text:
# How to convert a recurrence relation to a non recursive function

How does one convert a recurrence relation to a non-recursive function?

I am not sure about the substitution, recursion tree, and master method. Is there an easy way to do this?

-
For some recurrence relations this is very easy; for some it is very hard. What recurrence are you interested in, and what havce you tried? – Michael Lugo Mar 24 '11 at 22:18
Just to give a general idea, you may want to check out Wilf's Generatingfunctionology at math.upenn.edu/~wilf/DownldGF.html. Are you talking about the master theorom in computer science, by any chance? – Matt Groff Mar 24 '11 at 22:48
If you are referring to the methods in this MIT lecture and notes then the way to learn (as with most algorithms) is practice, initially following the book or notes. – Henry Mar 24 '11 at 23:07
Please make your posts self-contained; the body should have all the information necessary to understand them, not relying on the title for important information. – Arturo Magidin Mar 25 '11 at 3:17

The substitution method works especially well for recursions like $$f(n) = f(n-1) + g(n), \quad f(n) = f(n/2) + g(n),$$ and in any other situation in which $f(n)$ depends on at most one earlier value of $f$. In some cases you can look up the answer. For example, there is the "polynomial method", which is a recipe for solving recurrences of the type $$f(n) = \begin{cases} f(n-1) + P(n) & n > n_0, \\ C & n = n_0. \end{cases}$$ For example, if you define $f(0) = 0$ and $f(n) = f(n-1) + 2n - 1$ then you get using this method that $$f(n) = n^2.$$ A vast generalization is Gosper's algorithm, see also A=B.

format: plain


text:
Categories

## Proving linearity of the differential operator $a_n(x)D^n+a_{n-1}(x)D^{n-1}+\ldots+a_1(x)D+a_0(x)$.

I am taking a first course on ordinary differential equations(with background in linear algebra). I would like to know, if the proof to the below claim is correct (and that I am not using advanced facts to prove basic facts). Prove that every expression of the form: $$a_n(x)D^n+a_{n-1}(x)D^{n-1}+\ldots+a_1(x)D+a_0(x)$$ defines a linear transformation from $C^n[a,b]$ to […]

Categories

## Let X be a space and C a connected subset of X. Suppose $C \subset D\subset\bar C$. Then D is connected.

This is how i showed it works. Let X be a space and C a connected subset of X. Suppose, C $\subset$ D $\subset$ $\overline{C}$, Then D is connected.\ Let, C $\subset$ X be connected and suppose C $\subset$ D $\subset$ $\overline{C}$. Then, if U,V $\not=$ $\emptyset$ form a separation of D, then we have […]

Categories

## For non-negative reals such that $a+b+c\geq x+y+z$, $ab+bc+ca\geq xy+yz+zx$, and $abc\geq xyz$, show $a^k+b^k+c^k\geq x^k+y^k+z^k$ for $0 Let$a$,$b$,$c$,$x$,$y$,$z$be non-negative real numbers such that $$a+b+c \geq x+y+z,$$ $$ab+bc+ca \geq xy+yz+zx,$$ $$abc \geq xyz$$ Show that $$a^k+b^k+c^k \geq x^k+y^k+z^k, \quad 0 < k < 1$$ This is case$n=3$of a problem posted by Ji Chen in the Art of Problem Solving forums, 2008. I have […] Categories ## For non-negative reals such that$a+b+c\geq x+y+z$,$ab+bc+ca\geq xy+yz+zx$, and$abc\geq xyz$, show$a^k+b^k+c^k\geq x^k+y^k+z^k$for$0

Let $a$, $b$, $c$, $x$, $y$, $z$ be non-negative real numbers such that $$a+b+c \geq x+y+z,$$ $$ab+bc+ca \geq xy+yz+zx,$$ $$abc \geq xyz$$ Show that $$a^k+b^k+c^k \geq x^k+y^k+z^k, \quad 0 < k < 1$$ This is case $n=3$ of a problem posted by Ji Chen in the Art of Problem Solving forums, 2008. I have […]


text:
cuda-api-wrappers Thin C++-flavored wrappers for the CUDA Runtime API
cuda::kernel_t Class Reference

A non-owning wrapper class for CUDA __global__ functions. More...

#include <kernel.hpp>

## Public Member Functions

const void * ptr () const noexcept

device_t device () const noexcept

operator const void * () noexcept

kernel::attributes_t attributes () const

grid::dimension_t maximum_active_blocks_per_multiprocessor (grid::block_dimension_t num_threads_per_block, memory::shared::size_t dynamic_shared_memory_per_block, bool disable_caching_override=false)
Calculates the number of grid blocks which may be "active" on a given GPU multiprocessor simultaneously (i.e. More...

void set_attribute (kernel::attribute_t attribute, kernel::attribute_value_t value)

void opt_in_to_extra_dynamic_memory (cuda::memory::shared::size_t amount_required_by_kernel)
Change the hardware resource carve-out between L1 cache and shared memory for launches of the kernel to allow for at least the specified amount of shared memory. More...

grid::complete_dimensions_t min_grid_params_for_max_occupancy (memory::shared::size_t dynamic_shared_memory_size=no_dynamic_shared_memory, grid::block_dimension_t block_size_limit=0, bool disable_caching_override=false) const

template<typename UnaryFunction >
grid::complete_dimensions_t min_grid_params_for_max_occupancy (UnaryFunction block_size_to_dynamic_shared_mem_size, grid::block_dimension_t block_size_limit=0, bool disable_caching_override=false) const

void set_preferred_shared_mem_fraction (unsigned shared_mem_percentage)
Indicate the desired carve-out between shared memory and L1 cache when launching this kernel - with fine granularity. More...

void set_cache_preference (multiprocessor_cache_preference_t preference)
Indicate the desired carve-out between shared memory and L1 cache when launching this kernel - with coarse granularity. More...

void set_shared_memory_bank_size (multiprocessor_shared_memory_bank_size_option_t config)
Sets a device function's preference of shared memory bank size preference (for the current device probably) More...

## Detailed Description

A non-owning wrapper class for CUDA __global__ functions.

Note
The association of a kernel_t with an individual device is somewhat tenuous. That is, the same function pointer could be used with any other device (provided the kernel was compiled appropriately). However, many/most of the features, attributes and settings are device-specific.

## ◆ maximum_active_blocks_per_multiprocessor()

 grid::dimension_t cuda::kernel_t::maximum_active_blocks_per_multiprocessor ( grid::block_dimension_t num_threads_per_block, memory::shared::size_t dynamic_shared_memory_per_block, bool disable_caching_override = false )
inline

Calculates the number of grid blocks which may be "active" on a given GPU multiprocessor simultaneously (i.e.

with warps from any of these block being schedulable concurrently)

Parameters
 num_threads_per_block dynamic_shared_memory_per_block disable_caching_override On some GPUs, the choice of whether to cache memory reads affects occupancy. But what if this caching results in 0 potential occupancy for a kernel? There are two options, controlled by this flag. When it is set to false - the calculator will assume caching is off for the purposes of its work; when set to true, it will return 0 for such device functions. See also the "Unified L1/Texture Cache" section of the Maxwell tuning guide.

## ◆ min_grid_params_for_max_occupancy()

 grid::complete_dimensions_t cuda::kernel_t::min_grid_params_for_max_occupancy ( memory::shared::size_t dynamic_shared_memory_size = no_dynamic_shared_memory, grid::block_dimension_t block_size_limit = 0, bool disable_caching_override = false ) const
Parameters
 dynamic_shared_memory_size The amount of dynamic shared memory each grid block will need. block_size_limit do not return a block size above this value; the default, 0, means no limit on the returned block size. disable_caching_override On platforms where global caching affects occupancy, and when enabling caching would result in zero occupancy, the occupancy calculator will calculate the occupancy as if caching is disabled. Setting this to true makes the occupancy calculator return 0 in such cases. More information can be found about this feature in the "Unified L1/Texture Cache" section of the Maxwell tuning guide.
Returns
A pair, with the second element being the maximum achievable block size (1-dimensional), and the first element being the minimum number of such blocks necessary for keeping the GPU "busy" (again, in a 1-dimensional grid).

## ◆ opt_in_to_extra_dynamic_memory()

 void cuda::kernel_t::opt_in_to_extra_dynamic_memory ( cuda::memory::shared::size_t amount_required_by_kernel )
inline

Change the hardware resource carve-out between L1 cache and shared memory for launches of the kernel to allow for at least the specified amount of shared memory.

On several nVIDIA GPU micro-architectures, the L1 cache and the shared memory in each symmetric multiprocessor (=physical core) use the same hardware resources. The carve-out between the two uses has a device-wide value (which can be changed), but can also be set on the individual device-function level, by specifying the amount of shared memory the kernel may require.

## ◆ set_cache_preference()

 void cuda::kernel_t::set_cache_preference ( multiprocessor_cache_preference_t preference )
inline

Indicate the desired carve-out between shared memory and L1 cache when launching this kernel - with coarse granularity.

On several nVIDIA GPU micro-architectures, the L1 cache and the shared memory in each symmetric multiprocessor (=physical core) use the same hardware resources. The carve-out between the two uses has a device-wide value (which can be changed), but the driver can set another value for a specific function. This function doesn't make a demand from the CUDA runtime (as in opt_in_to_extra_dynamic_memory), but rather indicates what is the fraction of L1 to shared memory it would like the kernel scheduler to carve out.

Parameters
 preference one of: as much shared memory as possible, as much L1 as possible, or no preference (i.e. using the device default).
Note
similar to set_preferred_shared_mem_fraction() - but with coarser granularity.

## ◆ set_preferred_shared_mem_fraction()

 void cuda::kernel_t::set_preferred_shared_mem_fraction ( unsigned shared_mem_percentage )
inline

Indicate the desired carve-out between shared memory and L1 cache when launching this kernel - with fine granularity.

On several nVIDIA GPU micro-architectures, the L1 cache and the shared memory in each symmetric multiprocessor (=physical core) use the same hardware resources. The carve-out between the two uses has a device-wide value (which can be changed), but the driver can set another value for a specific function. This function doesn't make a demand from the CUDA runtime (as in opt_in_to_extra_dynamic_memory), but rather indicates what is the fraction of L1 to shared memory it would like the kernel scheduler to carve out.

Parameters
 shared_mem_percentage The percentage - from 0 to 100 - of the combined L1/shared memory space the user wishes to assign to shared memory.
Note
similar to set_cache_preference() - but with finer granularity.

## ◆ set_shared_memory_bank_size()

 void cuda::kernel_t::set_shared_memory_bank_size ( multiprocessor_shared_memory_bank_size_option_t config )
inline

Sets a device function's preference of shared memory bank size preference (for the current device probably)

Parameters
 config bank size setting to make

The documentation for this class was generated from the following files:


text:
# nth term fractional sequence

• January 31st 2010, 10:46 AM
nth term fractional sequence
Can anyone see the nth term of this...I've been at it for ages but can't crack it

2/3, 3/4, 4/5, 5/6, .....

Thanks
• January 31st 2010, 10:56 AM
1005
Quote:

Can anyone see the nth term of this...I've been at it for ages but can't crack it

2/3, 3/4, 4/5, 5/6, .....

Thanks

$\frac{1+n}{2+n}$
where $n = 1, 2, 3, ...$
• January 31st 2010, 11:07 AM


text:
# Shrinkage Limit

Written by Jerry Ratzlaff on . Posted in Geotechnical Engineering

Shrinkage limit, abbreviated as SL, a dimensionless number, is the maximum water content at which further reduction in the water content will not cause decrease in volume of soil. Other words it goes from semi-solid to a solid state.

