The limiting distribution of the trace of a random plane partition

We study the asymptotic behaviour of the trace (the sum of the diagonal parts) τ _n = τ _n (ω) of a plane partition ω of the positive integer n, assuming that ω is chosen uniformly at random from the set of all such partitions. We prove that (τ _n − c ₀ n ^2/3)/c ₁ n ^1/3 log^1/2 n converges weakly, as n → ∞, to the standard normal distribution, where c ₀ = ζ(2)/ [2ζ(3)]^2/3, c ₁ = √(1/3/) [2ζ(3)]^1/3 and ζ(s) = Σ _j=1^∞ j ^−s . Partial support given by the National Science Fund of the Bulgarian Ministry of Education and Science, grant No. VU-MI-105/2005.     

A bound for the representability of large numbers by ternary quadratic forms and nonhomogeneous waring equations

The solvability of the equation n = x ² + y ² + 6pz ² (p is a fixed large prime) is proved under some natural congruential conditions and the assumption nm ¹² > p ²¹. As an implication, the solvability of the equation n = x ² + y ² + u ³ + v ³ + z ⁴ + w ¹⁶ + t ^4k+1 for all sufficiently large n is established. Bibliography: 13 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 5–21.     

On Estimates of Factorial Quotients

We consider the factorial quotients (2n − 1)!!/(2n)!! in connection with the Wallis formula n ⁻¹(2n)!!²/(2n − 1)!!² → π. We improve the Wallis inequalities (n + 1/2)⁻¹(2n)!!²/(2n − 1)!!² < π < n ⁻¹(2n)!!²/(2n − 1)!!² for π and obtain new estimates of factorial quotients with error order not worse than 1/n ². __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 3, pp. 349–358, July–September, 2005.     

On (2<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript> + 1)-variable symmetric Boolean functions with submaximum algebraic immunity 2<Superscript><Emphasis Type="Italic">m</Emphasis>−1</Superscript>

All (2 ^m + 1)-variable symmetric Boolean functions with submaximal algebraic immunity 2 ^m−1 are described and constructed. The total number of such Boolean functions is 3² · 2^2m−3 +3 ^m−2 · 2⁴ − 2 for m ⩾ 2. This work was supported by the Major State Basic Research Development Program of China (Grant No. 2004CB3180004) and National Natural Science Foundation of China (Grant No. 60433050)     

Random Subgraphs Of Finite Graphs: III. The Phase Transition For The n-Cube

We study random subgraphs of the n-cube {0,1}ⁿ, where nearest-neighbor edges are occupied with probability p. Let p_c(n) be the value of p for which the expected size of the component containing a fixed vertex attains the value λ2^n/3, where λ is a small positive constant. Let ε=n(p−p_c(n)). In two previous papers, we showed that the largest component inside a scaling window given by |ε|=Θ(2^−n/3) is of size Θ(2^2n/3), below this scaling window it is at most 2(log 2)nε⁻², and above this scaling window it is at most O(ε2ⁿ). In this paper, we prove that for the size of the largest component is at least Θ(ε2ⁿ), which is of the same order as the upper bound. The proof is based on a method that has come to be known as “sprinkling,” and relies heavily on the specific geometry of the n-cube.     

On the Minimum Length of some Linear Codes of Dimension 5

In this paper, we shall prove that the minimum length n_q(5,d) is equal to g_q(5,d) +1 for q⁴−2q²−2q+1≤ d≤ q⁴ − 2q² − q and 2q⁴ − 2q³ − q² − 2q+1 ≤ d ≤ 2q⁴−2q³−q²−q, where g_q(5,d) means the Griesmer bound . Communicated by: J.D. Key     

On generalized energy equality of the Navier-Stokes equations

We show that for every initial dataa εL ²(Ω) there exists a weak solutionu of the Navier-Stokes equations satisfying the generalized energy inequality introduced by Caffarelli-Kohn-Nirenberg forn=3. We also show that if a weak solutionu εL ^s (0,T;L ^q (Ω)) with 2/q + 2/s ≤ 1 and 3/q + 1/s ≤ 1 forn=3, or 2/q + 2/s ≤ 1 andq ≥ 4 forn ≥ 4, thenu satisfies both the generalized and the usual energy equalities. Moreover we show that the generalized energy equality holds only under the local hypothesis thatu εL ^s (ε, T; L ^q (K)) for all compact setsK ⊂ ⊂ Ω and all 0 <ε <T with the same (q, s) as above when 3 ≤n ≤ 10.     

Real Hypersurfaces with Isometric Reeb Flow in Complex Two-Plane Grassmannians

 We classify all real hypersurfaces with isometric Reeb flow in the complex Grassmann manifold G ₂ (ℂ ^m+2 ) of all 2-dimensional linear subspaces in ℂ ^m+2 , m ≥ 3.     

