One of the most interesting problems in the theory of Birkhoff billiards is the problem of integrability. In all known examples of integrable billiards, the billiard tables are either conics, quadrics (closed ellipsoids as well as unclosed quadrics like paraboloids or cones), or specific configurations of conics or quadrics. This leads to the natural question: are there other integrable billiards? The Birkhoff conjecture states that if the billiard inside a convex, smooth, closed curve is integrable, then the curve is an ellipse or a circle. In this paper we study the Birkhoff billiard inside a cone in $\mathbb{R}^n$. We prove that the billiard always admits a first integral of degree two in the components of the velocity vector. Using this fact, we prove that every trajectory inside a $C^3$ convex cone has a finite number of reflections. Here, by $C^3$ convex cone, we mean a cone whose section with some hyperplane is a strictly convex closed $C^3$ submanifold of the hyperplane with nondegenerate second fundamental form. The main result of this paper is the following. We prove that the Birkhoff billiard inside a convex $C^3$ cone is integrable. This is the first example of an integ
Detecting and understanding out-of-distribution (OOD) samples is crucial in machine learning (ML) to ensure reliable model performance. Current OOD studies primarily focus on extrapolatory (outside) OOD, neglecting potential cases of interpolatory (inside) OOD. In this study, we introduce a novel perspective on OOD by suggesting it can be divided into inside and outside cases. We examine the inside-outside OOD profiles of datasets and their impact on ML model performance, using normalized Root Mean Squared Error (RMSE) and F1 score as the performance metrics on syntetically-generated datasets with both inside and outside OOD. Our analysis demonstrates that different inside-outside OOD profiles lead to unique effects on ML model performance, with outside OOD generally causing greater performance degradation, on average. These findings highlight the importance of distinguishing between inside and outside OOD for developing effective counter-OOD methods.
This paper reports an end-to-end empirical evaluation of the deadline-Information Leakage Score (ILS-dl) extension introduced in the companion methodology paper. The deadline-ILS extends the original ILS to deadline-resolved prediction-market contracts, the dominant structural form of publicly documented insider trading on Polymarket. We anchor the evaluation in the 2026 U.S.-Iran conflict cluster of the ForesightFlow Insider Cases (FFIC) inventory, the largest documented deadline cluster. The evaluation has four parts: per-category exponential-hazard estimation, a single-case ILS-dl computation, cross-market wallet analysis, and methodological refinements. Hazard-rate estimation produces an adequate exponential fit for military-geopolitics markets (KS p = 0.426, half-life 2.9 days, n = 18) and a preliminary fit for corporate-disclosure markets (n = 5). The regulatory-decision category is rejected as bimodal (p = 0.023). On the largest applicable FFIC contract ("US forces enter Iran by April 30," $269M volume), the article-derived public-event timestamp yields ILS-dl = +0.113 versus a resolution-anchored proxy value of -0.331: a 0.444 shift in magnitude on opposite sides of zero, d
For a unitary description of an evaporating black hole, one usually chooses the time slices that cover only outside of the event horizon, which is mostly problem-free because the event horizon is not encountered. However, is there any justification for avoiding time slices that cover inside the event horizon? To answer the question, we investigate the Wheeler-DeWitt equation, where the time slices can cover both inside and outside the event horizon. We find that one can reasonably construct a wave packet that covers outside, but the wave function must be annihilated near the event horizon. This observation strongly suggests that we cannot choose a coherent state for a spacelike hypersurface that crosses the event horizon. To explain the unitary time evolution, we must keep the slices as coherent states; hence, they must always be outside the event horizon. In contrast, inside the horizon, we cannot have a single coherent state of a classical spacetime. Hence, the interior must be a superposition of several coherent states, which implies that there exists a horizon-scale uncertainty and a black hole should be viewed as a highly quantum macroscopic object. We provide a synthetic appr
In this work we study inside-out dissections of polygons and polyhedra. We first show that an arbitrary polygon can be inside-out dissected with $2n+1$ pieces, thereby improving the best previous upper bound of $4(n-2)$ pieces. Additionally, we establish that a regular polygon can be inside-out dissected with at most $6$ pieces. Lastly, we prove that any polyhedron that can be decomposed into finitely many regular tetrahedra and octahedra can be inside-out dissected.
