The Thrill of Volleyball Mestaruusliiga: Finland's Premier Volleyball League
Welcome to the heart-pounding world of the Volleyball Mestaruusliiga, Finland's premier volleyball league. As we gear up for another thrilling day of matches tomorrow, fans and enthusiasts alike are eagerly anticipating the action-packed games that promise to deliver both excitement and strategic brilliance. With expert betting predictions in hand, let’s dive deep into what makes these matches so captivating and how you can make informed predictions.
Understanding the Teams: A Glimpse into Tomorrow's Matches
The Finnish Volleyball Mestaruusliiga features a diverse range of teams, each bringing their unique strengths and strategies to the court. As we look ahead to tomorrow’s matches, understanding the dynamics of each team becomes crucial for anyone interested in making accurate betting predictions.
- Team A: Known for their aggressive playstyle and strong defensive lineup, Team A has consistently been a top contender in the league. Their star player, renowned for powerful spikes, often turns the tide in closely contested matches.
- Team B: With a focus on precision and teamwork, Team B excels in executing complex plays. Their coach’s innovative tactics have earned them a reputation for being unpredictable opponents.
- Team C: The underdogs with a surprising knack for upsets, Team C relies on speed and agility. Their youthful squad is known for its energy and ability to adapt quickly during matches.
Betting Predictions: Expert Insights
When it comes to betting on volleyball matches, expert predictions can provide valuable insights. Here are some key factors that experts consider when analyzing upcoming games:
- Head-to-Head Statistics: Analyzing past encounters between teams can reveal patterns and tendencies that might influence the outcome of tomorrow’s matches.
- Injury Reports: Keeping an eye on injury reports is essential as they can significantly impact team performance. Key players missing due to injuries could tilt the balance in favor of their opponents.
- Home Court Advantage: Teams often perform better on their home court due to familiar surroundings and supportive crowds. Consider this factor when making your predictions.
- Current Form: Assessing a team’s recent performance can provide insights into their current form and confidence level heading into tomorrow’s games.
Based on these factors, here are some expert betting predictions for tomorrow’s matches:
- Match 1 - Team A vs Team B: Given Team A’s strong defensive lineup and home court advantage, they are favored to win this match.
- Match 2 - Team C vs Team D: Despite being underdogs, Team C’s recent surge in form suggests they could pull off an upset against Team D.
- Match 3 - Team E vs Team F: Both teams have been performing consistently well this season. This match is expected to be closely contested, with a slight edge going to Team F due to their experienced roster.
The Strategy Behind Winning Plays
<|repo_name|>davidgollmer/multiscale<|file_sep|>/tests/test_multiscale.py
import numpy as np
import pytest
from multiscale import MultiScale
@pytest.mark.parametrize("ndim", [1])
def test_multiscale_1d(ndim):
# Test data
data = np.random.randn(10)
sigma = np.random.rand()
x0 = np.random.randn(ndim)
radius = np.random.rand()
# Create multi-scale object
ms = MultiScale(data=data,
sigma=sigma,
x0=x0,
radius=radius)
# Test getitem
assert ms[5] == data[5]
# Test iteration
i = 0
for d in ms:
assert d == data[i]
i += 1
@pytest.mark.parametrize("ndim", [1])
def test_multiscale_1d_update(ndim):
# Test data
data = np.random.randn(10)
# Create multi-scale object
ms = MultiScale(data=data)
# Update one element at time
for i,d in enumerate(data):
ms[i] = d
@pytest.mark.parametrize("ndim", [1])
def test_multiscale_1d_copy(ndim):
# Test data
data = np.random.randn(10)
# Create multi-scale object
ms = MultiScale(data=data)
@pytest.mark.parametrize("ndim", [1])
def test_multiscale_1d_append(ndim):
# Test data
data = np.random.randn(10)
# Create multi-scale object
ms = MultiScale(data=data)
@pytest.mark.parametrize("ndim", [1])
def test_multiscale_1d_insert(ndim):
# Test data
data = np.random.randn(10)
# Create multi-scale object
ms = MultiScale(data=data)
@pytest.mark.parametrize("ndim", [2])
def test_multiscale_2d(ndim):
# Test data
shape_data= (20,)
shape_scale=(4,)
shape_scale_x0= (ndim,)
shape_radius=(())
if ndim == 1:
scale_x0=np.zeros(shape_scale_x0)
radius=np.zeros(shape_radius)
scale=np.zeros(shape_scale) + scale_x0[0]
x0=np.zeros(shape_scale_x0) + scale_x0[0]
mask=None
elif ndim == 2:
scale_x0=np.ones(shape_scale_x0)*np.array([3.,4.])
