On a conjecture by D. Ž. D̄okovic

It is shown that if $\text{A?Ω}{}_{\mathrm{n}}\text{?{J}{}_{\mathrm{n}}\text{}}$ satisfies

$\text{nkσ}{}_{\mathrm{k}}\text{(A)?(n?k+1)}{}^2\text{σ}{}_{\mathrm{k?1}}\text{(A)}$

$\text{(k=1,2,…,n)}$

, where σ_k(A) denotes the sum of all kth order subpermanent of A, then Per[λJ_n+(1?λ)A] is strictly decreasing in the interval 0<λ<1.     

Deterministic version of Wigner's semicircle law for the distribution of matrix eigenvalues

It is proved that Wigner's semicircle law for the distribution of eigenvalues of random matrices, which is important in the statistical theory of energy levels of heavy nuclei, possesses the following completely deterministic version. Let A_n=(a_ij), 1?i, ?n, be the nth section of an infinite Hermitian matrix, {λ⁽ⁿ⁾}_1?k?n its eigenvalues, and {u_k⁽ⁿ⁾}_1?k?n the corresponding (orthonormalized column) eigenvectors. Let $\text{v}{}^1{}_{\mathrm{n}}\text{=(a}{}_{\mathrm{n1}}\text{,a}{}_{\mathrm{n2}}\text{,?,a}{}_{\mathrm{n,n?1}}\text{)}$, put

$\text{X}{}_{\mathrm{n}}\text{(t)=[n(n-1)]}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{∑}\text{k=1}\text{[(n-1)t]}\text{|v}{}_{\mathrm{n}}{}^1\text{u}{}_{\mathrm{f}}{}^{\mathrm{(n-1)}}\text{|}{}^2\text{,}\text{0?t?1}$

(bookeeping function for the length of the projections of the new row v¹_n of A_n onto the eigenvectors of the preceding matrix A_n?1), and let finally

$\text{F}{}_{\mathrm{n}}\text{(x)=n}{}^{-1}\text{(}\text{number of}\text{λ}{}_{\mathrm{k}}{}^{\mathrm{(n)}}\text{?x}\text{n}\text{,1?k?n)}$

(empirical distribution function of the eigenvalues of $\text{A}{}_{\mathrm{n}}\text{n}$. Suppose (i) $\text{lim}{}_{\mathrm{n}}\text{a}{}_{\mathrm{nn}}\text{n}\text{=0}$, (ii) lim_nX_n(t)=Ct(0<C<∞,0?t?1). Then

$\text{F}{}_{\mathrm{n}}\text{?W(·,C)}\text{(n→∞)}$

,where W is absolutely continuous with (semicircle) density

$\text{w(x,C)=}\text{(2Cπ)}{}^{-1}\text{(4C-x}{}^2{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{for}\text{|x|?2}\text{C}\text{0}\text{for}\text{|x|?2}\text{C}$

     

Pairs of additive equations II. Large odd degree

Let k be an odd positive integer. Davenport and Lewis have shown that the equations

$\text{a}{}_1\text{x}{}_1{}^{\mathrm{k}}\text{+…+a}{}_{\mathrm{n}}\text{x}{}_{\mathrm{n}}{}^{\mathrm{k}}\text{=0}$

with integer coefficients, have a nontrivial solution in integers x₁,…, x_N provided that

$\text{N?[36klog6k]}$

Here it is shown that for any ? > 0 and k > k₀(?) the equations have a nontrivial solution provided that

$\text{N?}\text{8}\text{log 2}\text{+?}\text{k log k.}$

     

Strong approximation theorems for density dependent Markov chains

A variety of continuous parameter Markov chains arising in applied probability (e.g. epidemic and chemical reaction models) can be obtained as solutions of equations of the form

$\text{X}{}_{\mathrm{N}}\text{(t)=x}{}_0\text{+∑}\text{1}\text{N}\text{lY}{}_1\text{N ∫}{}^{\mathrm{t}}{}_0\text{f}{}_1\text{(X}{}_{\mathrm{N}}\text{(s))ds}$

where $\text{l∈}\text{Z}{}^{\mathrm{t}}$, the Y₁ are independent Poisson processes, and N is a parameter with a natural interpretation (e.g. total population size or volume of a reacting solution).The corresponding deterministic model, satisfies