The gonality of smooth curves with plane models

LetC be the normalization of an integral plane curve of degreed with δ ordinary nodes or cusps as its singularities. If δ=0, then Namba proved that there is no linear seriesg _d ^−2/1 and that everyg _d ^−1/1 is cut out by a pencil of lines passing through a point onC. The main purpose of this paper is to generalize his result to the case δ>0. A typical one is as follows: Ifd≥2(k+1), and δ<kd−(k+1)²+3 for somek>0, thenC has no linear seriesg _d ^−3/1 . We also show that ifd≥2k+3 and δ<kd−(k+1)²+2, then each linear seriesg _d ^−2/1 onC is cut out by a pencil of lines. We have similar results forg _d ^−1/1 andg _2d ^−9/1 . Furthermore, we also show that all of our theorems are sharp.     

Algebraic relations for reciprocal sums of odd terms in Fibonacci numbers

In this paper, we prove the algebraic independence of the reciprocal sums of odd terms in Fibonacci numbers ∑ _n=1^∞ F _2n−1⁻¹, ∑ _n=1^∞ F _2n−1⁻², ∑ _n=1^∞ F _2n−1⁻³ and write each ∑ _n=1^∞ F _2n−1^−s (s≥4) as an explicit rational function of these three numbers over ℚ. Similar results are obtained for various series including the reciprocal sums of odd terms in Lucas numbers.      

On a cardinal group invariant related to decompositions of Abelian groups

For each Abelian groupG, a cardinal invariant χ(G) is introduced and its properties are studied. In the special caseG = ℤ ⁿ , the cardinalχ(ℤ ⁿ ) is equal to the minimal cardinality of an essential subset of ℤ ⁿ , i.e., a of a subsetA ⊂ ℤ ⁿ such that, for any coloring of the group ℤ ⁿ inn colors, there exists an infinite one-color subset that is symmetric with respect to some pointα ofA. The estimaten( ^{n +} l)/2 ≤χ(ℤ ⁿ ) < 2n is proved for alln and the relationχ(ℤ ⁿ ) =n(n + 1)/2 forn ≤ 3. The structure of essential subsets of cardinalityχ(ℤ ⁿ ) in ℤ ⁿ is completely described forn ≤ 3. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 341–350, September, 1998.     

Maximum principles for the polyharmonic equation on Lipschitz domains

The Agmon-Miranda maximum principle for the polyharmonic equations of all orders is shown to hold in Lipschitz domains in ℝ³. In ℝn,n≥4, the Agmon-Miranda maximum principle andL ^p-Dirichlet estimates for certainp>2 are shown to fail in Lipschitz domains for these equations. In particular if 4≤n≤2m+1 theL ^p Dirichlet problem for Δ ^m fails to be solvable forp>2(n−1)/(n−3). Supported in part by the NSF.     

Cantor-Bendixson degrees and convexity in ℝ2

We present an ordinal rank, δ³, which refines the standard classification of non-convexity among closed planar sets. The class of closed planar sets falls into a hierarchy of order type ω₁ + 1 when ordered by δ-rank. The rank δ³ (S) of a setS is defined by means of topological complexity of 3-cliques in the set. A 3-clique in a setS is a subset ofS all of whose unordered 3-tuples fail to have their convex hull inS. Similarly, δⁿ (S) is defined for alln>1. The classification cannot be done using δ², which considers only 2-cliques (known in the literature also as “visually independent subsets”), and in dimension 3 or higher the analogous classification is not valid.     

Uniqueness of Nelsons diffusions

Let ℒ≔Δ/2+(∇φ/φ) ·∇ be a generalized Schr?dinger operator or generator of Nelsons diffusion, defined on C ^∞ ₀(D) where φ is a continuous and strictly positive function on an open domain D⊂ℝ ^d such that ∇φ∈L _loc ²(D). Some results are given about the two questions below: (i) Whether does ℒ generate a unique semigroup in L ¹(D, φ² dx)? (ii) Whether the semigroup determined by ℒ is strong Feller? Received: 21 October 1997 / Revised version: 3 September 1998     

On the pontryagin-steenrod-wu theorem

We present a short and direct proof (based on the Pontryagin-Thom construction) of the following Pontryagin-Steenrod-Wu theorem: (a) LetM be a connected orientable closed smooth (n + 1)-manifold,n≥3. Define the degree map deg: π ⁿ (M) →H ⁿ (M; ℤ) by the formula degf =f*[S ⁿ ], where [S ⁿ ] εH ⁿ (M; ℤ) is the fundamental class. The degree map is bijective, if there existsβ εH ₂(M, ℤ/2ℤ) such thatβ ·w ₂(M) ε 0. If suchβ does not exist, then deg is a 2-1 map; and (b) LetM be an orientable closed smooth (n+2)-manifold,n≥3. An elementα lies in the image of the degree map if and only ifρ ₂ α ·w ₂(M)=0, whereρ ₂: ℤ → ℤ/2ℤ is reduction modulo 2.     