The properties of a star with constant positive energy density inside (as for the Schwarzschild interior geometry) and a negative pressure are investigated, using a static conformally flat spacetime. Because of the negative pressure, the gravitational field inside is repulsive. Ricci and Kretschmann curvature invariants are finite. The energy conditions for the stress tensor of the perfect fluid are satisfied, excepting the strong energy condition which is not obeyed for $r<R/\sqrt{2}$, where $R$ is the radius of the object. The Komar mass is calculated and discussed.
We consider the problem of optimal inside portfolio $π(t)$ in a financial market with a corresponding wealth process $X(t)=X^π(t)$ modelled by \begin{align}\label{eq0.1} \begin{cases} dX(t)&=π(t)X(t)[α(t)dt+β(t)dB(t)]; \quad t\in[0, T] X(0)&=x_0>0, \end{cases} \end{align} where $B(\cdot)$ is a Brownian motion. We assume that the insider at time $t$ has access to market information $\varepsilon_t>0$ units ahead of time, in addition to the history of the market up to time $t$. The problem is to find an insider portfolio $π^{*}$ which maximizes the expected logarithmic utility $J(π)$ of the terminal wealth, i.e. such that $$\sup_πJ(π)= J(π^{*}), \text {where } J(π)= \mathbb{E}[\log(X^π(T))].$$ The insider market is called \emph{viable} if this value is finite. We study under what inside information flow $\mathbb{H}$ the insider market is viable or not. For example, assume that for all $t<T$ the insider knows the value of $B(t+ε_t)$, where $t + ε_t \geq T$ converges monotonically to $T$ from above as $t$ goes to $T$ from below. Then (assuming that the insider has a perfect memory) at time $t$ she has the inside information $\mathcal{H}_t$, consisting of the history $\m
In this paper, we present a multi-period trading model in the style of Kyle (1985)'s inside trading model, by assuming that there are at least two insiders in the market with long-lived private information, under the requirement that each insider publicly discloses his stock trades after the fact. Based on this model, we study the influences of "public disclosure" and "competition among insiders" on the trading behaviors of insiders. We find that the "competition among insiders" leads to higher effective price and lower insiders' profits, and the "public disclosure" makes each insider play a mixed strategy in every round except the last one. An interesting find is that as the total number of auctions goes to infinity, the market depth and the trading intensity at the first auction are all constants with the requirement of "public disclosure", while the market depth at the first auction goes to zero and the trading intensity of the first period goes to infinity without the requirement of "public disclosure".Moreover, we give the exact speed of the revelation of the private information, and show that all information is revealed immediately and the market depth goes to infinity immedi
Nonlinear cascade of low-frequency Alfvenic fluctuations (AFs) is regarded as one candidate of the energy sources to heat plasma during the non-adiabatic expansion of interplanetary coronal mass ejections (ICMEs). However, AFs inside ICMEs were seldom reported in the literature. In this study, we investigate AFs inside ICMEs using observations from Voyager 2 between 1 and 6 au. It is found that AFs with high degree of Alfvenicity frequently occurred inside ICMEs, for almost all the identified ICMEs (30 out of 33 ICMEs), and 12.6% of ICME time interval. As ICMEs expand and move outward, the percentage of AF duration decays linearly in general. The occurrence rate of AFs inside ICMEs is much less than that in ambient solar wind, especially within 4 au. AFs inside ICMEs are more frequently presented in the center and at the boundaries of ICMEs. In addition, the proton temperature inside ICME has a similar distribution. These findings suggest significant contribution of AFs on local plasma heating inside ICMEs.
