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 Posted: Sat Jul 05, 2008 12:07 pm Post subject: What is a Dimension Anyway?: Scientific American |
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Timothy Golden BandTech
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| Posted: Sat Jul 05, 2008 12:07 pm Post subject: Re: What is a Dimension Anyway?: Scientific American |
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On Jun 25, 4:37 pm, Roger Bagula <rlbag...@sbcglobal.net> wrote:
[quote]http://www.sciam.com/article.cfm?id=what-is-a-dimension-anyway&print=....
SciAm.com logo
Features - June 25, 2008
What is a Dimension Anyway?
This story is a supplement to the feature "Using Causality to Solve the
Puzzle of Quantum Spacetime" which was printed in the July 2008 issue of
Scientific American.
A Whole New Dimension to Space
In everyday life the number of dimensions refers to the minimum number
of measurements required to specify the position of an object, such as
latitude, longitude and altitude. Implicit in this definition is that
space is smooth and obeys the laws of classical physics.
But what if space is not so well behaved? What if its shape is
determined by quantum processes in which everyday notions cannot be
taken for granted? For these cases, physicists and mathematicians must
develop more sophisticated notions of dimensionality. The number of
dimensions need not even be an integer, as in the case of
fractals—patterns that look the same on all scales.
Cantor Set : Take a line, chop out the middle third and repeat ad
infinitum. The resulting fractal is larger than a solitary point but
smaller than a continuous line. Its Hausdorff dimension [see next page]
is 0.6309.
Sierpinski Gasket: A triangle from which ever smaller subtriangles have
been cut, this figure is intermediate between a one-dimensional line and
a 2-D surface. Its Hausdorff dimension is 1.5850.
Menger Sponge: A cube from which subcubes have been cut, this fractal is
a surface that partially spans a volume. Its Hausdorff dimension is
2.7268, similar to that of the human brain.
Generalized Definitions Of Dimensions
Hausdorff Dimension
Formulated by the early 20th-century German mathematician Felix
Hausdorff, this definition is based on how the volume, V, of a region
depends on its linear size, r. For ordinary three-dimensional space, V
is proportional to r3. The exponent gives the number of dimensions.
“Volume” can also refer to other measures of total size, such as area..
For the Sierpi´nski gasket, V is proportional to r1.5850, reflecting the
fact that this figure does not even fully cover an area.
Spectral Dimension
This definition describes how things spread through a medium over time,
be it an ink drop in a tank of water or a disease in a population. Each
molecule of water or individual in the population has a certain number
of closest neighbors, which determines the rate at which the ink or
disease diffuses. In a three-dimensional medium, a cloud of ink grows in
size as time to the 3/2 power. In the Sierpi´nski gasket, ink must ooze
through a twisty shape, so it spreads more slowly—as time to the 0.6826
power, corresponding to a spectral dimension of 1.3652.
Applying the Definitions
In general, different ways to calculate the number of dimensions give
different numbers, because they probe different aspects of the geometry.
For some geometric figures, the number of dimensions is not fixed. For
instance, diffusion may be a more complicated function than time to a
certain power.
Quantum-gravity simulations focus on the spectral dimension. They
imagine dropping a tiny being into one building block in the quantum
spacetime. From there the being walks around at random. The total number
of spacetime building blocks it touches over a given period reveals the
spectral dimension.
[/quote]
The trouble with these interpretations is already exposed in this
brief. Within the description of spectral dimension we see the
statement
"In a three-dimensional medium, a cloud of ink grows in size as
time to the 3/2 power."
Thus the word dimension is being used to mean different things, some
of which are more fundamental than others. Even in standard usage if
we ask for the dimensions of a box we are not likely going to get the
answers two or three back. Instead we are likely to get back a series
of continuum based values. So our usage of this word is pretty badly
flawed since we are using it to mean two complementary things, where
the discrete and continuous properties within our space description
are being treated with the same term.
Here is a construction which stays within the traditional discrete
dimensional context yet allows an interpretation of more or less
space:
http://bandtechnology.com/ConicalStudy/conic.html
To state the level of consumption of a space is close, but this
construction poses that we can have as much of an n dimensional space
as we would like.
