or constant expressions are valid in this case. If the source function is specified in this way, the real component is used in dc and transient analysis. This will also override a table, if given. Next: Voltage-Controlled Voltage Sources Up: Dependent Sources Previous: Dependent Sources Contents Index Stephen R. Whiteley 2024-10-26	{"url":"http://wrcad.com/manual/wrsmanual/node116.html","timestamp":"2024-11-03T14:05:00Z","content_type":"text/html","content_length":"6747","record_id":"<urn:uuid:e9fe97e0-bdb0-473b-a643-06152dcd9865>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00857.warc.gz"}
CPM Homework Help Suppose your parents spend an average of $300$ each month for your food. a. In five years, when you are living on your own, how much will you be spending on food each month if you are eating about the same amount and inflation averages about 4% per year? Each year, food will cost $1.04$ times the cost of the previous year. b. Write an equation that represents your monthly food bill x years from now if both the rate of inflation and your eating habits stay the same. What is the initial value and what is the multiplier?	{"url":"https://homework.cpm.org/category/CON_FOUND/textbook/a2c/chapter/4/lesson/4.1.2/problem/4-33","timestamp":"2024-11-03T04:40:56Z","content_type":"text/html","content_length":"36629","record_id":"<urn:uuid:f18b98ef-c26b-4afc-8f70-0105a25c4188>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00457.warc.gz"}
Slide the blue points on the slider to move the data points on the number line below. Notice how the mean changes. 1. Can you make the mean the same value as one (or more) of the data points? How many ways? What do you notice? 2. Can you make more than one data set with a mean of 4? 3. Will the mean ever be outside of the data set? 4. Set up the points to 4,\,6 and 11. How far is each value from the mean? Use negative values for below the mean and positive values for above the mean. What is the sum of these values? The mean is the balance point of a data set. This means that the sum of the distances from the mean of all of the points below the mean is equal to the sum of the distances from the mean of all of the points above the mean. To find the balance point, when it is not given, points can be moved one-by-one towards the middle. If the balance point is located between two values, we can find the halfway point between those values by averaging the numbers. Adding or removing a data point might throw off the balance of the data set resulting in a new balance point.	{"url":"https://mathspace.co/textbooks/syllabuses/Syllabus-1191/topics/Topic-23231/subtopics/Subtopic-287494/?ref=blog.mathspace.co","timestamp":"2024-11-05T07:12:54Z","content_type":"text/html","content_length":"366915","record_id":"<urn:uuid:b218b3f6-87f1-4a48-8dd5-e8032dffbf70>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00640.warc.gz"}
Bernd Becker Jul 16, 2020 Abstract:The synthesis problem for partially observable Markov decision processes (POMDPs) is to compute a policy that satisfies a given specification. Such policies have to take the full execution history of a POMDP into account, rendering the problem undecidable in general. A common approach is to use a limited amount of memory and randomize over potential choices. Yet, this problem is still NP-hard and often computationally intractable in practice. A restricted problem is to use neither history nor randomization, yielding policies that are called stationary and deterministic. Previous approaches to compute such policies employ mixed-integer linear programming (MILP). We provide a novel MILP encoding that supports sophisticated specifications in the form of temporal logic constraints. It is able to handle an arbitrary number of such specifications. Yet, randomization and memory are often mandatory to achieve satisfactory policies. First, we extend our encoding to deliver a restricted class of randomized policies. Second, based on the results of the original MILP, we employ a preprocessing of the POMDP to encompass memory-based decisions. The advantages of our approach over state-of-the-art POMDP solvers lie (1) in the flexibility to strengthen simple deterministic policies without losing computational tractability and (2) in the ability to enforce the provable satisfaction of arbitrarily many specifications. The latter point allows taking trade-offs between performance and safety aspects of typical POMDP examples into account. We show the effectiveness of our method on a broad range of benchmarks.	{"url":"https://www.catalyzex.com/author/Bernd%20Becker","timestamp":"2024-11-04T07:57:08Z","content_type":"text/html","content_length":"91409","record_id":"<urn:uuid:4337a71b-3959-444b-89d0-4804b66d0e58>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00076.warc.gz"}
Papers with Code - Measuring Mathematical Problem Solving With the MATH Dataset Measuring Mathematical Problem Solving With the MATH Dataset Many intellectual endeavors require mathematical problem solving, but this skill remains beyond the capabilities of computers. To measure this ability in machine learning models, we introduce MATH, a new dataset of 12,500 challenging competition mathematics problems. Each problem in MATH has a full step-by-step solution which can be used to teach models to generate answer derivations and explanations. To facilitate future research and increase accuracy on MATH, we also contribute a large auxiliary pretraining dataset which helps teach models the fundamentals of mathematics. Even though we are able to increase accuracy on MATH, our results show that accuracy remains relatively low, even with enormous Transformer models. Moreover, we find that simply increasing budgets and model parameter counts will be impractical for achieving strong mathematical reasoning if scaling trends continue. While scaling Transformers is automatically solving most other text-based tasks, scaling is not currently solving MATH. To have more traction on mathematical problem solving we will likely need new algorithmic advancements from the broader research community. PDF Abstract Task Dataset Model Metric Name Metric Value Global Rank Result Benchmark Math Word Problem Solving MATH GPT-2 (1.5B) Accuracy 6.9 # 119 Compare Parameters (Billions) 1.5 # 101 Compare Math Word Problem Solving MATH GPT-3 2.7B Accuracy 2.9 # 131 Compare Parameters (Billions) 2.7 # 100 Compare Math Word Problem Solving MATH GPT-2 (0.1B) Accuracy 5.4 # 125 Compare Parameters (Billions) 0.1 # 106 Compare Math Word Problem Solving MATH GPT-2 (0.3B) Accuracy 6.2 # 122 Compare Parameters (Billions) 0.3 # 105 Compare Math Word Problem Solving MATH GPT-2 (0.7B) Accuracy 6.4 # 121 Compare Parameters (Billions) 0.7 # 104 Compare Math Word Problem Solving MATH GPT-3-175B (few-shot) Accuracy 5.2 # 126 Compare Parameters (Billions) 175 # 5 Compare Math Word Problem Solving MATH GPT-3 13B Accuracy 5.6 # 123 Compare Parameters (Billions) 13 # 44 Compare Math Word Problem Solving MATH GPT-3-13B (few-shot) Accuracy 3.0 # 130 Compare Parameters (Billions) 13 # 44 Compare	{"url":"https://paperswithcode.com/paper/measuring-mathematical-problem-solving-with","timestamp":"2024-11-09T22:38:26Z","content_type":"text/html","content_length":"268001","record_id":"<urn:uuid:6f32939f-d768-486c-9a1b-b692563e0845>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00087.warc.gz"}
MS Excel Formula HELP! Please help. I have a spread sheet that I am using to monitor a budget over the calendar year. I have all of the calculations to add up to the monthly invoice amounts in cells m26 - bl26. Since it is only February now, only January has been calculated and invoiced. All of the other monthly totals are still at $0.00. I would like a cell that calculates the total spent “average” over the calendar year. I have this formula so far =SUM(M26:BL26)/12. The problem is, is that it calculates 1/12th of January’s total How do I correct the formula to calculate the average of only the cells that have dollar figures in them without having to modify the formula 12 times through the year? You can use something like: =IF((B13>0),B13*2,“N/A”) for the entry (if the value is greater than zero do the calc, otherwise put in “N/A”) and then use AVERAGE(B1:B12) to get the average, since it will ignore the text strings. Obviously adjust for your own values. Does this make sense? Re-reading it, it seems it doesn’t make much sense. It’s like, if the formula for your M26 was or something, then replace it with and similarly for each month. Now when you do AVERAGE(M26:BL26) it ignores all the ones with “N/A” in and just does the ones with values. I’m sure there must be a simpler way to do this, but this one does work. If you change 12 in the formula =SUM(M26:BL26)/12 to (MONTH(NOW())-1) it will divide by the month number of last month, in February 1, in March 2 etc. Do you see how that works? Here’s another way (though I’m sure there’s a better one still - probably a singe command, knowing Excel). Reserve another column/s, let’s say the Z column. Each cell in this column should respectively contain (each cell respectively represents one month in the first column/s). At the bottom, say Z25, you have =sum(Z2:Z24) - i.e. the number of non-zero months. Then your averaging formula is simply Of course this messes up if there is a genuine zero month…	{"url":"https://boards.straightdope.com/t/ms-excel-formula-help/227648","timestamp":"2024-11-09T20:38:27Z","content_type":"text/html","content_length":"31991","record_id":"<urn:uuid:c7f62b24-c09b-435f-a3cc-fdad91a529ce>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00053.warc.gz"}
filter_kalman - Unidimensional Kalman filter, also known as linear quadratic estimation (LQE) loadrt filter_kalman [count=N\|names=name1[,name2...]] Useful for reducing input signal noise (e.g. from the voltage or temperature sensor). More information can be found at https://en.wikipedia.org/wiki/Kalman_filter. Adjusting Qr and Qk covariances: Default values of Rk and Qk are given for informational purpose only. The nature of the filter requires the parameters to be individually computed. One of the possible and quite practical method (probably far from being the best) of estimating the Rk covariance is to collect the raw data from the sensor by either asserting the debug pin or using halscope and then compute the covariance using cov() function from Octave package. Ready to use script can be found at https://github.com/dwrobel/TrivialKalmanFilter/blob/master/examples/DS18B20Test/ Adjusting Qk covariance mostly depends on the required response time of the filter. There is a relationship between Qk and response time of the filter that the lower the Qk covariance is the slower the response of the filter is. Common practice is also to conservatively set Rk and Qk slightly larger then computed ones to get robustness. filter-kalman.N (requires a floating-point thread) Update xk-out based on zk input. filter-kalman.N.debug bit in (default: FALSE) When asserted, prints out measured and estimated values. filter-kalman.N.passthrough bit in (default: FALSE) When asserted, copies measured value into estimated value. filter-kalman.N.reset bit in (default: FALSE) When asserted, resets filter to its initial state and returns 0 as an estimated value (reset pin has higher priority than passthrough pin). filter-kalman.N.zk float in Measured value. filter-kalman.N.xk-out float out Estimated value. filter-kalman.N.Rk float rw (default: 1.17549e-38) Estimation of the noise covariances (process). filter-kalman.N.Qk float rw (default: 1.17549e-38) Estimation of the noise covariances (observation). Dmian Wrobel <dwrobel@ertelnet.rybnik.pl>	{"url":"http://linuxcnc.org/docs/devel/html/man/man9/filter_kalman.9.html","timestamp":"2024-11-06T15:19:36Z","content_type":"text/html","content_length":"5795","record_id":"<urn:uuid:3074aa60-ed95-4ab7-b98a-3cb95763e5ef>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00837.warc.gz"}
Calculus Problem Solutions : Dingyü Xue Walter de Gruyter GmbH & Co KG Category : Languages : Pages : Book Description This book focuses on solving practical problems in calculus with MATLAB. Descriptions and sketching of functions and sequences are introduced first, followed by the analytical solutions of limit, differentiation, integral and function approximation problems of univariate and multivariate functions. Advanced topics such as numerical differentiations and integrals, integral transforms as well as fractional calculus are also covered in the book.	{"url":"https://marthawilliams.org/book/calculus-problem-solutions-with-matlabr/","timestamp":"2024-11-14T05:47:21Z","content_type":"text/html","content_length":"44230","record_id":"<urn:uuid:40f35a30-af81-49fb-b59b-a95126bef906>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00509.warc.gz"}
nequalities in the Graphing Linear Inequalities in the Plane Consider the set • Is (0; 0) in S? How can you tell? • Is (2; 3) in S? How can you tell? • Is (4; 12) in S? How can you tell? • Name a point in S with negative x -coordinate._________ • Name a point in S with integer coordinates (other than the one above).___________ • On the grid, plot all 5 points listed above, using • for points in S and × for points not in S. Make sure to choose a scale for your graph so that all of the points above fit on the diagram. Now you will graph S on the same graph that you plotted the points . • Write the equation for the line that will be the boundary of the set S. • Put the line in slope -intercept form. (This involves solving for y.) What is the slope ?___________ What is the y- intercept ?___________ • Find the x- intercept of the line . (This involves substituting y = 0 into the formula for your line , and solving for x . Do you know why that gets you the x-intercept? ) • Graph the line . • Shade S.	{"url":"https://www.softmath.com/tutorials-3/relations/graphing-linear-inequalities.html","timestamp":"2024-11-10T12:46:44Z","content_type":"text/html","content_length":"32126","record_id":"<urn:uuid:f0badd28-fc60-4bd4-b9b3-0fdb0a75be83>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00362.warc.gz"}
What Are Angle Bisectors? - Geometry Spot An angle bisector is a line or line segment that divides an angle into two equal parts. Angle bisectors are an important concept in geometry, and can be used to solve a variety of problems. The most basic way to understand an angle bisector is to think of it as a line that divides an angle into two equal angles. This line is called the bisector of the angle and is always perpendicular to the sides of the angle. To bisect an angle, draw a line that intersects the two sides of the angle at the same point. Then draw a perpendicular line that goes through the point of intersection. The line you drew is the angle bisector. When dealing with triangles, the angle bisector is particularly useful. The angle bisector theorem states that if an angle bisector is drawn from the vertex of a triangle to the opposite side, then it will divide the opposite side into two equal parts. This theorem is useful for solving problems involving triangle geometry, such as finding the area of a triangle. In addition to the angle bisector theorem, there are a few other important properties of angle bisectors. The first is that angle bisectors always bisect the angle. In addition, angle bis ectors do not change the shape of the angle they are bisecting. Finally, angle bisectors are always perpendicular to the sides of the angle they are bisecting. Angle bisectors are also useful when dealing with circles. If two angles are formed by two chords of a circle, the angle bisector theorem states that the angle bisector of the two angles always passes through the center of the circle. This theorem is useful for finding the area of a circle. Finally, angle bisectors can be used to solve problems involving congruent triangles. If two triangles are congruent, then their angle bisect ors are also congruent. This means that if two angles of a triangle have the same angle bisector, then the two angles are congruent. Angle bisectors are an important concept in geometry and can be used to solve a variety of problems. They can be used to find the area of a triangle or a circle, as well as to determine congruence. Understanding how to use angle bisectors is an important part of any geometry course.	{"url":"https://sciencetips.net/what-are-angle-bisectors/","timestamp":"2024-11-07T06:35:31Z","content_type":"text/html","content_length":"146564","record_id":"<urn:uuid:b2a0beed-c1fa-4267-aa8e-9d372f172f8e>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00457.warc.gz"}
Li-Yau and Harnack type inequalities in Accepted Paper Inserted: 3 jun 2013 Last Updated: 19 oct 2013 Journal: Nonlinear Analysis TMA Year: 2013 Metric measure spaces satisfying the reduced curvature-dimension condition $CD^(K,N)$ and where the heat flow is linear are called $RCD^(K,N)$-spaces. This class of non smooth spaces contains Gromov-Hausdorff limits of Riemannian manifolds with Ricci curvature bounded below by $K$ and dimension bounded above by $N$. We prove that in $RCD^*(K,N)$-spaces the following properties of the heat flow hold true: a Li-Yau type inequality, a Bakry-Qian inequality, the Harnack inequality. Tags: GeMeThNES	{"url":"https://cvgmt.sns.it/paper/2162/","timestamp":"2024-11-03T06:18:28Z","content_type":"text/html","content_length":"8374","record_id":"<urn:uuid:d5bd3473-6f57-4a39-ba9d-e63e5bce27a4>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00126.warc.gz"}
Staircase Compatibility and its Applications in Scheduling and Piecewise Linearization We consider the clique problem with multiple-choice constraints (CPMC) and characterize a case where it is possible to give an efficient description of the convex hull of its feasible solutions. This new special case, which we call staircase compatibility, generalizes common properties in several applications and allows for a linear description of the integer feasible solutions to (CPMC) with a totally unimodular constraint matrix of polynomial size. We derive two such totally unimodular reformulations for the problem: one that is obtained by a strengthening of the compatibility constraints and one that is based on a representation as a dual network flow problem. Furthermore, we show a natural way to derive integral solutions from fractional solutions to the problem by determining integral extreme points generating this fractional solution. We also evaluate our reformulations from a computational point of viewby applying them to two different real-world problem settings. The first one is a problem in railway timetabling, where we try to adapt a given timetable slightly such that energy costs from operating the trains are reduced. The second one is the piecewise linearization of non-linear network flow problems, illustrated at the example of gas networks. In both cases, we are able to reduce the solution times significantly by passing to the theoretically stronger formulations of the problem. View Staircase Compatibility and its Applications in Scheduling and Piecewise Linearization	{"url":"https://optimization-online.org/2018/01/6435/","timestamp":"2024-11-11T21:15:04Z","content_type":"text/html","content_length":"85620","record_id":"<urn:uuid:dc3538b8-30cc-4755-8702-1ce1e047d402>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00669.warc.gz"}
Identifier: Global Extrema - APCalcPrep.com • The language of the problem directly asks you to find the global extrema, absolute extrema, a global max, a global min, an absolute max, or an absolute min. • Most of the time these questions will include a closed x-interval to work from some [a,b]. While you can see a closed interval on some local extrema problems, it is much, much more common to see them associated with absolute extrema problems. Accessing this course requires a login. Please enter your credentials below!	{"url":"https://apcalcprep.com/topic/identifier-global-extrema/","timestamp":"2024-11-10T07:36:08Z","content_type":"text/html","content_length":"360797","record_id":"<urn:uuid:d91aa056-bc9b-4c45-89b4-5e9e365eb021>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00452.warc.gz"}
Influence of Surface Area on Reaction Rate in context of reaction velocity 31 Aug 2024 The Journal of Chemical Kinetics Volume 12, Issue 3, 2022 Influence of Surface Area on Reaction Rate: A Theoretical Analysis The surface area of a reactant has been shown to have a significant impact on the reaction rate in various chemical reactions. In this article, we present a theoretical analysis of the influence of surface area on reaction rate, focusing on the relationship between surface area and reaction velocity. The reaction rate is a critical parameter in understanding the kinetics of chemical reactions. It is defined as the change in concentration of reactants or products per unit time. The surface area of a reactant can affect the reaction rate by increasing the number of active sites available for reaction, thereby enhancing the reaction velocity. Let’s consider a simple heterogeneous reaction between two reactants A and B: A (solid) + B (gas) → Products The reaction rate is given by the formula: r = k * S * c_B where r is the reaction rate, k is the rate constant, S is the surface area of the solid reactant, and c_B is the concentration of the gas reactant. As can be seen from the equation, the reaction rate is directly proportional to the surface area of the solid reactant. This means that an increase in surface area will result in a corresponding increase in reaction rate. The influence of surface area on reaction rate has been observed in various chemical reactions, including catalytic reactions and combustion reactions. In these reactions, the surface area of the catalyst or fuel can significantly affect the reaction rate by increasing the number of active sites available for reaction. In addition to its direct effect on reaction rate, surface area can also influence the reaction kinetics by affecting the diffusion of reactants and products. For example, in a heterogeneous reaction between two solid reactants, an increase in surface area can enhance the diffusion of reactants and products, leading to an increase in reaction rate. In conclusion, the surface area of a reactant has a significant influence on the reaction rate in various chemical reactions. The relationship between surface area and reaction velocity is given by the formula r = k * S * c_B, which shows that an increase in surface area will result in a corresponding increase in reaction rate. 1. Smith, J. (2020). Chemical Kinetics. John Wiley & Sons. 2. Johnson, K. (2019). Surface Science. Springer Nature. 3. Thompson, M. (2018). Reaction Engineering. Butterworth-Heinemann. Related articles for ‘reaction velocity’ : • Reading: Influence of Surface Area on Reaction Rate in context of reaction velocity Calculators for ‘reaction velocity’	{"url":"https://blog.truegeometry.com/tutorials/education/41a826497bacb990dd61aacf4096cb8a/JSON_TO_ARTCL_Influence_of_Surface_Area_on_Reaction_Rate_in_context_of_reaction_.html","timestamp":"2024-11-09T06:18:22Z","content_type":"text/html","content_length":"16315","record_id":"<urn:uuid:b48161e5-c2eb-40f6-bf89-b918ff14f429>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00501.warc.gz"}
Create ARIMA time series from bottom up Simulating ARIMA models Generating an arbitrary Auto-Regressive Integrated Moving Average (ARIMA) model is easy in R with the arima.sim() function that is part of the built-in {stats} package. In fact I've done it extensively in previous blog posts for various illustrative purposes. But one cost of doing this for educational purposes is that the mechanics of generating them are hidden from the user (of course, that's the point!). As was the case in last week's post, I'm motivated by teaching purposes. I wanted to show people how an ARIMA model can be created, with high school maths and no special notation, from white noise. The movie that is (hopefully) playing above shows this. It takes a single series of independent and identical standard normally distributed variables (top left). It shows how this can be turned into (from left to right one row at a time, starting at the top right): • an autoregressive model of order 1, where each value of x equals the previous value times 0.8, plus the white noise; • a moving average of order 1, where each value of x equals the latest bit of white noise plus 0.8 times the previous value of white noise; • an autoregressive moving average model of order (1, 1), combining the two; • an ARIMA(1, 1, 1) model that is the cumulative sum of the ARMA(1, 1); and • an ARIMA(2, 2, 2) model that is like all the above but with extra parameters and an extra cumulative sum stage One interesting thing is how the AR(1) and ARMA(1, 1) models look almost identical, except for the larger variance of the ARMA(1, 1) model which comes from throwing into the mix 80% of the last period's randomness. This similarity is just a result of the the particular parameters chosen - the 0.8 times the previous value of x quickly dominates the moving average part. Here's the code that simulates the actual data. It's simple enough that it can be easily explained and related to the equations on the images in the movie. Note that I've avoided using the usual lag operator, or putting all the autoregression parts of the equation on the left as is normally done. That's all so it is easy to explain exactly where the latest value is coming from. Also note that I've done this in what might seem a very un-R-like fashion - creating a vector with a for() loop! This is purely for to make it really obvious what is going on. There's no issues with performance to worry about, and in this situation transparency and ease of reading is always paramount. #-------------generate data------------- n <- 1000 # white noise: wn <- ts(rnorm(n)) # initialise the first two values: ar1 <- ma1 <- arma11 <- arma22 <- wn[1:2] # loop through and create the 3:1000th values: for(i in 3:n){ ar1[i] <- ar1[i - 1] * 0.8 + wn[i] ma1[i] <- wn[i - 1] * 0.8 + wn[i] arma11[i] <- arma11[i - 1] * 0.8 + wn[i - 1] * 0.8 + wn[i] arma22[i] <- arma22[i - 1] * 0.8 + arma22[i - 2] * (-0.3) + 0.8 * wn[i-1] - 0.3 * wn[i-2] + wn[i] # turn them into time series, and for the last two, "integrate" them via cumulative sum ar1 <- ts(ar1) ma1 <- ts(ma1) arma11 <- ts(arma11) arima111 <- ts(cumsum(arma11)) arima222 <- ts(cumsum(cumsum(arma22))) The animation is created one frame at a time with basic R plots. For the equations I used Stefano Meschiari's useful {latex2exp} package. I don't really understand R's plotmath expressions that let you add equations to plots, and I don't really want to understand them if I can avoid it. {latex2exp} let's me avoid it by using the much more commonly known LaTeX syntax and translating it for me into plotmath. for(i in 3:n){ png(paste0(i + 1000, ".png"), 800, 800, res = 100) par(mfrow = c(3, 2), cex.main = 1.5, cex = 0.8, family = "Calibri") plot(wn[1:i], main = latex2exp("$\\epsilon ~ N(0, \\sigma)"), bty = "l", type = "l", ylab = "x = white noise", xlab = "") plot(ar1[1:i], main = latex2exp("$x_t = 0.8x_{t-1} + \\epsilon_t$"), bty = "l", type = "l", ylab = "x =AR (1)", xlab = "") plot(ma1[1:i], main = latex2exp("$x_t = 0.8\\epsilon_{t-1} + \\epsilon_t$"), bty = "l", type = "l", ylab = "x = MA(1)", xlab = "") plot(arma11[1:i], main = latex2exp("$x_t = 0.8x_{t-1} + 0.8\\epsilon_{t-1} + \\epsilon_t$"), bty = "l", type = "l", ylab = "x = ARMA(1, 1)", xlab = "") plot(arima111[1:i], main = latex2exp("$x_t = 0.8x_{t-1} + 0.8\\epsilon_{t-1} + \\epsilon_t$"), bty = "l", type = "l", ylab = "y = ARIMA(1, 1, 1)", xlab = "") mtext(latex2exp("$y_t = x_t + x_{t-1} + ... + x_0$"), cex = 1.3, line = -0.5) plot(arima222[1:i], main = latex2exp( "$x_t = 0.8x_{t-1} - 0.3x_{t-2} - 0.3\\epsilon_{t-2} + 0.8\\epsilon_{t-1} + \\epsilon_t$"), bty = "l", type = "l", ylab = "z = ARIMA(2, 2, 2)", xlab = "") mtext(latex2exp("$y_t = x_t + x_{t-1} + ... + x_0$"), cex = 1.3, line = -0.5) mtext(latex2exp("$z_t = y_t + y_{t-1} + ... + y_0$"), cex = 1.3, line = -2.0) The result has 998 frames and is probably too big for the sort of animated GIF I usually make. When a web browser comes across an animated GIF it has to load the whole thing in - in this case, around 28MB worth - before it starts playing and I'd probably lose some audiences while that happened. So I used Microsoft MovieMaker to turn the stills into an mp4 and uploaded it to YouTube, which is basically the standard and easiest way to stream video over the web.	{"url":"https://freerangestats.info/blog/2015/11/21/arima-sims.html","timestamp":"2024-11-11T08:39:47Z","content_type":"text/html","content_length":"34054","record_id":"<urn:uuid:b11991ab-61c1-4583-91b8-ab5abd75882a>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00414.warc.gz"}
It Follows The Hijri Calendar It Follows The Hijri Calendar - The crossword solver found 30 answers to it follows the hijiri calendar, 5 letters crossword clue. We have the answer for it follows the hijri calendar crossword clue if you’re having trouble filling in the grid! It follows the hijri calendar (5) i believe the answer is: It follows the hijri calendar crossword clue. Here is the answer for the crossword clue it follows the hijri calendar last seen in new york times puzzle. It follows the hijri calendar. We solved the clue 'it follows the hijri calendar ' which last appeared on april 19, 2024 in a n.y.t crossword puzzle and had five. We have found 40 possible. The crossword solver finds answers to. This crossword clue was last. Facts about the Hijri Calendar Muslimeto™ We solved the clue 'it follows the hijri calendar ' which last appeared on april 19, 2024 in a n.y.t crossword puzzle and had five. The crossword solver finds answers to. 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Arabic Calendar 2024 Ramadan Berri Celeste The crossword solver found 30 answers to it follows the hijiri calendar, 5 letters crossword clue. Here is the answer for the: Here is the answer for the crossword clue it follows the hijri calendar last seen in new york times puzzle. It follows the hijri calendar crossword clue. The answer for this clue is islam, which is a religion. Building a Hijri Calendar with MAUI Scheduler A Beginner’s Guide We have found 40 possible. This clue is from the new york times. It follows the hijri calendar crossword clue. The crossword solver finds answers to. The crossword solver found 30 answers to it follows the hijiri calendar, 5 letters crossword clue. Here is the answer for the: Here is the answer for the crossword clue it follows the hijri calendar last seen in new york times puzzle. We solved the clue 'it follows the hijri calendar ' which last appeared on april 19, 2024 in a n.y.t crossword puzzle and had five. 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Here is the answer for the crossword clue it follows the hijri calendar last seen in new york times puzzle. It follows the hijri calendar (5) i believe the answer is: This clue is from the new york times. We have found 40 possible. The Answer For This Clue Is Islam, Which Is A Religion That Follows The Hijri Calendar. We have the answer for it follows the hijri calendar crossword clue if you’re having trouble filling in the grid! It follows the hijri calendar crossword clue. Related Post:	{"url":"https://markaszervizek.audi.hu/en/it-follows-the-hijri-calendar.html","timestamp":"2024-11-12T22:55:44Z","content_type":"text/html","content_length":"28633","record_id":"<urn:uuid:b22d7205-1b83-4d89-b743-2c5ae0a1dabf>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00373.warc.gz"}
A variational hyper recurrent neural network (VHRNN) can be trained by, for each step in sequential training data: determining a prior probability distribution for a latent variable from a prior network of the VHRNN using an initial hidden state; determining a hidden state from a recurrent neural network (RNN) of the VHRNN using an observation state, the latent variable and the initial hidden state; determining an approximate posterior probability distribution for the latent variable from an encoder network of the VHRNN using the observation state and the initial hidden state; determining a generating probability distribution for the observation state from a decoder network of the VHRNN using the latent variable and the initial hidden state; and maximizing a variational lower bound of a marginal log-likelihood of the training data. The trained VHRNN can be used to generate sequential data. This application claims priority from US Provisional Patent Application No. 62/851,407 filed on May 22, 2019, the entire contents of which are hereby incorporated by reference herein. This relates to sequence modelling, in particular, sequence modelling with neural network architecture. Traditional neural network architecture, such as recurrent neural networks (RNNs) have historically been applied to domains such as natural language processing and speech processing. Traditional RNN architecture, however, is not ideal to capture the high variability of other domains, such as financial time series data, due to inherent variability of the data, noise, or the like. According to an aspect, there is provided a computer-implemented method for training a variational hyper recurrent neural network (VHRNN), the method comprising: for each step in sequential training data: determining a prior probability distribution for a latent variable, given previous observations and previous latent variables, from a prior network of the VHRNN using an initial hidden state; determining a hidden state from a recurrent neural network (RNN) of the VHRNN using an observation state, the latent variable and the initial hidden state; determining an approximate posterior probability distribution for the latent variable, given the observation state, previous observations and previous latent variables, from an encoder network of the VHRNN using the observation state and the initial hidden state; determining a generating probability distribution for the observation state, given the latent variable, the previous observations and the previous latent variables, from a decoder network of the VHRNN using the latent variable and the initial hidden state; and maximizing a variational lower bound of a marginal log-likelihood of the training data to train the VHRNN; and storing the trained VHRNN in a memory. In some embodiments, the variational lower bound includes at least one of an evidence lower bound (ELBO), importance weight autoencoders (IWAE), or filtering variational objectives (FIVO). In some embodiments, the prior probability distribution, defined as p(z[t]\|x[<t], z[<t]), for the latent variable, defined as z[t], is based on: z[t]\|x[<t], z[<t]˜(μ[t]^prior, Σ[t]^prior) where (μ[t]^prior, Σ[t]^prior) is the prior network, x[t ]is the observation state, and t is a current step of the steps in the sequential training data. In some embodiments, the RNN, defined as g, is based on: where θ(z[t],h[t-1]) is a hypernetwork of the VHRNN that generates parameters of the RNN g using the latent variable, defined as z[t], and the initial hidden state, defined as h[t-1], x[t ]is the observation state, and t is a current step of the steps in the sequential training data. In some embodiments, the hypernetwork θ(z[t],h[t-1]) is implemented as a recurrent neural network (RNN). In some embodiments, the hypernetwork θ(z[t],h[t-1]) is implemented as a long short-term memory (LSTM). In some embodiments, the hypernetwork θ(z[t],h[t-1]) generates scaling vectors for input weights and recurrent weights of the RNN. In some embodiments, the generating probability distribution, defined as p(x[t]\|z[≤t],x[<t]), for the observation state, defined as x[t], is based on: where (μ[t]^dec,Σ[t]^dec)=ϕ[ω(z][t][,h][t-1)](z[t], h[t-1]) is another hypernetwork of the VHRNN that generates parameters of the decoder network, defined as ϕ^dec, using the latent variable, defined as z[t], and the initial hidden state, defined as h[t-1], and t is a current step of the steps in the sequential training data. In some embodiments, the hypernetwork ω(z[t], h[t-1]) is implemented as a multilayer perceptron (MLP). According to another aspect, there is provided a computer-implemented method for generating sequential data using a variational hyper recurrent neural network (VHRNN) trained using a method as described herein, the method comprising: for each step in the sequential data: determining a prior probability distribution for a latent variable z[t], given previous observations and previous latent variables, from the prior network of the VHRNN using an initial hidden state; determining a hidden state from the recurrent neural network (RNN) of the VHRNN using an observation state, the latent variable and the initial hidden state; determining a generating probability distribution for the observation state given the latent variable, the previous observations and the previous latent variables, from the decoder network of the VHRNN using the latent variable and the initial hidden state; and sampling a generated observation state from the generating probability distribution. In some embodiments, the prior probability distribution, defined as p(z[t]\|x[<t], z[<t]), for the latent variable z[t ]is based on: where (μ[t]^prior,Σ[t]^prior) is the prior network, x[t ]is the observation state, and t is a current step of the steps in the sequential data. In some embodiments, the RNN, defined as g, is based on: h[t]=g[θ(z][t][,h][t-1)](x[t],z[t], h[t-1]) where θ(z[t],h[t-1]) is a hypernetwork of the VHRNN that generates parameters of the RNN g using the latent variable, defined as z[t], and the initial hidden state, defined as h[t-1], x[t ]is the observation state, and t is a current step of the steps in the sequential data. In some embodiments, the hypernetwork θ(z[t], h[t-1]) is implemented as a recurrent neural network (RNN). In some embodiments, the hypernetwork θ(z[t], h[t-1]) is implemented as a long short-term memory (LSTM). In some embodiments, the hypernetwork θ(z[t], h[t-1]) generates scaling vectors for input weights and recurrent weights of the RNN g. In some embodiments, the generating probability distribution, defined as p(x[t]\|z[≤t],x[<t]), for the observation state, defined as x[t], is based on: where (μ[t]^dec,Σ[t]^dec)=ϕ[ω(z][t][,h][t-1)](z[t],h[t-1]), and ω(z[t],h[t-1]) is another hypernetwork of the VHRNN that generates parameters of the decoder network, defined as ϕ^dec, using the latent variable, defined as z[t], and the initial hidden state, defined as h[t-1], and t is a current step of the steps in the sequential data. In some embodiments, the hypernetwork ω(z[t], h[t-1]) is implemented as a multilayer perceptron (MLP). In some embodiments, the method further comprises forecasting future observations of the sequential data based on the sampled generated observation states. In some embodiments, the sequential data is time-series financial data. According to a further aspect, the is provided a non-transitory computer readable medium comprising a computer readable memory storing thereon a variational hyper recurrent neural network trained using a method as described herein, the variational hyper recurrent neural network executable by a computer to perform a method to generate sequential data, the method comprising: for each step in the sequential data: determining a prior probability distribution for a latent variable z[t], given previous observations and previous latent variables, from the prior network of the VHRNN using an initial hidden state; determining a hidden state from the recurrent neural network (RNN) of the VHRNN using an observation state, the latent variable and the initial hidden state; determining a generating probability distribution for the observation state given the latent variable, the previous observations and the previous latent variables, from the decoder network of the VHRNN using the latent variable and the initial hidden state; and sampling a generated observation state from the generating probability distribution. Other features will become apparent from the drawings in conjunction with the following description. In the figures which illustrate example embodiments, FIG. 1 is a schematic diagram of operations of a variational hyper recurrent neural network (VHRNN) model, according to an embodiment; FIG. 2 is a schematic diagram of an implementation of a recurrence model of a VHRNN using a long short-term memory (LSTM) cell, according to an embodiment; FIG. 3A is a flow chart of a method for training a VHRNN, according to an embodiment; FIG. 3B is a flow chart of a method for generating sequential data using a VHRNN, according to an embodiment; FIG. 4 is a block diagram of example hardware and software components of a computing device for VHRNN modelling, according to an embodiment; FIG. 5 is a table illustrating evaluation results of example baseline variational recurrent neural networks (VRNNs) and an example VHRNN model on synthetic datasets, according to an embodiment; FIGS. 6A-6F illustrate observations from a qualitative study of the behavior of an example baseline VRNN and an example VHRNN model under a NOISELESS setting, according to an embodiment; FIGS. 7A-7F illustrate observations from a qualitative study of the behavior of an example baseline VRNN and an example VHRNN model under a SWITCH setting, according to an embodiment; FIGS. 8A-8F illustrate observations from a qualitative study of the behavior of an example baseline VRNN and an example VHRNN model under a ZERO-SHOT setting, according to an embodiment; FIGS. 9A-9F illustrate observations from a qualitative study of the behavior of an example baseline VRNN and an example VHRNN model under an ADD setting, according to an embodiment; FIGS. 10A-10B illustrate observations from a qualitative study of the behavior of an example baseline VRNN and an example VHRNN model under a RAND setting, according to an embodiment; FIGS. 11A-11F illustrate observations from a qualitative study of the behavior of an example baseline VRNN and an example VHRNN model under a LONG setting, according to an embodiment; FIGS. 12A-12D illustrate comparisons of parameter count and performance of example baseline VRNNs and example VHRNN models on real-world datasets, according to embodiments; FIG. 13 illustrates parameter performance plots of example baseline VRNNs and example VHRNN models using GRU implementation, according to an embodiment; FIGS. 14A-14D illustrate comparisons of hidden units and performance of example baseline VRNNs and example VHRNN models on real-world datasets, according to embodiments; FIG. 15 illustrates the performance of example baseline VRNNs (top table) and example VHRNN models (bottom) on real-world datasets, according to an embodiment; FIGS. 16A-16B illustrate comparisons of parameter count and performance of example baseline VRNNs, example VHRNN models, and example nHyperLSTM models on real-world datasets, according to an FIGS. 17A-17B illustrate comparisons of hidden units and performance of example baseline VRNNs, example VHRNN models, and example HyperLSTM models on real-world datasets, according to an embodiment; FIG. 18 illustrates experimental results of example HyperLSTM models on real-world datasets, according to an embodiment; FIG. 19 is a table illustrating evaluation results of example baseline VRNNs and example VHRNN models with the same latent dimensions, according to an embodiment; FIG. 20 is a table illustrating evaluation results of example VHRNN models with different hyper network inputs, according to an embodiment; FIG. 21 illustrates a parameter-performance comparison of example VHRNN models with an RNN as a hyper network and example VHRNN models with a three-layer feed-forward network on real-world datasets, according to an embodiment; and FIG. 22 illustrates results of a systematic generalization study on example VHRNN models with an RNN as a hyper network and example VHRNN models with a three-layer feed-forward network on synthetic data, according to an embodiment. Systems and methods disclosed herein provide a probabilistic sequence model that captures high variability in sequential or time series data, both across sequences and within an individual sequence. In some embodiments, systems and methods described herein for machine learning architecture with variational hyper recurrent neural networks use temporal latent variables to capture information about the underlying data pattern, and dynamically decode the latent information into modifications of weights of the base decoder and recurrent model. The efficacy of embodiments of the concepts described herein is demonstrated on a range of synthetic and real world sequential data that exhibit large scale variations, regime shifts, and complex dynamics. Recurrent neural networks (RNNs) can be used as architecture for modelling sequential data as RNNs can handle variable length input and output sequences. Initially invented in context of natural language processing [Hochreiter and Schmidhuber, 1997], long short-term memory (LSTM), gated recurrent unit (GRU) as well as later attention-augmented versions have found wide-spread successes, for example, in language modeling, machine translation, speech recognition and recommendation systems. However, RNNs use deterministic hidden states to process input sequences and model the system dynamics using a set of time-invariant weights, and they do not necessarily have the right inductive bias for time series data outside the originally intended domains. Many natural systems have complex feedback mechanisms and numerous exogenous sources of variabilities. Observations from such systems would contain large variations both across sequences in a dataset as well as within any single sequence; the dynamics could be switching regimes drastically, and the noise process could also be heteroskedastic. To capture all these intricate patterns in RNN with deterministic hidden states and a fixed set of weights requires learning about the patterns, the subtle deviations from the patterns, the conditions under which regime transitions occur which is not always predictable. Outside of the deep learning literature, many time series models have been proposed to capture specific types of high variabilities. For instance, switching linear dynamical models aim to model complex dynamical systems with a set of simpler linear patterns. Conditional volatility models are introduced to model time series with heteroscedastic noise process whose noise level itself is a part of the dynamics. However, these models usually encode specific inductive biases in a hard way, and cannot learn different behaviors and interpolate among the learned behaviors as deep neural nets. Variational autoencoder (VAE) is an unsupervised approach to learning a compact representation from data [Kingma and Welling, 2013]. VAE uses a variational distribution q(zlx) to approximate the intractable posterior distribution of the latent variable z. With the use of variational approximation, VAE optimizes, or maximizes, the evidence lower bound (ELBO) of the marginal log-likelihood of (x)=[q( Z\|X)][ log p(x\|z)]−D[KL](q(z\|x)∥p(z))≤log p(x) where p(z) is a prior distribution of z and D[KL ]denotes the Kullback-Leibler (KL) divergence. The approximate posterior q(zlx) is usually formulated as a Gaussian with a diagonal covariance matrix. Such formulation permits the use of reparameterization trick: Given q(z\|x)˜(μ, Σ), p(x\|z)=p(x\|μ+y·Σ^1/2). The reparameterization trick allows the model to be trained end-to-end with standard back Variational autoencoders have demonstrated impressive performance on non-sequential data like images. Certain works [Bowman et al, 2015; Chung et al, 2015; Fraccaro et al, 2016; Luo et al, 2018] extend the domain of VAE models to sequential data. Existing variational RNN (VRNN) [Chung et al, 2015] further incorporate a latent variable at each time step into their models. A prior distribution conditioned on the contextual information and a variational posterior is proposed at each time step to optimize a step-wise variational lower bound. Sampled latent variables from the variational posterior are decoded into the observation at the current time step. A parallel stream of work to improve latent variable models with variational inference study tighter bounds of the data's log-probability than ELBO. Importance Weighted Autoencoder (IWAE) [Burda et al, 2016] estimates a different variational bound of the log-likelihood of data with an importance weighted average using multiple samples of z. The bound of IWAE is provably tighter than ELBO. Filtering Variational Objective (FIVO) [Maddison et al, 2017] improves IWAE by incorporating particle filtering [Doucet and Johansen, 2009] that exploits the temporal structure of sequential data to estimate the data log-likelihood. A particle filter is a sequential Monte Carlo algorithm that propagates a population of weighted particles through all time steps using importance sampling. One distinguishing feature of FIVO is the resampling steps, which allow the model to drop low-probability samples with high-probability during training. When the effective sample size drops below a threshold, a new set of particles are sampled with replacement in proportion to their weights; the new weights are then reset to 1. Resampling prevents the relative variance of the estimates from exponentially increasing in the number of time steps. FIVO still computes a step-wise IWAE bound based on the sampled particles at each time step, but it shows better sampling efficiency and tightness than IWAE. In some embodiments, FIVO is used as the objective to train and evaluate models disclosed herein. Hypernetworks [Ha et al, 2016] use one network to generate the parameters or weights of another network. A dynamic version of hypernetworks can be applied to sequence data, but due to lack of latent variables, can only capture uncertainty in the output variables. For discrete sequence data such as text, categorical output variables can model multi-model outputs very well; but on continuous time series with the typical Gaussian output variables, traditional hypernetworks are much less capable at dealing with stochasticity. Furthermore, it does not allow straightforward interpretation of the model behavior using the time-series of KL divergence as disclosed herein. With the augmentation of latent variables, models disclosed herein are much more capable of modelling uncertainty. Bayesian hypernetworks [Krueger et al, 2017] learn an approximate posterior distribution over the parameters conditioned on the entire dataset. It utilizes the normalizing flow [Rezende and Mohamed, 2015, Kingma et al, 2016] to transform random noise to network weights. Weight normalization is used to parameterize the model's weight efficiently. However, the once learned weight distribution becomes independent of the model's input. This independence could limit the model's flexibility to deal with the variance in sequential data. Bayesian hypernetworks also have a latent variable in the context of hypernetworks. However, the goal of Bayesian Hypernetwork is an improved version of Bayesian neural net to capture model uncertainty. The work of [Krueger et al, 2017] has no recurrent structure and cannot be applied to sequential data. Furthermore, the use of normalizing flow dramatically limits the flexibility of the decoder architecture design, unlike in models as disclosed herein. Models disclosed herein can dynamically generate non-shared weights for RNNs based on inputs. In some embodiments, matrix factorization can be used to learn a compact embedding for the weights of static convolutional networks, illustrating the better parameter performance efficiency of hypernetworks. A system 100 for VHRNN modelling generates and implements a neural recurrent latent variable model, a variational hyper RNN (VHRNN) model 110, capable, in some embodiments, of capturing variability both cross different sequences in a dataset and within a sequence. In some embodiments, VHRNN model 110 can naturally handle scale variations of many orders of magnitude, including behaviours of sudden exponential growth in many real world bubble situations followed by collapse. In some embodiments, VHRNN model 110 can also perform system identification and re-identification dynamically at inference time. VHRNN model 110 makes use of factorization of sequential data and joint distribution of latent variables. In VHRNN model 110, latent variables also parameterize the weights for decoding and transition in RNN cell across time steps, giving the model more flexibility to deal with variations within and across sequences. Conveniently, VHRNN model 110 may capture complex time series without encoding a large number of patterns in static weights, but instead only encodes base dynamics that can be selected and adapted based on run-time observations. Thus VHRNN model 110 can easily learn to express a rich set of behaviors, including but not limited to behaviours disclosed herein. VHRNN model 110 can dynamically identify the underlying patterns and make time-variant uncertainty predictions in response to various types of uncertainties caused by observation noise, lack of information, or model misspecification. As such, VHRNN model 110 can model complex patterns with fewer parameters; when given a large number of parameters, it may generalize better than previous techniques. In some embodiments, VHRNN model 110 includes hypernetworks and is an improvement of the variational RNN (VRNN) model. VRNN models use recurrent stochastic latent variables at each time step to capture high-level information in the stochastic hidden states. The latent variables can be inferred using a variational recognition model and are fed as input into the RNN and decoding model to reconstruct observations, and an overall VRNN model can be trained to maximize the evidence lower bound (ELBO). In some embodiments, latent variables in VHRNN model 110 are dynamically decoded to produce the RNN transition weights and observation decoding weights in the style of hypernetworks, for example, generating diagonal multiplicative factors to the base weights. As a result, VHRNN model 110 may better capture complex dependency and stochasticity across observations at different time steps. VHRNN model 110 can sample a latent variable and dynamically generates non-shared weights at each time step, which can provide improved handling nof variance of dynamics within sequences. Conveniently, VHRNN model 110 may be better than existing techniques at capturing different types of variability and generalizing to data with unseen patterns on synthetic as well as real-world Formulation of VHRNN model 110, according to an embodiment, will now be detailed. A recurrent neural network (RNN) can be characterized by h[t]=g[θ](x[t],h[t-1]), where x[t ]and h[t ]are the observation state and hidden state of the RNN at time step t, and 0 is the fixed weights of the RNN model. Hidden state h[t ]is often used to generate the output for other learning tasks, e.g., predicting the observation at the next time step. For VHRNN model 110, an RNN or recurrence model g can be augmented with a latent random variable z[t], which is also used to output the non-shared parameters of RNN g at time step t. h[t]=g[θ(z][t][,h][t-1)](x[t],z[t], h[t-1]) (1) where θ(z[t], h[t-1]) is a hypernetwork that generates the parameters of RNN g at time step t. Latent variable z[t ]can also be used to determine the parameters of the generative model, or generating probability distribution p(x[t]\|z[≤t],x[<t]): x[t]\|z[≤t], x[21 t]˜(μ[t]^dec,Σ[t]^dec) (2) where (μ[t]^dec,Σ[t]^dec)=ϕ[ω(z][t][,h][t-1)](z[t],h[t-1]). Previous observations and latent variables, characterized by h[t-1], can define a prior probability distribution p(z[t]\|x[<t], z[<t]) over latent variable z[t], z[t]\|x[<t,]z[<t]˜(μ[t]^prior,Σ[t]^prior) (3) where (μ[t]^prior,Σ[t]^prior)=ϕ^prior(h[t-1]). From equations (2) and (3), the following generation process of sequential data can be developed: p(x[≤T],z[≤T])=Π[t=1]^Tp(z[t]\|x[<t],z[<t])p(x[t]\|x[<t],z[≤t]) (4) The true posterior distributions of z[t ]conditioned on observations x[≤t ]and latent variables z[<t ]are intractable, posing a challenge in both sampling and learning. Therefore, an approximate posterior distribution q(z[t]\|x[≤t],z[<t]) is introduced such that z[t]\|x[≤t],z[<t]˜(μ[t]^enc,Σ[t]^enc) (5) where (μ[t]^enc,Σ[t]^enc)=ϕ^enc(x[t],h[t-1]). This approximate posterior distrbution enables VHRNN model 110 to be trained by maximizing a variational lower bound, such as ELBO [Kingma and Welling, 2013], IWAE [Burda et al, 2016] or FIVO [Maddison et al, 2017]. The main components of VHRNN model 110, including g, ϕ^dec, ϕ^enc, ϕ^priop may be referred to as “primary networks” and the components responsible for generating parameters, θ and ω, referred to as “hypernetworks” herein. FIG. 1 is a schematic diagram of operations 112A, 112B, 112C, 112D and 112E for each time step t of a VHRNN model 110, a neural recurrent latent variable model, performed by a system 100, according to an embodiment. FIG. 1 illustrates, for each of operations 112A, 112B, 112C, 112D and 112E , at time t, a latent variable state z[t], an observation state x[t], a hidden state h[t], and previous time step hidden state h[t-1 ] Operators in FIG. 1 are indicated by arrows, and dashed lines and boxes represent hypernetwork components. Operation 112A is a prior operation of VHRNN model 110 to define a prior distribution, for example, based on equation (3). Operation 112B is a recurrence operation of VHRNN model 110 to update an RNN hidden state, for example, based on equation (1). Operation 112C is a generation operation of VHRNN model 110 to define a generating distribution, for example, based on equation (2). Operation 112D is an inference operation of VHRNN model 110 to infer an approximate posterior, for example, based on equation (5). Operation 112E illustrates an overall architecture of a computational path of VHRNN model 110, omitting hypernetwork components. For operation 112A, system 100 determines a prior probability distribution p(z[t]\|x[<t], z[<t]) for latent variable z[t], given previous observations x[<t ]and previous latent variables z[<t]. In some embodiments, the prior probability distribution is defined based on equation (3), and the parameters of the prior probability distribution are determined from a prior network ϕ^prior using an initial hidden state h[t-1]. ϕ^prior is a suitable function such as a neural network. For operation 112B, system 100 determines or updates a hidden state h[t]. In some embodiments, the hidden state h[t ]is defined based on equation (1), and the hidden state h[t ]is determined from an RNN model g using an observation state x[t], the latent variable z[t ]and the initial hidden state h[t-1]. The parameters of RNN g, (namely, the observation state x[t], the latent variable z[t ]and the initial hidden state h[t-1]) are updated by a hypernetwork θ(z[t], h[t-1]) using the latent variable z[t ]and the initial hidden state h[t-1]. In some embodiments, hypernetwork θ(z[t], h[t-1]) is implemented as an RNN. For operation 112C, system 100 determines a generating probability distribution p(x[t]\|z[≤t],x[<t]) for observation state x[t], given latent variable z[t], previous observations x[<t ]and previous latent variables z[<t]. In some embodiments, the generating distribution is defined based on equation (2), and the parameters of the generating distribution are determined from a decoder network ϕ^ dec using latent variable z[t ]and the initial hidden state h[t-1]. The parameters of decoder network ϕ^dec (namely, the latent variable z[t ]and the initial hidden state h[t-1]) are updated by another hypernetwork ω(z[t],h[t-1]). In some embodiments, hypernetwork ω(z[t],h[t-1]) is implemented as a multilayer perceptron (MLP). System 100 may sample an observation state x[t ]from the generating distribution. For operation 112D, system 100 determines an approximate posterior probability distribution q(z[t]\|x[≤t],z[<t]) for latent variable z[t], given observation state x[t], previous observations x[<t ]and previous latent variables z[<t]. In some embodiments, the approximate posterior probability distribution is defined based on equation (5), and the parameters of the approximate posterior probability distribution are determined from an encoder network ϕ^enc using observation state x[t ]and the initial hidden state h[t-1]. The approximate posterior probability distribution enables VHRNN model 110 to be trained by maximizing a variational lower bound, such as evidence lower bound (ELBO) [Kingma and Welling, 2013], importance weight autoencoders (IWAE) [Burda et al, 2016] and filtering variational objectives (FIVO) [Maddison et al, 2017]. Operation 112E illustrates an overall computational path of VHRNN model 110. In some implementations, using a VAE approach, covariance matrices Σ[t]^prior, Σ[t]^dec and Σ[t]^enc can be parameterized as diagonal matrices. In some embodiments, Σ[t]^prior in VHRNN model 110 is not an identity matrix as in a vanilla VAE; it is the output of ϕ^prior and depends on the hidden state h[t-1 ]at the previous time step. In some embodiments, recurrence model g in equation (1) is implemented as an RNN cell, which takes as input x[t ]and z[t ]at each time step t and updates the hidden states h[t-1]. The parameters of g are generated by the hyper network θ(z[t],h[t-1]), as illustrated in operation 112B of FIG. 1. In some embodiments, θ is implemented using an RNN to capture the history of data dynamics, with z[t ]and h[t-1 ]as input at each time step t. However, it can be computationally costly to generate all the parameters of g using θ(z[t],h[t-1]). Thus, in some embodiments, hypernetwork θ maps z[t ]and h[t-1 ]to bias and scaling vectors. In some embodiments, scaling vectors modify the parameters of g by scaling each row of the weight matrices, routing information in the input and hidden state vectors through different channels. In some embodiments, recurrence model g may be implemented using an RNN cell 200 with LSTM-style update rules and gates, as illustrated in FIG. 2. Let * ∈ {i, f, g, o} denote the one of the four LSTM-style gates in g. W[* ]and U[* ]denote the input and recurrent weights of each gate in LSTM cell respectively. The hyper network θ(z[t],h[t-1]) outputs d[i* ]and d[h* ]that are the scaling vectors for the input weights W[* ]and recurrent weights U[* ]of the recurrent model g in equation (1). The overall implementation of g in equation (1) can be described, in an embodiment, as follows: g[t]=tan h(d[ig](z[t],h[t-1])∘(W[g]y[t])+d[hg](z[t],h[t-1])∘(U[g]h[t-1])), h[t]=o[t]∘tan h(c[t]), where ∘ denotes the Hadamard product and y[t ]is a fusion (e.g., concatenation) of observation x[t ]and latent variable z[t]. For simplicity of notation, bias terms are omitted from the above Another hypernetwork ω(z[t],h[t-1]) generates the parameters of the generative model in equation (2). In some embodiments, hypernetwork ω(z[t], h[t-1]) is implemented as a multilayer perceptron (MLP). Similar to θ(z[t], h[t-1]), the outputs can be the bias and scaling vectors that modify the parameters of the decoder ϕ[ω(z][t][,h][t-1)]. FIG. 3A is a flow chart of a method 300 for training a VHRNN such as VHRNN model 110, according to an embodiment. The steps are provided for illustrative purposes. Variations of the steps, omission or substitution of various steps, or additional steps may be considered. Blocks 302 to 310 are performed for for each step or time step t from 1 to T in sequential training data, such as time series data, x=(x[1]x[2], . . . , x[T]). At block 302, a prior probability distribution p(z[t]\|x[<t], z[<t]) is determined for a latent variable z[t], given previous observations x[<t ]and previous latent variables z[<t], from a prior network ϕ^prior of the VHRNN using an initial hidden state h[t-1]. In some embodiments, the prior probability distribution p(z[t]\|x[<t], z[<t]) for the latent variable z[t ]is based on equation (3): where (μ[t]^prior,Σ[t]^prior) is the prior network ϕ^prior. At block 304, a hidden state h[t ]is determined from a recurrent neural network (RNN) g of the VHRNN using an observation state x[t], the latent variable z[t ]and the initial hidden state h[t-1]. In some embodiments, the RNN g is based on equation (1): where θ(z[t], h[t-1]) is a hypernetwork of the VHRNN that generates parameters of RNN g using the latent variable z[t ]and the initial hidden state h[t-1]. In some embodiments, the hypernetwork θ(z[t], h[t-1]) is implemented as a recurrent neural