Search Results (178)

In this article, we introduce and study two new subclasses of analytic univalent functions defined via the Ma–Minda function. Specifically, we consider the class ${\mathcal{C}}_{{\xi }_q}$ of q-convex functions involving a suitable Ma–Minda function ${\xi }_q\left(z\right),$ together with its classical counterpart ${\mathcal{C}}_{\xi }$ corresponding to $\xi \left(z\right),$ where $0<q<1.$ We determine bounds for the first few Taylor–Maclaurin coefficients and deduce Fekete–Szegö and Kruskal inequality. Moreover, we obtain the associated Toeplitz determinants related to this class. We highlight several new consequences of our results that are of independent interest in geometric function theory. Full article

Training-free LLM compression avoids fine-tuning by approximating weight matrices from a small unlabelled calibration set; however, existing methods assign each layer an independent target rank with no global budget coordination, leaving parameter distribution across layers systematically suboptimal. We propose CLEAR (Convex-optimal Layer Energy Allocation and Representation), a training-free compression framework that combines an activation-whitened structured-dictionary representation with a provably optimal, globally coordinated budget allocation across layers. Ranks are distributed across all layers simultaneously via a convex water-filling optimization, whose KKT solution is provably optimal and achieves <${10}^{-3}$ pp precision on the dictionary-path parameter budget (the total realized retention ratio, including EoRA and outlier bypass parameters, deviates from the target by at most 0.02 pp in practice). Each layer is then compressed using activation-whitened structured sparse dictionaries ($W\approx DC$, k-sparse columns), supplemented by an EoRA low-rank residual correction and an outlier input-channel bypass. Evaluated on five LLMs spanning 600 M to 8 B parameters across four model families, CLEAR outperforms the current state of the art on three of four evaluated models at compression ratio (CR) = 0.2. On LLaMA-3.2-1B, average accuracy across eight zero-shot benchmarks reaches 50.4% versus 42.7% for CoSpaDi and 37.6% for SVD-LLM, with perplexity reduced from 63.7 to 22.4; on LLaMA-3-8B, 65.5% versus 61.8% for CoSpaDi. Component ablations confirm +5.22 pp from sparse dictionaries and +1.12 pp from global rank allocation. KFAC Fisher covariance proves counterproductive in this setting, degrading accuracy by 8.1 pp due to numerical overflow in SiLU gating layers and consequent budget misallocation. CLEAR completes in 8–30 min on a single GPU, requires no gradient computation at any stage (forward-pass only, including the cascade activation refresh in Phase 3), and reduces peak deployment memory by approximately 20% at BFloat16 precision. Full article

We develop a terminal-explicit geometric theory for the algebraic zero sets associated with the Riemann–Liouville and Caputo fractional derivatives of a complex polynomial. Expanding the polynomial about an arbitrary complex terminal a reduces both operators to gamma-weighted polynomial transforms. The coordinate shift $w=z-a$ is algebraically equivalent to a zero-terminal formulation; the terminal dependence re-enters through the shifted coefficients and through the pullback of the resulting geometry to the original z-plane. We derive exact barycenter identities, terminal-centered disk bounds, multiset convergence at the endpoint orders, first-order deformation formulas for simple roots, and solvable examples. The disk bounds provide a star-shaped localization framework rather than a convex-hull theorem. A regular-polygon family yields an exact homothety for the Riemann–Liouville roots, while numerical examples show curved generic trajectories and a nondegenerate Caputo evolution. The terminal therefore acts as a distinguished geometric anchor, with radial attraction occurring under additional algebraic symmetry. Full article

Hybrid conjugate gradient methods are considered as an efficient family of conjugate gradient (CG) methods used to solve unconstrained optimization problems. In this paper, on account of the outstanding performance of the PRP (Polak–Ribière–Polyak) conjugate gradient method and its exceptional numerical computational stability, we propose a hybrid conjugate gradient method for solving unconstrained optimization problems. By combining two PRP-type directions via convex combination, the proposed search direction dynamically adjusts to gradient change rates and satisfies the sufficient descent property. Under mild conditions, the global convergence of the proposed method is established. Numerical computations are presented to display the efficacy of the proposed algorithm compared to some existing algorithms. It is indicated that the proposed method is more effective in dealing with non-convex optimization problems. Finally, the applicability of the proposed method is shown in image restoration problems with noise, and preliminary experimental results demonstrate its effectiveness compared to some other methods. Full article