## Shrinkage limit formula

$$\large{ SL = \frac{ \left(m_1 \;-\; m_2\right) \;-\; \left(V_1 \;-\; V_2\right) \; PL }{ m_2 } }$$
Symbol English Metric
$$\large{ SL }$$ = shrinkage limit $$\large{dimensionless}$$
$$\large{ m_1 }$$  = inlet wet mass of soil $$\large{lbm}$$ $$\large{kg}$$
$$\large{ m_2 }$$  = final dry mass of soil $$\large{lbm}$$ $$\large{kg}$$
$$\large{ V_1 }$$  = inlet volume of soil $$\large{ft^3}$$ $$\large{m^3}$$
$$\large{ V_2 }$$  = final volume of dry soil $$\large{ft^3}$$ $$\large{m^3}$$
$$\large{ PL }$$  =  plastic limit $$\large{dimensionless}$$


text:
Control of over-tip leakage flow between turbine blade tips and the stationary shroud is one of the major challenges facing gas turbine designers today. The flow imposes large thermal loads on unshrouded high pressure (HP) turbine blades and is significantly detrimental to turbine blade life. This paper presents results from a computational study performed to investigate the detailed blade tip heat transfer on a sharp-edged, flat tip HP turbine blade. The tip gap is engine representative at 1.5% of the blade chord. Nusselt number distributions on the blade tip surface have been obtained from steady flow simulations and are compared with experimental data carried out in a superscale cascade, which allows detailed flow and heat transfer measurements in stationary and engine representative conditions. Fully structured, multiblock hexahedral meshes were used in the simulations performed in the commercial solver FLUENT. Seven industry-standard turbulence models and a number of different tip gridding strategies are compared, varying in complexity from the one-equation Spalart–Allmaras model to a seven-equation Reynolds stress model. Of the turbulence models examined, the standard $k-ω$ model gave the closest agreement to the experimental data. The discrepancy in Nusselt number observed was just 5%. However, the size of the separation on the pressure side rim was underpredicted, causing the position of reattachment to occur too close to the edge. Other turbulence models tested typically underpredicted Nusselt numbers by around 35%, although locating the position of peak heat flux correctly. The effect of the blade to casing motion was also simulated successfully, qualitatively producing the same changes in secondary flow features as were previously observed experimentally, with associated changes in heat transfer with the blade tip.

1.
Bunker
,
R. S.
, 2001, “
A Review of Turbine Blade Tip Heat Transfer
,”
0077-8923,
934
(
1
), pp.
64
79
.
2.
Mayle
,
R. E.
, and
Metzger
,
D. E.
, 1982, “
Heat Transfer at the Tip of an Unshrouded Turbine Blade
,”
Heat Transfer 1982—Proceedings of the Seventh International Heat Transfer Conference
, München, Federal Republic of Germany, Sept. 6–10,
U.
Grigull
,
E.
Hahne
,
K.
Stephan
, and
J.
Straub
, eds., Vol.
3
, pp.
87
92
.
3.
Yaras
,
M. I.
, and
Sjolander
,
S. A.
, 1992, “
Effects of Simulated Rotation on Tip Leakage in a Planar Cascade of Turbine Blades: Part I—Tip Gap Flow
,”
ASME J. Turbomach.
0889-504X,
114
(
3
), pp.
652
659
.
4.
,
Gm. S.
,
Han
,
J. C.
,
Teng
,
S.
, and
Boyle
,
R. J.
, 2000, “
Heat Transfer and Pressure Distributions on a Gas Turbine Blade Tip
,”
ASME J. Turbomach.
0889-504X,
122
(
4
), pp.
717
724
.
5.
,
Gm. S.
,
Han
,
J. C.
, and
Boyle
,
R. J.
, 2000, “
Heat Transfer and Flow on the Squealer Tip of a Gas Turbine Blade
,”
ASME J. Turbomach.
0889-504X,
122
(
4
), pp.
725
732
.
6.
Kwak
,
J. S.
,
Ahn
,
J.
,
Han
,
J. C.
,
Lee
,
C. P.
,
Bunker
,
R. S.
,
Boyle
,
R.
, and
Gaugler
,
R.
, 2003, “
Heat Transfer Coefficients on the Squealer Tip and Near-Tip Regions of a Gas Turbine Blade With Single or Double Squealer
,” ASME Paper No. GT2003-38907.
7.
Bunker
,
R. S.
,
Bailey
,
J. C.
, and
Ameri
,
A. A.
, 2000, “
Heat Transfer and Flow on the First-Stage Blade Tip of a Power Generation Gas Turbine: Part 1—Experimental Results
,”
ASME J. Turbomach.
0889-504X,
122
(
2
), pp.
263
271
.
8.
Ameri
,
A. A.
, and
Bunker
,
R. S.
, 2000, “
Heat Transfer and Flow on the First-Stage Blade Tip of a Power Generation Gas Turbine: Part 1—Simulation Results
,”
ASME J. Turbomach.
0889-504X,
122
(
2
), pp.
272
277
.
9.
Wilcox
,
D. C.
, 1994, “
Simulation of Transition With a Two-Equation Turbulence Model
,”
AIAA J.
0001-1452,
32
(
2
), pp.
247
255
.
10.
Menter
,
F. R.
, 1993, “
Zonal Two-Equation k-ω Turbulence Models for Aerodynamic Flows
,”
24th Fluid Dynamics, Plasmadynamics, and Lasers Conference
, Orlando, FL, Jul. 6–9, Paper No. AIAA-93-2906.
11.
Yang
,
H.
,
Acharya
,
S.
,
,
S. V.
,
Prakash
,
C.
, and
Bunker
,
R.
, 2002, “
Flow and Heat Transfer Predictions for a Flat-Tip Turbine Blade
,” ASME Paper No. GT2002-30190.
12.
Yang
,
H.
,
Acharya
,
S.
,
,
S. V.
,
Prakash
,
C.
, and
Bunker
,
R.
, 2002, “
Numerical Simulation of Flow and Heat Transfer Past a Turbine Blade With a Squealer-Tip
,” ASME Paper No. GT2002-30193.
13.
Palafox
,
P.
, 2006, “
Gas Turbine Tip Leakage Flow and Heat Transfer
,” DPhil thesis, Oxford University, Oxford, UK.
14.
Palafox
,
P.
,
Oldfield
,
M. L. G.
,
LaGraff
,
J. E.
, and
Jones
,
T. V.
, 2008, “
PIV Maps of Tip Leakage and Secondary Flow Fields on a Low-Speed Turbine Blade Cascade With Moving End Wall
,”
ASME J. Turbomach.
0889-504X,
130
(
1
), p.
011001
.
15.
Palafox
,
P.
,
Oldfield
,
M. L. G.
,
Ireland
,
P. T.
,
Jones
,
T. V.
, and
Lagraff
,
J. E.
, 2006, “
Blade Tip Heat Transfer and Aerodynamics in a Large Scale Turbine Cascade With Moving Endwall
,” ASME Paper No. GT2006-90425.
16.
Moffatt
,
R. J.
, 1982, “
Contributions to the Theory of Single-Sample Uncertainty Analysis
,”
ASME J. Fluids Eng.
0098-2202,
104
, pp.
250
260
.


text:
# Find a function for which integral does not exist but converges as a limit of a sequence

Find an example of a function $$f:[0,\infty)\rightarrow \mathbb{R}$$ integrable on all intervals such that $$\lim_{n\rightarrow \infty}\int_0^n f$$ converges as a limit of a sequence, but such that $$\int_0^\infty f$$ does not exist.

I think a kind of function whose integral oscillates between positive and negative region as $$n\rightarrow \infty$$ may do the job here but I am not sure. Appreciate your help.

• The statement is surely missing an $n$ somewhere; presumably the limit is of the integral over $[0,n]$? Otherwise, you have the limit of a constant. – Arturo Magidin Feb 3 at 23:45
• Yes there was a typo, fixed it. – Sher Afghan Feb 3 at 23:47
• Your idea is fine. Take $f(x) = \sin(2\pi x)$. – Arturo Magidin Feb 3 at 23:51

Just define $$f$$ separately on the intervals $$(n,n+\frac 1 2)$$ and $$(n+\frac 1 2,n+1)$$, $$n=1,2,...,$$ so that $$\int_n^{n+\frac 1 2} f(x)dx=1$$ and $$\int_{n+\frac 1 2} ^{n+1} f(x)dx=-1$$.
• E.g. $f(x)=2$ for $n\le x<n+1/2$ and $f(x)=-2$ for $n+1/2\le x<n+1.$ – DanielWainfleet Feb 4 at 3:03


text:
Outlook: Skyworks Solutions Inc. Common Stock is assigned short-term Ba1 & long-term Ba1 estimated rating.
Dominant Strategy : Wait until speculative trend diminishes
Time series to forecast n: 11 Feb 2023 for (n+3 month)
Methodology : Modular Neural Network (Emotional Trigger/Responses Analysis)

## Abstract

Skyworks Solutions Inc. Common Stock prediction model is evaluated with Modular Neural Network (Emotional Trigger/Responses Analysis) and Stepwise Regression1,2,3,4 and it is concluded that the SWKS stock is predictable in the short/long term. According to price forecasts for (n+3 month) period, the dominant strategy among neural network is: Wait until speculative trend diminishes

## Key Points

2. Technical Analysis with Algorithmic Trading
3. Is now good time to invest?

## SWKS Target Price Prediction Modeling Methodology

We consider Skyworks Solutions Inc. Common Stock Decision Process with Modular Neural Network (Emotional Trigger/Responses Analysis) where A is the set of discrete actions of SWKS stock holders, F is the set of discrete states, P : S × F × S → R is the transition probability distribution, R : S × F → R is the reaction function, and γ ∈ [0, 1] is a move factor for expectation.1,2,3,4

F(Stepwise Regression)5,6,7= $\begin{array}{cccc}{p}_{a1}& {p}_{a2}& \dots & {p}_{1n}\\ & ⋮\\ {p}_{j1}& {p}_{j2}& \dots & {p}_{jn}\\ & ⋮\\ {p}_{k1}& {p}_{k2}& \dots & {p}_{kn}\\ & ⋮\\ {p}_{n1}& {p}_{n2}& \dots & {p}_{nn}\end{array}$ X R(Modular Neural Network (Emotional Trigger/Responses Analysis)) X S(n):→ (n+3 month) $∑ i = 1 n r i$

n:Time series to forecast

p:Price signals of SWKS stock

j:Nash equilibria (Neural Network)

k:Dominated move

a:Best response for target price

For further technical information as per how our model work we invite you to visit the article below:

How do AC Investment Research machine learning (predictive) algorithms actually work?

## SWKS Stock Forecast (Buy or Sell) for (n+3 month)

Sample Set: Neural Network
Stock/Index: SWKS Skyworks Solutions Inc. Common Stock
Time series to forecast n: 11 Feb 2023 for (n+3 month)

According to price forecasts for (n+3 month) period, the dominant strategy among neural network is: Wait until speculative trend diminishes

X axis: *Likelihood% (The higher the percentage value, the more likely the event will occur.)

Y axis: *Potential Impact% (The higher the percentage value, the more likely the price will deviate.)

