scholarly journals Polynomization of the Bessenrodt–Ono Inequality

2020 ◽  
Vol 24 (4) ◽  
pp. 697-709
Author(s):  
Bernhard Heim ◽  
Markus Neuhauser ◽  
Robert Tröger
Keyword(s):  
Number Theory ◽  
Partition Function ◽  
Real Numbers ◽  
Special Values ◽  
Colored Partitions ◽  
Roots Of Polynomials

Abstract In this paper, we investigate a generalization of the Bessenrodt–Ono inequality by following Gian–Carlo Rota’s advice in studying problems in combinatorics and number theory in terms of roots of polynomials. We consider the number of k-colored partitions of n as special values of polynomials $$P_n(x)$$ P n ( x ) . We prove for all real numbers $$x >2 $$ x > 2 and $$a,b \in \mathbb {N}$$ a , b ∈ N with $$a+b >2$$ a + b > 2 the inequality: $$\begin{aligned} P_a(x) \, \cdot \, P_b(x) > P_{a+b}(x). \end{aligned}$$ P a ( x ) · P b ( x ) > P a + b ( x ) . We show that $$P_n(x) < P_{n+1}(x)$$ P n ( x ) < P n + 1 ( x ) for $$x \ge 1$$ x ≥ 1 , which generalizes $$p(n) < p(n+1)$$ p ( n ) < p ( n + 1 ) , where p(n) denotes the partition function. Finally, we observe for small values, the opposite can be true, since, for example: $$P_2(-3+ \sqrt{10}) = P_{3}(-3 + \sqrt{10})$$ P 2 ( - 3 + 10 ) = P 3 ( - 3 + 10 ) .

scholarly journals Distinct partitions and overpartitions

2021 ◽  
Vol 38 (1) ◽  
pp. 149-158
Author(s):  
MIRCEA MERCA ◽  
Keyword(s):  
Partition Function ◽  
Recurrence Relation ◽  
Recurrence Relations ◽  
Convergent Series ◽  
The Other ◽  
Large Integer ◽  
Linear Recurrence ◽  
Fast Computation ◽  
Real Numbers ◽  
Very High

In 1963, Peter Hagis, Jr. provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the partition function $Q(n)$ which counts partitions of $n$ into distinct parts. Computing $Q(n)$ by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we investigate new connections between partitions into distinct parts and overpartitions and obtain a surprising recurrence relation for the number of partitions of $n$ into distinct parts. By particularization of this relation, we derive two different linear recurrence relations for the partition function $Q(n)$. One of them involves the thrice square numbers and the other involves the generalized octagonal numbers. The recurrence relation involving the thrice square numbers provide a simple and fast computation of the value of $Q(n)$. This method uses only (large) integer arithmetic and it is simpler to program. Infinite families of linear inequalities involving partitions into distinct parts and overpartitions are introduced in this context.

scholarly journals A system of axiomatic set theory. Part III. Infinity and enumerability. Analysis

1942 ◽  
Vol 7 (2) ◽  
pp. 65-89 ◽  
Author(s):  
Paul Bernays
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Real Numbers ◽  
Full Generality ◽  
Axiomatic Set Theory

The foundation of analysis does not require the full generality of set theory but can be accomplished within a more restricted frame. Just as for number theory we need not introduce a set of all finite ordinals but only a class of all finite ordinals, all sets which occur being finite, so likewise for analysis we need not have a set of all real numbers but only a class of them, and the sets with which we have to deal are either finite or enumerable.We begin with the definitions of infinity and enumerability and with some consideration of these concepts on the basis of the axioms I—III, IV, V a, V b, which, as we shall see later, are sufficient for general set theory. Let us recall that the axioms I—III and V a suffice for establishing number theory, in particular for the iteration theorem, and for the theorems on finiteness.

