Rank and border rank of Kronecker powers of tensors and Strassen's laser method

We prove that the border rank of the Kronecker square of the little Coppersmith–Winograd tensor $$T_{cw,q}$$ T c w , q is the square of its border rank for $$q > 2$$ q > 2 and that the border rank of its Kronecker cube is the cube of its border rank for $$q > 4$$ q > 4 . This answers questions raised implicitly by Coppersmith & Winograd (1990, §11) and explicitly by Bläser (2013, Problem 9.8) and rules out the possibility of proving new upper bounds on the exponent of matrix multiplication using the square or cube of a little Coppersmith–Winograd tensor in this range.In the positive direction, we enlarge the list of explicit tensors potentially useful for Strassen's laser method, introducing a skew-symmetric version of the Coppersmith–Winograd tensor, $$T_{skewcw,q}$$ T s k e w c w , q . For $$q = 2$$ q = 2 , the Kronecker square of this tensor coincides with the $$3\times 3$$ 3 × 3 determinant polynomial, $$\det_{3} \in \mathbb{C}^{9} \otimes \mathbb{C}^{9} \otimes \mathbb{C}^{9}$$ det 3 ∈ C 9 ⊗ C 9 ⊗ C 9 , regarded as a tensor. We show that this tensor could potentially be used to show that the exponent of matrix multiplication is two.We determine new upper bounds for the (Waring) rank and the (Waring) border rank of $$\det_3$$ det 3 , exhibiting a strict submultiplicative behaviour for $$T_{skewcw,2}$$ T s k e w c w , 2 which is promising for the laser method.We establish general results regarding border ranks of Kronecker powers of tensors, and make a detailed study of Kronecker squares of tensors in $$\mathbb{C}^{3} \otimes \mathbb{C}^{3} \otimes \mathbb{C}^{3}$$ C 3 ⊗ C 3 ⊗ C 3 .

Bifurcations of quasi-periodic solutions from relative equilibria in the Lennard–Jones 2-body problem

We propose the general method of proving the bifurcation of new solutions from relative equilibria in N-body problems. The method is based on a symmetric version of Lyapunov center theorem. It is applied to study the Lennard–Jones 2-body problem, where we have proved the existence of new periodic or quasi-periodic solutions.

QKD Based on Symmetric Entangled Bernstein-Vazirani

This paper introduces a novel entanglement-based QKD protocol, that makes use of a modified symmetric version of the Bernstein-Vazirani algorithm, in order to achieve secure and efficient key distribution. Two variants of the protocol, one fully symmetric and one semi-symmetric, are presented. In both cases, the spatially separated Alice and Bob share multiple EPR pairs, each one qubit of the pair. The fully symmetric version allows both parties to input their tentative secret key from their respective location and acquire in the end a totally new and original key, an idea which was inspired by the Diffie-Hellman key exchange protocol. In the semi-symmetric version, Alice sends her chosen secret key to Bob (or vice versa). The performance of both protocols against an eavesdroppers attack is analyzed. Finally, in order to illustrate the operation of the protocols in practice, two small scale but detailed examples are given.

Evaluation of effectiveness of chosen-plaintext attacks on the Rao-Nam cryptosystem over a finite Abelian group

The Rao-Nam cryptosystem is a symmetric version of the McEliece code-based cryptosystem proposed to get rid of the shortcomings inherent in the first symmetric code-based encryption schemes. Almost immediately after the publication of this cryptosystem, attacks on it based on selected plaintexts appeared, which led to the emergence of various improvements and modifications of the original cryptosystem. The secret key in the traditional Rao-Nam scheme is a certain Boolean matrix and a set of binary vectors used to generate distortions during encryption. Such vectors must have different syndromes, that is, be different modulo of the code generated by the rows of the specified matrix. The original work of Rao and Nam considered two methods of forming the set of these vectors, the first of which consists in using predetermined vectors of sufficiently large weight, and the second is random selection of these vectors according to the equiprobable scheme. It is known that the first option does not provide the proper security of the Rao – Nam cryptosystem (due to the small number and simple structure of these vectors), but the second option is more meaningful and requires additional research. The purpose of this paper is to obtain estimates of the effectiveness (time complexity for a given upper bound of the error probability) of attacks on a cryptosystem, which generalizes the traditional Rao – Nam scheme to the case of a finite Abelian group (note that the need to study such versions of the Rao – Nam cryptosystem is due to their consideration in recent publications). Two attacks, based on selected plaintext, are presented. The first of them is not mentioned in the works known to the authors of this article and, under certain well-defined conditions, it allows recovering the secret key of the cryptosystem with quadratic complexity. The second attack is a generalized and simplified version of the well-known Struik-van Tilburg attack. It is shown that the complexity of this attack depends on the power of the stabilizer of the set of vectors, which forms the second part of the key, in the translation group of the Abelian group, over which the Rao – Nam cryptosystem is considered. In this paper, a bound is obtained for the probability of triviality of the stabilizer under the condition of random choice of this set. From the obtained bound, it follows that Struik-van Tilburg attack is, on average, noticeably more efficient than the worst case considered earlier.

