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1. On the arithmetic of a family of degree - two K3 surfaces

   https://doi.org/10.1017/S0305004118000087  

   BOUYER, FLORIAN; COSTA, EDGAR; FESTI, DINO; NICHOLLS, CHRISTOPHER; WEST, MCKENZIE (May 2019, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract Let ℙ denote the weighted projective space with weights (1, 1, 1, 3) over the rationals, with coordinates x , y , z and w ; let $$\mathcal{X}$$ be the generic element of the family of surfaces in ℙ given by \begin{equation*}X\colon w^2=x^6+y^6+z^6+tx^2y^2z^2.\end{equation*} The surface $$\mathcal{X}$$ is a K3 surface over the function field ℚ( t ). In this paper, we explicitly compute the geometric Picard lattice of $$\mathcal{X}$$ , together with its Galois module structure, as well as derive more results on the arithmetic of $$\mathcal{X}$$ and other elements of the family X . 

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2. VALUES OF RANDOM POLYNOMIALS IN SHRINKING TARGETS

   https://doi.org/https://doi.org/10.1090/tran/8204  

   Kelmer, Dubi Yu (January 2020, Transactions of the American Mathematical Society)

   null (Ed.)

   Relying on the classical second moment formula of Rogers we give an effective asymptotic formula for the number of integer vectors v in a ball of radius t, with value Q(v) in a shrinking interval of size t^{−λ}, that is valid for almost all indefinite quadratic forms in n variables for any λ 

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3. Realizing trees of configurations in thin sets

   https://doi.org/10.2140/pjm.2025.335.355  

   Greenleaf, Allan; Iosevich, Alex; Taylor, Krystal (January 2025, Pacific Journal of Mathematics)

   Let $$\phi(x,y)$$ be a continuous function, smooth away from the diagonal, such that, for some $$\alpha>0$$, the associated generalized Radon transforms \begin{equation} \label{Radon} R_t^{\phi}f(x)=\int_{\phi(x,y)=t} f(y) \psi(y) d\sigma_{x,t}(y) \end{equation} map $$L^2({\mathbb R}^d) \to H^{\alpha}({\mathbb R}^d)$$ for all $t>0$. Let $$E$$ be a compact subset of $${\mathbb R}^d$$ for some $$d \ge 2$$, and suppose that the Hausdorff dimension of $$E$$ is $$>d-\alpha$$. We show that any tree graph $$T$$ on $k+1$ ($$k \ge 1$$) vertices is realizable in $$E$$, in the sense that there exist distinct $$x^1, x^2, \dots, x^{k+1} \in E$$ and $t>0$ such that the $$\phi$$-distance $$\phi(x^i, x^j)$$ is equal to $$t$$ for all pairs $(i,j)$ corresponding to the edges of the graph $$T$$. 

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4. Individual differences and long-term consequences of tDCS augmented cognitive training

   Katz, B.; Au, J.; Buschkuehl, M.; Abagis, T.; Zabel, C.; Jaeggi, S.; Jonides, J. (October 2017, Journal of cognitive neuroscience)

