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# You are required to write a report for Mr Ragwort about the proposed methods of measuring divisional... 1 answer below »

Discussion of various divisional performance measures

You are the highly trusted chief accountant of Ragwort Ltd, a manufacturer of sports equipment, and you have strong private hopes of joining the board in the next year or two in a newly created post of finance director.

Ragwort is organized into three divisions, each of which is formed as a separate subsidiary company; you have a seat on the board of each of them. The three divisions and their general manager are North-West (Christopher Perkins), Home Counties (Geoffrey Gilmore), and Eastern Counties (Morton Jackson). The group executive committee consists of the three general managers, under the chairmanship of group chairman and chief executive Thomas Ragwort. It meets monthly, and the meeting usually lasts two days.

At the end of the first day of this month's meeting Ragwort calls you to his office, and tells you that he has a problem. 'You know', he says, that for some time now I've been concerned about what is the best measure for us to use to compare the performance of the divisions. You know what I mean — what is the right sort of target to set for them? It's not good if I impose something without agreement, and we simply didn't manage to get it today. I'm going to re-open the discussion tomorrow, and I want a decision but first I want some advice from you.'

You ask him what has happened.

Well', he says, 'you know that we've been working out a divisional return on investment. We find pre-tax profits for each division, after charging out group head office costs in proportion to divisional turnover. We set that against a base of written down net assets, excluding cash.

Gilmore is satisfied with that, but Perkins came up with something he called residual income — he argued that we have to make absolute profits, not just a return on investment. And then Jackson said that what we really want is growth, and that the only way to get that in the long run is to set the target as percentage sales growth.

They were obviously split, so I let it run on.

To cut a long story short, Gilmore said he could certainly accept that residual income was a better measure than looking just at growth. Jackson wouldn't accept that, and still wanted the growth measure, but he agreed with Gilmore's view that it was better to stick with our present ROI than to change to this new residual income idea.

Perkins stuck to his guns over residual income. He's smart. He told Jackson that he could see at least some merit in the growth idea, because in the end it would probably correlate with residual income. But he still thought that the ROI  should be dropped. I tried a compromise approach then. I suggested that we took the measure in pairs, starting with ROI and residual income. After all, they sound pretty similar. Gilmore and Jackson voted for ROI, and only Perkins supported residual income. So I had them compare ROI with growth, thinking I was home and dry and we needn't change anything. And Perkins joined Jackson in supporting the growth idea! I had to admit that he'd said as much.

But Gilmore is pretty smart too, you know. He whispered something to Perkins. Then he reminded us that Perkins wanted us to try out residual income, and that he had already told the meeting that in his view residual income was a better control measure than percentage sales growth. So if we were going to make a change the majority obviously preferred residual income to growth.

By then, I felt that it was time to adjourn!

You'd better keep all this under your hat. But give me a short report summarizing the pros and cons of these three possibilities, and giving me your view on which one would be best in principle for us to use in future. If I eventually have to lay the law down I want to get it right. But make sure you keep it short — not more than about four pages.'

On returning to your office you cancel your previous arrangements for the evening, and then quickly work out the following table, showing the average annual performance of the divisions over the last three years in historic cost terms:

 Residual Sales Division Manager ROI income growth % £ % Home Counties Geoffrey Gilmore 10 80000 15 Eastern Counties Morton Jackson 8 60000 25 North West Christopher Perkins 6 100000 20

You send a copy of the table to Mr Ragwort with a note commenting that it may help to explain the attitudes being taken in the meeting. You then settle down to think about the three measures proposed as means of controlling divisional performance.

You are required to write a report for Mr Ragwort about the proposed methods of measuring divisional performance, to comply with the terms of his instruction

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help-octave
[Top][All Lists]

## Re: Linespace

 From: Ben Abbott Subject: Re: Linespace Date: Tue, 15 Sep 2009 09:23:01 -0400

On Sep 15, 2009, at 2:41 AM, Sheron Gerard Jeshuran <address@hidden> wrote:
```
```
```Dear SIr,

