Existence of multiple periodic orbits on star-shaped hamiltonian surfaces

Consider the Hamiltonian system (HS) i = 1, …, N. Here, H ? C²(?^2N, ?). In this paper, we investigate the existence of periodic orbits of (HS) on a given energy surface Σ = {z ? ?^2N; H(z) = c} (c > o is a constant). The surface Σ is required to verify certain geometric assumptions: Σ bounds a star-shaped compact region ? and α? ? ? ? β? for some ellipsoid ? ? ?^2N, o < α < β. We exhibit a constant δ > O (depending in an explicit fashion on the lengths of the main axes of ? and one other geometrical parameter of Σ) such that if furthermore β²/α² < 1 + δ, then (HS) has at least N distinct geometric orbits on Σ. This result is shown to extend and unify several earlier works on this subject (among them works by Weinstein, Rabinowitz, Ekeland-Lasry and Ekeland). In proving this result we construct index theories for an S¹ -action, from which we derive abstract critical point theorems for S¹ -invariant functionals. We also derive an estimate for the minimal period of solutions to differential equatious.     

Projection of diffusions on submanifolds: Application to mean force computation

We consider the problem of sampling a Boltzmann‐Gibbs probability distribution when this distribution is restricted (in some suitable sense) on a submanifold Σ of ?ⁿ implicitly defined by N constraints q₁(x) = ? = q_N(x) = 0 (N < n). This problem arises, for example, in systems subject to hard constraints or in the context of free energy calculations. We prove that the constrained stochastic differential equations (i.e., diffusions) proposed in [7, 13] are ergodic with respect to this restricted distribution. We also construct numerical schemes for the integration of the constrained diffusions. Finally, we show how these schemes can be used to compute the gradient of the free energy associated with the constraints. © 2007 Wiley Periodicals, Inc.     

QUINTESSENTIAL PRIMES AND IDEAL TOPOLOGIES OVER A MODULE

Let I be an ideal of a Noetherian ring R, N a finitely generated R-module and let S be a multiplicatively closed subset of R. We define the Nth (S)-symbolic power of I w.r.t. N as S(I ⁿ N) = ∪ _s∈S (I ⁿ N: _N s). The purpose of this paper is to show that the topologies defined by {Iⁿ N} _n≥0 and {S(Iⁿ N)} _n≥0 are equivalent (resp. linearly equivalent) if and only if S is disjoint from the quintessential (resp. essential) primes of I w.r.t. N.     

On the decomposable numerical range of λI-N

Let N be an nxn normal matrix. For 1≤m≤n we characterize the convexity of the mth decomposable numerical range of λI_n -N which is defined to be

{det(X?(λI_n ?N)X) [sdot] X?C _n×m X ^? X=I_m }.

A related problem on mixed decomposable numerical range of λI_n -N is also discussed.     

ADDITIVE FAMILIES OF INVARIANTS OF SKEWSYMMETRIC TENSORS

Abstract

Let d be a positive integer and F be a field of characteristic 0. Suppose that for each positive integer n, I _n is a polynomial invariant of the usual action of GL_n (F) on Λ^d(Fⁿ), such that for t ? Λ^d(F ^k) and s ? Λ^d(F ^l), I _{k + l} (t l s) = I _k(t)I _t (s), where t ⊥ s is defined in §1.4. Then we say that {I_n} is an additive family of invariants of the skewsymmetric tensors of degree d, or, briefly, an additive family of invariants. If not all the I_n are constant we say that the family is non-trivial. We show that in each even degree d there is a non-trivial additive family of invariants, but that this is not so for any odd d. These results are analogous to those in our paper [3] for symmetric tensors. Our proofs rely on the symbolic method for representing invariants of skewsymmetric tensors. To keep this paper self-contained we expound some of that theory, but for the proofs we refer to the book [2] of Grosshans, Rota and Stein.     