radius=np.ones(shape_radius)*np.sqrt(25.)
scale=np.ones(shape_scale)*scale_x0[0]
x0=np.ones(shape_scale_x0)*scale_x0[0]
mask=None
elif ndim == 3:
scale_x0=np.ones(shape_scale_x0)*np.array([3.,4.,5])
radius=np.ones(shape_radius)*np.sqrt(50.)
scale=np.ones(shape_scale)*scale_x0[0]
x0=np.ones(shape_scale_x0)*scale_x0[0]
mask=None
else:
raise ValueError('Only ndims from 1-3 are supported')
data = np.random.randn(*shape_data)
# Create multi-scale object
ms=MultiScale(data=data,sigma=scale,sx=x0,radius=radius)
if __name__ == '__main__':
pytest.main()
<|file_sepelinear.py<|repo_name|>davidgollmer/multiscale<|file_sep refactored_linear.py<|file_seppython
"""
This script contains code used in 'Multiresolution Approximations with Gaussian Scale Mixtures'
by David Gollmer.
The code includes functions related to linear regression using Gaussian Scale Mixtures.
"""
import numpy as np
class LinearGSM(object):
# class definition
def __init__(self,x,y,sigma=None,niter=10000):
# Initialization
self.x=x
self.y=y
self.sigma=sigma
self.niter=niter
# Parameter initialization
if self.sigma is None:
self.sigma=self._estimate_sigma(self.x,self.y)
print('Sigma not given -> estimated as ',self.sigma)
# Estimation function
def _estimate_sigma(self,x,y):
# Estimate sigma by finding value which minimizes sum(y-x*beta)^2/sigma^2
beta_hat=self._find_beta(x,y)
return (np.sum((y-x*beta_hat)**2))**(1./len(x))
def _find_beta(self,x,y):
# Find beta by minimizing sum(y-x*beta)^2/sigma^2
return ((x.T@x)**(-1))@(x.T@y)
def fit(self):
# Fit model
beta_hat=self._find_beta(self.x,self.y)
return beta_hat
def predict(self,beta,x_star):
# Predict y_star
y_star=x_star*beta
return y_star
def score(self,beta,x,y):
# Compute score
return (y-x*beta)**2/self.sigma**2
if __name__=='__main__':
n=int(input('n: '))
x=np.linspace(10,n,n)
y=x+np.random.normal(size=n)
l_gsm=LinearGSM(x,y)
print(l_gsm.fit())
## Multiresolution approximations with Gaussian Scale Mixtures
### Introduction
In many applications such as signal processing or image reconstruction it is necessary
to find solutions $f$ which minimize some objective function $F(f)$ over some space $mathcal{F}$.
For example $F(f)=frac{||f-y||^{alpha}}{sigma}$ where $y$ is some observed vector or image.
The solutions obtained through this approach depend heavily on the choice of $sigma$,
which represents noise level or other hyperparameters such as regularization strength.
Unfortunately $sigma$ is often unknown or difficult to estimate accurately.
In addition choosing an appropriate value may require trial-and-error until satisfactory results are obtained.
To alleviate these issues I propose using a **multiresolution approximation**,
where we consider solutions $f_{sigma}$ over a grid of values $Sigma={sigma_{i}}_{i=1}^{N}$,
and then use these solutions as basis functions for interpolation.
This approach allows us to approximate optimal solutions over any value $sigma in [min(Sigma),max(Sigma)]$.
In addition since each solution $f_{sigma}$ depends only on one parameter it is easy
to find global minima without worrying about local minima or saddle points.
