This is the second part of a work devoted to the study of the set of integer solutions to a system of inequalities determined by homogeneous forms. According to the heuristics that the number of such solutions should match the volume of the set of real solutions, and among other results, sharp estimates for the volume of the semialgebraic domain under consideration were established in the first part. Here, these estimations are employed to establish (a set of) statements providing a global count on the number of solutions to the Diophantine inequalities under consideration. The results thus obtained go beyond the usual assumptions of smoothness and of nonvanishing of the Gaussian curvature. Specifically, the introduction of a semialgebraic level of flatness emerging from geometric tomography enables one to characterise the cases when an accurate counting estimate can be obtained in thin neighbourhoods of an algebraic variety. This is done with a specific view towards applications to Diophantine approximation.
The Lq-spectrum of a Borel measure is a fundamental concept in multifractal analysis. It is widely recognized that the Lq-spectrum associated with a fractal measure provides significant insights into its underlying dynamics and geometry. Consequently, the study of the Lq-spectrum is crucial for understanding dynamical systems and fractal measures. Our objective in this paper is to determine the exact rate of convergence of the Lq-spectra for Moran measures satisfying the Set Strong Separation Condition. As an application, we demonstrate that the empirical multifractal moment measures converge weakly to the normalized multifractal measures. Finally, we reexamine the analysis using tube formulas, and we try to show that the multifractal and fractal dimensions of the overlaps in a Moran set satisfying the Strong Open Set Condition are strictly smaller than the dimension of the set itself.
This is the second part of a work devoted to the study of the set of integer solutions to a system of inequalities determined by homogeneous forms. According to the heuristics that the number of such solutions should match the volume of the set of real solutions, and among other results, sharp estimates for the volume of the semialgebraic domain under consideration were established in the first part. Here, these estimations are employed to establish (a set of) statements providing a global count on the number of solutions to the Diophantine inequalities under consideration. The results thus obtained go beyond the usual assumptions of smoothness and of nonvanishing of the Gaussian curvature. Specifically, the introduction of a semialgebraic level of flatness emerging from geometric tomography enables one to characterise the cases when an accurate counting estimate can be obtained in thin neighbourhoods of an algebraic variety. This is done with a specific view towards applications to Diophantine approximation.
We investigate the existence and regularity of locally invariant manifolds near an approximately invariant set that satisfies a geometric hyperbolicity condition with respect to an abstract "generalized" dynamical system in Banach spaces. This hyperbolicity framework, which we term partial normal hyperbolicity, bridges the gap between normal hyperbolicity and partial hyperbolicity-concepts previously studied in finite dimensions and specific PDE contexts. Our generalized dynamical system accommodates non-smooth, non-Lipschitz, and even "non-mapping" dynamics, making it applicable to both well-posed and ill-posed differential equations. As an illustrative application, we employ our results to analyze the dynamics of whiskered tori.
We determine the maximal domain of meromorphy of any Euler product f(s1, . . . , sk) = p h(p-s1 ,...,p-sk ), where h is the quotient of two polynomials in k variables, with integer coefficients and with constant term equal to 1. More precisely, we define a domain Gamma subset of Ck, on which f admits a meromorphic extension, and such that for any z in partial derivative Gamma, there is no neighborhood of z on which f admits a meromorphic extension. This maximal domain Gamma is described as K & ring; +iRk, where K is an element of Rk is a rational cone, computed from the set of exponents of the two polynomials defining h. We also describe the divisor of f over Gamma, which comes from the local factors h(p-s1, ... , p-sk ), and from the zeta factors zeta(alpha 1s1 + & centerdot; & centerdot; & centerdot;+alpha ksk)-c alpha, where zeta denotes the Riemann zeta-function, corresponding to terms in the expansion of h as a formal infinite product li alpha(1-X1 alpha 1 & centerdot; & centerdot; & centerdot; X alpha k k )c alpha .We focus our study on the hyperplanes in the divisor, allowing us to use tools developed for the single variable case. We complete our study by giving a geometric and arithmetic description of the set of exponents occurring in the infinite product expansion of h, and by showing a new result on the geometric nature of the set of singular points of a holomorphic function defined over a tubular domain.