- Tim |
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Roger Bagula
Guest
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| Posted: Sun Jul 13, 2008 5:01 pm Post subject: Re: What is a Dimension Anyway?: Scientific American |
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Timothy Golden BandTechnology.com wrote:
[quote]
The trouble with these interpretations is already exposed in this
brief. Within the description of spectral dimension we see the
statement
"In a three-dimensional medium, a cloud of ink grows in size as
time to the 3/2 power."
Thus the word dimension is being used to mean different things, some
of which are more fundamental than others. Even in standard usage if
we ask for the dimensions of a box we are not likely going to get the
answers two or three back. Instead we are likely to get back a series
of continuum based values. So our usage of this word is pretty badly
flawed since we are using it to mean two complementary things, where
the discrete and continuous properties within our space description
are being treated with the same term.
Here is a construction which stays within the traditional discrete
dimensional context yet allows an interpretation of more or less
space:
http://bandtechnology.com/ConicalStudy/conic.html
To state the level of consumption of a space is close, but this
construction poses that we can have as much of an n dimensional space
as we would like.
- Tim
Your conical diagrams / picture remind me of a scientific American[/quote]
article a friend sent me several years back on Thom>s Catastrophe theory
( Sci Am Arr 1976):
http://en.wikipedia.org/wiki/Catastrophe_theory
"In mathematics, catastrophe theory is a branch of bifurcation theory in
the study of dynamical systems."
Specifically the Swallow tail:
F(x)=x^5/5 -c*x^3/2-b*x^2/2-a*x
Swallowtail catastrophe
V = x^5 + ax^3 + bx^2 + cx \,
The control parameter space is three dimensional. The bifurcation set in
parameter space is made up of three surfaces of fold bifurcations, which
meet in two lines of cusp bifurcations, which in turn meet at a single
swallowtail bifurcation point.
As the parameters go through the surface of fold bifurcations, one
minimum and one maximum of the potential function disappear. At the cusp
bifurcations, two minima and one maximum are replaced by one minimum;
beyond them the fold bifurcations disappear. At the swallowtail point,
two minima and two maxima all meet at a single value of x. For values of
a>0, beyond the swallowtail, there is either one maximum-minimum pair,
or none at all, depending on the values of b and c. Two of the surfaces
of fold bifurcations, and the two lines of cusp bifurcations where they
meet for a<0, therefore disappear at the swallowtail point, to be
replaced with only a single surface of fold bifurcations remaining.
Salvador Dalí's last painting, The Swallow>s Tail, was based on this
catastrophe. |
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Andor
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| Posted: Tue Jul 15, 2008 12:33 pm Post subject: Re: Simple question - non-linear filters |
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[quote]One interesting non-linear recursion that I once came up with is this:
y[n] = y[n-1] (2 - x[n] y[n-1]).
DSP riddle: what is the output y[n] in relation to the input x[n]?
Assume that the sequence x[n] lies in between 0 and 1. Hint: the
"filter" only works well for slowly varying input sequences x[n].
[/quote]
You are all linear sissies :-)! I>m crossposting this to
sci.nonlinear, perhaps they have an idea. |
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dbell
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| Posted: Tue Jul 15, 2008 7:53 pm Post subject: Re: Simple question - non-linear filters |
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On Jul 15, 8:33 am, Andor <andor.bari...@gmail.com> wrote:
[quote]One interesting non-linear recursion that I once came up with is this:
y[n] = y[n-1] (2 - x[n] y[n-1]).
DSP riddle: what is the output y[n] in relation to the input x[n]?
Assume that the sequence x[n] lies in between 0 and 1. Hint: the
"filter" only works well for slowly varying input sequences x[n].
You are all linear sissies :-)! I>m crossposting this to
sci.nonlinear, perhaps they have an idea.
[/quote]
Andor,
Your answer is that this implements the reciprocal function.
y[n] may approach/converge to the reciprocal of x[n] for sufficiently
slowly varying signals and suitable initial conditions on y[n].
Definitely don>t want ever want y[n] =0 (for any n) for it to work.
It can also work for some values of x[n] >1.
This is not a complete analysis, but I think it answers the question.
Dirk |
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Andor
Guest
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| Posted: Tue Jul 15, 2008 8:39 pm Post subject: Re: Simple question - non-linear filters |
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Dirk wrote:
[quote]On Jul 15, 8:33 am, Andor wrote:
One interesting non-linear recursion that I once came up with is this:
y[n] = y[n-1] (2 - x[n] y[n-1]).
