Representations of integers as sums of powers with increasing exponents

Let(n) be the least integer such thatn may be represented in the formn=x ₁ ² +x ₂ ³ +...+x _(n) ⁽ⁿ⁾⁺¹ wherex ₁,x ₂, ...,x _(n) are natural numbers. We computed(n) forn 250 000 and found that(n) 5 for all thesen exceptn=56, 160 for which(n)=6. Also(n) 4 for 41542<n<=250 000.     

Formulas for best extrapolation

We considern-point Lagrange-Hermite extrapolation forf(x), x>1, based uponf(x _i ),i=1(1)n, –1x _i 1, including non-distinct pointsx _i in confluent formulas involving derivatives. The problem is to find the pointsx _i that minimize the factor in the remainderP _n (x)f ⁽ⁿ⁾()/n, –1<<x subject to the condition|P _n (x)|M, –1x1,2^–n+1M2 ⁿ . The solution is significant only when a single set of pointsx _i suffices for everyx>1. The problem is here completely solved forn=1(1)4. Forn>4 it may be conjectured that there is a single minimal , 0 rn, whererr(M) is a non-decreasing function ofM, P _n (–1)=(–1) ⁿ M, and for 0rn–2, thej-th extremumP _n (x _{e, j} )=(–1) ^n–j M,j=1(1)n–r–1 (except forM=M _r ,r=1(1)n–1, whenj=1(1)n–r).     

Hodge decompositions of Loday symbols inK-theory and cyclic homology

In this paper, we study the Hodge decompositions ofK-theory and cyclic homology induced by the operations ^k and ^k , and in particular the decomposition of the Loday symbols x,y, ...z. Except in special cases, these Loday symbols do not have pure Hodge index. InK _n (A) they can project into every componentK _n ⁽ⁱ⁾ for 2in, and the projection of the Loday symbol x,y, ...,z intoK _n ⁽ⁿ⁾ is a multiple of the generalized Dennis-Stein symbol x,y, ...,z. Our calculations disprove conjectures of Beilinson and Soulé inK-theory, and of Gerstenhaber and Schack in Hochschild homology.Partially supported by National Security Agency grant MDA904-90-H-4019.Partially supported by National Science Foundation grant DMS-8803497.     

On quadratic spline interpolation

Using the quadratic spline interpolates(x) fitting the data (x _i,y _i), 0in and satisfying the end conditions_o=y_o, we give formulae approximatingy andy at selected knots by orders up toO(h ⁴).     

On a problem of Erdös and Alladi

We give an estimate for the quantity {f(n):nx, p(n)y}, wherep(n) denotes the greatest prime factor ofn andf belongs to a certain class of multiplicative functions. As an application, we show that for the Moebius function, ({(n):nx, p(n)y}) ({1:nx, p(n)y})^–1 tends to zero, asx, uniformly iny2, and thus settle a conjecture of Erdös.Supported by a grant from the Deutsche Forschungsgesellschaft.     

How non-uniform can a uniform sample be: a histogram approach

Summary Let _n be the empirical probability measure associated with n i.i.d. random vectors each having a uniform distribution in the unit square S of the plane. After _n is known, take the worst partition of the square into kn rectangles R _i, each with its short side at least times as long as the long side, and let Z= n|_n(R _j)–(R _j)|. We prove distribution inequalities for Z implying the right half of c _p,(n,k)^p/2 EZ ^p C _p,(n,k ^p/2, p > 0. (The left half follows easily by considering non-random partitions.) Similar results are obtained in other dimensions, and for population distributions other than uniform, and our results are related to data based histogram density estimation.Supported by NSF Grant MCS 8201128Supported by NSF Grant DMS-8401996     

A relation between Chung's and Strassen's laws of the iterated logarithm

Summary Let W(t) be a standard Wiener process and let f(x) be a function from the compact class in Strassen's law of the iterated logarithm. We investigate the lim inf behavior of the variable sup ¦W(xT)(2T loglog T)^–1/2–f(x)¦, 0x1 suitably normalized as T.This extends Chung's result valid for f(x)0, stating that lim inf.[ sup ¦(2T loglogT)^–1/2 W(xT)¦(loglog T)^–1]=/4 a.s. T 0x1     

Remarks on ergodicity of stationary irreducible transient Markov chains

Summary Consider a stationary process {X _n(), – < n < . If the measure of the process is finite (the measure of the whole sample space finite), it is well known that ergodicity of the process {X _n(), - < n < and of each of the subprocesses {X _n(), 0 n < , {X _n(), – < n 0 are equivalent (see [3]). We shall show that this is generally not true for stationary processes with a sigma-finite measure, specifically for stationary irreducible transient Markov chains. An example of a stationary irreducible transient Markov chain {X _n(), - < n <} with {itX_n(), 0 n < < ergodic but {X _n(), < n 0 nonergodic is given. That this can be the case has already been implicitly indicated in the literature [4]. Another example of a stationary irreducible transient Markov chain with both {X _n(), 0 n < and {itX _n(),-< < n 0} ergodic but {X _n(), - < n < nonergodic is presented. In fact, it is shown that all stationary irreducible transient Markov chains {X _n(), - < n < < are nonergodic.This research was supported in part by the Office of Naval Research.John Simon Guggenheim Memorial Fellow.     

