on the general linear group and hochschild homology our main result here is a rational computation of the homology of the adjoint action of the infinite general linear group of an arbitrary ring before stating the result we establish some notation and conventions rings are associative and with unit if a is a ring then gl a uk 0glk a is its infinite general linear group an a bimodule is an abelian group b which is both a left a module and a right a module and satisfies alb a2 al ba2 for ai ea be b for example b a if b is an a bimodule then m b uk omk b is the infinite additive group of matrices with entries in b conjugation defines an action the adjoint action of gl a on m b note that an a q bimodule is just an a bimodule which is also a rational vector space if b is an a q bimodule then hj a c q b denotes the hochschild homology of a q with coefficients in b our main result it appears in slightly more detailed form as theorem v 3 is

stochastic phase resetting of two coupled phase oscillators stimulated at different times a model of two coupled phase oscillators is presented where the oscillators are subject to random forces and are stimulated at different times transient phase dynamics synchronization and desynchronization which are stimulus locked i e tightly time locked to a repetitively administered stimulus are investigated complex coordinated responses in terms of a noise induced switching across trials between qualitatively different responses may occur when the two oscillators are reset close to an unstable fixed point of their relative phases this can be achieved with an appropriately chosen delay between the two stimuli the switching of the responses shows up as a coordinated cross trial ct response clustering of the oscillators where the two oscillators produce two different pairs of responses by varying noise amplitude and coupling strength we observe a stochastic resonance and a coupling mediated resonance of the ct response clustering respectively the presented data analysis method makes it possible to detect such processes in numerical and experimental signals its time resolution is enormous since it is only restricted by the time resolution of the preprocessing necessary for extracting the phases from experimental data in contrast standard data analysis tools applied across trials relative to stimulus onset such as ct averaging where an ensemble of poststimulus responses is simply averaged ct standard deviation and ct cross correlation fail in detecting complex coordinated responses and lead to severe misinterpretations and artifacts the consequences for the analysis of evoked responses in medicine and neuroscience are significant and are discussed in detail

stochastic resonance noise enhanced phase coherence we study stochastic resonance in periodically driven stochastic bistable systems in terms of phase synchronization by introduction of an instantaneous phase for the output we show explicitly the effect of phase locking between the input and output the stochastic dynamics of the phase difference between input and output appears to be similar to that of synchronized classical self sustained oscillators the degree of phase coherence is estimated by employing the effective diffusion constant for the phase difference this coherence becomes maximal for optimal noise intensities however phase synchronization effects can only be observed for sufficiently large magnitude of the periodic inputting signal

hochschild and cyclic homology of hypersurfaces the object of this paper is to compute the hochschild homology and cohomology of the ring a k x x f where k is an arbitrary commutative ring with unit andf is a polynomial in k x x we also compute the cyclic homology for this ring when k 2 q f is homogeneous and the cone it defines is not singular but at the origin the paper is divided into four sections in the first section a quick review of some basic notions in the hochschild and cyclic theories is made see for instance m chap 10

contragredient equivalence a canonical form and some applications abstract let a and c be m by n complex matrices and let b and d be n by m complex matrices the pair a b is contragrediently equivalent to the pair c d if there are square nonsingular complex matrices x and y such that xay 1 c and ybx 1 d contragredient equivalence is a common generalization of four basic equivalence relations similarity consimilarity complex orthogonal equivalence and unitary equivalence we develop a complete set of invariants and an explicit canonical form for contragredient equivalence and show that a a t is contragrediently equivalent to c c t if and only if there are complex orthogonal matrices p and q such that c paq using this result we show that the following are equivalent for a given n by n complex matrix a 1 1 a qs for some complex orthogonal q and some complex symmetric s 2 2 a t a is similar to aa t 3 3 a a t is contragrediently equivalent to a t a 4 4 a q 1 a t q 2 for some complex orthogonal q 1 q 2 5 5 a pa t p for some complex orthogonal p we then consider a linear operator on n by n complex matrices that shares the following properties with transpose operators for every pair of n by n complex matrices a and b a preserves the spectrum of a b a a and c ab b a we show that a a is contragrediently similar to b b if and only if a x 1 bx 2 for some nonsingular x 1 x 2 that satisfy x 1 1 x 1 and x 1 2 x 2 we also consider a factorization of the form a xy where x 1 x and y y we use the canonical form for the contragredient equivalence relation to give a new proof of a theorem of flanders concerning the relative sizes of the nilpotent jordan blocks of ab and ba we present a sufficient condition for the existence of square roots of ab and ba and close with a canonical form for complex orthogonal equivalence