DSP riddle: what is the output y[n] in relation to the input x[n]?
Assume that the sequence x[n] lies in between 0 and 1. Hint: the
"filter" only works well for slowly varying input sequences x[n].
You are all linear sissies :-)! I>m crossposting this to
sci.nonlinear, perhaps they have an idea.
Andor,
Your answer is that this implements the reciprocal function.
[/quote]
Aha, Dirk is a man with DSP in his heart! The answer is correct,
although not the first one. About one hour ago I recieved the first
correct entry off the list.
[quote]
y[n] may approach/converge to the reciprocal of x[n] for sufficiently
slowly varying signals and suitable initial conditions on y[n].
Definitely don>t want ever want y[n] =0 (for any n) for it to work.
It can also work for some values of x[n] >1.
[/quote]
Yes, indeed. However, when I tested this recursion I found that for
some sequences it would blow skyhigh. As you say, it works if x[n]
"varies slowly".
[quote]
This is not a complete analysis, but I think it answers the question.
[/quote]
The question of its stability is interesting. This seems to depend on
the frequency content of the input sequence x[n]. I never did a
thorough analysis myself because I never got to the point where the
filter would have been useful to me (modern DSPs have a reciprocal
instruction for seeding a Newton iteration algorithm that works more
accurate, stable and faster).
Regards,
Andor |
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Timothy Golden BandTech
Guest
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| Posted: Fri Jul 18, 2008 3:12 pm Post subject: Re: What is a Dimension Anyway?: Scientific American |
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On Jul 13, 10:28 am, Roger Bagula <rlbag...@sbcglobal.net> wrote:
[quote]Timothy Golden BandTechnology.com wrote:
The trouble with these interpretations is already exposed in this
brief. Within the description of spectral dimension we see the
statement
"In a three-dimensional medium, a cloud of ink grows in size as
time to the 3/2 power."
Thus the word dimension is being used to mean different things, some
of which are more fundamental than others. Even in standard usage if
we ask for the dimensions of a box we are not likely going to get the
answers two or three back. Instead we are likely to get back a series
of continuum based values. So our usage of this word is pretty badly
flawed since we are using it to mean two complementary things, where
the discrete and continuous properties within our space description
are being treated with the same term.
Here is a construction which stays within the traditional discrete
dimensional context yet allows an interpretation of more or less
space:
http://bandtechnology.com/ConicalStudy/conic.html
To state the level of consumption of a space is close, but this
construction poses that we can have as much of an n dimensional space
as we would like.
- Tim
Your conical diagrams / picture remind me of a scientific American
article a friend sent me several years back on Thom>s Catastrophe theory
( Sci Am Arr 1976):http://en.wikipedia.org/wiki/Catastrophe_theory
"In mathematics, catastrophe theory is a branch of bifurcation theory in
the study of dynamical systems."
Specifically the Swallow tail:
F(x)=x^5/5 -c*x^3/2-b*x^2/2-a*x
Swallowtail catastrophe
V = x^5 + ax^3 + bx^2 + cx \,
The control parameter space is three dimensional. The bifurcation set in
parameter space is made up of three surfaces of fold bifurcations, which
meet in two lines of cusp bifurcations, which in turn meet at a single
swallowtail bifurcation point.
As the parameters go through the surface of fold bifurcations, one
minimum and one maximum of the potential function disappear. At the cusp
bifurcations, two minima and one maximum are replaced by one minimum;
beyond them the fold bifurcations disappear. At the swallowtail point,
two minima and two maxima all meet at a single value of x. For values of
a>0, beyond the swallowtail, there is either one maximum-minimum pair,
or none at all, depending on the values of b and c. Two of the surfaces
of fold bifurcations, and the two lines of cusp bifurcations where they
meet for a<0, therefore disappear at the swallowtail point, to be
replaced with only a single surface of fold bifurcations remaining.
Salvador Dalí's last painting, The Swallow>s Tail, was based on this
catastrophe.
[/quote]
Thanks for the reply Roger.
I>ve been writing to Richard Morris and he goes toward singularity
theory as well.
The superplane (inverse cone) has a simplicity about it that is
physically achievable and flat everywhere except at the vertex. It
does take on curves but how the mechanical nature of the material
influences the math makes that a specialized topic. If it can be a
source of dynamics then that would be wonderful.
