efficient discrete approximations of quantum gates quantum compiling addresses the problem of approximating an arbitrary quantum gate with a string of gates drawn from a particular finite set it has been shown that this is possible for almost all choices of base sets and furthermore that the number of gates required for precision e is only polynomial in log 1 e here we prove that using certain sets of base gates quantum compiling requires a string length that is linear in log 1 e a result which matches the lower bound from counting volume up to constant factor

a generalisation to birkhoff von neumann theorem abstract the classic birkhoff von neumann theorem states that the set of doubly stochastic matrices is the convex hull of the permutation matrices in this paper we study a generalisation of this theorem in the type ii 1 setting namely we replace a doubly stochastic matrix with a collection of measure preserving partial isomorphisms of the unit interval with similar properties we show that a weaker version of this theorem still holds

on the spectral gap for finitely generated subgroups of su 2 we prove the spectral gap property for free subgroups of su 2 generated by elements satisfying a noncommutative diophantine property in particular for free subgroups generated by elements with algebraic entries

exact synthesis of single qubit unitaries over clifford cyclotomic gate sets we generalize an efficient exact synthesis algorithm for single qubit unitaries over the clifford t gate set which was presented by kliuchnikov maslov and mosca quantum inf comput 13 7 8 607 630 2013 their algorithm takes as input an exactly synthesizable single qubit unitary one which can be expressed without error as a product of clifford and t gates and outputs a sequence of gates which implements it the algorithm is optimal in the sense that the length of the sequence measured by the number of t gates is smallest possible in this paper for each positive even integer n we consider the clifford cyclotomic gate set consisting of the clifford group plus a z rotation by n we present an efficient exact synthesis algorithm which outputs a decomposition using the minimum number of n z rotations for the clifford t case n 4 the group of exactly synthesizable unitaries was shown to be equal to the group of unitaries with entries over the ring z e i n 1 2 we prove that this characterization holds for a handful of other small values of n but the fraction of positive even integers for which it fails to hold is 100

finite time blow up of solutions of an aggregation equation in rn we consider the aggregation equation u t nabla cdot u nabla k u 0 in rn n 2 where k is a rotationally symmetric nonnegative decaying kernel with a lipschitz point at the origin e g k x e x we prove finite time blow up of solutions from specific smooth initial data for which the problem is known to have short time existence of smooth solutions

super golden gates for pu 2 abstract to each of the symmetry groups of the platonic solids we adjoin a carefully designed involution yielding topological generators of pu 2 which have optimal covering properties as well as efficient navigation these are a consequence of optimal strong approximation for integral quadratic forms associated with certain special quaternion algebras and their arithmetic groups the generators give super efficient 1 qubit quantum gates and are natural building blocks for the design of universal quantum gates

formation of clumps and patches in self aggregation of finite size particles abstract new model equations are derived for the dynamics of self aggregation of finite size particles differences from standard debye huckel p debye e huckel zur theorie der elektrolyte 2 das grenzgesetz fur die elektrische leiftfahrigkeit on the theory of electrolytes 2 limiting law of electrical conductivity phys z 24 1923 305 325 and keller segel e f keller l a segel j theoret biol 26 1970 399 415 e f keller l a segel j theoret biol 30 1971 225 234 models are a the mobility of particles depends on the locally averaged particle density and b linear diffusion acts on that locally averaged particle density the cases both with and without diffusion are considered here surprisingly these simple modifications of standard models allow progress in the analytical description of evolution as well as the complete analysis of stationary states when remains positive the evolution of collapsed states in our model reduces exactly to finite dimensional dynamics of interacting particle clumps simulations show these collapsed clumped states emerging from smooth initial conditions even in one spatial dimension if vanishes for some averaged density the evolution leads to the spontaneous formation of jammed patches a weak solution with the density having compact support simulations confirm that a combination of these patches forms the final state for the system