We consider, in the complex domain, the Euler-Poisson equations describing the motion of a heavy rigid body about a fixed point. The real domain is a particular case of it. The Euler-Poisson equations admit three functionally independent first integrals H1, H2, H3, i.e. the area, geometrical and energy first integrals. In four cases (Euler, Lagrange, Kovalevskaya, kinetic symmetry) a fourth functionally independent first integral appears. It can be found among polynomials that do not depend on all variables. We study when, apart from the four cases above, the Euler-Poisson equations, restricted to the level manifolds of H1, H2 and H3 and all their mutual intersections, admit a new first integral which does not depend on all variables. In this way we cover the partially integrable Goryachev-Chaplygin case and describe, in the complex domain, a new class of partially integrable cases on the level manifold {H1 = 0, H2 = 0}. We also deduce their uniqueness. We also cover the Sretenskii case of partial integrability of the gyrostat equations (which generalizes the Goryachev-Chaplygin case) and describe a new class of their integrable cases in the complex domain. We provide a general quasi-algorithmic method to find all these cases and corresponding partial integrals. The use of computer algebra is unavoidable to carry out our investigations. By following the link (https://sdrive.cnrs.fr/public.php/dav/files/bKmGokMQnM5Jo5f/?accept=zip), the reader can verify all reported computations using EPEPcomp.zip available at that link. Note that a public user can only download files without the edition option.
Let (X, & micro;) be a space of homogeneous type satisfying & micro;(X) = infinity, the doubling property and the reverse doubling condition. Let L be a nonnegative self-adjoint operator on L2(X)whose heat kernel enjoys a Gaussian upper bound. We introduce the weighted homogeneous Bourgain-Morrey-Besov type spaces and Triebel-Lizorkin type spaces associated with the operator L. We obtain their continuous characterizations in terms of Peetre maximal functions, noncompactly supported functional calculus, and heat kernel. Atomic and molecular decompositions of these spaces are also given. As an application, we obtain the boundedness of the fractional power of L, the spectral multiplier of L on these spaces.
The paper is an extensive and systematic study of cardinal invariants we call slalom numbers, describing the combinatorics of sequences of sets of natural numbers. Our general approach, based on relational systems, covers many such cardinal characteristics, including localization and anti-localization cardinals. We show that most of the slalom numbers are connected to topological selection principles, in particular, we obtain the representation of the uniformity of meager and the cofinality of measure. Considering instances of slalom numbers parametrized by ideals on natural numbers, we focus on monotonicity properties with respect to ideal orderings and computational formulas for the disjoint sum of ideals. Hence, we get such formulas for several pseudo-intersection numbers as well as for the bounding and dominating numbers parametrized with ideals. Based on the effect of adding a Cohen real, we get many consistent constellations of different values of slalom numbers.
We investigate the existence and regularity of locally invariant manifolds near an approximately invariant set that satisfies a geometric hyperbolicity condition with respect to an abstract “generalized" dynamical system in Banach spaces. This hyperbolicity framework, which we term partial normal hyperbolicity, bridges the gap between normal hyperbolicity and partial hyperbolicity–concepts previously studied in finite dimensions and specific PDE contexts. Our generalized dynamical system accommodates non-smooth, non-Lipschitz, and even “non-mapping" dynamics, making it applicable to both well-posed and ill-posed differential equations. As an illustrative application, we employ our results to analyze the dynamics of whiskered tori.
We give a new characterization of a continuous embedding between two function spaces of type G Gamma. Such spaces are governed by functionals of type IL ( 1 t )q/r )1/q parallel to f parallel to G Gamma(r,q;w,delta) :=f & lowast; (s)r delta(s) ds w(t) dt , triangle(t) 0 0 where f & lowast; is the nonincreasing rearrangement of f, L is an element of (0, infinity], r, q is an element of (0, infinity), w, delta are weights on (0, L) and triangle(t) = St0 delta(s) ds for t is an element of (0, L). To characterize the embedding of such a space, say G Gamma(r1, q1; w1, delta 1), into another, G Gamma(r2, q2; w2, delta 2), means to find a balance condition on the four positive real parameters and the four weights in order that an appropriate inequality holds for every admissible function. We develop a new discretization technique which enables us to get rid of restrictions on parameters imposed in earlier work such as the nondegeneracy conditions or certain relations between the r's and the q's. Such restrictions were caused mainly by the use of duality techniques, which we avoid in this paper. On the other hand, we consider here only the case when q1 <= q2, leaving the reverse case to future work.