cyclic homology and equivariant homology the purpose of this paper is to explore the relationship between the cyclic homology and cohomology theories of connes 9 11 see also loday and quillen 20 and if equivariant homology and cohomology theories here ii is the circle group the most general results involve the definitions of the cyclic homology of cyclic chain complexes and the notions of cyclic and cocyclic spaces so precise statements will be postponed until w 3 in this introduction we explain some of the formal similarities between the cyclic theory and the equivariant theory and give two examples where the general results apply let a be an associative algebra over a commutative ring k then one can form the cyclic homology hc a and cohomology hc a of a these groups have periodicity operators

synchronization of stochastic bistable systems by biperiodic signals we study the nonlinear response of a noisy bistable system to a biperiodic signal through experiments with an electronic circuit schmitt trigger the signal we use is a biharmonic one i e a superposition of low and high frequency harmonic components it is shown that the mean switching frequency msf of the system can be locked at both low and high frequencies moreover the phenomenon of msf locking at the lower frequency can be induced and enhanced by the higher frequency excitation thus high frequency bias can control synchronization at the low frequency

phase synchronization and stochastic resonance effects in the crayfish caudal photoreceptor we study the nonlinear response of the crayfish caudal photoreceptor to periodic mechanical stimuli in terms of stochastic synchronization the amplitude and frequency of the mechanical stimuli and the light level are used as control parameters the system shows multiple locking regions as the stimulus frequency is varied we find that the synchronization index increases as the signal to noise ratio snr of the periodic drive in response to increasing light levels this effect exhibits features similar to stochastic resonance we demonstrate a nonlinear rectification effect in which the snr of the second harmonic of the input stimulus increases as the light level is raised and show that the corresponding synchronization index increases as the snr of the second harmonic

array enhancement of stochastic synchronization and signal to noise ratio gain in the nonlinear regime of signal transmission the nonlinear transformation of an external noisy signal by an array of noninteracting elements with internal noise is considered the array simulates a neuronal system that processes spike trains it is shown that increasing the number of array elements entails significant extending of the stochastic synchronization region and improvement of the signal to noise ratio snr the effects are demonstrated for arrays of triggers overdamped bistable oscillators and fitzhugh nagumo systems the interrelation between snr improvement and the efficiency of information processing is discussed

experimental study of stochastic resonance in a chua s circuit operating in a chaotic regime abstract we present results of an experimental study of stochastic resonance in an electronic chua s circuit whose dynamics switches between two different stable chaotic attractors when it is driven by a periodic signal and a gaussian white noise due to the internal dynamics of the attractors the minimum amplitude for the external forcing to induce jumps strongly depends on the external frequency we determine from the fourier transform of the output signal the amplification factor of the input signal and study its dependence on the external frequency and the noise intensity we show that the envelope of the distribution of switching times follows a gamma distribution typical from bistable systems and that the mean switching time decays exponentially with the noise intensity we propose a simple method for obtaining the optimal noise intensity from the residence and switching times probability distributions and show that it coincides with the value obtained from the maximum of the amplification factor

finite generation of hochschild homology algebras we prove converses of the hochschild kostant rosenberg theorem in particular if a commutative algebra s is flat and essentially of finite type over a noetherian ring mathbb k and the hochschild homology hh s mathbb k is a finitely generated s algebra for shuffle products then s is smooth over mathbb k