formation and evolution of singularities in anisotropic geometric continua evolutionary pdes for geometric order parameters that admit propagating singular solutions are introduced and discussed these singular solutions arise as a result of the competition between nonlinear and nonlocal processes in various familiar vector spaces several examples are given the motivating example is the directed self assembly of a large number of particles for technological purposes such as nano science processes in which the particle interactions are anisotropic this application leads to the derivation and analysis of gradient flow equations on lie algebra valued densities the riemannian structure of these gradient flow equations is also discussed

subconvex bounds for hecke maass forms on compact arithmetic quotients of semisimple lie groups let h be a semisimple algebraic group k a maximal compact subgroup of g h mathbb r and gamma subset h mathbb q a congruence arithmetic subgroup in this paper we generalize existing subconvex bounds for hecke maass forms on the locally symmetric space gamma backslash g k to corresponding bounds on the arithmetic quotient gamma backslash g for cocompact lattices using the spectral function of an elliptic operator the bounds obtained extend known subconvex bounds for automorphic forms to non trivial k types yielding such bounds for new classes of automorphic representations they constitute subconvex bounds for eigenfunctions on compact manifolds with both positive and negative sectional curvature we also obtain new subconvex bounds for holomorphic modular forms in the weight aspect

quadratic forms in unitary operators abstract let u 1 u n be unitary matrices on l 2 denote by the matrix a defined by a i i j j k 1 n u k i j u k i j acting as a bounded operator on l 2 n n in other words a is the sum of the kronecker products of u k with their complex conjugates we show the following sharp inequality as an application we show that the natural representation of u n n 1 acting on l 2 of the unit sphere in c n and restricted to mean zero functions satisfies for any choice 1 n in u n the lower bound this extends a result due to lubotzky phillips and sarnak who proved this with so 3 in the place of u n

expander graphs in pure and applied mathematics expander graphs are highly connected sparse finite graphs they play an important role in computer science as basic building blocks for network constructions error correcting codes algorithms and more in recent years they have started to play an increasing role also in pure mathematics number theory group theory geometry and more this expository article describes their constructions and various applications in pure and applied mathematics

the asymptotic distribution of lattice points in euclidean and non euclidean spaces the counting numbers for discrete subgroups of motions in euclidean and non euclidean spaces are obtained using the wave equation as the principal tool in dimensions 2 and 3 the error estimates are close to the best known

spherical designs discrepancy and numerical integration a spherical design is a point configuration on the sphere which yields exact equal weight quadrature formulae for polynomials up to a given de gree until now only very specific constructions for spherical designs are known we establish connections to spherical cap discrepancy and show some general discrepancy bounds furthermore we reformulate the problem of construct ing designs as an optimization problem and develop an algorithm for finding practical designs

trigonometric approximation and a general form of the erdos turan inequality there exists a positive function psi t on t 0 with fast decay infinity such that for every measurable set omega in the euclidean space and r 0 there exist entire functions a x and b x of exponential type r satisfying a x chi omega x b x and b x a x psi r dist x boundary omega this leads to erdos turan estimates for discrepancy of point set distributions in the multi dimensional torus analogous results hold for approximations by eigenfunctions of differential operators and discrepancy on compact manifolds

pattern generation in space and aspect a survey is presented of theories for the generation and maintenance of spatial pattern in reaction diffusion equations and their generalizations applications are selected from the biological sciences and physical chemistry special emphasis is placed on nonlocal interaction as manifested by the inclusion of terms involving higher derivatives or integrals it is stressed that traditional ideas of spatial pattern generation can usefully be extended to the study of pattern in general descriptive aspect variables particularly in understanding ecological diversity and heterogeneity

coarsening kinetics from a variable mobility cahn hilliard equation application of a semi implicit fourier spectral method an efficient semi implicit fourier spectral method is implemented to solve the cahn hilliard equation with a variable mobility the method is orders of magnitude more efficient than the conventional forward euler finite difference method thus allowing us to simulate large systems for longer times we studied the coarsening kinetics of interconnected two phase mixtures using a cahn hilliard equation with its mobility depending on local compositions in particular we compared the kinetics of bulk diffusion dominated and interface diffusion dominated coarsening in two phase systems results are compared with existing theories and previous computer simulations

