Herbrand consistency and bounded arithmetic

Zofia Adamowicz

Displaying similar documents to “Herbrand consistency and bounded arithmetic”

A note on Δ₁ induction and Σ₁ collection

Neil Thapen (2005)

Fundamenta Mathematicae

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Slaman recently proved that Σₙ collection is provable from Δₙ induction plus exponentiation, partially answering a question of Paris. We give a new version of this proof for the case n = 1, which only requires the following very weak form of exponentiation: " $x^{y}$ exists for some y sufficiently large that x is smaller than some primitive recursive function of y".

On the weak pigeonhole principle

Jan Krajíček (2001)

Fundamenta Mathematicae

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We investigate the proof complexity, in (extensions of) resolution and in bounded arithmetic, of the weak pigeonhole principle and of the Ramsey theorem. In particular, we link the proof complexities of these two principles. Further we give lower bounds to the width of resolution proofs and to the size of (extensions of) tree-like resolution proofs of the Ramsey theorem. We establish a connection between provability of WPHP in fragments of bounded arithmetic and cryptographic assumptions...

On integers of the form $p + 2^{k}$

Laurent Habsieger, Xavier-François Roblot (2006)

Acta Arithmetica

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Some properties of the class of arithmetic functions $T_{r} (N)$

R. P. Pakshirajan (1963)

Annales Polonici Mathematici

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Aposyndesis in $ℕ$

José del Carmen Alberto-Domínguez, Gerardo Acosta, Maira Madriz-Mendoza (2023)

Commentationes Mathematicae Universitatis Carolinae

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We consider the Golomb and the Kirch topologies in the set of natural numbers. Among other results, we show that while with the Kirch topology every arithmetic progression is aposyndetic, in the Golomb topology only for those arithmetic progressions $P (a, b)$ with the property that every prime number that divides $a$ also divides $b$ , it follows that being connected, being Brown, being totally Brown, and being aposyndetic are all equivalent. This characterizes the arithmetic progressions which are...

Product sets cannot contain long arithmetic progressions

Dmitrii Zhelezov (2014)

Acta Arithmetica

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Let B be a set of complex numbers of size n. We prove that the length of the longest arithmetic progression contained in the product set B.B = bb’ | b,b’ ∈ B cannot be greater than O((nlog²n)/(loglogn)) and present an example of a product set containing an arithmetic progression of length Ω(nlogn). For sets of complex numbers we obtain the upper bound $O (n^{3 / 2})$ .

An inconsistency equation involving means

Roman Ger, Tomasz Kochanek (2009)

Colloquium Mathematicae

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We show that any quasi-arithmetic mean $A_{φ}$ and any non-quasi-arithmetic mean M (reasonably regular) are inconsistent in the sense that the only solutions f of both equations $f (M (x, y)) = A_{φ} (f (x), f (y))$ and $f (A_{φ} (x, y)) = M (f (x), f (y))$ are the constant ones.

Numerical characterization of nef arithmetic divisors on arithmetic surfaces

Atsushi Moriwaki (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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In this paper, we give a numerical characterization of nef arithmetic $ℝ$ -Cartier divisors of $C^{0}$ -type on an arithmetic surface. Namely an arithmetic $ℝ$ -Cartier divisor $\bar{D}$ of $C^{0}$ -type is nef if and only if $\bar{D}$ is pseudo-effective and $\hat{\deg} ({\bar{D}}^{2}) = \hat{vol} (\bar{D})$ .

On a certain class of arithmetic functions

Antonio M. Oller-Marcén (2017)

Mathematica Bohemica

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A homothetic arithmetic function of ratio $K$ is a function $f : ℕ \to R$ such that $f (K n) = f (n)$ for every $n \in ℕ$ . Periodic arithmetic funtions are always homothetic, while the converse is not true in general. In this paper we study homothetic and periodic arithmetic functions. In particular we give an upper bound for the number of elements of $f (ℕ)$ in terms of the period and the ratio of $f$ .

On arithmetic progressions on Edwards curves

Enrique González-Jiménez (2015)

Acta Arithmetica

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Let $m \in ℤ_{> 0}$ and a,q ∈ ℚ. Denote by $_{m} (a, q)$ the set of rational numbers d such that a, a + q, ..., a + (m-1)q form an arithmetic progression in the Edwards curve $E_{d} : x ² + y ² = 1 + d x ² y ²$ . We study the set $_{m} (a, q)$ and we parametrize it by the rational points of an algebraic curve.

