Introduction.Let H be a complex HILBERT space and let L(H) be the algebra of all bounded linear operators on H . Write I for the identity and 0 for the zero operator of L(H). Use similar notations with respect to other spaces.An idempotent SEL(H) is an operator which satisfies the equation S l = S . Assume, in addition, that S is non-trivial, i.e. that S i O , I , and that S is non-HERMITian if nothing else is specified. Then H has the decomposition H = kerS i r a n S , where ranS=ranS. \Vrite rpS for the range projection of S , and denote ReS and ImS its HERMITian components: ReS= ~ (8+8*), I m 8 = ~. (8-8*). The HERMITian components lead to the following CARTESian decomposition of S : S=ReS+iImS. -If S is an idempotent, then (1-5) is an idempotent, the complementary idempotent of S . Moreover, the adjoints S* and ( I -5)* are idempotents too. The involution corresponding to S is (2s -1). One has ( 2 5 -I)2 = I . If s is HERMITian, i.e. a projection, then (25-1) is a symmetry. Let BCL(H) be an arbitrary operator. Let a(B) denote the spectrum of B. I n particular, write up for the point spectrum, u, for the continuous spectrum and zz for the approximate point spectrum. Let r(B) denote the spectral radius and let j IBl 1 denote the norm of B : llB[I=H~ip {IlBfIl 1 ilfil= 1, f C H } . Moreover, let IBl denote the root (B*B)'12 and let B+ = ~ (/BI + B ) , B-= -( j B [ -B ) be the definite components of B. Observe also the decompositions H = kerB@ranB* = kerB* @ @ r a i i . Assume now that B is invertible and denote, in addition, by cond(B) the condition number of B : cond(B) =IIB[I ~~B -~~~, where IIB-lll=(inf {I[Bf[l I llf[l= 1, f c H } ) -* . Further, if B is Hermitian denote its lower bound m(B), where m(B) = =inf { ( B f , f ) I ]lfil= 1 , f C H } and its upper bound M ( B ) , where M ( B ) =sup {(Bf, f )
Correlations between the feeding activity of termites and solar influences. Time series conducted for several months of the average amounts of wood eaten daily by termites which were kept under constant conditions of nutrition, temperature, moisture and darkness were compared by means of a special method of harmonic analysis with the corresponding total daily values of the disturbancies of the earth's magnetic field as a characteristic of the changing solar activity. Definite correlations were found. Periods when the termites' feeding activity does not deviate from an average value calculated for some time generally coincide with extreme data of the magnetic disturbancies. It remains undecided whether the changes of the magnetic field alone or other solar influences affect the termites. A non‐mathematical examination of several test series carried out with termites from 1965 till 1970 showed extreme feeding data at 27‐to‐28‐day rhythms corresponding to the solar revolutions. A correlation can, however, more easily be established with the relative sun spot numbers than with the data of the magnetic disturbancies. Feeding maxima were pronounced with falling relative sun spot numbers and minima with rising ones. A comparison of the feeding activity with the influence of atmospherics and other meteorological factors which shall be dealt with in another paper also revealed a definite influence on the termites' activity. The interaction of solar influences and meteorological conditions may result in particularly pronounced extrema of termite activity. Zusammenfassung Zeitreihen über mehrere Monate, bestehend einerseits aus Tagesmittelwerten für die von Termitengruppen bei konstanten Ernährungs‐, Temperatur‐, Feuchtigkeits‐ und Dunkelheitsbedingungen gefressenen Holzmengen, andererseits den entsprechenden Tagessummenwerten für die Magnetische Unruhe als einem Merkmal der wechselnden Sonnenaktivität, wurden miteinander an Hand von speziellen Methoden der Harmonischen Analyse verglichen. Dabei ergaben sich eindeutige Zusammenhänge zwischen beiden Wertegruppen. In der Regel fallen Zeiten, in denen die Fraßaktivität der Termiten nicht von einem für längere Zeit errechneten Durchschnittswert abweicht, ungefähr mit Extremwerten der Magnetischen Unruhe zusammen. Es bleibt dahingestellt, inwieweit nur die Magnetfeldänderungen selbst oder auch andere solare Einflüsse auf die Termiten wirken. Eine nichtmathematische Betrachtung mehrerer Versuchsreihen mit Termiten aus den Jahren 1965 bis 1970 läßt jeweils einige 27‐ bis 28‐Tage‐Rhythmen bei Extremwerten der Fraßaktivität, entsprechend der Sonnenumdrehung, feststellen. Beziehungen sind zu den Sonnenflecken‐Relativzahlen leichter als zu den Werten der Magnetischen Unruhe zu erkennen. Maxima der Fraßaktivität treten besonders bei fallenden, Minima bei steigenden Sonnenflecken‐Relativzahlen auf. Nach einem Vergleich der Fraßmengen mit atmospherics und anderen Wetterfaktoren, der in einer weiteren Arbeit behandelt werden soll, haben auch diese einen eindeutigen Einfluß auf di...
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