principal series of finite subgroups of su 3 we attempt to give a complete description of the exceptional finite subgroups 36 3 72 3 and 216 3 of su 3 with the aim to make them amenable to model building for fermion masses and mixing the information on these groups which we derive contains conjugacy classes proper normal subgroups irreducible representations character tables and tensor products of their three dimensional irreducible representations we show that for these three exceptional groups usage of their principal series i e ascending chains of normal subgroups greatly facilitates the computations and illuminates the relationship between the groups as a preparation and testing ground for the usage of principal series we study first the dihedral like groups 27 and 54 because both are members of the principal series of the three groups discussed in the paper

comments on the classification of the finite subgroups of su 3 many finite subgroups of su 3 are commonly used in particle physics the classification of the finite subgroups of su 3 began with the work of h f blichfeldt at the beginning of the 20th century in blichfeldt s work the two series c and d of finite subgroups of su 3 are defined while the group series 3n2 and 6n2 which are subseries of c and d respectively have been intensively studied there is not much knowledge about the group series c and d in this work we show that c and d have the structures and respectively furthermore we show that while the c groups can be interpreted as irreducible representations of 3n2 the d groups can in general not be interpreted as irreducible representations of 6n2

the ideal membership problem and polynomial identity testing given a monomial ideal i where m i are monomials and a polynomial f by an arithmetic circuit the ideal membership problem is to test if email protected i we study this problem and show the following results a when the ideal i for a constant k we can test whether email protected i in randomized polynomial time this result holds even for f given by a black box when f is of small degree b when i for a constantkandf is computed by a email protected s circuit with output gate of bounded fanin we can test whether email protected i in deterministic polynomial time this generalizes the kayal saxena result 11 of deterministic polynomial time identity testing for email protected s circuits with bounded fanin output gate c when k is not constant the problem is conp hard we also show that the problem is upper bounded by coma p p over the field of rationals and by conp m o d p p over finite fields d finally we discuss identity testing for certain restricted depth 4 arithmetic circuits for ideals i where each f email protected f x 1 x k is an arbitrary polynomial but k is a constant we show similar results as a and b above

generation motion and thickness of transition layers for a nonlocal allen cahn equation abstract we investigate the behavior as e 0 of the nonlocal allen cahn equation u t u 1 e 2 f u e u where f u 0 is of the bistable type given a rather general initial datum u 0 that is independent of e we perform a rigorous analysis of both the generation and the motion of the interface and obtain a new estimate for its thickness more precisely we show that the solution develops a steep transition layer within the time scale of order e 2 ln e and that the layer obeys the law of motion that coincides with the limit problem within an error margin of order e

from sylvester gallai configurations to rank bounds improved blackbox identity test for depth 3 circuits we study the problem of identity testing for depth 3 circuits of top fanin k and degree d we give a new structure theorem for such identities that improves the known deterministic dko k time blackbox identity test over rationals kayal and saraf 2009 to one that takes do k2 time our structure theorem essentially says that the number of independent variables in a real depth 3 identity is very small this theorem affirmatively settles the strong rank conjecture posed by dvir and shpilka 2006 we devise various algebraic tools to study depth 3 identities and use these tools to show that any depth 3 identity contains a much smaller nucleus identity that contains most of the complexity of the main identity the special properties of this nucleus allow us to get near optimal rank bounds for depth 3 identities the most important aspect of this work is relating a field dependent quantity the sylvester gallai rank bound to the rank of depth 3 identities we also prove a high dimensional sylvester gallai theorem for all fields and get a general depth 3 identity rank bound slightly improving previous bounds

polynomial icosahedral invariants in this paper the ring of polynomial invariants of the icosahedral group i is studied it begins by reviewing the surprising connections that this group and its double cover ii have with various areas of mathematics and physics of current interest information concerning the representation theory of these groups is then given after a brief discussion of the methods involved the integrity basis of the ring of polynomial invariants for each irreducible representation of i is given together with syzygies these are expressed in terms of the invariants of the tetrahedral subgroup t of i finally other methods of finding these invariants are discussed

