even minkowski valuations a new integral representation of smooth translation invariant and rotation equivariant even minkowski valuations is established explicit formulas relating previously obtained descriptions of such valuations with the new more accessible one are also derived moreover the action of alesker s hard lefschetz operators on these minkowski valuations is explored in detail

a generalization of n matrices abstract ky fan defines an n matrix to be a matrix of the form a ti b b 0 t b where b is the spectral radius of b and is the maximum of the spectral radii of all principal submatrices of b of order n 1 in this paper we define the closure n 0 matrices of n matrices by letting t it is shown that if a z and a 1 a n certain inequalities of n matrices are shown to hold for n 0 matrices and a method for constructing an n matrix from an m matrix is given

valuations on sobolev spaces all affinely covariant convex body valued valuations on the sobolev space w 1 1 bbb r n are completely classified it is shown that there is a unique such valuation for blaschke addition this valuation turns out to be the operator which associates with each function f in w 1 1 bbb r n the unit ball of its optimal sobolev norm

a characterization of blaschke addition a characterization of blaschke addition as a map between origin symmetric con vex bodies is established this results from a new characterization of minkowski addition as a map between origin symmetric zonoids combined with the use of l evy prokhorov metrics a full set of examples is provided that show the results are in a sense the best possible

convolutions transforms and convex bodies the paper studies convex bodies and star bodies in bbb r n by using radon transforms on grassmann manifolds p cosine transforms on the unit sphere and convolutions on the rotation group of bbb r n it presents dual mixed volume characterizations of i intersection bodies and l p balls which are related to certain volume inequalities for cross sections of convex bodies it considers approximations of special convex bodies by analytic bodies and various finite sums of ellipsoids which preserve special geometric properties convolution techniques are used to derive formulas for mixed volumes mixed surface measures and p cosine transforms they are also used to prove characterizations of geometric functionals such as surface area and dual quermassintegrals

intrinsic circular motions in stochastic pairwise epidemic models abstract the main purpose of the paper is to show that there exists intrinsically stochastic circular motion in a pairwise epidemic model which does not exist in the classical mean field susceptible infective susceptible sis models here a basic pairwise sis epidemic model is adopted by the method of scale separation in the case of a large population we can get a deterministic dynamical system which represents the temporal evolution of averaged densities of the system plus a diffusion process which is centered at the orbit of the aforementioned deterministic system it is discovered that there is a mode of circular motion in the diffusion process we consider that this intrinsic circular motion must originate from the fact that the original stochastic pairwise process is time irreversible since the intrinsic periods which are calculated from the two systems respectively have similar forms

convolutions and multiplier transformations of convex bodies rotation intertwining maps from the set of convex bodies in into itself that are continuous linear operators with respect to minkowski and blaschke addition are investigated the main focus is on blaschke minkowski homomorphisms we show that such maps are represented by a spherical convolution operator an application of this representation is a complete classification of all even blaschke minkowski homomorphisms which shows that these maps behave in many respects similar to the well known projection body operator among further applications is the following result if an even blaschke minkowski homomorphism maps a convex body to a polytope then it is a constant multiple of the projection body operator

nonnegative matrices with power invariant zero patterns abstract if a is a nonsingular m matrix the elements of the sequence a k all have the same zero pattern using the drazin inverse we show that a similar zero pattern invariance property holds for a class of matrices which is larger than the generalized m matrices

structure of stochastic dynamics near fixed points we analyze the structure of stochastic dynamics near either a stable or unstable fixed point where the force can be approximated by linearization we find that a cost function that determines a boltzmann like stationary distribution can always be defined near it such a stationary distribution does not need to satisfy the usual detailed balance condition but might have instead a divergence free probability current in the linear case the force can be split into two parts one of which gives detailed balance with the diffusive motion whereas the other induces cyclic motion on surfaces of constant cost function by using the jordan transformation for the force matrix we find an explicit construction of the cost function we discuss singularities of the transformation and their consequences for the stationary distribution this boltzmann like distribution may be not unique and nonlinear effects and boundary conditions may change the distribution and induce additional currents even in the neighborhood of a fixed point

