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The Minimal superstring unification, assuming orbifold compactification, provides interesting and rather detailed implications on physics at low energy. The interesting feature of this model is that the masses of the spectrum are related…

We demonstrate that commutativity of numerous one-dimensional subalgebras in $W_{1+\infty}$ algebra, i.e. the existence of many non-trivial integrable systems described in recent arXiv:2303.05273 follows from the subset of relations in…

A classic result, originally due to Kluberg-Stern and Zuber, states that operators that vanish by the classical equation of motion (eom) do not mix into "physical" operators. Here we show that and explain why this result does not hold in…

We define the domain of a linear fractional transformation in a space of operators and show that both the affine automorphisms and the compositions of symmetries act transitively on these domains. Further, we show that Liouville's theorem…

We provide an explicit family of pairs $(\alpha, \beta) \in \mathbb{R}^k \times \mathbb{R}^k$ such that for sufficiently regular $f$, there is a constant $C>0$ for which the theta sum bound…

Let $F$ be a non--Archimedean locally compact field (${\rm car}(F)\geq 0$), ${\bf G}$ be a connected reductive group defined over $F$, $\theta$ be an $F$--automorphism of ${\bf G}$, and $\omega$ be a character of ${\bf G}(F)$. We fix a Haar…

The boundary beta-function generates the renormalization group acting on the universality classes of one-dimensional quantum systems with boundary which are critical in the bulk but not critical at the boundary. We prove a gradient formula…

Understanding and manipulating work fluctuations in microscale and nanoscale systems are of both fundamental and practical interest. For example, in considering the Jarzynski equality $\langle e^{-\beta W} \rangle=e^{-\beta \Delta F}$, a…

We study the fractal uncertainty principle in the joint time-frequency representation, and we prove a version for the Short-Time Fourier transform with Gaussian window on the modulation spaces. This can equivalently be formulated in terms…

The Gromoll-Meyer's generalized Morse lemma (so called splitting lemma) near degenerate critical points on Hilbert spaces, which is one of key results in infinite dimensional Morse theory, is usually stated for at least $C^2$-smooth…

Let $U$ be a unitary operator defined on some infinite-dimensional complex Hilbert space ${\cal H}$. Under some suitable regularity assumptions, it is known that a local positive commutation relation between $U$ and an auxiliary…

We perform the spectral analysis of a zero temperature Pauli-Fierz system for small coupling constants. Under the hypothesis of Fermi golden rule, we show that the embedded eigenvalues of the uncoupled system disappear and establish a…

A $7$-tuple of commuting bounded operators $\mathbf{T} = (T_1, \dots, T_7)$ defined on a Hilbert space $\mathcal{H}$ is said to be a \textit{$\Gamma_{E(3; 3; 1, 1, 1)}$-contraction} if $\Gamma_{E(3; 3; 1, 1, 1)}$ is a spectral set for…

Four dimensional scalar-tensor theory is considered within two conformal frames, the Jordan frame (JF) and the Einstein frame (EF). The actions for the theory are equivalent and equations of motion can be obtained from each action. It is…

A $7$-tuple of commuting bounded operators $\textbf{T} = (T_1, \dots, T_7)$ on a Hilbert space $\mathcal{H}$ is called a \textit{$\Gamma_{E(3; 3; 1, 1, 1)} $-contraction} if $\Gamma_{E(3; 3; 1, 1, 1)}$ is a spectral set for $\textbf{T}. $…

The Quantum Ergodic Conjecture equates the Wigner function for a typical eigenstate of a classically chaotic Hamiltonian with a delta-function on the energy shell. This ensures the evaluation of classical ergodic expectations of simple…

The article studies the convergence of trigonometric Fourier series via a new Tauberian theorem for Ces\`{a}ro summable series in abstract normed spaces. This theorem generalizes some known results of Hardy and Littlewood for number series.…

We generalize some characterizations of uniformly rectifiable (UR) sets to sets whose Hausdorff content is lower regular (and in particular, do not need to be Ahlfors regular). For example, David and Semmes showed that, given an Ahlfors…

We study a family of mutually commutative difference operators associated with the affine root systems. These operators act on the space of meromorphic functions on the Cartan subalgebra of the affine Lie algebra. We show that the space…

We consider the third order differential equation derived from the deformed Seiberg-Witten differential for pure ${\cal N}=2$ SYM with gauge group $SU(3)$ in Nekrasov-Shatashvili limit of $\Omega$-background. We show that this is the same…