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About 6,090 papers for “Parity function”

Parity of Schur's partition function

Shi-Chao Chen · Comptes Rendus Mathématique · 2019

Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> be the number of Schur's partitions of n , i.e. the number of partitions of n into distinct parts congruent to 1, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>2</mml:mn> <mml:mspace width="0.40em"/> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi mathvariant="normal">mod</mml:mi> </mml:mrow> <mml:mspace width="0.25em"/> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> . We prove <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"> <mml:mfrac> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow> <mml:msup> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">log</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:mn>47</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>48</mml:mn> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mo>≪</mml:mo> <mml:mo>♯</mml:mo> <mml:mo stretchy="false">{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>n</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>x</mml:mi> <mml:mo>:</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo linebreak="badbreak" linebreakstyle="after">+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mtext> is odd</mml:mtext> <mml:mo stretchy="false">}</mml:mo> <mml:mo>≪</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow> <mml:msup> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">log</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mo>.</mml:mo> </mml:math>

Cited by 3

Evans function, parity and nonautonomous bifurcations

Christian Pötzsche, Robert Skiba · Proceedings of the Royal Society of Edinburgh Section A Mathematics · 2025

Abstract The concept of parity due to Fitzpatrick, Pejsachowicz and Rabier is a central tool in the abstract bifurcation theory of nonlinear Fredholm operators. In this paper, we relate the parity to the Evans function, which is widely used in the stability analysis for traveling wave solutions to evolutionary PDEs. In contrast, we use Evans function as a flexible tool yielding general sufficient condition for local bifurcations of specific bounded entire solutions to (Carathéodory) differential equations. These bifurcations are intrinsically nonautonomous in the sense that the assumptions implying them cannot be fulfilled for autonomous or periodic temporal forcings. In addition, we demonstrate that Evans functions are strictly related to the dichotomy spectrum and hyperbolicity, which play a crucial role in studying the existence of bounded solutions on the whole real line and therefore the recent field of nonautonomous bifurcation theory. Finally, by means of non-trivial examples we illustrate the applicability of our methods.

Cited by 0Open access

Evans function, parity and nonautonomous bifurcations

Pötzsche, Christian, Skiba, Robert · arXiv (Cornell University) · 2025

The concept of parity due to Fitzpatrick, Pejsachowicz and Rabier is a central tool in the abstract bifurcation theory of nonlinear Fredholm operators. In this paper, we relate the parity to the Evans function, which is widely used in the stability analysis for traveling wave solutions to evolutionary PDEs. As application we obtain a flexible and general condition yielding local bifurcations of specific bounded entire solutions to (Carath{é}odory) differential equations. These bifurcations are intrinsically nonautonomous in the sense that the assumptions implying them cannot be fulfilled for autonomous or periodic temporal forcings. In addition, we demonstrate that Evans functions are strictly related to the dichotomy spectrum and hyperbolicity, which play a crucial role in studying the existence of bounded solutions on the whole real line and therefore the recent field of nonautonomous bifurcation theory. Finally, by means of non-trivial examples we illustrate the applicability of our methods.

Cited by 0Open access

Parity, education, and postmenopausal cognitive function

Sybil L. Crawford · Menopause The Journal of The North American Menopause Society · 2020

University of Massachusetts Medical School, Worcester, MA. Address correspondence to: Sybil L. Crawford, PhD, University of Massachusetts Medical School, Graduate School of Nursing, 55 Lake Avenue North, S1-853, Worcester, MA 01655. E-mail: [email protected] Received 31 August, 2020 Revised 1 September, 2020 Accepted 1 September, 2020 Funding/support: None reported. Financial disclosure/conflicts of interest: None reported.

Cited by 2

The parity function in optical waves propagation

B. Ruiz, Héctor Rabal, Sergio Granieri · Greater South Information System · 2003

We propose a generalization of the concept of parity as a continuous function and show that it is conserved in optical waves free propagation. Some properties are commented and an experimental implementation to optically generate a representation is suggested.

Cited by 0Open access

The parity function in optical waves propagation

B. Ruiz, Héctor Rabal, Sergio Granieri · Greater South Information System · 2003

We propose a generalization of the concept of parity as a continuous function and show that it is conserved in optical waves free propagation. Some properties are commented and an experimental implementation to optically generate a representation is suggested.

Cited by 0Open access

Design of an Energy-Efficient Parity Function Circuit

Jeong Beom Kim · 2023

In this paper, we introduce an energy-efficient parity function circuit that is implemented using XOR (exclusive-OR) and XNOR (exclusive-NOR) circuits. The proposed circuit reduces the internal load capacitance on the critical path and is designed with only 8 transistors. The circuit produces perfect output signals for all input combinations. The proposed parity function circuit demonstrates a 6.4<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">%</sup> reduction in average propagation delay time, a 29.7<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">%</sup> decrease in power dissipation, a 36.6% decrease in power-delay product (PDP), a 43.0% decrease in energy-delay product (EDP), and a 55.6% decrease in energy-delay-delay product (EDDP), respectively. Compared with the previous circuits, the proposed circuit exhibits the improved characteristics in terms of average propagation delay time, power dissipation, PDP, EDP, and EEDP. The proposed circuits are implemented with standard CMOS 0.18um technology. Computer simulations using SPICE show that the proposed circuit performs the expected logic functions and achieves a reasonable performance.

Cited by 0

Parity of the partition function in quadratic progressions

Ono, Ken · ArXiv.org · 2025

The parity of the partition function $p(n)$ remains strikingly mysterious. Beyond a handful of fragmentary results, essentially nothing is known about the distribution of parity. We prove a uniform result on quadratic progressions. If $1<D\equiv 23\pmod{24}$ is square-free and only divisible by primes $\ell\equiv 1, 7\pmod 8$, then both parities occur infinitely often among $$ p\left(\frac{Dm^2+1}{24}\right), $$ with $(m,6)=1.$ The argument takes place on the modular curve $X_0(6)$ and shows that parity along these thin orbits is \emph{not constant}. The proof connects classical identities for the partition generating function, through the method of (twisted) Borcherds products, to the arithmetic geometry of {\it ordinary} CM fibers on the Deligne-Rapoport model of $X_0(6)$ in characteristic 2. This result is a special case of a general theorem for the coefficients of suitable vector-valued weight 1/2 harmonic Maass forms that satisfy a "Heegner packet'' condition.

Cited by 0Open access

An improvement on the parity of Schur's partition function

Yiwen Lu, Tao Wei, Xuejun Guo · arXiv (Cornell University) · 2022

We improve S.-C. Chen's result on the parity of Schur's partition function. Let $A(n)$ be the number of Schur's partitions of $n$, i.e., the number of partitions of $n$ into distinct parts congruent to $1, 2 \mod{3}$. S.-C. Chen \cite{MR3959837} shows $\small \frac{x}{(\log{x})^{\frac{47}{48}}} \ll \sharp \{0\le n\le x:A(2n+1)\; \text{is odd}\}\ll \frac{x}{(\log{x})^{\frac{1}{2}}}$. In this paper, we improve Chen's result to $\frac{x}{(\log{x})^{\frac{11}{12}}} \ll \sharp \{0\le n\le x:A(2n+1)\; \text{is odd}\}\ll \frac{x}{(\log{x})^{\frac{1}{2}}}.$

Cited by 0Open access