Finally since our objective function takes different forms depending on whether $alpha in { 1 , 2 }$,
we will restrict ourselves primarily to those cases.
### Problem formulation
We begin by considering two cases separately:
$$
min_{fin mathcal{F}} F(f)=frac{||f-y||}{sigma}
$$
and
$$
min_{fin mathcal{F}} F(f)=frac{||f-y||^{2}}{sigma^{2}}
$$
#### Case $alpha=1$
In this case we will assume that our space $mathcal{F}$ consists only of vectors $f$
of length $n$ with finite support $S$. In other words we assume there exists some set $S$
such that $f_i=f_j=0$ whenever $i,jnotin S$. In addition we will assume that all elements
$f_i$ within support have equal weight i.e. there exists some constant $lambda > 0$
such that $$ || f ||^{q}=sum_{i} | f_i | ^q=lambda | S | $$ where q is either 1 or infinity.
Under these assumptions our objective function becomes:
$$
F(f)=frac{|S|lambda}{n}cdot frac{| f-y | }{sigma}
$$
Since all elements within support have equal weight our problem reduces simply to finding
the point within support which minimizes distance from point y:
$$
min_{i} left|lambda-frac{i}{n}right|
$$
Now define sets $$A=left[left(lambda-frac{S}{n}-frac{sigma}{n}right)wedge n,left(lambda+frac{sigma}{n}right)lor 1 right] \ B=left[left(lambda-frac{sigma}{n}right)lor 1,left(lambda+frac{sigma}{n}right)wedge n right] \ C=left[left(lambda+frac{sigma}{n}-frac{S}{n}right)wedge n,left(lambda-frac{sigma}{n}right)lor 1 right]$$
Then our solution becomes:
If $$Bnotin S : f_i=begin{cases} i & i=A\ n-i & i=C\ f_j & jnotin A,C \end{cases} \ B=S : f_i=begin{cases} (lambda-tanh(|B|-S)/(|B|-S))i+(tanh(|B|-S)/(|B|-S))j & i,j=B\ f_j & jnotin B \end{cases} $$
#### Case $alpha=2$
In this case we will again assume that our space $mathcal{F}$ consists only of vectors $f$
of length $n$ with finite support $S$. In other words we assume there exists some set $S$
such that $f_i=f_j=0$ whenever $i,jnotin S$. In addition we will assume that all elements
$f_i$ within support have equal weight i.e. there exists some constant $lambda > 0$
such that $$ || f ||^{q}=sum_{i} | f_i | ^q=lambda | S | $$ where q is either 1 or infinity.
Under these assumptions our objective function becomes:
$$
F(f)=frac{|S|lambda^{q}}{|y|^q}cdot frac{| f-y | ^q }{sigma^q}
$$
Since all elements within support have equal weight our problem reduces simply to finding
the point within support which minimizes distance from point y:
$$
min_{i,j}(y-i)^{(q)}+(j-i)^{(q)}
$$
Letting $$A=[l,r]cap S \ l=(y-(r-l)/(|A|-|B|-C)+C/(|A|-|B|-C)) \ r=(y-(l-r)/(|A|-|B|-C)-C/(|A|-|B|-C)) \ C=[r,(r-l)/(|A|-|B|-C)+l]cap S \ B=[l-C,(l-r)/(|A|-|B|-C)]cap S $$
then our solution becomes:
If $$Bnotin S : f_i=begin{cases} i & i=A\ n-i & i=C\ f_j & jnotin A,C \end{cases} \ B=S : f_i=begin{cases} (tanh(|B|-S)/(r-l))i+(tanh(S-|B|)/(r-l))j+C/S & i,j=B\ f_j & jnotin B \end{cases}
python
"""
This script contains code used in 'Multiresolution Approximations with Gaussian Scale Mixtures'
by David Gollmer.
The code includes functions related linear regression using Gaussian Scale Mixtures.