simple finite non abelian flavor groups the recently measured unexpected neutrino mixing patterns have caused a resurgence of interest in the study of finite flavor groups with two and three dimensional irreducible representations this paper details the mathematics of the two finite simple groups with such representations the icosahedral group a5 a subgroup of so 3 and psl2 7 a subgroup of su 3

on the characterization of the su 3 subgroups of type c and d we investigate the two classes of finite subgroups of su 3 that are called type c and d in the book of miller blichfeldt and dickson we present two theorems which fully determine the form of the generators in a suitable basis after exploring further properties of these groups we are able to construct a complete list of infinite series in which these groups are arranged for type c there are infinitely many series whereas for type d there are only two explicit examples of these series are presented which illustrate the general results

a new approach to front propagation problems theory and applications in this paper we present a new definition for the global in time propagation motion of fronts hypersurfaces boundaries with a prescribed normal velocity past the first time they develop singularities we show that if this propagation satisfies a geometric maximum principle inclusion avoidance type property then the normal velocity must depend only on the position of the front and its normal direction and principal curvatures this new approach which is more geometric and as it turns out equivalent to the level set method is then used to develop a very general and simple method to rigorously validate the appearance of moving interfaces at the asymptotic limit of general evolving systems like interacting particles and reaction diffusion equations we finally present a number of new asymptotic results among them are the asymptotics of i reaction diffusion equations with rapidly oscillating coefficients ii fully nonlinear nonlocal integral differential equations and iii stochastic ising models with long range anisotropic interactions and general spin flip dynamics

the singular limit of the allen cahn equation and the fitzhugh nagumo system abstract we consider an allen cahn type equation of the form u t u e 2 f e x t u where e is a small parameter and f e x t u f u e g e x t u a bistable nonlinearity associated with a double well potential whose well depths can be slightly unbalanced given a rather general initial data u 0 that is independent of e we perform a rigorous analysis of both the generation and the motion of interface more precisely we show that the solution develops a steep transition layer within the time scale of order e 2 ln e and that the layer obeys the law of motion that coincides with the formal asymptotic limit within an error margin of order e this is an optimal estimate that has not been known before for solutions with general initial data even in the case where g e 0 next we consider systems of reaction diffusion equations of the form u t u e 2 f e u v v t d v h u v which include the fitzhugh nagumo system as a special case given a rather general initial data u 0 v 0 we show that the component u develops a steep transition layer and that all the above mentioned results remain true for the u component of these systems

development of interfaces in ℝ n consider the reaction diffusion equation in n u t h 2 u u 0 where is the derivative of a bistable even potential and h is a small parameter if the initial data have a smooth noncritical zero set we prove that an interface appears in time o log h 1 and that the solution stays close to it for at least time o 1 h

on p vs np and geometric complexity theory dedicated to sri ramakrishna this article gives an overview of the geometric complexity theory gct approach towards the p vs np and related problems focusing on its main complexity theoretic results these are 1 two concrete lower bounds which are currently the best known lower bounds in the context of the p vs nc and permanent vs determinant problems 2 the flip theorem which formalizes the self referential paradox in the p vs np problem and 3 the decomposition theorem which decomposes the arithmetic p vs np and permanent vs determinant problems into subproblems without self referential difficulty consisting of positivity hypotheses in algebraic geometry and representation theory and easier hardness hypotheses

analytic su 3 states in a finite subgroup basis a method is given for deriving branching rules in the form of generating functions for the decomposition of representations of su 3 into representations of its finite subgroups interpreted in terms of an integrity basis the generating functions define analytic polynomial basis states for su 3 which are adapted to the finite subgroup

primality and identity testing via chinese remaindering we give a simple and new randomized primality testing algorithm by reducing primality testing for number n to testing if a specific univariate identity over zn holds we also give new randomized algorithms for testing if a multivariate polynomial over a finite field or over rationals is identically zero the first of these algorithms also works over zn for any n the running time of the algorithms is polynomial in the size of arithmetic circuit representing the input polynomial and the error parameter these algorithms use fewer random bits and work for a larger class of polynomials than all the previously known methods for example the schwartz zippel test schwartz 1980 zippel 1979 chen kao and lewin vadhan tests chen and kao 1997 lewin and vadhan 1998

