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The method of Least Square Approximation is an important topic in some linear algebra classes. Despite this, little is known about how students come to understand it, particularly in a Realistic Mathematics Education setting. Here, we report on how students used literal symbols and equations when solving a least squares problem in a travel scenario, as well as their reflections on the least squares equation in an open-ended written question. We found students used unknowns and parameters in a variety of ways. We highlight how their use of dot product equations can be helpful towards supporting their understanding of the least squares equation. 

The method of Least Square Approximation is an important topic in some linear algebra classes. Despite this, little is known about how students come to understand it, particularly in a Realistic Mathematics Education setting. Here, we report on how students used literal symbols and equations when solving a least squares problem in a travel scenario, as well as their reflections on the least squares equation in an open-ended written question. We found students used unknowns and parameters in a variety of ways. We highlight how their use of dot product equations can be helpful towards supporting their understanding of the least squares equation. 

We investigate the influence of the host matrix on the photothermally driven actuation performance of negatively photochromic, donor−acceptor Stenhouse adduct (DASA)-based polymers. Using a modular Diels−Alder “click” platform, we designed polymeric materials with varying DASA incorporation and investigated the relationships between the material composition and the resulting physical, mechanical, and photoswitching properties. We demonstrate that increasing the DASA concentration in polymer conjugates has a dramatic effect on the material’s physical and mechanical properties, such as the glass transition temperature (Tg) and elastic modulus, as well as the photoswitching properties, which are found to be highly dependent on Tg. We establish using a simple photoresponsive bilayer that actuation performance is controlled by the bilayer stiffness rather than the photochrome incorporation of DASA. Finally, we report and compare the light-induced property changes in Tg and the elastic modulus between the materials comprising the open or closed forms of DASAs. Our results demonstrate the importance of designing a material that is stiff enough to provide the mechanical strength required for actuation under load, but soft enough to reversibly switch at the operational temperature and provide key considerations for the development of application-geared photoswitchable materials. 

A first search for beyond the standard model physics in jet multiplicity patterns of multilepton events is presented, using a data sample corresponding to an integrated luminosity of $138{\mathrm{fb}}^{-1}$of 13 TeV proton-proton collisions recorded by the CMS detector at the LHC. The search uses observed jet multiplicity distributions in one-, two-, and four-lepton events to explore possible enhancements in jet production rate in three-lepton events with and without bottom quarks. The data are found to be consistent with the standard model expectation. The results are interpreted in terms of supersymmetric production of electroweak chargino-neutralino superpartners with cascade decays terminating in prompt hadronic $R$-parity violating interactions. 

A search for the rare decay $D^0\to {\mu }^{+}{\mu }^{-}$is reported using proton-proton collision events at $\sqrt{s}=13.6\mathrm{TeV}$collected by the CMS detector in 2022–2023, corresponding to an integrated luminosity of $64.5{\mathrm{fb}}^{-1}$. This is the first analysis to use a newly developed inclusive dimuon trigger, expanding the scope of the CMS flavor physics program. The search uses $D^0$mesons obtained from $D^{*+}\to D^0{\pi }^{+}$decays. No significant excess is observed. A limit on the branching fraction of $\mathcal{B}(D^0\to {\mu }^{+}{\mu }^{-})<2.4\times {10}^{-9}$at 95% confidence level is set. This is the most stringent upper limit set on any flavor changing neutral current decay in the charm sector. 

