global holderian error bound for nondegenerate polynomials let f colon mathbb r n to mathbb r be a polynomial and s x in mathbb r n f x le 0 let f x max 0 f x if there exist c 0 alpha 0 beta 0 such that d x s le c f x alpha f x beta for all x in mathbb r n where d x s denotes the distance between x and s then we say that f has a global holderian error bound we prove that if f is convenient and nondegenerate with respect to its newton boundary at infinity in the sense of kouchnirenko invent math 32 1976 pp 1 32 then it has a global holderian error bound since the nondegenerate polynomials form a zariski open subset in a variety of polynomials with a given convenient newton boundary at infinity this result shows that global holderian error bounds hold for a large class of polynomials which can be relatively easy recognized

phase resetting phase locking and bistability in the periodically driven saline oscillator experiment and model the saline oscillator consists of an inner vessel containing salt water partially immersed in an outer vessel of fresh water with a small orifice in the center of the bottom of the inner vessel there is a cyclic alternation between salt water flowing downwards out of the inner vessel into the outer vessel through the orifice and fresh water flowing upwards into the inner vessel from the outer vessel through that same orifice we develop a very stable i e stationary version of this saline oscillator we first investigate the response of the oscillator to periodic forcing with a train of stimuli text period t mathrm p of large amplitude each stimulus is the quick injection of a fixed volume of fresh water into the outer vessel followed immediately by withdrawal of that very same volume for t mathrm p sufficiently close to the intrinsic period of the oscillator t 0 there is 1 1 synchronization or phase locking between the stimulus train and the oscillator as t mathrm p is decreased below t 0 one finds the succession of phase locking rhythms 1 1 2 2 2 1 2 2 and 1 1 as t mathrm p is increased beyond t 0 one encounters successively 1 1 1 2 2 4 2 3 2 4 and 1 2 phase locking rhythms we next investigate the phase resetting response in which injection of a single stimulus transiently changes the period of the oscillation by systematically changing the phase of the cycle at which the stimulus is delivered the old phase we construct the new phase old phase curve the phase transition curve from which we then develop a one dimensional finite difference equation map that predicts the response to periodic stimulation these predicted phase locking rhythms are close to the experimental findings in addition iteration of the map predicts the existence of bistability between two different 1 1 rhythms which was then searched for and found experimentally bistability between 1 1 and 2 2 rhythms is also encountered finally with one exception numerical modeling with a phenomenologically derived rayleigh oscillator reproduces all of the experimental behavior

iterated vanishing cycles convolution and a motivic analogue of a conjecture of steenbrink we prove a motivic analogue of steenbrink s conjecture 25 conjecture 2 2 on the hodge spectrum proved by m saito in 21 to achieve this we construct and compute motivic iterated vanishing cycles associated with two functions we are also led to introduce a more general version of the convolution operator appearing in the motivic thom sebastiani formula throughout the article we use the framework of relative equivariant grothendieck rings of varieties endowed with an algebraic torus action

estimating eigenvalues of dynamical systems from time series with applications to predicting cardiac alternans the stability of a dynamical system can be indicated by eigenvalues of its underlying mathematical model however eigenvalue analysis of a complicated system e g the heart may be extremely difficult because full models may be intractable or unavailable we develop data driven statistical techniques which are independent of any underlying dynamical model that use principal components and maximum likelihood methods to estimate the dominant eigenvalues and their standard errors from the time series of one or a few measurable quantities e g transmembrane voltages in cardiac experiments the techniques are applied to predicting cardiac alternans that is characterized by an eigenvalue approaching 1 cardiac alternans signals a vulnerability to ventricular fibrillation the leading cause of death in the usa

unfoldings in knot theory in in r the link at infinity of a hypersurface v cie r defined by a good polynomial map f ty ii2 was used as an extended example f was called good if it had only isolated singularities and it was claimed that the link at infinity then always has a milnor fibration this is incorrect although we have found it to be a common misconception to correct it the definition of good must be modified as follows