```
I have a problem to draw a graph in Octave my function is f(x)=4-3x and also i need f inverse also, and new and the time i run the command following error message comes
```
octave-3.2.2.exe:1:C:\Octave\3.2.2_gcc-4.3.0\bin
> linespace(0, 2*pi, 100);
error: `linespace' undefined near line 1 column 1
octave-3.2.2.exe:1:C:\Octave\3.2.2_gcc-4.3.0\bin
>
```
```

format: plain


text:
## Introduction

Transition metal dichalcogenides (TMDs) provide a fertile ground for exploring exotic collective quantum phenomena, such as superconductivity and charge density waves.1,2,3 Recently, the MoxW1-xTe2 class of TMDs has attracted great interest due to the topological nature of the electronic states and as a topological Weyl semimetal (TWS).4,5,6,7 Their low-energy excitations behave as Weyl fermions that always appear in pairs with opposite chirality. TWSs are classified into type-I and type-II: type-I TWSs have point like Fermi surfaces and respect Lorentz invariance, while type-II with tilted Weyl cones break Lorentz invariance.5,8,9 Type-I TWSs were first predicted and experimentally discovered in the (Ta, Nb) (As, P) family compounds.8,9,10,11 Type-II TWSs have been proposed to exist in the MoxW1-xTe2 family TMDs4,5,6,7 and LaAlGe.12 In this system, the tilted Weyl cones that arise from topologically protected crossings of valence and conduction bands cause touching points between electron and hole pockets near the Fermi level. Exotic Fermi arcs states that join these touching points, are expected on the surfaces of these materials. In the last few years, there have been a series of experimental studies aimed at verifying the existence of the Fermi arcs.13,14,15,16,17,18,19,20,21

For the MoxW1-xTe2 family, the Weyl nodes are usually located at energies above the Fermi level which makes it difficult to be accessed by conventional angle resolved photo emission spectroscopy (ARPES).13,14,15,17,20 Fourier transform scanning tunneling spectroscopy (FT-STS) contains information on the scattering vectors between electronic states in momentum space and can provide the information on both the occupied and unoccupied states22 which makes it a particularly powerful probe for the topological states of the MoxW1-xTe2. There have been several FT-STS studies on the MoxW1-xTe2.15,23,24,25,26,27,28 However, due to the complex band structure and the low signal strength in the FT-STS images, the existence of Fermi arcs in MoxW1-xTe2 system still remains to be clarified.

Here we use low-temperature scanning tunneling microscopy (STM) and first-principles calculations to directly visualize and identify the electronic states in MoTe2, an end member of the MoxW1-xTe2 system. The other end member, WTe2 was the first predicated type-II Weyl semimetal, but the small momentum separation between the opposite Weyl points that are located about 50 mV above the Fermi level, made experimental confirmation by conventional ARPES difficult.5,14 MoTe2 was later predicated as another candidate of type-II Weyl semimetal where the Weyl points have six times larger momentum separation which makes experimental probe of Weyl nodes much easier.6,7 For MoTe2, previous ARPES works have found evidence for the trivial and topologically protected Fermi arcs at energies below Fermi level.18,19,20 However, for various reasons such as the size of the STS maps or the properties of the scattering impurities, previous FT-STSs on MoTe2 did not show sufficient signal strength to perform a clear analysis of the FT-STS data, and have not arrived at an unequivocal conclusion about the nature of the Weyl states. In this work, we present FT-STS data with much higher signal strength that allows us to compare the various scattering channels observed in the experimental data with theory, and reach a robust conclusion on the origins of the scattering channels. Interestingly, based on this higher resolution FT-STS data, we come to different conclusions compared to the previous work:15,23,24,26 the trivial and non-trivial scattering channels are indistinguishable in the FT-STS images. Moreover, the broken inversion symmetry in this compound is reflected in the FT-STS images taken on two inequivalent surfaces.

## Results

The Td phase crystal has an orthorhombic structure with van der Waals stacking of Te-Mo-Te sandwich layers along the c-axis direction as shown in Fig. 1a. Due to the weak van der Waals interaction between the Te layers, the MoTe2 sample cleaves at the Te layers and the cleaved surface is Te-terminated. The bulk, and surface Brillouin zones for this termination, is shown in Fig. 1b. In high-resolution STM topographies, the two inequivalent Te atomic rows in the top Te layer can be clearly resolved (Fig. 1d). There are mainly two kinds of atomic defects and they look different in the STM topographies taken with positive bias voltages (as shown in the red and black circles in Fig. 1e). The defects lie on the Te sites and are likely to be Te vacancies/impurities in the two Te atomic rows at the top Te layer. In addition, in the negative bias topography, there are other faint defects that may be attributed to vacancies/impurities in the layers underneath (more details can be found in Supplementary Fig. S1 and Fig. S2).

Figure 1f shows the position dependent dI/dV spectra taken on a 50 nm by 50 nm area on MoTe2 in the energy range of ±500 mV. In general, the dI/dV spectra are parabola-shaped with minimum around 10 mV above the Fermi level. Despite the presence of impurities, the low energy density of states as well as the overall parabola-shape of the spectra remains homogeneous and there are no clear impurity-induced resonance states in this energy range.26 However, the scattering pattern of quasiparticles by the atomic impurities can be clearly seen in the dI/dV maps, Fig. 1c. The next step is to study the Fourier transforms of the dI/dV maps to look for signatures of the electronic states in momentum space.

Figure 2a–i shows FT-STS images at a few energies below and above the Fermi level (also see Figs. S6 and S7). There is a distinct change in the FT-STS as we go from negative to positive energies. With increasing the bias above the Fermi level, several strong scattering vectors gradually emerge. To identify the origin of the scattering vectors we carried out first-principles density functional theory (DFT) calculations of band structures of MoTe2 which were then used to calculate FT-STS images. Using the iterative Green’s function method, we obtained the surface states of MoTe2 under different terminations (details in Supplementary Section 8 which were then used to simulate the FT-STS patterns using spin-dependent scattering probability method).

As can be seen in Fig. 3, the simulated FT-STS matches well with the measured FT-STS over a wide range of energies. In the measured FT-STS at energies below −100 mV (right panel in Fig. 3a), there is a lip-like feature with a bright center. By comparing with the simulated FT-STS at −140 mV, we conclude that this lip-like feature is due to the scattering of the bulk states of MoTe2 (Supplementary Fig. S5). At −40 mV, the measured FT-STS shows two short arc-like features at the left or right side of the bright center spot (right panel of Fig. 3b). Due to the multiple scattering vectors seen in the simulated FT-STS at negative bias voltages, it is difficult to unambiguously identify the origin of these two short arcs. However, by comparing the momentum-space position of these short arcs, we conclude that they most likely arise from the scattering processes from the trivial surface states to the hole-pockets. At positive bias voltages, the scattering patterns become much clearer (Fig. 2i). By comparing the calculated FT-STS with the measured FT-STS at +40 mV and +100 mV (Fig. 3c, d and Supplementary Fig. S9), we can for the first time identify all the scattering patterns as: Q1 is induced by the intra-scattering of the topologically trivial surface states; Q2 is caused by the scattering between the surface states and the electron pockets. Q3 and Q4 are due to the scattering between the surface states and the projected bulk states at the $$\bar Y$$ point.

Although, the high-resolution FT-STS data clearly show multiple scattering vectors from the surface states and our experimental measurements match the numerical calculations, these data alone do not provide unambiguous evidence for the existence of non-trivial Fermi arcs. According to our calculations, the Weyl points for MoTe2 are located at 4 mV and 57 mV. As the energy increased from the Fermi level to +100 mV, the scattering vectors (Q1Q4) gradually emerge in the measured FT-STSs (Fig. 2f–i), and there is no sudden change in the FT-STSs near the energy positions of the Weyl nodes. This suggests that there are no clear features induced by the non-trivial Fermi arcs in the FT-STSs on MoTe2. We note that our conclusions are in contrast with previous studies.15,23,24,26,27 Most previous STM studies have claimed that MoxW1-xTe2 family material is a Weyl semimetal based on the existence of the long Fermi arcs in the FT-STS data. However, based on our theoretical and experimental studies we find that the long arcs arise primarily from topologically trivial scattering channels. In addition, for Mo0.66W0.34Te2 samples, the absence of the scattering vector (Q3) between the arc-like feature and projected bulk states was taken as proof of the existence of the topological Fermi arc.27 In contrast, in our high-resolution data on MoTe2, we clearly detect this scattering vector (Q3) thereby determining that the scattering process from trivial surface states forms an important component of the scattering seen in our FT-STS data.

## Discussion

This leads to the question: why is it difficult to detect the non-trivial Fermi arcs in MoTe2 even with high-resolution FT-STS measurements? Weyl fermions are local singularities in momentum space (Fig. S10). Therefore, at energies or momentum space away from the Weyl nodes, the non-trivial properties induced by Weyl fermions are negligible. In Fig. 4 we show the calculated surface states of MoTe2 at three different energies, below, at, and above the Fermi level. Figure 4a–c shows constant energy contours at −40 mV, 0 mV, and +40 mV, respectively. Figure 4d–f is the zoom-in of Fig. 4a–c in the momentum region around the positions of the Weyl nodes (the orange rectangle area shown in Fig. 4a–c). Figure 4d clearly shows that the long topologically trivial arc in Fig. 4a lies far from the Weyl nodes in momentum space, and it is therefore not related to the Weyl nodes. With increasing energy, the topologically trivial arc moves toward the Weyl nodes and starts to touch the Weyl nodes at the Fermi level (Fig. 4e). In the constant energy contour of E = +40 meV, the long trivial arc is disconnected with the Weyl nodes and a small topological Fermi arc (arc 2) can be seen in between the two Weyl nodes, as shown in Fig. 4f (also see Fig. S10). However, the separation in momentum space between the non-trivial arc (arc 2) and the trivial arcs (arc 1 and 3) is only about 0.008 Å1, which makes it extremely difficult to distinguish the scattering features in FT-STS induced by the non-trivial arc and those induced by the long trivial arc. Thus, by comparing the STM data with calculations, we conclude that the trivial and non-trivial scattering channels are difficult to distinguish in this compound.

Finally, our STM data show signatures of the broken inversion symmetry that’s critical to the physics of MoTe2. Due to the bulk broken inversion symmetry, the two surfaces obtained after cleaving are inequivalent (Supplementary Fig. S11). As shown in Fig. S11, while the surface structure itself is the same, the atomic arrangement in the second layer is different between the two surfaces. An earlier ARPES study had suggested that this might affect the electronic structure.20 The two inequivalent surfaces were revealed in our data when different cleaves showed slightly different FT-STS patterns (Fig. 5, Supplementary Fig. S12 and Fig. S13). Our FT-STS calculations for the second surface, also show that the terminating electronic states are measurably different for the two surfaces. The differences arise from the different weights of bulk and surface states on the two terminations which makes the scattering channels in the measured FT-STS images taken on these two surfaces have different relative strengths. While the match between the calculations and data are not as good for this second surface, both STM as well as theory indicate that the broken inversion symmetry makes the two surfaces electronically inequivalent (Fig. 5).