Z axis (Grey to Black): *Technical Analysis%

## IFRS Reconciliation Adjustments for Skyworks Solutions Inc. Common Stock

1. For the purposes of applying the requirement in paragraph 5.7.7(a), credit risk is different from asset-specific performance risk. Asset-specific performance risk is not related to the risk that an entity will fail to discharge a particular obligation but instead it is related to the risk that a single asset or a group of assets will perform poorly (or not at all).
2. If the holder cannot assess the conditions in paragraph B4.1.21 at initial recognition, the tranche must be measured at fair value through profit or loss. If the underlying pool of instruments can change after initial recognition in such a way that the pool may not meet the conditions in paragraphs B4.1.23–B4.1.24, the tranche does not meet the conditions in paragraph B4.1.21 and must be measured at fair value through profit or loss. However, if the underlying pool includes instruments that are collateralised by assets that do not meet the conditions in paragraphs B4.1.23–B4.1.24, the ability to take possession of such assets shall be disregarded for the purposes of applying this paragraph unless the entity acquired the tranche with the intention of controlling the collateral.
3. For the purpose of recognising foreign exchange gains and losses under IAS 21, a financial asset measured at fair value through other comprehensive income in accordance with paragraph 4.1.2A is treated as a monetary item. Accordingly, such a financial asset is treated as an asset measured at amortised cost in the foreign currency. Exchange differences on the amortised cost are recognised in profit or loss and other changes in the carrying amount are recognised in accordance with paragraph 5.7.10.
4. An entity is not required to restate prior periods to reflect the application of these amendments. The entity may restate prior periods if, and only if, it is possible without the use of hindsight and the restated financial statements reflect all the requirements in this Standard. If an entity does not restate prior periods, the entity shall recognise any difference between the previous carrying amount and the carrying amount at the beginning of the annual reporting period that includes the date of initial application of these amendments in the opening retained earnings (or other component of equity, as appropriate) of the annual reporting period that includes the date of initial application of these amendments.

*International Financial Reporting Standards (IFRS) adjustment process involves reviewing the company's financial statements and identifying any differences between the company's current accounting practices and the requirements of the IFRS. If there are any such differences, neural network makes adjustments to financial statements to bring them into compliance with the IFRS.

## Conclusions

Skyworks Solutions Inc. Common Stock is assigned short-term Ba1 & long-term Ba1 estimated rating. Skyworks Solutions Inc. Common Stock prediction model is evaluated with Modular Neural Network (Emotional Trigger/Responses Analysis) and Stepwise Regression1,2,3,4 and it is concluded that the SWKS stock is predictable in the short/long term. According to price forecasts for (n+3 month) period, the dominant strategy among neural network is: Wait until speculative trend diminishes

### SWKS Skyworks Solutions Inc. Common Stock Financial Analysis*

Rating Short-Term Long-Term Senior
Outlook*Ba1Ba1
Income StatementBaa2Ba3
Balance SheetCB2
Leverage RatiosCaa2Baa2
Cash FlowCC
Rates of Return and ProfitabilityCaa2Caa2

*Financial analysis is the process of evaluating a company's financial performance and position by neural network. It involves reviewing the company's financial statements, including the balance sheet, income statement, and cash flow statement, as well as other financial reports and documents.
How does neural network examine financial reports and understand financial state of the company?

### Prediction Confidence Score

Trust metric by Neural Network: 73 out of 100 with 816 signals.

## References

1. Çetinkaya, A., Zhang, Y.Z., Hao, Y.M. and Ma, X.Y., Is DOW Stock Expected to Go Up?(Stock Forecast). AC Investment Research Journal, 101(3).
2. Bottou L. 2012. Stochastic gradient descent tricks. In Neural Networks: Tricks of the Trade, ed. G Montavon, G Orr, K-R Müller, pp. 421–36. Berlin: Springer
3. J. Harb and D. Precup. Investigating recurrence and eligibility traces in deep Q-networks. In Deep Reinforcement Learning Workshop, NIPS 2016, Barcelona, Spain, 2016.
4. Vilnis L, McCallum A. 2015. Word representations via Gaussian embedding. arXiv:1412.6623 [cs.CL]
5. Swaminathan A, Joachims T. 2015. Batch learning from logged bandit feedback through counterfactual risk minimization. J. Mach. Learn. Res. 16:1731–55
6. V. Mnih, A. P. Badia, M. Mirza, A. Graves, T. P. Lillicrap, T. Harley, D. Silver, and K. Kavukcuoglu. Asynchronous methods for deep reinforcement learning. In Proceedings of the 33nd International Conference on Machine Learning, ICML 2016, New York City, NY, USA, June 19-24, 2016, pages 1928–1937, 2016
7. Bessler, D. A. S. W. Fuller (1993), "Cointegration between U.S. wheat markets," Journal of Regional Science, 33, 481–501.
Frequently Asked QuestionsQ: What is the prediction methodology for SWKS stock?
A: SWKS stock prediction methodology: We evaluate the prediction models Modular Neural Network (Emotional Trigger/Responses Analysis) and Stepwise Regression
Q: Is SWKS stock a buy or sell?
A: The dominant strategy among neural network is to Wait until speculative trend diminishes SWKS Stock.
Q: Is Skyworks Solutions Inc. Common Stock stock a good investment?
A: The consensus rating for Skyworks Solutions Inc. Common Stock is Wait until speculative trend diminishes and is assigned short-term Ba1 & long-term Ba1 estimated rating.
Q: What is the consensus rating of SWKS stock?
A: The consensus rating for SWKS is Wait until speculative trend diminishes.
Q: What is the prediction period for SWKS stock?
A: The prediction period for SWKS is (n+3 month)


text:
# Explaining why the slope varies in varying slope model?

When I fit a multilevel varying slope model, it is easy to summarize the variation in slope. However, I have not yet seen any materials that discusses how to explain such variation (i.e. what about this group that allows it to have a greater effect than the rest?)

For example, we want to test whether meal voucher helps student's performance. We fit a multilevel model of students within schools, allowing for the effect of meal voucher to vary across schools.

\begin{align} performance_i &\sim N(\alpha_{j[i]} + \beta_{j[i]} * voucher_i \\ \alpha_j &\sim N(\text{school level covariates}, \sigma_\alpha) \\ \beta_j &\sim N(\text{school level covariates}, \sigma_\beta) \end{align}

Now we need to explain why the variation in $\beta_j$ occurs. Is this causal inference goal possible with multi-level model?

• For clarification: what do you mean by "this group" and also "greater effect"? Does group = an aspect of grouping or clustering in your data (eg hospitals, schools, blocks, individuals with repeated measures) and for "greater effect" do you mean "Why does some aspect of clustering have a random slope but not others" in a model, with say, repeated measurements on students clustered within classrooms clustered within schools. – N Brouwer Nov 17 '14 at 3:42
• By group I mean clusters (school is a cluster of students). By "greater effect" I mean that, for example, meal voucher helps students in one school a lot more than in another school. In other words, the slope of "meal voucher" varies across school - how do we explain this? – Heisenberg Nov 17 '14 at 3:48
• I don't quite understand. If you have other measured school-level covariates that might explain some of the among-school variation (e.g. average income level -- ideally you would have this at the student level, but maybe it wouldn't be available), then you could include them as fixed effects and see how much of the among-school variation was explained/reduced in the model that incorporated them. If you don't have any other information, then I don't see how you can make any inferences to explain the variation ... – Ben Bolker Nov 17 '14 at 23:34
• @BenBolker I'm interested in what you just said. If I do have measured school-level covariates, where do I include them in the model to explain the varying effect of meal voucher? What kind of assessment can I make after running that model? (I'll edit my questions with a defined model). – Heisenberg Nov 17 '14 at 23:37


text:
Home OALib Journal OALib PrePrints Submit Ranking News My Lib FAQ About Us Follow Us+
 Title Keywords Abstract Author All
Search Results: 1 - 10 of 100 matches for " "
 Page 1 /100 Display every page 5 10 20 Item
 Physics , 2011, DOI: 10.1088/0004-637X/746/2/179 Abstract: Reports of the death of the precursor of Supernova (SN) 1961V in NGC 1058 are exaggerated. Consideration of the best astrometric data shows that the star, known as "Object 7," lies at the greatest proximity to SN 1961V and is the likely survivor of the "SN impostor" super-outburst. SN 1961V does not coincide with a neighboring radio source and is therefore not a radio SN. Additionally, the current properties of Object 7, based on data obtained with the Hubble Space Telescope, are consistent with it being a quiescent Luminous Blue Variable (LBV). Furthermore, post-explosion non-detections by the Spitzer Space Telescope do not necessarily and sufficiently rule out a surviving LBV. We therefore consider, based on the available evidence, that it is yet a bit premature to reclassify SN 1961V as a bona fide SN. The inevitable demise of this star, though, may not be too far off.
 Physics , 2009, DOI: 10.1002/asna.200911238 Abstract: New and existing photometry for the G0 Ia supergiant HD 18391 is analyzed in order to confirm the nature of the variablity previously detected in the star, which lies off the hot edge of the Cepheid instability strip. Small-amplitude variability at a level of \Delta V = 0.016+-0.002 is indicated, with a period of P=123.04+-0.06 d. A weaker second signal may be present at P=177.84+-0.18 with \Delta V = 0.007+-0.002, likely corresponding to fundamental mode pulsation if the primary signal represents overtone pulsation (123.04/177.84 = 0.69). The star, with a spectroscopic reddening of E(B-V) = 1.02, is associated with heavily-reddened B-type stars in its immediate vicinity that appear to be outlying members of an anonymous young cluster centered ~10 arcmin to the west and 1661+-73 pc distant. The cluster has nuclear and coronal radii of r_n=3.5 arcmin and R_c=14 arcmin, respectively, while the parameters for HD 18391 derived from membership in the cluster with its outlying B stars are consistent with those implied by its Cepheid-like pulsation, provided that it follows the semi-period-luminosity relation expected of such objects. Its inferred luminosity as a cluster member is M_V=-7.76+-0.10, its age (9+-1)x10^6 years, and its evolutionary mass ~19 M_{\sun}. HD 18391 is not a classical Cepheid, yet it follows the Cepheid period-luminosity relation closely, much like another Cepheid impostor, V810 Cen.
 Physics , 2012, Abstract: We propose a "Higgs impostor" model for the 125 GeV boson, $X$, recently discovered at the LHC. It is a technipion, $\eta_T$, with $I^G J^{PC} = 0^- 0^{-+}$ expected in this mass region in low-scale technicolor. Its coupling to pairs of standard-model gauge bosons are dimension-five operators whose strengths are determined within the model. It is easy for the gluon fusion rate $\sigma B(gg \to \eta_T \to \gamma\gamma)$ to agree with the measured one, but $\eta_T \to ZZ^*,\,WW^*$ are greatly suppressed relative to the standard-model Higgs rates. This is a crucial test of our proposal. In this regard, we assess the most recent data on $X$ decay modes, with a critical discussion of $X \to ZZ^* \to 4\ell$. In our model the $\eta_T$ mixes almost completely with the isovector $\pi^0_T$, giving two similar states, $\eta_L$ at 125 Gev and $\eta_H$ higher, possibly in the range 170--190 Gev. Important consequences of this mixing are (1) the only associated production of et al is via $\to \to W \eta_L$, and this could be sizable; (2) $\eta_H$ may soon be accessible in $gg \to \eta_H \to \gamma\gamma$; and (3) LSTC phenomenology at the LHC is substantially modified.
 Carsten Gottschlich Computer Science , 2015, Abstract: Fingerprint verification systems are becoming ubiquitous in everyday life. This trend is propelled especially by the proliferation of mobile devices with fingerprint sensors such as smartphones and tablet computers, and fingerprint verification is increasingly applied for authenticating financial transactions. In this study we describe a novel attack vector against fingerprint verification systems which we coin skilled impostor attack. We show that existing protocols for performance evaluation of fingerprint verification systems are flawed and as a consequence of this, the system's real vulnerability is systematically underestimated. We examine a scenario in which a fingerprint verification system is tuned to operate at false acceptance rate of 0.1% using the traditional verification protocols with random impostors (zero-effort attacks). We demonstrate that an active and intelligent attacker can achieve a chance of success in the area of 89% or more against this system by performing skilled impostor attacks. We describe a new protocol for evaluating fingerprint verification performance in order to improve the assessment of potential and limitations of fingerprint recognition systems. This new evaluation protocol enables a more informed decision concerning the operating threshold in practical applications and the respective trade-off between security (low false acceptance rates) and usability (low false rejection rates). The skilled impostor attack is a general attack concept which is independent of specific databases or comparison algorithms. The proposed protocol relying on skilled impostor attacks can directly be applied for evaluating the verification performance of other biometric modalities such as e.g. iris, face, ear, finger vein, gait or speaker recognition.
 Physics , 2014, Abstract: The frequently discussed strongly interacting gauge theory with a fermion flavor doublet in the two-index symmetric (sextet) representation of the SU(3) color gauge group is investigated. In previous studies the chiral condensate and the mass spectrum were shown to be consistent with chiral symmetry breaking ($\chi$SB) at vanishing fermion mass. The recently reported $\beta$-function is not inconsistent with this observation, suggesting that the model is very close to the conformal window and a light "Higgs impostor" could emerge as a composite state. In this work we describe the methodology and preliminary results of studying the emergence of the light composite scalar with $0^{++}$ quantum numbers.
 Physics , 2012, DOI: 10.1088/2041-8205/764/1/L6 Abstract: We propose that the energetic major outburst of the supernova (SN) impostor SN 2009ip in September 2012 (outburst 2012b) was a mergerburst event, where two massive stars merged. The previous outbursts of 2009 and 2011 might have occurred near periastron passages of the binary system prior to the merger, in a similar manner to the luminosity peaks in the nineteenth century Great Eruption of the massive binary system Eta Carinae. The major 2012b outburst and the 2012a pre-outburst, resemble the light curve of the mergerburst event V838 Mon. A merger of an evolved star with a mass of M1~60-100Mo and a secondary main sequence star of M2~0.2-0.5M1 can account for the energy of SN 2009ip and for the high velocities of the ejected gas. The ejected nebula is expected to have a non-spherical structure, e.g. bipolar or even a more complicated morphology.
 Indian Journal of Dermatology , 2001, Abstract: A case of Facial Hemiatrophy is being reported for its rarity and clinical interest.
 Surveys in Approximation Theory , 2010, Abstract: This is a survey on the subject of strong uniqueness in approximation theory.
 Physics , 2012, DOI: 10.1016/j.physletb.2012.10.079 Abstract: New results are reported from large scale lattice simulations of a frequently discussed strongly interacting gauge theory with a fermion flavor doublet in the two-index symmetric (sextet) representation of the SU(3) color gauge group. We find that the chiral condensate and the mass spectrum of the sextet model are consistent with chiral symmetry breaking in the limit of vanishing fermion mass. In contrast, sextet fermion mass deformations of spectral properties are not consistent with leading conformal scaling behavior near the critical surface of a conformal theory. A recent paper could not resolve the conformal fixed point of the gauge coupling from the slowly walking scenario of a very small nearly vanishing \beta-function (DeGrand:2012yq). It is argued that overall consistency with our new results is resolved if the sextet model is close to the conformal window, staying outside with a very small non-vanishing \beta-function. The model would exhibit then the simplest composite Higgs mechanism leaving open the possibility of a light scalar state with quantum numbers of the Higgs impostor. It would emerge as the pseudo-Goldstone dilaton state from spontaneous symmetry breaking of scale invariance. We will argue that even without association with the dilaton, the scalar Higgs-like state can be light very close to the conformal window. A new Higgs project of sextet lattice simulations is outlined to resolve these important questions.
 Mathematics , 2010, Abstract: This is a survey paper on the subject of strong uniqueness in approximation theory.
 Page 1 /100 Display every page 5 10 20 Item


