arXiv Math @arxiv_math@qoto.org

In this work, we study an optimal boundary control for the stochastic Allen Cahn Navier Stokes system. The governing system of nonlinear partial differential equations consists of the stochastic Navier Stokes equations with non homogeneous Navier slip boundary condition coupled with a phase-field equation, which is the convective Allen Cahn equation type. We investigate the well posedness of the nonlinear system. More precisely, the existence and uniqueness of global strong solution in dimension two is established and we prove the existence of an optimal solution to the control problem.

Modulo a prime number, we define semi-primitive roots as the square of primitive roots. We present a method for calculating primitive roots from quadratic residues, including semi-primitive roots. We then present progressions that generate primitive and semi-primitive roots, and deduce an algorithm to obtain the full set of primitive roots without any GCD calculation. Next, we present a method for determining irreducible quadratic forms with arbitrarily large conjectured asymptotic density of primes (after Shanks, [1][2]). To this end, we propose an algorithm for calculating the square root modulo p, based on the Tonelli-Shanks algorithm [4].

Here we construct infinitely many Hölder continuous global-in-time and stationary solutions to the stochastic Euler and hypodissipative Navier-Stokes equations in the space $C(\mathbb{R};C^{\vartheta})$ for $0<\vartheta<\frac{5}{7}β$, with $0<β< \frac{1}{24}$ and $0<β<\min\left\{\frac{1-2α}{3},\frac{1}{24}\right\}$ respectively. A modified stochastic convex integration scheme, using Beltrami flows as building blocks and propagating inductive estimates both pathwise and in expectation, plays a pivotal role to improve the regularity of Hölder continuous solutions for the underlying equations. As a main novelty with respect to the related literature, our result produces solutions with noteworthy Hölder exponents.

We prove local well-posedness of the Cahn-Hilliard equation with additive noise. Our method relies on paracontrolled calculus and the Da Prato-Debussche trick.

In this paper we show that a countable structure admitting a finite monomorphic decomposition has finite big Ramsey degrees if and only if so does every monomorphic part in its minimal monomorphic decomposition. The necessary prerequisite for this result is the characterization of monomorphic structures with finite big Ramsey degrees: a countable monomorphic structure has finite big Ramsey degrees if and only if it is chainable by a chain with finite big Ramsey degrees. Interestingly, both characterizations require deep structural properties of chains. Fraïssé's Conjecture (actually, its positive resolution due to Laver) is instrumental in the characterization of monomorphic structures with finite big Ramsey degrees, while the analysis of big Ramsey combinatorics of structures admitting a finite monomorphic decomposition requires a product Ramsey theorem for big Ramsey degrees. We find this last result particularly intriguing because big Ramsey degrees misbehave notoriously when it comes to general product statements.

We exhibit an example of a discontinuity point for the Lyapunov exponents as a function of the cocycle in the Hölder topology. The linear cocycle taking values in SL$(2,\mathbb{R})$ is locally constant and defined over a Bernoulli shift. Our example extends Bocker-Viana's and Butler's results.

We give analogues in the finite general linear group of two elementary results concerning long cycles and transpositions in the symmetric group: first, that the long cycles are precisely the elements whose minimum-length factorizations into transpositions yield a generating set, and second, that a long cycle together with an appropriate transposition generates the whole symmetric group.

The Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations have a rich structure related to the theory of Frobenius manifolds, with many known families of solutions. A Legendre transformation is a symmetry of the WDVV equations, introduced by Dubrovin. We explicitly compute the results of a Legendre transformation applied to $A_n$- and $B_n$-type multi-parameter rational solutions, relating them to known and new trigonometric solutions.