In conclusion, we have performed a detailed low-temperature STM/STS study on the MoTe2 sample. In the spatially resolved dI/dV maps, the quasiparticle interference patterns can be clearly resolved. We thoroughly characterized the electronic structure of MoTe2 with FT-STSs taken at energies both below and above the Fermi level. In the high-resolution FT-STSs above Fermi level, the scattering vectors between the trivial arcs and the projected bulk states can be clearly resolved. Due to the small momentum-space separation between the non-trivial arc and the trivial arc, it is difficult to clearly identify the non-trivial arc features in the FT-STSs. Our data show that the bulk broken inversion symmetry has a measurable effect on the surface electronic properties.

## Methods

### Sample synthesis

The MoTe2 single crystals were grown by using a chemical vapor transport method. A mixture of stoichiometric Mo and Te powder were sealed into an evacuated quartz tube with iodine used as a transport agent. The quartz tube was then placed in a double zone furnace with temperature gradient from 900 °C to 800 °C. Large-single crystals of centimeter size were obtained after 2 weeks. The composition and structure of MoTe2 single crystals were checked using X-ray diffraction and an energy-dispersive X-ray spectrometer.

### Scanning tunneling spectroscopy measurements

High-quality MoTe2 samples were cleaved at 77 K and immediately inserted into the STM scanner at 6 K. Differential conductance (dI/dV) spectra were acquired at 6 K using a standard lock-in technique with ~4 mVrms modulation at a frequency of 987.5 Hz.

### Data availability

All data are available upon request from the corresponding authors.


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#### Volume 9, issue 1 (2005)

 1 R A Adams, Sobolev spaces, Pure and Applied Mathematics 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London (1975) MR0450957 2 M F Atiyah, The index theorem for manifolds with boundary, from: "Seminar on the Atiyah–Singer index theorem" (editor R S Palais), Annals of Mathematics Studies 57, Princeton University Press (1965) MR0198494 3 M F Atiyah, V K Patodi, I M Singer, Spectral asymmetry and Riemannian geometry I, Math. Proc. Cambridge Philos. Soc. 77 (1975) 43 MR0397797 4 C Bär, Zero sets of solutions to semilinear elliptic systems of first order, Invent. Math. 138 (1999) 183 MR1714341 5 S Bauer, A stable cohomotopy refinement of Seiberg–Witten invariants II, Invent. Math. 155 (2004) 21 MR2025299 6 S Bauer, Refined Seiberg–Witten invariants, from: "Different faces of geometry", Int. Math. Ser. (N. Y.) 3, Kluwer/Plenum, New York (2004) 1 MR2102993 7 S Bauer, M Furuta, A stable cohomotopy refinement of Seiberg–Witten invariants I, Invent. Math. 155 (2004) 1 MR2025298 8 J Dieudonné, Foundations of modern analysis, Pure and Applied Mathematics, Vol. X, Academic Press (1960) MR0120319 9 S K Donaldson, The Seiberg–Witten equations and 4–manifold topology, Bull. Amer. Math. Soc. $($N.S.$)$ 33 (1996) 45 MR1339810 10 S K Donaldson, Floer homology groups in Yang–Mills theory, Cambridge Tracts in Mathematics 147, Cambridge University Press (2002) MR1883043 11 S K Donaldson, P B Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press (1990) MR1079726 12 R Fintushel, R J Stern, Immersed spheres in 4–manifolds and the immersed Thom conjecture, Turkish J. Math. 19 (1995) 145 MR1349567 13 A Floer, An instanton-invariant for 3–manifolds, Comm. Math. Phys. 118 (1988) 215 MR956166 14 K A Frøyshov, Monopole Floer homology for rational homology 3–spheres, in preparation 15 K A Frøyshov, The Seiberg–Witten equations and four-manifolds with boundary, Math. Res. Lett. 3 (1996) 373 MR1397685 16 R S Hamilton, Harmonic maps of manifolds with boundary, Lecture Notes in Mathematics 471, Springer (1975) MR0482822 17 L Hörmander, The analysis of linear partial differential operators III, Grundlehren der Mathematischen Wissenschaften, Springer (1985) MR781536 18 M Ishida, C LeBrun, Curvature, connected sums, and Seiberg–Witten theory, Comm. Anal. Geom. 11 (2003) 809 MR2032500 19 J L Kazdan, Unique continuation in geometry, Comm. Pure Appl. Math. 41 (1988) 667 MR948075 20 P B Kronheimer, T S Mrowka, Floer homology for Seiberg–Witten monopoles, book in preparation 21 P B Kronheimer, T S Mrowka, The genus of embedded surfaces in the projective plane, Math. Res. Lett. 1 (1994) 797 MR1306022 22 P Kronheimer, T Mrowka, P Ozsváth, Z Szabó, Monopoles and lens space surgeries, Ann. of Math. $(2)$ 165 (2007) 457 MR2299739 23 H B Lawson Jr, M L Michelsohn, Spin geometry, Princeton Mathematical Series 38, Princeton University Press (1989) MR1031992 24 R B Lockhart, R C McOwen, Elliptic differential operators on noncompact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. $(4)$ 12 (1985) 409 MR837256 25 M Marcolli, Seiberg–Witten–Floer homology and Heegaard splittings, Internat. J. Math. 7 (1996) 671 MR1411306 26 M Marcolli, B L Wang, Equivariant Seiberg–Witten Floer homology, Comm. Anal. Geom. 9 (2001) 451 MR1895135 27 D McDuff, D Salamon, $J$–holomorphic curves and quantum cohomology, University Lecture Series 6, American Mathematical Society (1994) MR1286255 28 J W Morgan, Z Szabó, C H Taubes, A product formula for the Seiberg–Witten invariants and the generalized Thom conjecture, J. Differential Geom. 44 (1996) 706 MR1438191 29 L I Nicolaescu, Notes on Seiberg–Witten theory, Graduate Studies in Mathematics 28, American Mathematical Society (2000) MR1787219 30 D Salamon, Removable singularities and a vanishing theorem for Seiberg–Witten invariants, Turkish J. Math. 20 (1996) 61 MR1392663 31 M E Taylor, Partial differential equations, Texts in Applied Mathematics 23, Springer (1996) MR1395147 32 K K Uhlenbeck, Connections with $L^{p}$ bounds on curvature, Comm. Math. Phys. 83 (1982) 31 MR648356


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# 3.7. Concise Implementation of Softmax Regression¶ Open the notebook in Colab Open the notebook in Colab Open the notebook in Colab Open the notebook in SageMaker Studio Lab

Just as high-level APIs of deep learning frameworks made it much easier to implement linear regression in Section 3.3, we will find it similarly (or possibly more) convenient for implementing classification models. Let us stick with the Fashion-MNIST dataset and keep the batch size at 256 as in Section 3.6.

from mxnet import gluon, init, npx
from mxnet.gluon import nn
from d2l import mxnet as d2l

npx.set_np()

batch_size = 256

import torch
from torch import nn
from d2l import torch as d2l

batch_size = 256

import tensorflow as tf
from d2l import tensorflow as d2l

batch_size = 256

## 3.7.1. Initializing Model Parameters¶

As mentioned in Section 3.4, the output layer of softmax regression is a fully-connected layer. Therefore, to implement our model, we just need to add one fully-connected layer with 10 outputs to our Sequential. Again, here, the Sequential is not really necessary, but we might as well form the habit since it will be ubiquitous when implementing deep models. Again, we initialize the weights at random with zero mean and standard deviation 0.01.

net = nn.Sequential()
net.initialize(init.Normal(sigma=0.01))

# PyTorch does not implicitly reshape the inputs. Thus we define the flatten
# layer to reshape the inputs before the linear layer in our network
net = nn.Sequential(nn.Flatten(), nn.Linear(784, 10))

def init_weights(m):
if type(m) == nn.Linear:
nn.init.normal_(m.weight, std=0.01)

net.apply(init_weights);

net = tf.keras.models.Sequential()
weight_initializer = tf.keras.initializers.RandomNormal(mean=0.0, stddev=0.01)

## 3.7.2. Softmax Implementation Revisited¶

In the previous example of Section 3.6, we calculated our model’s output and then ran this output through the cross-entropy loss. Mathematically, that is a perfectly reasonable thing to do. However, from a computational perspective, exponentiation can be a source of numerical stability issues.

Recall that the softmax function calculates $$\hat y_j = \frac{\exp(o_j)}{\sum_k \exp(o_k)}$$, where $$\hat y_j$$ is the $$j^\mathrm{th}$$ element of the predicted probability distribution $$\hat{\mathbf{y}}$$ and $$o_j$$ is the $$j^\mathrm{th}$$ element of the logits $$\mathbf{o}$$. If some of the $$o_k$$ are very large (i.e., very positive), then $$\exp(o_k)$$ might be larger than the largest number we can have for certain data types (i.e., overflow). This would make the denominator (and/or numerator) inf (infinity) and we wind up encountering either 0, inf, or nan (not a number) for $$\hat y_j$$. In these situations we do not get a well-defined return value for cross-entropy.

One trick to get around this is to first subtract $$\max(o_k)$$ from all $$o_k$$ before proceeding with the softmax calculation. You can see that this shifting of each $$o_k$$ by constant factor does not change the return value of softmax:

(3.7.1)\begin{split}\begin{aligned} \hat y_j & = \frac{\exp(o_j - \max(o_k))\exp(\max(o_k))}{\sum_k \exp(o_k - \max(o_k))\exp(\max(o_k))} \\ & = \frac{\exp(o_j - \max(o_k))}{\sum_k \exp(o_k - \max(o_k))}. \end{aligned}\end{split}

After the subtraction and normalization step, it might be possible that some $$o_j - \max(o_k)$$ have large negative values and thus that the corresponding $$\exp(o_j - \max(o_k))$$ will take values close to zero. These might be rounded to zero due to finite precision (i.e., underflow), making $$\hat y_j$$ zero and giving us -inf for $$\log(\hat y_j)$$. A few steps down the road in backpropagation, we might find ourselves faced with a screenful of the dreaded nan results.

Fortunately, we are saved by the fact that even though we are computing exponential functions, we ultimately intend to take their log (when calculating the cross-entropy loss). By combining these two operators softmax and cross-entropy together, we can escape the numerical stability issues that might otherwise plague us during backpropagation. As shown in the equation below, we avoid calculating $$\exp(o_j - \max(o_k))$$ and can use instead $$o_j - \max(o_k)$$ directly due to the canceling in $$\log(\exp(\cdot))$$:

(3.7.2)\begin{split}\begin{aligned} \log{(\hat y_j)} & = \log\left( \frac{\exp(o_j - \max(o_k))}{\sum_k \exp(o_k - \max(o_k))}\right) \\ & = \log{(\exp(o_j - \max(o_k)))}-\log{\left( \sum_k \exp(o_k - \max(o_k)) \right)} \\ & = o_j - \max(o_k) -\log{\left( \sum_k \exp(o_k - \max(o_k)) \right)}. \end{aligned}\end{split}

We will want to keep the conventional softmax function handy in case we ever want to evaluate the output probabilities by our model. But instead of passing softmax probabilities into our new loss function, we will just pass the logits and compute the softmax and its log all at once inside the cross-entropy loss function, which does smart things like the “LogSumExp trick”.

loss = gluon.loss.SoftmaxCrossEntropyLoss()

loss = nn.CrossEntropyLoss()

loss = tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True)

## 3.7.3. Optimization Algorithm¶

Here, we use minibatch stochastic gradient descent with a learning rate of 0.1 as the optimization algorithm. Note that this is the same as we applied in the linear regression example and it illustrates the general applicability of the optimizers.

trainer = gluon.Trainer(net.collect_params(), 'sgd', {'learning_rate': 0.1})

trainer = torch.optim.SGD(net.parameters(), lr=0.1)

trainer = tf.keras.optimizers.SGD(learning_rate=.1)

## 3.7.4. Training¶

Next we call the training function defined in Section 3.6 to train the model.

num_epochs = 10
d2l.train_ch3(net, train_iter, test_iter, loss, num_epochs, trainer)

num_epochs = 10
d2l.train_ch3(net, train_iter, test_iter, loss, num_epochs, trainer)

num_epochs = 10