Through two exact solution families to the Einstein equation and the one-to-one correspondence between their free parameters, we show that the ensemble of collapsars with only close-to-implementing horizon in the Schwarzschild time definition and the over-cross-oscillatory solid-balls in the Lemaitre time definition constitute two complementary description for the microscopic state of black holes formed through gravitational collapse. We quantise the solutions in the Schwarzschild time definition and show that the area law formula of Bekenstein-Hawking entropy follows naturally from the wave-functional's degeneracy of the collapsing materials. In two companion works, supports from the gravitational wave of binary merger observation and predictions for the fast radio burst of single body perturbations of this complementarity will be reported independently.
We give a necessary and sufficient condition for two circles, each with finitely many points added inside, to be betweenness isomorphic. We fully characterize the betweenness isomorphism classes in the family consisting of all circles with three collinear points inside.
We use a white noise approach to study the problem of optimal inside control of a stochastic delay equation driven by a Brownian motion B and a Poisson random measure N. In particular, we use Hida-Malliavin calculus and the Donsker delta functional to study the problem. We establish a sufficient and a necessary maximum principle for the optimal control when the trader from the beginning has inside information about the future value of some random variable related to the system.These results are applied to the problem of optimal inside harvesting control in a population modelled by a stochastic delay equation. Next, we apply a direct white noise method to find the optimal insider portfolio in a financial market where the risky asset price is given by a stochastic delay equation. A classical result of Pikovski and Karatzas shows that when the inside information is B(T), where T is the terminal time of the trading period, then the market is not viable. Our results show that with this inside information the market is not viable even if there is delay in the equations.
We develop a data-driven, {\em Partial Differential Equation-Ordinary Differential Equation} (PDE-ODE) model that describes the response of the {\em Carbon Dioxide} (\cotwon) dynamics inside a conference room, due to the presence of humans, or of a user-controlled exogenous source of \cotwon. We conduct two controlled experiments in order to develop and tune a model whose output matches the measured output concentration of \cotwo inside the room, when known inputs are applied to the model. In the first experiment, a controlled amount of \cotwo gas is released inside the room from a regulated supply, and in the second, a known number of humans produce a certain amount of \cotwo inside the room. For the estimation of the exogenous inputs, we design an observer, based on our model, using measurements of \cotwo concentrations at two locations inside the room. Parameter identifiers are also designed, based on our model, for the online estimation of the parameters of the model. We perform several simulation studies for the illustration of our designs.
Near-zero-index (NZI) materials, i.e. materials having a phase refractive index close to zero, are known to enhance or inhibit light-matter interactions. Most theoretical derivations of fundamental radiative processes rely on energetic considerations and detailed balance equations, but not on momentum considerations. Because momentum exchange should also be incorporated into theoretical models, we investigate momentum inside the three categories of NZI materials, i.e. inside epsilon-and-mu near-zero (EMNZ), epsilon-near-zero (ENZ) and mu-near-zero (MNZ) materials. In the context of Abraham-Minkowski debate in dispersive materials, we show that Minkowski-canonical momentum of light is zero inside all categories of NZI materials while Abraham-kinetic momentum of light is zero in ENZ and MNZ materials but nonzero inside EMNZ materials. We theoretically demonstrate that momentum recoil, transfer momentum from the field to the atom and Doppler shift are inhibited in NZI materials. Fundamental radiative processes inhibition is also explained due to those momentum considerations inside three-dimensional NZI materials. Lastly, absence of diffraction pattern in slits experiments is seen as
We analyze the physics of a type of homopolar motor comprising an AA battery with two cylindrical neodymium magnets on each end that roll inside a metal cylindrical tube. The motion of the motor results from the interaction between the magnetic field of the magnets and the magnetic field created by the current inside the magnets. We develop a model to describe the dynamics of the system, including the calculation of the terminal velocity of the motor due to eddy currents.
