conversation_id string | domain string | subDomain string | author_id string | question string | answer string | format string | images images list |
|---|---|---|---|---|---|---|---|
306510 | Math | Field theory and polynomials | 168 | Determine the number of monic primitive irreducible polynomials of degree
\[
d=11ci^{+e}_{2r}(G_4)+ci^{+e}_{2r}(G_1)+ci^{v}_{2r}(G_3)+ci^{+e}_{2r}(G_2)+ci^{-e}_{2r}(G_3)
\]
over the finite field
\[
\mathbb F_q,
\]
where
\[
q=2^{ci^{+e}_{2r}(G_1)+ci^{v}_{2r}(G_3)-ci^{+e}_{2r}(G_2)-ci^{-e}_{2r}(G_3)},
\]
and \(G_1,G_2,G_... | \[120\] | multi_panel | |
306490 | Math | Geometry | 168 | Consider the figure shown. Let $a$ denote the area of the shaded region. Define two vectors in $\mathbb{R}^3$ by \[ \mathbf{A} = [1,\ a,\ 2], \qquad \mathbf{B} = [c,\ 3,\ 1]. \] Determine the value of the constant $c$ such that the vectors $\mathbf{A}$ and $\mathbf{B}$ are orthogonal. | \[-\frac{71}{4} \] | single_image | |
306449 | Math | Number theory | 1718 | Determine the order of the ideal class group
\[
\mathrm{Cl}\!\left(\mathbb{Q}(\sqrt{-N})\right),
\]
where
\[
N=
\lambda_{3,2,1}(G_1\times G_3)^2+
\lambda_{3,2,1}(G_1\times G_2)\,
\lambda_{3,2,1}(G_1\times G_3)-
\lambda_{3,2,1}(G_1\times G_2)^2,
\]
and \(G_1,G_2,G_3\) denote the graphs shown in Figure 1, Figure 2, and F... | \[12\] | multi_panel | |
306427 | Math | Linear and multilinear algebra | 1718 | Let $G = (V,E)$ be a simple connected graph. For an ordered set of vertices
$S = \{u_1, u_2, \dots, u_t\} \subseteq V(G)$, the metric representation
of a vertex $v \in V(G)$ with respect to $S$ is defined by
\[
r(v \mid S) = \big(d(v,u_1), d(v,u_2), \dots, d(v,u_t)\big),
\]
where $d(u,v)$ denotes the length of a shor... | \[5\] | multi_image | |
306425 | Math | Differential geometry | 1718 | Determine the value of the surface integral
\[
\iint_{S}\left(x^2+y^2+z^2\right)\,dA,
\]
where
\[
S:\quad x^2+y^2+z^2=N,
\]
and
\[
N=C_4(G)\cdot C_5(G)+1,
\]
with \(G\) denoting the graph shown in the attached image.
Consider a simple and undirected graph \(G\) with vertex set \(V=V(G)\). For an integer \(k\ge1\), a \... | \[7396\pi\] | single_image | |
306423 | Math | Group theory | 1718 | Let $G=(V(G),E(G))$ be a finite, simple, connected graph. A function
$f:V(G)\to \{0,1,2,3\}$ is called a modern Roman dominating function if every vertex labeled $0$ is adjacent to two vertices, one labeled $2$ and one labeled $3$, and every vertex labeled $1$ is adjacent to at least one vertex labeled $2$ or $3$. The... | \[4423680\] | multi_image | |
306419 | Math | Combinatorics | 1428 |
Let \(\lambda\) be the Young diagram shown on the left in the figure, and let \(\mu\) be the Young diagram shown on the right in the figure. For a Young diagram \(\nu\), let \(\nu^\vee_j\) denote the length of its \(j\)-th column, with \(\nu^\vee_j=0\) for all sufficiently large \(j\). Define
\[
[z]:=z^{-1/2}-z^{1/2}.... |
\[
[u]^2\,[u q^{-1}]\,[u q]\,[u q\,\kappa^{-2}]\,[u q^{2}\kappa^{-2}]\,[u q^{-2}\kappa^{2}]
\] | multi_image | |
306415 | Math | Number theory | 1718 | Determine the number of primitive Dirichlet characters modulo
\[
N=\chi'_{NK}(G)^{\,3}+2,
\]
where \(G\) denotes the graph shown in the attached image.
In the attached image, the colored points denote the vertices of the graph and the line segments joining pairs of points denote the edges of the graph.
Let \(G=(V(G),... | \[125\] | single_image | |
306407 | Math | Differential geometry | 1428 | Use the right-hand panel of the given figure.
