Papers

A strong edge coloring is a proper edge coloring in which every color class is an induced matching; the least number of colors is the strong chromatic index $\chi'_s(G)$. Lin and Lin proved that every claw-free subcubic graph other than the triangular prism satisfies $\chi'_s(G) \le 7$, with all their tight examples containing diamonds. Kardos (Problem 4.1 of the open-problem collection of the 33rd Workshop on Cycles and Colourings) asked whether every diamond-free claw-free cubic graph is strongly 6-edge-colorable, equivalently whether $\chi'_s(T(G))=6$ for every cubic graph $G$, where $T(G)$ is the truncation of $G$. We exhibit an explicit connected, simple, diamond-free, claw-free cubic graph $H$ on 18 vertices with $\chi'_s(H)=7$, and show that it has the fewest vertices possible for such a non-prism example. Hence the first formulation, over simple cubic graphs, is false even after excluding the prism; and since $H$ is the truncation of a cubic multigraph with parallel edges, the two formulations are not equivalent unless "cubic graph" is allowed to mean loopless multigraph, under which reading the problem is answered negatively. The narrower version restricted to truncations of simple base graphs remains open.

The main object of study of this paper is $B_k(\Sigma_1)$, the unordered configuration space of $k$ points in the torus $\Sigma_1$. First, we produce a marked-point transfer argument which, combined with Chen–Zhang's theorem, establishes that $H_*(B_k (\Sigma_1);\mathbb{Z})$ has no $p$-torsion for $k\leq 2p-1$. Secondly, at the threshold $k=2p$, we prove that $H_{2p-2}(B_{2p}(\Sigma_1);\mathbb{Z})$ has $p$-torsion if and only if $p\geq 5$. For $p\geq5$, the class is the image under puncture filling of a generator of the Bianchi–Stavrou torsion group $\mathbb{Z}/p$ of the once-punctured torus; for $p=3$, Napolitano's calculation shows that this punctured class dies after filling. Finally, we also prove that $H_{2p}(B_{2p}(\Sigma_1);\mathbb{Z})$ and $H_{2p+1}(B_{2p}(\Sigma_1);\mathbb{Z})$ have no $p$-torsion for every odd $p$.

The multiple mixed values (MMVs) are level two variants of multiple zeta values produced by restricting the summation indices to fixed parity patterns. One particularly interesting problem is to determine exactly when such values are actually in level one, namely, expressible in terms of multiple zeta values. To solve this completely is beyond our current knowledge since it calls for new ideas from transcendental number theory. However, much progress has been made on the motivic level. Previously, using the descent theory developed by Brown et al. we have tackled this problem for a few special classes of MMVs with regular parity patterns among the summation indices, including Hoffman's multiple $t$-values, Kaneko and Tsumura's multiple $T$-values, and our own multiple $S$-values. In this paper, we turn to the general case and determine completely all the unramified motivic MMVs of depth less than four as well as a few families of unbounded depths. At the end of the paper, we will present some general conjectures to describe the unramified MMVs at all depths greater than three.

We prove that if a triangle is cut into $N$ congruent triangles and $N$ is prime, then $N=2$, $N=3$, or $N\equiv 1 \pmod 4$, and every such $N$ occurs. In particular, if $N > 3$ then $N$ is not a prime congruent to $3$ modulo $4$. All the proofs in this paper have been formalized in Lean. The argument divides into cases according to possible shapes of the tiled triangle. Let $\alpha,\beta,\gamma$ be the angles of the tile. Prior to this posting, there remained several open shapes. The two most difficult shapes were $(2\alpha,\beta,\alpha+\beta)$, the triquadratic shape, and $(\beta,\beta,3\alpha)$, the {\em isosceles-$\beta$} shape, in which $3\alpha+2\beta = \pi$ and $\alpha$ is not a rational multiple of $\pi$. A previous version of this paper relied on incorrect proofs of those cases, which have been retracted, and are replaced here by correct proofs. For each of those two shapes the number of tiles has the form $N = j^2 N_0$, where $N_0 > 1$ is determined by the shape of the tile and $j$ is a positive integer; so $N$ is prime, or even squarefree, only if $j = 1$. We prove by geometric reasoning that neither shape admits a tiling with $j = 1$, which shows that for both shapes $N$ is never prime, and indeed never squarefree. The shape $(\beta,\beta,3\alpha+\beta)$ (where $3\alpha + 3\beta = \pi$) has been settled by Bonfioli. In this paper, we settle the remaining cases, by number-theoretic arguments based on the area equation. Those proofs rule out $N$ prime, but not $N$ squarefree, and we give examples to show that tilings with squarefree $N$ do sometimes exist.