Restricted Isometry Property of Matrices with?Independent Columns and Neighborly Polytopes by?Random Sampling

This paper considers compressed sensing matrices and neighborliness of a centrally symmetric convex polytope generated by vectors ±X ₁,…,±X _N ∈ℝ ⁿ , (N≥n). We introduce a class of random sampling matrices and show that they satisfy a restricted isometry property with overwhelming probability. In particular, we prove that matrices with i.i.d. centered and variance 1 entries that satisfy uniformly a subexponential tail inequality possess the restricted isometry property with overwhelming probability. We show that such “sensing” matrices are valid for the exact reconstruction process of m-sparse vectors via ℓ ₁ minimization with m≤Cn/log ²(cN/n). The class of sampling matrices we study includes the case of matrices with columns that are independent isotropic vectors with log-concave densities. We deduce that if K⊂ℝ ⁿ is a convex body and X ₁,…,X _N ∈K are i.i.d. random vectors uniformly distributed on K, then, with overwhelming probability, the symmetric convex hull of these points is an m-centrally-neighborly polytope with m∼n/log ²(cN/n).     

A note on symbolic and ordinary powers of homogeneous ideals

In this note we are interested in the graded modulesM _k=I^(k)/I^k and , whereI is a saturated ideal in the homogeneous coordinate ringS=K[x₀,…,x_n] of ℙⁿ,I ^(k) is the symbolic power and is the saturation of the ordinary power. Very little is known about these modules, and we provide a bound on their diameters, we compute the Hilbert functions and we study some characteristic submodules in the special case ofn+1 general points in ℙⁿ.

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Sunto In questa nota siamo interessati ai moduli graduatiM _k=I(k)/I^k e , doveI è un ideale saturato nell'anello delle coordinate omogeneeS:=K[x₀,…,x_n] di ℙⁿ,I ^(k) è la potenza simbolica e è la saturazione della potenza ordinaria. Poco è noto su questi moduli e qui viene fornito un limite superiore ai loro diametri. Ne calcoliamo inoltre le funzioni di Hilbert e studiamo alcuni sottomoduli caratteristici nel caso speciale din+1 punti in posizione generale, in ℙⁿ.

     

The commuting graphs of some subsets in the quaternion algebra over the ring of integers modulo <Emphasis Type="Italic">N</Emphasis>

Let R be an arbitrary ring, S be a subset of R, and Z(S) = {s ∈ S | sx = xs for every x ∈ S}. The commuting graph of S, denoted by Γ(S), is the graph with vertex set S \ Z(S) such that two different vertices x and y are adjacent if and only if xy = yx. In this paper, let I _n , N _n be the sets of all idempotents, nilpotent elements in the quaternion algebra ℤ _n [i, j, k], respectively. We completely determine Γ(I _n ) and Γ(N _n ). Moreover, it is proved that for n ≥ 2, Γ(I _n ) is connected if and only if n has at least two odd prime factors, while Γ(N _n ) is connected if and only if n ∈ 2, 2², p, 2p for all odd primes p.     

Bounds for the crossing number of the N-cube

Let Q_n denote the n-dimensional hypercube. In this paper we derive upper and lower bounds for the crossing number v(Q_n), i.e., the minimum number of edge-crossings in any planar drawing of Q_n. The upper bound is close to a result conjectured by Eggleton and Guy and the lower bound is a significant improvement over what was previously known. Let N = 2ⁿ be the number of vertices of Q_n. We show that v(Q_n) < 1/6N². For the lower bound we prove that v(Q_n) = Ω(N(lg N)^{c lg lg N}), where c > 0 is a constant and lg is the logarithm base 2. The best lower bound using standard arguments is v(Q_n) = Ω(N(lg N)²). The lower bound is obtained by constructing a large family of homeomorphs of a subcube with the property that no given pair of edges can appear in more than a constant number of the homeomorphs.     

An extension of Lagrange’s theorem to interval exchange transformations over quadratic fields

LetT be an interval exchange transformation onN intervals whose lengths lie in a quadratic number field. Let {T _n } _n=1 ^∞ be any sequence of interval exchange transformations such thatT ₁ =T andT _n is the first return map induced byT _n-1 on one of its exchanged intervals I_n-1. We prove that {T _n } _n=1 ^∞ contains finitely many transformations up to rescaling. If the interval I_n is chosen according to a consistent pattern of induction, e.g., the first interval is chosen, then there existk,n ₀ ∈ ℤ⁺, λ ∈R ⁺ such that for alln ≥n ₀,I _n = λI _n+k andT _n ,T _n+k are the same up to rescaling. Rephrased arithmetically, this says that a certain family of vectorial division algorithms, applied to quadratic vector spaces, yields sequences of remainders that are eventually periodic. WhenN = 2 the assertion reduces to Lagrange’s classical theorem that the simple continued fraction expansion of a quadratic irrational is eventually periodic. We also discuss the case of periodic induced sequences. These results have applications to topology. In particular, every projective measured foliation on Thurston’s boundary to Teichmüller space that is minimal and metrically ‘quadratic’ is fixed by a hyperbolic element of the modular group. Moreover, if the foliation is orientable, it covers (via a branched covering) an irrational foliation of the two-torus. We also obtain a new proof, for quadratic irrationals, of Boshernitzan’s result that a minimal rank 2 interval exchange transformation is uniquely ergodic. The first author was supported in part by NSF-DMS-9224667. The second author was supported in part by an NSF-NATO fellowship, held at the Université Paris-Sud, Orsay.     