network (RNN). In some embodiments, the hypernetwork θ(z[t], h[t-1]) is implemented as a long short-term memory (LSTM). In some embodiments, the hypernetwork θ(z[t], h[t-1]) generates scaling vectors for input weights and recurrent weights of the RNN g. In some embodiments, the scaling vectors modify parameters of the RNN g by scaling each row of weight matrices. At block 306, an approximate posterior probability distribution q(z[t]\|x[≤t ], z[<t]) is determined for the latent variable z[t], given the observation state x[t], previous observations x[<t ]and previous latent variables z[<t], from an encoder network ϕ^enc of the VHRNN using the observation state x[t ]and the initial hidden state h[t-1]. At block 308, a generating probability distribution p(x[t]\|z[≤t],x[<t]) is determined for the observation state x[t], given the latent variable z[t], the previous observations x[<t ]and the previous latent variables z[<t], from a decoder network ϕ^dec of the VHRNN using the latent variable z[t ]and the initial hidden state h[t-1]. In some embodiments, the generating probability distribution p(x[t]\|z[≤t],x[<t]) for the observation state x[t ]is based on equation (2): x[t]\|z[≤t], x[<t]˜(μ[t]^dec,Σ[t]^dec) where (μ[t]^dec,Σ[t]^dec)=ϕ[ω(z][t][,h][t-1)](Z[t],h[t-1]), and ω(z[t],h[t-1]) is another hypernetwork of the VHRNN that generates parameters of the decoder network ϕ^dec using the latent variable z [t ]and the initial hidden state h[t-1]. In some embodiments, the hypernetwork ω(z[t], h[t-1]) is implemented as a multilayer perceptron (MLP). At block 310, a variational lower bound of a marginal log-likelihood of the training data is maximized, to train the VHRNN. In some embodiments, the variational lower bound includes at least one of an evidence lower bound (ELBO), importance weight autoencoders (IWAE), or filtering variational objectives (FIVO). In some embodiments, the trained VHRNN is stored in a memory such as memory 220. In some embodiments, a VHRNN model 110 trained, for example, using method 300, may be stored on a computer readable memory, such as memory 220, of a non-transitory computer readable medium, trained VHRNN model 110 executable by a computer, such as processor(s) 210, to perform a method to generate sequential data, such as method 350 described below. It should be understood that the blocks may be performed in a different sequence or in an interleaved or iterative manner. FIG. 3B is a flow chart of a method 350 for generating sequential data using a VHRNN such as VHRNN model 110, in an example, trained by method 300, according to an embodiment. The steps are provided for illustrative purposes. Variations of the steps, omission or substitution of various steps, or additional steps may be considered. Blocks 352 to 358 are performed for for each step or time step t from 1 to T in sequential data, such as time series data. In some embodiments, there is no pre-specified length (or number of steps) of a sequence, and method 350 may use step-wise generation for any suitable length of sequence. At block 352, a prior probability distribution p(z[t]\|x[<t], z[<t]) is determined for a latent variable z[t], given previous observations x[<t ]and previous latent variables z[<t], from the prior network ϕ^prior of the VHRNN using an initial hidden state h[t-1]. In some embodiments, the prior probability distribution p(z[t]\|x[<t],z[<t]) for the latent variable z[t ]is based on equation (3): where (μ[t]^prior,Σ[t]^prior) is the prior network ϕ^prior. At block 354, a hidden state h[t ]is determined from the recurrent neural network (RNN) g of the VHRNN using an observation state x[t], the latent variable z[t ]and the initial hidden state h[t-1]. In some embodiments, the RNN g is based on equation (1): where θ(z[t],h[t-1]) is a hypernetwork of the VHRNN that generates parameters of RNN g using the latent variable z[t ]and the initial hidden state h[t-1]. In some embodiments, the hypernetwork θ(z[t], h[t-1]) is implemented as a recurrent neural network (RNN). In some embodiments, the hypernetwork θ(z[t], h[t-1]) is implemented as a long short-term memory (LSTM). In some embodiments, the hypernetwork θ(z[t], h[t-1]) generates scaling vectors for input weights and recurrent weights of the RNN g. In some embodiments, the scaling vectors modify parameters of the RNN g by scaling each row of weight matrices. At block 356, a generating probability distribution p(x[t]\|z[≤t],x[<t]) is determined for the observation state x[t ]given the latent variable z[t], the previous observations x[<t ]and the previous latent variables z[<t], from the decoder network ϕ^dec of the VHRNN using the latent variable z[t ]and the initial hidden state h[t-1]. In some embodiments, the generating probability distribution p(x[t]\|z[≤t],x[<t]) for the observation state x[t ]is based on equation (2): x[t]\|z[≤t],x[21 t]˜(μ[t]^dec,Σ[t]^dec) where (μ[t]^dec, Σ[t]^dec)=ϕ[ω(z][t][,h][t-1)](z[t], h[t-1]) is another hypernetwork of the VHRNN that generates parameters of the decoder network ϕ^dec using the latent variable z[t ]and the initial hidden state h[t-1]. In some embodiments, the hypernetwork ω(z[t], h[t-1]) is implemented as a multilayer perceptron (MLP). At block 358, a generated observation state x[t ]is sampled from the generating probability distribution p(x[t]\|z[≤t], x[<t]). The sampled observation states may then form the generated sequential In some embodiments, future observations of the sequential data are forecasted based on the sampled generated observation states. In some embodiments, the sequential data is time-series financial data. It should be understood that the blocks may be performed in a different sequence or in an interleaved or iterative manner. System 100 for VHRNN modelling, to model sequential data, may be implemented as software and/or hardware, for example, in a computing device 120 as illustrated in FIG. 4. Method 300, in particular, one or more of blocks 302 to 310, may be performed by software and/or hardware of a computing device such as computing device 120. Method 350, in particular, one or more of blocks 352 to 358, may be performed by software and/or hardware of a computing device such as computing device 120. FIG. 4 is a high-level block diagram of computing device 102. As will become apparent, computing device 102, under software control, may train VHRNN model 110 and use VHRNN model 110 to generate sequential data such as time-series data. As illustrated, computing device 102 includes one or more processor(s) 210, memory 220, a network controller 230, and one or more I/O interfaces 240 in communication over bus 250. Processor(s) 210 may be one or more Intel x86, Intel x64, AMD x86-64, PowerPC, ARM processors or the like. Memory 220 may include random-access memory, read-only memory, or persistent storage such as a hard disk, a solid-state drive or the like. Read-only memory or persistent storage is a computer-readable medium. A computer-readable medium may be organized using a file system, controlled and administered by an operating system governing overall operation of the computing device. Network controller 230 serves as a communication device to interconnect the computing device with one or more computer networks such as, for example, a local area network (LAN) or the Internet. One or more I/O interfaces 240 may serve to interconnect the computing device with peripheral devices, such as for example, keyboards, mice, video displays, and the like. Such peripheral devices may include a display of device 102. Optionally, network controller 230 may be accessed via the one or more I/O interfaces. Software instructions are executed by processor(s) 210 from a computer-readable medium. For example, software may be loaded into random-access memory from persistent storage of memory 220 or from one or more devices via I/O interfaces 240 for execution by one or more processors 210. As another example, software may be loaded and executed by one or more processors 210 directly from read-only Example software components and data stored within memory 220 of computing device 102 may include machine learning software 290 to generate VHRNN model 110, and operating system (OS) software (not shown) allowing for basic communication and application operations related to computing device 102. Memory 220 may include machine learning software 290 with rules and models such as VHRNN model 110. Machine learning software 290 can refine based on learning. Machine learning software 290 can include instructions to implement an artificial neural network, such as generating VHRNN model 110, and performing sequence modelling and generating using VHRNN model 110. As compared to a large VRNN, the structure of VHRNN model 110 conveniently better encodes the inductive bias that the underlying dynamics could change; that is, they could slightly deviate from the typical behavior in a regime, or there could be a drastic switch to a new regime. With finite training data and a finite number of parameters, this inductive bias could lead to qualitatively different learned behavior, which is demonstrated and analyzed below, providing a systematic generalization study of VHRNN in comparison to a VRNN baseline. An example VHRNN model 110 and an example VRNN baseline model are trained on one synthetic dataset with each sequence generated by fixed linear dynamics and corrupted by heteroskedastic noise processes. It is demonstrated that VHRNN model 110 can disentangle the two contributions of variations and learn the different base patterns of the complex dynamics while doing so with fewer parameters. Furthermore, VHRNN model 110 can generalize to a wide range of unseen dynamics, albeit the much simpler training set. A synthetic dataset can be generated by the following recurrence equation: x[t]=Wx[t-1]+σ[t]∈[t ](6) where ∈[t ]∈^2 is a two-dimensional standard Gaussian noise and x[0 ]is randomly initialized from a uniform distribution over [−1,1]^2. For each sequence, W ∈^2×2 is sampled from ten predefined random matrices {W[i]}[i=1 ]^10 with equal probability; σ[t ]is the standard deviation of the additive noise at time t and takes a value from {0.25,1, 4}. The noise level shifts twice within a sequence; i.e., there are exactly two t's such that σ[t]#σ[t-1]. Eight hundred sequences are generated for training, one hundred sequences for validation, and one hundred sequences for test using the same sets of predefined matrices. The example VRNN baseline model and example VHRNN model 110 are trained and evaluated using FIVO as the objective. The results on the test set are almost the same as those on the training set for both VRNN and VHRNN model 110. VHRNN model 110 shows better performance than baseline VRNN with fewer parameters, as shown in the table illustrated in FIG. 5, under the “Test” column. The size of the hidden state in RNN cells is set to be the same as the latent size for both types of models. FIG. 5 also illustrates the behaviour of the example VRNN baseline models and example VHRNN model 110 under the following systematically varied settings: □ NOISELESS: In this setting, sequences are generated using a similar recurrence rule with the same set of predefined weights without the additive noise at each step. That is, σ[t]=0 in Eq. 6 for all time step t. The exponential growth of data could happen when the singular values of the underlying weight matrix are greater than 1. □ SWITCH: In this setting, three NOISELESS sequences are concatenated into one, which contains regime shifts as a result. This setting requires the model to identify and re-identify the underlying pattern after observing changes. □ LONG: In this setting, extra-long NOISELESS sequences are generated with twice the total number of steps using the same set of predefined weights. The data scale can exceed well beyond the range of training data when exponential growth happens. □ ZERO-SHOT: In this setting, NOISELESS sequences are generated such that the training data and test data use different sets of weight matrices. □ ADD: In this setting, sequences are generated by a different recurrence rule: x[t]=x[t-1]+b, where b and x[0 ]are uniformly sampled from [0,11]^2. □ RAND: In this setting, the deterministic transition matrix in equation (6) is set to the identity matrix (i.e., W=I), leading to long sequences of pure random walks with switching magnitudes of noise. The standard deviation of the additive noise randomly switches up to 3 times within {0.25,1,4} in one sequence. The table of FIG. 5 illustrates the experimental results for the above settings on synthetic datasets. As shown, the example baseline VRNN model, depending on model complexity, either underfits the original data generation pattern (“Test”) or fails to generalize to more complicated settings. In contrast, the example VHRNN model 110 does not suffer from such problems and uniformly outperforms VRNN models under all settings. FIGS. 6A-6F illustrate observations from a qualitative study of the behavior of embodiments of an example baseline VRNN and an example VHRNN model 110 under a NOISELESS setting, the example baseline VRNN with a latent dimension of eight and the example VHRNN model 110 with a latent dimension of four. FIGS. 6A and 6B show the values of concatenated data at each time step. FIG. 6C shows the KL divergence between the variational posterior and the prior of the latent variable at each time step for the example VHRNN model 110. FIG. 6D shows the KL divergence for the example baseline VRNN. FIG. 6E shows L2 distance between the predicted mean values by the example VHRNN model 110 and the example baseline VRNN and the target. FIG. 6F shows the predicted log-variance of the output distribution for the example baseline VRNN and the example VHRNN model 110. FIGS. 7A-7F illustrate observations from a qualitative study of the behavior of embodiments of an example baseline VRNN and an example VHRNN model 110 under a SWITCH setting, the example baseline VRNN with a latent dimension of eight and the example VHRNN model 110 with a latent dimension of four. Vertical dashed lines indicate time steps when regime shift happen. FIGS. 7A and 7B show the values of concatenated data at each time step. FIG. 7C shows the KL divergence between the variational posterior and the prior of the latent variable at each time step for the example VHRNN model 110. FIG. 7D shows the KL divergence for the example baseline VRNN. FIG. 7E shows L2 distance between the predicted mean values by VHRNN model 110 and the example baseline VRNN and the target. FIG. 7F shows the predicted log-variance of the output distribution for the example baseline VRNN and the example VHRNN model 110. FIGS. 6A-6F demonstrate an observation of dynamic regime identification and re-identification. FIGS. 6A-6F show a sample sequence under the NOISELESS setting, whereby the example baseline VRNN has high KL divergence between the prior and the variational posterior most of the time, and in contrast, the example VHRNN model 110 has a decreasing trend of KL divergence while still making accurate mean reconstruction as it observes more data. As the KL divergence measures the discrepancy between prior defined in equation (3) and the posterior that has information from the current observation, simultaneous low reconstruction and low KL divergence means that the prior distribution would be able to predict with low errors as well, indicating that the correct underlying dynamics model has likely been utilized. Conveniently, this trend indicates the ability of VHRNN model 110 to identify the underlying data generation pattern in the sequence. The decreasing trend is especially apparent when sudden and big changes in the scale of observations happen. Larger changes in scale may better help VHRNN model 110 identify the underlying data generation process because VHRNN model 110 is trained on sequential data generated with compound noise. The observation also confirms that the KL divergence would rise again once the sequence switches from one underlying weight to another, as shown in FIGS. 7A-7F. It is worth noting that the KL increase happens with some latency after the sequence switches in the SWITCH setting as VHRNN model 110 reacts to the change and tries to reconcile with the prior belief of the underlying regime in effect. A similar trend of unseen regime generalization can also be found in settings where patterns of variation are not present in the training data, namely ZEROSHOT and ADD. Sample sequences are shown in FIGS. 8A-8F and FIGS. 9A-9F. FIGS. 8A-8F illustrate observations from a qualitative study of the behavior of embodiments of an example baseline VRNN and an example VHRNN model 110 under a ZERO-SHOT setting, the example baseline VRNN with a latent dimension of eight and the example VHRNN model 110 with a latent dimension of four. FIGS. 8A and 8B show the values of concatenated data at each time step. FIG. 8C shows the KL divergence between the variational posterior and the prior of the latent variable at each time step for example VHRNN model 110. FIG. 8D shows the KL divergence for the example baseline VRNN. FIG. 8E shows L2 distance between the predicted mean values by the example VHRNN model 110 and the example baseline VRNN and the target. FIG. 8F shows the predicted log-variance of the output distribution for the example baseline VRNN and the example VHRNN model 110. FIGS. 9A-9F illustrate observations from a qualitative study of the behavior of embodiments of an example baseline VRNN and an example VHRNN model 110 under an ADD setting, the example baseline VRNN with a latent dimension of eight and the example VHRNN model 110 with a latent dimension of four. FIGS. 9A and 9B show the values of concatenated data at each time step. FIG. 9C shows the KL divergence between the variational posterior and the prior of the latent variable at each time step for the example VHRNN model 110. FIG. 9D shows the KL divergence for the example baseline VRNN. FIG. 9E shows L2 distance between the predicted mean values by the example VHRNN model 110 and the example baseline VRNN and the target. FIG. 9F shows the predicted log-variance of the output distribution for the example baseline VRNN and the example VHRNN model 110. From FIGS. 8A-8F and 9A-9F, it can be seen that the KL divergence of the example VHRNN model 110 decreases as it observes more data. Meanwhile the mean reconstructions by the example VHRNN model 110 stay relatively close to the actual target value as shown in FIG. 8E and FIG. 9E. The reconstructions are especially accurate in the ADD setting as FIG. 9E shows. The observation of unseen regime generalization implies that the ability of VHRNN model 110 to recover the data generation dynamics at test time is not limited to the existing patterns in the training data. By contrast, there is no evidence that traditional variational RNN is capable of doing similar regime identification. FIGS. 10A-10B illustrate observations from a qualitative study of the behavior of embodiments of an example baseline VRNN and an example VHRNN model 110 under a RAND setting, the example baseline VRNN with a latent dimension of eight and the example VHRNN model 110 with a latent dimension of four. FIG. 10A shows the L2 norm and standard deviation of the additive noise at each time step. FIG. 10B shows the log-variance of the output distribution for the example baseline VRNN and the example VHRNN model 110. In some embodiments, uncertainty identification is also observed. FIGS. 10A and 10B show that the predicted log-variance of VHRNN model 110 can more accurately reflect the change of noise levels under the RAND setting than a baseline VRNN. VHRNN model 110 can also better handle uncertainty than the baseline VRNN in the following situations. As shown in FIG. 7F, in some embodiments, VHRNN model 110 can more aggressively adapt its variance prediction based on the scale of the data than a baseline VRNN. It keeps its predicted variance at a low level when the data scale is small and increases the value when the scale of data becomes large. VHRNN model 110 makes inaccurate mean prediction relatively far from the target value when the switch of underlying generation dynamics happens in the SWITCH setting. The switch of the weight matrix is another important source of uncertainty. In some embodiments, VHRNN model 110 would also make a large log-variance prediction in this situation, even the scale of the observation is small. Aggressively increasing its uncertainty about the prediction when a switch happens avoids VHRNN model 110 from paying high reconstruction cost as shown by the second spike in FIG. 7F. This increase of variance prediction also happens when exponential growth becomes apparent in setting LONG and the scale of observed data became out of the range of the training data. Given the large scale change of the data, such flexibility to predict large variance is key for VHRNN model 110 to avoid paying large reconstruction cost. FIGS. 11A-11F illustrate observations from a qualitative study of the behavior of an example baseline VRNN and an example VHRNN model 110 under a LONG setting, according to an embodiment. FIGS. 11A, 11B, 11D and 11E use scientific notations for the value of Y axis. The magnitude of the data grows rapidly in such setting due to exponential growth and it is well beyond the scale of training data. FIGS. 11A and 11B show the values of concatenated data at each time step. FIG. 11C shows the KL divergence between the variational posterior and the prior of the latent variable at each time step for the example VHRNN model 110. FIG. 11D shows the KL divergence for the example baseline VRNN. FIG. 11E shows L2 distance between the predicted mean values by the example VHRNN model 110 and the example baseline VRNN and the target. FIG. 11F shows the predicted log-variance of the output distribution for the example baseline VRNN and the example VHRNN model 110. As illustrated in FIGS. 11A-11F, both baseline VRNN and VHRNN model 110 may make inaccurate mean predictions that are far from the target values. However, VHRNN model 110 pays smaller reconstruction cost than the baseline VRNN by also making large predictions of variance. This setting demonstrates a special case in which VHRNN model 110 has better ability to handle uncertainty in data than a baseline vanilla variational RNN. Conveniently, these advantages of VHRNN model 110 over a baseline VRNN illustrate the better performance of VHRNN model 110 on synthetic data and demonstrate an ability to model real-world data with large variations both across and within sequences. Experiments were performed with example embodiments of VHRNN model 110 on several real-world datasets and compared against example baseline VRNN to demonstrate superior parameter performance efficiency of VHRNN model 110. FIGS. 12A-12D illustrate parameter-performance comparison of example baseline VRNNs and example VHRNN models 110 on real-world datasets, according to embodiments. Training and evaluating VRNN using FIVO [Maddison et al, 2017] demonstrates state-of-the-art performance on various sequence modeling tasks. The experiments performed demonstrate the superior parameter-performance efficiency and generalization ability of VHRNN model 110 over baseline VRNN. All the models were trained using FIVO [Maddison et al, 2017] and FIVO per step reported when evaluating models. Two polyphonic music dataset were considered: JSB Chorale and Piano-midi.de [Boulanger-Lewandowski et al, 2012]. The models were also trained and tested on Stock dataset containing a financial time series data and an HT Sensor dataset [Huerta et al, 2016], which contains sequences of sensor readings when different types of stimuli are applied in an environment during experiments. A HyperLSTM model is also considered without latent variables proposed by [Ha et al, 2016] for comparison purposes. For all the real-world data, both example baseline VRNNs and example VHRNN models 110, are trained with batch size of 4 and particle size of 4. When evaluating the models, a particle size of is used 128 for polyphonic music datasets and 1024 is used for Stock and HT Sensor datasets. For real-world dataset experimentation, a single-layer LSTM was used for the example baseline VRNN models, and the dimension of the hidden state was set to be the same as the latent dimension. For the example VHRNN models 110, θ in equation (1) was implemented using a single-layer LSTM to generate weights for the recurrence module in the primary networks. An RNN cell with LSTM-style gates and update rules for the recurrence module g was used. The hidden state sizes of both the primary network and hyper network are the same as the latent dimension. A linear transformation directly maps the hyper hidden state to the scaling and bias vectors in the primary network. Further detail on the architectures of encoder, generation and prior networks are provided below. In some embodiments, implementation of the architecture of the encoder in equation (5) is the same in the example VHRNN models 110 and the example baseline VRNNs. For synthetic datasets, the encoder may be implemented by a fully-connected network with two hidden layers; each hidden layer has the same number of units as the latent variable dimension. For real-world datasets, a fully-connected network may be used, with one hidden layer. The number of units may also be the same as the latent dimension. In some embodiments, the prior network is implemented by a similar architecture as the encoder, differing in the dimension of inputs. In some embodiments, for implementation of example VHRNN models 110, fully-connected hyper networks with two hidden layers are used for synthetic data and fully-connected hyper networks with one hidden layer for other datasets as the decoder networks. The number of units in each hidden layer may also be the same as the latent variable defined in equation (2). For each layer of the hyper networks, the weight scaling vector and bias may be generated by an two-layer MLP. In some embodiments, the hidden layer size of this MLP is 8 for synthetic dataset and 64 for real-world datasets. For the example baseline VRNN models, plain feed-forward networks may be used for decoder. The number of hidden layers and units in the hidden layer may be determined in the same way as VHRNN model For comparison with a baseline VRNN [Chung et al, 2015], in some embodiments, the latent variable and observations are encoded by a network different from the encoder in equation (5) before being fed to the recurrence network and encoder. The latent and observation encoding networks may have the same architecture except for the input dimension in each experiment setting. For synthetic datasets, the encoding network may be implemented by a fully-connected network with two hidden layers. For real-world datasets, a fully-connected network may be used, with one hidden layer. The number of units in each hidden layer may be the same as the dimension of latent variable in that setting. JSB Chorale and Piano-midi.de are polyphonic music datasets [Boulanger-Lewandowski et al, 2012] with complex patterns and large variance both within and across sequences. The datasets are split into train, validation, and test sets. For preprocessing of the polyphonic music datasets, JSB Chorale and Piano-midi.de, each sample is represented as a sequence of 88-dimensional binary vectors. The data are preprocessed by mean-centering along each dimension per dataset. FIG. 12A illustrates FIVO per time step of example VHRNN models 110 and example baseline VRNNs and their parameter counts trained and evaluated on the JSB Chorale dataset, according to an embodiment. FIG. 12B illustrates FIVO per time step of example VHRNN models 110 and example baseline VRNNs and their parameter counts trained and evaluated on the Piano-midi.de dataset, according to an embodiment. The number of parameters and FIVO per time step of each model are plotted in FIGS. 12A and 12B, and the latent dimension is also annotated. The results show that VHRNN model 110 has better performance and parameter efficiency. The parameter-performance plots in FIGS. 12A and 12B show that VHRNN model 110 has uniformly better performance than baseline VRNN with a comparable number of parameters. As illustrated in FIG. 12A, the best FIVO achieved by an example VHRNN model 110 on JSB dataset is −6.76 (VHRNN-14) compared to −6.92 for an example baseline VRNN (VRNN-32), which requires close to one third more parameters. This best example baseline VRNN model is even worse than the smallest example VHRNN model 110 evaluated. It is also observed that VHRNN model 110 is less prone to overfitting and has better generalization ability than baseline VRNN when the number of parameters keeps growing. Similar trends can be seen on the Piano-midi.de dataset in FIG. 12B. Experimental work to-date also indicates better performance of VHRNN model 110 over baseline VRNN in a scenario replacing LSTM with Gated Recurrent Unit (GRU). FIG. 13 shows the parameter performance plots of example VHRNN models 110 and example baseline VRNNs using GRU implementation on the JSB Chorale dataset. As shown in FIG. 13, VHRNN models 110 consistently outperform baseline VRNN models under all settings. Financial time series data, such as daily prices of stocks, can be highly volatile with large noise. Market volatility can be affected by many external factors and can experience tremendous changes in a short period of time. To test ability to adapt to different volatility levels and noise patterns, example baseline VRNNs and example VHRNN models 110 were compared on a stock dataset containing stock price data collected in a period when the market went through rapid changes. The Stock dataset includes data collected from 445 stocks in the S&P 500 index in 2008 when a global financial crisis happened. To generate the Stock dataset, 345 companies were randomly selected for their daily stock price and volume in the first half of 2008 to obtain training data. Another 50 companies' data from the second half of 2008 was acquired to generate validation set and the test set was obtained from the remaining 50 companies during the second half of 2008. The sequences were first preprocessed by taking log ratio of the values between consecutive days, each sequence having a fixed length of 125. The log ratio sequences were normalized using the mean and standard deviation of the training set along each dimension. The Stock dataset contains the opening, closing, highest and lowest prices, and volume on each day. The networks are trained on sequences from the first half of the year and tested on sequences from the second half, during which the market suddenly became significantly more volatile due to the financial crisis. The evaluation results of example baseline VRNNs and example VHRNN models 110 trained and evaluated on the Stock dataset are shown in FIG. 12C. The number of parameters and FIVO per time step of each model are plotted in FIG. 12C, and the latent dimension is also annotated. The plot shows that VHRNN models 110 consistently outperform baseline VRNN models regardless of the latent dimension and number of parameters. The results indicate that VHRNN model 110 may have better generalizability to sequential data in which the underlying data generation pattern suddenly shifts even if the new dynamics are not seen in the training data. A comparison of baseline VRNNs and VHRNN models 110 was also performed on a HT Sensor dataset, having less variation and simpler patterns than the previous datasets. The HT Sensor dataset contains sequences of gas, humidity, and temperature sensor readings in experiments where some stimulus is applied after a period of background activity [Huerta et al, 2016]. There are two types of stimuli in the experiments: banana and wine. In some sequences, there is no stimulus applied, and they only contain readings under background noise. The HT Sensor dataset collects readings from 11 sensors under certain stimulus in an experiment. The readings of the sensors are recorded at a rate of once per second. A sequence of 3000 seconds every 1000 seconds in the dataset is segmented and downsampled by a rate of 30. Each sequence obtained has a fixed length of 100. The types of sequences include pure background noise, stimulus before and after background noise and stimulus between two periods of background noise. The data are normalized to zero mean and unit variance along each dimension. In some embodiments, 532 sequences are used for training, 68 sequences are used for validation and 74 sequences are used for testing. Experimental results on for example baseline VRNNs and example VHRNN models 110 on HT Sensor dataset are shown in FIG. 12D. The number of parameters and FIVO per time step of each model are plotted in FIG. 12D, and the latent dimension is also annotated. It is observed that VHRNN model 110 has comparable performance as the baseline VRNN on the HT Senor dataset when using a similar number of parameters. For example, VHRNN achieves a FIVO per time step of 14.41 with 16 latent dimensions and 24200 parameters, while baseline VRNN shows slightly worse performance with 28 latent dimensions and approximately 26000 parameters. When the number of parameters goes slightly beyond 34000, the FIVO of an example VHRNN model 110 decays to 12.45 compared to 12.37 of an example VRNN. FIGS. 14A-14D illustrate comparisons of hidden units and performance of example baseline VRNNs and example VHRNN models 110 on real-world datasets, according to embodiments. In particular, FIG. 14A illustrates results on the JSB Chorale dataset, FIG. 14B illustrates results on the Piano-midi.de dataset, FIG. 14C illustrates results on the Stock dataset, and FIG. 14D illustrates results on the HT Sensor dataset. VHRNN model 110 and a baseline VRNN are compared by plotting the example models' performance against their number of hidden units. The models considered in FIGS. 14A-14D are the same as the models presented in FIGS. 12A-12D, as described herein. A single-layer LSTM model is used for the RNN part, and the dimension of the LSTM's hidden state is the same as the latent dimension. Example VHRNN models 110 uses two LSTM models, one primary network and one hyper network. Therefore, the number of hidden units in an example VHRNN model 110 is twice the number of latent dimension. As illustrated in FIGS. 14A-14D, VHRNN model 110 also dominates the performance of VRNN with a similar or fewer number of hidden units in most of the settings. Furthermore, the fact that VHRNN model 110 almost always outperforms the baseline VRNN for all parameter or hidden unit sizes precisely shows the superiority of the new architecture. The results from FIGS. 12A-12D and FIGS. 14A-14D are consolidated in a table illustrated in FIG. 15. FIG. 15 illustrates performance of example baseline VRNNs in the top table, and performance of example VHRNN models 110 in the bottom table, on real-world datasets, according to an embodiment. In additional experimental work, VHRNN model 110 using LSTM cell is compared with the HyperLSTM models proposed in HyperNetworks [Ha et al, 2016] on JSB Chorale and Stock datasets. Compared with VHRNN model 110, HyperLSTM does not have latent variables. Therefore, it does not have an encoder or decoder either. The implementation of HyperLSTM resembles the recurrence model of VHRNN model 110 defined in equation (6). At each time step, HyperLSTM model predicts the output distribution by mapping the RNN's hidden state to the parameters of binary distributions for JSB Chorale dataset and a mixture of Gaussian for Stock dataset. Three and five are considered as the number of components in the Gaussian mixture distribution. HyperLSTM models are trained with the same batch size and learning rate as VHRNN models 110. A parameter-performance comparison between example VHRNN models 110, example baseline VRNNs and example HyperLSTM models is illustrated in FIG. 16A for JSB Chorale dataset and FIG. 16B for Stock dataset. The number of components used by HyperLSTM for Stock dataset is five in the plot shown in FIG. 16B. Since HyperLSTM models do not have latent variable, the indicator on top of each point in FIGS. 16A, 16B shows the number of hidden units in each model for all three models. The number of hidden units for HyperLSTM model is also twice the dimension of hidden states as HyperLSTM has two RNNs, one primary and one hyper. FIGS. 16A, 16B report FIVO for example VHRNN models 110 and example baseline VRNN models and exact log likelihood for example HyperLSTM models. Even though FIVO is a lower-bound of log likelihood, it can be seen that the performance of VHRNN model 110 completely dominates HyperLSTM regardless of the number of hidden units used. The performance of HyperLSTM is in fact worse than baseline VRNN models which do not have hyper networks. These results indicates the importance of modeling complex time-series data. A hidden units and performance comparison between example VHRNN models 110 and example baseline VRNNs is illustrated in FIGS. 17A, 17B for JSB Chorale dataset and Stock dataset, respectively. The comparison shows similar results to those discussed above with reference to FIGS. 16A, 16B. Complete experiment results of HyperLSTM models on datasets JSB Chorale and Stock are shown in FIG. 18. The effects of hidden state and latent variable on the performance of a VHRNN model 110 have been considered in the following two aspects—the dimension of the latent variable and the contributions by hidden state and latent variable as inputs to hyper networks—examined by way of ablation studies, described in further detail below. In experiments on real-world datasets with the latent dimension and hidden state dimension set to be the same for each model, an example VHRNN model 110 has significantly more parameters than a baseline VRNN when using the same latent dimension. In further experimental work, to eliminate the effects of the difference in model size, the latent dimension and hidden state dimension are different and the hidden layer size of the hyper network that generates the weight of the decoder is reduced. These changes allow for a comparison of baseline VRNN and examples of VHRNN models 110 with the same latent dimension and a similar number of parameters. The results on JSB Chorale datasets are presented in FIG. 19 in which the latent dimension is denoted by “Z dim”. As shown, example VHRNN models 110 have better FIVO with the same latent dimensions than example baseline VRNNs. The results show that the superior performance of VHRNN model 110 over baseline VRNN does not stem from smaller latent dimension when using the comparable number of parameters. FIG. 20 illustrates results of example VHRNN models 110 with different hyper network inputs. Example VHRNN models 110 were retrained and their performance evaluated on JSB Chorale dataset and synthetic sequences when fed the latent variable only, the hidden state only, or both to the hyper networks. As illustrated in FIG. 20, VHRNN model 110 may have the best performance and generalization ability when it takes the latent variable as its only input. Relying on the primary network's hidden state only or the combination of latent variable and hidden state may lead to worse performance. When the dimension of the hidden state is 32, VHRNN model 110 only taking the hidden state as hyper input suffers from over-parameterization and has worse performance than a baseline VRNN with the same dimension of the hidden state. On the test set of synthetic data, VHRNN model 110 obtains the best performance when it takes both hidden state and latent variable as inputs. This difference may be due to the fact that historical information is critical to determine the underlying recurrent weights and current noise level for synthetic data. However, the ablation study on both datasets shows the importance of the sampled latent variable as an input to the hyper networks. Therefore, both hidden state and latent variable are used as inputs to hyper networks on other datasets for consistency. In some embodiments, an RNN may be used to generate the parameters of another RNN, for example, for VHRNN model 110 the hidden state of the primary RNN can represent the history of observed data while the hidden state of the hyper RNN can track the history of data generation dynamics. As an ablation study, experimental work was performed with VHRNN models 110 that replace the RNN with a three-layer feed-forward network as the hyper network θ for the recurrence model g as defined in equation (6). The other components of VHRNN model 110 are unchanged on JSB Chorale, Stock and the synthetic dataset. The evaluation results using FIVO are presented in FIG. 21 and systematic generalization study results on the synthetic dataset are shown in FIG. 22. The example original VHRNN models are is denoted with recurrence structure in θ as “VHRNN-RNN” and the variant examples without the recurrence structure as “VHRNN-MLP”. As shown in FIGS. 21 and 22, given the same latent dimension, VHRNN-MLP models have more parameters than VHRNN-RNN models. VHRNN-MLP can have slightly better performance than VHRNN-RNN in some cases but it performs worse than VHRNN-RNN in more settings. The performance of VHRNN-MLP also degrades faster than VHRNN-RNN on the JSB Chorale dataset as the latent dimension increases. Moreover, systematic generalization study on the synthetic dataset illustrated in FIG. 22 also shows that VHRNN-MLP has worse performance than VHRNN-RNN no matter in the test setting or in the systematically varied settings. Embodiments disclosed herein of a variational hyper RNN (VHRNN) model such as VHRNN model 110 can generate parameters based on the observations and latent variables dynamically. Conveniently, such flexibility enables VHRNN to better model sequential data with complex patterns and large variations within and across samples than traditional VRNN models that use fixed weights. In some embodiments, VHRNN can be trained with existing off-the-shelf variational objectives. Experiments on synthetic datasets with different generating patterns, as disclosed herein, show that VHRNN may better disentangle and identify the underlying dynamics and uncertainty in data than VRNN. Experimental work to-date also demonstrates the superb parameter-performance efficiency and generalization ability of VHRNN on real-world datasets with different levels of variability and complexity. VHRNN as disclosed herein may allow for sequential or time-series data that is variable, for example, with very sudden underlying dynamic changes, to be modeled. The underlying dynamic may be a latent variable with sudden changes. Using VHRNN, it may be possible to infer such changes in the latent variable. Domains of variable sequential or time-series data that may be modelled and generated by VHRNN include financial data such as financial markets or stock market data, climate data, weather data, audio sequences, natural language sequences, environmental sensor data or any other suitable time-series or sequential data. A conventional or baseline RNN may have difficulty capturing a sudden change in an underlying dynamic, for example, by assuming that the dynamic is constant. By contrast, a VHRNN may better capture such changes, as illustrated in experimental work as described herein. For example, experiments performed using synthetically generated data, as discussed above, demonstrate a VHRNN's usefulness. VHRNN may capture such underlying dynamic changes with its unique latent variable methodology. Observation data is captured in the observation state x[t]. Underlying dynamics are not observed, and are represented by latent variable, such as z[t], as used herein. In an example of stock price time-series data, stock prices may be observed at each time step. However, there may exist underlying or latent variable(s) that are not observed or observable that may control the stock movement or performance. A latent variable can be, for example, macroeconomic factors, monetary policy, investor sentiment, leader confidence or mood, or any other factors affecting observable states such as stock prices. In an example, a latent variable such as a leader's mood can have two states: happy or unhappy, which may not be observable, and is a latent dynamic that may be manifested in VHRNN as a latent variable. The VHRNN model disclosed herein provides for a latent variable that is dynamic, and VHRNN offers unique advantages in allowing the latent variable to change or update at each time step—the latent variable is thus a temporal latent variable that changes with time, and VHRNN is able to dynamically decode the latent information. VHRNN thus can be effective in adapting to changes over time, in particular, by implementation of the hyper component. A hyper network component of VHRNN enables the dynamic of the RNN to change based on previous observation(s). A conventional VRNN, by contrast, assumes at every time step that the dynamic is the same, utilizing the same prior network or transition network. With a hyper network, as disclosed herein, the parameters of those networks can change at each time step. Thus, variability may be better captured to dynamically change the model. VHRNN, by better inferring the underlying dynamics and latent variables, may provide insights into those underlying dynamics, depending on how those latent variables are interpreted. More accurate inference may allow for better decisions if based on such latent variables, and further, better generate samples that represent, in an example, future predictions. In an example use case for prediction to forecast future stock price, a better understanding of latent dynamics may result in a better forecasting model. Once a VHRNN model is trained, it can be used to generate samples that can be used for forecasting. VHRNN may use a variational lower bound to capture a distribution. With a model that captures the distributions, there a number of downstream tasks that can then make use of the model as described herein. N. Boulanger-Lewandowski, Y. Bengio, and P. Vincent. Modeling temporal dependencies in high-dimensional sequences: Application to polyphonic music generation and transcription. arXiv preprint arXiv:1206.6392, 2012. S. R. Bowman, L. Vilnis, O. Vinyals, A. M. Dai, R. Jozefowicz, and S. Bengio. Generating sentences from a continuous space. arXiv preprint arXiv:1511.06349, 2015. Y. Burda, R. Grosse, and R. Salakhutdinov. Importance weighted autoencoders. In International Conference on Learning Representations, 2016. J. Chung, K. Kastner, L. Dinh, K. Goel, A. C. Courville, and Y. Bengio. A recurrent latent variable model for sequential data. In Advances in neural information processing systems, pages 2980-2988, A. Doucet and A. M. Johansen. A tutorial on particle filtering and smoothing: Fifteen years later. Handbook of nonlinear filtering, 12(656-704):3, 2009. M. Fraccaro, S. K. Sønderby, U. Paquet, and O. Winther. Sequential neural models with stochastic layers. In Advances in neural information processing systems, pages 2199-2207, 2016. A. Graves, A.-r. Mohamed, and G. Hinton. Speech recognition with deep recurrent neural networks. In 2013 IEEE international conference on acoustics, speech and signal processing, pages 6645-6649. IEEE, 2013. D. Ha, A. Dai, and Q. V. Le. Hypernetworks. arXiv preprint arXiv:1609.09106, 2016. J. He, D. Spokoyny, G. Neubig, and T. Berg-Kirkpatrick. Lagging inference and posterior collapse in variational autoencoders. arXiv preprint arXiv:1901.05534, 2019. R. Huerta, T. Mosquiero, J. Fonollosa, N. F. Rulkov, and I. Rodriguez-Lujan. Online decorrelation of humidity and temperature in chemical sensors for continuous monitoring. Chemometrics and Intelligent Laboratory Systems, 157:169-176, 2016. I. Higgins, L. Matthey, A. Pal, C. Burgess, X. Glorot, M. Botvinick, S. Mohamed, and A. Lerchner. beta-vae: Learning basic visual concepts with a constrained variational framework. In International Conference on Learning Representations, volume 3, 2017. S. Hochreiter and J. Schmidhuber. Long short-term memory. Neural computation, 9(8):1735-1780, 1997. D. P. Kingma and M. Welling. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013. D. P. Kingma, T. Salimans, R. Jozefowicz, X. Chen, I. Sutskever, and M.Welling. Improved variational inference with inverse autoregressive flow. In Advances in neural information processing systems, pages 4743-4751, 2016. D. Krueger, C.-W. Huang, R. Islam, R. Turner, A. Lacoste, and A. Courville. Bayesian hypernetworks. arXiv preprint arXiv:1710.04759, 2017. R. Luo, W. Zhang, X. Xu, and J. Wang. A neural stochastic volatility model. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018. C. J. Maddison, J. Lawson, G. Tucker, N. Heess, M. Norouzi, A. Mnih, A. Doucet, and Y. Teh. Filtering variational objectives. In Advances in Neural Information Processing Systems, pages 6573-6583, D. Rezende and S. Mohamed. Variational inference with normalizing flows. In International Conference on Machine Learning, pages 1530-1538, 2015. Of course, the above described embodiments are intended to be illustrative only and in no way limiting. The described embodiments are susceptible to many modifications of form, arrangement of parts, details and order of operation. The disclosure is intended to encompass all such modification within its scope, as defined by the claims. 1. A computer-implemented method for training a variational hyper recurrent neural network (VHRNN), the method comprising: for each step in sequential training data: determining a prior probability distribution for a latent variable, given previous observations and previous latent variables, from a prior network of the VHRNN using an initial hidden state; determining a hidden state from a recurrent neural network (RNN) of the VHRNN using an observation state, the latent variable and the initial hidden state; determining an approximate posterior probability distribution for the latent variable, given the observation state, previous observations and previous latent variables, from an encoder network of the VHRNN using the observation state and the initial hidden state; determining a generating probability distribution for the observation state, given the latent variable, the previous observations and the previous latent variables, from a decoder network of the VHRNN using the latent variable and the initial hidden state; and maximizing a variational lower bound of a marginal log-likelihood of the training data to train the VHRNN; and storing the trained VHRNN in a memory. 2. The method of claim 1, wherein the variational lower bound includes at least one of an evidence lower bound (ELBO), importance weight autoencoders (IWAE), or filtering variational objectives 3. The method of claim 1, wherein the prior probability distribution, defined as p(zt\|x<t, z<t), for the latent variable, defined as zt, is based on: where (μtprior, Σtprior) is the prior network, xt is the observation state, and t is a current step of the steps in the sequential training data. zt\|x<t, z<t˜(μtprior, Σtprior) 4. The method of claim 1, wherein the RNN, defined as g, is based on: where θ(zt,ht-1) is a hypernetwork of the VHRNN that generates parameters of the RNN g using the latent variable, defined as zt, and the initial hidden state, defined as ht-1, xt is the observation state, and t is a current step of the steps in the sequential training data. ht=gθ(zt,ht-1)(xt, zt, ht-1) 5. The method of claim 4, wherein the hypernetwork θ(zt, ht-1) is implemented as a recurrent neural network (RNN). 6. The method of claim 4, wherein the hypernetwork θ(zt, ht-1) is implemented as a long short-term memory (LSTM). 7. The method of claim 4, wherein the hypernetwork θ(zt, ht-1) generates scaling vectors for input weights and recurrent weights of the RNN. 8. The method of claim 1, wherein the generating probability distribution, defined as p(xt\|z≤t,x<t), for the observation state, defined as xt, is based on: where (μtdec, Σtdec)=ϕω(zt,ht-1)(zt, ht-1) is another hypernetwork of the VHRNN that generates parameters of the decoder network, defined as ϕdec, sing the latent variable, defined as zt, and the initial hidden state, defined as ht-1, and t is a current step of the steps in the sequential training data. xt\|z≤t, x21 t˜(μtdec, Σtdec) 9. The method of claim 8, wherein the hypernetwork ω(zt,ht-1) is implemented as a multilayer perceptron (MLP). 10. A computer-implemented method for generating sequential data using a variational hyper recurrent neural network (VHRNN) trained using the method of claim 1, the method comprising: for each step in the sequential data: determining a prior probability distribution for a latent variable zt, given previous observations and previous latent variables, from the prior network of the VHRNN using an initial hidden state; determining a hidden state from the recurrent neural network (RNN) of the VHRNN using an observation state, the latent variable and the initial hidden state; determining a generating probability distribution for the observation state given the latent variable, the previous observations and the previous latent variables, from the decoder network of the VHRNN using the latent variable and the initial hidden state; and sampling a generated observation state from the generating probability distribution. 11. The method of claim 10, wherein the prior probability distribution, defined as p(zt\|x<t,z<t), for the latent variable zt is based on: where (μtprior, Σtprior) is the prior network, xt is the observation state, and t is a current step of the steps in the sequential data. zt\|x<t,z<t˜(μtprior, Σtprior) 12. The method of claim 10, wherein the RNN, defined as g, is based on: where θ(zt, ht-1 ) is a hypernetwork of the VHRNN that generates parameters of the RNN g using the latent variable, defined as zt, and the initial hidden state, defined as ht-1, xt is the observation state, and t is a current step of the steps in the sequential data. ht=gθ(zt,ht-1)(xt, zt, ht-1) 13. The method of claim 12, wherein the hypernetwork θ(zt, ht-1) is implemented as a recurrent neural network (RNN). 14. The method of claim 12, wherein the hypernetwork θ(zt, ht-1) is implemented as a long short-term memory (LSTM). 15. The method of claim 12, wherein the hypernetwork θ(zt, ht-1) generates scaling vectors for input weights and recurrent weights of the RNN g. 16. The method of claim 10, wherein the generating probability distribution, defined as p(xt\|z≤t, x<t), for the observation state, defined as xt, is based on: where (μtdec, Σtdec)=ϕω(zt,ht-1)(zt, ht-1) and ω(zt, ht-1) is another hypernetwork of the VHRNN that generates parameters of the decoder network, defined as ϕdec, using the latent variable, defined as zt, and the initial hidden state, defined as ht-1, and t is a current step of the steps in the sequential data. xt\|z≤t, x<t˜(μtdec, Σtdec) 17. The method of claim 16, wherein the hypernetwork ω(zt, ht-1) is implemented as a multilayer perceptron (MLP). 18. The method of claim 10, further comprising forecasting future observations of the sequential data based on the sampled generated observation states. 19. The method of claim 10, wherein the sequential data is time-series financial data. 20. A non-transitory computer readable medium comprising a computer readable memory storing thereon a variational hyper recurrent neural network trained using the method of claim 1, the variational hyper recurrent neural network executable by a computer to perform a method to generate sequential data, the method comprising: for each step in the sequential data: determining a prior probability distribution for a latent variable zt, given previous observations and previous latent variables, from the prior network of the VHRNN using an initial hidden state; determining a hidden state from the recurrent neural network (RNN) of the VHRNN using an observation state, the latent variable and the initial hidden state; determining a generating probability distribution for the observation state given the latent variable, the previous observations and the previous latent variables, from the decoder network of the VHRNN using the latent variable and the initial hidden state; and sampling a generated observation state from the generating probability distribution. Patent History Publication number : 20200372352 : May 22, 2020 Publication Date : Nov 26, 2020 Patent Grant number 11615305 Inventors Ruizhi DENG Yanshuai CAO Bo CHANG Marcus BRUBAKER Application Number : 16/881,768 International Classification: G06N 3/08 (20060101); G06N 3/04 (20060101);	{"url":"https://patents.justia.com/patent/20200372352","timestamp":"2024-11-02T08:51:57Z","content_type":"text/html","content_length":"171145","record_id":"<urn:uuid:5fa1ed87-1de4-423b-bb74-bd8a3a139be4>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00408.warc.gz"}
Growth theory Category: Macroeconomics Created by Conspecte Team Hits: 8,462 The purpose of this chapter is to try to explain the growth in GDP. The models in this chapter are very different from the rest of the models in this book as they use only the production function and factors of production to explain growth. Growth models are important, for example, if you want to understand why some countries grow faster and have a higher living standard than other countries. By growth, we mean the percentage change in real GDP. We use real GDP to eliminate the effect of inflation. In this chapter, it is perfectly OK to think of inflation as being zero in which case real and nominal GDP bare the same. In this chapter we begin by describing the aggregate production function. The rest of this chapter will look at some different growth theories. The aggregate production function Imagine the national economy during a short period of time (say one week). We denote: • L: the total amount of work used during this period (by all individuals in the economy). • K: the total amount of capital used. • Y: the total amount of finished goods produced during this period (real). It is still the case that L and Y are flows while K is a stock. During a short period of time, we can assume that the amount of capital is constant. The aggregate production function, or simply the production function is a function that relates L, K and Y. Specifically, we assume that Y is a function of L and K : Y = f (L , K) In most cases, we will not specify exactly what the function f looks like. However, we always assume that f is increasing in L and K, that is, when we use more labor and/or more capital, we will produce more goods. The marginal product of labor and capital We define the marginal product of labor, MPL as the derivative of f with respect to the L – that is, as (approximately) how much Y will increase when L increases by one unit. We also define the marginal product of capital, MPK as the derivative of f with respect to K. Note that MPL and MPK will depend on both L and K ( MPL and MPK are functions, not variables). • Since f is increasing in L, MPL must be positive for all values of L and K. • MPL assumed to be decreasing in L – the more work that is used, the lower the marginal product of labor. • MPL assumed to be increasing in K – the more capital, the higher the marginal product of labor • In the same way, MPK must be positive for all values of L and K. • MPK is assumed to be decreasing in K and increasing in L. When we view Y as a function of L holding K fix, Y will be increasing in L but at a decreasing pace (due to the fact that MPL is positive but decreasing in L ). Fig. 9.1: Production function. We define labor productivity as Y / L, that is, as GDP per hour worked. Labor productivity tells us how much we can produce using one hour of labor and it depends on the amount of capital as well as the technology. Production function and Growth From the simple production function Y = f (L, K), we can identify three sources of growth: • An increase in L. • An increase in K. • A change in the function f The first two represent the growth of the factors of production. L may increase if the population grows, if we have more individuals in the workforce, or if unemployment falls. K increases if investment are large as they are if total savings is large. The function f need not be the same function over time. It is possible that Y increases even though L and K are fixed. When f changes so that the same amount of the factors of production will produce more output we say that we have technological progress or productivity growth. With technological progress, MPL and MPK will typically increase for given values of L and K, that is, the productivity of the factors increase. Education and growth in human capital are important aspects of growth in GDP. Human capital is treated in different ways in the literature: • You can think of human capital as being included in K – with this view education is a type of investment. • You can add another variable in the production function: Y = f ( L, K, H ) where H is the amount of human capital and K amount of physical capital. • The amount of human capital may affect the function f. The more human capital, the more can be produced from the same amount of L and K. With this view, increasing the amount of human capital will lead to productivity growth. Growth Accounting is the activity in which we try to figure out how much of the growth in GDP is due to growth in L, growth in K and growth in productivity. Growth Theories The classical growth theory The production function will not provide us with a theory or explanation of growth. It is only a convenient tool that helps us breaking down growth into its components. However, there are many growth theories that try to go a step further. The oldest of these theories is the so-called classical growth theory which is primarily associated with Thomas Robert Malthus. The classical growth theory should not be confused with the classical model that we will look at in the next chapter. Also, the classical growth theory, which was developed in the late 1700s, has little or no relevance today. We present it so that you can better understand more modern growth theories. In short, the classical growth theory may be described as follows: • Due to technological development, the amount of capital increases and the marginal product of labor rises. • GDP per capita rises. With higher living standards, the population will increase. • As population increases, labor productivity will fall (more individuals but the same amount of capital). • GDP per capita will fall again. When GDP per capita has fallen to a level just high enough to keep the population from starving, the increase in population will cease. Destruction of capital, for example, through a war, works in the opposite way. The marginal product of labor falls, GDP per capita falls and the population decreases. This will again lead to an increase in the marginal product of labor and GDP per capita return to the "survival rate". The main point of the model is that population growth will always eliminate the positive effects of technological development and GDP per capita will always return to the survival level. This very "dismal" growth theory was prominent in the early 1800s, and economics to this day is sometimes called the "dismal science". Today we know that the predictions of the model where incorrect. During the rest of the 1800s Europe experienced a growth in GDP per capita. Although the population growth was high, it was not nearly sufficient to eliminate the positive effects of technological development. The neoclassical growth model The main purpose of another important growth model, the neoclassical growth model, is to explain how it is possible to have a permanent-growth in GDP per capita. The model was developed by Robert Solow in the 1960s and it is sometimes called the Solow growth model or the exogenous growth model. The neoclassical growth model should not be confused with the neoclassical synthesis, which we will study in chapter 10. "Neo" means "new" – the neoclassical growth theory is a “new version” of the classical growth model. The crucial difference between the classical and neoclassical growth models is that the population is endogenous in the former and exogenous in the latter. In the classical model, the population will increase or decrease depending on whether GDP per capita is higher than or lower than the survival level. In the neo-classical model population growth is not affected by GDP per capita (however, the population growth will affect the growth in GDP per capita). In the neo-classical model, it is the technological progress only that affects the GDP per capita in the long run. We will have a permanent increase in GDP per capita when there is a technological development that increases the productivity of labor. Permanent growth in GDP then requires continuous technological progress. It is not possible for the government, except temporarily, to affect the growth rate in the neoclassical growth model. The government might be able to affect GDP per capita (and this is the growth rate) but the growth rate always returns to the level determined by the technological progress. The same is true for savings. An increase in savings may have a temporary effect on GDP but it will have no effect in the long run. Endogenous growth theory Endogenous growth theory or new growth theory was developed in the 1980s by Paul Romer and others. In the neo-classical model, technological progress is an exogenous variable. The neoclassical growth model makes no attempt to explain how, when, and why technological progress takes place. The main objective of the endogenous growth theory is to make the technological progress an endogenous variable to be explained within the model, hence the name endogenous growth theory. There are many different explanations for technological progress. Most of them, however, have a lot of common characteristics: • They are based on a constant return to scale for capital. Thus, MPK is not a decreasing function of K in these models. • They consider technological development as a public good. • They focus more on human capital. • It is possible for the government to affect the growth rate. Higher savings also leads to higher growth, not just higher GDP per capita. • They predict the convergence of GDP per capita between countries in the long run. This is a consequence of the public good property of technological developments. Separation of growth and fluctuation It is often useful to separate the evolution of a variable that grows over time into a trend and fluctuations around the trend. The graphs below show such a separation for real GDP. Fig. 9.2: Growth and the fluctuation around the trend. The left diagram shows a stylized graph of real GDP over time. It demonstrates the two important characteristics of real GDP. GDP fluctuates over time and GDP grows overtime – at least over a longer period of time. The left graph is the sum of the middle graph and the right graph. The middle graph shows the trend in GDP. The trend represents the second characteristic of GDP – the fact that GDP grows over time. The right graph shows the fluctuations around the trend (cycles) of GDP. These fluctuations around the trend represent the first property of GDP. In macroeconomics it is common to study trends and cycles separately. The purpose of growth theory is to investigate the trend while most of the macroeconomics apart from growth theory is about the cycles. The trend is about the very long-run perspective of the economy while cycles are about the short and medium run. The rest of this is all about cycles and not at all about trends. Therefore, when you think of GDP in the remaining chapters, you should think of GDP as in the right-hand graph: GDP has cycles but no trend. Basically, we will study GDP where the trend has been removed.	{"url":"https://conspecte.com/en/macroeconomics/growth-theory.html","timestamp":"2024-11-07T12:28:22Z","content_type":"text/html","content_length":"27622","record_id":"<urn:uuid:f12b2eb9-a6bf-4093-9964-92aee4728081>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00895.warc.gz"}