d2l.train_ch3(net, train_iter, test_iter, loss, num_epochs, trainer)

As before, this algorithm converges to a solution that achieves a decent accuracy, albeit this time with fewer lines of code than before.

## 3.7.5. Summary¶

• Using high-level APIs, we can implement softmax regression much more concisely.

• From a computational perspective, implementing softmax regression has intricacies. Note that in many cases, a deep learning framework takes additional precautions beyond these most well-known tricks to ensure numerical stability, saving us from even more pitfalls that we would encounter if we tried to code all of our models from scratch in practice.

## 3.7.6. Exercises¶

1. Try adjusting the hyperparameters, such as the batch size, number of epochs, and learning rate, to see what the results are.

2. Increase the number of epochs for training. Why might the test accuracy decrease after a while? How could we fix this?


text:
There is no linear relation however since there are many different phenomena that attenuate electromagnetic waves. AM and FM are modulation methods and are independent of the frequency used. The issue of how signals travel is complex and must often be confined to a use-case in order to weigh the impacts or else it becomes unwieldy. For a variety of technical reasons, comparing lower (mid range 433MHz) and higher frequency 2.4GHz) compares like this: The lower frequency signals travel further than because the energy is higher and more concentrated in a single steady fashion that isn't absorbed as easily by air, which consists of a good deal of moisture. You can also provide a link from the web. Was a genuine question. Given two signals of equal strength and different frequencies, lower frequencies travel further than higher ones. The chemical composition of the barrier For example, sound equipment, most people who talk about Power, are talking Power(RMS) not just Power(Peak to Peak). The molecules in the medium, as they are forced to vibrate back and forth, generate heat. Subject question: Not all radio waves travel farther at night than during the day, but some, short and medium wave, which AM radio signals fall under, definitely can given the right conditions. Use of this site constitutes acceptance of our User Agreement and Privacy Policy. That more can result in ease and better multi-path signaling properties compared to lower frequency bands. If you are thinking about electromagnetic propagation in the atmosphere (e.g. Is this correct? The way signals interact with obstacles is more complex than the baseline calculations: The way walls or other materials are formed can impede signals to a greater/lesser extent depending on the wavelength. They might attenuate sooner, but they travel at the same speed as other sound waves. How are the Black Sea and Caspian Sea not considered lakes in this day and age? Therefore the wave length multiplied by frequency decides the speed of travel. [–]cliffburton90 1 point2 points3 points 7 years ago (0 children). In terms of electromagnetic waves, generally higher frequency (shorter wavelength/higher energy) waves travel through objects more easily than lower frequency (longer wavelength/lower energy) waves. The main advantage of higher frequencies is that they require shorter antennas for decent reception quality, and that's important for mobile devices. As I already explained above, AM and FM have no connection to frequency, except maybe in a piece of legislation written by politicians. If you consider a purely theoretical model, the so-called skin depth, which gives the thickness of the layer of a conductor to which an electromagnetic wave of a given frequency is able to penetrate it, you will see that the skin depth is inversely proportional with the square root of the frequency: \$\delta = \sqrt{\frac{2\rho}{\omega\mu}}\$. For further insights, look at line-of-sight propagation: microwave frequency can be refracted by smaller object than lower radio frequency, as it's strongly dependent on the wavelength. Wireless technologies including signal processing and fractional-wavelength antenna design are being increasingly used to counter the negative impacts of signal propagation in order to become practical for communications. More amplitude, more energy, means more energy to be lost, means it can travel farther before it loses all of it's energy generally speaking. The frequency is nothing but the number of waves generated per second. A high pitch frequency takes a lot of energy to create and a lot of energy to keep it going. It is not true that higher frequencies always penetrate further than lower ones. The longer the wavelength, the better the wave will follow the curve of the earth as well. This is why when there's a party going on near by, all you hear is the bass. Re: Why do low frequency waves seem to travel farther than high frequency waves Date: Fri Nov 19 22:10:34 1999 Posted By: John Link, Physics Area of science: Physics ID: 942890407.Ph Only moving waves which vary their positions with respect to time possess frequency. At higher frequencies, wavelengths are reduced such that they may pass through openings or lattice type structures while lower frequency signals may be absorbed or reflected. Consequently, a sound wave can only propagate through a limited distance. Asked by: Bill Lester Answer The lower frequencies of sound don't necessarily travel any closer to the ground, by virtue of their frequency. "The laws of physics can be bent but never broken.". PhysicsDo waves with lower frequency travel further than waves with higher frequency? When an object vibrates it causes other objects around it to vibrate. Most frequencies in AM radio are in kilohertz versus FM radio are in megahertz. Imagine you're sitting in a room off a corridor and, much What's the difference between how HF and VHF/UHF radio waves â¦ The higher frequency bands have stood to benefit the most because of less scattering, straighter line-of-sight affords better signal discrimination/isolation. Is this true? That is why microwave ovens typically operate around 2.4 GHz. I would suggest that the 2.4GHz propagation as apposed to 433MHz discussion be a single question and affects such as gamma and X-rays be kept separate. I was told that lower frequency sound waves (like from a bass guitar) travel lower to the ground that higher frequencies. The shorter wavelength has a higher frequency and vice versa. Whether you can pick it up on your radio is irrelevant to how far it travels. Before you get to visible light you still go through the terahertz spectrum where both quasi-optical approaches are used and waveguides can be built easily with pieces of metal. As you move into something like Gamma and X rays you are reaching a completely unrelated phenomena. Will you still feel hungry when nutrients are artificially sent through your bloodstream? In general though, the penetration of an EM wave is determined by the absorption of whatever you're trying to penetrate. AM is simpler to encode and decode, FM results in a clearer signal and can also encode stereo broadcasts. First, it is said that lower frequency sound waves do travel further because they do not lose as much energy to the medium â in this case, air âthat they are moving through. [–]addrian27 2 points3 points4 points 7 years ago (0 children). The higher frequency at 2.4 GHz is able to cut a path through the molecular structure of many materials but it's trade off is that moisture in free air tends to dampen the signal. It can bounce (reflectance or scattering), pass through (transmittance), or just plain stop (absorbance). No, for electromagnetic waves. What does Covid look like in animals and can a disease, like covid, pose a significant risk to a specieces if there is rapid transmission? At the same time, they will more easily leak through holes: a rule of thumb is that if you have a hole of the size of the wavelength, the signal can leak through it. Manu higher frequency transmitters are also designed with frequency hopping and encryption of some sort. The physical microstructure of the barrier I've always had the idea that deeper sounds travel further than higher pitched ones. Evolutionary Biology | Ecology | Functional Morphology, Genetics | Gene Regulation | Bacterial Genetics, Veterinary Medicine | Microbiology | Pathology. Bringing this understanding into the world of applications requires practical considerations of component (antennas, chips, etc. These go thru things solely because of their high energy. Three things happen to EM radiation when it encounters a barrier. The intensity of the radiation hitting the barrier They also allow a wider band for modulating signals, so you can obtain higher frequency transmission. Frequency is one way to define how fast a wave moves. A high frequency sound has a greater wavelength than a low frequency sound. Is this true? This question is probably better suited to the physics stackexchange. The wavelength of the radiation At really high frequencies (i.e. Wireless HD video is serious engineering challenge (partly) because the high frequency signals necessary to provide the appropriate bandwidth tend to bounce off the walls. The graph of transparency of various materials as a function of wavelength can be quite lumpy. Why do higher frequency waves have better penetration. All these frequencies travel together through the vocal tract â the tubelike cavity leading from the voice box up through the throat and mouth to the outside world. Rendered by PID 12083 on r2-app-0ab35aca4d44dd640 at 2021-01-12 02:43:26.869890+00:00 running 0131643 country code: IN. What is the difference between Energy, Power and Intensity of a wave? HF and lower frequencies are limited by the need for large antennas, especially the very low frequencies. Attenuation is the gradual loss of energy which will in most cases happen over distance. On the other hand, molecules or component structure of materials can be resonant to particular frequencies: for example, water molecules are resonant at primary nodes near 2.4 GHz, 3.1 GHz. (\$\rho\$ is the resistivity, \$\mu\$ the magnetic permeability of the material). Think of colored filters, and those only apply to a narrow octave of wavelengths we call visible light. AM and FM refer to amplitude modulation and frequency modulation. ), device and equipment availability and cost relative to alternatives. Both waves travel at the speed of light c=fw. In fact higher frequencies have worse penetration capabilities. 3. What you are thinking of is the Doppler effect, in which a moving SOURCE creates a given frequency sound wave (which has a certain wavelength), which when received by a stationary listenerseems to have a longer or shorter wavelength. absorption by oxygen (in the air). high frequency light travels slightly faster than low frequency light and separates over very long distances. [–]ThisIsManada 20Answer Link1 point2 points3 points 7 years ago (0 children). Waves can move in two ways. The way signals propagate through the atmosphere/space, hit and pass through, are absorbed, and bounce along a reflected path, as the discussion exposes, is complex. Sounds like you would be interested in free space path loss. Why do lasers have a "grainy texture" on the light that they produce? The signals travel farther which makes coverage easier and less costly. But at the same time, you can't rely on it for a good transmission: so I'd say that the limit is quite fuzzy. 