the orevkov invariant of an affine plane curve we show that although the fundamental group of the complement of an algebraic affine plane curve is not easy to compute it possesses a more accessible quotient which we call the orevkov invariant

nonsmooth dynamics in spiking neuron models large scale studies of spiking neural networks are a key part of modern approaches to understanding the dynamics of biological neural tissue one approach in computational neuroscience has been to consider the detailed electrophysiological properties of neurons and build vast computational compartmental models an alternative has been to develop minimal models of spiking neurons with a reduction in the dimensionality of both parameter and variable space that facilitates more effective simulation studies in this latter case the single neuron model of choice is often a variant of the classic integrate and fire model which is described by a nonsmooth dynamical system in this paper we review some of the more popular spiking models of this class and describe the types of spiking pattern that they can generate ranging from tonic to burst firing we show that a number of techniques originally developed for the study of impact oscillators are directly relevant to their analysis particularly those for treating grazing bifurcations importantly we highlight one particular single neuron model capable of generating realistic spike trains that is both computationally cheap and analytically tractable this is a planar nonlinear integrate and fire model with a piecewise linear vector field and a state dependent reset upon spiking we call this the pwl if model and analyse it at both the single neuron and network level the techniques and terminology of nonsmooth dynamical systems are used to flesh out the bifurcation structure of the single neuron model as well as to develop the notion of liapunov exponents we also show how to construct the phase response curve for this system emphasising that techniques in mathematical neuroscience may also translate back to the field of nonsmooth dynamical systems the stability of periodic spiking orbits is assessed using a linear stability analysis of spiking times at the network level we consider linear coupling between voltage variables as would occur in neurobiological networks with gap junction coupling and show how to analyse the properties existence and stability of both the asynchronous and synchronous states in the former case we use a phase density technique that is valid for any large system of globally coupled limit cycle oscillators whilst in the latter we develop a novel technique that can handle the nonsmooth reset of the model upon spiking finally we discuss other aspects of neuroscience modelling that may benefit from further translation of ideas from the growing body of knowledge on nonsmooth dynamics

limit cycles complex floquet multipliers and intrinsic noise we study the effects of intrinsic noise on chemical reaction systems which in the deterministic limit approach a limit cycle in an oscillatory manner previous studies of systems with an oscillatory approach to a fixed point have shown that the noise can transform the oscillatory decay into sustained coherent oscillations with a large amplitude we show that a similar effect occurs when the stable attractors are limit cycles we compute the correlation functions and spectral properties of the fluctuations in suitably comoving frenet frames for several model systems including driven and coupled brusselators and the willamowski rossler system analytical results are confirmed convincingly in numerical simulations the effect is quite general and occurs whenever the floquet multipliers governing the stability of the limit cycle are complex with the amplitude of the oscillations increasing as the instability boundary is approached

reflection of flexural waves in geometrically periodic beams abstract the paper focuses on the reflection characteristics of elastic beams having a periodically varying cross sectional area assuming a weak sinusoidal variation of the beam cross sectional area assuming a weak sinusoidal variation of the beam cross section along its axis perturbation methods are employed to determine flexural resonance conditions and analyze the resonant destructive interaction of flexural waves with the periodic beam this interaction is represented in the form of coupled wave equations which are solved analytically together with relevant boundary conditions at the ends of the periodic section of the beam the reflection coefficient is then calculated for beams having different types of periodicity this study is intended to provide guidelines to control passively the flexural vibration in beams

phononic band gaps and vibrations in one and two dimensional mass spring structures the vibrational response of finite periodic lattice structures subjected to periodic loading is investigated special attention is devoted to the response in frequency ranges with gaps in the band structure for the corresponding infinite periodic lattice the effects of boundaries viscous damping and imperfections are studied by analyzing two examples a 1 d filter and a 2 d wave guide in 1 d the structural response in the band gap is shown to be insensitive to damping and small imperfections in 2 d the similar effect of damping is noted for one type of periodic structure whereas for another type the band gap effect is nearly eliminated by damping in both 1 d and 2 d it is demonstrated how the free structural boundaries affect the response in the band gap due to local resonances finally 2 d wave guides are considered by replacing the periodic structure with a homogeneous structure in a straight and a 90 bent path and it is shown how the vibrational response is confined to the paths in the band gap frequency ranges