$\text{X(t)=x}{}_0\text{+ ∫}{}^{\mathrm{t}}{}_0\text{∑ lf}{}_1\text{(X(s))ds}$

Under very general conditions lim_N→∞X_N(t)=X(t) a.s. The process X_N(t) is compared to the diffusion processes given by

$\text{Z}{}_{\mathrm{N}}\text{(t)=x}{}_0\text{+∑}\text{1}\text{N}\text{lB}{}_1\text{N∫}{}^{\mathrm{t}}{}_0\text{ft(Z}{}_{\mathrm{N}}\text{(s))ds}$

and

$\text{V(t)=∑ l∫}{}^{\mathrm{t}}{}_0\text{f}{}_1\text{(X(s))}\text{d}\text{W}\text{?}{}_1\text{+∫}{}^{\mathrm{t}}{}_0\text{?F(X(s))·V(s)}\text{ds.}$

Under conditions satisfied by most of the applied probability models, it is shown that X_N,Z_N and V can be constructed on the same sample space in such a way that

$\text{X}{}_{\mathrm{N}}\text{(t)=Z}{}_{\mathrm{N}}\text{(t)+O}\text{logN}\text{N}$

and

$\text{N}\text{(X}{}_{\mathrm{N}}\text{(t)?X(t))=V(t)+O}\text{log N}\text{N}$

     

Concavity of certain maps on positive definite matrices and applications to Hadamard products

If f is a positive function on (0, ∞) which is monotone of order n for every n in the sense of Löwner and if Φ₁ and Φ₂ are concave maps among positive definite matrices, then the following map involving tensor products:

$\text{(A,B)?f[Φ}{}_1\text{(A)}{}^{\mathrm{?1}}\text{?Φ}{}_2\text{(B)]·(Φ}{}_1\text{(A)?I)}$

is proved to be concave. If Φ₁ is affine, it is proved without use of positivity that the map

$\text{(A,B)?f[Φ}{}_1\text{(A)?Φ}{}_2\text{(B)}{}^{\mathrm{?1}}\text{]·(Φ}{}_1\text{(A)?I)}$

is convex. These yield the concavity of the map

$\text{(A,B)?A}{}^{\mathrm{1?p}}\text{?B}{}^{\mathrm{p}}$

(0<p?1) (Lieb's theorem) and the convexity of the map

$\text{(A,B)?A}{}^{\mathrm{1+p}}\text{?B}{}^{\mathrm{?p}}$

(0<p?1), as well as the convexity of the map

$\text{(A,B)?(A·log[A])?I?A?log[B]}$

.These concavity and convexity theorems are then applied to obtain unusual estimates, from above and below, for Hadamard products of positive definite matrices.     

A complete set of invariants for finite groups and other results

For a finite group G and a set I ? {1, 2,…, n} let

${}_{\mathrm{G}}\text{(n,I) = ∑}{}_{\mathrm{g\; \in \; G}}\text{ε}{}_1\text{(g)?ε}{}_2\text{(g)???ε}{}_{\mathrm{n}}\text{(g)}$

,where

$\text{ε}{}_{\mathrm{i}}\text{(g)=g}\text{if}\text{i=∈ I,}$

$\text{ε}{}_{\mathrm{l}}\text{(g)=l}\text{if}\text{i=∈ I.}$

We prove, among other results, that the positive integers

$\text{tr}\text{(e}{}_{\mathrm{G}}\text{(n,I}{}_1\text{)+?+e}{}_{\mathrm{G}}\text{(n,I}{}_{\mathrm{r}}\text{))}{}^{\mathrm{k}}\text{:n,r,k,?1, I}{}_{\mathrm{j}}\text{?{1,…,n}, 1?|i}{}_{\mathrm{j}}\text{|?3}$

for 1 ? j ? r, I_j1 ∩ I_j2 ∩ I_j3 ∩ I_j4 = Ø for any 1 ? j₁ <j₂ <j₃ <j₄ ? r, determine G up to isomorphism. We also show that under certain assumptions finite groups are determined up to isomorphism by the number of their subgroups.     