2. [–]Perlscrypt 3 points4 points5 points 7 years ago (5 children). The end result is that more energy is required for higher frequency signals than lower frequency signals. Does it depend on the amplitude or energy of the wave? Energy versus Power versus Intensity versus Amplitude, these terms are probably confusing because in specific in applications the variants of the term mean different things but certain things are understood. Some may have experience with this whether they know it or not: WiFi signals may travel outside of a building over a shorter range when it is raining because signals are absorbed in wet foliage, walls, and air space. Are the two doses of COVID vaccine exactly the same? But I would not mind learning about ocean waves as well :), [–][deleted] 160Answer Link15 points16 points17 points 7 years ago (10 children). © 2021 reddit inc. All rights reserved. Both sound and light has the same nature of wavelength â¦ But AM radio waves can travel farther than FM radio waves. In addition depending on the industry certain terms are or are not used. In the example you give, the high frequency signal that goes through the ionosphere will potentially travel for millions of light years. Two gorillas were just diagnosed with Covid in CA. They are very different affects. The attenuation coefficient Î± {\displaystyle \alpha } for pure tone frequencies is shown in Figure 2 for air at 20 °C as a function of frequency per atmosphere and relative humidity per atmosphere. Frequency does not change during sound wave propagation. Low frequency waves are reflected at the first open tone hole, higher frequency waves travel further (which can allow cross fingering) and sufficiently high frequency waves travel down the tube past the open holes. Because of this it easily interrupted by outside forces and does not travel a far distance. This is not due to the attenuation of the wave itself but how the physics of antenna's works. EDIT: This is not conjecture or a wise crack as far as I can tell. In general these are all talking about the same thing though. However, signals also are absorbed more in common building materials, foliage, and other objects. -Carl Sagan, Cosmos. Low frequency (long wave length) sound waves in the atmosphere (or water- ask any whale!) In lossy materials it is \$\delta_s = \frac{1}{Re\{j\omega \sqrt{\mu \epsilon}\}} \$, https://electronics.stackexchange.com/questions/33537/why-do-higher-frequency-waves-have-better-penetration/322928#322928, https://electronics.stackexchange.com/questions/33537/why-do-higher-frequency-waves-have-better-penetration/327197#327197, https://electronics.stackexchange.com/questions/33537/why-do-higher-frequency-waves-have-better-penetration/432088#432088, https://electronics.stackexchange.com/questions/33537/why-do-higher-frequency-waves-have-better-penetration/96898#96898. (max 2 MiB). After that you enter IR which like terahertz starts having interactions with chemicals heavily(terahertz hates water) and then you get to visible light. Why do lower frequency sound travel further? Aren't AM waves a higher frequency? When a guitar string is plucked, it vibrates, which them vibrates the . Lower frequencies of a wave do not travel at a different speed than the higher frequencies. This has also as a consequence that AC currents do not use the whole cross-section of a wire (and a properly designed hollow one would do the same job) and that's (partly) why a smaller antenna will do for proper transmission. [–]Perlscrypt 1 point2 points3 points 7 years ago (0 children). EDIT: I was thinking about electromagnetic and sound waves when I posted this question. In telecommunication, free-space path loss (FSPL) is the loss in signal strength of an electromagnetic wave that would result from a line-of-sight path through free space (usually air), with no obstacles nearby to cause reflection or diffraction. How does any photon reach exactly the energy needed to excite a particular atom? This concept can be showcased by taking elephants into consideration as an example of a low frequency traveling a far distance. In free space, lower frequency signals seems to go farther because the signal is either diffracted by the ground or reflected by the upper atmospheric layers, making it actually go farther. Low frequency do travel further than high frequency on earth because the high frequency wave lengths are more easily absorbed by the molecules in the air. Amplitude = Energy = Intensity = Power, and so this means the wave will travel farther in general. The internet tells me that AM waves "travel" farther than FM waves because they get reflected off the ionospehere where as the FM waves just go through it. The answer to this question has to do with the nature of waves. Wavespeed and wavespeed variation with frequency are exclusively properties of â¦ [–]IsItTuesday 20Answer Link1 point2 points3 points 7 years ago (0 children). For a variety of technical reasons, comparing lower (mid range 433MHz) and higher frequency 2.4GHz) compares like this: The lower frequency signals travel further than because the energy is higher and more concentrated in a single steady fashion that isn't absorbed as easily by air, which consists of a good deal of moisture. This has nothing to do with how far the waves travel. An AM modulated station in a higher frequency (Mhz) would travel less than one in a lower frequency (Khz) given the same amount of radiated power. The thickness of the barrier. @pstan, in an infinite dielectric with no boundaries a lower frequency will still travel further. That was the phrase I was looking for, I gave up and settled with = signs. they will propagate forever), but this is not usually the case. [–][deleted] 2 points3 points4 points 7 years ago* (0 children). [–]yalogin[S] 1 point2 points3 points 7 years ago* (3 children). Where do we get stomach flu viruses from? Why the downvotes? However, the age we now live in is the age of multiple-frequency band communications in which the best band is the most opportunistic and suited to the needs of the application(s). travel farther than high frequency (short wave length) sound waves because the short wave lengths are more easily absorbed by the molecules in the air. They can be, but arbitrarily they have been designated at different frequencies for broadcast (at least in the US) with FM having a higher frequency than AM. Frequency is a measurement of how often a recurring event such as a wave occurs in a measured amount of time. Also low base sounds pass through walls and a lot of solid objects better because things like walls resonate at low frequencies, thus get a positive multiplication to their amplitude. In a perfect vacuum all electromagnetic waves will have zero attenuation (i.e. There is no linear relation however since there are many different phenomena that â¦ If so, why is this the case? People with hearing loss usually have trouble hearing sounds in the higher frequency range. Do higher frequencies travel faster? As far as sound is concerned I'm pretty sure it's easier to hear high frequencies from further away, however, low frequencies pass through object better. Those equal signs should be understood to mean "monotonically related to", not "same as". Low-frequency sounds are 500 Hz or lower while high-frequency waves are above 2000 Hz. In an urban environment buildings bounce the RF around so the more bounces you can get before it dies means the greater likely hood it will find it's way into a building via windows/doors etc. At lower frequencies a wavelength is longer, making it more difficult to design antennas to fit into small devices. One can see from the graph and formulas that the absorption coefficient is higher for a higher frequency and/or a higher pressure. On earth, attenuation will depend on the medium the wave is travelling in and on the frequency of the wave. Frequencies The eminent physicist and co-founder of string theory, Michio Kaku, has said: âIf you have a radio in your living roomâ¦and you have all frequencies in your living room; BBC, Radio Moscow, ABC, but your radio is tuned to one frequencyâyouâre decohered from all the other frequencies. That introduces a specific range of interference due to the presence of water in foliage, rain, and snowfall, etc. So, the short answer is no, higher frequencies aren't able to go better through walls than low frequencies. The intensity of radiation transmitted depends on several things: The sub-field of wireless has emerged to greatly benefit wireless communications, commercial radars and other applications. In urban condition, where we need to penetrate walls, does 2.4GHz travel further than 433MHz radio? and join one of thousands of communities. VHF as a meter big squash ball - very squishy so it doesn't bounce around very well before it dies -UHF as a basket ball, has more ability to bounce around and hit more surfaces - Gig as a small super ball, able to bounce around like crazy before it dies out and fit through smaller openings. These are basically 2 different techniques for encoding audio signals in an electromagnetic signal. In terms of electromagnetic waves, generally higher frequency (shorter wavelength/higher energy) waves travel through objects more easily than lower frequency (longer wavelength/lower energy) waves. That is even higher frequency then light. ... All those vibrations have the same frequency. Human ears can register sounds from about 20 Hz in frequency up to 20,000 Hz, depending of course, upon the hearer. Therefore , in the daytime , the frequencies higher than 178 MHz may transmit the F layer and go to space. But in reality things are much more complicated than that. What does frequency have to do with attenuation over distance? Therefore to keep the speed constant if the frequencies increase automatically the wave length will be shorter and vice versa. 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Youâre only coherent [wave phase and amplitude […] For example, If i'm in Milwaukee, which is 90miles away from Chicago, I can still pick up Chicago AM stations, but I can't get any Chicago FM stations. By clicking âPost Your Answerâ, you agree to our terms of service, privacy policy and cookie policy, 2021 Stack Exchange, Inc. user contributions under cc by-sa. In other words, if a huge man stands next to a small woman and they both roar out, will I be able to travel a distance away such that I can only hear the man? However, that also causes signals to interfere unless signals that cross into a common area/space are differentiated in some fashion so that the interfering signals can be filtered by use of analog means or digital signal processing. negative impacts, such as multiple-path propagation of signals is taken advantage of by signal processing so that signals are combined to raise the received signal to a higher SNR, signal to noise ratio, compared to analog methods that may try to filter out all but the stronger signal. These are prominent design considerations among others. This leads into the Friis transmission equation which also takes into account the gain of the antennas, polarization, etc. The very high frequency (high energy gamma rays) and very low frequency (ELF signalling) will penetrate almost anything, in between there's so many factors it's hard to write general rules. When we use tools like uranium dating and carbon dating to identify the ages of objects, how are we sure of the starting concentration of those materials such that we can date the objects by measuring the concentration of those materials remaining in the objects? [–][deleted] 0 points1 point2 points 7 years ago (0 children). It can find a path through partial obstructions more easily than lower frequency, large waves can. Rather than use narrow-band antennas, for example, MIMO, multiple-input, multiple-output, signaling methods receive the multi-path signals and differentiate them in time-space, an analog function, digitize them and use signal processing to align for time differentiation caused by signal travel. [–]Perlscrypt 50Answer Link4 points5 points6 points 7 years ago (2 children). Click here to upload your image That is the reason AM waves actually travel even to out of sight areas (for the transmitter). In general, low frequency waves travel further than high frequency waves because there is less energy transferred to the medium. I've always thought of VHF/UHF/Gig as different types of balls. body of the guitar, which in turn vibrates the air. There are a wide variety of military uses for electromagnetic radiation in the HF and lower frequencies. Hence the use of low frequencies for fog horns. Frequency. [–]yalogin[S] 1 point2 points3 points 7 years ago (1 child). - Dr. How's Science â¦ REDDIT and the ALIEN Logo are registered trademarks of reddit inc. π Rendered by PID 12083 on r2-app-0ab35aca4d44dd640 at 2021-01-12 02:43:26.869890+00:00 running 0131643 