   A gr e at d e al of i nt er e st s urr o u n d s t h e u s e of tr a n s cr a ni al dir e ct c urr e nt sti m ul ati o n (t D C S) t o a u g m e nt c o g niti v e tr ai ni n g. H o w e v er, eff e ct s ar e i n c o n si st e nt a cr o s s st u di e s, a n d m et aa n al yti c e vi d e n c e i s mi x e d, e s p e ci all y f o r h e alt h y, y o u n g a d ult s. O n e m aj or s o ur c e of t hi s i n c o n si st e n c y i s i n di vi d u al diff er e n c e s a m o n g t h e p arti ci p a nt s, b ut t h e s e diff er e n c e s ar e r ar el y e x a mi n e d i n t h e c o nt e xt of c o m bi n e d tr ai ni n g/ sti m ul ati o n st u di e s. I n a d diti o n, it i s u n cl e ar h o w l o n g t h e eff e ct s of sti m ul ati o n l a st, e v e n i n s u c c e s sf ul i nt er v e nti o n s. S o m e st u di e s m a k e u s e of f oll o w- u p a s s e s s m e nt s, b ut v er y f e w h a v e m e a s ur e d p erf or m a n c e m or e t h a n a f e w m o nt hs aft er a n i nt er v e nti o n. H er e, w e utili z e d d at a fr o m a pr e vi o u s st u d y of t D C S a n d c o g niti v e tr ai ni n g [ A u, J., K at z, B., B u s c h k u e hl, M., B u n arj o, K., S e n g er, T., Z a b el, C., et al. E n h a n ci n g w or ki n g m e m or y tr ai ni n g wit h tr a n scr a ni al dir e ct c urr e nt sti m ul ati o n. J o u r n al of C o g niti v e N e u r os ci e n c e, 2 8, 1 4 1 9 – 1 4 3 2, 2 0 1 6] i n w hi c h p arti ci p a nts tr ai n e d o n a w or ki n g m e m or y t as k o v er 7 d a y s w hil e r e c ei vi n g a cti v e or s h a m t D C S. A n e w, l o n g er-t er m f oll o w- u p t o a ss es s l at er p erf or m a n c e w a s c o n d u ct e d, a n d a d diti o n al p arti ci p a nt s w er e a d d e d s o t h at t h e s h a m c o n diti o n w a s b ett er p o w er e d. W e a s s e s s e d b a s eli n e c o g niti v e a bilit y, g e n d er, tr ai ni n g sit e, a n d m oti v ati o n l e v el a n d f o u n d si g nifi c a nt i nt er a cti o ns b et w e e n b ot h b as eli n e a bilit y a n d m oti v ati o n wit h c o n diti o n ( a cti v e or s h a m) i n m o d els pr e di cti n g tr ai ni n g g ai n. I n a d diti o n, t h e i m pr o v e m e nt s i n t h e a cti v e c o nditi o n v er s u s s h a m c o n diti o n a p p e ar t o b e st a bl e e v e n a s l o n g a s a y e ar aft er t h e ori gi n al i nt er v e nti o n. ■ 

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5. Field tunable magnetic transitions of CsCo ₂ (MoO ₄ ) ₂ (OH): a triangular chain structure with a frustrated geometry

   https://doi.org/10.1039/d2qm01272c  

   Sanjeewa, Liurukara D.; Garlea, V. Ovidiu; Fishman, Randy S.; Foroughian, Mahsa; Yin, Li; Xing, Jie; Parker, David S.; Smith Pellizzeri, Tiffany M.; Sefat, Athena S.; Kolis, Joseph W. (March 2023, Materials Chemistry Frontiers)

   The sawtooth chain compound CsCo 2 (MoO 4 ) 2 (OH) is a complex magnetic system and here, we present a comprehensive series of magnetic and neutron scattering measurements to determine its magnetic phase diagram. The magnetic properties of CsCo 2 (MoO 4 ) 2 (OH) exhibit a strong coupling to the crystal lattice and its magnetic ground state can be easily manipulated by applied magnetic fields. There are two unique Co 2+ ions, base and vertex, with J bb and J bv magnetic exchange. The magnetism is highly anisotropic with the b -axis (chain) along the easy axis and the material orders antiferromagnetically at T N = 5 K. There are two successive metamagnetic transitions, the first at H c 1 = 0.2 kOe into a ferrimagnetic structure, and the other at H c 2 = 20 kOe to a ferromagnetic phase. Heat capacity measurements in various fields support the metamagnetic phase transformations, and the magnetic entropy value is intermediate between S = 3/2 and 1/2 states. The zero field antiferromagnetic phase contains vertex magnetic vectors (Co(1)) aligned parallel to the b -axis, while the base vectors (Co(2)) are canted by 34° and aligned in an opposite direction to the vertex vectors. The spins in parallel adjacent chains align in opposite directions, creating an overall antiferromagnetic structure. At a 3 kOe applied magnetic field, adjacent chains flip by 180° to generate a ferrimagnetic phase. An increase in field gradually induces the Co(1) moment to rotate along the b -axis and align in the same direction with Co(2) generating a ferromagnetic structure. The antiferromagnetic exchange parameters are calculated to be J bb = 0.028 meV and J bv = 0.13 meV, while the interchain exchange parameter is considerably weaker at J ch = (0.0047/ N ch ) meV. Our results demonstrate that the CsCo 2 (MoO 4 ) 2 (OH) is a promising candidate to study new physics associated with sawtooth chain magnetism and it encourages further theoretical studies as well as the synthesis of other sawtooth chain structures with different magnetic ions. 

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