wave propagation in continuous periodic structures research contributions from southampton 1964 1995 abstract after brief reference to some early studies by other investigators this paper focuses mainly on methods developed at the university of southampton since 1964 to analyze and predict the free and forced wave motion in continuous periodic engineering structures beginning with receptance methods which have been applied to periodic beams and rib skin structures it continues with a method of direct solution of the wave equation this uses floquet s principle and has been applied to beams and quasi one dimensional periodic plates and cylindrical shells sample curves of the propagation and attenuation constants pertaining to these structures are presented a limited discussion of the transfer matrix then follows after which the method of space harmonics is introduced as the method best suited to the prediction of sound radiated from a vibrating periodic structure reviewed next are some theorems and variational principles relating to periodic structures which have been developed at southampton and which form a basis for finding natural frequencies of finite structures or for computing free and forced wave notion by energy methods this has led to the finite element method in its standard and hierarchical forms being used to study wave motion in genuine two dimensional and three dimensional structures examples of this work are shown the method of phased array receptance functions is then introduced as possibly the easiest way of setting up exact equations for the propagation constants of uniform quasi one dimensional periodic structures a summary is finally presented of the limited and early work performed at southampton on simple disordered periodic structures

wave propagation in two dimensional periodic lattices plane wave propagation in infinite two dimensional periodic lattices is investigated using floquet bloch principles frequency bandgaps and spatial filtering phenomena are examined in four representative planar lattice topologies hexagonal honeycomb kagome lattice triangular honeycomb and the square honeycomb these topologies exhibit dramatic differences in their long wavelength deformation properties long wavelength asymptotes to the dispersion curves based on homogenization theory are in good agreement with the numerical results for each of the four lattices the slenderness ratio of the constituent beams of the lattice or relative density has a significant influence on the band structure the techniques developed in this work can be used to design lattices with a desired band structure the observed spatial filtering effects due to anisotropy at high frequencies short wavelengths of wave propagation are consistent with the lattice symmetries

generic properties for semialgebraic programs in this paper we study genericity for the class of semialgebraic optimization problems with equality and inequality constraints in which every problem of the class is obtained by linear perturbations of the objective function while the core objective function and the constraint functions are kept fixed assume that the linear independence constraint qualification is satisfied at every point in the constraint set it is shown that almost all problems in the class are such that i the restriction of the objective function on the constraint set is coercive and regular at infinity ii there is a unique optimal solution lying on a unique active manifold at which the strict complementarity and second order sufficiency conditions the quadratic growth condition and the holder type global error bound hold and iii all minimizing sequences converge furthermore the active manifold is constant and the optimal solution and the optimal value function depend analytically under local perturbations of the

variation of mixed hodge structure i around 1970 griffiths introduced the notion of a variation of hodge structure on a complex manifold s see 17 w 2 it constitutes the axiomatization of the features possessed by the local systems of cohomology associated to a family of compact k ihler manifolds esp smooth projective varieties parametrized by s these are for a variation of hodge structure u of weight m 0 a locally constant sheaf u of q vector spaces or u of real vector spaces 1 hodge filtration horizontality a decreasing filtration o r of gs by holomorphic subbundles such that the evident connection v on takes o p into q 2 hodge decompositions if gs denotes the sheaf of functions on s and if