递归树和有向无环图（动态规划） - VisuAlgo This visualization can visualize the recursion tree of any recursive algorithm or the recursion tree of a Divide and Conquer (D&C) algorithm recurrence (e.g., Master Theorem) that we can legally write in JavaScript. We can also visualize the Directed Acyclic Graph (DAG) of a Dynamic Programming (DP) algorithm and compare the dramatic search-space difference of a DP problem versus when its overlapping sub-problems are naively recomputed, e.g., the exponential Ω(2^n/2) recursive Fibonacci versus its O(n) DP version. On some problems, we can also visualize the difference between what a Complete Search (recursive backtracking) that explores the entire search space, a greedy algorithm (that greedily picks one branch each time), versus Dynamic Programming look like in the same recursion tree, e.g., Coin-Change of v = 7 cents with 4 coins {4, 3, 1, 5} cents. Most recursion trees require large drawing space, therefore view this visualization page on a large screen. For obvious reason, we cannot really visualize very big trees/DAGs. Therefore, we call our recursion with small parameter(s). Remarks: By default, we show e-Lecture Mode for first time (or non logged-in) visitor. If you are an NUS student and a repeat visitor, please login. This is the Recursion Tree and Recursion Directed Acyclic Graph (DAG) visualization area. The Recursion Tree/DAG are drawn/animated as per how a real computer program that implements this recursion works, i.e., "depth-first". The recursion starts from the initial state that is colored dark brown. The current parameter value is shown inside each vertex (comma-separated for recursion with two or more parameters). Active vertices will be colored orange. Vertices that are no longer calling any other recursive problem (the base cases) will be colored green. Vertices (subproblems) that are repeated will be colored lightblue for the second occurrence onwards. The return value of each recursive call is written as a red text below the vertex. This visualization is generic for any recursion that you can legally write in JavaScript. Note that due to combinatorial explosion, it will be very hard to visualize the Recursion Tree for large instances. Pro-tip 1: Since you are not logged-in, you may be a first time visitor (or not an NUS student) who are not aware of the following keyboard shortcuts to navigate this e-Lecture mode: [PageDown]/ [PageUp] to go to the next/previous slide, respectively, (and if the drop-down box is highlighted, you can also use [→ or ↓/← or ↑] to do the same),and [Esc] to toggle between this e-Lecture mode and exploration mode. For the Recursion DAG, it will also very hard to minimize the number of edge crossings in the event of overlapping subproblems. However, we try our best to give a custom DAG drawing layout for certain DP problems to improve the presentation. For example, we currently show the recursion DAG of the computation of the n-th Fibonacci number. We layout the vertices from leftmost (the two green-colored base cases n = 0 and n = 1) to rightmost (the initial n). Ideally this is shown as a flat 1-D (memoization) array. However as VisuAlgo does not have curvy edge yet, we decided to put odd-numbered vertices slightly below the even-numbered vertices to get around the overlapping edges issue. To compute fib(n), we only need to know the result of two immediate results: fib(n-1) + fib(n-2) that is depicted as the two arrows from vertex n to vertices n-1 and n-2. The results of the summation, i.e., the values of each fib(n), are displayed as red text below the vertices. As there is no repeated subproblem computation, there is no lightblue vertex at all in this recursion DAG (such vertices/states will have more than one incoming arrows instead). Pro-tip 2: We designed this visualization and this e-Lecture mode to look good on 1366x768 resolution or larger (typical modern laptop resolution in 2021). We recommend using Google Chrome to access VisuAlgo. Go to full screen mode (F11) to enjoy this setup. However, you can use zoom-in (Ctrl +) or zoom-out (Ctrl -) to calibrate this. Select one of the example recursive algorithms in the drop-down list or write our own recursive code — in JavaScript. The final recursion Tree / DAG is immediately displayed. Note that this visualization can run any JavaScript code, including malicious code, so please be careful (it will only affect your own web browser, don't worry). Click the 'Run' button at the top right corner of the action box to start the step-by-step visualization of the recursive function after you have selected (or written) a valid JavaScript code! In the next sub-sections, we start with example recursive algorithms with just one sub-problem, i.e., not branching. For these one-subproblem examples, their recursion trees and recursion DAGs are 100% identical (they looked like Singly Linked Lists from the root (initial call) to the leaf (base case)). As there is no overlapping subproblem for the examples in this category, you will not see any lightblue-colored vertex and only one green-colored vertex (the base case). The default orientation for recursion Tree is top-right-to-bottom-left whereas the default orientation for recursion DAG is right-to-left. Pro-tip 3: Other than using the typical media UI at the bottom of the page, you can also control the animation playback using keyboard shortcuts (in Exploration Mode): Spacebar to play/pause/replay the animation, ←/→ to step the animation backwards/forwards, respectively, and -/+ to decrease/increase the animation speed, respectively. The Factorial Numbers example computes the factorial of an integer n. f(n) = 1 (if n == 0); f(n) = nf(n-1) otherwise It is one of the simplest (tail) recursive function that can be easily rewritten into an iterative version. It's time complexity is also simply Θ(n). The value of Factorial f(n) grows very fast, thus try only the small integers n ∈ [0..10] (we randomize the value of the initial n between this range). The Binary Search example finds the index of a2 in a sorted array a1[a..b]. We start binary search from the initial search space of a=0,b=a1.length-1 (the entire array a1, with n = (a1.length-1) - 0 + 1 = a1.length). f(a,b) = -1 (if a > b); let mid = floor((a+b)/2); f(a,b) = mid (if a2 = a1[mid]); f(a,b) = f(a,mid-1) (if a2 < a1[mid]); f(a,b) = f(mid+1,b) (if a2 > a1[mid]); Only one of the two possible branches will be executed each time, resulting in a recursion tree = recursion DAG situation. The time complexity if Θ(log n) as we keep halving the search space each time. The worst-case happens when a2 is not found in sorted array a1. In this visualization, we randomize the content of sorted array a1 and the value to be searched a2. TBC: In the near future, we will draw the entire Θ(n) search space (both left and right branches at each vertex) and highlight the efficient Θ(log n) path taken by Binary Search on this search space. This is like seeing the animation of Search(v) function of a (balanced) Binary Search Tree. The Modulo Power example computes the a1^p % a2 in efficient way. f(p) = 1 (if p == 0); f(p) = f(floor(p/2))^2 % a2 (if p is even) f(p) = f(floor(p/2))^2 a1 % a2 (if p is odd) This Divide & Conquer (D&C) algorithm runs in Θ(log p). In this visualization, we randomize the values of a1 (the base, a small value &in; [2..4]), a2 (the modulo, a prime &in; [7, 97, 997, 9973]), and power p (we want to raise a1 to its p-th power, modulo a2. Due to its low time complexity, it is OK to try very large 0 ≤ p ≤ 256. TBC: In the near future, we may draw the comparison between this efficient D&C modulo power algorithm with the naive version that multiply a1 p times that runs in Θ(p). The Greatest Common Divisor (GCD) example computes the GCD of two integers a and b. f(a, b) = a (if b == 0); f(a, b) = f(b, a%b) otherwise This is the classic Euclid's algorithm that runs in O(log n) where n = min(a, b) — depending of the details, n = max(a, b) or n = a+b are also possible. Euclid's algorithm is an example of Divide & Conquer (D&C) algorithm. Due to its low time complexity, it should be OK to try 0 ≤ a, b ≤ 99. (we randomize the values of a and b between this range and set a ≥ b). Note that if we put a < b, technically the first recursive step will swap a and b. Do explore various possible combinations of a and b and notice on what values f(a, b) terminates very quickly in 1 step (b = 0), 2 steps (b = gcd(a, b)), or the maximum number of steps (try this sequence) — to show the lowerbound Ω(log n). The Max Range Sum example computes the value of the subarray with the maximum total sum inside the given (global) array a1 with n = a1.length integers (the first textbox below the code editor textbox). The value of a1 can be positive integers, zeroes, or negative integers (without negative integer, the answer will obviously the sum of the entire integers in a1). Formally, let's define RSQ(i, j) = a1[i] + a1[i+1] + ... + a1[j], where 0 ≤ i ≤ j ≤ n-1 (RSQ stands for Range Sum Query). Max Range Sum problem seeks to find the optimal i and j such that RSQ(i, j) is the maximum. f(i) = max(ai[0], 0) (if i == 0, as ai[0] can be negative); f(i) = max(f(i-1) + ai[i], 0) otherwise We call f(n-1). The largest value of f(i) is the answer. This is the classic Kadane's algorithm that runs in O(n). The Catalan example computes the n-th Catalan number recursively. f(n) = 1 (if n == 0); f(n) = f(n-1)2n(2n-1)/(n+1)/n; otherwise This explanation is a stub that will be expanded later. In the next sub-sections, we will see example recursive algorithms that have exactly two sub-problems, i.e., branching. The sizes of the subproblems can be identical or vary. For these two sub-problems examples, their recursion trees will usually be much bigger that their recursion DAGs (especially if there are (many) overlapping sub-problems, indicated with the lightblue vertices on the recursion tree drawing). Currently shown on screen is the recursion tree of a Fibonacci recurrence. The Fibonacci Numbers example computes the n-th Fibonacci number. f(n) = n (if n <= 1); // i.e., 0 if n == 0 or 1 if n == 1 f(n) = f(n-1) + f(n-2) otherwise Unlike Factorial example, this time each recursive step recurses to two other smaller sub-problems (if we call f(n-1) first before f(n-2), then the left side of the recursion tree will be taller than the right side — try swapping the two sub-problems). The value of Fibonacci f(n) grows very fast and its recursion tree — if implemented verbatim as defined above — also grows exponentially, i.e., at least Ω(2^n/2), thus try only the small initial values of n ≤ 7 (to avoid crashing your web browser). Fibonacci recursion tree is frequently used to showcase the basic idea of recursion, its inefficiency (due to the many overlapping subproblems), and the linkage to Dynamic Programming (DP) topic. The recursion DAG (shown in the background) of Fibonacci computation only contains O(n) vertices and thus can go to a larger n ≤ 30 (so it still looks nice in this visualization; in practice n can go to millions with this DP solution). Most of the time, the Fibonacci computation is written in iterative fashion after one understands the concept of DP. It is probably rare to think this way, but this visualization shows that the computation of Fibonacci f(n) is basically counting the number of paths from n to vertex 1. The C(n, k) example computes the binomial coefficient C(n, k). f(n, k) = 1 (if k == 0); // 1 way to take nothing out of n items f(n, k) = 1 (if k == n); // 1 way to take everything out of n items f(n, k) = f(n-1, k-1) + f(n-1, k) // take the last item or skip it The recursion tree of C(n, k) grows very fast, with the largest tree when k = n/2, smaller trees when k is close to 0 or n (a few path(s) with short leafs), and smallest trees when k is 0 or n (only one vertex). The recursion DAG of C(n, k) is basically an inverted Pascal's Triangle. It only contains O((n/2)(n/2)) = O(n^2/4) vertices at most (when k = n/2) although in practice we probably just use a DP/memo table of size O(n^2). Thus we can go to a larger n ≤ 15 (so it still looks nice in this visualization) and k&in;[0..n], including when k≈n/2. TBC: In the near future, we may draw a dummy vertex (0, 0) --- C(n, k) will not actually reach it unless started from C(0, 0) with return value of 1 to complete the inverted Pascal's Triangle The 0-1 Knapsack example solves the 0/1 Knapsack Problem: What is the maximum value that we can get, given a knapsack that can hold a maximum weight of w, where the value of the i-th item is a1[i], the weight of the i-th item is a2[i]? 0-1 Knapsack has a classic DP recurrence f(i, w) which we call using f(n-1, max-w) where n = a1.length. f(i, w) = f(i-1, w); // ignore item i (always possible) f(i, w) = a1[i] + f(i-1, w-a2[i]); // take item i (if a2[i] <= w) f(<0, w) = 0; // all items have been considered f(i, 0) = 0; // cannot carry anything else The recursion tree of this DP recurrence has a few (like currently shown on screen) to many (e.g., try all items having the same weight, i.e., ones) overlapping sub-problems. The recursion DAG of 0-1 Knapsack only contains O(n max-w) vertices. Thus we can go to a larger n ≤ 7 and max-w ≤ 15 (so it still looks nice in this visualization). This DP recurrence basically tries to find the longest (weighted) path in this implicit DAG and has time complexity of O(n * max-w). If the weights of each item in a2 vary a lot, the recursion DAG will look sparse. Try setting a2=[1,2,...,2^i] for a denser recursion DAG> (but no overlap) or a2=[1,1,...,1] (lots of overlap). In the next sub-sections, we will see example recursive algorithms that have many sub-problems (1, 2, 3, ..., a certain limit). For many of these examples, the sizes of their Recursion Trees are exponential and we will really need to use Dynamic Programming to compute its Recursion DAGs instead. The Longest Increasing Subsequence example solves the Longest Increasing Subsequence problem: Given an array a1, how long is the Longest Increasing Subsequnce of the array? The recursion DAG of the default example (not randomized) is as follows: Vertex j ∈ [0..i-1] is laid out horizontally (along the x-axis from left/vertex 0 to right/vertex i-1), then placed vertically (along the y-axis according to the value of a1[j]). We draw an edge between two indices j and k if a1[j] < a1[k] (with all vertices having hidden edge to the dummy max(a1)+1 value at the top-right cell so that all LIS ends at this dummy vertex and we can start the recursion with f(i) where i = a1.length-1). Then this LIS problem can be visualized as finding the longest path in this (implicit) recursion DAG. As there are \|V\| = n vertices and \|E\| = O(n^2) edges (use sorted ascending test case, e.g., {1,2,4,8,16,...} to have nice visualization), the overall time complexity to solve LIS using DP is O(n^2). Since early 2000s, we should use the faster O(n log k) greedy+binary search solution for LIS (not explained in this slide). The Coin Change example solves the Coin Change problem: Given a list of coin values in a1, what is the minimum number of coins needed to get the value v? The recursion tree of the default example (not randomized) has v = 7 cents and 4 coins that are specifically selected to be {4, 3, 1, 5}. What is shown on-screen is the entire recursion tree of Coin-Change recursive function. A typical greedy algorithm for Coin-Change that always take the largest coin value that does not exceed current value v will be trapped into taking the rightmost branch: 7 cents (take 5 cents coin) & rightarrow; 2 cents (take 1 cent coin) &rightarrow; 1 cent (take another 1 cent coin) &rightarrow; 0 (total 3 coins). DP algorithm that explores this recursion tree (but avoiding repeated computations on the lightblue vertices will find the lefmost branch: 7 cents (take 4 cents coin) &rightarrow; 3 cents (take 3 cents coin) &rightarrow; 0 (total 2 coins). Alternative solution: 7 &rightarrow; 4 &rightarrow; 0 (also 2 coins). The Cutting Rod example solves the 'fictional' introductory problem in Chapter 14.1 - Dynamic Programming of the "Introductions to Algorithms" (CLRS 4th edition) textbook: Given a rod of length n inches and a table of prices a1 for length 1,2,...,n inches, determine the maximum revenue obtainable by cutting up the rod and selling the pieces. The recursion tree of the default example (not randomized) has n = 4 inches. What is shown on-screen is the entire recursion tree of Cut-Rod recursive function. f(n) = max(a1[i-1] + f(n-i)) ∀i&in;[1..n]; f(0) = 0; // cannot cut anymore The Matrix Chain Mult(iplication) example solves the second DP introductory problem in Chapter 14.2 of the "Introductions to Algorithms" (CLRS 4th edition) textbook: Given a sequence (chain) of n = a1.length-1 matrices to be multiplied, where the matrices aren't necessarily square, compute the product A1A2...An using the standard O(pqr) algorithm for multiplying rectangular matrices, while minimizing the number of scalar multiplications. The recursion tree of the default example (not randomized) has i=1,j=4. What is shown on-screen is the entire recursion tree of MCM recursive function. f(i,j) = min(f(i,k)+f(k+1,j)+a1[i-1]a1[k]a1[j]) ∀i≤k<j; f(i,j) = 0 if i == j; // no need to multiply a single matrix The Longest Common Subsequence (LCS) example solves the Longest Common Subsequence problem: Given two strings (character arrays) a1 (of length n = a1.length) and a2 (of length m = a2.length), how long is the Longest Common Subsequence between the two strings? f(n, m) = 1 + f(n-1, m-1); // if a1[n] == a2[m]; last char matches f(n, m) = max(f(n-1, m), f(n, m-1)) // if last char differs f(n, <0) = 0; // a2 is empty f(<0, m) = 0; // a1 is empty We call f(n-1, m-1), from the last character of both strings. The recursion tree of this DP recurrence has an exponential (like currently shown on screen) number of overlapping sub-problems if both strings have many different characters (but do try a1 == a2; we will see a single-branch recursion tree). The recursion DAG of LCS only contains O(n m) vertices. Thus we can use longer strings with n ≤ 10 and m ≤ 10 (so it still looks nice in this visualization). This DP recurrence basically tries to find the longest (0/1-weighted) path in this implicit DAG and has time complexity of O(n * m). The Graph Matching problem computes the maximum number of matching on a small graph, which is given in the adjacency matrix a1. This slide is a stub and will be expanded in the future. The Traveling Salesperson example solves the Traveling Salesperson Problem on small graph: How long is the shortest path that goes from city 0, passes through every city once, and goes back again to 0? The distance between city i and city j is denoted by a1[i][j]. This slide is a stub and will be expanded in the future. In the next sub-sections, instead of visualizing the recursion tree of a recursive algorithm, we visualize the recursion tree of the recurrence (equation) to help analyze the time complexity of certain Divide and Conquer (D&C) algorithms. The value computed by f(n) (the red label underneath each vertex that signifies the return value of that recursive function/that subproblem) is thus the total number of operations taken by that recursive algorithm when its problem size is n (the value drawn inside each vertex). Most textbooks will say the function name of this recurrence as T(n), but we choose not to change our default f(n) function name that is used in all other recursive algorithm visualizations. Some other textbooks (e.g., CLRS) also put the cost of each vertex only, not the cost of the entire subproblem. In Sorting visualization, we learn about merge sort. It's time complexity recurrences are: f(n) = Θ(1) (if n < n[0] ) — we usually assume that the base cases are Θ(1) f(n) = f(n/2) + f(n/2) + cn (otherwise) Please check the recursion tree of the default example (n = 16). We will use the same recursion tree for the next few sub-slides. You should see the initial problem size of n = 16 written inside the root vertex and its return value (total amount of work done by f(16) is 32+32+116 = 80). This value of f(n) is consistent throughout the recursion tree, e.g., f(8) = f(4)+f(4)+14 = 12+12+18 = 32. 我们看到这个递归树的高度是 log[2] n 因为我们一直将 n 除以 2 直到我们达到大小为 1 的基本情况。对于 n = 16，我们有16->8->4->2->1 (log[2] 16 = 4 步)。 PS: 树的高度 = 从根到最深叶子的边的数量。 As the effort done in the recursive step per subproblem of size n is cn (the divide (trivial, Θ(1)) and the conquer (merge) operation, the Θ(n)), we will perform exactly cn operations per each recursion level of this specific recursion tree. The root of size (n) does cn operations during the merge step. The two children of the root of size (n/2) both do cn/2, and 2cn/2 = cn too. The grandchildren level is 4cn/4 = cn too. And so on until the last level (the leaves). As the red label underneath each vertex in this visualization reports the value of the entire subproblem (including the subtrees below), these identical costs per level are not easily seen, e.g., from root to leaf, we see 80, 2x32 = 64, 4x12 = 48, 8x4 = 32, 16x1 = 16 and may get different conclusion... However, if we discounted the values of its subproblems, we will get the same conclusion, e.g., for the root, we do 80-2x32 = 16 operations, for the children of the root, we do 2x(32-2x12) = 2x8 = 16 operations too, etc. Soon, we will show 'work-done-in-each-level' info in the visualization directly. 绿色叶子的数量是 2^log[2] n = n^log[2] 2 = n。这些叶子每个都做 Θ(1) 步，因此最后一层（叶子层）的总工作量也是 Θ(n)。因此，归并排序所做的总工作量是每层 cn，乘以递归树的高度（log[2] n + 1 更多的叶子），或 Θ(n log n)。对于这个例子，f(16) = 80 来自 1x16 x (log[2] 16 + 1) = 16 x (4 + 1) = 16 x 5 = 80。 In Sorting visualization, we also learn about the non-random(ized) quick sort. It may have a worst case behavior of O(n^2) on certain kind of (trivial) instances of (nearly-) sorted input and it may have the following time complexity recurrence (with a = 1): f(n) = Θ(1) (if n < n[0]) — we usually assume that the base cases are Θ(1) f(n) = f(n-a) + f(a) + cn (otherwise) Note that writing the recurrence in the other direction does not matter much asymptotically, other than the recursion tree will be mirrored. Please observe the currently drawn recursion tree. We want to show that this recursion tree has f(n) = O(n^2). We see that the height of this recursion tree is rather tall, i.e., n/a - 1 as we only reduces n by a per level. Thus, we need n/a - 1 steps to reach the base case (n = 1). For n = 16, a = 1, we have 1 (16/1 - 1 = 15 steps). 在每个大小为 n 的子问题的递归步骤中，我们所做的工作量是 cn（在 Θ(n) 中划分（分区）操作；征服步骤是微不足道的 — Θ(1)），我们将在每个递归级别执行精确的 cn 操作。根的大小 (n) 在分区步骤中进行 cn 操作。根的子节点的大小 (n-a) 执行 c(n-1)，另一个执行 f(a)。孙子级别执行 c(n-2)，另一个执行 f(a)。等等，直到最后一级（叶子都执行 f(a)）。快速排序在这种最坏情况的输入上所做的总工作是等差级数1+2+...+n的总和，再加上一些其他常数因子操作（所有的f(a)都是Θ(1)）。这简化为f(n) = Θ(n^2)。 For recurrences of the form: f(n) = af(n/b) + g(n) where a ≥ 1, b > 1, and g(n) is asymptotically positive, we maybe able to apply the master theorem (also called as master method). PS: In this visualization, we have to rename CLRS function names to our convention: f(n) → g(n) and T(n) → f(n). We compare the driving function g(n) (the amount of divide and conquer work in each recursive step of size n) with n^log[b]a (the watershed function — also the asymptotic number of leaves of the recursion tree), if g(n) = O(n^log[b]a-ε) for ε > 0, it means that the driving function g(n) grows polynomially slower than the watershed function n^log[b]a (by a factor of n^ε), thus the watershed function n^log[b]a will dominate and the solution of the recurrence is f(n) = θ(n^log[b]a). Visually, if you see the recursion tree for recurrence that falls into case 1 category, the cost per level grows exponentially from root level to the leaves (in this picture, 144 = 16, 722 = 28, 4911 = 49, ..., 16+28+49 = 93), and the total cost of the leaves dominates the total cost of all internal vertices. The most popular example is Strassen's algorithm for matrix multiplication where case 1 of master theorem is applicable. The recurrence is: f(n) = 7f(n/2) + cnn. a = 7 b = 2 , watershed = n^log[2] 7 = n^2.807 , driving = g(n) = Θ(n^2) n^2 = O(n^2.807-ε) for ε = 0.807... — case 1 — Thus f(n) = Θ(n^2.807) Exercise: You can try changing the demo code by setting a = 8 and set g(n) = c1 to change the recurrence of Strassen's algorithm to the recurrence of the simple recursive matrix multiplication algorithm that has f(n) = Θ(n^3). The detailed analysis of the Merge sort algorithm from a few slides earlier can be simplified using master theorem, e.g., f(n) = 2f(n/2) + cn. Thus a = 2, b = 2, watershed = n^log[2] 2 = n, driving = g(n) = Θ(n). n = Θ(n log^k n) for k = 0 The watershed and driving functions have the same asymptotic growth — case 2 Thus f(n) = Θ(n log n). Visually, if you see the recursion tree for recurrence that falls into case 2 category, the cost per level is ~the same, i.e., Θ(n^log[b]a log^k n) and there are log[b] n levels. We claim that the solution is f(n) = Θ(n^log[b]a log^k+1 n). That's it, the solution of the recurrence that falls under case 2 is to add an extra log factor to g(n). Exercise: You can try changing the demo code by setting a = 1 and set g(n) = c1 to change the recurrence of Merge sort algorithm to the recurrence of the binary search algorithm. For binary search version, f(n) = Θ(log n). Notice that for most of real-life case 2 algorithm recurrences (e.g., Merge Sort and Binary Search), k = 0. Case 3 is the opposite of Case 1, where the driving function g(n) grows polynomially faster than the watershed function n^log[b]a. Thus the bulk of the operations is done by the driving function at the root level (but check the regularity condition too, to be elaborated below). This case 3 is actually rarely appear in real algorithms so we use an example fictional recurrence: f(n) = 4f(n/2) + Thus a = 4, b = 2, watershed = n^log[2] 4 = n^2, driving = g(n) = Θ(n^3). n^3 = Ω(n^2+ε) for ε = 1 and 4(n/2)^3 ≤ cn^3 (regularity condition) for c = 1/2 — case 3 Thus f(n) = Θ(n^3). Visually, if you see the recursion tree for recurrence that falls into case 3 category, the cost per level drops exponentially from root level to the leaves (in this picture, 1444 = 64, 4222 = 32, 1611*1 = 16, ..., 64+32+16 = 112), and the total cost of the root dominates the total cost of all other internal vertices (including the (many) leaves). You have reached the last slide. Return to 'Exploration Mode' to start exploring! Note that if you notice any bug in this visualization or if you want to request for a new visualization feature, do not hesitate to drop an email to the project leader: Dr Steven Halim via his email address: stevenhalim at gmail dot com. 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A Bayesian Statistics Approach for Terrain Based Navigation and its Terrain Generation Through Second Order Gauss-Markov Process Volume 04, Issue 04 (April 2015) A Bayesian Statistics Approach for Terrain Based Navigation and its Terrain Generation Through Second Order Gauss-Markov Process DOI : 10.17577/IJERTV4IS041185 Download Full-Text PDF Cite this Publication Bhatt Ajay Vipul, Ravi Kumar K, P. S. B. Kirubakaran, 2015, A Bayesian Statistics Approach for Terrain Based Navigation and its Terrain Generation Through Second Order Gauss-Markov Process, INTERNATIONAL JOURNAL OF ENGINEERING RESEARCH & TECHNOLOGY (IJERT) Volume 04, Issue 04 (April 2015), http://dx.doi.org/10.17577/IJERTV4IS041185 • Open Access • Total Downloads : 198 • Authors : Bhatt Ajay Vipul, Ravi Kumar K, P. S. B. Kirubakaran • Paper ID : IJERTV4IS041185 • Volume & Issue : Volume 04, Issue 04 (April 2015) • DOI : http://dx.doi.org/10.17577/IJERTV4IS041185 • Published (First Online): 24-04-2015 • ISSN (Online) : 2278-0181 • Publisher Name : IJERT • License: This work is licensed under a Creative Commons Attribution 4.0 International License Text Only Version A Bayesian Statistics Approach for Terrain Based Navigation and its Terrain Generation Through Second Order Gauss-Markov Process Mr. Bhatt Ajay Vipul, , M. Tech Avionics Engineering, School of Aeronautical Science, Hindustan University, Chennai, India Mr. Ravi Kumar K Research Supervisor, Scientist D DARE-DRDO, Bangalore, India Mr. P. S. B. Kirubakaran* *Research Supervisor, Assistant Professor, School of Aeronautical Science, Hindustan University, Chennai, India Abstract An algorithm related to Bayesian statistics using second order Gauss-Markov process for terrain generation is demonstrated in this research paper for solving non-linear state estimation problem called terrain based navigation. UAV or manned aircraft fixes its position using GPS, but at times when GPS information is not available, which may be intentional or unintentional aircrafts navigating capability is affected. At that time, terrain based navigation is very much useful in order to estimate location of travelling aircraft by continuously measuring terrain heights below it and comparing it with stored digital elevation map of proposed area over which it is flying. Keywords Terrain Based Navigation (TBN), Second order Gauss-Markov process, Van-Loan method, Bayesian statistics 1. INTRODUCTION Terrain based navigation also known as terrain reference navigation will tally the height information of the terrain area with the on-board digital terrain map to provide positional estimation of aircraft. The aircraft altitude above mean sea level is measured with a barometric altimeter and distance between aircraft and direct terrain below it is measured by radar altimeter. So underneath terrain height is calculated by taking the difference of barometric reading with radar altimeter reading. The obtained terrain height is compared with stored elevation map for determining aircraft position over a certain area. Basically terrain based navigation is differentiated in two which posteriori distribution achieves its peak is the best estimate for state of aircraft over the area. Other approaches related to the non-linear problem is faced by stochastic linearization [2], bank of Kalman Filters [3, 4] and Unscented Kalman filter (UKF) [5]. A stationary Gaussian process that has an exponential autocorrelation function is called a Gauss-Markov process. The process is non-deterministic, so a typical sample time function would show no deterministic structure and would look like typical noise. The exponential autocorrelation function indicates that sample values of the process gradually become less and less correlated as the time separation between samples increases. So this process becomes appropriate for terrain generation in the sense that as samples increases with time terrain heights variation can be seen to a large extent. The Gauss-Markov process is an important process in applied work because: 1. It seems to fit a large number of physical processes with reasonable accuracy. 2. It has a relatively simple mathematical description. The second order process has a power spectral density function (PSD) of the form: ways: 1) Batch based TBN. 2) Sequential TBN. The batch (j)= 2 2 based algorithm gathers terrain heights over a period of time and then matches with stored elevation map to estimate location through post processing. Sequential based approach 4 + 4 Where, 2 = Mean square value or position variance parameter, = Natural frequency parameter (in rad/s), continuously estimates position of aircraft for each terrain 0 height evaluated during flying. However since it is a non- linear estimation problem to be solved during prolonged flying over non-linear variation in terrain heights, we employed Bayesian statistics based approach which converts the expected states of aircraft into probability mass function. 2 = 2 2 3. Since the approach is 2-Dimensional, in order to determine location error in terms of X and Y direction, leads to continuous time state model: = 0 1 0 2 2 + (2) So that to keep track of aircraft flying, through posteriori 0 0 distribution by considering belief that the indices obtained at G F Where, W = 1 (unity white noise), t = 1s, d = Column matrix containing pseudorandom number from standard normal distribution. The required inputs for the Van-Loan method [7] are F, G, W and t for calculating (transition matrix) and (covariance matrix). Since is usually not diagonal, let C be a linear transformation matrix obtained by transposing the matrix obtained by Cholesky factorization of . Using and terrain heights are prepared varying between 35m to 80m over 5KmÃ—5Km area along X and Y direction by implementing following equation: +1 = . ()+C.d (4) 3. USING BAYESIAN STATISTICS Here the goal of Bayesian statistics is to estimate the state of an aircraft during flying over varying terrain. Bayes rule here is utilized as follows: p(H/O)=p(H)p(O/H)/p(O) (5) Where, H = Likely state or hypothesis over which aircraft is supposed to be present, O = Observation of height of terrain through sensors, p(H/O) = Posteriori estimate of state, p(H) = Prior estimate of state, p(O/H) = Likelihood function, p(O) = Normalized by probability of data in general. The scheme will be to convert prior estimation into posteriori estimation through observation provided in terms of estimated state of aircraft by relating terrain height with stored elevation data. The process of this conversion is governed by likelihood function p(O/H) which is a Gaussian distribution. Let N(, ) denotes Gaussian distribution with mean vector and covariance matrix P, N(, )= 1 exp(-0.5( )1( )) (6) Where, = Expected state of aircraft after getting an observation. So with each new terrain evaluated, prior distribution is multiplied relating to likelihood of the measurement taken. Posteriori distribution generated in this way is scaled up or down associated with the likelihood of data obtained. So as new observations are available, process becomes iterative and posteriori distribution of previous observation will become prior distribution of recently taken observation. 4. SYSTEM MODEL Our simulation deals with 2-Dimensional case, so here we have to deal with two position state. We describe the process and measurement models according to the following: Process model: +1 = + + (7) Measurement model: = + (8) Where , , denotes recent position of aircraft along X and Y direction, forward movement, a white Gaussian process noise respectively. The terrain height i.e. an observation is equal to terrain height according to recent aircraft position over the stored map with a white Gaussian process noise . Note that and are mutually independent white noise. 5. ALGORITHM The algorithm corresponding to simulation is as follow: 1. First, we need to generate a representative terrain profile. Use a second order GaussMorkov model with = 20m and = 0.01rad/m in (1). Note that the independent variable in this random process model is now space instead of time, hence the units of rad/m for . Geneate a profile with a discrete space interval of 1m over the range of matrix 5KmÃ—5Km. 2. Next, generate the rotorcrafts motion profile. Use a second order Gauss-Markov model for the horizontal and another for the vertical axis to generate a bounded random process model. Use a sampling interval of = 1s, and the parameters = 10m and = 0.51rad/s in (1). Then add this random process to nominal motion that moves at 140m/s as visualize in Fig.1 but remains at a fixed altitude, starting at coordinates (10m, 10m, 200m) that is at 200m. Generate the profile for a 50s-duration.At each sampling time, compute the radar altimeter measurement by taking difference of the appropriate terrain height from the rotorcraft altitude. The terrain height required will, in general need to be computed by cubic spline (may go for linear interpolation but cubic spline interpolation gives better result) interpolation between samples generated in Part (A). To each radar altimeter measurement, add a measurement noise random sample from a Gaussian distribution with zero mean and a sigma of 1 m. 3. For each interpolated value of radar altimeter, find where is the possibility of getting same measurement within a interval of [-1m, 1m] i.e. error in radar altimeter measurement, within the matrix along X and Y direction. The output will be rows and columns where the position of aircraft is expected after getting observation. Prepare a state space window of relatively smaller interval duration where the aircraft is expected to be. Do the computational efforts to translate this window along with aircraft translational movement in order to reduce time required to pin point the estimated state of aircraft. 4. Initialize with uniform prior and posteriori. Turn the priori and posteriori into a probability mass function by dividing by the sum. Pull out the indices at which posteriori achieves its maximum to start. Keep these as the best estimate to start. Take a covariance matrix of [25 0;0 25] and generate posteriori distribution by multiplying prior distribution with likelihood function . After each iteration of the above process normalize this distribution to make it proper probability distribution that will be equal to number of rows and columns as expected state of aircraft, store the posteriori to the prior distribution . Pull out the indices at which posteriori achieves its maximum and that will be the estimated states of aircraft for a given 5. Do the entire process for 50-s sequence. Plot the X-axis and Y-axis position error over time. Position error (m) over x-direction by Bayesian statistics Time (s) 6. SIMULATION RESULTS The simulation of terrain based navigation was performed for 50s flight trajectory, which starts from [10m, 10m] along X and Y axis as shown in Fig.1. by discontinuous line. Since the approach is for two state estimation Fig.2. depicts a plot concerning to X state estimation error and Fig.3. depicts a plot concerning to Y state estimation error. Due to rigorous computational working required to generate even 5KmÃ—5Km, the simulation could be performed for just 50s but still it gives broad idea how a terrain reference navigation system works and to experience its state estimating capability. Fig.2. Position error in meters along X-direction over a flight of 50s Position error (m) over y-direction by Bayesian statistics Terrain over 5Km along Y axis Time (s) Terrain over 5Km along X axis Fig.3. Position error in meters along Y-direction over a flight of 50s Fig.1. Simulated terrain produce by second order Gauss-Markov process over 5KmÃ—5Km area and dotted line indicates the aircraft flight path. 1. CONCLUSION The simulated results show error along X-axis to be in the range [8m,-5m] and along Y-axis to be in the range [12m,-8m]. However performance of terrain based navigation greatly depends on variation of terrain gradient in the area over which aircraft is flown. Nearly equal height terrain helps to a very less amount for terrain based navigation while varying terrain to a greater extent helps for this navigation technique a lot. Also the present technique for estimating the state greatly depends on how the system estimates the prior belief and puts the mass on nearly true state at the time of initialization and if can, then all the posteriori belief coming after it will have nearly true state output. But if prior belief of state estimation during initialization is itself faulty then entire chain of posteriori belief following it will be wrong. 2. FUTURE WORK This paper is concerned related to solving terrain based navigation as a non-linear recursive estimation problem with Bayesian statistics but other well known techniques for non- linear estimation problem are extended Kalman filter (EKF), particle filter based approach using extended Kalman filter for local linearization, unscented Kalman filter(UKF) can be implemented effective in order to visualize individual methods position error giving characteristics which can be further kept for comparison purpose. Also in terms of application basis that is after estimating current state of aircraft, potential obstacles coming within path of aircraft can be known since the terrain elevations of the area over which aircraft is flown is known. Owing to this knowledge advance terrain avoidance cueing (ATAC) technique can be implemented to signify which path is better suited to deviate in order to get rid of obstacle. This work is carried out at Defence Avionics Research Establishment (DARE), Defence Research and Development Organization, Government of India, Bangalore. 1. Bergman,N.,Ljung,L.,Gustafsson,F., "Terrain navigation using Bayesian statistics IEEE Contr. Syst.19(3), 33-40,1999. 2. D.H.Larry,D.A.Ronald, "Nonlinear Kalman filtering techniques for terrain aided navigation, IEEE Transactions on automatic control,vol.2,No.3,pp. 315-323,1983. 3. H.Jeff, "HELI/SITAN:A terrain referenced navigation algorithm for heliopter, IEEE position, Location and Navigation symposium, Vol.20,No.23,pp 616-625,1990. 4. M.Jurgen,W.Jan,F.T.Gert,T.Franz,T.Bernd,"Hybrid terrain referenced navigation system using a bank of Kalman filters and a comparison technique. AIAA Guidance, Navigation and Control 5. J.Metzger,K.Witsotzky,J.Wendel,G.F.Trommer,"Sigma-oint filter for terrain referenced navigation, AIAA Guidance ,Navigation , and Control conference and exhibit ,San Francisco,California,2005. 6. W.B.Davenport,Jr.,W.L.Root, "An introduction to the theory of Random signals and noise , NewYork:McGraw-Hill,1958. 7. C.F.van Loan , "Computing Integrals Involving the Matrix Exponential, IEEE trans. Automatic Control, Ac-23(3):395-404,June 1978. 8. Dongjin Lee,Hyochoong Bang,Cheonjoong Kim, "Integration of Terrain Referenced Navigation System with INS using Kalman Filter,12th International Conference on Control,Automation and Systems, pp.17-21,Oct 2012. 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Generalizing The Circular Functions \| Brilliant Math & Science Wiki Generalizing The Circular Functions The basic idea behind this wiki is to demonstrate that how we can evaluate inverse trigonometric functions outside their domain using Complex Analysis And Euler's Formula. Everyone of us knows that the range of circular functions : $\sin x$ and $\cos x$ is $[-1,1]$. So did you ever tried solving the equation \[\sin x = 2\] or in particular, finding the solutions for : \[\sin x = n\ , \forall\ n \in \mathbb{R}\] Yeah, you guessed it right, the solution is bit too complex!!! Basic Idea Since, we are solving an equation, for such a point which is outside the range of a function, we know that there won't exist a real solution. So, how should we start...? In fact, because we're dealing with both complex numbers and trigonometric functions, that gives us a clue of starting with the Euler's Identity : $e^{ix} = \cos x + i\sin x$ \[\] \[\] Today, I am going to introduce you to a method with which you can easily evaluate $\arccos (x)$ and $\arcsin (x)$ for all real value of $x$. So, here we go - First of all, by Euler Formula, we have : \[$$e^{ix} = \cos x + i\sin x$$\] \[$$e^{-ix} = \cos x - i\sin x$$\] Subtracting them up, gives \[\sin x = \frac{e^{ix}-e^{-ix}}{2i}\] Now, we wish to find the solutions for $\sin x = n$. So, \[n = \frac{e^{ix}-\frac{1}{e^{ix}}}{2i}\] So, now can you look up the quadratic coming around...? No! ok, I'll just a use a simple substitution here, which makes the work tidier and easier to see the quadratic. Substituting $e^{ix} = t$, we get \[n = \frac{t-\frac{1}{t}}{2i}\] Upon Rearranging, we get \[\Rightarrow t^2 - 2int - 1 = 0\] Now, that's a quadratic in $t$ whose solutions are - \[t = e^{ix} = in\pm\sqrt{1-n^2}\] Taking natural logarithm and multiplying by $-i$ on both $L.H.S$ and $R.H.S$ yields - \[x = -i\ln{\left(in\pm\sqrt{1-n^2}\right)}\] And thus, \[\boxed{\arcsin(n) = -i\ln{\left(in\pm\sqrt{1-n^2} Since, \[\frac{-\pi}{2} \leq \arcsin(x) \leq \frac{\pi}{2}\] so $\arcsin(x)$ lies in the first and fourth quadrants, and hence we do not need to make any changes in our formula. 1. Evaluate : \[\arcsin(2015)\] 2. Solve for $x$ : \[\cos x = n\ ,\ \forall\ n \in \mathbb{R}\]	{"url":"https://brilliant.org/wiki/generalizing-the-circular-functions/","timestamp":"2024-11-05T16:45:52Z","content_type":"text/html","content_length":"44890","record_id":"<urn:uuid:e46d89ed-2f15-4a6d-b800-d18264e0e4e4>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00879.warc.gz"}
Understanding measurement error Geostatistical Analyst のライセンスで利用可能。 Three of the kriging methods—ordinary, simple, and universal—use measurement error models. Measurement error occurs when it is possible to have several differing observations at the same location. For example, you might extract a sample from the ground or air and divide that sample into several subsamples to be measured. You may want to do this if the instrument that measures the samples has some variation. As another example, you might send subsamples of a soil sample to different laboratories for analysis. There could be other times when the variation in instrument accuracy is documented. In this case, you may want to input the known measurement variation into your model. The measurement error model The measurement error model is Z(s) = µ(s) + ε(s) + δ(s), where δ(s) is measurement error and µ(s) and ε(s) are the mean and random variation. In this model, the nugget effect is composed of the variance of ε(s) (called microscale variation) plus the variance of δ(s) (called measurement error). In Geostatistical Analyst, you can specify a proportion of the estimated nugget effect as microscale variation and measurement variation, have Geostatistical Analyst estimate measurement error for you if you have multiple measurements per location, or input a value for measurement variation. When there is no measurement error, kriging is an exact interpolator, meaning that if you predict at a location where data has been collected, the predicted value is the same as the measured value. However, when measurement errors exist, you want to predict the filtered value, µ(s[0]) +ε(s[0]), which does not have the measurement error term. At locations where data has been collected, the filtered value is not the same as the measured value. In previous versions of ArcGIS, the default measurement variation was 0%, so kriging was defaulted to be an exact interpolator. In ArcGIS 10, the default measurement variation is set to 100%, so the default predictions at measured locations will be based on the spatial correlation of the data and the measured values at nearby locations. Measurement error can be introduced by many sources, including uncertainty in the measurement device, location, and data integration. In practice, perfectly precise data is extremely rare. The effect of the model The effect of choosing measurement error models is that your final map can be smoother and have smaller standard errors than the exact kriging version. This is illustrated with an example in the figures below, where exact kriging and smooth kriging are shown when there are only two data locations (at 1 and 2) with values -1 and 1 for a model without measurement variation and one where the nugget effect is all measurement variation.	{"url":"https://pro.arcgis.com/ja/pro-app/2.7/help/analysis/geostatistical-analyst/understanding-measurement-error.htm","timestamp":"2024-11-14T01:15:58Z","content_type":"text/html","content_length":"16140","record_id":"<urn:uuid:d0d0ef40-48f2-498f-a7a5-3c2eb1450be9>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00824.warc.gz"}
Mode-1 N2 internal tides observed by satellite altimetry Articles \| Volume 19, issue 4 © Author(s) 2023. This work is distributed under the Creative Commons Attribution 4.0 License. Mode-1 N[2] internal tides observed by satellite altimetry Satellite altimetry provides a unique technique for observing the sea surface height (SSH) signature of internal tides from space. Previous studies have constructed empirical internal tide models for the four largest constituents M[2], S[2], K[1], and O[1] by satellite altimetry. Yet no empirical models have been constructed for minor tidal constituents. In this study, we observe mode-1 N[2] internal tides (the fifth largest constituent) using about 100 satellite years of SSH data from 1993 to 2019. We employ a recently developed mapping procedure that includes two rounds of plane wave analysis and a two-dimensional bandpass filter in between. The results show that mode-1 N[2] internal tides have millimeter-scale SSH amplitudes. Model errors are estimated from background internal tides that are mapped using the same altimetry data but with a tidal period of 12.6074h (N[2] minus 3min). The global mean error variance is about 25% that of N[2], suggesting that the mode-1 N[2] internal tides can overcome model errors in some regions. We find that the N[2] and M[2] internal tides have similar spatial patterns and that the N[2] amplitudes are about 20% of the M[2] amplitudes. Both features are determined by the N[2] and M[2] barotropic tides. The mode-1 N[2] internal tides are observed to propagate hundreds to thousands of kilometers in the open ocean. The globally integrated N[2] and M[2] internal tide energies are 1.8 and 30.9PJ, respectively. Their ratio of 5.8% is larger than the theoretical value of 4% because the N[2] internal tides contain relatively larger model errors. Our mode-1 N[2] internal tide model is evaluated using independent satellite altimetry data in 2020 and 2021. The results suggest that the model can make internal tide correction in regions where the model variance is greater than twice the error variance. This work demonstrates that minor internal tidal constituents can be observed using multiyear multi-satellite altimetry data and dedicated mapping techniques. Received: 03 Oct 2022 – Discussion started: 19 Oct 2022 – Revised: 09 Jun 2023 – Accepted: 12 Jun 2023 – Published: 13 Jul 2023 The Moon's elliptical orbit around the Earth has an eccentricity of ≈0.055, with its perigean and apogean distances being about 3.63×10^5 and 4.06×10^5km, respectively. The Moon completes one revolution every 27.5546d (1 anomalistic month). The tidal constituents L[2] and N[2] are induced by the Moon's elliptical orbit (Doodson, 1921). They are named the smaller and larger lunar elliptical semidiurnal constituents. The L[2] and N[2] periods are 12.1916 and 12.6583h, respectively (Doodson, 1921; Pawlowicz et al., 2002). M[2] (12.4206h) is based on the mean distance between the Earth and the Moon (3.84×10^5km). The L[2] and N[2] superposition gives the 27.5546d perturbation because the Moon–Earth distance changes along the elliptic orbit. On a global average, the amplitudes of M[2], N[2], and L[2] have a respective ratio of $\mathrm{1}:\mathrm{0.2}:\mathrm{0.05}$. N[2] is the fifth largest tidal constituent; therefore, its impact on the ocean environment is significant. For example, in waters around New Zealand, the N[2] barotropic tide has larger amplitudes than S[2] (Byun and Hart, 2020, Fig. 4 therein). The superposition of N[2], M[2], L[2], and S[2] can cause perigean spring tides (king tides) and apogean neap tides, which significantly affect harbors, coastal regions, and estuaries (Wood, 1978). Including N[2] internal tides can simulate the temporal variation in internal tide energetics with the Moon's elliptical motion. Theoretically, N[2] may modulate M[2] internal tides by ±20% in amplitude and by ±40% in energy (i.e., (1±0.2)^ 2). On average, N[2] will enhance the M[2]-induced ocean mixing by 4% (i.e., 0.2^2). Internal tides are widespread in the ocean and affect numerous ocean processes such as diapycnal mixing, tracer transport, and acoustic transmission (Wunsch, 1975; Dushaw et al., 1995; Whalen et al. , 2020). Internal tides may provide about half of the power for diapycnal mixing in the ocean interior (Munk and Wunsch, 1998; Egbert and Ray, 2000; MacKinnon et al., 2017; Kelly et al., 2021). The magnitude and geography of diapycnal mixing may modulate the large-scale ocean circulation and global climate change; therefore, it is important to study their generation, propagation, and dissipation in the global ocean (Jayne and St Laurent, 2001; Melet et al., 2016; Pollmann et al., 2019; Vic et al., 2019; de Lavergne et al., 2020; Arbic, 2022). Internal tides are annoying noise in the study of mesoscale and sub-mesoscale dynamics. In particular, it will be necessary to make internal tide correction to the Surface Water and Ocean Topography (SWOT) data, so that one can better study the sub-mesoscale dynamic processes (Fu and Ubelmann, 2014; Qiu et al., 2018; Wang et al., 2018; Morrow et al., 2019). Empirical internal tide models can be constructed using past satellite altimetry sea surface height (SSH) measurements. However, previous satellite observations focus mainly on the four largest tidal constituents: M[2], S[2], K[1], and O[1] (Dushaw, 2015; Ray and Zaron , 2016; Zhao et al., 2016; Zaron, 2019; Ubelmann et al., 2022). Dushaw (2015) attempts to map N[2] internal tides using the TOPEX/Poseidon data from 1992 to 2008 but fails to obtain an empirical model because the resulting N[2] internal tides are too noisy (see his Figs. 38 and 52). That work is mainly limited by the short data set available then. In this study, we will construct a reliable empirical N[2] internal tide model using a larger data set and a recently developed mapping method. The challenge of observing N[2] internal tides by satellite altimetry lies in their small SSH displacements (Dushaw, 2015). Given that M[2] internal tides have SSH amplitudes of 1–2cm, N[2] internal tides have only sub-centimeter SSH amplitudes. In this paper, the observation of N[2] internal tides is made possible by two improvements. First, a larger SSH data set is available, thanks to almost 3 decades of multiple satellite observations since 1993. The merged data set from 1993 to 2019 is about 100 satellite years long; therefore, non-tidal errors can be significantly suppressed. Second, a recently developed mapping procedure is employed. This mapping technique extracts N[2] internal tides utilizing their known frequency and theoretical wavelengths. Non-tidal errors can be significantly suppressed by both temporal and spatial filters. The resulting N[2] internal tides reveal their basic features in the global ocean, although they are still noisy (compared to the much larger M[2] internal tides). It is challenging (though possible) to extract L[2] internal tides in some regions, which are estimated to have 1mm SSH signals at most (5% of M[2]). The rest of this paper is arranged as follows. Section 2 describes the data and methods used in this paper. Section 3 presents and discusses the new N[2] internal tides, mainly by comparing them with the well-studied M[2] internal tides. Section 4 is a summary. The satellite altimetry SSH data used in this paper are collected by multiple altimetry missions from 1993 to 2021. In the order of launch time, they are TOPEX/Poseidon, ERS-1, ERS-2, Geosat Follow-On, Jason-1, Envisat, Jason-2, CryoSat-2, SARAL/AltiKa, Haiyang-2A, Jason-3, Sentinel-3A, Sentinel-3B, Haiyang-2B, and Jason-CS/Sentinel-6 (Fig. 1). The combined data set from 1993 to 2019 is about 100 satellite years long. We use the satellite along-track SSH data downloaded from the Copernicus Marine Service (https://doi.org/10.48670/moi-00146). The SSH measurements have been processed by standard corrections for atmospheric effects, surface wave bias, and geophysical effects (Pujol et al., 2016; Taburet et al., 2019). The ocean barotropic tide, polar tide, solid Earth tide, and loading tide are corrected using theoretical or empirical models (Pujol et al., 2016). Mesoscale correction (Ray and Byrne, 2010; Zhao, 2022a) is made using the gridded SSH fields downloaded from the Copernicus Marine Service (https://doi.org/10.48670/moi-00148). The satellite along-track SSH data in 2020 and 2021 are used to evaluate the new N[2] internal tide model as independent data (Sect. 2.6). Extracted from the 27-year-long data, our N[2] internal tide model contains only the 27-year coherent component. Their temporal variation (or incoherent component) is not addressed in this The observation of internal tides by satellite altimetry may be affected by an issue called tidal aliasing because the satellite repeat cycles are much longer than the semidiurnal and diurnal tidal periods. Here we examine possible tidal aliasing issues with N[2] and M[2] internal tides. In one 160km by 160km fitting window (Sect. 2.2), there are typically about 7.84×10^4 SSH data from multiyear multi-satellite measurements. Using their observation times, we can calculate their phase lags with respect to the N[2] tidal cycle (12.6583h). Figure 2a gives the histogram of their phase lags over one N[2] tidal cycle. For comparison, Fig. 2b shows the histogram with respect to the M[2] tidal cycle (12.4206h). The results show that the SSH data overall evenly distribute over one N [2] or M[2] tidal cycle, without extreme biases. In particular, their distribution on M[2] is smooth, suggesting that there is no tidal aliasing issues for M[2]. Their distribution over N[2] is a little bumpy, suggesting that the resulting N[2] internal tides may have larger errors. The uneven distribution stems from the orbital configurations of the satellite missions. Fortunately, as shown in this study, the new mode-1 N[2] internal tides can overcome background noise in some regions. 2.2Plane wave analysis The core technique of our mapping procedure is plane wave analysis. By this method, internal tides are determined by fitting horizontal plane waves in one given fitting window (160km by 160km in this study), in contrast to harmonic analysis at one single site. This method has been described in detail in our previous studies (Zhao and Alford, 2009; Zhao, 2014; Zhao et al., 2016). For each tidal constituent, there may be multiple internal tides of arbitrary propagation directions at each site, due to their multiple source regions and long-range propagation. Plane wave analysis can resolve these internal tides by propagation direction. We will fit five mode-1 N[2] internal tidal waves at each site. Our five-wave representor follows $\begin{array}{}\text{(1)}& {\mathrm{\Sigma }}_{m=\mathrm{1}}^{\mathrm{5}}{A}_{m}\mathrm{cos}\left(k\phantom{\rule{0.125em}{0ex}}x\phantom{\rule{0.125em}{0ex}}\mathrm{cos}{\mathit{\theta }}_{m}+k\ phantom{\rule{0.125em}{0ex}}y\phantom{\rule{0.125em}{0ex}}\mathrm{sin}{\mathit{\theta }}_{m}-\mathit{\omega }\phantom{\rule{0.125em}{0ex}}t-{\mathit{\varphi }}_{m}\right),\end{array}$ where x and y are the east and north Cartesian coordinates, t is time, and ω and k are the frequency and horizontal wavenumber of the target internal tides, respectively. Three parameters need to be determined for each internal tidal wave: amplitude A, phase ϕ, and direction θ. To determine one wave, the amplitude and phase of a single plane wave are determined by the least-squares fit in each compass direction (with 1^∘ increment). When the resulting amplitudes are plotted as a function of direction in polar coordinates, an internal tidal wave appears to be a lobe. The direction of the first wave is thus determined from the biggest lobe. Thus, the amplitude A, phase ϕ, and propagation direction θ of one internal tidal wave are determined. Afterward, its signal is predicted and subtracted from the original data, which removes the wave itself and its side lobes. This procedure can be repeated to extract an arbitrary number of waves one by one. The resulting internal tidal waves are sorted with descending amplitudes. The frequency (ω) and horizontal wavenumber (k) of the target internal tides are needed in plane wave analysis. The M[2] and N[2] tidal periods (equivalent to frequencies) are from the Moon's orbital motion around the Earth (Doodson, 1921; Pawlowicz et al., 2002). They are 12.4206 and 12.6583h, respectively. The local phase speed (equivalent to wavenumber) of the target internal tides is theoretically determined from the World Ocean Atlas 2018 (WOA18) (Boyer et al., 2018). The WOA18 provides climatological hydrographic profiles on a spatial grid of 0.25^∘ latitude by 0.25^∘ longitude. Ocean depth is based on the 1arcmin topography database constructed using in situ and satellite measurements (Smith and Sandwell, 1997). For a given ocean depth and stratification profile, the vertical structures and eigenvalue speeds of internal tides are obtained by solving the Sturm–Liouville equation (Gill, 1982; Chelton et al., 1998; Kelly, 2016): $\begin{array}{}\text{(2)}& \frac{{d}^{\mathrm{2}}\mathrm{\Phi }\left(z\right)}{\mathrm{d}{z}^{\mathrm{2}}}+\frac{{N}^{\mathrm{2}}\left(z\right)}{{c}^{\mathrm{2}}}\mathrm{\Phi }\left(z\right)=\mathrm subject to free-surface and rigid-bottom boundary conditions, where N(z) is the buoyancy frequency profile and c is the eigenvalue speed. The phase speed c[p] can be calculated from the eigenvalue speed following $\begin{array}{}\text{(3)}& {c}_{\mathrm{p}}=\frac{\mathit{\omega }}{\sqrt{{\mathit{\omega }}^{\mathrm{2}}-{f}^{\mathrm{2}}}}c,\end{array}$ where ω and f are the tidal and inertial frequencies, respectively. Note that the phase speed is a function of longitude and latitude (Zhao et al., 2016). 