Primes in short intervals

Let (x) stand for the number of primes not exceedingx. In the present work it is shown that if 23/421,yx andx>x() then (x)–(x–y)>y/(100 logx). This implies for the difference between consecutive primes the inequalityp _n+1–p _n p _n ^23/42 .     

Two-sided polynomial approximation of functions

Conditions are established when the collocation polynomials P_m(x) and P_M(x), m M, constructed respectively using the system of nodes x_j of multiplicities a_j 1, j = O,, n, and the system of nodes x_-r,,x_o,,x_n,,x_n+r1, r O, r₁ O, of multiplicities a_-r,,(a_o + y_o),,(a_n + y_n),,a_n+r1, a_j + y_j 1, are two sided-approximations of the function f on the intervals , x_j[, j = O,...,n + 1, and on unions of any number of these intervals. In this case, the polynomials P_m (x), P_M ^(l) (x) with l a_j are two-sided approximations of the function f⁽¹⁾ in the neighborhood of the node x_j and the integrals of the polynomials P_m(x), P_M(x) over D_j are two-sided approximations of the integral of the function f (over D_j). If the multiplicities a_j a_j + y_j of the nodes x_j are even, then this is also true for integrals over the set _j= _µ ^k D_j µ 1, k n. It is shown that noncollocation polynomials (Fourier polynomials, etc.) do not have these properties.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, pp. 31–37, 1989.     

Small potential corrections for the discrete eigenvalues of the Sturm-Liouville problem

Summary The inverse Sturm-Liouville problem is the problem of finding a good approximation of a potential functionq such that the eigenvalue problem (^*)–y +qy=y holds on (0, ) fory(0)=y()=0 and a set of given eigenvalues . Since this problem has to be solved numerically by discretization and since the higher discrete eigenvalues strongly deviate from the corresponding Sturm-Liouville eigenvalues , asymptotic corrections for the 's serve to get better estimates forq. Let _k (1kn) be the first eigenvalues of (^*), let _k be the corresponding discrete eigenvalues obtained by the finite element method for (^*) and let _{k k} for the special caseq=0. Then, starting from an asymptotic correction technique proposed by Paine, de Hoog and Anderssen, new estimates for the errors of the corrected discrete eigenvalues are obtained and confirm and improve the knownO(kh ²)(h:=/(n+1)) behaviour. The estimates are based on new Sobolev inequalities and on Fourier analysis and it is shown that for 4+c ₂ k(n+1)/2, wherec ₁ andc ₂ are constants depending onq which tend to 0 for vanishingq.     

Intuitionistic weak arithmetic

We construct -framed Kripke models of i₁ and i₁ non of whose worlds satisfies xy(x=2yx=2y+1) and x,yzExp(x, y, z) respectively. This will enable us to show that i₁ does not prove ¬¬xy(x=2yx=2y+1) and i₁ does not prove ¬¬x, yzExp(x, y, z). Therefore, i₁¬¬lop and i₁¬¬i₁. We also prove that HAl₁ and present some remarks about i₂. Mathematics Subject Classification (2000):03F30, 03F55, 03H15.     

Positive definite dot product kernels in learning theory

In the classical support vector machines, linear polynomials corresponding to the reproducing kernel K(x,y)=xy are used. In many models of learning theory, polynomial kernels K(x,y)=_l=0^Na_l(xy)^l generating polynomials of degree N, and dot product kernels K(x,y)=_l=0⁺a_l(xy)^l are involved. For corresponding learning algorithms, properties of these kernels need to be understood. In this paper, we consider their positive definiteness. A necessary and sufficient condition for the dot product kernel K to be positive definite is given. Generally, we present a characterization of a function f:RR such that the matrix [f(xⁱx^j)]_i,j=1^m is positive semi-definite for any x¹,x²,...,x^mRⁿ, n2. Supported by CERG Grant No. CityU 1144/01P and City University of Hong Kong Grant No. 7001342.AMS subject classification 42A82, 41A05     

Some extensions of the principles of idealization transfer and choice in the relative internal set theory

The results established in this paper are in connection with the Relative Internal Set Theory (R.I.S.T.). The main result is the general principle of choice: Let be a level and let (x, y) be anexternalbounded formula of the language of R.I.S.T.. Suppose that to each elementx, dominated by , corresponds an elementy _x such that (x, y _x ) holds, then there exists a function of choice such that, which is a very general principle of choice, for everyx dominated by , (x, (x)) holds. More than that, we establish that if all the elementsy _x are uniformly dominated by a level then we can prescribe that the function of choice is also dominated by .     