Quantum phase transitions in nonhermitian harmonic oscillator

The Stone theorem requires that in a physical Hilbert space $${{{\mathcal {H}}}}$$ H the time-evolution of a stable quantum system is unitary if and only if the corresponding Hamiltonian H is self-adjoint. Sometimes, a simpler picture of the evolution may be constructed in a manifestly unphysical Hilbert space $${{{\mathcal {K}}}}$$ K in which H is nonhermitian but $${{\mathcal {PT}}}$$ PT -symmetric. In applications, unfortunately, one only rarely succeeds in circumventing the key technical obstacle which lies in the necessary reconstruction of the physical Hilbert space $${{{\mathcal {H}}}}$$ H . For a $${{\mathcal {PT}}}$$ PT -symmetric version of the spiked harmonic oscillator we show that in the dynamical regime of the unavoided level crossings such a reconstruction of $${{{\mathcal {H}}}}$$ H becomes feasible and, moreover, obtainable by non-numerical means. The general form of such a reconstruction of $${{{\mathcal {H}}}}$$ H enables one to render every exceptional unavoided-crossing point tractable as a genuine, phenomenologically most appealing quantum-phase-transition instant.

Solution formulas for differential Sylvester and Lyapunov equations

The differential Sylvester equation and its symmetric version, the differential Lyapunov equation, appear in different fields of applied mathematics like control theory, system theory, and model order reduction. The few available straight-forward numerical approaches when applied to large-scale systems come with prohibitively large storage requirements. This shortage motivates us to summarize and explore existing solution formulas for these equations. We develop a unifying approach based on the spectral theorem for normal operators like the Sylvester operator $${\mathcal {S}}(X)=AX+XB$$S(X)=AX+XB and derive a formula for its norm using an induced operator norm based on the spectrum of A and B. In view of numerical approximations, we propose an algorithm that identifies a suitable Krylov subspace using Taylor series and use a projection to approximate the solution. Numerical results for large-scale differential Lyapunov equations are presented in the last sections.

Existence of solutions for a nonhomogeneous Dirichlet problem involving $p(x)$-Laplacian operator and indefinite weight

We obtain multiplicity and uniqueness results in the weak sense for the following nonhomogeneous quasilinear equation involving the $p(x)$ p ( x ) -Laplacian operator with Dirichlet boundary condition: $$ -\Delta _{p(x)}u+V(x) \vert u \vert ^{q(x)-2}u =f(x,u)\quad \text{in }\varOmega , u=0 \text{ on }\partial \varOmega , $$ − Δ p ( x ) u + V ( x ) | u | q ( x ) − 2 u = f ( x , u ) in Ω , u = 0 on ∂ Ω , where Ω is a smooth bounded domain in $\mathbb{R}^{N}$ R N , V is a given function with an indefinite sign in a suitable variable exponent Lebesgue space, $f(x,t)$ f ( x , t ) is a Carathéodory function satisfying some growth conditions. Depending on the assumptions, the solutions set may consist of a bounded infinite sequence of solutions or a unique one. Our technique is based on a symmetric version of the mountain pass theorem.

Testing the Symmetric Assumption of Complementary Relationship: A Comparison between the Linear and Nonlinear Advection-Aridity Models in a Large Ephemeral Lake

The accuracy of a complementary relationship (CR) evapotranspiration (ET) model depends on how to parameterize the relationship between apparent potential ET and actual ET as the land surface changes from wet to dry. Yet, the validity of its inherent symmetric assumption of the original CR framework, i.e., the B value equal to one, is controversial. In this study, we conduct a comparative study between a linear, symmetric version (B = 1) and a nonlinear, asymmetric version (B is not necessarily equal to 1) of the advection-aridity (AA) CR model in a large ephemeral lake, which experiences dramatic changes in surface/atmosphere humidity. The results show that B was typically 1.1 ± 1.4 when ET ≤ ETPT ≤ ETPM, where ETPM and ETPT are estimated using the Penman (PM) and Priestley–Taylor (PT) equations, respectively; the AA model performed reasonably well in this case. However, the value of B can be negative and deviate from 1 significantly if the inequality ET ≤ ETPT ≤ ETPM is violated, which is quite common in humid environments. Because the actual ET can be negatively (B > 0) or positively (B < 0) related to the evaporative demand of the air, the nonlinear AA model generally performs better than the AA model if ET ≤ ETPM is satisfied. Although B is not significantly correlated with the atmospheric relative humidity (RH), both models, especially the nonlinear AA model, resulted in negative biases when ET > ETPM, which generally occur at high RH conditions. Both the linear and the nonlinear AA models performed better under higher water level conditions, however, our study highlights the need for higher-order (≥3) polynomial functions when CR models are applied in humid environments.

Beyond general relativity models, magnetosphere structure and dark matter stars

The magnetosphere structure of compact objects is considered in the context of a theory of gravity with dynamical torsion field beyond standard General Relativity (GR). To this end, a new spherically symmetric solution is obtained in this theoretical framework, physically representing a compact object of pseudoscalar fields (for example, axion field). The axially symmetric version of the Grad–Shafranov equation (GSE) is also derived in this context, and used to describe the magnetosphere dynamics of the obtained “axion star”. The interplay between high-energy processes and the seed magnetic field with respect to the global structure of the magnetosphere is briefly discussed.