Search Results (30)

This paper is devoted to the study of the algebra of bounded-type block-symmetric analytic functions on the Banach space ${\mathcal{l}}_1\left({\mathbb{C}}^s\right).$ In particular, it presents a description of the spectrum of this algebra in terms of exceptional characters ${\varphi }_{\alpha }$ and characters that can be associated with exponential-type functions of several variables with plane zeros. Due to this representation, it is proven that every element of the spectrum is a convolution of an exceptional character with a point evaluation functional. Full article

This manuscript introduces and investigates a new class of operators, termedn-quasi-$[m,\mathcal{C}]$-symmetric operators, which generalize and extend the existing notions of $[m,\mathcal{C}]$-symmetric and n-quasi-$[m,\mathcal{C}]$-isometric operators. Specifically, given a conjugation $\mathcal{C}$ on a Hilbert space, an operator $\mathcal{Q}\in \mathcal{B}\left(\mathcal{K}\right)$ is said to be n-quasi-$[m,\mathcal{C}]$-symmetric if it satisfies the relation${\mathcal{Q}}^{\ast n}\left({\displaystyle \sum _{0\le j\le m}}{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{m}{j}}\right)\mathcal{C}{\mathcal{Q}}^{m-j}\mathcal{C}{\mathcal{Q}}^j\right){\mathcal{Q}}^n=0$. Our study systematically explores the algebraic properties and structural characterization of n-quasi-$[m,\mathcal{C}]$-symmetric operators through matrix representations, providing a deeper understanding of their internal structure. Moreover, we establish sufficient conditions under which the powers and products of such operators inherit the n-quasi-$[m,\mathcal{C}]$-symmetric property. Additionally, we investigate the tensor products of n-quasi-$[m,\mathcal{C}]$-symmetric operators. Finally, we identify conditions that distinguish n-quasi-$[m,\mathcal{C}]$-symmetric operators from n-quasi-$[m-1;\mathcal{C}]$-symmetric operators. Full article

We developed a self-consistent double-null description of an evaporating white-hole spacetime by embedding the outgoing Vaidya solution in a coordinate system that remains regular across the future horizon. Starting from the radiation-coordinate form, we specialize in retarded time so that a monotonically decreasing mass function $M\left(u\right)$ encodes outgoing positive-energy flux. Expressing the metric in null coordinates $(u,v)$, Einstein’s equations for a single-directional null-dust stress–energy tensor, $T_{uu}=\rho \left(u\right)$, then reduce to one first-order PDE for the areal radius: ${\mathsf{\partial }}_{v}r=B\left(u\right)\left(1-2M\left(u\right)/r\right)$. Its integral, $r+2M\left(u\right)ln|r-2M(u\left)\right|=v-C\left(u\right)$, defines an implicit foliation of outgoing null cones. The metric coefficient follows algebraically as $f(u,v)=1-2M\left(u\right)/r$. Residual gauge freedom in $B\left(u\right)$ and $C\left(u\right)$ is fixed so that u matches the Bondi retarded time at null infinity, while v remains analytic at the apparent horizon, generalizing the Kruskal prescription to dynamical mass loss. In the limit $M\left(u\right)\to M$, the construction reduces to the familiar Eddington–Finkelstein and Kruskal forms. Our solution, therefore, provides a compact analytic framework for studying white-hole evaporation, Hawking-like energy fluxes, and back-reaction in spherically symmetric settings without encountering coordinate singularities. Full article

In this paper, the use of nilpotent Lie algebras as the basis for homomorphic encryption based on additive operations is explored. The $g$-setting is set up over $g{l}_n\left({\mathbb{Z}}_q\right)$) and the group $G=exp\left(g\right)$, and it is noted that the exponential and logarithm series are truncated by nilpotency in a natural way. From this, an additive symmetric conjugation scheme is constructed: given a message element $M$ and a central randomizer $U\in z\left(g\right),$ we encrypt $=Kexp\left(M+U\right)K^{-1}$ and decrypt to $M=log\left(K^{-1}CK\right)-U$. The scheme is additive in nature, with the security defined in the IND-CPA model. Integrity is ensured using an encrypt-then-MAC construction. These properties together provide both confidentiality and robustness while preserving the homomorphic functionality. The scheme realizes additive homomorphism through a truncated BCH-sum, so it is suitable for ciphertext summations. We implemented a prototype and took reproducible measurements (Python 3.11/NumPy) of the series $\{{10,10}^2,{10}^3,{10}^4,{10}^5\}$ over 10 iterations, reporting the medians and 95% confidence intervals. The graphs exhibit that the latency per operation remains constant at fixed values, and the total time scales approximately linearly with the batch size; we also report the throughput, peak memory usage, $\mid C\mid /\mid M\mid$ expansion rate, and achievable aggregation depth. The applications are federated reporting, IoT telemetry, and privacy-preserving aggregations in DBMS; the limitations include its additive nature (lacking general multiplicative homomorphism), IND-CPA (but not CCA), and side-channel resistance requirements. We place our approach in contrast to the standard FHE building blocks BFV/BGV/CKKS nd the emerging NIST PQC standards (FIPS 203/204/205), as a well-established security model with future engineering optimizations. Full article