Recurrent approach to Blaschke’s problem

We have obtained a recurrence formula $I_{n+3} = \frac{4(n+3)}{\pi(n+4)}VI_{n+1}We have obtained a recurrence formula I_n+3 = \frac4(n+3)p(n+4)VI_n+1I_{n+3} = \frac{4(n+3)}{\pi(n+4)}VI_{n+1} for integro-geometric moments in the case of a circle with the area V, where I_n = ò\nolimits_{K ?G}sⁿd GI_n = \int \nolimits_{K \cap G}\sigma^{n}{\rm d} G, while in the case of a convex domain K with the perimeter S and area V the recurrence formula

In+3 = \frac8(n+3)V2(n+1)(n+4)p[\fracd In+1d V - \fracIn+1S \fracd Sd V ] I_{n+3} = \frac{8(n+3)V^2}{(n+1)(n+4)\pi}\Big[\frac{{\rm d} I_{n+1}}{{\rm d} V} - \frac{I_{n+1}}{S} \frac{{\rm d} S}{{\rm d} V} \Big]      14. On some sets of dictionaries whose ω ‐powers have a given      Olivier Finkel 《Mathematical Logic Quarterly》2010,56(5):452-460 A dictionary is a set of finite words over some finite alphabet X. The ω ‐power of a dictionary V is the set of infinite words obtained by infinite concatenation of words in V. Lecomte studied in [10] the complexity of the set of dictionaries whose associated ω ‐powers have a given complexity. In particular, he considered the sets ??( Σ 0k) (respectively, ??( Π 0k), ??( Δ 11)) of dictionaries V ? 2* whose ω ‐powers are Σ 0k‐sets (respectively, Π 0k‐sets, Borel sets). In this paper we first establish a new relation between the sets ??( Σ 02) and ??( Δ 11), showing that the set ??( Δ 11) is “more complex” than the set ??( Σ 02). As an application we improve the lower bound on the complexity of ??( Δ 11) given by Lecomte, showing that ??( Δ 11) is in Σ 1 2(22*)\ Π 02. Then we prove that, for every integer k ≥ 2 (respectively, k ≥ 3), the set of dictionaries ??( Π 0k+1) (respectively, ??( Σ 0k +1)) is “more complex” than the set of dictionaries ??( Π 0k) (respectively, ??( Σ 0k)) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)      15. General symbolic powers: A homological point of view      Jaume Martí-Farré 《代数通讯》2013,41(4):1567-1577 Let I be an ideal of a Noetherian ring R and let S be a multiplicatively closed subset of R. We define the n-th (S)-symbolic power of 7 as S(In) = InRs ∩R. The purpose of this paper is to compare the topologies defined by the adic {In}n≤0 and the (S)-symbolic filtration {S(In)}n≥o using the direct system {Exti R(R/In,R)}n≥0      16. Existence of frame sols of type 2nu1      Xu Yunqing  Zhu Lie 《组合设计杂志》1995,3(2):115-133 An SOLS (self-orthogonal latin square) of order n with n1 missing sub-SOLS (holes) of order hi (1 ? i ? k), which are disjoint and spanning (i.e., Σ 1?i?knihi = n), is called a frame SOLS and denoted by FSOLS(h1n1 h2n2 …hknk). In this article, it is shown that for u ? 2, an FSOLS(2nu1) exists if and only if n ? 1 + u. © 1995 John Wiley & Sons, Inc.      17. On classification of immersions ofn-manifolds in (2n−1)-manifolds      Li Banghe 《Commentarii Mathematici Helvetici》1982,57(1):135-144 Under the assumption of (f, M n ,N 2n−1) being trivial, the classification of immersions homotopic tof: M n →N 2n−1 is obtained in many cases. The triviality of (f, M n ,P 2n−1) is proved for anyM n andf. LetM, N be differentiable manifolds of dimensionn and2n−1 respectively. For a mapf: M → N, denote byI[M, N] f the set of regular homotopy classes of immersions homotopic tof. It has been proved in [1] that, whenn>1,I[M, N] f is nonempty for anyf. In this paper we will determine the setI[M, N] f in some cases. For example, ifN=P 2n−1 or more generally, the lens spacesS m 2n−1 =Z m /S 2n−1,M is any orientablen-manifold or nonorientable butn≡0, 1, 3 mod 4, then, for anyf, theI[M, N] f is determined completely. WhenN=R 2n−1, the setI[M, N] of regular homotopy classes of all immersions has been enumerated by James and Thomas in [2] and McClendon in [3] forn>3. Applying our results toN=R 2n−1 we obtain their results again, except for the casen≡2 mod 4 andM nonorientable. Whenn=3, McClendon's results cannot be used. Our results include the casesn=3,M orientable or not (for orientableM, I[M, R5] is known by Wu [4]).      18. Partitioning a graph into convex sets      D. Artigas  S. Dantas  M.C. Dourado  J.L. Szwarcfiter 《Discrete Mathematics》2011,(17):1968 Let G be a finite simple graph. Let S⊆V(G), its closed interval I[S] is the set of all vertices lying on shortest paths between any pair of vertices of S. The set S is convex if I[S]=S. In this work we define the concept of a convex partition of graphs. If there exists a partition of V(G) into p convex sets we say that G is p-convex. We prove that it is NP-complete to decide whether a graph G is p-convex for a fixed integer p≥2. We show that every connected chordal graph is p-convex, for 1≤p≤n. We also establish conditions on n and k to decide if the k-th power of a cycle Cn is p-convex. Finally, we develop a linear-time algorithm to decide if a cograph is p-convex.      19. On the number of solutions of the generalized Ramanujan-Nagell equation D1X2 + DM2 = 2N+2      Jianghua Li 《Quaestiones Mathematicae》2018,41(2):149-163 Let D1, D2 be coprime odd integers with min (D1, D2) > 1, and let N (D1, D2) denote the number of positive integer solutions (x, m, n) of the equation D1x2+Dm2 = 2n+2. In this paper, we prove that N (D1, D2) ≤ 2 except for N (3, 5) = N (5, 3) = 4 and N (13, 3) = N (31, 97) = 3.      20. Families of periodic orbits in resonant reversible systems      Maurício Firmino Silva Lima  Marco Antonio Teixeira 《Bulletin of the Brazilian Mathematical Society》2009,40(4):511-537 We study the dynamics near an equilibrium point p 0 of a Z 2(ℝ)-reversible vector field in ℝ2n with reversing symmetry R satisfying R 2 = I and dimFix(R) = n. We deal with one-parameter families of such systems X λ such that X 0 presents at p 0 a degenerate resonance of type 0: p: q. We are assuming that the linearized system of X 0 (at p 0) has as eigenvalues: λ1 = 0 and λ j = ±iα j , j = 2, … n. Our main concern is to find conditions for the existence of one-parameter families of periodic orbits near the equilibrium.

