Mixing Times of Critical Two‐Dimensional Potts Models

We study dynamical aspects of the q‐state Potts model on an n × n box at its critical β_c(q). Heat‐bath Glauber dynamics and cluster dynamics such as Swendsen–Wang (that circumvent low‐temperature bottlenecks) are all expected to undergo “critical slowdowns” in the presence of periodic boundary conditions: the inverse spectral gap, which in the subcritical regime is O(1), should at criticality be polynomial in n for 1 < q ≤ 4, and exponential in n for q > 4 in accordance with the predicted discontinuous phase transition. This was confirmed for q = 2 (the Ising model) by the second author and Sly, and for sufficiently large q by Borgs et al. Here we show that the following holds for the critical Potts model on the torus: for q=3, the inverse gap of Glauber dynamics is n^O(1); for q = 4, it is at most n^{O(log n)}; and for every q > 4 in the phase‐coexistence regime, the inverse gaps of both Glauber dynamics and Swendsen‐Wang dynamics are exponential in n. For free or monochromatic boundary conditions and large q, we show that the dynamics at criticality is faster than on the torus (unlike the Ising model where free/periodic boundary conditions induce similar dynamical behavior at all temperatures): the inverse gap of Swendsen‐Wang dynamics is exp(n^o(1)). © 2017 Wiley Periodicals, Inc.     

On the ternary goldbach problem with primes in independent arithmetic progressions

We show that for every fixed A > 0 and θ > 0 there is a ϑ = ϑ(A, θ) > 0 with the following property. Let n be odd and sufficiently large, and let Q ₁ = Q ₂:= n ^1/2(log n)^−ϑ and Q ₃:= (log n) ^θ . Then for all q ₃ ≦ Q ₃, all reduced residues a ₃ mod q ₃, almost all q ₂ ≦ Q ₂, all admissible residues a ₂ mod q ₂, almost all q ₁ ≦ Q ₁ and all admissible residues a ₁ mod q ₁, there exists a representation n = p ₁ + p ₂ + p ₃ with primes p _i ≡ a _i (q _i ), i = 1, 2, 3.      

Asymptotic Properties of N-th Order Differential Equations with Delayed Argument

The aim of this paper is to deduce oscillatory and asymptotic behaviour of delay differential equation L_nu(t)– P(t)u(τ(t))= 0 from the oscillation of a set of the first order delay differential equations with larger deviating argument of the form y′(t)+ q_i(t) y(w(t)) = 0.     

Commuting polarities and maximal partial ovoids of H(4,q2)

In PG(4,q²), q odd, let Q(4,q²) be a non‐singular quadric commuting with a non‐singular Hermitian variety H(4,q²). Then these varieties intersect in the set of points covered by the extended generators of a non‐singular quadric Q₀ in a Baer subgeometry Σ₀ of PG(4,q²). It is proved that any maximal partial ovoid of H(4,q²) intersecting Q₀ in an ovoid has size at least 2(q²+1). Further, given an ovoid O of Q₀, we construct maximal partial ovoids of H(4,q²) of size q³+1 whose set of points lies on the hyperbolic lines 〈P,X〉 where P is a fixed point of O and X varies in O\{P}. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 307–313, 2009     

Minimum deformations of commutative algebra and linear groupGL(n)

In the algebra of formal seriesM _q (x ⁱ ), the relations of generalized commutativity that preserve the tensorI _q grading and depend on parametersq(i, k) are considered. A norm of the differential calculus onM _q consistent with theI _q grading is chosen. A new construction of a symmetrized tensor product of algebras of the typeM _q (x ⁱ ) and a corresponding definition of the minimally deformed linear groupQGL(n) and Lie algebraqgl(n) are proposed. A study is made of the connection ofQGL(n) andqgl(n) with the special matrix algebra Mat(n, Q), which consists of matrices with noncommuting elements. The deformed determinant in the algebra Mat(n, Q) is defined. The exponential mapping in the algebra Mat(n, Q) is considered on the basis of the Campbell-Hausdorff formula.Institute of Applied Physics, Tashkent State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 3, pp. 403–417, June, 1993.     