This paper primarily investigates the solutions to the system of equations $A_{1}X{B}_1=C_1$ and $A_{2}X{B}_2=C_2$. This system generalizes the classical equation $AXB=C$, as well as the system of equations $AX=B$ and $XC=D$, and finds broad applications in control theory, signal processing, networking, optimization, and other related fields. Various methods for solving this system are introduced, including the generalized inverse method, the $\mathrm{vec}$-operator method, matrix decomposition techniques, Cramer’s rule, and iterative algorithms. Based on these approaches, the paper discusses general solutions, symmetric solutions, Hermitian solutions, and other special types of solutions over different algebraic structures, such as number fields, the real field, the complex field, the quaternion division ring, principal ideal domains, regular rings, strongly *-reducible rings, and operators on Banach spaces. In addition, matrix systems related to the system $A_{1}X{B}_1=C_1$ and $A_{2}X{B}_2=C_2$ are also explored. Full article

Causal discovery involves searching intractably large spaces. Decomposing the search space into classes of observationally equivalent causal models is a well-studied avenue to making discovery tractable. This paper studies the topological structure underlying causal equivalence to develop a categorical formulation of Chickering’s transformational characterization of Bayesian networks. A homotopic generalization of the Meek–Chickering theorem on the connectivity structure within causal equivalence classes and a topological representation of Greedy Equivalence Search (GES) that moves from one equivalence class of models to the next are described. Specifically, this work defines causal models as propable symmetric monoidal categories (cPROPs), which define a functor category ${\mathcal{C}}^P$ from a coalgebraic PROP P to a symmetric monoidal category $\mathcal{C}$. Such functor categories were first studied by Fox, who showed that they define the right adjoint of the inclusion of Cartesian categories in the larger category of all symmetric monoidal categories. cPROPs are an algebraic theory in the sense of Lawvere. cPROPs are related to previous categorical causal models, such as Markov categories and affine CDU categories, which can be viewed as defined by cPROP maps specifying the semantics of comonoidal structures corresponding to the “copy-delete” mechanisms. This work characterizes Pearl’s structural causal models (SCMs) in terms of Cartesian cPROPs, where the morphisms that define the endogenous variables are purely deterministic. A higher algebraic K-theory of causality is developed by studying the classifying spaces of observationally equivalent causal cPROP models by constructing their simplicial realization through the nerve functor. It is shown that Meek–Chickering causal DAG equivalence generalizes to induce a homotopic equivalence across observationally equivalent cPROP functors. A homotopic generalization of the Meek–Chickering theorem is presented, where covered edge reversals connecting equivalent DAGs induce natural transformations between homotopically equivalent cPROP functors and correspond to an equivalence structure on the corresponding string diagrams. The Grothendieck group completion of cPROP causal models is defined using the Grayson–Quillen construction and relate the classifying space of cPROP causal equivalence classes to classifying spaces of an induced groupoid. A real-world domain modeling genetic mutations in cancer is used to illustrate the framework in this paper. Full article