We prove that every $(P_2\cup P_3)$-free graph $G$, with $k=\omega(G)$, satisfies \[ \chi(G)=O\!\left(k^3\frac{\ln\ln k}{\ln k}\right). \] Thus the class admits an $o(k^3)$ binding function. We also prove the explicit bound \[ \chi(G)\le \frac{k^3}{8}+\frac{5k^2}{4}-4k+8 \qquad(k\ge8), \] with a sharper formula for odd $k$. A further refinement gives an explicit cubic bound with leading coefficient $104/837<1/8$.

Newton's method is one of the most effective second-order algorithms for smooth optimization because of its fast local convergence. However, existing globally convergent Newton-type methods typically require convexity or strong convexity of the objective function, while approaches for nonconvex optimization often rely on Hessian regularization at every iteration. In this paper, we propose a general line-search Newton framework for unconstrained optimization that avoids repeated Hessian regularization by exploiting the Newton direction only when it is well-defined and suitable. The proposed framework encompasses several existing hybrid gradient–Newton methods as special cases and naturally yields a new extragradient Newton method. We establish global convergence under mild assumptions, including the Polyak–Lojasiewicz–Kurdyka (PLK) condition, allowing both isolated and nonisolated accumulation points. We further prove local superlinear and quadratic convergence under appropriate regularity assumptions. Finally, we apply the proposed framework to strongly quasiconvex optimization and provide, to the best of our knowledge, the first Newton-type algorithm together with a comprehensive convergence analysis for this important class of nonconvex optimization problems. Numerical experiments demonstrate the effectiveness of the proposed methods.

In this paper, we are concerned with the existence and stability of temporal periodic solutions to a class of the Benjamin-Bona-Mahony (BBM) equation with damping in the torus $\mathbb{T}$, whose interior is applied with periodic force $f(x, t)$ with temporal period $\theta$. These types of solutions are established in $H^{\ell}, \ell\ge 0$, and it is noteworthy to see that the I-energy method is invoked when low regularity of $\ell\in[0, 1)$ is pursued.

Categorification—the process of constructing a categorical model of a piece of mathematics—often identifies a common abstraction that connects formerly unrelated but known structures. In the case of de Paiva's categorification of Gödel's Dialectica interpretation, we find that its specialization to partial orders produces (functorial) embeddings of Heyting algebras into residuated lattices that appear to have been overlooked. For the non-categorical audience, we present this specialization and take care to reproduce the original proofs in the algebraic setting. Along the way we obtain results particular to this algebraic setting: an embedding lacking an evident adjoint in de Paiva's general construction acquires a definable one here; a single Dialectica tensor validates contraction in the intuitionistic construction D yet refutes it in the classical variant G; and, over ZF, the poset reflection PD(Set) collapses onto the four-element algebra PD(2) exactly when the Axiom of Choice holds.

We study the porous medium equation with a combustion-type reaction, \[ u_t=\Delta u^m+f(u),\qquad x\in\mathbb R^N,\ t>0, \] for radial, nonnegative, compactly supported initial data. A complete classification of the long-time behaviour of bounded solutions is established. In dimension $N=2$, every such solution converges locally uniformly to one of the constants $0$, $\theta$, or $1$; for ordered families of initial data there exists a unique threshold parameter separating vanishing from spreading, and the critical solution converges to the ignition temperature $\theta$. In dimensions $N\ge3$, under a natural total-disconnectedness condition on the set of central values of ground states, every bounded solution converges locally uniformly to either $0$, $\theta$, $1$, or a radial ground state $U\in\mathcal S$. Moreover, the ignition state $\theta$ is excluded as a transition limit in $N\ge3$ via a novel normalized annular perturbation argument. For transition solutions, we provide estimates for the propagation speed of the free boundary: in dimension two, \[ b(t)\asymp \frac{\sqrt t}{(\log t)^{\frac{m-1}{2m}}}, \] and in dimensions $N\ge3$, whenever the limit is a ground state, \[ b(t)\asymp t^{\frac{m}{N(m-1)+2}}. \] These results reveal a sharp dimensional dichotomy in the degenerate combustion dynamics.

