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The excess measures a coalition's complaint against an allocation: a positive excess means its own attainable value exceeds its assigned total. The core of a cooperative game requires every excess to be nonpositive at an efficient allocation. The nucleolus lexicographically minimizes the decreasingly sorted excess vector over imputations.

An imputation is an efficient, individually rational allocation in a transferable utility game. Its total is the grand-coalition value and each player receives at least its singleton value. The set is nonempty exactly when $v(N)\ge {\sum }_{i}v(\{i\})$, and is then compact. This individual-rationality condition distinguishes the nucleolus from the prenucleolus.

Can a specialized donor still supply information for other cell types? The 1962 The Developmental Capacity of Nuclei taken from Intestinal Epithelium Cells of Feeding Tadpoles by J. B. Gurdon used mid-intestinal cells of feeding Xenopus laevis tadpoles. Donors were recognized by columnar shape, basal cell nuclei and a striated absorptive border. Dissociation used Versene solution; this made individual donors accessible. The readout was subsequent development, including normal feeding tadpoles, rather than just division or survival. ^[ref]

Apparatus and intervention. A microscope with ultraviolet illumination directed light at the surface region containing the unfertilized ovum's chromosomes, inactivating the recipient nuclear contribution. Irradiation also softened the surrounding jelly so a glass micropipette could enter. The pipette delivered donor nuclear material into recipient cytoplasm. In the technique's validation, fertilizing irradiated eggs produced haploid embryos, supporting destruction of the maternal nuclear contribution. This is inactivation, not a diagram of surgically removing an intact cell nucleus. ^[ref]

Controls ask different questions. Donor cell nuclei carried a marker absent from recipients; post-blastula embryos carried that marker. The marker distinguished diploid donors with one nucleolus per cell nucleus from the usual two-nucleolus recipient background, rather than mistaking one nucleolus for proof of a haploid state. ^[ref] Interspersed transfers of blastula or later embryonic cell nuclei checked variable recipient quality. A positive embryonic-donor control thus tests the preparation, while the marker tests donor origin. Abnormal development remained common. Neither control establishes universal transfer competence. ^[ref]

Inference and limits. A successful marked donor demonstrates that specialization had not irreversibly removed all information required to build other cell types. Failure is less decisive: transfer damage, recipient condition and incomplete cellular reprogramming can also prevent development. The marker identifies nuclear origin; it is not a complete DNA sequence comparison. The experiment tests donor nuclear potential in an altered cytoplasm, rather than normal donor cell fate, a universal capacity of every donor, or development from a mature mammalian red blood cell. The Nobel account also describes serial transfers through embryonic intermediates; these must not be silently combined with direct transfers into one success rate. ^[ref]

When the imputation set is nonempty, the nucleolus is the unique imputation lexicographically minimizing the vector of excesses of a coalition sorted decreasingly. It minimizes the largest complaint first, then subsequent complaints among ties. It can be computed through successive linear programs that minimize the current maximum excess and retain constraints fixed by prior stages. Its uniqueness is the standard nucleolus theorem. If the core of a cooperative game is nonempty, the nucleolus belongs to it; an empty core does not preclude a nucleolus.

chromatin has organization across overlapping length scales, and each assay measures a particular aspect rather than a universal “openness”. Local DNA exposure, histone occupancy, chemical state, long-range proximity and position within the nucleus are related but distinct. A strong experimental design chooses the relevant scale and combines a genomic assay with an independent structural or functional measurement.

At the molecular scale, DNA wraps around a histone octamer to form a nucleosome core, with roughly 147 base pairs protected; linker DNA connects neighbouring units, and linker histones can affect their organization. The core contains two copies each of H2A, H2B, H3 and H4. Crystallography established the detailed DNA–histone arrangement using a defined nucleosome core particle.^[ref] Such structures explain local contacts, but a purified core does not specify the arrangement of an entire chromosome in a living nucleus. histone variants, tail modifications and ATP-dependent remodelling change stability and access without requiring DNA-sequence changes.