In this paper, we introduce and investigate a new subclass of analytic functions in the open unit disk by combining the $(\lambda ,q)$-fractional differintegral operator with the Euler generating function through the technique of subordination. This class, denoted by ${\mathcal{K}}_{q,\lambda }\left(\tau \right)$, unifies and extends several families of q-starlike and q-convex functions as special cases corresponding to particular choices of the parameter $\lambda$. For this class, coefficient estimates for the initial Taylor coefficients and a Fekete–Szegő inequality are established. Furthermore it is shown that the transformed function $D_q^{\lambda }f$ is q-starlike in the unit disk under suitable assumptions on the parameters. In addition, first-order distortion estimates are obtained for the associated q-starlike and q-convex subclasses. Several consequences are derived for the special cases $\lambda =0$ and $\lambda =1$, corresponding, respectively, to q-starlike and q-convex function classes. The classical limit $q\to 1^{-}$ is also investigated and establish a new connection between Euler generating functions, fractional q-operators, and subclasses of analytic and univalent functions. Full article

Neural metric learning is often assessed by retrieval accuracy, but a learned dissimilarity can rank examples well while failing to have norm-like algebraic structure. This paper studies a precise finite question: within a Euclidean-anchored residual family of neural dissimilarities, can one reduce sampled defects of homogeneity, subadditivity, and dyadic reconstruction on latent differences without destroying retrieval performance? The construction is inspired by asymptotically metrically normable (AMN) vector spaces, but its claims are finite, sampled, and latent: it does not prove global AMN rigidity or certify a metric on the input space. The framework is motivated by the observation that many learned similarities have the form $K=\exp (-E/\tau )$ and therefore encode an unbounded distance-like quantity or squared distance-like quantity behind a bounded affinity. The AMN-relevant object is this cost, not the bounded kernel value. We formalize bounded-perturbation stability of the large-scale specific energy $E(nv,0)/n$, the conversion of subadditivity into multiplicative affinity consistency, and the quotient interpretation in which directions of zero large-scale cost are collapsed. The mathematical development then introduces finite dyadic diagnostics, learned-gauge and convex-unit-ball interpretations, finite norm-envelope witnesses, dyadic stability bounds, and refinement towers of witness norms. The empirical part reports full official Fashion-MNIST experiments with supervised-contrastive and proxy-anchor-style Euclidean baselines, post hoc audits for shrinkage, residual flexibility, off-training scales, and latent extrapolation, and a ten-seed full-query/full-gallery UCI Human Activity Recognition benchmark. The results show that Euclidean objectives can be stronger for Recall@1, whereas AMN-inspired residual regularization substantially reduces finite norm-like defects inside the residual family. The contribution is therefore a finite diagnostic and regularization framework for learned latent dissimilarities, not a state-of-the-art retrieval objective. Full article

Explicit formulas for graphical Stirling and Bell numbers are known for relatively few graph families. We derive exact expressions for three classes whose independence structure admits a complete combinatorial description: complete multipartite graphs, the graph obtained from a balanced complete bipartite graph by deleting a perfect matching, and the Mycielskian of a star. For complete multipartite graphs we express the graphical Stirling number as a convolution of classical Stirling numbers across the partite classes, and we recover the known factorization of the graphical Bell number as a product of classical Bell numbers. For the matching-deleted graph we show that its graphical Bell number is a binomial convolution of squared Bell numbers, which we identify as a moment of a product of two independent Poisson random variables with unit mean. This representation yields log-convexity of the sequence, a sharp exponential lower bound, a two-sided estimate, and a Laplace-transform identity. For the Mycielskian of a star, a decomposition according to the block containing the original center vertex, together with Vandermonde’s convolution and a Stirling recurrence, gives a single-sum closed form for the graphical Stirling numbers, from which two explicit evaluations follow. Several resulting integer sequences appear in the OEIS, and one Bell-number sequence appears not to be currently recorded there. Full article

