Crossing antisymmetric Polyakov blocks + dispersion relation

Many CFT problems, e.g. ones with global symmetries, have correlation functions with a crossing antisymmetric sector. We show that such a crossing antisymmetric function can be expanded in terms of manifestly crossing antisymmetric objects, which we call the ‘+ type Polyakov blocks’. These blocks are built from AdSd+1 Witten diagrams. In 1d they encode the ‘+ type’ analytic functionals which act on crossing antisymmetric functions. In general d we establish this Witten diagram basis from a crossing antisymmetric dispersion relation in Mellin space. Analogous to the crossing symmetric case, the dispersion relation imposes a set of independent ‘locality constraints’ in addition to the usual CFT sum rules given by the ‘Polyakov conditions’. We use the Polyakov blocks to simplify more general analytic functionals in d > 1 and global symmetry functionals.

Highly excited pure gauge SU(3) flux tubes

Flux tube spectra are expected to have full towers of levels due to the quantization of the string vibrations. We study a spectrum of flux tubes with static quark and antiquark sources with pure gauge SU(3) lattice QCD in 3+1 dimensions up to a significant number of excitations. To go high in the spectrum, we specialize in the most symmetric case Σg+, use a large set of operators, solve the generalized eigenvalue and compare different lattice QCD gauge actions and anisotropies.

On the Existence of Linearly Oscillating Galaxies

We consider two classes of steady states of the three-dimensional, gravitational Vlasov-Poisson system: the spherically symmetric Antonov-stable steady states (including the polytropes and the King model) and their plane symmetric analogues. We completely describe the essential spectrum of the self-adjoint operator governing the linearized dynamics in the neighborhood of these steady states. We also show that for the steady states under consideration, there exists a gap in the spectrum. We then use a version of the Birman-Schwinger principle first used by Mathur to derive a general criterion for the existence of an eigenvalue inside the first gap of the essential spectrum, which corresponds to linear oscillations about the steady state. It follows in particular that no linear Landau damping can occur in the neighborhood of steady states satisfying our criterion. Verification of this criterion requires a good understanding of the so-called period function associated with each steady state. In the plane symmetric case we verify the criterion rigorously, while in the spherically symmetric case we do so under a natural monotonicity assumption for the associated period function. Our results explain the pulsating behavior triggered by perturbing such steady states, which has been observed numerically.

A transport equation approach for deep neural networks with quenched random weights

We consider a multi-layer Sherrington-Kirkpatrick spin-glass as a model for deep restricted Boltzmann machines with quenched random weights and solve for its free energy in the thermodynamic limit by means of Guerra's interpolating techniques under the RS and 1RSB ansatz. In particular, we recover the expression already known for the replica-symmetric case. Further, we drop the restriction constraint by introducing intra-layer connections among spins and we show that the resulting system can be mapped into a modular Hopfield network, which is also addressed via the same techniques up to the first step of replica symmetry breaking.

Study of Forward and Backward Modes in Double-Sided Dielectric-Filled Corrugated Waveguides

This work studies the propagation characteristics of a rectangular waveguide with aligned/ misaligned double-sided dielectric-filled metallic corrugations. Two modes are found to propagate in the proposed double-sided configuration below the hollow-waveguide cutoff frequency: a quasi-resonant mode and a backward mode. This is in contrast to the single-sided configuration, which only allows for backward propagation. Moreover, the double-sided configuration can be of interest for waveguide miniaturization on account of the broader band of its backward mode. The width of the stopband between the quasi-resonant and backward modes can be controlled by the misalignment of the top and bottom corrugations, being null for the glide-symmetric case. The previous study is complemented with numerical results showing the impact of the height of the corrugations, as well as the filling dielectric permittivity, on the bandwidth and location of the appearing negative-effective-permeability band. The multi-modal transmission-matrix method has also been employed to estimate the rejection level and material losses in the structure and to determine which port modes are associated with the quasi-resonant and backward modes. Finally, it is shown that glide symmetry can advantageously be used to reduce the dispersion and broadens the operating band of the modes.