Towards a consistent set-theory

1951 ◽  
Vol 16 (2) ◽  
pp. 130-136 ◽  
Author(s):  
John Myhill
Keyword(s):  
Number Theory ◽  
Real Number ◽  
Set Theory ◽  
Mathematical Practice ◽  
Axiom Of Choice ◽  
Small Fragment ◽  
Real Numbers ◽  
The Real ◽  
Classical Construction

In a previous paper, I proved the consistency of a non-finitary system of logic based on the theory of types, which was shown to contain the axiom of reducibility in a form which seemed not to interfere with the classical construction of real numbers. A form of the system containing a strong axiom of choice was also proved consistent.It seems to me now that the real-number approach used in that paper, though valid, was not the most fruitful one. We can, on the lines therein suggested, prove the consistency of axioms closely resembling Tarski's twenty axioms for the real numbers; but this, from the standpoint of mathematical practice, is a pitifully small fragment of analysis. The consistency of a fairly strong set-theory can be proved, using the results of my previous paper, with little more difficulty than that of the Tarski axioms; this being the case, it would seem a saving in effort to derive the consistency of such a theory first, then to strengthen that theory (if possible) in such ways as can be shown to preserve consistency; and finally to derive from the system thus strengthened, if need be, a more usable real-number theory. The present paper is meant to achieve the first part of this program. The paragraphs of this paper are numbered consecutively with those of my previous paper, of which it is to be regarded as a continuation.

scholarly journals On Roots of Polynomials and Algebraically Closed Fields

2017 ◽  
Vol 25 (3) ◽  
pp. 185-195 ◽  
Author(s):  
Christoph Schwarzweller
Keyword(s):  
Algebraic Theory ◽  
Polynomial Rings ◽  
Multiple Roots ◽  
Real Numbers ◽  
The Real ◽  
Closed Fields ◽  
Roots Of Polynomials ◽  
Algebraically Closed Fields ◽  
Finite Domains ◽  
Identity Theorem

Summary In this article we further extend the algebraic theory of polynomial rings in Mizar [1, 2, 3]. We deal with roots and multiple roots of polynomials and show that both the real numbers and finite domains are not algebraically closed [5, 7]. We also prove the identity theorem for polynomials and that the number of multiple roots is bounded by the polynomial’s degree [4, 6].

scholarly journals Higher Order Oscillation and Uniform Distribution

2019 ◽  
Vol 14 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Shigeki Akiyama ◽  
Yunping Jiang
Keyword(s):  
Number Theory ◽  
Real Number ◽  
Uniform Distribution ◽  
Differentiable Function ◽  
Mild Condition ◽  
Higher Order ◽  
Real Numbers ◽  
Mobius Function ◽  
Real Polynomials ◽  
Almost All

AbstractIt is known that the Möbius function in number theory is higher order oscillating. In this paper we show that there is another kind of higher order oscillating sequences in the form (e2πiαβn g(β))n∈𝕅, for a non-decreasing twice differentiable function g with a mild condition. This follows the result we prove in this paper that for a fixed non-zero real number α and almost all real numbers β> 1 (alternatively, for a fixed real number β> 1 and almost all real numbers α) and for all real polynomials Q(x), sequences (αβng(β)+ Q(n)) n∈𝕅 are uniformly distributed modulo 1.

Commutative version of the local Hamiltonian problem and common eigenspace problem

2005 ◽  
Vol 5 (3) ◽  
pp. 187-215
Author(s):  
S. Bravyi ◽  
M. Vyalyi
Keyword(s):  
Hilbert Space ◽  
Quantum Systems ◽  
Real Numbers ◽  
Positive Dimension ◽  
Special Values ◽  
Quantum Analogue ◽  
Complete Problems ◽  
The Common ◽  
Np Complete ◽  
Special Case

We study the complexity of a problem Common Eigenspace --- verifying consistency of eigenvalue equations for composite quantum systems. The input of the problem is a family of pairwise commuting Hermitian operators H_1,\ldots,H_r on a Hilbert space (\CC^d)^{\otimes n} and a string of real numbers \lambda=(\lambda_1,\ldots,\lambda_r). The problem is to determine whether the common eigenspace specified by equalities H_a|\psi\ra=\lambda_a|\psi\ra, a=1,\ldots,r has a positive dimension. We consider two cases: (i) all operators H_a are k-local; (ii) all operators H_a are factorized. It can be easily shown that both problems belong to the class \QMA --- quantum analogue of \NP, and that some \NP-complete problems can be reduced to either (i) or (ii). A non-trivial question is whether the problems (i) or (ii) belong to \NP? We show that the answer is positive for some special values of k and d. Also we prove that the problem (ii) can be reduced to its special case, such that all operators H_a are factorized projectors and all \lambda_a=0.