Inspired by the Simon conjecture for minimal surfaces in spheres, we study closed surfaces with parallel mean curvature vector and positive Gaussian curvature immersed in unit spheres. Let $h$ be the second fundamental form, let $\mathbf{H}$ be the mean curvature vector field, and set $\tilde h=h-\mathbf{H}g$ and $\tilde S=|\tilde h|^2=|h|^2-2H^2$, where $H=|\mathbf{H}|$ and $g$ is the induced Riemannian metric on the surface $M$. We establish three Simons-type integral identities for $\tilde S$, which extend the first, second and third gap identities in the minimal case. As applications, we obtain the first two sharp endpoint gaps and several rigidity and oscillation estimates in the third interval. We further characterize the endpoint cases by combining these identities with the classification theorems of Calabi and Yau.

In this note, we consider the Sherrington–Kirkpatrick model with deterministic external field. Let $q=q(\beta,h)$ denote the solution of the replica-symmetric self-consistency equation \[ q=\mathbb E\tanh^2\!\left(h+\beta\sqrt q\,Z\right), \qquad Z\sim N(0,1), \] where $\beta$ and $h$ are inverse temperature and external field, respectively. By refining Lata\la' s argument, previously limited to \(\beta < \frac{1}{2}\), and using the Kearns–Saul inequality, we prove overlap concentration and convergence of the free energy to the replica symmetric formula with error \(O(N^{-1})\) whenever \[ \beta^2\frac{q}{{\rm arctanh}q}<1. \] Note that for any $\beta<1$ and $h\in \mathbb R$, the condition above is satisfied. Moreover, for every nonzero $h$, this region contains a nonempty interval with $\beta>1$.

We disprove the Polycirculant Conjecture, which states that every transitive 2-closed permutation group is non-elusive, i.e. contains a derangement of prime order. In fact, we prove a stronger result, answering a long-standing question of Marušič and Jordan: there exists a vertex-transitive graph admitting no semiregular automorphism. To do so, we employ recently developed methods of Chen et al. for constructing elusive groups via non-split extensions, allowing us to construct an elusive group $7^6.\mathrm{PSU}_3(3)$ of degree 16,464. We show that this group is the full automorphism group of seven of its orbital graphs and hence is 2-closed. Our example extends to infinitely many counterexamples of the Polycirculant Conjecture, and infinitely many vertex-transitive graphs admitting no semiregular automorphism.

Let ${\mathbb F}_q^n$ denote the vector space of dimension $n$ over ${\mathbb F}_q$ and AG$(n,q)$ denote the corresponding affine space. An $\textit{affine vector space partition}$ of AG$(n,q)$ is a collection ${\mathcal P}$ of affine subspaces that partition the points of AG$(n,q)$. If all subspaces in ${\mathcal P}$ have the same dimension $d$, then ${\mathcal P}$ is called an $\textit{affine $d$-spread}$. We say that an affine partition ${\mathcal P}$ is $\textit{completely tight}$ if for any pair $C,C'\in{\mathcal P}$ with $C=v+S$, $C'=v'+S'$, where $v,v'\in{\rm AG}(n,q)$ and $S\neq S'$ are linear subspaces of ${\mathbb F}_q^n$, we have $ S\cap S'=\{{\bf 0}\}$. An affine partition ${\mathcal P}$ is said to be $\textit{irreducible}$ if there is no subset ${\mathcal P}'\subset {\mathcal P}$ such that $1<|{\mathcal P}'|<|{\mathcal P}|$ and the union of all subspaces in ${\mathcal P}'$ is a subspace of AG$(n,q)$. For all $d \geq 1$ and $n>2d$, we construct a completely tight irreducible affine $d$-spread of AG$(n,q)$. This also settles a recent conjecture of Bamberg et al. on the existence of tight irreducible affine $d$-spreads.

We derive a general distribution formula applicable to a wide variety of Gibbs-type partitions and use it to obtain large sample results for linear combinations of the component frequency spectrum $(M_{jn})_{1\le j\le n}$ (in genetics, the allele frequency spectrum) associated with a random partitioning of $\{1,2,\ldots, n\}$. The two-parameter Pitman-Yor sampling model is analysed in detail and asymptotic distributions of sums of the form $\sum _{j=\lf \lambda n\rf}^{\lf \mu n\rf} M_{jn}$, $0<\lambda\le \mu\le 1$, are obtained. Our results suggest a possible functional limit theorem for $\sum _{j=\lf \lambda n\rf}^{n} M_{jn}$. Useful connections with limit shapes for random structures on the set of partitions and other applications are suggested.

