Boundary value problem with normal derivatives for a higher-order elliptic equation on the plane

For an elliptic operator of order 2l with constant (and only leading) real coefficients, we consider a boundary value problem in which the normal derivatives of order (k _j ?1), j = 1,..., l, where 1 ≤ k ₁ < ··· < k _l, are specified. It becomes the Dirichlet problem for kj = j and the Neumann problem for k _j = j + 1. We obtain a sufficient condition for the Fredholm property of which problem and derive an index formula.     

Generalized triangulations and diagonal-free subsets of stack polyominoes

For n?3, let Ω_n be the set of line segments between vertices in a convex n-gon. For j?1, a j-crossing is a set of j distinct and mutually intersecting line segments from Ω_n such that all 2j endpoints are distinct. For k?1, let Δ_n,k be the simplicial complex of subsets of Ω_n not containing any (k+1)-crossing. For example, Δ_n,1 has one maximal set for each triangulation of the n-gon. Dress, Koolen, and Moulton were able to prove that all maximal sets in Δ_n,k have the same number k(2n-2k-1) of line segments. We demonstrate that the number of such maximal sets is counted by a k×k determinant of Catalan numbers. By the work of Desainte-Catherine and Viennot, this determinant is known to count quite a few other objects including fans of non-crossing Dyck paths. We generalize our result to a larger class of simplicial complexes including some of the complexes appearing in the work of Herzog and Trung on determinantal ideals.     

Isomorphism Checking of I-graphs

We consider the class of I-graphs, which is a generalization of the class of the generalized Petersen graphs. We show that two I-graphs I(n, j, k) and I(n, j ₁, k ₁) are isomorphic if and only if there exists an integer a relatively prime to n such that either {j ₁, k ₁} =? {a j mod n, a k mod n } or {j ₁, k ₁} =? {a j mod n, ? a k mod n }. This result has an application in the enumeration of non-isomorphic I-graphs and unit-distance representations of generalized Petersen graphs.     

On the rational approximation ofe −x on [0, ∞)

In recent years the problem of uniform approximation ofe ^?x on [0, ∞) by rational functions has found wide interest. In this paper we present a method for the construction of rational approximations toe ^?x that seem to come arbitrarily near to the asymptotically best one. We start with a generalization of the integral form of the Padé approximant by introducing certain real parametersa _j ,b _i ,k and?. The corresponding error function has again an integral representation and is estimated for everyx∈[0,∞) by the Laplace method. This leads to the consideration of a finite number of new error functionsφ _i·j whose maxima are determined by a set of nonlinear equations and, under some restrictions on thea _j ,b _i ,k, and?, are also unique. Variation of these parameters according to a special algorithm and computation of the corresponding maxima of the functionsφ _i·j shows that forn→∞ the order of rational approximation ofe ^?x on [0, ∞) is at least 9.03^?n .     

A combinatorial problem involving graphs and matrices

In this paper we discuss a combinatorial problem involving graphs and matrices. Our problem is a matrix analogue of the classical problem of finding a system of distinct representatives (transversal) of a family of sets and relates closely to an extremal problem involving 1-factors and a long standing conjecture in the dimension theory of partially ordered sets. For an integer n ?1, let n denote the n element set {1,2,3,…, n}. Then let A be a k×t matrix. We say that A satisfies property P(n, k) when the following condition is satisfied: For every k-taple (x₁,x₂,…,x_k?n^k there exist k distinct integers j₁,j₂,…,j_k so that x_i= a_ii for i= 1,2,…,k. The minimum value of t for which there exists a k × t matrix A satisfying property P(n,k) is denoted by f(n,k). For each k?1 and n sufficiently large, we give an explicit formula for f(n, k): for each n?1 and k sufficiently large, we use probabilistic methods to provide inequalities for f(n,k).     

Restricted <Emphasis Type="Italic">k</Emphasis>-color partitions

We generalize overpartitions to (k, j)-colored partitions: k-colored partitions in which each part size may have at most j colors. We find numerous congruences and other symmetries. We use a wide array of tools to prove our theorems: generating function dissections, modular forms, bijections, and other combinatorial maps. In the process of proving certain congruences, we find results of independent interest on the number of partitions with exactly 2 sizes of part in several arithmetic progressions. We find connections to divisor sums, the Han/Nekrasov–Okounkov hook length formula and a possible approach to finitization, and other topics, suggesting that a rich mine of results is available. We pose several immediate questions and conjectures.     