A<sc>bstract</sc> A search for a heavy pseudoscalar Higgs boson, A, decaying to a 125 GeV Higgs boson h and a Z boson is presented. The h boson is identified via its decay to a pair of tau leptons, while the Z boson is identified via its decay to a pair of electrons or muons. The search targets the production of the A boson via the gluon-gluon fusion process, gg → A, and in association with bottom quarks,$$\text{b}\overline{\text{b}}\text{A }$$. The analysis uses a data sample corresponding to an integrated luminosity of 138 fb⁻¹collected with the CMS detector at the CERN LHC in proton-proton collisions at a centre-of-mass energy of$$\sqrt{s}=13$$TeV. Constraints are set on the product of the cross sections of the A production mechanisms and the A → Zh decay branching fraction. The observed (expected) upper limit at 95% confidence level ranges from 0.049 (0.060) pb to 1.02 (0.79) pb for the gg → A process and from 0.053 (0.059) pb to 0.79 (0.61) pb for the$$\text{b}\overline{\text{b}}\text{A }$$process in the probed range of the A boson mass,m_A, from 225 GeV to 1 TeV. The results of the search are used to constrain parameters within the$${\text{M}}_{\text{h},\text{EFT}}^{125}$$benchmark scenario of the minimal supersymmetric extension of the standard model. Values of tanβbelow 2.2 are excluded in this scenario at 95% confidence level for allm_Avalues in the range from 225 to 350 GeV. 

The polarization of the $\mathrm{\Lambda }$and $\overline{\mathrm{\Lambda }}$hyperons along the beam direction has been measured in proton-lead ( $p\text{-Pb}$) collisions at a center-of-mass energy per nucleon pair of 8.16 TeV. The data were obtained with the CMS detector at the LHC and correspond to an integrated luminosity of $186.0\pm 6.5{\mathrm{nb}}^{-1}$. A significant azimuthal dependence of the hyperon polarization, characterized by the second-order Fourier sine coefficient $P_{z,\mathrm{s}2}$, is observed. The $P_{z,\mathrm{s}2}$values decrease as a function of charged particle multiplicity, but increase with transverse momentum. A hydrodynamic model that describes the observed $P_{z,\mathrm{s}2}$values in nucleus-nucleus collisions by introducing vorticity effects does not reproduce either the sign or the magnitude of the $p\text{-Pb}$results. These observations pose a challenge to the current theoretical implementation of spin polarization in heavy ion collisions and offer new insights into the origin of spin polarization in hadronic collisions at LHC energies. 

A<sc>bstract</sc> Inclusive and differential cross sections for Higgs boson production in proton-proton collisions at a centre-of-mass energy of 13.6 TeV are measured using data collected with the CMS detector at the LHC in 2022, corresponding to an integrated luminosity of 34.7 fb⁻¹. Events with the diphoton final state are selected, and the measured inclusive fiducial cross section is$${\sigma }_{\text{fid}}={74}\pm {11}{\left({\text{stat}}\right)}_{-4}^{+5}\left({\text{syst}}\right)$$fb, in agreement with the standard model prediction of 67.8 ± 3.8 fb. Differential cross sections are measured as functions of several observables: the Higgs boson transverse momentum and rapidity, the number of associated jets, and the transverse momentum of the leading jet in the event. Within the uncertainties, the differential cross sections agree with the standard model predictions. 

Incoherent $J/\psi$photoproduction in heavy ion ultraperipheral collisions (UPCs) provides a sensitive probe of localized, fluctuating gluonic structures within heavy nuclei. This Letter reports the first measurement of the photon-nucleon center-of-mass energy ( $W_{\gamma \mathrm{N}}$) dependence of this process in PbPb UPCs at a nucleon-nucleon center-of-mass energy of 5.02 TeV, using $1.52{\mathrm{nb}}^{-1}$of data recorded by the CMS experiment. The measurement covers a wide $W_{\gamma \mathrm{N}}$range of $\approx 40–400\mathrm{GeV}$, probing gluons carrying a fraction $x$of nucleon momentum down to an unexplored regime of $6.5\times {10}^{-5}$. Compared to baseline predictions neglecting nuclear effects, the measured cross sections exhibit significantly greater suppression at lower $x$. Additionally, the ratio of incoherent to coherent photoproduction is found to be constant across the probed $W_{\gamma \mathrm{N}}$and $x$range, disfavoring the establishment of the black disk limit. This Letter provides critical insights into the $x$-dependent evolution of fluctuating gluonic structures within nuclei and calls for further advancements in theoretical models incorporating nuclear shadowing and gluon saturation.