text:
In [1, p. 94 exercise 4.2] we are asked to consider the estimator
 $\hat{P}_{AVPER}(0)=\frac{1}{N}\sum\limits_{m=0}^{N-1}\hat{P}_{PER}^{m}(0)$ (1)

where
 $\hat{P}_{PER}^{m}(0)= x^{2}[m]$ (2)

for the process of Problem 4.1. We are informed that this estimator may be viewed as an averaged periodogram. In this point of view the data record is sectioned into blocks (in this case, of length 1) and the periodograms for each block are averaged. We are asked to find the mean and variance of $\hat{P}_{AVPER}(0)$ and compare the result to that obtained in [2]. read the conclusion >


text:
Question
A professional boxer hits his opponent with a 1000-N horizontal blow that lasts for 0.150 s. (a) Calculate the impulse imparted by this blow. (b) What is the opponent’s final velocity, if his mass is 105 kg and he is motionless in midair when struck near his center of mass? (c) Calculate the recoil velocity of the opponent’s 10.0-kg head if hit in this manner, assuming the head does not initially transfer significant momentum to the boxer’s body. (d) Discuss the implications of your answers for parts (b) and (c).
1. $150\textrm{ kg}\cdot\textrm{m/s}$
2. $1.43 \textrm{ m/s}$
3. $15.0\textrm{ m/s}$
4. The velocity when impacting only the head is significant and would result in concussion injury when imparted in such a short time. To achieve a knockout, boxers should strategically aim for the head.
Solution Video


text:
# Is there an in-line comment system, convention or program?

When writing papers such as this one https://github.com/geneura-papers/2015-Evostar/blob/b9474f66cd45aee316136831c4a9b4acaef4e89f/GPbot/GPbot-evostar2015.tex, we use LaTeX comments for, well, commenting on the paper itself and making suggestion. There are a couple of issues

• When issues are resolved, we erase the comments by hand. Except when we don't, which is often.
• When we submit the paper, we also have to erase all comments by hand. Sometimes we've submitted sources to places that allow downloading with all our comments, including in-jokes.
• We have no way to know unresolved issues or, in some cases, who's created it. We usually sign them, but sometimes we forget.

We would need a tool that, maybe using it after compilation, would allow

1. Use some simple markup for issues, like note important or solved (in which case it could be erased)
2. Erase all issues for production (there's probably something that already does that, I'm sure), except maybe some marked as comments that should be left in the source (meta data such as file names or whatever)
3. It would be perfect if it could be tied with git and GitHub
4. List unresolved issued

It would be weird that there's no such tool, but I've been unable to find it. On the other hand, if we set down to write it, would anybody else be interested in using it? Has somebody found similar problems and how have you solved them?

• Welcome to TeX.SX!. To be honest, I do not have a direct solution but some ideas or hints. Could you please tell on which operating systems you are going to work? – albert Nov 26 '14 at 17:33
• Use a version control system. – jub0bs Nov 26 '14 at 17:38
• If you have followed the link I put above, we do use a version control system. However, GitHub does not allow inline comments on the source, only on the commits. We did think about it, but it is not what we are looking for, since, in fact, it's not inline. We use GitHub issues (just check the site) for general discussion on the paper not tied to a particular line or phrase. – jjmerelo Nov 26 '14 at 17:42
• @jjmerelo There is no link in your post, at least that I can find... I would also recommend the todonotes package. You can have a "Table of To-dos" at the beginning of your document, and each user can be assigned a color. These would be marked inline in the source and also show up in the PDF (unless you set the "final" option, in which case none would show in the PDF) – darthbith Nov 26 '14 at 17:48
• @jjmerelo: If there is really no "out-of-the-box" solution for your problem, you could try to use a tag-like syntax. In general, you could imagine something like % TODO: stuff to be done and read those lines by a e.g. a python-script using regular expressions and summarize found lines in a additional file which would be kind of a ToDo-list. I'm not sure, but maybe you could even pass the linenumber to the summarized list which would give a you a nice overview. In addition, you could think about a command, providing the needed %TODO-line and passing options like editor and comment massage. – albert Nov 26 '14 at 18:30

This might be stating the obvious, but you could define a command that does nothing and use that as a marker in the input file.

This command ignores all input it receives. Its only function is as a comment in the input file. The first argument is optional, so you can type \FIX[note]{comment} or just \FIX{comment}.

\documentclass{article}

\newcommand{\FIX}[2][\relax]{\relax}
\newcommand{\SOLVED}{\FIX}

\begin{document}

The sentunz may not be spelld korrectly.
%%
\FIX[important]{Fire the employee who wrote this.}
\SOLVED[2014-11-26]{Employee fired}
%%

\end{document}

You could later redefine these commands to do something else, such as print a version with annotations in the margin:

\renewcommand{\FIX}[2][\relax]{\marginpar{{\footnotesize \textbf{#1}\par #2}}}

You could also create a counter for FIX commands, \newcounter{fix}, and increment it each time the command is called, \stepcounter{fix}. At the end of the document you could call a command to print \thefix in the annotated version.

• That's more or less what the todonotes package does. It's a start, so thanks! – jjmerelo Nov 26 '14 at 19:06

The FiXme package: CTAN Link Documentation FiXme is a collaborative annotation tool for LATEX documents. Seems to be a comprehensive solution.


text:
## Trigonometry (11th Edition) Clone

We can find the domain of $sin ~\theta$: domain: $-\infty \lt \theta \lt \infty$ We can find the domain of $cos ~\theta$: domain: $-\infty \lt \theta \lt \infty$ We can find the domain of $tan ~\theta$: domain: $\theta \neq \frac{\pi}{2}+\pi~n$, for any integer $n$ We can find the domain of $csc ~\theta$: domain: $\theta \neq \pi~n$, for any integer $n$ We can find the domain of $sec ~\theta$: domain: $\theta \neq \frac{\pi}{2}+\pi~n$, for any integer $n$ We can find the domain of $cot ~\theta$: domain: $\theta \neq \pi~n$, for any integer $n$
We can find the domain of $sin ~\theta$: $sin~\theta = \frac{x}{r}$ Since $r$ is always positive, $sin ~\theta$ is defined for all real numbers. domain: $-\infty \lt \theta \lt \infty$ We can find the domain of $cos ~\theta$: $cos~\theta = \frac{y}{r}$ Since $r$ is always positive, $cos ~\theta$ is defined for all real numbers. domain: $-\infty \lt \theta \lt \infty$ We can find the domain of $tan ~\theta$: $tan~\theta = \frac{sin~\theta}{cos~\theta}$ $tan ~\theta$ is undefined when $cos~\theta = 0$ $tan ~\theta$ is undefined when $\theta = \frac{\pi}{2}+\pi~n$, for any integer $n$ domain: $\theta \neq \frac{\pi}{2}+\pi~n$, for any integer $n$ We can find the domain of $csc ~\theta$: $csc~\theta = \frac{1}{sin~\theta}$ $csc ~\theta$ is undefined when $sin~\theta = 0$ $csc ~\theta$ is undefined when $\theta = \pi~n$, for any integer $n$ domain: $\theta \neq \pi~n$, for any integer $n$ We can find the domain of $sec ~\theta$: $sec~\theta = \frac{1}{cos~\theta}$ $sec ~\theta$ is undefined when $cos~\theta = 0$ $sec ~\theta$ is undefined when $\theta = \frac{\pi}{2}+\pi~n$, for any integer $n$ domain: $\theta \neq \frac{\pi}{2}+\pi~n$, for any integer $n$ We can find the domain of $cot ~\theta$: $cot~\theta = \frac{cos~\theta}{sin~\theta}$ $cot ~\theta$ is undefined when $sin~\theta = 0$ $cot ~\theta$ is undefined when $\theta = \pi~n$, for any integer $n$ domain: $\theta \neq \pi~n$, for any integer $n$


text:
# Inverse Matrix

## Matrices are an important topic in terms of class 11 mathematics. In that, most weightage is given to inverse matrix problems. Let us first define the inverse of a matrix.

### According to the inverse of a matrix definition, a square matrix A of order n is said to be invertible if there exists another square matrix B of order n such that AB = BA = I.

where I is the identity of order n*n. Identity matrix of order 2 is denoted by

## The inverse matrix can be found only with the square matrix. The square matrix has to be non-singular, i.e, its determinant has to be non-zero. A common question arises, how to find the inverse of a square matrix? By inverse matrix definition in math, we can only find inverses in square matrices.