We prove that there exist two Bloch functions $f_1$ and $f_2$ on $\mathbb D$ such that $$ |f_1(z)|+|f_2(z)| \geq \left(\log\frac{1}{1-|z|}\right)^{1/2}, \qquad z\in\mathbb D, $$ thereby resolving an open problem posed in 2008 by Girela, Peláez, Pérez-González and Rättyä. Our proof is based on a new Szegő-type recursion involving $\mathbb C^2$-valued polynomials and their reciprocal polynomials.

Hierarchical neural networks are widely used in artificial intelligence, yet their mathematical properties remain incompletely understood. In the infinite-width limit, two different theoretical frameworks have been proposed. One reduces deep learning to kernel regression with a fixed kernel by assuming that the parameters remain close to their initialization, whereas the other allows the parameters to move away from their initialization, requiring the kernel itself to be optimized. In this paper, we study a three-layer neural network with a finite but large number of hidden units. We show that training the input-to-hidden weights yields a smaller generalization error than keeping them fixed. Furthermore, the latter setting exhibits singularities in the parameter space, whereas the former does not. These findings indicate that singularities play an essential role even in wide neural networks.

We prove monotonicity properties for medians of gamma sums and differences of the form $Z_{\alpha} = \alpha X_1 + (2 - \alpha)X_2$, where $X_1$ and $X_2$ are independent gamma random variables with common shape parameter. By combining these monotonicity properties with known bounds for the median of the gamma distribution, we establish sharp bounds for the median of the variance-gamma and McKay Type I distributions. Also, by exploiting the normal variance-mean mixture representation of the generalized hyperbolic distribution together with bounds for a ratio of modified Bessel functions of the second kind, we obtain sharp bounds for the median of the generalized hyperbolic distribution. We thus resolve all five conjectures of Gaunt and Merkle (2021). As a by-product of our analysis, we show that the variance-gamma distributions with positive asymmetry parameter and the McKay Type I distributions satisfy the “mode-median-mean” inequality for all admissible parameter values, and that the same is true of the generalized hyperbolic distribution with positive asymmetry parameter.

Let $n\ge 4$ be an even integer. We show that $\{1^{\sigma(1)},2^{\sigma(2)},\ldots,n^{\sigma(n)}\}$ is a complete residue system modulo $n$ for some permutation $\sigma$ of $\{1,2,\ldots,n\}$ if and only if $p:=n/2$ is prime and $p-1$ is squarefree. This complements the results in [Int. J. Number Theory 14 (2018), 653–660], where an analogue characterization has been proved for odd integers.

Large language models produce prompt-dependent probabilities over words, whereas scientific systems require uncertainty over meaningful states that can be updated as evidence arrives. We develop an observable framework for determining when language-derived probabilities support such a sequential state representation. Theoretically, we define typed measurable transformations of contextual language, construct a minimal closed representation, and give necessary and sufficient conditions for semantic updates to exist uniquely. We bound irreducible nonclosure and accumulated error, and under average contraction prove existence, uniqueness and stability of an external random recursion on a probability simplex. These results define a stochastic lexical calculus without attributing an internal calculus to the language model. Empirically, frozen experiments test the observable implications. Raw prompt-conditioned probabilities fail the prespecified invariance gate; after prompt-specific calibration, a common three-state representation passes the stability gates and covers 28 of 30 untouched eight-step paths, or 0.933 at nominal level 0.90. Accordingly, language probabilities support a stochastic state only conditionally on verified closure, stability and coverage within a declared operating domain.

Let $\mathsf{d}$, $\mathsf{d}^{\ast}$, and $\mathsf{d}_{\ast}$ denote, respectively, the asymptotic density, the upper asymptotic density, and the lower asymptotic density on $\mathbb N$. We show that there are no constants $0<\nu<1$ and $c>0$ such that $$ \mathsf{d}_{\ast}(A+A)\ge c\bigl(\mathsf{d}^{\ast}(A+A)\bigr)^{1-\nu}\mathsf{d}(A)^{\nu} $$ for every set $A\subseteq \mathbb{N}$ which admits asymptotic density. This answers in the negative a question of I.Z. Ruzsa.

We exhibit two finite orthogonality hypergraphs that admit faithful orthogonal representations by rays in complex three-dimensional space but admit none in real three-dimensional space. Thus, although each orthogonality diagram is unchanged, its coordinatizability in dimension three depends on whether the scalar field is complex or real. The first hypergraph has 20 vertices and 12 contexts. The second hypergraph adds one vertex and two contexts. Phase-adjusted realification also gives explicit faithful orthogonal representations of both hypergraphs in real six-dimensional space.