Let $({\rm CP}^{2},\omega)$ be complex projective $2$-space with the Fubini--Study form normalized by
$$
\int_{{\rm CP}^{1}}\omega=1.
$$
Define
$$
\mu([z_{1}:z_{2}:z_{3}])
=
\frac{1}{2\sum_{j=1}^{3}|z_{j}|^{2}}
\bigl(|z_{1}|^{2},|z_{2}|^{2}\bigr),
$$
let
$$
\Delta=\mu({\rm... |
\[
\left(\frac{2}{3},\,4\right)
\] | multi_image | |
306296 | Biology | Neurobiology | 899 | In this multi-compartment model of leech T-cells, different spike initiation zone (SIZ) distributions are tested on reconstructed morphologies while holding ion channel parameters fixed.
Using the Figure $1$, identify which SIZ placement produces the second lowest height across the reconstructed morphologies? | Anterior | single_image | |
306284 | Biology | Evolutionary Biology | 899 | In the three-figure presentation of chromosome fusion dynamics (represented in figures $1$, $2$, and $3$) driving rediploidization in $Schizothoracinae$ polyploid fishes, the process is characterized by progressive waves of homeologous chromosome fusions that are visualized through comparative synteny in a circos plot,... | $23$ and $20$ | multi_image | |
306283 | Biology | Molecular Biology | 880 | The leptospiral ClpP system comprises of two isoforms ($ClpP1$ and $ClpP2$). The $ClpP1$ and $ClpP2$ associate to make hetero-complex. Based on the Figure$1$ and Figure$2$, identify the minimal functional state enabling proteolysis. Answer in $3-4$ words. | $ClpXP1P2$ complex + ATP | multi_image | |
306278 | Biology | Molecular Biology | 899 | Examine the figure from the HLA immunopeptidomics analysis of non-canonical open reading frames. The top panels display detected total peptides and predicted HLA-binding non-canonical peptides across sample groups.
The central heatmap shows predicted binders for peptides from different ncORF biotypes with aggregate... | uORFs | single_image | |
306274 | Biology | Neurobiology | 936 | Based on the longitudinal data in Figures $1$ and $2$, identify the specific anatomical locus where the experimental intervention stabilizes dopaminergic terminal integrity, thereby providing a putative neuroanatomical substrate for the observed improvement in the "Seven series" neuropsychological subscore.
| Nucleus accumbens | multi_image | |
306270 | Biology | Molecular Biology | 936 | In the provided pathway in the figure, if an experimental pharmacological agent simultaneously induces angiogenesis and inhibits endocytosis, while the $\text{RAF-MEK-ERK}$ axis remains inactive, which specific membrane-proximal complex is the most likely direct target being allosterically stabilized?
| Active GTP-bound RAS | single_image | |
306269 | Biology | Cell biology | 936 | Analyze the protein mapping in panel C in the attached figure. Identify the exact amino acid residues (range) of the intermediary protein required to inhibit the proteasome-mediated turnover of the effector, and name the specific ubiquitin linkage type that this interaction serves to suppress.
| Residues $85-271$; $\text{K48}$-linked polyubiquitination | multi_panel | |
306253 | Biology | Neurobiology | 936 | According to the $\text{CC-GWAS}$ specific enrichment results shown in the figure, if Schizophrenia’s unique genetic signature in adulthood is primarily vascular in nature, what specific fetal cell class identifies the distinct, early-life biological substrate for Major Depression?
| Intermediate progenitor | multi_panel | |
306248 | Biology | Evolutionary Biology | 936 | Based on the comparative expression of the single-copy $\text{Pax2}$ homolog in $\text{Phalangium opilio}$ and the expression of its duplicates in $\text{Centruroides sculpturatus}$ in the figure, what evolutionary mechanism is specifically illustrated by the loss of $Pax2a$ expression specifically in spider median eye... | Developmental system drift | multi_panel | |
306241 | Biology | Molecular Biology | 936 | In the figure $1$, the loading plot distinguishes Group $\text{O}$ from Group $\text{P}$ using multiple classes of metabolites. While the OPLS-DA model identifies broad amino acid changes, the study's Biosigner algorithm (specifically the Random Forest classifier), visualized in the Venn diagram in figure $2$, identifi... | $SM \ C24:1$ | multi_image | |
306136 | Chemistry | Organic Chemistry | 982 | Based on the attached images, please identify the IUPAC name of the final compound.
