colored group field theory random matrix models generalize to group field theories gft whose feynman graphs are dual to higher dimensional topological spaces the perturbative development of the usual gft s is rather involved combinatorially and plagued by topological singularities which we discuss in great detail in this paper thus very difficult to control and unsatisfactory

on the lower central quotients of the free groups of finite rank of a certain variety abstract let f n be a free group of rank n and k f n the k th term of the lower central series of f n for l 1 we denote by f n q l the quotient group of f n by a normal subgroup 2 3 f n l 2 2 f n in this paper we show that each of the graded quotients of the lower central series of the group f n q l for any l 1 is a free abelian group and give a basis of it by using a generalized chen s integration in free groups

approximations for expected generation number a deterministic formula is commonly used to approximate the expected generation number of a population of growing cells however this can give misleading results because it does not allow for natural variation in the times that individual cells take to reproduce here we present more accurate approximations for both symmetric and asymmetric cell division based on the first two moments of the generation time distribution these approximations are also robust we illustrate the improved approximations using data that arise from monitoring individual yeast cells under a microscope and also demonstrate how the approximations can be used when such detailed data are not available

memory effects in a nonequilibrium growth model we study memory effects in a kinetic roughening model for d 1 a different dynamic scaling is uncovered in the memory dominated phases the kardar parisi zhang scaling is restored in the absence of noise dc 2 represents the critical dimension where memory is shown to smoothen the roughening front a 0 studies on a discrete atomistic model in the same universality class reconfirm the analytical results in the large time limit while a different scaling behavior shows up for t t with t being the memory characteristic of the atomistic model results can be generalized for other nonconservative systems

on the large n limit of schwinger dyson equations of a rank 3 tensor field theory we analyze in this paper the large n limit of the schwinger dyson equations in a rank 3 tensor quantum field theory which are derived with the help of ward takahashi identities in order to have a well defined large n limit appropriate scalings in powers of n for the various terms present in the action are explicitly found a perturbative check of our results is done up to second order in the coupling constant we analyze in this paper the large n limit of the schwinger dyson equations in a rank 3 tensor quantum field theory which are derived with the help of ward takahashi identities in order to have a well defined large n limit appropriate scalings in powers of n for the various terms present in the action are explicitly found a perturbative check of our results is done up to second order in the coupling constant

the full ward takahashi identity for colored tensor models colored tensor models ctm is a random geometrical approach to quantum gravity we scrutinize the structure of the connected correlation functions of general ctm interactions and organize them by boundaries of feynman graphs for rank d interactions including but not restricted to all melonic varphi 4 vertices to wit solely those quartic vertices that can lead to dominant spherical contributions in the large n expansion the aforementioned boundary graphs are shown to be precisely all possibly disconnected vertex bipartite regularly edge d colored graphs the concept of ctm compatible boundary graph automorphism is introduced and an auxiliary graph calculus is developed with the aid of these constructs certain u invariance of the path integral measure is fully exploited in order to derive a strong ward takahashi identity for ctms with a symmetry breaking kinetic term for the rank 3 varphi 4 theory we get the exact integral like equation for the 2 point function similarly exact equations for higher multipoint functions can be readily obtained departing from this full ward takahashi identity our results hold for some group field theories as well altogether our non perturbative approach trades some graph theoretical methods for analytical ones we believe that these tools can be extended to tensorial syk models

scaling behaviour of the time fractional kardar parisi zhang equation the scaling behaviour of the time fractional kardar parisi zhang tfkpz equation in 1 1 dimensions is investigated by scaling analysis and numerical simulations the surface morphology and critical exponents with different fractional orders are obtained the analytical results are consistent with the corresponding numerical solutions based on a caputo type fractional derivative we find that similar to the normal kardar parisi zhang equation anomalous behaviour does not appear in the tfkpz model according to the scaling idea of local slope and numerical evidence however there exists significant finite time effect of local scaling exponents in the tfkpz system our results also imply that memory effects can affect the scaling behaviour of evolving fractional surface growth

ward takahashi identities for the colored boulatov model ward takahashi wt identities of the colored boulatov model are derived using a generic unitary field transformation in a specific instance this generic transformation turns out to be a symmetry of the interaction so that particular classes of reduced wt identities for that symmetry are consequently identified and interpreted