Generalised Holley-Preston inequalities on measure spaces and their products

Summary It is shown that if (X, ) is a product of totally ordered measure spaces andf _j (j=1,2,3,4) are measurable non-negative functions onX satisfyingf ₁(x)f₂(y)f₃(xy)f₄(xy), where (, ) are the lattice operations onX, then (f ₁ d)(f ₂ d)(f ₃ d)(f ₄ d). This generalises results of Ahlswede and Daykin (for counting measure on finite sets) and Preston (for special choices off _j).     

Stochastically monotone Markov Chains

Summary A real-valued discrete time Markov Chain {X _n} is defined to be stochastically monotone when its one-step transition probability function pr {X _n+1y¦ X _n=x} is non-increasing in x for every fixed y. This class of Markov Chains arises in a natural way when it is sought to bound (stochastically speaking) the process {X _n} by means of a smaller or larger process with the same transition probabilities; the class includes many simple models of applied probability theory. Further, a given stochastically monotone Markov Chain can readily be bounded by another chain {Y _n}, with possibly different transition probabilities and not necessarily stochastically monotone, and this is of particular value when the latter process leads to simpler algebraic manipulations. A stationary stochastically monotone Markov Chain {X _n} has cov(f(X ₀), f(X _n)) cov(f(X ₀), f(X _n+1))0 (n =1, 2,...) for any monotonic function f(·). The paper also investigates the definition of stochastic monotonicity on a more general state space, and the properties of integer-valued stochastically monotone Markov Chains.     

Integrability of certain solutions of differential equations

Let L=P_o(d/d_t)ⁿ+P₁(d/d_t)^n–1+...+P_n denote a formally self-adjoint differential expression on an open intervalI=(a, b) (–a. Here the P_k are complex valued with (n — k) continuous derivatives onI, and P₀(t) 0 onI. We discuss integrability of functions which are adjoint to certain fundamental solutions ofLy=y, and a related consequence.     

Laws of Iterated Logarithm and Related Asymptotics for Estimators of Conditional Density and Mode

Let (X _i , Y _i ) be a sequence of i.i.d. random vectors in R with an absolutely continuous distribution function H and let g _x (y), y R denote the conditional density of Y given X = x(F), the support of F, assuming that it exists. Also let M(x) be the (unique) conditional mode of Y given X = x defined by M(x) = arg max _y (y)). In this paper new classes of smoothed rank nearest neighbor (RNN) estimators of g _x (y), its derivatives and M(x) are proposed and the laws of iterated logarithm (pointwise), uniform a.s. convergence over – < y < and x a compact C(F) and the asymptotic normality for the proposed estimators are established. Our results and proofs also cover the Nadayara-Watson (NW) case. It is shown using the concept of the relative efficiency that the proposed RNN estimator is superior (asymtpotically) to the corresponding NW type estimator of M(x), considered earlier in literature.     

A Local Approach to 1-Homogeneous Graphs

Let bea distance-regular graph with diameter d. For vertices x and y of at distancei, 1 i d, we define the setsC _i(x,y) = i–1(x) (y), A _i (x,y) = _i (x) (y) and B _i (x,y) = _i+1(x) (y).Then we say has the CAB_j property,if the partition CAB _i (x,y) = {C _i (x,y),A _i (x,y),B _i (x,y)}of the local graph of y is equitable for each pairof vertices x and y of at distance i j. We show that in with the CAB_j property then the parameters ofthe equitable partitions CAB i(x,y) do not dependon the choice of vertices x and y atdistance i for all i j. The graph has the CAB property if it has the CAB _d property. We show the equivalence of the CAB property and the1-homogeneous property in a distance-regular graph with a ₁ 0. Finally, we classify the 1-homogeneous Terwilligergraphs with c ₂ 2.     

On Large Oscillations of the Remainder of the Prime Number Theorems

Under the assumption of the appropriate Ricmann hypothesis it is shown that max _tx min _l1 t ^-1/2 ((x, q, 1) – (x, q, l)) > ( – ) log₃ x and min _tx max _l1 t ^-1/2 ((x, q, 1) – (x, q, l)) < –( – ) log₃ x for x > x ₀(q, ). The proof is quite elementary, and x ₀ can be estimated effectively. As a by-product a formula for the k-th power moment of certain normed error terms is obtained.