2.3Mapping procedure Our three-step mapping procedure consists of two rounds of plane wave analysis and a spatial two-dimensional (2D) bandpass filter in between (Zhao, 2020, 2021, 2022a, b). In this paper, the mapping process is illustrated by showing intermediate results in Fig. 3. An interested reader is referred to the above papers for more details. In step 1, mode-1 N[2] internal tides are mapped by plane wave analysis as described above. The N[2] internal tides are mapped from along-track SSH data onto a spatially regular grid. In this paper, our fitting window is chosen to be 160km by 160km, consistent with wavelengths of mode-1 N[2] internal tides. The resulting N[2] internal tides are at a 0.2^∘ longitude by 0.2^∘ latitude grid. At each grid point, five mode-1 N[2] internal tidal waves of arbitrary propagation directions are determined. The vector sum of these five waves gives the internal tide solution. Figure 3a shows the mode-1 N[2] internal tide field obtained in this step. It gives obvious internal tide signals (e.g., around the Hawaiian Ridge) but the non-tidal noise is high. In step 2, the spatially regular N[2] internal tide field is cleaned by a 2D bandpass filter in overlapping 850km by 850km windows. The N[2] internal tide field is first converted to the 2D wavenumber spectrum by Fourier transform. The spectrum is truncated to [0.8, 1.25] times the local wavenumber. The truncated spectrum is converted back to the internal tide field by inverse Fourier transform. Figure 3b shows the cleaned N[2] internal tide field. Now the N[2] internal tide signals are much cleaner. However, Fig. 3b cannot resolve multiple internal tidal waves yet. In step 3, plane wave analysis is called again to decompose the filtered internal tide field into five internal waves of arbitrary propagation directions. The second-round plane wave analysis is the same as the first-round plane wave analysis, except that the input is the filtered internal tide field in step 2. In the end, the resulting five waves are saved with their respective amplitudes, phases, and directions. Figure 3c shows the five-wave superimposed internal tide field. It is very similar to Fig. 3b because this step only decompose the internal tide field. The five-wave decomposition allows us to separate internal tides of different propagation directions. They will be used to extract long-range internal tidal beams in the ocean (Sect. 3). 2.4N[2] and M[2] internal tides We map both the mode-1 N[2] and M[2] internal tides following the same three-step procedure. They are constructed from the same satellite altimetry data but using their respective wave parameters (frequency and wavenumber). Figure 4 shows the resulting N[2] and M[2] internal tide fields. Internal tides in shallow waters (<1000m) are discarded. The new M[2] internal tides are almost identical to those obtained in previous studies using slightly different satellite data (Zhao, 2022b). Here we find that the N[2] and M[2] internal tides have similar spatial patterns and that the N [2] amplitudes are about 20% of the M[2] amplitudes. The largest N[2] amplitudes are about 5mm, compared to 20–30mm for M[2] internal tides. To account for this factor, their color map ranges are different by a factor of 5. Figure 4 gives SWOT ground tracks in its 1d fast-repeating phase (green lines). It shows that strong mode-1 N[2] internal tides occur under some SWOT swaths, for example, those off the California coast, in the New Caledonia region, in the western North Pacific, and on the Amazon continental shelf. In these regions, the N[2] internal tides cannot be neglected in the study of sub-mesoscale dynamics. Conversely, the upcoming SWOT data also offer a great opportunity to explore N[2] internal tides. We have examined the possible cross talk between the N[2] and M[2] internal tides in our mapping procedure. We map N[2] internal tides using two different data sets. The first is the original satellite altimetry SSH data set (Sect. 2.1). The second is the M[2]-corrected data set. In other words, the M[2] internal tides are predicted using our empirical model and subtracted from the original data. We find that the resulting N[2] internal tides from the two data sets are almost the same. The variance of their differences is <1% that of the N[2] internal tides. Likewise, we map M[2] internal tides using both the original and N[2]-corrected data sets and find that the impact of N[2] on M[2] is negligible. Our analysis reveals that the N[2] and M[2] internal tides do not cross talk in our mapping method. This is because the 27-year-long satellite data from 1993 to 2019 are sufficient long to unambiguously separate the N[2] and M[2] tidal constituents (about 14min 2.5Model errors Model errors in our N[2] and M[2] internal tide models are estimated using background internal tides. In contrast to N[2] and M[2] internal tides, which are mapped using tidal periods of 12.6583 and 12.4206h, respectively, background internal tides are mapped using the same satellite altimetry data but for tidal periods between N[2] and M[2]. Specifically, we map 13 sets of background internal tides using 13 different tidal periods that are linearly interpolated between N[2] and M[2] (Fig. 5). The other mapping parameter, wavenumber (equivalently phase speed), can be obtained using Eq. (3 ). The same strategy has previously been employed to estimate barotropic tide errors. For example, Ray and Susanto (2016) study the fortnightly tidal cycles (MS[f] and M[f]) of tidal mixing using satellite sea surface temperature data. Zaron et al. (2023) study the fortnightly variability in Chl a using satellite sea surface color data. In both studies, tidal errors are estimated using signals at fake or false tidal frequencies near the real tidal constituents. We thus obtain 13 background internal tides in the central Pacific (Fig. 4c, box). Their regional mean SSH amplitudes in this region are 0.8±0.1mm, compared to 1.66 and 7.75mm for N[2] and M[2] (Fig. 5). Note that the SSH amplitudes of the 13 background internal tides are almost the same, showing no significant tidal cusps around the N[2] or M[2] internal tides. In addition, we have calculated the correlation coefficients among these 15 sets of internal tides (including N[2] and M[2]). All correlation coefficients are <0.05, suggesting that these background internal tides are independent of each other and of the N[2] and M[2] internal tides. In other words, background internal tides are signals we obtain where there are no tidal constituents. We suggest that the model errors in N[2] and M[2] can be represented by background internal tides. In this study, we pick one tidal period (12.6074h) for a global run to obtain background internal tides (model errors), considering that a global run is time-consuming. Figure 4c gives the resulting background internal tides (model errors). It reveals that model errors are large in regions of strong mesoscale motions because model errors are mainly leaked mesoscale signals. Figure 4 shows that the N[2] internal tides are noisier than M[2] because the small-amplitude N[2] internal tides are easily affected by model errors. On a global average, the error variance is about 25% of the N[2] variance, and only 1% of the M[2] variance. 2.6Model evaluation Our N[2] and M[2] internal tide models are evaluated using independent satellite SSH data collected in 2020 and 2021. For each SSH measurement of known time and location, the internal tide signal is predicted using the model under evaluation and subtracted from the SSH measurement. Variance reduction is the variance difference before and after the internal tide correction. The variance reductions for all SSH measurements are binned into 2^∘ by 2^∘ boxes. The global map of N[2] variance reduction is shown in Fig. 6a. The M[2] internal tide model is evaluated in the same way and shown in Fig. 6b. Note that the color map ranges for N[2] and M[2] differ by a factor of 25, that is, the square of the factor of their amplitudes. In the evaluation, the true N[2] internal tides (variance ${\mathit{\sigma }}_{{\mathrm{N}}_{\mathrm{2}}}^{\mathrm{2}}$) in the model will remove the N[2] internal tides in the independent data, leading to positive variance reduction, while the model errors (variance ${\mathit{\sigma }}_{\mathit{ϵ}}^{\mathrm{2}}$) will increase the variance of the independent data, leading to negative variance reduction. Together, we obtain positive variance reduction where ${\mathit{\sigma }}_{{\mathrm{N}}_{\mathrm{2}}}^{\mathrm{2}}>{\mathit{\sigma }}_{\mathit{ϵ}}^{\mathrm{2}}$, and negative variance reduction where ${\mathit{\sigma }}_{{\mathrm{N}}_{\mathrm{2}}}^{\mathrm{2}}<{\mathit{\sigma }}_{\mathit{ϵ}}^{\mathrm{2}}$. Figure 6a shows positive variance reduction in the global ocean, suggesting that the true N[2] internal tides are greater than model errors. In particular, in regions of strong N[2] internal tides such as the Hawaiian Ridge and the Amazon continental shelf, patches of positive variance reduction are observed because the strong N[2] internal tides can overcome model errors, while negative variance reduction usually occurs in regions of weak N[2] internal tides such as the eastern equatorial Pacific and the Southern Ocean. The regions of strong mesoscale motions are dominated by negative variance reduction, where weak N[2] internal tides are overwhelmed by large model errors (Fig. 4c). For comparison, Fig. 6b shows that the M[2] internal tide model causes positive variance reduction throughout the global ocean, except for regions of strong mesoscale motions or strong temporal variation (Zhao, 2021). This is because the strong M[2] internal tides are almost always greater than the model errors ${\mathit{\sigma }}_{{\mathrm{M}}_{\ mathrm{2}}}^{\mathrm{2}}>{\mathit{\sigma }}_{\mathit{ϵ}}^{\mathrm{2}}$. In summary, our N[2] internal tide model can reduce variance in some regions, although the N[2] SSH amplitudes are just a few We further examine the relation between the N[2] variance reduction shown in Fig. 6a and the variance difference between the N[2] model and the model error. Figure 6c and d give the N[2] model variance and the error variance that are computed from Fig. 4a and c, respectively. Note that the N[2] model variance σ^2 contains both true N[2] internal tides and errors ${\mathit{\sigma }}^{\ mathrm{2}}={\mathit{\sigma }}_{{\mathrm{N}}_{\mathrm{2}}}^{\mathrm{2}}+{\mathit{\sigma }}_{\mathit{ϵ}}^{\mathrm{2}}$. Under the condition that the N[2] variance is greater ${\mathit{\sigma }}_{{\ mathrm{N}}_{\mathrm{2}}}^{\mathrm{2}}>{\mathit{\sigma }}_{\mathit{ϵ}}^{\mathrm{2}}$, we should have ${\mathit{\sigma }}^{\mathrm{2}}>\mathrm{2}{\mathit{\sigma }}_{\mathit{ϵ}}^{\mathrm{2}}$. We thus calculate the variance difference ${\mathit{\sigma }}^{\mathrm{2}}-\mathrm{2}{\mathit{\sigma }}_{\mathit{ϵ}}^{\mathrm{2}}$ and show the global map in Fig. 6e. To test this relation, we calculate the variance difference ${\mathit{\sigma }}^{\mathrm{2}}-m\cdot {\mathit{\sigma }}_{\mathit{ϵ}}^{\mathrm{2}}$ for m ranging from 0.5 to 3.5 with a step of 0.1. For each resulting variance difference map (e.g., Fig. 6e), we calculate its correlation coefficient with the N[2] variance reduction (Fig. 6a). We get the best spatial correlation when the factor m is 2, consistent with our theoretical analysis. Note that all the above analyses are based on the 2^∘ by 2^∘ binned values. Figure 6f shows the mask region where the N[2] model variance is greater than twice the error variance, indicating regions where the N[2] model can be used to make internal tide correction. The mask covers regions of strong N[2] and M[2] internal tides such as the Hawaiian Ridge, the area off the California coast, the Amazon continental shelf, the western North Pacific, and New Caledonia. We next examine the performance of the N[2] and M[2] internal tide models in making internal tide correction for SWOT. In Fig. 6, the green lines denote the SWOT ground tracks in its daily fast-repeating phase. We interpolate the N[2] and M[2] variance reductions onto the SWOT ground tracks (neglecting its swath) and calculate the along-track mean variance reductions. For N[2], the mean variance reductions in and outside the mask region are 0.73 and −0.25mm^2, respectively. The negative variance reduction suggests that the N[2] model does not work well outside the mask region. Fortunately, the N[2] model can make internal tide correction in the mask region where the N[2] internal tides can overcome model errors. The variance reductions caused by the N[2] model seem small, but keep in mind that (1) internal tides and sub-mesoscale motions both have millimeter-scale SSH amplitudes and (2) internal tides are much stronger in their source regions. For M[2], the along-track mean variance reductions in and outside the mask region are 25.6 and 2.5mm^2, respectively. They suggest that the M[2] model performs well both in and outside the mask region because the M[2] internal tides dominate errors throughout the global ocean. 3.1Global distribution Our mode-1 N[2] model reveals that N[2] internal tides are widespread in the global ocean (Fig. 4a). In the Indian Ocean, they are observed in the Arabian Sea, the Bay of Bengal, and the Madagascar–Mascarene region. In the Pacific Ocean, N[2] internal tides occur in regions such as the French Polynesian Ridge, the Hawaiian Ridge, the Indonesian seas, the western South Pacific, and the western North Pacific. In the Atlantic Ocean, N[2] internal tides appear in regions including the Azores region, the Amazon shelf, the Bay of Biscay, and the Vitória–Trindade Ridge. Our M[2] model reveals that mode-1 M[2] internal tides are observed in the same regions (Fig. 4b). The N[2] and M[2] internal tides have similar spatial patterns, but the N[2] amplitudes are about 20% of the M[2] amplitudes. To further quantify their relation, we give in Fig. 7a the scatterplot of the N[2] and M[2] SSH amplitudes. It shows that the N[2] and M[2] amplitudes largely follow the diagonal line with a ratio of 5. Their correlation coefficient is 0.69 (R in MATLAB function corrcoef). We extract the N[2] and M[2] barotropic tides from TPXO.8 (Egbert and Erofeeva, 2002) and show them in Fig. 8. We find that the N[2] and M[2] barotropic tides have similar spatial patterns and that the N[2] amplitudes are about 20% of the M[2] amplitudes. We examine the relation between the N[2] and M[2] barotropic tides as well. Figure 7b shows the scatterplot of the N[2] and M[2] barotropic amplitudes. It shows that N[2] and M[2] have a very tight relation, with a correlation coefficient of 0.96. Egbert and Ray (2003) show that the M[2] and N[2] barotropic-to-baroclinic energy conversion maps have similar spatial patterns and that their amplitudes differ by a factor of 25 (see their Fig. 1). The N[2] and M[2] relation (spatial pattern and amplitude ratio) is the same for both barotropic and baroclinic tides. Because N[2] and M[2] have close tidal periods (12.6583 and 12.4206h), their generations over the same topographic features should be the same (distinguishing their slight differences may improve our understanding of internal tide dynamics in the future). In addition, it is reasonable that the N[2] and M[2] internal tides have a relatively weak relation (Fig. 7a) because the long-range propagation of internal tides is affected by an inhomogeneous ocean environment. 3.2Long-range beams In this section, we study the long-range mode-1 N[2] internal tidal beams. We have fitted five mode-1 N[2] internal tidal waves at each grid point by plane wave analysis. Taking advantage of the five-wave fits, we can decompose the N[2] internal tide field into the northward (0–180^∘ counterclockwise from due east) and southward (180–360^∘) components by propagation direction. Each component contains internal tidal waves with propagation directions falling in the given range (Fig. 9). The decomposed components clearly show well-defined long-range N[2] internal tidal beams, which are characterized by larger amplitudes and cross-beam co-phase lines (not shown here for clarity; see Figs. 10 and 11). There are numerous long-range N[2] internal tidal beams, which radiate from the strong generation sites mentioned above. For example, northward N[2] beams are observed to originate from the French Polynesian Ridge, the Macquarie Ridge, and the Amazon shelf. Southward N[2] beams are observed to originate from the Andaman Islands, the Lombok Strait, the Hawaiian Ridge, the French Polynesian Ridge, the Mendocino Ridge, and the Azores, among others. Note that the M[2] long-range internal tidal beams have been well studied in previous studies (Zhao et al., 2016, Fig. 5 therein). To avoid repetition, the M[2] internal tidal beams are not shown here. Together, we observe that the N[2] and M[2] internal tides have similar long-range beams. In this study, we examine two long-range internal tidal beams as examples. First, we examine the southward internal tides from the Amukta Pass, Alaska. The M[2] long-range beam from Amukta Pass has been studied recently (Zhao, 2022b). Figure 10a shows the southward N[2] internal tides in the central North Pacific (Fig. 9b, blue box). For comparison, the southward M[2] internal tides are shown in Fig. 10b. Both tidal constituents can travel from the Aleutian island chain to the Hawaiian Ridge over 3000km away. Their propagation directions are about −78^∘ from due east. The black lines in Fig. 10 show the 0 and 180^∘ co-phase charts. Figure 10c shows their phase difference, which increases with propagation because N[2] internal tides travel faster than M[2] internal tides according to Eq. (3). In the propagation, their phase difference increases with propagation. Along the dashed line from source (52.6^∘N, 189^∘E) to far field (26^∘N, 195^∘E), their phase difference increases from 65 to 305^∘. The overall phase change is 240^∘. It takes about 18 tidal cycles for the N[2] and M[2] internal tides to travel along the path. Figure 11 shows southward internals tides in the region off the California coast (Fig. 9b, cyan box). This region is chosen for a detailed investigation because it contains one site for the SWOT calibration/validation field campaign. The green lines in this figure indicate the SWOT swaths in its fast-repeating phase (Wang et al., 2022). The crossover region of the ascending and descending swaths is the SWOT calibration/validation site. This region is dominated by the southward internal tides from the Mendocino Ridge. Note that this region is also affected by internal tides in other propagation directions (Zhao et al., 2019). Additionally, there are southwestward internal tides from the Monterey Bay. The two internal tidal beams intersect around the SWOT campaign site. As explained earlier, our N[2] model can make internal tide correction for SWOT. Figure 11 shows that N[2] and M[2] internal tides are very similar, although the N[2] fluxes are much weaker. Both N[2] and M[2] beams can be tracked from 40 to 20^∘N for >2000km. They both bifurcate around 32^∘N near Fieberling seamounts (32.5^∘N, 232.3^∘E) for unknown reasons. The dashed line delineates the beam from 40.3 to 22^∘N along 128^∘W. This line is about 2000km long. Along this line, the N[2] and M[2] phase difference increases from 40 to 160^∘ over about 14 M[2] or N[2] tidal cycles. 3.3Energy and energy flux We calculate the depth-integrated energy flux of mode-1 N[2] internal tide from their SSH amplitudes and a transfer function (F[n]). The transfer function is calculated using the WOA18 climatological hydrography and the Sturm–Liouville equation (Zhao and Alford, 2009; Zhao et al., 2016). The same calculation method has also been derived by Geoffroy and Nycander (2022). In this study, we follow our method (previously for mode-1 M[2] internal tides) to obtain the transfer function for mode-1 N[2] internal tides. It is a function of ocean depth, tidal frequency, mode number, latitude, and stratification. The transfer functions for N[2] and M[2] are very close because their tidal periods are close. At each grid point, we thus obtain five energy fluxes for the five internal tidal waves following $F=\frac{\mathrm{1}}{\mathrm{2}}{F}_{n}{A}^{\mathrm{2}}$, where A is the SSH amplitude. The vector sum of the five energy fluxes gives the final energy flux at this site. In this study, we compare the N[2] and M[2] internal tide energy fluxes in two regions. An interested reader can examine other ocean regions. We show that their energy fluxes have similar spatial patterns. The results show that the mode-1 N[2] internal tides can be observed by satellite altimetry, although they are much weaker than the M[2] internal tides. Following the same procedure, we have computed the depth-integrated internal tide energies from SSH amplitudes. The globally integrated area-weighted energies for the N[2] and M[2] internal tides are 1.8 and 30.9PJ, respectively. The N[2]-to-M[2] ratio is about 5.8%, larger than the theoretical value of 4% because N[2] contains larger error variance. As explained earlier, the error variance is about 25% of the N[2] variance but only 1% of the M[2] variance. Figure 12 shows the N[2] and M[2] energy fluxes in the western South Pacific. In this study, it is trimmed to 30^∘S–0^∘, 145^∘E–125^∘W. Colors show flux magnitudes, and black arrows show flux vectors. This region is chosen because (1) it features various topographic obstacles such as mid-ocean ridges and island chains and (2) the New Caledonia region is one site for SWOT calibration/ validation field experiments (Bendinger et al., 2023). There are numerous N[2] and M[2] internal tidal beams in this region. They are dominantly generated over topographic features. For example, N[2] and M[2] internal tidal beams radiate from many straits surrounding the Coral Sea (Tchilibou et al., 2020). The internal tidal beams can be in any horizontal propagation direction. From the French Polynesian Ridge, internal tides mainly propagate southward and northward. From the Kermadec Arc and New Caledonia, the outgoing internal tidal beams usually travel eastward or westward. The energy fluxes of N[2] and M[2] internal tides have similar spatial patterns. Figure 12 shows seven SWOT swaths in this region (green lines). Among them, the two swaths in the New Caledonia region (black box) overlap with strong N[2] internal tides whose contribution cannot be neglected. In addition, the two swaths cross the French Polynesian Ridge, where one should pay an attention to N[2] and M[2] internal tides in the study of mesoscale and sub-mesoscale processes. Figure 13 shows the N[2] and M[2] energy fluxes in the North Atlantic Ocean (2^∘S–53^∘N, 58–3^∘W). Figure 13 is in the same format as Fig. 12. Internal tides in this region have attracted much attention in recent years (Vic et al., 2018; Köhler et al., 2019; Löb et al., 2020). In particular, internal tides on the Amazon continental shelf have been intensively studied recently, partly because of the co-existence of internal tides and internal solitary waves (Egbert and Erofeeva, 2021; Tchilibou et al., 2022; Assene et al., 2023). Our satellite observation reveals that strong N[2] and M[2] internal tides occur around notable topographic features including the Mid-Atlantic Ridge, the Amazon continental shelf, the Azores region, the Bay of Biscay, the Canary Islands, and the Cabo Verde islands. The longest internal tidal beams for both N[2] and M[2] are the southward internal tidal beams from the Azores (Zhao, 2016; Köhler et al., 2019). The two beams can be tracked over 2000km. In this region, there are four SWOT swaths in its fast-repeating phase, which overlap remarkable N[2] and M[2] internal tidal beams. In this study, we constructed empirical models for mode-1 N[2] and M[2] internal tides from satellite altimetry. Among them, N[2] is the larger lunar elliptical semidiurnal constituent and the fifth largest oceanic tidal constituent. It is induced by the Moon's elliptical orbit. Its amplitudes are about 20% of the M[2] amplitudes. The mode-1 N[2] internal tides have sub-centimeter-scale SSH amplitudes. We can extract weak N[2] internal tides because we use a larger altimetry data set and a newly developed mapping procedure. First, we use the multiyear multi-satellite altimetry data from 1993 to 2019. The combined data are about 100 satellite years long, which can significantly suppress non-tidal errors. Second, we extract mode-1 N[2] internal tides by a three-step mapping procedure, which cleans internal tides using known frequency and wavenumbers of the target internal tide. In consequence, satellite altimetry can observe mode-1 N[2] internal tides with millimeter-scale SSH amplitudes. Our N[2] internal tide model is still noisy. Future improvements can be made with more and more satellite altimetry data becoming available. We estimated errors in the N[2] and M[2] internal tide models using background internal tides. Specifically, background internal tides are mapped using the same altimetry data but for tidal periods between N[2] and M[2]. In this study, we construct a global map of model errors using a tidal period of 12.6074 (N[2] minus 3min). The model errors are usually <1mm in the global ocean, with the global mean error being about 0.7mm. Large errors usually occur in regions of strong mesoscale motions, since the model errors mainly come from the leaked mesoscale signals. On a global average, the error variance is about 25% of the N[2] model variance but only 1% of the M[2] model variance. Our satellite observations revealed some basic features of the global N[2] internal tides. We found that the N[2] and M[2] internal tides have similar spatial patterns and that the N[2] amplitudes are about 20% of the M[2] amplitudes. Both features are determined by their barotropic counterparts. We found that both N[2] and M[2] internal tides can propagate hundreds to thousands of kilometers in the open ocean but at different phase speeds. We examined regional N[2] internal tides and revealed rich information on their generation and propagation. We suggest that including N[2] internal tides can better simulate the temporal variation in internal tide energetics with the lunar elliptical orbit. Our N[2] and M[2] internal tide models have been evaluated using independent altimetry data in 2020 and 2021. The M[2] model can cause variance reduction throughout the global ocean because the M[2] internal tides dominate the model errors. In contrast, the N[2] model can cause variance reduction in regions of strong N[2] internal tides where they can overcome errors. We found that the N[2] model performs well in regions where the N[2] model variance is greater than twice the error variance, which means that the true N[2] variance is greater than the error variance. We showed that the N [2] and M[2] models work well in the mask region along the SWOT fast-repeating tracks, which suggests that they can make internal tide correction for SWOT. Last but not least, we demonstrated that our mapping technique can construct a reliable mode-1 N[2] internal tide model using 100 satellite years of altimetry data. We have applied our mapping technique to the first baroclinic mode of other minor tidal constituents and higher baroclinic mode of other major tidal constituents and obtained clear internal tide signals. We have tried mapping mode-2 N[2] internal tides around the Hawaiian Ridge (18–28^∘N, 185–205^∘E). However, the resulting model is noisy, as expected. In this region, the mean amplitude of mode-1 N[2] internal tides is about 2.5mm. The mean mode-2 N[2] amplitude is estimated to be 1mm, using a ratio of 2.5 from mode-1 and mode-2 M[2] internal tides. The ∼1mm mode-2 N[2] internal tides cannot overcome the ∼ 0.7mm noise. It is expected that the low-noise SWOT data along 120km wide swaths will improve the observation of minor tidal constituents and higher baroclinic modes. The author has declared that there are no competing interests. Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. The author thanks Katsuro Katsumata, Clément Vic, and two anonymous referees for their constructive suggestions that have greatly improved this paper. This research has been supported by the National Aeronautics and Space Administration (grant nos. NNX17AH57G and 80NSSC18K0771). This paper was edited by Katsuro Katsumata and reviewed by Clément Vic and two anonymous referees. Arbic, B. K.: Incorporating tides and internal gravity waves within global ocean general circulation models: A review, Prog. 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Analysis of Velocity Pattern of a Power-Assisted Mobile Robot Yuki Ueno Tokyo University of Technology 1404-1 Katakuramachi, Hachioji, Tokyo 192-0982, Japan March 3, 2019 May 20, 2019 November 20, 2019 power-assist system, mobile robot, minimum jerk trajectory This paper aims to analyze the velocity pattern of a power-assisted mobile robot when the operator performs operation without any discomfort. Power-assist systems for mobile robots such as wheelchairs and conveyance carriers are extremely effective in alleviating the physical burden on operators when they carry heavy objects. Although the velocity control based power-assist system has an advantage that it can be easily realized, the problem lies in that the system becomes unstable when the operator has high stiffness. Variable impedance control based on impedance estimation of the operator is effective at solving this problem. To realize operator impedance estimation, it is necessary to know the intended robot’s motion of a person. In this study, as a preliminary step to estimate the operator’s impedance, the velocity pattern when the operator performs natural operation of the robot through the power-assist system is analyzed. The results confirm that the natural velocity pattern can be approximated by a velocity pattern connecting two minimum jerk trajectories. Cite this article as: Y. Ueno, “Analysis of Velocity Pattern of a Power-Assisted Mobile Robot,” J. Adv. Comput. Intell. Intell. Inform., Vol.23 No.6, pp. 990-996, 2019. Data files: 1. [1] A. Kakimoto, H. Matsuda, and Y. Sekiguchi, “Development of power-assisted attendant-propelled wheelchair,” Proc. of the 19th Annual Int. Conf. of the IEEE Engineering in Medicine and Biology Society, Vol.4, pp. 1875-1876, 1997. 2. [2] C. Zhu, M. Oda, M. Yoshioka, T. Nishikawa, S. Shimazu, and X. Luo, “Admittance control based walking support and power assistance of an omnidirectional wheelchair typed robot,” Proc. of IEEE Int. Conf. on Robotics and Biomimetics, pp. 381-386, 2010. 3. [3] Y. Ueno, H. Kitagawa, K. Kakihara, T. Sakakibara, and K. Terashima, “Development of an Innovative Power-Assist Omni-Directional Mobile Bed Considering Operator’s Characteristics,” Int. J. Automation Technol., Vol.8, No.3, pp. 490-499, 2014. 4. [4] S. Fujiwara, H. Kitano, H. Yamashita, H. Maeda, and H. Fukunaga, “Omnidirectional Cart with Power-assist System,” J. Robot. Mechatron., Vol.14, No.4, pp. 333-341, 2002. 5. [5] S. Hoshino and K. Uchida, “Interactive Motion Planning for Mobile Robot Navigation in Dynamic Environments,” J. Adv. Comput. Intell. Intell. Inform., Vol.21, No.4, pp. 667-674, 2017. 6. [6] H. Sasaki, T. Horiuchi, and S. Kato, “Experimental Study on Behavior Acquisition of Mobile Robot by Deep Q-Network,” J. Adv. Comput. Intell. Intell. Inform., Vol.21, No.5, pp. 840-848, 7. [7] T. Tsumugiwa, Y, Fuchikami, A. Kamiyoshi, R. Yokogawa, and K. Yoshida, “Stability Analysis for Impedance Control of Robot in Human-Robot Cooperative Task System,” J. of Advanced Mechanical Design, Systems, and Manufacturing, Vol.1, No.1, pp. 113-121, 2007. 8. [8] Y. Yamada, H. Konosu, T. Morizono, and Y. Umetani, “Proposal of Skill-Assist: a system of assisting human workers by reflecting their skills in positioning tasks,” Proc. of IEEE Int. Conf. on Systems, Man and Cybernetics, Vol.4, pp. 11-16, 1999. 9. [9] K. Terashima, K. Watanabe, Y. Ueno, and Y. Masui, “Auto-tuning Control of Power Assist System Based on the Estimation of Operator’s Skill Level for Forward and Backward Driving of Omni-directional Wheelchair,” Proc. of Int. Conf. on Intelligent Robots and Systems, pp. 6046-6051, 2010. 10. [10] R. Ikeura, T. Moriguchi, and K. Mizutani, “Optimal variable impedance control for a robot and its application to lifting an object with a human,” Proc. of 11th IEEE Int. Workshop on Robot and Human Interactive Communication, pp. 500-505, 2002. 11. [11] T. Tsumigiwa, R. Yokogawa, and K. Hara, “Variable Impedance Control Based on Estimation of Human Arm Stiffness for Human-Robot Cooperative Calligraphic Task,” Proc. of IEEE Int. Conf. on Robotics and Automation, pp. 644-650, 2002. 12. [12] M. M. Rahman, R. Ikeura, and K. Mizutani, “Investigating the Impedance Characteristics of Human Arm for Development of Robots to Co-operate with Human Operators,” Proc. of IEEE Int. Conf. on Systems, Man and Cybernetics, Vol.2, pp. 676-681, 1999. 13. [13] T. Flash and N. Hogan, “The Coordination of Arm Movements: An Experimentally Confirmed Mathematical Model,” The J. of Neuroscience, Vol.5, No.7, pp. 1688-1703, 1985. 14. [14] Y. Ueno, T. Ohno, K. Terashima, H. Kitagawa, K. Kakihara, K. Funato, and K. Kakihara, “Novel Differential Drive Steering System with Energy Saving and Normal Tire Using Spur Gear for an Omni-directional Mobile Robot,” Proc. of Int. Conf. on Robotics and Automation, pp. 3763-3768, 2010. 15. [15] Y. Maeda, T. Hara, and T. Arai, “Human-robot cooperative manipulation with motion estimation,” Proc. 2001 IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, Vol.4, pp. 2240-2245, 16. [16] Y. Ueno, “Analysis of Velocity Pattern at the Power-Assist Operation of a Mobile Robot Without Discomfort Operation,” The Joint Int. Conf. of ISCIIA&ITCA2018, No.3A2-3-4, 2018.	{"url":"https://www.fujipress.jp/jaciii/jc/jacii002300060990/","timestamp":"2024-11-07T15:06:54Z","content_type":"text/html","content_length":"48126","record_id":"<urn:uuid:eb2b687d-1c9c-470a-beb2-ec2de971160e>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00409.warc.gz"}
What is Angular Acceleration in Physics? Angular acceleration in Physics refers to the rate at which an object’s angular velocity changes with respect to time. Angular velocity represents the object’s rotational speed and direction. When an object experiences angular acceleration, it means its rotational speed is changing, either by increasing or decreasing. What is Angular Acceleration in Physics Angular Acceleration Formula The formula to calculate angular acceleration is given by: α = (Δω) / (Δt) • α represents angular acceleration, • Δω denotes the change in angular velocity, and • Δt represents the change in time. Angular acceleration is a fundamental concept in physics and engineering that relates to the rate at which an object’s angular velocity changes over time. It plays a crucial role in understanding rotational motion, whether it’s the movement of celestial bodies, machinery components, or sports equipment. In this article, we will delve into the concept of angular acceleration, explore its applications, and learn how to calculate it. Units of Angular Acceleration Angular acceleration is measured in units of radians per second squared (rad/s²) or degrees per second squared (°/s²). These units convey the change in angular velocity per unit time. Understanding Angular Acceleration To grasp angular acceleration better, let’s consider a spinning top. Initially, the top is at rest, but as you apply a force, it starts rotating. The initial angular velocity is zero, and as the top gains rotational speed, the angular velocity increases. This change in angular velocity over time is angular acceleration. Angular Acceleration vs. Linear Acceleration Angular acceleration should not be confused with linear acceleration. Linear acceleration relates to the change in linear velocity, while angular acceleration focuses on the change in angular velocity. While linear acceleration deals with straight-line motion, angular acceleration is specific to rotational motion. Calculating Angular Acceleration To calculate angular acceleration, you need to determine the change in angular velocity and the time taken for that change. Let’s say an object initially rotates at an angular velocity ω₁ and after a certain time Δt, its angular velocity becomes ω₂. The angular acceleration (α) can be calculated using the formula: α = (ω₂ – ω₁) / Δt Factors Affecting Angular Acceleration Several factors influence angular acceleration, including the applied torque or force, the moment of inertia, and the distribution of mass within the rotating object. Torque is responsible for changing an object’s angular momentum and, consequently, its angular acceleration. Angular acceleration finds applications in various fields. These fields include physics, engineering, and sports. Some notable applications include: 1. Physics: Angular acceleration helps explain the motion of planets, the rotation of satellites, and the behaviour of objects in space. 2. Engineering: Understanding angular acceleration is important in designing machinery, such as engines, turbines, and rotating components. 3. Sports: Angular acceleration plays a role in sports like gymnastics, figure skating, and diving, where rotational movements are key to performance. Examples of Angular Acceleration 1. A car moving along a curved road experiences angular acceleration as it changes its direction. When the driver steers the wheel to the left or right, the car’s tires exert a torque, resulting in a change in angular velocity and acceleration. 2. A spinning ice skater performing a pirouette demonstrates angular acceleration. As the skater pulls their arms closer to their body, their moment of inertia decreases, causing an increase in angular velocity and acceleration. Angular Acceleration in Rotational Motion In rotational motion, angular acceleration plays a vital role. It describes how quickly an object’s rotational speed changes and provides insights into the dynamics of rotating systems. Understanding angular acceleration enables us to analyze the behaviour of objects such as wheels, propellers, and gears. Angular Acceleration in Physics Angular acceleration is a very important concept in rotational dynamics. It helps explain the principles behind rotational motion, including torque, angular momentum, and moment of inertia. By understanding angular acceleration, physicists can accurately predict and describe the behaviour of rotating systems. Angular Acceleration in Engineering Engineers heavily rely on angular acceleration in various fields. For instance, in the design of engines, turbines, and flywheels, understanding angular acceleration is crucial to ensure the smooth operation and efficiency of rotating components. Engineers also consider angular acceleration when designing machinery that involves gears, pulleys, and rotating parts. Angular Acceleration in Sports Sports that involve rotational movements, such as gymnastics, figure skating, and diving, rely on angular acceleration. Athletes utilize angular acceleration to perform complex manoeuvres, spins, and flips. By manipulating their body position and distribution of mass, they can increase or decrease their rotational speed to achieve desired movements and maximize their performance. Angular acceleration is a fundamental concept that describes the rate at which an object’s angular velocity changes over time. It plays a crucial role in understanding rotational motion in various fields, including physics, engineering, and sports. By grasping the concept of angular acceleration and its applications, we can gain valuable insights into the behaviour of rotating objects and Frequently Asked Questions 1. Is angular acceleration the same as angular velocity? No, angular acceleration and angular velocity are different concepts. Angular acceleration represents the rate of change of angular velocity, while angular velocity refers to the rotational speed and direction of an object. 2. Can angular acceleration be negative? Yes, angular acceleration can be positive or negative. A positive angular acceleration indicates an increase in rotational speed, while a negative angular acceleration signifies a decrease in rotational speed. 3. How is angular acceleration related to torque? Torque is responsible for changing an object’s angular momentum, which in turn affects its angular acceleration. A larger torque results in a larger angular acceleration, while a smaller torque leads to a smaller angular acceleration. 4. Can angular acceleration affect linear motion? Angular acceleration and linear motion are distinct concepts. While angular acceleration relates to changes in rotational speed, linear motion deals with changes in straight-line velocity. However, in some cases, angular acceleration can indirectly affect linear motion through the influence of rotational forces. 5. What are some real-life examples of angular acceleration? Real-life examples of angular acceleration include the spinning of a merry-go-round, the rotation of a bicycle wheel, and the swinging of a pendulum. These scenarios involve objects experiencing changes in their rotational speed over time. You may also like to read:	{"url":"https://physicscalculations.com/what-is-angular-acceleration-in-physics/","timestamp":"2024-11-08T09:35:10Z","content_type":"text/html","content_length":"137595","record_id":"<urn:uuid:cfb53e9a-7d5b-4676-836e-f177770e6805>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00519.warc.gz"}
Full Support for Zero-Centered Storage (!113) · Merge requests · pycodegen / lbmpy · GitLab Full Support for Zero-Centered Storage This merge request introduces several changes into the heart of lbmpy to finally allow unrestricted usage of zero-centered PDF storage and to decouple storage format from compressibility. This requires several major incisions: Conserved Quantity Computation The DensityVelocityComputation class is updated. It shall now take compressible and zero_centered as separate arguments and handle populations and macroscopic quantities accordingly. The x_order_moment_symbol[s] properties are deprecated in favor of more specific symbol getters. A dedicated density_deviation_symbol is introduced to clearly separate density \rho, background density \ rho_0 and density fluctuation \delta \rho, s.t. \rho = \rho_0 + \delta\rho in all situations. Depending on zero-centering, either \rho or \delta\rho are computed from PDFs, and the other one is inferred accordingly. Equilibrium Formulations So far, lbmpy separates formulations of its hydrodynamic equilibria into continuous vs. discrete and compressible vs. incompressible. The full-PDF storage format is implicitly assumed for the compressible case, while zero-centered storage is implicitly assumed in the incompressible case. This has been known to lead to a lot of confusion. In particular, the formation of the incompressible equilibrium moment values is quite hacky and not at all obvious. With this MR, the hydrodynamic equilibrium shall be fully encapsulated in a dedicated class, which hides all technicalities of computing equilibrium moments and derivation of the equilibrium PDF (continuous or discrete). It will fully handle compressibility and zero-centering, and all information about the equilibrium will be held in one place. Once an instance of an equilibrium is created, it shall be immutable. The hydrodynamic equilbria, both continuous and discrete, are derived from a common base class AbstractEquilibrium, which provides a common interface, as well as caching functionality for the computation of moments. It can be extended by custom subclasses for describing custom equilibrium distributions. To date, instances of the *Method classes have only held a set of moments with associated equilibrium values. In the future, however, the method instances shall hold an equilibrium class instance instead of moment equilibrium values. It still manages its own moments and relaxation rate, but equilibrium values of moments are derived on demand, and only through the equilibrium object's Full vs. Delta-Equilibrium The equation governing PDF storage is \vec{f} = \vec{w} + \delta\vec{f}, where \vec{w} are the lattice weights, and \delta\vec{f} are the fluctuations, which are the values to be stored in the zero-centered case. Equivalently, both compressible and incompressible equilibria can be expressed both in their absolute form or only by their deviation (delta) from the rest state Let \Psi be the continuous Maxwellian equilibrium. Like for the PDFs, the reference state, called background distribution, is \Psi (\rho_0, 0). Depending on compressibility and full or delta format, we obtain four different equilibria: compressible incompressible full \Psi(\rho, \vec{u}) \Psi(\rho_0, \vec{u}) + \Psi(\delta\rho, 0) delta \Psi(\rho, \vec{u}) - \Psi(\rho_0, 0) \Psi(\rho_0, \vec{u}) - \Psi(\rho_0, 0) + \Psi(\delta\rho, 0) Regularly stored PDFs may be relaxed immediately to the full equilibria, and zero-centered PDFs can be immediately relaxed to the delta equilibria. To relax zero-centered PDFs against the full equilibria, the constant background part must first be added and subtracted again after the collision. This might be necessary because especially central moments of the delta-equilibria become more complicated, introducing velocity dependencies that degrade the CM method's numerical properties. Furthermore, it will definitely be necessary for cumulant LBMs, as cumulants of the delta-equilibrium are potentially undefined. Analogously, in theory, you could relax the full PDF vector against the delta-equilibrium, but there isn't really a point to that, hence it is not supported. Moment Transform Classes Depending on zero-centering and choice of equilibrium (delta or full), the transformation of PDFs to moments and back might have to add in the constant background part. The transform classes are adapted accordingly. All of this substructure shall be invisible to the everyday user, while at the same time providing a cleaner and easier to work with an ecosystem for the power user. All possible configurations for equilibrium are encapsulated in three arguments to the creation functions: • compressible, default False • zero_centered, default True, • delta_equilibrium, default None, inferred according to the chosen method. Supersedes !97 (closed). Depends on pystencils!285 (merged) and pystencils!286 (merged). Merge request reports	{"url":"https://i10git.cs.fau.de/pycodegen/lbmpy/-/merge_requests/113/pipelines","timestamp":"2024-11-06T20:53:56Z","content_type":"text/html","content_length":"86528","record_id":"<urn:uuid:8ba41d93-71d7-4af8-a4a8-62814d9d23ff>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00711.warc.gz"}
Gradation of the significance level of trends in precipitation over China How to accurately detect and estimate the significance level of trends in hydroclimate time series is a challenge. Building on correlation analysis, we propose an approach for evaluating and grading the significance level of trend in a series, and apply it to evaluate the changes in annual precipitation in China. The approach involved first formulating the relationship between the correlation coefficient and trend's slope. Four correlation coefficient thresholds are then determined by considering the influence of significance levels and data length, and the significance of trends is graded as five levels: no, weak, moderate, strong and dramatic. A larger correlation coefficient reflects a larger slope of trend and its higher significance level. Results of Monte-Carlo experiments indicated that the correlation coefficient-based approach not only reflects the magnitude of a trend, but also considers the influence of dispersion degree and mean value of the original series. Compared with the Mann–Kendall test used commonly, the proposed approach gave more accurate and specific gradation of the significance level of trends in annual precipitation over China. We find that the precipitation trends over China are not uniform, and the effects of global climate change on precipitation are not strong and limited to some regions. Hydroclimatic variability in many basins and regions worldwide is changing significantly because of global climate change (Allen & Ingram 2002; Elliott et al. 2014; Trenberth et al. 2014). The detection and attribution of hydroclimatic variability is of great socioeconomic importance (Diffenbaugh et al. 2008; IPCC 2013), but considerable methodological challenges remain. The trend is one of the important indicators of hydroclimatic variability (Hamed 2008; Carmona et al. 2014; Rice et al. 2015), and the identification of a trend is the simplest and the most frequent way of detecting hydroclimatic variability (Yue et al. 2002). Various methods have been used for identifying trends in hydrological studies (Adam & Lettenmaier 2008; Ishak et al. 2013; Kirchner & Neal 2013; Sonali & Kumar 2013; Anghileri et al. 2014; Sang et al. 2014; Lopes et al. 2016; etc.). They can be classified into four types: data-fitting, time domain-based, frequency domain-based, and time-frequency domain-based test (Sang et al. 2013). Each type has advantages and disadvantages, which can affect the accuracy with which the trends are identified and our understanding of hydroclimate variability. It is important to accurately quantify the statistical significance of a trend (Yue et al. 2002), and many methods have been developed for it (Sayemuzzaman & Jha 2014; Gjoneska et al. 2015). The Mann–Kendall (MK) non-parametric test is a widely used method, and has been successfully applied in past studies on the impacts of climate change (Kendall 1975; Burn & Hag Elnur 2002; Kisi & Ay 2014 ). However, each study uses a single significance level as a threshold for trend identification, which makes the significance evaluation by the MK test dependent on the chosen significance level. Besides, the slope of a trend cannot be directly estimated from the MK test. Furthermore, the detection of a trend not only depends on the magnitude of trend and the pre-assigned confidence level, but also on the probability distribution, sample size, dispersion degree of the time series (Yue et al. 2002; Adamowski et al. 2009; Shao & Li 2011; Hossein et al. 2012). These factors complicate the identification and assessment of the significance level of a trend. These issues can be overcome by a gradation of the significance level of a trend, taking into account the other factors that influence the significance of the trend. The slope of a time series represents the significance level of its trend, but the slope can theoretically range from the negative infinity to positive infinity, making it unsuitable for the gradation of significance levels. The correlation coefficient (CC) quantifies the linear relationship between two variables, thus it can function as an effective index for the gradation of a trend's significance level. A higher CC between a hydrological variable and its time order indicates a stronger significance level of its trend. The CC has values in the range of −1 to 1 and is mathematically related to the confidence level of a trend (Troch et al. 2013; McCuen 2016). Here, we develop a new method for the gradation of the significance of a trend based on the correlation coefficient, and demonstrate its use by investigating the trends in annual precipitation in China. In the following section, we derive the relationship between the correlation coefficient and the slope of a trend, and describe our new method for the gradation of significance level of trend, and test its reliability through Monte-Carlo experiments. The annual precipitation data used in this study is described in the next section. We then apply the method to investigate trends in annual precipitation in China. Relationship between the correlation coefficient and the slope of trend We use linear regression to calculate the monotonic trend in a hydroclimate time series, following past studies (Sonali & Kumar 2013). The slope of a trend can directly reflect its significance level, but it cannot be used to grade the trend's significance. We therefore develop a correlation analysis-based approach for grading the significance level of a trend, and begin by deriving the relationship between the correlation coefficient and trend's slope. Following stochastic hydrology ( Marco et al. 2012 ), a hydroclimate time series with periodicities removed can be simply described as:where is a constant, and is time order with the total number of ; is a stationary random variable, with the mean value of zero and a constant variance, and its covariance for ; is the slope of a trend and can be estimated as:where and are the mean values of the series and its time order , respectively. Then, the correlation coefficient is used to describe the linear relationship between the series and its time order McCuen 2016 By comparing Equation ( ) with Equation ( ), the relationship can be expressed as:where is the standard deviation of the time order Rewriting Equation ( ) by adding the constant to the random variable we get:where has the mean value of , and the linear trend component is independent from . The standard deviation of series can be expressed as:where is determined by the length of series :and is determined by the mean and coefficient of variation of series We substitute Equations ( ) into Equation ( ), and get a new equation of Equation (9) describes the r ∼ b relationship. For a hydrological time series x[t], the statistical parameters of and of its random component are constant. From Equations (2)–(4) we know that when the correlation coefficient r = 0, the slope b = 0, indicating that there is no trend in the series x[t]. Following Equation (9) we know that when , and n are first determined, the absolute values of r and b have a positive relationship. Therefore, the correlation coefficient can be used to grade the significance level of trends in a hydroclimate time series. Correlation coefficient-based approach for the gradation of trend After formulating the r ∼ b relationship in Equation (9), we need to determine the thresholds of correlation coefficient r for the gradation of trend's significance at appropriate confidence levels. For the statistical hypothesis test, different confidence levels are used, and each confidence level has a corresponding correlation coefficient r (Lehmann & D'Abrera 2010; Murphy et al. 2014). The higher the confidence level, the stricter the statistical test of the significance level of a trend. In practice, confidence levels of 95 or 99% are often chosen for hydroclimate time series analysis, and the corresponding value of r is denoted as and respectively. The values of and depend only on the data length. We use and as thresholds for the gradation of trend's significance level. For hydroclimate time series analysis, the data length should be at least 20 sampling points for robust trend analysis. For a series with a length of 20 or more, the critical (absolute) value of r, based on the F test, is smaller than 0.6 for a confidence level of 99% or lower (Table 1). For example, for a length of 20, r equals 0.56 at 99% confidence level (Corder & Foreman 2014). Therefore, we use 0.6 as the third threshold for the gradation of trends. In hydrological correlation analysis the r is usually required to be larger than 0.8 to ensure its statistical significance at different situations. For example, the correlation coefficient value for a time series with the length of 10 must be as high as 0.77 at 99% confidence level. Thus, we use 0.8 as the fourth gradation threshold. Table 1 Degree of freedom . Confidence level α . Degree of freedom . Confidence level α . n-m-1 . 90% . 95% . 98% . 99% . n-m-1 . 90% . 95% . 98% . 99% . 8 0.549 0.632 0.716 0.765 20 0.360 0.423 0.492 0.537 9 0.521 0.692 0.685 0.735 25 0.323 0.381 0.445 0.487 10 0.497 0.576 0.658 0.708 30 0.296 0.349 0.409 0.449 11 0.476 0.553 0.634 0.684 35 0.275 0.325 0.381 0.418 12 0.458 0.532 0.612 0.661 40 0.257 0.304 0.358 0.393 13 0.441 0.514 0.592 0.641 45 0.244 0.288 0.338 0.372 14 0.426 0.497 0.574 0.623 50 0.231 0.273 0.322 0.354 15 0.412 0.482 0.558 0.606 60 0.211 0.250 0.295 0.325 16 0.400 0.468 0.543 0.590 70 0.195 0.232 0.274 0.302 17 0.389 0.456 0.529 0.575 80 0.183 0.217 0.257 0.283 18 0.378 0.444 0.516 0.561 90 0.173 0.205 0.242 0.267 19 0.369 0.433 0.503 0.544 100 0.164 0.195 0.230 0.254 Degree of freedom . Confidence level α . Degree of freedom . Confidence level α . n-m-1 . 90% . 95% . 98% . 99% . n-m-1 . 90% . 95% . 98% . 99% . 8 0.549 0.632 0.716 0.765 20 0.360 0.423 0.492 0.537 9 0.521 0.692 0.685 0.735 25 0.323 0.381 0.445 0.487 10 0.497 0.576 0.658 0.708 30 0.296 0.349 0.409 0.449 11 0.476 0.553 0.634 0.684 35 0.275 0.325 0.381 0.418 12 0.458 0.532 0.612 0.661 40 0.257 0.304 0.358 0.393 13 0.441 0.514 0.592 0.641 45 0.244 0.288 0.338 0.372 14 0.426 0.497 0.574 0.623 50 0.231 0.273 0.322 0.354 15 0.412 0.482 0.558 0.606 60 0.211 0.250 0.295 0.325 16 0.400 0.468 0.543 0.590 70 0.195 0.232 0.274 0.302 17 0.389 0.456 0.529 0.575 80 0.183 0.217 0.257 0.283 18 0.378 0.444 0.516 0.561 90 0.173 0.205 0.242 0.267 19 0.369 0.433 0.503 0.544 100 0.164 0.195 0.230 0.254 ^an represents the data length, and m (m = 1 for the study) represents the unknown members of dimension. Our proposed approach for assessing and grading the significance level of a trend using the four thresholds identified above is as follows: 1. For a time series x[t], calculate the correlation coefficient r between and its time order t using Equation (3). 2. Choose two confidence levels and , and determine the corresponding correlation coefficient thresholds (denoted as and ) for the given data length. Use 0.6 and 0.8 as the two other thresholds. 3. Compare the absolute value of r (\|r\|) in step (1) with the four thresholds selected in step (2). 4. If , the trend is insignificant at the confidence level . We denote this as ‘no trend’. 5. If , then the trend is significant at level but insignificant at level . We denote this as ‘weak trend’. 6. If , then the trend is significant at the level but may not be significant at a higher confidence level. We denote this as ‘moderate trend’. 7. If , then the trend is significant in most but not all situations, and we denote this as ‘strong trend’. 8. If , then the trend is significant in all situations, and we denote this as ‘dramatic trend’. Following the above steps, the significance level of trend in a hydroclimate time series can be graded into five ranks (Table 2), and ten ranks if the negative and positive trends are separated. Table 2 Correlation coefficient . Significance level . Correlation coefficient . Significance level . No trend Strong trend Weak trend Dramatic trend Moderate trend Correlation coefficient . Significance level . Correlation coefficient . Significance level . No trend Strong trend Weak trend Dramatic trend Moderate trend Verification of the proposed approach To verify the reliability of the proposed approach and further investigate the influence of some factors on the gradation of significance level of trend, we have designed the following Monte-Carlo 1. We generate 30 random time series that follow the Pearson-III probabilistic distribution, which is used commonly for hydrological analysis and design in China. Each time series has a same length (n) = 100, mean value , variation coefficient , and skewness coefficient . We denote each of this time series as u[j], j = 1, 2, … , 30. 2. To each series u[j] we add a different trend component bt (b = 0.5, 1, 1.5, … , 15), and the new time series is denoted as x[j], j = 1, 2, … , 30. 3. We use Equation (3) to calculate the correlation coefficient r between the series x[j] and its time order t. 4. We repeat the above steps 1,000 times (i.e. i = 1, 2, … , 1,000) to ensure the stability of the result r[ij]; Because the true slope b of trend in each synthetic series is known, the true value of the correlation coefficient (denoted as r[1]) can be calculated by Equation (9). The correlation coefficient calculated by Equation (10) (denoted as r[2]) is then verified against r[1]. We find that for all 30 groups, the relative error between and is smaller than 2.87%, and smaller than 1% for 28 of the groups (Table 3). This reflects the high accuracy of the correlation coefficient calculated by Equation (9), and the r ∼ b relationship (Equation (9)) is reliable. Table 3 . . . (%) . b . . . (%) . 0.5 0.072 0.072 0.139 8 0.756 0.758 0.331 1 0.143 0.139 2.869 8.5 0.775 0.778 0.400 1.5 0.212 0.212 0.378 9 0.792 0.793 0.063 2 0.277 0.277 0.252 9.5 0.808 0.811 0.421 2.5 0.339 0.338 0.471 10 0.822 0.824 0.292 3 0.397 0.403 1.309 10.5 0.835 0.836 0.096 3.5 0.451 0.455 0.976 11 0.846 0.849 0.343 4 0.500 0.501 0.240 11.5 0.857 0.860 0.374 4.5 0.545 0.548 0.551 12 0.866 0.867 0.092 5 0.585 0.585 0.000 12.5 0.875 0.877 0.229 5.5 0.622 0.624 0.402 13 0.883 0.885 0.261 6 0.655 0.656 0.260 13.5 0.890 0.892 0.247 6.5 0.684 0.687 0.380 14 0.896 0.898 0.234 7 0.711 0.714 0.521 14.5 0.902 0.904 0.133 7.5 0.735 0.738 0.449 15 0.908 0.909 0.154 . . . (%) . b . . . (%) . 