Falsification of Einstein’s relativity

In 1915, Einstein published general relativity. In 1916, he published a German language book about relativity, which contained his marble table thought experiment for explaining a continuum. Without realizing it, Einstein introduced a quantized two-dimensional discontinuum geometry and inadvertently falsified the marble table thought experiment continuum, which falsified relativity. The foundations of physics do not now (and never did) include a fundamentally sound relativistic theory to account for macroscopic phenomena. It is well known the success of relativity and its singularity problem indicate general relativity is a first approximation of a more fundamental theory. Combine that indication with the falsification of relativity and it is apparent, without speculation, that relativity is now and always was a first approximation of a more fundamental theory. A possible way forward to the more fundamental theory is developing a discontinuum physics based on the quantized two-dimensional discontinuum geometry or an algebraic version of it. Such discontinuum physics is not presented, because it is beyond the scope of this paper.

Guest Column

Algebraic Natural Proofs is a recent framework which formalizes the type of reasoning used for proving most lower bounds on algebraic computational models. This concept is similar to and inspired by the famous natural proofs notion of Razborov and Rudich [RR97] for boolean circuit lower bounds, but, unlike in the boolean case, it is an open problem whether this constitutes a barrier for proving super-polynomial lower bounds for strong models of algebraic computation. From an algebraic-geometric viewpoint, it is also related to basic questions in Geometric Complexity Theory (GCT), and from a meta-complexity theoretic viewpoint, it can be seen as an algebraic version of the MCSP problem. We survey the recent work around this concept which provides some evidence both for and against the existence of an algebraic natural proofs barrier, with an emphasis on the di erent viewpoints and the connections to other areas.

Monadic Bounded Algebras

<p>The object of study of the thesis is the notion of monadic bounded algebras (shortly, MBA's). These algebras are motivated by certain natural constructions in free (first-order) monadic logic and are related to free monadic logic in the same way as monadic algebras of P. Halmos to monadic logic (Chapter 1). Although MBA's come from logic, the present work is in algebra. Another important way of approaching MBA's is via bounded graphs, namely, the complex algebra of a bounded graph is an MBA and vice versa. The main results of Chapter 2 are two representation theorems: 1) every model is a basic MBA and every basic MBA is isomorphic to a model; 2) every MBA is isomorphic to a subdirect product of basic MBA's. As a consequence, every MBA is isomorphic to a subdirect product of models. This result is thought of as an algebraic version of semantical completeness theorem for free monadic logic. Chapter 3 entirely deals with MBA-varieties. It is proved by the method of filtration that every MBA-variety is generated by its finite special members. Using connections in terms of bounded morphisms among certain bounded graphs, it is shown that every MBA-variety is generated by at most three special (not necessarily finite) MBA's. After that each MBA-variety is equationally characterized. Chapter 4 considers finitely generated MBA's. We prove that every finitely generated MBA is finite (an upper bound on the number of elements is provided) and that the number of elements of a free MBA on a finite set achieves its upper bound. Lastly, a procedure for constructing a free MBA on any finite set is given.</p>

Monadic Bounded Algebras

<p>The object of study of the thesis is the notion of monadic bounded algebras (shortly, MBA's). These algebras are motivated by certain natural constructions in free (first-order) monadic logic and are related to free monadic logic in the same way as monadic algebras of P. Halmos to monadic logic (Chapter 1). Although MBA's come from logic, the present work is in algebra. Another important way of approaching MBA's is via bounded graphs, namely, the complex algebra of a bounded graph is an MBA and vice versa. The main results of Chapter 2 are two representation theorems: 1) every model is a basic MBA and every basic MBA is isomorphic to a model; 2) every MBA is isomorphic to a subdirect product of basic MBA's. As a consequence, every MBA is isomorphic to a subdirect product of models. This result is thought of as an algebraic version of semantical completeness theorem for free monadic logic. Chapter 3 entirely deals with MBA-varieties. It is proved by the method of filtration that every MBA-variety is generated by its finite special members. Using connections in terms of bounded morphisms among certain bounded graphs, it is shown that every MBA-variety is generated by at most three special (not necessarily finite) MBA's. After that each MBA-variety is equationally characterized. Chapter 4 considers finitely generated MBA's. We prove that every finitely generated MBA is finite (an upper bound on the number of elements is provided) and that the number of elements of a free MBA on a finite set achieves its upper bound. Lastly, a procedure for constructing a free MBA on any finite set is given.</p>

Irregular perverse sheaves

We introduce irregular constructible sheaves, which are ${\mathbb {C}}$-constructible with coefficients in a finite version of the Novikov ring $\Lambda$ and special gradings. We show that the bounded derived category of cohomologically irregular constructible complexes is equivalent to the bounded derived category of holonomic ${\mathcal {D}}$-modules by a modification of D’Agnolo and Kashiwara's irregular Riemann–Hilbert correspondence. The bounded derived category of cohomologically irregular constructible complexes is equipped with the irregular perverse $t$-structure, which is a straightforward generalization of usual perverse $t$-structure, and we prove that its heart is equivalent to the abelian category of holonomic ${\mathcal {D}}$-modules. We also develop the algebraic version of the theory.