New Graph Polynomials from the Bethe Approximation of the Ising Partition Function

2010 ◽  
Vol 20 (2) ◽  
pp. 299-320 ◽  
Author(s):  
Y. WATANABE ◽  
K. FUKUMIZU
Keyword(s):  
Performance Analysis ◽  
Partition Function ◽  
The Other ◽  
Cambridge Philos ◽  
Graph Polynomials ◽  
Bethe Approximation ◽  
Special Values ◽  
Spanning Subgraphs

We introduce two graph polynomials and discuss their properties. One is a polynomial of two variables whose investigation is motivated by the performance analysis of the Bethe approximation of the Ising partition function. The other is a polynomial of one variable that is obtained by the specialization of the first one. It is shown that these polynomials satisfy deletion–contraction relations and are new examples of the V-function, which was introduced by Tutte (Proc. Cambridge Philos. Soc.43, 1947, p. 26). For these polynomials, we discuss the interpretations of special values and then obtain the bound on the number of sub-coregraphs,i.e., spanning subgraphs with no vertices of degree one. It is proved that the polynomial of one variable is equal to the monomer–dimer partition function with weights parametrized by that variable. The properties of the coefficients and the possible region of zeros are also discussed for this polynomial.

Approximating sequences and Hausdorff measure

1974 ◽  
Vol 76 (1) ◽  
pp. 161-172 ◽  
Author(s):  
R. J. Gardner
Keyword(s):  
Number Theory ◽  
Numerical Analysis ◽  
Hausdorff Measure ◽  
Metric Space ◽  
Complex Analysis ◽  
Hausdorff Measures ◽  
Real Numbers ◽  
Majorizing Sequences ◽  
Approximating Sequence ◽  
Separable Metric Space

Approximating sequences have been extensively studied in many branches of mathematics, for example, in number theory (approximating real numbers by rationals) and in numerical analysis (approximations to functions by polynomials). In (1), A. Hyllengren introduced a type of approximating sequence ‘majorizing sequences’ which he used in solving a problem in complex analysis. In this note we study a very similar concept, which is general enough to be applicable to any separable metric space, and which turns out to have strong connexions with the theory of Hausdorff measures (as did Hyllengren's majorizing sequences).

scholarly journals Stability analysis of an ensemble of simple harmonic oscillators

2021 ◽  
pp. 2150034
Author(s):  
R. K. Thakur ◽  
B. N. Tiwari ◽  
R. Nigam ◽  
Y. Xu ◽  
P. K. Thiruvikraman
Keyword(s):  
Partition Function ◽  
Harmonic Oscillator ◽  
Geometric Phase ◽  
Hessian Matrix ◽  
Harmonic Oscillators ◽  
Real Numbers ◽  
Oscillator Potential ◽  
Harmonic Oscillator Potential ◽  
The Stability ◽  
Embedding Function

In this paper, we investigate the stability of the configurations of harmonic oscillator potential that are directly proportional to the square of the displacement. We derive expressions for fluctuations in partition function due to variations of the parameters, viz. the mass, temperature and the frequency of oscillators. Here, we introduce the Hessian matrix of the partition function as the model embedding function from the space of parameters to the set of real numbers. In this framework, we classify the regions in the parameter space of the harmonic oscillator fluctuations where they yield a stable statistical configuration. The mechanism of stability follows from the notion of the fluctuation theory. In Secs. ?? and ??, we provide the nature of local and global correlations and stability regions where the system yields a stable or unstable statistical basis, or it undergoes into geometric phase transitions. Finally, in Sec. ??, the comparison of results is provided with reference to other existing research.

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