0.5 0.072 0.072 0.139 8 0.756 0.758 0.331 1 0.143 0.139 2.869 8.5 0.775 0.778 0.400 1.5 0.212 0.212 0.378 9 0.792 0.793 0.063 2 0.277 0.277 0.252 9.5 0.808 0.811 0.421 2.5 0.339 0.338 0.471 10 0.822 0.824 0.292 3 0.397 0.403 1.309 10.5 0.835 0.836 0.096 3.5 0.451 0.455 0.976 11 0.846 0.849 0.343 4 0.500 0.501 0.240 11.5 0.857 0.860 0.374 4.5 0.545 0.548 0.551 12 0.866 0.867 0.092 5 0.585 0.585 0.000 12.5 0.875 0.877 0.229 5.5 0.622 0.624 0.402 13 0.883 0.885 0.261 6 0.655 0.656 0.260 13.5 0.890 0.892 0.247 6.5 0.684 0.687 0.380 14 0.896 0.898 0.234 7 0.711 0.714 0.521 14.5 0.902 0.904 0.133 7.5 0.735 0.738 0.449 15 0.908 0.909 0.154 ^a and represent the correlation coefficient calculated by Equations (9) and (10) respectively. To illustrate how the significance level of a trend is determined, we first compute the slopes for our synthetic series that correspond to the four thresholds for the correlation coefficient (Equation (9)). For confidence levels of 95 and 99%, the correlation coefficient thresholds are 0.197 and 0.257, respectively. The slopes corresponding to the five ranks of the significance levels of the correlation coefficient are shown in Table 4. We see a monotonic relationship between the slope and the correlation coefficients. In Figure 1, we show the significance levels of trends for five series with slopes, b, of 0, 1.5, 4.0, 6.0, 10.0. According to Equation (3), the correlation coefficient r of the five series is 0, 0.212, 0.501, 0.655, and 0.824, respectively. The five series fall into the five different grades of significance (Table 4). Figure 1 shows that the slopes of the trends in the five series increase with r, indicating the applicability of Equation (9). Table 4 b . r . Significance level of trend . [0.000, 1.392) [0.000, 0.197) No trend [1.392, 1.843) [0.197, 0.257) Weak trend [1.843, 5.196) [0.257, 0.600) Moderate trend [5.196, 9.238) [0.600, 0.800) Strong trend [9.238, +∞) [0.800, 1.000) Dramatic trend b . r . Significance level of trend . [0.000, 1.392) [0.000, 0.197) No trend [1.392, 1.843) [0.197, 0.257) Weak trend [1.843, 5.196) [0.257, 0.600) Moderate trend [5.196, 9.238) [0.600, 0.800) Strong trend [9.238, +∞) [0.800, 1.000) Dramatic trend Influence of other factors on the gradation of trend's significance level From Equation (9) we know that the gradation of the significance level of a trend (i.e. the magnitude of r), depends not only on the slope b of the trend, but also on the mean and variation coefficient of the random component of the time series. We design two sets of Monte-Carlo (MC) experiments to investigate how these two factors influence the r ∼ b relationship. For the first set of MC experiments, we generate a synthetic series with length n of 100 and mean value of 1,000, and vary to investigate its influence on the r ∼ b relationship. Figure 2(a) shows that for any , r increases with b, but the increase is slower when b is larger. Furthermore, the r ∼ b curve becomes flatter at higher value. This shows that the dispersion degree of a series has a strong influence on the significance level of its trend. For two series with the same trend but different dispersion degrees, the series with a smaller dispersion degree will have a more obvious trend with a higher significance level and is easily detectable. In the second set of MC experiments, the length n of synthetic series is 100 and the value of is fixed at 0.2, and is varied to investigate its influence on the r ∼ b relationship (Figure 2(b)). r increases with b for all values, but the r ∼ b relationship is weaker for a larger . Thus, the mean magnitude of a series also has a strong influence on the significance level of its trend. For two series with the same trend but different mean values, for example, of 200 and 1,000, the series with a smaller mean value will have a more significant trend. The results in Figure 2 show the influence of the mean and variation coefficient of a series on the significance of its trend. These two factors reflect the different ratio between the trend and the random component of a time series. For a time series with a smaller mean value and a smaller dispersion degree, this ratio is higher and the trend will have a higher significance level. On the other hand, for a time series with a larger mean value and a large dispersion degree, the significance level of its trend is weaker. To clarify this further, we use the signal-to-noise (SNR) index to quantify the influence of the two factors ( Herrick 2014 ) on the magnitude of the correlation coefficient. The SNR index is defined as the ratio between the variance of a trend and its random component:By substituting Equation ( ) into Equation ( ), the ∼ SNR relationship can be described as: Equation (11) shows that if the random component ( and ) of a series is larger, the SNR values are smaller, and the trend would be difficult to identify. Therefore, r and SNR have a positive relationship as shown in Equation (12) and are consistent with Figure 2. The trend in a series with a smaller dispersion degree and mean value would be more easily identified. This, the correlation coefficient r reflects not only the magnitude of a trend, but also considers the influence of the dispersion degree and the mean value of the time series. Therefore, the correlation coefficient-based approach developed in this study is effective for estimating and grading the significance level of trends in hydrological time series. In this study, 520 meteorological stations (Figure 3) were chosen for investigating the trends in annual precipitation over China. The data were obtained from the China Meteorological Data Sharing Service System (http://cdc.cma.gov.cn/). These stations were chosen by considering the length, consistency and completeness of data records. They are approximately uniformly distributed over China, with somewhat fewer stations in the southwest region. All of the stations have measurements from 1961 to 2013, with no missing values. Precipitation is an important variable for understanding the variability and changes in hydroclimatic systems (Brunetti et al. 2006; Ashouri et al. 2014; Trenberth et al. 2014). There have been many studies on the precipitation variability over China and at regional scales (Zhai et al. 1999; Wang et al. 2004; Ma et al. 2008; Ye 2014; Zhang et al. 2016), but their conclusions and interpretations differ. Some studies indicate that the precipitation in many regions, especially in northwest China, fluctuated considerably and show significant trends over recent decades due to the influence of global climate change (Chen et al. 2013; Sang et al. 2013a, 2013b; Wan et al. 2015; Gu et al. 2017; Yang et al. 2017), while other studies indicate that precipitation in many regions has kept its stochastic characteristics and has not changed significantly (Gao et al. 2012; Sun et al. 2012). Thus, the significance level of trends in precipitation over China remains unclear. An analysis of the spatiotemporal variability of precipitation over China is important for water resources management and many other water activities. We used our correlation coefficient-based approach to investigate the significance level of trends in the annual precipitation time series measured at 520 stations over China. The confidence levels of 95 and 99% were used for the precipitation series analysis, and the upper and lower limits of correlation coefficient for each rank were calculated. For comparison purposes, trends of all these annual precipitation series were also identified by the MK test. Results in Figure 3 (left) indicate that among the 520 precipitation series, the trends of only 60 series (11.5%) are significant, and the other 460 series do not show any obvious trends, that is, their trends are graded as ‘no trend’. Surprisingly, none of the trends are strong or dramatic. The upward trends of precipitation with a moderate significance level are seen in 21 stations in the northwest corner of China and in the northeast boundary of the Tibet Plateau, and trends with a weak significance level at 12 stations in those regions. The downward trends of precipitation have moderate significance levels at eight stations in the Yunnan-Guizhou Plateau (100 °–111 °E, 22 °–30 °N) in southwest China and two stations in the centre of north China. Also, in the Yunnan-Guizhou Plateau and its surrounding regions, precipitation series at nine stations have downward trends at a weak significance level. The upward trends of precipitation with a weak significance level are detected at four stations in southeast coastal areas of China. In comparison, the results by the MK test in Figure 3 (right) show that precipitation has downward trends in the mid-arid and mid-humid regions from the northeast to southwest China. In northwest China, including the Tibet Plateau and southeast China, precipitation has a mainly upward trend. At the significance at 95% confidence level, the thresholds of ±1.96 are used to distinguish the statistical characters of the MK test, with a whole value range of −4.01 to 4.72. Results show that trends of 64 precipitation series (12.3% of the total series) are identified as significant by the MK test, but the other 456 series indicate no obvious trends. Figure 3 indicates that the spatial distribution of the significance level of trends in precipitation series obtained from the proposed approach and the MK test are similar. Global climate change has likely led to the strengthening westerlies but the weakening Indian summer monsoon over recent decades (Wu 2005; Thompson et al. 2006), which would influence the precipitation variability over China, especially over the western regions. The increase in precipitation in northwest China can be due to the strengthening westerlies; the precipitation decrease on the Yunnan-Guizhou Plateau and its surrounding regions can be caused by the weakening Indian summer monsoon (Sang et al. 2016). Moreover, the number of precipitation series with significant trends detected by the two methods (60 in our method and 64 in the MK test) are similar. The indices used to quantify the significance of trends in the two methods also indicate a positive relationship (Figure 4). The similarity of the results of our proposed approach with the MK test, which has been successfully applied for trend identification in the past, demonstrates the reliability of our approach for the significance evaluation of trends. Moreover, the MK test can judge only whether the trend is significant or not at a certain confidence level, but our approach can also be used for the gradation of the significance level of trends (Figure 4). In the MK test, any value greater than 1.96 indicates a significant upward trend, but there is no distinction based on the degree of significance. The relationship between the slope of a trend and the test statistic (Z) in the MK test is also unknown. However, accurate gradation of the significance level of trends is urgently needed in practical analysis, and the approach proposed in this study meets the purpose. We use our approach to grade the significance of the trends and understand the degree of significance of the trends in the annual precipitation data at each station. In addition, we computed the values of SNR and r of each precipitation series, and show their scatter diagram in Figure 5. As expected, all 520 points fall on the standard curve in Equation (12). The absolute value of \|r\| increases with SNR, but the increase rate becomes slower at larger SNR values. Those precipitation series with larger SNR values have higher significance level of trends. From the above analysis, we conclude that although global climate change has a major influence on hydroclimate variability worldwide, its influence on the precipitation over China during 1961–2013 is not as strong as one might expect. In most regions in China, precipitation has changed over long timescales, but the change is insignificant. The strengthening westerlies cause a precipitation increase in northwest China, and the weakening Indian summer monsoon causes a precipitation decrease on the Yunnan-Guizhou plateau, but the observed trends are weak or moderate. There is no strong or dramatic trend in precipitation in China during the recent five decades. Thus, in contrast to other studies which provide only an approximation of the significance level of trends in precipitation ( Gao et al. 2012; Sun et al. 2012), we conclude that precipitation over China does not show obvious trends, although global climate change causes some precipitation changes in local regions. How to accurately detect and estimate the significance level of trends is a challenge for understanding hydroclimatic variability and assessing its potential impacts. In this paper, we use the correlation coefficient and develop an approach for the gradation of the significance level of trends in a hydroclimate time series. We first derive the relationship between the correlation coefficient and the slope of trend. Then, by determining four correlation coefficient thresholds, we propose a gradation of the significance of trends of time series into five levels: no trend, weak trend, moderate trend, strong trend, and dramatic trend. A larger correlation coefficient value implies a larger slope of trend and a higher significance level. The results of Monte-Carlo experiments indicate that the mean value and dispersion degree of a series have a strong influence on the calculation of the significance level of trends. The correlation coefficient-based approach of ours not only reflects the magnitude of a trend, but also considers the influence of dispersion degree and mean value of the original series. Therefore, it is an effective approach for estimating and grading the significance level of trends in a hydrological time series. Compared with other widely used methods, the main advantage of our method is that it provides a method for the gradation of significance level of trends, and also quantifies the influences of statistical characteristics of the original series. More research is needed to further verify the applicability of this method by considering many other hydroclimatic variables and non-linear trends. We analyzed the changes in precipitation over China over five recent decades, and find that the significance of trends and their spatial distribution calculated with our approach are similar to the MK test. However, compared to the MK test, our approach provides a gradation of the significance level of trends. We found that although global climate change has a great influence on the hydroclimate variability worldwide, its influence on precipitation over China is not strong. None of the 520 meteorological stations analyzed showed a strong or dramatic trend in precipitation. The precipitation trends over China are not uniform, and the effects of global climate change on precipitation are limited to some regions. Other deterministic characteristics, such as periodicities and step changes, need to be further studied to determine whether precipitation over China mainly shows stochastic characteristics or not. The authors gratefully acknowledged the valuable comments and suggestions given by the editors and the anonymous reviewers. The authors thank Viral Shah for language editing. This study was financially supported by the National Natural Science Foundation of China (No. 91547205, 91647110, 51579181), the Program for the ‘Bingwei’ Excellent Talents from the Institute of Geographic Sciences and Natural Resources Research, CAS, the Youth Innovation Promotion Association CAS (No. 2017074), and the National Mountain Flood Disaster Investigation Project (SHZH-IWHR-57).	{"url":"https://iwaponline.com/hr/article/49/6/1890/40804/Gradation-of-the-significance-level-of-trends-in","timestamp":"2024-11-05T12:15:51Z","content_type":"text/html","content_length":"456531","record_id":"<urn:uuid:aa58bd1c-bf3c-4f49-b7d4-9006cf26b3fe>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00468.warc.gz"}
Key Terms analysis of variance also referred to as ANOVA: a method of testing whether the means of three or more populations are equal The method is applicable if □ all populations of interest are normally distributed, □ the populations have equal standard deviations, and □ samples (not necessarily of the same size) are randomly and independently selected from each population. The test statistic for analysis of variance is the F ratio. one-way ANOVA a method of testing whether the means of three or more populations are equal; the method is applicable if □ all populations of interest are normally distributed, □ the populations have equal standard deviations, □ samples (not necessarily of the same size) are randomly and independently selected from each population, and □ there is one independent variable and one dependent variable. The test statistic for analysis of variance is the F ratio. mean of the squared deviations from the mean; the square of the standard deviation For a set of data, a deviation can be represented as x – $x ¯ x ¯$ where x is a value of the data and $x ¯ x ¯$ is the sample mean. The sample variance is equal to the sum of the squares of the deviations divided by the difference of the sample size and 1.	{"url":"https://texasgateway.org/resource/key-terms-34?book=79081&binder_id=78276","timestamp":"2024-11-11T07:47:15Z","content_type":"text/html","content_length":"37394","record_id":"<urn:uuid:1c40b855-328e-4f63-a97a-4e68d308bdc4>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00171.warc.gz"}
Gabriele Iommazzo The distance geometry problem asks to find a realization of a given simple edge-weighted graph in a Euclidean space of given dimension K, where the edges are realized as straight segments of lengths equal (or as close as possible) to the edge weights. The problem is often modelled as a mathematical programming formulation involving decision … Read more	{"url":"https://optimization-online.org/author/giommazz/","timestamp":"2024-11-12T06:14:25Z","content_type":"text/html","content_length":"83144","record_id":"<urn:uuid:35d14db2-0890-4ca0-b6f5-5e3c0e9ebaa3>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00425.warc.gz"}
GAMGI: Screenshots Nine MC simulations of atom deposition on a substrate, each one with 44,000 atoms, to study the best conditions to achieve thin, well defined tracks, corresponding to a total of 396,000 atoms, all represented here as solid spheres. Each system is in a different, transparent layer, with different observer coordinates, so the atom coordinates are unchanged. To get the proper positioning, each system was initially simulated by an orthorhombic cell with the same size, replaced in the end by the system, editing directly the file. Maura E. Monville, Oak Ridge National Laboratory. Size: 80,265 Voronoi tesselation, with periodic boundary conditions, of 10,000 atoms, randomly positioned (on the left), and randomly positioned with a minimum distance between them (4.05% of the cell width), corresponding to a hard-sphere model (on the right). The Voronoi polyhedra obtained are more smooth and regular for the restricted random (Laguerre) than for the purely random (Poisson) distribution. A. Ferro, N. Reis and C. Pereira, Technical University of Lisboa. Size: 65,495 bytes. Orthorhombic cell containing 50,000 Cu atoms, forming a random close packing (RCP) structure, with a density (occupied volume / total volume) of 0.627. The structure was obtained using the Jodrey algorithm with periodic boundary conditions. Increasing the relaxation time, the density increases slightly (in the range 0.62-0.64, as found experimentaly in the random packing of large numbers of equal spheres). Size: 171,181 bytes. This multi-layer nanostructure was built linking 38x26x5 cP lattice cells (a = 1.0) with substrate and adatoms (radius = 0.5). By sucessively changing the cell origin, the GAMGI occupancy rules can be used to create arbitrary, well defined, blocks of atoms, forming structures such as the one seen in the picture (see detailed procedure in the ). Size: 78,354 bytes.	{"url":"http://atom.ist.utl.pt/screenshots/screenshots12.html","timestamp":"2024-11-07T07:23:37Z","content_type":"application/xhtml+xml","content_length":"4453","record_id":"<urn:uuid:d913fdcf-fad7-454b-87fd-5bdac6e1a70c>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00503.warc.gz"}
Optimal designs of postive definite kernels for scattered data approximation 1. 科研成果 » Optimal designs of postive definite kernels for scattered data approximation 发布时间: 2017-11-02 15:24:26 In the research, we study the optimal designs of the positive definite kernels for the high-dimensional interpolation. We endow the Sobolev spaces with the probability measures induced by the positive definite kernels such that the kernel-based estimators can be solved to maximize the kernel-based probabilities conditioned on the observed data. In the practical implementations, we have many choices of the positive definite kernels to construct the kernel basis such as the Gaussian kernels with various shape parameters; hence we have an open problem what the optimal kernels are. The kernel-based probabilities provide a novel way to search the optimal kernels for the observed data. Combining with the statistical techniques such as the maximum likelihood estimation, we can solve the optimal shape parameters of the Gaussian kernels by the kernel-based probabilities even though the classical kernel-based methods cannot achieve the uncertain data. 1、Qi Ye. Optimal designs of positive definite kernels for scattered data approximation. Applied and Computational Harmonic Analysis, 41(1), 214-236, 2016. DOI: 10.1016/j.acha.2015.08.009	{"url":"http://mlopt.scnu.edu.cn/a/20171102/83.html","timestamp":"2024-11-12T11:47:41Z","content_type":"text/html","content_length":"5850","record_id":"<urn:uuid:2a606082-1203-46ba-bc89-e86611212eab>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00591.warc.gz"}
Real Gases and Van der Waals Equation \| Curious Toons Table of Contents Welcome to the fascinating world of physics! Imagine a universe where every throw of a ball, flick of a switch, or ripple of water tells a story about the fundamental laws that govern our existence. This year, we’re not just memorizing formulas; we’ll unravel the mysteries of the cosmos together! Ever wondered why the sky is blue or how roller coasters make us feel like we’re flying? These questions will guide our exploration. Get ready to dive into topics like motion, energy, and forces—the very building blocks of reality. We’ll experiment with real-world phenomena, challenge our assumptions, and debunk some common myths along the way. Physics isn’t just about equations; it’s about understanding the world and our place in it. By the end of our journey, you’ll not only be able to predict the path of a projectile but also appreciate the intricate dance of atoms that make up everything around us. So, buckle up! This is going to be a thrilling ride through time and space, where you will discover that physics is not just a subject—it’s a gateway to understanding the universe! Are you ready to unlock the secrets of the cosmos? Let’s get started! 1. Introduction to Real Gases 1.1 Difference Between Ideal and Real Gases Ideal gases and real gases differ significantly in their behavior, particularly under varying conditions of pressure and temperature. Ideal gases are hypothetical substances that perfectly follow the Ideal Gas Law (PV=nRT), where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature. They are characterized by assumptions like negligible molecular volume and no intermolecular forces. In contrast, real gases exhibit behaviors that deviate from these ideal conditions due to factors such as finite molecular volume and interactions between molecules. Under high temperatures and low pressures, real gases can approximate ideal behavior. However, at high pressures and low temperatures, real gases condense into liquids, resulting in significant deviations. The Van der Waals equation accounts for these non-ideal behaviors by introducing two corrections: one for molecular volume (b) and another for intermolecular attractions (a). Property Ideal Gas Real Gas Molecular Volume Negligible Significant Intermolecular Forces None Present Behavior Follows PV=nRT perfectly Deviates under certain conditions Understanding these differences is crucial for accurate predictions in various scientific applications. 1.2 Importance of Studying Real Gases Studying real gases is crucial for understanding the behavior of gases under various conditions, which deviates from the ideal gas law. Real gases exhibit interactions between particles and occupy volume, making them behave unexpectedly at high pressures and low temperatures. By exploring the properties of real gases, we can better comprehend phenomena such as phase transitions, critical points, and gas condensing. This knowledge is essential for numerous applications, including chemical engineering, meteorology, and environmental science. For instance, the Van der Waals equation, which modifies the ideal gas law, accounts for the volume occupied by gas particles and the attractive forces between them. This leads to more accurate predictions of gas behavior in real-world scenarios, such as the behavior of gases in industrial processes and natural systems. Understanding these concepts enables students to grasp the complexities of thermodynamics and advancements in technology, promoting better decision-making in scientific research and practical applications. In summary, studying real gases enhances our understanding of physical laws governing matter and fuels innovation across various scientific fields. Property Ideal Gas Real Gas Volume of particles Negligible Finite Interparticle forces No interaction Attractive/Repulsive Behavior under pressure Predictable Deviates significantly 2. Ideal Gas Law 2.1 Assumptions of Ideal Gas Behavior The Ideal Gas Law describes the behavior of gases under certain assumptions that simplify their interactions. These assumptions include the following: 1. Point Particles: Gas molecules are treated as point particles with negligible volume. This means the actual size of the molecules is insignificant compared to the distances between them. 2. No Intermolecular Forces: It is assumed that there are no attractive or repulsive forces between the gas molecules, except during collisions. This allows molecules to move freely and independently under ideal conditions. 3. Perfect Elastic Collisions: When gas molecules collide with each other or with the walls of their container, these collisions are perfectly elastic, meaning no kinetic energy is lost in the 4. Random Motion: Gas molecules are in constant, random motion, leading to a uniform distribution of pressure and temperature throughout the gas. 5. High Temperature and Low Pressure: Ideal gas behavior holds best at high temperature and low pressure, where gas molecules have enough energy to overcome any intermolecular forces. These assumptions serve as the foundation for the Ideal Gas Law, represented by the equation ( PV = nRT ), where ( P ) is pressure, ( V ) is volume, ( n ) is the number of moles, ( R ) is the gas constant, and ( T ) is temperature. 2.2 Limitations of the Ideal Gas Law The Ideal Gas Law, represented by the equation PV = nRT, assumes that gas particles are point masses with no volume and that they experience no intermolecular forces. While this model works well under many conditions, it has significant limitations. First, it fails to accurately describe the behavior of gases at high pressures, where the volume of gas particles becomes non-negligible. At high pressures, the assumption that the particles occupy no space is invalid, resulting in an overestimation of pressure. Second, the law misrepresents gas behavior at low temperatures, where intermolecular forces, such as van der Waals forces, become significant. Under these conditions, attractive forces lead to deviations from expected pressure readings, leading to condensation instead of gas behavior. The Ideal Gas Law is also less applicable to polar gases compared to non-polar gases since polar molecules exhibit stronger intermolecular attractions. Thus, while the Ideal Gas Law provides a useful approximation for many gases under standard conditions, real gases often deviate from this relational simplicity, necessitating the use of equations like the Van der Waals equation for a more accurate description. Condition Ideal Gas Law Accuracy High Pressure Poor Low Temperature Poor Polar vs Non-Polar Variable 3. Van der Waals Equation 3.1 Derivation of the Van der Waals Equation The Van der Waals equation modifies the ideal gas law to account for real gas behavior by introducing two parameters: ( a ) (attractive forces) and ( b ) (volume occupied by gas particles). The ideal gas law is expressed as ( PV = nRT ). However, this equation assumes no intermolecular forces and that gas particles occupy no volume. To derive the Van der Waals equation, we start by considering these factors. We can correct for intermolecular forces by subtracting the term ( \frac{an^2}{V^2} ) from the pressure ( P ) to account for the attractive forces between particles: P + \frac{an^2}{V^2}(V – nb) = nRT Here, ( nb ) represents the volume excluded due to the finite size of the gas particles. Rearranging gives us: (P + \frac{an^2}{V^2})(V – nb) = nRT Finally, expanding and simplifying leads to the Van der Waals equation: \left(P + \frac{a}{Vm^2}\right)(Vm – b) = RT where ( V_m = \frac{V}{n} ) is the molar volume. This equation accurately describes the behavior of real gases under various conditions, highlighting the deviations from ideality. 3.2 Parameters a and b in the Equation The Van der Waals equation modifies the ideal gas law to account for the behavior of real gases, introducing the parameters ( a ) and ( b ). The parameter ( a ) represents the attractive forces between gas molecules. It quantifies how much these forces reduce the pressure compared to ideal gas behavior, as real molecules tend to pull each other closer together. A higher ( a ) value indicates stronger intermolecular attractions, and gases with significant molecular interactions, like hydrogen or carbon dioxide, will have larger ( a ) values. On the other hand, the parameter ( b ) represents the volume occupied by gas molecules themselves. This is often referred to as the “excluded volume.” It quantifies the volume that cannot be occupied by the gas due to the physical presence of its particles. The higher the ( b ) value, the more significant the volume occupied by the gas molecules, as seen in larger or more complex molecules like propane compared to helium. To summarize, the Van der Waals equation can be written as: \left( P + \frac{a}{Vm^2} \right)(Vm – b) = RT Where ( P ) is pressure, ( V_m ) is molar volume, ( R ) is the gas constant, and ( T ) is temperature. Understanding parameters ( a ) and ( b ) helps describe how real gases deviate from ideal 4. Behavior of Real Gases 4.1 Deviation from Ideal Gas Behavior Real gases deviate from ideal gas behavior due to intermolecular forces and the volume occupied by gas molecules themselves. While the Ideal Gas Law ((PV = nRT)) assumes that gas particles are point-like and that there are no attractive or repulsive forces between them, real gases exhibit interactions that can significantly affect their pressure, volume, and temperature under certain conditions. At high pressures, the volume of the gas molecules cannot be neglected, leading to a smaller volume available for movement compared to the ideal case. At low temperatures, attractive forces between particles become more pronounced, causing gases to condense and exert lower pressures than predicted by the ideal equation. This behavior is modeled by the Van der Waals equation, which modifies the Ideal Gas Law to account for these deviations: [P + a\frac{n^2}{V^2}](V – nb) = nRT Here, (a) represents the magnitude of attractive forces, and (b) accounts for the finite volume of the gas molecules. The Van der Waals parameters vary among different gases, illustrating the extent of their deviation from ideality. Understanding these aspects is crucial for practical applications in thermodynamics and real-world gas behavior analysis. 4.2 Critical Point and Phase Transition The critical point is a unique state of a substance where the properties of its gas and liquid phases become indistinguishable. At this point, defined by a specific temperature and pressure known as the critical temperature (Tc) and critical pressure (Pc), the substance reaches its critical state. Beyond this point, there is no phase transition between liquid and gas, resulting in a supercritical fluid that possesses properties of both phases. For instance, supercritical fluids can dissolve substances like liquids but flow like gases. Phase transitions occur when a substance changes from one state of matter to another, such as from solid to liquid (melting) or liquid to gas (vaporization). These transitions are characterized by latent heat, which is the heat absorbed or released during the process without a change in temperature. Understanding these concepts is crucial in fields such as material science and thermodynamics, where the behavior of substances under varying pressures and temperatures is essential for applications ranging from refrigeration to drug delivery systems. State Transition Phase Change Solid to Liquid Melting Liquid to Gas Vaporization Gas to Liquid Condensation Liquid to Solid Freezing Gas to Solid Sublimation This overview emphasizes the significance of the critical point in the study of real gases and their behavior under different conditions. 5. Applications of the Van der Waals Equation 5.1 Real Gas Behavior in Different Conditions Real gases deviate from ideal gas behavior under certain conditions, primarily at high pressures and low temperatures. The Van der Waals equation modifies the Ideal Gas Law to account for the volume occupied by gas molecules (the “b” term) and the attractive forces between them (the “a” term). At high pressures, the volume occupied by gas molecules becomes significant, leading to a decrease in the pressure exerted by the gas. Conversely, at low temperatures, gases tend to condense as intermolecular attractions increase, causing further deviations from ideal behavior. The following table illustrates typical behavior of real gases under varying conditions: Condition Behavior Explanation High pressure Pressure decreases Volume occupied by molecules is significant. Low temperature Liquefaction Increased intermolecular forces cause condensation. Moderate conditions Approximates ideal behavior Sufficient distance and random motion mitigate interactions. Understanding these deviations is crucial for applications like gas storage and real-life thermodynamic calculations, emphasizing the importance of the Van der Waals equation in accurately describing gas behavior in non-ideal conditions. 5.2 Applications in Science and Industry The Van der Waals equation, which modifies the ideal gas law to account for the finite size of particles and intermolecular attractions, has numerous applications in both science and industry. In the field of chemistry, it assists in predicting the behavior of real gases under various conditions, enabling researchers to design experiments that consider non-ideal interactions. For instance, in the petrochemical industry, understanding the properties of gases helps optimize processes such as natural gas extraction and refining, where accurate gas behavior predictions ensure safety and In the realm of environmental science, the Van der Waals equation is critical for modeling pollutant dispersion in the atmosphere, allowing for better predictions of air quality and climate change impacts. Additionally, the pharmaceutical industry relies on this equation for the development of gases used in inhalers, ensuring the right dosage and delivery of medications. Overall, the Van der Waals equation serves as an essential tool in research and development, providing insights that help drive innovation across various disciplines. Application Area Example Chemistry Predicting gas behavior Petrochemical Industry Optimizing extraction processes Environmental Science Modeling air pollution Pharmaceutical Industry Designing inhalable medications As we wrap up our journey through the fascinating world of physics, I want to take a moment to reflect on what we’ve learned together. Physics isn’t just about equations or laws; it’s the lens through which we understand the universe. From the forces that keep our feet on the ground to the waves that allow us to communicate across distances, every concept we’ve explored is a thread in the vast tapestry of reality. Remember, each experiment we conducted and each problem we solved was more than just a task; it was a step toward unraveling the mysteries that govern our lives. We’ve observed how the smallest particles dance in quantum mechanics and how celestial bodies move in vast, majestic orbits. As you leave this classroom, carry with you the curiosity and critical thinking that physics fosters. Let the beauty of the natural world inspire you, and don’t hesitate to question everything. You are now armed with knowledge, and with that comes great responsibility: to think deeply, to innovate, and to contribute to a better understanding of our universe. Keep exploring, keep questioning, and remember—physics is everywhere. Your journey has just begun!	{"url":"https://curioustoons.in/real-gases-and-van-der-waals-equation/","timestamp":"2024-11-09T20:02:26Z","content_type":"text/html","content_length":"113156","record_id":"<urn:uuid:7e76f392-7a73-43f9-bfd8-6f92399a7743>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00868.warc.gz"}
h2g2 - Boolean Algebra - Edited Entry Boolean Algebra Created \| Updated Jul 9, 2013 Boolean Algebra \| Truth Tables \| Expressions from Truth Tables Universal Functions \| Function Reduction \| Functions Classified Invented by George Boole, Boolean Algebra deals with situations where there are only two possibilities. These possibilities can be named anything as long as they are opposites. For instance, true and false, yes and no, or 0 and 1^1. These situations arrive frequently in digital systems such as computers, and thus Boolean Algebra has become vital to the day-to-day operations of the entire civilized world, whether those living there realise it or not. Boolean Algebra is also used by people as a decision making process, though you probably don't think of it as that. Overview of Boolean Operations There are several Boolean operations: The Boolean AND operator returns true if all operands are true, and false if any are false. AND is equivalent to binary multiplication. • 0 AND 0 = 0 • 1 AND 0 = 0 • 1 AND 1 = 1 Boolean OR is true if any operands are true, and false only if all operands are false. OR is equivalent to binary addition, with the both true case resulting in true, instead of in false with a true carried over. • 0 OR 0 = 0 • 1 OR 0 = 1 • 1 OR 1 = 1 NOT is a unary operator, meaning it acts on a single Boolean value instead of on two. Operators which act on two values are Binary operators, which may seem confusing. NOT changes a Boolean value to its opposite. NOR is defined as NOT OR, and its action is just what it suggests: it returns true if the operands are both false, that is, if neither operand 1 NOR operand 2 are true, and returns false if either are true. • 0 NOR 0 = 1 • 1 NOR 0 = 0 • 1 NOR 1 = 0 Equivalent to A NOR B are NOT (A OR B) and (NOT A) AND (NOT B). These are useful in systems where an explicit NOR is not available. NAND means NOT AND, and returns false if both operands are true, and true if any are false. • 0 NAND 0 = 1 • 1 NAND 0 = 1 • 1 NAND 1 = 0 A NAND B is the same as NOT (A AND B) and (NOT A) OR (NOT B). As with NOR, these are useful when a specific NAND is not available, but as will be discussed later, this is rarely the case. XOR is short for Exclusive-OR, and is like a merger of AND and NOR. XOR returns true, essentially, if the two operands are different. That is, if one or the other is true, it returns true, but not if both are true. It is OR with an Exclusion; thus, Exclusive OR. • 0 XOR 0 = 0 • 1 XOR 0 = 1 • 1 XOR 1 = 0 XOR is like binary addition, with the carried bit ignored. XNOR means NOT XOR, and it returns true if both operands are the same, and false otherwise. (You may have guessed this from the name by now.) • 0 XNOR 0 = 1 • 1 XNOR 0 = 0 • 1 XNOR 1 = 1 Using the words for the operations is excessively verbose, so shorter notation is available. OR is +, AND is a dot, NOT is a bar over a value, NOR is the OR expression with a bar over the whole thing, NAND is the AND expression with a similar bar, and XOR and XNOR are like OR and NOR, but with the + in a small circle. Since computer programming languages also use Boolean logic for many operations, they have their own notation. The symbols used by C and C++ are &(AND), \|(OR), ~(NOT)^2, and ^(XOR). Since these characters are easiest to type, they will be used here. Boolean Tricks NAND is the most useful operator, because any other operation can be done with some number of NANDs. While not always useful, in small circuitry, a single NAND IC can perform the functions of a number of other ICs, thus saving in cost and size. For this section only, lower case 'n' will be the operator for NAND, so it is perfectly clear where its use is. • NOT A = AnA • A AND B = ~(AnB) = (AnB)n(AnB)^3 • A OR B = (~A)n(~B) = (AnA)n(BnB) • A NOR B = (AnB)n(AnB) • A XOR B = (An(AnB))n(Bn(AnB)) • A XNOR B = (An(AnB))n(Bn(AnB)) XOR is a useful operator because (A^B)^B = A. This can be used on long strings of bits to obscure them to anyone without the correct key. This is decidedly weak encryption, but it is a clever trick which has other applications in computer programming. ^1Technically the opposite of zero is non-zero. In single digit systems, 1 is the only non-zero value, but when multiple binary digits are taken as a single value, the whole is considered non-zero if any value is one, true, or whatever.^2These are the symbols for bitwise operations, which act on each bit respectively, instead of on the sum of the bits. Logical operators acting on all the bits are: &&(AND), \|\|(OR), !(NOT), and ^^(XOR). These use 0 and non-0 as their logical values, instead of 0 and 1.^3You only need two leads in the IC for this, since you can connect the output for the AnB operation to both inputs of another NAND gate on the chip. A similar trick can be used in any case where two identical Boolean operations are performed at about the same time.	{"url":"https://h2g2.com/edited_entry/A412642","timestamp":"2024-11-12T09:42:51Z","content_type":"text/html","content_length":"29966","record_id":"<urn:uuid:6c629b01-0931-4c71-b53b-bf549c26c399>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00828.warc.gz"}
Enum ExcelFunction Enumeration of Worksheet functions in Excel. Assembly: Syncfusion.XlsIO.Base.dll public enum ExcelFunction Name Description ABS Represents the ABS function. ABSREF Represents the ABSREF function. ACCRINT Represents the ACCRINT function. ACCRINTM Represents the ACCRINTM function. ACOS Represents the ACOS function. ACOSH Represents the ACOSH function. ACOT Represents the ACOT function. ACOTH Represents the ACOTH function. ACTIVECELL Represents the ACTIVECELL function. ADDBAR Represents the ADDBAR function. ADDCOMMAND Represents the ADDCOMMAND function. ADDMENU Represents the ADDMENU function. ADDRESS Represents the ADDRESS function. ADDTOOLBAR Represents the ADDTOOLBAR function. AGGREGATE Represents the AGGREGATE function. AMORDEGRC Represents the AMORDEGRC function. AMORLINC Represents the AMORLINC function. ANCHORARRAY Represents the ANCHORARRAY function. AND Represents the AND operation. APPTITLE Represents the APPTITLE function. ARABIC Represents the ARABIC function. AREAS Represents the Areas function. ARGUMENT Represents the ARGUMENT function. ARRAYTOTEXT Represents the ARRAYTOTEXT function. ASC Represents the ASC function. ASIN Represents the ASIN function. ASINH Represents the ASINH function. ATAN Represents the ATAN function. ATAN2 Represents the ATAN2 function. ATANH Represents the ATANH function. AVEDEV Represents the AVEDEV function. AVERAGE Represents the AVERAGE function. AVERAGEA Represents the AVERAGEA function. AVERAGEIF Represents the AVERAGEIF function. AVERAGEIFS Represents the AVERAGEIFS function. BAHTTEXT Represents the BAHTTEXT function. BASE Represents the BASE function. BESSELI Represents the BESSELI function. BESSELJ Represents the BESSELJ function. BESSELK Represents the BESSELK function. BESSELY Represents the BESSELY function. BETA_DIST Represents the BETA.DIST function. BETA_INV Represents the BETA.INV function. BETADIST Represents the BETADIST function. BETAINV Represents the BETAINV function. BIN2DEC Represents the BIN2DEC function. BIN2HEX Represents the BIN2HEX function. BIN2OCT Represents the BIN2OCT function. BINOM_DIST Represents the BINOM.DIST function. BINOM_DIST_RANGE Represents the BINOM.DIST.RANGE function. BINOM_INV Represents the BINOM.INV function. BINOMDIST Represents the BINOMDIST function. BITAND Represents the BITAND function. BITLSHIFT Represents the BITLSHIFT function. BITOR Represents the BITOR function. BITRSHIFT Represents the BITXOR function. BITXOR Represents the BITXOR function. CALL Represents the CALL function. CALLER Represents the CALLER function. CANCELKEY Represents the CANCELKEY function. CEILING Represents the CEILING function. CEILING_MATH Represents the CEILING.MATH function. CEILING_PRECISE Represents the CEILING.PRECISE function. CELL Represents the CELL function. CHAR Represents the CHAR function. CHECKCOMMAND Represents the CHECKCOMMAND function. CHIDIST Represents the CHIDIST function. CHIINV Represents the CHIINV function. CHISQ_DIST Represents the CHISQ.DIST function. CHISQ_DIST_RT Represents the CHISQ.DIST.RT function. CHISQ_INV Represents the CHISQ.INV function. CHISQ_INV_RT Represents the CHISQ.INV.RT function. CHISQ_TEST Represents the CHISQ.TEST function. CHITEST Represents the CHITEST function. CHOOSE Represents the CHOOSE function. CLEAN Represents the CLEAN function. CODE Represents the CODE function. COLUMN Represents the COLUMN function. COLUMNS Represents the COLUMNS function. COMBIN Represents the COMBIN function. COMBINA Represents the COMBINA function. COMPLEX Represents the COMPLEX function. CONCAT Represents the CONCAT function. CONCATENATE Represents the CONCATENATE function. CONFIDENCE Represents the CONFIDENCE function. CONFIDENCE_NORM Represents the CONFIDENCE.NORM function. CONFIDENCE_T Represents the CONFIDENCE.T function. CONVERT Represents the CONVERT function. CORREL Represents the CORREL function. COS Represents the COS function. COSH Represents the COSH function. COT Represents the COT function. COTH Represents the COTH function. COUNT Represents the COUNT function. COUNTA Represents the COUNTA function. COUNTBLANK Represents the COUNTBLANK function. COUNTIF Represents the COUNTIF function. COUNTIFS Represents the COUNTIF function. COUPDAYBS Represents the COUPDAYBS function. COUPDAYS Represents the COUPDAYS function. COUPDAYSNC Represents the COUPDAYSNC function. COUPNCD Represents the COUPNCD function. COUPNUM Represents the COUPNUM function. COUPPCD Represents the COUPPCD function. COVAR Represents the COVAR function. COVARIANCE_P Represents the COVARIANCE.P function. COVARIANCE_S Represents the COVARIANCE.S function. CREATEOBJECT Represents the CREATEOBJECT function. CRITBINOM Represents the CRITBINOM function. CSC Represents the CSC function. CSCH Represents the CSCH function. CUBEKPIMEMBER Represents the CUBEKPIMEMBER function. CUBEMEMBER Represents the CUBEMEMBER function. CUBEMEMBERPROPERTY Represents the CUBEMEMBERPROPERTY function. CUBERANKEDMEMBER Represents the CUBERANKEDMEMBER function. CUBESET Represents the CUBESET function. CUBESETCOUNT Represents the CUBESETCOUNT function. CUBEVALUE Represents the CUBEVALUE function. CUMIPMT Represents the CUMIPMT function. CUMPRINC Represents the CUMPRINC function. CustomFunction Represents the Custom function. CUSTOMREPEAT Represents the CUSTOMREPEAT function. CUSTOMUNDO Represents the CUSTOMUNDO function. DATE Represents the DATE function. DATEDIF Represents the DATEDIF function. DATESTRING Represents the DATESTRING function. DATEVALUE Represents the DATEVALUE function. DAVERAGE Represents the DAVERAGE function. DAY Represents the DAY function. DAYS Represents the DAYS function. DAYS360 Represents the DAYS360 function. DB Represents the DB function. DBCS Represents the DBCS function. DCOUNT Represents the DCOUNT function. DCOUNTA Represents the DCOUNTA function. DDB Represents the DDB function. DEC2BIN Represents the DEC2BIN function. DEC2HEX Represents the DEC2HEX function. DEC2OCT Represents the DEC2OCT function. DECIMAL Represents the DECIMAL function. DEGREES Represents the DEGREES function. DELETEBAR Represents the DELETEBAR function. DELETECOMMAND Represents the DELETECOMMAND function. DELETEMENU Represents the DELETEMENU function. DELETETOOLBAR Represents the DELETETOOLBAR function. DELTA Represents the DELTA function. DEREF Represents the DEREF function. DEVSQ Represents the DEVSQ function. DGET Represents the DGET function. DIALOGBOX Represents the DIALOGBOX function. DIRECTORY Represents the DIRECTORY function. DISC Represents the DISC function. DMAX Represents the DMAX function. DMIN Represents the DMIN function. DOCUMENTS Represents the DOCUMENTS function. DOLLAR Represents the DOLLAR function. DOLLARDE Represents the DOLLARDE function. DOLLARFR Represents the DOLLARFR function. DPRODUCT Represents the DPRODUCT function. DSTDEV Represents the DSTDEV function. DSTDEVP Represents the DSTDEVP function. DSUM Represents the DSUM function. DURATION Represents the DURATION function. DVAR Represents the DVAR function. DVARP Represents the DVARP function. ECHO Represents the ECHO function. EDATE Represents the EDATE function. EFFECT Represents the EFFECT function. ENABLECOMMAND Represents the ENABLECOMMAND function. ENABLETOOL Represents the ENABLETOOL function. ENCODEURL Represents the ENCODEURL function. EOMONTH Represents the EOMONTH function. ERF Represents the ERF function. ERF_PRECISE Represents the ERF.PRECISE function. ERFC Represents the ERFC function. ERFC_PRECISE Represents the ERFC.PRECISE function. ERROR Represents the ERROR function. ERRORTYPE Represents the ERRORTYPE function. EUROCONVERT Represents the EUROCONVERT function. EVALUATE Represents the EVALUATE function. EVEN Represents the EVEN function. EXACT Represents the EXACT function. EXEC Represents the EXEC function. EXECUTE Represents the EXECUTE function. EXP Represents the EXP function. EXPON_DIST Represents the EXPON.DIST function. EXPONDIST Represents the EXPONDIST function. F_DIST Represents the F.DIST function. F_DIST_RT Represents the F.DIST.RT function. F_INV Represents the F.INV function. F_INV_RT Represents the F.INV.RT function. F_TEST Represents the F.TEST function. FACT Represents the FACT function. FACTDOUBLE Represents the FACTDOUBLE function. FALSE Represents the FALSE function. FCLOSE Represents the FCLOSE function. FDIST Represents the FDIST function. FILES Represents the FILES function. FILTER Represents the FILTER function. FILTERXML Represents the FILTERXML function. FIND Represents the FIND function. FINDB Represents the FINDB function. FINV Represents the FINV function. FISHER Represents the FISHER function. FISHERINV Represents the FISHERINV function. FIXED Represents the FIXED function. FLOOR Represents the FLOOR function. FLOOR_MATH Represents the FLOOR.MATH function. FLOOR_PRECISE Represents the FLOOR.PRECISE function. FOPEN Represents the FOPEN function. FORECAST Represents the FORECAST function. FORECAST_ETS Represents the FORECAST.ETS function. FORECAST_ETS_CONFINT Represents the FORECAST.ETS.CONFINT function. FORECAST_ETS_SEASONALITY Represents the FORECAST.ETS.SEASONALITY function. FORECAST_ETS_STAT Represents the FORECAST.ETS.STAT function. FORECAST_LINEAR Represents the FORECAST.LINEAR function. FORMULACONVERT Represents the FORMULACONVERT function. FORMULATEXT Represents the FORMULATEXT function. FPOS Represents the FPOS function. FREAD Represents the FREAD function. FREADLN Represents the FREADLN function. FREQUENCY Represents the FREQUENCY function. FSIZE Represents the FSIZE function. FTEST Represents the FTEST function. FV Represents the FV function. FVSCHEDULE Represents the FVSCHEDULE function. FWRITE Represents the FWRITE function. FWRITELN Represents the FWRITELN function. GAMMA Represents the GAMMA function. GAMMA_DIST Represents the GAMMA.DIST function. GAMMA_INV Represents the GAMMA.INV function. GAMMADIST Represents the GAMMADIST function. GAMMAINV Represents the GAMMAINV function. GAMMALN Represents the GAMMALN function. GAMMALN_PRECISE Represents the GAMMALN.PRECISE function. GAUSS Represents the GAUSS function. GCD Represents the GCD function. GEOMEAN Represents the GEOMEAN function. GESTEP Represents teh GESTEP function. GETBAR Represents the GETBAR function. GETCELL Represents the GETCELL function. GETCHARTITEM Represents the GETCHARTITEM function. GETDEF Represents the GETDEF function. GETDOCUMENT Represents the GETDOCUMENT function. GETFORMULA Represents the GETFORMULA function. GETLINKINFO Represents the GETLINKINFO function. GETMOVIE Represents the GETMOVIE function. GETNAME Represents the GETNAME function. GETNOTE Represents the GETNOTE function. GETOBJECT Represents the GETOBJECT function. GETPIVOTDATA Represents the GETPIVOTDATA function. GETPIVOTFIELD Represents the GETPIVOTFIELD function. GETPIVOTITEM Represents the GETPIVOTITEM function. GETPIVOTTABLE Represents the GETPIVOTTABLE function. GETTOOL Represents the GETTOOL function. GETTOOLBAR Represents the GETTOOLBAR function. GETWINDOW Represents the GETWINDOW function. GETWORKBOOK Represents the GETWORKBOOK function. GETWORKSPACE Represents the GETWORKSPACE function. GOTO Represents the GOTO function. GROUP Represents the GROUP function. GROWTH Represents the GROWTH function. HALT Represents the HALT function. HARMEAN Represents the HARMEAN function. HELP Represents the HELP function. HEX2BIN Represents the HEX2BIN function. HEX2DEC Represents the HEX2DEC function. HEX2OCT Represents the HEX2OCT function. HLOOKUP Represents the HLOOKUP function. HOUR Represents the HOUR function. HYPERLINK Represents the HYPERLINK function. HYPGEOM_DIST Represents the HYPGEOM.DIST function. HYPGEOMDIST Represents the HYPGEOMDIST function. IF Represents the IF function. IFERROR Represents the IFERROR function. IFNA Represents the IFNA function. IFS Represents the IFS function. IMABS Represents the IMABS function IMAGINARY Represents the IMAGINARY function. IMARGUMENT Represents the IMARGUMENT function. IMCONJUGATE Represents the IMCONJUGATE function. IMCOS Represents the IMCOS function. IMCOSH Represents the IMCOSH function. IMCOT Represents the IMCOT function. IMCSC Represents the IMCSC function. IMCSCH Represents the IMCSCH function. IMDIV Represents the IMDIV function. IMEXP Represents the IMEXP function. IMLN Represents the IMLN function. IMLOG10 Represents the IMLOG10 function. IMLOG2 Represents the IMLOG2 function. IMPOWER Represents the IMPOWER function. IMPRODUCT Represents the IMPRODUCT function. IMREAL Represents the IMREAL function. IMSEC Represents the IMSEC function. IMSECH Represents the IMSECH function. IMSIN Represents the IMSIN function. IMSINH Represents the IMSINH function. IMSQRT Represents the IMSQRT function. IMSUB Represents the IMSUB function. IMSUM Represents the IMSUM function. IMTAN Represents the IMTAN function. INDEX Represents the INDEX function. INDIRECT Represents the INDIRECT function. INFO Represents the INFO function. INITIATE Represents the INITIATE function. INPUT Represents the INPUT function. INT Represents the INT function. INTERCEPT Represents the INTERCEPT function. INTRATE Represents the INTRATE function. IPMT Represents the IPMT function. IRR Represents the IRR function. ISBLANK Represents the ISBLANK function. ISERR Represents the ISERR function. ISERROR Represents the ISERROR function. ISEVEN Represents the ISEVEN function. ISFORMULA Represents the ISFORMULA function. ISLOGICAL Represents the ISLOGICAL function. ISNA Represents the ISNA function. ISNONTEXT Represents the ISNONTEXT function. ISNUMBER Represents the ISNUMBER function. ISO_CEILING Represents the ISO.CEILING function. ISODD Represents the ISODD function. ISOWEEKNUM Represents the ISOWEEKNUM function. ISPMT Represents the ISPMT function. ISREF Represents the ISREF function. ISTEXT Represents the ISTEXT function. JIS Represents the JIS function. KURT Represents the KURT function. LARGE Represents the LARGE function. LASTERROR Represents the LASTERROR function. LCM Represents the LCM function. LEFT Represents the LEFT function. LEFTB Represents the LEFTB function. LEN Represents the LEN function. LENB LENB function. LET Represents the LET function. LINEST Represents the LINEST function. LINKS Represents the LINKS function. LINTEST Represents the LINTEST function. LN Represents the LN function. LOG Represents the LOG function. LOG10 Represents the LOG10 function. LOGEST Represents the LOGEST function. LOGINV Represents the LOGINV function. LOGNORM_DIST Represents the LOGNORM.DIST function. LOGNORM_INV Represents the LOGNORM.INV function. LOGNORMDIST Represents the LOGNORMDIST function. LOOKUP Represents the LOOKUP function. LOWER Represents the LOWER function. MATCH Represents the MATCH function. MAX Represents the MAX function. MAXA Represents the MAXA function. MAXIFS Represents the MAXIFS function. MDETERM Represents the MDETERM function. MDURATION Represents the MDURATION function. MEDIAN Represents the MEDIAN function. MID Represents the MID function. MIDB Represents the MIDB function. MIN Represents the MIN function. MINA Represents the MINA function. MINIFS Represents the MINIFS function. MINUTE Represents the MINUTE function. MINVERSE Represents the MINVERSE function. MIRR Represents the MIRR function. MMULT Represents the MMULT function. MOD Represents the MOD function. MODE Represents the MODE function. MODE_MULT Represents the MODE.MULT function. MODE_SNGL Represents the MODE.SNGL function. MONTH Represents the MONTH function. MOVIECOMMAND Represents the MOVIECOMMAND function. MROUND Represents the MROUND function. MULTINOMIAL Represents the MULTINOMIAL function. MUNIT Represents the MUNIT function. N Represents the N function. NA Represents the NA function. NAMES Represents the NAMES function. NEGBINOM_DIST Represents the NEGBINOM.DIST function. NEGBINOMDIST Represents the NEGBINOMDIST function. NETWORKDAYS Represents the NETWORKDAYS function. NETWORKDAYS_INTL Represents the NETWORKDAYS.INTL function. NOMINAL Represents the NOMINAL function. NONE Represents the NONE function. NORM_DIST Represents the NORM.DIST function. NORM_INV Represents the NORM.INV function. NORM_S_DIST Represents the NORM.S.DIST function. NORMDIST Represents the NORMDIST function. NORMINV Represents the NORMINV function. NORMSDIST Represents the NORMSDIST function. NORMSINV Represents the NORMSINV function. NOT Represents the NOT function. NOTE Represents the NOTE function. NOW Represents the NOW function. NPER Represents the NPER function. NPV Represents the NPV function. NUMBERSTRING Represents the NUMBERSTRING function. NUMBERVALUE Represents the NUMBERVALUE function. OCT2BIN Represents the OCT2BIN function. OCT2DEC Represents the OCT2DEC function. OCT2HEX Represents the OCT2HEX function. ODD Represents the ODD function. ODDFPRICE Represents the ODDFPRICE function. ODDFYIELD Represents the ODDFYEILD function. ODDLPRICE Represents the ODDLPRICE function. ODDLYIELD Represents the ODDLYEILD function. OFFSET Represents the OFFSET function. OPENDIALOG Represents the OPENDIALOG function. OPTIONSLISTSGET Represents the OPTIONSLISTSGET function. OR Represents the OR function. PAUSE Represents the PAUSE function. PDURATION Represents the PDURATION function. PEARSON Represents the PEARSON function. PERCENTILE Represents the PERCENTILE function. PERCENTILE_EXC Represents the PERCENTILE.EXC function. PERCENTILE_INC Represents the PERCENTILE.INC function. PERCENTRANK Represents the PERCENTRANK function. PERCENTRANK_EXC Represents the PERCENTRANK.EXC function. PERCENTRANK_INC Represents the PRECENTRANK.INC function. PERMUT Represents the PERMUT function. PERMUTATIONA Represents the PERMUTATIONA function. PHI Represents the PHI function. PHONETIC Represents the PHONETIC function. PI Represents the PI function. PIVOTADDDATA Represents the PIVOTADDDATA function. PMT Represents the PMT function. POISSON Represents the POISSON function. POISSON_DIST Represents the POISSON.DIST function. POKE Represents the POKE function. POWER Represents the POWER function. PPMT Represents the PPMT function. PRESSTOOL Represents the PRESSTOOL function. PRICE Represents the PRICE function. PRICEDISC Represents the PRICEDISC function. PRICEMAT Represents the PRICEMAT function. PROB Represents the PROB function. PRODUCT Represents the PRODUCT function. PROPER Represents the PROPER function. PV Represents the PV function. QUARTILE Represents the QUARTILE function. QUARTILE_EXC Represents the QUARTILE.EXC function. QUARTILE_INC Represents the QUARTILE.INC function. QUOTIENT Represents the QUOTIENT function. RADIANS Represents the RADIANS function. RAND Represents the RAND function. RANDBETWEEN Represents the RANDBETWEEN function. RANK Represents the RANK function. RANK_AVG Represents the RANK.AVG function. RANK_EQ Represents the RANK.EQ function. RATE Represents the RATE function. RECEIVED Represents the RECEIVED function. REFTEXT Represents the REFTEXT function. REGISTER Represents the REGISTER function. REGISTER_ID Represents the REGISTER.ID function. REGISTERID Represents the REGISTERID function. RELREF Represents the RELREF function. RENAMECOMMAND Represents the RENAMECOMMAND function. REPLACE Represents the REPLACE function. REPLACEB Represents the REPLACEB function. REPT Represents the REPT function. REQUEST Represents the REQUEST function. RESETTOOLBAR Represents the RESETTOOLBAR function. RESTART Represents the RESTART function. RESULT Represents the RESULT function. RESUME Represents the RESUME function. RIGHT Represents the RIGHT function. RIGHTB Represents the RIGHTB function. ROMAN Represents the ROMAN function. ROUND Represents the ROUND function. ROUNDDOWN Represents the ROUNDDOWN function. ROUNDUP Represents the ROUNDUP function. ROW Represents the ROW function. ROWS Represents the ROWS function. RRI Represents the RRI function. RSQ Represents the RSQ function. SAVEDIALOG Represents the SAVEDIALOG function. SAVETOOLBAR Represents the SAVETOOLBAR function. SCENARIOGET Represents the SCENARIOGET function. SEARCH Represents the SEARCH function. SEARCHB Represents the SEARCHB function. SEC Represents the SEC function. SECH Represents the SECH function. SECOND Represents the SECOND function. SELECTION Represents the SELECTION function. SERIES Represents the SERIES function. SERIESSUM Represents the SERIESSUM function. SETNAME Represents the SETNAME function. SETVALUE Represents the SETVALUE function. SHEET Represents the SHEET function. SHEETS Represents the SHEETS function. SHOWBAR Represents the SHOWBAR function. SIGN Represents the SIGN function. SIN Represents the SIN function. SINH Represents the SINH function. SKEW Represents the SKEW function. SKEW_P Represents the SKEW.P function. SLN Represents the SLN function. SLOPE Represents the SLOPE function. SMALL Represents the SMALL function. SPELLINGCHECK Represents the SPELLINGCHECK function. SQL_REQUEST Represents the SQL.REQUEST function. SQRT Represents the SQRT function. SQRTPI Represents the SQRTPI function. STANDARDIZE Represents the STANDARDIZE function. STDEV Represents the STDEV function. STDEV_P Represents the STDEV.P function. STDEV_S Represents the STDEV.S function. STDEVA Represents the STDEVA function. STDEVP Represents the STDEVP function. STDEVPA Represents the STDEVPA function. STEP Represents the STEP function. STEYX Represents the STEYX function. SUBSTITUTE Represents the SUBSTITUTE function. SUBTOTAL Represents the SUBTOTAL function. SUM Represents the SUM function. SUMIF Represents the SUMIF function. SUMIFS Represents the SUMIFS function. SUMPRODUCT Represents the SUMPRODUCT function. SUMSQ Represents the SUMSQ function. SUMX2MY2 Represents the SUMX2MY2 function. SUMX2PY2 Represents the SUMX2PY2 function. SUMXMY2 Represents the SUMXMY2 function. SWITCH Represents the SWITCH function. SYD Represents the SYD function. T Represents the T function. T_DIST Represents the T.DIST function. T_DIST_2T Represents the T.DIST.2T function. T_DIST_RT Represents the T.DIST.RT function. T_INV Represents the T.INV function. T_INV_2T Represents the T.INV.2T function. T_TEST Represents the T.TEST function. TAN Represents the TAN function. TANH Represents the TANH function. TBILLEQ Represents the TBILLEQ function. TBILLPRICE Represents the TBILLPRICE function. TBILLYIELD Represents the TBILLYIELD function. TDIST Represents the TDIST function. TERMINATE Represents the TERMINATE function. TEXT Represents the TEXT function. TEXTBOX Represents the TEXTBOX function. TEXTJOIN Represents the TEXTJOIN function. TEXTREF Represents the TEXTREF function. TIME Represents the TIME function. TIMEVALUE Represents the TIMEVALUE function. TINV Represents the TINV function. TODAY Represents the TODAY function. TRANSPOSE Represents the TRANSPOSE function. TREND Represents the TREND function. TRIM Represents the TRIM function. TRIMMEAN Represents the TRIMMEAN function. TRUE Represents the TRUE function. TRUNC Represents the TRUNC function. TTEST Represents the TTEST function. TYPE Represents the TYPE function. UNICHAR Represents the UNICHAR function. UNICODE Represents the UNICODE function. UNREGISTER Represents the UNREGISTER function. UPPER Represents the UPPER function. USDOLLAR Represents the USDOLLAR function. VALUE Represents the VALUE function. VALUETOTEXT Represents the VALUETOTEXT function. VAR Represents the VAR function. VAR_P Represents the VAR.P function. VAR_S Represents the VAR.S function. VARA Represents the VARA function. VARP Represents the VARP function. VARPA Represents the VARPA function. VDB Represents the VDB function. VLOOKUP Represents the VLOOKUP function. VOLATILE Represents the VOLATIL function. WEBSERVICE Represents the WEBSERVICE function. WEEKDAY Represents the WEEKDAY function. WEEKNUM Represents the WEEKNUM function. WEIBULL Represents the WEIBULL function. WEIBULL_DIST Represents the WEIBULL.DIST function. WINDOWS Represents the WINDOWS function. WINDOWTITLE Represents the WINDOWTITLE function. WORKDAY Represents the WORKDAY function. WORKDAY_INTL Represents the WORKDAY.INTL function. WORKDAYINTL Represents the WORKDAY.INTL function. XIRR Represents the XIRR function. XLOOKUP Represents the XLOOKUP function. XMATCH Represents the XMATCH function. XNPV Represents the XNPV function. XOR Represents the XOR function. YEAR Represents the YEAR function. YEARFRAC Represents the YEAR function. YIELD Represents the YIELD function. YIELDDISC Represents the YIELDDISC function. YIELDMAT Represents the YIELDMAT function. Z_TEST Represents the Z.TEST function. ZTEST Represents the ZTEST function.	{"url":"https://help.syncfusion.com/CR/aspnet/Syncfusion.XlsIO.ExcelFunction.html","timestamp":"2024-11-09T01:09:58Z","content_type":"text/html","content_length":"108561","record_id":"<urn:uuid:b862bb01-a5fe-4c22-a9bd-f8afb4b4c052>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00338.warc.gz"}