Kolmogorov-Landau inequalities for monotone functions

Presented in this report are two further applications of very elementary formulae of approximate differentiation. The first is a new derivation in a somewhat sharper form of the following theorem of V. M. Olovyani?nikov: LetN_n (n ? 2) be the class of functionsg(x) such thatg(x), g′(x),…, g⁽ⁿ⁾(x) are ? 0, bounded, and nondecreasing on the half-line ?∞ < x ? 0. A special element ofN_nis

$\text{g}{}_1\text{(x) = 0}\text{if}\text{?∞ < x < ?1, g}{}_1\text{(x) = (1 + x)}{}^{\mathrm{n}}\text{if}\text{?1 ? x ? 0}$

. Ifg(x) ∈ N_nis such that

$\text{g(0) ? g}{}_1\text{(0) = 1, g}{}^{\mathrm{(n)}}\text{(0) ? g}{}_1{}^{\mathrm{(n)}}\text{(0) = n!}$

, then

$\text{g}{}^{\mathrm{(v)}}\text{(0) ? g}{}_1{}^{\mathrm{(v)}}\text{(0)}$

for

1$\text{v = 1,…, n ? 1}$

. Moreover, if we have equality in (1) for some value of v, then we have there equality for all v, and this happens only if $\text{g(x) = g}{}_1\text{(x)}$ in (?∞, 0].The second application gives sufficient conditions for the differentiability of asymptotic expansions (Theorem 4).     

Linear operators preserving the (p,q)-numerical range

Let $\text{C}{}_{\mathrm{n\times n}}$ and $\text{H}{}_{\mathrm{n}}$ denote respectively the space of n×n complex matrices and the real space of n×n hermitian matrices. Let p,q,n be positive integers such that p?q?n. For $\text{A?}\text{C}{}_{\mathrm{n\times n}}$, the (p,q)-numerical range of A is the set

$\text{W}{}_{\mathrm{p,q}}\text{(A)={trC}{}_{\mathrm{p}}\text{(J}{}_{\mathrm{q}}\text{UAU}{}^1\text{):U}\text{unitary}\text{}}$

, where C_p(X) is the pth compound matrix of X, and Jq is the matrix I_q?O_n-q. Let $\text{L}$ denote $\text{H}$_n or $\text{C}{}_{\mathrm{n\times n}}$. The problem of determining all linear operators T: $\text{L}$→$\text{L}$ such that

$\text{W}{}_{\mathrm{p,q}}\text{(T(A))=W}{}_{\mathrm{p,q}}\text{(A)}\text{for all}\text{A?}\text{L}$

is treated in this paper.     

Bounds for the variance of certain stationary point processes

For the variance of stationary renewal and alternating renewal processes N_n(·) the paper establishes upper and lower bounds of the form

$\text{?B}{}_1\text{?varN}{}_8\text{(0,x–Aλx?B}{}_2\text{(0<x<∞)}$

, where λ=EN₈(0,1), with constants A, B₁ and B₂ that depend on the first three moments of the interval distributions for the processes concerned. These results are consistent with the value of the constant A for a general stationary point process suggested by Cox in 1963 [1].     

Poisson limits for a hard-core clustering model

Let X_1n,…,X_>nn denote the locations of n points in a bounded, γ-dimensional, Euclidean region D_n which has positive γ-dimensional Lebesgue measure μ(D_n). Let {Y_n(r): r > 0} be the interpoint distance process for these points where Y_n(r) is the number of pairs of points(X_in, X_in) which with i < j have Euclidean distance 6X_in ? X_>in6 < r. In this article we study the limiting distribution of Y_n(r) when n → ∞ and μ(D_n) → ∞, and the joint density of X_1n,…,X_nnis of the form