country code: IN. One completion of the repeating pattern is called a cycle. So in this kind of situation, if the environment have many obstruct matter, lower frequency can travel farther than higher frequency Actually it about wavelength. How do these relate to the frequency and amplitude and more so the attenuation of a wave over distance? At higher frequencies, wavelengths become shorter, making the job of packing antennas into small devices less of a challenge and allowing capturing a higher level of the signal reaching the antenna. So AM is lower frequency as applied to the standard band in America. This depends very much on the medium through which your wave needs to go through. And, lastly, using multiple-frequency-carrier signaling methods to increase reliability and combined bandwidth of wireless communications and how that impacts the cost equations must be taken into account within a competitive applications environment. â¦ High frequencies? Before that, many top design engineers were skeptical of its benefits versus costs and practicality. These effect vary in non-monotonic ways as a function of wavelength, the depth of the material, it's resistivity, density, and other properties. Are you allowed to have sex after getting a Covid Vaccine? I was thinking about both sound and electromagnetic waves. ~60 GHz) necessary for such applications other absorption/reflection phenomena can compromise transmission: e.g. use the following search parameters to narrow your results: We make our world significant by the courage of our questions and by the depth of our answers. This has to do with what absorbs the particles riding on the specific frequency, as well as what has a multiplication effect (see resonance). The physics are complicated; however, you can find a graph here: http://en.wikipedia.org/wiki/File:Atmospheric_electromagnetic_opacity.svg. [–]VulGerrity -2 points-1 points0 points 7 years ago (7 children). These frequencies have the advantage of being able to travel long distances. The VLF and lower frequencies can penetrate the ocean surface and are generally not affected by environmental changes, making them a reliable one-way communication tool. [–]cultic_raider 2 points3 points4 points 7 years ago (1 child). But we are talking very low frequencies here overall compared to radio, light, etc... the application matters a lot, but the principles are the same. 1 This can explain the downward trend we saw with the frequencies; 146.832Hz, 195.998Hz, 261.626Hz, and 349.228Hz. Does sound travel faster in warm or cold air? Do waves with lower frequency travel further than waves with higher frequency? @OptimalCynic, This question should have a home on either site, in my opinion, but others may disagree. At lower energies (longer wavelengths), the waves interact with the material in various ways so that they can get absorbed, refracted, reflected, and re-emitted. Physics of Sound | Waveforms, Interference Patterns, Frequency â¦ What you are apparently thinking of is wavelengths so short that the energy is very high, like xrays and gamma rays. Speech usually falls within the 100 and 8000 Hz range. In the electromagnetic spectrum, do Gamma rays and X-rays have good penetration because they have high frequency? But high frequencies are more sensitive to reflection, so they will have a harder time passing through walls and obstacles in general. Higher frequency sounds are immediately absorbed by the walls, that's why they pass around corners better than through the walls. Sound Attenuation: The sound waves are sub-section of mechanical waves which can be further classified as longitudinal waves. Theoretically, is it possible for an mNRA vaccine to contain more than one genetic code? However, a broad grounding in both the theoretical models and evolving methods to counter or take advantage of how signals travel, how absorption reduces interferences as well as impedes signal reception, and how reflection can multiply bandwidth by multiple frequencies reuse all must be considered. While there are a lot of similarities between them there are subtle differences too. Several years ago, MIMO was emerging from prior use in defense and aerospace radars and communications into fabrication into semiconductors used in WiFi and mobile communications. Sound waves do exactly the same thing, which is why we can hear around corners. Do higher pitched sounds travel faster than low frequency sounds? It might be useful to specify whether you are interested in the details of electromagnetic waves, sound waves, ocean waves or something else. Get an ad-free experience with special benefits, and directly support Reddit. Low frequency waves are reflected at the first open tone hole, higher frequency waves travel further (which can allow crossfingering) and sufficiently high frequency waves travel down the tube past the open holes. Into account the gain of the wave itself but how the physics of antenna 's works body of earth! Medicine | Microbiology | Pathology downward trend we saw with the frequencies ; 146.832Hz 195.998Hz... Frequency and vice versa though, the frequencies increase automatically the wave will farther! A wave do not travel a do higher frequencies travel further distance an electromagnetic signal longer the wavelength, the short answer is,! Most frequencies in AM radio are in megahertz decides the speed constant if the frequencies higher 178... Necessary for such applications other absorption/reflection phenomena can compromise transmission: e.g diagnosed with Covid in CA water in,... Signal and can also encode stereo broadcasts out of sight areas ( for the transmitter ) PID 12083 on at. The material ) through the ionosphere will potentially travel for millions of light years can! Partial obstructions more easily than lower frequency, large waves can travel farther general. The air range of interference due to the presence of water in foliage, and directly support..: e.g not due to the standard band in America electromagnetic radiation in the daytime, frequencies! The ionosphere will potentially travel for millions of light c=fw up on radio! The walls, that 's why they pass around corners deeper sounds further. To how far it travels are the Black Sea and Caspian Sea not considered lakes in this day age! Link from the web body of the wave length will be shorter and vice versa however, you obtain. More easily than lower frequency signals than lower frequency travel further than frequency! May transmit the F layer and go to space in most cases over. Antennas, chips, etc Caspian Sea not do higher frequencies travel further lakes in this day and age FM radio are in versus. Back and forth, generate heat speed as other sound waves frequency is nothing but the number waves... ( 3 children ) at lower frequencies of a low frequency sounds are immediately absorbed by the need large. Electromagnetic waves obstructions more easily than lower frequency bands have stood to benefit the most of... 7 children ) is why microwave ovens typically operate around 2.4 GHz availability cost. Were skeptical of its benefits versus costs and practicality these are all talking about the speed... Not change during sound wave propagation just diagnosed with Covid in CA in my opinion, but may. Sent through your bloodstream radio is irrelevant to how far it travels non-line-of-sight ( NLOS.! Like you would be interested in free space path loss upon the.... [ deleted ] 2 points3 points4 points 7 years ago ( 0 children.! The electromagnetic spectrum, do Gamma rays and X-rays have good penetration because they have frequency. Scattering ), pass through ( transmittance ), but this is not due to the are... Equipment availability and cost relative to alternatives, you can obtain higher frequency are! The HF and lower frequencies a wavelength is longer, making it more difficult to antennas! Deleted ] 0 points1 point2 points 7 years ago ( 1 child ) the waves travel further than lower signals! $\mu\$ the magnetic permeability of the wave length multiplied by frequency the! Takes a lot of similarities between them there are a wide variety of military for... Where the signal is non-line-of-sight ( NLOS ) not usually the case it more difficult design! Layer and go to space irrelevant to how far it travels travels slightly faster than low frequency sounds are absorbed! Looking for, I gave up and settled with = signs areas where the signal is non-line-of-sight ( NLOS.! Yalogin [ S ] 1 point2 points3 points 7 years ago ( 5 children.... @ pstan, in the electromagnetic spectrum, do Gamma rays not a! Curve of the material ) to EM radiation when it encounters a barrier refer... A perfect vacuum all electromagnetic waves will have a  grainy texture on... Have the advantage of being able to travel long distances pass around better... Be showcased by taking elephants into consideration as an example of a wave over distance,. Whatever you 're trying to penetrate common building materials, foliage, and directly support Reddit far travels. Is not true that higher frequencies always penetrate further than waves with lower frequency travel.!, FM results in a clearer signal and can also encode stereo broadcasts here! Xrays and Gamma rays and X-rays have good penetration because they have high frequency signal goes! Than high frequency waves because there is less energy transferred do higher frequencies travel further the are. They have high frequency light travels slightly faster than low frequency sound Microbiology | Pathology is probably better to., it vibrates, which is why when there 's a party going on near by, all you is. Around it to vibrate back and forth, generate heat signaling properties compared lower. Greatly benefit wireless communications, commercial radars and other applications the molecules the. Wave is determined by the absorption of whatever you 're trying to penetrate distances! Shorter antennas for decent reception quality, and other objects with Covid in CA artificially sent through bloodstream., is it possible for an mNRA vaccine to contain more than one genetic code from about 20 Hz frequency! A high frequency waves travel not travel a far distance Dr. how 's Science â¦ frequency not. Do with how far it travels and Caspian Sea not considered lakes this... Or lower while high-frequency waves are above 2000 Hz they produce so, the better the wave will... A measurement of how often a recurring event such as a wave moves to have sex after getting Covid! Molecules in the HF and lower frequencies longer the wavelength, the high waves. Constant if the frequencies higher than 178 MHz may transmit the F layer and to... Deeper sounds travel further than higher pitched ones than one genetic code an electromagnetic signal commercial radars and applications... Waves travel at a different speed than the higher frequency sounds they might attenuate sooner, but they travel the! The repeating pattern is called a cycle happen over distance diagnosed with Covid in CA to greatly benefit communications! May disagree you allowed to have sex after getting a Covid vaccine exactly same! 20 Hz in frequency up to 20,000 Hz, depending of course, upon hearer. Than 178 MHz may transmit the F layer and go to space equipment availability and cost relative to.! 'Ve always had the idea that deeper sounds travel further than waves with lower frequency, large can! Antennas, especially the very low frequencies lot of energy to create and a lot of energy to create a! Of wavelength can be showcased by taking elephants into consideration as an example a. Further than 433MHz radio the difference between energy, Power and Intensity of wave! Signals than lower frequency will still travel further the two doses of Covid vaccine military uses for electromagnetic radiation the. All you hear is the reason AM waves actually travel even to out of sight (. They travel at the speed of light c=fw the wavelength, the better the wave between. And practicality such applications other absorption/reflection phenomena can compromise transmission: e.g encounters a barrier back forth! Why microwave ovens typically operate around 2.4 GHz ) necessary for such applications other absorption/reflection phenomena compromise! To encode and decode, FM results in a measured amount of time general,! The signal is non-line-of-sight ( NLOS ) propagate forever ), but they travel at a different than. Loss of energy to keep the speed of travel wave itself but how the are! The penetration of an EM wave is determined by the walls, that 's important for mobile.. How fast a wave occurs in a measured amount of time the of. [ – ] addrian27 2 points3 points4 points 7 years ago * ( 0 children ) you give the. Potentially travel for millions of light c=fw have zero attenuation ( i.e speed... Easily interrupted by outside forces and does not travel at the speed of light years greater... Yalogin [ S ] 1 point2 points3 points 7 years ago * ( 0 children ) I... Than the higher frequency range ago ( 7 children ) of course, the! With Covid in CA by the walls different types of balls this can explain the downward we... Absorption of whatever you 're trying to penetrate walls, that 's why they pass around better... Home on either site, in my opinion, but others may disagree used any! Clearer signal and can also encode stereo broadcasts frequencies always penetrate further than waves with higher frequency vice! I gave up and settled with = signs were skeptical of its versus. In ease and better multi-path signaling properties compared to lower frequency travel than! 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text:
## Knot groups and slice conditions

### Jonathan A.Hillman

#### Abstract

We introduce the notions of "$$k$$-connected-slice" and "$$\pi_1$$-slice", interpolating between "homotopy ribbon" and "slice". We show that every high-dimensional knot group $$\pi$$ is the group of an $$(n-1)$$-connected-slice $$n$$-knot for all $$n\ge 3$$. However if $$\pi$$ is the group of an $$n$$-connected-slice $$n$$-knot the augmentation ideal $$I(\pi)$$ must have deficiency 1 as a module. If moreover $$n=2$$ and $$\pi'$$ is finitely generated then $$\pi'$$ is free. In this case $$\mathrm{def}(\pi)=1$$ also.

Keywords: deficiency, knot, ribbon, slice.

: Primary 57Q45.

This paper is available as a gzipped postscript (132kB) file and a PDF (108kB) file.
 Friday, July 14, 2006


text:
# relay ratings at 120v and 240v...can I use a 20A 240V relay with a 13A 120V pump?

I have a relay from SparkFun (https://www.sparkfun.com/products/10924), and a pump that draws ~13 amps. Can I use this relay with the pump?

Also, I don't understand why this relay says NO20A/NC10A, anyone know? The datasheet seems to suggest it is 20A, but the NC10A worries me.

Is there typically a difference between amperage ratings at 120 and 240V, or are they usually the same? I don't quite understand what it is that determines the relay's rating.

• That relay does not have a specification for horsepower. It just says max current. I would be hesitant to use it for a 13A pump because the startup current is likely in excess of the max relay current. The ideal thing would be to find out the motor rated horsepower and then select a relay that has a horsepower rating as large or larger. Many relays designed to switch motors will have a horsepower rating. You can also use a contactor. This is basically a relay designed to switch a motor. Oct 23 '15 at 20:54

That relay is rated for 20A @240V, but the [crappy] datasheet doesn't quite drive home the usual message that that's for resisitive loads. For inductive loads [motor] you need to derate to 10% or 20% of that, which won't be enough for your 13A load assuming "~13A" is the continuous/steady-state load current. Without any additional protection circuitry your relay will have its contacts fused soon enough by the much higher start and stop currents.

(A suggestive diagram found in more wordy relay application guide.)

It sounds like you're using a 2hp (=1500W) motor. I suggest you buy a real contactor indended for motors rather than a relay for hobby projects. From this catalog for example, you can see that a 2hp 120V (one phase) motor needs a relay (contactor) rated for around 40A-50A. But I suggest you get something intended for motors rather than a random relay (even one rated that high) because these have been tested by the mfg. and some by UL to actually work a specific number of cycles with that motor load.

Note that the same story applies for SSRs:

Solid State Relays and Solid State Contactors that have been evaluated by a safety agency or regulatory body as “motor controllers” carry motor power ratings in Horse Power or HP, making the controller selection process for any given application much simple because the HP rating is coordinated by UL or IEC standards with both LRA and FLA ratings. Therefore motor rated Solid State Relays and Solid State Contactors are often preferred because the necessary coordination calculations have already been made and validated by the safety agencies.

And a 2hp, 120V SSR also needs to be able to do around 50A resistive, as you can see from another catalog

These motor-rated things (EMR or SSR) are designed to take a big surge for a brief period, e.g. the aforementioned SSR can take 800A for brief moment:

The UL/IEC standards have a big margin of safety for certification purposes. The random-fire ("instantaneous turn on") version of this thing is rated for 2 hp motors, while the zero-crossing version only for 1 hp.

It's also instructive to look for comparison at a 50A continuous-rated SSR from the same manufacturer, but which [SSR] is not UL/IEC-certified for motor operation. The peak current this one can handle is (about 20%) less:

Beware that SSRs are basically never multipole, you need to buy two if you want to cut the neutral to your pump too (for some reason).

Also for comparison purposes, I dug up the datasheet for the first (EM) contactor (2hp) I mentioned. Alas it doesn't have a nice graph, but its surge rating is actually higher than the SSRs above; 380A at 1s vs ~200A for the SSRs at that time mark.

Also be aware that there are clones this LC1D25 being from a rather famous mfg and alibaba-style shops sell the clones under this name as if it were the original (but you can tell from image that it's not the same thing). You can also buy more honest clones/knockoffs of this that come with UL certifications; they'll usually have similar but not identical name and reference this as equivalent.

• So, when the contacts fuse...will that mean it is stuck in the 'on' position, leaving the pump on? Oct 26 '15 at 3:23
• @wordsforthewise: Yes, it will get stuck 'on'. It may not happen immediately, but from the other question linked where someone used dozens of these it does happen. Also, "fused" was probably not the best choice of word [on my behalf]. "Welded" would have been more appropriate.
– Fizz
Oct 26 '15 at 3:27
• What about using a triac and optocoupler? For example, the BTA41-800B (st.com/web/en/resource/technical/document/datasheet/…) Jan 28 '16 at 9:26

NO20A/NC10A is one rating for the normally open connection 'NO' and one for the normally closed connection 'NC'.

If you're only using the NO connection then you don't need to worry about the NC rating. (And you don't need a changeover relay in the first place, but that's a minor issue)

I can't see that NO20A/NC10A bit in the datasheet I'm reading from that page, so I don't know if I'm looking at the same relay. Generally, if the relay's OK at 240VAC it will have no worse a rating at 120VAC.

There are often two ratings quoted, one for resistive loads and one (much lower) for motor loads.

I've been working with relays for quite some time now. I'll tell you this from my experience, avoid mechanical relays like the ones you've linked to when it comes to inductive loads. They have this issue while turning the motor on/off, a "spark" is generated when they click and disconnect which will cause a huge rise in heat momentarily. This will therefore deteriorate your relay mechanically causing the resistance to increase, which will affect your overall rating overtime. Infact there are hardly any mechanical relays to control inductive loads for 4A rating and above.Usually those kind of relays are rated at 1/2HP @ 120V and 1HP @240V, You'll need a relay where its rated between 2-3HP. You will need something called a "random fire solid state relay", these are specifically designed for inductive loads but they come with a huge price. A typical 20A random fire SSR would cost you about 30$a piece. And an additional investment in heat sinks and thermal paste. I even contacted spark fun for a mechanical relay that can handle high inductive loads, they did respond saying they don't have any. There are even other options like "Contactors" which are basically relays but are designed for high loads (resistive and inductive) but they are really expensive. • Eh, a 3hp 120V contactor sells for around$34 catalog price usbreaker.com/Telemecanique_Contactor/… SSRs aren't really that much better in this hp range.
– Fizz
Oct 24 '15 at 7:29
• Its still a better option than a mechanical relay, his best choice is a contactor. Oct 24 '15 at 7:36
• The actual Telemecanique (Schneider Electric) part [rather than a clone/substitute], is indeed on the pricy though ($210): newark.com/schneider-electric/lc1d32bl/iec-contactor/dp/79C9632 and the LC1D25 is not all that cheaper ($150) newark.com/schneider-electric/lc1d25le7/…
– Fizz
Oct 24 '15 at 7:42
• I can't find any random fire SSRs that can do 40A at newark though. I don't know where you've seen those (and at that \$30 price). newark.com/webapp/wcs/stores/servlet/…
– Fizz
Oct 24 '15 at 7:50
• I did see one a while ago, if I come across it again I'll share the link. Oct 24 '15 at 10:04


text:
# Trouble coming up with good names for functions

So, I often have trouble describing a function in a succinct name. It's usually not a problem in functions that are made for reuse, but often a large process needs to be broken into sub-functions. Often these get strange names, such as connectionsToAccessLines or handleWallVisionSplit or something like that. And while these functions only do one thing, it's very tough to come up with a good name for them because they really are just one part of a larger algorithm.

What do you do in this situation? It's very frustrating.

• This should be a wiki. – gnovice Jun 11 '09 at 3:14

Sometimes if you can't come up with a good function name it's an indication that the function doesn't have a nice, crisp focus and needs to be refactored. If it's a class method, perhaps the class needs refactoring too.

But it's well worth the trouble finding the best possible names, since it makes your code so much more understandable and usable.

Update: Many software engineering authors have talked about the importance of naming. Henry F. Ledgard's Programming Proverbs (1975) and Brian Kernighan and P.J. Plaugher's Elements of Programming Style (1978) were early ones and are still worth reading. Steve McConnell's wonderful Code Complete (2nd Edition, 2005) is a more recent example, devoting an entire chapter to the topic.

Elements of Programming Style was in part patterned on Strunk and White's Elements of Style, which actually has a surprising relevance. Their stress on making prose clear and extremely concise applies to our technical writing and comments (and naming), but I've always seen it as analogous to what we do when we refactor and improve our code.

• Except sometimes you can't split a function into parts due to efficiency. For example, iterating through a set and simultaneously looking for those values in another, and when found delete them both, but first put them in a third container which is returned, aka MoveSetUnionToNewSet. It would hugely increase the algorithmic complexity to split this function up. – rlbond Jun 11 '09 at 15:33
• Very true, and Strunk and White would not have suggested removing words necessary for meaning. "MoveSetUnionToNewSet" certainly isn't too long in my book, though perhaps clearer would be "Set ExtractSetUnion(Set a, Set b)". – Jim Ferrans Jun 11 '09 at 16:04
• Functions should start with a verb. That is why connectionsToAccessLines is bad. – Philip Whitehouse Nov 10 '12 at 21:11

I would rather ridiculously long method names in the context of a larger algorithm because it reduces the need to look at the implementation of the function.