A functional equation involving f and $f^{- 1}$ .

J. S. Ratti, Y. -F. Lin (1990)

Colloquium Mathematicae

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Generalized weighted quasi-arithmetic means and the Kolmogorov-Nagumo theorem

Janusz Matkowski (2013)

Colloquium Mathematicae

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A generalization of the weighted quasi-arithmetic mean generated by continuous and increasing (decreasing) functions $f ₁, . . ., f_{k} : I \to ℝ$ , k ≥ 2, denoted by $A^{[f ₁, . . ., f_{k}]}$ , is considered. Some properties of $A^{[f ₁, . . ., f_{k}]}$ , including “associativity” assumed in the Kolmogorov-Nagumo theorem, are shown. Convex and affine functions involving this type of means are considered. Invariance of a quasi-arithmetic mean with respect to a special mean-type mapping built of generalized means is applied in solving a functional equation. For...

A structure theorem for sets of small popular doubling

Przemysław Mazur (2015)

Acta Arithmetica

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We prove that every set A ⊂ ℤ satisfying $\sum_{x} m i n (1_{A} * 1_{A} (x), t) \leq (2 + δ) t | A |$ for t and δ in suitable ranges must be very close to an arithmetic progression. We use this result to improve the estimates of Green and Morris for the probability that a random subset A ⊂ ℕ satisfies |ℕ∖(A+A)| ≥ k; specifically, we show that $ℙ (| ℕ ∖ (A + A) | \geq k) = Θ (2^{- k / 2})$ .

Lambert series and Liouville's identities

A. Alaca, Ş. Alaca, E. McAfee, K. S. Williams

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The relationship between Liouville’s arithmetic identities and products of Lambert series is investigated. For example it is shown that Liouville’s arithmetic formula for the sum $\sum_{\binom{(a, b, x, y) \in ℕ ⁴}{a x + b y = n}} (F (a - b) - F (a + b))$ , where n ∈ ℕ and F: ℤ → ℂ is an even function, is equivalent to the Lambert series for $(\sum_{n = 1}^{\infty} (q ⁿ / (1 - q ⁿ)) s i n n θ) ²$ (θ ∈ ℝ, |q| < 1) given by Ramanujan.

Automorphisms of models of bounded arithmetic

Ali Enayat (2006)

Fundamenta Mathematicae

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We establish the following model-theoretic characterization of the fragment IΔ₀ + Exp + BΣ₁ of Peano arithmetic in terms of fixed points of automorphisms of models of bounded arithmetic (the fragment IΔ₀ of Peano arithmetic with induction limited to Δ₀-formulae). Theorem A. The following two conditions are equivalent for a countable model of the language of arithmetic: (a) satisfies IΔ₀ + BΣ₁ + Exp; (b) $= I_{f i x} (j)$ for some nontrivial automorphism j of an end extension of that satisfies IΔ₀. Here...

О моделях арифметики в $A$ -изолиях.

В.Л. Михеев (1991)

Algebra i Logika

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Iterated quasi-arithmetic mean-type mappings

Paweł Pasteczka (2016)

Colloquium Mathematicae

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We work with a fixed N-tuple of quasi-arithmetic means $M ₁, . . ., M_{N}$ generated by an N-tuple of continuous monotone functions $f ₁, . . ., f_{N} : I \to ℝ$ (I an interval) satisfying certain regularity conditions. It is known [initially Gauss, later Gustin, Borwein, Toader, Lehmer, Schoenberg, Foster, Philips et al.] that the iterations of the mapping $I^{N} ∋ b \mapsto (M ₁ (b), . . ., M_{N} (b))$ tend pointwise to a mapping having values on the diagonal of $I^{N}$ . Each of [all equal] coordinates of the limit is a new mean, called the Gaussian product of the means $M ₁, . . ., M_{N}$ taken...

Precompactness in the uniform ergodic theory

Yu. Lyubich, J. Zemánek (1994)

Studia Mathematica

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We characterize the Banach space operators T whose arithmetic means ${n^{- 1} (I + T + . . . + T^{n - 1})}_{n \geq 1}$ form a precompact set in the operator norm topology. This occurs if and only if the sequence ${n^{- 1} T^{n}}_{n \geq 1}$ is precompact and the point 1 is at most a simple pole of the resolvent of T. Equivalent geometric conditions are also obtained.