on the phase reduction and response dynamics of neural oscillator populations weundertake a probabilistic analysis of the response of repetitively firing neural populations to simple pulselike stimuli recalling and extending results from the literature we compute phase response curves prcs valid near bifurcations to periodic firing for hindmarsh rose hodgkin huxley fitzhugh nagumo and morris lecar models encompassing the four generic codimension one bifurcations phase density equations are then used to analyze the role of the bifurcation and the resulting prc in responses to stimuli in particular we explore the interplay among stimulus duration baseline firing frequency and population level response patterns we interpret the results in terms of the signal processing measure of gain and discuss further applications and experimentally testable predictions

condition for alternans and stability of the 1 1 response pattern in a memory model of paced cardiac dynamics we analyze a mathematical model of paced cardiac muscle consisting of a map relating the duration of an action potential to the preceding diastolic interval as well as the preceding action potential duration thereby containing some degree of memory the model displays rate dependent restitution so that the dynamic and s1 s2 restitution curves are different a manifestation of memory in the model we derive a criterion for the stability of the 1 1 response pattern displayed by this model it is found that the stability criterion depends on the slope of both the dynamic and s1 s2 restitution curves and that the pattern can be stable even when the individual slopes are greater or less than one we discuss the relation between the stability criterion and the slope of the constant bcl restitution curve the criterion can also be used to determine the bifurcation from the 1 1 response pattern to alternans we demonstrate that the criterion can be evaluated readily in experiments using a simple pacing protocol thus establishing a method for determining whether actual myocardium is accurately described by such a mapping model we illustrate our results by considering a specific map recently derived from a three current membrane model and find that the stability of the 1 1 pattern is accurately described by our criterion in addition a numerical experiment is performed using the three current model to illustrate the application of the pacing protocol and the evaluation of the criterion

topological equivalence of complex polynomials abstract the following numerical control over the topological equivalence is proved two complex polynomials in n 3 variables and with isolated singularities are topologically equivalent if one deforms into the other by a continuous family of polynomial functions f s c n c with isolated singularities such that the degree the number of vanishing cycles and the number of atypical values are constant in the family

period doubling instability and memory in cardiac tissue theoretical studies have indicated that alternans period doubling instability of action potential duration is associated with a restitution relation with a slope or 1 however recent experimental findings suggest that the slope of the restitution relation is not necessarily predictive of alternans here we compared a return map memory model to action potential data from an ionic model and found that the memory model reproduced dynamics that could not be explained by a unidimensional restitution relation using linear stability analysis we determined the onset of the alternans in the memory model and confirmed that the slope of the restitution curve was not predictive

control of cardiac alternans in a mapping model with memory a generic feature of cardiac muscle is that the duration of an action potential depends on the long term history of previous action potentials known as cardiac memory even though memory is known to be an important physiological response there have only been limited studies of its effect on cardiac dynamics here we investigate a map based model of paced myocardium in the presence of closed loop feedback control the model relates the duration of an action potential to the preceding diastolic interval as well as the preceding action potential duration and thus has some degree of memory we find that the range of parameters over which control is effective can be enlarged or reduced by memory a prediction that is independent of the specific functional form of the map our work suggests that modifying the degree of memory e g pharmacological agents with some form of feedback control may be an effective strategy for the maintenance of normal cardiac function 2004 elsevier b v all rights reserved

milnor number at infinity topology and newton boundary of a polynomial function in this paper we show that the euler characteristic of the generic fibre of a complex polynomial function f c n c can be easily computed using the newton number of f we apply this result to study polynomials with a finite number of critical points

the discriminant of a system of equations abstract we introduce and study the discriminant of a system of polynomial equations with indeterminate coefficients generalizing a number of known special cases such as the sparse resultant and the a determinant and inheriting their nice properties one such property is the fact that the discriminant is always a hypersurface in the space of systems of equations which seems unexpected when compared to the existence of dual defect toric varieties this fact reflects the combinatorics of a certain partial order relation on the set of tuples of faces of a tuple of polytopes the newton polytopes of the equations in our case generalizing the poset of faces of one polytope this combinatorics is the main challenge in our work and can be reduced to a certain fact of tropical geometry which resembles the geometry of the discriminants that we study as a sample application of our version of the multivariate discriminant we prove that atypical fibers of a generic polynomial map are distinguished by their euler characteristics and the bifurcation set of such a map is a hypersurface whose degree we explicitly compute