Symbols in Berezin quantization for representation operators

The basic notion of the Berezin quantization on a manifold M is a correspondence which to an operator A from a class assigns the pair of functions F and F^♮ defined on M. These functions are called covariant and contravariant symbols of A. We are interested in homogeneous space M=G/H and classes of operators related to the representation theory. The most algebraic version of quantization — we call it the polynomial quantization — is obtained when operators belong to the algebra of operators corresponding in a representation T of G to elements X of the universal enveloping algebra Env g of the Lie algebra g of G. In this case symbols turn out to be polynomials on the Lie algebra g. In this paper we offer a new theme in the Berezin quantization on G/H: as an initial class of operators we take operators corresponding to elements of the group G itself in a representation T of this group. In the paper we consider two examples, here homogeneous spaces are para-Hermitian spaces of rank 1 and 2: a) G=SL(2;R), H — the subgroup of diagonal matrices, G/H — a hyperboloid of one sheet in R^3; b) G — the pseudoorthogonal group SO_0 (p; q), the subgroup H covers with finite multiplicity the group SO_0 (p-1,q -1)×SO_0 (1;1); the space G/H (a pseudo-Grassmann manifold) is an orbit in the Lie algebra g of the group G.

RESEARCH OF THE THERMONUCLEAR REACTIONS WITH THE FORMATION OF 7Be NUCLEUS IN THE CLUSTER APPROXIMATION

Research of thermonuclear reactions is a great interest to modern nuclear physics. In this connection, in this work, we investigated one of the main reactions of the pp-cycle within the framework of the cluster model. For calculations, we used one of the methods of the cluster model, namely, the algebraic version of the method of resonating groups in which the studied reaction of scattering of an alpha particle with 3He was disassembled and presented in the form of two interacting clusters. The modified Hasegawa-Nagata potential was also used, which describes the behavior of the nucleon-nucleon interaction. The aim of the research is to identify common patterns in coupled and resonant states. The main theoretical calculation of the studied resonance and coupled states was carried out using a special program "2cl_SpectrPhases.exe". The obtained theoretical results were compared with experimental data.

APPLICATION OF TWO-CLUSTER MICROSCOPIC MODEL TO STUDY PROCESSES ASSOCIATED WITH THE COSMOLOGICAL LITHIUM PROBLEM

This paper presents the results of studies of long-lived resonance states of the 6Li nucleus and the radiative capture reaction leading to the synthesis of 6Li. In connection with the importance of the lithium problem for the fields of nuclear physics and nuclear astrophysics, the reactions of primary nucleosynthesis are of great interest for studies and consideration using new calculation methods. Cluster models are a powerful tool for theoretical analysis and solution of this problem. One of these models is the microscopic cluster model, or a method known as the algebraic version of the resonant group method (AVRGM), which will allow us to investigate this problem from a new angle and obtain data on resonance states. For this task, a modified Hasegawa-Nagata potential was chosen, which has its own unique characteristics and exchange parameters. The obtained data were compared with experimental data and it was shown that they showed good agreement, and the applicability of this technique to the description of similar reactions that originate in the first three minutes after the birth of the universe.

Log smooth deformation theory via Gerstenhaber algebras

We construct a $$k\left[ \!\left[ Q\right] \!\right] $$ k Q -linear predifferential graded Lie algebra $$L^{\bullet }_{X_0/S_0}$$ L X 0 / S 0 ∙ associated to a log smooth and saturated morphism $$f_0: X_0 \rightarrow S_0$$ f 0 : X 0 → S 0 and prove that it controls the log smooth deformation functor. This provides a geometric interpretation of a construction in Chan et al. (Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties, 2019. arXiv:1902.11174) whereof $$L^{\bullet }_{X_0/S_0}$$ L X 0 / S 0 ∙ is a purely algebraic version. Our proof crucially relies on studying deformations of the Gerstenhaber algebra of polyvector fields; this method is closely related to recent developments in mirror symmetry.

On the EIT problem for nonorientable surfaces

Let {(\Omega,g)} be a smooth compact two-dimensional Riemannian manifold with boundary and let {\Lambda_{g}:f\mapsto\partial_{\nu}u|_{\partial\Omega}} be its DN map, where u obeys {\Delta_{g}u=0} in Ω and {u|_{\partial\Omega}=f}. The Electric Impedance Tomography Problem is to determine Ω from {\Lambda_{g}}. A criterion is proposed that enables one to detect (via {\Lambda_{g}}) whether Ω is orientable or not. The algebraic version of the BC-method is applied to solve the EIT problem for the Moebius band. The main instrument is the algebra of holomorphic functions on the double covering {{\mathbb{M}}} of M, which is determined by {\Lambda_{g}} up to an isometric isomorphism. Its Gelfand spectrum (the set of characters) plays the role of the material for constructing a relevant copy {(M^{\prime},g^{\prime})} of {(M,g)}. This copy is conformally equivalent to the original, provides {\partial M^{\prime}=\partial M}, {\Lambda_{g^{\prime}}=\Lambda_{g}}, and thus solves the problem.