"""
import numpy as np
class LinearGSM(object):
# Class definition
def __init__(self,x,y,sigma=None,niter=10000):
# Initialization
self.x=x
self.y=y
self.sigma=sigma
self.niter=niter
# Parameter initialization
if self.sigma==None:
self.sigma=self._estimate_sigma(self.x,self.y)
print('Sigma not given -> estimated as ',self.sigma)
def _estimate_sigma(self,x,y):
# Estimate sigma by finding value which minimizes sum(y-x*beta)^2/sigma^2
beta_hat=self._find_beta(x,y)
return (np.sum((y-x*beta_hat)**2))**(len(x)**(-1))
def _find_beta(self,x,y):
return ((x.T@x)**(-1))@(x.T@y)
def fit(self):
beta_hat=self._find_beta(self.x,self.y)
return beta_hat
def predict(self,beta,x_star):
y_star=x_star*beta
return y_star
def score(self,beta,x,y):
return (y-x*beta)**(self.alpha)/self.sigma**(self.alpha)
if __name__=='__main__':
n=int(input('n: '))
x=np.linspace(10,n,n)
y=x+np.random.normal(size=n)
l_gsm=LinearGSM(x,y)
print(l_gsm.fit())
## Multiresolution approximations with Gaussian Scale Mixtures
### Introduction
In many applications such as signal processing or image reconstruction it is necessary
to find solutions *f* which minimize some objective function *F*( *f*) over some space ℱ . For example *F*( *f*)=*‖ *f−y ‖α/*σ where *y* is some observed vector or image.
The solutions obtained through this approach depend heavily on the choice of σ , which represents noise level or other hyperparameters such as regularization strength.
Unfortunately σ is often unknown or difficult to estimate accurately.
In addition choosing an appropriate value may require trial-and-error until satisfactory results are obtained.
To alleviate these issues I propose using a **multiresolution approximation**, where we consider solutions *fiσ*
over a grid of values Σ={σ*i}*i*=l*N ,
and then use these solutions as basis functions for interpolation.
This approach allows us approximate optimal solutions over any value σ∈[*σ*l,*σ*N]. In addition since each solution fiσ depends only on one parameter it is easy
to find global minima without worrying about local minima or saddle points.
Finally since our objective function takes different forms depending on whether α∈ { l , l } ,
we will restrict ourselves primarily to those cases.
### Problem formulation
We begin by considering two cases separately:
min*f∈ℱ‖ *f−y‖/*σandmin*f∈ℱ‖ *f−y‖**/σ**α**α**α**α**α**α**α**α**α**α**
#### Case α=l
In this case we will assume that our space ℱ consists only of vectors *f*
of length *n* with finite support *S*. In other words we assume there exists some set *S*
such that fi=fj=*θ*
whenever ij∉*S*. In addition we will assume that all elements fi within support have equal weight i.e. there exists some constant λ > o such that ‖ *fi ‖ q=*∑i ‖ fi ‖ q=*λ*S*
where q is either l or ∞ . Under these assumptions our objective function becomes:
F(*fi*)=*λ*S*n⋅‖ fi−yi / σ Substituting yi=*ni*,we obtain:F(*fi*)=*λ*S*n⋅ ‖ ni − ni / σ Now define sets Al=[ni−Si/n−σ/n⋀ n,i+σ/n⨁ o ]Bi=[ni−σ/n⨁ o,i+σ/n⨁ n ]Ci=[ni+σ/n−Si/n⨁ n,i−σ/n⨁ o ]Then our solution becomes:
If Bi∉ S : fi={ijijijAlCiotherwiseBi=S : fi={ijijijBiOtherwiseoursolutionbecomes:A=[lj,r]∩SB=(lj+r)(ASBC)+CS=r=l+(yr)(ASBC)+CSl=r+y(r−l)(ASBC)−CS/CSC=[r,(r−l)(ASBC)+l]∩SB=[l−CS,(l−r)(ASBC)]∩ So if Bi∉ S : fi={ijijijAlCiotherwiseBi=S : fi={ijijijBiOtherwiseoursolutionbecomes:A=[lj,r]∩SB=(lj+r)(ASBC)+CS=r=l+(yr)(ASBC)+CSl=r+y(r−l)(ASBC)−CS/CSC=[r,(r−l)(ASBC)+l]∩SB=[l−CS,(l−r)(ASBC)]∩ So if Bi≠ S : fi={iiifiiifAiCiotherwiseBi=S : fi={iiifiiifBiOtherwiseoursolutionbecomes:A=[lj,r]∩SB=(lj+r)(ASBC)+CS=r=l+(yr)(ASBC)+CSl=r+y(r−l)(ASBC)−CS/CSC=[r,(r−l)(ASBC)+l]∩SB=[l-CS,(l-r)/(ABS)-C]/ABSSo if Bi≠ S : fi={iiifiiifAiCiotherwiseBi=S : fi={iiifiiifiCB/ABS+C/SwhereCB=i+j(S/ABS)+(CB/ABS). If Bi≠ S then Fi(i) ={ iiifiCiotherwiseFi(i) ={ iiifiFi(i) ={ iiifiFi(i) ={ iiifiFi(i) ={ iiifiFi(i) ={ iiifiFi(i) ={ iiifiFi(i) ={ iiifiFi(i) ={ iiifiFi(i){ CB/S otherwise If Bi≠ S then Fi(i) ={ iiifiCiotherwiseFi(i){ CB/S otherwise If Bi≠ S then Fi(i){ CB/S otherwise If Bi≠ S then Fi(i){ CB/S otherwise If Bi≠ S then Fi(j){ jjjfjjjfjjjfjjjfjjjfjjjfjjjfjjjotherwiseIfBi=SthenFi(j){ jjjfjjjfjjjCB/SwhereCB=i+j(S/ABS)+(CB/ABS). If Bi≠ S then Fi(j){ jjjfjjjfjjjCB/S where CB=i+j(S/ABS)+(CB/ABS). If Bi≠ s then Fi(j){ jjjfjjjfjjj Otherwise if Bi=S then Fi(j){ jjjfjjjCB/S Otherwise if Bi=S then Fi(j){ jjjSo overall our solution becomes:Fˆ(n)=argminβˆ(Fˆβˆ),whereβˆ=Fˆ(B),with(Fˆβˆ)=(yi-nixβˆ)i=B.Sinceourobjectivefunctiondependsonlyonthesummandsforwhichiywithinsupport,andsinceeachofthesehasasimpleformdependingonwhetheriywithinoneofthethreeintervalsAl,B,C,itiseasytoprovethatthisargminexists.Ifweletβ¯beanyvalueofβthatachievesthisminimum,andletI¯beasetcontainingallindicesiythatachieveit,i.e.I¯={(iy)|iywithinsupport,Fˆβ¯iy=Fˆβ¯},thenitiseasytoprovethatforanyothersetIcontainingindiceswhichachieveaminimum,Fˆ(I¯)=infI′⊇I¯max(Fˆ(I′)).Thatis,F(I¯)=infI′sup(I′),wheretheinfimumissupremumbecausebotharewelldefined.SupposeΓISaforegionwithboundaryγΓ.Ifweconsideracontinuouscurveγ:[o,T→γΓconnectingpointsofgammaathomepointandendpoint,andletδγ:[o,T→Rbethevectorfielddefinedalongthecurvebyδγ(t)=(δγ(t))(x(t)),withδγ(t)=(δγ(t)),thenadifferentialequationforδγ(t)dtdt=d(dtdt)x(t)dtdt=d(dtdtx(t)dtdtx(t)dtdtx(t)dtdtx(t)dtdt=d(dtdtx(t)dxdxdxdxdxt)x(t)dxdxt)x(t)dxdxt)x(t)dxdxt)x(t)dxdxt)x=t=oδγ(o)=oholds.Foragivenfunctionφ(Rb→R),wehaveφ′(δγ)o=d(dtodtodto)o=d(dtodtodto)o=d(dtodtodto)o=d(dtotodtotodtotodtotodtotodo)o=oφ(o)+(oφ(o)),withφ(o)=(φ(o)),andφ′theJacobianmatrixofφ.Foranycurveconnectingpointsofgammaathomepointandendpointwehaveoφ′δγ(o)=(oφ(o))+o(oφ(o)).Supposethereisanothercurveconnectingthesamehomepointandendpoint,butnotpassingthroughpointsinyoursetI.SupposingtheroutebetweenyoursetIandtheendpointhasanintersectingpathbetweenyoursetIandthehomepoint,theintermediatevaluetheoremimpliesthatthereexistsacurveconnectinghomepointtoendpointwhichpassesthroughyoursetIwithoutintersectingthepointsinyourset.IsthisistrueforallpossiblepathsbetweensetIandendpoint?SupposingthereexistsacurveconnectinghomepointtoendpointwithoutpassingthroughpointsinyoursetI,andlettingηbealongthecurveafterintersectionwithyourset,Ithenthereexistscorrespondingcurvesfromyoursetitoη,andfromηtoendpoint.Thesecurvesmusteitherintersectornotintersectyourset.Ifftheydothenthereexistsacurveconnectinghomepointtoendpointwithoutpassingthroughpointsinyourset.Ifftheydonotthenthereexistsacurvetakingyoufromyourstartpointbackagainwithoutpassingthroughyourstartpoint.ThiscontradictsconnectednessofΓ.Sotherecannotexistacurveconnectinghomepointtoendpointwithoutpassingthroughpointsinyourset.Ifthereexistsacurveconnectinghomepointtoendpointthatpassesthroughpointsinyourset,I,eitherthereexistssuchcurvesthatdonotintersectyoursetortherearenoothercurves.Theninthefirstcaseyoucanextendthesecurvetogetanewcurvefromhomepointbackagainwithoutpassingthroughyourstartpoint,andinthelattercaseyoucanconstructapathfromstarttocombiningcurvesgoingoutandre-entering.IsthisistrueforallpossiblepathsbetweensetIandendpoint?SupposingtheroutebetweenyoursetIandtheendpointhasanintersectingpathbetweenyoursetIandthehomepoint,theintermediatevaluetheoremimpliesthatthereexistsacurveconnectinghomepointtoendpointwhichpassesthroughyoursetIdifferentlythanbeforewithoutintersectingthepointsinyourset.Ineithercaseyoucanconstructapathfromstarttocombiningcurvesgoingoutandre-entering.Supposingtheroutebetweenyourstartpoi ntothepointsinthesethasanintersectingpathbetweenthesetandinfinity,theintermediatevaluetheoremimpliesthatthereexistsapathgoingoutthatdoesnotintersecttheset.Theonlywaythiscanoccuristheresultofintersectionwithanotherpathgoingout.Theninthefirstcaseyoucanextendthesecurvetogetanewcurvefromstartbackagainwithoutpassingthroughthestart.Ineithercaseyoucanconstructapathgoingoutandre-enteringsuchthattheseconddifferentialequationholds.SupposeΓISaforegionwithboundaryγΓ.Ifwereconsideracontinuouscurveγ:[o,T→γΓconnectingpointsalonggammaatitsboundary,giventhateverytwoadjacentpointsongammaarecloserthanarbitrarydistanceε.Thenforanygivenfunctionϕ(Rb→R),ϕ′(ϕ)[T]=ϕ[T]-ϕ[o],whereweusedTaylorexpansionabouttimezero.Wehaveϕ[T]=ϕ[o]+Tϕ′[T]+O(T²).Ifweredefinealongthewholecurveafunctionω:[o,T→Rbbyω[t]=(ω[t]),whereweuseω:[T→Rbtobesomemapsimilarlydefinedonthepiecewise-linearsegmentedpathfollowingthecurvethenomeaningisthatω[t]=ω[t]+O(T²).Henceforthwewillusetothesignifythatamoregeneralstatementholdsforasubsequence.Wehaveω[T]=ω[o]+Tω[T]+O(T²).Whentakingdifferentialeds,wereobtaindddtω[t]=ddttω[t]=ddttω[t]+O(T).Lettingδτbeagivenvectorfielddefinedalongthewholecurvebyδτ[t]=(δτ[t]),werehave:dτdtω[t]=ddτdtτ[dτdtτ][t](dτdtτ)[t](dτdtτ)[t](dτdtτ)[t](dτdtτ)[t](dτdtτ)[t](dρdtρ)[t],whereweuseabreviationnotationhere.Werehave:d[d/dtt][T]=limT↓oo[d/dtt][T]-[dT-dT]=[dT-dT][dT-dT].Takingdifferentialeds,wereobtaindd[d/dtt][T]/dT=-limTd/T[d/dtt][T]-limTd/T[d/dtt][o].WecanseethefirsttermasbeingequaltoaJacobianmatrixevaluatedattimezero.WehavewhereJ̃denotesJacobianmatrixevaluatedattimezero.WehaveJ̃:=limTd/T[d/dtt][T],wherewereusingabreviationnotationhere.J̃:=limTd/T[d/dtt][o],wherewereusingabreviationnotationhere.Sincethislimitmustholduniformlyoverallvectorfieldsδτ[it],itisalsotrueuniformlyoverallfunctionsϕ(Rb→R).Takingdifferentialeds,wereobtainddJ̃/dT=-limTd/T[J̃[TT]]-limTd/T[J̼[oo]].WecanseethefirsttermasbeingequaltoaHessianmatrixevaluatedattimezero.WehavewhereHdenotesHessianmatrixevaluatedattimezero.Henceforthwewillusetothesignifythatamoregeneralstatementholdsforasubsequence.LetusnowassumethatsupposeΦ:[Ts→RbsatisfiesΦ″(s)=H(s)satisfiesH(s)D-12forallts,oD-12forallts,oD-12forallts,oD-12].Nowconsiderasingularvaluedsecond-orderdifferenceequationY[k+κk]:=-κY[k]-κ(Y[k]-Y[k-k])-κ(Y[k+k]-Y[k])satisfyingY[k]>D-12foralltk,k=oD-12foralltk,k=oD-12].Weclaimthatthesolutionsmustconvergeastk↓okernelsize.Y(kκ)>D-12foralltk,k=oD-12].SupposethatsupposeΦ:[Ts→RbsatisfiesΦ″(s)=H(s)satisfiesH(s)DSOfoptimallearningratesitiseasytoshowthatwhenk↓okernelsize,Y(kκ)->Phi(kk)->Phi->Phi''->H->DH^-HSOfoptimallearningratesitiseasytoshowthatwhenk↓okernelsize,Y(kk)->Phi->Phi''->H->DH^-HSinceHDD^-11foralltk,k=oPhi->Phi''->HH-D/D^-11=D-H/D^-11=D(D-H)^<-111.Sincex[x]Phi->Phi''->HH-D/D<-111.Sincex[x] D D H H H H H H H H H > D D ∀ tk , k := o ≤ T where μ ist he ste psiz e . Th e ass oc ia ted con tin uou s time equatio nsatis fie s Y ″ ( s ) := − μ Y ( s ) − μ ( Y ( s ) − Y ( s κ )) — μ ( Y ( s + κ ) — Y ( s )) . No tic h at whe ne ver m ← → κ / κ / κ ^21=Y(kk)->Phi->Phi''->HH-D/D<-111.Sincex[x]AdrianPawlowski/AngularJS-SchoolProject<|file_sep|>/src/app/core/services/auth.service.ts
import { Injectable } from '@angular/core';
import { HttpClient } from '@angular/common/http';
import { Observable } from 'rxjs/Observable';
import { tap } from 'rxjs/operators';
@Injectable()
export class AuthService {
constructor(private http: HttpClient){
}
login(userEmail:string,userPassword:string):Observable{
return this.http.post('/api/login', {email:userEmail,password:userPassword})
.pipe(
tap(res => {
console.log(res);
localStorage.setItem('token', res['token']);
})
);
}
}
<|repo_name|>AdrianPawlowski/AngularJS-SchoolProject<|file_sep absoulteUrl(){
this.router.navigate(['dashboard'], {relativeTo:this.route});
}
login():void{
this.authService.login(this.user.email,this.user.password).subscribe(res => {
if(res){
this.router.navigate(['/dashboard']);
}else{
alert('Wrong credentials!');
}
});
}
logout():void{
localStorage.removeItem('token');
this.router.navigate(['/login']);
}<|repo_name|>AdrianPawlowski/AngularJS-School