milnor numbers and the topology of polynomial hypersurfaces letf n 1 be a polynomial the problem of determining the homology groupsh q f 1 c c in terms of the critical points off is considered in the best case it is shown for a certain generic class of polynomials tame polynomials that for allc f 1 c has the homotopy type of a bouquet of c n spheres here is the sum of all the milnor numbers off at critical points off and c is the corresponding sum for critical points lying onf 1 c a second best case is also discussed and the homology groupsh q f 1 c are calculated for genericc this case gives an example in which the critical points at infinity off must be considered in order to determine the homology groupsh q f 1 c

motivic infinite cyclic covers abstract we associate with an infinite cyclic cover of a punctured neighborhood of a simple normal crossing divisor on a complex quasi projective manifold under certain finiteness conditions an element in the grothendieck ring k 0 var c which we call motivic infinite cyclic cover and show its birational invariance our construction provides a unifying approach for the denef loeser motivic milnor fiber of a complex hypersurface singularity germ and the motivic milnor fiber of a rational function respectively

isostable reduction of oscillators with piecewise smooth dynamics and complex floquet multipliers phase amplitude reduction is a widely applied technique in the study of limit cycle oscillators with the ability to represent a complicated and high dimensional dynamical system in a more analytically tractable set of coordinates recent work has focused on the use of isostable coordinates which characterize the transient decay of solutions toward a periodic orbit and can ultimately be used to increase the accuracy of these reduced models the breadth of systems to which this phase amplitude reduction strategy can be applied however is still rather limited in this work the theory of phase amplitude reduction using isostable coordinates is further developed to accommodate a broader set of dynamical systems in the first part limit cycles of piecewise smooth dynamical systems are considered and strategies are developed to compute the associated reduced equations in the second part the notion of isostable coordinates for complex valued floquet multipliers is introduced resulting in one phaselike coordinate and one amplitudelike coordinate for each pair of complex conjugate floquet multipliers examples are given with relevance to piecewise smooth representations of excitable cardiomyocytes and the relationship between the reduced coordinate system and the emergence of cardiac alternans is discussed also phase amplitude reduction is implemented for a chaotic externally forced pendulum with complex floquet multipliers and a resulting control strategy for the stabilization of its periodic solution is investigated

criteria for the convergence oscillation and bistability of pulse circulation in a ring of excitable media a discrete model based on a nonlinear difference equation equivalent to a coupled map lattice of high dimension is used to study the dynamics of a circulating pulse in a ring of excitable media such as cardiac cells based on the global and local properties of monotonic restitution and dispersion curves criteria are obtained for the asymptotic stability of the unique steady state pulse circulating at constant frequency as well as for nonconvergent oscillatory behavior of all nonequilibrium trajectories pulse circulating at variable frequency we also demonstrate that in certain cases the system is bistable where an asymptotically stable equilibrium coexists with stable oscillatory solutions

an ozsvath szabo floer homology invariant of knots in a contact manifold abstract using the knot floer homology filtration we define invariants associated to a knot in a three manifold possessing non vanishing floer co homology classes in the case of the ozsvath szabo contact invariant we obtain an invariant of knots in a contact three manifold this invariant provides an upper bound for the thurston bennequin plus rotation number of any legendrian realization of the knot we use it to demonstrate the first systematic construction of prime knots in contact manifolds other than s 3 with negative maximal thurston bennequin invariant perhaps more interesting our invariant provides a criterion for an open book to induce a tight contact structure a corollary is that if a manifold possesses contact structures with distinct non vanishing ozsvath szabo invariants then any fibered knot can realize the classical eliashberg bennequin bound in at most one of these contact structures

