mean square values of hecke l series formed with r th order characters let l be a number field containing the r th roots of unity starting with the rankin selberg convolution of a metaplectic eisenstein series on the r fold cover of gl 2 with itself we construct a dirichlet series defined over l whose coefficients involve the r th order twists of a fixed hecke l function we then observe that a group of functional equations can be naturally associated with this construction combining this with the convexity theorem for holomorphic functions of several complex variables we show that this object as a function of two complex variables admits meromorphic continuation to 2 as an application we obtain asymptotic formulae for mean square values of the r th order twists of an arbitrary hecke l function defined over l

monotonic continuous time random walks with drift and stochastic reset events in this paper we consider a stochastic process that may experience random reset events which suddenly bring the system to the starting value and analyze the relevant statistical magnitudes we focus our attention on monotonic continuous time random walks with a constant drift the process increases between the reset events either by the effect of the random jumps or by the action of the deterministic drift as a result of all these combined factors interesting properties emerge like the existence for any drift strength of a stationary transition probability density function or the faculty of the model to reproduce power law like behavior general formulas for two extreme statistics the survival probability and the mean exit time are also derived to corroborate in an independent way the results of the paper monte carlo methods were used these numerical estimations are in full agreement with the analytical predictions

linear transformations that preserve immanants abstract we characterize the linear transformations on matrices that preserve the immanants i e d t x d x for all x where is an irreducible character of degree greater than one on s n we present a necessary and sufficient condition for t to be invertible if it preserves the generalized matrix function d c and prove that the preservers of immanants are always nonsingular

spectral content of fractional brownian motion with stochastic reset we analyse the power spectral density psd s t f with t being the observation time and f the frequency of a fractional brownian motion fbm with an arbitrary hurst index undergoing a stochastic resetting to the origin at a constant rate r the resetting process introduced some time ago as an example of an efficient optimisable search algorithm to this end we first derive an exact expression for the covariance function of an arbitrary not necessarily a fbm process with a reset expressing it through the covariance function of the parental process without a reset which yields the desired result for the fbm in a particular case we then use this result to compute exactly the power spectral density for fbm for all frequency f the asymptotic large frequency f behaviour of the psd turns out to be distinctly different for sub and super diffusive fbms we show that for large f the psd has a power law tail where the exponent for sub diffusive fbm while for all thus somewhat unexpectedly the exponent in the superdiffusive case h 1 2 sticks to its brownian value and does not depend on h

linear groups associated with elements of a group algebra let c be an arbitrary nonzero function from sm into a field k if a aij is an m square matrix over k define dc a smc i 1m ai i we are concerned with the characterization of the group of nonsingular matrices a satisfying dc ax dc x for all m square matrices x over k the author previously defined 13 a galois connection between the set of left ideals of the group algebra k sm ordered by inclusion and the set of subgroups of the full linear group also ordered by inclusion we characterize the closed elements of by the closure operation arising from this connection

matrices with prescribed eigenvalues and principal submatrices abstract we give a necessary and sufficient condition for the existence of a square matrix with prescribed eigenvalues and prescribed complementary principal blocks

matrices with prescribed nonprincipal blocks and the trace function abstract let a and b be fixed matrices we discuss the existence of matrices u and v such that vau 1 ubv 1 is nonscalar and simultaneously has a prescribed trace

interlacing inequalities matrix groups abstract this paper has two parts 1 interlacing inequalities in linear algebra eigenvalues singular values invariant polynomials attempts to construct a unifying theory 2 subgroups of the full linear group defined by polynomial identities

quantum unique ergodicity for half integral weight automorphic forms we investigate the analogue of the quantum unique ergodicity que conjecture for half integral weight automorphic forms assuming the generalized riemann hypothesis grh we establish que for both half integral weight holomorphic hecke cusp forms for 0 4 lying in kohnen s plus subspace and for half integral weight hecke maa cusp forms for 0 4 lying in kohnen s plus subspace by combining the former result along with an argument of rudnick it follows that under grh the zeros of these holomorphic hecke cusp equidistribute with respect to hyperbolic measure on 0 4 h as the weight tends to infinity

diffusion with stochastic resetting at power law times what happens when a continuously evolving stochastic process is interrupted with large changes at random intervals ensuremath tau distributed as a power law ensuremath sim ensuremath tau ensuremath 1 ensuremath alpha phantom rule 0 28em 0ex ensuremath alpha g0 modeling the stochastic process by diffusion and the large changes as abrupt resets to the initial condition we obtain exact closed form expressions for both static and dynamic quantities while accounting for strong correlations implied by a power law our results show that the resulting dynamics exhibits a spectrum of rich long time behavior from an ever spreading spatial distribution for ensuremath alpha l1 to one that is time independent for ensuremath alpha g1 the dynamics has strong consequences on the time to reach a distant target for the first time we specifically show that there exists an optimal ensuremath alpha that minimizes the mean time to reach the target thereby offering a step towards a viable strategy to locate targets in a crowded environment