Let $\mathcal{H}$ be a separable complex Hilbert space, $\mathcal{B}\left(\mathcal{H}\right)$ the algebra of bounded linear operators on $\mathcal{H}$, $\mu$ a finite Borel measure on ${\mathbb{R}}_{+}$ with the finite $(n+1)$-th moment, $f\left(z\right):={\int }_{{\mathbb{R}}_{+}}{e}^{-tz}d\mu \left(t\right)$ for all $\Re z⩾0,{\mathcal{C}}_{\mathrm{\Psi }}\left(\mathcal{H}\right)$, and $\left|\right|·{\left|\right|}_{\mathrm{\Psi }}$ the ideal of compact operators and the norm associated to a symmetrically norming function $\mathrm{\Psi },$ respectively. If $A,D\in \mathcal{B}\left(\mathcal{H}\right)$ are accretive, then the Laplace transformer on $\mathcal{B}\left(\mathcal{H}\right),$$X\mapsto {\int }_{{\mathbb{R}}_{+}}{e}^{-tA}X{e}^{-tD}d\mu \left(t\right)$ is well defined for any $X\in \mathcal{B}\left(\mathcal{H}\right)$ as is the newly introduced Taylor remainder transformer $R_n(f;D,A)X:=f\left(A\right)X-\sum _{k=0}^n\frac{1}{k!}\sum _{i=0}^k{(-1)}^i\left(\genfrac{}{}{0pt}{}{k}{i}\right)A^{k-i}X{D}^i{f}^{\left(k\right)}\left(D\right).$ If $A,D^{*}$ are also $(n+1)$-accretive, ${\sum }_{k=0}^{n+1}{(-1)}^k\left(\genfrac{}{}{0pt}{}{n+1}{k}\right)A^{n+1-k}X{D}^k\in {\mathcal{C}}_{\mathrm{\Psi }}\left(\mathcal{H}\right)$ and $\left|\right|·{\left|\right|}_{\mathrm{\Psi }}$ is Q* norm, then $\left|\right|·{\left|\right|}_{\mathrm{\Psi }}$ norm estimates for ${\left({\sum }_{k=0}^{n+1}\left(\genfrac{}{}{0pt}{}{n+1}{k}\right)A^k{A}^{n+1-k}\right)}^{\frac{1}{2}}{R}_n(f;D,A)X{\left({\sum }_{k=0}^{n+1}\left(\genfrac{}{}{0pt}{}{n+1}{k}\right)D^{n+1-k}{D}^{*k}\right)}^{\frac{1}{2}}$ are obtained as the spacial cases of the presented estimates for (also newly introduced) Taylor remainder transformers related to a pair of Laplace transformers, defined by a subclass of accretive operators. Full article

Under certain assumptions, the existence of a unique solution of mixed integral equation (MIE) of the second type with a symmetric kernel is discussed, in $L_2\left[\Omega \right]\times C\left[0,T\right],T<1,$$\Omega$ is the position domain of integration and T is the time. The convergence error and the stability error are considered. Then, after using the separation technique, the MIE transforms into a system of Hammerstein integral equations (SHIEs) with time-varying coefficients. The nonlinear algebraic system (NAS) is obtained after using the degenerate method. New and special cases are derived from this work. Moreover, numerical results are computed using MATLAB R2023a software. Full article

Since bipolar quantum linear algebra (BQLA), under two operations–-addition and multiplication—demonstrates the properties of semirings, and since ideals play an important role in abstract algebra, our results are compelling for the ideals of a semiring. In this article, we investigate the characteristics of ideals, principal ideals, prime ideals, maximal ideals, and the smallest ideal containing any nonempty subset. By applying elementary real analysis, particularly the infimum, our main result is stated as follows: for any closed set I in BQLA, I is a nontrivial proper ideal if and only if there exists $c\in (0,1]$ such that $I=\left\{(-x,y)\in {\mathbb{R}}^2|cx\le y\le {\displaystyle \frac{x}{c}}\mathrm{and}x,y\ge 0\right\}$. This shows that its shape has to be symmetric with the graph $y=-x$. Full article

Substitution boxes, or S-boxes, are one of the most important mathematical primitives in modern symmetric cryptographic algorithms. Given their importance, in the past decades, they have been thoroughly analyzed and evaluated by the academic world. Thus, a lot of desirable characteristics a given S-box should possess have been found. This includes, as much as possible, higher nonlinearity and algebraic degrees as well as, as much as possible, lower values of differential uniformity, autocorrelation and sum of squares indicator values. In this work, we use power mappings over $GF(2^8)$ to generate, enumerate and evaluate all bijective S-boxes yielded by pentanomials of the form $f(x)=x^a+x^b+x^c+x^d+x^e$ given $0<a<b<c<d<e<256$. We find a total of 152,320 different bijective S-boxes, which are further classified into 41,458 different groups in terms of the aforementioned characteristics as well as the number of their fixed points. Having this data, an S-box designer can easily generate a bijective substitution S-box with parameters of their choice. By using pentanomials, we show how we can easily construct S-boxes with cryptographic properties similar to those found in some popular S-boxes like the Kuznyechik S-box proposed by the Russian Federation’s standardization agency as well as the Skipjack S-box proposed by the National Security Agency of the USA. Full article