We construct a strong Markov process X corresponding to the Dirichlet form of Servadei and Valdinoci and use the process to solve the corresponding Neumann boundary problem for the fractional Laplacian and the half-line. When started in (0, infinity), the process behaves like the isotropic alpha-stable process. At the first exit time from (0, infinity), X jumps to a point y on the negative half-line, spends an exponential time at y, then jumps back to the positive half-line and starts afresh. The asymptotic behavior of X strongly depends on the parameter alpha. In particular, its lifetime is finite if alpha is an element of (1, 2), and infinite if alpha is an element of (0, 1]. Our construction is based on concatenation of Markov processes. We identify the Dirichlet form corresponding to X with the form of Servadei and Valdinoci, prove a Hardy inequality, and propose various characterizations of the domain for the form. Under suitable assumptions, the solution of the Neumann boundary problem is given by the Green operator corresponding to the process X.
Non-collision singularities of the n-body problem are initial conditions without global solution, that however do not lead to collision in the limit. The question whether the set of non-collision singularities in the n-body problem is improbable, is open and the first in Barry Simon's list of fifteen problems in mathematical physics from the year 1984. By now this question is only answered affirmatively in the case of n = 4 bodies. We cannot answer the full question for five or more particles, but can give two different kinds of partial answers proving that some suitable subsets of the set of non-collision singularities are improbable. One case is similar to the case of a non-collision singularity with four particles, but there might be more particles that do not come close to the diverging subsystem close to the escape time and have at most binary collisions at the escape time. The second improbability result is for orbits similar to the first examples of non-collision singular orbits, constructed by Xia in 1992. They consist of five particles with two outer binaries and a messenger commuting between the binaries. Under a suitable mass restriction we can prove the improbability of these orbits. The tool for proving these improbability results is the so-called Poincar & eacute; surface method developed by Fleischer (2019). The argument is based on a fine analysis of the orbits and deriving quantitative estimates at suitable reference times during the passages. The results are some application and extension of results proved in the author's 2023 dissertation.
We construct a strong Markov process corresponding to the Dirichlet form of Servadei and Valdinoci and use the process to solve the corresponding Neumann boundary problem for the fractional Laplacian and the half-line.
We analyze the problem of determining Waring decompositions of the powers of any quadratic form over the field of complex numbers. Our main goal is to provide information about their rank and also to obtain decompositions whose size is as close as possible to this value. This is a classical problem and these forms assume importance especially because of their invariance under the action of the special orthogonal group. We give the detailed procedure to prove that the apolar ideal of the s-th power of a quadratic form is generated by the harmonic polynomials of degree s+1. We also generalize and improve some of the results on real decompositions given by B. Reznick in his notes of 1992, focusing on possibly minimal decompositions and providing new ones, both real and complex. We investigate the rank of the second power of a non-degenerate quadratic form in n variables, which in most cases is equal to (n^2+n+2)/2, and also give some results on powers of ternary quadratic forms.
For 1 <= p <= q <= infinity and a locally convex space E, we introduce and study the (V-& lowast;) subsets of order (p, q) of E and the (V) subsets of order (p, q) of the topological dual E ' of E. Using these sets we define and study (sequential) Pe & lstrok;czy & nacute;ski's property V-& lowast; of order (p, q), (sequential) Pe & lstrok;czy & nacute;ski's property V of order (p,q), and Pe & lstrok;czy & nacute;ski's property (u) of order p in the class of all locally convex spaces. To this end, we also introduce and study several new completeness-type properties, weak barrelledness conditions, Schur-type properties, the Gantmacher property for locally convex spaces, and (q, p)-summing operators between locally convex spaces. Applications to some classical function spaces are given.