Logical Contradictions in the One-Way ANOVA and Tukey–Kramer Multiple Comparisons Tests with More Than Two Groups of Observations

We show that the one-way ANOVA and Tukey–Kramer (TK) tests agree on any sample with two groups. This result is based on a simple identity connecting the Fisher–Snedecor and studentized probabilistic distributions and is proven without any additional assumptions; in particular, the standard ANOVA assumptions (independence, normality, and homoscedasticity (INAH)) are not needed. In contrast, it is known that for a sample with k>2 groups of observations, even under the INAH assumptions, with the same significance level α, the above two tests may give opposite results: (i) ANOVA rejects its null hypothesis H0A:μ1=…=μk, while the TK one, H0TK(i,j):μi=μj, is not rejected for any pair i,j∈{1,…,k}; (ii) the TK test rejects H0TK(i,j) for a pair (i,j) (with i≠j), while ANOVA does not reject H0A. We construct two large infinite pseudo-random families of samples of both types satisfying INAH: in case (i) for any k≥3 and in case (ii) for some larger k. Furthermore, case (ii) ANOVA, being restricted to the pair of groups (i,j), may reject equality μi=μj with the same α. This is an obvious contradiction, since μ1=…=μk implies μi=μj for all i,j∈{1,…,k}. Such contradictions appear already in the symmetric case for k=3, or in other words, for three groups of d,d, and c observations with sample means +1,−1, and 0, respectively. We outline conditions necessary and sufficient for this phenomenon. Similar contradictory examples are constructed for the multivariable linear regression (MLR). However, for these constructions, it seems difficult to verify the Gauss–Markov assumptions, which are standardly required for MLR. Mathematics Subject Classification: 62 Statistics.

BPS Skyrme submodels of the five-dimensional Skyrme model

In this paper, we search for the BPS skyrmions in some BPS submodels of the generalized Skyrme model in five-dimensional spacetime using the BPS Lagrangian method. We focus on the static solutions of the Bogomolny’s equations and their corresponding energies with topological charge B > 0 is an integer. We consider two main cases based on the symmetry of the effective Lagrangian of the BPS submodels, i.e. the spherically symmetric and non-spherically symmetric cases. For the spherically symmetric case, we find two BPS submodels. The first BPS submodels consist of a potential term and a term proportional to the square of the topological current. The second BPS submodels consist of only the Skyrme term. The second BPS submodel has BPS skyrmions with the same topological charge B > 1, but with different energies, that we shall call “topological degenerate” BPS skyrmions. It also has the usual BPS skyrmions with equal energies, if the topological charge is a prime number. Another interesting feature of the BPS skyrmions, with B > 1, in this BPS submodel, is that these BPS skyrmions have non-zero pressures in the angular direction. For the non-spherically symmetric case, there is only one BPS submodel, which is similar to the first BPS submodel in the spherically symmetric case. We find that the BPS skyrmions depend on a constant k and for a particular value of k we obtain the BPS skyrmions of the first BPS submodel in the spherically symmetric case. The total static energy and the topological charge of these BPS skyrmions also depend on this constant. We also show that all the results found in this paper satisfy the full field equations of motions of the corresponding BPS submodels.

Goodness-of-fit test for $$\alpha$$-stable distribution based on the quantile conditional variance statistics

The class of $$\alpha$$ α -stable distributions is ubiquitous in many areas including signal processing, finance, biology, physics, and condition monitoring. In particular, it allows efficient noise modeling and incorporates distributional properties such as asymmetry and heavy-tails. Despite the popularity of this modeling choice, most statistical goodness-of-fit tests designed for $$\alpha$$ α -stable distributions are based on a generic distance measurement methods. To be efficient, those methods require large sample sizes and often do not efficiently discriminate distributions when the corresponding $$\alpha$$ α -stable parameters are close to each other. In this paper, we propose a novel goodness-of-fit method based on quantile (trimmed) conditional variances that is designed to overcome these deficiencies and outperforms many benchmark testing procedures. The effectiveness of the proposed approach is illustrated using extensive simulation study with focus set on the symmetric case. For completeness, an empirical example linked to plasma physics is provided.

Reflection of Functions and Linear Prognosis

Periodicity of wide class of functions as a result of reflection of even and arbitrary regular functions is proved. By consideration of new scalar work in space of linear shell of initial n vectors the equivalence of values of two different scalar productions is proved. The example of linear transformation is considered on plane for the symmetric case, resulting in possibility to make to use the orthogonal sides of rhombus at projection on the plane of its parties.