We introduce computational strategies for measuring the ``size'' of the spectrum of bounded self-adjoint operators using various metrics such as the Lebesgue measure, fractal dimensions, the number of connected components (or gaps), and other spectral characteristics. Our motivation comes from the study of almost-periodic operators, particularly those that arise as models of quasicrystals. Such operators are known for intricate hierarchical patterns and often display delicate spectral properties, such as Cantor spectra, which are significant in studying quantum mechanical systems and materials science. We propose a series of algorithms that compute these properties under different assumptions and explore their theoretical implications through the Solvability Complexity Index (SCI) hierarchy. This approach provides a rigorous framework for understanding the computational feasibility of these problems, proving algorithmic optimality, and enhancing the precision of spectral analysis in practical settings. For example, we show that our methods are optimal by proving certain lower bounds (impossibility results) for the class of limit-periodic Schrödinger operators. We demonstrate our methods through state-of-the-art computations for aperiodic systems in one and two dimensions, effectively capturing these complex spectral characteristics. The results contribute significantly to connecting theoretical and computational aspects of spectral theory, offering insights that bridge the gap between abstract mathematical concepts and their practical applications in physical sciences and engineering. Based on our work, we conclude with conjectures and open problems regarding the spectral properties of specific models.

Fixed point ratios for primitive permutation groups have been extensively studied. Relying on a recent work of Burness and Guralnick, we obtain further results in the area. For a prime $p$ and a finite group $G$, we use fixed point ratios to study the number of Sylow $p$-subgroups of $G$ and the minimal size of a covering by proper subgroups of the set of $p$-elements of $G$.

In the present paper, we determine the algebraic relations among the tractable coordinates of logarithms of Anderson $t$-modules constructed by taking the tensor product of Drinfeld modules of rank $r$ defined over the algebraic closure of the rational function field and their $(r-1)$-st exterior powers with the Carlitz tensor powers. Our results, in the case of the tensor powers of the Carlitz module, generalize the work of Chang and Yu on the algebraic independence of polylogarithms.

The insertion, deletion, substitution (IDS) correcting code has garnered increased attention due to significant advancements in DNA storage that emerged recently. Despite this, the pursuit of optimal solutions in IDS-correcting codes remains an open challenge, drawing interest from both theoretical and engineering perspectives. This work introduces a pioneering approach named THEA-code. The proposed method follows a heuristic idea of employing an end-to-end autoencoder for the integrated encoding and decoding processes. To address the challenges associated with deploying an autoencoder as an IDS-correcting code, we propose innovative techniques, including the differentiable IDS channel, the entropy constraint on the codeword, and the auxiliary reconstruction of the source sequence. These strategies contribute to the successful convergence of the autoencoder, resulting in a deep learning-based IDS-correcting code with commendable performance. Notably, THEA-Code represents the first instance of a deep learning-based code that is independent of conventional coding frameworks in the IDS-correcting domain. Comprehensive experiments, including an ablation study, provide a detailed analysis and affirm the effectiveness of THEA-Code.

Many multi-dimensional (M-D) graph signals appear in the real world, such as digital images, sensor network measurements and temperature records from weather observation stations. It is a key challenge to design a transform method for processing these graph M-D signals in the linear canonical transform domain. This paper proposes the two-dimensional graph linear canonical transform based on the central discrete dilated Hermite function (2-D CDDHFs-GLCT) and the two-dimensional graph linear canonical transform based on chirp multiplication-chirp convolution-chirp multiplication decomposition (2-D CM-CC-CM-GLCT). Then, extending 2-D CDDHFs-GLCT and 2-D CM-CC-CM-GLCT to M-D CDDHFs-GLCT and M-D CM-CC-CM-GLCT. In terms of the computational complexity, additivity and reversibility, M-D CDDHFs-GLCT and M-D CM-CC-CM-GLCT are compared. Theoretical analysis shows that the computational complexity of M-D CM-CC-CM-GLCT algorithm is obviously reduced. Simulation results indicate that M-D CM-CC-CM-GLCT achieves comparable additivity to M-D CDDHFs-GLCT, while M-D CM-CC-CM-GLCT exhibits better reversibility. Finally, M-D GLCT is applied to data compression to show its application advantages. The experimental results reflect the superiority of M-D GLCT in the algorithm design and implementation of data compression.

In this paper, we deal with the growth and oscillation of solutions of higher order linear differential equations. Under the conditions that there exists a coefficient which dominates the other coefficients by its lower $% (α,β,γ)$-order and lower $(α,β,γ)$-type, we obtain some growth and oscillation properties of solutions of such equations which improve and extend some recently results of the author and Biswas \cite{b8}.