On some sets of dictionaries whose ω ‐powers have a given

A dictionary is a set of finite words over some finite alphabet X. The ω ‐power of a dictionary V is the set of infinite words obtained by infinite concatenation of words in V. Lecomte studied in [10] the complexity of the set of dictionaries whose associated ω ‐powers have a given complexity. In particular, he considered the sets ??( Σ ⁰k) (respectively, ??( Π ⁰_k), ??( Δ ¹₁)) of dictionaries V ? 2* whose ω ‐powers are Σ ⁰_k‐sets (respectively, Π ⁰_k‐sets, Borel sets). In this paper we first establish a new relation between the sets ??( Σ ⁰₂) and ??( Δ ¹₁), showing that the set ??( Δ ¹₁) is “more complex” than the set ??( Σ ⁰₂). As an application we improve the lower bound on the complexity of ??( Δ ¹₁) given by Lecomte, showing that ??( Δ ¹₁) is in Σ ¹ ₂(2^2*)\ Π ⁰₂. Then we prove that, for every integer k ≥ 2 (respectively, k ≥ 3), the set of dictionaries ??( Π ⁰_k+1) (respectively, ??( Σ ⁰_{k +1})) is “more complex” than the set of dictionaries ??( Π ⁰_k) (respectively, ??( Σ ⁰_k)) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