Strict minimum message length (SMML) is an information-theoretic coding principle that represents a continuous statistical model by a finite set of assertions and a partition of the sample space. We show that the SMML objective decomposes into assertion entropy and conditional cross-entropy, balancing the cost of identifying an assertion against the cost of encoding data under the assigned model. For any fixed partition, the optimal codepoint for each cell is the model distribution that minimises Kullback–Leibler (KL) divergence from the data distribution restricted to that cell. Using the local Fisher–Rao geometry of regular parametric models, we show that, under a high-resolution LAN-scale regime, SMML partitions are asymptotically the pullback, through the maximum-likelihood estimator, of weighted Fisher–Rao Voronoi tessellations in parameter space, with assertion probabilities appearing as additive weights. For regular canonical exponential families, SMML codepoints satisfy a moment-matching condition and admit an interpretation as KL/Bregman centroids, while exact SMML cells are pullbacks of convex polyhedra in sufficient-statistic space. Together, these results show that SMML induces a natural information-geometric quantisation linking entropy-based coding, KL projection, and divergence-based Voronoi geometry. Full article

Background and Clinical Significance: Peristomal pyoderma gangrenosum (PPG) is a rare subtype of pyoderma gangrenosum, most commonly associated with inflammatory bowel disease or haematologic disorders. Its occurrence in patients with solid malignancies is uncommon. PPG in an oncologic setting poses diagnostic and therapeutic challenges because systemic immunosuppressive therapy, wound care, and ongoing chemotherapy must be carefully balanced; Case Presentation: We report the case of a Japanese man in his 50s with stage IVB rectal adenocarcinoma who developed rapidly progressive peristomal ulceration clinically consistent with PPG around a colostomy 12 weeks after initiation of panitumumab-containing systemic chemotherapy. The diagnosis was made on clinical grounds and was strongly supported by the clinical morphology, exclusion of major mimickers, and response to systemic corticosteroid therapy, although histopathological confirmation was not obtained. Because existing diagnostic criteria for pyoderma gangrenosum are not specifically designed for peristomal disease, they were used as supportive rather than definitive diagnostic tools. Skin biopsy was avoided due to the risk of pathergy at the peristomal site. Superficial cultures were not obtained because frequent cleansing and faecal contamination were likely to compromise diagnostic accuracy. To minimise mechanical pathergy, the stoma appliance was changed from a one-piece soft convex system to a two-piece flat system. Multidisciplinary management, including systemic corticosteroids, meticulous stoma care, and selective ultrasonic debridement, resulted in complete epithelialisation by Week 26. Chemotherapy was temporarily withheld during the active inflammatory phase and later resumed. Despite successful control of the peristomal ulceration, the patient died from progressive malignancy at Week 34; Conclusions: This case highlights the clinical challenge of balancing immunosuppressive therapy for clinically suspected PPG with ongoing oncologic treatment. Mechanical pathergy related to stoma appliance use was considered a more likely precipitating factor than chemotherapy alone, although panitumumab may have contributed to impaired cutaneous repair. Close collaboration among dermatologists, oncologists, surgeons, WOC nurses, and family caregivers is essential for multidisciplinary decision-making in complex oncologic settings. Full article

We classify the continuous reciprocally symmetric cost functions $J:(0,\infty )\to \mathbb{R}$ with $J\left(1\right)=0$ and strictly convex log-substitution $G\left(t\right):=J\left(e^t\right)$ (admissible costs) for which the symmetric compound $J\left(xy\right)+J(x/y)$ depends on $(x,y)$ only through $\left(J\right(x),J(y\left)\right)$. We first prove that this dependence is automatic: for every admissible J, there exists a unique continuous combiner (the auxiliary function P that encodes the compound) $P:{[0,\infty )}^2\to \mathbb{R}$ with $J\left(xy\right)+J(x/y)=P\left(J\right(x),J(y\left)\right)$ for all $x,y>0$ (Theorem 1); P is symmetric, non-negative, satisfies $P(u,0)=2u$, and inherits monotonicity and coercivity from admissibility. When P is required to be a polynomial, a growth rate comparison between two recursions for G forces $degP\le 2$ (Theorem 4), so $P(u,v)=cuv+2u+2v$ with $c\ge 0$, and the corresponding admissible costs are exhausted by two explicit families (Theorem 8)—the hyperbolic family $J\left(x\right)=c^{-1}(x^{\lambda }+x^{-\lambda })-2c^{-1}$ ($c,\lambda >0$) and the degenerate quadratic family $J\left(x\right)=a{(lnx)}^2$ ($a>0$)—with the latter arising as the Inönü–Wigner contraction $\lambda \to 0^{+}$, ${\lambda }^2/c\to a$ of the former (Theorem 9). Two regularity extensions are obtained: a Lebesgue-measurable cost satisfying explicit regularity hypotheses admits a continuous representative (Theorem 5), and in the entire finite-order regime, the diagonal combiner $Q\left(u\right):=P(u,u)$, when polynomial of degree $d^{\prime }$, obeys the sharp bound $d^{\prime }\ge 2^{\rho }$ (Theorem 6), attained with equality in both classified families. The normalisations $P(1,1)=6$ and $G^{″}\left(0\right)=1$ single out the canonical representative $J_{\mathrm{cost}}\left(x\right)=\frac{1}{2}(x+x^{-1})-1$. Full article