Given a square matrix A

Its determinant value is given by [(a*d)-(c*d)].

Some important results –

• The inverse of a square matrix, if exists, is unique

• AA-1= I= A-1a

• If A and B are invertible then (AB)-1=  B-1A-1

• Every orthogonal matrix is invertible

• If A is symmetric then its inverse is also symmetric.

Broadly there are two ways to find the inverse of a matrix:

1. ### Using Determinants –

This matrix inversion method is suitable to find the inverse of the 2 by 2 matrix.

where adj(A) refers to the adjoint matrix A, |A| refers to the determinant of a matrix A.

adjoint of a matrix is found by taking the transpose of the cofactor matrix.

Cij = (-1)ij det (Mij),  Cij is the cofactor matrix

1. ### By Elementary Transformation –

This method is suitable to find the inverse of the n*n matrix.

In this method first, write A=IA if you are considering row operations, and A=AI if you are considering column operation.

Let us take 3 matrices X, A, and B such that X = AB. To find out the required identity matrix we find out using elementary operations and reduce to an identity matrix

If A-1 exists then to find A-1 using elementary row operations is as follows:

1.      Write A = IA, where I is the identity matrix as order as A.

2.      Validate the sum by performing the necessary row operations on LHS to get I in LHS. Make sure to perform the same operations on RHS so that you get I=BA.

If A-1 exists then to find A-1 using elementary column operations is as follows:

1.      Write A = AI, where I is the identity matrix as order as A.

2.      Validate the sum by performing the necessary column operations on LHS to get I in LHS. Make sure to perform the same operations on RHS so that you get I=AB.

The identity matrix of n*n is represented in the figure below

The identity matrix In is a n x n square matrix with the main diagonal of 1’s and all other elements are O’s.

If A is a m x n matrix, then

ImA = A and AIn = A

If A is a n x n matrix, then

AIn = InA = A

### Properties of Inverse of a Square Matrix

• Square matrix A is invertible if and only if |A|≠ 0

• (A-1)-1=A

• (A’)-1 = (A-1)’

• (AB)-1 = B-1A-1 In general ( A1A1A1 … An)-1 =  An-1An – 1-1   …  A3-1A2-1A1-1

• Also, if a non singular square matrix A is symmetric, then A-1 is also symmetric.

• |A-1| = |A|-1

• AA-1 = A-1A = I

• (Ak)-1 = (A-1) Akk ∈ N

• If  abc ≠ 0, then A-1 = $\begin{bmatrix} 1/a & 0 & 0\\ 0 & 1/b & 0 \\ 0 & 0 & 1/c\end {bmatrix}$

Here’s Something For More Clarity –

How can we find the inverse of a 3 x 3 matrix?

In order to figure out the inverse of the 3 x 3 matrix, first of all, we need to determine the determinant of the matrix. If the determinant will be zero, the matrix will not be having any inverse. Then move the matrix by re-writing the first row as the first column, the middle row as the main column, and the third row as the third column.

Similarly,

What’s The Inverse of a 4 x 4 Matrix?

The inverse of the ‘n’ x ‘n’ matrix ‘A’ is the ‘n’ x ‘n’ matrix ‘B.’ Like ‘AB’ = ‘BA’ = ‘I.’ And if we get the inverse of the 4 x 4 matrix ‘A’ to be ‘B,’ then we’ll only have to multiply ‘AB’ and ‘BA’ to test our work.

How Are We Going To Measure The Inverse?

We can calculate the inverse of the matrix in the following steps-

1st Step – Calculate a minor matrix.

2nd Step – Then convert it to a cofactor matrix.

4th Step  – Finally, multiply with 1 / Determinant.

1.  How important is this topic?

Answer  – This topic plays a major role in this chapter. The inverse of a matrix is a definite 4 mark question which you can attempt easily once you have mastered it. Also one has to be very careful while using the elementary transformation. Keeping this in mind double-check whether you are applying row or column operation. Read the question twice before applying the solution.

2. What are the important questions in the matrix chapter?


text:
### 25Deconstructing Loops

#### 25.1Setup: Two Functions

Let’s look at two functions we wrote earlier in Factoring Numbers:

fun gcd(a, b):
if b == 0:
a
else:
gcd(b, num-modulo(a, b))
end
end

fun pr(n):
fun g(x): num-modulo((x * x) + 1, n) end
fun iter(x, y, d):
new-x = g(x)
new-y = g(g(y))
new-d = gcd(num-abs(new-x - new-y), n)
| new-d == 1 then:
iter(new-x, new-y, new-d)
| new-d == n then:
none
| otherwise:
some(new-d)
end
end
iter(2, 2, 1)
end

We’ve written both recursively: gcd by calling itself and pr with recursion on its inner function. But if you’ve programmed before, you’ve probably written similar programs with loops.

Exercise

Because we don’t have loops in Pyret, the best we can do is to use a higher-order function; which ones would you use?

But let’s see if we can do something “better”, i.e., get closer to a traditional-looking program.

Before we start changing any code, let’s make sure we have some tests for gcd:

check:
gcd(4, 5) is 1
gcd(5, 7) is 1
gcd(21, 21) is 21
gcd(12, 24) is 12
gcd(12, 9) is 3
end

#### 25.2Abstracting a Loop

Now let’s think about how we can create a loop. At each iteration, a loop has a status: whether it’s done or whether it should continue. Since we have two parameters here, let’s record two parameters for continuing:

data LoopStatus:
| done(final-value)
| next-2(new-arg-1, new-arg-2)
end

Now we can write a function that does the actual iteration:

fun loop-2(f, arg-1, arg-2):
r = f(arg-1, arg-2)
cases (LoopStatus) r:
| done(v) => v
| next-2(new-arg-1, new-arg-2) => loop-2(f, new-arg-1, new-arg-2)
end
end

Note that this is completely generic: it has nothing to do with gcd. (It is generic in the same way that higher-order functions like map and filter are generic.) It just repeats if f says to repeat, stops if f says to stop. This is the essence of a loop.

Exercise

Observe also that we could, if we wanted, stage [Staging] loop-2, because f never changes. Rewrite it that way.

With loop-2, we can rewrite gcd:

fun gcd(p, q):
loop-2(
{(a, b):
if b == 0:
done(a)
else:
next-2(b, num-modulo(a, b))
end},
p,
q)
end

Now it might seem to you we haven’t done anything useful at all. In fact, this looks like a significant step backward. At least before we just had simple, clean recursion, the way Euclid intended it. Now we have a higher-order function and we’re passing it the erstwhile gcd code as a function and there’s this LoopStatus datatype and…everything’s gotten much more complicated.

But, not really. The reason we put it in this form is because we’re about to exploit a feature of Pyret. The for construct in Pyret actually rewrites as follows:

for F(a from a_i, b from b_i, …): BODY end

gets rewritten to

F({(a, b, …): BODY}, a_i, b_i, …)

For example, if we write

for map(i from range(0, 10)): i + 1 end

this becomes

map({(i): i + 1}, range(0, 10))

Now you may see why we rewrote gcd. Going in reverse, we can rewrite

F({(a, b, …): BODY}, a_i, b_i, …)

as

for F(a from a_i, b from b_i, …): BODY end

so the function becomes just

fun gcd(p, q):
for loop-2(a from p, b from q):
if b == 0:
done(a)
else:
next-2(b, num-modulo(a, b))
end
end
end

and now closely resembles a traditional “loop” program.

#### 25.3Is It Really a Loop?

This whole section should be considered an aside for people with more advanced computing knowledge.

If you know something about language implementation, you may know that loops have the property that the iteration does not consume extra space (beyond what the program already needs), and the repetition takes place very quickly (a “jump instruction”). In principle, our loop-2 function does not have this property: every iteration is a function call, which is more expensive and builds additional stack context. However, one or both of these does not actually occur in practice.

In terms of space, the recursive call to loop-2 is the last thing that a call to loop-2 does. Furthermore, nothing in loop-2 consumes and manipulates the return from that recursive call. This is therefore called a tail call. Pyret—like some other languages—causes tail calls to not take any extra stack space. In principle, Pyret can also turn some tail calls into jumps. Therefore, this version has close to the same performance as a traditional loop.

#### 25.4Re-Examining for

The definition of for given above should make you suspicious: Where’s the loop?!? In fact, Pyret’s for does not do any looping at all: it’s simply a fancy way of writing lam. Any “looping” behavior is in the function written after for. To see that, let’s use for with a non-looping function.

Recall that

for F(a from a_i, b from b_i, …): BODY end

gets rewritten to

F({(a, b, …): BODY}, a_i, b_i, …)

Thus, suppose we have this function (from Functions as Data):

delta-x = 0.0001
fun d-dx-at(f, x):
(f(x + delta-x) - f(x)) / delta-x
end

We can call it like this to get approximately 20:

d-dx-at({(n): n * n}, 10)

That means we can also call it like this:

for d-dx-at(n from 10): n * n end

Indeed:

check:
for d-dx-at(n from 10): n * n end
is
d-dx-at({(n): n * n}, 10)
end

Since d-dx-at has no iterative behavior, no iteration occurs. The looping behavior is given entirely by the function specified after for, such as map, filter, or loop-2 above.

#### 25.5Rewriting Pollard-Rho

Now let’s tackle Pollard-rho. Notice that it’s a three-parameter function, so we can’t use the loop-2 we had before: that’s only a suitable loop when we have two arguments that change on each iteration (often the iteration variable and an accumulator). It would be easy to design a 3-argument version of loop, say loop-3, but we could also have a more general solution, using a tuple:

data LoopStatus:
| done(v)
| next–2(new-x, new-y)
| next-n(new-t)
end

fun loop-n(f, t):
r = f(t)
cases (LoopStatus) r:
| done(v) => v
| next-n(new-t) => loop-n(f, new-t)
end
end

where t is a tuple.

So now we can rewrite pr. Let’s first rename the old pr function as pr-old so we can keep it around for testing. Now we can define a “loop”-based pr:

fun pr(n):
fun g(x): num-modulo((x * x) + 1, n) end
for loop-n({x; y; d} from {2; 2; 1}):
new-x = g(x)
new-y = g(g(y))
new-d = gcd(num-abs(new-x - new-y), n)
| new-d == 1 then:
next-n({new-x; new-y; new-d})
| new-d == n then:
done(none)
| otherwise:
done(some(new-d))
end
end
end

Indeed, we can test that the two behave in exactly the same way:

check:
ns = range(2, 100)
l1 = map(pr-old, ns)
l2 = map(pr, ns)
l1 is l2
end

#### 25.6Nested Loops

We can also write a nested loop this way. Suppose we have a list like

lol = [list: [list: 1, 2], [list: 3], [list:], [list: 4, 5, 6]]

and we want to sum the whole thing by summing each sub-list. Here it is:

for loop-2(ll from lol, sum from 0):
cases (List) ll:
| empty => done(sum)
l-sum =
for loop-2(es from l, sub-sum from 0):
cases (List) es:
| empty => done(sub-sum)
| link(e, r) => next-2(r, e + sub-sum)
end
end
next-2(rl, sum + l-sum)
end
end

We can simplify this by writing it as two functions:

fun sum-a-lon(lon :: List<Number>):
for loop-2(es from lon, sum from 0):
cases (List) es:
| empty => done(sum)
next-2(r, e + sum)
end
end
end

fun sum-a-lolon(lolon :: List<List<Number>>):
for loop-2(l from lolon, sum from 0):
cases (List) l:
| empty => done(sum)
next-2(r, sum-a-lon(lon) + sum)
end
end
end

check:
sum-a-lolon(lol) is 21
end

Notice that the two functions are remarkably similar. This suggests an abstraction:

fun sum-a-list(f, L):
for loop-2(e from L, sum from 0):
cases (List) e:
| empty => done(sum)
next-2(r, f(elt) + sum)
end
end
end

Using this, we can rewrite the two previous functions as:

fun sum-a-lon(lon :: List<Number>):
sum-a-list({(e): e}, lon)
end

fun sum-a-lolon(lolon :: List<List<Number>>):
sum-a-list(sum-a-lon, lolon)
end

check:
sum-a-lolon(lol) is 21
end

With the annotations, it becomes clear what each function does. In sum-a-lon, each element is a number, so it “contributes itself” to the overall sum. In sum-a-lolon, each element is a list of numbers, so it “contributes its sum-a-lon” to the overall sum.