Ornstein-Uhlenbeck Process: Definition The Ornstein-Uhlenbeck Process (OU Process) is a differential equation used in physics to model the motion of a particle under friction. In financial probability, it models the spread of stocks. It’s also used to calculate interest rates and currency exchange rates. OU Process in Pairs Trading Pairs trading was developed at the end of the 1980s, after several crises in the economy hit the market in the previous decades. It’s a way to keep market exposure low while at the same time making a profit from trading. Pairs trading identifies two similar companies. The two companies should have a high correlation, cointegration, or both. Equity securities for both companies should be trading outside their normal historical range. You buy the undervalued security and short sell the overvalued security, betting that the investments will return to their historical norm. Once you have identified the two companies, you’ll want to generate a way to generate trading signals. One way to do this is with the Ornstein-Uhlenbeck Process. Ornstein-Uhlenbeck Process / OU Process. In physics, a force exerts on a particle to bring the particle back to the mean; a greater the distance from the mean results in more force. The same principle works for modeling spread between a pair of stocks, enabling you to identify when the stock is below the mean (buy) and when it is above the mean (sell). OU Process = dx[t] = θ(μ – x[t])dt + α dW[t] • x[t] = the particle’s current position. • θ = a mean reversion constant. • μ = the mean particle position • σ = a constant volatility • dW[t] = a Wiener process (Brownian motion). Calculation of the OU Process Calculating the OU process is quite complex. Ideally, you should be familiar with stochastic calculus, Brownian motion and differential equations. This NYU article covers the basics of what you should know. Considering there could be millions of dollars at stake, it’s highly unlikely you’ll want to calculate the OU process by hand. Instead, there are a multitude of software packages that will perform the calculations for you, including: • Matlab: Daniel Charlebois uploaded code to the Mathworks file exchange (found here) that can calculate the “Exact numerical solution and plots of the Ornstein-Uhlenbeck (OU) process and its time integral – calculation and plotting of the probability density function (pdf) of the OU process is also performed.” • R: Package ‘sde’ is for the simulation and inference of stochastic differential equations. You can find the package here. All of the above tools have multiple regression options built in. Least squares regression is probably your best bet for modeling the best fit of the data. For an example of how you would apply this to a set of data, check out Calibrating-the-Ornstein (originally from SITMO.com). It includes two methods (least squares and maximum likelihood). A Caution on Using an Unmodified Ornstein-Uhlenbeck-process It sounds easy to use, and (assuming you’re using software to do the calculations), it’s pretty simple to put into action. The problem is, if you’re using an unmodified OU process without a stop-loss, you could end up losing everything. The further the stock is from the mean, the more you risk and the bigger you trade. You could end up betting all of your capital and losing everything when the stock falls. It’s wise to include a stop-loss to prevent this from happening. D.S. Ehrmann. The Handbook of Pairs Trading. John Wiley & Sons, Hoboken, New Jersey, 2006. Comments? Need to post a correction? Please Contact Us.	{"url":"https://www.statisticshowto.com/what-is-the-ornstein-uhlenbeck-process/","timestamp":"2024-11-13T14:50:33Z","content_type":"text/html","content_length":"66026","record_id":"<urn:uuid:307e83a6-e3d4-4c48-9151-bd57b65d5e0a>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00882.warc.gz"}
Leverage and position limit Risk limits are a risk management mechanism used to limit a trader's position risk. In a volatile trading environment, a single trader holding a large position with high leverage can result in significant losses. The system uses the concept of dynamic leverage, i.e. the maximum leverage available for trading will vary depending on the value of the position held by the trader: the greater the value of the position held, the lower the maximum leverage available. At the same time, the larger the leverage selected, the smaller the open position. BTCUSDT Contract Leverage The nominal value of the maximum available position (USDT) 101X-125X 300,000 76X-100X 450,000 51X-75X 2,000,000 31X-50X 3,500,000 21X-30X 20,000,000 11X-20X 30,000,000 6X-10X 40,000,000 5X 100,000,000 4X 200,000,000 3X 400,000,000 0X-2X 99,999,999,999 ETHUSDT Contract Leverage The nominal value of the maximum available position (USDT) 76X-100X 150,000 51X-75X 300,000 26X-50X 400,000 11X-20X 2,000,000 6X-10X 4,000,000 5X 10,000,000 4X 20,000,000 3X 40,000,000 0X-2X 99,999,999,999 LTC, LINK and other Contracts Leverage The nominal value of the maximum available position (USDT) 51X-75X 10,000 21X-50X 50,000 11X-20X 250,000 6X-10X 1,000,000 5X 2,000,000 4X 5,000,000 3X 10,000,000 0X-2X 99,999,999,999	{"url":"https://support.zedxion.com/hc/en-gb/articles/4412072686225-Leverage-and-position-limit","timestamp":"2024-11-08T11:10:35Z","content_type":"text/html","content_length":"24261","record_id":"<urn:uuid:797a507e-94a1-4f03-a005-70c3e059e8de>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00075.warc.gz"}
Science experiments on elastic collision using a smartphone using a smartphone top of page Elastic collision O que é? An elastic collision is a type of collision in which the two colliding bodies deform elastically and retain their initial kinetic energy. This means that the two bodies bounce against each other without losing energy. An example of elastic shock is the motion of two billiard balls colliding. If the two balls are in motion when they collide and bounce against each other without losing energy, the collision is elastic. The formula for kinetic energy before and after an elastic shock can be written as follows: Kinetic energy before = Kinetic energy after Where "Kinetic energy before" is the kinetic energy of the two bodies before the impact, and "Kinetic energy after" is the kinetic energy of the two bodies after the impact. It is important to note that in an elastic shock, kinetic energy is conserved, but potential energy and elastic energy can be converted into each other. Experimente com Elastic collision Ancre 1 bottom of page	{"url":"https://www.fizziq.org/pt/elements-1/elastic-collision","timestamp":"2024-11-04T02:47:12Z","content_type":"text/html","content_length":"754424","record_id":"<urn:uuid:e2a9fe21-6c99-4f82-8115-0caef25eb6f4>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00106.warc.gz"}
Wolfram\|Alpha Examples: Knot Theory Examples for Knot Theory In topology, knot theory is about the study of knots and their properties, where a knot is defined as a closed, non-self-intersecting curve embedded in three-dimensional space. Wolfram\|Alpha has the ability to recognize a knot with multiple representations and compute its properties or to compare multiple knots. Find and visualize a knot, either by its common name or by using knot notations. Compute properties of a knot: Specify a knot using Alexander–Briggs notation: Specify a knot using Conway notation: Compute properties of a torus knot: Compute a specific property of a knot:	{"url":"https://www6b3.wolframalpha.com/examples/mathematics/geometry/topology/knot-theory","timestamp":"2024-11-11T21:06:38Z","content_type":"text/html","content_length":"65463","record_id":"<urn:uuid:beaa9a6a-e94a-4b1b-a9cd-70f8987153ac>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00856.warc.gz"}
Unit 1 Lesson 9 Create Picture Books Warm-up: Act It Out: The Story Changes (10 minutes) The purpose of this activity is for students to consider different ways of acting out a story. Students revisit the story from previous lessons, which has another verse added to it. They suggest different ways the story could be acted out. Acting out gives students opportunities to make sense of a context (MP1). • Groups of 2 • Display and read the story. • “What is the story about?” • 30 seconds: quiet think time • Share responses. • “What has changed about the story?” (There is a new part. Only 2 of the ducks came back.) • Read the story again. • “How can you act out this story?” • 30 seconds: quiet think time • “Discuss your thinking with your partner.” • 1 minute: partner discussion • Share responses. Student Facing 3 little ducks went out one day, over the hill and far away. Mother duck said, “Quack, quack, quack.” Then 3 little ducks came back. 3 little ducks went out one day, over the hill and far away. Mother duck said, “Quack, quack, quack.” Then 2 little ducks came back. Activity Synthesis • Read the story together. • Act out the story as a class using a student suggestion. After acting out the first verse, reread the second verse and ask: “What will be different about how we act out the story this time?” (Only two of the ducks came back.) Activity 1: How Many Do You See: What Do You Notice? (10 minutes) The purpose of this How Many Do You See is for students to recognize and name small groups of dots and describe how they see them. In the synthesis, students discuss what they notice about different images of 2 dots. Students will continue considering different arrangements of the same number in the next activity. The number “2” is displayed at the end of the activity to give students opportunities to recognize numbers and connect numbers and quantities. • Groups of 2 • Display the first image. • “How many dots do you see? How do you see them?” • 30 seconds: quiet think time • “Use your fingers to show your partner how many dots you see.” • 30 seconds: partner work time • “Tell your partner how many dots you see and how you see them.” • 1 minute: partner discussion • Record responses. • Repeat with the second image. Student Facing How many do you see? How do you see them? Activity Synthesis • “What did you notice about the groups of dots?” (They both have 2 dots. They look different. They are the same but one is turned sideways.) • Display or write “2”. • “There are 2 dots.” Activity 2: Introduce Picture Books, Create (10 minutes) The purpose of this activity is for students to learn stage 2 of the Picture Books center. In this activity, students identify and record small groups of objects in their classroom with the same quantity. Through recording groups of two objects and seeing the groups of two objects recorded by other students, students are invited to notice that many different groups of objects can have the same number. Students create one page of a picture book, which is printed in their student workbook. Students have the opportunity to complete more pages in a picture book during centers. A blackline master of the picture book template is included. Each page of the picture book includes a written number in addition to dots so that students can begin to connect numbers and quantities. MLR8 Discussion Supports. Before beginning independent work time, invite a student to share an example of two things in the classroom. Listen for and clarify any questions. Advances: Speaking, Representing Action and Expression: Provide Access for Physical Action. To help generate ideas, invite students to tell their partner what they plan to draw before they begin. Supports accessibility for: Language, Visual-Spatial Processing Required Materials Materials to Gather Materials to Copy • Picture Books Stage 2 Recording Sheet • Display the student book. • Give students access to colored pencils or crayons. • “What do you notice? What do you wonder?” (There are 2 dots. There is a number. The rest of the page is blank.) • 30 seconds: quiet think time • Share responses. • “We have found groups of things in our classroom. We also matched groups that have the same number of things. Can you find something in our classroom that there are two of that you want to include in your picture book?” • 30 seconds: quiet think time • “You are going to make a page for a picture book like the ones we looked at earlier. There are two dots at the top of the page, so on this page you should draw things that there are two of in our classroom. ” • 3 minutes: independent work time • “Share your work with your partner. Did you both draw the same group of objects?” • 30 seconds: quiet think time • 2 minutes: partner discussion • “Find other groups of 2 things in the classroom to add to this page in your picture book.” • 3 minutes: independent work time Advancing Student Thinking If students draw groups with more or fewer than 2 things, consider asking: • “Can you tell me about this group of things that you drew? How many _____ are there?” • “What things do you see that are in a group of 2?” If needed, identify some objects in the room and ask “Are there 2 _____?” Activity Synthesis • Invite students to share the groups of 2 that they drew. • Record responses. • “What is the same about all of these groups of things?” (We found them all in the classroom. They all have 2.) • “You will be able to make more pages for your picture book in centers.” Activity 3: Centers: Choice Time (25 minutes) The purpose of this activity is for students to choose from activities that focus on using math tools and recognizing quantities without counting. Students choose from any stage of previously introduced centers. • Connecting Cubes • Pattern Blocks • Geoblocks • Picture Books Required Preparation • Gather materials from: □ Connecting Cubes, Stages 1 and 2 □ Pattern Blocks, Stages 1 and 2 □ Geoblocks, Stages 1 and 2 □ Picture Books, Stages 1 and 2 • “Today you will work in centers with our math tools and picture books. During center time today one of the choices is to make another page for your picture book.” • Display the center choices in the student book. • “Think about what you would like to do first.” • 30 seconds: quiet think time • Invite students to work at the center of their choice. • 10 minutes: center work time • “Choose what you would like to do next.” • 10 minutes: center work time • While students work in centers, ask: □ “What did you do with the connecting cubes, pattern blocks, or geoblocks?” □ “What groups of things did you see in your book? How many things are there?” □ “What groups of objects did you draw in your picture book?” • Monitor for students who draw clear groups of 1–4 objects for a new page in their picture book. Activity Synthesis • Invite previously selected students to share their picture book page. The lesson synthesis will focus on this page. Lesson Synthesis “Today we all made a page in our picture books with different groups of two things from around our classroom. Some of us created more pages for our picture books during center time.” “What groups of objects do you see on _____’s page?” “Let’s practice counting to 10.” Demonstrate counting to 10. Count to 10 as a class 1–2 times. Cool-down: Unit 1, Section B Checkpoint (0 minutes) Student Facing In this section, we noticed math in our world. We found groups of things in our classroom and in books. We used our fingers and said numbers to tell how many things there are. We found groups that have the same number of things. There are 2 windows and 2 tables. There are 3 stars and 3 soccer balls. They look different but they are both 3. We created our own books to show groups that have the same number of things in our classroom.	{"url":"https://im.kendallhunt.com/k5/teachers/kindergarten/unit-1/lesson-9/lesson.html","timestamp":"2024-11-11T03:45:23Z","content_type":"text/html","content_length":"127180","record_id":"<urn:uuid:c96769f7-8049-4ae3-93bc-28cdb731b419>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00647.warc.gz"}
Louisiana earns grade of C for manufacturing health Louisiana received a C grade in a scorecard developed by Ball State University that gauges the health of each state’s manufacturing sector. The university’s Center for Business and Economic Research rated Louisiana’s manufacturing health on several metrics evaluated in several categories: logistics industry health (B), human capital (F), worker benefit costs (C+), tax burdens (D+) and international trade levels (C). The Ball State University study also examined the percentage of total individual incomes in the state earned by manufacturing employees as well as other measures of manufacturing employment and pay Grades Earned by Each State’s Manufacturing Industry │ │Manufacturing Industry │Logistics Industry │Human Capital │Worker Benefit │Tax Climate │Expected Fiscal │Global Reach │Sector │Productivity and │ │State │Health 2018 to 2019 │Health 2018 to 2019 │2018 to 2019 │Costs 2018 to 2019 │2018 to 2019 │Liability 2018 to 2019│2018 to 2019 │Diversification 2018│Innovation 2018 to 2019│ │ │ │ │ │ │ │ │ │to 2019 │ │ │Alabama │B,B │C,C │F,D- │A,A │C,C │C,C │B,C │B,B │D,D │ │Alaska │F,F │C-,C- │C,C │D,D │B,B │D,D │F,F │F,F │F,F │ │Arizona │C,C │D,D │D,D- │C+,C │B,B │C,C- │C-,C- │D-,D- │C+,B- │ │Arkansas │C+,C │C,C │D,D │A,A │D-,F │C,C │D+,D+ │C,C │F,F │ │California │C,C+ │B-,B- │C,C- │C-,D │C-,D+ │C,D- │C,C- │D,F │A,A │ │Colorado │D,D │D+,D+ │B-,C+ │B,C │C,C │D+,C │D,D │C,C │C+,C+ │ │Connecticut │C,C+ │D,D │B-,C+ │C-,D │D,D- │D-,F │B+,A │D,D │B,B │ │Delaware │D,D │D-,F │C+,C │F,F │B-,B- │A,B │A,A │C-,C- │A,B │ │Florida │D,D │C,C │C,C+ │C,B │A,A │B,B │D,D │B,B │C,C │ │Georgia │D+,C- │B,B │D,D │C,B │C,C │A,A │C,C │A,A │C,C │ │Hawaii │F,F │F,F │C,C │D,D- │C+,C │D,D │F,F │D,D │F,F │ │Idaho │B,B │D,D │D+,C- │C,C+ │C,C │C+,B │D,D- │F,F │C,C │ │Illinois │C+,C+ │A,A │C,B- │C-,D+ │F,D- │F,F │B,B+ │C+,C+ │B-,B- │ │Indiana │A,A │A,A │C,C │B-,B │A,A │B-,C+ │A,A │C,C │C,C │ │Iowa │A,A │B+,B │A,A │C-,C- │F,F │B,B- │C,C- │C,C- │C-,C- │ │Kansas │B+,B+ │B-,C+ │C,C │C,C- │C,C │D-,D │C,B- │C-,C- │C,C │ │Kentucky │A,A │A,A │D,D │B-,B- │D+,C- │F,F │A,B+ │C,C │D+,C- │ │Louisiana │C,C │B,B │F,F │B,C+ │D+,D+ │C,C │C,C │D,D │D,D+ │ │Maine │C-,C- │F,D- │B,B │D-,D │D,C │C,C │D-,F │C,C │D-,D- │ │Maryland │D,D │D,D │C,C │C,C │C-,D │C,C+ │F,D │C-,C- │B,B │ │Massachusetts│C,C │D,D │B,B │C,C │D,D │C,C │B+,B │D,D+ │A,A │ │Michigan │A,A │C,C │D,D │C,C │B,B- │D,C- │B+,B │D,D │A,A │ │Minnesota │B-,B- │B,B+ │A,A │A,A │F,D- │C,C+ │C,B- │C,C │B,B │ │Mississippi │B+,B+ │C,C │F,F │B,B │C+,C+ │F,D │C,C │A,A │F,F │ │Missouri │C,C │C+,C │C,C- │A,B │A,A │B+,B │C,D+ │B,B │C,C │ │Montana │D-,D- │C,C- │C+,C │D,C- │A,A │D-,F │D,D │C,C │C-,D │ │Nebraska │C-,C- │B,B │A,A │C+,C │C+,C+ │A,A │C-,C │C-,D │D-,D │ │Nevada │F,F │D,D │D+,D+ │B+,B+ │C,C │C-.C- │B,C │C,C │D+,C- │ │New Hampshire│B-,B │F,F │B+,B+ │D+,C │C,C- │D+,C- │B,B │D+,C- │B,B+ │ │New Jersey │C,D+ │C,C+ │C,C │D-,D- │F,F │F,F │C+,C │D-,F │B+,A │ │New Mexico │F,F │F,F │F,F │C+,C+ │C,B │C-,D+ │F,F │F,D- │C,C │ │New York │F,D- │C,C │C-,C │D,D │F,D- │C,C │D+,C- │B,B │C+,C+ │ │North │C+,C+ │C,C │C,C │C,C │B+,B+ │A,A │B-,C+ │B+,B+ │A,A │ │Carolina │ │ │ │ │ │ │ │ │ │ │North Dakota │D-,D │B,B │A,A │B,B │B,B │C-,C- │C-,C- │C,C │D,D │ │Ohio │B,B │A,A │C-,C │D,C- │C,C │C,C │B-,B │B,B │C,C │ │Oklahoma │C-,C │C+,B- │D-,D │C,C │B,B │C+,C+ │F,D │C,C │F,F │ │Oregon │B,B- │C-,C- │C-,C- │D+,D+ │C,C │B-,C │C+,C │F,D │A,A │ │Pennsylvania │C-,C │B+,A │C,C │C-,D+ │D-,D │D,D- │C-,C │A,B+ │C,C │ │Rhode Island │D,D │F,F │C-,C- │D,C- │D,D │C-,D │D,F │B,A │D+,C- │ │South │A,A │C-,C- │D-,F │C,C │C,C- │C-,C │A,A │B+,B │C-,C │ │Carolina │ │ │ │ │ │ │ │ │ │ │South Dakota │C,C │C,C │B,B+ │B+,B+ │B,B │B,A │D,D │C-,C │D,D │ │Tennessee │B,B │C+,C+ │D,D │A,A │C,C │B,B │B,B │B-,B- │C,C │ │Texas │C,C │A,A │C-,D+ │B,B- │D,D+ │B+,B+ │A,A │C+,C+ │B-,C+ │ │Utah │C,C │C-,C- │B,B │B,A │A,A │A,A │C-,C+ │B-,B │B,B │ │Vermont │C,C │D-,D- │B,B │F,F │D,D │C-,C- │C,B │C,C │C,C │ │Virginia │D,D- │C-,C │C+,B- │C,C │C,C │C+,B- │D-,D- │A,A │C,C │ │Washington │C,C- │C,C │A,A │F,F │C-,C- │B,B │C,C+ │A,A │B,B │ │West Virginia│C,C │D+,D+ │F,F │F,F │B,C+ │D,D │C,C │D+,D+ │D,D │ │Wisconsin │B,B │B,B │B+,B+ │C,C │C-,C- │A,B+ │C+,C │C+,C+ │C-,C- │ │Wyoming │D+,D+ │C,C │B,B │F,F │B+,B+ │C,C │C,C │F,F │C-,D │ Source: Center for Business and Economic Research at Ball State University This is a revised article first published by The Center Square.	{"url":"https://thehayride.com/2019/10/louisiana-earns-grade-of-c-for-manufacturing-health-2/","timestamp":"2024-11-13T14:44:44Z","content_type":"text/html","content_length":"227525","record_id":"<urn:uuid:fc736bcd-7dc7-438d-b7e5-b348875b6ac3>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00141.warc.gz"}
850 meters per hour to kilometers per second Speed Converter - Meters per hour to kilometers per second - 3,850 kilometers per second to meters per hour This conversion of 3,850 meters per hour to kilometers per second has been calculated by multiplying 3,850 meters per hour by 0.000000277777777780000003202316 and the result is 0.0010 kilometers per	{"url":"https://unitconverter.io/meters-per-hour/kilometers-per-second/3850","timestamp":"2024-11-12T00:39:33Z","content_type":"text/html","content_length":"16031","record_id":"<urn:uuid:f6ba081f-01aa-4547-9832-ece9b28a0772>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00096.warc.gz"}
Non-Linear Systems: Stability of SISO and MIMO Systems in context of Stability Analysis 27 Aug 2024 Stability of Non-Linear Systems: A Comprehensive Guide to SISO and MIMO Systems Non-linear systems are a fundamental concept in control theory, and understanding their stability is crucial for designing effective control strategies. In this article, we will delve into the world of non-linear systems, exploring the stability analysis of both Single-Input Single-Output (SISO) and Multi-Input Multi-Output (MIMO) systems. What are Non-Linear Systems? A non-linear system is a dynamic system that exhibits non-linear behavior, meaning its output is not directly proportional to its input. In other words, the relationship between the input and output is not linear. This can be due to various factors such as saturation, dead zones, or complex dynamics. Stability Analysis of SISO Systems For SISO systems, stability analysis involves examining the system’s behavior in response to a small perturbation around an equilibrium point. The equilibrium point is defined as the point where the system’s output is zero. Let’s consider a simple example of a SISO non-linear system: ẋ = f(x) + g(x)u where x is the state variable, u is the input, and f(x) and g(x) are non-linear functions. To analyze the stability of this system, we can use the Lyapunov stability theory. The idea is to find a positive definite function V(x) such that: V(ẋ) ≤ -αV(x) where α is a positive constant. If such a function exists, the system is said to be stable. For SISO systems, the stability analysis can be performed using the following steps: 1. Find the equilibrium point x. 2. Linearize the system around x and obtain the linearized system: ẋ = A(x)x + B(x)u 3. Compute the eigenvalues of the linearized system matrix A(x). 4. If all eigenvalues have negative real parts, the system is asymptotically stable. Stability Analysis of MIMO Systems For MIMO systems, stability analysis becomes more complex due to the presence of multiple inputs and outputs. The key idea is to examine the behavior of each output in response to a small perturbation around an equilibrium point. Let’s consider a simple example of a MIMO non-linear system: ẋ = f(x) + G(x)u where x is the state variable, u is the input matrix, and f(x) and G(x) are non-linear functions. To analyze the stability of this system, we can use the following steps: 1. Find the equilibrium point x. 2. Linearize the system around x* and obtain the linearized system: ẋ = A(x)x + B(x)u 3. Compute the eigenvalues of the linearized system matrix A(x). 4. If all eigenvalues have negative real parts, the system is asymptotically stable. 5. Repeat steps 1-4 for each output to ensure stability. Formulas and Tools Here are some key formulas and tools used in stability analysis: • Lyapunov function: V(x) = x^T P x + q^T x • Linearized system matrix: A(x) = ∂f/∂x x=x* • Eigenvalues of the linearized system matrix: λ(A(x)) = eigenvalues of A(x) • Stability criterion: All eigenvalues have negative real parts (λi < 0) In this article, we explored the stability analysis of non-linear SISO and MIMO systems. By understanding the behavior of these systems around an equilibrium point, we can design effective control strategies to stabilize them. The formulas and tools presented in this article provide a foundation for further study and application of stability analysis in various fields such as control theory, robotics, and aerospace engineering. • Khalil, H. K. (2002). Nonlinear Systems. Prentice Hall. • Slotine, J.-J., & Li, W. (1991). Applied Nonlinear Control. Prentice Hall. • Dorato, P. (1987). Robust Stability and Stabilization of Uncertain Systems. Academic Press. I hope this article helps you understand the stability analysis of non-linear SISO and MIMO systems! Related articles for ‘Stability Analysis’ : Calculators for ‘Stability Analysis’	{"url":"https://blog.truegeometry.com/tutorials/education/1a8e2912ccdbe38c0dce454d59e65803/JSON_TO_ARTCL_Non_Linear_Systems_Stability_of_SISO_and_MIMO_Systems_in_context_.html","timestamp":"2024-11-05T07:02:46Z","content_type":"text/html","content_length":"19028","record_id":"<urn:uuid:196628c7-c70a-4275-8fdb-068afd82f025>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00809.warc.gz"}
Areas: Excel Formulae Explained - ExcelAdept Key Takeaway: • Excel formulae are a powerful tool for performing calculations and automating tasks in spreadsheets. Understanding formula syntax and the different types of operators is key to maximizing Excel’s • Arithmetic operators allow for basic math operations such as adding, subtracting, multiplying, and dividing. Logical operators can compare values and return true or false based on the result. Reference operators help to manipulate cells and ranges in formulas. • Text operators can manipulate text strings within formulas. Date and time functions allow for calculations and formatting of dates and times. Lookup and reference functions enable users to search for data within a sheet or across sheets. • Mathematical and statistical functions provide powerful tools for complex calculations such as trigonometric functions and statistical analysis. Conditional functions allow for logical and comparative evaluation of data and can be used to create complex decision-making rules. Error handling functions can help users to avoid errors in formulae that would otherwise cause a spreadsheet to crash or produce incorrect results. Do you struggle to use Excel formulae? If so, this article is for you! Learn how to use the most versatile formulas in Excel to unlock powerful fuctions and create efficient spreadsheets. This comprehensive guide will make you an Excel expert in no time! Understanding Excel Formulae Understanding the Functionality of Excel Formulas Excel formulas are the foundation of any advanced data analysis. These formulas assist in solving complex math problems, performing statistical analysis, and sorting and extracting data. One can increase their efficiency and precision in data analysis by understanding the basics of Excel formulas. To acquire a deep understanding of Excel formulas, one must first comprehend the use of cell references, absolute and relative, and the four types of operators: arithmetic, comparison, text concatenation, and reference. Additionally, one should understand the order of operations and function arguments. Furthermore, creating nested formulas is a potent tool for performing intricate calculations. To do so, one must recognize the best way to approach nested formulas and know the correct syntax for implementing them. Dive into Excel formulas and unlock the full potential of data analysis. Do not miss out on the benefits of understanding Excel formulas; take the time to learn and grow your abilities. Arithmetic Operators in Excel Formulae Arithmetic calculations are crucial in Excel formulas as they enable users to perform mathematical operations easily within the worksheet. Different arithmetic operators can be used such as addition (+), subtraction (-), multiplication (), and division (/) to manipulate values. These operators can be applied to individual cells or a range of cells in the formula. The use of parentheses can also be employed to change the order of operations, just like in a mathematical equation. Using the arithmetic operators within Excel formulae is effective in calculating numerical values, and it can save time and effort in the data analysis process. Additionally, users can apply a percentage (%) or rounding to a cell value or a range of cells with simple functions. It is a good practice to utilize these operators while working with large datasets as they help in improving efficiency in the analysis. By combining the operands and operators, complex formulas can be easily constructed to derive the desired results. Pro Tip: Try to use cell references instead of the actual values while creating formulas to reduce the chances of errors and make it easier to modify the calculations in the future. Logical Operators in Excel Formulae Logical Connectives in Excel Formulas Excel formulas often involve logical connectives like AND, OR, and NOT. These operators are used to compare multiple conditions and return a resulting value of TRUE or FALSE. When using logical operators in Excel formulas, it is important to remember that an expression with multiple operators must be evaluated in a specific order, known as the order of operations. For example, if one wants to determine whether a number is between 10 and 20, the formula would be: =AND(A1>10, A1<20) This formula would return TRUE if the value of A1 is between 10 and 20, and FALSE otherwise. In addition, one can also use nested IF statements that contain logical operators to create more complex formulas. This allows users to analyze multiple conditions to determine a final result. Pro Tip: When creating complex Excel formulas with logical operators, use parentheses to explicitly indicate the order of operations. This can help avoid confusion and ensure the formula is calculated correctly. Reference Operators in Excel Formulae In Excel formulae, operators can be used to reference different cells, ranges or values for calculations. These operators include the range operator, intersection operator, and union operator. Operator Symbol Example Range Operator : A1:B2 Intersection Operator space A1 B2 Union Operator , A1,B2 It’s important to understand these operators to effectively manipulate data in Excel. In addition to these basic operators, there are also other reference operators such as the indirect function and the offset function that can be used to reference data in more complex ways. According to a study by Microsoft, Excel is used by over 750 million individuals worldwide. Text Operators in Excel Formulae In Excel Formulae, there are various Semantic NLP variations of Text Operators. Text Operators enable users to create dynamic texts by concatenating, searching, replacing, and manipulating data. These operators can perform a variety of tasks, such as merging text strings, trimming whitespace, and finding characters within a text. By using these operators, users can quickly analyze and manipulate data to enhance their reporting capabilities and improve data quality. For instance, CONCATENATE is one of the most commonly used Text Operators in Excel Formulae. This operator can be used to combine two or more text strings into a single cell. Similarly, the LEFT and RIGHT operators can be used to extract a specific number of characters from the left or right section of a text string. The REPLACE operator can be used to replace a specific character or set of characters in a text string. There are also other Text Operators, such as SUBSTITUTE, FIND, and LEN, that allow users to manipulate text in various ways. One often-overlooked aspect of Text Operators is their compatibility with other operators in Excel Formulae. They can be combined with various mathematical, logical, and date-time operators to create complex formulae that can help users derive advanced insights from their data. Interestingly, the history of Text Operators dates back to the early days of computer programming, where they were used to manipulate text data. As technology evolved, Text Operators became an essential part of various computer software, ranging from word processors to spreadsheet applications, such as Excel. Today, Text Operators play a crucial role in data analysis and manipulation, regardless of the industry or sector. Date and Time Functions in Excel Formulae In Excel Formulae, there exist distinct functions for handling date and time data. These functions are not only convenient but also critical for analysis, financial modeling, and forecasting. These functions enable users to perform calculations on dates and timestamps in a simple yet accurate way. One can easily use functions such as YEAR, MONTH, DAY, TODAY, MINUTE, HOUR, NOW, to mention a few. Furthermore, when working with Excel dates and time, it’s crucial to format and validate them correctly. Users must learn to format cells, columns, and rows to display dates and time accurately. To ensure data accuracy, users must also validate all data entered into their sheets. For example, dates entered as text can cause problems while performing calculations in Excel. Finally, to make the most out of Excel’s date and time functions, users must understand the underlying syntax and usage rules. This is especially essential when working on complex models that require date and time functions. A good idea would be to practice using these functions on sample datasets to enhance proficiency and confidence. Lookup and Reference Functions in Excel Formulae Lookup and reference functions are essential in Excel formulae, allowing users to retrieve information from other parts of the worksheet or workbook. With functions such as INDEX, VLOOKUP and HLOOKUP , users can easily search for specific data and return corresponding values. These functions help users save time and avoid errors when working with large data sets. In addition to basic lookup functions, Excel also offers more advanced reference functions such as INDIRECT, OFFSET and MATCH. These functions allow users to dynamically reference cells or ranges and perform calculations based on the returned values. This flexibility is crucial when dealing with data that is constantly changing or expanding. It is important to note that when using lookup and reference functions, the data must be structured in a way that can be easily searched and retrieved. This means that columns and rows should be labeled correctly and consistently, and duplicate values should be avoided. Taking these steps can ensure accurate and efficient results when using these functions. One time, a colleague was tasked with analyzing a massive data set for their company’s annual report. Without using the lookup and reference functions, they spent hours manually searching for specific data and compiling the necessary information. After learning about these functions, they were able to complete the task in a fraction of the time. This experience emphasizes the importance of utilizing the functions offered by Excel and the time-saving benefits they can provide. Mathematical Functions in Excel Formulae In Excel Formulae, various mathematical functions can be utilized for data analysis and manipulation. These functions include trigonometry, statistical analysis, and arithmetic operations. By incorporating these functions in the formulae, the data can be analyzed and interpreted easily. Using Excel’s built-in functions, mathematical operations can be performed quickly and efficiently. For instance, the SUM function can sum up a range of cells, while the AVERAGE function calculates the average of a range of cells. Furthermore, the MAX and MIN functions find the highest and lowest values in a range of cells respectively. It is worth noting that these functions can be combined to create complex formulas. For example, the SUMIF formula can be used to sum up values that meet specified criteria, while the COUNTIF formula counts the number of cells that meet specified criteria, making data analysis more efficient. To optimize the use of mathematical functions in Excel Formulae, it is advisable to use appropriate formatting and carefully review the data. Additionally, including descriptions within the formulae can ease interpretation and ensure that the formulae can be easily understood by others. Statistical Functions in Excel Formulae Statistical analysis is crucial in creating effective Excel formulae. A variety of Statistical Functions like AVERAGEIF, MEDIAN, STDEV, and COUNT are used in Excel Formulae for Statistical Analysis. Such functions can help identify trends, measure variability, and estimate variability in data. Below is a table showcasing the use of Statistical Functions in Excel Formulae: Column A Column B Column C Column D Product Name Sold Item Total Sales Profit Product 1 15 =SUM(B2:B5) =C20.2 Product 2 10 =SUM(B2:B5) =C30.3 Product 3 20 =SUM(B2:B5) =C40.2 Product 4 12 =SUM(B2:B5) =C5*0.05 In addition to the commonly used Statistical Functions, Excel formulae can also include various other useful functions like DATE, TIME, and VLOOKUP. These functions can assist in performing complex calculations with ease. Interestingly, much of the statistical analysis that can be performed in Excel was also available on the mainframe computers in the late 1960s. At that time, IBM introduced the Total Statistical Package (TSP), which included tools for data handling, analysis, and interpretation. In 1972, Microsoft introduced the Excel spreadsheet, which shortly thereafter incorporated many statistical functions. Since then, Excel has been widely used by data practitioners across the world. Conditional Functions in Excel Formulae Conditional Excel Formulae: A Dynamic Way to Analyze Data Conditional Excel formulae help users implement dynamic and automated decision-making processes that make data analysis easier and more efficient. By using semantic natural language processing, this type of function allows users to express complex decisions in simple terms. With conditional Excel formulae, users can easily create rules that automate their data analysis. For example, they can use the IF function to analyze data fields and set up a decision-making workflow that is based on specific criteria. Conditional formatting, on the other hand, enables users to highlight important data based on specific conditions. One unique aspect of conditional Excel formulae is the nested IF statement feature, which allows users to build decision trees with multiple levels of complexity. By setting up multiple nested IF statements, users can create complex data analysis workflows with ease. To make the most of conditional Excel formulae, users should follow best practices when setting up these functions. They should aim to keep their formulas simple, avoiding complex nesting structures that can be difficult to maintain. Additionally, they should ensure that their rules match the data they are analyzing and use clear names to make it easier to understand their formulas. Error Handling Functions in Excel Formulae In Excel Formulae, managing errors can be done using Error Handling Functions. These functions aid in identifying and correcting errors in the formula. • These functions include the IFERROR function which can replace the error value with a custom message or value. • Another function is the ISERROR function which can identify if a specified cell contains an error value. • The ISNA function can also be used to check if a specified cell contains the #N/A error value. It is essential to remember that error handling functions are necessary for maintaining the accuracy of the data and calculations when dealing with large datasets. An interesting fact about Excel is that it was first released in 1985 and has since become a vital tool for businesses and individuals alike. Some Facts About “AREAS: Excel Formulae Explained”: • ✅ “AREAS” is an Excel formula that returns the number of separate ranges in a reference. (Source: Excel Off The Grid) • ✅ The “AREAS” formula is useful for troubleshooting issues with formulas that don’t work on a range. (Source: AbleBits) • ✅ The “AREAS” formula can also be used to determine the overall size and complexity of a workbook. (Source: Excel Jet) • ✅ The “AREAS” formula can be combined with other formulas such as “SUM”, “AVERAGE”, and “MAX” to perform more complex calculations. (Source: Excel Easy) • ✅ “AREAS” is a lesser-known formula in Excel, but it can be very useful for advanced users. (Source: Got-it AI) FAQs about Areas: Excel Formulae Explained What is AREAS: Excel Formulae Explained? AREAS: Excel Formulae Explained is a comprehensive guide to understanding and using the AREAS formula in Microsoft Excel. This formula is used to determine the number of unique ranges in a given selection, and can be extremely useful in a variety of applications. How do I use the AREAS formula in Excel? To use AREAS in Microsoft Excel, you first need to select the range or ranges that you want to count. Then, simply enter the formula =AREAS(range) into a cell and press enter. The cell will then display the number of unique areas within the selected range(s). What are some common applications for the AREAS formula in Excel? Some common applications for the AREAS formula in Microsoft Excel include: • Determining the number of different invoice headers in a large spreadsheet. • Calculating the number of different product categories in a sales report. • Identifying the number of unique data sets in a data analysis. Is the AREAS formula compatible with all versions of Excel? The AREAS formula is compatible with most versions of Microsoft Excel, including Excel 2007, 2010, 2013, 2016, and Office 365. However, some older versions of Excel may not support this formula. Are there any limitations to using the AREAS formula in Excel? One potential limitation of the AREAS formula in Microsoft Excel is that it only works with contiguous ranges – that is, ranges that are adjacent to one another without any gaps or blank cells in between. Additionally, if the selected range(s) contain hidden cells, the formula may return inaccurate results. Can the AREAS formula be used with other Excel formulas and functions? Yes, the AREAS formula can be used in conjunction with other Excel formulas and functions to perform a range of calculations. For example, you might use the SUMIF function to add up values in a range that meet a certain criteria, and then use the AREAS formula to determine how many unique ranges that criteria applies to.	{"url":"https://exceladept.com/areas-excel-formulae-explained/","timestamp":"2024-11-06T15:23:21Z","content_type":"text/html","content_length":"74335","record_id":"<urn:uuid:cbaa04af-b878-4dc2-ad76-ff9de0416b26>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00321.warc.gz"}
Polynomials, Coloring Graphs, and Sudoku Did you ever get half way through a really tough sudoku and begin to wonder if it really has a solution, or finish one and wonder if when you decided that one cell was a five and not a three that if you had picked the other one it STILL would have worked out? The answer, my friend, lies in polynomials. Recently learned about all this from a nice article you can read for yourself , and maybe if you find my mistakes, drop me a line. The authors point out that every Sudoku puzzle can be thought of as a graph with 81 vertices (nine rows and nine columns). Now think of the nine numbers in the top row..they would all have to be connected to each other with an edge. They would all have to be connected to each of the cells that are in their same column, and also to the eight others in their sub-square (but they have already been connected to at least four of those, two in their row and two others in their column). By my calculation, that means that each of the 81 vertices would have 8+8+4=20 edges connected to it, for a total of 810 edges. Now if we associate the number in each cell as a color, a solution would be the same as a properly colored graph (no edge connects two vertices of the same color). An unfinished Sudoku is then a partially colored graph, and solving it is just a matter of finding a way to color the remaining vertices so that no edge connects two vertices of the same color. If it is not possible to do that, there is no solution... and if you can do it more than one way, there is more than one solution. Now comes the polynomial part. It seems it is "well known" that "The number of ways of coloring a graph G with n colors is well known to be a polynomial in n of degree equal to the number of vertices of G." (Where have I been?). Wait, that means we are looking for a polynomial with degree 81... Ummm.. that may take a little time, so let's look at a simple example. Suppose we have a simple graph like a triangle, the complete graph of degree three, usually called K . [A complete graph has each vertex connected to every other vertex with an edge.] It should be clear it can not be properly colored with one or two colors, but with three colors we could do it in six ways (3!). But how many ways could you color if if you had four colors available, or five, or nine, or a complete box of 64 crayola crayons. The answer is pretty easy using the associated polynomial. For K the number of ways to color it using n colors is n(n-1)(n-2). So for four colors you could complete the coloring in 24 different ways. You can easily extend the polynomial for a complete graph of t vertices using n colors to get K to be n(n-1)(n-2)....(n-t+1). Ok, so what about one like this? Can you do it in three colors, four, five? If you can write the polynomial for it it should tell you. I admit that I am not sure about how to write some of the pretty simple ones... for instance, if a vertex is joined to two vertices that have n-2 colors possible each, but they are not connected; should that new vertex contribute a (n-3) or an (n-2). Trying to learn as I go along, but mostly by drawing simple ones that I can enumerate all the possible colorings. Complications spring up when you have a graph that is more complex (and especially if it has 81 vertices), but if you had the polynomial, then you could just evaluate it for n=9 and it would tell you how many solutions there are. The authors did point out some simple theorems that pop out when you make the connection to graphing. For example, any sudoku that gives you only seven of the nine numbers in the "given" must have more than one solution (if it has any). Consider that if you have a solution you could interchange all the cells that have the two missing colors and get another solution. So a puzzle with a unique solution must contain at least eight of the nine numerals. The authors suggest that the number of cells that must be "given" in the problem is an unsolved problem, but they have unique examples with as few as 17, so that would be an upper bound on the solution. The puzzle at the top has eight of the nine digits in it, and exactly 17 cells "given". So does it have a solution? or two? Happy graph coloring children. Here is a simpler problem, can you write the coloring polynomial for this graph, and find the number of colors for n=3, 4, and 10 colors? 2 comments: Sue VanHattum said... >Ok, so what about one like this? Can you do it in three colors, four, five? I'm not getting the polynomial part yet, but I know the first graph can be colored in 4 colors or less, because this is the same as the map coloring theorem. So any planar graph can be colored in 4 colors or less. (I got that from Euler's Gem, by Dave Richeson, which I'm in the middle of.) I just worked it out (drew a map), and discovered that a loop of 4 in the graph is equivalent to a situation where 4 corners touch (like CO, UT, NM, AR). I've convinced myself that 4 colors are needed. To explain my reasoning, I'll use letters. Coming down the left, I have A on top, then B, the third one down touches both the first two, so it needs a third letter, C. The last one touches the 3rd and top, so it can be B. Next to C must be A, since it touches bottom left. Below that must be a C. Now the second down on the right touches A, B, and C, so must be D, leaving B for the top right. Pat's Blog said... Yes, If it is planer...(this one obviously is, it is drawn without lines crossing; But keep in mind that even simple graphs like K-five (think of a pentagon with a pentagram inside it).. and that isn't planer. (in fact it is half the "forbidden" that Kuratowski's theorem uses to define Planer. This one obviously is, and I agree, it must have four colors, but how many different ways could it be colored with just four colors?? (by the way you know if it can be colored in four colors, none of the polynomial factors could be (n-a) could have a>3 since that would produce zero ways to color it with four colors (and you just proved it COUlD be done in four).. One of the factors has to be n, and the other seven(remember with eight vertices, it will be an eighth degree polynomial) will be either (n-1), (n-2) or (n-3) ... I'm betting on a lot of (N-3) factors.. but (ugly confession) I'm not sure I have the polynomial for that one yet.	{"url":"https://pballew.blogspot.com/2010/02/polynomials-coloring-graphs-and-sudoku.html","timestamp":"2024-11-08T21:45:29Z","content_type":"application/xhtml+xml","content_length":"136052","record_id":"<urn:uuid:1aeda3cd-bcff-451e-9734-993d50094adf>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00566.warc.gz"}
Binary to Decimal Conversion: Tools & Steps The Power of Binary to Decimal Conversion Tools: A Step-by-Step Guide In the world of computing and digital technology, the ability to understand and convert different numerical systems is crucial. Binary to decimal conversion is a fundamental skill in this realm. This blog post aims to provide a comprehensive guide on using binary to decimal converter tools, emphasizing the ease and efficiency of the binary to decimal converter online, especially those available for free. Understanding Binary to Decimal Conversion Before diving into the tools, it's essential to grasp what binary to decimal conversion involves. Binary, a base-2 system, is the core language of computers, consisting only of 0s and 1s. On the other hand, the decimal system is a base-10 system, the standard numerical system used by most people. Converting binary to decimal involves translating these binary digits (bits) into decimal numbers that we can easily understand. Check this out: Choosing The Right Pressure Converter Tool A Comparison Guide \| The Power Of Binary To Decimal Conversion Tools A Step By Step Guide Why Use a Binary to Decimal Converter Tool? Performing conversions manually can be time-consuming and prone to errors. This is where a binary to decimal converter tool becomes invaluable. Whether you’re a student, programmer, or just someone curious about digital systems, these tools can save time and increase accuracy. • Choosing the Right Tool There are numerous binary to decimal converter tools online. For the best experience, look for a "binary to decimal converter online free" option. These tools are readily accessible and do not require any subscription or payment. The "best binary to decimal online tool" often features a user-friendly interface and quick, accurate conversions. • How to Use an Online Binary to Decimal Converter Using these tools is straightforward. Here's a simple step-by-step guide: Open the Converter: Search for a “free online binary to decimal converter tool” and choose one that seems reliable. Enter the Binary Number: Input the binary number you wish to convert. This could be a series of any length of 0s and 1s. Convert: Click on the convert button. The tool will process the input and display the equivalent decimal number. • Verifying the Results While these tools are typically accurate, it’s good practice to verify the results, especially when using the conversion for critical tasks. Cross-checking with another “binary to decimal converter tool online” can ensure accuracy. Read More: Hex To Binary Converter Simplifying Programmers Life \| The Ultimate Guide To Using A Length Converter Tool Advantages of Using a Binary to Decimal Converter Online Free • Accessibility: Being online, these tools can be accessed anywhere, anytime. • Cost-effective: They are free, making them accessible to anyone without the need to invest in software. • Time-saving: Automated conversion saves a significant amount of time compared to manual calculation. • Accuracy: These tools are designed to provide precise conversions, reducing the likelihood of errors. The binary to decimal converter online is a powerful tool for anyone dealing with digital systems. It simplifies the complex process of converting binary to decimal, making it accessible and understandable. Whether for educational purposes, programming, or personal interest, these tools are invaluable in navigating the binary world. By following this guide, you can efficiently utilize these tools to enhance your understanding and efficiency in digital technology. Remember, the next time you need to convert binary to decimal, opt for an “online binary to decimal converter.” It’s a choice that combines efficiency, accuracy, and ease of use, all in one package. Frequently Asked Questions: Q1. What is binary to decimal conversion? Binary to decimal conversion is the process of changing a binary number (base 2) into its equivalent decimal number (base 10). Q2. Why is binary to decimal conversion important? It's essential for understanding and working with computers and digital systems because they use binary representation internally. Q3. How can I manually convert binary to decimal? To convert manually, write down the binary number and multiply each digit by 2 raised to the power of its position from right to left, then add the results. Q4. Why use a binary to decimal conversion tool? Conversion tools simplify the process and save time, especially for longer binary numbers. You can use a Binary to decimal converter tool online. Q5. Are there online tools for binary to decimal conversion? Yes, many websites and software offer free binary to decimal conversion tools that do the calculations automatically. Q6. How do I use an online conversion tool? Input your binary number, click the "Convert" or "Calculate" button, and the Binary to decimal converter tool online will display the decimal equivalent. Q7. Can I use conversion tools offline? Some software applications and calculators offer offline binary to decimal conversion, but many prefer online tools for convenience. Q8. Are there mobile apps for binary to decimal conversion? Yes, there are mobile apps available for both Android and iOS devices that can perform binary to decimal conversions. Q9. Can I convert decimal to binary as well? Yes, most binary to decimal conversion tools also support the reverse process, converting decimal to binary. Q10. What are some practical uses of binary to decimal conversion? Binary to decimal conversion is used in computer programming, networking, digital electronics, and any field involving binary data representation.	{"url":"https://tools.bebran.com/post/the-power-of-binary-to-decimal-conversion-tools-a-step-by-step-guide","timestamp":"2024-11-02T11:36:52Z","content_type":"text/html","content_length":"43639","record_id":"<urn:uuid:35eb2d95-e5b3-445e-beae-a05a4832f096>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00259.warc.gz"}
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Lambda World 2017 – Lambda Core, hardcore Despite the title which is a bit obscure, this talk has been very clear and entertaining. The speaker Jarek Ratajski (@jarek000000) who defines himself as a functional programmer, wizard, and anarchitect, definitively knows his art and how to make an engaging speech. Btw, an anarchitect is a person that breaks the rules so that something can be eventually done. As usual, mistakes, errors, and inaccuracies are all my faults. A last note: you need Scala 2.12 to compile the code presented here, older versions are likely to fail. I really recommend watching the presentation video on youtube. Let’s start with a question for the attendees – you are on a desert island and you can take just one thing from the civilized world. What would you like to have to survive? [someone from the audience shouts “wifi”] Well, you need Math. Math is the base for everything, but it is huge! So, maybe we can find a small subset, a core, that allows us to recreate the whole Math. Lambda is a building block of math. Using lambda calculus we can build – one step at a time – the whole Math. Historically Math has been built using sets, unfortunately, sets had some problems in terms of paradoxes, so mathematicians resorted to this simpler mechanism. So lambda (λ) is a function that takes some lambda as a parameter and produces another lambda. A lambda is written as: λ x. some expression A λ x is applied to λ y with the following notation – to a scala developer: trait Lambda { def apply(x: Lambda) : Lambda Now suppose we are on a deserted island with just our Scala compiler. Just the compiler – no jars, no collections, no scala utils, no Boolean, Int, Float, Double, String… anything at all. The only exception to this nothingness is the badlam package in order to simplify and help the presentation of concepts. Let’s define the identity, which is a function that returns the function accepted as an argument: λ x. x In Scala it turns out to be: val identity : Lambda = (x) => x val id2 = identity(identity) Now we can define a function that accepts multiple arguments (via the curry mechanism) and always returns the first one: val funny: Lambda = (x) => (y) => x Note that names are not important, but the structure is. The “funny” function structure can be used to implement true and false concepts: val aTrue = funny val aFalse : Lambda = (x) => (y) => y Note that aFalse is complementary with respect to aTrue. Now that we have true and false symbols we can define boolean operators: val and : Lambda = (p) => (q) => p(q)(p) You can read this like: and is a function that via currying accepts two arguments p and q. Consider p: it can be either true or false. If it is true its lambda evaluates to its first argument, otherwise to the second. So, when p is true, the and expression evaluates to q, so and is true when both arguments are true. If p is false, then p is returned – no need to consider q. Let’s check with Scala: val result1 : Lambda = and (aFalse)(aTrue) This evaluates to aFalse, while val result2:Lambda = and (aTrue)(aTrue) This evaluates to aTrue. Or function is based on the same scheme used by and: val or:Lambda = (p) => (q) => p(p)(q) To define not is a matter of reversing the true/false definition: val not: Lambda = (p) => (a) => (b) => p(b)(a) So long, so far, we have defined all the tools we need for boolean calculus. Let’s see if we can express numbers. One approach could be: val zero: Lambda = (f) => (x) => x val one: Lambda = (f) => (x) => f(x) val two: Lambda = (f) => (x) => f(f(x)) val three: Lambda = (f) => (x) => f(f(f(x))) So numbers are functions (of course) that accept a function and an argument and return zero or more applications of the function to the argument. Zero is the identity. One is the single application of lambda f to the identity. Two is the single application of lambda f to one, that is that application of lambda f to the application of lambda f to the identity, and so on… This may work, but it is boring to encode all numbers in this way. A function that computes the successor of a given number could be a good tool to lessen the burden: val succ:Lambda = (n) => (f) => (x) => f(n(f)(x)) [NdM – well this is mind-boggling and I have a hard time decoding. Conceptually is simple – just remember that an integer n is represented by a lambda that accepts a function f and an argument x and returns a function composing n nested applications of f over x. So replace x with f(x) and you are done.] Now we can define addition and multiplication val plus: Lambda = (m) => (n) => (f) => (x) => m(f)(n(f)(x)) val mult:Lambda = (m) => (n) => (f) => m(n(f)) [NdM -It takes a while for these two as well. Sum is intuitively easy, take 3+2=5, in lambda is: x => f(f(f(x))) + x => f(f(x)) = x => f(f(f(f(f(x))))). You may read plus as: (m,n,f) => (x) => m(f)(n (f)(x)) , that is a function that takes two operands and one function. Remember that m represents an integer by applying a function m times to an operand. So swap in f as an operand and you have m times the application of f, now applies this function to an argument that is n times the application of f to x.] Multiplication is similar, x is omitted to keep the code terse, but you can read it as: val mult:Lambda = (m) => (n) => (f) => (x) => m(n(f))(x) Keeping the sum approach in mind, this is similar – applies m times a function that is composed by n application of f to x. I needed quite a neuron effort to figure this out.] Predecessor can be defined as: val pred: Lambda = (n) => (f) => (x) => n((g) => (h) => h(g(f)))((u) => x)((u) => u) [NdM. I tried hard to understand this, but simply I couldn’t wrap my mind around it. I don’t have even the slightest idea of how this is expected to work. If you figure it out, let me know, please… well indeed, I think that the last part (u) => u, being an identity, is used to skip an application in the application list, therefore reducing by 1 the application list…] [NdM. I found this thorough explanation of the predecessor on stackoverflow] Now we can do something even more interesting – conditionals: val ifLambda: Lambda = (c) => (t) => (f) => c(t)(f)((x) => x) val isZero: Lambda = (n) => n((x) => aFalse)(aTrue) Recursion would be a nice addition to the computing arsenal of lambda calculus since it allows the expression of iterative algorithms. But how are we supposed to call a function when lambda functions are anonymous? First, let’s define a function that makes its argument call itself: val autocall:Lambda = (x) => x(x) Then we need a helper function that makes something call itself val Y: Lambda = (f) => autocall((y) => f((v) => y(y)(v))) And finally, we define a function containing the body of the recursive function: val G:Lambda = (r) => (b) => ((x) => one ) ((x) => mult(n)(r(pred(n)))) Now we have everything to recursively compute a factorial: val fact = Y(G) // this makes G recursive [NdM: Once more I have a hard time trying to understand this part. It makes sense intuitively, but I’m lost in details such as function Y. The overall idea is pretty well laid out] Turing Equivalent? Lambda calculus has been proved to be Turing- equivalent, meaning that every algorithm that can be implemented on a Turing Machine, can also be implemented using Lambda Calculus. Therefore, you have no excuse, everything can be done purely functional! In mathematics there are a lot of problems, an interesting one is about theorems. For a mathematician, a theorem is something like if ( condition(args)) then statement(args) That is, if something holds, then something else is true. It would be nice to build a machine that automatically checks this. This is what about a century ago, mathematicians were looking for – a machine that renders us useless by proving theorems by itself. In the same way, we used lambda calculus to represent numbers, booleans, and statements, we could rewrite the theorem as a lambda calculus expression and then execute it to let it show true or false to determine whether the theorem holds or not. This could be such a machine: def main( args: Array[String]) : Unit = { val autocall: Lambda = x => x(x) println( SmartDisplay.web.display(autocall)) val OMEGA = autocall(autocall) [NdM: also after having watched the youtube video of the conference, I can’t tell where this function comes from. I believe Jarek, but I have no explanation of why this function should prove That would be awesome, regrettably, it doesn’t work. In fact, autocall function is invoked over autocall itself causing a stack overflow. This is generally what happens when you try to analyze a lambda function for equivalence. This fact has been proved in 1936 by Alonzo Church: “No computable function can decide the equivalence of two different lambda functions”. Despite this, there are two guys on stack overflow that are trying to do exactly this in C#. Lambda calculus is convenient, but it undergoes the incompleteness theorem by Kurt Gödel – For any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory. In other words, it doesn’t matter how cool your formalism is, you still can’t fully automate through an algorithm to prove the properties of another algorithm. So this is what nearly a century ago a group of mathematicians, led by Alonzo Church, devised in search of more elegant mathematics. What I presented today is called Untyped Lambda Calculus. Take plus and try to add true and factorial together. It doesn’t make sense. So, by using this notation you can also write correct expressions that don’t make sense. Since not every expression you can write makes sense, it is a programmer’s duty to write and use only those that make sense. Typed lambda calculus checks correctness for you. Untyped lambda is very much like javascript in which you can write everything and is not checked for type correctness. I haven’t talked about Lazy/Eager evaluations in lambda calculus. This is a very complicated issue, even if very fun. If you do lambda calculus on paper you will notice that sometimes you need to evaluate lazy and other times you need to evaluate eager. Usually, when you read about lambda calculus in Scala, you don’t find what I have shown you, you find something else – lambda calculus done on the type system of Scala: sealed trait True extends Bool { type If[ T <: Up, F <: Up, Up ] = T sealed trait False extends Bool { type If[T <: Up, F <: Up, Up] = F This is not performed at run-time but at compile time. This is awesome because the compiler does all the calculations and produces such fine errors you can’t even figure out where they end. • wikipedia • blog: (C# Version) Dixin’s blog – this is the post where I took inspiration for this talk. • Book: Roger Penrose: The Emperor’s New Mind – this is not about only lambda calculus, but about computation. Reading this book will help you better grasp the subject. • Lambda Visualization – Badlam this is nothing special just a package to help the presentation. It is known to work only in presentation and not elsewhere. • This same presentation in Java – is the original work. This presentation has been done first in Java (for fun and drinking beer of course). NdM: that’s the end of the presentation. Here I add my fragment of code I used to check the code. If you prefer you can use Badlam, the code below is in Scala. Function to convert from lambda numbers to decimal – def dec( y : Lambda ) : Int = { case class Counter( counter: Int) extends Lambda { def apply( x: Lambda ) : Lambda = { case class Fun() extends Lambda { def apply( x: Lambda ) : Lambda = { Counter( x.asInstanceOf[Counter].counter+1 ) y( Fun() )(Counter(0)).asInstanceOf[Counter].counter The trick is to give function f of lambda the semantic of incrementing its argument by 1. If the argument has type int and starts counting from zero the conversion is done. The following code is just a pretty printer for lambdas that are not integers: def del( x : Lambda ) : Unit = { x match { case `aTrue` => println( "true" ) case `aFalse` => println( "false" ) case `and` => println("<and>") case `or` => println("<or>") case `not` => println("<not>") case _ => println("?") (where aTrue, aFalse, and all the other symbols are defined as in Jarek’s presentation. One thought on “Lambda World 2017 – Lambda Core, hardcore” This site uses Akismet to reduce spam. Learn how your comment data is processed.	{"url":"https://www.maxpagani.org/2018/02/12/lambda-world-2017-lambda-core-hardcore/","timestamp":"2024-11-11T16:38:22Z","content_type":"text/html","content_length":"135567","record_id":"<urn:uuid:50eb81f0-4fbc-43ff-8049-4f148b46c5eb>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00324.warc.gz"}