interlacing inequalities and control theory abstract given a p by p matrix a we solve the problem of the existence of a p by q matrix b such that a b has prescribed controllability indices and i p a b has prescribed invariant polynomials the solution of this problem together with an earlier theorem of the author s is used to provide a new proof of the sa thompson interlacing inequalities for invariant polynomials

on the spectrum of fractional brownian motions fractional brownian motions fbms provide useful models for a number of physical phenomena whose empirical spectra obey power laws of fractional order however due to the nonstationary nature of these processes the precise meaning of such spectra remains generally unclear two complementary approaches are proposed which are intended to clarify this point the first one based on a time frequency analysis takes into account the nonstationary nature of fbm and puts emphasis on time averaged measurements the second one based on a time scale analysis is matched to self similarity properties of fbm and reveals an underlying stationary structure relative to each time scaling

on quadratic families of cm elliptic curves given a cm elliptic curve with weierstrass equation y 2 f x and a positive definite binary quadratic form q v v we show that there are infinitely many reduced integer pairs u v such that the twisted elliptic curve q u v y 2 f x has analytic rank and consequently mordell weil rank one in fact it follows that the number of such pairs with u v x is at least x 2 e for any e 0

matrices with prescribed rows and invariant factors abstract in this paper we solve the problem of the existence of an n n matrix over an arbitrary field when its invariant polynomials and either some rows or columns are prescribed the solution is given in terms of invariant factor inequalities and of majorization inequalities involving controllability indices and the degrees of the invariant polynomials

optimal stochastic restart renders fluctuations in first passage times universal stochastic restart may drastically reduce the expected run time of a computer algorithm expedite the completion of a complex search process or increase the turnover rate of an enzymatic reaction these diverse first passage time fpt processes seem to have very little in common but it is actually quite the other way around here we show that the relative standard deviation associated with the fpt of an optimally restarted process i e one that is restarted at a constant nonzero rate which brings the mean fpt to a minimum is always unity we interpret further generalize and discuss this finding and the implications arising from it

on the interlacing property for singular values and eigenvalues abstract a new proof of the complete interlacing theorem for singular values is presented the technique used works equally well to obtain the interlacing theorem for eigenvalues of hermitian matrices

equality of decomposable symmetrized tensors and matrix groups let g be a subgroup of the full symmetric group sn and a character of g a matrix can be defined as an n n matrix b which satisfies dgx bx dgx x for every n n matrix x they form a multiplicative group denoted s g x which plays a fundamental role in the study of equality of two decomposable symmetrized tensors the main result of this paper theorems 2 2 2 3 and 2 4 is a complete description of the matrices in s g x this description has many consequences that we present there are also results on related questions

quadratic families of elliptic curves and unirationality of degree 1 conic bundles we study families of elliptic curves whose coefficients are degree 2 polynomials in a variable t all such curves together form an algebraic surface which is birational to a conic bundle with 7 singular fibers we prove that such conic bundles are unirational as a consequence one obtains that for infinitely many values of t the resulting elliptic curve has rank at least 1

aging wiener khinchin theorem and critical exponents of 1 f β noise the power spectrum of a stationary process may be calculated in terms of the autocorrelation function using the wiener khinchin theorem we here generalize the wiener khinchin theorem for nonstationary processes and introduce a time dependent power spectrum ensuremath langle s t m ensuremath omega ensuremath rangle where t m is the measurement time for processes with an aging autocorrelation function of the form ensuremath langle i t i t ensuremath tau ensuremath rangle t mathrm ensuremath upsilon ensuremath phi mathrm ea ensuremath tau t where ensuremath phi mathrm ea x is a nonanalytic function when x is small we find aging 1 f ensuremath beta noise aging 1 f ensuremath beta noise is characterized by five critical exponents we derive the relations between the scaled autocorrelation function and these exponents we show that our definition of the time dependent spectrum retains its interpretation as a density of fourier modes and discuss the relation to the apparent infrared divergence of 1 f ensuremath beta noise we illustrate our results for blinking quantum dot models single file diffusion and brownian motion in a logarithmic potential