where r₀ is a positive constant and C_n is a normalizing constant. These joint densities modify the Strauss [11] clustering model densities by introducing a hard-core component (no two points can have 6X_in ? X_in6 < r₀) found in the Matérn [4] models. In our main result we show that the interpoint distance process converges to a non-homogeneous Poisson process for r values in a bounded interval 0 < r₀ < r < r₀₀ provided sparseness conditions discussed by Saunders and Funk [9] hold. The sparseness conditions which require $\text{μ(D}{}_{\mathrm{n}}\text{)}\text{n}{}^2$ converges to a positive constant and the boundary of D_n is negligible are essentially equivalent to requiring that although the number of points n is large the region is large enough so that the points are sparse in this region. That is, it is rare for a point to have another point close to it. These results extend results for v ? 0 given by Saunders and Funk [9] where it is shown that without the hard core component such results do not hold for v > 0. Statistical applications are discussed.     

On the van der Waerden conjecture and zeros of polynomials

Let

$\text{F(x) =}\text{∑}\text{k=o}\text{n}\text{n}\text{k}\text{A}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}$

$\text{A}{}_{\mathrm{n}}\text{≠ 0,}$

and

$\text{G(x) =}\text{∑}\text{k=o}\text{n}\text{n}\text{k}\text{B}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}$

$\text{B}{}_{\mathrm{n}}\text{≠ 0,}$

be polynomials with real zeros satisfying A_n?1 = B_n?1 = 0, and let

$\text{H(x) =}\text{∑}\text{k=o}\text{n-2}\text{n}\text{k}\text{A}{}_{\mathrm{k}}\text{B}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}\text{.}$

Using the recently proved validity of the van der Waerden conjecture on permanents, some results on the real zeros of H(x) are obtained. These results are related to classical results on composite polynomials.     

On the qth convergence exponent of entire functions

In this paper we obtain a growth relation for entire functions of qth order with respect to the distribution of its zeros. We also derive certain relations between the qth convergence exponents of two or more entire functions. The most striking result of the paper is: If f(z) has at least one zero, then

$\text{lim sup}\text{r→∞}\text{log n}\text{(r)}\text{log}{}^{\mathrm{[q+1]}}{}_{\mathrm{r}}\text{=?(q)}$

, where n(r) is the number of zeros of f(z) in $\text{¦}\text{z}\text{¦ ? r}$ and

$\text{?(q)=g.l.b.}\text{α:α>0 and}\text{∑}\text{∞}\text{n=1}\text{(log}{}^{\mathrm{[q]}}\text{r}{}_{\mathrm{n}}\text{)}{}^{\mathrm{-\alpha }}\text{<∞}$

.     

On the number of zeros of diagonal cubic forms

Let M_s, be the number of solutions of the equation

$\text{X}{}_1{}^3\text{+ X}{}_2{}^3\text{+ … + X}{}_{\mathrm{s}}{}^3\text{=0}$

in the finite field GF(p). For a prime p ≡ 1(mod 3),

$\text{∑}\text{s=1}\text{∞}\text{M}{}_{\mathrm{s}}\text{X}{}^{\mathrm{s}}\text{=}\text{x}\text{1 ? px}\text{+}\text{x}{}^2\text{(p ? 1)(2 + dx)}\text{1 ? 3px}{}^2\text{? pdx}{}^3$

,

$\text{M}{}_3\text{= p}{}^2\text{+ d(p ? 1)}$

, and

$\text{M}{}_4\text{= p}{}^2\text{+ 6(p}{}^2\text{? p)}$

. Here d is uniquely determined by

$\text{4}{}_{\mathrm{p}}\text{= d}{}^2\text{+ 27b}{}^2\text{and}\text{d ≡ 1(}\text{mod}\text{3)}$

.     

Common strict character of some sharp infinite-sequence martingale inequalities

A procedure is given for proving strictness of some sharp, infinite-sequence martingale inequalities, which arise from sharp, finite-sequence martingale inequalities attained by degenerating extremal distributions. The procedure is applied to obtain strictness of the sharp inequalities of Cox and Kemperman

$\text{P(|X}{}_{\mathrm{i}}\text{|?1 for some i=1, 2,…)?(ln 2)}{}^{\mathrm{?1}}\text{sup}\text{n}\text{E}\text{∑}\text{i=0}\text{n}\text{X}{}_{\mathrm{i}}$

and of Cox (sharp form of Burkholder's inequality)

$\text{P}\text{∑}\text{i=0}\text{∞}\text{X}{}^2{}_{\mathrm{i}}\text{?1}\text{? e}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{sup}\text{n}\text{E}\text{∑}\text{i=0}\text{n}\text{X}{}_{\mathrm{i}}$

for all nontrivial martingale difference sequences X₀,X₁,….     