The square $G^2$ of a graph joins two vertices at distance at most two; a proper coloring of $G^2$ is a 2-distance coloring of $G$. For a cactus $G$ (every edge on at most one cycle) we determine both the 2-distance chromatic number $\chi(G^2)$ and the choice number $\mathrm{ch}(G^2)$ exactly: they are always equal, and the common value is $\Delta+1$ if $\Delta\ge4$, is 4 if $\Delta=3$ and $G$ has no block equal to $C_5$, is 5 if $\Delta=3$ and $G$ has a $C_5$ block, and is the classical value if $\Delta\le2$. For $\Delta\ge6$ the value $\Delta+1$ is already known, since cacti are outerplanar and hence $K_{2,3}$-minor-free (Hetherington-Woodall; Agnarsson-Halldorsson). Our contribution is the small-degree regime. For subcubic cacti we obtain a complete classification in both the ordinary and list settings, with the 5-cycle as the unique obstruction; the list statement has no prior analogue and is a genuine positive instance of the List Square Coloring Conjecture, which is false in general. A single elimination order then handles all $\Delta\ge4$ uniformly and, in particular, settles the two cases $\Delta\in\{4,5\}$ that the superclass bounds leave at $\Delta+2$. The number 5 turns out to be one obstruction wearing three disguises: $C_5^2=K_5$, the Frobenius number of $\{3,4\}$, and a degenerate $K_4$ list-coloring instance.

Let $p$ be a prime, and let $B$ be a $p$-block of a finite group $G$. A $p$-block $B$ is called real if the set of irreducible ordinary characters contained in $B$ is invariant under complex conjugation. Motivated by Harris' classification of the finite groups with a unique $p$-block for an arbitrary prime $p$, and by McHugh and Schaeffer Fry's classification of the finite quasi-simple groups with a unique real $2$-block, we classify, in this paper, all finite groups admitting exactly one real $2$-block.

At the critical Hardy Gaussian weight for the Schrödinger equation in one space dimension on $[0,1]$, the known nonzero scalar example in the weighted $L^2$ class carries a complex-valued potential. Cassano and Fanelli observed that the existence of a real-valued scalar endpoint example was open, and produced examples with real electric and magnetic potentials only after introducing a magnetic potential. We construct a nonzero smooth solution of $i\partial_t u+\partial_x^2 u=Vu$ on $\mathbb{R}\times[0,1]$ such that $e^{x^2/4}u(\cdot,0),e^{x^2/4}u(\cdot,1)\in L^2(\mathbb{R})$ and the potential is bounded, smooth, real-valued, and purely scalar and is supported in one fixed compact spatial interval for all times. This compact-support endpoint example is the main result. We also record the explicit rational-tail realization underlying the construction. The main results of this paper were obtained by the multi-agent system Eureka and have subsequently been verified by the authors.

We study exact moments and algebraic sampling of a reciprocal quadratic Mellin integral. A finite elementary formula determines every nonzero rational-weight first moment with specified branches. For $|b|<2\sqrt{c}$, mixed half-weight moments admit a dilogarithmic formula, and all even half-weight moments follow from one polynomial recurrence with a geometric error bound. At zero Mellin weight with $b=0$, half-integer deformations form an explicit two-function elliptic space. At the singular boundary, a two-parameter reflection gives a radical integral equal to $\pi$ for every parameter, a cubic logarithmic integral in dilogarithms, and an elementary golden-ratio specialization. All odd zero-weight boundary moments also have explicit formulas. For a reduced rational weight $p/q$, the sampling function generates a field of degree $q^2$ and genus $(q-1)(q-2)/2$, outside the stated degeneracies. Explicit cubic maps and a Picard–Fuchs certificate fix its elliptic period normalization and yield CM companion series with polynomial-weight closure. At denominator four, we determine the three elliptic quotient maps and their differential pullbacks. For the original kernel $(c,b)=(5,-2)$, the Jacobian is isogenous to $E_{1728}\times E_0^2$, and homological transfer gives gamma-valued periods of the sampling differential. Classical identities are credited and used as ingredients in these family-specific formulas.