The van Kampen obstruction and its relatives

We review a cochain-free treatment of the classical van Kampen obstruction ϑ to embeddability of an n-polyhedron in ℝ²ⁿ and consider several analogs and generalizations of ϑ, including an extraordinary lift of ϑ, which has been studied by J.-P. Dax in the manifold case. The following results are obtained: (1) The mod 2 reduction of ϑ is incomplete, which answers a question of Sarkaria. (2) An odd-dimensional analog of ϑ is a complete obstruction to linkless embeddability (=“intrinsic unlinking”) of a given n-polyhedron in ℝ²ⁿ⁺¹. (3) A “blown-up” one-parameter version of ϑ is a universal type 1 invariant of singular knots, i.e., knots in ℝ³ with a finite number of rigid transverse double points. We use it to decide in simple homological terms when a given integer-valued type 1 invariant of singular knots admits an integral arrow diagram (= Polyak-Viro) formula. (4) Settling a problem of Yashchenko in the metastable range, we find that every PL manifold N nonembeddable in a given ℝ ^m , m ≥ $ \frac{{3(n + 1)}} {2} $ \frac{{3(n + 1)}} {2} , contains a subset X such that no map N → ℝ ^m sends X and N \ X to disjoint sets. (5) We elaborate on McCrory’s analysis of the Zeeman spectral sequence to geometrically characterize “k-co-connected and locally k-co-connected” polyhedra, which we embed in ℝ2 ^n−k for k < $ \frac{{n - 3}} {2} $ \frac{{n - 3}} {2} , thus extending the Penrose-Whitehead-Zeeman theorem.     

Completely symmetric configurations for <Emphasis Type="Italic">σ</Emphasis>-games on grid graphs

The paper deals with σ-games on grid graphs (in dimension 2 and more) and conditions under which any completely symmetric configuration of lit vertices can be reached – in particular the completely lit configuration – when starting with the all-unlit configuration. The answer is complete in dimension 2. In dimension ≥3, the answer is complete for the σ ⁺-game, and for the σ ⁻-game if at least one of the sizes is even. The case σ ⁻, dimension ≥3 and all sizes odd remains open.     

Linear,hydrodynamic flow in a rotating saturated porous medium

Linear, steady, axisymmetric flow of a homogeneous fluid in a rigid, bounded, rotating, saturated porous medium is analyzed. The fluid motions are driven by differential rotation of horizontal boundaries. The dynamics of the interior region and vertical boundary layers are investigated as functions of the Ekman number E(=v/ΩL ²) and rotational Darcy 3 numberN(=kΩ/v) which measures the ratio between the Coriolis force and the Darcy frictional term. IfN≫E ^−1/2, the permeability is sufficiently high and the flow dynamics are the same as those of the conventional free flow problem with Stewartson'sE ^1/3 andE ^1/4 double layer structure. For values ofN≤E ^−1/2 the effect of porous medium is felt by the flow; the Taylor-Proudman constraint is no longer valid. ForN≪E ^−1/3 the porous medium strongly affects the flow; viscous side wall layer is absent to the lowest order and the fluid pumped by the Ekman layer, returns through a region of thicknessO(N ⁻¹). The intermediate rangeE ^−1/3≪N≪E ^−1/2 is characterized by double side wall layer structure: (1)E ^1/3 layer to return the mass flux and (ii) (NE)^1/2 layer to adjust the interior azimuthal velocity to that of the side wall. Spin-up problem is also discussed and it is shown that the steady state is reached quickly in a time scaleO(N).     

Cubic Diophantine Inequalities

Let λ₁, λ₂,..., λ₇ be real numbers satisfying λ _i ≥ 1. In this paper, we prove there are integers x ₁,..., x ₇ such that the inequalities |λ₁ x ³ ₁ + λ₂ x ³ ₂ + ⋯ + λ₇ x ³ ₇| < 1 and hold simultaneously. Received November 18, 1997, Accepted October 23, 1998     

Explicit expression of Cartan’s connection for Levi-nondegenerate 3-manifolds in complex surfaces,and identification of the Heisenberg sphere

We study effectively the Cartan geometry of Levi-nondegenerate C ⁶-smooth hypersurfaces M ³ in ℂ². Notably, we present the so-called curvature function of a related Tanaka-type normal connection explicitly in terms of a graphing function for M, which is the initial, single available datum. Vanishing of this curvature function then characterizes explicitly the local biholomorphic equivalence of such M ³ ⊂ ℂ² to the Heisenberg sphere ℍ³, such M’s being necessarily real analytic.