The topic of dimension is really puzzling. I think our modern teaching
has us building up dimension from a one dimensional structure upward
so that we think of 3D space as RxRxR. This could be a misnomer.
Plenty of physics challenges the Euclidean interpretation but within
that physics (and mathematics) we still build from the real line. I am
fairly certain that this is wrong and helps account for some of the
conundrums of physics. Time for instance is a unidirectional feature
and we do not witness any freedom to position ourselves in time. This
is a zero dimensional quality. So the idea that something more
primitive than one dimensional (with more dynamics than the Euclidean
point) exists in nature is ground to reconsider the entire puzzle. In
your singularity link the reliance upon functional analysis poses a
similar dependency and this could then bleed into calculus as well.
The phonomenon as I see it is merely a recurrence of n+1 dimensional
behaviors. I don>t mean to say that all of math must be wrong, but
that maybe there is another way to go about it. The chaos theory
people are doing just that so maybe we are in similar boats. The
question really simply is
What are we overlooking?
It seems difficult to believe that we are missing something
fundamental yet this superplane construction is an instance of such a
thing, so it can be taken as consistent that there are still more such
fundamental things which remain to be found.
- Tim
- Tim |
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Visa Schafer
Guest
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| Posted: Fri Nov 21, 2008 1:41 am Post subject: Visual pleasures of physics |
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Mäntyankeroinen (Bursaphelenchus xylophilus)
Oireet
Männyt kuolevat nopeasti. Pihkaneritys tyrehtyy. Latvukset tulevat
punaruskeiksi.
*Koska tämä mäntyankeroinenKIN tuntuu olevan ydinhallintomme määritelyssä
100% TABU on aihetta toki syytä pitää jatkuvasti esillä. Samoin kuin
maailmanlaajuinen tritiumsadevesilammikkoihinsa kuolevien matelijoiden ja
sammakoiden hätätilajulistukset ja muut vastaavat säteilyn mehiläistapot
toki! Näin tällaisen ilmon poistetun nettihausta:"Suomessa metsätaloutta
uhkaa eniten mäntyankeroinen, joka on luokiteltu vaaralliseksi ...
Mäntyankeroisen leviämisen riski Suomeen on varsin todellinen, ...!"
*Aika hurjaa tekstiä muuten, vai mitä ja SUPO/IAEA mitä ilmeisemmin kielsi
noin kaikenkertovan materiaalin julkaisun? .. .. No yhtä kaikki herättää
HETI epäilyksiä virannomaistemme ja ydinhallinnon mitä hämärimmistä
syy-yhteyskytköksistä!
Tuhonaiheuttaja
Tuskin paljain silmin erottuva nematodi eli sukkulamato (alle 1 mm).
Mäntyankeroinen on hyvin pieni, aikusena alle 1 mm pitkä, läpikuultava
sukkulamato. Ennen aikuistumista on neljä toukkamaista kehitysastetta,
elelevät pääasiassa mäntyjen pihkatiehyeissä. Lisääntyminen
optimioloissa on hyvin vilkasta ja saastunut puu kuolee nopeasti.
Mäntyankeroinen voi levitä kuolevaan puuhun
*Äärimmäisen tärkeä ydinkytkös. Kun jo viime kesänä ilmoitettiin mm. Ranskan
metsien kuolevan kesäisin enemmän kuin kasvavan säteilyn tuhottua
metsänpohmjien ionitasapainot on pöytä jo katettu Portugalista, Siperiasta
vääjäämättä lähestyville mäntyankeroisille. Kiitos ydinvoimateollisuutemme
kasvien luontainen vastustus on nyt nollissa, pon apetiit!
lisääntymään tulleiden
hyönteisten välityksellä. Mäntyankeroisen arvellaan olevan kotoisin
Pohjois-Amerikasta, jossa mäntylajit ovat kehittyneet melko
vastustuskykyisiksi. Kotoinen mäntymme on herkkä mäntyankeroiselle.
*Bulseye! Mihin siis tarvitaan jatkossa ydinsähköämme, jota suunnitellaan jo
härskisti 60v periodeille?Ei siis mikään ihme, että ydinherroillamme on
kiirus ydinmmyllyineen, ennen kun kansa tajuaa ydinjymäytyksen suuruudet.
nou metasää, ja ei tarvetta ydinvoimalle, tämä setti muuten pitää ja toimii!
