Minimum Energy Problem on the Hypersphere

We consider the minimum energy problem on the unit sphere \(\mathbb {S}^{d-1}\) in the Euclidean space \(\mathbb {R}^{d}\), d = 3, in the presence of an external field Q, where the charges are assumed to interact according to Newtonian potential 1/r ^d-2, with r denoting the Euclidean distance. We solve the problem by finding the support of the extremal measure, and obtaining an explicit expression for the density of the extremal measure. We then apply our results to an external field generated by a point charge of positive magnitude, placed at the North Pole of the sphere, and to a quadratic external field.     

Riesz extremal measures on the sphere for axis-supported external fields

We investigate the minimal Riesz s-energy problem for positive measures on the d-dimensional unit sphere Sd in the presence of an external field induced by a point charge, and more generally by a line charge. The model interaction is that of Riesz potentials |x−y|^−s with d−2?s<d. For a given axis-supported external field, the support and the density of the corresponding extremal measure on Sd is determined. The special case s=d−2 yields interesting phenomena, which we investigate in detail. A weak^∗ asymptotic analysis is provided as s→⁺(d−2).     

Voter model in a random environment in ? d

We consider the voter model with flip rates determined by {?? _e , e ?? E _d }, where E _d is the set of all non-oriented nearest-neighbour edges in the Euclidean lattice ? ^d . Suppose that {?? _e , e ?? E _d } are independent and identically distributed (i.i.d.) random variables satisfying ?? _e ? 1. We prove that when d = 2, almost surely for all random environments, the voter model has only two extremal invariant measures: ?? ₀ and ?? ₁.     

On point energies, separation radius and mesh norm for -extremal configurations on compact sets in

We investigate bounds for point energies, separation radius, and mesh norm of certain arrangements of N points on sets A from a class of d-dimensional compact sets embedded in R^d′, 1dd^′. We assume that these points interact through a Riesz potential V=|·|^-s, where s>0 and |·| is the Euclidean distance in . With and denoting, respectively, the separation radius and mesh norm of s-extremal configurations, which are defined to yield minimal discrete Riesz s-energy, we show, in particular, the following.(A) For the d-dimensional unit sphere and s<d-1, and, moreover, if sd-2. The latter result is sharp in the case s=d-2. In addition, point energies for s-extremal configurations are asymptotically equal. This observation relates to numerical experiments on observed scar defects in certain biological systems.(B) For and s>d, and the mesh ratio is uniformly bounded for a wide subclass of . We also conclude that point energies for s-extremal configurations have the same order, as N→∞.     

Infinitely many Gerretsen-Blundon style quadratic inequalities, all strongest in Blundon’s sense

Blundon has proved that if R, r and s are respectively the circumradius, the inradius and the semiperimeter of a triangle, then the strongest possible inequalities of the form q(R, r) ≤ s ² ≤ Q(R, r) that hold for all triangles becoming equalities for the equilaterals where q, Q real quadratic forms, occur for the Gerretsen forms q _B (R, r) = 16Rr ? 5r ² and Q _B (R, r) = 4R ² + 4Rr + 3r ²; strongest in the sense that if Q is a quadratic form and s ² ≤ Q(R, r) ≤ Q _B (R, r) for all triangles then Q(R, r) = Q _B (R, r), and similarly for q _B (R, r). In this paper we prove that Q _B (resp. q _B ) is just one of infinitely many forms that appear as minimal (resp. maximal) elements in the partial order induced by the comparability relation in a certain set of forms, and we conclude that all these minimal forms are strongest in Blundon’s sense. We actually find all possible such strongest forms. Moreover we find all possible quadratic forms q, Q for which q(R, r) ≤ s ² ≤ Q(R, r) for all triangles and which hold as equalities for the equilaterals.     

Construction of nonseparable orthonormal compactly supported wavelet bases for L2（Rd）

Suppose M and N are two r × r and s × s dilation matrices,respectively.Let ΓM and ΓN represent the complete sets of representatives of distinct cosets of the quotient groups M-TZr/Zr and N-TZs/Zs,respe...     

Riesz Spherical Potentials with External Fields and Minimal Energy Points Separation

In this paper we consider the minimal energy problem on the sphere S ^d for Riesz potentials with external fields. Fundamental existence, uniqueness, and characterization results are derived about the associated equilibrium measure. The discrete problem and the corresponding weighted Fekete points are investigated. As an application we obtain the separation of the minimal s-energy points for d – 2 < s < d. The explicit form of the separation constant is new even for the classical case of s = d – 1. Research supported, in part, by a National Science Foundation Research grant DMS 0532154.     