For local accessibility, ATAC-seq uses transposase-mediated insertion of sequencing adapters into accessible DNA. It maps accessible promoters and regulatory elements, and its fragment-length distribution can contain signatures of nucleosome-free regions and neighbouring nucleosomes. The original assay combined rapid accessibility mapping with information about nucleosome organization.^[ref] DNase I hypersensitivity provides a related cleavage-based measure. Both have sequence bias and depend on preparation and enzyme exposure; a footprint can reflect enzyme preferences or protected binding, and is not a direct identification of the occupying transcription factor. Compare biological replicates, enzyme-dose controls, appropriate background and biological cell composition. Accessibility does not prove that an element regulates a particular gene.

nucleosome mapping by MNase-seq addresses occupancy and spacing more directly: partial micrococcal-nuclease digestion preferentially removes exposed linker DNA, and sequencing protected fragments maps nucleosome-associated regions. Digestion titration is essential because over-digestion can preferentially lose fragile nucleosomes and create apparent depletion; MNase also has sequence preferences and can recover other protected particles. Compare fragment sizes, histone enrichment and an independent accessibility assay. Chemical mapping or defined in vitro reconstitution can refine particular nucleosome positions, while single-molecule measurements can expose heterogeneity hidden in a population average.

Chemical states require different assays. histone acetylation and methylation at particular residues can correlate with distinct regulatory states; they are not interchangeable with physical accessibility. chromatin immunoprecipitation followed by sequencing, or targeted CUT&RUN, maps association of a histone modification or protein with genomic regions. antibody specificity, histone abundance, input/background and calibrated normalization matter, especially when the modification changes globally. In CUT&RUN, an antibody recruits a nuclease to release nearby DNA, reducing some backgrounds associated with bulk chromatin fragmentation.^[ref] DNA methylation can be measured by appropriately controlled bisulfite-based or direct-read approaches; standard bisulfite measurements do not automatically distinguish every modified cytosine species. These assays locate a chemical mark, not the geometric folding of the fibre.

At the level of nucleosome arrays, regular compact fibres can form in some reconstituted or specialized conditions. A universal hierarchy in which every interphase chromosome passes through a uniform 30-nm fibre is not supported. electron tomography and related imaging can examine the distribution and packing of chromatin directly. ChromEMT combined DNA-selective staining with electron tomography and revealed irregular chromatin chains and variable compaction in biological cells.^[ref] Fixation, staining, sectioning and limited sampled volumes must be considered; cryogenic and super-resolution approaches provide complementary views. Dense staining also need not imply a distinct biochemical chromatin state.

Longer-range folding produces loops and domains of enhanced contact. chromosome conformation capture crosslinks, digests and ligates spatially associated fragments; targeted variants test selected loci, whereas Hi-C samples contacts genome-wide. Contact matrices reveal local domains and broad A/B compartmentalization. High-resolution in situ Hi-C mapped chromatin loops and their association with CTCF orientation and cohesin.^[ref] A topologically associating domain is a region with preferential internal contacts under a stated analysis, not necessarily a stable, sharply bounded object in every biological cell. Contact frequency is neither an absolute distance nor proof that two sites directly interact. Restriction-site distribution, crosslinking, ligation, coverage and bin size affect resolution and interpretation.

Perturbation can distinguish mechanisms underlying similar-looking domains. Acute cohesin loss eliminated loop domains while compartment organization persisted, showing that local loop organization and large-scale compartmental segregation are separable.^[ref] Imaging by DNA fluorescence in situ hybridization can measure distance distributions between chosen loci and test heterogeneity; live labels track dynamics but can perturb loci. Agreement between contact enrichment and a shift in distance distributions is stronger than either assay alone, although the observables still differ.

At nuclear scale, chromosomes occupy territories and regions associate preferentially with the nuclear lamina or nucleolus. DNA imaging maps territories and radial positions, while lamina-association assays such as DamID map sequences near a labelled nuclear compartment over an integration interval. The original DamID approach tethered a DNA adenine methyltransferase to a chromatin protein to record nearby genomic regions.^[ref] Such maps record proximity and history rather than a single fixed structure. Mitotic condensation is a distinct organization with additional constraints and chromosome-shaping machinery, not simply maximally closed interphase chromatin. The appropriate pairing is exposure assays for local access, protected-fragment assays for nucleosomes, protein/mark profiling for composition, contact assays for genomic proximity and imaging for spatial organization, with perturbations to connect organization to function.

For a coalition $S$, its coalition excess at an allocation is $e(S,x)=v(S)-x(S)$, measuring its complaint. Sort all these excesses of a coalition in decreasing order. The nucleolus is the imputation whose sorted vector is smallest in lexicographic order: minimize the greatest complaint first, then the next among ties, and continue. Including the empty and grand coalitions merely inserts two fixed zero entries and does not change the minimizing allocation.