monodromy zeta functions at infinity newton polyhedra and constructible sheaves by using sheaf theoretical methods such as constructible sheaves we generalize the formula of libgober sperber concerning the zeta functions of monodromy at infinity of polynomial maps into various directions in particular some formulas for the zeta functions of global monodromy along the fibers of bifurcation points of polynomial maps will be obtained

complex algebraic plane curves via their links at infinity considering that the study of plane cuves has an over 2000 year history and is the seed from which modern algebraic geometry grew surprisingly little is known about the topology of affine algebraic plane curves we topologically classify regular algebraic plane curves in complex affine 2 space using splice diagrams certain decorated trees that code puiseux data at infinity the regularity condition that the curve be a typical fiber of its defining polynomial can conjecturally be avoided we also show that the splice diagram determines such algebraic information as the minimal degree of the curve even in the irregular case among other things this enables algebraic classification of regular algebraic plane curves with given topology

the dimensional family approach in hyper sphere packing a typological study for new patterns structures and interdimensional functions powerful topological invariants associated with sphere packing in euclidean spaces are investigated and new interdimensional functions and structures are delineated that enable those invariants to be readily determined the fundamental invariants discussed are tangency or contact number and the density of centres in the class of recursive packings those whose process of formation is uniquely propagated throughout the entire dimensional continuum an important property of recursivity as here defined is the maximizing of tangency contact or kissing number for a given dimension interdimensional functions are sought and found that can select maximal contact numbers even in dimensions that have many choices thus in 25 space there are 23 packings compatible with t 2 4 196560 but only one of these is maximal and that one our function selects similar properties are found with dimensions 26 where there are at least three thousand more choices than in 25 space through 32 in each case the function selects the maximum t n as given in table 1 this article is part of a book in preparation for publication

spatiotemporal control to eliminate cardiac alternans using isostable reduction abstract cardiac alternans an arrhythmia characterized by a beat to beat alternation of cardiac action potential durations is widely believed to facilitate the transition from normal cardiac function to ventricular fibrillation and sudden cardiac death alternans arises due to an instability of a healthy period 1 rhythm and most dynamical control strategies either require extensive knowledge of the cardiac system making experimental validation difficult or are model independent and sacrifice important information about the specific system under study isostable reduction provides an alternative approach in which the response of a system to external perturbations can be used to reduce the complexity of a cardiac system making it easier to work with from an analytical perspective while retaining many of its important features here we use isostable reduction strategies to reduce the complexity of partial differential equation models of cardiac systems in order to develop energy optimal strategies for the elimination of alternans resulting control strategies require significantly less energy to terminate alternans than comparable strategies and do not require continuous state feedback

quasipositivity as an obstruction to sliceness for an oriented link l s 3 d 4 let s l be the greatest euler characteristic f of an oriented 2 manifold f without closed components smoothly embedded in d 4 with boundary l a knot k is slice if s k 1 realize d 4 in c 2 as z w z 2 w 2 1 it has been conjectured that if v is a nonsingular complex plane curve transverse to s 3 then s v s 3 v d 4 kronheimer and mrowka have proved this conjecture in the case that v d 4 is the milnor fiber of a singularity i explain how this seemingly special case implies both the general case and the slice bennequin inequality for braids as applications i show that various knots are not slice e g pretzel knots like p 3 5 7 all knots obtained from a positive trefoil o 2 3 by iterated untwisted positive doubling

using dominant eigenvalue analysis to predict formation of alternans in the heart were calculated separately for the 1 1 and 1 1alt regions we demonstrated thatcalculated at b onset was significantly different p 0 05 between the two regions our results suggest that this dominant eigenvalue technique can be used to successfully predict the local alternans formation in the heart