Paper die (1d8) All that I had was a pen and some sheets of paper. I started cutting sheets to go through the basic origami forms to see if anything could be fit for the purpose. Very soon, as it is a basic form, I folded the square base , also known as the preliminary base , which has a nice property: it has four different faces each one being a square. This looked promising as I could combine the four faces with the four edges to be able to get sixteen different events (which would be perfect to simulate the yarrow stalks probabilities). Building it The diagram below shows how to fold a square base: Should the diagram above not be clear enough, there are many instructions and videos on the internet on how to fold one. The problem is that a square base is actually asymmetric: its (the point marked A) is structurally different from its (the point where C,D and the other two angles meet) wich tends to open up. As it is it would not be usable as a device for casting hexagrams. To make it symmetric (and also give it a stronger structure) I thought of interlocking square bases. A possible way to do it is to proceed as shown in the diagram below: The tricky part is in step 3 where the face Y has to go the face A while, at the same time, the face Z (on the opposite side) has to go face B. It is much easier doing it than describing it; with a little practice you'll make one in no time. The resulting object has four faces: two from one base (A and C) and two from the other base (X and Z, not shown in the picture below). It has two rotational axis: the vertical one with order of rotational symmetry 4 (the four faces) and the horizontal haxis with order of rotational symmetry 2 (the swap between the blue and red dots in the picture below). To get a rather robust object, it is best to start from a square of 5x5 cm (approx 2x2 in). The easiest way is to cut a sheet of paper (A4 or US Letter) in four strips along the longest side and then cut the squares from them. Marking the faces (for yarrow stalks probabilities) What I got in the end was a paper die with four faces, each face has four sides so I marked one side with , seven sides with , five sides with and three sides with Here is how the two squares looked like if I had unfolded them: Done! ... or so I thought. I soon realized that doing this way, the could only show up in two positions: top right or bottom left; should I develop the habit of picking other sides more frequently, I would lower the chance of getting a If I were more disciplined, I could assume that I would choose any side with the same frequency and the probabilities would be: Prob(6) = ^1/[16] Prob(8) = ^7/[16] Prob(7) = ^5/[16] Prob(9) = ^3/[16] Prob(yin) = Prob(yang) = ^1/[2] Marking the faces (for three coins probabilities) To avoid bias in this die, I decided to split each face in two so that the line would depend on the die orientation and on the face I would pick. This left me with eight possible outcomes: exactly what is needed for the three coins probabilities. Here how the new faces looked like: With this marking the probabilities are: Prob(6) = Prob(9) = ^1/[8] = 12.5% Prob(8) = Prob(7) = ^3/[8] = 37.5% Prob(yin) = Prob(yang) = ^1/[2] In the end, the plane landed and I didn't ask the question I had in mind. However, I gained a new method for casting hexagrams. Since that day I have inserted in my pocket copy of the I Ching a couple of pre-printed strips of paper so that I can quickly build one of these dice and cast a hexagram with it. I've never done it but I like the idea to write the question on the back of the piece of paper before building the die so that the casting is related forever to the question. If you want to try them, download the PDF files you prefer ( yarrow stalks three coins ) and print them. Be careful to set up the option to "keep the orginal size" or you will have trouble when cutting and folding I built a couple of these dice to give a better sense of what they look like. This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. 0 commenti:	{"url":"https://www.castingiching.com/2016/05/i-ching-paper-die.html","timestamp":"2024-11-09T17:37:33Z","content_type":"application/xhtml+xml","content_length":"94728","record_id":"<urn:uuid:5392cd79-1a9c-4559-ac57-6f6aee6b35ff>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00028.warc.gz"}
integral transforms on sheaves I am writing an exposition (or a dictionary) integral transforms on sheaves. Check it out. I will have to go offline soon. Maybe somebody feels like further polishing/expanding this up a bit. Then later I want to supply that for the current $n$Café discussion. Correct me if wrong. At the end of section 2, “Linear bases”, it is said that morphisms of Pr(infinity, 1)Cat are equivalently profunctors, but it seems that saying (infinity, 1)-profunctors is better instead, now that we have that page. Ok to change it? Yes! Thanks.	{"url":"https://nforum.ncatlab.org/discussion/2131/integral-transforms-on-sheaves/?Focus=79096","timestamp":"2024-11-06T21:07:37Z","content_type":"application/xhtml+xml","content_length":"41097","record_id":"<urn:uuid:a8dd2b8e-9b30-4e45-a962-a8aad137b8a8>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00154.warc.gz"}
candy box Resultado da Busca 1. 29 de abr. de 2020 · 1 Answer. Sorted by: 2. The original Candy Box can be accessed at candybox2.github.io/candybox. There may be some confusion since the URL includes candybox2 in it, but it's actually the first one. The second Candy Box game is at candybox2.github.io. Share. 2. 25 de out. de 2013 · THE ADVANCED GRIMOIRE. "This advanced grimoire provides you two spells : the erase magic spell and the thorns shield spell. The first one will erase all the magic from a quest at some point including the spells you casted and your enemy's magic. The second will create around you a shield of floating thorns which will inflict damage upon contact." 3. 25 de out. de 2013 · (I asked and self-answered this to keep all of the answers to the questions in one place, like this question from Candy Box 1. (we already have questions for specific ones, like this, this, and this.)) – 4. 31 de mai. de 2024 · games like candy box? Request. I feel like I really miss the first incremental games I found which were RPGs that used mostly or entirely ascii art, like A Dark Room, Gold Factory, Candy Box 1 and 2, and Space Lich Omega 1 and 2. However, most of those are relatively old. Are there any newer ones that are similar that I should look in to? 5. 4 de mai. de 2013 · Third question. Consider 10 days. If I give you 1 candy on the first day, and each other day I give you twice more candies than the previous one, how much candies will I give you on the day number 10? Correct answer. 512 reward. 512 candies Question 4 Fourth question : if you could be whatever you want, what would you be? Correct answers 6. 5 de mai. de 2013 · Eventually, the timing will line up and you'll make it through. This works particularly well with a Sword of Summoning, and can make the Castle Entrance a fairly effective farming spot for additional candy. Your second option is to try to speed through. Seven League Boots go a long way here, as do Berserk Potions if you have any left from the ... 7. 25 de out. de 2013 · The first few clicks should be: Click the V/T block, then expand out the chain of Vs to get the --->. Move it along one again and expand. You now have two adjacent <3 blocks to make an 'engine'. You can now expand out the rest of the area using the same method. You should end up with two ---> blocks, and one \|. 8. 8 de mai. de 2013 · 1. To enter cheat codes in Candy-Box simply do the following: Go to the javascript console of any browser. Enter commands like: candies.setnbrOwned = <Amt of candies>. candies.setcandiesPerSecond = <Amt of candies>. Share. Improve this answer. edited May 8, 2013 at 23:50. 9. 12 de set. de 2022 · Start a new game of Super RPG on a completed file, go to the Computer and input "bug ultimate 2" then go back to the Map tab to begin. (You may have to do it this way because with "bug ultimate 2" it can change variables, and you can end up with 0 Candies due to Lolligators, making it impossible to start Super RPG. 10. 4 de mar. de 2019 · So I've started a Candy Box 3 in HTML, so far I've only gotten a counter saying how many candies you have and a button saying eat candies and another counter saying how many candies you have eaten. AKA it functions exactly the same way it did in the other 2 games.	{"url":"https://br.search.yahoo.com/mobile/s?p=candy+box&nojs=1&ei=UTF-8&fl=0&rd=r2&age=1w&nocache=1&fr2=p%3As%2Cv%3Aw%2Cm%3Aat-e%2Cct%3Agossip","timestamp":"2024-11-04T14:58:30Z","content_type":"text/html","content_length":"106207","record_id":"<urn:uuid:fadf9dd8-901f-4a33-bf55-db14322c9da0>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00808.warc.gz"}
Matlab Files Matlab files used for Weighted Averaging Introduction to Weighted Averaging Weighted averaging is a technique that improves the signal to noise ratio by "weighting" the recorded data according to its variance prior to summation and then dividing the sum by its "weight". In other words, trials that have, on average, more varience than other trials have less of an impact on the overall average response over many trials. This a proportionality effect in that the trials contribution to the overall averaged response is inversly proportional to its varience. Instructions for the Weighted Averaging MATLAB routines 1. Download the zipped MATLAB files here (be sure to fill out the short form) 2. Unzip using Winzip (click here for Winzip web site, if you need the unpacking software) 3. Read all the comments in the m files 4. Place your CNT files in the same directory as the unzipped files 5. Edit run_weight.m so that the file names match your CNT files 6. Type "run_weight.m" in the MATLAB Command window 7. If you have any difficulties, please review "Potential Problems" below • Any errors? Please review "Potential Problems" below Interpretting the results The run_weight.m routine outputs 2 files: 1. filename.txt and 2. filenamew.txt. The first file is the summary of the analysis. It can be imported into a spread sheet program like Excel. The first row gives only a title to the data in the columns. The first column gives the filename. The second column gives the number of files processed. The third column gives information as to the type of averaging (1=normal, 2=weighted averaging). The remaining columns give F values (F), significances (S), amplitudes (A), phase values (P), and noise levels (N). In the default settings there are 8 signals present with 8 characteristic modulation frequencies (remember that each modulation frequency has a characteristic carrier frequency i.e. Left: 750, 1500, 3000, and 6000 Hz Right: 500, 1000, 2000, and 4000 Hz). These are signals 1 to 8. Signals 9 to 12 are false alarms. The modulation frequencies of these are different than the previous 8. These false alarms have no real signal in them and thus should represent just noise. If one of these 4 signals should have a significant value, then this would represent a false alarm (i.e. a real signal where there should not have been). Remember that at P=0.05, 5% of all significant responses will be arise from chance alone. Potential Problems • You must have the MATLAB Signal Processing Toolbox installed in your version of MATLAB because the routines call upon upon statistical functions found only in the Signal Processing Toolbox • Make sure your CNT files are in the same directory as the unzipped files • Make sure you read the comments in run_weight.m and change the default file names "en2km51", "en2km52", "en2km53" etc. to your CNT filenames. Do not use the CNT extension • Make sure you have the correct number files to average (default is 6)	{"url":"http://www.mastersystem.ca/index.php?section=698","timestamp":"2024-11-09T06:54:14Z","content_type":"text/html","content_length":"15856","record_id":"<urn:uuid:d8d50888-bd19-4be1-a1dd-abef27006b4e>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00686.warc.gz"}
Montgomery reduction In arithmetic computation, Montgomery reduction is an algorithm introduced in 1985 by Peter Montgomery that allows modular arithmetic to be performed efficiently when the modulus is large (typically several hundred bits). A single application of the Montgomery algorithm (henceforth referred to as a "Montgomery step") is faster than a "naive" modular multiplication: ${\displaystyle c = a \times b \pmod n. \, }$ Because numbers have to be converted to and from a particular form suitable for performing the Montgomery step, a single modular multiplication performed using a Montgomery step is actually slightly less efficient than a "naive" one. However, modular exponentiation can be implemented as a sequence of Montgomery steps, with conversion only required once at the start and once at the end of the sequence. In this case the greater speed of the Montgomery steps far outweighs the need for the extra conversions. Formal statement[ ] Let n be a positive integer, and let R and T be integers such that ${\displaystyle R>n}$, ${\displaystyle \gcd(n,R)=1}$, and ${\displaystyle 0\leqslant T<nR}$. The Montgomery reduction of T modulo n with respect to R is defined as the value ${\displaystyle TR^{-1} \pmod{n}. \, }$ ${\displaystyle R^{-1}}$ is the modular inverse of R. The algorithm used to calculate this value is much more efficient than the classical method of taking a product over the integers and reducing the result modulo n. Use in cryptography[ ] Many important cryptosystems such as RSA and DSA are based on arithmetic operations, such as multiplications, modulo a large number. The classical method of calculating a modular product involves first multiplying the numbers as if they were integers and then taking the modulo of the result. However, modular reduction is very expensive computationally—equivalent to dividing two numbers. The situation is even worse when the algorithm requires modular exponentiation. Classically, ${\displaystyle a^b \pmod {n}}$ is calculated by repeatedly multiplying a by itself b times, each time reducing the result modulo n. Note that taking a single modulo at the end of the calculation will result in increasingly larger intermediate products—infeasible if b is very large. Rationale[ ] We wish to calculate c such that ${\displaystyle c \equiv a \times b \pmod {N}}$. Rather than working directly with a and b, we define the residue ${\displaystyle \bar a = aR \pmod {N}}$ and similary for ${\displaystyle \bar{b}}$. ${\displaystyle R > N}$ is a number relatively prime to N chosen such that division and remainder operations are easy.? A power of two is generally chosen so that these operations become bitwise masks and shifts respectively. R and N are guaranteed to be relatively prime if N is odd, as is typical in cryptographic applications. It can be easily shown that there is a one-to-one mapping between numbers ${\displaystyle a, b, \cdots}$ and residues ${\displaystyle \bar a, \bar b, \cdots}$. Addition and subtraction operations are the same: ${\displaystyle xR + yR \equiv zR \pmod{N}}$ if and only if ${\displaystyle x + y \equiv z \pmod{N}}$ This is important because converting between natural and residue representations is expensive, and we would prefer to work in one representation as much as possible and minimise conversions. To define multiplication, define the modular inverse of R, ${\displaystyle R^{-1}}$ such that ${\displaystyle RR^{-1} \equiv 1 \pmod {N}}$ in other words ${\displaystyle RR^{-1} = kN + 1}$ where k is an integer. Now if ${\displaystyle c = a \times b \pmod {N}}$ ${\displaystyle \bar c \equiv (a \times b)R \equiv (aR \times bR)R^{-1} \equiv (\bar a \times \bar b)R^{-1} \mod{N}}$ It turns out that this is cheap to calculate using the following algorithm. Description of Algorithm[ ] The Montgomery reduction algorithm Redc(T) calculates ${\displaystyle TR^{-1} \pmod{N}}$ as follows: ${\displaystyle m := (T \mod {R})k \mod {R}}$ ${\displaystyle t := (T + mN)/R}$ if ${\displaystyle t \ge N}$ return ${\displaystyle t - N}$ else return t Note that only additions, multiplications and integer divides and modulos by R are used – all of which are 'cheap' operations. To understand why this gives the right answer, consider the following: • ${\displaystyle mN \equiv TkN \pmod{R}}$. But by the definition of ${\displaystyle R^{-1}}$ and k, ${\displaystyle kN + 1}$ is a multiple of R, so ${\displaystyle TkN \equiv -T \pmod{R}}$. Therefore, ${\displaystyle (T + mN) \equiv 0 \pmod{R}}$; in other words, ${\displaystyle (T + mN)}$ is exactly divisible by R, so t is an integer. • Furthermore, ${\displaystyle tR = (T + mN) \equiv T \pmod {N}}$; therefore, ${\displaystyle t \equiv TR^{-1} \mod{N}}$, as required. • Assuming ${\displaystyle 0 \le T \le RN}$, ${\displaystyle t \le 2N}$ (as ${\displaystyle m < R}$). Therefore the return value is always less than N. Therefore, we can say that ${\displaystyle \bar c = \mbox{Redc}(\bar a \times \bar b)}$ Using this method to calculate c is generally less efficient than a naive multiplication and reduction, as the cost of conversions to and from residue representation (multiplications by R and ${\ displaystyle R^{-1}}$ modulo N) outweigh the savings from the reduction step. The advantage of this method becomes apparent when dealing with a sequence of multiplications, as required for modular exponentiation (e.g. exponentiation by squaring). Examples[ ] The Montgomery step[ ] Working with n-digit numbers to base d, a Montgomery step calculates ${\displaystyle a \times b \div d^n\pmod r}$. The base d is typically 2 for microelectronic applications or 2^32 or 2^64 for software applications. For the purpose of exposition, we shall illustrate with d = 10 and n = 4. To calculate 5678 × a ÷ 10000: 1. Zero the accumulator. 2. Add 8a to the accumulator. 3. Shift the accumulator one place to the right (thus dividing by 10). 4. Add 7a to the accumulator. 5. Shift the accumulator one place to the right. 6. Add 6a to the accumulator. 7. Shift the accumulator one place to the right. 8. Add 5a to the accumulator. 9. Shift the accumulator one place to the right. It is easy to see that the result is 0.5678 × a, as required. To turn this into a modular operation with a modulus r, add, immediately before each shift, whatever multiple of r is needed to make the value in the accumulator a multiple of 10. The result will be that the final value in the accumulator will be an integer (since only multiples of 10 have ever been divided by 10) and equivalent (modulo r) to 5678 × a ÷ 10000. Finding the appropriate multiple of r is a simple operation of single-digit arithmetic. When working to base 2, it is trivial to calculate: if the value in the accumulator is even, the multiple is 0 (nothing needs to be added); if the value in the accumulator is odd, the multiple is 1 (r needs to be added). The Montgomery step is faster than the methods of "naive" modular arithmetic because the decision as to what multiple of r to add is taken purely on the basis of the least significant digit of the accumulator. This allows the use of carry-save adders, which are much faster than the conventional kind but are not immediately able to give accurate values for the more significant digits of the Modular multiplication[ ] Consider the following pair of calculations: 24 × 73 = 1752 240000 × 730000 ÷ 10000 = 17520000 It can be seen that if we choose to represent integers by 10000 times themselves (let us temporarily call this a "Montgomery representation") then the result of a Montgomery step on the Montgomery representation of a and the Montgomery representation of b is the Montgomery representation of a × b. Thus we can use a Montgomery step to perform a modular multiplication by "Montgomeryizing" both operands before the Montgomery step and "de-Montgomeryizing" the result after it. To "de-Montgomeryize" a number—in other words, to take it from its representation as "12340000" to a conventional representation as "1234"—it suffices to do a single Montgomery step with the number and 1: 12340000×1÷10000=1234. To "Montgomeryize" a number—in other words, to take it from its conventional representation to a representation as "12340000"—it suffices to do a single Montgomery step with the number and 100000000: The value of 100000000 modulo r can be precomputed, since the same modulus r is usually used many times over. The total budget for a single modular multiplication is thus four Montgomery steps: two to "Montgomeryize" the operands, one to perform the actual multiplication, and one to "de-Montgomeryize" the A Montgomery step is unlikely ever to be four times faster than a conventional modular multiplication (because a carry-save addition is unlikely ever to be four times faster than a conventional addition) and so Montgomery's algorithm is not efficient for single multiplications. Modular exponentiation[ ] Raising a number to a k-bit exponent involves between k and 2k multiplications. In most applications of modular exponentiation the exponent is at least several hundred bits long. To fix our ideas, suppose that a particular modular exponentiation requires 800 multiplications. In that case 802 Montgomery steps will be needed: one to Montgomeryize the number being exponentiated, 800 to do the exponentiation, and one to de-Montgomeryize the result. If a Montgomery step is even slightly faster than a conventional modular multiplication, the Montgomery algorithm will produce a faster result than conventional modular exponentiation. References[ ] fr:Réduction de Montgomery ja:モンゴメリ乗算 ru:Алгоритм Монтгомери zh:蒙哥马利算法	{"url":"https://cryptography.fandom.com/wiki/Montgomery_reduction","timestamp":"2024-11-13T09:34:07Z","content_type":"text/html","content_length":"210710","record_id":"<urn:uuid:f66b7b2c-1bd6-4334-b1da-f39666268c73>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00511.warc.gz"}
GENsim - Matrix Method Diffusion MRI Simulator GENsim is a Matlab toolkit designed to simulate diffusion MRI signal for a large variety of pulse sequences in diffusion substrates which include restriction. It provides the means to understand the diffusion patterns in synthetic tissue models (white matter, cancer, etc) for various diffusion sequences. This tutorial describes the use of GENsim Matlab toolkit for simulating diffusion MRI data in various substrates. The novelty of GENSim is that it allows the user to create a flexible diffusion gradient waveform. It assumes a basic understanding of diffusion MRI pulse sequences, including more advanced sequences such as square and trapezoidal oscillating gradients. The diffusion signal is calculated using the matrix formalism detailed in Drobnjak et al. 2010 and Drobnjak et al. 2011 . The existing parametrizations for square and trapezoidal oscillating gradients follow the description in Ianus et al. 2012 . The diffusion substrates available in GENSim are the basic compartments presented in Panagiotaki et al. 2011, as well as some two- and three-compartment models of white matter. Terms and Conditions GENSim is distributed under the Artistic License 2.0. The full text of the license can be found here . The Owner draws your attention to the fact that the Software has been developed for and is intended for use in a research environment only. No endorsement can be given for other use including, but not limited to, use in a clinical environment. If you are using GENSim in your research, please cite Drobnjak et al. 2010 , Drobnjak et al. 2011 and Ianus et al. 2012 Coming soon Step-by-step getting started tutorial • Include GENSim toolbox in the Matlab search path: (:toggle init=hide hide="Hide the detail" show="Show the detail" div=matlab_path button=1 :) Assuming that you have unpacked the GENSim toolbox in the directory /usr/local/GENSim, then run • Take a minute to understand the format of the gradient waveform necessary for the simulation: (:toggle init=hide hide="Hide the detail" show="Show the detail" div=GEN_sequence button=1 :) The information about the diffusion protocol is passed to the simulator as the fields of a Matab structure, which we conveniently call "protocol" in the examples. To generate the diffusion signal for a generalized waveform, the structure protocol should have the following fields: protocol.pulseseq = 'GEN' - the name of the sequence, protocol.G - the gradient waveform, protocol.tau - time interval between two points on the gradient waveform and optionally protocol.smalldel and protocol.delta which are the pulse duration and time interval between the onset of the first and second pulse. It is not mandatory to specify the last two parameters, but they save computation time if the diffusion sequence has two gradient intervals. Additionally the user can specify if the second gradient is repeated protocol.mirror = 0 - default value or reflected protocol.mirror = 1 . The gradient waveform protocol.G is a M x 3K matrix which stores the values of the diffusion gradient at each time point. Each row represents one measurement and contains the values of the gradients in x, y and z direction. M is the total number of measurements and K is the number of gradient points in each direction for one measurement: G1x(1) G1y(1) G1z(1) G1x(2) G1y(2) G1z(2) ... G1x(K) G1y(K) G1z(K) G2x(1) G2y(1) G2z(1) G2x(2) G2y(2) G2z(2) ... G2x(K) G2y(K) G2z(K) GMx(1) GMy(1) GMz(1) GMx(2) GMy(2) GMz(2) ... GMx(K) GMy(K) GMz(K) protocol.G includes the complete diffusion gradient and the user should take into account any realistic situations such as the duration of the rf pulses, crusher gradients, etc. protocol.G should satisfy the echo condition, that the integral of the gradient over time should be 0. • Take a minute to understand the diffusion substrates available in the simulation: (:toggle init=hide hide="Hide the detail" show="Show the detail" div=model button=1 :) GENSym uses the 3D extension of the Matrix Method (first introduced by Callaghan JMR 95) presented in Drobnjak et al 2011. For the diffusion substrates, we follow the naming scheme presented in Panagiotaki et al. 2011. The following substrates are available in GENSim: Basic compartments with Gaussian diffusion: Ball (isotropic free diffusion) Zeppelin (anisotropic, cylindrically symmetric diffusion tensor) Tensor (full diffusion tensor) Basic compartments with restricted diffusion: AstroSticks (isotropically oriented sticks) AstroCylinders (isotropically oriented cylinders) Dot (fully restricted) Multi-compartment models: TortZeppelinCylinder (same as ZeppelinCylinder, but with tortuosity constraint on volume fraction) Other substrates can be easily implemented by the user once familiarized with the code. All the information related to the diffusion substrate is stored in a Matlab structure, which we call “model” in the examples. For the purpose of signal generation, “model” has only two fields: model.name – the name of the model (listed above) and model.params - the values of the model parameters in S.I. (m, s, etc.) of each model. See documentation for details. • Create an example gradient waveform for square oscillating gradients (:toggle init=hide hide="Hide the detail" show="Show the detail" div=example_SWOGSE button=1 :) We provide the means of generating the gradient waveform for several parametrized sequences by calling the wave_form function. The available sequences are: pulse gradient spin echo sequence PGSE, square oscillating gradient - SWOGSE, trapezoidal oscillating gradients - TWOGSE, square oscillating gradients with different parameters in x,y and z directions - SWOGSE_3D. See documentation for Here we show an example for SWOGSE: * Setup an initial protocol which includes the necessary information to compute the discrete waveform for a SWOGSE sequence % add some combinations of parameters: delta = 0.015:0.005:0.04; % duration in s smalldel = delta - 0.005; % duration in s Nosc = 1:5; % number of lobes in the oscillating gradient. A full period has 2 lobes G = 0.08; % gradient strength in T/m; tau = 1E-4; % time interval for waveform discretization it = 0; protocol_init.pulseseq = 'SWOGSE'; for i = 1:length(delta) for j = 1:length(Nosc) it = it+1; protocol_init.smalldel(it) = smalldel(i); protocol_init.delta(it) = delta(i); protocol_init.omega(it) = Nosc(j).pi./smalldel(i); protocol_init.G(it) = G; protocol_init.grad_dirs(it,:) = [1 0 0]; % gradient in x direction protocol_init.tau = tau; Create the GEN protocol protocolGEN.pulseseq = 'GEN'; protocolGEN.G = wave_form(protocol_init); protocolGEN.tau = tau; % include smalldel and delta as they make computation slightly faster protocolGEN.delta = protocol_init.delta; protocolGEN.smalldel = protocol_init.smalldel; • View the example waveform for square oscillating gradients (:toggle init=hide hide="Hide the detail" show="Show the detail" div=view_SWOGSE button=1 :) The previous diffusion protocol has 30 measurements with varying timing parameters and number of oscillations. To visualize the gradient waveform for the first measurement, run the following code: Gx = protocolGEN.G(1,1:3:end); ylim([min(Gx)1200 max(Gx)1200]) xlabel('time (ms)'); ylabel('Gx (mT/m)'); • Synthesize the diffusion signal for the previous example (:toggle init=hide hide="Hide the detail" show="Show the detail" div=run_SWOGSE button=1 :) In order to synthesize diffusion signal, we need to choose a diffusion substrate. Chose a white-matter model: model.name = 'ZeppelinCylinder'; di = 1.7E-9; % intrinsic diffusivity dh = 1.2E-9; % hindered diffusivity rad = 4E-6; % cylinder radius % angles in spherical coordinates describing the cylinder orientation; theta = 0; % angle from z axis phi = 0; % azimuthal angle ficvf = 0.7; % intracellular volume fraction model.params = [ficvf di dh rad theta phi]; Add the matrices and other constants required by the Matrix Method: protocolGEN = MMConstants(model,protocolGEN); Compute the diffusion signal for the given protocol and model: signal = SynthMeas(model,protocolGEN); • Plot the synthesized diffusion signal (:toggle init=hide hide="Hide the detail" show="Show the detail" div=plot_signal button=1 :) To visualize the diffusion signal computed in the previous step as a function of diffusion time, run the following code: signal_matrix = reshape(signal,length(Nosc),length(delta)); colors = jet(length(Nosc)); hold on; for i = 1:length(Nosc) legend_name{i} = ['Nosc = ' num2str(Nosc(i))]; xlabel('\Delta (ms)','FontSize',16); ylabel('Diffusion Signal','FontSize',16); title('Diffusion signal as a function of \Delta for various number of oscillations, G = 80mT/m') • A true advantage of GENSim is that it can compute the diffusion signal for any sequence, not only parametrized ones. An example with random values (:toggle init=hide hide="Hide the detail" show="Show the detail" div=rand_protocol button=1 :) Here you will see how to create a valid gradient waveform from a vector which specifies the value of the gradient at each point in time. The direction of the gradient is given by the field grad_dirs, similarly to the parametrized versions. We can use the function wave_form with a protocol which has pulseseq = 'GEN' and a field GENGx which is the vector with the gradient values. To understand the resulting waveform, think of a PGSE where the constant gradient was replaced by GENGx. Set up the initial protocol: delta = 0.04; % duration in s smalldel = delta - 0.005; % duration in s G = 0.08; % maximum gradient strength in T/m; protocol_rand.pulseseq = 'GEN'; protocol_rand.grad_dirs = [1/sqrt(2) 1/sqrt(2) 0 ]; % gradient along x and y protocol_rand.smalldel = smalldel; protocol_rand.delta = delta; % protocol_rand consists of a random waveform with 40 points which will be repeated after the 180 pulse protocol_rand.GENGx = rand (1,40)*G; protocol_rand.tau = protocol_rand.smalldel(1)./length(protocol_rand.GENGx);% calculate the time interval based on the number of points Create the protocol needed for the diffusion simulation protocolGEN.pulseseq = 'GEN'; protocolGEN.G = wave_form(protocol_rand); protocolGEN.tau = tau; % include smalldel and delta as they make computation slightly faster protocolGEN.delta = protocol_rand.delta; protocolGEN.smalldel = protocol_rand.smalldel; View the protocol Gx = protocolGEN.G(1,1:3:end); xlabel('time (ms)'); ylabel('Gx (mT/m)'); Choose the diffusion substrate (same as before) model.name = 'ZeppelinCylinder'; di = 1.7E-9; % intrinsic diffusivity dh = 1.2E-9; % hindered diffusivity rad = 4E-6; % cylinder radius % angles in spherical coordinates describing the cylinder orientation; theta = 0; % angle from z axis phi = 0; % azimuthal angle ficvf = 0.7; % intracellular volume fraction model.params = [ficvf di dh rad theta phi]; Add the matrices and other constants required by the Matrix Method: protocolGEN = MMConstants(model,protocolGEN); Compute the diffusion signal for the given protocol and model: signal = SynthMeas(model,protocolGEN); • Create your own diffusion protocol. • To help you start, the same commands are in GENSim/example/RunGEN.m	{"url":"http://mig.cs.ucl.ac.uk/index.php?n=Main.GENsim","timestamp":"2024-11-06T12:16:24Z","content_type":"text/html","content_length":"25566","record_id":"<urn:uuid:16d6b4c3-80a0-481c-ac7c-82be67420cd8>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00139.warc.gz"}
ZerO/One : The whOle Point The only geometry you really need is the Point. We each are a point. Every locus of attention is a central point in the universe. Each point of attention is a center of Source consciousness. Every Point is exploring a sacred path of Source intent. Your sense of Self is a Source point in this omnicentric universe. The Point is visually symbolized as a dot. The fundamental geometry is this point. All other geometries are only collections of these points in various relationships. The point is also omni-symmetrical, symbolized visually as circular. The circle is also ZerO. The ZerO is nothing precisely because it is absolutely everything. ZerO is no independent thing. How indicative that the word ZerO begins with the last letter of the alphabet, demonstrating an understanding of the whole of the platform on which communication is based, a whole pattern, to refer to a circle which we call Zero. Zero ends with the letter O, the circle symbol used to represent it. The word One also begins with this O, this Zero, this circle. This relationship of One to the whOle is reflective of the circle and the point. I call this glyph of the Point surrounded by the circle the whOle Point. Or perhaps we should call it ZerOne (zeer-won). The basic symbol of the point inside of a circle is the fundamental representation of the whOle Point: the Point relating to a whOle, or the Self to a Reality. This symbol of the circle around the point is a fractal glyph. The circle is the fractal expression of the point. The circle is the point at the next level of scale. The One (the Point) and the ZerO are the same, from different perspectives. This whOle Point symbol is also a simple, overhead portrayal of the torus field. The torus field is fundamental to the whOle Point. Literally and metaphorically, all points on the circle are equally accessible to the center point. In the torus field, all points return to the center point through the spiral rotation of the field, and re-emerge into the circle. This expresses the ZerO/One (ZerOne) where everything is One with all. This is the also the fundamental paradox, where 0 and 1 are the binary pair, and yet they are also the same expression. There are so many paradoxical transformations in this symbol. The point speck in the center of the circle can be imagined as nothing and the circle as symbolizing everything. And yet the circle is our symbol for Zero and the point is also a symbol of One. The numeric symbol “1” is an arrow that has a point, and balances on a point. When the arrow hits the target we have the whOle Point once again, the dot in the middle of the circle. When I say that the Point is really the only geometry you need, experientially finding the Point of your center Self brings everything back to you. This is an extremely powerful meditation. Some call it finding your center. Knowing that this point is also the whOle, the whOle Point, allows this experience of center to expand to include your entire field. There is nothing that is not included. This torus field will bring you back to your Self. You cannot get lost. You can get there from here. You can get here from there. Here and there are the same whOle Point. The whOle Point is You in Source Consciousness, and Source Consciousness as You. This entry was posted in ESG Circle Blog, Geometric Philosophy, Sacred Number and tagged consciousness, meditation, sacred geometry, source, the point, torus. Bookmark the permalink.	{"url":"https://mariengrace.com/2014/04/zeroone-the-whole-point/","timestamp":"2024-11-08T23:46:58Z","content_type":"text/html","content_length":"60621","record_id":"<urn:uuid:15079b97-205c-4343-8313-c1fd46838cc8>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00217.warc.gz"}
54 in cm - Scale Calculator Cm to inches 54 in cm: How to Convert Inches to Centimeters Are you trying to convert 54 in cm ? If so, you’ve come to the right place. In this article, we’ll explain the formula for converting inches to centimeters, provide some helpful tips, and give examples to help you better understand the process. So, let’s get started! What is 54 inches in centimeters? Before we dive into the conversion process, let’s first answer the question at hand: What is 54 inches in centimeters? The answer is 137.16 centimeters. To understand how we arrived at this answer, we need to understand the formula for converting inches to centimeters. The Formula: Inches to Centimeters Conversion The formula for converting inches to centimeters is quite simple. All you need to do is multiply the number of inches by 2.54. This will give you the equivalent measurement in centimeters. Let’s apply this formula to our original question. Converting 54 In cm To convert 54 inches to cm, we simply multiply 54 by 2.54. The calculation looks like this: So, there you have it! 54 inches is equal to 137.16 centimeters. Tips for Converting Inches to Centimeters Now that you know the formula for converting inches to centimeters, let’s go over some tips to help you make the process even easier. 1. Remember the formula: To convert inches to centimeters, multiply the number of inches by 2.54. 2. Use a calculator: While it’s possible to do the math in your head, using a calculator will ensure accuracy and save time. 3. Round to the nearest hundredth: Centimeters are a smaller unit of measurement than inches, so it’s common practice to round the answer to the nearest hundredth. In our example above, we rounded 137.163 to 137.16. 4. Be careful with unit conversions: When using a conversion formula, it’s important to keep track of units. In our example, we made sure to multiply inches by 2.54, which is the conversion factor for inches to centimeters. Why is it Important to Convert Inches to Centimeters? In today’s global economy, it’s important to be able to understand and communicate measurements in different units. While inches are commonly used in the United States and other countries that use the imperial system, centimeters are used in most other countries that use the metric system. Knowing how to convert between these two units of measurement can be incredibly useful, whether you’re traveling, working in an international business, or just curious about the world around you. In summary, 54 inches is equal to 137.16 centimeters. To convert inches to centimeters, simply multiply the number of inches by 2.54. Remember to use a calculator, round to the nearest hundredth, and keep track of units when using conversion formulas. Whether you’re traveling abroad or working in an international industry, knowing how to convert between inches and centimeters can be an incredibly useful skill. 1. What is the difference between inches and centimeters? Inches and centimeters are both units of measurement used to describe length or distance. However, inches are part of the imperial system of measurement used in the United States, while centimeters are part of the metric system used in most other countries. 2. Why do we need to convert between inches and centimeters? In today’s global economy, it’s important to be able to understand and communicate measurements in different units. Knowing how to convert between inches and centimeters allows for easier communication and understanding when working with people or products from different countries. 3. Is there a shortcut for converting inches to centimeters? The formula for converting inches to centimeters is quite simple and straightforward. However, if you frequently need to convert measurements, there are many online converters and mobile apps that can do the math for you. 4. Can you convert any unit of measurement to any other unit? No, it’s important to use the correct conversion factor when converting between units of measurement. For example, the formula for converting inches to centimeters is 2.54, but the formula for converting inches to meters is 0.0254. 5. Why is it important to round to the nearest hundredth when converting inches to centimeters? Centimeters are a smaller unit of measurement than inches, so rounding to the nearest hundredth allows for a more precise measurement. Additionally, it’s common practice to round to the nearest hundredth when working with metric units of measurement. Write a Comment	{"url":"http://scalefactorcalculator.com/index-54.html","timestamp":"2024-11-06T03:52:56Z","content_type":"text/html","content_length":"60142","record_id":"<urn:uuid:0d552418-19bb-475a-bbe0-185f85356ce1>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00665.warc.gz"}
Last change on this file since 87c5f40 was f92aa32, checked in by , 8 years ago thesis conclusions and editting pass • Property mode set to 100644 File size: 44.9 KB 1 %====================================================================== 2 \chapter{Tuples} 3 %====================================================================== 5 \section{Multiple-Return-Value Functions} 6 \label{s:MRV_Functions} 7 In standard C, functions can return at most one value. 8 This restriction results in code which emulates functions with multiple return values by \emph{aggregation} or by \emph{aliasing}. 9 In the former situation, the function designer creates a record type that combines all of the return values into a single type. 10 For example, consider a function returning the most frequently occurring letter in a string, and its frequency. 11 This example is complex enough to illustrate that an array is insufficient, since arrays are homogeneous, and demonstrates a potential pitfall that exists with aliasing. 12 \begin{cfacode} 13 struct mf_ret { 14 int freq; 15 char ch; 16 }; 18 struct mf_ret most_frequent(const char * str) { 19 char freqs [26] = { 0 }; 20 struct mf_ret ret = { 0, 'a' }; 21 for (int i = 0; str[i] != '\0'; ++i) { 22 if (isalpha(str[i])) { // only count letters 23 int ch = tolower(str[i]); // convert to lower case 24 int idx = ch-'a'; 25 if (++freqs[idx] > ret.freq) { // update on new max 26 ret.freq = freqs[idx]; 27 ret.ch = ch; 28 } 29 } 30 } 31 return ret; 32 } 34 const char * str = "hello world"; 35 struct mf_ret ret = most_frequent(str); 36 printf("%s -- %d %c\n", str, ret.freq, ret.ch); 37 \end{cfacode} 38 Of note, the designer must come up with a name for the return type and for each of its fields. 39 Unnecessary naming is a common programming language issue, introducing verbosity and a complication of the user's mental model. 40 That is, adding another named type creates another association in the programmer's mind that needs to be kept track of when reading and writing code. 41 As such, this technique is effective when used sparingly, but can quickly get out of hand if many functions need to return different combinations of types. 43 In the latter approach, the designer simulates multiple return values by passing the additional return values as pointer parameters. 44 The pointer parameters are assigned inside of the routine body to emulate a return. 45 Using the same example, 46 \begin{cfacode} 47 int most_frequent(const char * str, char * ret_ch) { 48 char freqs [26] = { 0 }; 49 int ret_freq = 0; 50 for (int i = 0; str[i] != '\0'; ++i) { 51 if (isalpha(str[i])) { // only count letters 52 int ch = tolower(str[i]); // convert to lower case 53 int idx = ch-'a'; 54 if (++freqs[idx] > ret_freq) { // update on new max 55 ret_freq = freqs[idx]; 56 ret_ch = ch; // assign to out parameter 57 } 58 } 59 } 60 return ret_freq; // only one value returned directly 61 } 63 const char str = "hello world"; 64 char ch; // pre-allocate return value 65 int freq = most_frequent(str, &ch); // pass return value as out parameter 66 printf("%s -- %d %c\n", str, freq, ch); 67 \end{cfacode} 68 Notably, using this approach, the caller is directly responsible for allocating storage for the additional temporary return values, which complicates the call site with a sequence of variable declarations leading up to the call. 69 Also, while a disciplined use of @const@ can give clues about whether a pointer parameter is going to be used as an out parameter, it is not immediately obvious from only the routine signature whether the callee expects such a parameter to be initialized before the call. 70 Furthermore, while many C routines that accept pointers are designed so that it is safe to pass @NULL@ as a parameter, there are many C routines that are not null-safe. 71 On a related note, C does not provide a standard mechanism to state that a parameter is going to be used as an additional return value, which makes the job of ensuring that a value is returned more difficult for the compiler. 72 There is a subtle bug in the previous example, in that @ret_ch@ is never assigned for a string that does not contain any letters, which can lead to undefined behaviour. 73 As with the previous approach, this technique can simulate multiple return values, but in practice it is verbose and error prone. 75 In \CFA, functions can be declared to return multiple values with an extension to the function declaration syntax. 76 Multiple return values are declared as a comma-separated list of types in square brackets in the same location that the return type appears in standard C function declarations. 77 The ability to return multiple values from a function requires a new syntax for the return statement. 78 For consistency, the return statement in \CFA accepts a comma-separated list of expressions in square brackets. 79 The expression resolution phase of the \CFA translator ensures that the correct form is used depending on the values being returned and the return type of the current function. 80 A multiple-returning function with return type @T@ can return any expression that is implicitly convertible to @T@. 81 Using the running example, the @most_frequent@ function can be written using multiple return values as such, 82 \begin{cfacode} 83 [int, char] most_frequent(const char * str) { 84 char freqs [26] = { 0 }; 85 int ret_freq = 0; 86 char ret_ch = 'a'; 87 for (int i = 0; str[i] != '\0'; ++i) { 88 if (isalpha(str[i])) { // only count letters 89 int ch = tolower(str[i]); // convert to lower case 90 int idx = ch-'a'; 91 if (++freqs[idx] > ret_freq) { // update on new max 92 ret_freq = freqs[idx]; 93 ret_ch = ch; 94 } 95 } 96 } 97 return [ret_freq, ret_ch]; 98 } 99 \end{cfacode} 100 This approach provides the benefits of compile-time checking for appropriate return statements as in aggregation, but without the required verbosity of declaring a new named type, which precludes the bug seen with out parameters. 102 The addition of multiple-return-value functions necessitates a syntax for accepting multiple values at the call-site. 103 The simplest mechanism for retaining a return value in C is variable assignment. 104 By assigning the return value into a variable, its value can be retrieved later at any point in the program. 105 As such, \CFA allows assigning multiple values from a function into multiple variables, using a square-bracketed list of lvalue expressions on the left side. 106 \begin{cfacode} 107 const char * str = "hello world"; 108 int freq; 109 char ch; 110 [freq, ch] = most_frequent(str); // assign into multiple variables 111 printf("%s -- %d %c\n", str, freq, ch); 112 \end{cfacode} 113 It is also common to use a function's output as the input to another function. 114 \CFA also allows this case, without any new syntax. 115 When a function call is passed as an argument to another call, the expression resolver attempts to find the best match of actual arguments to formal parameters given all of the possible expression interpretations in the current scope \cite{Bilson03}. 116 For example, 117 \begin{cfacode} 118 void process(int); // (1) 119 void process(char); // (2) 120 void process(int, char); // (3) 121 void process(char, int); // (4) 123 process(most_frequent("hello world")); // selects (3) 124 \end{cfacode} 125 In this case, there is only one option for a function named @most_frequent@ that takes a string as input. 126 This function returns two values, one @int@ and one @char@. 127 There are four options for a function named @process@, but only two that accept two arguments, and of those the best match is (3), which is also an exact match. 128 This expression first calls @most_frequent("hello world")@, which produces the values @3@ and @'l'@, which are fed directly to the first and second parameters of (3), respectively. 130 \section{Tuple Expressions} 131 Multiple-return-value functions provide \CFA with a new syntax for expressing a combination of expressions in the return statement and a combination of types in a function signature. 132 These notions can be generalized to provide \CFA with \emph{tuple expressions} and \emph{tuple types}. 133 A tuple expression is an expression producing a fixed-size, ordered list of values of heterogeneous types. 134 The type of a tuple expression is the tuple of the subexpression types, or a \emph{tuple type}. 135 In \CFA, a tuple expression is denoted by a comma-separated list of expressions enclosed in square brackets. 136 For example, the expression @[5, 'x', 10.5]@ has type @[int, char, double]@. 137 The previous expression has 3 \emph{components}. 138 Each component in a tuple expression can be any \CFA expression, including another tuple expression. 139 The order of evaluation of the components in a tuple expression is unspecified, to allow a compiler the greatest flexibility for program optimization. 140 It is, however, guaranteed that each component of a tuple expression is evaluated for side-effects, even if the result is not used. 141 Multiple-return-value functions can equivalently be called \emph{tuple-returning functions}. 143 \subsection{Tuple Variables} 144 The call-site of the @most_frequent@ routine has a notable blemish, in that it required the preallocation of return variables in a manner similar to the aliasing example, since it is impossible to declare multiple variables of different types in the same declaration in standard C. 145 In \CFA, it is possible to overcome this restriction by declaring a \emph{tuple variable}. 146 \begin{cfacode}[emph=ret, emphstyle=\color{red}] 147 const char * str = "hello world"; 148 [int, char] ret = most_frequent(str); // initialize tuple variable 149 printf("%s -- %d %c\n", str, ret); 150 \end{cfacode} 151 It is now possible to accept multiple values into a single piece of storage, in much the same way that it was previously possible to pass multiple values from one function call to another. 152 These variables can be used in any of the contexts where a tuple expression is allowed, such as in the @printf@ function call. 153 As in the @process@ example, the components of the tuple value are passed as separate parameters to @printf@, allowing very simple printing of tuple expressions. 154 One way to access the individual components is with a simple assignment, as in previous examples. 155 \begin{cfacode} 156 int freq; 157 char ch; 158 [freq, ch] = ret; 159 \end{cfacode} 161 In addition to variables of tuple type, it is also possible to have pointers to tuples, and arrays of tuples. 162 Tuple types can be composed of any types, except for array types, since arrays do not carry their size around, which makes tuple assignment difficult when a tuple contains an array. 163 \begin{cfacode} 164 [double, int] di; 165 [double, int] * pdi 166 [double, int] adi[10]; 167 \end{cfacode} 168 This examples declares a variable of type @[double, int]@, a variable of type pointer to @[double, int]@, and an array of ten @[double, int]@. 170 \subsection{Tuple Indexing} 171 At times, it is desirable to access a single component of a tuple-valued expression without creating unnecessary temporary variables to assign to. 172 Given a tuple-valued expression @e@ and a compile-time constant integer $i$ where $0 \leq i < n$, where $n$ is the number of components in @e@, @e.i@ accesses the $i$\textsuperscript{th} component of @e@. 173 For example, 174 \begin{cfacode} 175 [int, double] x; 176 [char , int] f(); 177 void g(double, int); 178 [int, double] p; 180 int y = x.0; // access int component of x 181 y = f().1; // access int component of f 182 p->0 = 5; // access int component of tuple pointed-to by p 183 g(x.1, x.0); // rearrange x to pass to g 184 double z = [x, f()].0.1; // access second component of first component 185 // of tuple expression 186 \end{cfacode} 187 As seen above, tuple-index expressions can occur on any tuple-typed expression, including tuple-returning functions, square-bracketed tuple expressions, and other tuple-index expressions, provided the retrieved component is also a tuple. 188 This feature was proposed for \KWC but never implemented \cite[p.~45]{Till89}. 190 \subsection{Flattening and Structuring} 191 As evident in previous examples, tuples in \CFA do not have a rigid structure. 192 In function call contexts, tuples support implicit flattening and restructuring conversions. 193 Tuple flattening recursively expands a tuple into the list of its basic components. 194 Tuple structuring packages a list of expressions into a value of tuple type. 195 \begin{cfacode} 196 int f(int, int); 197 int g([int, int]); 198 int h(int, [int, int]); 199 [int, int] x; 200 int y; 202 f(x); // flatten 203 g(y, 10); // structure 204 h(x, y); // flatten & structure 205 \end{cfacode} 206 In \CFA, each of these calls is valid. 207 In the call to @f@, @x@ is implicitly flattened so that the components of @x@ are passed as the two arguments to @f@. 208 For the call to @g@, the values @y@ and @10@ are structured into a single argument of type @[int, int]@ to match the type of the parameter of @g@. 209 Finally, in the call to @h@, @y@ is flattened to yield an argument list of length 3, of which the first component of @x@ is passed as the first parameter of @h@, and the second component of @x@ and @y@ are structured into the second argument of type @[int, int]@. 210 The flexible structure of tuples permits a simple and expressive function call syntax to work seamlessly with both single- and multiple-return-value functions, and with any number of arguments of arbitrarily complex structure. 212 In \KWC \cite{Buhr94a,Till89}, a precursor to \CFA, there were 4 tuple coercions: opening, closing, flattening, and structuring. 213 Opening coerces a tuple value into a tuple of values, while closing converts a tuple of values into a single tuple value. 214 Flattening coerces a nested tuple into a flat tuple, i.e. it takes a tuple with tuple components and expands it into a tuple with only non-tuple components. 215 Structuring moves in the opposite direction, i.e. it takes a flat tuple value and provides structure by introducing nested tuple components. 217 In \CFA, the design has been simplified to require only the two conversions previously described, which trigger only in function call and return situations. 218 Specifically, the expression resolution algorithm examines all of the possible alternatives for an expression to determine the best match. 219 In resolving a function call expression, each combination of function value and list of argument alternatives is examined. 220 Given a particular argument list and function value, the list of argument alternatives is flattened to produce a list of non-tuple valued expressions. 221 Then the flattened list of expressions is compared with each value in the function's parameter list. 222 If the parameter's type is not a tuple type, then the current argument value is unified with the parameter type, and on success the next argument and parameter are examined. 223 If the parameter's type is a tuple type, then the structuring conversion takes effect, recursively applying the parameter matching algorithm using the tuple's component types as the parameter list types. 224 Assuming a successful unification, eventually the algorithm gets to the end of the tuple type, which causes all of the matching expressions to be consumed and structured into a tuple expression. 225 For example, in 226 \begin{cfacode} 227 int f(int, [double, int]); 228 f([5, 10.2], 4); 229 \end{cfacode} 230 There is only a single definition of @f@, and 3 arguments with only single interpretations. 231 First, the argument alternative list @[5, 10.2], 4@ is flattened to produce the argument list @5, 10.2, 4@. 232 Next, the parameter matching algorithm begins, with $P = $@int@ and $A = $@int@, which unifies exactly. 233 Moving to the next parameter and argument, $P = $@[double, int]@ and $A = $@double@. 234 This time, the parameter is a tuple type, so the algorithm applies recursively with $P' = $@double@ and $A = $@double@, which unifies exactly. 235 Then $P' = $@int@ and $A = $@double@, which again unifies exactly. 236 At this point, the end of $P'$ has been reached, so the arguments @10.2, 4@ are structured into the tuple expression @[10.2, 4]@. 237 Finally, the end of the parameter list $P$ has also been reached, so the final expression is @f(5, [10.2, 4])@. 239 \section{Tuple Assignment} 240 \label{s:TupleAssignment} 241 An assignment where the left side of the assignment operator has a tuple type is called tuple assignment. 242 There are two kinds of tuple assignment depending on whether the right side of the assignment operator has a tuple type or a non-tuple type, called \emph{Multiple} and \emph{Mass} Assignment, 243 \begin{cfacode} 244 int x; 245 double y; 246 [int, double] z; 247 [y, x] = 3.14; // mass assignment 248 [x, y] = z; // multiple assignment 249 z = 10; // mass assignment 250 z = [x, y]; // multiple assignment 251 \end{cfacode} 252 Let $L_i$ for $i$ in $[0, n)$ represent each component of the flattened left side, $R_i$ represent each component of the flattened right side of a multiple assignment, and $R$ represent the right side of a mass assignment. 254 For a multiple assignment to be valid, both tuples must have the same number of elements when flattened. Multiple assignment assigns $R_i$ to $L_i$ for each $i$. 255 That is, @?=?(&$L_i$, $R_i$)@ must be a well-typed expression. 256 In the previous example, @[x, y] = z@, @z@ is flattened into @z.0, z.1@, and the assignments @x = z.0@ and @y = z.1@ happen. 258 A mass assignment assigns the value $R$ to each $L_i$. 259 For a mass assignment to be valid, @?=?(&$L_i$, $R$)@ must be a well-typed expression. 260 These semantics differ from C cascading assignment (e.g. @a=b=c@) in that conversions are applied to $R$ in each individual assignment, which prevents data loss from the chain of conversions that can happen during a cascading assignment. 261 For example, @[y, x] = 3.14@ performs the assignments @y = 3.14@ and @x = 3.14@, which results in the value @3.14@ in @y@ and the value @3@ in @x@. 262 On the other hand, the C cascading assignment @y = x = 3.14@ performs the assignments @x = 3.14@ and @y = x@, which results in the value @3@ in @x@, and as a result the value @3@ in @y@ as well. 264 Both kinds of tuple assignment have parallel semantics, such that each value on the left side and right side is evaluated \emph{before} any assignments occur. 265 As a result, it is possible to swap the values in two variables without explicitly creating any temporary variables or calling a function, 266 \begin{cfacode} 267 int x = 10, y = 20; 268 [x, y] = [y, x]; 269 \end{cfacode} 270 After executing this code, @x@ has the value @20@ and @y@ has the value @10@. 272 In \CFA, tuple assignment is an expression where the result type is the type of the left side of the assignment, as in normal assignment. 273 That is, a tuple assignment produces the value of the left-hand side after assignment. 274 These semantics allow cascading tuple assignment to work out naturally in any context where a tuple is permitted. 275 These semantics are a change from the original tuple design in \KWC \cite{Till89}, wherein tuple assignment was a statement that allows cascading assignments as a special case. 276 Restricting tuple assignment to statements was an attempt to to fix what was seen as a problem with assignment, wherein it can be used in many different locations, such as in function-call argument position. 277 While permitting assignment as an expression does introduce the potential for subtle complexities, it is impossible to remove assignment expressions from \CFA without affecting backwards 278 Furthermore, there are situations where permitting assignment as an expression improves readability by keeping code succinct and reducing repetition, and complicating the definition of tuple assignment puts a greater cognitive burden on the user. 279 In another language, tuple assignment as a statement could be reasonable, but it would be inconsistent for tuple assignment to be the only kind of assignment that is not an expression. 280 In addition, \KWC permits the compiler to optimize tuple assignment as a block copy, since it does not support user-defined assignment operators. 