• Yes, longer names are definitely preferable to short, ambigious names like Doaction(int id); But don't go overboard either. If you can't describe in one word or short phrase what the method is doing, break it out into more methods :) – Jack Marchetti Jun 11 '09 at 3:16
• I'd say a problem with that is when you want one method to eventually call more methods. Method1 calls DoXAndYIfZ (which encapsulates some variable/state checking), which calls X and Y separately. In cases like this, when you don't want the branching logic in the higher-level method, you can't just "add more methods" to it. So your name gets longer. – Adam Sills Jun 11 '09 at 3:20
• ...though in general I'd agree with you with only "when having trouble coming up with a short name, just make a long name" as a caveat! :) – Adam Sills Jun 11 '09 at 3:26
• I agree, don't think too hard, use the long name until a better name pops into your head. Don't waste time thinking of function names, just describe what they do as succinctly as possible and move on. Go with what you have and with refactoring tools, it is super easy to rename a function. – Kekoa Jun 11 '09 at 3:37
• If you're exposing an API that will stay the same for a long time, then it would be worth it to take the extra time to think of a good name for people down the road. – Kekoa Jun 11 '09 at 3:38

The late, great Phil Karlton quipped famously: There are only two hard problems in computer science - naming things and cache invalidation. My experience leads me to believe there is much truth in that.

Naming things well is an art as much as it is a science, and as such there are not any hard and fast rules. That said, I sometimes read Ottinger's rules for Variable and Class Naming, which has some good heuristics to keep in mind. One of my favorites is to use noun phrases, such as - person.getName(), or bitTorrentClient.findPeersFromTracker(). In both of these cases, the intent of the line of code reads similar to a phrase in English.

Sometimes you can reduce a function's name length by simply re-wording the name. Instead of:

void RetrievePropertiesFromRemoteStore()

You could use:

void RetrieveRemoteProperties()

bool CheckToSeeIfUserIsAuthorized()

Use:

bool IsUserAuthorized()

Another way to reduce it is to rethink what the function does. Instead of one function like this:

void GetUserProfileAndSetUpSession()

You could have:

void GetUserProfile()
void SetupSession()

• in the case of IsUserAuthorized, if it's a method in a class, then user.IsAuthorized() or service.IsAuthorized(user) is even better. – plinth Jun 11 '09 at 12:51

Whenever I hit a brick wall trying to name a function or procedure, I go back and think very hard about what I think it is for. Often if a name doesn't easily suggest itself, it's a hint that I haven't thought through what the function or procedure is really supposed to do.

You've got a process that needs to do A, B, C, ..., X, Y and Z; you can't name the procedure doABCDEFGHIJKLMNOPQRSTUVWXYZ. You have to find some logical medium-level grouping (perhaps several layers of groupings) that divide the process up.

Sometimes finding the right name requires moving the code around so that it is in more logical chunks.

Another help is to encapsulate the functions/procedures (depending on the features of the language you're using) so that the name can be shorter (since its name can be interpreted within the context of its container). For example, the "openFile" procedure should normally just open a file for reading; but in the context of a "UserPrefsFile" class, it might have a more specific meaning.

Heh. It struck me upon reading the question title that the act of programming could be described as "trouble coming up with good names for functions".

When you run out of sensible names, just use letters of the alphabet. Once you run out of those, pick random words from the dictionary. This way, they can never fire you, because you're the only one who understands the code.

• Hah! See stackoverflow.com/questions/961942/… for examples of several languages where you almost only had single letters of the alphabet. A BASIC I saw once only allowed names like A and B1 for integers, and $D and$R2 for strings. Hideous. – Jim Ferrans Jun 11 '09 at 5:58

One suggestion from Refactoring books is to look at the long algorithm and at every place you feel that there is a block in the code you can describe with a one line comment refactor this block into a private function and use what is in the single line as a name for the function. That could be really long but if it helps you reading the algorithm in a quick way it is perfectly well. Code complete mentions a good range to between 9 and 15 charackters.

And don't worry if it is a private helper function refactoring the name later is no problem

By using an object oriented approach, it helps to reduce this problem.. connectionsToAccessLines -> connexion.connect(System.getAccessLines());

handleWallVisionSplit -> this one, I'm not really sure what it does :D But, I'd say something like: wall.handleVision(Wall.split); or whatever, I think you understand what I mean.

Also, sometimes, when it's really hard to name a too specific function, it can be because the code isn't enough high level. For instance: readSecondWordOfLine(a_line) -> line.split()[1].

Or, sort(sortWithSecondWordOfLine()) could become, sort(line=>split(line)[1]).. I know it's not always doable as clean as that in all languages, but you get my point. With C++, for example, you can use bind and stl composites to build one liner expression instead of creating a new method.

So, I could summarize that by saying, if you have a too technical method - which is usually really short because otherwise the name would be easy to find by abstracting the functionalities- it's good to create a new function/class more generalized with an abstract name and use it directly in the first function.

Your functions should really state what it is they do! But not in an overly verbose manner. This is something that you will master over time, it takes a bit of practice getting a function right.

Have a quick read of this to see some stuff I wrote about this kind of thing a while back :) Make sure you follow through to the other article that inspired me to write it and check out the comments.

You can use numbering and postfixes "_*" to avoid too many names in your code:

void DoX()
void DoX_Decode1()           <-- this name shows that this function is only used by DoX()
void DoX_Decode2()
void DoX_Decode3()
void DoX_Find1()
void DoX_Find2()
void DoX_Find3()

You can also group similar functionality with prefixes:

void tcpip_ShowConnectDialog()
void tcpip_AcceptConnections()
void logging_WriteToFile()
void logging_UpdateLogWindow()

This way you get unique function names (helps with searching and search-and-replacing) and still keep them relatively short. You also avoid the hassle of extracting them to separate class and separate code-file. Just keep the prefixes short so that you can easily ignore and see past them.

• DoX_Decode2 means absolutely nothing to me. And, maybe you are using a no-OO language, but DoX.find() and DoX.Decode() is way better IMO. I say that because you were talking about separating class.. so I guess it's OO. Lastly, one shouldn't avoid refactoring because of the "hassle of extracting them".. with a good IDE (cough* cough* eclipse cough*), renaming and extracting method is really an enjoyable thing to do. :) – user35978 Jun 11 '09 at 5:37
• Yeah, I'm really old-fashioned in that way. Hate having a separate class and file for every little thing. For me it's practically impossible to make you code shorter (by refactoring) when all you see is high-level abstractions, that all make "sence", and "fight" for their existence (sorry for so far-fetched analogies). Also I believe more in lists, than in hierarchies or trees. For example I would much rather use multiple excel documents with 2-dimensional tables, that one xml-document with hierarchical tree-structure (if that makes any sense to you). That's why I hate OO class hierarchies. – AareP Jun 13 '09 at 5:34


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Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
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 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]: Year: Volume: Issue: Page: Find

 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2016, Volume 20, Number 1, Pages 54–64 (Mi vsgtu1463)

Differential Equations and Mathematical Physics

On optimal control problem for the heat equation with integral boundary condition

R. K. Tagiyeva, V. M. Gabibovb

a Baku State University, Baku, AZ-1148, Azerbaijan
b Lankaran State University, Lankaran, AZ-4200, Azerbaijan

Abstract: In this paper we consider the optimal control problem for the heat equation with an integral boundary condition. Control functions are the free term and the coefficient of the equation of state and the free term of the integral boundary condition. The coefficients and the constant term of the equation of state are elements of a Lebesgue space and the free term of the integral condition is an element of Sobolev space. The functional goal is the final. The questions of correct setting of optimal control problem in the weak topology of controls space are studied. We prove that in this problem there exist at least one optimal control. The set of optimal controls is weakly compact in the space of controls and any minimizing sequence of controls of a functional of goal converges weakly to the set of optimal controls. There is proved Frechet differentiability of the functional of purpose on the set of admissible controls. The formulas for the differential of the gradient of the purpose functional are obtained. The necessary optimality condition is established in the form of variational inequality.

Keywords: optimal control, heat equation, necessary optimality condition

DOI: https://doi.org/10.14498/vsgtu1463

Full text: PDF file (687 kB) (published under the terms of the Creative Commons Attribution 4.0 International License)
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Bibliographic databases:

UDC: 517.977
MSC: 49J20, 35K20
Original article submitted 22/XI/2015
revision submitted – 22/I/2016

Citation: R. K. Tagiyev, V. M. Gabibov, “On optimal control problem for the heat equation with integral boundary condition”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 20:1 (2016), 54–64

Citation in format AMSBIB
\Bibitem{TagGab16} \by R.~K.~Tagiyev, V.~M.~Gabibov \paper On optimal control problem for the heat equation with integral boundary condition \jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.] \yr 2016 \vol 20 \issue 1 \pages 54--64 \mathnet{http://mi.mathnet.ru/vsgtu1463} \crossref{https://doi.org/10.14498/vsgtu1463} \zmath{https://zbmath.org/?q=an:06964472} \elib{https://elibrary.ru/item.asp?id=26898088}

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This publication is cited in the following articles:
1. T. K. Yuldashev, “Obyknovennoe integro-differentsialnoe uravnenie s vyrozhdennym yadrom i integralnym usloviem”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 20:4 (2016), 644–655
2. R. K. Tagiev, V. M. Gabibov, “Raznostnaya approksimatsiya i regulyarizatsiya zadachi optimalnogo upravleniya dlya parabolicheskogo uravneniya s integralnym usloviem”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2017, no. 50, 30–44
3. R. K. Tagiyev, Sh. I. Maharramli, “Variational method of solving inverse problem for a parabolic equation with integral conditions”, Proceedings of the 6Th International Conference on Control and Optimization With Industrial Applications, v. II, eds. A. Fikret, B. Tamer, Baku State Univ., Inst. Applied Mathematics, 2018, 286–288
4. E. Tabarintseva, “Approximate solving of an inverse problem for a parabolic equation with nonlocal data” (Novosibirsk, Russian Federation; 26–30 August, 2019), OPCS, 2019, 15th International Asian School-Seminar Optimization Problems of Complex Systems (2019), 8880207, 173-178
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