We lay down the foundations of the theory of spaces of distributions on the product X1 x X2 of doubling metric measure spaces X1, X2 in the presence of non-negative self-adjoint operators L1, L2, whose heat kernels have Gaussian localization and the Markov property. This theory includes the development of two-parameter functional calculus induced by L1, L2, integral operators with highly localized kernels, test functions and distributions associated to L1, L2, and spectral spaces accompanied by maximal Peetre and Nikolski type inequalities. Hardy spaces are developed in this two-parameter product setup. Two types of Besov and Triebel-Lizorkin spaces are introduced and studied: ordinary spaces and spaces with dominating mixed smoothness, with emphasis on the latter. Embedding results are obtained and spectral multipliers are developed.
We investigate the quotients of Banach manifolds with respect to free actions of pseudogroups of local diffeomorphisms. These quotient spaces are called H-manifolds since the corresponding simply transitive action of the pseudogroup on its orbits is regarded as a homogeneity condition. The importance of these structures stems from the fact that for every regular foliation without holonomy of a Banach manifold, the corresponding leaf space has the natural structure of an H-manifold. This is our main technical result, and one of its remarkable consequences is an infinite-dimensional version of Sophus Lie's third fundamental theorem, to the effect that every real Banach-Lie algebra can be integrated to an H-group, that is, a group object in the category of H-manifolds. In addition to these general results we discuss a wealth of examples of H-groups which are not Banach-Lie groups.
We introduce, investigate and compare several order type relations on the set of tripotents in a JB^*-triple. The main two relations we address are ≤_h and ≤_n. We say that u≤_h e (or u≤_n e) if u is a self-adjoint (or normal) element of the Peirce-2 subspace associated to e considered as a unital JB^*-algebra with unit e. It turns out that these relations need not be transitive, so we consider their transitive hulls as well. Properties of these transitive hulls appear to be closely connected with types of von Neumann algebras, with the results on products of symmetries, with determinants in finite-dimensional Cartan factors, with finiteness and other structural properties of JBW^*-triples.
In this paper, we consider a two-phase problem for two immiscible, viscous, incompressible fluids in the presence of a uniform gravitational field acting vertically downward in the N-dimensional Euclidean space R (N) , N >= 2 . The two fluids are separated from one another by the sharp interface Gamma( t ) = { ( x ' , x (N) ) : x ' is an element of R N - 1 , x (N) = eta ( x ' , t ) } at time t >= 0 and surface tension is included on Gamma( t ) . The fluid occupying the region x (N) > eta ( x ' , t ) is called the upper fluid, while the other fluid occupying the region x (N) < eta ( x ' , t ) is called the lower fluid. It is well-known that the trivial steady state, i.e., the motionless state with the flat interface x (N )= 0 , is unstable if the upper fluid is heavier than the lower one due to gravity. This instability is called the Rayleigh- Taylor instability. On the other hand, the present paper treats the following two cases: (i) the lower fluid is heavier than the upper one; (ii) the two fluids have equal density. For these two cases, we prove time decay estimates of L (p)-L (q) type for the Stokes semigroup associated with a linearized system of the above two-phase problem. We emphasize that the decay rate of the semigroup generated by the fractional Laplacian appears in the L (p)-L (q) time decay estimates of the Stokes semigroup.
In this paper we study the elementary theory of graph products of groups and show that under natural conditions on the vertex groups we can recover (the core of) the underlying graph and the associated vertex groups. More precisely, we require the vertex groups to satisfy a non-generic almost positive sentence, a condition which generalizes a range of natural ``non-freeness conditions"such as the satisfaction of a group law, having nontrivial center or being boundedly simple. As a corollary, we determine an invariant of the elementary theory of a right-angled Artin group, the core of the defining graph, which we conjecture to determine the elementary class of the RAAG. We further combine our results with the results of Sela on free products of groups to describe all finitely generated groups elementarily equivalent to certain RAAGs. We also deduce rigidity results on the elementary classification of graph products of groups for specific types of vertex groups, such as finite, nilpotent or classical linear groups.