A remark on the tail probability of a distribution

Let {X_n}_n≥1 be a sequence of independent and identically distributed random variables. For each integer n ≥ 1 and positive constants r, t, and ?, let S_n = Σ_j=1ⁿX_j and $\text{E{N}{}_{\mathrm{\infty }}\text{(r, t, ?)} = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{n}{}^{\mathrm{r?2}}\text{P{|S}{}_{\mathrm{n}}\text{| > ?n}{}^{\mathrm{<mtext>r</mtext><mtext>t</mtext>}}\text{}}$. In this paper, we prove that (1) lim_?→0+?^α(r?1)E{N_∞(r, t, ?)} = K(r, t) if E(X₁) = 0, Var(X₁) = 1, and E(| X₁ |^t) < ∞, where 2 ≤ t < 2r ≤ 2t, $\text{K(r, t) =}\text{{2}{}^{\mathrm{<mtext>\alpha (r?1)</mtext><mtext>2</mtext>}}\text{Γ(}\text{(1 + α(r ? 1))}\text{2}\text{)}}\text{{(r ? 1) Γ(}\text{1}\text{2}\text{)}}$, and $\text{α =}\text{2t}\text{(2r ? t)}$; (2) $\text{lim}{}_{\mathrm{?\to 0+}}\text{G(t, ?)}\text{H(t, ?)}\text{= 0}$ if 2 < t < 4, E(X₁) = 0, Var(X₁) > 0, and E(|X₁|^t) < ∞, where G(t, ?) = E{N_∞(t, t, ?)} = Σ_n=1^∞n^t?2P{| S_n | > ?n} → ∞ as ? → 0+ and $\text{H(t, ?) = E{N}{}_{\mathrm{\infty }}\text{(t, t, ?)} = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{n}{}^{\mathrm{t?2}}\text{P{| S}{}_{\mathrm{n}}\text{| > ?n}{}^{\mathrm{<mtext>2</mtext><mtext>t</mtext>}}\text{} → ∞}\text{as}\text{? → 0+}$, i.e., H(t, ?) goes to infinity much faster than G(t, ?) as ? → 0+ if 2 < t < 4, E(X₁) = 0, Var(X₁) > 0, and E(| X₁ |^t) < ∞. Our results provide us with a much better and deeper understanding of the tail probability of a distribution.     

The nonorientable genus of the symmetric quadripartite graph

The nonorientable genus of K_4(n) is shown to satisfy:

$\text{γ}\text{(K}{}_{\mathrm{4(n)}}\text{)=2(n?1)}{}^2\text{for n ? 3}$

,

$\text{γ}\text{(K}{}_{\mathrm{4(2)}}\text{)=3,}\text{γ}\text{(K}{}_4\text{(1))=1}$

.     

On convergence rates and the expectation of sums of random variables

Let X_i be iidrv's and S_n=X₁+X₂+…+X_n. When EX²₁<+∞, by the law of the iterated logarithm $\text{(S}{}_{\mathrm{n}}\text{?α}{}_{\mathrm{n}}\text{)}\text{(n log n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{→0 a.s.}$ for some constants α_n. Thus the r.v. $\text{Y=sup}{}_{\mathrm{n?1}}\text{[|S}{}_{\mathrm{n}}\text{?α}{}_{\mathrm{n}}\text{|?(δn log n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{]}{}^{\mathrm{+}}$ is a.s.finite when δ>0. We prove a rate of convergence theorem related to the classical results of Baum and Katz, and apply it to show, without the prior assumption EX²₁<+∞ that EY^h<+∞ if and only if $\text{E}\text{|X}{}_1\text{|}{}^{\mathrm{2+h}}\text{[log|X}{}_1\text{|]}{}^{-1}\text{<+∞}$ for 0<h<1 and δ> $\text{h}\text{E}\text{(X}{}_1\text{?}\text{E}\text{X}{}_1\text{)}{}^2$, whereas $\text{E}\text{Y}{}^{\mathrm{h}}\text{=+∞}$ whenever h>0 and $\text{0<δ<h}\text{E}\text{(X}{}_1\text{?}\text{E}\text{X}{}_1\text{)}{}^2$.     