In this paper, we introduce and study a Szász-type family of positive linear operators generated by Euler polynomials of negative order on $[0,\infty )$. The construction is based on an explicit finite representation of these polynomials with non-negative terms, which ensures the positivity of the corresponding kernel. We prove the basic properties of the operators and show that they can be represented as finite convex combinations of shifted classical Szász operators. We also provide a probabilistic representation of the kernel as a finite mixture of Poisson distributions, which clarifies the role of the parameter k and the resulting moment structure. The corresponding algebraic and central moment identities are derived and used to establish convergence on compact intervals and to obtain quantitative estimates in terms of the modulus of continuity, Lipschitz-type classes, and Peetre’s K-functional. Furthermore, Voronovskaya-type asymptotic results are obtained, including a quantitative form and a second-order asymptotic formula. Numerical tables and a graphical illustration are presented for selected test functions and parameter values, and the results are consistent with the theoretical convergence behaviour. The paper shows that Euler polynomials of negative order provide a positive and structurally tractable framework for constructing Szász-type approximation operators on the positive real axis. Full article

A fundamental objective in the study of convex cones is the description and analysis of their extreme rays. In the case of the copositive cone, these rays are generated by extremal copositive matrices, which encode the boundary structure of the cone and are closely related to challenging instances of copositive and completely positive programming. In this work, we propose a constructive framework for generating copositive matrices from a given extremal copositive matrix of a smaller order and establish conditions under which copositivity and extremality are preserved. This approach highlights the interplay between the zero structure of a copositive matrix, its minimal zeros, and the facial geometry of the copositive cone. The results obtained allow one to generate new families of extremal copositive matrices in higher dimensions. Full article

The article derives sufficient conditions under which the normalized Rabotnov function becomes q-close-to-convex relative to specific starlike functions on the open unit disk. To enhance the impact of our results, we include some consequences derived from the main theorems, along with graphical illustrations. The starlikeness of the Rabotnov function with respect to different aspects also falls within the scope of this study. Full article

Let $\mathcal{H}$ be a complex separable Hilbert space. We study the closure of the numerical range of the generalized pencil $T=P+\alpha Q+\beta PQ$, where $(P,Q)$ is a pair of orthogonal projections and $(\alpha ,\beta )\in {\mathbb{R}}^2$. Using Halmos’ two-subspace theorem, it is shown that, under suitable assumptions, $\overline{W\left(T\right)}$ is the closed convex hull of a family of ellipses $\mathcal{E}\left(\lambda \right)$ parametrized by $\lambda \in \sigma \left(PQ\right)$. Moreover, the spectrum $\sigma \left(T\right)$ coincides with the set of all foci of this elliptic family, revealing a precise geometric relation between the spectrum and the numerical range of such operators. Full article

We extend the classical Euler-type integral representations for the Appell functions $F_1$, $F_2$, and $F_3$, to the appropriate Kampé de Fériet functions by using integration against the Meijer–Nørlund G-function. In particular, these representations provide analytic continuation of the corresponding Kampé de Fériet functions. We further focus on the following two applications. First, we obtain sufficient conditions for complete monotonicity on the positive quadrant for three families of the Kampé de Fériet functions. These conditions can be expressed directly in terms of parameters and imply, among other things, joint log-convexity and related inequalities for partial derivatives of the Kampé de Fériet functions. Second, we show how known reduction and transformation formulas for the Appell and the generalized hypergeometric functions can be lifted to Kampé de Fériet functions by concatenating parameter arrays via the integral representations. This yields several reduction formulas, including extensions of some classical and new product identities. Further combining integration against the Meijer–Nørlund G-function with Slater’s double series transformation we obtain several exotic identities for infinite sums of the generalized hypergeometric functions. Full article