Sur la singularité des produits de Riesz et des mesures spectrales associées à la somme des chiffres

Using a property of generalized characters, we first prove that two Riesz products with constant coefficients and distinct Fourier spectra are mutually singular. IfS _r (n) denotes the sum of digits in ther-adic representation of the integern, the same technique allows us to establish the mutual singularity of the spectral measures of the sequences: α(n)=exp[2iπaS _p (n)],β(n)=exp[2iπbS _q (n)], for every pair of integersp≠q, a, b being real numbers such thata(p−1)∉ {tiZ} andb(q−1)∉Z. This result has been proved by T. Kamae whenp andq are two relatively prime integers.      

Maximal partial spreads in PG (3, q)

We transfer the whole geometry of PG(3, q) over a non-singular quadric Q_4,q of PG(4, q) mapping suitably PG(3, q) over Q_4,q. More precisely the points of PG(3, q) are the lines of Q_4,q; the lines of PG(3, q) are the tangent cones of Q_4,q and the reguli of the hyperbolic quadrics hyperplane section of Q_4,q. A plane of PG(3, q) is the set of lines of Q_4,q meeting a fixed line of Q_4,q. We remark that this representation is valid also for a projective space over any field K and we apply the above representation to construct maximal partial spreads in PG(3, q). For q even we get new cardinalities for For q odd the cardinalities are partially known.     

The scaling window of the 2‐SAT transition

We consider the random 2‐satisfiability (2‐SAT) problem, in which each instance is a formula that is the conjunction of m clauses of the form x∨y, chosen uniformly at random from among all 2‐clauses on n Boolean variables and their negations. As m and n tend to infinity in the ratio m/n→α, the problem is known to have a phase transition at α_c=1, below which the probability that the formula is satisfiable tends to one and above which it tends to zero. We determine the finite‐size scaling about this transition, namely the scaling of the maximal window W(n, δ)=(α_?(n,δ), α₊(n,δ)) such that the probability of satisfiability is greater than 1?δ for α<α_? and is less than δ for α>α₊. We show that W(n,δ)=(1?Θ(n^?1/3), 1+Θ(n^?1/3)), where the constants implicit in Θ depend on δ. We also determine the rates at which the probability of satisfiability approaches one and zero at the boundaries of the window. Namely, for m=(1+ε)n, where ε may depend on n as long as |ε| is sufficiently small and |ε|n^1/3 is sufficiently large, we show that the probability of satisfiability decays like exp(?Θ(nε³)) above the window, and goes to one like 1?Θ(n^?1|ε|^?3 below the window. We prove these results by defining an order parameter for the transition and establishing its scaling behavior in n both inside and outside the window. Using this order parameter, we prove that the 2‐SAT phase transition is continuous with an order parameter critical exponent of 1. We also determine the values of two other critical exponents, showing that the exponents of 2‐SAT are identical to those of the random graph. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 201–256 2001     

Varietà di Segre e ovoidi dello spazio polare Q+(7, q)

Summary In this paper, for q even, we construct an ovoid O ₃ and a spread S of the finite classical polar space Q⁺(7, q) determinated by a hyperbolic quadric Q⁺ of PG(7, q) such that there is a subgroup of PGO ₊ ⁸ (q) isomorphic to PGL₂(q ³), which maps O ₃ in itself and S in S and is 3-transitive on O ₃ and on S; for q>2, S is not a Desarguesian spread of Q⁺(7, q) and O ₃ is a Desarguesian ovoid.

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Varietà di Segre e ovoidi dello spazio polare Q⁺(7, q)

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Al Prof. Adriano Barlotti in occasione del suo 60^o compleanno     

Onq-difference equations andZ n decompositions of exp q function

Theq-extended hyperbolic functions ofn-th order {h _q,s(z)}s∈ Z _n which areZ _n-components of expq function form the set fundamental solutions of a simpleq-difference equation. Against the background ofq-deformed hyperbolic functions ofn-th order other natural extensions and related topics are considered. Apart from easy general solution of homogenous ordinaryq-difference equations ofn-th order main trigonometric-like identity known for hyperbolic functions ofn-th order is given itsq-commutative counterpart. Hint how to arrive at other identities is implicit in theq-treatment proposed. The paper constitutes an example of the application of the method of projections presented in Advances in Applied Clifford Algebras publication [19]; see also references to Ben Cheikh’s papers.     