| 3-bromo-12-(2-methoxyphenyl)-10,12-dihydro-11H-benzo[5,6]chromeno[2,3-d]pyrimidin-11-one | multi_image | |
306127 | Chemistry | Analytical Chemistry | 994 | A researcher is conducting a comparative study to detect Prostate Specific Antigen (PSA) using two types of label-free sensors: an Aptasensor (represented by panels A and B in Fig A) and an Immunosensor (represented by panels C and D in Fig A). Both sensors are developed using graphene quantum dots-gold nanorods (GQDs-... | 1.05 | multi_panel | |
306123 | Chemistry | Inorganic Chemistry | 961 | The attached image shows the synthetic scheme a macrocyclic Ligand L and subsequent reactions for the formation of Complexes A and B. Given that uridine monophosphate acts as a bridging ligand, determine the sum of coordination numbers for all Cu-centers in complex B. | 20 | single_image | |
306122 | Chemistry | Inorganic Chemistry | 961 | The attached images, Figure1 and Figure2, represent the synthetic scheme for a neutral Complex D. Provide the full systematic IUPAC name for Complex D accurately representing all ligands and their binding modes. (Use \(\kappa\), \(\eta\) and \(\mu\) notations as necessary). | \(\text{pentacarbonyl[(2-(dimethylamino)-5-methylphenyl)}(1\textit{H}\text{-indol-3-yl)phenylphosphane-}\kappa P]\text{tungsten(0)}\)
| multi_image | |
306120 | Chemistry | Organic Chemistry | 982 | Using the attached images, identify the IUPAC name of the final compound.
Note :
\(TfOH\) is Trifluoromethanesulfonic acid
\(DCE\) is 1,2-Dichloroethane
\(Pd(OAc)_2\) is Palladium(II) acetate
| (3-(2,2-diphenylvinyl)-6-(10H-phenoxazin-10-yl)-2-phenyl-2H-isoindol-1-yl)diphenylphosphine oxide | multi_image | |
306118 | Chemistry | Inorganic Chemistry | 959 | The attached multi-panel image includes the reaction schemes showing the formation of complexes X, X', Y and Y' along with the positive-ion ESI mass spectra of (a) Complex X and (b) Complex Y; and the $^{31}\text{P}\{^1\text{H}\}$ NMR spectra of (c) Complex X and (d) Complex Y recorded in $\text{CD}_2\text{Cl}_2$.
Det... | 34 | multi_panel | |
306116 | Chemistry | Organic Chemistry | 982 | Using the attached reaction scheme in the multi-panel image, determine the SMILES string of compound P.
Note :
1. For reaction a), \(H_2SO_4\), was added dropwise at \(0 ^\circ C\). After complete addition, the mixture was stirred for \(5\ h\) at \(40 ^\circ C\)
2. For reaction b), Initially mixture was stirred at \... | \(OC1=CC=C2C(OC(C(C3=CC=C(C4NC(NC(C)=C4C(OC)=O)=O)C=C3)C2C)=O)=C1\) | multi_panel | |
306115 | Chemistry | Inorganic Chemistry | 961 | The attached images outline the synthetic process to obtain Ligand C. When an ethanolic solution of Ligand C is treated with an equimolar amount of copper(II) nitrate in an acetonitrile/water mixture, a discrete dicationic complex, D, is formed.
Derive the full, detailed systematic IUPAC name for complex D, utilizing... | [4-(4-methoxyphenyl)-3,5-dimethyl-1,7-di(pyridin-2-yl)-1,7-dihydrodipyrazolo[3,4-b:4',3'-e]pyridine-$\kappa^3N^{1'},N^{1''},N^8$]copper(II) ion | multi_image | |
306113 | Chemistry | Inorganic Chemistry | 961 | The attached multi-panel image shows the reaction schemes for the synthesis of Ligand L, Complex X and Complex Y along with their $^1\text{H}$ NMR spectra (500 MHz, \(CDCl_3\)).
Based on this information, determine the sum total of chloride ligands in complexes X and Y. | 4 | multi_panel | |
306110 | Chemistry | Organic Chemistry | 982 | Determine the IUPAC name of compound \(A\) formed in the reaction scheme shown in the attached file.
Note: \(I_2\) is added at room temperature and then the temperature is elevated to \(95 ^\circ C\). Reaction is carried out in a sealed tube.
\(1,2-DCE\) is 1,2-Dichloroethane
\(BF_3. OEt_2\) is Boron trifluoride di... | (E)-4-(7,8-Dichloro-11H-indeno[1,2-b]quinoxalin-11-ylidene)-2,3-di-p-tolylcyclobut-2-en-1-one | single_image | |
305996 | Physics | High-energy particle physics | 1683 | Using the physical mechanisms and parameter-space structures illustrated by Fig. 1, Fig. 2, Fig. 3, and Fig. 4.
Work in the TS5 heavy-light regime with
\[
M_{G^*}=1.5~\text{TeV},\qquad
m_T=m_B=m_{\tilde T}=m_{\tilde B}=1.0~\text{TeV},
\]
\[
Y_*=3,\qquad m_t=173~\text{GeV},\qquad v=246~\text{GeV},
\]
and impose
\[
s... | \[1.172\times 10^{-2}
\] | multi_image | |
305995 | Physics | Physics (general) | 1683 |
Use the one-dimensional temporal-gauge reduction of the lattice geometry in Figure first. In units \(m=1\), place \(\phi_s^n\) on vertices and \(A_{s+1/2}^n\) on spatial links, with lattice spacings
\[
h=0.04,\qquad \tau=0.005 .