valuations and busemann petty type problems projection and intersection bodies define continuous and gl n contravariant valuations they played a critical role in the solution of the shephard problem for projections of convex bodies and its dual version for sections the busemann petty problem we consider the question whether fkfl implies v k v l where f is a homogeneous continuous operator on convex or star bodies which is an so n equivariant valuation important previous results for projection and intersection bodies are extended to a large class of valuations

volume inequalities of convex bodies from cosine transforms on grassmann manifolds abstract the l p cosine transform on grassmann manifolds naturally induces finite dimensional banach norms whose unit balls are origin symmetric convex bodies in r n reverse isoperimetric type volume inequalities for these bodies are established which extend results from the sphere to grassmann manifolds

predator prey cycles from resonant amplification of demographic stochasticity we present the simplest individual level model of predator prey dynamics and show via direct calculation that it exhibits cycling behavior the deterministic analogue of our model recovered when the number of individuals is infinitely large is the volterra system with density dependent prey reproduction which is well known to fail to predict cycles this difference in behavior can be traced to a resonant amplification of demographic fluctuations which disappears only when the number of individuals is strictly infinite our results indicate that additional biological mechanisms such as predator satiation may not be necessary to explain observed predator prey cycles in real finite populations

the 3 ball is a local pessimum for packing abstract it was conjectured by ulam that the ball has the lowest optimal packing fraction out of all convex three dimensional solids here we prove that any origin symmetric convex solid of sufficiently small asphericity can be packed at a higher efficiency than balls we also show that in dimensions 4 5 6 7 8 and 24 there are origin symmetric convex bodies of arbitrarily small asphericity that cannot be packed using a lattice as efficiently as balls can be

log concavity properties of minkowski valuations new orlicz brunn minkowski inequalities are established for rigid motion compatible minkowski valuations of arbitrary degree these extend classical log concavity properties of intrinsic volumes and generalize seminal results of lutwak and others two different approaches which refine previously employed techniques are explored it is shown that both lead to the same class of minkowski valuations for which these inequalities hold an appendix by semyon alesker contains the proof of a new classification of generalized translation invariant valuations

subword complexity and projection bodies abstract a polytope p 0 1 d and an 0 1 d induce a so called hartman sequence h p 0 1 z which is by definition 1 at the kth position if k mod 1 p and 0 otherwise k z we prove an asymptotic formula for the subword complexity of such a hartman sequence this result establishes a connection between symbolic dynamics and convex geometry if the polytope p is convex then the subword complexity of h p asymptotically equals the volume of the projection body p of p for almost all 0 1 d

the affine sobolev zhang inequality on bv rn abstract the affine sobolev zhang inequality is extended to b v r n the space of functions of bounded variation on r n and the equality cases are characterized as a consequence the petty projection inequality for sets of finite perimeter which implies the isoperimetric inequality for sets of finite perimeter is established

contravariant minkowski valuations a complete classification of all continuous gl n contravariant minkowski valuations is established as an application we present a family of sharp isoperimetric inequalities for such valuations which generalize the classical petty projection inequality

inequalities for mixed projection bodies mixed projection bodies are related to ordinary projection bodies zonoids in the same way that mixed volumes are related to ordinary volume analogs of the classical inequalities from the brunn minkowski theory such as the minkowski brunn minkowski and aleksandrov fenchel inequalities are developed for projection and mixed projection bodies

projection functions area measures and the alesker fourier transform abstract dual to koldobsky s notion of j intersection bodies the class of j projection bodies is introduced generalizing minkowski s classical notion of projection bodies of convex bodies a fourier analytic characterization of j projection bodies in terms of their area measures of order j is obtained in turn this yields an equivalent characterization of j projection bodies involving alesker s fourier type transform on translation invariant smooth spherical valuations as applications of these results several basic properties of j projection bodies are established and new non trivial examples are constructed