emergent singular solutions of nonlocal density magnetization equations in one dimension we investigate the emergence of singular solutions in a nonlocal model for a magnetic system we study a modified gilbert type equation for the magnetization vector and find that the evolution depends strongly on the length scales of the nonlocal effects we pass to a coupled density magnetization model and perform a linear stability analysis noting the effect of the length scales of nonlocality on the system s stability properties we carry out numerical simulations of the coupled system and find that singular solutions emerge from smooth initial data the singular solutions represent a collection of interacting particles clumpons by restricting ourselves to the two clumpon case we are reduced to a two dimensional dynamical system that is readily analyzed and thus we classify the different clumpon interactions possible

coarsening dynamics of the one dimensional cahn hilliard model the dynamics of one dimensional cahn hilliard model is studied the stationary and particle type solutions the bubbles are perused as a function of initial conditions boundary conditions and system size we characterize the bubble solutions which are involved in the coarsening dynamics and establish the bifurcation scenarios of the system a set of ordinary differential equation permits us to describe the coarsening dynamics in very good agreement with numerical simulations we also compare these dynamics with the bubble dynamics deduced from the classical kink interaction computation where our model seems to be more appropriated in the case of two bubbles we deduce analytical expressions for the bubble s position and the bubble s width besides a simple description of the ulterior dynamics is presented

numerical methods for the landau lifshitz equation we discuss numerical schemes for various forms of the landau lifshitz equation a new simple projection method is introduced and is shown to be unconditionally stable the advantages over other schemes are also demonstrated numerically

singular solutions of a modified two component camassa holm equation the camassa holm ch equation is a well known integrable equation describing the velocity dynamics of shallow water waves this equation exhibits spontaneous emergence of singular solutions peakons from smooth initial conditions the ch equation has been recently extended to a two component integrable system ch2 which includes both velocity and density variables in the dynamics although possessing peakon solutions in the velocity the ch2 equation does not admit singular solutions in the density profile we modify the ch2 system to allow a dependence on the average density as well as the pointwise density the modified ch2 system mch2 does admit peakon solutions in the velocity and average density we analytically identify the steepening mechanism that allows the singular solutions to emerge from smooth spatially confined initial data numerical results for the mch2 system are given and compared with the pure ch2 case these numerics show that the modification in the mch2 system to introduce the average density has little short time effect on the emergent dynamical properties however an analytical and numerical study of pairwise peakon interactions for the mch2 system shows a different asymptotic feature namely besides the expected soliton scattering behavior seen in overtaking and head on peakon collisions the mch2 system also allows the phase shift of the peakon collision to diverge in certain parameter regimes

best possible rates of distribution of dense lattice orbits in homogeneous spaces this paper establishes upper and lower bounds on the speed of approximation in a wide range of natural diophantine approximation problems the upper and lower bounds coincide in many cases giving rise to optimal results in diophantine approximation which were inaccessible previously our approach proceeds by establishing more generally upper and lower bounds for the rate of distribution of dense orbits of a lattice subgroup in a connected lie or algebraic group g acting on suitable homogeneous spaces g h the upper bound is derived using a quantitative duality principle for homogeneous spaces reducing it to a rate of convergence in the mean ergodic theorem for a family of averaging operators supported on h and acting on g in particular the quality of the upper bound on the rate of distribution we obtain is determined explicitly by the spectrum of h in the automorphic representation on l 2 g we show that the rate is best possible when the representation in question is tempered and show that the latter condition holds in a wide range of examples

geometric gradient flow dynamics with singular solutions the gradient flow dynamics of an arbitrary geometric quantity is derived using a generalization of darcy s law we consider flows in both lagrangian and eulerian formulations the lagrangian formulation includes a dissipative modification of fluid mechanics eulerian equations for self organization of scalars 1 forms and 2 forms are shown to reduce to nonlocal characteristic equations we identify singular solutions of these equations corresponding to collapsed clumped states and discuss their evolution

determinant formula for the six vertex model the partition function of a six vertex model with domain wall boundary conditions is considered on the finite lattice the authors show that the partition function satisfies a recursive relation they solve the recursion relation by a determinant formula this gives a determinant representation for the partition function they use the quantum inverse scattering method qism