This paper provides some new characterizations of the diamond partial order for rectangular matrices by using properties of inner inverses, minus order, and SVD decompositions. In addition, the recently introduced 1MP generalized inverse and its dual are used to characterize the diamond partial order as a ${\cal G}$-based one. Finally, the one-sided star partial order is investigated by using 1MP- and MP1-inverses.

We generalize a previous result of Stevenson to the category of dendroidal sets, yielding the right cancellation property of dendroidal inner anodynes within the class of normal monomorphisms. As an application of this property, we show how to construct a symmetric monoidal $\infty$-category $\mathsf{Env}(X)^\otimes$ from a dendroidal $\infty$-operad $X$, in a way that generalizes the symmetric monoidal envelope of a coloured operad.

This work develops a robust and efficient framework of the adjoint gradient-based aerodynamic shape optimization (ASO) using high-order discontinuous Galerkin methods (DGMs) as the CFD solver. The adjoint-enabled gradients based on different CFD solvers or solution representations are derived in detail, and the potential advantage of DG representations is discovered that the adjoint gradient computed by the DGMs contains a modification term which implies information of higher-order moments of the solution as compared with finite volume methods (FVMs). A number of numerical cases are tested for investigating the impact of different CFD solvers (including DGMs and FVMs) on the evaluation of the adjoint-enabled gradients. The numerical results demonstrate that the DGMs can provide more precise adjoint gradients even on a coarse mesh as compared with the FVMs under coequal computational costs, and extend the capability to explore the design space, further leading to acquiring the aerodynamic shapes with more superior aerodynamic performance.

Recently, in Axioms 10(2): 119 (2021), a nonclassical first-order theory T of sets and functions has been introduced as the collection of axioms we have to accept if we want a foundational theory for (all of) mathematics that is not weaker than ZF, that is finitely axiomatized, and that does not have a countable model (if it has a model at all, that is). Here we prove that T is relatively consistent with ZF. We conclude that this is an important step towards showing that T is an advancement in the foundations of mathematics.

Pseudocolimits are formal gluing constructions that combine objects in a category indexed by a pseudofunctor. When the objects are categories and the domain of the pseudofunctor is small and filtered it has been known since Exppose 6 in SGA4 that the pseudocolimit can be computed by taking the Grothendieck construction of the pseudofunctor and inverting the class of cartesian arrows with respect to the canonical fibration. This paper is a reformatted version of a MSc thesis submitted and defended at Dalhousie University in August 2022. The first part presents a set of conditions for defining an internal category of elements of a diagram of internal categories and proves it is the oplax colimit. The second part presents a set of conditions on an ambient category and an internal category with an object of weak-equivalences that allows an internal description of the axioms for a category of (right) fractions and a definition of the internal category of (right) fractions when all the conditions and axioms are satisfied. These are combined to present a suitable context for computing the pseudocolimit of a small filtered diagram of internal categories.

Discussions surrounding the nature of the infinite in mathematics have been underway for two millennia. Mathematicians, philosophers, and theologians have all taken part. The basic question has been whether the infinite exists only in potential or exists in actuality. Only at the end of the 19th century, a set theory was created that works with the actual infinite. Initially, this theory was rejected by other mathematicians. The creator behind the theory, the German mathematician Georg Cantor, felt all the more the need to challenge the long tradition that only recognised the potential infinite. In this, he received strong support from the interest among German neothomist philosophers, who, under the influence of the Encyclical of Pope Leo XIII, Aeterni Patris, began to take an interest in Cantor's work. Gradually, his theory even acquired approval from the Vatican theologians. Cantor was able to firmly defend his work and at the turn of the 20th century, he succeeded in gaining its acceptance. The storm that had accompanied its original rejection now accompanied its acceptance. The theory became the basis on which modern mathematics was and is still founded, even though the majority of mathematicians know nothing of its original theological justification. Set theory, which today rests on an axiomatic foundation, no longer poses the question of the existence of actual infinite sets. The answer is expressed in its basic axiom: natural numbers form an infinite set. No substantiation has been discovered other than Cantor's: the set of all natural numbers exists from eternity as an idea in God's intellect.