Finally, to bring this full circle, we can rewrite the above the functions as follows:

fun sum-a-lon(lon :: List<Number>):
for sum-a-list(e :: Number from lon): e end
end

fun sum-a-lolon(lolon :: List<List<Number>>):
for sum-a-list(l :: List<Number> from lolon): sum-a-lon(l) end
end

Arguably this makes even clearer what each element contributes. In sum-a-lon each element is a number, so it contributes just that number. In sum-a-lolon, each element is a list of numbers, so it must contribute sum-a-lon of that list.

#### 25.7Loops, Values, and Customization

Observe two important ways in which the loops above differ from traditional loops:
1. Every loop produces a value. This is consistent with the rest of the language, where—as much as possible—computations try to produce answers. We don’t have to produce a value; for instance, the following program, reminiscent of looping programs in many other languages, will work just fine in Pyret:

for each(i from range(0, 10)): print(i) end

However, this is the unusual case. In general, we want expressions to produce values so that we can compose them together.

2. Many languages have strong opinions on exactly how many looping constructs there should be: two? three? four? In Pyret, there are no built-in looping constructs at all; there’s just a syntax (for) that serves as a proxy for creating a specific lam. With it, we can reuse existing iterative functions (like map and filter), but also define new ones. Some can be very generic, like loop-2 or loop-n, but others can be very specific, like sum-a-list. The language designers don’t prevent you from writing a loop that is useful to your situation, and sometimes the loop can be very expressive, as we see from rewriting sum-a-lon and sum-a-lolon atop for and sum-a-list.


text:
# FlymakeTex

The following are hints for using FlyMake with TeX and LaTeX.

Note: If your latex file has digits immediately before the dot (e.g. `design2.tex`), flymake will assume it is a portion of a larger work and try to find another file that `\include`s it!

Is it possible to use latex with flymake on linux machines? I try it today and it seems that the mode works only under miktex.

This mode works on linux machines with customization. The default program (texify) is incorrect, so I used chktex:

```  (defun flymake-get-tex-args (file-name)
(list "chktex" (list "-q" "-v0" file-name)))```

If you use a recent version of LaTeX having the option “-file-line-error-style”, you can use:

```  (defun flymake-get-tex-args (file-name)
(list "latex" (list "-file-line-error-style" file-name)))```

## ChkTeX

Another setting to use ChkTeX: After installing the checking tool, you can put the following .chktexrc on your home direcotry:

```  # ~/.chktexrc
OutFormat
{

# -v0; silent mode : change the default setting of -v0 as follows:
"%f%b%l%b%c%b%k: %m!n"```

Then, write .emacs for flymake as follows:

```  ;; Flymake for LaTeX
(defun flymake-get-tex-args (file-name)
(list "chktex" (list "-g0" "-r" "-l" (expand-file-name "~/.chktexrc") "-I" "-q" "-v0" file-name)))
(push
'("^\\(\.+\.tex\\):\\([0-9]+\\):\\([0-9]+\\):\\(.+\\)"
1 2 3 4) flymake-err-line-patterns)```

– ike

For Japanese, unfortunately, chktex doesn’t understand 2byte code e.g. Japanese character. Trick: we can pass the tex file to the kakasi filter, then check it.

Here is another solution for Japanese TeX users:

```  ;; Flymake for LaTeX
(defun flymake-get-tex-args (file-name)
(list (expand-file-name "~/bin/flycheck_latex.sh") (list file-name)))
(push
'("^\\(\.+\.tex\\):\\([0-9]+\\):\\([0-9]+\\):\\(.+\\)"
nil 2 3 4) flymake-err-line-patterns)```
```  #!/bin/sh

real_file_name=\$@
file_name=`basename "\$real_file_name"`
tmp_file=`mktemp`
tmp_dir=`mktemp -d`
fake_name="\$tmp_dir/\$file_name"

# assumed that the source code has EUC-JP encoding
cat "\$real_file_name" | kakasi -ieuc -Ha -Ka -Ja -Ea -ka -s > "\$tmp_file"

mkdir -p \$tmp_dir
cp \$tmp_file \$fake_name
chktex -g0 -r -l ~/.chktexrc -I -q -v0 \$fake_name