mac lane co homology of the second kind and wieferich primes abstract in this paper we investigate the connection between the mac lane co homology and wieferich primes in finite localizations of global number rings following the ideas of polishchuk positselski 29 we define the mac lane co homology of the second kind of an associative ring with a central element we compute these invariants for finite localizations of global number rings with an element w and obtain that the result is closely related to the wieferich primes to the base w in particular for a given non zero integer w the infiniteness of wieferich primes to the base w turns out to be equivalent to the following for any positive integer n we have h m l ii 0 z 1 n w q as an application of our technique we identify the ring structure on the mac lane cohomology of a global number ring and compute the adams operations introduced in this case by mccarthy 26 on its mac lane homology

cross correlation of instantaneous phase increments in pressure flow fluctuations applications to cerebral autoregulation we investigate the relationship between the blood flow velocities bfv in the middle cerebral arteries and beat to beat blood pressure bp recorded from a finger in healthy and post stroke subjects during the quasisteady state after perturbation for four different physiologic conditions supine rest head up tilt hyperventilation and co2 rebreathing in upright position to evaluate whether instantaneous bp changes in the steady state are coupled with instantaneous changes in the bfv we compare dynamical patterns in the instantaneous phases of these signals obtained from the hilbert transform as a function of time we find that in post stroke subjects the instantaneous phase increments of bp and bfv exhibit well pronounced patterns that remain stable in time for all four physiologic conditions while in healthy subjects these patterns are different less pronounced and more variable we propose an approach based on the cross correlation of the instantaneous phase increments to quantify the coupling between bp and bfv signals we find that the maximum correlation strength is different for the two groups and for the different conditions for healthy subjects the amplitude of the cross correlation between the instantaneous phase increments of bp and bfv is small and attenuates within 3 5 heartbeats in contrast for post stroke subjects this amplitude is significantly larger and cross correlations persist up to 20 heartbeats further we show that the instantaneous phase increments of bp and bfv are cross correlated even within a single heartbeat cycle we compare the results of our approach with three complementary methods direct bp bfv cross correlation transfer function analysis and phase synchronization analysis our findings provide insight into the mechanism of cerebral vascular control in healthy subjects suggesting that this control mechanism may involve rapid adjustments within a heartbeat of the cerebral vessels so that bfv remains steady in response to changes in peripheral bp

on ptak s derivation of the jordan normal form some readers of 1 might appreciate the following comments that make more explicit how ptak s beautiful insight there leads to a trivial proof of the basics of the jordan normal form the proof of theorem 1 of 1 can also be based on the observation that x being finite dimensional the sequence 0 ker a ker a2 must eventually be stationary i e ker a ker aq p for some q and all p 0 for such q let xr and xs be the range and the kernel respectively of a hence dim x dim xr dim xs further for any x xr xs x az for some z and so z ker a2q ker a hence x 0 therefore x is the direct sum of the two a invariant subspaces xs and xr and a is regular on xr since a is and nilpotent on xs in the setup and notation of theorem 2 of 1 there must be by duality some y0 in y for which x0aq 1 y0 0 hence the q order matrix x0aj 1 y0a q i i j 1 q is triangular with nonzero diagonal entries therefore invertible and this guarantees that x is the direct sum x0 x with x0 the linear span of x0aj 1 j 1 q and x the annihilator of y0a q i i 1 q both of which are ainvariant moreover it shows x0aj 1 j 1 q to be a basis for x0 and the matrix representation with respect to this basis of a restricted to x0 has the familiar form of a jordan block for the eigenvalue 0 now x being finite dimensional there are a invariant direct sum decompositions x x1 xm that are minimal in the sense that none of its summands is the direct sum of two nontrivial a invariant subspaces take any one such then the matrix representation for a with respect to any basis made up from bases for the summands xi is block diagonal with the ith block the matrix representation of the restriction ai of a to xi with respect to the chosen basis for xi assuming the underlying field to be algebraically closed the restriction ai of a to xi has some eigenvalue i and in view of the minimality of xi theorem 1 ensures