the nonfinite generation of aut g g free metabelian of rank 3 the group of automorphisms of the free metabelian group of rank 3 is not finitely generated let h be a free solvable group of rank n 2 aut h the automorphism group of h and inn h the group of inner automorphisms of h for n 2 it was shown by the authors and e formanek 4 theorem 1 that aut h inn h gl2 z z the ring of integers one consequence of this fact is that aut h is finitely generated f g the purpose of this paper is to prove the following contrasting theorem theorem if g is the free metabelian group of rank three then aut g is not finitely generated this result may also be compared with a theorem of l auslander 1 which states that the automorphism group of a polycyclic group is finitely presented if g denotes the commutator subgroup of g then the kernel of the natural homomorphism aut g 4 aut g g is called the group of ia automorphisms of g and will be denoted by ia g ia g which contains inn g was shown by the authors 3 to be non f g for g as in the theorem to prove that aut g is not f g we need to show that ia g is not f g as an aut g g gl3 z operator group the question arises concerning aut h where h is free metabelian of rank n 3 the immediate feeling that aut h is also non f g may be incorrect for definiteness let n 4 suppose g is free metabelian of rank 3 and consider aut g as embedded in aut h in an obvious manner the nonfinite generatiion of aut g comes from the existence of nontame automorphisms in aut g i e automorphisms of g f f which are not induced by automorphisms of the free group f of rank three and the necessity to include infinitely many nontame automorphisms in any generating set for aut g however the authors have discovered that many of the nontame automorphisms become tame when considered as elements in aut h whether this phenomenon is true for all nontame elements of aut g is unknown as yet but the question of the finite or nonfinite generation of aut h must be considered a difficult open problem with the interesting possibility of finite generation as the answer received by the editors december 31 1980 1980 mathematics subject classification primary 20f28

the cokernel of the johnson homomorphisms of the automorphism group of a free metabelian group in this paper we determine the cokernel of the k th johnson homomorphisms of the automorphism group of a free metabelian group for k 2 and n 4 as a corollary we obtain a lower bound on the rank of the graded quotient of the johnson filtration of the automorphism group of a free group furthermore by using the second johnson homomorphism we determine the image of the cup product map in the rational second cohomology group of the ia automorphism group of a free metabelian group and show that it is isomorphic to that of the ia automorphism group of a free group which is already determined by pettet finally by considering the kernel of the magnus representations of the automorphism group of a free group and a free metabelian group we show that there are non trivial rational second cohomology classes of the ia automorphism group of a free metabelian group which are not in the image of the cup product map

diagrammatics of a colored syk model and of an syk like tensor model leading and next to leading orders the sachdev ye kitaev syk model is a model of q interacting fermions gross and rosenhaus have proposed a generalization of the syk model which involves fermions with different flavors in terms of feynman graphs those flavors are reminiscent of the colors used in random tensor theory this gives us the opportunity to apply some modern yet elementary tools developed in the context of random tensors to one particular instance of such colored syk models we illustrate our method by identifying all diagrams which contribute to the leading and next to leading orders of the 2 point and 4 point functions in the large n expansion and argue that our method can be further applied if necessary in the second part we focus on the recently introduced gurau witten tensor model and also extract the leading and next to leading orders of the 2 point and 4 point functions this analysis turns out to be remarkably more involved than in the colored syk model

kardar parisi zhang equation with temporally correlated noise a self consistent approach in this paper we discuss the well known kardar parisi zhang kpz equation driven by temporally correlated noise we use a self consistent approach to derive the scaling exponents of this system we also draw general conclusions about the behavior of the dynamic structure factor ensuremath phi q t as a function of time the approach we use here generalizes the well known self consistent expansion sce that was used successfully in the case of the kpz equation driven by white noise but unlike sce it is not based on a fokker planck form of the kpz equation but rather on its langevin form a comparison to two other analytical methods as well as to the only numerical study of this problem is made and a need for an updated extensive numerical study is identified we also show that a generalization of this method to any spatiotemporal correlations in the noise is possible and two examples of this kind are considered

some classes of renormalizable tensor models we identify new families of renormalizable tensor models from anterior renormalizable tensor models via a mapping capable of reducing or increasing the rank of the theory without having an effect on the renormalizability property mainly a version of the rank 3 tensor model as defined by ben geloun and samary ann henri poincare 14 1599 2013 e print arxiv 1201 0176 hep th 10 1007 s00023 012 0225 5 and the grosse wulkenhaar model in 4d and 2d generate three different classes of renormalizable models the proof of the renormalizability is fully performed for the first reduced model the same procedure can be applied for the remaining cases interestingly we find that due to the peculiar behavior of anisotropic wave function renormalizations the rank 3 tensor model reduced to a matrix model generates a simple super renormalizable vector model

e2 sl2 for most laurent polynomial rings it is not known whether or not sl2 r and e2 r are equal for r z t t 1 zthe ring of rational integers orforr k t1 t 1 t2 t2j1 k a field e2 s is the subgroup of sl2 s generated by the elementary matrices o 1 a e s we discuss this question for related laurent polynomial rings r s tj tl 1 tk tk our results show that for most laurent polynomial rings r sl2 r e2 r and moreover any generating set for s12 r must contain infinitely many elements not in e2 r suslin 9 has proved that for n 2 3 and arbitrary k sln r e r forr equal to z tl t11 tk tkjl ork tj t1 1 tk tk by contrast we state

multi scaling of the n point density function for coalescing brownian motions this paper gives a derivation for the large time asymptotics of the n point density function of a system of coalescing brownian motions on r