# clean up
rm -fr "\$tmp_file"
rm -fr "\$tmp_dir"```

--ike

## Tex Live!

To use flymake with latex or pdflatex from the texlive distribution on linux (as e.g., included in OpenSuse?):

```  (defun flymake-get-tex-args (file-name)
(list "pdflatex" (list "-file-line-error" "-draftmode" "-interaction=nonstopmode" file-name)))```

--Lars

CategoryProgrammerUtils


text:
# MLE for a homogeneous Poisson process?

If we have a data set consisting of event times $\{t_1, t_2, \ldots, t_N\}$ and would like to model this as a Poisson process with intensity $\lambda$, how do we do it? Intuitively, I would expect that we can calculate the average waiting time $w = \frac{1}{N-1}\sum_{i=2}^N (t_i-t_{i-1})$ and then set $\hat{\lambda} = 1/w$?

Is this correct? If so, how is it justified? Do we have to consider fitting a Poisson distribution to the number of events after doing some sort of binning?

• Look at interarrival times $\{ t_1, t_2 - t_1, ..., t_N - t_{N-1} \}$ and fit exponential distribution with parameter $\lambda$ to them. – stans Aug 5 '18 at 13:43
• Thanks. Can we do something similar if the process is non-homogeneous? – theQman Aug 5 '18 at 14:36
• Similar: yes. But not exactly the same. Naturally, if $\lambda$ changes with time, you need to make up your mind on the exact form of time dependence and the number of parameters involved. The best course of action would be for you to read up on non-homogeneous Poisson process. Ross is a good reference. – stans Aug 5 '18 at 16:15

You can use Maximum Likelihood Estimation, either with synchronous data (time-binned data) or asynchronous data (time-stamped data). The likelihood function changes accordingly.

For time-binned (or synchronous) data, you can simply use the joint Poisson probability mass function for your observed counts as the likelihood function:

$L = \prod^{K}_{i=1} \frac{\lambda^{x_i}}{x_i!} \exp(-\lambda)$,

where $K$ is the number of bins, $x_i$ the count of events in bin $i$, and $\lambda$ the constant intensity that you want to estimate.

For asynchronous data, the likelihood is specified as follows:

$L = \left[ \prod^{N(T)}_{i=1} \lambda^*(t_i) \right] \exp\left[-\int^{T}_{0}\lambda^*(s) ds \right]$,

where $N(T)$ is the number of points at end-of-sample time $T$, and $\lambda^*(t)$ is the conditional intensity function, which is simply the constant $\lambda^*(t)=\lambda$ for the homogeneous Poisson process.

In some cases including the homogeneous Poisson process, there are closed-form solutions for both cases (take logs, set derivative with respect to $\lambda$ equal to zero, and solve for $\lambda$). Otherwise the log-likelihood can be optimised numerically.

For more background on theory and estimation, these are good references:

• "For synchronous data (i.e. counts in regular time bins), you can simply use the joint Poisson probability mass function for your observed counts as the likelihood function." Can you explain this a bit more? If time is divided into bins, then what are fitting to a Poisson distribution? The number of times $k$ events occurs in a bin? – theQman Aug 7 '18 at 0:24
• For each bin, we count the number of events and fit \lambda to that joint distribution of counts. Please also see updated question. I hope this makes it clearer! – mloning Aug 7 '18 at 6:33


text:
The Optical Morphology of the Young Planetary Nebula M1-5
Session 51 -- Planetary Nebulae
Display presentation, Thursday, January 13, 9:30-6:45, Salons I/II Room (Crystal Gateway)

## [51.05] The Optical Morphology of the Young Planetary Nebula M1-5

M. Bobrowsky (CTA)

HST images of the young planetary nebula (PN) M1-5 are presented. Although this object was identified as a PN several decades ago, until recently its small size precluded optical investigations of its morphology. Radio observations indicated an apparent angular size of 2" which, by itself, could simply have been the result of a large distance. But optical spectra showed that the electron density is high (1-3 x $10^{4}$) and radio data indicated a high emission measure, implying that the PN is young, compact, and perhaps, ionization-bounded.

Narrow-band images in $H\beta$ and [O III]$5007\AA$ show a mildly bipolar nebula. The dimensions of the nebula are approximately 1."9 x 2."6, with the major axis running roughly north-south. However, the two poles appear (in projection) $155\deg$ apart, not the $180\deg$ expected from an axially symmetric bipolar nebula. The deviation from axial symmetry could be the result of asymmetric mass loss or it could result from the interaction of the nebula with the interstellar medium.

At an estimated distance of 2.1 kpc, the size of M1-5 is only 0.019 x 0.026 pc. Small, young PNe evolve very quickly. (E.g., a similar object, Hen 1357, became a PN only during the past few decades.) Therefore it is not surprising that our observed [O III]$5007\AA$/$H\beta$ line ratio is higher (at 4.8) than previously observations indicated.

Based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for this work was provided by NASA through grant number GO-3603.01-91A from the Space Telescope Science Institute.


text:
# Existence of only one bounded linear functional.

Given $X$ a linear normed space over $K$ . $I$ be arbitrary indexing set , $\{f_\alpha: \alpha \in I\}\subset X$ and a family $\{c_\alpha: \alpha\in I\} \subset K$, I want to know that there exists exactly one bounded linear functional $f'\in X'$ with

• $f'(f)=c_\alpha\quad \mbox{ for all}\quad \alpha \in I$

• $||f||\le M\quad \mbox{ for some}\quad M\geq 0$

exists when for every finite subfamily $J \subset I$ and every choice of the member $\{\beta_\alpha :\alpha \in J\} \subset K$ the following inequality holds: $$\left|\sum_{\alpha \in J}\beta_\alpha c_\alpha\right|\le M\left\|\sum_{\alpha\in J} \beta_\alpha f_\alpha\right\|_X$$ I have been trying to solve this problem , to be frank i don't understand the question fully , I need a big deal of help to solve and understand this problem . Thank you very much .

-
Are the $f_\alpha$ linearly independent? And $J$ assumed finite? –  Davide Giraudo Nov 29 '12 at 21:27
@DavideGiraudo : As per question i found , it says nothing about the dependency of $f_\alpha$ , but yes $J$ is finite . –  Theorem Nov 29 '12 at 21:30

I assume that $\{f_\alpha:\alpha\in I\}$ are linearly independent. Consider $X_0=\mathrm{span}\{f_\alpha:\alpha\in I\}$ and define $$f'':X_0\to K: \sum\limits_{\alpha\in I}\beta_\alpha f_\alpha\mapsto \sum\limits_{\alpha\in I}\beta_\alpha c_\alpha f_\alpha$$ Take arbitrary $x\in X_0$, then $x=\sum\limits_{\alpha\in J}\beta_\alpha f_\alpha$ for some finite $J\subset I$ and $\{\beta_\alpha:\alpha\in J\}\subset K$. In this case $$|f''(x)|= \left|f''\left(\sum\limits_{\alpha\in J}\beta_\alpha f_\alpha\right)\right|= \left|\sum\limits_{\alpha\in J}\beta_\alpha c_\alpha\right|\leq M\left\Vert \sum\limits_{\alpha\in J}\beta_\alpha f_\alpha\right\Vert=M\Vert x\Vert$$ Since $x\in X_0$ is arbitrary we conclude that $f''\in X_0^*$ and $\Vert f''\Vert\leq M$. Then by Hahn-Banach theorem you can extend $f''$ to the $f'\in X^*$ with $\Vert f'\Vert\leq M$ and $f'|_{X_0}=f''$. In this case $f'(f_\alpha)=f''(f_\alpha)=c_\alpha$, so we have built the desired functional $f'$.


text:
Verification of the depletion calculation scheme of an highly heterogeneous PWR core design - Archive ouverte HAL Access content directly
Conference Papers Year : 2016

## Verification of the depletion calculation scheme of an highly heterogeneous PWR core design

(1) , (1) , (1)
1
S. Mengelle
• Function : Correspondent author
• PersonId : 1063081

Connectez-vous pour contacter l'auteur

#### Abstract

In order to enhance the fuel utilization in Light Water Reactors and save resources, it is necessary to increase the conversion of fertile material ($^238$U) and ($^{239}$Pu). This can be obtained by designing low moderation PWR fuel assemblies which enhance the conversion of $^{238}$U into $^{239}$Pu. To ensure a high level of performance, the introduction of fertile elements is mandatory. This kind of design would therefore be complex to model. The flux gradient at the boundary between fissile fuel and fertile blankets is difficult to estimate by using a standard neutronic model based on transport-diffusion calculation schemes. The presence of fissile and fertile zones in the core leads to modify the standard calculation scheme based on a transport-diffusion calculation to take into account this heterogeneity. As a consequence, such transport calculations (first step of the calculation) which are performed in fundamental mode on a 3x3 cluster of fuel assemblies representative of the core loading pattern needs to be verified. Up to now, such verification was performed on static configurations by comparisons with the Monte-Carlo code TRIPOLI-4. New capability for depletion calculation has been recently introduced in the reference Monte-Carlo code TRIPOLI-4 by coupling it with the depletion module of the MENDEL code. This allows now to model the burnup depletion of a complex 3D geometry with TRIPOLI-4. This type of calculation is used as a reference to verify deterministic multi-groups calculations. This paper presents the comparisons between these two types of calculations in order to verify the performances and evaluate the safety criteria of highly heterogeneous core designs.

### Dates and versions

cea-02442330 , version 1 (16-01-2020)

### Identifiers

• HAL Id : cea-02442330 , version 1

### Cite

S. Mengelle, S. Douce, F. Damian. Verification of the depletion calculation scheme of an highly heterogeneous PWR core design. ICAPP 2016 - International Congress on Advances in Nuclear Power Plants, Apr 2016, San Francisco, United States. ⟨cea-02442330⟩

### Export

BibTeX TEI Dublin Core DC Terms EndNote Datacite

13 View


text:
Touch feeling belongs to the 5 human senses and it is implemented via force/position and pressure receptors of the skin, related to manipulation, perception, and communication. It helps us to interact with, and navigate through the environment (physical or virtual). Touch supplies information about shape, hardness, roughness, texture, and temperature via direct contact with an object.

Artificially created sense of touch or tactile feedback is an area of haptic technology or kinesthetic communication1. Demand for such technologies is appearing due to fast developments in robotics, remote control in medicine, virtual reality, and smart electronics2,3.

Materials and devices for haptic application of such technologies as mechanical vibration4, responsive polymers5,6 microelectromechanical system (MEMS)7, Peltier elements8, lasers9, ultrasound10, pneumatics11, air jets12, surface acoustic waves13, electrostatics14 etc.

Despite significant progress, high-quality haptic feedback devices remain challenging mainly due to the lack of stability and spatiotemporal resolution. One way to address these issues is the application of dynamic coatings. They adapt upon an external stimulus that changes their properties and provide on-screen tactile feedback at the position of touch by localizing it with a high resolution (micrometers) in various perception forms, such as surfaces that change their topography15 or switch between “dry” and ‘’wet’’16. Direct change of the material provides a real rather than simulated sensation of touch.

In this work we study such dynamic coatings with adaptive modulus based on a liquid crystal network (LCN)17,18, which is a widely studied material for active surfaces19. It permits predefining, freezing and subsequent tuning the molecular order, which, in turn, influences the macroscopic properties of the material20,21,22. Low molar mass liquid crystal (LC) acrylates23 are an attractive choice as building blocks for LCNs, as they are easy to orient and more applicable for coatings, especially for those with complex geometries, due to their low processing viscosity. As soon as the desired molecular alignment is established during the coating process, the LC monomers are cured by photopolymerization fixing the molecular organization in a network structure. Triggered manipulation of the molecular order changes the polymer properties such as wettability, elastic modulus, glass transition temperature, density etc. Artificially created adaptive modulus materials24 are known and based on molecular rearrangement upon application of an external stimulus such as heat, light, or electricity. LCN-based photoactive materials are intensively studied25,26,27,28,29,30,31,32. Almost invariably, their photo-responsiveness relies on the usage of azobenzene-containing molecules33 either covalently incorporated into the network structure or as guest additive. These examples comprise a shape-memory LCN-based material capable of dynamic modulus change upon photoactivation, previously developed by our group34. Such effect was obtained due to the incorporation of a low amount (2 wt%) of azobenzene-containing diacrylate in the LCN structure. Reversible isomerization of this photo-responsive agent, triggered by light with wavelengths close to azobenzene stereoisomers absorption maximum, ~ 365 nm (trans to cis) and ~ 450 nm (cis to trans), leads to molecular oscillations, distortion of the molecular order, free volume generation and, finally, photo-softening35. Excess free volume is energetically unfavorable15, thus the effect disappears when an external stimulus is removed. Modulus change can be amplified by dual-wavelength illumination34, which leads, however to substantial overheating and is less convenient for practical usage. One potential way to overcome this issue is to incorporate a small amount of fluorescent dye36. It emits photons with wavelength within cis-azobenzene absorption band after excitation with 365 nm light which was explored for temporal free volume generation37.

In the present study, we demonstrate a strong amplification (280% for a coating on a glass surface) of modulus change due to the incorporation of a fluorescent dye in the material structure. We also demonstrate the translation of the adaptive modulus effect from a free-standing film to a coating and localize it in a designed location of the sample. We envision that this unique approach enables the enhanced performance of haptic communication devices due to the possibility of fine-tuning the elasticity change (via UV power intensity control) and complex haptic information communication due to very high resolution (~ 2 μm). We believe that our findings permit a significant step toward the real-life application of haptic technology as such coatings can be applied to modify objects, enabling them with a haptic communication feature. Finally, the localization of modulus change on designed locations on the surface is essential for the communication of spatially resolved complex haptic information.

## Results and discussion

### Changing LCN-based material properties via incorporation of fluorescent dye

We prepared a LCN, with the composition shown in Fig. 6, in the presence and absence of fluorescent compound 7 to determine its influence on the isomerization reaction of the azobenzene moiety. We have chosen a material with a chiral-nematic molecular order. Due to the helicoidal organization of the mesogens with the helix axis orthogonal to the film/coating surface, an in-plane isodeformation occurs upon a change of the molecular order. Consequently, the azobenzene moieties are predominantly aligned with their transition moment in the plane of the film/coating, which enhances their absorption of incoming light.

Absorbance spectra, recorded before and after UV illumination are shown in Fig. 1. As the fluorescent dye is strongly absorbing below 400 nm (see Fig. 2), we did not analyze the trans azobenzene absorbance region. The data indicate that the trans to cis conversion for the dye-free LCN film is 4.8 times larger than for the sample with dye. Hence, there is coexistence of considerable amounts of the trans and cis azobenzene-containing mesogens at equilibrium conditions (reached within 5 min).

The coexistence of trans and cis azobenzene is an indication of intense oscillatory transitions between stereoisomers induced by combined irradiation with UV (from LED) and blue light (from fluorescent dye). Indeed, this reaction is reversible and, consequently, should be considered as a dynamic process. It leads to a stronger plasticizing effect of polyacrylate chains and amplification of the photo-softening effect (Fig. 2).

Importantly, only a small fraction (~ 2%) of 365 nm light is transmitted through the sample coating at equilibrium for the dye-containing sample, while for the dye-free sample it is as high as ~ 20% (Fig. S1 and supplementary video 2). These observations signify that the dye-containing LCN film, placed in between the light source and human skin, in a real-life application drastically reduces the harmful impact of UV irradiation.