Distribution of the coefficients of modular forms and the partition function

Let ? be an odd prime and j, s be positive integers. We study the distribution of the coefficients of integer and half-integral weight modular forms modulo an odd positive integer M. As an application, we investigate the distribution of the ordinary partition function p(n) modulo ? ^j and prove that for each integer 1?≤ r?<?? ^j , $$\sharp\{1\le n\le X\ |\ p(n)\equiv r\pmod{\ell^j} \}\gg_{s,r,\ell^j} \frac{\sqrt X}{\log X}(\log\log X)^s.$$      

Rationality of Dedekind sums in function fields

In the previous paper we introduced the higher dimensional Dedekind sum in a function field (Bayad and Hamahata, Acta Arith. 152:71–80, 2012). The purpose of this paper is to present a criterion for the rationality of our Dedekind sum. To do so, we establish a connection between the field of definition of the Drinfeld module ? and the field of definition of the higher dimensional Dedekind sum s _Λ (a ₀;a ₁,…,a _d ) associated to the A-lattice Λ, which corresponds to ?.     

Bounds for consecutive kth power residues in the Eisenstein integers

This paper extends to the Eisenstein integers a + b? (a, bτZ, ?² + ? + 1 = 0) the problem of the existence of a bound on the size of a sequence of m consecutive kth powe residues of p, for all but a finite number of primes p and independent of p. The least such bound is denoted by Λ_E(k, m). It is shown that Λ_E(k, 2) is finite for k = 2, 3, 4 or 6n + 1. On the other hand, for every k, Λ_E(2k, 3) = Λ_E(3k, 4) = Λ_E(k, 6) = ∞. Similar results are obtained for the related bound for m consecutives all in the same coset modulo the subgroup of kth power residues.     

Split Malcev algebras

We study the structure of split Malcev algebras of arbitrary dimension over an algebraically closed field of characteristic zero. We show that any such algebras M is of the form $M={\mathcal U} +\sum_{j}I_{j}$ with ${\mathcal U}$ a subspace of the abelian Malcev subalgebra H and any I _j a well described ideal of M satisfying [I _j ,I _k ]?=?0 if j????k. Under certain conditions, the simplicity of M is characterized and it is shown that M is the direct sum of a semisimple split Lie algebra and a direct sum of simple non-Lie Malcev algebras.     

Nonvanishing cohomology and classes of Gorenstein rings

We give counterexamples to the following conjecture of Auslander: given a finitely generated module M over an Artin algebra Λ, there exists a positive integer n_M such that for all finitely generated Λ-modules N, if Ext_Λⁱ(M,N)=0 for all i?0, then Ext_Λⁱ(M,N)=0 for all i?n_M. Some of our examples moreover yield homologically defined classes of commutative local rings strictly between the class of local complete intersections and the class of local Gorenstein rings.     

The Relaxed Edge-Coloring Game and <Emphasis Type="Italic">k</Emphasis>-Degenerate Graphs

The (r, d)-relaxed edge-coloring game is a two-player game using r colors played on the edge set of a graph G. We consider this game on forests and more generally, on k-degenerate graphs. If F is a forest with Δ(F)=Δ, then the first player, Alice, has a winning strategy for this game with r=Δ?j and d≥2j+2 for 0≤j≤Δ?1. This both improves and generalizes the result for trees in Dunn, C. (Discret. Math. 307, 1767–1775, 2007). More broadly, we generalize the main result in Dunn, C. (Discret. Math. 307, 1767–1775, 2007) by showing that if G is k-degenerate with Δ(G)=Δ and j∈[Δ+k?1], then there exists a function h(k,j) such that Alice has a winning strategy for this game with r=Δ+k?j and d≥h(k,j).     

Sequences of Hopf-Swan subgroups

Let l be an odd prime which satisfies Vandiver's conjecture, let n?1 be an integer, and let K=Q(ζn) where ζn is a primitive lnth root of unity. Let Cl denote the cyclic group of order l. For each j, j=1,…,ln−1, there exists an inclusion of Larson orders in KCl: Λj−1⊆Λj and a corresponding surjection of Hopf-Swan subgroups T(Λj−1)→T(Λj). For the cases n=1,2 we investigate the structure of various terms in the sequence of Hopf-Swan subgroups including the Swan subgroup T(Λ₀).     