Distribution of values of additive functions on short intervals

We consider a family of completely additive functions β_q(n) defined on the set of natural numbers. We find an asymptotic expression for the summation function Σ _n≤x β _q (n study its distribution on short intervals.     

The largest connected component in a random mapping

A random mapping (T; q) of a finite set V = {1, 2,…,n} into itself assigns independently to each i ? V its unique image j = TT(i)E V with probability q for i = j and with probability $ \frac{{1 - q}}{{n - 1}} $ for j ≠ i. The purpose of the article is to determine the asymptotic behaviour of the size of the largest connected component of the random digraph G_T(q) representing thes mapping as n–x, regarding all possible values of the parameter q = q(n). © 1994 John Wiley & Sons, Inc.     

A continuous function space with a Faber basis

Let be compact with #S=∞ and let C(S) be the set of all real continuous functions on S. We ask for an algebraic polynomial sequence (P_n)_n=0^∞ with deg P_n=n such that every fC(S) has a unique representation f=∑_i=0^∞ α_iP_i and call such a basis Faber basis. In the special case of , 0<q<1, we prove the existence of such a basis. A special orthonormal Faber basis is given by the so-called little q-Legendre polynomials. Moreover, these polynomials state an example with A(S_q)≠U(S_q)=C(S_q), where A(S_q) is the so-called Wiener algebra and U(S_q) is the set of all fC(S_q) which are uniquely represented by its Fourier series.     

Direct factors of polynomial rings over finite fields

Let G_q denote the multiplicative semigroup of all monic polynomials in one indeterminate over a finite field F_q with q elements. By a direct factor of G_q is understood a subset B₁ of G_q such that, for some subset B₂ of G_q, every polynomial w G_q has a unique factorization in the form w = b₁b₂ for b_i B_i. An asymptotic formula B₁^#(n) c₁qⁿ as n → ∞ is derived for the total number B₁^#(n) of polynomials of degree n in an arbitrary direct factor B₁ of G_q, c₁ a constant depending on B₁.     

Unbounded divergence of simple quadrature formulas

Given a sequence of real or complex coefficients c_i and a sequence of distinct nodes t_i in a compact interval T, we prove the divergence and the unbounded divergence on superdense sets in the space C(T) of the simple quadrature formulas ∝_Tx(t)du(t) = Q_n(x) + R_n(x) and ∝_Tw(t)x(t)dt = Q_n(x) + R_n(x), where Q_n(x)=∑_i=1^m_n c_ix(t_i), ε C(T).The divergence (not certainly unbounded) for at most one continuous function of the first simple quadrature formula, with m_n = n and u(t) = t, was established by P. J. Davis in 1953.     

A note on asymptotic expansions for Markov chains using operator theory

We consider asymptotic expansions for sums S_n on the form S_n = ƒ₀(X₀) + ƒ(X₁, X₀) + … + ƒ(X_n, X_n−1), where X_i is a Markov chain. Under different ergodicity conditions on the Markov chain and certain conditional moment conditions on ƒ(X_i, X_i−1), a simple representation of the characteristic function of S_n is obtained. The representation is in term of the maximal eigenvalue of the linear operator sending a function g(x) into the function x → E(g(X_i)exp[itƒ(X_i, x)]|X_i−1 = x).     

Translating IΔ0 + exp Proofs into Weaker Systems

The purpose of this paper is to explore the relationship between IΔ₀ + exp and its weaker subtheories. We give a method of translating certain classes of IΔ₀ + exp proofs into weaker systems of arithmetic such as Buss' systems S₂. We show if IE_i (exp) ⊢ A with a proof P of expind‐rank(P) ≤ n + 1where all (∀ ≤: right) or (∃ ≤: left) have bounding terms not containing function symbols, then Sⁱ ₂ ⊇ IE_i,2 ⊢ Aⁿ. Here A is not necessarily a bounded formula. For IOpen(exp) we prove a similar result. Using our translations we show IOpen(exp) ⊊ IΔ₀(exp). Here IΔ₀(exp) is a conservative extension of IΔ₀ + exp obtained by adding to IΔ₀ a symbol for 2 ^x to the language as well as defining axioms for it.