crofton measures and minkowski valuations a description of continuous rigid motion compatible minkowski valuations is established as an application we present a brunn minkowski type inequality for intrinsic volumes of these valuations

affine vs euclidean isoperimetric inequalities abstract it is shown that every even zonal measure on the euclidean unit sphere gives rise to an isoperimetric inequality for sets of finite perimeter which directly implies the classical euclidean isoperimetric inequality the strongest member of this large family of inequalities is shown to be the only affine invariant one among them the petty projection inequality as an application a family of sharp sobolev inequalities for functions of bounded variation is obtained each of which is stronger than the classical sobolev inequality moreover corresponding families of l p isoperimetric and sobolev type inequalities are also established

blaschke and minkowski endomorphisms of convex bodies we consider maps of the family of convex bodies in euclidean d dimensional space into itself that are compatible with certain structures on this family a minkowski endomorphism is a continuous minkowski additive map that commutes with rotations for d 3 a representation theorem for such maps is given showing that they are mixtures of certain prototypes these prototypes are obtained by applying the generalized spherical radon transform to support functions we give a complete characterization of weakly monotonic minkowski endomorphisms a corresponding theory is developed for blaschke endomorphisms where additivity is now understood with respect to blaschke addition using a special mixed volume an adjoining operator can be introduced this operator allows one to identify the class of blaschke endomorphisms with the class of weakly monotonic non degenerate and translation covariant minkowski endomorphisms the following application is also shown if a weakly monotonic and non trivial endomorphism maps a convex body to a homothet of itself then this body must be a ball

on markham s m matrix properties abstract if a is an n n irreducible singular m matrix where n 1 it is shown that there is no positive integer m such that a m is triangular related results are proven for those matrices pap t where a is a reducible singular m matrix and p is a particular permutation matrix and for certain n 0 matrices an analogue of the minkowski determinant theorem is extended to a larger subclass of the m matrices

lutwak petty projection inequalities for minkowski valuations and their duals abstract lutwak s volume inequalities for polar projection bodies of all orders are generalized to polarizations of minkowski valuations generated by even zonal measures on the euclidean unit sphere this is based on analogues of mixed projection bodies for such minkowski valuations and a generalization of the notion of centroid bodies a new integral representation is used to single out lutwak s inequalities as the strongest among these families of inequalities which in turn are related to a conjecture on affine quermassintegrals in the dual setting a generalization of volume inequalities for intersection bodies of all orders by leng and lu is proved these results are related to grinberg s inequalities for dual affine quermassintegrals

on powers of nonnegative matrices we investigate the structure of powers of nonnegative matrices a and in particular characterize those a for which some power is essentially triangular we also show how the number of irreducible components of a can be determined from its powers

mathematical formalism for isothermal linear irreversibility we prove the equivalence among symmetricity time reversibility and zero entropy production of the stationary solutions of linear stochastic differential equations a sufficient and necessary rever

the steiner formula for minkowski valuations a steiner type formula for continuous translation invariant minkowski valuations is established in combination with a recent result on the symmetry of rigid motion invariant homogeneous bivaluations this new steiner type formula is used to obtain a family of brunn minkowski type inequalities for rigid motion intertwining minkowski valuations

projection bodies and valuations let be the projection operator which maps every polytope to its projection body it is well known that maps the set of polytopes pn in rn into pn that it is a valuation and that for every p pn p is affinely associated to p it is shown that these properties characterize the projection operator this proves a conjecture by lutwak

extremum problems for the cone volume functional of convex polytopes abstract lutwak yang and zhang defined the cone volume functional u over convex polytopes in r n containing the origin in their interiors and conjectured that the greatest lower bound on the ratio of this centro affine invariant u to volume v is attained by parallelotopes in this paper we give affirmative answers to the conjecture in r 2 and r 3 some new sharp inequalities characterizing parallelotopes in r n are established moreover a simplified proof for the conjecture restricted to the class of origin symmetric convex polytopes in r n is provided