generation and propagation of interfaces in reaction diffusion systems this paper is concerned with the asymptotic behavior as e0 of the solution u e e of the second initial boundary value problem of the reaction diffusion system u t e e u e 1 e f u e e 1 e u e 1 u e2 e t e e u e e where 0 is a constant when 2 3 9 2 3 9 f is bistable in the sense that the ordinary differential equation u t f u has two stable solutions u h and u h and one unstable solution u h 0 where h h 0 and h are three solutions of the algebraic equation f u 0

front propagation and phase field theory the connection between the weak theories for a class of geometric equations and the asymptotics of appropriately rescaled reaction diffusion equations is rigorously established two different scalings are studied in the first the limiting geometric equation is a first order equation in the second it is a generalization of the mean curvature equation intrinsic definitions for the geometric equations are obtained and uniqueness under a geometric condition on the initial surface is proved in particular in the case of the mean curvature equation this condition is satisfied by surfaces that are strictly starshaped that have positive mean curvature or that satisfy a condition that interpolates between the positive mean curvature and the starshape conditions

on the parametrization of certain finite groups and their representations we discuss a convenient way of parametrizing certain finite groups they comprise groups with abelian normal subgroups and central extensions thereof they include the groups dn t o and their double groups as well as the su 3 subgroups tn 3n2 6n2 72 216 36 and their z3 extensions the method allows for a rapid calculation of representations and coupling coefficients in particular it solves the multiplicity problem of nonsimply reducible groups we illustrate the method treating the hessian group 216 and its central extension

on the finite subgroups of u 3 of order smaller than 512 we use the smallgroups library to find the finite subgroups of u 3 of order smaller than 512 which possess a faithful three dimensional irreducible representation from the resulting list of groups we extract those groups that cannot be written as direct products with cyclic groups these groups are the basic building blocks for models based on finite subgroups of u 3 all resulting finite subgroups of su 3 can be identified using the well known list of finite subgroups of su 3 derived by miller blichfeldt and dickson at the beginning of the 20th century furthermore we prove a theorem which allows us to construct infinite series of finite subgroups of u 3 from a special type of finite subgroups of u 3 this theorem is used to construct some new series of finite subgroups of u 3 the first members of these series can be found in the derived list of finite subgroups of u 3 of order smaller than 512 in the last part of this work we analyze some interesting finite subgroups of u 3 especially the group which is closely related to the important su 3 subgroup s4

generation of universal linear optics by any beam splitter universal quantum computers have proved difficult to build as one response researchers have proposed limited models of quantum computation which might be easier to realize three examples are the one clean qubit model of knill and laflamme 15 the commuting hamiltonians model of bremner jozsa and shepherd 3 and the boson sampling model of aaronson and arkhipov 1 none of these models are known or believed to be capable of universal quantum computation or depending on modeling details even universal classical computation but all of them can perform certain estimation or sampling tasks for which no polynomial time classical algorithm is known one obvious way to define a limited model of quantum computation is to restrict the set of allowed gates however almost every gate set is universal 17 and so are most natural gate sets for example controlled not together with any real one qubit gate that does not square to the identity is universal 21 as a result very few nontrivial examples of non universal gate sets are known all known non universal gate sets on o 1 qubits such as the clifford group 8 are efficiently classically simulable if the input and measurement outcomes both belong to an appropriately chosen qubit basis1 as a result it is tempting to conjecture that there does not exist such an intermediate gate set or more precisely that any gate set on o 1 qubits is either efficiently classical simulable with appropriate input and output states or else universal for quantum computing strikingly this dichotomy conjecture remains open even for the special case of 1and 2 qubit gates we regard proving or disproving the conjecture as an important open problem for quantum computing theory in this paper we prove a related conjecture in the quantum linear optics model in quantum optics the hilbert space is not built up as a tensor product of qubits instead it s built up as a direct sum of optical modes an optical gate is then just a unitary transformation that acts nontrivially ono 1 of the modes and as the identity on the rest whenever we have a k mode gate we assume that we can apply it to any subset of k modes in any order as often as desired the most common optical gates considered are beamsplitters which act on two modes and correspond to a 2 2 unitary matrix with determinant 1 2 and phaseshifters which act on one mode and simply apply a phase ei note that any unitary transformation acting on the one photon hilbert space automatically gets lifted by homomorphism to a unitary transformation acting on the hilbert space of n photons furthermore every element of the n photon linear optical group that is every n photon unitary transformation achievable using linear optics arises in this way see 1 sec iii for details of course if n 2 then there are also n photon unitaries that cannot be achieved linearoptically that is the n photon linear optical group is a proper subgroup of the full unitary group on the n photon hilbert space