For every efficient allocation, the excesses of a coalition of $\{2\}$ and $\{1,3\}$ sum to zero:The largest proper-coalition coalition excess is therefore at least zero. Since the core of a cooperative game from part (b) is nonempty, this first-stage minimum is exactly zero and its minimizers are precisely that core of a cooperative game. Restrict to $x=(t,2,10-t)$, $1\le t\le 6$. The six proper-coalition excesses of a coalition areThe two zero excesses of a coalition are fixed. Among the remaining excesses of a coalition, $t-7<t-6$, so the next largest complaint is $\max \{1-t,t-6\}$. It is minimized by equalizing those two affine expressions:This minimizer is unique, so later lexicographic stages cannot change it. ThusIts sorted proper-coalition coalition excess vector is $(0,0,-5/2,-5/2,-5/2,-7/2)$. The calculation both identifies the nucleolus and proves uniqueness for this game without replacing the requested computation by the general uniqueness theorem.

For a transferable utility game, the characteristic function of a coalitional game assigns a coalition $S$ its largest total attainable revenue $v(S)$, with $v(\varnothing )=0$. The core of a cooperative game consists of payoff vectors $x$ satisfying efficiency, $x(N)=v(N)$, and coalition stability, $x(S)\ge v(S)$ for every $S$. A coalition receiving less could operate alone and distribute its extra value to make each member better off.

Use thousands of pounds as the monetary unit. The physical directed arcs are $A\to B$ and $C\to D$, owned by $i$, with capacities $3,2$; $A\to C$ and $B\to D$, owned by $j$, with capacities $1,1$; and $B\to C$, owned by $k$, with capacity $2$.

The physical network is acyclic. There are exactly three $A$–$D$ graph paths and one $B$–$C$ graph path. Let $u,v,w$ be the $A$–$D$ flows on $ABD$, $ACD$, $ABCD$, respectively, and let $z$ be the direct $B$–$C$ demand. Their capacities and revenue areEvery feasible traffic pattern decomposes into these graph paths, so these inequalities are exact. The first is redundant, since $u\le 1$ and $w\le 2$ imply it.

Formulate this as a minimum-cost flow through the two-resource path packing as minimum-cost circulation reduction. Use auxiliary vertices $s,L,R,t$ and the following arcs, whose entries are (capacity, cost):The first forward graph path carries $u$, $sLt$ carries $v$, $sLRt$ carries $w$, and $sRt$ carries $z$. Conservation at $L$ imposes $v+w\le 2$, and at $R$ imposes $w+z\le 2$; the individual finite-capacity arcs give the other bounds. The return arc closes the circulation. Its cost is exactly $-R$, so this is an ordinary minimum-cost circulation with precisely the required feasible traffic set. Unlimited capacities may all be replaced by $5$, the maximum possible total graph path flow.

An linear programming optimality certificate is particularly short:The feasible choice $u=v=w=z=1$ attains equality. Thus the actual traffic is three units from $A$ to $D$ and one from $B$ to $C$, andAll four positive terms in the bound must be tight at any optimum, so these graph path amounts are unique.

The two traffic demands must retain their identities. Merely adding fictitious return arcs $D\to A$ and $C\to B$ to the physical graph can create an invalid graph cycle combining both rewards: $A\to C\to B\to D\to A$ would credit the $j$-only network with revenue even though it carries neither requested demand. The resource-flow formulation above avoids that problem.

For the characteristic function, only complete physical graph paths owned by a coalition are available. Company $i$ or $j$ alone has no requested graph path; company $k$ carries two units of $B$–$C$ traffic. The pair $\{i,j\}$ carries one unit on each of $ABD,ACD$; the pair $\{i,k\}$ sends two units on $ABCD$ rather than selling them at the lower $B$–$C$ rate; and $\{j,k\}$ has only the two direct $B$–$C$ units. ThereforeWriting payments in thousands, all stable divisions of the revenue are exactlyEquivalently, $8\le x_k\le 10$, $0\le x_j\le 10$ and $x_i=22-x_j-x_k$. This is the entire core, not just one acceptable allocation.

The nucleolus minimizes in lexicographic order the sorted decreasing excesses of a coalition $e(S,x)=v(S)-x(S)$ over imputations. Empty and grand-coalition coalition excesses are fixed zeros and can be omitted. For the six other coalitions, efficiency givesThe two $k$-related coalition excesses sum to $-2$, so their maximum is at least $-1$. This bound is attainable, and attaining it forces $x_k=9$. All six coalition excesses are then at most $-1$ precisely when $1\le x_j\le 9$ and $x_i=13-x_j$. The fixed pair of coalition excesses equals $-1$ throughout these first-stage minimizers.