### Amplification of bulk modulus change

We performed dynamic mechanical analysis on the samples with and without the fluorescent dye. In the absence of an external stimulus their elasticities are similar and in the range of 1.6–1.8 GPa at room temperature (Fig. 3a). Under UV illumination a clear contrast in elastic properties for the samples was observed. Specifically, for the dye-containing sample the modulus decreases by ~ 970 MPa. For dye-free sample this effect is more moderate: a decrease of ~ 230 MPa is observed. Hence, the addition of fluorescent dye leads to significant amplification of the modulus change in line with our initial assumption. Both response time and relaxation time (restoring initially high modulus value) are within 15 s indicating that mechanical equilibrium is reached faster than photochemical equilibrium. Sequential photoactivation leads to a repetitive identical decrease of the modulus for both samples.

Temperature-related modulus changes are minor (Fig. 3b) for the investigated power range. Sample illumination with the sequentially increasing power of UV light leads to a rising storage modulus change in line with UV intensity (Fig. 3c). Relative change of the modulus is most significant for the first illumination event which is related to the relaxation of initially present stresses in the polymer structure. The temperature rises linearly with irradiation intensity (Fig. 3d). Linear changes in temperature and modulus as functions of power permit prediction and precise control of the material elasticity by will.

### Translating adaptive modulus from free-standing film to a coating

The prospect of transferring the concept of reversible modulus change from bulk material to a coating was studied via AFM. For this purpose, force-indentation curves of LCN coatings on glass surfaces with the same composition as for bulk modulus measurements were recorded. Identical measurements were conducted for the sample with and without dye in dark, during UV illumination (~ 90 mW/cm2), and after switching off UV light (Fig. 4).

For the same applied force (stress), an increase in tip indentation (strain) during UV light is observed for both samples. After illumination, the indentation recovers back to its initial level. These data reflect the evolution of elastic modulus as it is inversely proportional to strain. It is also an indication of the photo-softening effect on the investigated surfaces. Modulus values calculated from these data prove substantial amplification of elasticity reduction by the presence of the fluorescent dye in the LCN coating. Indeed, modulus decreased by 31% and 87% for the samples without and with dye respectively. Accordingly, we have demonstrated 280% amplification of modulus change due to the presence of the fluorescent dye in the material. Such modulus change (from 2.2 to 0.3 GPa) can be sensed by human37. The temperature during illumination is estimated to be 38 °C and 33 °C for the samples with and without dye respectively, which proves that modulus change occurs due to the photo-softening effect.

Remarkably, we observed a significant modulus change regardless of the direction of applied force. Indeed, during bulk modulus measurements using DMTA, the free-standing LCN film was stretched perpendicularly to the helix director. While for AFM experiments, the force was applied parallel to the helix axis (perpendicularly to the surface). Such tolerance to the direction of the applied force can be explained by the concept of free volume. It is known for LCNs that during photoactivation the intermolecular spaces increase38. This enables plasticized polyacrylate main chains to be deformed in any direction.

UV penetration depth could become an issue for the thicker samples (due to the strong absorbance of trans-azobenzene and the fluorescencent dye). However, similar changes of the modulus upon illumination for the dye-containing film were observed by AFM and DMTA signifying that the UV light reaches throughout the entire sample.

The increasing lag between retraction and extension curves during photoactivation indicates amplification of adhesion upon illumination. For the dye-containing sample, this effect is much more pronounced (at 0.2 µN the lag is 45.2 nm and 25.7 nm for the sample with and without dye respectively). Indeed, a lower modulus increases the indentation depth and therefore the contact area between the AFM tip leading to higher adhesive forces and adhesive work.

### Localization modulus change

To enable complex and instructive haptic information transfer, patterned modulus change is a prerequisite. We, therefore, studied localized softening effects by means of AFM via recording a force map under UV illumination. By using a photomask containing 15 µm linear patterns (Fig. 5c), we could guarantee the presence of illuminated and “dark” areas (Fig. 5) and measure corresponding modulus values for a 25 × 25 µm region of the coating surface. The modulus of the same area was recorded before and during illumination (Fig. 5a).

A clear contrast with a gradual transition of ~ 2 µm between low and high modulus zones was observed in coherence with illuminated and dark regions of the sample (Fig. 5a2). The same area, recorded in dark has a homogeneously distributed high modulus zone. These data indicate a spatially restricted photo-softening effect with high (micrometer scale) resolution. Hence, by tuning the size and shape of photomask patterns, desired haptic information can theoretically be communicated.

It should be noted, that despite the clear contrast between low and high modulus, the average modulus values of the sample region that is covered by the mask during the exposure procedure are lower than those measured for the samples in absence of any irradiation. This is related to partial illumination of these regions due to diffraction, scattering, and light reflection from the cantilever surface (which is in nanometer-scale proximity to the analyzed spot).

The difference in absolute modulus values recorded for force mapping (Fig. 5) and single force (Fig. 4) experiments is related to high (125 s-1) and low (0.25 s-1) strain rates applied respectively, as elastic modulus is higher for larger strain rates39.

## Conclusions

In the present study, we developed adaptive modulus surfaces based on chiral-nematic LCN. The material is significantly photo-softened (87% decrease of the modulus for a coating on glass) by bringing covalently connected azobenzene-containing moieties into an oscillating mode by exposure with low-intensity UV light (~ 90 mW/cm2) triggering trans to cis isomerization. The backward, cis to trans, reaction is simultaneously triggered by additional exposure to blue light, due to the presence of a small amount (0.5%) of fluorescent dye avoiding the use of two light sources. Due to a substantial amount of trans isomer at the thermodynamic equilibrium, only a small fraction of 365 nm light (~ 2% for a 10 μm sample) is transmitted, which reduces the harmful effect of UV light upon human contact. Moreover, higher energy efficiency and minor overheating were achieved in comparison to previously studied dual-wavelength illumination of dye-free samples. In addition, we have demonstrated successful localization of modulus change with a high resolution (around 2 μm). We envision that our original approach enables the fabrication of haptic devices with enhanced performance in the near future.

## Materials and methods

### Materials

Figure 6 shows the mixture used to produce an adaptive modulus LCN material.

Monomers 1 to 3 were obtained from Merck UK. Monomer 4 was obtained from BASF. Monomer 5 was custom-synthesized by Syncom (Groningen, the Netherlands). Photoinitiator 6 was purchased from Ciba Specialty Chemicals. Fluorescent dye 7, 2,5-bis (5-tertbutyl-benzoxazol-2-yl) thiophene, was obtained from Sigma Aldrich. Polymer films and coatings were fabricated using a mixture containing 21.9 (or 22.4 for the reference sample) wt% monomer 1, 40.6 wt% monomer 2, 31 wt% monomer 3, 4 wt% monomer 4, 2 wt% monomer 5, 1 wt% photoinitiator 6 and 0.5 (or 0 for the reference sample) wt% fluorescent dye 7. The constituents were mixed by dissolving in dichloromethane.

### Sample preparation

Glass substrates are cleaned by a 5-min dip in acetone and 2-propanol sequentially while stirring followed by flushing with demi water and drying with a nitrogen flow. A1051 (Sunever, Nissan Chemical, Japan) was used to obtain a planar alignment of the liquid crystal monomer mixture. It was spin coated on cleaned glass followed by baking at oven for 90 min at 180 °C. Gentle manual rubbing on a velvet cloth gave the substrate the surface structure needed for desired LC alignment. A cell provided with alignment layers was capillary filled with the LC monomer mixture and subsequently cured by UV exposure at 42 °C for 30 min with an intensity of 400 mW/cm2 using a mercury lamp (EXPR Omnicure S2000) equipped with a cut-off filter transmitting light > 400 nm to prevent premature isomerization of the azobenzene group during polymerization. The samples were post-cured at 120 °C to ensure full cure of the acrylate monomers.

### Sample characterization

UV-Vis spectroscopy was performed on a LAMBDA™ 750 UV/Vis/NIR spectrophotometer (PerkinElmer) equipped with a 150 mm integrating sphere. Sample films with a thickness of 6 um were illuminated with 365 nm LED (Mouser Electronics) between measurements with intensity ~ 50 mW/cm2 and 1–5 min duration.

The macroscopic mechanical properties of the films were acquired using dynamic mechanical thermal analysis (DMTA) (Q800 Dynamic Mechanical Analyzer from TA Instruments) at a frequency of 1 Hz. For this, a sample of 10 × 3 × 0.01 mm dimensions was fixed at both ends in clamps and characterized in a strain-controlled mode. 365 nm LED (Mouser Electronics) is used to provide monochromatic light with power is ~ 50 mW/cm2. Temperature was monitored remotely using an infrared camera (Fluke Ti-32).

Atomic force microscopy images and force measurements were recorded in tapping and contact mode, respectively, using a Cypher Environmental Scanner (ES) equipped with a closed cell and a normal sample stage. Force measurements were carried out using a super luminescent diode in order to reduce the signal-to-noise ratio. Silicon (100) AC160TS(-R3) probes (Oxford Instruments, spring constant k = 8.77–29.51 N/m) were used for all measurements and calibrated using the ‘Get Real’ function in the Igor Pro software. It should be noted that spring constants below 20 N/m were desirable for proper measurements. 365 nm LED (Mouser Electronics) embedded in home-made 3D printed holder was used to provide monochromatic light with power ~ 90 mW/cm2. Temperature was monitored remotely by reproducing experimental conditions outside the AFM compartment using an infrared camera (Fluke Ti-32). Processing of the images were done to enhance the contrast using plane fit and flattening at order 1 using Gwyddion v2.60. For the AFM force spectroscopy (AFM-FS), force-distance curves were recorded in a 20 × 20 µm area by approaching and retracting the cantilever tip to the sample on various spots before, during and after illumination with 365 nm light at a constant velocity of 1 µm/s. Additional force maps of 30 × 30 µm (128 × 128 pixels) were recorded before and during illumination with 365 nm light at a scan rate of 0.39 Hz and a Z-rate of 125 Hz. All AFM measurements were acquired in air and at the machine’s temperature (~ 30 °C). For our AFM force-distance curves and maps, we employed Sneddon’s model for cone-shaped AFM tips in order to fit the experimental data and obtain the elastic modulus of the sample:

$$F=\frac{2}{\pi }*\frac{{E}_{s}}{1-{V}_{s}^{2}}*\mathit{tan}\alpha *{\delta }^{2}(Eq.X)$$

where F is the applied force, Es is the elastic modulus, Vs is the Poisson’s ratio and α the opening angle of the AFM tip and δ the indentation depth.


text:
# Absolute convergence help

miglo

## Homework Statement

$$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{\sqrt{n}+1}{n+1}$$

## Homework Equations

absolute convergence test

## The Attempt at a Solution

by book says that the series converges because $$\sum_{n=1}^{\infty}\frac{\sqrt{n}+1}{n+1}$$ converges
but they don't show how the absolute value of the original series converges, and I've tried showing it myself but i keep getting divergence
i know that as n grows larger and larger the behavior of $$\frac{\sqrt{n}+1}{n+1}$$ is similar to that of $$\frac{\sqrt{n}}{n}$$ so i tried using limit comparison and direct comparison with $\frac{1}{n}$ but i keep getting divergence
i tried the integral test but i kept getting divergence also
ive been trying this for far too long so any help would be greatly appreciated

Homework Helper
Gold Member
You are correct that the positive term series diverges.

miglo
but my book says that the original series converges by the absolute convergence test
so wouldn't that mean that $$\sum_{n=1}^{\infty}\frac{\sqrt{n}+1}{n+1}$$ converges also? or is this an error in the book?

so wouldn't that mean that $$\sum_{n=1}^{\infty}\frac{\sqrt{n}+1}{n+1}$$ converges also? or is this an error in the book?


text:
Accelerating the pace of engineering and science

# Documentation

## Akaike's Criteria for Model Validation

### Definition of FPE

Akaike's Final Prediction Error (FPE) criterion provides a measure of model quality by simulating the situation where the model is tested on a different data set. After computing several different models, you can compare them using this criterion. According to Akaike's theory, the most accurate model has the smallest FPE.

 Note:   If you use the same data set for both model estimation and validation, the fit always improves as you increase the model order and, therefore, the flexibility of the model structure.

Akaike's Final Prediction Error (FPE) is defined by the following equation:

$FPE=V\left(\frac{1+d}{N}}{1-d}{N}}\right)$

where V is the loss function, d is the number of estimated parameters, and N is the number of values in the estimation data set.

The toolbox assumes that the final prediction error is asymptotic for d<<N and uses the following approximation to compute FPE:

$FPE=V\left(1+2d}{N}\right)$

The loss function V is defined by the following equation:

$V=\mathrm{det}\left(\frac{1}{N}\sum _{1}^{N}\epsilon \left(t,{\stackrel{^}{\theta }}_{N}\right){\left(\epsilon \left(t,{\stackrel{^}{\theta }}_{N}\right)\right)}^{T}\right)$

where ${\theta }_{N}$ represents the estimated parameters.

### Computing FPE

You can compute Akaike's Final Prediction Error (FPE) criterion for linear and nonlinear models.

 Note:   FPE for nonlinear ARX models that include a tree partition nonlinearity is not supported.

To compute FPE, use the fpe command, as follows:

`FPE = fpe(m1,m2,m3,...,mN)`

According to Akaike's theory, the most accurate model has the smallest FPE.

You can also access the FPE value of an estimated model , m, as follows:

• If m is a linear model, type m.Report.Fit.FPE

• If m is a nonlinear model, type m.EstimationInfo.FPE

### Definition of AIC

Akaike's Information Criterion (AIC) provides a measure of model quality by simulating the situation where the model is tested on a different data set. After computing several different models, you can compare them using this criterion. According to Akaike's theory, the most accurate model has the smallest AIC.

 Note:   If you use the same data set for both model estimation and validation, the fit always improves as you increase the model order and, therefore, the flexibility of the model structure.

Akaike's Information Criterion (AIC) is defined by the following equation:

$AIC=\mathrm{log}V+\frac{2d}{N}$

where V is the loss function, d is the number of estimated parameters, and N is the number of values in the estimation data set.

The loss function V is defined by the following equation:

$V=\mathrm{det}\left(\frac{1}{N}\sum _{1}^{N}\epsilon \left(t,{\stackrel{^}{\theta }}_{N}\right){\left(\epsilon \left(t,{\stackrel{^}{\theta }}_{N}\right)\right)}^{T}\right)$

where ${\theta }_{N}$ represents the estimated parameters.

For d<<N:

$AIC=\mathrm{log}\left(V\left(1+\frac{2d}{N}\right)\right)$

 Note:   AIC is approximately equal to log(FPE).

### Computing AIC

Use the aic command to compute Akaike's Information Criterion (AIC) for one or more linear or nonlinear models, as follows:

`AIC = aic(m1,m2,m3,...,mN)`

According to Akaike's theory, the most accurate model has the smallest AIC.