On generalized square-full numbers in an arithmetic progression

Angkana Sripayap, Pattira Ruengsinsub, Teerapat Srichan (2022)

Czechoslovak Mathematical Journal

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Let $a$ and $b \in ℕ$ . Denote by $R_{a, b}$ the set of all integers $n > 1$ whose canonical prime representation $n = p_{1}^{α_{1}} p_{2}^{α_{2}} \dots p_{r}^{α_{r}}$ has all exponents $α_{i}$ $(1 \leq i \leq r)$ being a multiple of $a$ or belonging to the arithmetic progression $a t + b$ , $t \in ℕ_{0} : = ℕ \cup {0}$ . All integers in $R_{a, b}$ are called generalized square-full integers. Using the exponent pair method, an upper bound for character sums over generalized square-full integers is derived. An application on the distribution of generalized square-full integers in an arithmetic progression is given. ...

Lower and upper bounds for the provability of Herbrand consistency in weak arithmetics

Zofia Adamowicz, Konrad Zdanowski (2011)

Fundamenta Mathematicae

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We prove that for i ≥ 1, the arithmetic $I Δ ₀ + Ω_{i}$ does not prove a variant of its own Herbrand consistency restricted to the terms of depth in $(1 + ε) l o g^{i + 2}$ , where ε is an arbitrarily small constant greater than zero. On the other hand, the provability holds for the set of terms of depths in $l o g^{i + 3}$ .

On the equation $n (n + d) \dots (n + (i ₀ - 1) d) (n + (i ₀ + 1) d) \dots (n + (k - 1) d) = y^{l}$ with 0 < i₀ < k-1

N. Saradha, T. N. Shorey (2007)

Acta Arithmetica

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Algebraic and topological structures on the set of mean functions and generalization of the AGM mean

Bakir Farhi (2013)

Colloquium Mathematicae

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We present new structures and results on the set of mean functions on a given symmetric domain in ℝ². First, we construct on a structure of abelian group in which the neutral element is the arithmetic mean; then we study some symmetries in that group. Next, we construct on a structure of metric space under which is the closed ball with center the arithmetic mean and radius 1/2. We show in particular that the geometric and harmonic means lie on the boundary of . Finally, we give...

On embedding models of arithmetic of cardinality ℵ₁ into reduced powers

Juliette Kennedy, Saharon Shelah (2003)

Fundamenta Mathematicae

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In the early 1970’s S. Tennenbaum proved that all countable models of PA₁¯ + ∀₁ -Th(ℕ) are embeddable into the reduced product $ℕ^{ω} / ℱ$ , where ℱ is the cofinite filter. In this paper we show that if M is a model of PA¯ + ∀₁ - Th(ℕ), and |M| = ℵ₁, then M is embeddable into $ℕ^{ω} / D$ , where D is any regular filter on ω.

Matroids over a ring

Alex Fink, Luca Moci (2016)

Journal of the European Mathematical Society

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We introduce the notion of a matroid $M$ over a commutative ring $R$ , assigning to every subset of the ground set an $R$ -module according to some axioms. When $R$ is a field, we recover matroids. When $R = ℤ$ , and when $R$ is a DVR, we get (structures which contain all the data of) quasi-arithmetic matroids, and valuated matroids, i.e. tropical linear spaces, respectively. More generally, whenever $R$ is a Dedekind domain, we extend all the usual properties and operations holding for matroids (e.g., duality),...

Arithmetic of 0-cycles on varieties defined over number fields

Yongqi Liang (2013)

Annales scientifiques de l'École Normale Supérieure

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Let $X$ be a rationally connected algebraic variety, defined over a number field $k .$ We find a relation between the arithmetic of rational points on $X$ and the arithmetic of zero-cycles. More precisely, we consider the following statements: (1) the Brauer-Manin obstruction is the only obstruction to weak approximation for $K$ -rational points on $X_{K}$ for all finite extensions $K / k;$ (2) the Brauer-Manin obstruction is the only obstruction to weak approximation in some sense that we define for zero-cycles...