Invariant and dual subtraction games resolving the Duchêne–Rigo conjecture

We prove a recent conjecture of Duchêne and Rigo, stating that every complementary pair of homogeneous Beatty sequences represents the solution to an invariant impartial game. Here invariance means that each available move in a game can be played anywhere inside the game board. In fact, we establish such a result for a wider class of pairs of complementary sequences, and in the process generalize the notion of a subtraction game. Given a pair of complementary sequences $\left(a_n\right)$ and $\left(b_n\right)$ of positive integers, we define a game $G$ by setting $\left\{\{a_n,b_n\}\right\}$ as invariant moves. We then introduce the invariant game $G^{?}$, whose moves are all non-zero $P$-positions of $G$. Provided the set of non-zero $P$-positions of $G^{?}$ equals $\left\{\{a_n,b_n\}\right\}$, this is the desired invariant game. We give sufficient conditions on the initial pair of sequences for this ‘duality’ to hold.     

Hardness of approximating the Shortest Vector Problem in high ℓp norms

We present a new hardness of approximation result for the Shortest Vector Problem in ${\ell }_p$ norm (denoted by ${\mathrm{SVP}}_p$). Assuming NP $⊈$ ZPP, we show that for every $\epsilon >0$, there is a constant $p(\epsilon )$ such that for all integers $p⩾p(\epsilon )$, the problem ${\mathrm{SVP}}_p$ has no polynomial time approximation algorithm with approximation ratio $p^{1-\epsilon }$.     

Fractional Wishart processes and ε-fractional Wishart processes with applications

In this paper, we introduce two new matrix stochastic processes: fractional Wishart processes and $\epsilon$-fractional Wishart processes with integer indices which are based on the fractional Brownian motions and then extend $\epsilon$-fractional Wishart processes to the case with non-integer indices. Both processes include classic Wishart processes (if the Hurst index $H$ equals $\frac{1}{2}$) and present serial correlation of stochastic processes. Applying $\epsilon$-fractional Wishart processes to financial volatility theory, the financial models account for the stochastic volatilities of the assets and for the stochastic correlations not only between the underlying assets’ returns but also between their volatilities and for stochastic serial correlation of the relevant assets.     

Testing hypergraph colorability

We study the problem of testing properties of hypergraphs. The goal of property testing is to distinguish between the case whether a given object has a certain property or is “far away” from the property. We prove that the fundamental problem of $\ell$-colorability of k-uniform hypergraphs can be tested in time independent of the size of the hypergraph. We present a testing algorithm that examines only $(k\ell /\epsilon {)}^{\mathrm{O}(k)}$ entries of the adjacency matrix of the input hypergraph, where $\epsilon$ is a distance parameter independent of the size of the hypergraph. The algorithm tests only a constant number of entries in the adjacency matrix provided that $\ell$, k, and $\epsilon$ are constants. This result is a generalization of previous results about testing graph colorability.     

Information transmittal,relativity and gravitation

Special relativity considered in [Albert Einstein, Zur Elektrodynamik der bewegte Körper, Ann. Phys. 17 (1905) 891–921], and gravitation, studied in a series of papers, notably in [Albert Einstein, Zum gegenwärtigen Stände des Gravitationsproblemen, Phys. Z. 14 (1913) 1249–1262], are further analyzed regarding the principle of relativity, gravitation, and the notion of mass. The energy relation derived by Einstein from the relativistic Maxwell equations is applied to potential energy $W\left(x\right)$ of the gravitational field along the right line for which Einstein’s transformations are valid. This defines the intensity $G\left(x\right)=dW/dx$ of the relativistic force of gravity along a right line of observation in the gravitational field. The force is proportional to the observed acceleration according to the formula $\epsilon G\left(x\right)=\mu {\xi }_{\tau \tau }=\mu x_{tt}{\beta }^3$ where $\mu$ is the inert mass in the second Newton’s law of motion and $\epsilon$ is the charge (mass) in the relativistic electromagnetic (gravitational) field. In everyday life, we see that all bodies visually fall under gravity (i.e. in a common gravitational field) with the same observed acceleration ${\xi }_{\tau \tau }$ as if having equal inert and gravitational masses: $\mu /\epsilon =1$, with respect to the synchronized time $\tau$. However, if the principle of relativity extended by Einstein to the case of the uniformly accelerated rectilinear motion is valid, then this relation should also be true with respect to $x_{tt}$, that is, $(\mu /\epsilon ){\beta }^3=1$, in proper time $t$ of a still observer and of the carrying system (falling body), thus, depending on velocity $v$ at which the acceleration ${\xi }_{\tau \tau }$ is measured. This means that the inert mass $\mu$ and the gravitational mass $\epsilon$ can be considered equal only at $v=0$, and otherwise are related by the equation $\epsilon =\mu {\beta }^3\ge \mu$, where Einstein’s calibration factor $\beta ={[1?{(v/V)}^2]}^{?0.5}\ge 1,\left|v\right|<V$, and $\beta ?1$ for small $\left|v\right|$ compared with the speed of light $V=300000\mathrm{km/s}$ at which we see the falling bodies. If $v>0$, then the observed gravitational mass $\epsilon$ is greater than the inert mass $\mu$. The increase of mass is concurrent with the increase of tensions that at high velocities $v\to V$ induce overheating in the particle accelerators and colliders. To comply with the nature of observation, the information transmittal signals are incorporated in the Lorentz invariant of the $4D$ geometry, leading to the local invariants of relativistic dynamics that include gravitation and the speed of signals used in observation of moving bodies. With the same communication signals, those invariants hold for the synchronized time and coordinates of moving systems irrespective of their relative velocities. A procedure is developed for measurement and computation of the accelerations produced by variable gravitational and/or electromagnetic fields through the measurements of velocities of a moving body, so that the motion of the body and the field of forces acting on it can be fully identified. The results open new avenues for research in the theory of relativity and its applications.     

Radial Basis Function generated Finite Differences for option pricing problems

In this paper we present a numerical method to price options based on Radial Basis Function generated Finite Differences (RBF-FD) in space and the Backward Differentiation Formula of order 2 (BDF-2) in time. We use Gaussian RBFs that depend on a shape parameter $\epsilon$. The choice of this parameter is crucial for the performance of the method. We chose $\epsilon$ as $\mathrm{const}?h^{?1}$ and we derive suitable values of the constant for different stencil sizes in 1D and 2D. This constant is independent of the problem parameters such as the volatilities of the underlying assets and the interest rate in the market. In the literature on option pricing with RBF-FD, a constant value of the shape parameter is used. We show that this always leads to ill-conditioning for decreasing $h$, whereas our proposed method avoids such ill-conditioning. We present numerical results for problems in 1D, 2D, and 3D demonstrating the useful features of our method such as discretization sparsity, flexibility in node placement, and easy dimensional extendability, which provide high computational efficiency and accuracy.