oscillations of fourier coefficients of modular forms a p 2p cos 0 p since we know the truth of the ramanujan petersson conjecture it follows that the 0 p s are real inspired by the sato tate conjecture for elliptic curves serre 14 conjectured that the 0 p s are uniformly distributed in the interval 0 rc with respect to the 1 measure sin2odo following serre we shall refer to this as the sato tate r conjecture there being no room for confusion it has been known for a long time that the truth of this conjecture implies much about the oscillatory behaviour of the fourier coefficients in particular the following is implied by the sato tate conjecture theorem 1 for any normalized hecke eigenform

anomalous diffusion under stochastic resettings a general approach we present a general formulation of the resetting problem which is valid for any distribution of resetting intervals and arbitrary underlying processes we show that in such a general case a stationary distribution may exist even if the reset free process is not stationary as well as a significant decreasing in the mean first passage time we apply the general formalism to anomalous diffusion processes which allow simple and explicit expressions for poissonian resetting events

diffusion with optimal resetting we consider the mean time to absorption by an absorbing target of a diffusive particle with the addition of a process whereby the particle is reset to its initial position with rate r we consider several generalisations of the model of m r evans and s n majumdar 2011 diffusion with stochastic resetting phys rev lett 106 160601 i a space dependent resetting rate r x ii resetting to a random position z drawn from a resetting distribution p z iii a spatial distribution for the absorbing target pt x as an example of i we show that the introduction of a non resetting window around the initial position can reduce the mean time to absorption provided that the intial position is sufficiently far from the target we address the problem of optimal resetting that is minimising the mean time to absorption for a given target distribution for an exponentially decaying target distribution centred at the origin we show that a transition in the optimal resetting distribution occurs as the target distribution narrows

random walk with random resetting to the maximum position we study analytically a simple random walk model on a one dimensional lattice where at each time step the walker resets to the maximum of the already visited positions to the rightmost visited site with a probability r and with probability 1 r it undergoes symmetric random walk i e it hops to one of its neighboring sites with equal probability 1 r 2 for r 0 it reduces to a standard random walk whose typical distance grows as n for large n in the presence of a nonzero resetting rate 0 r 1 we find that both the average maximum and the average position grow ballistically for large n with a common speed v r moreover the fluctuations around their respective averages grow diffusively again with the same diffusion coefficient d r we compute v r and d r explicitly we also show that the probability distribution of the difference between the maximum and the location of the walker becomes stationary as n however the approach to this stationary distribution is accompanied by a dynamical phase transition characterized by a weakly singular large deviation function we also show that r 0 is a special critical point for which the growth laws are different from the r 0 case and we calculate the exact crossover functions that interpolate between the critical r 0 and the off critical r 0 behavior for finite but large n

dynamical transition in the temporal relaxation of stochastic processes under resetting a stochastic process when subject to resetting to its initial condition at a constant rate generically reaches a nonequilibrium steady state we study analytically how the steady state is approached in time and find an unusual relaxation mechanism in these systems we show that as time progresses an inner core region around the resetting point reaches the steady state while the region outside the core is still transient the boundaries of the core region grow with time as power laws at late times with new exponents alternatively at a fixed spatial point the system undergoes a dynamical transition from the transient to the steady state at a characteristic space dependent timescale t x we calculate analytically in several examples the large deviation function associated with this spatiotemporal fluctuation and show that generically it has a second order discontinuity at a pair of critical points characterizing the edges of the inner core these singularities act as separatrices between typical and atypical trajectories our results are verified in the numerical simulations of several models such as simple diffusion and fluctuating one dimensional interfaces

large deviations for markov processes with resetting markov processes restarted or reset at random times to a fixed state or region in space have been actively studied recently in connection with random searches foraging and population dynamics here we study the large deviations of time additive functions or observables of markov processes with resetting by deriving a renewal formula linking generating functions with and without resetting we are able to obtain the rate function of such observables characterizing the likelihood of their fluctuations in the long time limit we consider as an illustration the large deviations of the area of the ornstein uhlenbeck process with resetting other applications involving diffusions random walks and jump processes with resetting or catastrophes are discussed

minimax and interlacing thoerems for matrices abstract an exposition is given of some recent and important work by e marques de sa and r c thompson which characterized the relationship between the similarity invariants of a square matrix over a field and those of a principal submatrix of that matrix this work is then related to results on eigenvalues of complex hermitian matrices the courant fischer minimax theorem and the cauchy interlacing theorem on singular values of rectangular complex matrices due to thompson and on invariant factors of rectangular matrices over a principal ideal domain due to sa and thompson a partial unification of these theories is presented and several open questions stated