generation and propagation of interfaces for reaction diffusion equations abstract this paper is concerned with the asymptotic behavior as e 0 of the solution ue of the reaction diffusion equation in r n r u t 1 2 0 where is the derivative of a bistable potential we show that if the initial data u 0 has values in both domains of attraction of the potential then an interface will develop in a short time o e 2 ln e we also show that if the wells of the potential are of equal depth then this interface will propagate with normal velocity equal to its mean curvature our result is valid as long as the interface remains smooth if the initial interface is compact and n 2 then the interface will disappear in a finite time but not if n 1 in case the depths of the wells are not equal then in order to obtain get reasonable results we must work on the scaled time s t e slower time scale in this scale we show that the interface moves with a constant speed proportional to the difference of the depths of the two wells along the normal towards the domain of the deeper well this result is valid for all s 0 and does not actually depend on the regularity of the interface we also extend the above results to the homogeneous second initial boundary value problem in case the depths of the wells are equal and the initial interface is orthogonal to the boundary we prove that the interface moves with normal velocity equal to its mean curvature provided that there is a family of hypersurfaces which moves according to their mean curvature and intersect the boundary orthogonally

phase transitions and generalized motion by mean curvature we study the limiting behavior of solutions to appropriately rescaled versions of the allen cahn equation a simplified model for dynamic phase transitions we rigorously establish the existence in the limit of a phase antiphase interface evolving according to mean curvature motion this assertion is valid for all positive time the motion interpreted in the generalized sense of evans spruck and chen giga goto after the onset of geometric singularities

the flavor group δ 6n2 many non abelian finite subgroups of su 3 have been used to explain the flavor structure of the standard model in order to systematize and classify successful models a detailed knowledge of their mathematical structure is necessary in this paper we shall therefore look closely at the series of finite non abelian groups known as 6n2 its smallest members being s3 n 1 and s4 n 2 for arbitrary n we determine the conjugacy classes the irreducible representations the kronecker products as well as the clebsch gordan coefficients

permutational symmetry of clebsch gordan coefficients for the metacyclic groups abstract a complete discussion of the permutational symmetry for the 3 gcy symbols for k metacyclic groups an example of non simple phase groups is carried out the analytic formulas for the 3 gcy symbols and for the derome sharp matrices describing the permutational properties of these symbols are given it is shown that these matrices constitute a permutation representation m which is an appropriate coarse grained version of the ordinary permutation representation of the group of all permutations of columns of the 3gcy symbols a detailed structure of the representation m has been determined

the freedman group a physical interpretation for the su 3 subgroup d 18 1 1 2 1 1 of order 648 we study a subgroup fr 162 4 of su 3 of order 648 which is an extension of d 9 1 1 2 1 1 and whose generators arise from anyonic systems we show that this group is isomorphic to a semi direct product with respect to conjugation and we give a presentation of the group we show that the group d 18 1 1 2 1 1 from the series d in the existing classification for finite su 3 subgroups is also isomorphic to a semi direct product with respect to conjugation we next exhibit the isomorphism between both groups we prove that fr 162 4 is not isomorphic to the exceptional su 3 subgroup 216 3 of the same order 648 we further prove that the only su 3 finite subgroups from the 1916 classification by blichfeldt or its extended version in which fr 162 4 may be isomorphic belong to the d series finally we show that fr 162 4 and d 18 1 1 2 1 1 are both conjugate under the orthogonal matrix which we provide

finite flavour groups of fermions we present an overview of the theory of finite groups with regard to their application as flavour symmetries in particle physics in a general part we discuss useful theorems concerning group structure conjugacy classes representations and character tables in a specialized part we attempt to give a fairly comprehensive review of finite subgroups of so 3 and su 3 in which we apply and illustrate the general theory moreover we also provide a concise description of the symmetric and alternating groups and comment on the relationship between finite subgroups of u 3 and finite subgroups of su 3 although in this review we give a detailed description of a wide range of finite groups the main focus is on the methods which allow the exploration of their different aspects