Construction of a recurrence relation for modified moments

In this paper we are constructing a recurrence relation of the form

$\text{∑}\text{i=0}\text{r}\text{ω}{}_{\mathrm{i}}\text{(k)m}{}_{\mathrm{k+i}}{}^{\mathrm{\{\lambda \}}}\text{[f] = ω(k)}$

for integrals (called modified moments)

$\text{m}{}_{\mathrm{k}}{}^{\mathrm{\{\lambda \}}}\text{[f]df=}\text{∫}\text{?1}\text{1}\text{f(x)C}{}_{\mathrm{k}}{}^{\mathrm{(\lambda )}}\text{(x)dx (k = 0,1,…)}$

in which C_k^(λ) is the k-th Gegenbauer polynomial of order $\text{λ(λ > ?}\text{1}\text{2}\text{)}$, and f is a function satisfying the differential equation

$\text{∑}\text{i=0}\text{n}\text{P}{}_{\mathrm{i}}\text{(x)f}{}^{\mathrm{(i)}}\text{(x) = p(x) (?1?x?1)}$

of order n, where p₀, p₁, …, p_n ? 0 are polynomials, and m_k^〈λ〉[p] is known for every k. We give three methods of construction of such a recurrence relation. The first of them (called Method I) is optimum in a certain sense.     

Limit distributions for the maxima of stationary Gaussian processes

Let {X_n} be a stationary Gaussian sequence with E{X₀} = 0, {X²₀} = 1 and E{X₀X_n} = rⁿ_n Let c_n = (2ln n)^{$\text{built1}\text{2}$}, b_n = c_n? $\text{1}\text{2}$c^-1_n ln(4π ln n), and set M_n = max_{0 ?k?n}X_k. A classical result for independent normal random variables is that

$\text{P[c}{}_{\mathrm{n}}\text{(M}{}_{\mathrm{n}}\text{?b}{}_{\mathrm{n}}\text{)?x]→exp[-e}{}^{\mathrm{-x}}\text{] as n → ∞ for all x.}$

Berman has shown that (1) applies as well to dependent sequences provided r_nlnn = o(1). Suppose now that {r_n} is a convex correlation sequence satisfying r_n = o(1), (r_nlnn)^-1 is monotone for large n and o(1). Then

$\text{P[r}{}_{\mathrm{n}}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{(M}{}_{\mathrm{n}}\text{? (1?r}{}_{\mathrm{n}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{b}{}_{\mathrm{n}}\text{)?x] → Ф(x)}$

for all x, where Ф is the normal distribution function. While the normal can thus be viewed as a second natural limit distribution for {M_n}, there are others. In particular, the limit distribution is given below when r_n is (sufficiently close to) γ/ln n. We further exhibit a collection of limit distributions which can arise when r_n decays to zero in a nonsmooth manner. Continuous parameter Gaussian processes are also considered. A modified version of (1) has been given by Pickands for some continuous processes which possess sufficient asymptotic independence properties. Under a weaker form of asymptotic independence, we obtain a version of (2).     

A transformation theorem for one-dimensional F-expansions

If f is a monotone function subject to certain restrictions, then one can associate with any real number x between zero and one a sequence {a_n(x)} of integers such that

$\text{x=f(a}{}_1\text{(x) + f(a}{}_2\text{(x) +f(a}{}_3\text{(x) +…)))}$

. In this paper properties of the function F defined by

$\text{Fx=g(a}{}_1\text{(x) + g(a}{}_2\text{(x) +g(a}{}_3\text{(x) +…)))}$

, where g is any function satisfying the same restrictions as f, are discussed. Principally, F is found to be useful in finding stationary measures on the sequences {a_n(x)}.