In this paper, the authors propose a new algorithm to hide data inside image using steganography technique. The proposed algorithm uses binary codes and pixels inside an image. The zipped file is used before it is converted to binary codes to maximize the storage of data inside the image. By applying the proposed algorithm, a system called Steganography Imaging System (SIS) is developed. The system is then tested to see the viability of the proposed algorithm. Various sizes of data are stored inside the images and the PSNR (Peak signal-to-noise ratio) is also captured for each of the images tested. Based on the PSNR value of each images, the stego image has a higher PSNR value. Hence this new steganography algorithm is very efficient to hide the data inside the image.
We study the problem of optimal inside control of an SPDE (a stochastic evolution equation) driven by a Brownian motion and a Poisson random measure. Our optimal control problem is new in two ways: (i) The controller has access to inside information, i.e. access to information about a future state of the system, (ii) The integro-differential operator of the SPDE might depend on the control. In the first part of the paper, we formulate a sufficient and a necessary maximum principle for this type of control problem, in two cases: (1) When the control is allowed to depend both on time t and on the space variable x. (2) When the control is not allowed to depend on x. In the second part of the paper, we apply the results above to the problem of optimal control of an SDE system when the inside controller has only noisy observations of the state of the system. Using results from nonlinear filtering, we transform this noisy observation SDE inside control problem into a full observation SPDE insider control problem. The results are illustrated by explicit examples.
Using 3D numerical hydrodynamical simulations we show that a type Ia supernova (SN Ia) explosion inside a planetary nebula (PN) can explain the observed shape of the G1.9+0.3 supernova remnant (SNR) and its X-ray morphology. The SNR G1.9+0.3 morphology can be generally described as a sphere with two small and incomplete lobes protruding on opposite sides of the SNR, termed "ears", a structure resembling many elliptical PNe. Observations show the synchrotron X-ray emission to be much stronger inside the two ears than in the rest of the SNR. We numerically show that a spherical SN Ia explosion into a circumstellar matter (CSM) with the structure of an elliptical PN with ears and clumps embedded in the ears can explain the X-ray properties of SNR G1.9+0.3. While the ejecta has already collided with the PN shell in most of the SNR and its forward shock has been slowed down, the ejecta is still advancing inside the ears. The fast forward shock inside the ears explains the stronger X-ray emission there. SN Ia inside PNe (SNIPs) seem to comprise a non-negligible fraction of resolved SN Ia remnants.
We show using molecular dynamics simulation that spatial confinement of water inside carbon nanotubes (CNT) substantially increases its boiling temperature and that a small temperature growth above the boiling point dramatically raises the inside pressure. Capillary theory successfully predicts the boiling point elevation down to 2 nm, below which large deviations between the theory and atomistic simulation take place. Water behaves qualitatively different inside narrow CNTs, exhibiting transition into an unusual phase, where pressure is gas-like and grows linearly with temperature, while the diffusion constant is temperature-independent. Precise control over boiling by CNT diameter, together with the rapid growth of inside pressure above the boiling point, suggests a novel drug delivery protocol. Polar drug molecules are packaged inside CNTs; the latter are delivered into living tissues and heated by laser. Solvent boiling destroys CNT capping agents and releases the drug.
Recently we have reported that the lattice QCD can not study the physical hadron formation from the unphysical quarks and gluons because it operates the unphysical QCD Hamiltonian of all the partons inside the hadron on the physical energy eigenstate of the hadron to obtain the physical energy eigenvalue of the hadron. However, since the parton distribution function (PDF) inside the hadron is unphysical (although it is well defined in QCD), we find in this paper that the unphysical energy of all the partons inside the hadron (instead of the physical energy of the hadron) can be used to study the PDF using the lattice QCD. Hence we find that the lattice QCD can study the parton distribution function inside the hadron from the first principle.