Mäntyankeroinen on hyvin vaarallinen tuholainen ja aiheuttaa vahinkoa
etenkin levittyään alueille, missä mäntylajit eivät ole ehtineet kehittää
vastustuskykyä, kuten Japanissa ja Kiinassa. Euroopassa laji on tavattu
Portugalista. Suomessa lajia ei ole havaittu luonnossa, mutta saastuneilta
alueilta tulleessa puutavarassa ja pakkausmateriaalissa sitä on tavattu.
*Itse asiassa jo 40 invaasiota (-08). Ja lisää tulee kunnes metsämme menee.
Ydinalamme tietää tämän jo NYT!
Kehitys
Lisääntyminen optimioloissa (25 - 30 °C) on hyvin nopeata ja kehitys munasta
aikuiseksi kestää vain kolmisen vuorokautta. Ankeroinen vaatii suhteellisen
korkeata lämpötilaa (yli 20°C) voidakseen kehittyä, mutta kykenee säilymään
hengissä useita kuukausia -17 °C lämpötilassa.
*Lokoisaa ruoka-apetta Suomessa silmän kantamattomiin ankeroismadoille. Ja
vauhti leviämistarpeelle on toki silmitön. Mihin tarvitsemme ydinvoimaa
pian, tai edes NYT? Ehkä ydinvoiman voisi valjastaa nykyisen 70% meren
kiehutuksensa lisäksi 100% energian haaskaamisiin ja pelkiksi
kampamaneettikasvattamoiksi läpi vuoden!) Jaa niin 05.06.-08 YLE kauhisteli,
että kasvihuonekaasuja tulee eniten kuin kansa viemäröi litroittain kuumaa
käsienpesuvettä ja kummuttaa sitten Itämerestä metaanit, typpioksiduulit ja
ties mitkä hiilipäästöt.
*Toki ydinvoimaloittemme tuhansien ja tuhansien ja tuhansien megawattien
lauhdekumpuamisista ydintyyliin taas 100% vaiettiin? Miksi ihmeessä ei
totuus kelpaa. Ydinvoima esim. TVO:n lauhteissa kun todistetusti tuottaa
kummuttaen metaanisedimenttikerrostumia Itämerestä tauotta jo näillä
kahdella voimalallaan 5 000MW hiilipolton edestä! Miksei näitä tutkita
julkisuuteen? Ai niin, kun "-08 alkuun hämäräperäisesti" Ilmatieteenlaitos
lopetti koko Suomen keskeisimmät merentutkimuslaitoksen arandoinnit
ydinlauhdevesissä, kun alkoi ydinkytkösluu paistaa!!
Leviäminen
Vaikka ankeroiset voivat säilyä hengissä kuolleella puulla vuosikausia,
tarvitsevat ne siirtäjäeliön eli vektorin kuljettamaan ne uuteen
lisääntymispaikkaan lähistön terveisiin mäntyihin. Vektoreina toimivat
sellaiset hyönteiset, jotka iskeytyvät lisääntymään kuoleviin puihin ja
kaivautuvat puuaineeseen, näin ankeroiset voivat niihin tunkeutua. Lisäksi
hyönteisten täytyy aikuistuttuaan käydä nakertamassa terveiden mäntyjen
vuosikasvaimia, oksia, runkoa tai juuria, jotta ankeroiset pääsisivät uuden
puun solukoihin. Mahdollisia vektoreita Suomessa ovat pääasiassa
Monochamus-suvun sarvijäärät: suutari ja ranskanräätäli, mutta muidenkin
hyönteisten arvellaan voivan toimia satunnaisesti vektoreina.
Tuhokohteet
Mäntyankeroisen tuhoja on esiintynyt kotoisilta puulajeilta vain männyllä,
joka on varsin altis tuhoille. Muualla maailmassa ankeroisen on todettu
tuhonneen erityisesti sylvestris-ryhmän mäntyjä: punamäntyjä (Pinus
resinosa), japaninmustamäntyjä (P. thunbergii), japaninpunamäntyjä (P.
densiflora) ja P. luchuensis -mäntyjä. Lisäksi se on todettu lukuisilta
mänty-, kuusi- ja muilta havupuulajeilta.
Tuhon eteneminen
Suotuisten olosuhteiden vallitessa mäntyankeroisten määrä
kymmentuhatkertaistuu parin viikon aikana.