\]
The discrete action is
\[
S=h\tau\sum_{n,s}
\left[
\left|\frac{\phi_s^{n+1}-\phi_s^n... | \[
R=21.261231584
\]
| multi_image | |
305988 | Physics | Physics (general) | 1683 | In figure second, the (H=0.5) one-stream grey-soliton family is close to the maximum speed where localized solutions still exist; figure first encodes that exact one-stream threshold; figure third shows that for essentially the same Lorentz factor, a detached high-(K) quantum instability lobe is born in the symmetric t... | \[1.20844\] | multi_image | |
305983 | Physics | Electromagnetism and Photonics | 866 | Consider figure with numerical input. A conditional-drive source is fixed at $11.272~\mathrm{GHz}$. Work at the unique listed calibration setting for which that source is closest to the actual driven resonance obtained from the displayed KPO 1 frequency, the fixed KPO 2 frequency, and the displayed AC shift at the same... | \[
(\delta_m^\ast,\; P_2,\; K_{\mathrm{eff}}/2\pi,\; \varepsilon)
=\left(
80~\mathrm{m}\Phi_0,\;
1.0715\times 10^{-3},\;
-6.3863~\mathrm{MHz},\;
199.435~\mathrm{kHz}
\right).
\] | single_image | |
305977 | Physics | Astrophysics | 1670 | Using the provided data in Figures 3 and 4, for PSR J0740+6620 (solid red curves) and the framework of a generalized polytropic Equation of State $p_r = 0.3\rho + K\rho^2$ within a relativistic anisotropic stellar model, what is the exact value of the local anisotropy factor $\Delta(10)$ at $r = 10 \text{ km}$—rounded ... | $0.0074$ | multi_image | |
305967 | Physics | Condensed Matter Physics | 1030 |
Refer to the attached image (Figure-\(1\)) and use its boundary identifications and notation throughout.
Let the momentum-space base manifold be obtained from the fundamental domain
\[
\tau_{1/2}=[-\pi,\pi)\times[-\pi,0]
\]
by the identification
\[
(k_x,0)\sim(-k_x,-\pi),
\]
as encoded by the attached image.
On thi... | $$1.85$$ | multi_panel | |
305966 | Physics | Physics (general) | 1670 |
You are analyzing the ideal MHD equilibrium of an advanced steady-state tokamak scenario. You are provided with three visual diagnostics characterizing this plasma state:
* Figure A: The poloidal cross-section of the outermost closed flux surface in $(X, Z)$ coordinates, where $X$ is the major radius in meters.
* Fi... | $$\ \mathbf{341,000 \text{ Pa}}$$ | multi_image | |
305965 | Physics | Quantum Mechanics | 864 | The figure labeled Topology-III is part of the mathematical specification. Without the colored incidence pattern and the printed $c_X, q_X$ labels in Topology-III, the graph is not defined. Path crossings are not vertices, and grey endpoints have degree one.
Let
$$S(C)=S_0(C-C_p)^{-\gamma},\qquad C=C_p+\varepsilon,\qq... | $21-\sqrt{313}$ | single_image | |
305964 | Physics | Quantum Mechanics | 864 | As illustrated in the Boundary-Shell Support Geometry for the Araki–Gibbs Operator (Fig: BoundaryShell), consider a one-dimensional open quantum spin register of $n$ qubits, with $n$ larger than every support length appearing in the analysis. Let $H=\sum_{j=1}^{n-K+1}H_j$, where $K\ge2$, each $H_j$ is Hermitian, $\|H_j... | $$m_{\mathrm{cert}}(\beta, K, \gamma) = \left\lceil \exp\left( \max\{1, \beta\} \left( \frac{20}{\gamma} \right)^K \right) \right\rceil$$ | single_image | |
305961 | Physics | Astrophysics | 1670 | Consider a spatially flat FLRW cosmology in which the total effective fluid is a two-component mixture: a Brans-Dicke scalar field $\phi$ non-minimally coupled to gravity with parameter $\omega_{\mathrm{BD}}$, and a quintessence field $\psi$ with power-law potential $V(\psi) = \tfrac{1}{n+1}\psi^{n+1}$, with no additio... | $$\frac{40}{81}$$ | multi_image | |
306295 | Biology | Molecular Biology | 880 | In the fusion-substrate experiments, one construct carrying the full adaptor-derived tail is efficiently eliminated even when attached to a heterologous folded domain, whereas another construct lacking the extreme C-terminal segment resists turnover despite retaining most of the adaptor sequence. Considering the degrad... | $MecA96-121$ | multi_image |
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