Let $n\ge 2$ be a fixed integer and $\mathcal{A}$ be a $C^{\ast }$-algebra. A permuting n-linear map $\mathcal{G}:{\mathcal{A}}^n\to \mathcal{A}$ is known to be symmetric generalized n-derivation if there exists a symmetric n-derivation $\mathfrak{D}:{\mathcal{A}}^n\to \mathcal{A}$ such that $\mathcal{G}\left({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_i{\varsigma }_i^{\prime },\dots ,{\varsigma }_n\right)=\mathcal{G}\left({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_i,\dots ,{\varsigma }_n\right){\varsigma }_i^{\prime }+{\varsigma }_i\mathfrak{D}({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_i^{\prime },\dots ,{\varsigma }_n)$ holds ∀${\varsigma }_i,{\varsigma }_i^{\prime }\in \mathcal{A}$. In this paper, we investigate the structure of $C^{\ast }$-algebras involving generalized linear n-derivations. Moreover, we describe the forms of traces of linear n-derivations satisfying certain functional identity. Full article

The rational field $\mathbb{Q}$ is highly desired in many applications. Algorithms using the rational number field $\mathbb{Q}$ algebraic number fields use only integer arithmetics and are easy to implement. Therefore, studying and designing systems and expansions with coefficients in $\mathbb{Q}$ or algebraic number fields is particularly interesting. This paper discusses constructing quasi-tight framelets with symmetry over an algebraic field. Compared to tight framelets, quasi-tight framelets have very similar structures but much more flexibility in construction. Several recent papers have explored the structure of quasi-tight framelets. The construction of symmetric quasi-tight framelets directly applies the generalized spectral factorization of $2\times 2$ matrices of Laurent polynomials with specific symmetry structures. We adequately formulate the latter problem and establish the necessary and sufficient conditions for such a factorization over a general subfield $\mathbb{F}$ of $\mathbb{C}$, including algebraic number fields as particular cases. Our proofs of the main results are constructive and thus serve as a guideline for construction. We provide several examples to demonstrate our main results. Full article

In this paper, we consider a supersymmetric version of block-symmetric polynomials on a Banach space of two-sided absolutely summing series of vectors in ${\mathbb{C}}^s$ for some positive integer $s>1.$ We describe some sequences of generators of the algebra of block-supersymmetric polynomials and algebraic relations between the generators for the finite-dimensional case and construct algebraic bases of block-supersymmetric polynomials in the infinite-dimensional case. Furthermore, we propose some consequences for algebras of block-supersymmetric analytic functions of bounded type and their spectra. Finally, we consider some special derivatives in algebras of block-symmetric and block-supersymmetric analytic functions and find related Appell-type sequences of polynomials. Full article

The aim of this work is to investigate some algebraic structures of objects which are defined and related to a manifold. Consider L to be a smooth manifold and ${\Gamma }^{\infty }\left(TL\right)$ to be the module of smooth vector fields over the ring of smooth functions $C^{\infty }\left(L\right)$. We prove that the module ${\Gamma }^{\infty }\left(TL\right)$ is projective and finitely generated, but it is not semisimple. Therefore, it has a proper socle and nonzero Jacobson radical. Furthermore, we prove that this module is reflexive by showing that it is isomorphic to its bidual. Additionally, we investigate the structure of the Lie algebra of smooth vector fields. We give some questions and open problems at the end of the paper. We believe that our results are important because they link two different disciplines in modern pure mathematics. Full article

We consider algebras of polynomials and analytic functions that are invariant with respect to semidirect products of groups of bounded operators on Banach spaces with symmetric bases. In particular, we consider algebras of so-called block-symmetric and double-symmetric analytic functions on Banach spaces ${\mathit{\ell }}_p\left({\mathbb{C}}^n\right)$ and the homomorphisms of these algebras. In addition, we describe an algebraic basis in the algebra of double-symmetric polynomials and discuss a structure of the spectrum of the algebra of double-symmetric analytic functions on ${\mathit{\ell }}_p\left({\mathbb{C}}^n\right).$ Full article