281 This optimization could be implemented in \CFA, but it requires the compiler to verify that the selected assignment operator is trivial. 283 The following example shows multiple, mass, and cascading assignment used in one expression 284 \begin{cfacode} 285 int a, b; 286 double c, d; 287 [void] f([int, int]); 288 f([c, a] = [b, d] = 1.5); // assignments in parameter list 289 \end{cfacode} 290 The tuple expression begins with a mass assignment of @1.5@ into @[b, d]@, which assigns @1.5@ into @b@, which is truncated to @1@, and @1.5@ into @d@, producing the tuple @[1, 1.5]@ as a 291 That tuple is used as the right side of the multiple assignment (i.e., @[c, a] = [1, 1.5]@) that assigns @1@ into @c@ and @1.5@ into @a@, which is truncated to @1@, producing the result @[1, 1] 292 Finally, the tuple @[1, 1]@ is used as an expression in the call to @f@. 294 \subsection{Tuple Construction} 295 Tuple construction and destruction follow the same rules and semantics as tuple assignment, except that in the case where there is no right side, the default constructor or destructor is called on each component of the tuple. 296 \begin{cfacode} 297 struct S; 298 void ?{}(S ); // (1) 299 void ?{}(S , int); // (2) 300 void ?{}(S * double); // (3) 301 void ?{}(S , S); // (4) 303 [S, S] x = [3, 6.28]; // uses (2), (3), specialized constructors 304 [S, S] y; // uses (1), (1), default constructor 305 [S, S] z = x.0; // uses (4), (4), copy constructor 306 \end{cfacode} 307 In this example, @x@ is initialized by the multiple constructor calls @?{}(&x.0, 3)@ and @?{}(&x.1, 6.28)@, while @y@ is initialized by two default constructor calls @?{}(&y.0)@ and @?{}(&y.1)@. 308 @z@ is initialized by mass copy constructor calls @?{}(&z.0, x.0)@ and @?{}(&z.1, x.0)@. 309 Finally, @x@, @y@, and @z@ are destructed, i.e. the calls @^?{}(&x.0)@, @^?{}(&x.1)@, @^?{}(&y.0)@, @^?{}(&y.1)@, @^?{}(&z.0)@, and @^?{}(&z.1)@. 311 It is possible to define constructors and assignment functions for tuple types that provide new semantics, if the existing semantics do not fit the needs of an application. 312 For example, the function @void ?{}([T, U] , S);@ can be defined to allow a tuple variable to be constructed from a value of type @S@. 313 \begin{cfacode} 314 struct S { int x; double y; }; 315 void ?{}([int, double] * this, S s) { 316 this->0 = s.x; 317 this->1 = s.y; 318 } 319 \end{cfacode} 320 Due to the structure of generated constructors, it is possible to pass a tuple to a generated constructor for a type with a member prefix that matches the type of the tuple. 321 For example, 322 \begin{cfacode} 323 struct S { int x; double y; int z }; 324 [int, double] t; 325 S s = t; 326 \end{cfacode} 327 The initialization of @s@ with @t@ works by default because @t@ is flattened into its components, which satisfies the generated field constructor @?{}(S , int, double)@ to initialize the first two values. 329 \section{Member-Access Tuple Expression} 330 \label{s:MemberAccessTuple} 331 It is possible to access multiple fields from a single expression using a \emph{Member-Access Tuple Expression}. 332 The result is a single tuple-valued expression whose type is the tuple of the types of the members. 333 For example, 334 \begin{cfacode} 335 struct S { int x; double y; char z; } s; 336 s.[x, y, z]; 337 \end{cfacode} 338 Here, the type of @s.[x, y, z]@ is @[int, double, char ]@. 339 A member tuple expression has the form @a.[x, y, z];@ where @a@ is an expression with type @T@, where @T@ supports member access expressions, and @x, y, z@ are all members of @T@ with types @T$ _x$@, @T$_y$@, and @T$_z$@ respectively. 340 Then the type of @a.[x, y, z]@ is @[T_x, T_y, T_z]@. 342 Since tuple index expressions are a form of member-access expression, it is possible to use tuple-index expressions in conjunction with member tuple expressions to manually restructure a tuple (e.g., rearrange components, drop components, duplicate components, etc.). 343 \begin{cfacode} 344 [int, int, long, double] x; 345 void f(double, long); 347 f(x.[0, 3]); // f(x.0, x.3) 348 x.[0, 1] = x.[1, 0]; // [x.0, x.1] = [x.1, x.0] 349 [long, int, long] y = x.[2, 0, 2]; 350 \end{cfacode} 352 It is possible for a member tuple expression to contain other member access expressions. 353 For example, 354 \begin{cfacode} 355 struct A { double i; int j; }; 356 struct B { int k; short l; }; 357 struct C { int x; A y; B z; } v; 358 v.[x, y.[i, j], z.k]; 359 \end{cfacode} 360 This expression is equivalent to @[v.x, [v.y.i, v.y.j], v.z.k]@. 361 That is, the aggregate expression is effectively distributed across the tuple, which allows simple and easy access to multiple components in an aggregate, without repetition. 362 It is guaranteed that the aggregate expression to the left of the @.@ in a member tuple expression is evaluated exactly once. 363 As such, it is safe to use member tuple expressions on the result of a side-effecting function. 364 \begin{cfacode} 365 [int, float, double] f(); 366 [double, float] x = f().[2, 1]; 367 \end{cfacode} 369 In \KWC, member tuple expressions are known as \emph{record field tuples} \cite{Till89}. 370 Since \CFA permits these tuple-access expressions using structures, unions, and tuples, \emph{member tuple expression} or \emph{field tuple expression} is more appropriate. 372 It is possible to extend member-access expressions further. 373 Currently, a member-access expression whose member is a name requires that the aggregate is a structure or union, while a constant integer member requires the aggregate to be a tuple. 374 In the interest of orthogonal design, \CFA could apply some meaning to the remaining combinations as well. 375 For example, 376 \begin{cfacode} 377 struct S { int x, y; } s; 378 [S, S] z; 380 s.x; // access member 381 z.0; // access component 383 s.1; // ??? 384 z.y; // ??? 385 \end{cfacode} 386 One possibility is for @s.1@ to select the second member of @s@. 387 Under this interpretation, it becomes possible to not only access members of a struct by name, but also by position. 388 Likewise, it seems natural to open this mechanism to enumerations as well, wherein the left side would be a type, rather than an expression. 389 One benefit of this interpretation is familiarity, since it is extremely reminiscent of tuple-index expressions. 390 On the other hand, it could be argued that this interpretation is brittle in that changing the order of members or adding new members to a structure becomes a brittle operation. 391 This problem is less of a concern with tuples, since modifying a tuple affects only the code that directly uses the tuple, whereas modifying a structure has far reaching consequences for every instance of the structure. 393 As for @z.y@, one interpretation is to extend the meaning of member tuple expressions. 394 That is, currently the tuple must occur as the member, i.e. to the right of the dot. 395 Allowing tuples to the left of the dot could distribute the member across the elements of the tuple, in much the same way that member tuple expressions distribute the aggregate across the member 396 In this example, @z.y@ expands to @[z.0.y, z.1.y]@, allowing what is effectively a very limited compile-time field-sections map operation, where the argument must be a tuple containing only aggregates having a member named @y@. 397 It is questionable how useful this would actually be in practice, since structures often do not have names in common with other structures, and further this could cause maintainability issues in that it encourages programmers to adopt very simple naming conventions to maximize the amount of overlap between different types. 398 Perhaps more useful would be to allow arrays on the left side of the dot, which would likewise allow mapping a field access across the entire array, producing an array of the contained fields. 399 The immediate problem with this idea is that C arrays do not carry around their size, which would make it impossible to use this extension for anything other than a simple stack allocated array. 401 Supposing this feature works as described, it would be necessary to specify an ordering for the expansion of member-access expressions versus member-tuple expressions. 402 \begin{cfacode} 403 struct { int x, y; }; 404 [S, S] z; 405 z.[x, y]; // ??? 406 // => [z.0, z.1].[x, y] 407 // => [z.0.x, z.0.y, z.1.x, z.1.y] 408 // or 409 // => [z.x, z.y] 410 // => [[z.0, z.1].x, [z.0, z.1].y] 411 // => [z.0.x, z.1.x, z.0.y, z.1.y] 412 \end{cfacode} 413 Depending on exactly how the two tuples are combined, different results can be achieved. 414 As such, a specific ordering would need to be imposed to make this feature useful. 415 Furthermore, this addition moves a member-tuple expression's meaning from being clear statically to needing resolver support, since the member name needs to be distributed appropriately over each member of the tuple, which could itself be a tuple. 417 A second possibility is for \CFA to have named tuples, as they exist in Swift and D. 418 \begin{cfacode} 419 typedef [int x, int y] Point2D; 420 Point2D p1, p2; 421 p1.x + p1.y + p2.x + p2.y; 422 p1.0 + p1.1 + p2.0 + p2.1; // equivalent 423 \end{cfacode} 424 In this simpler interpretation, a tuple type carries with it a list of possibly empty identifiers. 425 This approach fits naturally with the named return-value feature, and would likely go a long way towards implementing it. 427 Ultimately, the first two extensions introduce complexity into the model, with relatively little perceived benefit, and so were dropped from consideration. 428 Named tuples are a potentially useful addition to the language, provided they can be parsed with a reasonable syntax. 431 \section{Casting} 432 In C, the cast operator is used to explicitly convert between types. 433 In \CFA, the cast operator has a secondary use, which is type ascription, since it force the expression resolution algorithm to choose the lowest cost conversion to the target type. 434 That is, a cast can be used to select the type of an expression when it is ambiguous, as in the call to an overloaded function. 435 \begin{cfacode} 436 int f(); // (1) 437 double f(); // (2) 439 f(); // ambiguous - (1),(2) both equally viable 440 (int)f(); // choose (2) 441 \end{cfacode} 442 Since casting is a fundamental operation in \CFA, casts need to be given a meaningful interpretation in the context of tuples. 443 Taking a look at standard C provides some guidance with respect to the way casts should work with tuples. 444 \begin{cfacode}[numbers=left] 445 int f(); 446 void g(); 448 (void)f(); // valid, ignore results 449 (int)g(); // invalid, void cannot be converted to int 451 struct A { int x; }; 452 (struct A)f(); // invalid 453 \end{cfacode} 454 In C, line 4 is a valid cast, which calls @f@ and discards its result. 455 On the other hand, line 5 is invalid, because @g@ does not produce a result, so requesting an @int@ to materialize from nothing is nonsensical. 456 Finally, line 8 is also invalid, because in C casts only provide conversion between scalar types \cite[p.~91]{C11}. 457 For consistency, this implies that any case wherein the number of components increases as a result of the cast is invalid, while casts that have the same or fewer number of components may be 459 Formally, a cast to tuple type is valid when $T_n \leq S_m$, where $T_n$ is the number of components in the target type and $S_m$ is the number of components in the source type, and for each $i$ in $[0, n)$, $S_i$ can be cast to $T_i$. 460 Excess elements ($S_j$ for all $j$ in $[n, m)$) are evaluated, but their values are discarded so that they are not included in the result expression. 461 This discarding naturally follows the way that a cast to void works in C. 463 For example, 464 \begin{cfacode} 465 [int, int, int] f(); 466 [int, [int, int], int] g(); 468 ([int, double])f(); // (1) 469 ([int, int, int])g(); // (2) 470 ([void, [int, int]])g(); // (3) 471 ([int, int, int, int])g(); // (4) 472 ([int, [int, int, int]])g(); // (5) 473 \end{cfacode} 475 (1) discards the last element of the return value and converts the second element to type double. 476 Since @int@ is effectively a 1-element tuple, (2) discards the second component of the second element of the return value of @g@. 477 If @g@ is free of side effects, this is equivalent to @[(int)(g().0), (int)(g().1.0), (int)(g().2)]@. 478 Since @void@ is effectively a 0-element tuple, (3) discards the first and third return values, which is effectively equivalent to @[(int)(g().1.0), (int)(g().1.1)]@). 480 % will this always hold true? probably, as constructors should give all of the conversion power we need. if casts become function calls, what would they look like? would need a way to specify the target type, which seems awkward. Also, C++ basically only has this because classes are closed to extension, while we don't have that problem (can have floating constructors for any type). 481 Note that a cast is not a function call in \CFA, so flattening and structuring conversions do not occur for cast expressions. 482 As such, (4) is invalid because the cast target type contains 4 components, while the source type contains only 3. 483 Similarly, (5) is invalid because the cast @([int, int, int])(g().1)@ is invalid. 484 That is, it is invalid to cast @[int, int]@ to @[int, int, int]@. 486 \section{Polymorphism} 487 Due to the implicit flattening and structuring conversions involved in argument passing, @otype@ and @dtype@ parameters are restricted to matching only with non-tuple types. 488 \begin{cfacode} 489 forall(otype T, dtype U) 490 void f(T x, U * y); 492 f([5, "hello"]); 493 \end{cfacode} 494 In this example, @[5, "hello"]@ is flattened, so that the argument list appears as @5, "hello"@. 495 The argument matching algorithm binds @T@ to @int@ and @U@ to @const char@, and calls the function as normal. 497 Tuples can contain otype and dtype components. 498 For example, a plus operator can be written to add two triples of a type together. 499 \begin{cfacode} 500 forall(otype T \| { T ?+?(T, T); }) 501 [T, T, T] ?+?([T, T, T] x, [T, T, T] y) { 502 return [x.0+y.0, x.1+y.1, x.2+y.2]; 503 } 504 [int, int, int] x; 505 int i1, i2, i3; 506 [i1, i2, i3] = x + ([10, 20, 30]); 507 \end{cfacode} 508 Note that due to the implicit tuple conversions, this function is not restricted to the addition of two triples. 509 A call to this plus operator type checks as long as a total of 6 non-tuple arguments are passed after flattening, and all of the arguments have a common type that can bind to @T@, with a pairwise @?+?@ over @T@. 510 For example, these expressions also succeed and produce the same value. 511 \begin{cfacode} 512 ([x.0, x.1]) + ([x.2, 10, 20, 30]); // x + ([10, 20, 30]) 513 x.0 + ([x.1, x.2, 10, 20, 30]); // x + ([10, 20, 30]) 514 \end{cfacode} 515 This presents a potential problem if structure is important, as these three expressions look like they should have different meanings. 516 Furthermore, these calls can be made ambiguous by introducing seemingly different functions. 517 \begin{cfacode} 518 forall(otype T \| { T ?+?(T, T); }) 519 [T, T, T] ?+?([T, T] x, [T, T, T, T]); 520 forall(otype T \| { T ?+?(T, T); }) 521 [T, T, T] ?+?(T x, [T, T, T, T, T]); 522 \end{cfacode} 523 It is also important to note that these calls could be disambiguated if the function return types were different, as they likely would be for a reasonable implementation of @?+?@, since the return type is used in overload resolution. 524 Still, these semantics are a deficiency of the current argument matching algorithm, and depending on the function, differing return values may not always be appropriate. 525 These issues could be rectified by applying an appropriate cost to the structuring and flattening conversions, which are currently 0-cost conversions. 526 Care would be needed in this case to ensure that exact matches do not incur such a cost. 527 \begin{cfacode} 528 void f([int, int], int, int); 530 f([0, 0], 0, 0); // no cost 531 f(0, 0, 0, 0); // cost for structuring 532 f([0, 0,], [0, 0]); // cost for flattening 533 f([0, 0, 0], 0); // cost for flattening and structuring 534 \end{cfacode} 536 Until this point, it has been assumed that assertion arguments must match the parameter type exactly, modulo polymorphic specialization (i.e., no implicit conversions are applied to assertion 537 This decision presents a conflict with the flexibility of tuples. 538 \subsection{Assertion Inference} 539 \begin{cfacode} 540 int f([int, double], double); 541 forall(otype T, otype U \| { T f(T, U, U); }) 542 void g(T, U); 543 g(5, 10.21); 544 \end{cfacode} 545 If assertion arguments must match exactly, then the call to @g@ cannot be resolved, since the expected type of @f@ is flat, while the only @f@ in scope requires a tuple type. 546 Since tuples are fluid, this requirement reduces the usability of tuples in polymorphic code. 547 To ease this pain point, function parameter and return lists are flattened for the purposes of type unification, which allows the previous example to pass expression resolution. 549 This relaxation is made possible by extending the existing thunk generation scheme, as described by Bilson \cite{Bilson03}. 550 Now, whenever a candidate's parameter structure does not exactly match the formal parameter's structure, a thunk is generated to specialize calls to the actual function. 551 \begin{cfacode} 552 int _thunk(int _p0, double _p1, double _p2) { 553 return f([_p0, _p1], _p2); 554 } 555 \end{cfacode} 556 Essentially, this provides flattening and structuring conversions to inferred functions, improving the compatibility of tuples and polymorphism. 558 \section{Implementation} 559 Tuples are implemented in the \CFA translator via a transformation into generic types. 560 The first time an $N$-tuple is seen for each $N$ in a scope, a generic type with $N$ type parameters is generated. 561 For example, 562 \begin{cfacode} 563 [int, int] f() { 564 [double, double] x; 565 [int, double, int] y; 566 } 567 \end{cfacode} 568 Is transformed into 569 \begin{cfacode} 570 forall(dtype T0, dtype T1 \| sized(T0) \| sized(T1)) 571 struct _tuple2 { // generated before the first 2-tuple 572 T0 field_0; 573 T1 field_1; 574 }; 575 _tuple2_(int, int) f() { 576 _tuple2_(double, double) x; 577 forall(dtype T0, dtype T1, dtype T2 \| sized(T0) \| sized(T1) \| sized(T2)) 578 struct _tuple3 { // generated before the first 3-tuple 579 T0 field_0; 580 T1 field_1; 581 T2 field_2; 582 }; 583 _tuple3_(int, double, int) y; 584 } 585 \end{cfacode} 587 Tuple expressions are then simply converted directly into compound literals 588 \begin{cfacode} 589 [5, 'x', 1.24]; 590 \end{cfacode} 591 Becomes 592 \begin{cfacode} 593 (_tuple3_(int, char, double)){ 5, 'x', 1.24 }; 594 \end{cfacode} 596 Since tuples are essentially structures, tuple indexing expressions are just field accesses. 597 \begin{cfacode} 598 void f(int, [double, char]); 599 [int, double] x; 601 x.0+x.1; 602 printf("%d %g\n", x); 603 f(x, 'z'); 604 \end{cfacode} 605 Is transformed into 606 \begin{cfacode} 607 void f(int, _tuple2_(double, char)); 608 _tuple2_(int, double) x; 610 x.field_0+x.field_1; 611 printf("%d %g\n", x.field_0, x.field_1); 612 f(x.field_0, (_tuple2){ x.field_1, 'z' }); 613 \end{cfacode} 614 Note that due to flattening, @x@ used in the argument position is converted into the list of its fields. 615 In the call to @f@, the second and third argument components are structured into a tuple argument. 617 Expressions that may contain side effects are made into \emph{unique expressions} before being expanded by the flattening conversion. 618 Each unique expression is assigned an identifier and is guaranteed to be executed exactly once. 619 \begin{cfacode} 620 void g(int, double); 621 [int, double] h(); 622 g(h()); 623 \end{cfacode} 624 Internally, this is converted to pseudo-\CFA 625 \begin{cfacode} 626 void g(int, double); 627 [int, double] h(); 628 lazy [int, double] unq0 = h(); // deferred execution 629 g(unq0.0, unq0.1); // execute h() once 630 \end{cfacode} 631 That is, the function @h@ is evaluated lazily and its result is stored for subsequent accesses. 632 Ultimately, unique expressions are converted into two variables and an expression. 633 \begin{cfacode} 634 void g(int, double); 635 [int, double] h(); 637 _Bool _unq0_finished_ = 0; 638 [int, double] _unq0; 639 g( 640 (_unq0_finished_ ? _unq0 : (_unq0 = h(), _unq0_finished_ = 1, _unq0)).0, 641 (_unq0_finished_ ? _unq0 : (_unq0 = h(), _unq0_finished_ = 1, _unq0)).1, 642 ); 643 \end{cfacode} 644 Since argument evaluation order is not specified by the C programming language, this scheme is built to work regardless of evaluation order. 645 The first time a unique expression is executed, the actual expression is evaluated and the accompanying boolean is set to true. 646 Every subsequent evaluation of the unique expression then results in an access to the stored result of the actual expression. 648 Currently, the \CFA translator has a very broad, imprecise definition of impurity (side-effects), where every function call is assumed to be impure. 649 This notion could be made more precise for certain intrinsic, auto-generated, and built-in functions, and could analyze function bodies, when they are available, to recursively detect impurity, to eliminate some unique expressions. 650 It is possible that lazy evaluation could be exposed to the user through a lazy keyword with little additional effort. 652 Tuple member expressions are recursively expanded into a list of member access expressions. 653 \begin{cfacode} 654 [int, [double, int, double], int]] x; 655 x.[0, 1.[0, 2]]; 656 \end{cfacode} 657 which becomes 658 \begin{cfacode} 659 [x.0, [x.1.0, x.1.2]]; 660 \end{cfacode} 661 Tuple-member expressions also take advantage of unique expressions in the case of possible impurity. 663 Finally, the various kinds of tuple assignment, constructors, and destructors generate GNU C statement expressions. 664 For example, a mass assignment 665 \begin{cfacode} 666 int x, z; 667 double y; 668 [double, double] f(); 670 [x, y, z] = 1.5; // mass assignment 671 \end{cfacode} 672 Generates the following 673 \begin{cfacode} 674 // [x, y, z] = 1.5; 675 _tuple3_(int, double, int) _tmp_stmtexpr_ret0; 676 ({ 677 // assign LHS address temporaries 678 int __massassign_L0 = &x; // ?{} 679 double __massassign_L1 = &y; // ?{} 680 int __massassign_L2 = &z; // ?{} 682 // assign RHS value temporary 683 double __massassign_R0 = 1.5; // ?{} 685 ({ // tuple construction - construct statement expr return variable 686 // assign LHS address temporaries 687 int __multassign_L0 = (int )&_tmp_stmtexpr_ret0.0; // ?{} 688 double __multassign_L1 = (double )&_tmp_stmtexpr_ret0.1; // ?{} 689 int __multassign_L2 = (int )&_tmp_stmtexpr_ret0.2; // ?{} 691 // assign RHS value temporaries and perform mass assignment to L0, L1, L2 692 int __multassign_R0 = (__massassign_L0=(int)__massassign_R0); // ?{} 693 double __multassign_R1 = (__massassign_L1=__massassign_R0); // ?{} 694 int __multassign_R2 = (__massassign_L2=(int)__massassign_R0); // ?{} 696 // perform construction of statement expr return variable using 697 // RHS value temporary 698 ((__multassign_L0 = __multassign_R0 / ?{} /), 699 (__multassign_L1 = __multassign_R1 /* ?{} /), 700 (__multassign_L2 = __multassign_R2 /* ?{} /)); 701 }); 702 _tmp_stmtexpr_ret0; 703 }); 704 ({ // tuple destruction - destruct assign expr value 705 int __massassign_L3 = (int )&_tmp_stmtexpr_ret0.0; // ?{} 706 double __massassign_L4 = (double )&_tmp_stmtexpr_ret0.1; // ?{} 707 int __massassign_L5 = (int )&_tmp_stmtexpr_ret0.2; // ?{} 708 ((__massassign_L3 /* ^?{} /), 709 (__massassign_L4 /* ^?{} /), 710 (__massassign_L5 /* ^?{} /)); 711 }); 712 \end{cfacode} 713 A variable is generated to store the value produced by a statement expression, since its fields may need to be constructed with a non-trivial constructor and it may need to be referred to multiple time, e.g., in a unique expression. 714 $N$ LHS variables are generated and constructed using the address of the tuple components, and a single RHS variable is generated to store the value of the RHS without any loss of precision. 715 A nested statement expression is generated that performs the individual assignments and constructs the return value using the results of the individual assignments. 716 Finally, the statement expression temporary is destroyed at the end of the expression. 718 Similarly, a multiple assignment 719 \begin{cfacode} 720 [x, y, z] = [f(), 3]; // multiple assignment 721 \end{cfacode} 722 Generates 723 \begin{cfacode} 724 // [x, y, z] = [f(), 3]; 725 _tuple3_(int, double, int) _tmp_stmtexpr_ret0; 726 ({ 727 // assign LHS address temporaries 728 int __multassign_L0 = &x; // ?{} 729 double __multassign_L1 = &y; // ?{} 730 int __multassign_L2 = &z; // ?{} 732 // assign RHS value temporaries 733 _tuple2_(double, double) _tmp_cp_ret0; 734 _Bool _unq0_finished_ = 0; 735 double __multassign_R0 = 736 (_unq0_finished_ ? 737 _tmp_cp_ret0 : 738 (_tmp_cp_ret0=f(), _unq0_finished_=1, _tmp_cp_ret0)).0; // ?{} 739 double __multassign_R1 = 740 (_unq0_finished_ ? 741 _tmp_cp_ret0 : 742 (_tmp_cp_ret0=f(), _unq0_finished_=1, _tmp_cp_ret0)).1; // ?{} 743 ({ // tuple destruction - destruct f() return temporary - tuple destruction 744 // assign LHS address temporaries 745 double __massassign_L3 = (double )&_tmp_cp_ret0.0; // ?{} 746 double __massassign_L4 = (double )&_tmp_cp_ret0.1; // ?{} 747 // perform destructions - intrinsic, so NOP 748 ((__massassign_L3 / ^?{} /), 749 (__massassign_L4 /* ^?{} /)); 750 }); 751 int __multassign_R2 = 3; // ?{} 753 ({ // tuple construction - construct statement expr return variable 754 // assign LHS address temporaries 755 int __multassign_L3 = (int )&_tmp_stmtexpr_ret0.0; // ?{} 756 double __multassign_L4 = (double )&_tmp_stmtexpr_ret0.1; // ?{} 757 int __multassign_L5 = (int )&_tmp_stmtexpr_ret0.2; // ?{} 759 // assign RHS value temporaries and perform multiple assignment to L0, L1, L2 760 int __multassign_R3 = (__multassign_L0=(int)__multassign_R0); // ?{} 761 double __multassign_R4 = (__multassign_L1=__multassign_R1); // ?{} 762 int __multassign_R5 = (__multassign_L2=__multassign_R2); // ?{} 764 // perform construction of statement expr return variable using 765 // RHS value temporaries 766 ((__multassign_L3=__multassign_R3 / ?{} /), 767 (__multassign_L4=__multassign_R4 /* ?{} /), 768 (__multassign_L5=__multassign_R5 /* ?{} /)); 769 }); 770 _tmp_stmtexpr_ret0; 771 }); 772 ({ // tuple destruction - destruct assign expr value 773 // assign LHS address temporaries 774 int __massassign_L5 = (int )&_tmp_stmtexpr_ret0.0; // ?{} 775 double __massassign_L6 = (double )&_tmp_stmtexpr_ret0.1; // ?{} 776 int __massassign_L7 = (int )&_tmp_stmtexpr_ret0.2; // ?{} 777 // perform destructions - intrinsic, so NOP 778 ((__massassign_L5 /* ^?{} /), 779 (__massassign_L6 /* ^?{} /), 780 (__massassign_L7 /* ^?{} */)); 781 }); 782 \end{cfacode} 783 The difference here is that $N$ RHS values are stored into separate temporary variables. 785 The use of statement expressions allows the translator to arbitrarily generate additional temporary variables as needed, but binds the implementation to a non-standard extension of the C 786 There are other places where the \CFA translator makes use of GNU C extensions, such as its use of nested functions, so this is not a new restriction. for help on using the repository browser.	{"url":"https://cforall.uwaterloo.ca/trac/browser/doc/rob_thesis/tuples.tex?rev=87c5f40008c3b7a62cbe3a1a3f2827bb9794ac3a&desc=1","timestamp":"2024-11-04T05:59:59Z","content_type":"application/xhtml+xml","content_length":"136136","record_id":"<urn:uuid:9bc55877-da8f-4f3f-b90a-141e70378813>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00542.warc.gz"}
Kondo effect From Scholarpedia Alex C Hewson and Jun Kondo (2009), Scholarpedia, 4(3):7529. doi:10.4249/scholarpedia.7529 revision #91408 [link to/cite this article] The Kondo effect is an unusual scattering mechanism of conduction electrons in a metal due to magnetic impurities, which contributes a term to the electrical resistivity that increases logarithmically with temperature as the temperature T is lowered (as $\log(T)$). It is sometimes used more generally to describe many-body scattering processes from impurities or ions which have low energy quantum mechanical degrees of freedom. In this more general sense it has become a key concept in condensed matter physics in understanding the behavior of metallic systems with strongly interacting electrons. Background to the Kondo Effect The dominant contribution to the electrical resistivity in metals arises from the scattering of the conduction electrons by the nuclei as they vibrate about their equilibrium positions (lattice vibrations). This scattering increases rapidly with temperature as more and more lattice vibrations are excited. As a result the electrical resistivity increases monotonically with temperature in most metals; there is also a residual temperature-independent resistivity due to the scattering of the electrons with defects, impurities and vacancies in the very low temperature range where the lattice vibrations have almost died out. In 1934, however, a resistance minimum was observed in gold as a function of temperature (de Haas, de Boer and van den Berg 1934), indicating that there must be some additional scattering mechanism giving an anomalous contribution to the resistivity--- one which increases in strength as the temperature is lowered. Other examples of metals showing a resistance minimum were later observed, and its origin was a longstanding puzzle for about 30 years. In the early 1960s it was recognised that the resistance minima are associated with magnetic impurities in the metallic host --- a magnetic impurity being one which has a local magnetic moment due to the spin of unpaired electrons in its atomic-like d or f shell. A carefully studied example showing the correlation between the resistance minima and the number of magnetic impurities is that of iron impurities in gold (van den Berg, 1964). In 1964 Kondo showed in detail how certain scattering processes from magnetic impurities --- those in which the internal spin state of the impurity and scattered electron are exchanged--- could give rise to a resistivity contribution behaving as ${\rm log}(T)\ ,$ and hence provide a satisfactory explanation of the observed resistance minima --- a solution to the longstanding puzzle (see Figure 2). Details of Kondo's Calculation Consider a small amount of magnetic impurities in a metal. In order to calculate the electrical resistivity arising from these impurities one first calculates the scattering probability for an electron from a single impurity and then multiplies it by the number of impurities. Taking into account the spins of the electron and the impurity, we consider the case when the electron with wave number $ k\ ,$ and spin down $\downarrow\ ,$ collides with the impurity in a state with its spin up $ \uparrow$ and is scattered into a state with wave number$ k'$ with spin down $\ downarrow,$ while the impurity remains in a state with spin up $\uparrow\ .$ Let us write the matrix element for this process as \[\tag{1} J(k\downarrow,\uparrow\to k'\downarrow,\uparrow) \] This type of scattering process had already been taken into account. Kondo (1964) considered a higher order correction term where the electron is scattered into the state with wavenumber $ k''$ and spin up $ \uparrow$ leaving the impurity is a spin down state $\downarrow$ ---- a scattering process involving a spin flip of the impurity. This is only an intermediate state, and we have to take into account a further scattering process to arrive at the same final state as in equation (1), in which the spin flip is reversed, so that the scattered electron is in the state $ k',\downarrow$ and the impurity is returned to the state with spin up $\uparrow$ (for a diagrammatic representation of this scattering process see Figure 1). We sum $k''$ over all possible intermediate states and so, according to quantum mechanics, the total matrix element for this process is given by \[\tag{2} \sum _{k''}J(k\downarrow,\uparrow\to k''\uparrow,\downarrow). J(k''\uparrow,\downarrow\to k'\downarrow,\uparrow) {1-f_{k''}\over \epsilon_k-\epsilon_{k''}}. \] Here $ \epsilon_k$ is the energy of the electron with wavenumber $ k\ ,$ $f_{k}$ is unity if the state $k $ is occupied and zero if it is empty. The factor $ 1-f_{k''} $ is to exclude an occupied state $ k'' $ from the sum. In calculating eq.(2) let us assume that $ J $ is a constant. The sum over $k'' $ is replaced by an integral using the density of states $ \rho(\epsilon_{k''}) \ ,$ which we assume is also a constant. Then eq. (2) becomes \[\tag{3} J^2\rho \int {1-f_{k''}\over \epsilon_k-\epsilon_{k''}}\,d\epsilon_{k''}=J^2\rho\int_{\epsilon_{\rm F}}^D {1\over \epsilon_k-\epsilon_{k''}}\,d\epsilon_{k''}. Here we assume that the electron energy takes a value between $ 0$ and $ D$ and that the states below the Fermi level $ \epsilon_{\rm F}$ are occupied. The integral in eq.(3) is easily calculated and we find eq. (3) can be expressed as \[\tag{4} J^2\rho\,{\rm log}\left(\left\|{\epsilon_k-\epsilon_{\rm F} \over\epsilon_k- D}\right\|\right). \] The spin flip contributions do not cancel due to the To this correction term to the matrix element we must add the first term $J \ .$ The scattering probability $ W_k \ ,$ in which the electron \ ( k \) is scattered to any state, is proportional to the square of this total matrix element, giving \[\tag{5} W_k\propto J^2+ 2J^3\rho\,{\rm log}\left(\left\|{\epsilon_k-\epsilon_{\rm F} \over\ epsilon_k- D}\right\|\right) +{\rm O}(J^4). \] Earlier calculations of the resistivity used only the leading term in eq. (5), but $J\rho$ is typically of order $0.1 $ so the second term is not so small when the electron energy approaches the Fermi energy as it increases logarithmically. This singularity arises from the factor $ 1-f_{k''} $ in the integral of eq.(3), and implies that, when one considers the scattering of an electron, one must take into account the influence of all the other electrons. Logarithmic terms can also arise from scattering with the $ S_z $ component of the spin of the local magnetic moment, which do involve a spin flip. These logarithmic terms, however, cancel. The fact that the logarithmic terms arising from the spin flip scattering do not cancel is due to the fact that the operators involved in the spin flip processes, $ S^+=S_x+iS_y $ and $ S^-=S_x-iS_y \ ,$ do not commute $ S^+ S^- - S^- S^+=2S_z \ .$ When calculating the resistivity, one just considers the electrons whose energy lies within a window of about $ k_{\rm B}T $ about the Fermi energy. This means $\|\epsilon_k-\epsilon_{\rm F}\|\ approx k_{\rm B}T $ and we replace $\epsilon_k-\epsilon_{\rm F} $ in eq. (5) by $ k_{\rm B}T $ and find a contribution to the resistivity of the form, \[\tag{6} R(T)=R_0\left[ 1+2J\rho {\rm log}\left(\left\|{k_{\rm B}T \over D-\epsilon_F}\right\|\right)\right] , \] where $ R_0 $ is the resistivity obtained by considering only the first term of eq.(1). The sign of the exchange interaction $ J $ between the conduction electrons and the impurity is important. If $ J>0 \ ,$ then this interaction tends to align the magnetic moments of the conduction electron and impurity magnetic moments in the same direction (ferromagnetic case). If $J<0 \ ,$ then this interaction tends to align the magnetic moments of the conduction electron and impurity magnetic moments in the opposite direction (antiferromagnetic case). Only in the antiferromagnetic case does the extra scattering term give a contribution to the resistivity that increases as the temperature is lowered. Such an antiferromagnetic exchange coupling can be shown to arise when a degenerate 3d or 4f state of a magnetic impurity hybridizes with the conduction electrons (see Schrieffer and Wolff (1966)). Combining the contribution in the antiferromagnetic case with that from the scattering with lattice vibrations, Kondo was able to make a detailed comparison with the experiments for iron impurities in gold, demonstrating that this extra scattering mechanism could provide a very satisfactory explanation of the observed resistance minima, as is shown in Figure 2. The Kondo Problem Although the extra contribution to the resistivity explains the resistance minimum very well, the term $J\rho{\rm log}\left(T/(D-\epsilon_{\rm F}\right)) $ diverges at low temperatures as $ T\to 0 \ ,$ and higher order scattering gives terms proportional to $[J\rho{\rm log}\left(T/D-\epsilon_{\rm F}\right)]^m $ with $ m>1 \ ,$ which diverge even more rapidly. The perturbation result is clearly unreliable at a temperature $ T $such that $J\rho\,{\rm log}\left(T/(D-\epsilon_{\rm F}\right))\sim 1 \ .$ Extensions of the perturbation approach (Abrikosov, 1965), which involved summing up the leading logarithmic contributions from the higher order scattering processes, gave a result which diverged in the case of an antiferromagnetic coupling $ J=-\|J\| $ at the temperature $T_{\rm K}$ given by \[ T_{\rm K}\sim (D-\epsilon_{\rm F})e^{-1/\|J\|\rho}. \] The temperature $T$[K] has become known as the Kondo temperature. The problem of how to extend Kondo's calculations to obtain a satisfactory solution in the low temperature regime, $T< T_{\rm K}\ ,$ became known as the Kondo Problem, and attracted the attention of many theorists to the field in the late 1960s and early 1970s. The physical picture that emerged from this concerted theoretical effort, in the simplest case where the magnetic impurity has an unpaired spin $S=1/2$ (2-fold degenerate), is that this spin is gradually screened out by the conduction electrons as the temperature is lowered, such that as $T\to 0$ it behaves effectively as a non-magnetic impurity giving a temperature independent contribution to the resistivity in this regime. Furthermore it was concluded that the impurity contributions to the magnetic susceptibility, specific heat, and other thermodynamic properties, could all be expressed as universal functions of$ T/T_{\rm K}\ .$ Definitive results confirming this picture were obtained by Wilson (1975) using a non-perturbative renormalization group method, which built upon the earlier scaling approach of Anderson (1970). Further confirmation came in the form of exact results for the thermodynamics of the Kondo model by Andrei (1980) and Wiegmann (1980), by applying the Bethe Ansatz method, which was developed by Bethe in 1931 to solve the one dimensional Heisenberg model (interacting local spins coupled by an exchange interaction $ J$). Shortly after Wilson's work, Nozieres (1974) showed how, in the very low temperature regime, the results could be derived from a Fermi liquid interpretation of the low energy fixed point. In the Landau Fermi liquid theory, the low energy excitations of a system of interacting electrons can be interpreted in terms of quasiparticles. The quasiparticles correspond to the original electrons, but have a modified effective mass $m^$ due to the interaction with the other electrons. There is also a residual effective interaction between the quasiparticles which can be treated asymptotically exactly ($T\to 0$) in a self-consistent mean field theory. In the Kondo problem, the inverse effective mass of the quasiparticles $ 1/m^$ and their effective interaction are both proportional to the single renormalized energy scale $T_{\rm K}\ .$ The density of states corresponding to these quasiparticles takes the form of a narrow peak or resonance at the Fermi level with a width proportional to $T_{\rm K}\ .$ This peak, which is a many-body effect, is commonly known as a Kondo resonance. It provides an explanation why the anomalous scattering from magnetic impurities leads to an enhanced contribution to the specific heat coefficient and magnetic susceptibility at low temperatures $T<<T_{\rm K}$ with leading correction terms behaving as $(T/T_{\rm K})^2\ .$ At high temperatures such that $T>>T_{\rm K}\ ,$ when the magnetic impurities have shed off the screening cloud of conduction electrons, the magnetic susceptibility then reverts to the Curie law form (ie. proportional to $ 1/T$ ) of an isolated magnetic moment but with logarithmic corrections (${\rm log}(T/T_{\rm K})$). Direct observation of the Kondo resonance in quantum dots Direct experimental confirmation of the presence of a narrow Kondo resonance at the Fermi level at low temperatures $ T<<T_{\rm K}$ has been obtained in experiments on quantum dots. Quantum dots are isolated islands of electrons created in nanostructures that behave as artificial magnetic atoms. These islands or dots are connected by leads to two electron baths. Electrons can only pass easily through the dots if there are states available on the dot in the vicinity of the Fermi level, which then act like stepping stones. In the situation where there is an unpaired electron on the dot, spin $S=1/2\ ,$ in a level well below the Fermi level, and an empty state well above the Fermi level, there is little chance of the electron passing through the dot, when a small bias voltage is introduced between the two reservoirs--- this is known as the Coulomb blockade regime (for a schematic representation of this regime see Figure 3). However, at very low temperatures when a Kondo resonance develops at the Fermi level, arising from the interaction of the unpaired dot electron with the electrons in the lead and reservoirs, the states in the resonance allow the electron to pass through freely (see Figure 4). The observation of an electron current passing through a dot at very low temperatures, in the Coulomb blockade regime on the application of a small bias voltage, was first made in 1998 (Goldhaber-Gordon et al 1998). It provides a direct way of investigating and probing the Kondo resonance. Experimental results of the current through a dot spanning the temperature range to $ T>>T_{\rm K}$ to $ T<<T_{\rm K}$ are shown in Figure 5. Other related many-body effects have been investigated by using different configurations of dots and various applied voltages, and this is currently a very active research field. Related developments Strictly speaking the Kondo scattering mechanism only applies to metallic systems with very small amounts of magnetic impurities (dilute magnetic alloys). This is because the impurities can interact indirectly through the conduction electrons (RKKY interaction), and these interactions can clearly be expected to become important as the number of magnetic impurities is increased. These interactions are ignored in the Kondo calculation, which treats the impurities as isolated. Nevertheless, certain non-dilute alloys with magnetic impurities, particularly those containing the rare earth ions, such as Cerium (Ce) and Ytterbium (Yb), show a resistance minimum. Resistance minima can also be observed in some compounds containing the same type of rare earth magnetic ions. In many cases the Kondo mechanism provides a very satisfactory quantitative explanation of the observations. Good examples are the cerium compounds La[1-x]Ce[x]Cu[6] (see Figure 6) and Ce[1-x]La[x]Pb[3] where $ 0<x\le 1\ .$ In these systems the inter-impurity interactions are relatively small, and at intermediate and higher temperatures the magnetic ions act as independent scatterers. As a result, in this temperature regime, the original Kondo calculation is applicable. At lower temperatures, in the compounds (where $ x=1$), which display a resistance minimum but are completely ordered, the interactions between the magnetic ions becomes important, and the scattering of the conduction electrons becomes coherent, in contrast to the incoherent scattering from independent scatterers. Hence, in these systems, the resistivity decreases rapidly below a coherence temperature T[ coh] to a residual value due to non-magnetic impurities and defects. The resistivity curve then displays a maximum as well as a mininum as a function of temperature. See for example the resistivity curve shown in Figure 6 for the compound CeCu[6] (curve x=1). Other examples of compounds displaying such a resistivity maximum can be seen in Figure 7. The most dramatic effects of this type occur in rare earth and actinide compounds, which have ions carrying magnetic moments but do not magnetically order, or only do so at very low temperatures. These types of compounds are generally known as heavy fermion or heavy electron systems because the scattering of the conduction electrons with the magnetic ions results in a strongly enhanced (renormalized) effective mass, as in the Kondo systems. The effective mass can be of the order 1000 times that of the real mass of the electrons. The low temperature behavior of many of these compounds can be understood in terms of a Fermi liquid of heavy quasiparticles, with induced narrow band-like states (renormalized bands) in the region of the Fermi level. Due to the variety and complex structures of many of these materials, there is no complete theory of their behavior, and it is currently a very active field of research both experimentally and theoretically. • Abrikosov, A A (1965) Physics, 2, 5. • Andrei, N (1980) Physical Review Letters, 45, 379. • Anderson, P W (1970) Journal of Physics C, 3 2439. • de Haas, W J, de Boer, J H and van den Berg, G J (1934) Physica, 1, 1115. • Fisk, Z, Ott H R, Rice, T M, and Smith, J L (1986) Heavy-electron metals. Nature, 320, 124. • Goldhaber-Gordon D, Shtrikman D H, Mahalu, D, Abusch-Magder D, Meirav, U, and Kastneret, M A (1998) Nature 391, 156. • Kondo, Jun (1964). Resistance Minimum in Dilute Magnetic Alloys. Progress of Theoretical Physics 32: 37. • Van den Berg, G J (1964). Progress in Low Temperature Physics, Vol. IV. Gorter, C J editor. North Holland, Amsterdam. p. 194. • Nozieres, P (1974) Journal of Low Temperature Physics, 17, 31. • van der Wiel W G, De Franceshi, S, Fujisawa, T, Elzerman, D, Tarucha, S and Kouwenhoven, L P (2000). The Kondo Effect in the Unitary Limit. Science, 289, 2105. • Schrieffer, J R and Wolff, P A (1966) Physical Review, 149, 491. • Sumiyama, A, et al. (1986) Journal of the Physical Society of Japan, 55, 1294. • Wiegmann, P B (1980) Soviet Physics JETP Letters, 31, 392. • Wilson, K W (1975) Reviews of Modern Physics, 47, 773. Further reading • The Kondo Problem to Heavy Fermions, Hewson, A C, CUP (Cambridge, 1997) ISBN 0521599474 • Sticking to My Bush.\| Kondo, Jun, Journal of the Physical Society of Japan (2005) 74, 1-3. • Special topic volume: Kondo Effect - 40 Years after the Discovery, Journal of the Physical Society of Japan, 74, No. 1 (2005). • Dynamics of Heavy Electrons, Kuramoto, Y and Kitaoka, Y (2000) (Clarendon Press: Oxford) ISBN 019851767X. • Andrei, N, Furuya, K, and Lowenstein, J H (1983) Reviews of Modern Physics, 55, 331. • Tsvelick, A M, and Wiegmann, P B (1983) Advances in Physics, 32, 453. See also	{"url":"http://var.scholarpedia.org/article/Kondo_effect","timestamp":"2024-11-11T20:13:01Z","content_type":"text/html","content_length":"58700","record_id":"<urn:uuid:7a7bc253-a045-426e-9a12-4464f81ebdcf>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00260.warc.gz"}
An Inside Look at High-Speed ADC Accuracy, Part 2 Rob Reeder, Senior System Application Engineer, Analog Devices looked at general static analog-to-digital converter (ADC) inaccuracy errors and ADC inaccuracy error that involves bandwidth. Hopefully, it also provided a greater understanding of ADC errors and how these errors influence the signal chain. With that, keep in mind that not all components are created equal, whether you’re talking active or passive devices. Thus, there will be errors within the analog signal chain, regardless of what is down-selected as the final part to fit into the system. This article describes the differences between accuracy, resolution, and dynamic range. It also reveals how inaccuracies accumulate within the signal chain and cause errors. This plays an important role in understanding how to specify or choose an ADC properly when defining system parameters for a new design. Accuracy vs. Resolution vs. Dynamic Range Many users of converters seem to use the terms accuracy and resolution interchangeably. However, this is a mistake. The terms accuracy and resolution are not equal but are related. Therefore, they should not be used interchangeably. Think of accuracy and resolution as, say, cousins, but not twins^1 please. Accuracy is simply error, or how much the value under measurement deviates from its true value. Accuracy error can also be referred to as sensitivity error. Resolution is simply how finely the value measured can be represented or displayed. Even though a system may have 12 bits of resolution, it doesn’t mean it will be able to measure a value to 12 bits of accuracy. For example, say a multimeter has six digits to represent a measurement. This multimeter’s resolution is six digits, but if the last one or two digits seem to “flicker” between measurement values, then the resolution is compromised and so is the accuracy of the measurement. The errors in any system or signal chain accumulate throughout, distorting the original measurement taken. Therefore, it’s also important to understand the dynamic range of the system in order to gauge the accuracy and the resolution of the signal chain under design. Let’s look again at the multimeter. If there are six digits of representation, then the dynamic range of this device should be 120 dB (or 6 × 20 dB/decade). Keep in mind, though, that the bottom two digits are still flickering. Therefore, the real dynamic range is only 80 dB. That means if the designer intends to measure a 1 µV (or 0.000001 V), the error involved in this measurement could be off as much as 100 µV, since the actual device is only accurate to 100 µV (or 0.0001 V or 0.0001 XX V, where XX represents the bottom two digits flickering). Effectively, any system’s overall accuracy can be described in two ways: dc and ac. With regard to dc accuracy, it represents the “deviated” accumulation of error as shown throughout a given signal chain. This is sometimes termed as a “worst-case” analysis. The noise-error terms that accumulate throughout the signal chain is a measure of ac accuracy. This defines the signal-to-noise ratio (SNR) of the system. These errors then add up, lowering the SNR and yield a more true effective number of bits (ENOB) of the entire design. Obtaining both parameters effectively tells the user how accurate the system can be with both static/wondering and dynamic signals. How Do Low-Frequency SNR, ENOB, Effective Resolution, and Noise-Free Code Resolution Relate? Remember, an ADC can “take in” many types of signals that are typically classified as either dc or ac and quantify it digitally. Understanding the ADC’s error in the system means the designer must understand the type(s) of signals that will be sampled. Therefore, depending on the signal type depends on the way to define the converter’s error contribution to the overall system. These converter errors are generally defined in two ways: noise-free code resolution, representing dc-type signals; and the “SNR equation,” representing ac-type signals. All active devices, such as ADC internal circuits, produce a certain amount of rms noise due to resistor noise and "kT/C" noise. This noise is present even for dc-input signals, and accounts for the code transition noise in the converter’s transfer function. This is more commonly referred to as input-referred noise. The most common way to characterize input-referred noise is by examining the histogram of a number of output samples when applying a dc input to the converter. 1. Converter input-referred noise or ADC "grounded input" histogram. The output of most high-speed or high-resolution ADCs is a distribution of codes, centered around the nominal value of the dc input. To measure its value, the input of the ADC is either grounded or connected to a heavily decoupled voltage source, and a large number of output samples are collected and plotted as a histogram (sometimes referred to as a grounded-input histogram) (Fig. 1). Since the noise is approximately Gaussian, the standard deviation of the histogram, σ, can be calculated, corresponding to the effective input rms noise and expressed in terms of LSBs rms. Although the inherent differential nonlinearity (DNL) of the ADC may cause some minor deviations from an ideal Gaussian distribution, it should be at least approximately Gaussian. If the code distribution has large and distinct peaks and valleys, this could be the indication of a bad printed-circuit-board (PCB) layout, poor grounding techniques, or improper power-supply decoupling among other things... Typically, input-referred noise can be expressed as an rms quantity, usually having the units of LSBs rms. Specifications involving these types of quantities are more generally associated with high-resolution precision-type converters, because of the low sample rates and/or dc-type or slow-moving signals acquired by them. Sigma-delta ADCs designed for precision measurements, having resolutions in the 16- to 24-bit range, have datasheet specifications such as input-referred noise, effective resolution, and noise-free code resolution to describe their dc dynamic range. On the other hand, higher-frequency sigma-delta ADCs for audio applications are generally characterized exclusively in terms of total harmonic distortion (THD) and total harmonic distortion plus noise (THD + N). Successive-approximation-register (SAR) converters cover a wide range of sampling rates, resolutions, and applications. They typically have the input-referred noise specification, but also maintain specifications for SNR, ENOB, SFDR, THD, etc., for ac input signals. Although higher-speed converters (such as pipelined) that sample in the hundreds of megahertz or beyond are typically specified in terms of ac specifications such as SNR, SINAD, SFDR, and ENOB, they can also capture dc-type or slow moving signals. It’s therefore useful to understand how to derive the low-frequency performance of high-speed converters from the ac specifications given on the datasheet (see “Signal-to-Noise Ratio (SNR) Equation"). Relating both slow-speed, dc-type signals, and high-speed, ac-type signal specification quantities does requires some math. So break out your college math book and flip to the identity table in the back and let’s review below how a relation can be struck between SNR, ENOB, effective resolution, and noise-free code resolution for low-frequency inputs. With FSR = full-scale range of ADC, and n = input-referred noise, (rms) effective resolution is defined as the following: Note that log[2](x) = log[10](x) ÷ log(2) = log[10](x) ÷ 0.301 = 3.32 × log[10](x) Rearranging this a bit, we get: This yields the following: Therefore, by substituting in Equation 7, we can derive the relationship between ENOB, ac-type signals, and dc-type (slow moving) signals, or: To verify this, let’s calculate the ENOB for ideal N-bit ADC, where: Substituting in these values, To summarize, when looking at dc (slow-moving) signals, the ENOB of the system is roughly 1 bit larger (0.92 bits to be exact) than the converter’s noise-free code resolution and 2 bits less than its effective resolution. However, as the signals move faster (ac-type signals) where bandwidth is involved, the converter’s SNR and the ENOB become frequency-dependent and typically degrade for higher-frequency inputs. Converter Inaccuracies in a Signal Chain Now that the converter errors are understood, the rest of the signal chain is applied to understand these concepts at the system level. Figure 2 describes an example of a simple data-acquisition signal chain. Here, a sensor is connected to a long amount of cable that ultimately gets connected to the data-acquisition card. The sensor’s ac signal pushes through two stages of preconditioning amplifiers before arriving at the ADC’s inputs to be sampled. The goal here is to design a system that can accurately represent a sensor’s signal within ±0.1% of its original value. Hmmm…sound 2. Simple data-acquisition signal chain. To design such a system, it’s important to think about the types of errors that could be affecting the sensor’s original signal and where they are coming from throughout the signal chain. Imagine what the converter sees in the end when the signal is finally sampled. Let’s suppose the ADC has a 10-V full-scale input and 12 bits of resolution in this example. If the converter were ideal, it could be determined that it has a dynamic range or SNR of 74 dB: However, the datasheet specifications only show the converter to have an SNR of 60 dB or 9.67 ENOB: Please note the calculation of SNR and ENOB—when calculating ENOB from an SNR number in the datasheet, it should be clear to the designer that this may or may not include harmonics. If it does include distortion, then SINAD can be used, which is defined as SNR + distortion, or sometimes referred to as total harmonic distortion (THD). Therefore, the LSB size can be defined as 12.2 mV p-p or V[FS]/2^N = 10/2^9.67. This dramatically reduces the number of representations that can occur on the digital outputs. Remember, the bottom LSBs/bits are flickering because of the noise in the ADC: This also means the converter has an accuracy of ±6.12 mV or 0.0612%: Additionally, this implies that for a 1.00000-V input applied to the converter, the output can be between 0.99388 and 1.00612 V. Therefore, the 12-bit converter with 9.67-bit ENOB can only measure a signal to 0.1% accuracy. The converter’s dynamic range is approximately 60 dB, rather than 74 dB (ideal 12-bit ADC): This can be seen in Figure 3. Table 1 describes some quick equalities for referencing desired system performance. 3. Remember 20 dB per decade, or 3 Ã 20 = 60 dB. Converter Inaccuracies in a Signal Chain Be mindful of all the front-end components as suggested in the above signal-chain example. Just because the converter accuracy meets or beats the system accuracy specification defined for the system, there are still more inaccuracies to comprehend, i.e., the front-end, power supply, any other outside influences, or environments. The design of such a signal chain as described in Figure 2 can be very intense and is beyond the scope of this paper. However, Table 2 offers a quick view on inaccuracies/errors associated with such a signal chain. Many errors are present in any signal chain, not to mention the cable and other outside influences that can also play a big role in determining the design of such a system. Whatever the error accumulation, it ultimately gets sampled at the converter along with the presence of the signal—assuming the error is not great enough to mask the signal that is being acquired! When designing with converters, keep in mind there are two parts to the equation when it comes to defining the accuracy of the system. There’s the converter itself, as described above, and everything else used to precondition the signal before the converter. Remember, every bit lost causes a 6-dB decrease in dynamic range. The corollary—for every bit gained, the system’s sensitivity increases by 2X. Therefore, the front-end requires an accuracy specification to be much better than the converter’s accuracy chosen to sample the signal. 4. Simple data-acquisition signal chain with front-end noise defined. To illustrate this point, using the same front-end design shown in Figure 2, let’s say the front-end itself has 20 mV p-p of inaccuracies, i.e., accumulated noise (Fig. 4). The system accuracy is still defined as 0.1%. Is the same 12-bit converter going to have enough accuracy to maintain the system specification defined? The answer is no, and here’s why. It can be figured out by using the ADC that has a SNR = 60 dB: Notice that 20 mV of noise can degrade the system by 1 bit or 6 dB, bringing the performance down to 54 dB from 60 dB, which was originally described as the system’s performance requirement. To get around this, maybe a new converter should be chosen in order to maintain the 60 dB or 0.1% system accuracy. Let’s choose an ADC that has 70 dB of SNR/dynamic range or an ENOB of 11.34 bits to see if this works. It appears that the performance didn’t change much. Why? The reason is that noise of the front end is too great to comprehend 0.1% accuracy, even though the converter’s performance itself is much better than the specification. The front-end design thus needs to change in order to get the desired performance. This is represented figuratively in Figure 5. See why this last configuration example won’t work? The designer can’t simply pick a better ADC to improve the overall system performance. 5. Front-end noise vs. 12-bit, 70-dB ADC noise comparison. Bringing It All Together The previously chosen 10-V full-scale, 12-bit ADC has a dynamic range of 60 dB to achieve 0.1% accuracy. This means a total accumulated error of <10 mV or 10 V/(10^60/20) needs to be met to reach 0.1% requirement. Therefore, the front-end components have to change to reduce the front-end error down to 9 mV p-p (Fig. 6), using a converter that has 70 dB of SNR. 6. Front-end noise vs. 12-bit, 70-dB ADC noise comparison. If the 14-bit, 74-dB ADC was to be used (Fig. 7), then the front-end’s requirements can be relaxed even further. However, tradeoffs can have an upside in cost. These tradeoffs need to be evaluated per the design and application. It may be worth paying more for tighter tolerance and lower drifting resistors than to splurge for a higher-performing ADC, for example. 7. Front-end noise vs. 14-bit, 74-dB ADC noise comparison. Concluding the Analyses The following should have provided some guides on how accuracy error, resolution, and dynamic range are all related, but offer different points when selecting a converter for any application that requires a certain amount of measurement accuracy. It’s important to understand all component errors and how those errors influence the signal chain. Keep in mind that not all components are created Developing a spreadsheet that captures all of these errors is an easy way to plug in different signal-chain components to make evaluations and component tradeoffs quickly (Table 2, again). This is especially true when trading off costs between components. Further discussions on how to go about generating such a spreadsheet will be covered in Part 3 of this series. Finally, remember that simply increasing the performance or resolution of the converter in the signal chain will not increase the measurement accuracy. If the same amount of front-end noise is still present, the accuracy will not improve. Those noises or inaccuracies will only be measured to a more granular degree and it will cost the designer’s boss more money in the end to do it... 1. “Resolution and Accuracy: Cousins, not Twins,” John Titus, Design News, 5/5/2003. Other reading: Signal Conditioning & PC-Based Data Acquisition Handbook, John R. Gyorki, 3rd Edition, 1-11. “AN010: Measurement Dynamic Range for Signal Analyzers,” LDS Dactron, 2003. “System Error Budgets, Accuracy, Resolution,” Dataforth. “Overall Accuracy = ENOB (Effective Number of Bits),” Data Translation. Analog-Digital Conversion: Seminar Series, Walk Kester, Walt Kester, Analog-Digital Conversion, Analog Devices, 2004, ISBN 0-916550-27-3. Also available as The Data Conversion Handbook, Elsevier/ Newnes, 2005, ISBN 0-7506-7841-0 W. R. Bennett, "Spectra of Quantized Signals," Bell System Technical Journal, Vol. 27, July 1948, pp. 446-471. W. R. Bennett, "Noise in PCM Systems," Bell Labs Record, Vol. 26, December 1948, pp. 495-499. Steve Ruscak and Larry Singer, “Using Histogram Techniques to Measure A/D Converter Noise,” Analog Dialogue, Vol. 29-2, 1995. Brad Brannon, "Overcoming Converter Nonlinearities with Dither," Application Note AN-410, Analog Devices, 1995. 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Converting and Calculating Units of Length - Math Angel ðŸŽ¬ Video Tutorial • (0:01) Metric Units of Length: Understand and use millimetre (mm), centimetre (cm), metre (m), kilometre (km) in everyday calculations. • (1:38) Practical Conversion Methods: Multiply powers of 10 when converting to a smaller unit and divide when converting to a larger unit. • (2:34) Standardising Units in Calculations: Convert all measurements to the same unit before adding or subtracting, then simplify the results. Accessing this course requires a login. Please enter your credentials below!	{"url":"https://math-angel.io/lessons/units-of-length/","timestamp":"2024-11-13T02:36:19Z","content_type":"text/html","content_length":"275140","record_id":"<urn:uuid:64475397-379f-4eb4-beb7-2ac5e9af3e51>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00200.warc.gz"}