Monotonicity of the probability of percolation for Bernoulli random fields on periodic graphs

In this work, the problem of percolation of the Bernoulli random field on periodic graphs ?? of an arbitrary dimension d is studied. A theorem on nondecreasing dependence of the probability of percolation Q(c ₁ , ?? , c _n ) with respect to each of the parameters c _i , i = 1÷n, ?C concentration of the Bernoulli field is proved.     

A note on extremal metrics of non-constant scalar curvature

By working in ? ⁿ with potentials of the forma logu + s(u), u the square of the distant to the origin, we obtain extremal Kähler metrics of nonconstant scalar curvature on the blow-up of ? ⁿ at \(\vec 0\) . We then show that these metrics can be completed at ∞ by adding a ?? ^n?1, and reobtain the extremal Kähler metrics of non-constant scalar curvature constructed by Calabi on the blow-up of ?? ⁿ at one point. A similar construction produces this type of metrics on other bundles over ?? ^{n ? 1}.     

Picard dimension of signed radial Kato measures

The Picard dimension dimμ of a signed local Kato measure μ on the punctured unit ball in R^d, d ≥ 2, is the cardinal number of the set of extremal rays of the convex cone of all continuous solutions u ≥ 0 of the time-independent SchrSdinger equation Δu -- uμ = 0 on the punctured ball 0 〈 ｜｜x｜｜ 〈 1, with vanishing boundary values on the sphere ｜｜x｜｜ = 1. Using potential theory associated with the Schrodinger operator we prove, in this paper, that the dimμ for a signed radial Kato measure is 0, 1 or ＋∞. In particular, we obtain the Picard dimension of locally Holder continuous functions P proved by Nakai and Tada by other methods.     

Disjoint unions of complete minors

Given integers r and s, and n large compared to r and s, we determine the maximum size of a graph of order n having no minor isomorphic to sK_r, the union of s disjoint copies of K_r.The extremal function depends on the relative sizes of r and s. If s is small compared to r the extremal function is essentially independent of s. On the other hand, if s is large compared to r, there is a unique extremal graph ; this assertion is a generalization of the case r=3 which is a classical result of Erd?s and Pósa.     

Über s-reguläre Graphen

Let G be a graph of valency d = 1 + p^rn with p prime and n < p. It is shown that if the automorphism group of G contains a subgroup A which acts as a regular permutation group on the set of s-arcs in G then s ≤ 7 and s ≠ 6; if d = p^r + 1, r = 2^r, v ≥ 1 an integer, p ≠ 2, then s = 1.     

On inequalities for the norms of intermediate derivatives of multiply monotone functions defined on a finite segment

We study the following modification of the Landau?CKolmogorov problem: Let k; r ?? ?, 1 ?? k ?? r ? 1, and p, q, s ?? [1,??]. Also let MM ^m , m ?? ?; be the class of nonnegative functions defined on the segment [0, 1] whose derivatives of orders 1, 2,??,m are nonnegative almost everywhere on [0, 1]. For every ?? > 0, find the exact value of the quantity $$ \omega_{p,q,s}^{k,r}\left( {\delta; M{M^m}} \right): = \sup \left\{ {{{\left\| {{x^{(k)}}} \right\|}_q}:x \in M{M^m},{{\left\| x \right\|}_p} \leqslant \delta, {{\left\| {{x^{(k)}}} \right\|}_s} \leqslant 1} \right\}. $$ We determine the quantity $ \omega_{p,q,s}^{k,r}\left( {\delta; M{M^m}} \right) $ in the case where s = ?? and m ?? {r, r ? 1, r ? 2}. In addition, we consider certain generalizations of the above-stated modification of the Landau?CKolmogorov problem.     

Intersection theory of algebraic obstructions

Let A be a noetherian commutative ring of dimension d and L be a rank one projectiveA-module. For 1≤r≤d, we define obstruction groups E^r(A,L). This extends the original definition due to Nori, in the case r=d. These groups would be called Euler class groups. In analogy to intersection theory in algebraic geometry, we define a product (intersection) E^r(A,A)×E^s(A,A)→E^r+s(A,A). For a projective A-module Q of rank n≤d, with an orientation , we define a Chern class like homomorphism

w(Q,χ):E^d−n(A,L^′)→E^d(A,LL^′),     

Properties of Euclidean and non-Euclidean distance matrices

A distance matrix D of order n is symmetric with elements $\text{?}\text{1}\text{2}\text{d}{}_{\mathrm{ij}}{}^2$, where d_ii=0. D is Euclidean when the $\text{1}\text{2}\text{n(n?1)}$ quantities d_ij can be generated as the distances between a set of n points, X (n×p), in a Euclidean space of dimension p. The dimensionality of D is defined as the least value of p=rank(X) of any generating X; in general p+1 and p+2 are also acceptable but may include imaginary coordinates, even when D is Euclidean. Basic properties of Euclidean distance matrices are established; in particular, when ρ=rank(D) it is shown that, depending on whether e^TD^?e is not or is zero, the generating points lie in either p=ρ?1 dimensions, in which case they lie on a hypersphere, or in p=ρ?2 dimensions, in which case they do not. (The notation e is used for a vector all of whose values are one.) When D is non-Euclidean its dimensionality p=r+s will comprise r real and s imaginary columns of X, and (r, s) are invariant for all generating X of minimal rank. Higher-ranking representations can arise only from p+1=(r+1)+s or p+1=r+ (s+1) or p+2=(r+1)+(s+1), so that not only are r, s invariant, but they are both minimal for all admissible representations X.     

What Is the Complexity of Surface Integration?

We study the worst case complexity of computing ε-approximations of surface integrals. This problem has two sources of partial information: the integrand f and the function g defining the surface. The problem is nonlinear in its dependence on g. Here, f is an r times continuously differentiable scalar function of l variables, and g is an s times continuously differentiable injective function of d variables with l components. We must have d⩽l and s⩾1 for surface integration to be well-defined. Surface integration is related to the classical integration problem for functions of d variables that are min{r, s−1} times continuously differentiable. This might suggest that the complexity of surface integration should be Θ((1/ε)^{d/min{r, s−1}}). Indeed, this holds when d<l and s=1, in which case the surface integration problem has infinite complexity. However, if d⩽l and s⩾2, we prove that the complexity of surface integration is O((1/ε)^d/min{r, s}). Furthermore, this bound is sharp whenever d<l.     

The packing problem for projective geometries over GF(3) with dimension greater than five

An (n, d) set in the projective geometry PG(r, q) is a set of n points, no d of which are dependent. The packing problem is that of finding n(r, q, d), the largest size of an (n, d) set in PG(r, q). The packing problem for PG(r, 3) is considered. All of the values of n(r, 3, d) for r ? 5 are known. New results for r = 6 are n(6, 3, 5) = 14 and 20 ? n(6, 3, 4) ? 31. In general, upper bounds on n(r, q, d) are determined using a slightly improved sphere-packing bound, the linear programming approach of coding theory, and an orthogonal (n, d) set with the known extremal values of n(r, q, d)—values when r and d are close to each other. The BCH constructions and computer searches are used to give lower bounds. The current situation for the packing problem for PG(r, 3) with r ? 15 is summarized in a final table.     

Ternary codes of steiner triple systems

The code over a finite field F^q of a design ?? is the space spanned by the incidence vectors of the blocks. It is shown here that if ?? is a Steiner triple system on v points, and if the integer d is such that 3^d ≤ v < 3^d+1, then the ternary code C of ?? contains a subcode that can be shortened to the ternary generalized Reed-Muller code ?_F3(2(d ? 1),d) of length 3^d. If v = 3^d and d ≥ 2, then C_? ? ?_F3(1,d)? ? _F3(2(d ? 1),d) ? C. © 1994 John Wiley & Sons, Inc.     

Riesz s-equilibrium measures on d-dimensional fractal sets as s approaches d

Let A be a compact set in Rp of Hausdorff dimension d. For s∈(0,d), the Riesz s-equilibrium measure μs,A is the unique Borel probability measure with support in A that minimizes the double integral over the Riesz s-kernel |x−y|^−s over all such probability measures. In this paper we show that if A is a strictly self-similar d-fractal, then μs,A converges in the weak-star topology to normalized d-dimensional Hausdorff measure restricted to A as s approaches d from below.