Minimize the largest of the remaining coalition excesses. They become $x_j-13,-x_j,x_j-10,-1-x_j$, whose maximum is $\max (-x_j,x_j-10)$. Its unique minimum is $-5$ at $x_j=5$. Hence $x_i=8$ and $x_k=9$. The sorted proper coalition excesses there are $(-1,-1,-5,-5,-6,-8)$, and

For an imputation $y$, the excess of a coalition is $e(S,y)=v(S)-{\sum }_{i\in S}{y}_i$. Arrange these excesses in decreasing order. The nucleolus is the imputation that lexicographically minimizes this ordered vector: minimize the largest complaint first, then the next largest among ties, and so on. The empty and grand-coalition excesses are fixed at zero for every efficient allocation, so they can be omitted without changing the lexicographic comparison.

For every efficient $y$ the three pair excesses have sumHence at least one pair has excess at least $-4/3$, and the largest proper-coalition excess is at least $-4/3$. At the proposed allocation all three pair excesses equal $-4/3$, while all three singleton excesses equal $-8/3$. It therefore attains the best possible largest proper-coalition excess.

Moreover, any allocation attaining this bound must have all three pair excesses equal $-4/3$, since they are individually at most that value and sum to $-4$. This forceswhose unique solution is $y=x$. It is already an imputation, so the first nonconstant lexicographic stage has a unique minimizer; there are no later choices to resolve. Therefore

For a transferable utility game with $v(\varnothing )=0$, an efficient allocation satisfies ${\sum }_{i\in N}{x}_i=v(N)$. An imputation is efficient and individually rational, $x_i\ge v(\{i\})$. Write $x(S)={\sum }_{i\in S}{x}_i$ and define the excess of a coalition as $e(S,x)=v(S)-x(S)$.

The core of a cooperative game isIt consists of allocations immune to a coalition's blocking: no coalition can obtain more for its members by leaving. The singleton inequalities imply individual rationality. The core may be empty.

The nucleolus, when the imputation set is nonempty, is the unique imputation that lexicographically minimizes the list of coalition excesses arranged from largest to smallest. It first minimizes the largest complaint, then the second largest among ties, and so on. Including the empty and grand coalitions adds constant zeros and does not change the solution. Minimization on efficient allocations without individual rationality instead defines the prenucleolus, a different convention that matters for this paper's game.

The Shapley value is the average marginal contribution of each player over all uniformly ordered player arrivals:Exactly $|S|!(n-|S|-1)!$ orderings have $S$ as the predecessor set of $i$. This is an average-contribution fairness rule, rather than a blocking-stability condition; it need not be individually rational for an arbitrary game.

For three players, the predecessor-set weights are $1/3$ for sizes zero and two, and $1/6$ for size one. ThusTherefore the Shapley value is $\boxed{(19/3,17/6,17/6)}$, whose coordinates sum to $12$.

For the nucleolus, use imputations $x_1+x_2+x_3=12$, $x_1\ge 4$, $x_2\ge 3$, $x_3\ge 2$. The excess of $\{1,3\}$ is $11-(12-x_2)=x_2-1\ge 2$. Thus the smallest possible largest excess is at least $2$. It is attained when $x_2=3$, $x_3=9-x_1$, and $5\le x_1\le 7$: the remaining proper-coalition excesses then areall at most $2$, while $e_{13}=2$. Conversely, a largest excess of $2$ forces $x_2=3$ and precisely this interval for $x_1$.

On this first-stage face the top excess $2$ is fixed. The next largest excess is $7-x_1\ge 0$, because the other varying excesses are nonpositive and $e_2=0$ is fixed. Its unique minimum is zero at $x_1=7$. No further lexicographic tie remains. Hence the nucleolus is $\boxed{(7,3,2)}$. Its proper-coalition excesses, sorted decreasingly, are $(2,0,0,0,0,-3)$.

Finally, a core allocation would require $x_2\ge 3$ and $x_1+x_3\ge 11$, whose sum contradicts the efficient total $12$. Thus the core of a cooperative game is empty. The game is not superadditive, so neither core nonemptiness nor individual rationality of the Shapley value should be presumed. Under the distinct prenucleolus convention the answer would be $(15/2,2,5/2)$. Balancing $e_2=3-x_2$ and $e_{13}=x_2-1$ gives a first-stage maximum of $1$ and forces $x_2=2$. The remaining first-stage constraints restrict $x_1$ to $[7,8]$. The next largest complaints are $e_{12}=8-x_1$ and $e_{23}=x_1-7$, balanced at $x_1=15/2$, with $x_3=5/2$. This is not an imputation and is not the nucleolus under the definition in (a).

The prenucleolus uses the same sorted excesses of a coalition but minimizes over all efficient payoff vectors, without imposing individual rationality. It can differ from the nucleolus when coalition values are not superadditive. Any computation must state which domain is used rather than silently relaxing imputation inequalities.