*On siinä ydinaavikoituvasa Suomessamme kohta rapinaa salomaillamme!
Uusien puiden valtaamiseen se
pystyy kuitenkin vain tietyssä elinkiertonsa vaiheessa (ns. kestotoukka).
Lisäksi se vaatii levitäkseen siirtäjäeliön, jona toimii useimmiten jokin
Monochamus-suvun jäärä. Ankeroisten lisääntymistä suosii kuiva ja lämmin
kesä.
Vaikutus puuhun
Mäntyankeroiset tukkivat lisääntyessään puun tiehyet. Ne valtaavat
saastuttamansa puun kokonaan siirtyessään puun sisällä pihkatiehyeiden
kautta niin oksiin, runkoon kuin juuristoonkin. Ankeroisten aiheuttaman,
usein nopean, puiden kuolemisen syytä ei aivan tarkkaan vielä tiedetä.
Vektorien mukana kulkeutuu usein myös sienitauteja ja pihkanerityksen
heikentyessä puut altistuvat monille muille tuhoeliöille. Lakastuminen
saattaakin olla seurausta useiden eri tekijöiden yhteisvaikutuksesta.
Elinkierto
Vektorit
Ankeroisten tunnistaminen on vaikeata koska kuolevassa ja kuolleessa puussa
suorastaan kuhisee erilaisia ankeroisia.
Vahingot metsätaloudessa
Mäntyankeroisesiintymät vaativat aina välittömiä toimia. Mäntyankeroinen
pystyy lisääntymään ja tappamaan puita nopeasti. Ankeroinen tappaa Japanissa
vuosittain miljoonia kuutiometrejä puuta.
*90% metsästään katosi muutamissa vuosissa Siperian pakkasista ja tuulista
huolimatta läpi maan.
Tuhoriskin arviointi
Sääolot, ainakin Etelä-Suomessa, mahdollistavat mäntyankeroisen
lisääntymisen ja kotoinen mäntymme on todettu alttiiksi ankeroiselle. Vaara
mäntyankeroisen leviämiselle Suomeen on melko suuri,
*Saivartelua, koska leviäminen on SELVIÖ!
koska kaikkia
puutavaraeriä ja pakkausmateriaaleja saastuneilta alueilta ei pystytä
tarkastamaan. Lisäksi on mahdollista, että Aasiasta peräisin olevan
puutavaran mukana kulkeutuvat ankeroiset pääsisivät vakiintumaan
lähialueille. Lämmin ilma ja kuivuus altistavat puita.
*Ankeroinen trodellisuudessa lähestyy Suomea jo kahdelta maayhteydeltä
ongelmitta. Metsäteollisuus ja ydinala ankeroinen tulee, oletteko valmiina
pysäyttämään koneenne!?***
Torjuntamenetelmät
Tehokasta keinoa ankeroisen torjumiseksi ei ole toistaiseksi pystytty
kehittämään.
*Osuma ja upotus! Miksi haaskata ydintekniikkaan, jolle ei ole edes jatkossa
TARVETTA!!
Toimivimmaksi on osoittautunut vektorihyönteisten hävittäminen
ja saastuneiden puiden polttaminen ennenkuin vektorit poistuvat niistä.
Suomessa tiukat tuontipuurajoitukset ovat paras keino leviämisen
estämiseksi. Tuontipuu saastuneilta alueilta tulee olla kuumakäsiteltyä 56°C
30 minuutin ajan siten, että käsittely ulottuu puun keskustaan asti.
Mäntyankeroinen on Euroopan ja Välimeren Kasvinsuojelujärjestön
karanteenilistoilla (European and Mediterranean Plant Protection
Organization, EPPO). Lisäksi EU:lla on säännökset sen leviämisen
ehkäisemiseksi. Mikäli epäillään mäntyankeroisen levinneen metsään, on syytä
ottaa yhteyttä Kasvintuotannon tarkastuskeskukseen, missä asia pystytään
varmistamaan.
Teksti: Kuvat: Lähteet:
Pouttu, A.
Tomminen, J. Tomminen, J.
Kyushu Research Center
EPPO
Pouttu, A. Tomminen, J. ja Nuorteva, M. (1987)
EU:n lainsäädäntö
05.08.2003 / APou |
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