feynman diagrams of generalized matrix models and the associated manifolds in dimension four the problem of constructing a quantum theory of gravity has been tackled with very different strategies most of which rely on the interplay between ideas from physics and from advanced mathematics on the mathematical side a central role is played by combinatorial topology often used to recover the space time manifold from the other structures involved an extremely attractive possibility is that of encoding all possible space times as specific feynman diagrams of a suitable field theory in this work we analyze how exactly one can associate combinatorial four manifolds with the feynman diagrams of certain tensor theories

reproduction time statistics and segregation patterns in growing populations pattern formation in microbial colonies of competing strains under purely space limited population growth has recently attracted considerable research interest we show that the reproduction time statistics of individuals has a significant impact on the sectoring patterns generalizing the standard eden growth model we introduce a simple one parameter family of reproduction time distributions indexed by the variation coefficient d 0 1 which includes deterministic d 0 and memory less exponential distribution d 1 as extreme cases we present convincing numerical evidence and heuristic arguments that the generalized model is still in the kpz universality class and the changes in patterns are due to changing prefactors in the scaling relations which we are able to predict quantitatively at the example of saccharomyces cerevisiae we show that our approach using the variation coefficient also works for more realistic reproduction time distributions

a reduction of the target of the johnson homomorphisms of the automorphism group of a free group let f n be a free group of rank n and f n n the quotient group of f n by a subgroup n 3 n 3 n 2 n 2 n 2 where n k denotes the k th subgroup of the lower central series of the free group f n in this paper we determine the group structure of the graded quotients of the lower central series of the group f n n by using a generalized chen s integration in free groups then we apply it to the study of the johnson homomorphisms of the automorphism group of f n in particular under taking a reduction of the target of the johnson homomorphism induced from a quotient map f n f n n we see that there appear only two irreducible components the morita obstruction s k h q and the schur weyl module of type h k 2 12 q in the cokernel of the rational johnson homomorphism k q k id q for k 5 and n k 2

scale invariant growth processes in expanding space many growth processes lead to intriguing stochastic patterns and complex fractal structures which exhibit local scale invariance properties such structures can often be described effectively by space time trajectories of interacting particles and their large scale behavior depends on the overall growth geometry we establish an exact relation between statistical properties of structures in uniformly expanding and fixed geometries which preserves the local scale invariance and is independent of other properties such as the dimensionality this relation generalizes standard conformal transformations as the natural symmetry of self affine growth processes we illustrate our main result numerically for various structures of coalescing levy flights and fractional brownian motions including also branching and finite particle sizes one of the main benefits of this approach is a full explicit description of the asymptotic statistics in expanding domains which are often nontrivial and random due to amplification of initial fluctuations

on the graded quotients of the rings of fricke characters of free groups abstract for a free group f n on n generators we consider the aut f n invariant ideal j generated by tr x 2 for all x f n in the ring x q f n of fricke characters of f n this ideal defines a descending filtration j j 2 j 3 of x q f n for each k 1 we introduce the normal subgroup e n k consisting of all automorphisms of f n which act on j j k 1 trivially these normal subgroups define a central filtration of aut f n this is a fricke character analogue of the andreadakis johnson filtration a n k of aut f n the main purpose of the paper is to show that e n 1 coincides with inn f n a n 2 where inn f n is the inner automorphism group of the free group f n and that a n 2 k e n k for any k 1

the surface statistics of a granular aggregate the problem of the surface fluctuations in a settled granular material is posed a simple model is given which describes the process by which a particle settles and comes to rest on the existing surface of the packing and from this a set of langevin equations for the fourier modes of the surface are derived these equations imply that the fourier amplitudes behave like the velocities of a set of independent brownian particles we show that this results in logarithmically divergent surface fluctuations if the flux of particles onto the surface is random the divergence being removed by a more accurate description of the settling material for example by having the granules fall through a sieve

morphology and dynamic scaling analysis of cell colonies with linear growth fronts the growth of linear cell colony fronts is investigated from the morphology of cell monolayer colonies the cell size and shape distribution the front displacement velocity and the dynamic scaling analysis of front roughness fluctuations at the early growth stages colony patterns consist of rather ordered compact domains of small cells whereas at advanced stages an uneven distribution of cells sets in and some large cells and cells exhibiting large filopodia are produced colony front profiles exhibit overhangs and behave as fractals with the dimension d f 1 25 0 05 the colony fronts shift at 0 22 0 02 m min 1 average constant linear velocity and their roughness w increases with time t dynamic scaling analysis of experimental and overhang corrected growth profile data shows that w versus system width l log log plots collapse to a single curve when l exceeds a certain threshold value l o a width corresponding to the average diameter of few cells then the influence of overhangs on the roughness dynamics becomes negligible and a growth exponent 0 33 0 02 is derived from the structure factor analysis of overhang corrected profiles a global roughness exponent s 0 50 0 05 is obtained for l 200 m this set of exponents fulfills the family vicsek relationship it is consistent with the predictions of the continuous kardar parisi zhang model