On the effective determination of Maass cusp forms by L-values

Let f be a Hecke–Maass cusp form of Laplace eigenvalue 1/4+μ ² with |μ|≤Λ for \(\mathit{SL}_{2}(\mathbb{Z})\) . We show that f is uniquely determined by the central values of Rankin–Selberg L-functions L(s,f?g), where g runs over the set of holomorphic cusp forms of weight k? _? Λ ^1+3θ+? for any ?>0 for \(\mathit{SL}_{2}(\mathbb{Z})\) .     

Representation dimension and tilting

Let Λ be a finite dimensional k-algebra over an algebraically closed field k and let _ΛT be a splitting tilting module of projective dimension at most 1. Let Γ=End_ΛT. If the representation dimension of Λ is at most 3 then the main result asserts that the representation dimension of Γ does not exceed that of Λ.     

Supersingular parameters of the Deuring normal form

It is proved that the supersingular parameters α of the elliptic curve E ₃(α): Y ²+αXY+Y=X ³ in Deuring normal form satisfy α=3+γ ³, where γ lies in the finite field $\mathbb{F}_{p^{2}}$ . This is accomplished by finding explicit generators for the normal closure N of the finite extension k(α)/k(j(α)), where α is an indeterminate over $k=\mathbb{F}_{p^{2}}$ , and j(α) is the j-invariant of E ₃(α). Computing an explicit algebraic form for the elements of the Galois group of the extension N/k(j) leads to some new relationships between supersingular parameters for the Deuring normal form. The function field N, which contains the function field of the cubic Fermat curve, is then used to show how the results of Fleckinger for the Deuring normal form are related to cubic theta functions.     

Theta Series of Quaternion Algebras over Function Fields

Let K be the function field over a finite field of odd order, and let H be a definite quaternion algebra over K. If Λ is an order of level M in H, we define theta series for each ideal I of Λ using the reduced norm on H. Using harmonic analysis on the completed algebra H_∞ and the arithmetic of quaternion algebras, we establish a transformation law for these theta series. We also define analogs of the classical Hecke operators and show that in general, the Hecke operators map the theta series to a linear combination of theta series attached to different ideals, a generalization of the classical Eichler Commutation Relation.     

A weighted geometric regularity from order restricted statistical inference

In Euclideank-space, the cone of vectors x = (x ₁,x ₂,...,x _k ) satisfyingx ₁ ≤x ₂ ≤ ... ≤x _k and $\sum\nolimits_{j = 1}^k {x_j } = 0$ is generated by the vectorsv _j = (j ?k, ...,j ?k,j, ...,j) havingj ?k’s in its firstj coordinates andj’s for the remainingk ?j coordinates, for 1 ≤j <k. In this equal weights case, the average angle between v _i and v _j over all pairs (i, j) with 1 ≤i <j <k is known to be 60°. This paper generalizes the problem by considering arbitrary weights with permutations.     

Solutions of Semilinear Elliptic Equations in Tubes

Given a smooth compact k-dimensional manifold Λ embedded in ? ^m , with m≥2 and 1≤k≤m?1, and given ?>0, we define B _? (Λ) to be the geodesic tubular neighborhood of radius ? about Λ. In this paper, we construct positive solutions of the semilinear elliptic equation $$\begin{cases} \Delta u + u^p = 0 &\mbox{in } B_{\epsilon}(\varLambda) \\ u = 0 & \mbox{on } \partial B_{\epsilon}(\varLambda) , \end{cases} $$ when the parameter ? is chosen small enough. In this equation, the exponent p satisfies either p>1 when n:=m?k≤2 or $p\in(1, \frac{n+2}{n-2})$ when n>2. In particular, p can be critical or supercritical in dimension m≥3. As ? tends to 0, the solutions we construct have Morse index tending to infinity. Moreover, using a Pohozaev type argument, we prove that our result is sharp in the sense that there are no positive solutions for $p>\frac{n+2}{n-2}$ , n≥3, if ? is sufficiently small.     

A mod <Emphasis Type="Italic">ℓ</Emphasis> Atkin–Lehner theorem and applications

If f(z) is a weight \({k\in \frac{1}{2}\mathbb {Z}}\) meromorphic modular form on Γ₀(N) satisfying

$f(z)=\sum_{n\geq n_0} a_ne^{2\pi i mnz}, $

where \({m \nmid N,}\) then f is constant. If k ≠ 0, then f = 0. Atkin and Lehner [2] derived the theory of integer weight newforms from this fact. We use the geometric theory of modular forms to prove the analog of this fact for modular forms modulo ?. We show that the same conclusion holds if gcd(N ?,m) = 1 and the nebentypus character is trivial at ?. We use this to study the parity of the partition function and the coefficients of Klein’s j-function.