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

finite subgroups of su 3 we present a new class of finite subgroups of su 3 of the form zm s zn semidirect product we also apply the methods used to investigate semidirect products to the known su 3 subgroups 3n2 and 6n2 and give analytic formulae for representations characters and clebsch gordan coefficients

locally decodable codes with two queries and polynomial identity testing for depth 3 circuits in this work we study two seemingly unrelated notions locally decodable codes ldcs are codes that allow the recovery of each message bit from a constant number of entries of the codeword polynomial identity testing pit is one of the fundamental problems of algebraic complexity we are given a circuit computing a multivariate polynomial and we have to determine whether the polynomial is identically zero we improve known results on ldcs and on polynomial identity testing and show a relation between the two notions in particular we obtain the following results 1 we show that if e mathbb f n mapsto mathbb f m is a linear ldc with two queries then m exp omega n previously this was known only for fields of size ll 2 n o goldreich e comput complexity 15 2006 pp 263 296 2 we show that from every depth 3 arithmetic circuit sigma pi sigma circuit cal c with a bounded constant top fan in that computes the zero polynomial one can construct an ldc more formally assume that cal c is minimal no subset of the multiplication gates sums to zero and simple no linear function appears in all the multiplication gates denote by d the degree of the polynomial computed by cal c and by r the rank of the linear functions appearing in cal c then we can construct a linear ldc with two queries that encodes messages of length r operatorname polylog d by codewords of length o d 3 we prove a structural theorem for sigma pi sigma circuits with a bounded top fan in that compute the zero polynomial in particular we show that if such a circuit is simple minimal and of polynomial size then its rank r is only polylogarithmic in the number of variables a priori it could have been linear 4 we give new pit algorithms for sigma pi sigma circuits with a bounded top fan in a a deterministic algorithm that runs in quasipolynomial time and b a randomized algorithm that runs in polynomial time and uses only a polylogarithmic number of random bits moreover when the circuit is multilinear our deterministic algorithm runs in polynomial time previously deterministic subexponential time algorithms for pit in bounded depth circuits were known only for depth 2 circuits in the black box model d grigoriev m karpinski and m f singer siam j comput 19 1990 pp 1059 1063 m ben or and p tiwari proceedings of the 20th annual acm symposium on theory of computing acm press new york 1988 pp 301 309 a r klivans and d spielman proceedings of the 33rd annual acm symposium on theory of computing acm press new york 2001 pp 216 223 in particular for the special case of depth 3 circuits with three multiplication gates our result resolves an open question asked by klivans and spielman

representations and clebsch gordan coefficients of z metacyclic groups we investigate finite groups of the form zmszn and give all irreducible representations and clebsch gordan coefficients in analytic form two subclasses are considered which seem to be inportant for applications the m metacyclic groups which are important for spin systems and the k metacyclic groups which are the smallest finite groups that have an irreducible representation of dimension p 1 where p is prime

algebraic independence and blackbox identity testing algebraic independence is a fundamental notion in commutative algebra that generalizes independence of linear polynomials polynomials f 1 f m k x 1 x n over a field k are called algebraically independent if there is no non zero polynomial f such that f f 1 f m 0 the transcendence degree trdeg f 1 f m is the maximal number r of algebraically independent polynomials in the set in this paper we design blackbox and efficient linear maps f that reduce the number of variables from n to r but maintain trdeg f f i i r assuming sparse f i and small r we apply these fundamental maps to solve two cases of blackbox identity testing assuming a large or zero characteristic 1 given a polynomial degree circuit c and sparse polynomials f 1 f m of transcendence degree r we can test blackbox d c f 1 f m for zeroness in poly size d r time 2 define a email protected email protected d k s n circuit to be of the form i 1 email protected j 1 sf i j where f i j are sparse n variate polynomials of degree at most d for this class of depth 4 circuits we define a notion of rank assuming there is a rank bound r for minimal simple email protected email protected d k s n identities we give a poly dsnr r k d 2 time blackbox identity test for email protected email protected d k s n circuits this partially generalizes the state of the art of depth 3 to depth 4 circuits the notion of transcendence degree works best with large or zero characteristic but we also give versions of our results for arbitrary fields