We resolve the two-colour three-component conjecture of Fernández, Pavez-Signé and Stein for random bipartite graphs. More precisely, we prove that if $G\sim G(n,n,p)$ and $p\gg\sqrt{\log n/n}$, then with high probability every red–blue edge-colouring of $G$ admits a cover of its vertex set by at most three monochromatic connected components. The proof is based on a uniform expansion lemma for unions of common neighbourhoods and an alternating common-neighbourhood expansion argument.

In this paper, we establish an explicit boundary-corrected Li–Yau type gradient estimate for positive solutions of the Dirichlet heat equation on the Euclidean half-space. The main idea is to exploit the reflection structure of the Dirichlet heat kernel and introduce a normalized kernel-induced probability measure. Under this representation, logarithmic derivatives of the heat kernel become expectations of explicit kernel quantities. The reflected Gaussian component generates a hyperbolic correction term involving \[ \coth\left(\frac{x_n y_n}{2t}\right), \] which has no analogue in the whole Euclidean heat equation. Using Jensen's inequality and the sharp estimate \[ 0<z\,\csch z<1 , \qquad z>0, \] we prove that every positive solution satisfies \[ \Delta\log w(x,t) \geq -\frac n{2t} -\frac1{x_n^2}. \] The first term represents the classical Euclidean Li–Yau diffusion scaling, while the second term is an explicit inverse-square correction determined by the distance to the Dirichlet boundary. Our approach provides a direct kernel interpretation of the boundary effect and suggests possible extensions to more general domains.

Let $S$ be the unit forward shift, and let $$ \mathcal{A} = \mathbb{K}[x]\langle S,S^{-1}\rangle \big/ \bigl(Sx-(x+1)S\bigr) $$ be the algebra of finite Laurent-shift operators with polynomial coefficients over a characteristic-zero field $\mathbb{K}$. We classify all $\mathfrak{sl}_2$-triples in $\mathcal{A}$ with fixed lowering operator $F=1-S$. Writing $D=S-1$ and $X=xS^{-1}$, every completion is uniquely determined by $\lambda\in\mathbb{K}$ and $g\in\mathbb{K}[S,S^{-1}]$, with $$ H=2(X+g)D-\lambda,\qquad E=(X+g)^2D-\lambda(X+g). $$ We give an intrinsic recognition and reconstruction from $H$ and determine the exact extreme shifts of $H$ and $E$. The associated monic eigenpolynomials form a $\Delta$-Appell sequence. Multiplication by $x$ has finite lower recurrence bandwidth exactly when $g\in S^{-1}\mathbb{K}[S]$; in this case, the bandwidth equals the right endpoint of the Cartan stencil. Otherwise, $H$ and $E$ remain finite-order, while the degree recurrence has an infinite tail whose eventual signed coefficients form a polynomial recovering the lowest Laurent term of $g$. Over $\mathbb{R}$, positive-measure orthogonality occurs exactly for translated monic Charlier systems.

This paper establishes a bidirectional relationship between isomorphisms of semirings and isomorphisms of their induced valuation algebras. We show that a semiring isomorphism is sufficient for a valuation algebra isomorphism. Conversely, for additively idempotent semirings without multiplicatively idempotent zero divisors, the existence of a valuation algebra isomorphism implies that the underlying semirings are isomorphic.

The rapid advancement of sixth-generation (6G) networks is accelerating the convergence of media intelligence and communication intelligence, driving media communication beyond conventional bit-level delivery toward intelligent, semantic-aware, and generative paradigms. Emerging media services require not only high data rates and low latency, but also semantic awareness, perceptual quality assurance, adaptive resource orchestration, trustworthy content processing, and personalized media generation. Meanwhile, media technologies are evolving from handcrafted signal processing and conventional coding toward artificial intelligence (AI)-driven representation learning, content understanding, and generative reconstruction. Motivated by these trends, this paper presents a systematic survey of media communication technologies for 6G vision communication by revisiting the evolution of communication and media technologies and clarifying the intrinsic relationship between media content processing and wireless transmission. We introduce a unified framework consisting of four key dimensions: AI-driven media technologies, media-aware wireless transmission, large model-enabled media communication, and intelligent network infrastructures. Specifically, AI-driven media technologies encompass media coding, content understanding, quality assessment, security and compliance detection, and AIGC-enabled media generation, while media-aware wireless transmission is examined from three complementary perspectives: semantic joint source-channel optimization, which jointly encodes task-relevant semantic information; source-aware transmission optimization, which leverages media characteristics for channel adaptation, prediction, and compensation; and channel-aware source optimization, which adapts media coding and reconstruction based on real-time channel conditions.