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Explicit Factorization of $x^{p+1}-1$ over $\mathbb{Z}_{p^e}$: A Structural Approach via Dickson Polynomials

arXiv:2604.19038v2 Announce Type: replace Abstract: Let $p$ be an odd prime. The factorization of the polynomial $x^{p+1}-1$ over the integer residue ring $\mathbb{Z}_{p^e}$ is pivotal for constructing cyclic codes with Hermitian symmetry, a critical resource for Linear Complementary Dual (LCD) codes and Entanglement-Assisted Quantum Error-Correcting Codes (EAQECC). Traditionally, lifting factorizations relies on the generic Hensel's Lemma, masking the underlying algebraic structure.

arXiv CS 9d ago

England’s Freya Kemp makes mark with bat and ball to level series against India

2nd T20i: Eng, 168-5, bt India, 142-9, by 26 runsIndia’s decision to retire out Bhatia in run chase backfiresFreya Kemp smashed an unbeaten 39 from 13 balls as England finally unleashed some batting firepower to post 168 for five and beat India by 26 runs in the second T20 international. Kemp found able support from Dani Gibson as England’s two finishers added 39 from the final two overs to level the series. The pressure told on India’s fielders, who – after earlier offering Sophia Dunkley a...

The Guardian Sport 11d ago

England’s Freya Kemp makes mark with bat and ball to level series against India

2nd T20i: Eng, 168-5, bt India, 142-9, by 26 runsIndia’s decision to retire out Bhatia in run chase backfiresFreya Kemp smashed an unbeaten 39 from 13 balls as England finally unleashed some batting firepower to post 168 for five and beat India by 26 runs in the second T20 international. Kemp found able support from Dani Gibson as England’s two finishers added 39 from the final two overs to level the series. The pressure told on India’s fielders, who – after earlier offering Sophia Dunkley a...

The Guardian UK 11d ago

Dmitry Bivol reclaims top spot at light heavyweigh...

Dmitry Bivol returned to the rankings after a 15-month absence, reclaiming the No. 1 spot at light heavyweight with a dominant performance against Michael Eifert . The fight was less about the opponent and more about Bivol knocking the rust off with a one-sided victory that was never in doubt. After knocking down Eifert in the opening round with a left hook, Bivol remained in control to retain the WBA and IBF light heavyweight titles by unanimous decision.

ESPN 8d ago

Kemp stars as England level T20 series against India

Freya Kemp stars as hosts win second T20I to level series in Bristol England beat India by 26 runs in Bristol to level T20I series and set up decider in Taunton; watch England's third T20 international versus India, live on Sky Sports Cricket on Tuesday from 6pm (first ball 6.30pm) Saturday 30 May 2026 17:42, UK Freya Kemp starred with bat and ball as England secured a series-levelling T20 victory over India by 26 runs in Bristol. Kemp came to the crease with England floundering on 116-4...

Sky Sports Football 11d ago

'Facing as many balls as possible' - should England promote Kemp up the order?

Freya Kemp: Should England promote all-rounder up the order after impressing in T20 against India? After her dazzling display with bat and ball against India, should Freya Kemp be promoted up England's batting order?; watch England's third T20 international versus India, live on Sky Sports Cricket on Tuesday from 6pm (first ball 6.30pm) Sunday 31 May 2026 07:06, UK Are England extracting all they can from Freya Kemp, or should she be moved up the order?

Sky Sports Football 10d ago

30 years of Pokémon: The 'immersive' phenomenon still finding new fans

feel good The enduring appeal of Pokémon, still finding new fans 30 years after its release Sun 7 Jun 2026 at 6:10am For Jez Sundqvist, the Pokémon experience is still unforgettable almost three decades after he first encountered it. "It just brings back all those memories of getting probably, you know, Pokémon Red or Pokémon Blue for the Game Boy and cracking that out of the packet," he said. It is a joy he now experiences with his 11-year-old son, Auden, in their hometown of Port Lincoln...

ABC Australia 3d ago

Diverse binding poses of agonistic neurotoxins on human Na<sub>v</sub>1.6

Abstract Voltage-gated sodium (Nav) channels are key targets of various venomous toxins. Deciphering the binding poses and mechanisms of action of representative toxins will help to dissect the functional mechanism of the channels and facilitate therapeutic development targeting Nav channels1,2. Here we present cryo-electron microscopy (cryo-EM) structures of distinct binding poses of three agonistic peptide toxins on the human Nav1.6–β1 channel complex.

Nature 17h ago

Mutation-dependent responses to sleep and exercise in clonal haematopoiesis

Abstract Clonal haematopoiesis (CH) activates inflammation and increases the risk of atherosclerosis1,2. Whether lifestyle alters CH clone expansion or the phenotypic programming of CH mutant cells, thereby affecting atherosclerosis, is unknown. Here, in humans and mice and across mutations in Jak2, Tet2, Trp53 and Dnmt3a, we demonstrate mutation-dependent responses to sleep and exercise in CH and show that mutant cells are uniquely sensitive to lifestyle.

Nature 17h ago

A thalamus–brainstem attractor network drives history-biased decisions

Abstract Natural environments often change gradually, making it adaptive to bias decisions on the basis of the recent past — a phenomenon known as serial dependence1,2,3. Large-scale recordings during behaviour have identified that serial dependence is a common motif for decision-making, with neural representations of past experiences found throughout the brain4,5,6,7,8,9,10,11. However, it remains unclear whether this bias arises from dedicated neural circuits with history-specific...

Nature 17h ago