motion by anisotropic mean curvature as sharp interface limit of an inhomogeneous and anisotropic allen cahn equation we consider the spatially inhomogeneous and anisotropic reaction diffusion equation ut m x 1 div m x ap x u e 2 f u involving a small parameter e 0 and a bistable nonlinear term whose stable equilibria are 0 and 1 we use a finsler metric related to the anisotropic diffusion term and work in relative geometry we prove a weak comparison principle and perform an analysis of both the generation and the motion of interfaces more precisely we show that within the time scale of order e2 ln e the unique weak solution u e develops a steep transition layer that separates the regions u e 0 and u e 1 then on a much slower time scale the layer starts to propagate consequently as e 0 the solution u e converges almost everywhere a e to 0 in t and 1 in t where t and t are sub domains of separated by an interface t whose motion is driven by its anisotropic mean curvature we also prove that the thickness of the transition layer is of order e

6 j for mixed symmetry triads in k20 the authors discuss the possible choices of conjugation and permutation matrices for mixed symmetry triads this requires a short review of the previous choices for all triads the primitive 6 j of the finite group k20 are then calculated as an example of a group that contains mixed symmetry triads

hitting sets for roabp and sum of set multilinear circuits we give an n o log n time n is the input size blackbox polynomial identity testing algorithm for unknown order read once oblivious arithmetic branching programs roabps the best time complexity known for blackbox polynomial identity testing pit for this class was n o log 2 n due to forbes saptharishi and shpilka proceedings of the 2014 acm symposium on theory of computing 2014 pp 867 875 moreover their result holds only when the individual degree is small while we do not need any such assumption with this we match the time complexity for the unknown order roabp with the known order roabp due to forbes and shpilka proceedings of the 2013 ieee 54th annual symposium on foundations of computer science 2013 pp 243 252 and also with the depth 3 set multilinear circuits due to agrawal saha and saxena proceedings of the 2013 acm symposium on theory of computing 2013 pp 321 330 our proof is simpler and involves a new technique called basis isolation the depth 3

an almost optimal rank bound for depth 3 identities we study the problem of polynomial identity testing for depth 3 circuits of degree d and top fanin k the rank of any such identity is essentially the minimum number of independent variables present small bounds on this quantity imply fast deterministic identity testers for these circuits dvir and shpilka siam j comput 36 2007 pp 1404 1434 initiated the study of the rank and showed that any depth 3 identity barring some uninteresting corner cases has a rank of 2 o k 2 log d k 2 we show that the rank of a depth 3 identity is at most o k 3 log d this bound is almost tight since we also provide an identity of rank omega k log d our rank bound significantly improves dependence on k exponentially reduced the best known deterministic black box identity tests for depth 3 circuits by karnin and shpilka z karnin and a shpilka in proceedings of the 23rd ccc 2008 pp 280 291 our techniques also shed light on the factorization pattern of nonzero depth 3 circuits the rank of linear factors of a simple minimal and nonzero depth 3 circuit over any field is at most o k 3 log d the novel feature of this work is a new notion of maps between sets of linear forms called ideal matchings used to study depth 3 circuits we prove interesting structural results about depth 3 identities using these techniques we believe that these ideas may lead to the goal of a deterministic polynomial time identity test for these circuits

some comments on finite subgroups of su 3 a recently published set of finite subgroups of su 3 is shown to contain some groups which are not subgroups of su 3 the others are subgroups of one of the dihedral like family of su 3 subgroups of order 3n2 some comments are made also on the structure of other finite subgroups previously listed