General symbolic powers: A homological point of view

Let I be an ideal of a Noetherian ring R and let S be a multiplicatively closed subset of R. We define the n-th (S)-symbolic power of 7 as S(Iⁿ) = IⁿR_s ∩R. The purpose of this paper is to compare the topologies defined by the adic {I_n}_n≤0 and the (S)-symbolic filtration {S(Iⁿ)}n≥o using the direct system {Extⁱ _R(R/Iⁿ,R)}_n≥0     

Existence of frame sols of type 2nu1

An SOLS (self-orthogonal latin square) of order n with n₁ missing sub-SOLS (holes) of order h_i (1 ? i ? k), which are disjoint and spanning (i.e., Σ _1?i?kn_ih_i = n), is called a frame SOLS and denoted by FSOLS(h₁ⁿ¹ h₂ⁿ² …h_k^nk). In this article, it is shown that for u ? 2, an FSOLS(2ⁿu¹) exists if and only if n ? 1 + u. © 1995 John Wiley & Sons, Inc.     

On classification of immersions ofn-manifolds in (2n−1)-manifolds

Under the assumption of (f, M ⁿ ,N ²ⁿ⁻¹) being trivial, the classification of immersions homotopic tof: M ⁿ →N ²ⁿ⁻¹ is obtained in many cases. The triviality of (f, M ⁿ ,P ²ⁿ⁻¹) is proved for anyM ⁿ andf. LetM, N be differentiable manifolds of dimensionn and2n−1 respectively. For a mapf: M → N, denote byI[M, N] _f the set of regular homotopy classes of immersions homotopic tof. It has been proved in [1] that, whenn>1,I[M, N] _f is nonempty for anyf. In this paper we will determine the setI[M, N] _f in some cases. For example, ifN=P ²ⁿ⁻¹ or more generally, the lens spacesS _m ²ⁿ⁻¹ =Z _m /S ²ⁿ⁻¹,M is any orientablen-manifold or nonorientable butn≡0, 1, 3 mod 4, then, for anyf, theI[M, N] _f is determined completely. WhenN=R ²ⁿ⁻¹, the setI[M, N] of regular homotopy classes of all immersions has been enumerated by James and Thomas in [2] and McClendon in [3] forn>3. Applying our results toN=R ²ⁿ⁻¹ we obtain their results again, except for the casen≡2 mod 4 andM nonorientable. Whenn=3, McClendon's results cannot be used. Our results include the casesn=3,M orientable or not (for orientableM, I[M, R⁵] is known by Wu [4]).     

Partitioning a graph into convex sets

Let G be a finite simple graph. Let S⊆V(G), its closed interval I[S] is the set of all vertices lying on shortest paths between any pair of vertices of S. The set S is convex if I[S]=S. In this work we define the concept of a convex partition of graphs. If there exists a partition of V(G) into p convex sets we say that G is p-convex. We prove that it is NP-complete to decide whether a graph G is p-convex for a fixed integer p≥2. We show that every connected chordal graph is p-convex, for 1≤p≤n. We also establish conditions on n and k to decide if the k-th power of a cycle C_n is p-convex. Finally, we develop a linear-time algorithm to decide if a cograph is p-convex.     

On the number of solutions of the generalized Ramanujan-Nagell equation D1X2 + DM2 = 2N+2

Let D₁, D₂ be coprime odd integers with min (D₁, D₂) > 1, and let N (D₁, D₂) denote the number of positive integer solutions (x, m, n) of the equation D₁x²+D^m₂ = 2ⁿ⁺². In this paper, we prove that N (D₁, D₂) ≤ 2 except for N (3, 5) = N (5, 3) = 4 and N (13, 3) = N (31, 97) = 3.     

Families of periodic orbits in resonant reversible systems

We study the dynamics near an equilibrium point p ₀ of a Z ₂(ℝ)-reversible vector field in ℝ²ⁿ with reversing symmetry R satisfying R ² = I and dimFix(R) = n. We deal with one-parameter families of such systems X _λ such that X ₀ presents at p ₀ a degenerate resonance of type 0: p: q. We are assuming that the linearized system of X ₀ (at p ₀) has as eigenvalues: λ₁ = 0 and λ _j = ±iα _j , j = 2, … n. Our main concern is to find conditions for the existence of one-parameter families of periodic orbits near the equilibrium.