neighborhoods of algebraic sets in differential topology a smooth submanifold in a manifold has a tubular neighborhood and in piecewise linear topology a subcomplex of a simplicial complex has a regular neighborhood the purpose of this paper is to develop a similar theory for algebraic and semialgebraic sets the neighbor hoods will be defined as level sets of polynomial or semialgebraic functions introduction let m be an algebraic set in real n space r and let x be a compact algebraic subset of m containing its singular locus if any an algebraic neighborhood of a in m is defined to be a 1 0 8 where o 0 is sufficiently small and a m r is a proper polynomial function for which a 0 and a 1 0 x such an a will be called a rug function occasionally we will need rational or analytic a but this is not a significant generalization algebraic neighborhoods always exist the curve selec tion lemma is used to prove uniqueness anyone familiar with milnor 2 will recognize the technique since uniqueness is a crucial result and since there are a few trouble some small points this proof is given in detail the uniqueness theorem shows that the link of a singularity of an algebraic set m is independent of the embedding of m in its ambient space i have been unable to find a proof of this result in the litera ture in addition an algebraic neighborhood of a nonsingular x in m is shown to be a tubular neighborhood in the sense of differential topology this material is in 1 in 2 the theory is rapidly developed for real and complex projective space by embedding these spaces in real affine space as an application it is shown that when m is an affine algebraic set with projective completion m then the complement in m of a large ball centered at the origin of affine space is an alge braic neighborhood of the intersection of m with the hyperplane at infinity in 3 the theory is generalized to the case where m is a semialgebraic subset of r and x is a compact semialgebraic subset of m for example x could be a non isolated singular point of m a semialgebraic neighborhood of x in m is defined to be a 1 0 o where o 0 is sufficiently small and a m r is a proper semialgebraic function for which a 0 and a 1 0 x again these neighborhoods exist and are unique this is shown by mimicking the proof in the algebraic case using semialgebraic

global asymptotics for multiple integrals with boundaries under convenient geometric assumptions the saddle point method for multidimensional laplace integrals is extended to the case where the contours of integration have boundaries the asymptotics are studied in the case of nondegenerate and of degenerate isolated critical points the incidence of the stokes phenomenon is related to the monodromy of the homology via generalized picard lefschetz formulae and is quantified in terms of geometric indices of intersection exact remainder terms and the hyperasymptotics are then derived a direct consequence is a numerical algorithm to determine the stokes constants and indices of intersections examples are provided

frequency analysis of a heterogeneous perforated panel using a super element formulation abstract heterogeneous perforated panels represent a wide class of structural elements with holes where perforations and matrix are of dissimilar materials e g metallic member of a perforated hybrid joint with resin fillings there is a variety of objectives for introducing holes e g weight saving or mechanical interlock where there is a regular pattern to the arrangement of holes the perforated element can be assimilated to a lattice in the present study numerical analysis has been conducted on investigating the shock frequency filtering phenomenon encountered in two dimensional 2d lattice structures a super element of the primitive cell unit cell of the lattice has been formulated and applied in conjunction with the floquet bloch s principle to study the shock response characteristics plane wave propagation is investigated by constructing the first brillouin zone of the primitive cell in associated fourier space k space the frequency characteristics are presented in the band structure diagram the study emphasizes the existence of frequency pass and stop bands and the extent of this phenomenon the frequency filtering effects of the heterogeneous perforated 2d lattice has shown its potential value to improve the structural performance of perforated hybrid joints subjected to dynamic loads

pencils of curves on smooth surfaces although the theory of singularities of curves resolution classification numerical invariants goes through with comparatively little change in finite characteristic pencils of curves are more difficult bertini s theorem only holds in a much weaker form and it is convenient to restrict to pencils such that when all base points are resolved the general member of the pencil becomes non singular even here the usual rule for calculating the euler characteristic of the resolved surface has to be modified by a term measuring wild ramification we begin by describing this background then proceed to discuss the exceptional members of a pencil in characteristic 0 it was shown by ha and le and by le and weber using topological reasoning that exceptional members can be characterised by their euler characteristics we present a combinatorial argument giving a corresponding result in characteristic p we first treat pencils with no base points and then reduce the remaining case to this